{"text": "[GOAL]\n\u03b1 \u03b2 : FinBddDistLatCat\ne : \u2191\u03b1.toBddDistLatCat.toDistLatCat \u2243o \u2191\u03b2.toBddDistLatCat.toDistLatCat\n\u22a2 ((let src :=\n {\n toSupHom :=\n { toFun := \u2191e, map_sup' := (_ : \u2200 (a b : \u2191\u03b1.toBddDistLatCat.toDistLatCat), \u2191e (a \u2294 b) = \u2191e a \u2294 \u2191e b) },\n map_inf' := (_ : \u2200 (a b : \u2191\u03b1.toBddDistLatCat.toDistLatCat), \u2191e (a \u2293 b) = \u2191e a \u2293 \u2191e b) };\n {\n toLatticeHom :=\n {\n toSupHom :=\n { toFun := \u2191e,\n map_sup' :=\n (_ :\n \u2200 (a b : \u2191\u03b1.toBddDistLatCat.toDistLatCat),\n SupHom.toFun\n {\n toSupHom :=\n { toFun := \u2191e,\n map_sup' :=\n (_ : \u2200 (a b : \u2191\u03b1.toBddDistLatCat.toDistLatCat), \u2191e (a \u2294 b) = \u2191e a \u2294 \u2191e b) },\n map_inf' :=\n (_ : \u2200 (a b : \u2191\u03b1.toBddDistLatCat.toDistLatCat), \u2191e (a \u2293 b) = \u2191e a \u2293 \u2191e b) }.toSupHom\n (a \u2294 b) =\n SupHom.toFun\n {\n toSupHom :=\n { toFun := \u2191e,\n map_sup' :=\n (_ : \u2200 (a b : \u2191\u03b1.toBddDistLatCat.toDistLatCat), \u2191e (a \u2294 b) = \u2191e a \u2294 \u2191e b) },\n map_inf' :=\n (_ : \u2200 (a b : \u2191\u03b1.toBddDistLatCat.toDistLatCat), \u2191e (a \u2293 b) = \u2191e a \u2293 \u2191e b) }.toSupHom\n a \u2294\n SupHom.toFun\n {\n toSupHom :=\n { toFun := \u2191e,\n map_sup' :=\n (_ : \u2200 (a b : \u2191\u03b1.toBddDistLatCat.toDistLatCat), \u2191e (a \u2294 b) = \u2191e a \u2294 \u2191e b) },\n map_inf' :=\n (_ : \u2200 (a b : \u2191\u03b1.toBddDistLatCat.toDistLatCat), \u2191e (a \u2293 b) = \u2191e a \u2293 \u2191e b) }.toSupHom\n b) },\n map_inf' :=\n (_ :\n \u2200 (a b : \u2191\u03b1.toBddDistLatCat.toDistLatCat),\n SupHom.toFun src.toSupHom (a \u2293 b) = SupHom.toFun src.toSupHom a \u2293 SupHom.toFun src.toSupHom b) },\n map_top' := (_ : \u2191e \u22a4 = \u22a4), map_bot' := (_ : \u2191e \u22a5 = \u22a5) }) \u226b\n let src :=\n {\n toSupHom :=\n { toFun := \u2191(OrderIso.symm e),\n map_sup' :=\n (_ :\n \u2200 (a b : \u2191\u03b2.toBddDistLatCat.toDistLatCat),\n \u2191(OrderIso.symm e) (a \u2294 b) = \u2191(OrderIso.symm e) a \u2294 \u2191(OrderIso.symm e) b) },\n map_inf' :=\n (_ :\n \u2200 (a b : \u2191\u03b2.toBddDistLatCat.toDistLatCat),\n \u2191(OrderIso.symm e) (a \u2293 b) = \u2191(OrderIso.symm e) a \u2293 \u2191(OrderIso.symm e) b) };\n {\n toLatticeHom :=\n {\n toSupHom :=\n { toFun := \u2191(OrderIso.symm e),\n map_sup' :=\n (_ :\n \u2200 (a b : \u2191\u03b2.toBddDistLatCat.toDistLatCat),\n SupHom.toFun\n {\n toSupHom :=\n { toFun := \u2191(OrderIso.symm e),\n map_sup' :=\n (_ :\n \u2200 (a b : \u2191\u03b2.toBddDistLatCat.toDistLatCat),\n \u2191(OrderIso.symm e) (a \u2294 b) = \u2191(OrderIso.symm e) a \u2294 \u2191(OrderIso.symm e) b) },\n map_inf' :=\n (_ :\n \u2200 (a b : \u2191\u03b2.toBddDistLatCat.toDistLatCat),\n \u2191(OrderIso.symm e) (a \u2293 b) = \u2191(OrderIso.symm e) a \u2293 \u2191(OrderIso.symm e) b) }.toSupHom\n (a \u2294 b) =\n SupHom.toFun\n {\n toSupHom :=\n { toFun := \u2191(OrderIso.symm e),\n map_sup' :=\n (_ :\n \u2200 (a b : \u2191\u03b2.toBddDistLatCat.toDistLatCat),\n \u2191(OrderIso.symm e) (a \u2294 b) = \u2191(OrderIso.symm e) a \u2294 \u2191(OrderIso.symm e) b) },\n map_inf' :=\n (_ :\n \u2200 (a b : \u2191\u03b2.toBddDistLatCat.toDistLatCat),\n \u2191(OrderIso.symm e) (a \u2293 b) =\n \u2191(OrderIso.symm e) a \u2293 \u2191(OrderIso.symm e) b) }.toSupHom\n a \u2294\n SupHom.toFun\n {\n toSupHom :=\n { toFun := \u2191(OrderIso.symm e),\n map_sup' :=\n (_ :\n \u2200 (a b : \u2191\u03b2.toBddDistLatCat.toDistLatCat),\n \u2191(OrderIso.symm e) (a \u2294 b) = \u2191(OrderIso.symm e) a \u2294 \u2191(OrderIso.symm e) b) },\n map_inf' :=\n (_ :\n \u2200 (a b : \u2191\u03b2.toBddDistLatCat.toDistLatCat),\n \u2191(OrderIso.symm e) (a \u2293 b) =\n \u2191(OrderIso.symm e) a \u2293 \u2191(OrderIso.symm e) b) }.toSupHom\n b) },\n map_inf' :=\n (_ :\n \u2200 (a b : \u2191\u03b2.toBddDistLatCat.toDistLatCat),\n SupHom.toFun src.toSupHom (a \u2293 b) = SupHom.toFun src.toSupHom a \u2293 SupHom.toFun src.toSupHom b) },\n map_top' := (_ : \u2191(OrderIso.symm e) \u22a4 = \u22a4), map_bot' := (_ : \u2191(OrderIso.symm e) \u22a5 = \u22a5) }) =\n \ud835\udfd9 \u03b1\n[PROOFSTEP]\next\n[GOAL]\ncase w\n\u03b1 \u03b2 : FinBddDistLatCat\ne : \u2191\u03b1.toBddDistLatCat.toDistLatCat \u2243o \u2191\u03b2.toBddDistLatCat.toDistLatCat\nx\u271d : (forget FinBddDistLatCat).obj \u03b1\n\u22a2 \u2191((let src :=\n {\n toSupHom :=\n { toFun := \u2191e, map_sup' := (_ : \u2200 (a b : \u2191\u03b1.toBddDistLatCat.toDistLatCat), \u2191e (a \u2294 b) = \u2191e a \u2294 \u2191e b) },\n map_inf' := (_ : \u2200 (a b : \u2191\u03b1.toBddDistLatCat.toDistLatCat), \u2191e (a \u2293 b) = \u2191e a \u2293 \u2191e b) };\n {\n toLatticeHom :=\n {\n toSupHom :=\n { toFun := \u2191e,\n map_sup' :=\n (_ :\n \u2200 (a b : \u2191\u03b1.toBddDistLatCat.toDistLatCat),\n SupHom.toFun\n {\n toSupHom :=\n { toFun := \u2191e,\n map_sup' :=\n (_ : \u2200 (a b : \u2191\u03b1.toBddDistLatCat.toDistLatCat), \u2191e (a \u2294 b) = \u2191e a \u2294 \u2191e b) },\n map_inf' :=\n (_ : \u2200 (a b : \u2191\u03b1.toBddDistLatCat.toDistLatCat), \u2191e (a \u2293 b) = \u2191e a \u2293 \u2191e b) }.toSupHom\n (a \u2294 b) =\n SupHom.toFun\n {\n toSupHom :=\n { toFun := \u2191e,\n map_sup' :=\n (_ : \u2200 (a b : \u2191\u03b1.toBddDistLatCat.toDistLatCat), \u2191e (a \u2294 b) = \u2191e a \u2294 \u2191e b) },\n map_inf' :=\n (_ :\n \u2200 (a b : \u2191\u03b1.toBddDistLatCat.toDistLatCat), \u2191e (a \u2293 b) = \u2191e a \u2293 \u2191e b) }.toSupHom\n a \u2294\n SupHom.toFun\n {\n toSupHom :=\n { toFun := \u2191e,\n map_sup' :=\n (_ : \u2200 (a b : \u2191\u03b1.toBddDistLatCat.toDistLatCat), \u2191e (a \u2294 b) = \u2191e a \u2294 \u2191e b) },\n map_inf' :=\n (_ :\n \u2200 (a b : \u2191\u03b1.toBddDistLatCat.toDistLatCat), \u2191e (a \u2293 b) = \u2191e a \u2293 \u2191e b) }.toSupHom\n b) },\n map_inf' :=\n (_ :\n \u2200 (a b : \u2191\u03b1.toBddDistLatCat.toDistLatCat),\n SupHom.toFun src.toSupHom (a \u2293 b) = SupHom.toFun src.toSupHom a \u2293 SupHom.toFun src.toSupHom b) },\n map_top' := (_ : \u2191e \u22a4 = \u22a4), map_bot' := (_ : \u2191e \u22a5 = \u22a5) }) \u226b\n let src :=\n {\n toSupHom :=\n { toFun := \u2191(OrderIso.symm e),\n map_sup' :=\n (_ :\n \u2200 (a b : \u2191\u03b2.toBddDistLatCat.toDistLatCat),\n \u2191(OrderIso.symm e) (a \u2294 b) = \u2191(OrderIso.symm e) a \u2294 \u2191(OrderIso.symm e) b) },\n map_inf' :=\n (_ :\n \u2200 (a b : \u2191\u03b2.toBddDistLatCat.toDistLatCat),\n \u2191(OrderIso.symm e) (a \u2293 b) = \u2191(OrderIso.symm e) a \u2293 \u2191(OrderIso.symm e) b) };\n {\n toLatticeHom :=\n {\n toSupHom :=\n { toFun := \u2191(OrderIso.symm e),\n map_sup' :=\n (_ :\n \u2200 (a b : \u2191\u03b2.toBddDistLatCat.toDistLatCat),\n SupHom.toFun\n {\n toSupHom :=\n { toFun := \u2191(OrderIso.symm e),\n map_sup' :=\n (_ :\n \u2200 (a b : \u2191\u03b2.toBddDistLatCat.toDistLatCat),\n \u2191(OrderIso.symm e) (a \u2294 b) = \u2191(OrderIso.symm e) a \u2294 \u2191(OrderIso.symm e) b) },\n map_inf' :=\n (_ :\n \u2200 (a b : \u2191\u03b2.toBddDistLatCat.toDistLatCat),\n \u2191(OrderIso.symm e) (a \u2293 b) =\n \u2191(OrderIso.symm e) a \u2293 \u2191(OrderIso.symm e) b) }.toSupHom\n (a \u2294 b) =\n SupHom.toFun\n {\n toSupHom :=\n { toFun := \u2191(OrderIso.symm e),\n map_sup' :=\n (_ :\n \u2200 (a b : \u2191\u03b2.toBddDistLatCat.toDistLatCat),\n \u2191(OrderIso.symm e) (a \u2294 b) =\n \u2191(OrderIso.symm e) a \u2294 \u2191(OrderIso.symm e) b) },\n map_inf' :=\n (_ :\n \u2200 (a b : \u2191\u03b2.toBddDistLatCat.toDistLatCat),\n \u2191(OrderIso.symm e) (a \u2293 b) =\n \u2191(OrderIso.symm e) a \u2293 \u2191(OrderIso.symm e) b) }.toSupHom\n a \u2294\n SupHom.toFun\n {\n toSupHom :=\n { toFun := \u2191(OrderIso.symm e),\n map_sup' :=\n (_ :\n \u2200 (a b : \u2191\u03b2.toBddDistLatCat.toDistLatCat),\n \u2191(OrderIso.symm e) (a \u2294 b) =\n \u2191(OrderIso.symm e) a \u2294 \u2191(OrderIso.symm e) b) },\n map_inf' :=\n (_ :\n \u2200 (a b : \u2191\u03b2.toBddDistLatCat.toDistLatCat),\n \u2191(OrderIso.symm e) (a \u2293 b) =\n \u2191(OrderIso.symm e) a \u2293 \u2191(OrderIso.symm e) b) }.toSupHom\n b) },\n map_inf' :=\n (_ :\n \u2200 (a b : \u2191\u03b2.toBddDistLatCat.toDistLatCat),\n SupHom.toFun src.toSupHom (a \u2293 b) = SupHom.toFun src.toSupHom a \u2293 SupHom.toFun src.toSupHom b) },\n map_top' := (_ : \u2191(OrderIso.symm e) \u22a4 = \u22a4), map_bot' := (_ : \u2191(OrderIso.symm e) \u22a5 = \u22a5) })\n x\u271d =\n \u2191(\ud835\udfd9 \u03b1) x\u271d\n[PROOFSTEP]\nexact e.symm_apply_apply _\n[GOAL]\n\u03b1 \u03b2 : FinBddDistLatCat\ne : \u2191\u03b1.toBddDistLatCat.toDistLatCat \u2243o \u2191\u03b2.toBddDistLatCat.toDistLatCat\n\u22a2 ((let src :=\n {\n toSupHom :=\n { toFun := \u2191(OrderIso.symm e),\n map_sup' :=\n (_ :\n \u2200 (a b : \u2191\u03b2.toBddDistLatCat.toDistLatCat),\n \u2191(OrderIso.symm e) (a \u2294 b) = \u2191(OrderIso.symm e) a \u2294 \u2191(OrderIso.symm e) b) },\n map_inf' :=\n (_ :\n \u2200 (a b : \u2191\u03b2.toBddDistLatCat.toDistLatCat),\n \u2191(OrderIso.symm e) (a \u2293 b) = \u2191(OrderIso.symm e) a \u2293 \u2191(OrderIso.symm e) b) };\n {\n toLatticeHom :=\n {\n toSupHom :=\n { toFun := \u2191(OrderIso.symm e),\n map_sup' :=\n (_ :\n \u2200 (a b : \u2191\u03b2.toBddDistLatCat.toDistLatCat),\n SupHom.toFun\n {\n toSupHom :=\n { toFun := \u2191(OrderIso.symm e),\n map_sup' :=\n (_ :\n \u2200 (a b : \u2191\u03b2.toBddDistLatCat.toDistLatCat),\n \u2191(OrderIso.symm e) (a \u2294 b) = \u2191(OrderIso.symm e) a \u2294 \u2191(OrderIso.symm e) b) },\n map_inf' :=\n (_ :\n \u2200 (a b : \u2191\u03b2.toBddDistLatCat.toDistLatCat),\n \u2191(OrderIso.symm e) (a \u2293 b) = \u2191(OrderIso.symm e) a \u2293 \u2191(OrderIso.symm e) b) }.toSupHom\n (a \u2294 b) =\n SupHom.toFun\n {\n toSupHom :=\n { toFun := \u2191(OrderIso.symm e),\n map_sup' :=\n (_ :\n \u2200 (a b : \u2191\u03b2.toBddDistLatCat.toDistLatCat),\n \u2191(OrderIso.symm e) (a \u2294 b) = \u2191(OrderIso.symm e) a \u2294 \u2191(OrderIso.symm e) b) },\n map_inf' :=\n (_ :\n \u2200 (a b : \u2191\u03b2.toBddDistLatCat.toDistLatCat),\n \u2191(OrderIso.symm e) (a \u2293 b) =\n \u2191(OrderIso.symm e) a \u2293 \u2191(OrderIso.symm e) b) }.toSupHom\n a \u2294\n SupHom.toFun\n {\n toSupHom :=\n { toFun := \u2191(OrderIso.symm e),\n map_sup' :=\n (_ :\n \u2200 (a b : \u2191\u03b2.toBddDistLatCat.toDistLatCat),\n \u2191(OrderIso.symm e) (a \u2294 b) = \u2191(OrderIso.symm e) a \u2294 \u2191(OrderIso.symm e) b) },\n map_inf' :=\n (_ :\n \u2200 (a b : \u2191\u03b2.toBddDistLatCat.toDistLatCat),\n \u2191(OrderIso.symm e) (a \u2293 b) =\n \u2191(OrderIso.symm e) a \u2293 \u2191(OrderIso.symm e) b) }.toSupHom\n b) },\n map_inf' :=\n (_ :\n \u2200 (a b : \u2191\u03b2.toBddDistLatCat.toDistLatCat),\n SupHom.toFun src.toSupHom (a \u2293 b) = SupHom.toFun src.toSupHom a \u2293 SupHom.toFun src.toSupHom b) },\n map_top' := (_ : \u2191(OrderIso.symm e) \u22a4 = \u22a4), map_bot' := (_ : \u2191(OrderIso.symm e) \u22a5 = \u22a5) }) \u226b\n let src :=\n {\n toSupHom :=\n { toFun := \u2191e, map_sup' := (_ : \u2200 (a b : \u2191\u03b1.toBddDistLatCat.toDistLatCat), \u2191e (a \u2294 b) = \u2191e a \u2294 \u2191e b) },\n map_inf' := (_ : \u2200 (a b : \u2191\u03b1.toBddDistLatCat.toDistLatCat), \u2191e (a \u2293 b) = \u2191e a \u2293 \u2191e b) };\n {\n toLatticeHom :=\n {\n toSupHom :=\n { toFun := \u2191e,\n map_sup' :=\n (_ :\n \u2200 (a b : \u2191\u03b1.toBddDistLatCat.toDistLatCat),\n SupHom.toFun\n {\n toSupHom :=\n { toFun := \u2191e,\n map_sup' :=\n (_ : \u2200 (a b : \u2191\u03b1.toBddDistLatCat.toDistLatCat), \u2191e (a \u2294 b) = \u2191e a \u2294 \u2191e b) },\n map_inf' :=\n (_ : \u2200 (a b : \u2191\u03b1.toBddDistLatCat.toDistLatCat), \u2191e (a \u2293 b) = \u2191e a \u2293 \u2191e b) }.toSupHom\n (a \u2294 b) =\n SupHom.toFun\n {\n toSupHom :=\n { toFun := \u2191e,\n map_sup' :=\n (_ : \u2200 (a b : \u2191\u03b1.toBddDistLatCat.toDistLatCat), \u2191e (a \u2294 b) = \u2191e a \u2294 \u2191e b) },\n map_inf' :=\n (_ : \u2200 (a b : \u2191\u03b1.toBddDistLatCat.toDistLatCat), \u2191e (a \u2293 b) = \u2191e a \u2293 \u2191e b) }.toSupHom\n a \u2294\n SupHom.toFun\n {\n toSupHom :=\n { toFun := \u2191e,\n map_sup' :=\n (_ : \u2200 (a b : \u2191\u03b1.toBddDistLatCat.toDistLatCat), \u2191e (a \u2294 b) = \u2191e a \u2294 \u2191e b) },\n map_inf' :=\n (_ : \u2200 (a b : \u2191\u03b1.toBddDistLatCat.toDistLatCat), \u2191e (a \u2293 b) = \u2191e a \u2293 \u2191e b) }.toSupHom\n b) },\n map_inf' :=\n (_ :\n \u2200 (a b : \u2191\u03b1.toBddDistLatCat.toDistLatCat),\n SupHom.toFun src.toSupHom (a \u2293 b) = SupHom.toFun src.toSupHom a \u2293 SupHom.toFun src.toSupHom b) },\n map_top' := (_ : \u2191e \u22a4 = \u22a4), map_bot' := (_ : \u2191e \u22a5 = \u22a5) }) =\n \ud835\udfd9 \u03b2\n[PROOFSTEP]\next\n[GOAL]\ncase w\n\u03b1 \u03b2 : FinBddDistLatCat\ne : \u2191\u03b1.toBddDistLatCat.toDistLatCat \u2243o \u2191\u03b2.toBddDistLatCat.toDistLatCat\nx\u271d : (forget FinBddDistLatCat).obj \u03b2\n\u22a2 \u2191((let src :=\n {\n toSupHom :=\n { toFun := \u2191(OrderIso.symm e),\n map_sup' :=\n (_ :\n \u2200 (a b : \u2191\u03b2.toBddDistLatCat.toDistLatCat),\n \u2191(OrderIso.symm e) (a \u2294 b) = \u2191(OrderIso.symm e) a \u2294 \u2191(OrderIso.symm e) b) },\n map_inf' :=\n (_ :\n \u2200 (a b : \u2191\u03b2.toBddDistLatCat.toDistLatCat),\n \u2191(OrderIso.symm e) (a \u2293 b) = \u2191(OrderIso.symm e) a \u2293 \u2191(OrderIso.symm e) b) };\n {\n toLatticeHom :=\n {\n toSupHom :=\n { toFun := \u2191(OrderIso.symm e),\n map_sup' :=\n (_ :\n \u2200 (a b : \u2191\u03b2.toBddDistLatCat.toDistLatCat),\n SupHom.toFun\n {\n toSupHom :=\n { toFun := \u2191(OrderIso.symm e),\n map_sup' :=\n (_ :\n \u2200 (a b : \u2191\u03b2.toBddDistLatCat.toDistLatCat),\n \u2191(OrderIso.symm e) (a \u2294 b) = \u2191(OrderIso.symm e) a \u2294 \u2191(OrderIso.symm e) b) },\n map_inf' :=\n (_ :\n \u2200 (a b : \u2191\u03b2.toBddDistLatCat.toDistLatCat),\n \u2191(OrderIso.symm e) (a \u2293 b) =\n \u2191(OrderIso.symm e) a \u2293 \u2191(OrderIso.symm e) b) }.toSupHom\n (a \u2294 b) =\n SupHom.toFun\n {\n toSupHom :=\n { toFun := \u2191(OrderIso.symm e),\n map_sup' :=\n (_ :\n \u2200 (a b : \u2191\u03b2.toBddDistLatCat.toDistLatCat),\n \u2191(OrderIso.symm e) (a \u2294 b) =\n \u2191(OrderIso.symm e) a \u2294 \u2191(OrderIso.symm e) b) },\n map_inf' :=\n (_ :\n \u2200 (a b : \u2191\u03b2.toBddDistLatCat.toDistLatCat),\n \u2191(OrderIso.symm e) (a \u2293 b) =\n \u2191(OrderIso.symm e) a \u2293 \u2191(OrderIso.symm e) b) }.toSupHom\n a \u2294\n SupHom.toFun\n {\n toSupHom :=\n { toFun := \u2191(OrderIso.symm e),\n map_sup' :=\n (_ :\n \u2200 (a b : \u2191\u03b2.toBddDistLatCat.toDistLatCat),\n \u2191(OrderIso.symm e) (a \u2294 b) =\n \u2191(OrderIso.symm e) a \u2294 \u2191(OrderIso.symm e) b) },\n map_inf' :=\n (_ :\n \u2200 (a b : \u2191\u03b2.toBddDistLatCat.toDistLatCat),\n \u2191(OrderIso.symm e) (a \u2293 b) =\n \u2191(OrderIso.symm e) a \u2293 \u2191(OrderIso.symm e) b) }.toSupHom\n b) },\n map_inf' :=\n (_ :\n \u2200 (a b : \u2191\u03b2.toBddDistLatCat.toDistLatCat),\n SupHom.toFun src.toSupHom (a \u2293 b) = SupHom.toFun src.toSupHom a \u2293 SupHom.toFun src.toSupHom b) },\n map_top' := (_ : \u2191(OrderIso.symm e) \u22a4 = \u22a4), map_bot' := (_ : \u2191(OrderIso.symm e) \u22a5 = \u22a5) }) \u226b\n let src :=\n {\n toSupHom :=\n { toFun := \u2191e, map_sup' := (_ : \u2200 (a b : \u2191\u03b1.toBddDistLatCat.toDistLatCat), \u2191e (a \u2294 b) = \u2191e a \u2294 \u2191e b) },\n map_inf' := (_ : \u2200 (a b : \u2191\u03b1.toBddDistLatCat.toDistLatCat), \u2191e (a \u2293 b) = \u2191e a \u2293 \u2191e b) };\n {\n toLatticeHom :=\n {\n toSupHom :=\n { toFun := \u2191e,\n map_sup' :=\n (_ :\n \u2200 (a b : \u2191\u03b1.toBddDistLatCat.toDistLatCat),\n SupHom.toFun\n {\n toSupHom :=\n { toFun := \u2191e,\n map_sup' :=\n (_ : \u2200 (a b : \u2191\u03b1.toBddDistLatCat.toDistLatCat), \u2191e (a \u2294 b) = \u2191e a \u2294 \u2191e b) },\n map_inf' :=\n (_ : \u2200 (a b : \u2191\u03b1.toBddDistLatCat.toDistLatCat), \u2191e (a \u2293 b) = \u2191e a \u2293 \u2191e b) }.toSupHom\n (a \u2294 b) =\n SupHom.toFun\n {\n toSupHom :=\n { toFun := \u2191e,\n map_sup' :=\n (_ : \u2200 (a b : \u2191\u03b1.toBddDistLatCat.toDistLatCat), \u2191e (a \u2294 b) = \u2191e a \u2294 \u2191e b) },\n map_inf' :=\n (_ :\n \u2200 (a b : \u2191\u03b1.toBddDistLatCat.toDistLatCat), \u2191e (a \u2293 b) = \u2191e a \u2293 \u2191e b) }.toSupHom\n a \u2294\n SupHom.toFun\n {\n toSupHom :=\n { toFun := \u2191e,\n map_sup' :=\n (_ : \u2200 (a b : \u2191\u03b1.toBddDistLatCat.toDistLatCat), \u2191e (a \u2294 b) = \u2191e a \u2294 \u2191e b) },\n map_inf' :=\n (_ :\n \u2200 (a b : \u2191\u03b1.toBddDistLatCat.toDistLatCat), \u2191e (a \u2293 b) = \u2191e a \u2293 \u2191e b) }.toSupHom\n b) },\n map_inf' :=\n (_ :\n \u2200 (a b : \u2191\u03b1.toBddDistLatCat.toDistLatCat),\n SupHom.toFun src.toSupHom (a \u2293 b) = SupHom.toFun src.toSupHom a \u2293 SupHom.toFun src.toSupHom b) },\n map_top' := (_ : \u2191e \u22a4 = \u22a4), map_bot' := (_ : \u2191e \u22a5 = \u22a5) })\n x\u271d =\n \u2191(\ud835\udfd9 \u03b2) x\u271d\n[PROOFSTEP]\nexact e.apply_symm_apply _\n", "meta": {"mathlib_filename": "Mathlib.Order.Category.FinBddDistLatCat", "llama_tokens": 8337, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.6039318337259584, "lm_q2_score": 0.3007455914759599, "lm_q1q2_score": 0.18162983654507442}} {"text": "[GOAL]\nC : Type u\ninst\u271d\u00b9\u2070 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\u2081\ninst\u271d\u2079 : Category.{max v u, w\u2081} D\nE : Type w\u2082\ninst\u271d\u2078 : Category.{max v u, w\u2082} E\nF : D \u2964 E\ninst\u271d\u2077 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst\u271d\u2076 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst\u271d\u2075 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u2074 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 E\ninst\u271d\u00b3 : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 F\ninst\u271d\u00b2 : (X : C) \u2192 (W : Cover J X) \u2192 (P : C\u1d52\u1d56 \u2964 D) \u2192 PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\nP\u271d P : C\u1d52\u1d56 \u2964 D\ninst\u271d\u00b9 : (F : D \u2964 E) \u2192 (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 F\ninst\u271d :\n (F : D \u2964 E) \u2192\n (X : C) \u2192 (W : Cover J X) \u2192 (P : C\u1d52\u1d56 \u2964 D) \u2192 PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\n\u22a2 (whiskeringLeft C\u1d52\u1d56 D E).obj (sheafify J P) \u2245 (whiskeringLeft C\u1d52\u1d56 D E).obj P \u22d9 sheafification J E\n[PROOFSTEP]\nrefine' J.plusFunctorWhiskerLeftIso _ \u226a\u226b _ \u226a\u226b Functor.associator _ _ _\n[GOAL]\nC : Type u\ninst\u271d\u00b9\u2070 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\u2081\ninst\u271d\u2079 : Category.{max v u, w\u2081} D\nE : Type w\u2082\ninst\u271d\u2078 : Category.{max v u, w\u2082} E\nF : D \u2964 E\ninst\u271d\u2077 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst\u271d\u2076 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst\u271d\u2075 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u2074 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 E\ninst\u271d\u00b3 : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 F\ninst\u271d\u00b2 : (X : C) \u2192 (W : Cover J X) \u2192 (P : C\u1d52\u1d56 \u2964 D) \u2192 PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\nP\u271d P : C\u1d52\u1d56 \u2964 D\ninst\u271d\u00b9 : (F : D \u2964 E) \u2192 (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 F\ninst\u271d :\n (F : D \u2964 E) \u2192\n (X : C) \u2192 (W : Cover J X) \u2192 (P : C\u1d52\u1d56 \u2964 D) \u2192 PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\n\u22a2 (whiskeringLeft C\u1d52\u1d56 D E).obj (plusObj J P) \u22d9 plusFunctor J E \u2245\n ((whiskeringLeft C\u1d52\u1d56 D E).obj P \u22d9 plusFunctor J E) \u22d9 plusFunctor J E\n[PROOFSTEP]\nrefine' isoWhiskerRight _ _\n[GOAL]\nC : Type u\ninst\u271d\u00b9\u2070 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\u2081\ninst\u271d\u2079 : Category.{max v u, w\u2081} D\nE : Type w\u2082\ninst\u271d\u2078 : Category.{max v u, w\u2082} E\nF : D \u2964 E\ninst\u271d\u2077 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst\u271d\u2076 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst\u271d\u2075 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u2074 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 E\ninst\u271d\u00b3 : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 F\ninst\u271d\u00b2 : (X : C) \u2192 (W : Cover J X) \u2192 (P : C\u1d52\u1d56 \u2964 D) \u2192 PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\nP\u271d P : C\u1d52\u1d56 \u2964 D\ninst\u271d\u00b9 : (F : D \u2964 E) \u2192 (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 F\ninst\u271d :\n (F : D \u2964 E) \u2192\n (X : C) \u2192 (W : Cover J X) \u2192 (P : C\u1d52\u1d56 \u2964 D) \u2192 PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\n\u22a2 (whiskeringLeft C\u1d52\u1d56 D E).obj (plusObj J P) \u2245 (whiskeringLeft C\u1d52\u1d56 D E).obj P \u22d9 plusFunctor J E\n[PROOFSTEP]\nrefine' J.plusFunctorWhiskerLeftIso _\n[GOAL]\nC : Type u\ninst\u271d\u00b9\u2070 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\u2081\ninst\u271d\u2079 : Category.{max v u, w\u2081} D\nE : Type w\u2082\ninst\u271d\u2078 : Category.{max v u, w\u2082} E\nF\u271d : D \u2964 E\ninst\u271d\u2077 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst\u271d\u2076 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst\u271d\u2075 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u2074 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 E\ninst\u271d\u00b3 : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 F\u271d\ninst\u271d\u00b2 : (X : C) \u2192 (W : Cover J X) \u2192 (P : C\u1d52\u1d56 \u2964 D) \u2192 PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\u271d\nP\u271d P : C\u1d52\u1d56 \u2964 D\nF : D \u2964 E\ninst\u271d\u00b9 : (F : D \u2964 E) \u2192 (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 F\ninst\u271d :\n (F : D \u2964 E) \u2192\n (X : C) \u2192 (W : Cover J X) \u2192 (P : C\u1d52\u1d56 \u2964 D) \u2192 PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\n\u22a2 NatTrans.app (sheafificationWhiskerLeftIso J P).hom F = (sheafifyCompIso J F P).hom\n[PROOFSTEP]\ndsimp [sheafificationWhiskerLeftIso, sheafifyCompIso]\n[GOAL]\nC : Type u\ninst\u271d\u00b9\u2070 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\u2081\ninst\u271d\u2079 : Category.{max v u, w\u2081} D\nE : Type w\u2082\ninst\u271d\u2078 : Category.{max v u, w\u2082} E\nF\u271d : D \u2964 E\ninst\u271d\u2077 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst\u271d\u2076 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst\u271d\u2075 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u2074 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 E\ninst\u271d\u00b3 : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 F\u271d\ninst\u271d\u00b2 : (X : C) \u2192 (W : Cover J X) \u2192 (P : C\u1d52\u1d56 \u2964 D) \u2192 PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\u271d\nP\u271d P : C\u1d52\u1d56 \u2964 D\nF : D \u2964 E\ninst\u271d\u00b9 : (F : D \u2964 E) \u2192 (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 F\ninst\u271d :\n (F : D \u2964 E) \u2192\n (X : C) \u2192 (W : Cover J X) \u2192 (P : C\u1d52\u1d56 \u2964 D) \u2192 PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\n\u22a2 (plusCompIso J F (plusObj J P)).hom \u226b plusMap J (plusCompIso J F P).hom \u226b \ud835\udfd9 (plusObj J (plusObj J (P \u22d9 F))) =\n (plusCompIso J F (plusObj J P)).hom \u226b plusMap J (plusCompIso J F P).hom\n[PROOFSTEP]\nrw [Category.comp_id]\n[GOAL]\nC : Type u\ninst\u271d\u00b9\u2070 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\u2081\ninst\u271d\u2079 : Category.{max v u, w\u2081} D\nE : Type w\u2082\ninst\u271d\u2078 : Category.{max v u, w\u2082} E\nF\u271d : D \u2964 E\ninst\u271d\u2077 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst\u271d\u2076 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst\u271d\u2075 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u2074 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 E\ninst\u271d\u00b3 : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 F\u271d\ninst\u271d\u00b2 : (X : C) \u2192 (W : Cover J X) \u2192 (P : C\u1d52\u1d56 \u2964 D) \u2192 PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\u271d\nP\u271d P : C\u1d52\u1d56 \u2964 D\nF : D \u2964 E\ninst\u271d\u00b9 : (F : D \u2964 E) \u2192 (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 F\ninst\u271d :\n (F : D \u2964 E) \u2192\n (X : C) \u2192 (W : Cover J X) \u2192 (P : C\u1d52\u1d56 \u2964 D) \u2192 PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\n\u22a2 NatTrans.app (sheafificationWhiskerLeftIso J P).inv F = (sheafifyCompIso J F P).inv\n[PROOFSTEP]\ndsimp [sheafificationWhiskerLeftIso, sheafifyCompIso]\n[GOAL]\nC : Type u\ninst\u271d\u00b9\u2070 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\u2081\ninst\u271d\u2079 : Category.{max v u, w\u2081} D\nE : Type w\u2082\ninst\u271d\u2078 : Category.{max v u, w\u2082} E\nF\u271d : D \u2964 E\ninst\u271d\u2077 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst\u271d\u2076 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst\u271d\u2075 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u2074 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 E\ninst\u271d\u00b3 : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 F\u271d\ninst\u271d\u00b2 : (X : C) \u2192 (W : Cover J X) \u2192 (P : C\u1d52\u1d56 \u2964 D) \u2192 PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\u271d\nP\u271d P : C\u1d52\u1d56 \u2964 D\nF : D \u2964 E\ninst\u271d\u00b9 : (F : D \u2964 E) \u2192 (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 F\ninst\u271d :\n (F : D \u2964 E) \u2192\n (X : C) \u2192 (W : Cover J X) \u2192 (P : C\u1d52\u1d56 \u2964 D) \u2192 PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\n\u22a2 (\ud835\udfd9 (plusObj J (plusObj J (P \u22d9 F))) \u226b plusMap J (plusCompIso J F P).inv) \u226b (plusCompIso J F (plusObj J P)).inv =\n plusMap J (plusCompIso J F P).inv \u226b (plusCompIso J F (plusObj J P)).inv\n[PROOFSTEP]\nerw [Category.id_comp]\n[GOAL]\nC : Type u\ninst\u271d\u2078 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\u2081\ninst\u271d\u2077 : Category.{max v u, w\u2081} D\nE : Type w\u2082\ninst\u271d\u2076 : Category.{max v u, w\u2082} E\nF : D \u2964 E\ninst\u271d\u2075 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst\u271d\u2074 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst\u271d\u00b3 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 E\ninst\u271d\u00b9 : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 F\ninst\u271d : (X : C) \u2192 (W : Cover J X) \u2192 (P : C\u1d52\u1d56 \u2964 D) \u2192 PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\nP : C\u1d52\u1d56 \u2964 D\n\u22a2 sheafification J D \u22d9 (whiskeringRight C\u1d52\u1d56 D E).obj F \u2245 (whiskeringRight C\u1d52\u1d56 D E).obj F \u22d9 sheafification J E\n[PROOFSTEP]\nrefine' Functor.associator _ _ _ \u226a\u226b _\n[GOAL]\nC : Type u\ninst\u271d\u2078 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\u2081\ninst\u271d\u2077 : Category.{max v u, w\u2081} D\nE : Type w\u2082\ninst\u271d\u2076 : Category.{max v u, w\u2082} E\nF : D \u2964 E\ninst\u271d\u2075 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst\u271d\u2074 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst\u271d\u00b3 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 E\ninst\u271d\u00b9 : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 F\ninst\u271d : (X : C) \u2192 (W : Cover J X) \u2192 (P : C\u1d52\u1d56 \u2964 D) \u2192 PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\nP : C\u1d52\u1d56 \u2964 D\n\u22a2 plusFunctor J D \u22d9 plusFunctor J D \u22d9 (whiskeringRight C\u1d52\u1d56 D E).obj F \u2245\n (whiskeringRight C\u1d52\u1d56 D E).obj F \u22d9 sheafification J E\n[PROOFSTEP]\nrefine' isoWhiskerLeft (J.plusFunctor D) (J.plusFunctorWhiskerRightIso _) \u226a\u226b _\n[GOAL]\nC : Type u\ninst\u271d\u2078 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\u2081\ninst\u271d\u2077 : Category.{max v u, w\u2081} D\nE : Type w\u2082\ninst\u271d\u2076 : Category.{max v u, w\u2082} E\nF : D \u2964 E\ninst\u271d\u2075 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst\u271d\u2074 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst\u271d\u00b3 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 E\ninst\u271d\u00b9 : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 F\ninst\u271d : (X : C) \u2192 (W : Cover J X) \u2192 (P : C\u1d52\u1d56 \u2964 D) \u2192 PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\nP : C\u1d52\u1d56 \u2964 D\n\u22a2 plusFunctor J D \u22d9 (whiskeringRight C\u1d52\u1d56 D E).obj F \u22d9 plusFunctor J E \u2245\n (whiskeringRight C\u1d52\u1d56 D E).obj F \u22d9 sheafification J E\n[PROOFSTEP]\nrefine' _ \u226a\u226b Functor.associator _ _ _\n[GOAL]\nC : Type u\ninst\u271d\u2078 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\u2081\ninst\u271d\u2077 : Category.{max v u, w\u2081} D\nE : Type w\u2082\ninst\u271d\u2076 : Category.{max v u, w\u2082} E\nF : D \u2964 E\ninst\u271d\u2075 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst\u271d\u2074 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst\u271d\u00b3 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 E\ninst\u271d\u00b9 : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 F\ninst\u271d : (X : C) \u2192 (W : Cover J X) \u2192 (P : C\u1d52\u1d56 \u2964 D) \u2192 PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\nP : C\u1d52\u1d56 \u2964 D\n\u22a2 plusFunctor J D \u22d9 (whiskeringRight C\u1d52\u1d56 D E).obj F \u22d9 plusFunctor J E \u2245\n ((whiskeringRight C\u1d52\u1d56 D E).obj F \u22d9 plusFunctor J E) \u22d9 plusFunctor J E\n[PROOFSTEP]\nrefine' (Functor.associator _ _ _).symm \u226a\u226b _\n[GOAL]\nC : Type u\ninst\u271d\u2078 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\u2081\ninst\u271d\u2077 : Category.{max v u, w\u2081} D\nE : Type w\u2082\ninst\u271d\u2076 : Category.{max v u, w\u2082} E\nF : D \u2964 E\ninst\u271d\u2075 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst\u271d\u2074 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst\u271d\u00b3 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 E\ninst\u271d\u00b9 : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 F\ninst\u271d : (X : C) \u2192 (W : Cover J X) \u2192 (P : C\u1d52\u1d56 \u2964 D) \u2192 PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\nP : C\u1d52\u1d56 \u2964 D\n\u22a2 (plusFunctor J D \u22d9 (whiskeringRight C\u1d52\u1d56 D E).obj F) \u22d9 plusFunctor J E \u2245\n ((whiskeringRight C\u1d52\u1d56 D E).obj F \u22d9 plusFunctor J E) \u22d9 plusFunctor J E\n[PROOFSTEP]\nexact isoWhiskerRight (J.plusFunctorWhiskerRightIso _) (J.plusFunctor E)\n[GOAL]\nC : Type u\ninst\u271d\u2078 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\u2081\ninst\u271d\u2077 : Category.{max v u, w\u2081} D\nE : Type w\u2082\ninst\u271d\u2076 : Category.{max v u, w\u2082} E\nF : D \u2964 E\ninst\u271d\u2075 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst\u271d\u2074 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst\u271d\u00b3 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 E\ninst\u271d\u00b9 : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 F\ninst\u271d : (X : C) \u2192 (W : Cover J X) \u2192 (P : C\u1d52\u1d56 \u2964 D) \u2192 PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\nP : C\u1d52\u1d56 \u2964 D\n\u22a2 NatTrans.app (sheafificationWhiskerRightIso J F).hom P = (sheafifyCompIso J F P).hom\n[PROOFSTEP]\ndsimp [sheafificationWhiskerRightIso, sheafifyCompIso]\n[GOAL]\nC : Type u\ninst\u271d\u2078 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\u2081\ninst\u271d\u2077 : Category.{max v u, w\u2081} D\nE : Type w\u2082\ninst\u271d\u2076 : Category.{max v u, w\u2082} E\nF : D \u2964 E\ninst\u271d\u2075 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst\u271d\u2074 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst\u271d\u00b3 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 E\ninst\u271d\u00b9 : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 F\ninst\u271d : (X : C) \u2192 (W : Cover J X) \u2192 (P : C\u1d52\u1d56 \u2964 D) \u2192 PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\nP : C\u1d52\u1d56 \u2964 D\n\u22a2 \ud835\udfd9 (plusObj J (plusObj J P) \u22d9 F) \u226b\n (plusCompIso J F (plusObj J P)).hom \u226b\n (\ud835\udfd9 (plusObj J (plusObj J P \u22d9 F)) \u226b plusMap J (plusCompIso J F P).hom) \u226b \ud835\udfd9 (plusObj J (plusObj J (P \u22d9 F))) =\n (plusCompIso J F (plusObj J P)).hom \u226b plusMap J (plusCompIso J F P).hom\n[PROOFSTEP]\nsimp only [Category.id_comp, Category.comp_id]\n[GOAL]\nC : Type u\ninst\u271d\u2078 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\u2081\ninst\u271d\u2077 : Category.{max v u, w\u2081} D\nE : Type w\u2082\ninst\u271d\u2076 : Category.{max v u, w\u2082} E\nF : D \u2964 E\ninst\u271d\u2075 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst\u271d\u2074 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst\u271d\u00b3 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 E\ninst\u271d\u00b9 : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 F\ninst\u271d : (X : C) \u2192 (W : Cover J X) \u2192 (P : C\u1d52\u1d56 \u2964 D) \u2192 PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\nP : C\u1d52\u1d56 \u2964 D\n\u22a2 \ud835\udfd9 (plusObj J (plusObj J P) \u22d9 F) \u226b (plusCompIso J F (plusObj J P)).hom \u226b plusMap J (plusCompIso J F P).hom =\n (plusCompIso J F (plusObj J P)).hom \u226b plusMap J (plusCompIso J F P).hom\n[PROOFSTEP]\nerw [Category.id_comp]\n[GOAL]\nC : Type u\ninst\u271d\u2078 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\u2081\ninst\u271d\u2077 : Category.{max v u, w\u2081} D\nE : Type w\u2082\ninst\u271d\u2076 : Category.{max v u, w\u2082} E\nF : D \u2964 E\ninst\u271d\u2075 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst\u271d\u2074 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst\u271d\u00b3 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 E\ninst\u271d\u00b9 : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 F\ninst\u271d : (X : C) \u2192 (W : Cover J X) \u2192 (P : C\u1d52\u1d56 \u2964 D) \u2192 PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\nP : C\u1d52\u1d56 \u2964 D\n\u22a2 NatTrans.app (sheafificationWhiskerRightIso J F).inv P = (sheafifyCompIso J F P).inv\n[PROOFSTEP]\ndsimp [sheafificationWhiskerRightIso, sheafifyCompIso]\n[GOAL]\nC : Type u\ninst\u271d\u2078 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\u2081\ninst\u271d\u2077 : Category.{max v u, w\u2081} D\nE : Type w\u2082\ninst\u271d\u2076 : Category.{max v u, w\u2082} E\nF : D \u2964 E\ninst\u271d\u2075 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst\u271d\u2074 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst\u271d\u00b3 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 E\ninst\u271d\u00b9 : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 F\ninst\u271d : (X : C) \u2192 (W : Cover J X) \u2192 (P : C\u1d52\u1d56 \u2964 D) \u2192 PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\nP : C\u1d52\u1d56 \u2964 D\n\u22a2 ((\ud835\udfd9 (plusObj J (plusObj J (P \u22d9 F))) \u226b plusMap J (plusCompIso J F P).inv \u226b \ud835\udfd9 (plusObj J (plusObj J P \u22d9 F))) \u226b\n (plusCompIso J F (plusObj J P)).inv) \u226b\n \ud835\udfd9 (plusObj J (plusObj J P) \u22d9 F) =\n plusMap J (plusCompIso J F P).inv \u226b (plusCompIso J F (plusObj J P)).inv\n[PROOFSTEP]\nsimp only [Category.id_comp, Category.comp_id]\n[GOAL]\nC : Type u\ninst\u271d\u2078 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\u2081\ninst\u271d\u2077 : Category.{max v u, w\u2081} D\nE : Type w\u2082\ninst\u271d\u2076 : Category.{max v u, w\u2082} E\nF : D \u2964 E\ninst\u271d\u2075 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst\u271d\u2074 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst\u271d\u00b3 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 E\ninst\u271d\u00b9 : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 F\ninst\u271d : (X : C) \u2192 (W : Cover J X) \u2192 (P : C\u1d52\u1d56 \u2964 D) \u2192 PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\nP : C\u1d52\u1d56 \u2964 D\n\u22a2 (\ud835\udfd9 (plusObj J (plusObj J (P \u22d9 F))) \u226b plusMap J (plusCompIso J F P).inv) \u226b (plusCompIso J F (plusObj J P)).inv =\n plusMap J (plusCompIso J F P).inv \u226b (plusCompIso J F (plusObj J P)).inv\n[PROOFSTEP]\nerw [Category.id_comp]\n[GOAL]\nC : Type u\ninst\u271d\u2078 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\u2081\ninst\u271d\u2077 : Category.{max v u, w\u2081} D\nE : Type w\u2082\ninst\u271d\u2076 : Category.{max v u, w\u2082} E\nF : D \u2964 E\ninst\u271d\u2075 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst\u271d\u2074 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst\u271d\u00b3 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 E\ninst\u271d\u00b9 : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 F\ninst\u271d : (X : C) \u2192 (W : Cover J X) \u2192 (P : C\u1d52\u1d56 \u2964 D) \u2192 PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\nP : C\u1d52\u1d56 \u2964 D\n\u22a2 whiskerRight (toSheafify J P) F \u226b (sheafifyCompIso J F P).hom = toSheafify J (P \u22d9 F)\n[PROOFSTEP]\ndsimp [sheafifyCompIso]\n[GOAL]\nC : Type u\ninst\u271d\u2078 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\u2081\ninst\u271d\u2077 : Category.{max v u, w\u2081} D\nE : Type w\u2082\ninst\u271d\u2076 : Category.{max v u, w\u2082} E\nF : D \u2964 E\ninst\u271d\u2075 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst\u271d\u2074 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst\u271d\u00b3 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 E\ninst\u271d\u00b9 : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 F\ninst\u271d : (X : C) \u2192 (W : Cover J X) \u2192 (P : C\u1d52\u1d56 \u2964 D) \u2192 PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\nP : C\u1d52\u1d56 \u2964 D\n\u22a2 whiskerRight (toSheafify J P) F \u226b (plusCompIso J F (plusObj J P)).hom \u226b plusMap J (plusCompIso J F P).hom =\n toSheafify J (P \u22d9 F)\n[PROOFSTEP]\nerw [whiskerRight_comp, Category.assoc]\n[GOAL]\nC : Type u\ninst\u271d\u2078 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\u2081\ninst\u271d\u2077 : Category.{max v u, w\u2081} D\nE : Type w\u2082\ninst\u271d\u2076 : Category.{max v u, w\u2082} E\nF : D \u2964 E\ninst\u271d\u2075 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst\u271d\u2074 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst\u271d\u00b3 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 E\ninst\u271d\u00b9 : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 F\ninst\u271d : (X : C) \u2192 (W : Cover J X) \u2192 (P : C\u1d52\u1d56 \u2964 D) \u2192 PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\nP : C\u1d52\u1d56 \u2964 D\n\u22a2 whiskerRight (toPlus J P) F \u226b\n whiskerRight (plusMap J (toPlus J P)) F \u226b\n (plusCompIso J F (plusObj J P)).hom \u226b plusMap J (plusCompIso J F P).hom =\n toSheafify J (P \u22d9 F)\n[PROOFSTEP]\nslice_lhs 2 3 => rw [plusCompIso_whiskerRight]\n[GOAL]\ncase a.a\nC : Type u\ninst\u271d\u2078 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\u2081\ninst\u271d\u2077 : Category.{max v u, w\u2081} D\nE : Type w\u2082\ninst\u271d\u2076 : Category.{max v u, w\u2082} E\nF : D \u2964 E\ninst\u271d\u2075 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst\u271d\u2074 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst\u271d\u00b3 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 E\ninst\u271d\u00b9 : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 F\ninst\u271d : (X : C) \u2192 (W : Cover J X) \u2192 (P : C\u1d52\u1d56 \u2964 D) \u2192 PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\nP : C\u1d52\u1d56 \u2964 D\n| whiskerRight (plusMap J (toPlus J P)) F \u226b (plusCompIso J F (plusObj J P)).hom\ncase a.a\nC : Type u\ninst\u271d\u2078 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\u2081\ninst\u271d\u2077 : Category.{max v u, w\u2081} D\nE : Type w\u2082\ninst\u271d\u2076 : Category.{max v u, w\u2082} E\nF : D \u2964 E\ninst\u271d\u2075 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst\u271d\u2074 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst\u271d\u00b3 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 E\ninst\u271d\u00b9 : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 F\ninst\u271d : (X : C) \u2192 (W : Cover J X) \u2192 (P : C\u1d52\u1d56 \u2964 D) \u2192 PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\nP : C\u1d52\u1d56 \u2964 D\n| plusMap J (plusCompIso J F P).hom\ncase a\nC : Type u\ninst\u271d\u2078 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\u2081\ninst\u271d\u2077 : Category.{max v u, w\u2081} D\nE : Type w\u2082\ninst\u271d\u2076 : Category.{max v u, w\u2082} E\nF : D \u2964 E\ninst\u271d\u2075 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst\u271d\u2074 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst\u271d\u00b3 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 E\ninst\u271d\u00b9 : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 F\ninst\u271d : (X : C) \u2192 (W : Cover J X) \u2192 (P : C\u1d52\u1d56 \u2964 D) \u2192 PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\nP : C\u1d52\u1d56 \u2964 D\n| whiskerRight (toPlus J P) F\n[PROOFSTEP]\nrw [plusCompIso_whiskerRight]\n[GOAL]\ncase a.a\nC : Type u\ninst\u271d\u2078 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\u2081\ninst\u271d\u2077 : Category.{max v u, w\u2081} D\nE : Type w\u2082\ninst\u271d\u2076 : Category.{max v u, w\u2082} E\nF : D \u2964 E\ninst\u271d\u2075 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst\u271d\u2074 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst\u271d\u00b3 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 E\ninst\u271d\u00b9 : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 F\ninst\u271d : (X : C) \u2192 (W : Cover J X) \u2192 (P : C\u1d52\u1d56 \u2964 D) \u2192 PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\nP : C\u1d52\u1d56 \u2964 D\n| whiskerRight (plusMap J (toPlus J P)) F \u226b (plusCompIso J F (plusObj J P)).hom\ncase a.a\nC : Type u\ninst\u271d\u2078 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\u2081\ninst\u271d\u2077 : Category.{max v u, w\u2081} D\nE : Type w\u2082\ninst\u271d\u2076 : Category.{max v u, w\u2082} E\nF : D \u2964 E\ninst\u271d\u2075 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst\u271d\u2074 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst\u271d\u00b3 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 E\ninst\u271d\u00b9 : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 F\ninst\u271d : (X : C) \u2192 (W : Cover J X) \u2192 (P : C\u1d52\u1d56 \u2964 D) \u2192 PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\nP : C\u1d52\u1d56 \u2964 D\n| plusMap J (plusCompIso J F P).hom\ncase a\nC : Type u\ninst\u271d\u2078 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\u2081\ninst\u271d\u2077 : Category.{max v u, w\u2081} D\nE : Type w\u2082\ninst\u271d\u2076 : Category.{max v u, w\u2082} E\nF : D \u2964 E\ninst\u271d\u2075 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst\u271d\u2074 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst\u271d\u00b3 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 E\ninst\u271d\u00b9 : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 F\ninst\u271d : (X : C) \u2192 (W : Cover J X) \u2192 (P : C\u1d52\u1d56 \u2964 D) \u2192 PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\nP : C\u1d52\u1d56 \u2964 D\n| whiskerRight (toPlus J P) F\n[PROOFSTEP]\nrw [plusCompIso_whiskerRight]\n[GOAL]\ncase a.a\nC : Type u\ninst\u271d\u2078 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\u2081\ninst\u271d\u2077 : Category.{max v u, w\u2081} D\nE : Type w\u2082\ninst\u271d\u2076 : Category.{max v u, w\u2082} E\nF : D \u2964 E\ninst\u271d\u2075 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst\u271d\u2074 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst\u271d\u00b3 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 E\ninst\u271d\u00b9 : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 F\ninst\u271d : (X : C) \u2192 (W : Cover J X) \u2192 (P : C\u1d52\u1d56 \u2964 D) \u2192 PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\nP : C\u1d52\u1d56 \u2964 D\n| whiskerRight (plusMap J (toPlus J P)) F \u226b (plusCompIso J F (plusObj J P)).hom\ncase a.a\nC : Type u\ninst\u271d\u2078 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\u2081\ninst\u271d\u2077 : Category.{max v u, w\u2081} D\nE : Type w\u2082\ninst\u271d\u2076 : Category.{max v u, w\u2082} E\nF : D \u2964 E\ninst\u271d\u2075 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst\u271d\u2074 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst\u271d\u00b3 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 E\ninst\u271d\u00b9 : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 F\ninst\u271d : (X : C) \u2192 (W : Cover J X) \u2192 (P : C\u1d52\u1d56 \u2964 D) \u2192 PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\nP : C\u1d52\u1d56 \u2964 D\n| plusMap J (plusCompIso J F P).hom\ncase a\nC : Type u\ninst\u271d\u2078 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\u2081\ninst\u271d\u2077 : Category.{max v u, w\u2081} D\nE : Type w\u2082\ninst\u271d\u2076 : Category.{max v u, w\u2082} E\nF : D \u2964 E\ninst\u271d\u2075 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst\u271d\u2074 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst\u271d\u00b3 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 E\ninst\u271d\u00b9 : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 F\ninst\u271d : (X : C) \u2192 (W : Cover J X) \u2192 (P : C\u1d52\u1d56 \u2964 D) \u2192 PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\nP : C\u1d52\u1d56 \u2964 D\n| whiskerRight (toPlus J P) F\n[PROOFSTEP]\nrw [plusCompIso_whiskerRight]\n[GOAL]\nC : Type u\ninst\u271d\u2078 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\u2081\ninst\u271d\u2077 : Category.{max v u, w\u2081} D\nE : Type w\u2082\ninst\u271d\u2076 : Category.{max v u, w\u2082} E\nF : D \u2964 E\ninst\u271d\u2075 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst\u271d\u2074 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst\u271d\u00b3 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 E\ninst\u271d\u00b9 : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 F\ninst\u271d : (X : C) \u2192 (W : Cover J X) \u2192 (P : C\u1d52\u1d56 \u2964 D) \u2192 PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\nP : C\u1d52\u1d56 \u2964 D\n\u22a2 whiskerRight (toPlus J P) F \u226b\n ((plusCompIso J F P).hom \u226b plusMap J (whiskerRight (toPlus J P) F)) \u226b plusMap J (plusCompIso J F P).hom =\n toSheafify J (P \u22d9 F)\n[PROOFSTEP]\nrw [Category.assoc, \u2190 J.plusMap_comp, whiskerRight_toPlus_comp_plusCompIso_hom, \u2190 Category.assoc,\n whiskerRight_toPlus_comp_plusCompIso_hom]\n[GOAL]\nC : Type u\ninst\u271d\u2078 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\u2081\ninst\u271d\u2077 : Category.{max v u, w\u2081} D\nE : Type w\u2082\ninst\u271d\u2076 : Category.{max v u, w\u2082} E\nF : D \u2964 E\ninst\u271d\u2075 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst\u271d\u2074 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst\u271d\u00b3 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 E\ninst\u271d\u00b9 : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 F\ninst\u271d : (X : C) \u2192 (W : Cover J X) \u2192 (P : C\u1d52\u1d56 \u2964 D) \u2192 PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\nP : C\u1d52\u1d56 \u2964 D\n\u22a2 toPlus J (P \u22d9 F) \u226b plusMap J (toPlus J (P \u22d9 F)) = toSheafify J (P \u22d9 F)\n[PROOFSTEP]\nrfl\n[GOAL]\nC : Type u\ninst\u271d\u2078 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\u2081\ninst\u271d\u2077 : Category.{max v u, w\u2081} D\nE : Type w\u2082\ninst\u271d\u2076 : Category.{max v u, w\u2082} E\nF : D \u2964 E\ninst\u271d\u2075 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst\u271d\u2074 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst\u271d\u00b3 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 E\ninst\u271d\u00b9 : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 F\ninst\u271d : (X : C) \u2192 (W : Cover J X) \u2192 (P : C\u1d52\u1d56 \u2964 D) \u2192 PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\nP : C\u1d52\u1d56 \u2964 D\n\u22a2 toSheafify J (P \u22d9 F) \u226b (sheafifyCompIso J F P).inv = whiskerRight (toSheafify J P) F\n[PROOFSTEP]\nrw [Iso.comp_inv_eq]\n[GOAL]\nC : Type u\ninst\u271d\u2078 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\u2081\ninst\u271d\u2077 : Category.{max v u, w\u2081} D\nE : Type w\u2082\ninst\u271d\u2076 : Category.{max v u, w\u2082} E\nF : D \u2964 E\ninst\u271d\u2075 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst\u271d\u2074 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst\u271d\u00b3 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 E\ninst\u271d\u00b9 : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 F\ninst\u271d : (X : C) \u2192 (W : Cover J X) \u2192 (P : C\u1d52\u1d56 \u2964 D) \u2192 PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\nP : C\u1d52\u1d56 \u2964 D\n\u22a2 toSheafify J (P \u22d9 F) = whiskerRight (toSheafify J P) F \u226b (sheafifyCompIso J F P).hom\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\ninst\u271d\u00b9\u00b2 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\u2081\ninst\u271d\u00b9\u00b9 : Category.{max v u, w\u2081} D\nE : Type w\u2082\ninst\u271d\u00b9\u2070 : Category.{max v u, w\u2082} E\nF : D \u2964 E\ninst\u271d\u2079 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst\u271d\u2078 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst\u271d\u2077 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u2076 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 E\ninst\u271d\u2075 : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 F\ninst\u271d\u2074 : (X : C) \u2192 (W : Cover J X) \u2192 (P : C\u1d52\u1d56 \u2964 D) \u2192 PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\nP : C\u1d52\u1d56 \u2964 D\ninst\u271d\u00b3 : ConcreteCategory D\ninst\u271d\u00b2 : PreservesLimits (forget D)\ninst\u271d\u00b9 : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\ninst\u271d : ReflectsIsomorphisms (forget D)\n\u22a2 (sheafifyCompIso J F P).inv =\n sheafifyLift J (whiskerRight (toSheafify J P) F) (_ : Presheaf.IsSheaf J (sheafify J P \u22d9 F))\n[PROOFSTEP]\napply J.sheafifyLift_unique\n[GOAL]\ncase a\nC : Type u\ninst\u271d\u00b9\u00b2 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\u2081\ninst\u271d\u00b9\u00b9 : Category.{max v u, w\u2081} D\nE : Type w\u2082\ninst\u271d\u00b9\u2070 : Category.{max v u, w\u2082} E\nF : D \u2964 E\ninst\u271d\u2079 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst\u271d\u2078 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst\u271d\u2077 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u2076 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 E\ninst\u271d\u2075 : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 F\ninst\u271d\u2074 : (X : C) \u2192 (W : Cover J X) \u2192 (P : C\u1d52\u1d56 \u2964 D) \u2192 PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\nP : C\u1d52\u1d56 \u2964 D\ninst\u271d\u00b3 : ConcreteCategory D\ninst\u271d\u00b2 : PreservesLimits (forget D)\ninst\u271d\u00b9 : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\ninst\u271d : ReflectsIsomorphisms (forget D)\n\u22a2 toSheafify J (P \u22d9 F) \u226b (sheafifyCompIso J F P).inv = whiskerRight (toSheafify J P) F\n[PROOFSTEP]\nrw [Iso.comp_inv_eq]\n[GOAL]\ncase a\nC : Type u\ninst\u271d\u00b9\u00b2 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\u2081\ninst\u271d\u00b9\u00b9 : Category.{max v u, w\u2081} D\nE : Type w\u2082\ninst\u271d\u00b9\u2070 : Category.{max v u, w\u2082} E\nF : D \u2964 E\ninst\u271d\u2079 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst\u271d\u2078 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst\u271d\u2077 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u2076 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 E\ninst\u271d\u2075 : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 F\ninst\u271d\u2074 : (X : C) \u2192 (W : Cover J X) \u2192 (P : C\u1d52\u1d56 \u2964 D) \u2192 PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\nP : C\u1d52\u1d56 \u2964 D\ninst\u271d\u00b3 : ConcreteCategory D\ninst\u271d\u00b2 : PreservesLimits (forget D)\ninst\u271d\u00b9 : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\ninst\u271d : ReflectsIsomorphisms (forget D)\n\u22a2 toSheafify J (P \u22d9 F) = whiskerRight (toSheafify J P) F \u226b (sheafifyCompIso J F P).hom\n[PROOFSTEP]\nsimp\n", "meta": {"mathlib_filename": "Mathlib.CategoryTheory.Sites.CompatibleSheafification", "llama_tokens": 17326, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.6477982179521103, "lm_q2_score": 0.2720245569956929, "lm_q1q2_score": 0.17621702326102212}} {"text": "[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nA : Type u\u2083\ninst\u271d : Category.{v\u2083, u\u2083} A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nL : GrothendieckTopology A\nU\u271d : C\nS\u271d : Sieve U\u271d\nhS : S\u271d \u2208 GrothendieckTopology.sieves J U\u271d\n\u22a2 Sieve.functorPushforward (\ud835\udfed C) S\u271d \u2208 GrothendieckTopology.sieves J ((\ud835\udfed C).obj U\u271d)\n[PROOFSTEP]\nsimpa using hS\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nA : Type u\u2083\ninst\u271d : Category.{v\u2083, u\u2083} A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nL : GrothendieckTopology A\nF : C \u2964 D\nhF : CoverPreserving J K F\nG : D \u2964 A\nhG : CoverPreserving K L G\nU\u271d : C\nS\u271d : Sieve U\u271d\nhS : S\u271d \u2208 GrothendieckTopology.sieves J U\u271d\n\u22a2 Sieve.functorPushforward (F \u22d9 G) S\u271d \u2208 GrothendieckTopology.sieves L ((F \u22d9 G).obj U\u271d)\n[PROOFSTEP]\nrw [Sieve.functorPushforward_comp]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nA : Type u\u2083\ninst\u271d : Category.{v\u2083, u\u2083} A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nL : GrothendieckTopology A\nF : C \u2964 D\nhF : CoverPreserving J K F\nG : D \u2964 A\nhG : CoverPreserving K L G\nU\u271d : C\nS\u271d : Sieve U\u271d\nhS : S\u271d \u2208 GrothendieckTopology.sieves J U\u271d\n\u22a2 Sieve.functorPushforward G (Sieve.functorPushforward F S\u271d) \u2208 GrothendieckTopology.sieves L ((F \u22d9 G).obj U\u271d)\n[PROOFSTEP]\nexact hG.cover_preserve (hF.cover_preserve hS)\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nA : Type u\u2083\ninst\u271d : Category.{v\u2083, u\u2083} A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nL : GrothendieckTopology A\nG : C \u2964 D\nhG : CompatiblePreserving K G\n\u2131 : SheafOfTypes K\nZ : C\nT : Presieve Z\nx : FamilyOfElements (G.op \u22d9 \u2131.val) T\nh : Compatible x\n\u22a2 Compatible (FamilyOfElements.functorPushforward G x)\n[PROOFSTEP]\nrintro Z\u2081 Z\u2082 W g\u2081 g\u2082 f\u2081' f\u2082' H\u2081 H\u2082 eq\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nA : Type u\u2083\ninst\u271d : Category.{v\u2083, u\u2083} A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nL : GrothendieckTopology A\nG : C \u2964 D\nhG : CompatiblePreserving K G\n\u2131 : SheafOfTypes K\nZ : C\nT : Presieve Z\nx : FamilyOfElements (G.op \u22d9 \u2131.val) T\nh : Compatible x\nZ\u2081 Z\u2082 W : D\ng\u2081 : W \u27f6 Z\u2081\ng\u2082 : W \u27f6 Z\u2082\nf\u2081' : Z\u2081 \u27f6 G.obj Z\nf\u2082' : Z\u2082 \u27f6 G.obj Z\nH\u2081 : Presieve.functorPushforward G T f\u2081'\nH\u2082 : Presieve.functorPushforward G T f\u2082'\neq : g\u2081 \u226b f\u2081' = g\u2082 \u226b f\u2082'\n\u22a2 \u2131.val.map g\u2081.op (FamilyOfElements.functorPushforward G x f\u2081' H\u2081) =\n \u2131.val.map g\u2082.op (FamilyOfElements.functorPushforward G x f\u2082' H\u2082)\n[PROOFSTEP]\nunfold FamilyOfElements.functorPushforward\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nA : Type u\u2083\ninst\u271d : Category.{v\u2083, u\u2083} A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nL : GrothendieckTopology A\nG : C \u2964 D\nhG : CompatiblePreserving K G\n\u2131 : SheafOfTypes K\nZ : C\nT : Presieve Z\nx : FamilyOfElements (G.op \u22d9 \u2131.val) T\nh : Compatible x\nZ\u2081 Z\u2082 W : D\ng\u2081 : W \u27f6 Z\u2081\ng\u2082 : W \u27f6 Z\u2082\nf\u2081' : Z\u2081 \u27f6 G.obj Z\nf\u2082' : Z\u2082 \u27f6 G.obj Z\nH\u2081 : Presieve.functorPushforward G T f\u2081'\nH\u2082 : Presieve.functorPushforward G T f\u2082'\neq : g\u2081 \u226b f\u2081' = g\u2082 \u226b f\u2082'\n\u22a2 \u2131.val.map g\u2081.op\n (FunctorPushforwardStructure.casesOn (getFunctorPushforwardStructure H\u2081) fun Z_1 g h h\u2081 fac =>\n \u2131.val.map h.op (x g h\u2081)) =\n \u2131.val.map g\u2082.op\n (FunctorPushforwardStructure.casesOn (getFunctorPushforwardStructure H\u2082) fun Z_1 g h h\u2081 fac =>\n \u2131.val.map h.op (x g h\u2081))\n[PROOFSTEP]\nrcases getFunctorPushforwardStructure H\u2081 with \u27e8X\u2081, f\u2081, h\u2081, hf\u2081, rfl\u27e9\n[GOAL]\ncase mk\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nA : Type u\u2083\ninst\u271d : Category.{v\u2083, u\u2083} A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nL : GrothendieckTopology A\nG : C \u2964 D\nhG : CompatiblePreserving K G\n\u2131 : SheafOfTypes K\nZ : C\nT : Presieve Z\nx : FamilyOfElements (G.op \u22d9 \u2131.val) T\nh : Compatible x\nZ\u2081 Z\u2082 W : D\ng\u2081 : W \u27f6 Z\u2081\ng\u2082 : W \u27f6 Z\u2082\nf\u2082' : Z\u2082 \u27f6 G.obj Z\nH\u2082 : Presieve.functorPushforward G T f\u2082'\nX\u2081 : C\nf\u2081 : X\u2081 \u27f6 Z\nh\u2081 : Z\u2081 \u27f6 G.obj X\u2081\nhf\u2081 : T f\u2081\nH\u2081 : Presieve.functorPushforward G T (h\u2081 \u226b G.map f\u2081)\neq : g\u2081 \u226b h\u2081 \u226b G.map f\u2081 = g\u2082 \u226b f\u2082'\n\u22a2 \u2131.val.map g\u2081.op\n (FunctorPushforwardStructure.casesOn\n { preobj := X\u2081, premap := f\u2081, lift := h\u2081, cover := hf\u2081, fac := (_ : h\u2081 \u226b G.map f\u2081 = h\u2081 \u226b G.map f\u2081) }\n fun Z_1 g h h\u2081_1 fac => \u2131.val.map h.op (x g h\u2081_1)) =\n \u2131.val.map g\u2082.op\n (FunctorPushforwardStructure.casesOn (getFunctorPushforwardStructure H\u2082) fun Z_1 g h h\u2081 fac =>\n \u2131.val.map h.op (x g h\u2081))\n[PROOFSTEP]\nrcases getFunctorPushforwardStructure H\u2082 with \u27e8X\u2082, f\u2082, h\u2082, hf\u2082, rfl\u27e9\n[GOAL]\ncase mk.mk\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nA : Type u\u2083\ninst\u271d : Category.{v\u2083, u\u2083} A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nL : GrothendieckTopology A\nG : C \u2964 D\nhG : CompatiblePreserving K G\n\u2131 : SheafOfTypes K\nZ : C\nT : Presieve Z\nx : FamilyOfElements (G.op \u22d9 \u2131.val) T\nh : Compatible x\nZ\u2081 Z\u2082 W : D\ng\u2081 : W \u27f6 Z\u2081\ng\u2082 : W \u27f6 Z\u2082\nX\u2081 : C\nf\u2081 : X\u2081 \u27f6 Z\nh\u2081 : Z\u2081 \u27f6 G.obj X\u2081\nhf\u2081 : T f\u2081\nH\u2081 : Presieve.functorPushforward G T (h\u2081 \u226b G.map f\u2081)\nX\u2082 : C\nf\u2082 : X\u2082 \u27f6 Z\nh\u2082 : Z\u2082 \u27f6 G.obj X\u2082\nhf\u2082 : T f\u2082\nH\u2082 : Presieve.functorPushforward G T (h\u2082 \u226b G.map f\u2082)\neq : g\u2081 \u226b h\u2081 \u226b G.map f\u2081 = g\u2082 \u226b h\u2082 \u226b G.map f\u2082\n\u22a2 \u2131.val.map g\u2081.op\n (FunctorPushforwardStructure.casesOn\n { preobj := X\u2081, premap := f\u2081, lift := h\u2081, cover := hf\u2081, fac := (_ : h\u2081 \u226b G.map f\u2081 = h\u2081 \u226b G.map f\u2081) }\n fun Z_1 g h h\u2081_1 fac => \u2131.val.map h.op (x g h\u2081_1)) =\n \u2131.val.map g\u2082.op\n (FunctorPushforwardStructure.casesOn\n { preobj := X\u2082, premap := f\u2082, lift := h\u2082, cover := hf\u2082, fac := (_ : h\u2082 \u226b G.map f\u2082 = h\u2082 \u226b G.map f\u2082) }\n fun Z_1 g h h\u2081 fac => \u2131.val.map h.op (x g h\u2081))\n[PROOFSTEP]\nsuffices \u2131.val.map (g\u2081 \u226b h\u2081).op (x f\u2081 hf\u2081) = \u2131.val.map (g\u2082 \u226b h\u2082).op (x f\u2082 hf\u2082) by simpa using this\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nA : Type u\u2083\ninst\u271d : Category.{v\u2083, u\u2083} A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nL : GrothendieckTopology A\nG : C \u2964 D\nhG : CompatiblePreserving K G\n\u2131 : SheafOfTypes K\nZ : C\nT : Presieve Z\nx : FamilyOfElements (G.op \u22d9 \u2131.val) T\nh : Compatible x\nZ\u2081 Z\u2082 W : D\ng\u2081 : W \u27f6 Z\u2081\ng\u2082 : W \u27f6 Z\u2082\nX\u2081 : C\nf\u2081 : X\u2081 \u27f6 Z\nh\u2081 : Z\u2081 \u27f6 G.obj X\u2081\nhf\u2081 : T f\u2081\nH\u2081 : Presieve.functorPushforward G T (h\u2081 \u226b G.map f\u2081)\nX\u2082 : C\nf\u2082 : X\u2082 \u27f6 Z\nh\u2082 : Z\u2082 \u27f6 G.obj X\u2082\nhf\u2082 : T f\u2082\nH\u2082 : Presieve.functorPushforward G T (h\u2082 \u226b G.map f\u2082)\neq : g\u2081 \u226b h\u2081 \u226b G.map f\u2081 = g\u2082 \u226b h\u2082 \u226b G.map f\u2082\nthis : \u2131.val.map (g\u2081 \u226b h\u2081).op (x f\u2081 hf\u2081) = \u2131.val.map (g\u2082 \u226b h\u2082).op (x f\u2082 hf\u2082)\n\u22a2 \u2131.val.map g\u2081.op\n (FunctorPushforwardStructure.casesOn\n { preobj := X\u2081, premap := f\u2081, lift := h\u2081, cover := hf\u2081, fac := (_ : h\u2081 \u226b G.map f\u2081 = h\u2081 \u226b G.map f\u2081) }\n fun Z_1 g h h\u2081_1 fac => \u2131.val.map h.op (x g h\u2081_1)) =\n \u2131.val.map g\u2082.op\n (FunctorPushforwardStructure.casesOn\n { preobj := X\u2082, premap := f\u2082, lift := h\u2082, cover := hf\u2082, fac := (_ : h\u2082 \u226b G.map f\u2082 = h\u2082 \u226b G.map f\u2082) }\n fun Z_1 g h h\u2081 fac => \u2131.val.map h.op (x g h\u2081))\n[PROOFSTEP]\nsimpa using this\n[GOAL]\ncase mk.mk\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nA : Type u\u2083\ninst\u271d : Category.{v\u2083, u\u2083} A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nL : GrothendieckTopology A\nG : C \u2964 D\nhG : CompatiblePreserving K G\n\u2131 : SheafOfTypes K\nZ : C\nT : Presieve Z\nx : FamilyOfElements (G.op \u22d9 \u2131.val) T\nh : Compatible x\nZ\u2081 Z\u2082 W : D\ng\u2081 : W \u27f6 Z\u2081\ng\u2082 : W \u27f6 Z\u2082\nX\u2081 : C\nf\u2081 : X\u2081 \u27f6 Z\nh\u2081 : Z\u2081 \u27f6 G.obj X\u2081\nhf\u2081 : T f\u2081\nH\u2081 : Presieve.functorPushforward G T (h\u2081 \u226b G.map f\u2081)\nX\u2082 : C\nf\u2082 : X\u2082 \u27f6 Z\nh\u2082 : Z\u2082 \u27f6 G.obj X\u2082\nhf\u2082 : T f\u2082\nH\u2082 : Presieve.functorPushforward G T (h\u2082 \u226b G.map f\u2082)\neq : g\u2081 \u226b h\u2081 \u226b G.map f\u2081 = g\u2082 \u226b h\u2082 \u226b G.map f\u2082\n\u22a2 \u2131.val.map (g\u2081 \u226b h\u2081).op (x f\u2081 hf\u2081) = \u2131.val.map (g\u2082 \u226b h\u2082).op (x f\u2082 hf\u2082)\n[PROOFSTEP]\napply hG.Compatible \u2131 h _ _ hf\u2081 hf\u2082\n[GOAL]\ncase mk.mk\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nA : Type u\u2083\ninst\u271d : Category.{v\u2083, u\u2083} A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nL : GrothendieckTopology A\nG : C \u2964 D\nhG : CompatiblePreserving K G\n\u2131 : SheafOfTypes K\nZ : C\nT : Presieve Z\nx : FamilyOfElements (G.op \u22d9 \u2131.val) T\nh : Compatible x\nZ\u2081 Z\u2082 W : D\ng\u2081 : W \u27f6 Z\u2081\ng\u2082 : W \u27f6 Z\u2082\nX\u2081 : C\nf\u2081 : X\u2081 \u27f6 Z\nh\u2081 : Z\u2081 \u27f6 G.obj X\u2081\nhf\u2081 : T f\u2081\nH\u2081 : Presieve.functorPushforward G T (h\u2081 \u226b G.map f\u2081)\nX\u2082 : C\nf\u2082 : X\u2082 \u27f6 Z\nh\u2082 : Z\u2082 \u27f6 G.obj X\u2082\nhf\u2082 : T f\u2082\nH\u2082 : Presieve.functorPushforward G T (h\u2082 \u226b G.map f\u2082)\neq : g\u2081 \u226b h\u2081 \u226b G.map f\u2081 = g\u2082 \u226b h\u2082 \u226b G.map f\u2082\n\u22a2 (g\u2081 \u226b h\u2081) \u226b G.map f\u2081 = (g\u2082 \u226b h\u2082) \u226b G.map f\u2082\n[PROOFSTEP]\nsimpa using eq\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nA : Type u\u2083\ninst\u271d : Category.{v\u2083, u\u2083} A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nL : GrothendieckTopology A\nG : C \u2964 D\nhG : CompatiblePreserving K G\n\u2131 : SheafOfTypes K\nZ : C\nT : Presieve Z\nx : FamilyOfElements (G.op \u22d9 \u2131.val) T\nh : FamilyOfElements.Compatible x\nY : C\nf : Y \u27f6 Z\nhf : T f\n\u22a2 FamilyOfElements.functorPushforward G x (G.map f) (_ : Presieve.functorPushforward G T (G.map f)) = x f hf\n[PROOFSTEP]\nunfold FamilyOfElements.functorPushforward\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nA : Type u\u2083\ninst\u271d : Category.{v\u2083, u\u2083} A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nL : GrothendieckTopology A\nG : C \u2964 D\nhG : CompatiblePreserving K G\n\u2131 : SheafOfTypes K\nZ : C\nT : Presieve Z\nx : FamilyOfElements (G.op \u22d9 \u2131.val) T\nh : FamilyOfElements.Compatible x\nY : C\nf : Y \u27f6 Z\nhf : T f\n\u22a2 (FunctorPushforwardStructure.casesOn (getFunctorPushforwardStructure (_ : Presieve.functorPushforward G T (G.map f)))\n fun Z_1 g h h\u2081 fac => \u2131.val.map h.op (x g h\u2081)) =\n x f hf\n[PROOFSTEP]\nrcases e\u2081 : getFunctorPushforwardStructure (image_mem_functorPushforward G T hf) with \u27e8X, g, f', hg, eq\u27e9\n[GOAL]\ncase mk\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nA : Type u\u2083\ninst\u271d : Category.{v\u2083, u\u2083} A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nL : GrothendieckTopology A\nG : C \u2964 D\nhG : CompatiblePreserving K G\n\u2131 : SheafOfTypes K\nZ : C\nT : Presieve Z\nx : FamilyOfElements (G.op \u22d9 \u2131.val) T\nh : FamilyOfElements.Compatible x\nY : C\nf : Y \u27f6 Z\nhf : T f\nX : C\ng : X \u27f6 Z\nf' : G.obj Y \u27f6 G.obj X\nhg : T g\neq : G.map f = f' \u226b G.map g\ne\u2081 :\n getFunctorPushforwardStructure (_ : Presieve.functorPushforward G T (G.map f)) =\n { preobj := X, premap := g, lift := f', cover := hg, fac := eq }\n\u22a2 (FunctorPushforwardStructure.casesOn { preobj := X, premap := g, lift := f', cover := hg, fac := eq }\n fun Z_1 g h h\u2081 fac => \u2131.val.map h.op (x g h\u2081)) =\n x f hf\n[PROOFSTEP]\nsimpa using hG.Compatible \u2131 h f' (\ud835\udfd9 _) hg hf (by simp [eq])\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nA : Type u\u2083\ninst\u271d : Category.{v\u2083, u\u2083} A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nL : GrothendieckTopology A\nG : C \u2964 D\nhG : CompatiblePreserving K G\n\u2131 : SheafOfTypes K\nZ : C\nT : Presieve Z\nx : FamilyOfElements (G.op \u22d9 \u2131.val) T\nh : FamilyOfElements.Compatible x\nY : C\nf : Y \u27f6 Z\nhf : T f\nX : C\ng : X \u27f6 Z\nf' : G.obj Y \u27f6 G.obj X\nhg : T g\neq : G.map f = f' \u226b G.map g\ne\u2081 :\n getFunctorPushforwardStructure (_ : Presieve.functorPushforward G T (G.map f)) =\n { preobj := X, premap := g, lift := f', cover := hg, fac := eq }\n\u22a2 f' \u226b G.map g = \ud835\udfd9 (G.obj Y) \u226b G.map f\n[PROOFSTEP]\nsimp [eq]\n[GOAL]\nC\u271d : Type u\u2081\ninst\u271d\u2075 : Category.{v\u2081, u\u2081} C\u271d\nD\u271d : Type u\u2082\ninst\u271d\u2074 : Category.{v\u2082, u\u2082} D\u271d\nA : Type u\u2083\ninst\u271d\u00b3 : Category.{v\u2083, u\u2083} A\nJ : GrothendieckTopology C\u271d\nK\u271d : GrothendieckTopology D\u271d\nL : GrothendieckTopology A\nG\u271d : C\u271d \u2964 D\u271d\nhG : CompatiblePreserving K\u271d G\u271d\n\u2131 : SheafOfTypes K\u271d\nZ : C\u271d\nT : Presieve Z\nx : FamilyOfElements (G\u271d.op \u22d9 \u2131.val) T\nh : Compatible x\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} D\nK : GrothendieckTopology D\nG : C \u2964 D\ninst\u271d : RepresentablyFlat G\n\u22a2 CompatiblePreserving K G\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase Compatible\nC\u271d : Type u\u2081\ninst\u271d\u2075 : Category.{v\u2081, u\u2081} C\u271d\nD\u271d : Type u\u2082\ninst\u271d\u2074 : Category.{v\u2082, u\u2082} D\u271d\nA : Type u\u2083\ninst\u271d\u00b3 : Category.{v\u2083, u\u2083} A\nJ : GrothendieckTopology C\u271d\nK\u271d : GrothendieckTopology D\u271d\nL : GrothendieckTopology A\nG\u271d : C\u271d \u2964 D\u271d\nhG : CompatiblePreserving K\u271d G\u271d\n\u2131 : SheafOfTypes K\u271d\nZ : C\u271d\nT : Presieve Z\nx : FamilyOfElements (G\u271d.op \u22d9 \u2131.val) T\nh : Compatible x\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} D\nK : GrothendieckTopology D\nG : C \u2964 D\ninst\u271d : RepresentablyFlat G\n\u22a2 \u2200 (\u2131 : SheafOfTypes K) {Z : C} {T : Presieve Z} {x : FamilyOfElements (G.op \u22d9 \u2131.val) T},\n Compatible x \u2192\n \u2200 {Y\u2081 Y\u2082 : C} {X : D} (f\u2081 : X \u27f6 G.obj Y\u2081) (f\u2082 : X \u27f6 G.obj Y\u2082) {g\u2081 : Y\u2081 \u27f6 Z} {g\u2082 : Y\u2082 \u27f6 Z} (hg\u2081 : T g\u2081)\n (hg\u2082 : T g\u2082), f\u2081 \u226b G.map g\u2081 = f\u2082 \u226b G.map g\u2082 \u2192 \u2131.val.map f\u2081.op (x g\u2081 hg\u2081) = \u2131.val.map f\u2082.op (x g\u2082 hg\u2082)\n[PROOFSTEP]\nintro \u2131 Z T x hx Y\u2081 Y\u2082 X f\u2081 f\u2082 g\u2081 g\u2082 hg\u2081 hg\u2082 e\n[GOAL]\ncase Compatible\nC\u271d : Type u\u2081\ninst\u271d\u2075 : Category.{v\u2081, u\u2081} C\u271d\nD\u271d : Type u\u2082\ninst\u271d\u2074 : Category.{v\u2082, u\u2082} D\u271d\nA : Type u\u2083\ninst\u271d\u00b3 : Category.{v\u2083, u\u2083} A\nJ : GrothendieckTopology C\u271d\nK\u271d : GrothendieckTopology D\u271d\nL : GrothendieckTopology A\nG\u271d : C\u271d \u2964 D\u271d\nhG : CompatiblePreserving K\u271d G\u271d\n\u2131\u271d : SheafOfTypes K\u271d\nZ\u271d : C\u271d\nT\u271d : Presieve Z\u271d\nx\u271d : FamilyOfElements (G\u271d.op \u22d9 \u2131\u271d.val) T\u271d\nh : Compatible x\u271d\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} D\nK : GrothendieckTopology D\nG : C \u2964 D\ninst\u271d : RepresentablyFlat G\n\u2131 : SheafOfTypes K\nZ : C\nT : Presieve Z\nx : FamilyOfElements (G.op \u22d9 \u2131.val) T\nhx : Compatible x\nY\u2081 Y\u2082 : C\nX : D\nf\u2081 : X \u27f6 G.obj Y\u2081\nf\u2082 : X \u27f6 G.obj Y\u2082\ng\u2081 : Y\u2081 \u27f6 Z\ng\u2082 : Y\u2082 \u27f6 Z\nhg\u2081 : T g\u2081\nhg\u2082 : T g\u2082\ne : f\u2081 \u226b G.map g\u2081 = f\u2082 \u226b G.map g\u2082\n\u22a2 \u2131.val.map f\u2081.op (x g\u2081 hg\u2081) = \u2131.val.map f\u2082.op (x g\u2082 hg\u2082)\n[PROOFSTEP]\nlet c : Cone (cospan g\u2081 g\u2082 \u22d9 G) :=\n (Cones.postcompose (diagramIsoCospan (cospan g\u2081 g\u2082 \u22d9 G)).inv).obj\n (PullbackCone.mk f\u2081 f\u2082 e)\n /-\n This can then be viewed as a cospan of structured arrows, and we may obtain an arbitrary cone\n over it since `StructuredArrow W u` is cofiltered.\n Then, it suffices to prove that it is compatible when restricted onto `u(c'.X.right)`.\n -/\n[GOAL]\ncase Compatible\nC\u271d : Type u\u2081\ninst\u271d\u2075 : Category.{v\u2081, u\u2081} C\u271d\nD\u271d : Type u\u2082\ninst\u271d\u2074 : Category.{v\u2082, u\u2082} D\u271d\nA : Type u\u2083\ninst\u271d\u00b3 : Category.{v\u2083, u\u2083} A\nJ : GrothendieckTopology C\u271d\nK\u271d : GrothendieckTopology D\u271d\nL : GrothendieckTopology A\nG\u271d : C\u271d \u2964 D\u271d\nhG : CompatiblePreserving K\u271d G\u271d\n\u2131\u271d : SheafOfTypes K\u271d\nZ\u271d : C\u271d\nT\u271d : Presieve Z\u271d\nx\u271d : FamilyOfElements (G\u271d.op \u22d9 \u2131\u271d.val) T\u271d\nh : Compatible x\u271d\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} D\nK : GrothendieckTopology D\nG : C \u2964 D\ninst\u271d : RepresentablyFlat G\n\u2131 : SheafOfTypes K\nZ : C\nT : Presieve Z\nx : FamilyOfElements (G.op \u22d9 \u2131.val) T\nhx : Compatible x\nY\u2081 Y\u2082 : C\nX : D\nf\u2081 : X \u27f6 G.obj Y\u2081\nf\u2082 : X \u27f6 G.obj Y\u2082\ng\u2081 : Y\u2081 \u27f6 Z\ng\u2082 : Y\u2082 \u27f6 Z\nhg\u2081 : T g\u2081\nhg\u2082 : T g\u2082\ne : f\u2081 \u226b G.map g\u2081 = f\u2082 \u226b G.map g\u2082\nc : Cone (cospan g\u2081 g\u2082 \u22d9 G) :=\n (Cones.postcompose (diagramIsoCospan (cospan g\u2081 g\u2082 \u22d9 G)).inv).obj (PullbackCone.mk f\u2081 f\u2082 e)\n\u22a2 \u2131.val.map f\u2081.op (x g\u2081 hg\u2081) = \u2131.val.map f\u2082.op (x g\u2082 hg\u2082)\n[PROOFSTEP]\nlet c' := IsCofiltered.cone (c.toStructuredArrow \u22d9 StructuredArrow.pre _ _ _)\n[GOAL]\ncase Compatible\nC\u271d : Type u\u2081\ninst\u271d\u2075 : Category.{v\u2081, u\u2081} C\u271d\nD\u271d : Type u\u2082\ninst\u271d\u2074 : Category.{v\u2082, u\u2082} D\u271d\nA : Type u\u2083\ninst\u271d\u00b3 : Category.{v\u2083, u\u2083} A\nJ : GrothendieckTopology C\u271d\nK\u271d : GrothendieckTopology D\u271d\nL : GrothendieckTopology A\nG\u271d : C\u271d \u2964 D\u271d\nhG : CompatiblePreserving K\u271d G\u271d\n\u2131\u271d : SheafOfTypes K\u271d\nZ\u271d : C\u271d\nT\u271d : Presieve Z\u271d\nx\u271d : FamilyOfElements (G\u271d.op \u22d9 \u2131\u271d.val) T\u271d\nh : Compatible x\u271d\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} D\nK : GrothendieckTopology D\nG : C \u2964 D\ninst\u271d : RepresentablyFlat G\n\u2131 : SheafOfTypes K\nZ : C\nT : Presieve Z\nx : FamilyOfElements (G.op \u22d9 \u2131.val) T\nhx : Compatible x\nY\u2081 Y\u2082 : C\nX : D\nf\u2081 : X \u27f6 G.obj Y\u2081\nf\u2082 : X \u27f6 G.obj Y\u2082\ng\u2081 : Y\u2081 \u27f6 Z\ng\u2082 : Y\u2082 \u27f6 Z\nhg\u2081 : T g\u2081\nhg\u2082 : T g\u2082\ne : f\u2081 \u226b G.map g\u2081 = f\u2082 \u226b G.map g\u2082\nc : Cone (cospan g\u2081 g\u2082 \u22d9 G) :=\n (Cones.postcompose (diagramIsoCospan (cospan g\u2081 g\u2082 \u22d9 G)).inv).obj (PullbackCone.mk f\u2081 f\u2082 e)\nc' : Cone (Cone.toStructuredArrow c \u22d9 StructuredArrow.pre c.pt (cospan g\u2081 g\u2082) G) :=\n IsCofiltered.cone (Cone.toStructuredArrow c \u22d9 StructuredArrow.pre c.pt (cospan g\u2081 g\u2082) G)\n\u22a2 \u2131.val.map f\u2081.op (x g\u2081 hg\u2081) = \u2131.val.map f\u2082.op (x g\u2082 hg\u2082)\n[PROOFSTEP]\nhave eq\u2081 : f\u2081 = (c'.pt.hom \u226b G.map (c'.\u03c0.app left).right) \u226b eqToHom (by simp) :=\n by\n erw [\u2190 (c'.\u03c0.app left).w]\n dsimp\n simp\n[GOAL]\nC\u271d : Type u\u2081\ninst\u271d\u2075 : Category.{v\u2081, u\u2081} C\u271d\nD\u271d : Type u\u2082\ninst\u271d\u2074 : Category.{v\u2082, u\u2082} D\u271d\nA : Type u\u2083\ninst\u271d\u00b3 : Category.{v\u2083, u\u2083} A\nJ : GrothendieckTopology C\u271d\nK\u271d : GrothendieckTopology D\u271d\nL : GrothendieckTopology A\nG\u271d : C\u271d \u2964 D\u271d\nhG : CompatiblePreserving K\u271d G\u271d\n\u2131\u271d : SheafOfTypes K\u271d\nZ\u271d : C\u271d\nT\u271d : Presieve Z\u271d\nx\u271d : FamilyOfElements (G\u271d.op \u22d9 \u2131\u271d.val) T\u271d\nh : Compatible x\u271d\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} D\nK : GrothendieckTopology D\nG : C \u2964 D\ninst\u271d : RepresentablyFlat G\n\u2131 : SheafOfTypes K\nZ : C\nT : Presieve Z\nx : FamilyOfElements (G.op \u22d9 \u2131.val) T\nhx : Compatible x\nY\u2081 Y\u2082 : C\nX : D\nf\u2081 : X \u27f6 G.obj Y\u2081\nf\u2082 : X \u27f6 G.obj Y\u2082\ng\u2081 : Y\u2081 \u27f6 Z\ng\u2082 : Y\u2082 \u27f6 Z\nhg\u2081 : T g\u2081\nhg\u2082 : T g\u2082\ne : f\u2081 \u226b G.map g\u2081 = f\u2082 \u226b G.map g\u2082\nc : Cone (cospan g\u2081 g\u2082 \u22d9 G) :=\n (Cones.postcompose (diagramIsoCospan (cospan g\u2081 g\u2082 \u22d9 G)).inv).obj (PullbackCone.mk f\u2081 f\u2082 e)\nc' : Cone (Cone.toStructuredArrow c \u22d9 StructuredArrow.pre c.pt (cospan g\u2081 g\u2082) G) :=\n IsCofiltered.cone (Cone.toStructuredArrow c \u22d9 StructuredArrow.pre c.pt (cospan g\u2081 g\u2082) G)\n\u22a2 G.obj ((Cone.toStructuredArrow c \u22d9 StructuredArrow.pre c.pt (cospan g\u2081 g\u2082) G).obj left).right = G.obj Y\u2081\n[PROOFSTEP]\nsimp\n[GOAL]\nC\u271d : Type u\u2081\ninst\u271d\u2075 : Category.{v\u2081, u\u2081} C\u271d\nD\u271d : Type u\u2082\ninst\u271d\u2074 : Category.{v\u2082, u\u2082} D\u271d\nA : Type u\u2083\ninst\u271d\u00b3 : Category.{v\u2083, u\u2083} A\nJ : GrothendieckTopology C\u271d\nK\u271d : GrothendieckTopology D\u271d\nL : GrothendieckTopology A\nG\u271d : C\u271d \u2964 D\u271d\nhG : CompatiblePreserving K\u271d G\u271d\n\u2131\u271d : SheafOfTypes K\u271d\nZ\u271d : C\u271d\nT\u271d : Presieve Z\u271d\nx\u271d : FamilyOfElements (G\u271d.op \u22d9 \u2131\u271d.val) T\u271d\nh : Compatible x\u271d\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} D\nK : GrothendieckTopology D\nG : C \u2964 D\ninst\u271d : RepresentablyFlat G\n\u2131 : SheafOfTypes K\nZ : C\nT : Presieve Z\nx : FamilyOfElements (G.op \u22d9 \u2131.val) T\nhx : Compatible x\nY\u2081 Y\u2082 : C\nX : D\nf\u2081 : X \u27f6 G.obj Y\u2081\nf\u2082 : X \u27f6 G.obj Y\u2082\ng\u2081 : Y\u2081 \u27f6 Z\ng\u2082 : Y\u2082 \u27f6 Z\nhg\u2081 : T g\u2081\nhg\u2082 : T g\u2082\ne : f\u2081 \u226b G.map g\u2081 = f\u2082 \u226b G.map g\u2082\nc : Cone (cospan g\u2081 g\u2082 \u22d9 G) :=\n (Cones.postcompose (diagramIsoCospan (cospan g\u2081 g\u2082 \u22d9 G)).inv).obj (PullbackCone.mk f\u2081 f\u2082 e)\nc' : Cone (Cone.toStructuredArrow c \u22d9 StructuredArrow.pre c.pt (cospan g\u2081 g\u2082) G) :=\n IsCofiltered.cone (Cone.toStructuredArrow c \u22d9 StructuredArrow.pre c.pt (cospan g\u2081 g\u2082) G)\n\u22a2 f\u2081 =\n (c'.pt.hom \u226b G.map (NatTrans.app c'.\u03c0 left).right) \u226b\n eqToHom\n (_ :\n G.obj ((StructuredArrow.pre X (cospan g\u2081 g\u2082) G).obj (StructuredArrow.mk (f\u2081 \u226b \ud835\udfd9 (G.obj Y\u2081)))).right =\n G.obj Y\u2081)\n[PROOFSTEP]\nerw [\u2190 (c'.\u03c0.app left).w]\n[GOAL]\nC\u271d : Type u\u2081\ninst\u271d\u2075 : Category.{v\u2081, u\u2081} C\u271d\nD\u271d : Type u\u2082\ninst\u271d\u2074 : Category.{v\u2082, u\u2082} D\u271d\nA : Type u\u2083\ninst\u271d\u00b3 : Category.{v\u2083, u\u2083} A\nJ : GrothendieckTopology C\u271d\nK\u271d : GrothendieckTopology D\u271d\nL : GrothendieckTopology A\nG\u271d : C\u271d \u2964 D\u271d\nhG : CompatiblePreserving K\u271d G\u271d\n\u2131\u271d : SheafOfTypes K\u271d\nZ\u271d : C\u271d\nT\u271d : Presieve Z\u271d\nx\u271d : FamilyOfElements (G\u271d.op \u22d9 \u2131\u271d.val) T\u271d\nh : Compatible x\u271d\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} D\nK : GrothendieckTopology D\nG : C \u2964 D\ninst\u271d : RepresentablyFlat G\n\u2131 : SheafOfTypes K\nZ : C\nT : Presieve Z\nx : FamilyOfElements (G.op \u22d9 \u2131.val) T\nhx : Compatible x\nY\u2081 Y\u2082 : C\nX : D\nf\u2081 : X \u27f6 G.obj Y\u2081\nf\u2082 : X \u27f6 G.obj Y\u2082\ng\u2081 : Y\u2081 \u27f6 Z\ng\u2082 : Y\u2082 \u27f6 Z\nhg\u2081 : T g\u2081\nhg\u2082 : T g\u2082\ne : f\u2081 \u226b G.map g\u2081 = f\u2082 \u226b G.map g\u2082\nc : Cone (cospan g\u2081 g\u2082 \u22d9 G) :=\n (Cones.postcompose (diagramIsoCospan (cospan g\u2081 g\u2082 \u22d9 G)).inv).obj (PullbackCone.mk f\u2081 f\u2082 e)\nc' : Cone (Cone.toStructuredArrow c \u22d9 StructuredArrow.pre c.pt (cospan g\u2081 g\u2082) G) :=\n IsCofiltered.cone (Cone.toStructuredArrow c \u22d9 StructuredArrow.pre c.pt (cospan g\u2081 g\u2082) G)\n\u22a2 f\u2081 =\n ((Functor.fromPUnit c.pt).map (NatTrans.app c'.\u03c0 left).left \u226b\n ((Cone.toStructuredArrow c \u22d9 StructuredArrow.pre c.pt (cospan g\u2081 g\u2082) G).obj left).hom) \u226b\n eqToHom\n (_ :\n G.obj ((StructuredArrow.pre X (cospan g\u2081 g\u2082) G).obj (StructuredArrow.mk (f\u2081 \u226b \ud835\udfd9 (G.obj Y\u2081)))).right =\n G.obj Y\u2081)\n[PROOFSTEP]\ndsimp\n[GOAL]\nC\u271d : Type u\u2081\ninst\u271d\u2075 : Category.{v\u2081, u\u2081} C\u271d\nD\u271d : Type u\u2082\ninst\u271d\u2074 : Category.{v\u2082, u\u2082} D\u271d\nA : Type u\u2083\ninst\u271d\u00b3 : Category.{v\u2083, u\u2083} A\nJ : GrothendieckTopology C\u271d\nK\u271d : GrothendieckTopology D\u271d\nL : GrothendieckTopology A\nG\u271d : C\u271d \u2964 D\u271d\nhG : CompatiblePreserving K\u271d G\u271d\n\u2131\u271d : SheafOfTypes K\u271d\nZ\u271d : C\u271d\nT\u271d : Presieve Z\u271d\nx\u271d : FamilyOfElements (G\u271d.op \u22d9 \u2131\u271d.val) T\u271d\nh : Compatible x\u271d\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} D\nK : GrothendieckTopology D\nG : C \u2964 D\ninst\u271d : RepresentablyFlat G\n\u2131 : SheafOfTypes K\nZ : C\nT : Presieve Z\nx : FamilyOfElements (G.op \u22d9 \u2131.val) T\nhx : Compatible x\nY\u2081 Y\u2082 : C\nX : D\nf\u2081 : X \u27f6 G.obj Y\u2081\nf\u2082 : X \u27f6 G.obj Y\u2082\ng\u2081 : Y\u2081 \u27f6 Z\ng\u2082 : Y\u2082 \u27f6 Z\nhg\u2081 : T g\u2081\nhg\u2082 : T g\u2082\ne : f\u2081 \u226b G.map g\u2081 = f\u2082 \u226b G.map g\u2082\nc : Cone (cospan g\u2081 g\u2082 \u22d9 G) :=\n (Cones.postcompose (diagramIsoCospan (cospan g\u2081 g\u2082 \u22d9 G)).inv).obj (PullbackCone.mk f\u2081 f\u2082 e)\nc' : Cone (Cone.toStructuredArrow c \u22d9 StructuredArrow.pre c.pt (cospan g\u2081 g\u2082) G) :=\n IsCofiltered.cone (Cone.toStructuredArrow c \u22d9 StructuredArrow.pre c.pt (cospan g\u2081 g\u2082) G)\n\u22a2 f\u2081 = (\ud835\udfd9 X \u226b f\u2081 \u226b \ud835\udfd9 (G.obj Y\u2081)) \u226b \ud835\udfd9 (G.obj Y\u2081)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase Compatible\nC\u271d : Type u\u2081\ninst\u271d\u2075 : Category.{v\u2081, u\u2081} C\u271d\nD\u271d : Type u\u2082\ninst\u271d\u2074 : Category.{v\u2082, u\u2082} D\u271d\nA : Type u\u2083\ninst\u271d\u00b3 : Category.{v\u2083, u\u2083} A\nJ : GrothendieckTopology C\u271d\nK\u271d : GrothendieckTopology D\u271d\nL : GrothendieckTopology A\nG\u271d : C\u271d \u2964 D\u271d\nhG : CompatiblePreserving K\u271d G\u271d\n\u2131\u271d : SheafOfTypes K\u271d\nZ\u271d : C\u271d\nT\u271d : Presieve Z\u271d\nx\u271d : FamilyOfElements (G\u271d.op \u22d9 \u2131\u271d.val) T\u271d\nh : Compatible x\u271d\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} D\nK : GrothendieckTopology D\nG : C \u2964 D\ninst\u271d : RepresentablyFlat G\n\u2131 : SheafOfTypes K\nZ : C\nT : Presieve Z\nx : FamilyOfElements (G.op \u22d9 \u2131.val) T\nhx : Compatible x\nY\u2081 Y\u2082 : C\nX : D\nf\u2081 : X \u27f6 G.obj Y\u2081\nf\u2082 : X \u27f6 G.obj Y\u2082\ng\u2081 : Y\u2081 \u27f6 Z\ng\u2082 : Y\u2082 \u27f6 Z\nhg\u2081 : T g\u2081\nhg\u2082 : T g\u2082\ne : f\u2081 \u226b G.map g\u2081 = f\u2082 \u226b G.map g\u2082\nc : Cone (cospan g\u2081 g\u2082 \u22d9 G) :=\n (Cones.postcompose (diagramIsoCospan (cospan g\u2081 g\u2082 \u22d9 G)).inv).obj (PullbackCone.mk f\u2081 f\u2082 e)\nc' : Cone (Cone.toStructuredArrow c \u22d9 StructuredArrow.pre c.pt (cospan g\u2081 g\u2082) G) :=\n IsCofiltered.cone (Cone.toStructuredArrow c \u22d9 StructuredArrow.pre c.pt (cospan g\u2081 g\u2082) G)\neq\u2081 :\n f\u2081 =\n (c'.pt.hom \u226b G.map (NatTrans.app c'.\u03c0 left).right) \u226b\n eqToHom\n (_ :\n G.obj ((StructuredArrow.pre X (cospan g\u2081 g\u2082) G).obj (StructuredArrow.mk (f\u2081 \u226b \ud835\udfd9 (G.obj Y\u2081)))).right =\n G.obj Y\u2081)\n\u22a2 \u2131.val.map f\u2081.op (x g\u2081 hg\u2081) = \u2131.val.map f\u2082.op (x g\u2082 hg\u2082)\n[PROOFSTEP]\nhave eq\u2082 : f\u2082 = (c'.pt.hom \u226b G.map (c'.\u03c0.app right).right) \u226b eqToHom (by simp) :=\n by\n erw [\u2190 (c'.\u03c0.app right).w]\n dsimp\n simp\n[GOAL]\nC\u271d : Type u\u2081\ninst\u271d\u2075 : Category.{v\u2081, u\u2081} C\u271d\nD\u271d : Type u\u2082\ninst\u271d\u2074 : Category.{v\u2082, u\u2082} D\u271d\nA : Type u\u2083\ninst\u271d\u00b3 : Category.{v\u2083, u\u2083} A\nJ : GrothendieckTopology C\u271d\nK\u271d : GrothendieckTopology D\u271d\nL : GrothendieckTopology A\nG\u271d : C\u271d \u2964 D\u271d\nhG : CompatiblePreserving K\u271d G\u271d\n\u2131\u271d : SheafOfTypes K\u271d\nZ\u271d : C\u271d\nT\u271d : Presieve Z\u271d\nx\u271d : FamilyOfElements (G\u271d.op \u22d9 \u2131\u271d.val) T\u271d\nh : Compatible x\u271d\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} D\nK : GrothendieckTopology D\nG : C \u2964 D\ninst\u271d : RepresentablyFlat G\n\u2131 : SheafOfTypes K\nZ : C\nT : Presieve Z\nx : FamilyOfElements (G.op \u22d9 \u2131.val) T\nhx : Compatible x\nY\u2081 Y\u2082 : C\nX : D\nf\u2081 : X \u27f6 G.obj Y\u2081\nf\u2082 : X \u27f6 G.obj Y\u2082\ng\u2081 : Y\u2081 \u27f6 Z\ng\u2082 : Y\u2082 \u27f6 Z\nhg\u2081 : T g\u2081\nhg\u2082 : T g\u2082\ne : f\u2081 \u226b G.map g\u2081 = f\u2082 \u226b G.map g\u2082\nc : Cone (cospan g\u2081 g\u2082 \u22d9 G) :=\n (Cones.postcompose (diagramIsoCospan (cospan g\u2081 g\u2082 \u22d9 G)).inv).obj (PullbackCone.mk f\u2081 f\u2082 e)\nc' : Cone (Cone.toStructuredArrow c \u22d9 StructuredArrow.pre c.pt (cospan g\u2081 g\u2082) G) :=\n IsCofiltered.cone (Cone.toStructuredArrow c \u22d9 StructuredArrow.pre c.pt (cospan g\u2081 g\u2082) G)\neq\u2081 :\n f\u2081 =\n (c'.pt.hom \u226b G.map (NatTrans.app c'.\u03c0 left).right) \u226b\n eqToHom\n (_ :\n G.obj ((StructuredArrow.pre X (cospan g\u2081 g\u2082) G).obj (StructuredArrow.mk (f\u2081 \u226b \ud835\udfd9 (G.obj Y\u2081)))).right =\n G.obj Y\u2081)\n\u22a2 G.obj ((Cone.toStructuredArrow c \u22d9 StructuredArrow.pre c.pt (cospan g\u2081 g\u2082) G).obj right).right = G.obj Y\u2082\n[PROOFSTEP]\nsimp\n[GOAL]\nC\u271d : Type u\u2081\ninst\u271d\u2075 : Category.{v\u2081, u\u2081} C\u271d\nD\u271d : Type u\u2082\ninst\u271d\u2074 : Category.{v\u2082, u\u2082} D\u271d\nA : Type u\u2083\ninst\u271d\u00b3 : Category.{v\u2083, u\u2083} A\nJ : GrothendieckTopology C\u271d\nK\u271d : GrothendieckTopology D\u271d\nL : GrothendieckTopology A\nG\u271d : C\u271d \u2964 D\u271d\nhG : CompatiblePreserving K\u271d G\u271d\n\u2131\u271d : SheafOfTypes K\u271d\nZ\u271d : C\u271d\nT\u271d : Presieve Z\u271d\nx\u271d : FamilyOfElements (G\u271d.op \u22d9 \u2131\u271d.val) T\u271d\nh : Compatible x\u271d\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} D\nK : GrothendieckTopology D\nG : C \u2964 D\ninst\u271d : RepresentablyFlat G\n\u2131 : SheafOfTypes K\nZ : C\nT : Presieve Z\nx : FamilyOfElements (G.op \u22d9 \u2131.val) T\nhx : Compatible x\nY\u2081 Y\u2082 : C\nX : D\nf\u2081 : X \u27f6 G.obj Y\u2081\nf\u2082 : X \u27f6 G.obj Y\u2082\ng\u2081 : Y\u2081 \u27f6 Z\ng\u2082 : Y\u2082 \u27f6 Z\nhg\u2081 : T g\u2081\nhg\u2082 : T g\u2082\ne : f\u2081 \u226b G.map g\u2081 = f\u2082 \u226b G.map g\u2082\nc : Cone (cospan g\u2081 g\u2082 \u22d9 G) :=\n (Cones.postcompose (diagramIsoCospan (cospan g\u2081 g\u2082 \u22d9 G)).inv).obj (PullbackCone.mk f\u2081 f\u2082 e)\nc' : Cone (Cone.toStructuredArrow c \u22d9 StructuredArrow.pre c.pt (cospan g\u2081 g\u2082) G) :=\n IsCofiltered.cone (Cone.toStructuredArrow c \u22d9 StructuredArrow.pre c.pt (cospan g\u2081 g\u2082) G)\neq\u2081 :\n f\u2081 =\n (c'.pt.hom \u226b G.map (NatTrans.app c'.\u03c0 left).right) \u226b\n eqToHom\n (_ :\n G.obj ((StructuredArrow.pre X (cospan g\u2081 g\u2082) G).obj (StructuredArrow.mk (f\u2081 \u226b \ud835\udfd9 (G.obj Y\u2081)))).right =\n G.obj Y\u2081)\n\u22a2 f\u2082 =\n (c'.pt.hom \u226b G.map (NatTrans.app c'.\u03c0 right).right) \u226b\n eqToHom\n (_ :\n G.obj ((StructuredArrow.pre X (cospan g\u2081 g\u2082) G).obj (StructuredArrow.mk (f\u2082 \u226b \ud835\udfd9 (G.obj Y\u2082)))).right =\n G.obj Y\u2082)\n[PROOFSTEP]\nerw [\u2190 (c'.\u03c0.app right).w]\n[GOAL]\nC\u271d : Type u\u2081\ninst\u271d\u2075 : Category.{v\u2081, u\u2081} C\u271d\nD\u271d : Type u\u2082\ninst\u271d\u2074 : Category.{v\u2082, u\u2082} D\u271d\nA : Type u\u2083\ninst\u271d\u00b3 : Category.{v\u2083, u\u2083} A\nJ : GrothendieckTopology C\u271d\nK\u271d : GrothendieckTopology D\u271d\nL : GrothendieckTopology A\nG\u271d : C\u271d \u2964 D\u271d\nhG : CompatiblePreserving K\u271d G\u271d\n\u2131\u271d : SheafOfTypes K\u271d\nZ\u271d : C\u271d\nT\u271d : Presieve Z\u271d\nx\u271d : FamilyOfElements (G\u271d.op \u22d9 \u2131\u271d.val) T\u271d\nh : Compatible x\u271d\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} D\nK : GrothendieckTopology D\nG : C \u2964 D\ninst\u271d : RepresentablyFlat G\n\u2131 : SheafOfTypes K\nZ : C\nT : Presieve Z\nx : FamilyOfElements (G.op \u22d9 \u2131.val) T\nhx : Compatible x\nY\u2081 Y\u2082 : C\nX : D\nf\u2081 : X \u27f6 G.obj Y\u2081\nf\u2082 : X \u27f6 G.obj Y\u2082\ng\u2081 : Y\u2081 \u27f6 Z\ng\u2082 : Y\u2082 \u27f6 Z\nhg\u2081 : T g\u2081\nhg\u2082 : T g\u2082\ne : f\u2081 \u226b G.map g\u2081 = f\u2082 \u226b G.map g\u2082\nc : Cone (cospan g\u2081 g\u2082 \u22d9 G) :=\n (Cones.postcompose (diagramIsoCospan (cospan g\u2081 g\u2082 \u22d9 G)).inv).obj (PullbackCone.mk f\u2081 f\u2082 e)\nc' : Cone (Cone.toStructuredArrow c \u22d9 StructuredArrow.pre c.pt (cospan g\u2081 g\u2082) G) :=\n IsCofiltered.cone (Cone.toStructuredArrow c \u22d9 StructuredArrow.pre c.pt (cospan g\u2081 g\u2082) G)\neq\u2081 :\n f\u2081 =\n (c'.pt.hom \u226b G.map (NatTrans.app c'.\u03c0 left).right) \u226b\n eqToHom\n (_ :\n G.obj ((StructuredArrow.pre X (cospan g\u2081 g\u2082) G).obj (StructuredArrow.mk (f\u2081 \u226b \ud835\udfd9 (G.obj Y\u2081)))).right =\n G.obj Y\u2081)\n\u22a2 f\u2082 =\n ((Functor.fromPUnit c.pt).map (NatTrans.app c'.\u03c0 right).left \u226b\n ((Cone.toStructuredArrow c \u22d9 StructuredArrow.pre c.pt (cospan g\u2081 g\u2082) G).obj right).hom) \u226b\n eqToHom\n (_ :\n G.obj ((StructuredArrow.pre X (cospan g\u2081 g\u2082) G).obj (StructuredArrow.mk (f\u2082 \u226b \ud835\udfd9 (G.obj Y\u2082)))).right =\n G.obj Y\u2082)\n[PROOFSTEP]\ndsimp\n[GOAL]\nC\u271d : Type u\u2081\ninst\u271d\u2075 : Category.{v\u2081, u\u2081} C\u271d\nD\u271d : Type u\u2082\ninst\u271d\u2074 : Category.{v\u2082, u\u2082} D\u271d\nA : Type u\u2083\ninst\u271d\u00b3 : Category.{v\u2083, u\u2083} A\nJ : GrothendieckTopology C\u271d\nK\u271d : GrothendieckTopology D\u271d\nL : GrothendieckTopology A\nG\u271d : C\u271d \u2964 D\u271d\nhG : CompatiblePreserving K\u271d G\u271d\n\u2131\u271d : SheafOfTypes K\u271d\nZ\u271d : C\u271d\nT\u271d : Presieve Z\u271d\nx\u271d : FamilyOfElements (G\u271d.op \u22d9 \u2131\u271d.val) T\u271d\nh : Compatible x\u271d\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} D\nK : GrothendieckTopology D\nG : C \u2964 D\ninst\u271d : RepresentablyFlat G\n\u2131 : SheafOfTypes K\nZ : C\nT : Presieve Z\nx : FamilyOfElements (G.op \u22d9 \u2131.val) T\nhx : Compatible x\nY\u2081 Y\u2082 : C\nX : D\nf\u2081 : X \u27f6 G.obj Y\u2081\nf\u2082 : X \u27f6 G.obj Y\u2082\ng\u2081 : Y\u2081 \u27f6 Z\ng\u2082 : Y\u2082 \u27f6 Z\nhg\u2081 : T g\u2081\nhg\u2082 : T g\u2082\ne : f\u2081 \u226b G.map g\u2081 = f\u2082 \u226b G.map g\u2082\nc : Cone (cospan g\u2081 g\u2082 \u22d9 G) :=\n (Cones.postcompose (diagramIsoCospan (cospan g\u2081 g\u2082 \u22d9 G)).inv).obj (PullbackCone.mk f\u2081 f\u2082 e)\nc' : Cone (Cone.toStructuredArrow c \u22d9 StructuredArrow.pre c.pt (cospan g\u2081 g\u2082) G) :=\n IsCofiltered.cone (Cone.toStructuredArrow c \u22d9 StructuredArrow.pre c.pt (cospan g\u2081 g\u2082) G)\neq\u2081 :\n f\u2081 =\n (c'.pt.hom \u226b G.map (NatTrans.app c'.\u03c0 left).right) \u226b\n eqToHom\n (_ :\n G.obj ((StructuredArrow.pre X (cospan g\u2081 g\u2082) G).obj (StructuredArrow.mk (f\u2081 \u226b \ud835\udfd9 (G.obj Y\u2081)))).right =\n G.obj Y\u2081)\n\u22a2 f\u2082 = (\ud835\udfd9 X \u226b f\u2082 \u226b \ud835\udfd9 (G.obj Y\u2082)) \u226b \ud835\udfd9 (G.obj Y\u2082)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase Compatible\nC\u271d : Type u\u2081\ninst\u271d\u2075 : Category.{v\u2081, u\u2081} C\u271d\nD\u271d : Type u\u2082\ninst\u271d\u2074 : Category.{v\u2082, u\u2082} D\u271d\nA : Type u\u2083\ninst\u271d\u00b3 : Category.{v\u2083, u\u2083} A\nJ : GrothendieckTopology C\u271d\nK\u271d : GrothendieckTopology D\u271d\nL : GrothendieckTopology A\nG\u271d : C\u271d \u2964 D\u271d\nhG : CompatiblePreserving K\u271d G\u271d\n\u2131\u271d : SheafOfTypes K\u271d\nZ\u271d : C\u271d\nT\u271d : Presieve Z\u271d\nx\u271d : FamilyOfElements (G\u271d.op \u22d9 \u2131\u271d.val) T\u271d\nh : Compatible x\u271d\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} D\nK : GrothendieckTopology D\nG : C \u2964 D\ninst\u271d : RepresentablyFlat G\n\u2131 : SheafOfTypes K\nZ : C\nT : Presieve Z\nx : FamilyOfElements (G.op \u22d9 \u2131.val) T\nhx : Compatible x\nY\u2081 Y\u2082 : C\nX : D\nf\u2081 : X \u27f6 G.obj Y\u2081\nf\u2082 : X \u27f6 G.obj Y\u2082\ng\u2081 : Y\u2081 \u27f6 Z\ng\u2082 : Y\u2082 \u27f6 Z\nhg\u2081 : T g\u2081\nhg\u2082 : T g\u2082\ne : f\u2081 \u226b G.map g\u2081 = f\u2082 \u226b G.map g\u2082\nc : Cone (cospan g\u2081 g\u2082 \u22d9 G) :=\n (Cones.postcompose (diagramIsoCospan (cospan g\u2081 g\u2082 \u22d9 G)).inv).obj (PullbackCone.mk f\u2081 f\u2082 e)\nc' : Cone (Cone.toStructuredArrow c \u22d9 StructuredArrow.pre c.pt (cospan g\u2081 g\u2082) G) :=\n IsCofiltered.cone (Cone.toStructuredArrow c \u22d9 StructuredArrow.pre c.pt (cospan g\u2081 g\u2082) G)\neq\u2081 :\n f\u2081 =\n (c'.pt.hom \u226b G.map (NatTrans.app c'.\u03c0 left).right) \u226b\n eqToHom\n (_ :\n G.obj ((StructuredArrow.pre X (cospan g\u2081 g\u2082) G).obj (StructuredArrow.mk (f\u2081 \u226b \ud835\udfd9 (G.obj Y\u2081)))).right =\n G.obj Y\u2081)\neq\u2082 :\n f\u2082 =\n (c'.pt.hom \u226b G.map (NatTrans.app c'.\u03c0 right).right) \u226b\n eqToHom\n (_ :\n G.obj ((StructuredArrow.pre X (cospan g\u2081 g\u2082) G).obj (StructuredArrow.mk (f\u2082 \u226b \ud835\udfd9 (G.obj Y\u2082)))).right =\n G.obj Y\u2082)\n\u22a2 \u2131.val.map f\u2081.op (x g\u2081 hg\u2081) = \u2131.val.map f\u2082.op (x g\u2082 hg\u2082)\n[PROOFSTEP]\nconv_lhs => rw [eq\u2081]\n[GOAL]\nC\u271d : Type u\u2081\ninst\u271d\u2075 : Category.{v\u2081, u\u2081} C\u271d\nD\u271d : Type u\u2082\ninst\u271d\u2074 : Category.{v\u2082, u\u2082} D\u271d\nA : Type u\u2083\ninst\u271d\u00b3 : Category.{v\u2083, u\u2083} A\nJ : GrothendieckTopology C\u271d\nK\u271d : GrothendieckTopology D\u271d\nL : GrothendieckTopology A\nG\u271d : C\u271d \u2964 D\u271d\nhG : CompatiblePreserving K\u271d G\u271d\n\u2131\u271d : SheafOfTypes K\u271d\nZ\u271d : C\u271d\nT\u271d : Presieve Z\u271d\nx\u271d : FamilyOfElements (G\u271d.op \u22d9 \u2131\u271d.val) T\u271d\nh : Compatible x\u271d\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} D\nK : GrothendieckTopology D\nG : C \u2964 D\ninst\u271d : RepresentablyFlat G\n\u2131 : SheafOfTypes K\nZ : C\nT : Presieve Z\nx : FamilyOfElements (G.op \u22d9 \u2131.val) T\nhx : Compatible x\nY\u2081 Y\u2082 : C\nX : D\nf\u2081 : X \u27f6 G.obj Y\u2081\nf\u2082 : X \u27f6 G.obj Y\u2082\ng\u2081 : Y\u2081 \u27f6 Z\ng\u2082 : Y\u2082 \u27f6 Z\nhg\u2081 : T g\u2081\nhg\u2082 : T g\u2082\ne : f\u2081 \u226b G.map g\u2081 = f\u2082 \u226b G.map g\u2082\nc : Cone (cospan g\u2081 g\u2082 \u22d9 G) :=\n (Cones.postcompose (diagramIsoCospan (cospan g\u2081 g\u2082 \u22d9 G)).inv).obj (PullbackCone.mk f\u2081 f\u2082 e)\nc' : Cone (Cone.toStructuredArrow c \u22d9 StructuredArrow.pre c.pt (cospan g\u2081 g\u2082) G) :=\n IsCofiltered.cone (Cone.toStructuredArrow c \u22d9 StructuredArrow.pre c.pt (cospan g\u2081 g\u2082) G)\neq\u2081 :\n f\u2081 =\n (c'.pt.hom \u226b G.map (NatTrans.app c'.\u03c0 left).right) \u226b\n eqToHom\n (_ :\n G.obj ((StructuredArrow.pre X (cospan g\u2081 g\u2082) G).obj (StructuredArrow.mk (f\u2081 \u226b \ud835\udfd9 (G.obj Y\u2081)))).right =\n G.obj Y\u2081)\neq\u2082 :\n f\u2082 =\n (c'.pt.hom \u226b G.map (NatTrans.app c'.\u03c0 right).right) \u226b\n eqToHom\n (_ :\n G.obj ((StructuredArrow.pre X (cospan g\u2081 g\u2082) G).obj (StructuredArrow.mk (f\u2082 \u226b \ud835\udfd9 (G.obj Y\u2082)))).right =\n G.obj Y\u2082)\n| \u2131.val.map f\u2081.op (x g\u2081 hg\u2081)\n[PROOFSTEP]\nrw [eq\u2081]\n[GOAL]\nC\u271d : Type u\u2081\ninst\u271d\u2075 : Category.{v\u2081, u\u2081} C\u271d\nD\u271d : Type u\u2082\ninst\u271d\u2074 : Category.{v\u2082, u\u2082} D\u271d\nA : Type u\u2083\ninst\u271d\u00b3 : Category.{v\u2083, u\u2083} A\nJ : GrothendieckTopology C\u271d\nK\u271d : GrothendieckTopology D\u271d\nL : GrothendieckTopology A\nG\u271d : C\u271d \u2964 D\u271d\nhG : CompatiblePreserving K\u271d G\u271d\n\u2131\u271d : SheafOfTypes K\u271d\nZ\u271d : C\u271d\nT\u271d : Presieve Z\u271d\nx\u271d : FamilyOfElements (G\u271d.op \u22d9 \u2131\u271d.val) T\u271d\nh : Compatible x\u271d\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} D\nK : GrothendieckTopology D\nG : C \u2964 D\ninst\u271d : RepresentablyFlat G\n\u2131 : SheafOfTypes K\nZ : C\nT : Presieve Z\nx : FamilyOfElements (G.op \u22d9 \u2131.val) T\nhx : Compatible x\nY\u2081 Y\u2082 : C\nX : D\nf\u2081 : X \u27f6 G.obj Y\u2081\nf\u2082 : X \u27f6 G.obj Y\u2082\ng\u2081 : Y\u2081 \u27f6 Z\ng\u2082 : Y\u2082 \u27f6 Z\nhg\u2081 : T g\u2081\nhg\u2082 : T g\u2082\ne : f\u2081 \u226b G.map g\u2081 = f\u2082 \u226b G.map g\u2082\nc : Cone (cospan g\u2081 g\u2082 \u22d9 G) :=\n (Cones.postcompose (diagramIsoCospan (cospan g\u2081 g\u2082 \u22d9 G)).inv).obj (PullbackCone.mk f\u2081 f\u2082 e)\nc' : Cone (Cone.toStructuredArrow c \u22d9 StructuredArrow.pre c.pt (cospan g\u2081 g\u2082) G) :=\n IsCofiltered.cone (Cone.toStructuredArrow c \u22d9 StructuredArrow.pre c.pt (cospan g\u2081 g\u2082) G)\neq\u2081 :\n f\u2081 =\n (c'.pt.hom \u226b G.map (NatTrans.app c'.\u03c0 left).right) \u226b\n eqToHom\n (_ :\n G.obj ((StructuredArrow.pre X (cospan g\u2081 g\u2082) G).obj (StructuredArrow.mk (f\u2081 \u226b \ud835\udfd9 (G.obj Y\u2081)))).right =\n G.obj Y\u2081)\neq\u2082 :\n f\u2082 =\n (c'.pt.hom \u226b G.map (NatTrans.app c'.\u03c0 right).right) \u226b\n eqToHom\n (_ :\n G.obj ((StructuredArrow.pre X (cospan g\u2081 g\u2082) G).obj (StructuredArrow.mk (f\u2082 \u226b \ud835\udfd9 (G.obj Y\u2082)))).right =\n G.obj Y\u2082)\n| \u2131.val.map f\u2081.op (x g\u2081 hg\u2081)\n[PROOFSTEP]\nrw [eq\u2081]\n[GOAL]\nC\u271d : Type u\u2081\ninst\u271d\u2075 : Category.{v\u2081, u\u2081} C\u271d\nD\u271d : Type u\u2082\ninst\u271d\u2074 : Category.{v\u2082, u\u2082} D\u271d\nA : Type u\u2083\ninst\u271d\u00b3 : Category.{v\u2083, u\u2083} A\nJ : GrothendieckTopology C\u271d\nK\u271d : GrothendieckTopology D\u271d\nL : GrothendieckTopology A\nG\u271d : C\u271d \u2964 D\u271d\nhG : CompatiblePreserving K\u271d G\u271d\n\u2131\u271d : SheafOfTypes K\u271d\nZ\u271d : C\u271d\nT\u271d : Presieve Z\u271d\nx\u271d : FamilyOfElements (G\u271d.op \u22d9 \u2131\u271d.val) T\u271d\nh : Compatible x\u271d\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} D\nK : GrothendieckTopology D\nG : C \u2964 D\ninst\u271d : RepresentablyFlat G\n\u2131 : SheafOfTypes K\nZ : C\nT : Presieve Z\nx : FamilyOfElements (G.op \u22d9 \u2131.val) T\nhx : Compatible x\nY\u2081 Y\u2082 : C\nX : D\nf\u2081 : X \u27f6 G.obj Y\u2081\nf\u2082 : X \u27f6 G.obj Y\u2082\ng\u2081 : Y\u2081 \u27f6 Z\ng\u2082 : Y\u2082 \u27f6 Z\nhg\u2081 : T g\u2081\nhg\u2082 : T g\u2082\ne : f\u2081 \u226b G.map g\u2081 = f\u2082 \u226b G.map g\u2082\nc : Cone (cospan g\u2081 g\u2082 \u22d9 G) :=\n (Cones.postcompose (diagramIsoCospan (cospan g\u2081 g\u2082 \u22d9 G)).inv).obj (PullbackCone.mk f\u2081 f\u2082 e)\nc' : Cone (Cone.toStructuredArrow c \u22d9 StructuredArrow.pre c.pt (cospan g\u2081 g\u2082) G) :=\n IsCofiltered.cone (Cone.toStructuredArrow c \u22d9 StructuredArrow.pre c.pt (cospan g\u2081 g\u2082) G)\neq\u2081 :\n f\u2081 =\n (c'.pt.hom \u226b G.map (NatTrans.app c'.\u03c0 left).right) \u226b\n eqToHom\n (_ :\n G.obj ((StructuredArrow.pre X (cospan g\u2081 g\u2082) G).obj (StructuredArrow.mk (f\u2081 \u226b \ud835\udfd9 (G.obj Y\u2081)))).right =\n G.obj Y\u2081)\neq\u2082 :\n f\u2082 =\n (c'.pt.hom \u226b G.map (NatTrans.app c'.\u03c0 right).right) \u226b\n eqToHom\n (_ :\n G.obj ((StructuredArrow.pre X (cospan g\u2081 g\u2082) G).obj (StructuredArrow.mk (f\u2082 \u226b \ud835\udfd9 (G.obj Y\u2082)))).right =\n G.obj Y\u2082)\n| \u2131.val.map f\u2081.op (x g\u2081 hg\u2081)\n[PROOFSTEP]\nrw [eq\u2081]\n[GOAL]\ncase Compatible\nC\u271d : Type u\u2081\ninst\u271d\u2075 : Category.{v\u2081, u\u2081} C\u271d\nD\u271d : Type u\u2082\ninst\u271d\u2074 : Category.{v\u2082, u\u2082} D\u271d\nA : Type u\u2083\ninst\u271d\u00b3 : Category.{v\u2083, u\u2083} A\nJ : GrothendieckTopology C\u271d\nK\u271d : GrothendieckTopology D\u271d\nL : GrothendieckTopology A\nG\u271d : C\u271d \u2964 D\u271d\nhG : CompatiblePreserving K\u271d G\u271d\n\u2131\u271d : SheafOfTypes K\u271d\nZ\u271d : C\u271d\nT\u271d : Presieve Z\u271d\nx\u271d : FamilyOfElements (G\u271d.op \u22d9 \u2131\u271d.val) T\u271d\nh : Compatible x\u271d\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} D\nK : GrothendieckTopology D\nG : C \u2964 D\ninst\u271d : RepresentablyFlat G\n\u2131 : SheafOfTypes K\nZ : C\nT : Presieve Z\nx : FamilyOfElements (G.op \u22d9 \u2131.val) T\nhx : Compatible x\nY\u2081 Y\u2082 : C\nX : D\nf\u2081 : X \u27f6 G.obj Y\u2081\nf\u2082 : X \u27f6 G.obj Y\u2082\ng\u2081 : Y\u2081 \u27f6 Z\ng\u2082 : Y\u2082 \u27f6 Z\nhg\u2081 : T g\u2081\nhg\u2082 : T g\u2082\ne : f\u2081 \u226b G.map g\u2081 = f\u2082 \u226b G.map g\u2082\nc : Cone (cospan g\u2081 g\u2082 \u22d9 G) :=\n (Cones.postcompose (diagramIsoCospan (cospan g\u2081 g\u2082 \u22d9 G)).inv).obj (PullbackCone.mk f\u2081 f\u2082 e)\nc' : Cone (Cone.toStructuredArrow c \u22d9 StructuredArrow.pre c.pt (cospan g\u2081 g\u2082) G) :=\n IsCofiltered.cone (Cone.toStructuredArrow c \u22d9 StructuredArrow.pre c.pt (cospan g\u2081 g\u2082) G)\neq\u2081 :\n f\u2081 =\n (c'.pt.hom \u226b G.map (NatTrans.app c'.\u03c0 left).right) \u226b\n eqToHom\n (_ :\n G.obj ((StructuredArrow.pre X (cospan g\u2081 g\u2082) G).obj (StructuredArrow.mk (f\u2081 \u226b \ud835\udfd9 (G.obj Y\u2081)))).right =\n G.obj Y\u2081)\neq\u2082 :\n f\u2082 =\n (c'.pt.hom \u226b G.map (NatTrans.app c'.\u03c0 right).right) \u226b\n eqToHom\n (_ :\n G.obj ((StructuredArrow.pre X (cospan g\u2081 g\u2082) G).obj (StructuredArrow.mk (f\u2082 \u226b \ud835\udfd9 (G.obj Y\u2082)))).right =\n G.obj Y\u2082)\n\u22a2 \u2131.val.map\n ((c'.pt.hom \u226b G.map (NatTrans.app c'.\u03c0 left).right) \u226b\n eqToHom\n (_ :\n G.obj ((StructuredArrow.pre X (cospan g\u2081 g\u2082) G).obj (StructuredArrow.mk (f\u2081 \u226b \ud835\udfd9 (G.obj Y\u2081)))).right =\n G.obj Y\u2081)).op\n (x g\u2081 hg\u2081) =\n \u2131.val.map f\u2082.op (x g\u2082 hg\u2082)\n[PROOFSTEP]\nconv_rhs => rw [eq\u2082]\n[GOAL]\nC\u271d : Type u\u2081\ninst\u271d\u2075 : Category.{v\u2081, u\u2081} C\u271d\nD\u271d : Type u\u2082\ninst\u271d\u2074 : Category.{v\u2082, u\u2082} D\u271d\nA : Type u\u2083\ninst\u271d\u00b3 : Category.{v\u2083, u\u2083} A\nJ : GrothendieckTopology C\u271d\nK\u271d : GrothendieckTopology D\u271d\nL : GrothendieckTopology A\nG\u271d : C\u271d \u2964 D\u271d\nhG : CompatiblePreserving K\u271d G\u271d\n\u2131\u271d : SheafOfTypes K\u271d\nZ\u271d : C\u271d\nT\u271d : Presieve Z\u271d\nx\u271d : FamilyOfElements (G\u271d.op \u22d9 \u2131\u271d.val) T\u271d\nh : Compatible x\u271d\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} D\nK : GrothendieckTopology D\nG : C \u2964 D\ninst\u271d : RepresentablyFlat G\n\u2131 : SheafOfTypes K\nZ : C\nT : Presieve Z\nx : FamilyOfElements (G.op \u22d9 \u2131.val) T\nhx : Compatible x\nY\u2081 Y\u2082 : C\nX : D\nf\u2081 : X \u27f6 G.obj Y\u2081\nf\u2082 : X \u27f6 G.obj Y\u2082\ng\u2081 : Y\u2081 \u27f6 Z\ng\u2082 : Y\u2082 \u27f6 Z\nhg\u2081 : T g\u2081\nhg\u2082 : T g\u2082\ne : f\u2081 \u226b G.map g\u2081 = f\u2082 \u226b G.map g\u2082\nc : Cone (cospan g\u2081 g\u2082 \u22d9 G) :=\n (Cones.postcompose (diagramIsoCospan (cospan g\u2081 g\u2082 \u22d9 G)).inv).obj (PullbackCone.mk f\u2081 f\u2082 e)\nc' : Cone (Cone.toStructuredArrow c \u22d9 StructuredArrow.pre c.pt (cospan g\u2081 g\u2082) G) :=\n IsCofiltered.cone (Cone.toStructuredArrow c \u22d9 StructuredArrow.pre c.pt (cospan g\u2081 g\u2082) G)\neq\u2081 :\n f\u2081 =\n (c'.pt.hom \u226b G.map (NatTrans.app c'.\u03c0 left).right) \u226b\n eqToHom\n (_ :\n G.obj ((StructuredArrow.pre X (cospan g\u2081 g\u2082) G).obj (StructuredArrow.mk (f\u2081 \u226b \ud835\udfd9 (G.obj Y\u2081)))).right =\n G.obj Y\u2081)\neq\u2082 :\n f\u2082 =\n (c'.pt.hom \u226b G.map (NatTrans.app c'.\u03c0 right).right) \u226b\n eqToHom\n (_ :\n G.obj ((StructuredArrow.pre X (cospan g\u2081 g\u2082) G).obj (StructuredArrow.mk (f\u2082 \u226b \ud835\udfd9 (G.obj Y\u2082)))).right =\n G.obj Y\u2082)\n| \u2131.val.map f\u2082.op (x g\u2082 hg\u2082)\n[PROOFSTEP]\nrw [eq\u2082]\n[GOAL]\nC\u271d : Type u\u2081\ninst\u271d\u2075 : Category.{v\u2081, u\u2081} C\u271d\nD\u271d : Type u\u2082\ninst\u271d\u2074 : Category.{v\u2082, u\u2082} D\u271d\nA : Type u\u2083\ninst\u271d\u00b3 : Category.{v\u2083, u\u2083} A\nJ : GrothendieckTopology C\u271d\nK\u271d : GrothendieckTopology D\u271d\nL : GrothendieckTopology A\nG\u271d : C\u271d \u2964 D\u271d\nhG : CompatiblePreserving K\u271d G\u271d\n\u2131\u271d : SheafOfTypes K\u271d\nZ\u271d : C\u271d\nT\u271d : Presieve Z\u271d\nx\u271d : FamilyOfElements (G\u271d.op \u22d9 \u2131\u271d.val) T\u271d\nh : Compatible x\u271d\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} D\nK : GrothendieckTopology D\nG : C \u2964 D\ninst\u271d : RepresentablyFlat G\n\u2131 : SheafOfTypes K\nZ : C\nT : Presieve Z\nx : FamilyOfElements (G.op \u22d9 \u2131.val) T\nhx : Compatible x\nY\u2081 Y\u2082 : C\nX : D\nf\u2081 : X \u27f6 G.obj Y\u2081\nf\u2082 : X \u27f6 G.obj Y\u2082\ng\u2081 : Y\u2081 \u27f6 Z\ng\u2082 : Y\u2082 \u27f6 Z\nhg\u2081 : T g\u2081\nhg\u2082 : T g\u2082\ne : f\u2081 \u226b G.map g\u2081 = f\u2082 \u226b G.map g\u2082\nc : Cone (cospan g\u2081 g\u2082 \u22d9 G) :=\n (Cones.postcompose (diagramIsoCospan (cospan g\u2081 g\u2082 \u22d9 G)).inv).obj (PullbackCone.mk f\u2081 f\u2082 e)\nc' : Cone (Cone.toStructuredArrow c \u22d9 StructuredArrow.pre c.pt (cospan g\u2081 g\u2082) G) :=\n IsCofiltered.cone (Cone.toStructuredArrow c \u22d9 StructuredArrow.pre c.pt (cospan g\u2081 g\u2082) G)\neq\u2081 :\n f\u2081 =\n (c'.pt.hom \u226b G.map (NatTrans.app c'.\u03c0 left).right) \u226b\n eqToHom\n (_ :\n G.obj ((StructuredArrow.pre X (cospan g\u2081 g\u2082) G).obj (StructuredArrow.mk (f\u2081 \u226b \ud835\udfd9 (G.obj Y\u2081)))).right =\n G.obj Y\u2081)\neq\u2082 :\n f\u2082 =\n (c'.pt.hom \u226b G.map (NatTrans.app c'.\u03c0 right).right) \u226b\n eqToHom\n (_ :\n G.obj ((StructuredArrow.pre X (cospan g\u2081 g\u2082) G).obj (StructuredArrow.mk (f\u2082 \u226b \ud835\udfd9 (G.obj Y\u2082)))).right =\n G.obj Y\u2082)\n| \u2131.val.map f\u2082.op (x g\u2082 hg\u2082)\n[PROOFSTEP]\nrw [eq\u2082]\n[GOAL]\nC\u271d : Type u\u2081\ninst\u271d\u2075 : Category.{v\u2081, u\u2081} C\u271d\nD\u271d : Type u\u2082\ninst\u271d\u2074 : Category.{v\u2082, u\u2082} D\u271d\nA : Type u\u2083\ninst\u271d\u00b3 : Category.{v\u2083, u\u2083} A\nJ : GrothendieckTopology C\u271d\nK\u271d : GrothendieckTopology D\u271d\nL : GrothendieckTopology A\nG\u271d : C\u271d \u2964 D\u271d\nhG : CompatiblePreserving K\u271d G\u271d\n\u2131\u271d : SheafOfTypes K\u271d\nZ\u271d : C\u271d\nT\u271d : Presieve Z\u271d\nx\u271d : FamilyOfElements (G\u271d.op \u22d9 \u2131\u271d.val) T\u271d\nh : Compatible x\u271d\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} D\nK : GrothendieckTopology D\nG : C \u2964 D\ninst\u271d : RepresentablyFlat G\n\u2131 : SheafOfTypes K\nZ : C\nT : Presieve Z\nx : FamilyOfElements (G.op \u22d9 \u2131.val) T\nhx : Compatible x\nY\u2081 Y\u2082 : C\nX : D\nf\u2081 : X \u27f6 G.obj Y\u2081\nf\u2082 : X \u27f6 G.obj Y\u2082\ng\u2081 : Y\u2081 \u27f6 Z\ng\u2082 : Y\u2082 \u27f6 Z\nhg\u2081 : T g\u2081\nhg\u2082 : T g\u2082\ne : f\u2081 \u226b G.map g\u2081 = f\u2082 \u226b G.map g\u2082\nc : Cone (cospan g\u2081 g\u2082 \u22d9 G) :=\n (Cones.postcompose (diagramIsoCospan (cospan g\u2081 g\u2082 \u22d9 G)).inv).obj (PullbackCone.mk f\u2081 f\u2082 e)\nc' : Cone (Cone.toStructuredArrow c \u22d9 StructuredArrow.pre c.pt (cospan g\u2081 g\u2082) G) :=\n IsCofiltered.cone (Cone.toStructuredArrow c \u22d9 StructuredArrow.pre c.pt (cospan g\u2081 g\u2082) G)\neq\u2081 :\n f\u2081 =\n (c'.pt.hom \u226b G.map (NatTrans.app c'.\u03c0 left).right) \u226b\n eqToHom\n (_ :\n G.obj ((StructuredArrow.pre X (cospan g\u2081 g\u2082) G).obj (StructuredArrow.mk (f\u2081 \u226b \ud835\udfd9 (G.obj Y\u2081)))).right =\n G.obj Y\u2081)\neq\u2082 :\n f\u2082 =\n (c'.pt.hom \u226b G.map (NatTrans.app c'.\u03c0 right).right) \u226b\n eqToHom\n (_ :\n G.obj ((StructuredArrow.pre X (cospan g\u2081 g\u2082) G).obj (StructuredArrow.mk (f\u2082 \u226b \ud835\udfd9 (G.obj Y\u2082)))).right =\n G.obj Y\u2082)\n| \u2131.val.map f\u2082.op (x g\u2082 hg\u2082)\n[PROOFSTEP]\nrw [eq\u2082]\n[GOAL]\ncase Compatible\nC\u271d : Type u\u2081\ninst\u271d\u2075 : Category.{v\u2081, u\u2081} C\u271d\nD\u271d : Type u\u2082\ninst\u271d\u2074 : Category.{v\u2082, u\u2082} D\u271d\nA : Type u\u2083\ninst\u271d\u00b3 : Category.{v\u2083, u\u2083} A\nJ : GrothendieckTopology C\u271d\nK\u271d : GrothendieckTopology D\u271d\nL : GrothendieckTopology A\nG\u271d : C\u271d \u2964 D\u271d\nhG : CompatiblePreserving K\u271d G\u271d\n\u2131\u271d : SheafOfTypes K\u271d\nZ\u271d : C\u271d\nT\u271d : Presieve Z\u271d\nx\u271d : FamilyOfElements (G\u271d.op \u22d9 \u2131\u271d.val) T\u271d\nh : Compatible x\u271d\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} D\nK : GrothendieckTopology D\nG : C \u2964 D\ninst\u271d : RepresentablyFlat G\n\u2131 : SheafOfTypes K\nZ : C\nT : Presieve Z\nx : FamilyOfElements (G.op \u22d9 \u2131.val) T\nhx : Compatible x\nY\u2081 Y\u2082 : C\nX : D\nf\u2081 : X \u27f6 G.obj Y\u2081\nf\u2082 : X \u27f6 G.obj Y\u2082\ng\u2081 : Y\u2081 \u27f6 Z\ng\u2082 : Y\u2082 \u27f6 Z\nhg\u2081 : T g\u2081\nhg\u2082 : T g\u2082\ne : f\u2081 \u226b G.map g\u2081 = f\u2082 \u226b G.map g\u2082\nc : Cone (cospan g\u2081 g\u2082 \u22d9 G) :=\n (Cones.postcompose (diagramIsoCospan (cospan g\u2081 g\u2082 \u22d9 G)).inv).obj (PullbackCone.mk f\u2081 f\u2082 e)\nc' : Cone (Cone.toStructuredArrow c \u22d9 StructuredArrow.pre c.pt (cospan g\u2081 g\u2082) G) :=\n IsCofiltered.cone (Cone.toStructuredArrow c \u22d9 StructuredArrow.pre c.pt (cospan g\u2081 g\u2082) G)\neq\u2081 :\n f\u2081 =\n (c'.pt.hom \u226b G.map (NatTrans.app c'.\u03c0 left).right) \u226b\n eqToHom\n (_ :\n G.obj ((StructuredArrow.pre X (cospan g\u2081 g\u2082) G).obj (StructuredArrow.mk (f\u2081 \u226b \ud835\udfd9 (G.obj Y\u2081)))).right =\n G.obj Y\u2081)\neq\u2082 :\n f\u2082 =\n (c'.pt.hom \u226b G.map (NatTrans.app c'.\u03c0 right).right) \u226b\n eqToHom\n (_ :\n G.obj ((StructuredArrow.pre X (cospan g\u2081 g\u2082) G).obj (StructuredArrow.mk (f\u2082 \u226b \ud835\udfd9 (G.obj Y\u2082)))).right =\n G.obj Y\u2082)\n\u22a2 \u2131.val.map\n ((c'.pt.hom \u226b G.map (NatTrans.app c'.\u03c0 left).right) \u226b\n eqToHom\n (_ :\n G.obj ((StructuredArrow.pre X (cospan g\u2081 g\u2082) G).obj (StructuredArrow.mk (f\u2081 \u226b \ud835\udfd9 (G.obj Y\u2081)))).right =\n G.obj Y\u2081)).op\n (x g\u2081 hg\u2081) =\n \u2131.val.map\n ((c'.pt.hom \u226b G.map (NatTrans.app c'.\u03c0 right).right) \u226b\n eqToHom\n (_ :\n G.obj ((StructuredArrow.pre X (cospan g\u2081 g\u2082) G).obj (StructuredArrow.mk (f\u2082 \u226b \ud835\udfd9 (G.obj Y\u2082)))).right =\n G.obj Y\u2082)).op\n (x g\u2082 hg\u2082)\n[PROOFSTEP]\nsimp only [op_comp, Functor.map_comp, types_comp_apply, eqToHom_op, eqToHom_map]\n[GOAL]\ncase Compatible\nC\u271d : Type u\u2081\ninst\u271d\u2075 : Category.{v\u2081, u\u2081} C\u271d\nD\u271d : Type u\u2082\ninst\u271d\u2074 : Category.{v\u2082, u\u2082} D\u271d\nA : Type u\u2083\ninst\u271d\u00b3 : Category.{v\u2083, u\u2083} A\nJ : GrothendieckTopology C\u271d\nK\u271d : GrothendieckTopology D\u271d\nL : GrothendieckTopology A\nG\u271d : C\u271d \u2964 D\u271d\nhG : CompatiblePreserving K\u271d G\u271d\n\u2131\u271d : SheafOfTypes K\u271d\nZ\u271d : C\u271d\nT\u271d : Presieve Z\u271d\nx\u271d : FamilyOfElements (G\u271d.op \u22d9 \u2131\u271d.val) T\u271d\nh : Compatible x\u271d\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} D\nK : GrothendieckTopology D\nG : C \u2964 D\ninst\u271d : RepresentablyFlat G\n\u2131 : SheafOfTypes K\nZ : C\nT : Presieve Z\nx : FamilyOfElements (G.op \u22d9 \u2131.val) T\nhx : Compatible x\nY\u2081 Y\u2082 : C\nX : D\nf\u2081 : X \u27f6 G.obj Y\u2081\nf\u2082 : X \u27f6 G.obj Y\u2082\ng\u2081 : Y\u2081 \u27f6 Z\ng\u2082 : Y\u2082 \u27f6 Z\nhg\u2081 : T g\u2081\nhg\u2082 : T g\u2082\ne : f\u2081 \u226b G.map g\u2081 = f\u2082 \u226b G.map g\u2082\nc : Cone (cospan g\u2081 g\u2082 \u22d9 G) :=\n (Cones.postcompose (diagramIsoCospan (cospan g\u2081 g\u2082 \u22d9 G)).inv).obj (PullbackCone.mk f\u2081 f\u2082 e)\nc' : Cone (Cone.toStructuredArrow c \u22d9 StructuredArrow.pre c.pt (cospan g\u2081 g\u2082) G) :=\n IsCofiltered.cone (Cone.toStructuredArrow c \u22d9 StructuredArrow.pre c.pt (cospan g\u2081 g\u2082) G)\neq\u2081 :\n f\u2081 =\n (c'.pt.hom \u226b G.map (NatTrans.app c'.\u03c0 left).right) \u226b\n eqToHom\n (_ :\n G.obj ((StructuredArrow.pre X (cospan g\u2081 g\u2082) G).obj (StructuredArrow.mk (f\u2081 \u226b \ud835\udfd9 (G.obj Y\u2081)))).right =\n G.obj Y\u2081)\neq\u2082 :\n f\u2082 =\n (c'.pt.hom \u226b G.map (NatTrans.app c'.\u03c0 right).right) \u226b\n eqToHom\n (_ :\n G.obj ((StructuredArrow.pre X (cospan g\u2081 g\u2082) G).obj (StructuredArrow.mk (f\u2082 \u226b \ud835\udfd9 (G.obj Y\u2082)))).right =\n G.obj Y\u2082)\n\u22a2 \u2131.val.map\n (IsCofiltered.cone\n (Cone.toStructuredArrow\n ((Cones.postcompose (diagramIsoCospan (cospan g\u2081 g\u2082 \u22d9 G)).inv).obj (PullbackCone.mk f\u2081 f\u2082 e)) \u22d9\n StructuredArrow.pre\n ((Cones.postcompose (diagramIsoCospan (cospan g\u2081 g\u2082 \u22d9 G)).inv).obj (PullbackCone.mk f\u2081 f\u2082 e)).pt\n (cospan g\u2081 g\u2082) G)).pt.hom.op\n (\u2131.val.map\n (G.map\n (NatTrans.app\n (IsCofiltered.cone\n (Cone.toStructuredArrow\n ((Cones.postcompose (diagramIsoCospan (cospan g\u2081 g\u2082 \u22d9 G)).inv).obj (PullbackCone.mk f\u2081 f\u2082 e)) \u22d9\n StructuredArrow.pre\n ((Cones.postcompose (diagramIsoCospan (cospan g\u2081 g\u2082 \u22d9 G)).inv).obj (PullbackCone.mk f\u2081 f\u2082 e)).pt\n (cospan g\u2081 g\u2082) G)).\u03c0\n left).right).op\n (eqToHom\n (_ :\n \u2131.val.obj (op (G.obj Y\u2081)) =\n \u2131.val.obj\n (op\n (G.obj\n ((Cone.toStructuredArrow\n ((Cones.postcompose (diagramIsoCospan (cospan g\u2081 g\u2082 \u22d9 G)).inv).obj\n (PullbackCone.mk f\u2081 f\u2082 e)) \u22d9\n StructuredArrow.pre\n ((Cones.postcompose (diagramIsoCospan (cospan g\u2081 g\u2082 \u22d9 G)).inv).obj\n (PullbackCone.mk f\u2081 f\u2082 e)).pt\n (cospan g\u2081 g\u2082) G).obj\n left).right)))\n (x g\u2081 hg\u2081))) =\n \u2131.val.map\n (IsCofiltered.cone\n (Cone.toStructuredArrow\n ((Cones.postcompose (diagramIsoCospan (cospan g\u2081 g\u2082 \u22d9 G)).inv).obj (PullbackCone.mk f\u2081 f\u2082 e)) \u22d9\n StructuredArrow.pre\n ((Cones.postcompose (diagramIsoCospan (cospan g\u2081 g\u2082 \u22d9 G)).inv).obj (PullbackCone.mk f\u2081 f\u2082 e)).pt\n (cospan g\u2081 g\u2082) G)).pt.hom.op\n (\u2131.val.map\n (G.map\n (NatTrans.app\n (IsCofiltered.cone\n (Cone.toStructuredArrow\n ((Cones.postcompose (diagramIsoCospan (cospan g\u2081 g\u2082 \u22d9 G)).inv).obj (PullbackCone.mk f\u2081 f\u2082 e)) \u22d9\n StructuredArrow.pre\n ((Cones.postcompose (diagramIsoCospan (cospan g\u2081 g\u2082 \u22d9 G)).inv).obj (PullbackCone.mk f\u2081 f\u2082 e)).pt\n (cospan g\u2081 g\u2082) G)).\u03c0\n right).right).op\n (eqToHom\n (_ :\n \u2131.val.obj (op (G.obj Y\u2082)) =\n \u2131.val.obj\n (op\n (G.obj\n ((Cone.toStructuredArrow\n ((Cones.postcompose (diagramIsoCospan (cospan g\u2081 g\u2082 \u22d9 G)).inv).obj\n (PullbackCone.mk f\u2081 f\u2082 e)) \u22d9\n StructuredArrow.pre\n ((Cones.postcompose (diagramIsoCospan (cospan g\u2081 g\u2082 \u22d9 G)).inv).obj\n (PullbackCone.mk f\u2081 f\u2082 e)).pt\n (cospan g\u2081 g\u2082) G).obj\n right).right)))\n (x g\u2082 hg\u2082)))\n[PROOFSTEP]\napply congr_arg\n[GOAL]\ncase Compatible.h\nC\u271d : Type u\u2081\ninst\u271d\u2075 : Category.{v\u2081, u\u2081} C\u271d\nD\u271d : Type u\u2082\ninst\u271d\u2074 : Category.{v\u2082, u\u2082} D\u271d\nA : Type u\u2083\ninst\u271d\u00b3 : Category.{v\u2083, u\u2083} A\nJ : GrothendieckTopology C\u271d\nK\u271d : GrothendieckTopology D\u271d\nL : GrothendieckTopology A\nG\u271d : C\u271d \u2964 D\u271d\nhG : CompatiblePreserving K\u271d G\u271d\n\u2131\u271d : SheafOfTypes K\u271d\nZ\u271d : C\u271d\nT\u271d : Presieve Z\u271d\nx\u271d : FamilyOfElements (G\u271d.op \u22d9 \u2131\u271d.val) T\u271d\nh : Compatible x\u271d\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} D\nK : GrothendieckTopology D\nG : C \u2964 D\ninst\u271d : RepresentablyFlat G\n\u2131 : SheafOfTypes K\nZ : C\nT : Presieve Z\nx : FamilyOfElements (G.op \u22d9 \u2131.val) T\nhx : Compatible x\nY\u2081 Y\u2082 : C\nX : D\nf\u2081 : X \u27f6 G.obj Y\u2081\nf\u2082 : X \u27f6 G.obj Y\u2082\ng\u2081 : Y\u2081 \u27f6 Z\ng\u2082 : Y\u2082 \u27f6 Z\nhg\u2081 : T g\u2081\nhg\u2082 : T g\u2082\ne : f\u2081 \u226b G.map g\u2081 = f\u2082 \u226b G.map g\u2082\nc : Cone (cospan g\u2081 g\u2082 \u22d9 G) :=\n (Cones.postcompose (diagramIsoCospan (cospan g\u2081 g\u2082 \u22d9 G)).inv).obj (PullbackCone.mk f\u2081 f\u2082 e)\nc' : Cone (Cone.toStructuredArrow c \u22d9 StructuredArrow.pre c.pt (cospan g\u2081 g\u2082) G) :=\n IsCofiltered.cone (Cone.toStructuredArrow c \u22d9 StructuredArrow.pre c.pt (cospan g\u2081 g\u2082) G)\neq\u2081 :\n f\u2081 =\n (c'.pt.hom \u226b G.map (NatTrans.app c'.\u03c0 left).right) \u226b\n eqToHom\n (_ :\n G.obj ((StructuredArrow.pre X (cospan g\u2081 g\u2082) G).obj (StructuredArrow.mk (f\u2081 \u226b \ud835\udfd9 (G.obj Y\u2081)))).right =\n G.obj Y\u2081)\neq\u2082 :\n f\u2082 =\n (c'.pt.hom \u226b G.map (NatTrans.app c'.\u03c0 right).right) \u226b\n eqToHom\n (_ :\n G.obj ((StructuredArrow.pre X (cospan g\u2081 g\u2082) G).obj (StructuredArrow.mk (f\u2082 \u226b \ud835\udfd9 (G.obj Y\u2082)))).right =\n G.obj Y\u2082)\n\u22a2 \u2131.val.map\n (G.map\n (NatTrans.app\n (IsCofiltered.cone\n (Cone.toStructuredArrow\n ((Cones.postcompose (diagramIsoCospan (cospan g\u2081 g\u2082 \u22d9 G)).inv).obj (PullbackCone.mk f\u2081 f\u2082 e)) \u22d9\n StructuredArrow.pre\n ((Cones.postcompose (diagramIsoCospan (cospan g\u2081 g\u2082 \u22d9 G)).inv).obj (PullbackCone.mk f\u2081 f\u2082 e)).pt\n (cospan g\u2081 g\u2082) G)).\u03c0\n left).right).op\n (eqToHom\n (_ :\n \u2131.val.obj (op (G.obj Y\u2081)) =\n \u2131.val.obj\n (op\n (G.obj\n ((Cone.toStructuredArrow\n ((Cones.postcompose (diagramIsoCospan (cospan g\u2081 g\u2082 \u22d9 G)).inv).obj\n (PullbackCone.mk f\u2081 f\u2082 e)) \u22d9\n StructuredArrow.pre\n ((Cones.postcompose (diagramIsoCospan (cospan g\u2081 g\u2082 \u22d9 G)).inv).obj\n (PullbackCone.mk f\u2081 f\u2082 e)).pt\n (cospan g\u2081 g\u2082) G).obj\n left).right)))\n (x g\u2081 hg\u2081)) =\n \u2131.val.map\n (G.map\n (NatTrans.app\n (IsCofiltered.cone\n (Cone.toStructuredArrow\n ((Cones.postcompose (diagramIsoCospan (cospan g\u2081 g\u2082 \u22d9 G)).inv).obj (PullbackCone.mk f\u2081 f\u2082 e)) \u22d9\n StructuredArrow.pre\n ((Cones.postcompose (diagramIsoCospan (cospan g\u2081 g\u2082 \u22d9 G)).inv).obj (PullbackCone.mk f\u2081 f\u2082 e)).pt\n (cospan g\u2081 g\u2082) G)).\u03c0\n right).right).op\n (eqToHom\n (_ :\n \u2131.val.obj (op (G.obj Y\u2082)) =\n \u2131.val.obj\n (op\n (G.obj\n ((Cone.toStructuredArrow\n ((Cones.postcompose (diagramIsoCospan (cospan g\u2081 g\u2082 \u22d9 G)).inv).obj\n (PullbackCone.mk f\u2081 f\u2082 e)) \u22d9\n StructuredArrow.pre\n ((Cones.postcompose (diagramIsoCospan (cospan g\u2081 g\u2082 \u22d9 G)).inv).obj\n (PullbackCone.mk f\u2081 f\u2082 e)).pt\n (cospan g\u2081 g\u2082) G).obj\n right).right)))\n (x g\u2082 hg\u2082))\n[PROOFSTEP]\ninjection c'.\u03c0.naturality WalkingCospan.Hom.inl with _ e\u2081\n[GOAL]\ncase Compatible.h\nC\u271d : Type u\u2081\ninst\u271d\u2075 : Category.{v\u2081, u\u2081} C\u271d\nD\u271d : Type u\u2082\ninst\u271d\u2074 : Category.{v\u2082, u\u2082} D\u271d\nA : Type u\u2083\ninst\u271d\u00b3 : Category.{v\u2083, u\u2083} A\nJ : GrothendieckTopology C\u271d\nK\u271d : GrothendieckTopology D\u271d\nL : GrothendieckTopology A\nG\u271d : C\u271d \u2964 D\u271d\nhG : CompatiblePreserving K\u271d G\u271d\n\u2131\u271d : SheafOfTypes K\u271d\nZ\u271d : C\u271d\nT\u271d : Presieve Z\u271d\nx\u271d : FamilyOfElements (G\u271d.op \u22d9 \u2131\u271d.val) T\u271d\nh : Compatible x\u271d\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} D\nK : GrothendieckTopology D\nG : C \u2964 D\ninst\u271d : RepresentablyFlat G\n\u2131 : SheafOfTypes K\nZ : C\nT : Presieve Z\nx : FamilyOfElements (G.op \u22d9 \u2131.val) T\nhx : Compatible x\nY\u2081 Y\u2082 : C\nX : D\nf\u2081 : X \u27f6 G.obj Y\u2081\nf\u2082 : X \u27f6 G.obj Y\u2082\ng\u2081 : Y\u2081 \u27f6 Z\ng\u2082 : Y\u2082 \u27f6 Z\nhg\u2081 : T g\u2081\nhg\u2082 : T g\u2082\ne : f\u2081 \u226b G.map g\u2081 = f\u2082 \u226b G.map g\u2082\nc : Cone (cospan g\u2081 g\u2082 \u22d9 G) :=\n (Cones.postcompose (diagramIsoCospan (cospan g\u2081 g\u2082 \u22d9 G)).inv).obj (PullbackCone.mk f\u2081 f\u2082 e)\nc' : Cone (Cone.toStructuredArrow c \u22d9 StructuredArrow.pre c.pt (cospan g\u2081 g\u2082) G) :=\n IsCofiltered.cone (Cone.toStructuredArrow c \u22d9 StructuredArrow.pre c.pt (cospan g\u2081 g\u2082) G)\neq\u2081 :\n f\u2081 =\n (c'.pt.hom \u226b G.map (NatTrans.app c'.\u03c0 left).right) \u226b\n eqToHom\n (_ :\n G.obj ((StructuredArrow.pre X (cospan g\u2081 g\u2082) G).obj (StructuredArrow.mk (f\u2081 \u226b \ud835\udfd9 (G.obj Y\u2081)))).right =\n G.obj Y\u2081)\neq\u2082 :\n f\u2082 =\n (c'.pt.hom \u226b G.map (NatTrans.app c'.\u03c0 right).right) \u226b\n eqToHom\n (_ :\n G.obj ((StructuredArrow.pre X (cospan g\u2081 g\u2082) G).obj (StructuredArrow.mk (f\u2082 \u226b \ud835\udfd9 (G.obj Y\u2082)))).right =\n G.obj Y\u2082)\nleft_eq\u271d :\n (((Functor.const WalkingCospan).obj c'.pt).map Hom.inl).left \u226b (NatTrans.app c'.\u03c0 one).left =\n (NatTrans.app c'.\u03c0 left).left \u226b\n ((Cone.toStructuredArrow c \u22d9 StructuredArrow.pre c.pt (cospan g\u2081 g\u2082) G).map Hom.inl).left\ne\u2081 :\n (((Functor.const WalkingCospan).obj c'.pt).map Hom.inl).right \u226b (NatTrans.app c'.\u03c0 one).right =\n (NatTrans.app c'.\u03c0 left).right \u226b\n ((Cone.toStructuredArrow c \u22d9 StructuredArrow.pre c.pt (cospan g\u2081 g\u2082) G).map Hom.inl).right\n\u22a2 \u2131.val.map\n (G.map\n (NatTrans.app\n (IsCofiltered.cone\n (Cone.toStructuredArrow\n ((Cones.postcompose (diagramIsoCospan (cospan g\u2081 g\u2082 \u22d9 G)).inv).obj (PullbackCone.mk f\u2081 f\u2082 e)) \u22d9\n StructuredArrow.pre\n ((Cones.postcompose (diagramIsoCospan (cospan g\u2081 g\u2082 \u22d9 G)).inv).obj (PullbackCone.mk f\u2081 f\u2082 e)).pt\n (cospan g\u2081 g\u2082) G)).\u03c0\n left).right).op\n (eqToHom\n (_ :\n \u2131.val.obj (op (G.obj Y\u2081)) =\n \u2131.val.obj\n (op\n (G.obj\n ((Cone.toStructuredArrow\n ((Cones.postcompose (diagramIsoCospan (cospan g\u2081 g\u2082 \u22d9 G)).inv).obj\n (PullbackCone.mk f\u2081 f\u2082 e)) \u22d9\n StructuredArrow.pre\n ((Cones.postcompose (diagramIsoCospan (cospan g\u2081 g\u2082 \u22d9 G)).inv).obj\n (PullbackCone.mk f\u2081 f\u2082 e)).pt\n (cospan g\u2081 g\u2082) G).obj\n left).right)))\n (x g\u2081 hg\u2081)) =\n \u2131.val.map\n (G.map\n (NatTrans.app\n (IsCofiltered.cone\n (Cone.toStructuredArrow\n ((Cones.postcompose (diagramIsoCospan (cospan g\u2081 g\u2082 \u22d9 G)).inv).obj (PullbackCone.mk f\u2081 f\u2082 e)) \u22d9\n StructuredArrow.pre\n ((Cones.postcompose (diagramIsoCospan (cospan g\u2081 g\u2082 \u22d9 G)).inv).obj (PullbackCone.mk f\u2081 f\u2082 e)).pt\n (cospan g\u2081 g\u2082) G)).\u03c0\n right).right).op\n (eqToHom\n (_ :\n \u2131.val.obj (op (G.obj Y\u2082)) =\n \u2131.val.obj\n (op\n (G.obj\n ((Cone.toStructuredArrow\n ((Cones.postcompose (diagramIsoCospan (cospan g\u2081 g\u2082 \u22d9 G)).inv).obj\n (PullbackCone.mk f\u2081 f\u2082 e)) \u22d9\n StructuredArrow.pre\n ((Cones.postcompose (diagramIsoCospan (cospan g\u2081 g\u2082 \u22d9 G)).inv).obj\n (PullbackCone.mk f\u2081 f\u2082 e)).pt\n (cospan g\u2081 g\u2082) G).obj\n right).right)))\n (x g\u2082 hg\u2082))\n[PROOFSTEP]\ninjection c'.\u03c0.naturality WalkingCospan.Hom.inr with _ e\u2082\n[GOAL]\ncase Compatible.h\nC\u271d : Type u\u2081\ninst\u271d\u2075 : Category.{v\u2081, u\u2081} C\u271d\nD\u271d : Type u\u2082\ninst\u271d\u2074 : Category.{v\u2082, u\u2082} D\u271d\nA : Type u\u2083\ninst\u271d\u00b3 : Category.{v\u2083, u\u2083} A\nJ : GrothendieckTopology C\u271d\nK\u271d : GrothendieckTopology D\u271d\nL : GrothendieckTopology A\nG\u271d : C\u271d \u2964 D\u271d\nhG : CompatiblePreserving K\u271d G\u271d\n\u2131\u271d : SheafOfTypes K\u271d\nZ\u271d : C\u271d\nT\u271d : Presieve Z\u271d\nx\u271d : FamilyOfElements (G\u271d.op \u22d9 \u2131\u271d.val) T\u271d\nh : Compatible x\u271d\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} D\nK : GrothendieckTopology D\nG : C \u2964 D\ninst\u271d : RepresentablyFlat G\n\u2131 : SheafOfTypes K\nZ : C\nT : Presieve Z\nx : FamilyOfElements (G.op \u22d9 \u2131.val) T\nhx : Compatible x\nY\u2081 Y\u2082 : C\nX : D\nf\u2081 : X \u27f6 G.obj Y\u2081\nf\u2082 : X \u27f6 G.obj Y\u2082\ng\u2081 : Y\u2081 \u27f6 Z\ng\u2082 : Y\u2082 \u27f6 Z\nhg\u2081 : T g\u2081\nhg\u2082 : T g\u2082\ne : f\u2081 \u226b G.map g\u2081 = f\u2082 \u226b G.map g\u2082\nc : Cone (cospan g\u2081 g\u2082 \u22d9 G) :=\n (Cones.postcompose (diagramIsoCospan (cospan g\u2081 g\u2082 \u22d9 G)).inv).obj (PullbackCone.mk f\u2081 f\u2082 e)\nc' : Cone (Cone.toStructuredArrow c \u22d9 StructuredArrow.pre c.pt (cospan g\u2081 g\u2082) G) :=\n IsCofiltered.cone (Cone.toStructuredArrow c \u22d9 StructuredArrow.pre c.pt (cospan g\u2081 g\u2082) G)\neq\u2081 :\n f\u2081 =\n (c'.pt.hom \u226b G.map (NatTrans.app c'.\u03c0 left).right) \u226b\n eqToHom\n (_ :\n G.obj ((StructuredArrow.pre X (cospan g\u2081 g\u2082) G).obj (StructuredArrow.mk (f\u2081 \u226b \ud835\udfd9 (G.obj Y\u2081)))).right =\n G.obj Y\u2081)\neq\u2082 :\n f\u2082 =\n (c'.pt.hom \u226b G.map (NatTrans.app c'.\u03c0 right).right) \u226b\n eqToHom\n (_ :\n G.obj ((StructuredArrow.pre X (cospan g\u2081 g\u2082) G).obj (StructuredArrow.mk (f\u2082 \u226b \ud835\udfd9 (G.obj Y\u2082)))).right =\n G.obj Y\u2082)\nleft_eq\u271d\u00b9 :\n (((Functor.const WalkingCospan).obj c'.pt).map Hom.inl).left \u226b (NatTrans.app c'.\u03c0 one).left =\n (NatTrans.app c'.\u03c0 left).left \u226b\n ((Cone.toStructuredArrow c \u22d9 StructuredArrow.pre c.pt (cospan g\u2081 g\u2082) G).map Hom.inl).left\ne\u2081 :\n (((Functor.const WalkingCospan).obj c'.pt).map Hom.inl).right \u226b (NatTrans.app c'.\u03c0 one).right =\n (NatTrans.app c'.\u03c0 left).right \u226b\n ((Cone.toStructuredArrow c \u22d9 StructuredArrow.pre c.pt (cospan g\u2081 g\u2082) G).map Hom.inl).right\nleft_eq\u271d :\n (((Functor.const WalkingCospan).obj c'.pt).map Hom.inr).left \u226b (NatTrans.app c'.\u03c0 one).left =\n (NatTrans.app c'.\u03c0 right).left \u226b\n ((Cone.toStructuredArrow c \u22d9 StructuredArrow.pre c.pt (cospan g\u2081 g\u2082) G).map Hom.inr).left\ne\u2082 :\n (((Functor.const WalkingCospan).obj c'.pt).map Hom.inr).right \u226b (NatTrans.app c'.\u03c0 one).right =\n (NatTrans.app c'.\u03c0 right).right \u226b\n ((Cone.toStructuredArrow c \u22d9 StructuredArrow.pre c.pt (cospan g\u2081 g\u2082) G).map Hom.inr).right\n\u22a2 \u2131.val.map\n (G.map\n (NatTrans.app\n (IsCofiltered.cone\n (Cone.toStructuredArrow\n ((Cones.postcompose (diagramIsoCospan (cospan g\u2081 g\u2082 \u22d9 G)).inv).obj (PullbackCone.mk f\u2081 f\u2082 e)) \u22d9\n StructuredArrow.pre\n ((Cones.postcompose (diagramIsoCospan (cospan g\u2081 g\u2082 \u22d9 G)).inv).obj (PullbackCone.mk f\u2081 f\u2082 e)).pt\n (cospan g\u2081 g\u2082) G)).\u03c0\n left).right).op\n (eqToHom\n (_ :\n \u2131.val.obj (op (G.obj Y\u2081)) =\n \u2131.val.obj\n (op\n (G.obj\n ((Cone.toStructuredArrow\n ((Cones.postcompose (diagramIsoCospan (cospan g\u2081 g\u2082 \u22d9 G)).inv).obj\n (PullbackCone.mk f\u2081 f\u2082 e)) \u22d9\n StructuredArrow.pre\n ((Cones.postcompose (diagramIsoCospan (cospan g\u2081 g\u2082 \u22d9 G)).inv).obj\n (PullbackCone.mk f\u2081 f\u2082 e)).pt\n (cospan g\u2081 g\u2082) G).obj\n left).right)))\n (x g\u2081 hg\u2081)) =\n \u2131.val.map\n (G.map\n (NatTrans.app\n (IsCofiltered.cone\n (Cone.toStructuredArrow\n ((Cones.postcompose (diagramIsoCospan (cospan g\u2081 g\u2082 \u22d9 G)).inv).obj (PullbackCone.mk f\u2081 f\u2082 e)) \u22d9\n StructuredArrow.pre\n ((Cones.postcompose (diagramIsoCospan (cospan g\u2081 g\u2082 \u22d9 G)).inv).obj (PullbackCone.mk f\u2081 f\u2082 e)).pt\n (cospan g\u2081 g\u2082) G)).\u03c0\n right).right).op\n (eqToHom\n (_ :\n \u2131.val.obj (op (G.obj Y\u2082)) =\n \u2131.val.obj\n (op\n (G.obj\n ((Cone.toStructuredArrow\n ((Cones.postcompose (diagramIsoCospan (cospan g\u2081 g\u2082 \u22d9 G)).inv).obj\n (PullbackCone.mk f\u2081 f\u2082 e)) \u22d9\n StructuredArrow.pre\n ((Cones.postcompose (diagramIsoCospan (cospan g\u2081 g\u2082 \u22d9 G)).inv).obj\n (PullbackCone.mk f\u2081 f\u2082 e)).pt\n (cospan g\u2081 g\u2082) G).obj\n right).right)))\n (x g\u2082 hg\u2082))\n[PROOFSTEP]\nexact hx (c'.\u03c0.app left).right (c'.\u03c0.app right).right hg\u2081 hg\u2082 (e\u2081.symm.trans e\u2082)\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\nA : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nL : GrothendieckTopology A\nG : C \u2964 D\nhG : CompatiblePreserving K G\n\u2131 : SheafOfTypes K\nZ : C\nT : Presieve Z\nx : FamilyOfElements (G.op \u22d9 \u2131.val) T\nh : Compatible x\nF : C \u2964 D\ninst\u271d\u00b9 : Full F\ninst\u271d : Faithful F\nhF : {c : C} \u2192 {d : D} \u2192 (d \u27f6 F.obj c) \u2192 (c' : C) \u00d7 (F.obj c' \u2245 d)\n\u22a2 CompatiblePreserving K F\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase Compatible\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\nA : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nL : GrothendieckTopology A\nG : C \u2964 D\nhG : CompatiblePreserving K G\n\u2131 : SheafOfTypes K\nZ : C\nT : Presieve Z\nx : FamilyOfElements (G.op \u22d9 \u2131.val) T\nh : Compatible x\nF : C \u2964 D\ninst\u271d\u00b9 : Full F\ninst\u271d : Faithful F\nhF : {c : C} \u2192 {d : D} \u2192 (d \u27f6 F.obj c) \u2192 (c' : C) \u00d7 (F.obj c' \u2245 d)\n\u22a2 \u2200 (\u2131 : SheafOfTypes K) {Z : C} {T : Presieve Z} {x : FamilyOfElements (F.op \u22d9 \u2131.val) T},\n Compatible x \u2192\n \u2200 {Y\u2081 Y\u2082 : C} {X : D} (f\u2081 : X \u27f6 F.obj Y\u2081) (f\u2082 : X \u27f6 F.obj Y\u2082) {g\u2081 : Y\u2081 \u27f6 Z} {g\u2082 : Y\u2082 \u27f6 Z} (hg\u2081 : T g\u2081)\n (hg\u2082 : T g\u2082), f\u2081 \u226b F.map g\u2081 = f\u2082 \u226b F.map g\u2082 \u2192 \u2131.val.map f\u2081.op (x g\u2081 hg\u2081) = \u2131.val.map f\u2082.op (x g\u2082 hg\u2082)\n[PROOFSTEP]\nintrov hx he\n[GOAL]\ncase Compatible\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\nA : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nL : GrothendieckTopology A\nG : C \u2964 D\nhG : CompatiblePreserving K G\n\u2131\u271d : SheafOfTypes K\nZ\u271d : C\nT\u271d : Presieve Z\u271d\nx\u271d : FamilyOfElements (G.op \u22d9 \u2131\u271d.val) T\u271d\nh : Compatible x\u271d\nF : C \u2964 D\ninst\u271d\u00b9 : Full F\ninst\u271d : Faithful F\nhF : {c : C} \u2192 {d : D} \u2192 (d \u27f6 F.obj c) \u2192 (c' : C) \u00d7 (F.obj c' \u2245 d)\n\u2131 : SheafOfTypes K\nZ : C\nT : Presieve Z\nx : FamilyOfElements (F.op \u22d9 \u2131.val) T\nhx : Compatible x\nY\u2081 Y\u2082 : C\nX : D\nf\u2081 : X \u27f6 F.obj Y\u2081\nf\u2082 : X \u27f6 F.obj Y\u2082\ng\u2081 : Y\u2081 \u27f6 Z\ng\u2082 : Y\u2082 \u27f6 Z\nhg\u2081 : T g\u2081\nhg\u2082 : T g\u2082\nhe : f\u2081 \u226b F.map g\u2081 = f\u2082 \u226b F.map g\u2082\n\u22a2 \u2131.val.map f\u2081.op (x g\u2081 hg\u2081) = \u2131.val.map f\u2082.op (x g\u2082 hg\u2082)\n[PROOFSTEP]\nobtain \u27e8X', e\u27e9 := hF f\u2081\n[GOAL]\ncase Compatible.mk\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\nA : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nL : GrothendieckTopology A\nG : C \u2964 D\nhG : CompatiblePreserving K G\n\u2131\u271d : SheafOfTypes K\nZ\u271d : C\nT\u271d : Presieve Z\u271d\nx\u271d : FamilyOfElements (G.op \u22d9 \u2131\u271d.val) T\u271d\nh : Compatible x\u271d\nF : C \u2964 D\ninst\u271d\u00b9 : Full F\ninst\u271d : Faithful F\nhF : {c : C} \u2192 {d : D} \u2192 (d \u27f6 F.obj c) \u2192 (c' : C) \u00d7 (F.obj c' \u2245 d)\n\u2131 : SheafOfTypes K\nZ : C\nT : Presieve Z\nx : FamilyOfElements (F.op \u22d9 \u2131.val) T\nhx : Compatible x\nY\u2081 Y\u2082 : C\nX : D\nf\u2081 : X \u27f6 F.obj Y\u2081\nf\u2082 : X \u27f6 F.obj Y\u2082\ng\u2081 : Y\u2081 \u27f6 Z\ng\u2082 : Y\u2082 \u27f6 Z\nhg\u2081 : T g\u2081\nhg\u2082 : T g\u2082\nhe : f\u2081 \u226b F.map g\u2081 = f\u2082 \u226b F.map g\u2082\nX' : C\ne : F.obj X' \u2245 X\n\u22a2 \u2131.val.map f\u2081.op (x g\u2081 hg\u2081) = \u2131.val.map f\u2082.op (x g\u2082 hg\u2082)\n[PROOFSTEP]\napply (\u2131.1.mapIso e.op).toEquiv.injective\n[GOAL]\ncase Compatible.mk.a\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\nA : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nL : GrothendieckTopology A\nG : C \u2964 D\nhG : CompatiblePreserving K G\n\u2131\u271d : SheafOfTypes K\nZ\u271d : C\nT\u271d : Presieve Z\u271d\nx\u271d : FamilyOfElements (G.op \u22d9 \u2131\u271d.val) T\u271d\nh : Compatible x\u271d\nF : C \u2964 D\ninst\u271d\u00b9 : Full F\ninst\u271d : Faithful F\nhF : {c : C} \u2192 {d : D} \u2192 (d \u27f6 F.obj c) \u2192 (c' : C) \u00d7 (F.obj c' \u2245 d)\n\u2131 : SheafOfTypes K\nZ : C\nT : Presieve Z\nx : FamilyOfElements (F.op \u22d9 \u2131.val) T\nhx : Compatible x\nY\u2081 Y\u2082 : C\nX : D\nf\u2081 : X \u27f6 F.obj Y\u2081\nf\u2082 : X \u27f6 F.obj Y\u2082\ng\u2081 : Y\u2081 \u27f6 Z\ng\u2082 : Y\u2082 \u27f6 Z\nhg\u2081 : T g\u2081\nhg\u2082 : T g\u2082\nhe : f\u2081 \u226b F.map g\u2081 = f\u2082 \u226b F.map g\u2082\nX' : C\ne : F.obj X' \u2245 X\n\u22a2 \u2191(\u2131.val.mapIso (Iso.op e)).toEquiv (\u2131.val.map f\u2081.op (x g\u2081 hg\u2081)) =\n \u2191(\u2131.val.mapIso (Iso.op e)).toEquiv (\u2131.val.map f\u2082.op (x g\u2082 hg\u2082))\n[PROOFSTEP]\nsimp only [Iso.op_hom, Iso.toEquiv_fun, \u2131.1.mapIso_hom, \u2190 FunctorToTypes.map_comp_apply]\n[GOAL]\ncase Compatible.mk.a\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\nA : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nL : GrothendieckTopology A\nG : C \u2964 D\nhG : CompatiblePreserving K G\n\u2131\u271d : SheafOfTypes K\nZ\u271d : C\nT\u271d : Presieve Z\u271d\nx\u271d : FamilyOfElements (G.op \u22d9 \u2131\u271d.val) T\u271d\nh : Compatible x\u271d\nF : C \u2964 D\ninst\u271d\u00b9 : Full F\ninst\u271d : Faithful F\nhF : {c : C} \u2192 {d : D} \u2192 (d \u27f6 F.obj c) \u2192 (c' : C) \u00d7 (F.obj c' \u2245 d)\n\u2131 : SheafOfTypes K\nZ : C\nT : Presieve Z\nx : FamilyOfElements (F.op \u22d9 \u2131.val) T\nhx : Compatible x\nY\u2081 Y\u2082 : C\nX : D\nf\u2081 : X \u27f6 F.obj Y\u2081\nf\u2082 : X \u27f6 F.obj Y\u2082\ng\u2081 : Y\u2081 \u27f6 Z\ng\u2082 : Y\u2082 \u27f6 Z\nhg\u2081 : T g\u2081\nhg\u2082 : T g\u2082\nhe : f\u2081 \u226b F.map g\u2081 = f\u2082 \u226b F.map g\u2082\nX' : C\ne : F.obj X' \u2245 X\n\u22a2 \u2131.val.map (f\u2081.op \u226b e.hom.op) (x g\u2081 hg\u2081) = \u2131.val.map (f\u2082.op \u226b e.hom.op) (x g\u2082 hg\u2082)\n[PROOFSTEP]\nsimpa using hx (F.preimage <| e.hom \u226b f\u2081) (F.preimage <| e.hom \u226b f\u2082) hg\u2081 hg\u2082 (F.map_injective <| by simpa using he)\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\nA : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nL : GrothendieckTopology A\nG : C \u2964 D\nhG : CompatiblePreserving K G\n\u2131\u271d : SheafOfTypes K\nZ\u271d : C\nT\u271d : Presieve Z\u271d\nx\u271d : FamilyOfElements (G.op \u22d9 \u2131\u271d.val) T\u271d\nh : Compatible x\u271d\nF : C \u2964 D\ninst\u271d\u00b9 : Full F\ninst\u271d : Faithful F\nhF : {c : C} \u2192 {d : D} \u2192 (d \u27f6 F.obj c) \u2192 (c' : C) \u00d7 (F.obj c' \u2245 d)\n\u2131 : SheafOfTypes K\nZ : C\nT : Presieve Z\nx : FamilyOfElements (F.op \u22d9 \u2131.val) T\nhx : Compatible x\nY\u2081 Y\u2082 : C\nX : D\nf\u2081 : X \u27f6 F.obj Y\u2081\nf\u2082 : X \u27f6 F.obj Y\u2082\ng\u2081 : Y\u2081 \u27f6 Z\ng\u2082 : Y\u2082 \u27f6 Z\nhg\u2081 : T g\u2081\nhg\u2082 : T g\u2082\nhe : f\u2081 \u226b F.map g\u2081 = f\u2082 \u226b F.map g\u2082\nX' : C\ne : F.obj X' \u2245 X\n\u22a2 F.map (F.preimage (e.hom \u226b f\u2081) \u226b g\u2081) = F.map (F.preimage (e.hom \u226b f\u2082) \u226b g\u2082)\n[PROOFSTEP]\nsimpa using he\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nA : Type u\u2083\ninst\u271d : Category.{v\u2083, u\u2083} A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nL : GrothendieckTopology A\nG\u271d : C \u2964 D\nhG : CompatiblePreserving K G\u271d\n\u2131\u271d : SheafOfTypes K\nZ : C\nT : Presieve Z\nx : FamilyOfElements (G\u271d.op \u22d9 \u2131\u271d.val) T\nh : Compatible x\nG : C \u2964 D\nhG\u2081 : CompatiblePreserving K G\nhG\u2082 : CoverPreserving J K G\n\u2131 : Sheaf K A\n\u22a2 Presheaf.IsSheaf J (G.op \u22d9 \u2131.val)\n[PROOFSTEP]\nintro X U S hS x hx\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nA : Type u\u2083\ninst\u271d : Category.{v\u2083, u\u2083} A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nL : GrothendieckTopology A\nG\u271d : C \u2964 D\nhG : CompatiblePreserving K G\u271d\n\u2131\u271d : SheafOfTypes K\nZ : C\nT : Presieve Z\nx\u271d : FamilyOfElements (G\u271d.op \u22d9 \u2131\u271d.val) T\nh : Compatible x\u271d\nG : C \u2964 D\nhG\u2081 : CompatiblePreserving K G\nhG\u2082 : CoverPreserving J K G\n\u2131 : Sheaf K A\nX : A\nU : C\nS : Sieve U\nhS : S \u2208 GrothendieckTopology.sieves J U\nx : FamilyOfElements ((G.op \u22d9 \u2131.val) \u22d9 coyoneda.obj (op X)) S.arrows\nhx : Compatible x\n\u22a2 \u2203! t, IsAmalgamation x t\n[PROOFSTEP]\nchange FamilyOfElements (G.op \u22d9 \u2131.val \u22d9 coyoneda.obj (op X)) _ at x \n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nA : Type u\u2083\ninst\u271d : Category.{v\u2083, u\u2083} A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nL : GrothendieckTopology A\nG\u271d : C \u2964 D\nhG : CompatiblePreserving K G\u271d\n\u2131\u271d : SheafOfTypes K\nZ : C\nT : Presieve Z\nx\u271d : FamilyOfElements (G\u271d.op \u22d9 \u2131\u271d.val) T\nh : Compatible x\u271d\nG : C \u2964 D\nhG\u2081 : CompatiblePreserving K G\nhG\u2082 : CoverPreserving J K G\n\u2131 : Sheaf K A\nX : A\nU : C\nS : Sieve U\nhS : S \u2208 GrothendieckTopology.sieves J U\nx : FamilyOfElements (G.op \u22d9 \u2131.val \u22d9 coyoneda.obj (op X)) S.arrows\nhx : Compatible x\n\u22a2 \u2203! t, IsAmalgamation x t\n[PROOFSTEP]\nlet H := \u2131.2 X _ (hG\u2082.cover_preserve hS)\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nA : Type u\u2083\ninst\u271d : Category.{v\u2083, u\u2083} A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nL : GrothendieckTopology A\nG\u271d : C \u2964 D\nhG : CompatiblePreserving K G\u271d\n\u2131\u271d : SheafOfTypes K\nZ : C\nT : Presieve Z\nx\u271d : FamilyOfElements (G\u271d.op \u22d9 \u2131\u271d.val) T\nh : Compatible x\u271d\nG : C \u2964 D\nhG\u2081 : CompatiblePreserving K G\nhG\u2082 : CoverPreserving J K G\n\u2131 : Sheaf K A\nX : A\nU : C\nS : Sieve U\nhS : S \u2208 GrothendieckTopology.sieves J U\nx : FamilyOfElements (G.op \u22d9 \u2131.val \u22d9 coyoneda.obj (op X)) S.arrows\nhx : Compatible x\nH : IsSheafFor (\u2131.val \u22d9 coyoneda.obj (op X)) (Sieve.functorPushforward G S).arrows :=\n Sheaf.cond \u2131 X (Sieve.functorPushforward G S) (CoverPreserving.cover_preserve hG\u2082 hS)\n\u22a2 \u2203! t, IsAmalgamation x t\n[PROOFSTEP]\nlet hx' := hx.functorPushforward hG\u2081 (sheafOver \u2131 X)\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nA : Type u\u2083\ninst\u271d : Category.{v\u2083, u\u2083} A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nL : GrothendieckTopology A\nG\u271d : C \u2964 D\nhG : CompatiblePreserving K G\u271d\n\u2131\u271d : SheafOfTypes K\nZ : C\nT : Presieve Z\nx\u271d : FamilyOfElements (G\u271d.op \u22d9 \u2131\u271d.val) T\nh : Compatible x\u271d\nG : C \u2964 D\nhG\u2081 : CompatiblePreserving K G\nhG\u2082 : CoverPreserving J K G\n\u2131 : Sheaf K A\nX : A\nU : C\nS : Sieve U\nhS : S \u2208 GrothendieckTopology.sieves J U\nx : FamilyOfElements (G.op \u22d9 \u2131.val \u22d9 coyoneda.obj (op X)) S.arrows\nhx : Compatible x\nH : IsSheafFor (\u2131.val \u22d9 coyoneda.obj (op X)) (Sieve.functorPushforward G S).arrows :=\n Sheaf.cond \u2131 X (Sieve.functorPushforward G S) (CoverPreserving.cover_preserve hG\u2082 hS)\nhx' : Compatible (FamilyOfElements.functorPushforward G x) := Compatible.functorPushforward hG\u2081 (sheafOver \u2131 X) hx\n\u22a2 \u2203! t, IsAmalgamation x t\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase h\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nA : Type u\u2083\ninst\u271d : Category.{v\u2083, u\u2083} A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nL : GrothendieckTopology A\nG\u271d : C \u2964 D\nhG : CompatiblePreserving K G\u271d\n\u2131\u271d : SheafOfTypes K\nZ : C\nT : Presieve Z\nx\u271d : FamilyOfElements (G\u271d.op \u22d9 \u2131\u271d.val) T\nh : Compatible x\u271d\nG : C \u2964 D\nhG\u2081 : CompatiblePreserving K G\nhG\u2082 : CoverPreserving J K G\n\u2131 : Sheaf K A\nX : A\nU : C\nS : Sieve U\nhS : S \u2208 GrothendieckTopology.sieves J U\nx : FamilyOfElements (G.op \u22d9 \u2131.val \u22d9 coyoneda.obj (op X)) S.arrows\nhx : Compatible x\nH : IsSheafFor (\u2131.val \u22d9 coyoneda.obj (op X)) (Sieve.functorPushforward G S).arrows :=\n Sheaf.cond \u2131 X (Sieve.functorPushforward G S) (CoverPreserving.cover_preserve hG\u2082 hS)\nhx' : Compatible (FamilyOfElements.functorPushforward G x) := Compatible.functorPushforward hG\u2081 (sheafOver \u2131 X) hx\n\u22a2 (fun t => IsAmalgamation x t) ?w \u2227\n \u2200 (y : ((G.op \u22d9 \u2131.val) \u22d9 coyoneda.obj (op X)).obj (op U)), (fun t => IsAmalgamation x t) y \u2192 y = ?w\ncase w\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nA : Type u\u2083\ninst\u271d : Category.{v\u2083, u\u2083} A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nL : GrothendieckTopology A\nG\u271d : C \u2964 D\nhG : CompatiblePreserving K G\u271d\n\u2131\u271d : SheafOfTypes K\nZ : C\nT : Presieve Z\nx\u271d : FamilyOfElements (G\u271d.op \u22d9 \u2131\u271d.val) T\nh : Compatible x\u271d\nG : C \u2964 D\nhG\u2081 : CompatiblePreserving K G\nhG\u2082 : CoverPreserving J K G\n\u2131 : Sheaf K A\nX : A\nU : C\nS : Sieve U\nhS : S \u2208 GrothendieckTopology.sieves J U\nx : FamilyOfElements (G.op \u22d9 \u2131.val \u22d9 coyoneda.obj (op X)) S.arrows\nhx : Compatible x\nH : IsSheafFor (\u2131.val \u22d9 coyoneda.obj (op X)) (Sieve.functorPushforward G S).arrows :=\n Sheaf.cond \u2131 X (Sieve.functorPushforward G S) (CoverPreserving.cover_preserve hG\u2082 hS)\nhx' : Compatible (FamilyOfElements.functorPushforward G x) := Compatible.functorPushforward hG\u2081 (sheafOver \u2131 X) hx\n\u22a2 ((G.op \u22d9 \u2131.val) \u22d9 coyoneda.obj (op X)).obj (op U)\n[PROOFSTEP]\nswap\n[GOAL]\ncase w\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nA : Type u\u2083\ninst\u271d : Category.{v\u2083, u\u2083} A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nL : GrothendieckTopology A\nG\u271d : C \u2964 D\nhG : CompatiblePreserving K G\u271d\n\u2131\u271d : SheafOfTypes K\nZ : C\nT : Presieve Z\nx\u271d : FamilyOfElements (G\u271d.op \u22d9 \u2131\u271d.val) T\nh : Compatible x\u271d\nG : C \u2964 D\nhG\u2081 : CompatiblePreserving K G\nhG\u2082 : CoverPreserving J K G\n\u2131 : Sheaf K A\nX : A\nU : C\nS : Sieve U\nhS : S \u2208 GrothendieckTopology.sieves J U\nx : FamilyOfElements (G.op \u22d9 \u2131.val \u22d9 coyoneda.obj (op X)) S.arrows\nhx : Compatible x\nH : IsSheafFor (\u2131.val \u22d9 coyoneda.obj (op X)) (Sieve.functorPushforward G S).arrows :=\n Sheaf.cond \u2131 X (Sieve.functorPushforward G S) (CoverPreserving.cover_preserve hG\u2082 hS)\nhx' : Compatible (FamilyOfElements.functorPushforward G x) := Compatible.functorPushforward hG\u2081 (sheafOver \u2131 X) hx\n\u22a2 ((G.op \u22d9 \u2131.val) \u22d9 coyoneda.obj (op X)).obj (op U)\n[PROOFSTEP]\napply H.amalgamate (x.functorPushforward G)\n[GOAL]\ncase w\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nA : Type u\u2083\ninst\u271d : Category.{v\u2083, u\u2083} A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nL : GrothendieckTopology A\nG\u271d : C \u2964 D\nhG : CompatiblePreserving K G\u271d\n\u2131\u271d : SheafOfTypes K\nZ : C\nT : Presieve Z\nx\u271d : FamilyOfElements (G\u271d.op \u22d9 \u2131\u271d.val) T\nh : Compatible x\u271d\nG : C \u2964 D\nhG\u2081 : CompatiblePreserving K G\nhG\u2082 : CoverPreserving J K G\n\u2131 : Sheaf K A\nX : A\nU : C\nS : Sieve U\nhS : S \u2208 GrothendieckTopology.sieves J U\nx : FamilyOfElements (G.op \u22d9 \u2131.val \u22d9 coyoneda.obj (op X)) S.arrows\nhx : Compatible x\nH : IsSheafFor (\u2131.val \u22d9 coyoneda.obj (op X)) (Sieve.functorPushforward G S).arrows :=\n Sheaf.cond \u2131 X (Sieve.functorPushforward G S) (CoverPreserving.cover_preserve hG\u2082 hS)\nhx' : Compatible (FamilyOfElements.functorPushforward G x) := Compatible.functorPushforward hG\u2081 (sheafOver \u2131 X) hx\n\u22a2 Compatible (FamilyOfElements.functorPushforward G x)\n[PROOFSTEP]\nexact hx'\n[GOAL]\ncase h\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nA : Type u\u2083\ninst\u271d : Category.{v\u2083, u\u2083} A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nL : GrothendieckTopology A\nG\u271d : C \u2964 D\nhG : CompatiblePreserving K G\u271d\n\u2131\u271d : SheafOfTypes K\nZ : C\nT : Presieve Z\nx\u271d : FamilyOfElements (G\u271d.op \u22d9 \u2131\u271d.val) T\nh : Compatible x\u271d\nG : C \u2964 D\nhG\u2081 : CompatiblePreserving K G\nhG\u2082 : CoverPreserving J K G\n\u2131 : Sheaf K A\nX : A\nU : C\nS : Sieve U\nhS : S \u2208 GrothendieckTopology.sieves J U\nx : FamilyOfElements (G.op \u22d9 \u2131.val \u22d9 coyoneda.obj (op X)) S.arrows\nhx : Compatible x\nH : IsSheafFor (\u2131.val \u22d9 coyoneda.obj (op X)) (Sieve.functorPushforward G S).arrows :=\n Sheaf.cond \u2131 X (Sieve.functorPushforward G S) (CoverPreserving.cover_preserve hG\u2082 hS)\nhx' : Compatible (FamilyOfElements.functorPushforward G x) := Compatible.functorPushforward hG\u2081 (sheafOver \u2131 X) hx\n\u22a2 (fun t => IsAmalgamation x t) (IsSheafFor.amalgamate H (FamilyOfElements.functorPushforward G x) hx') \u2227\n \u2200 (y : ((G.op \u22d9 \u2131.val) \u22d9 coyoneda.obj (op X)).obj (op U)),\n (fun t => IsAmalgamation x t) y \u2192 y = IsSheafFor.amalgamate H (FamilyOfElements.functorPushforward G x) hx'\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase h.left\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nA : Type u\u2083\ninst\u271d : Category.{v\u2083, u\u2083} A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nL : GrothendieckTopology A\nG\u271d : C \u2964 D\nhG : CompatiblePreserving K G\u271d\n\u2131\u271d : SheafOfTypes K\nZ : C\nT : Presieve Z\nx\u271d : FamilyOfElements (G\u271d.op \u22d9 \u2131\u271d.val) T\nh : Compatible x\u271d\nG : C \u2964 D\nhG\u2081 : CompatiblePreserving K G\nhG\u2082 : CoverPreserving J K G\n\u2131 : Sheaf K A\nX : A\nU : C\nS : Sieve U\nhS : S \u2208 GrothendieckTopology.sieves J U\nx : FamilyOfElements (G.op \u22d9 \u2131.val \u22d9 coyoneda.obj (op X)) S.arrows\nhx : Compatible x\nH : IsSheafFor (\u2131.val \u22d9 coyoneda.obj (op X)) (Sieve.functorPushforward G S).arrows :=\n Sheaf.cond \u2131 X (Sieve.functorPushforward G S) (CoverPreserving.cover_preserve hG\u2082 hS)\nhx' : Compatible (FamilyOfElements.functorPushforward G x) := Compatible.functorPushforward hG\u2081 (sheafOver \u2131 X) hx\n\u22a2 (fun t => IsAmalgamation x t) (IsSheafFor.amalgamate H (FamilyOfElements.functorPushforward G x) hx')\n[PROOFSTEP]\nintro V f hf\n[GOAL]\ncase h.left\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nA : Type u\u2083\ninst\u271d : Category.{v\u2083, u\u2083} A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nL : GrothendieckTopology A\nG\u271d : C \u2964 D\nhG : CompatiblePreserving K G\u271d\n\u2131\u271d : SheafOfTypes K\nZ : C\nT : Presieve Z\nx\u271d : FamilyOfElements (G\u271d.op \u22d9 \u2131\u271d.val) T\nh : Compatible x\u271d\nG : C \u2964 D\nhG\u2081 : CompatiblePreserving K G\nhG\u2082 : CoverPreserving J K G\n\u2131 : Sheaf K A\nX : A\nU : C\nS : Sieve U\nhS : S \u2208 GrothendieckTopology.sieves J U\nx : FamilyOfElements (G.op \u22d9 \u2131.val \u22d9 coyoneda.obj (op X)) S.arrows\nhx : Compatible x\nH : IsSheafFor (\u2131.val \u22d9 coyoneda.obj (op X)) (Sieve.functorPushforward G S).arrows :=\n Sheaf.cond \u2131 X (Sieve.functorPushforward G S) (CoverPreserving.cover_preserve hG\u2082 hS)\nhx' : Compatible (FamilyOfElements.functorPushforward G x) := Compatible.functorPushforward hG\u2081 (sheafOver \u2131 X) hx\nV : C\nf : V \u27f6 U\nhf : S.arrows f\n\u22a2 ((G.op \u22d9 \u2131.val) \u22d9 coyoneda.obj (op X)).map f.op\n (IsSheafFor.amalgamate H (FamilyOfElements.functorPushforward G x) hx') =\n x f hf\n[PROOFSTEP]\nconvert H.isAmalgamation hx' (G.map f) (image_mem_functorPushforward G S hf)\n[GOAL]\ncase h.e'_3.h\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nA : Type u\u2083\ninst\u271d : Category.{v\u2083, u\u2083} A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nL : GrothendieckTopology A\nG\u271d : C \u2964 D\nhG : CompatiblePreserving K G\u271d\n\u2131\u271d : SheafOfTypes K\nZ : C\nT : Presieve Z\nx\u271d : FamilyOfElements (G\u271d.op \u22d9 \u2131\u271d.val) T\nh : Compatible x\u271d\nG : C \u2964 D\nhG\u2081 : CompatiblePreserving K G\nhG\u2082 : CoverPreserving J K G\n\u2131 : Sheaf K A\nX : A\nU : C\nS : Sieve U\nhS : S \u2208 GrothendieckTopology.sieves J U\nx : FamilyOfElements (G.op \u22d9 \u2131.val \u22d9 coyoneda.obj (op X)) S.arrows\nhx : Compatible x\nH : IsSheafFor (\u2131.val \u22d9 coyoneda.obj (op X)) (Sieve.functorPushforward G S).arrows :=\n Sheaf.cond \u2131 X (Sieve.functorPushforward G S) (CoverPreserving.cover_preserve hG\u2082 hS)\nhx' : Compatible (FamilyOfElements.functorPushforward G x) := Compatible.functorPushforward hG\u2081 (sheafOver \u2131 X) hx\nV : C\nf : V \u27f6 U\nhf : S.arrows f\ne_1\u271d : ((G.op \u22d9 \u2131.val) \u22d9 coyoneda.obj (op X)).obj (op V) = (\u2131.val \u22d9 coyoneda.obj (op X)).obj (op (G.obj V))\n\u22a2 x f hf = FamilyOfElements.functorPushforward G x (G.map f) (_ : Presieve.functorPushforward G S.arrows (G.map f))\n[PROOFSTEP]\nrw [hG\u2081.apply_map (sheafOver \u2131 X) hx]\n[GOAL]\ncase h.right\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nA : Type u\u2083\ninst\u271d : Category.{v\u2083, u\u2083} A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nL : GrothendieckTopology A\nG\u271d : C \u2964 D\nhG : CompatiblePreserving K G\u271d\n\u2131\u271d : SheafOfTypes K\nZ : C\nT : Presieve Z\nx\u271d : FamilyOfElements (G\u271d.op \u22d9 \u2131\u271d.val) T\nh : Compatible x\u271d\nG : C \u2964 D\nhG\u2081 : CompatiblePreserving K G\nhG\u2082 : CoverPreserving J K G\n\u2131 : Sheaf K A\nX : A\nU : C\nS : Sieve U\nhS : S \u2208 GrothendieckTopology.sieves J U\nx : FamilyOfElements (G.op \u22d9 \u2131.val \u22d9 coyoneda.obj (op X)) S.arrows\nhx : Compatible x\nH : IsSheafFor (\u2131.val \u22d9 coyoneda.obj (op X)) (Sieve.functorPushforward G S).arrows :=\n Sheaf.cond \u2131 X (Sieve.functorPushforward G S) (CoverPreserving.cover_preserve hG\u2082 hS)\nhx' : Compatible (FamilyOfElements.functorPushforward G x) := Compatible.functorPushforward hG\u2081 (sheafOver \u2131 X) hx\n\u22a2 \u2200 (y : ((G.op \u22d9 \u2131.val) \u22d9 coyoneda.obj (op X)).obj (op U)),\n (fun t => IsAmalgamation x t) y \u2192 y = IsSheafFor.amalgamate H (FamilyOfElements.functorPushforward G x) hx'\n[PROOFSTEP]\nintro y hy\n[GOAL]\ncase h.right\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nA : Type u\u2083\ninst\u271d : Category.{v\u2083, u\u2083} A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nL : GrothendieckTopology A\nG\u271d : C \u2964 D\nhG : CompatiblePreserving K G\u271d\n\u2131\u271d : SheafOfTypes K\nZ : C\nT : Presieve Z\nx\u271d : FamilyOfElements (G\u271d.op \u22d9 \u2131\u271d.val) T\nh : Compatible x\u271d\nG : C \u2964 D\nhG\u2081 : CompatiblePreserving K G\nhG\u2082 : CoverPreserving J K G\n\u2131 : Sheaf K A\nX : A\nU : C\nS : Sieve U\nhS : S \u2208 GrothendieckTopology.sieves J U\nx : FamilyOfElements (G.op \u22d9 \u2131.val \u22d9 coyoneda.obj (op X)) S.arrows\nhx : Compatible x\nH : IsSheafFor (\u2131.val \u22d9 coyoneda.obj (op X)) (Sieve.functorPushforward G S).arrows :=\n Sheaf.cond \u2131 X (Sieve.functorPushforward G S) (CoverPreserving.cover_preserve hG\u2082 hS)\nhx' : Compatible (FamilyOfElements.functorPushforward G x) := Compatible.functorPushforward hG\u2081 (sheafOver \u2131 X) hx\ny : ((G.op \u22d9 \u2131.val) \u22d9 coyoneda.obj (op X)).obj (op U)\nhy : IsAmalgamation x y\n\u22a2 y = IsSheafFor.amalgamate H (FamilyOfElements.functorPushforward G x) hx'\n[PROOFSTEP]\nrefine' H.isSeparatedFor _ y _ _ (H.isAmalgamation (hx.functorPushforward hG\u2081 (sheafOver \u2131 X)))\n[GOAL]\ncase h.right\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nA : Type u\u2083\ninst\u271d : Category.{v\u2083, u\u2083} A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nL : GrothendieckTopology A\nG\u271d : C \u2964 D\nhG : CompatiblePreserving K G\u271d\n\u2131\u271d : SheafOfTypes K\nZ : C\nT : Presieve Z\nx\u271d : FamilyOfElements (G\u271d.op \u22d9 \u2131\u271d.val) T\nh : Compatible x\u271d\nG : C \u2964 D\nhG\u2081 : CompatiblePreserving K G\nhG\u2082 : CoverPreserving J K G\n\u2131 : Sheaf K A\nX : A\nU : C\nS : Sieve U\nhS : S \u2208 GrothendieckTopology.sieves J U\nx : FamilyOfElements (G.op \u22d9 \u2131.val \u22d9 coyoneda.obj (op X)) S.arrows\nhx : Compatible x\nH : IsSheafFor (\u2131.val \u22d9 coyoneda.obj (op X)) (Sieve.functorPushforward G S).arrows :=\n Sheaf.cond \u2131 X (Sieve.functorPushforward G S) (CoverPreserving.cover_preserve hG\u2082 hS)\nhx' : Compatible (FamilyOfElements.functorPushforward G x) := Compatible.functorPushforward hG\u2081 (sheafOver \u2131 X) hx\ny : ((G.op \u22d9 \u2131.val) \u22d9 coyoneda.obj (op X)).obj (op U)\nhy : IsAmalgamation x y\n\u22a2 IsAmalgamation (FamilyOfElements.functorPushforward G x) y\n[PROOFSTEP]\nrintro V f\n \u27e8Z, f', g', h, rfl\u27e9\n -- porting note: didn't need coercion (S : Presieve U) in Lean 3\n[GOAL]\ncase h.right.intro.intro.intro.intro\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nA : Type u\u2083\ninst\u271d : Category.{v\u2083, u\u2083} A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nL : GrothendieckTopology A\nG\u271d : C \u2964 D\nhG : CompatiblePreserving K G\u271d\n\u2131\u271d : SheafOfTypes K\nZ\u271d : C\nT : Presieve Z\u271d\nx\u271d : FamilyOfElements (G\u271d.op \u22d9 \u2131\u271d.val) T\nh\u271d : Compatible x\u271d\nG : C \u2964 D\nhG\u2081 : CompatiblePreserving K G\nhG\u2082 : CoverPreserving J K G\n\u2131 : Sheaf K A\nX : A\nU : C\nS : Sieve U\nhS : S \u2208 GrothendieckTopology.sieves J U\nx : FamilyOfElements (G.op \u22d9 \u2131.val \u22d9 coyoneda.obj (op X)) S.arrows\nhx : Compatible x\nH : IsSheafFor (\u2131.val \u22d9 coyoneda.obj (op X)) (Sieve.functorPushforward G S).arrows :=\n Sheaf.cond \u2131 X (Sieve.functorPushforward G S) (CoverPreserving.cover_preserve hG\u2082 hS)\nhx' : Compatible (FamilyOfElements.functorPushforward G x) := Compatible.functorPushforward hG\u2081 (sheafOver \u2131 X) hx\ny : ((G.op \u22d9 \u2131.val) \u22d9 coyoneda.obj (op X)).obj (op U)\nhy : IsAmalgamation x y\nV : D\nZ : C\nf' : Z \u27f6 U\ng' : V \u27f6 G.obj Z\nh : S.arrows f'\n\u22a2 (\u2131.val \u22d9 coyoneda.obj (op X)).map (g' \u226b G.map f').op y =\n FamilyOfElements.functorPushforward G x (g' \u226b G.map f') (_ : \u2203 Z_1 g h, S.arrows g \u2227 g' \u226b G.map f' = h \u226b G.map g)\n[PROOFSTEP]\nerw [FamilyOfElements.comp_of_compatible (S.functorPushforward G) hx'\n (image_mem_functorPushforward G (S : Presieve U) h) g']\n[GOAL]\ncase h.right.intro.intro.intro.intro\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nA : Type u\u2083\ninst\u271d : Category.{v\u2083, u\u2083} A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nL : GrothendieckTopology A\nG\u271d : C \u2964 D\nhG : CompatiblePreserving K G\u271d\n\u2131\u271d : SheafOfTypes K\nZ\u271d : C\nT : Presieve Z\u271d\nx\u271d : FamilyOfElements (G\u271d.op \u22d9 \u2131\u271d.val) T\nh\u271d : Compatible x\u271d\nG : C \u2964 D\nhG\u2081 : CompatiblePreserving K G\nhG\u2082 : CoverPreserving J K G\n\u2131 : Sheaf K A\nX : A\nU : C\nS : Sieve U\nhS : S \u2208 GrothendieckTopology.sieves J U\nx : FamilyOfElements (G.op \u22d9 \u2131.val \u22d9 coyoneda.obj (op X)) S.arrows\nhx : Compatible x\nH : IsSheafFor (\u2131.val \u22d9 coyoneda.obj (op X)) (Sieve.functorPushforward G S).arrows :=\n Sheaf.cond \u2131 X (Sieve.functorPushforward G S) (CoverPreserving.cover_preserve hG\u2082 hS)\nhx' : Compatible (FamilyOfElements.functorPushforward G x) := Compatible.functorPushforward hG\u2081 (sheafOver \u2131 X) hx\ny : ((G.op \u22d9 \u2131.val) \u22d9 coyoneda.obj (op X)).obj (op U)\nhy : IsAmalgamation x y\nV : D\nZ : C\nf' : Z \u27f6 U\ng' : V \u27f6 G.obj Z\nh : S.arrows f'\n\u22a2 (\u2131.val \u22d9 coyoneda.obj (op X)).map (g' \u226b G.map f').op y =\n (sheafOver \u2131 X).val.map g'.op\n (FamilyOfElements.functorPushforward G x (G.map f') (_ : Presieve.functorPushforward G S.arrows (G.map f')))\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase h.right.intro.intro.intro.intro\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nA : Type u\u2083\ninst\u271d : Category.{v\u2083, u\u2083} A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nL : GrothendieckTopology A\nG\u271d : C \u2964 D\nhG : CompatiblePreserving K G\u271d\n\u2131\u271d : SheafOfTypes K\nZ\u271d : C\nT : Presieve Z\u271d\nx\u271d : FamilyOfElements (G\u271d.op \u22d9 \u2131\u271d.val) T\nh\u271d : Compatible x\u271d\nG : C \u2964 D\nhG\u2081 : CompatiblePreserving K G\nhG\u2082 : CoverPreserving J K G\n\u2131 : Sheaf K A\nX : A\nU : C\nS : Sieve U\nhS : S \u2208 GrothendieckTopology.sieves J U\nx : FamilyOfElements (G.op \u22d9 \u2131.val \u22d9 coyoneda.obj (op X)) S.arrows\nhx : Compatible x\nH : IsSheafFor (\u2131.val \u22d9 coyoneda.obj (op X)) (Sieve.functorPushforward G S).arrows :=\n Sheaf.cond \u2131 X (Sieve.functorPushforward G S) (CoverPreserving.cover_preserve hG\u2082 hS)\nhx' : Compatible (FamilyOfElements.functorPushforward G x) := Compatible.functorPushforward hG\u2081 (sheafOver \u2131 X) hx\ny : ((G.op \u22d9 \u2131.val) \u22d9 coyoneda.obj (op X)).obj (op U)\nhy : IsAmalgamation x y\nV : D\nZ : C\nf' : Z \u27f6 U\ng' : V \u27f6 G.obj Z\nh : S.arrows f'\n\u22a2 y \u226b \u2131.val.map ((G.map f').op \u226b g'.op) =\n FamilyOfElements.functorPushforward G x (G.map f') (_ : Presieve.functorPushforward G S.arrows (G.map f')) \u226b\n \u2131.val.map g'.op\n[PROOFSTEP]\nsimp [hG\u2081.apply_map (sheafOver \u2131 X) hx h, \u2190 hy f' h]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nA : Type u\u2083\ninst\u271d : Category.{v\u2083, u\u2083} A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nL : GrothendieckTopology A\nG\u271d : C \u2964 D\nhG : CompatiblePreserving K G\u271d\n\u2131\u271d : SheafOfTypes K\nZ : C\nT : Presieve Z\nx : FamilyOfElements (G\u271d.op \u22d9 \u2131\u271d.val) T\nh : Compatible x\nG : C \u2964 D\nhG\u2081 : CompatiblePreserving K G\nhG\u2082 : CoverPreserving J K G\n\u2131 : Sheaf K A\n\u22a2 { obj := fun \u2131 => pullbackSheaf hG\u2081 hG\u2082 \u2131,\n map := fun {X Y} f => { val := ((whiskeringLeft C\u1d52\u1d56 D\u1d52\u1d56 A).obj G.op).map f.val } }.map\n (\ud835\udfd9 \u2131) =\n \ud835\udfd9\n ({ obj := fun \u2131 => pullbackSheaf hG\u2081 hG\u2082 \u2131,\n map := fun {X Y} f => { val := ((whiskeringLeft C\u1d52\u1d56 D\u1d52\u1d56 A).obj G.op).map f.val } }.obj\n \u2131)\n[PROOFSTEP]\next1\n[GOAL]\ncase h\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nA : Type u\u2083\ninst\u271d : Category.{v\u2083, u\u2083} A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nL : GrothendieckTopology A\nG\u271d : C \u2964 D\nhG : CompatiblePreserving K G\u271d\n\u2131\u271d : SheafOfTypes K\nZ : C\nT : Presieve Z\nx : FamilyOfElements (G\u271d.op \u22d9 \u2131\u271d.val) T\nh : Compatible x\nG : C \u2964 D\nhG\u2081 : CompatiblePreserving K G\nhG\u2082 : CoverPreserving J K G\n\u2131 : Sheaf K A\n\u22a2 ({ obj := fun \u2131 => pullbackSheaf hG\u2081 hG\u2082 \u2131,\n map := fun {X Y} f => { val := ((whiskeringLeft C\u1d52\u1d56 D\u1d52\u1d56 A).obj G.op).map f.val } }.map\n (\ud835\udfd9 \u2131)).val =\n (\ud835\udfd9\n ({ obj := fun \u2131 => pullbackSheaf hG\u2081 hG\u2082 \u2131,\n map := fun {X Y} f => { val := ((whiskeringLeft C\u1d52\u1d56 D\u1d52\u1d56 A).obj G.op).map f.val } }.obj\n \u2131)).val\n[PROOFSTEP]\napply ((whiskeringLeft _ _ _).obj G.op).map_id\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nA : Type u\u2083\ninst\u271d : Category.{v\u2083, u\u2083} A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nL : GrothendieckTopology A\nG\u271d : C \u2964 D\nhG : CompatiblePreserving K G\u271d\n\u2131 : SheafOfTypes K\nZ : C\nT : Presieve Z\nx : FamilyOfElements (G\u271d.op \u22d9 \u2131.val) T\nh : Compatible x\nG : C \u2964 D\nhG\u2081 : CompatiblePreserving K G\nhG\u2082 : CoverPreserving J K G\nX\u271d Y\u271d Z\u271d : Sheaf K A\nf : X\u271d \u27f6 Y\u271d\ng : Y\u271d \u27f6 Z\u271d\n\u22a2 { obj := fun \u2131 => pullbackSheaf hG\u2081 hG\u2082 \u2131,\n map := fun {X Y} f => { val := ((whiskeringLeft C\u1d52\u1d56 D\u1d52\u1d56 A).obj G.op).map f.val } }.map\n (f \u226b g) =\n { obj := fun \u2131 => pullbackSheaf hG\u2081 hG\u2082 \u2131,\n map := fun {X Y} f => { val := ((whiskeringLeft C\u1d52\u1d56 D\u1d52\u1d56 A).obj G.op).map f.val } }.map\n f \u226b\n { obj := fun \u2131 => pullbackSheaf hG\u2081 hG\u2082 \u2131,\n map := fun {X Y} f => { val := ((whiskeringLeft C\u1d52\u1d56 D\u1d52\u1d56 A).obj G.op).map f.val } }.map\n g\n[PROOFSTEP]\next1\n[GOAL]\ncase h\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nA : Type u\u2083\ninst\u271d : Category.{v\u2083, u\u2083} A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nL : GrothendieckTopology A\nG\u271d : C \u2964 D\nhG : CompatiblePreserving K G\u271d\n\u2131 : SheafOfTypes K\nZ : C\nT : Presieve Z\nx : FamilyOfElements (G\u271d.op \u22d9 \u2131.val) T\nh : Compatible x\nG : C \u2964 D\nhG\u2081 : CompatiblePreserving K G\nhG\u2082 : CoverPreserving J K G\nX\u271d Y\u271d Z\u271d : Sheaf K A\nf : X\u271d \u27f6 Y\u271d\ng : Y\u271d \u27f6 Z\u271d\n\u22a2 ({ obj := fun \u2131 => pullbackSheaf hG\u2081 hG\u2082 \u2131,\n map := fun {X Y} f => { val := ((whiskeringLeft C\u1d52\u1d56 D\u1d52\u1d56 A).obj G.op).map f.val } }.map\n (f \u226b g)).val =\n ({ obj := fun \u2131 => pullbackSheaf hG\u2081 hG\u2082 \u2131,\n map := fun {X Y} f => { val := ((whiskeringLeft C\u1d52\u1d56 D\u1d52\u1d56 A).obj G.op).map f.val } }.map\n f \u226b\n { obj := fun \u2131 => pullbackSheaf hG\u2081 hG\u2082 \u2131,\n map := fun {X Y} f => { val := ((whiskeringLeft C\u1d52\u1d56 D\u1d52\u1d56 A).obj G.op).map f.val } }.map\n g).val\n[PROOFSTEP]\napply ((whiskeringLeft _ _ _).obj G.op).map_comp\n", "meta": {"mathlib_filename": "Mathlib.CategoryTheory.Sites.CoverPreserving", "llama_tokens": 45592, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.5078118642792044, "lm_q2_score": 0.3415824927356586, "lm_q1q2_score": 0.1734596424412326}} {"text": "[GOAL]\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\nI J : Box \u03b9\nc c\u2081 c\u2082 : \u211d\u22650\nr r\u2081 r\u2082 : (\u03b9 \u2192 \u211d) \u2192 \u2191(Set.Ioi 0)\n\u03c0 \u03c0\u2081 \u03c0\u2082 : TaggedPrepartition I\nl l\u2081 l\u2082 : IntegrationParams\n\u22a2 Henstock \u2264 Riemann\n[PROOFSTEP]\ntrivial\n[GOAL]\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\nI J : Box \u03b9\nc c\u2081 c\u2082 : \u211d\u22650\nr r\u2081 r\u2082 : (\u03b9 \u2192 \u211d) \u2192 \u2191(Set.Ioi 0)\n\u03c0 \u03c0\u2081 \u03c0\u2082 : TaggedPrepartition I\nl l\u2081 l\u2082 : IntegrationParams\n\u22a2 Henstock \u2264 McShane\n[PROOFSTEP]\ntrivial\n[GOAL]\n\u03b9\u271d : Type u_1\ninst\u271d : Fintype \u03b9\u271d\nI J : Box \u03b9\u271d\nc c\u2081 c\u2082 : \u211d\u22650\nr\u271d r\u2081 r\u2082 : (\u03b9\u271d \u2192 \u211d) \u2192 \u2191(Set.Ioi 0)\n\u03c0 \u03c0\u2081 \u03c0\u2082 : TaggedPrepartition I\nl\u271d l\u2081 l\u2082 : IntegrationParams\n\u03b9 : Type u_2\nl : IntegrationParams\nhl : l.bRiemann = false\nr : (\u03b9 \u2192 \u211d) \u2192 \u2191(Set.Ioi 0)\n\u22a2 RCond l r\n[PROOFSTEP]\nsimp [RCond, hl]\n[GOAL]\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\nI J : Box \u03b9\nc c\u2081 c\u2082 : \u211d\u22650\nr r\u2081 r\u2082 : (\u03b9 \u2192 \u211d) \u2192 \u2191(Set.Ioi 0)\n\u03c0 \u03c0\u2081 \u03c0\u2082 : TaggedPrepartition I\nl l\u2081 l\u2082 : IntegrationParams\nh\u2081 : MemBaseSet l I c\u2081 r\u2081 \u03c0\u2081\nh\u2082 : MemBaseSet l I c\u2082 r\u2082 \u03c0\u2082\nhU : TaggedPrepartition.iUnion \u03c0\u2081 = TaggedPrepartition.iUnion \u03c0\u2082\n\u22a2 \u2203 \u03c0,\n Prepartition.iUnion \u03c0 = \u2191I \\ TaggedPrepartition.iUnion \u03c0\u2081 \u2227\n (l.bDistortion = true \u2192 Prepartition.distortion \u03c0 \u2264 c\u2081) \u2227 (l.bDistortion = true \u2192 Prepartition.distortion \u03c0 \u2264 c\u2082)\n[PROOFSTEP]\nwlog hc : c\u2081 \u2264 c\u2082 with H\n[GOAL]\ncase inr\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\nI J : Box \u03b9\nc c\u2081 c\u2082 : \u211d\u22650\nr r\u2081 r\u2082 : (\u03b9 \u2192 \u211d) \u2192 \u2191(Set.Ioi 0)\n\u03c0 \u03c0\u2081 \u03c0\u2082 : TaggedPrepartition I\nl l\u2081 l\u2082 : IntegrationParams\nh\u2081 : MemBaseSet l I c\u2081 r\u2081 \u03c0\u2081\nh\u2082 : MemBaseSet l I c\u2082 r\u2082 \u03c0\u2082\nhU : TaggedPrepartition.iUnion \u03c0\u2081 = TaggedPrepartition.iUnion \u03c0\u2082\nH :\n \u2200 {\u03b9 : Type u_1} [inst : Fintype \u03b9] {I : Box \u03b9} {J : Box \u03b9} {c : \u211d\u22650} {c\u2081 c\u2082 : \u211d\u22650} {r : (\u03b9 \u2192 \u211d) \u2192 \u2191(Set.Ioi 0)}\n {r\u2081 r\u2082 : (\u03b9 \u2192 \u211d) \u2192 \u2191(Set.Ioi 0)} {\u03c0 : TaggedPrepartition I} {\u03c0\u2081 \u03c0\u2082 : TaggedPrepartition I} {l : IntegrationParams}\n {l\u2081 l\u2082 : IntegrationParams},\n MemBaseSet l I c\u2081 r\u2081 \u03c0\u2081 \u2192\n MemBaseSet l I c\u2082 r\u2082 \u03c0\u2082 \u2192\n TaggedPrepartition.iUnion \u03c0\u2081 = TaggedPrepartition.iUnion \u03c0\u2082 \u2192\n c\u2081 \u2264 c\u2082 \u2192\n \u2203 \u03c0,\n Prepartition.iUnion \u03c0 = \u2191I \\ TaggedPrepartition.iUnion \u03c0\u2081 \u2227\n (l.bDistortion = true \u2192 Prepartition.distortion \u03c0 \u2264 c\u2081) \u2227\n (l.bDistortion = true \u2192 Prepartition.distortion \u03c0 \u2264 c\u2082)\nhc : \u00acc\u2081 \u2264 c\u2082\n\u22a2 \u2203 \u03c0,\n Prepartition.iUnion \u03c0 = \u2191I \\ TaggedPrepartition.iUnion \u03c0\u2081 \u2227\n (l.bDistortion = true \u2192 Prepartition.distortion \u03c0 \u2264 c\u2081) \u2227 (l.bDistortion = true \u2192 Prepartition.distortion \u03c0 \u2264 c\u2082)\n[PROOFSTEP]\nsimpa [hU, _root_.and_comm] using @H _ _ I J c c\u2082 c\u2081 r r\u2082 r\u2081 \u03c0 \u03c0\u2082 \u03c0\u2081 _ l\u2082 l\u2081 h\u2082 h\u2081 hU.symm (le_of_not_le hc)\n[GOAL]\n\u03b9\u271d : Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\u271d\nI\u271d : Box \u03b9\u271d\nc\u2081\u271d c\u2082\u271d : \u211d\u22650\nr\u2081\u271d r\u2082\u271d : (\u03b9\u271d \u2192 \u211d) \u2192 \u2191(Set.Ioi 0)\n\u03c0\u2081\u271d \u03c0\u2082\u271d : TaggedPrepartition I\u271d\nl\u271d : IntegrationParams\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\nI J : Box \u03b9\nc c\u2081 c\u2082 : \u211d\u22650\nr r\u2081 r\u2082 : (\u03b9 \u2192 \u211d) \u2192 \u2191(Set.Ioi 0)\n\u03c0 \u03c0\u2081 \u03c0\u2082 : TaggedPrepartition I\nl l\u2081 l\u2082 : IntegrationParams\nh\u2081 : MemBaseSet l I c\u2081 r\u2081 \u03c0\u2081\nh\u2082 : MemBaseSet l I c\u2082 r\u2082 \u03c0\u2082\nhU : TaggedPrepartition.iUnion \u03c0\u2081 = TaggedPrepartition.iUnion \u03c0\u2082\nhc : c\u2081 \u2264 c\u2082\n\u22a2 \u2203 \u03c0,\n Prepartition.iUnion \u03c0 = \u2191I \\ TaggedPrepartition.iUnion \u03c0\u2081 \u2227\n (l.bDistortion = true \u2192 Prepartition.distortion \u03c0 \u2264 c\u2081) \u2227 (l.bDistortion = true \u2192 Prepartition.distortion \u03c0 \u2264 c\u2082)\n[PROOFSTEP]\nby_cases hD : (l.bDistortion : Prop)\n[GOAL]\ncase pos\n\u03b9\u271d : Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\u271d\nI\u271d : Box \u03b9\u271d\nc\u2081\u271d c\u2082\u271d : \u211d\u22650\nr\u2081\u271d r\u2082\u271d : (\u03b9\u271d \u2192 \u211d) \u2192 \u2191(Set.Ioi 0)\n\u03c0\u2081\u271d \u03c0\u2082\u271d : TaggedPrepartition I\u271d\nl\u271d : IntegrationParams\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\nI J : Box \u03b9\nc c\u2081 c\u2082 : \u211d\u22650\nr r\u2081 r\u2082 : (\u03b9 \u2192 \u211d) \u2192 \u2191(Set.Ioi 0)\n\u03c0 \u03c0\u2081 \u03c0\u2082 : TaggedPrepartition I\nl l\u2081 l\u2082 : IntegrationParams\nh\u2081 : MemBaseSet l I c\u2081 r\u2081 \u03c0\u2081\nh\u2082 : MemBaseSet l I c\u2082 r\u2082 \u03c0\u2082\nhU : TaggedPrepartition.iUnion \u03c0\u2081 = TaggedPrepartition.iUnion \u03c0\u2082\nhc : c\u2081 \u2264 c\u2082\nhD : l.bDistortion = true\n\u22a2 \u2203 \u03c0,\n Prepartition.iUnion \u03c0 = \u2191I \\ TaggedPrepartition.iUnion \u03c0\u2081 \u2227\n (l.bDistortion = true \u2192 Prepartition.distortion \u03c0 \u2264 c\u2081) \u2227 (l.bDistortion = true \u2192 Prepartition.distortion \u03c0 \u2264 c\u2082)\n[PROOFSTEP]\nrcases h\u2081.4 hD with \u27e8\u03c0, h\u03c0U, h\u03c0c\u27e9\n[GOAL]\ncase pos.intro.intro\n\u03b9\u271d : Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\u271d\nI\u271d : Box \u03b9\u271d\nc\u2081\u271d c\u2082\u271d : \u211d\u22650\nr\u2081\u271d r\u2082\u271d : (\u03b9\u271d \u2192 \u211d) \u2192 \u2191(Set.Ioi 0)\n\u03c0\u2081\u271d \u03c0\u2082\u271d : TaggedPrepartition I\u271d\nl\u271d : IntegrationParams\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\nI J : Box \u03b9\nc c\u2081 c\u2082 : \u211d\u22650\nr r\u2081 r\u2082 : (\u03b9 \u2192 \u211d) \u2192 \u2191(Set.Ioi 0)\n\u03c0\u271d \u03c0\u2081 \u03c0\u2082 : TaggedPrepartition I\nl l\u2081 l\u2082 : IntegrationParams\nh\u2081 : MemBaseSet l I c\u2081 r\u2081 \u03c0\u2081\nh\u2082 : MemBaseSet l I c\u2082 r\u2082 \u03c0\u2082\nhU : TaggedPrepartition.iUnion \u03c0\u2081 = TaggedPrepartition.iUnion \u03c0\u2082\nhc : c\u2081 \u2264 c\u2082\nhD : l.bDistortion = true\n\u03c0 : Prepartition I\nh\u03c0U : Prepartition.iUnion \u03c0 = \u2191I \\ TaggedPrepartition.iUnion \u03c0\u2081\nh\u03c0c : Prepartition.distortion \u03c0 \u2264 c\u2081\n\u22a2 \u2203 \u03c0,\n Prepartition.iUnion \u03c0 = \u2191I \\ TaggedPrepartition.iUnion \u03c0\u2081 \u2227\n (l.bDistortion = true \u2192 Prepartition.distortion \u03c0 \u2264 c\u2081) \u2227 (l.bDistortion = true \u2192 Prepartition.distortion \u03c0 \u2264 c\u2082)\n[PROOFSTEP]\nexact \u27e8\u03c0, h\u03c0U, fun _ => h\u03c0c, fun _ => h\u03c0c.trans hc\u27e9\n[GOAL]\ncase neg\n\u03b9\u271d : Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\u271d\nI\u271d : Box \u03b9\u271d\nc\u2081\u271d c\u2082\u271d : \u211d\u22650\nr\u2081\u271d r\u2082\u271d : (\u03b9\u271d \u2192 \u211d) \u2192 \u2191(Set.Ioi 0)\n\u03c0\u2081\u271d \u03c0\u2082\u271d : TaggedPrepartition I\u271d\nl\u271d : IntegrationParams\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\nI J : Box \u03b9\nc c\u2081 c\u2082 : \u211d\u22650\nr r\u2081 r\u2082 : (\u03b9 \u2192 \u211d) \u2192 \u2191(Set.Ioi 0)\n\u03c0 \u03c0\u2081 \u03c0\u2082 : TaggedPrepartition I\nl l\u2081 l\u2082 : IntegrationParams\nh\u2081 : MemBaseSet l I c\u2081 r\u2081 \u03c0\u2081\nh\u2082 : MemBaseSet l I c\u2082 r\u2082 \u03c0\u2082\nhU : TaggedPrepartition.iUnion \u03c0\u2081 = TaggedPrepartition.iUnion \u03c0\u2082\nhc : c\u2081 \u2264 c\u2082\nhD : \u00acl.bDistortion = true\n\u22a2 \u2203 \u03c0,\n Prepartition.iUnion \u03c0 = \u2191I \\ TaggedPrepartition.iUnion \u03c0\u2081 \u2227\n (l.bDistortion = true \u2192 Prepartition.distortion \u03c0 \u2264 c\u2081) \u2227 (l.bDistortion = true \u2192 Prepartition.distortion \u03c0 \u2264 c\u2082)\n[PROOFSTEP]\nexact \u27e8\u03c0\u2081.toPrepartition.compl, \u03c0\u2081.toPrepartition.iUnion_compl, fun h => (hD h).elim, fun h => (hD h).elim\u27e9\n[GOAL]\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\nI J : Box \u03b9\nc c\u2081 c\u2082 : \u211d\u22650\nr r\u2081 r\u2082 : (\u03b9 \u2192 \u211d) \u2192 \u2191(Set.Ioi 0)\n\u03c0 \u03c0\u2081 \u03c0\u2082\u271d : TaggedPrepartition I\nl l\u2081 l\u2082 : IntegrationParams\nh\u03c0\u2081 : MemBaseSet l I c r\u2081 \u03c0\u2081\nhle : \u2200 (x : \u03b9 \u2192 \u211d), x \u2208 \u2191Box.Icc I \u2192 r\u2082 x \u2264 r\u2081 x\n\u03c0\u2082 : Prepartition I\nhU : Prepartition.iUnion \u03c0\u2082 = \u2191I \\ TaggedPrepartition.iUnion \u03c0\u2081\nhc : l.bDistortion = true \u2192 Prepartition.distortion \u03c0\u2082 \u2264 c\nx\u271d : l.bDistortion = true\n\u22a2 Prepartition.iUnion \u22a5 = \u2191I \\ TaggedPrepartition.iUnion (unionComplToSubordinate \u03c0\u2081 \u03c0\u2082 hU r\u2082) \u2227\n Prepartition.distortion \u22a5 \u2264 c\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\nI J : Box \u03b9\nc c\u2081 c\u2082 : \u211d\u22650\nr r\u2081 r\u2082 : (\u03b9 \u2192 \u211d) \u2192 \u2191(Set.Ioi 0)\n\u03c0 \u03c0\u2081 \u03c0\u2082 : TaggedPrepartition I\nl l\u2081 l\u2082 : IntegrationParams\nh\u03c0 : MemBaseSet l I c r \u03c0\np : Box \u03b9 \u2192 Prop\n\u22a2 MemBaseSet l I c r (TaggedPrepartition.filter \u03c0 p)\n[PROOFSTEP]\nrefine'\n \u27e8fun J hJ => h\u03c0.1 J (\u03c0.mem_filter.1 hJ).1, fun hH J hJ => h\u03c0.2 hH J (\u03c0.mem_filter.1 hJ).1, fun hD =>\n (distortion_filter_le _ _).trans (h\u03c0.3 hD), fun hD => _\u27e9\n[GOAL]\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\nI J : Box \u03b9\nc c\u2081 c\u2082 : \u211d\u22650\nr r\u2081 r\u2082 : (\u03b9 \u2192 \u211d) \u2192 \u2191(Set.Ioi 0)\n\u03c0 \u03c0\u2081 \u03c0\u2082 : TaggedPrepartition I\nl l\u2081 l\u2082 : IntegrationParams\nh\u03c0 : MemBaseSet l I c r \u03c0\np : Box \u03b9 \u2192 Prop\nhD : l.bDistortion = true\n\u22a2 \u2203 \u03c0',\n Prepartition.iUnion \u03c0' = \u2191I \\ TaggedPrepartition.iUnion (TaggedPrepartition.filter \u03c0 p) \u2227\n Prepartition.distortion \u03c0' \u2264 c\n[PROOFSTEP]\nrcases h\u03c0.4 hD with \u27e8\u03c0\u2081, h\u03c0\u2081U, hc\u27e9\n[GOAL]\ncase intro.intro\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\nI J : Box \u03b9\nc c\u2081 c\u2082 : \u211d\u22650\nr r\u2081 r\u2082 : (\u03b9 \u2192 \u211d) \u2192 \u2191(Set.Ioi 0)\n\u03c0 \u03c0\u2081\u271d \u03c0\u2082 : TaggedPrepartition I\nl l\u2081 l\u2082 : IntegrationParams\nh\u03c0 : MemBaseSet l I c r \u03c0\np : Box \u03b9 \u2192 Prop\nhD : l.bDistortion = true\n\u03c0\u2081 : Prepartition I\nh\u03c0\u2081U : Prepartition.iUnion \u03c0\u2081 = \u2191I \\ TaggedPrepartition.iUnion \u03c0\nhc : Prepartition.distortion \u03c0\u2081 \u2264 c\n\u22a2 \u2203 \u03c0',\n Prepartition.iUnion \u03c0' = \u2191I \\ TaggedPrepartition.iUnion (TaggedPrepartition.filter \u03c0 p) \u2227\n Prepartition.distortion \u03c0' \u2264 c\n[PROOFSTEP]\nset \u03c0\u2082 := \u03c0.filter fun J => \u00acp J\n[GOAL]\ncase intro.intro\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\nI J : Box \u03b9\nc c\u2081 c\u2082 : \u211d\u22650\nr r\u2081 r\u2082 : (\u03b9 \u2192 \u211d) \u2192 \u2191(Set.Ioi 0)\n\u03c0 \u03c0\u2081\u271d \u03c0\u2082\u271d : TaggedPrepartition I\nl l\u2081 l\u2082 : IntegrationParams\nh\u03c0 : MemBaseSet l I c r \u03c0\np : Box \u03b9 \u2192 Prop\nhD : l.bDistortion = true\n\u03c0\u2081 : Prepartition I\nh\u03c0\u2081U : Prepartition.iUnion \u03c0\u2081 = \u2191I \\ TaggedPrepartition.iUnion \u03c0\nhc : Prepartition.distortion \u03c0\u2081 \u2264 c\n\u03c0\u2082 : TaggedPrepartition I := TaggedPrepartition.filter \u03c0 fun J => \u00acp J\n\u22a2 \u2203 \u03c0',\n Prepartition.iUnion \u03c0' = \u2191I \\ TaggedPrepartition.iUnion (TaggedPrepartition.filter \u03c0 p) \u2227\n Prepartition.distortion \u03c0' \u2264 c\n[PROOFSTEP]\nhave : Disjoint \u03c0\u2081.iUnion \u03c0\u2082.iUnion := by simpa [h\u03c0\u2081U] using disjoint_sdiff_self_left.mono_right sdiff_le\n[GOAL]\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\nI J : Box \u03b9\nc c\u2081 c\u2082 : \u211d\u22650\nr r\u2081 r\u2082 : (\u03b9 \u2192 \u211d) \u2192 \u2191(Set.Ioi 0)\n\u03c0 \u03c0\u2081\u271d \u03c0\u2082\u271d : TaggedPrepartition I\nl l\u2081 l\u2082 : IntegrationParams\nh\u03c0 : MemBaseSet l I c r \u03c0\np : Box \u03b9 \u2192 Prop\nhD : l.bDistortion = true\n\u03c0\u2081 : Prepartition I\nh\u03c0\u2081U : Prepartition.iUnion \u03c0\u2081 = \u2191I \\ TaggedPrepartition.iUnion \u03c0\nhc : Prepartition.distortion \u03c0\u2081 \u2264 c\n\u03c0\u2082 : TaggedPrepartition I := TaggedPrepartition.filter \u03c0 fun J => \u00acp J\n\u22a2 Disjoint (Prepartition.iUnion \u03c0\u2081) (TaggedPrepartition.iUnion \u03c0\u2082)\n[PROOFSTEP]\nsimpa [h\u03c0\u2081U] using disjoint_sdiff_self_left.mono_right sdiff_le\n[GOAL]\ncase intro.intro\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\nI J : Box \u03b9\nc c\u2081 c\u2082 : \u211d\u22650\nr r\u2081 r\u2082 : (\u03b9 \u2192 \u211d) \u2192 \u2191(Set.Ioi 0)\n\u03c0 \u03c0\u2081\u271d \u03c0\u2082\u271d : TaggedPrepartition I\nl l\u2081 l\u2082 : IntegrationParams\nh\u03c0 : MemBaseSet l I c r \u03c0\np : Box \u03b9 \u2192 Prop\nhD : l.bDistortion = true\n\u03c0\u2081 : Prepartition I\nh\u03c0\u2081U : Prepartition.iUnion \u03c0\u2081 = \u2191I \\ TaggedPrepartition.iUnion \u03c0\nhc : Prepartition.distortion \u03c0\u2081 \u2264 c\n\u03c0\u2082 : TaggedPrepartition I := TaggedPrepartition.filter \u03c0 fun J => \u00acp J\nthis : Disjoint (Prepartition.iUnion \u03c0\u2081) (TaggedPrepartition.iUnion \u03c0\u2082)\n\u22a2 \u2203 \u03c0',\n Prepartition.iUnion \u03c0' = \u2191I \\ TaggedPrepartition.iUnion (TaggedPrepartition.filter \u03c0 p) \u2227\n Prepartition.distortion \u03c0' \u2264 c\n[PROOFSTEP]\nrefine' \u27e8\u03c0\u2081.disjUnion \u03c0\u2082.toPrepartition this, _, _\u27e9\n[GOAL]\ncase intro.intro.refine'_1\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\nI J : Box \u03b9\nc c\u2081 c\u2082 : \u211d\u22650\nr r\u2081 r\u2082 : (\u03b9 \u2192 \u211d) \u2192 \u2191(Set.Ioi 0)\n\u03c0 \u03c0\u2081\u271d \u03c0\u2082\u271d : TaggedPrepartition I\nl l\u2081 l\u2082 : IntegrationParams\nh\u03c0 : MemBaseSet l I c r \u03c0\np : Box \u03b9 \u2192 Prop\nhD : l.bDistortion = true\n\u03c0\u2081 : Prepartition I\nh\u03c0\u2081U : Prepartition.iUnion \u03c0\u2081 = \u2191I \\ TaggedPrepartition.iUnion \u03c0\nhc : Prepartition.distortion \u03c0\u2081 \u2264 c\n\u03c0\u2082 : TaggedPrepartition I := TaggedPrepartition.filter \u03c0 fun J => \u00acp J\nthis : Disjoint (Prepartition.iUnion \u03c0\u2081) (TaggedPrepartition.iUnion \u03c0\u2082)\n\u22a2 Prepartition.iUnion (Prepartition.disjUnion \u03c0\u2081 \u03c0\u2082.toPrepartition this) =\n \u2191I \\ TaggedPrepartition.iUnion (TaggedPrepartition.filter \u03c0 p)\n[PROOFSTEP]\nsuffices \u2191I \\ \u03c0.iUnion \u222a \u03c0.iUnion \\ (\u03c0.filter p).iUnion = \u2191I \\ (\u03c0.filter p).iUnion by simp [*]\n[GOAL]\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\nI J : Box \u03b9\nc c\u2081 c\u2082 : \u211d\u22650\nr r\u2081 r\u2082 : (\u03b9 \u2192 \u211d) \u2192 \u2191(Set.Ioi 0)\n\u03c0 \u03c0\u2081\u271d \u03c0\u2082\u271d : TaggedPrepartition I\nl l\u2081 l\u2082 : IntegrationParams\nh\u03c0 : MemBaseSet l I c r \u03c0\np : Box \u03b9 \u2192 Prop\nhD : l.bDistortion = true\n\u03c0\u2081 : Prepartition I\nh\u03c0\u2081U : Prepartition.iUnion \u03c0\u2081 = \u2191I \\ TaggedPrepartition.iUnion \u03c0\nhc : Prepartition.distortion \u03c0\u2081 \u2264 c\n\u03c0\u2082 : TaggedPrepartition I := TaggedPrepartition.filter \u03c0 fun J => \u00acp J\nthis\u271d : Disjoint (Prepartition.iUnion \u03c0\u2081) (TaggedPrepartition.iUnion \u03c0\u2082)\nthis :\n \u2191I \\ TaggedPrepartition.iUnion \u03c0 \u222a\n TaggedPrepartition.iUnion \u03c0 \\ TaggedPrepartition.iUnion (TaggedPrepartition.filter \u03c0 p) =\n \u2191I \\ TaggedPrepartition.iUnion (TaggedPrepartition.filter \u03c0 p)\n\u22a2 Prepartition.iUnion (Prepartition.disjUnion \u03c0\u2081 \u03c0\u2082.toPrepartition this\u271d) =\n \u2191I \\ TaggedPrepartition.iUnion (TaggedPrepartition.filter \u03c0 p)\n[PROOFSTEP]\nsimp [*]\n[GOAL]\ncase intro.intro.refine'_1\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\nI J : Box \u03b9\nc c\u2081 c\u2082 : \u211d\u22650\nr r\u2081 r\u2082 : (\u03b9 \u2192 \u211d) \u2192 \u2191(Set.Ioi 0)\n\u03c0 \u03c0\u2081\u271d \u03c0\u2082\u271d : TaggedPrepartition I\nl l\u2081 l\u2082 : IntegrationParams\nh\u03c0 : MemBaseSet l I c r \u03c0\np : Box \u03b9 \u2192 Prop\nhD : l.bDistortion = true\n\u03c0\u2081 : Prepartition I\nh\u03c0\u2081U : Prepartition.iUnion \u03c0\u2081 = \u2191I \\ TaggedPrepartition.iUnion \u03c0\nhc : Prepartition.distortion \u03c0\u2081 \u2264 c\n\u03c0\u2082 : TaggedPrepartition I := TaggedPrepartition.filter \u03c0 fun J => \u00acp J\nthis : Disjoint (Prepartition.iUnion \u03c0\u2081) (TaggedPrepartition.iUnion \u03c0\u2082)\n\u22a2 \u2191I \\ TaggedPrepartition.iUnion \u03c0 \u222a\n TaggedPrepartition.iUnion \u03c0 \\ TaggedPrepartition.iUnion (TaggedPrepartition.filter \u03c0 p) =\n \u2191I \\ TaggedPrepartition.iUnion (TaggedPrepartition.filter \u03c0 p)\n[PROOFSTEP]\nhave h : (\u03c0.filter p).iUnion \u2286 \u03c0.iUnion := biUnion_subset_biUnion_left (Finset.filter_subset _ _)\n[GOAL]\ncase intro.intro.refine'_1\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\nI J : Box \u03b9\nc c\u2081 c\u2082 : \u211d\u22650\nr r\u2081 r\u2082 : (\u03b9 \u2192 \u211d) \u2192 \u2191(Set.Ioi 0)\n\u03c0 \u03c0\u2081\u271d \u03c0\u2082\u271d : TaggedPrepartition I\nl l\u2081 l\u2082 : IntegrationParams\nh\u03c0 : MemBaseSet l I c r \u03c0\np : Box \u03b9 \u2192 Prop\nhD : l.bDistortion = true\n\u03c0\u2081 : Prepartition I\nh\u03c0\u2081U : Prepartition.iUnion \u03c0\u2081 = \u2191I \\ TaggedPrepartition.iUnion \u03c0\nhc : Prepartition.distortion \u03c0\u2081 \u2264 c\n\u03c0\u2082 : TaggedPrepartition I := TaggedPrepartition.filter \u03c0 fun J => \u00acp J\nthis : Disjoint (Prepartition.iUnion \u03c0\u2081) (TaggedPrepartition.iUnion \u03c0\u2082)\nh : TaggedPrepartition.iUnion (TaggedPrepartition.filter \u03c0 p) \u2286 TaggedPrepartition.iUnion \u03c0\n\u22a2 \u2191I \\ TaggedPrepartition.iUnion \u03c0 \u222a\n TaggedPrepartition.iUnion \u03c0 \\ TaggedPrepartition.iUnion (TaggedPrepartition.filter \u03c0 p) =\n \u2191I \\ TaggedPrepartition.iUnion (TaggedPrepartition.filter \u03c0 p)\n[PROOFSTEP]\next x\n[GOAL]\ncase intro.intro.refine'_1.h\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\nI J : Box \u03b9\nc c\u2081 c\u2082 : \u211d\u22650\nr r\u2081 r\u2082 : (\u03b9 \u2192 \u211d) \u2192 \u2191(Set.Ioi 0)\n\u03c0 \u03c0\u2081\u271d \u03c0\u2082\u271d : TaggedPrepartition I\nl l\u2081 l\u2082 : IntegrationParams\nh\u03c0 : MemBaseSet l I c r \u03c0\np : Box \u03b9 \u2192 Prop\nhD : l.bDistortion = true\n\u03c0\u2081 : Prepartition I\nh\u03c0\u2081U : Prepartition.iUnion \u03c0\u2081 = \u2191I \\ TaggedPrepartition.iUnion \u03c0\nhc : Prepartition.distortion \u03c0\u2081 \u2264 c\n\u03c0\u2082 : TaggedPrepartition I := TaggedPrepartition.filter \u03c0 fun J => \u00acp J\nthis : Disjoint (Prepartition.iUnion \u03c0\u2081) (TaggedPrepartition.iUnion \u03c0\u2082)\nh : TaggedPrepartition.iUnion (TaggedPrepartition.filter \u03c0 p) \u2286 TaggedPrepartition.iUnion \u03c0\nx : \u03b9 \u2192 \u211d\n\u22a2 x \u2208\n \u2191I \\ TaggedPrepartition.iUnion \u03c0 \u222a\n TaggedPrepartition.iUnion \u03c0 \\ TaggedPrepartition.iUnion (TaggedPrepartition.filter \u03c0 p) \u2194\n x \u2208 \u2191I \\ TaggedPrepartition.iUnion (TaggedPrepartition.filter \u03c0 p)\n[PROOFSTEP]\nfconstructor\n[GOAL]\ncase intro.intro.refine'_1.h.mp\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\nI J : Box \u03b9\nc c\u2081 c\u2082 : \u211d\u22650\nr r\u2081 r\u2082 : (\u03b9 \u2192 \u211d) \u2192 \u2191(Set.Ioi 0)\n\u03c0 \u03c0\u2081\u271d \u03c0\u2082\u271d : TaggedPrepartition I\nl l\u2081 l\u2082 : IntegrationParams\nh\u03c0 : MemBaseSet l I c r \u03c0\np : Box \u03b9 \u2192 Prop\nhD : l.bDistortion = true\n\u03c0\u2081 : Prepartition I\nh\u03c0\u2081U : Prepartition.iUnion \u03c0\u2081 = \u2191I \\ TaggedPrepartition.iUnion \u03c0\nhc : Prepartition.distortion \u03c0\u2081 \u2264 c\n\u03c0\u2082 : TaggedPrepartition I := TaggedPrepartition.filter \u03c0 fun J => \u00acp J\nthis : Disjoint (Prepartition.iUnion \u03c0\u2081) (TaggedPrepartition.iUnion \u03c0\u2082)\nh : TaggedPrepartition.iUnion (TaggedPrepartition.filter \u03c0 p) \u2286 TaggedPrepartition.iUnion \u03c0\nx : \u03b9 \u2192 \u211d\n\u22a2 x \u2208\n \u2191I \\ TaggedPrepartition.iUnion \u03c0 \u222a\n TaggedPrepartition.iUnion \u03c0 \\ TaggedPrepartition.iUnion (TaggedPrepartition.filter \u03c0 p) \u2192\n x \u2208 \u2191I \\ TaggedPrepartition.iUnion (TaggedPrepartition.filter \u03c0 p)\n[PROOFSTEP]\nrintro (\u27e8hxI, hx\u03c0\u27e9 | \u27e8hx\u03c0, hxp\u27e9)\n[GOAL]\ncase intro.intro.refine'_1.h.mp.inl.intro\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\nI J : Box \u03b9\nc c\u2081 c\u2082 : \u211d\u22650\nr r\u2081 r\u2082 : (\u03b9 \u2192 \u211d) \u2192 \u2191(Set.Ioi 0)\n\u03c0 \u03c0\u2081\u271d \u03c0\u2082\u271d : TaggedPrepartition I\nl l\u2081 l\u2082 : IntegrationParams\nh\u03c0 : MemBaseSet l I c r \u03c0\np : Box \u03b9 \u2192 Prop\nhD : l.bDistortion = true\n\u03c0\u2081 : Prepartition I\nh\u03c0\u2081U : Prepartition.iUnion \u03c0\u2081 = \u2191I \\ TaggedPrepartition.iUnion \u03c0\nhc : Prepartition.distortion \u03c0\u2081 \u2264 c\n\u03c0\u2082 : TaggedPrepartition I := TaggedPrepartition.filter \u03c0 fun J => \u00acp J\nthis : Disjoint (Prepartition.iUnion \u03c0\u2081) (TaggedPrepartition.iUnion \u03c0\u2082)\nh : TaggedPrepartition.iUnion (TaggedPrepartition.filter \u03c0 p) \u2286 TaggedPrepartition.iUnion \u03c0\nx : \u03b9 \u2192 \u211d\nhxI : x \u2208 \u2191I\nhx\u03c0 : \u00acx \u2208 TaggedPrepartition.iUnion \u03c0\n\u22a2 x \u2208 \u2191I \\ TaggedPrepartition.iUnion (TaggedPrepartition.filter \u03c0 p)\ncase intro.intro.refine'_1.h.mp.inr.intro\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\nI J : Box \u03b9\nc c\u2081 c\u2082 : \u211d\u22650\nr r\u2081 r\u2082 : (\u03b9 \u2192 \u211d) \u2192 \u2191(Set.Ioi 0)\n\u03c0 \u03c0\u2081\u271d \u03c0\u2082\u271d : TaggedPrepartition I\nl l\u2081 l\u2082 : IntegrationParams\nh\u03c0 : MemBaseSet l I c r \u03c0\np : Box \u03b9 \u2192 Prop\nhD : l.bDistortion = true\n\u03c0\u2081 : Prepartition I\nh\u03c0\u2081U : Prepartition.iUnion \u03c0\u2081 = \u2191I \\ TaggedPrepartition.iUnion \u03c0\nhc : Prepartition.distortion \u03c0\u2081 \u2264 c\n\u03c0\u2082 : TaggedPrepartition I := TaggedPrepartition.filter \u03c0 fun J => \u00acp J\nthis : Disjoint (Prepartition.iUnion \u03c0\u2081) (TaggedPrepartition.iUnion \u03c0\u2082)\nh : TaggedPrepartition.iUnion (TaggedPrepartition.filter \u03c0 p) \u2286 TaggedPrepartition.iUnion \u03c0\nx : \u03b9 \u2192 \u211d\nhx\u03c0 : x \u2208 TaggedPrepartition.iUnion \u03c0\nhxp : \u00acx \u2208 TaggedPrepartition.iUnion (TaggedPrepartition.filter \u03c0 p)\n\u22a2 x \u2208 \u2191I \\ TaggedPrepartition.iUnion (TaggedPrepartition.filter \u03c0 p)\n[PROOFSTEP]\nexacts [\u27e8hxI, mt (@h x) hx\u03c0\u27e9, \u27e8\u03c0.iUnion_subset hx\u03c0, hxp\u27e9]\n[GOAL]\ncase intro.intro.refine'_1.h.mpr\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\nI J : Box \u03b9\nc c\u2081 c\u2082 : \u211d\u22650\nr r\u2081 r\u2082 : (\u03b9 \u2192 \u211d) \u2192 \u2191(Set.Ioi 0)\n\u03c0 \u03c0\u2081\u271d \u03c0\u2082\u271d : TaggedPrepartition I\nl l\u2081 l\u2082 : IntegrationParams\nh\u03c0 : MemBaseSet l I c r \u03c0\np : Box \u03b9 \u2192 Prop\nhD : l.bDistortion = true\n\u03c0\u2081 : Prepartition I\nh\u03c0\u2081U : Prepartition.iUnion \u03c0\u2081 = \u2191I \\ TaggedPrepartition.iUnion \u03c0\nhc : Prepartition.distortion \u03c0\u2081 \u2264 c\n\u03c0\u2082 : TaggedPrepartition I := TaggedPrepartition.filter \u03c0 fun J => \u00acp J\nthis : Disjoint (Prepartition.iUnion \u03c0\u2081) (TaggedPrepartition.iUnion \u03c0\u2082)\nh : TaggedPrepartition.iUnion (TaggedPrepartition.filter \u03c0 p) \u2286 TaggedPrepartition.iUnion \u03c0\nx : \u03b9 \u2192 \u211d\n\u22a2 x \u2208 \u2191I \\ TaggedPrepartition.iUnion (TaggedPrepartition.filter \u03c0 p) \u2192\n x \u2208\n \u2191I \\ TaggedPrepartition.iUnion \u03c0 \u222a\n TaggedPrepartition.iUnion \u03c0 \\ TaggedPrepartition.iUnion (TaggedPrepartition.filter \u03c0 p)\n[PROOFSTEP]\nrintro \u27e8hxI, hxp\u27e9\n[GOAL]\ncase intro.intro.refine'_1.h.mpr.intro\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\nI J : Box \u03b9\nc c\u2081 c\u2082 : \u211d\u22650\nr r\u2081 r\u2082 : (\u03b9 \u2192 \u211d) \u2192 \u2191(Set.Ioi 0)\n\u03c0 \u03c0\u2081\u271d \u03c0\u2082\u271d : TaggedPrepartition I\nl l\u2081 l\u2082 : IntegrationParams\nh\u03c0 : MemBaseSet l I c r \u03c0\np : Box \u03b9 \u2192 Prop\nhD : l.bDistortion = true\n\u03c0\u2081 : Prepartition I\nh\u03c0\u2081U : Prepartition.iUnion \u03c0\u2081 = \u2191I \\ TaggedPrepartition.iUnion \u03c0\nhc : Prepartition.distortion \u03c0\u2081 \u2264 c\n\u03c0\u2082 : TaggedPrepartition I := TaggedPrepartition.filter \u03c0 fun J => \u00acp J\nthis : Disjoint (Prepartition.iUnion \u03c0\u2081) (TaggedPrepartition.iUnion \u03c0\u2082)\nh : TaggedPrepartition.iUnion (TaggedPrepartition.filter \u03c0 p) \u2286 TaggedPrepartition.iUnion \u03c0\nx : \u03b9 \u2192 \u211d\nhxI : x \u2208 \u2191I\nhxp : \u00acx \u2208 TaggedPrepartition.iUnion (TaggedPrepartition.filter \u03c0 p)\n\u22a2 x \u2208\n \u2191I \\ TaggedPrepartition.iUnion \u03c0 \u222a\n TaggedPrepartition.iUnion \u03c0 \\ TaggedPrepartition.iUnion (TaggedPrepartition.filter \u03c0 p)\n[PROOFSTEP]\nby_cases hx\u03c0 : x \u2208 \u03c0.iUnion\n[GOAL]\ncase pos\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\nI J : Box \u03b9\nc c\u2081 c\u2082 : \u211d\u22650\nr r\u2081 r\u2082 : (\u03b9 \u2192 \u211d) \u2192 \u2191(Set.Ioi 0)\n\u03c0 \u03c0\u2081\u271d \u03c0\u2082\u271d : TaggedPrepartition I\nl l\u2081 l\u2082 : IntegrationParams\nh\u03c0 : MemBaseSet l I c r \u03c0\np : Box \u03b9 \u2192 Prop\nhD : l.bDistortion = true\n\u03c0\u2081 : Prepartition I\nh\u03c0\u2081U : Prepartition.iUnion \u03c0\u2081 = \u2191I \\ TaggedPrepartition.iUnion \u03c0\nhc : Prepartition.distortion \u03c0\u2081 \u2264 c\n\u03c0\u2082 : TaggedPrepartition I := TaggedPrepartition.filter \u03c0 fun J => \u00acp J\nthis : Disjoint (Prepartition.iUnion \u03c0\u2081) (TaggedPrepartition.iUnion \u03c0\u2082)\nh : TaggedPrepartition.iUnion (TaggedPrepartition.filter \u03c0 p) \u2286 TaggedPrepartition.iUnion \u03c0\nx : \u03b9 \u2192 \u211d\nhxI : x \u2208 \u2191I\nhxp : \u00acx \u2208 TaggedPrepartition.iUnion (TaggedPrepartition.filter \u03c0 p)\nhx\u03c0 : x \u2208 TaggedPrepartition.iUnion \u03c0\n\u22a2 x \u2208\n \u2191I \\ TaggedPrepartition.iUnion \u03c0 \u222a\n TaggedPrepartition.iUnion \u03c0 \\ TaggedPrepartition.iUnion (TaggedPrepartition.filter \u03c0 p)\ncase neg\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\nI J : Box \u03b9\nc c\u2081 c\u2082 : \u211d\u22650\nr r\u2081 r\u2082 : (\u03b9 \u2192 \u211d) \u2192 \u2191(Set.Ioi 0)\n\u03c0 \u03c0\u2081\u271d \u03c0\u2082\u271d : TaggedPrepartition I\nl l\u2081 l\u2082 : IntegrationParams\nh\u03c0 : MemBaseSet l I c r \u03c0\np : Box \u03b9 \u2192 Prop\nhD : l.bDistortion = true\n\u03c0\u2081 : Prepartition I\nh\u03c0\u2081U : Prepartition.iUnion \u03c0\u2081 = \u2191I \\ TaggedPrepartition.iUnion \u03c0\nhc : Prepartition.distortion \u03c0\u2081 \u2264 c\n\u03c0\u2082 : TaggedPrepartition I := TaggedPrepartition.filter \u03c0 fun J => \u00acp J\nthis : Disjoint (Prepartition.iUnion \u03c0\u2081) (TaggedPrepartition.iUnion \u03c0\u2082)\nh : TaggedPrepartition.iUnion (TaggedPrepartition.filter \u03c0 p) \u2286 TaggedPrepartition.iUnion \u03c0\nx : \u03b9 \u2192 \u211d\nhxI : x \u2208 \u2191I\nhxp : \u00acx \u2208 TaggedPrepartition.iUnion (TaggedPrepartition.filter \u03c0 p)\nhx\u03c0 : \u00acx \u2208 TaggedPrepartition.iUnion \u03c0\n\u22a2 x \u2208\n \u2191I \\ TaggedPrepartition.iUnion \u03c0 \u222a\n TaggedPrepartition.iUnion \u03c0 \\ TaggedPrepartition.iUnion (TaggedPrepartition.filter \u03c0 p)\n[PROOFSTEP]\nexacts [Or.inr \u27e8hx\u03c0, hxp\u27e9, Or.inl \u27e8hxI, hx\u03c0\u27e9]\n[GOAL]\ncase intro.intro.refine'_2\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\nI J : Box \u03b9\nc c\u2081 c\u2082 : \u211d\u22650\nr r\u2081 r\u2082 : (\u03b9 \u2192 \u211d) \u2192 \u2191(Set.Ioi 0)\n\u03c0 \u03c0\u2081\u271d \u03c0\u2082\u271d : TaggedPrepartition I\nl l\u2081 l\u2082 : IntegrationParams\nh\u03c0 : MemBaseSet l I c r \u03c0\np : Box \u03b9 \u2192 Prop\nhD : l.bDistortion = true\n\u03c0\u2081 : Prepartition I\nh\u03c0\u2081U : Prepartition.iUnion \u03c0\u2081 = \u2191I \\ TaggedPrepartition.iUnion \u03c0\nhc : Prepartition.distortion \u03c0\u2081 \u2264 c\n\u03c0\u2082 : TaggedPrepartition I := TaggedPrepartition.filter \u03c0 fun J => \u00acp J\nthis : Disjoint (Prepartition.iUnion \u03c0\u2081) (TaggedPrepartition.iUnion \u03c0\u2082)\n\u22a2 Prepartition.distortion (Prepartition.disjUnion \u03c0\u2081 \u03c0\u2082.toPrepartition this) \u2264 c\n[PROOFSTEP]\nhave : (\u03c0.filter fun J => \u00acp J).distortion \u2264 c := (distortion_filter_le _ _).trans (h\u03c0.3 hD)\n[GOAL]\ncase intro.intro.refine'_2\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\nI J : Box \u03b9\nc c\u2081 c\u2082 : \u211d\u22650\nr r\u2081 r\u2082 : (\u03b9 \u2192 \u211d) \u2192 \u2191(Set.Ioi 0)\n\u03c0 \u03c0\u2081\u271d \u03c0\u2082\u271d : TaggedPrepartition I\nl l\u2081 l\u2082 : IntegrationParams\nh\u03c0 : MemBaseSet l I c r \u03c0\np : Box \u03b9 \u2192 Prop\nhD : l.bDistortion = true\n\u03c0\u2081 : Prepartition I\nh\u03c0\u2081U : Prepartition.iUnion \u03c0\u2081 = \u2191I \\ TaggedPrepartition.iUnion \u03c0\nhc : Prepartition.distortion \u03c0\u2081 \u2264 c\n\u03c0\u2082 : TaggedPrepartition I := TaggedPrepartition.filter \u03c0 fun J => \u00acp J\nthis\u271d : Disjoint (Prepartition.iUnion \u03c0\u2081) (TaggedPrepartition.iUnion \u03c0\u2082)\nthis : distortion (TaggedPrepartition.filter \u03c0 fun J => \u00acp J) \u2264 c\n\u22a2 Prepartition.distortion (Prepartition.disjUnion \u03c0\u2081 \u03c0\u2082.toPrepartition this\u271d) \u2264 c\n[PROOFSTEP]\nsimpa [hc]\n[GOAL]\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\nI J : Box \u03b9\nc c\u2081 c\u2082 : \u211d\u22650\nr r\u2081 r\u2082 : (\u03b9 \u2192 \u211d) \u2192 \u2191(Set.Ioi 0)\n\u03c0\u271d \u03c0\u2081 \u03c0\u2082 : TaggedPrepartition I\nl l\u2081 l\u2082 : IntegrationParams\n\u03c0 : Prepartition I\n\u03c0i : (J : Box \u03b9) \u2192 TaggedPrepartition J\nh : \u2200 (J : Box \u03b9), J \u2208 \u03c0 \u2192 MemBaseSet l J c r (\u03c0i J)\nhp : \u2200 (J : Box \u03b9), J \u2208 \u03c0 \u2192 IsPartition (\u03c0i J)\nhc : l.bDistortion = true \u2192 Prepartition.distortion (Prepartition.compl \u03c0) \u2264 c\n\u22a2 MemBaseSet l I c r (Prepartition.biUnionTagged \u03c0 \u03c0i)\n[PROOFSTEP]\nrefine'\n \u27e8TaggedPrepartition.isSubordinate_biUnionTagged.2 fun J hJ => (h J hJ).1, fun hH =>\n TaggedPrepartition.isHenstock_biUnionTagged.2 fun J hJ => (h J hJ).2 hH, fun hD => _, fun hD => _\u27e9\n[GOAL]\ncase refine'_1\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\nI J : Box \u03b9\nc c\u2081 c\u2082 : \u211d\u22650\nr r\u2081 r\u2082 : (\u03b9 \u2192 \u211d) \u2192 \u2191(Set.Ioi 0)\n\u03c0\u271d \u03c0\u2081 \u03c0\u2082 : TaggedPrepartition I\nl l\u2081 l\u2082 : IntegrationParams\n\u03c0 : Prepartition I\n\u03c0i : (J : Box \u03b9) \u2192 TaggedPrepartition J\nh : \u2200 (J : Box \u03b9), J \u2208 \u03c0 \u2192 MemBaseSet l J c r (\u03c0i J)\nhp : \u2200 (J : Box \u03b9), J \u2208 \u03c0 \u2192 IsPartition (\u03c0i J)\nhc : l.bDistortion = true \u2192 Prepartition.distortion (Prepartition.compl \u03c0) \u2264 c\nhD : l.bDistortion = true\n\u22a2 distortion (Prepartition.biUnionTagged \u03c0 \u03c0i) \u2264 c\n[PROOFSTEP]\nrw [Prepartition.distortion_biUnionTagged, Finset.sup_le_iff]\n[GOAL]\ncase refine'_1\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\nI J : Box \u03b9\nc c\u2081 c\u2082 : \u211d\u22650\nr r\u2081 r\u2082 : (\u03b9 \u2192 \u211d) \u2192 \u2191(Set.Ioi 0)\n\u03c0\u271d \u03c0\u2081 \u03c0\u2082 : TaggedPrepartition I\nl l\u2081 l\u2082 : IntegrationParams\n\u03c0 : Prepartition I\n\u03c0i : (J : Box \u03b9) \u2192 TaggedPrepartition J\nh : \u2200 (J : Box \u03b9), J \u2208 \u03c0 \u2192 MemBaseSet l J c r (\u03c0i J)\nhp : \u2200 (J : Box \u03b9), J \u2208 \u03c0 \u2192 IsPartition (\u03c0i J)\nhc : l.bDistortion = true \u2192 Prepartition.distortion (Prepartition.compl \u03c0) \u2264 c\nhD : l.bDistortion = true\n\u22a2 \u2200 (b : Box \u03b9), b \u2208 \u03c0.boxes \u2192 distortion (\u03c0i b) \u2264 c\n[PROOFSTEP]\nexact fun J hJ => (h J hJ).3 hD\n[GOAL]\ncase refine'_2\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\nI J : Box \u03b9\nc c\u2081 c\u2082 : \u211d\u22650\nr r\u2081 r\u2082 : (\u03b9 \u2192 \u211d) \u2192 \u2191(Set.Ioi 0)\n\u03c0\u271d \u03c0\u2081 \u03c0\u2082 : TaggedPrepartition I\nl l\u2081 l\u2082 : IntegrationParams\n\u03c0 : Prepartition I\n\u03c0i : (J : Box \u03b9) \u2192 TaggedPrepartition J\nh : \u2200 (J : Box \u03b9), J \u2208 \u03c0 \u2192 MemBaseSet l J c r (\u03c0i J)\nhp : \u2200 (J : Box \u03b9), J \u2208 \u03c0 \u2192 IsPartition (\u03c0i J)\nhc : l.bDistortion = true \u2192 Prepartition.distortion (Prepartition.compl \u03c0) \u2264 c\nhD : l.bDistortion = true\n\u22a2 \u2203 \u03c0',\n Prepartition.iUnion \u03c0' = \u2191I \\ TaggedPrepartition.iUnion (Prepartition.biUnionTagged \u03c0 \u03c0i) \u2227\n Prepartition.distortion \u03c0' \u2264 c\n[PROOFSTEP]\nrefine' \u27e8_, _, hc hD\u27e9\n[GOAL]\ncase refine'_2\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\nI J : Box \u03b9\nc c\u2081 c\u2082 : \u211d\u22650\nr r\u2081 r\u2082 : (\u03b9 \u2192 \u211d) \u2192 \u2191(Set.Ioi 0)\n\u03c0\u271d \u03c0\u2081 \u03c0\u2082 : TaggedPrepartition I\nl l\u2081 l\u2082 : IntegrationParams\n\u03c0 : Prepartition I\n\u03c0i : (J : Box \u03b9) \u2192 TaggedPrepartition J\nh : \u2200 (J : Box \u03b9), J \u2208 \u03c0 \u2192 MemBaseSet l J c r (\u03c0i J)\nhp : \u2200 (J : Box \u03b9), J \u2208 \u03c0 \u2192 IsPartition (\u03c0i J)\nhc : l.bDistortion = true \u2192 Prepartition.distortion (Prepartition.compl \u03c0) \u2264 c\nhD : l.bDistortion = true\n\u22a2 Prepartition.iUnion (Prepartition.compl \u03c0) = \u2191I \\ TaggedPrepartition.iUnion (Prepartition.biUnionTagged \u03c0 \u03c0i)\n[PROOFSTEP]\nrw [\u03c0.iUnion_compl, \u2190 \u03c0.iUnion_biUnion_partition hp]\n[GOAL]\ncase refine'_2\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\nI J : Box \u03b9\nc c\u2081 c\u2082 : \u211d\u22650\nr r\u2081 r\u2082 : (\u03b9 \u2192 \u211d) \u2192 \u2191(Set.Ioi 0)\n\u03c0\u271d \u03c0\u2081 \u03c0\u2082 : TaggedPrepartition I\nl l\u2081 l\u2082 : IntegrationParams\n\u03c0 : Prepartition I\n\u03c0i : (J : Box \u03b9) \u2192 TaggedPrepartition J\nh : \u2200 (J : Box \u03b9), J \u2208 \u03c0 \u2192 MemBaseSet l J c r (\u03c0i J)\nhp : \u2200 (J : Box \u03b9), J \u2208 \u03c0 \u2192 IsPartition (\u03c0i J)\nhc : l.bDistortion = true \u2192 Prepartition.distortion (Prepartition.compl \u03c0) \u2264 c\nhD : l.bDistortion = true\n\u22a2 \u2191I \\ Prepartition.iUnion (Prepartition.biUnion \u03c0 fun J => (\u03c0i J).toPrepartition) =\n \u2191I \\ TaggedPrepartition.iUnion (Prepartition.biUnionTagged \u03c0 \u03c0i)\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\nI\u271d J : Box \u03b9\nc c\u2081 c\u2082 : \u211d\u22650\nr r\u2081 r\u2082 : (\u03b9 \u2192 \u211d) \u2192 \u2191(Set.Ioi 0)\n\u03c0 \u03c0\u2081\u271d \u03c0\u2082\u271d : TaggedPrepartition I\u271d\nl\u271d l\u2081 l\u2082 : IntegrationParams\nI : Box \u03b9\nl : IntegrationParams\n\u03c0\u2081 \u03c0\u2082 : Prepartition I\nh : Prepartition.iUnion \u03c0\u2081 = Prepartition.iUnion \u03c0\u2082\n\u22a2 toFilteriUnion l I \u03c0\u2081 = toFilteriUnion l I \u03c0\u2082\n[PROOFSTEP]\nsimp only [toFilteriUnion, toFilterDistortioniUnion, h]\n[GOAL]\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\nI\u271d J : Box \u03b9\nc c\u2081 c\u2082 : \u211d\u22650\nr r\u2081 r\u2082 : (\u03b9 \u2192 \u211d) \u2192 \u2191(Set.Ioi 0)\n\u03c0 \u03c0\u2081 \u03c0\u2082 : TaggedPrepartition I\u271d\nl\u271d l\u2081 l\u2082 l : IntegrationParams\nI : Box \u03b9\n\u03c0\u2080 : Prepartition I\n\u22a2 HasBasis (toFilteriUnion l I \u03c0\u2080) (fun r => \u2200 (c : \u211d\u22650), RCond l (r c)) fun r =>\n {\u03c0 | \u2203 c, MemBaseSet l I c (r c) \u03c0 \u2227 TaggedPrepartition.iUnion \u03c0 = Prepartition.iUnion \u03c0\u2080}\n[PROOFSTEP]\nhave := fun c => l.hasBasis_toFilterDistortioniUnion I c \u03c0\u2080\n[GOAL]\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\nI\u271d J : Box \u03b9\nc c\u2081 c\u2082 : \u211d\u22650\nr r\u2081 r\u2082 : (\u03b9 \u2192 \u211d) \u2192 \u2191(Set.Ioi 0)\n\u03c0 \u03c0\u2081 \u03c0\u2082 : TaggedPrepartition I\u271d\nl\u271d l\u2081 l\u2082 l : IntegrationParams\nI : Box \u03b9\n\u03c0\u2080 : Prepartition I\nthis :\n \u2200 (c : \u211d\u22650),\n HasBasis (toFilterDistortioniUnion l I c \u03c0\u2080) (RCond l) fun r =>\n {\u03c0 | MemBaseSet l I c r \u03c0 \u2227 TaggedPrepartition.iUnion \u03c0 = Prepartition.iUnion \u03c0\u2080}\n\u22a2 HasBasis (toFilteriUnion l I \u03c0\u2080) (fun r => \u2200 (c : \u211d\u22650), RCond l (r c)) fun r =>\n {\u03c0 | \u2203 c, MemBaseSet l I c (r c) \u03c0 \u2227 TaggedPrepartition.iUnion \u03c0 = Prepartition.iUnion \u03c0\u2080}\n[PROOFSTEP]\nsimpa only [setOf_and, setOf_exists] using hasBasis_iSup this\n[GOAL]\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\nI\u271d J : Box \u03b9\nc c\u2081 c\u2082 : \u211d\u22650\nr r\u2081 r\u2082 : (\u03b9 \u2192 \u211d) \u2192 \u2191(Set.Ioi 0)\n\u03c0 \u03c0\u2081 \u03c0\u2082 : TaggedPrepartition I\u271d\nl\u271d l\u2081 l\u2082 l : IntegrationParams\nI : Box \u03b9\n\u22a2 HasBasis (toFilteriUnion l I \u22a4) (fun r => \u2200 (c : \u211d\u22650), RCond l (r c)) fun r =>\n {\u03c0 | \u2203 c, MemBaseSet l I c (r c) \u03c0 \u2227 IsPartition \u03c0}\n[PROOFSTEP]\nsimpa only [TaggedPrepartition.isPartition_iff_iUnion_eq, Prepartition.iUnion_top] using l.hasBasis_toFilteriUnion I \u22a4\n[GOAL]\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\nI\u271d J : Box \u03b9\nc c\u2081 c\u2082 : \u211d\u22650\nr r\u2081 r\u2082 : (\u03b9 \u2192 \u211d) \u2192 \u2191(Set.Ioi 0)\n\u03c0 \u03c0\u2081 \u03c0\u2082 : TaggedPrepartition I\u271d\nl\u271d l\u2081 l\u2082 l : IntegrationParams\nI : Box \u03b9\n\u22a2 HasBasis (toFilter l I) (fun r => \u2200 (c : \u211d\u22650), RCond l (r c)) fun r => {\u03c0 | \u2203 c, MemBaseSet l I c (r c) \u03c0}\n[PROOFSTEP]\nsimpa only [setOf_exists] using hasBasis_iSup (l.hasBasis_toFilterDistortion I)\n[GOAL]\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\nI J : Box \u03b9\nc c\u2081 c\u2082 : \u211d\u22650\nr r\u2081 r\u2082 : (\u03b9 \u2192 \u211d) \u2192 \u2191(Set.Ioi 0)\n\u03c0 \u03c0\u2081 \u03c0\u2082 : TaggedPrepartition I\nl\u271d l\u2081 l\u2082 l : IntegrationParams\nh : I \u2264 J\n\u22a2 Tendsto (\u2191(embedBox I J h)) (toFilteriUnion l I \u22a4) (toFilteriUnion l J (Prepartition.single J I h))\n[PROOFSTEP]\nsimp only [toFilteriUnion, tendsto_iSup]\n[GOAL]\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\nI J : Box \u03b9\nc c\u2081 c\u2082 : \u211d\u22650\nr r\u2081 r\u2082 : (\u03b9 \u2192 \u211d) \u2192 \u2191(Set.Ioi 0)\n\u03c0 \u03c0\u2081 \u03c0\u2082 : TaggedPrepartition I\nl\u271d l\u2081 l\u2082 l : IntegrationParams\nh : I \u2264 J\n\u22a2 \u2200 (i : \u211d\u22650),\n Tendsto (\u2191(embedBox I J h)) (toFilterDistortioniUnion l I i \u22a4)\n (\u2a06 (c : \u211d\u22650), toFilterDistortioniUnion l J c (Prepartition.single J I h))\n[PROOFSTEP]\nintro c\n[GOAL]\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\nI J : Box \u03b9\nc\u271d c\u2081 c\u2082 : \u211d\u22650\nr r\u2081 r\u2082 : (\u03b9 \u2192 \u211d) \u2192 \u2191(Set.Ioi 0)\n\u03c0 \u03c0\u2081 \u03c0\u2082 : TaggedPrepartition I\nl\u271d l\u2081 l\u2082 l : IntegrationParams\nh : I \u2264 J\nc : \u211d\u22650\n\u22a2 Tendsto (\u2191(embedBox I J h)) (toFilterDistortioniUnion l I c \u22a4)\n (\u2a06 (c : \u211d\u22650), toFilterDistortioniUnion l J c (Prepartition.single J I h))\n[PROOFSTEP]\nset \u03c0\u2080 := Prepartition.single J I h\n[GOAL]\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\nI J : Box \u03b9\nc\u271d c\u2081 c\u2082 : \u211d\u22650\nr r\u2081 r\u2082 : (\u03b9 \u2192 \u211d) \u2192 \u2191(Set.Ioi 0)\n\u03c0 \u03c0\u2081 \u03c0\u2082 : TaggedPrepartition I\nl\u271d l\u2081 l\u2082 l : IntegrationParams\nh : I \u2264 J\nc : \u211d\u22650\n\u03c0\u2080 : Prepartition J := Prepartition.single J I h\n\u22a2 Tendsto (\u2191(embedBox I J h)) (toFilterDistortioniUnion l I c \u22a4) (\u2a06 (c : \u211d\u22650), toFilterDistortioniUnion l J c \u03c0\u2080)\n[PROOFSTEP]\nrefine' le_iSup_of_le (max c \u03c0\u2080.compl.distortion) _\n[GOAL]\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\nI J : Box \u03b9\nc\u271d c\u2081 c\u2082 : \u211d\u22650\nr r\u2081 r\u2082 : (\u03b9 \u2192 \u211d) \u2192 \u2191(Set.Ioi 0)\n\u03c0 \u03c0\u2081 \u03c0\u2082 : TaggedPrepartition I\nl\u271d l\u2081 l\u2082 l : IntegrationParams\nh : I \u2264 J\nc : \u211d\u22650\n\u03c0\u2080 : Prepartition J := Prepartition.single J I h\n\u22a2 Filter.map (\u2191(embedBox I J h)) (toFilterDistortioniUnion l I c \u22a4) \u2264\n toFilterDistortioniUnion l J (max c (Prepartition.distortion (Prepartition.compl \u03c0\u2080))) \u03c0\u2080\n[PROOFSTEP]\nrefine'\n ((l.hasBasis_toFilterDistortioniUnion I c \u22a4).tendsto_iff (l.hasBasis_toFilterDistortioniUnion J _ _)).2 fun r hr => _\n[GOAL]\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\nI J : Box \u03b9\nc\u271d c\u2081 c\u2082 : \u211d\u22650\nr\u271d r\u2081 r\u2082 : (\u03b9 \u2192 \u211d) \u2192 \u2191(Set.Ioi 0)\n\u03c0 \u03c0\u2081 \u03c0\u2082 : TaggedPrepartition I\nl\u271d l\u2081 l\u2082 l : IntegrationParams\nh : I \u2264 J\nc : \u211d\u22650\n\u03c0\u2080 : Prepartition J := Prepartition.single J I h\nr : (\u03b9 \u2192 \u211d) \u2192 \u2191(Set.Ioi 0)\nhr : RCond l r\n\u22a2 \u2203 ia,\n RCond l ia \u2227\n \u2200 (x : TaggedPrepartition I),\n x \u2208 {\u03c0 | MemBaseSet l I c ia \u03c0 \u2227 TaggedPrepartition.iUnion \u03c0 = Prepartition.iUnion \u22a4} \u2192\n \u2191(embedBox I J h) x \u2208\n {\u03c0 |\n MemBaseSet l J (max c (Prepartition.distortion (Prepartition.compl \u03c0\u2080))) r \u03c0 \u2227\n TaggedPrepartition.iUnion \u03c0 = Prepartition.iUnion \u03c0\u2080}\n[PROOFSTEP]\nrefine' \u27e8r, hr, fun \u03c0 h\u03c0 => _\u27e9\n[GOAL]\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\nI J : Box \u03b9\nc\u271d c\u2081 c\u2082 : \u211d\u22650\nr\u271d r\u2081 r\u2082 : (\u03b9 \u2192 \u211d) \u2192 \u2191(Set.Ioi 0)\n\u03c0\u271d \u03c0\u2081 \u03c0\u2082 : TaggedPrepartition I\nl\u271d l\u2081 l\u2082 l : IntegrationParams\nh : I \u2264 J\nc : \u211d\u22650\n\u03c0\u2080 : Prepartition J := Prepartition.single J I h\nr : (\u03b9 \u2192 \u211d) \u2192 \u2191(Set.Ioi 0)\nhr : RCond l r\n\u03c0 : TaggedPrepartition I\nh\u03c0 : \u03c0 \u2208 {\u03c0 | MemBaseSet l I c r \u03c0 \u2227 TaggedPrepartition.iUnion \u03c0 = Prepartition.iUnion \u22a4}\n\u22a2 \u2191(embedBox I J h) \u03c0 \u2208\n {\u03c0 |\n MemBaseSet l J (max c (Prepartition.distortion (Prepartition.compl \u03c0\u2080))) r \u03c0 \u2227\n TaggedPrepartition.iUnion \u03c0 = Prepartition.iUnion \u03c0\u2080}\n[PROOFSTEP]\nrw [mem_setOf_eq, Prepartition.iUnion_top] at h\u03c0 \n[GOAL]\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\nI J : Box \u03b9\nc\u271d c\u2081 c\u2082 : \u211d\u22650\nr\u271d r\u2081 r\u2082 : (\u03b9 \u2192 \u211d) \u2192 \u2191(Set.Ioi 0)\n\u03c0\u271d \u03c0\u2081 \u03c0\u2082 : TaggedPrepartition I\nl\u271d l\u2081 l\u2082 l : IntegrationParams\nh : I \u2264 J\nc : \u211d\u22650\n\u03c0\u2080 : Prepartition J := Prepartition.single J I h\nr : (\u03b9 \u2192 \u211d) \u2192 \u2191(Set.Ioi 0)\nhr : RCond l r\n\u03c0 : TaggedPrepartition I\nh\u03c0 : MemBaseSet l I c r \u03c0 \u2227 TaggedPrepartition.iUnion \u03c0 = \u2191I\n\u22a2 \u2191(embedBox I J h) \u03c0 \u2208\n {\u03c0 |\n MemBaseSet l J (max c (Prepartition.distortion (Prepartition.compl \u03c0\u2080))) r \u03c0 \u2227\n TaggedPrepartition.iUnion \u03c0 = Prepartition.iUnion \u03c0\u2080}\n[PROOFSTEP]\nrefine' \u27e8\u27e8h\u03c0.1.1, h\u03c0.1.2, fun hD => le_trans (h\u03c0.1.3 hD) (le_max_left _ _), fun _ => _\u27e9, _\u27e9\n[GOAL]\ncase refine'_1\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\nI J : Box \u03b9\nc\u271d c\u2081 c\u2082 : \u211d\u22650\nr\u271d r\u2081 r\u2082 : (\u03b9 \u2192 \u211d) \u2192 \u2191(Set.Ioi 0)\n\u03c0\u271d \u03c0\u2081 \u03c0\u2082 : TaggedPrepartition I\nl\u271d l\u2081 l\u2082 l : IntegrationParams\nh : I \u2264 J\nc : \u211d\u22650\n\u03c0\u2080 : Prepartition J := Prepartition.single J I h\nr : (\u03b9 \u2192 \u211d) \u2192 \u2191(Set.Ioi 0)\nhr : RCond l r\n\u03c0 : TaggedPrepartition I\nh\u03c0 : MemBaseSet l I c r \u03c0 \u2227 TaggedPrepartition.iUnion \u03c0 = \u2191I\nx\u271d : l.bDistortion = true\n\u22a2 \u2203 \u03c0',\n Prepartition.iUnion \u03c0' = \u2191J \\ TaggedPrepartition.iUnion (\u2191(embedBox I J h) \u03c0) \u2227\n Prepartition.distortion \u03c0' \u2264 max c (Prepartition.distortion (Prepartition.compl \u03c0\u2080))\n[PROOFSTEP]\nrefine' \u27e8_, \u03c0\u2080.iUnion_compl.trans _, le_max_right _ _\u27e9\n[GOAL]\ncase refine'_1\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\nI J : Box \u03b9\nc\u271d c\u2081 c\u2082 : \u211d\u22650\nr\u271d r\u2081 r\u2082 : (\u03b9 \u2192 \u211d) \u2192 \u2191(Set.Ioi 0)\n\u03c0\u271d \u03c0\u2081 \u03c0\u2082 : TaggedPrepartition I\nl\u271d l\u2081 l\u2082 l : IntegrationParams\nh : I \u2264 J\nc : \u211d\u22650\n\u03c0\u2080 : Prepartition J := Prepartition.single J I h\nr : (\u03b9 \u2192 \u211d) \u2192 \u2191(Set.Ioi 0)\nhr : RCond l r\n\u03c0 : TaggedPrepartition I\nh\u03c0 : MemBaseSet l I c r \u03c0 \u2227 TaggedPrepartition.iUnion \u03c0 = \u2191I\nx\u271d : l.bDistortion = true\n\u22a2 \u2191J \\ Prepartition.iUnion \u03c0\u2080 = \u2191J \\ TaggedPrepartition.iUnion (\u2191(embedBox I J h) \u03c0)\n[PROOFSTEP]\ncongr 1\n[GOAL]\ncase refine'_1.e_a\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\nI J : Box \u03b9\nc\u271d c\u2081 c\u2082 : \u211d\u22650\nr\u271d r\u2081 r\u2082 : (\u03b9 \u2192 \u211d) \u2192 \u2191(Set.Ioi 0)\n\u03c0\u271d \u03c0\u2081 \u03c0\u2082 : TaggedPrepartition I\nl\u271d l\u2081 l\u2082 l : IntegrationParams\nh : I \u2264 J\nc : \u211d\u22650\n\u03c0\u2080 : Prepartition J := Prepartition.single J I h\nr : (\u03b9 \u2192 \u211d) \u2192 \u2191(Set.Ioi 0)\nhr : RCond l r\n\u03c0 : TaggedPrepartition I\nh\u03c0 : MemBaseSet l I c r \u03c0 \u2227 TaggedPrepartition.iUnion \u03c0 = \u2191I\nx\u271d : l.bDistortion = true\n\u22a2 Prepartition.iUnion \u03c0\u2080 = TaggedPrepartition.iUnion (\u2191(embedBox I J h) \u03c0)\n[PROOFSTEP]\nexact (Prepartition.iUnion_single h).trans h\u03c0.2.symm\n[GOAL]\ncase refine'_2\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\nI J : Box \u03b9\nc\u271d c\u2081 c\u2082 : \u211d\u22650\nr\u271d r\u2081 r\u2082 : (\u03b9 \u2192 \u211d) \u2192 \u2191(Set.Ioi 0)\n\u03c0\u271d \u03c0\u2081 \u03c0\u2082 : TaggedPrepartition I\nl\u271d l\u2081 l\u2082 l : IntegrationParams\nh : I \u2264 J\nc : \u211d\u22650\n\u03c0\u2080 : Prepartition J := Prepartition.single J I h\nr : (\u03b9 \u2192 \u211d) \u2192 \u2191(Set.Ioi 0)\nhr : RCond l r\n\u03c0 : TaggedPrepartition I\nh\u03c0 : MemBaseSet l I c r \u03c0 \u2227 TaggedPrepartition.iUnion \u03c0 = \u2191I\n\u22a2 TaggedPrepartition.iUnion (\u2191(embedBox I J h) \u03c0) = Prepartition.iUnion \u03c0\u2080\n[PROOFSTEP]\nexact h\u03c0.2.trans (Prepartition.iUnion_single _).symm\n[GOAL]\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\nI J : Box \u03b9\nc c\u2081 c\u2082 : \u211d\u22650\nr\u271d r\u2081 r\u2082 : (\u03b9 \u2192 \u211d) \u2192 \u2191(Set.Ioi 0)\n\u03c0 \u03c0\u2081 \u03c0\u2082 : TaggedPrepartition I\nl\u271d l\u2081 l\u2082 l : IntegrationParams\n\u03c0\u2080 : Prepartition I\nhc\u2081 : Prepartition.distortion \u03c0\u2080 \u2264 c\nhc\u2082 : Prepartition.distortion (Prepartition.compl \u03c0\u2080) \u2264 c\nr : (\u03b9 \u2192 \u211d) \u2192 \u2191(Set.Ioi 0)\n\u22a2 \u2203 \u03c0, MemBaseSet l I c r \u03c0 \u2227 \u03c0.toPrepartition \u2264 \u03c0\u2080 \u2227 TaggedPrepartition.iUnion \u03c0 = Prepartition.iUnion \u03c0\u2080\n[PROOFSTEP]\nrcases \u03c0\u2080.exists_tagged_le_isHenstock_isSubordinate_iUnion_eq r with \u27e8\u03c0, hle, hH, hr, hd, hU\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\nI J : Box \u03b9\nc c\u2081 c\u2082 : \u211d\u22650\nr\u271d r\u2081 r\u2082 : (\u03b9 \u2192 \u211d) \u2192 \u2191(Set.Ioi 0)\n\u03c0\u271d \u03c0\u2081 \u03c0\u2082 : TaggedPrepartition I\nl\u271d l\u2081 l\u2082 l : IntegrationParams\n\u03c0\u2080 : Prepartition I\nhc\u2081 : Prepartition.distortion \u03c0\u2080 \u2264 c\nhc\u2082 : Prepartition.distortion (Prepartition.compl \u03c0\u2080) \u2264 c\nr : (\u03b9 \u2192 \u211d) \u2192 \u2191(Set.Ioi 0)\n\u03c0 : TaggedPrepartition I\nhle : \u03c0.toPrepartition \u2264 \u03c0\u2080\nhH : IsHenstock \u03c0\nhr : IsSubordinate \u03c0 r\nhd : distortion \u03c0 = Prepartition.distortion \u03c0\u2080\nhU : TaggedPrepartition.iUnion \u03c0 = Prepartition.iUnion \u03c0\u2080\n\u22a2 \u2203 \u03c0, MemBaseSet l I c r \u03c0 \u2227 \u03c0.toPrepartition \u2264 \u03c0\u2080 \u2227 TaggedPrepartition.iUnion \u03c0 = Prepartition.iUnion \u03c0\u2080\n[PROOFSTEP]\nrefine' \u27e8\u03c0, \u27e8hr, fun _ => hH, fun _ => hd.trans_le hc\u2081, fun _ => \u27e8\u03c0\u2080.compl, _, hc\u2082\u27e9\u27e9, \u27e8hle, hU\u27e9\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\nI J : Box \u03b9\nc c\u2081 c\u2082 : \u211d\u22650\nr\u271d r\u2081 r\u2082 : (\u03b9 \u2192 \u211d) \u2192 \u2191(Set.Ioi 0)\n\u03c0\u271d \u03c0\u2081 \u03c0\u2082 : TaggedPrepartition I\nl\u271d l\u2081 l\u2082 l : IntegrationParams\n\u03c0\u2080 : Prepartition I\nhc\u2081 : Prepartition.distortion \u03c0\u2080 \u2264 c\nhc\u2082 : Prepartition.distortion (Prepartition.compl \u03c0\u2080) \u2264 c\nr : (\u03b9 \u2192 \u211d) \u2192 \u2191(Set.Ioi 0)\n\u03c0 : TaggedPrepartition I\nhle : \u03c0.toPrepartition \u2264 \u03c0\u2080\nhH : IsHenstock \u03c0\nhr : IsSubordinate \u03c0 r\nhd : distortion \u03c0 = Prepartition.distortion \u03c0\u2080\nhU : TaggedPrepartition.iUnion \u03c0 = Prepartition.iUnion \u03c0\u2080\nx\u271d : l.bDistortion = true\n\u22a2 Prepartition.iUnion (Prepartition.compl \u03c0\u2080) = \u2191I \\ TaggedPrepartition.iUnion \u03c0\n[PROOFSTEP]\nexact Prepartition.compl_congr hU \u25b8 \u03c0.toPrepartition.iUnion_compl\n[GOAL]\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\nI\u271d J : Box \u03b9\nc c\u2081 c\u2082 : \u211d\u22650\nr\u271d r\u2081 r\u2082 : (\u03b9 \u2192 \u211d) \u2192 \u2191(Set.Ioi 0)\n\u03c0 \u03c0\u2081 \u03c0\u2082 : TaggedPrepartition I\u271d\nl\u271d l\u2081 l\u2082 l : IntegrationParams\nI : Box \u03b9\nhc : Box.distortion I \u2264 c\nr : (\u03b9 \u2192 \u211d) \u2192 \u2191(Set.Ioi 0)\n\u22a2 \u2203 \u03c0, MemBaseSet l I c r \u03c0 \u2227 IsPartition \u03c0\n[PROOFSTEP]\nrw [\u2190 Prepartition.distortion_top] at hc \n[GOAL]\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\nI\u271d J : Box \u03b9\nc c\u2081 c\u2082 : \u211d\u22650\nr\u271d r\u2081 r\u2082 : (\u03b9 \u2192 \u211d) \u2192 \u2191(Set.Ioi 0)\n\u03c0 \u03c0\u2081 \u03c0\u2082 : TaggedPrepartition I\u271d\nl\u271d l\u2081 l\u2082 l : IntegrationParams\nI : Box \u03b9\nhc : Prepartition.distortion \u22a4 \u2264 c\nr : (\u03b9 \u2192 \u211d) \u2192 \u2191(Set.Ioi 0)\n\u22a2 \u2203 \u03c0, MemBaseSet l I c r \u03c0 \u2227 IsPartition \u03c0\n[PROOFSTEP]\nhave hc' : (\u22a4 : Prepartition I).compl.distortion \u2264 c := by simp\n[GOAL]\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\nI\u271d J : Box \u03b9\nc c\u2081 c\u2082 : \u211d\u22650\nr\u271d r\u2081 r\u2082 : (\u03b9 \u2192 \u211d) \u2192 \u2191(Set.Ioi 0)\n\u03c0 \u03c0\u2081 \u03c0\u2082 : TaggedPrepartition I\u271d\nl\u271d l\u2081 l\u2082 l : IntegrationParams\nI : Box \u03b9\nhc : Prepartition.distortion \u22a4 \u2264 c\nr : (\u03b9 \u2192 \u211d) \u2192 \u2191(Set.Ioi 0)\n\u22a2 Prepartition.distortion (Prepartition.compl \u22a4) \u2264 c\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\nI\u271d J : Box \u03b9\nc c\u2081 c\u2082 : \u211d\u22650\nr\u271d r\u2081 r\u2082 : (\u03b9 \u2192 \u211d) \u2192 \u2191(Set.Ioi 0)\n\u03c0 \u03c0\u2081 \u03c0\u2082 : TaggedPrepartition I\u271d\nl\u271d l\u2081 l\u2082 l : IntegrationParams\nI : Box \u03b9\nhc : Prepartition.distortion \u22a4 \u2264 c\nr : (\u03b9 \u2192 \u211d) \u2192 \u2191(Set.Ioi 0)\nhc' : Prepartition.distortion (Prepartition.compl \u22a4) \u2264 c\n\u22a2 \u2203 \u03c0, MemBaseSet l I c r \u03c0 \u2227 IsPartition \u03c0\n[PROOFSTEP]\nsimpa [isPartition_iff_iUnion_eq] using l.exists_memBaseSet_le_iUnion_eq \u22a4 hc hc' r\n[GOAL]\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\nI\u271d J : Box \u03b9\nc c\u2081 c\u2082 : \u211d\u22650\nr r\u2081 r\u2082 : (\u03b9 \u2192 \u211d) \u2192 \u2191(Set.Ioi 0)\n\u03c0 \u03c0\u2081 \u03c0\u2082 : TaggedPrepartition I\u271d\nl\u271d l\u2081 l\u2082 l : IntegrationParams\nI : Box \u03b9\n\u22a2 NeBot (toFilterDistortion l I (Box.distortion I))\n[PROOFSTEP]\nsimpa using (l.toFilterDistortioniUnion_neBot' I \u22a4).mono inf_le_left\n", "meta": {"mathlib_filename": "Mathlib.Analysis.BoxIntegral.Partition.Filter", "llama_tokens": 17831, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.5234203340678567, "lm_q2_score": 0.3174262655876758, "lm_q1q2_score": 0.16614736197581348}}