| {"text": "[GOAL]\nα β : FinBddDistLatCat\ne : ↑α.toBddDistLatCat.toDistLatCat ≃o ↑β.toBddDistLatCat.toDistLatCat\n⊢ ((let src :=\n {\n toSupHom :=\n { toFun := ↑e, map_sup' := (_ : ∀ (a b : ↑α.toBddDistLatCat.toDistLatCat), ↑e (a ⊔ b) = ↑e a ⊔ ↑e b) },\n map_inf' := (_ : ∀ (a b : ↑α.toBddDistLatCat.toDistLatCat), ↑e (a ⊓ b) = ↑e a ⊓ ↑e b) };\n {\n toLatticeHom :=\n {\n toSupHom :=\n { toFun := ↑e,\n map_sup' :=\n (_ :\n ∀ (a b : ↑α.toBddDistLatCat.toDistLatCat),\n SupHom.toFun\n {\n toSupHom :=\n { toFun := ↑e,\n map_sup' :=\n (_ : ∀ (a b : ↑α.toBddDistLatCat.toDistLatCat), ↑e (a ⊔ b) = ↑e a ⊔ ↑e b) },\n map_inf' :=\n (_ : ∀ (a b : ↑α.toBddDistLatCat.toDistLatCat), ↑e (a ⊓ b) = ↑e a ⊓ ↑e b) }.toSupHom\n (a ⊔ b) =\n SupHom.toFun\n {\n toSupHom :=\n { toFun := ↑e,\n map_sup' :=\n (_ : ∀ (a b : ↑α.toBddDistLatCat.toDistLatCat), ↑e (a ⊔ b) = ↑e a ⊔ ↑e b) },\n map_inf' :=\n (_ : ∀ (a b : ↑α.toBddDistLatCat.toDistLatCat), ↑e (a ⊓ b) = ↑e a ⊓ ↑e b) }.toSupHom\n a ⊔\n SupHom.toFun\n {\n toSupHom :=\n { toFun := ↑e,\n map_sup' :=\n (_ : ∀ (a b : ↑α.toBddDistLatCat.toDistLatCat), ↑e (a ⊔ b) = ↑e a ⊔ ↑e b) },\n map_inf' :=\n (_ : ∀ (a b : ↑α.toBddDistLatCat.toDistLatCat), ↑e (a ⊓ b) = ↑e a ⊓ ↑e b) }.toSupHom\n b) },\n map_inf' :=\n (_ :\n ∀ (a b : ↑α.toBddDistLatCat.toDistLatCat),\n SupHom.toFun src.toSupHom (a ⊓ b) = SupHom.toFun src.toSupHom a ⊓ SupHom.toFun src.toSupHom b) },\n map_top' := (_ : ↑e ⊤ = ⊤), map_bot' := (_ : ↑e ⊥ = ⊥) }) ≫\n let src :=\n {\n toSupHom :=\n { toFun := ↑(OrderIso.symm e),\n map_sup' :=\n (_ :\n ∀ (a b : ↑β.toBddDistLatCat.toDistLatCat),\n ↑(OrderIso.symm e) (a ⊔ b) = ↑(OrderIso.symm e) a ⊔ ↑(OrderIso.symm e) b) },\n map_inf' :=\n (_ :\n ∀ (a b : ↑β.toBddDistLatCat.toDistLatCat),\n ↑(OrderIso.symm e) (a ⊓ b) = ↑(OrderIso.symm e) a ⊓ ↑(OrderIso.symm e) b) };\n {\n toLatticeHom :=\n {\n toSupHom :=\n { toFun := ↑(OrderIso.symm e),\n map_sup' :=\n (_ :\n ∀ (a b : ↑β.toBddDistLatCat.toDistLatCat),\n SupHom.toFun\n {\n toSupHom :=\n { toFun := ↑(OrderIso.symm e),\n map_sup' :=\n (_ :\n ∀ (a b : ↑β.toBddDistLatCat.toDistLatCat),\n ↑(OrderIso.symm e) (a ⊔ b) = ↑(OrderIso.symm e) a ⊔ ↑(OrderIso.symm e) b) },\n map_inf' :=\n (_ :\n ∀ (a b : ↑β.toBddDistLatCat.toDistLatCat),\n ↑(OrderIso.symm e) (a ⊓ b) = ↑(OrderIso.symm e) a ⊓ ↑(OrderIso.symm e) b) }.toSupHom\n (a ⊔ b) =\n SupHom.toFun\n {\n toSupHom :=\n { toFun := ↑(OrderIso.symm e),\n map_sup' :=\n (_ :\n ∀ (a b : ↑β.toBddDistLatCat.toDistLatCat),\n ↑(OrderIso.symm e) (a ⊔ b) = ↑(OrderIso.symm e) a ⊔ ↑(OrderIso.symm e) b) },\n map_inf' :=\n (_ :\n ∀ (a b : ↑β.toBddDistLatCat.toDistLatCat),\n ↑(OrderIso.symm e) (a ⊓ b) =\n ↑(OrderIso.symm e) a ⊓ ↑(OrderIso.symm e) b) }.toSupHom\n a ⊔\n SupHom.toFun\n {\n toSupHom :=\n { toFun := ↑(OrderIso.symm e),\n map_sup' :=\n (_ :\n ∀ (a b : ↑β.toBddDistLatCat.toDistLatCat),\n ↑(OrderIso.symm e) (a ⊔ b) = ↑(OrderIso.symm e) a ⊔ ↑(OrderIso.symm e) b) },\n map_inf' :=\n (_ :\n ∀ (a b : ↑β.toBddDistLatCat.toDistLatCat),\n ↑(OrderIso.symm e) (a ⊓ b) =\n ↑(OrderIso.symm e) a ⊓ ↑(OrderIso.symm e) b) }.toSupHom\n b) },\n map_inf' :=\n (_ :\n ∀ (a b : ↑β.toBddDistLatCat.toDistLatCat),\n SupHom.toFun src.toSupHom (a ⊓ b) = SupHom.toFun src.toSupHom a ⊓ SupHom.toFun src.toSupHom b) },\n map_top' := (_ : ↑(OrderIso.symm e) ⊤ = ⊤), map_bot' := (_ : ↑(OrderIso.symm e) ⊥ = ⊥) }) =\n 𝟙 α\n[PROOFSTEP]\next\n[GOAL]\ncase w\nα β : FinBddDistLatCat\ne : ↑α.toBddDistLatCat.toDistLatCat ≃o ↑β.toBddDistLatCat.toDistLatCat\nx✝ : (forget FinBddDistLatCat).obj α\n⊢ ↑((let src :=\n {\n toSupHom :=\n { toFun := ↑e, map_sup' := (_ : ∀ (a b : ↑α.toBddDistLatCat.toDistLatCat), ↑e (a ⊔ b) = ↑e a ⊔ ↑e b) },\n map_inf' := (_ : ∀ (a b : ↑α.toBddDistLatCat.toDistLatCat), ↑e (a ⊓ b) = ↑e a ⊓ ↑e b) };\n {\n toLatticeHom :=\n {\n toSupHom :=\n { toFun := ↑e,\n map_sup' :=\n (_ :\n ∀ (a b : ↑α.toBddDistLatCat.toDistLatCat),\n SupHom.toFun\n {\n toSupHom :=\n { toFun := ↑e,\n map_sup' :=\n (_ : ∀ (a b : ↑α.toBddDistLatCat.toDistLatCat), ↑e (a ⊔ b) = ↑e a ⊔ ↑e b) },\n map_inf' :=\n (_ : ∀ (a b : ↑α.toBddDistLatCat.toDistLatCat), ↑e (a ⊓ b) = ↑e a ⊓ ↑e b) }.toSupHom\n (a ⊔ b) =\n SupHom.toFun\n {\n toSupHom :=\n { toFun := ↑e,\n map_sup' :=\n (_ : ∀ (a b : ↑α.toBddDistLatCat.toDistLatCat), ↑e (a ⊔ b) = ↑e a ⊔ ↑e b) },\n map_inf' :=\n (_ :\n ∀ (a b : ↑α.toBddDistLatCat.toDistLatCat), ↑e (a ⊓ b) = ↑e a ⊓ ↑e b) }.toSupHom\n a ⊔\n SupHom.toFun\n {\n toSupHom :=\n { toFun := ↑e,\n map_sup' :=\n (_ : ∀ (a b : ↑α.toBddDistLatCat.toDistLatCat), ↑e (a ⊔ b) = ↑e a ⊔ ↑e b) },\n map_inf' :=\n (_ :\n ∀ (a b : ↑α.toBddDistLatCat.toDistLatCat), ↑e (a ⊓ b) = ↑e a ⊓ ↑e b) }.toSupHom\n b) },\n map_inf' :=\n (_ :\n ∀ (a b : ↑α.toBddDistLatCat.toDistLatCat),\n SupHom.toFun src.toSupHom (a ⊓ b) = SupHom.toFun src.toSupHom a ⊓ SupHom.toFun src.toSupHom b) },\n map_top' := (_ : ↑e ⊤ = ⊤), map_bot' := (_ : ↑e ⊥ = ⊥) }) ≫\n let src :=\n {\n toSupHom :=\n { toFun := ↑(OrderIso.symm e),\n map_sup' :=\n (_ :\n ∀ (a b : ↑β.toBddDistLatCat.toDistLatCat),\n ↑(OrderIso.symm e) (a ⊔ b) = ↑(OrderIso.symm e) a ⊔ ↑(OrderIso.symm e) b) },\n map_inf' :=\n (_ :\n ∀ (a b : ↑β.toBddDistLatCat.toDistLatCat),\n ↑(OrderIso.symm e) (a ⊓ b) = ↑(OrderIso.symm e) a ⊓ ↑(OrderIso.symm e) b) };\n {\n toLatticeHom :=\n {\n toSupHom :=\n { toFun := ↑(OrderIso.symm e),\n map_sup' :=\n (_ :\n ∀ (a b : ↑β.toBddDistLatCat.toDistLatCat),\n SupHom.toFun\n {\n toSupHom :=\n { toFun := ↑(OrderIso.symm e),\n map_sup' :=\n (_ :\n ∀ (a b : ↑β.toBddDistLatCat.toDistLatCat),\n ↑(OrderIso.symm e) (a ⊔ b) = ↑(OrderIso.symm e) a ⊔ ↑(OrderIso.symm e) b) },\n map_inf' :=\n (_ :\n ∀ (a b : ↑β.toBddDistLatCat.toDistLatCat),\n ↑(OrderIso.symm e) (a ⊓ b) =\n ↑(OrderIso.symm e) a ⊓ ↑(OrderIso.symm e) b) }.toSupHom\n (a ⊔ b) =\n SupHom.toFun\n {\n toSupHom :=\n { toFun := ↑(OrderIso.symm e),\n map_sup' :=\n (_ :\n ∀ (a b : ↑β.toBddDistLatCat.toDistLatCat),\n ↑(OrderIso.symm e) (a ⊔ b) =\n ↑(OrderIso.symm e) a ⊔ ↑(OrderIso.symm e) b) },\n map_inf' :=\n (_ :\n ∀ (a b : ↑β.toBddDistLatCat.toDistLatCat),\n ↑(OrderIso.symm e) (a ⊓ b) =\n ↑(OrderIso.symm e) a ⊓ ↑(OrderIso.symm e) b) }.toSupHom\n a ⊔\n SupHom.toFun\n {\n toSupHom :=\n { toFun := ↑(OrderIso.symm e),\n map_sup' :=\n (_ :\n ∀ (a b : ↑β.toBddDistLatCat.toDistLatCat),\n ↑(OrderIso.symm e) (a ⊔ b) =\n ↑(OrderIso.symm e) a ⊔ ↑(OrderIso.symm e) b) },\n map_inf' :=\n (_ :\n ∀ (a b : ↑β.toBddDistLatCat.toDistLatCat),\n ↑(OrderIso.symm e) (a ⊓ b) =\n ↑(OrderIso.symm e) a ⊓ ↑(OrderIso.symm e) b) }.toSupHom\n b) },\n map_inf' :=\n (_ :\n ∀ (a b : ↑β.toBddDistLatCat.toDistLatCat),\n SupHom.toFun src.toSupHom (a ⊓ b) = SupHom.toFun src.toSupHom a ⊓ SupHom.toFun src.toSupHom b) },\n map_top' := (_ : ↑(OrderIso.symm e) ⊤ = ⊤), map_bot' := (_ : ↑(OrderIso.symm e) ⊥ = ⊥) })\n x✝ =\n ↑(𝟙 α) x✝\n[PROOFSTEP]\nexact e.symm_apply_apply _\n[GOAL]\nα β : FinBddDistLatCat\ne : ↑α.toBddDistLatCat.toDistLatCat ≃o ↑β.toBddDistLatCat.toDistLatCat\n⊢ ((let src :=\n {\n toSupHom :=\n { toFun := ↑(OrderIso.symm e),\n map_sup' :=\n (_ :\n ∀ (a b : ↑β.toBddDistLatCat.toDistLatCat),\n ↑(OrderIso.symm e) (a ⊔ b) = ↑(OrderIso.symm e) a ⊔ ↑(OrderIso.symm e) b) },\n map_inf' :=\n (_ :\n ∀ (a b : ↑β.toBddDistLatCat.toDistLatCat),\n ↑(OrderIso.symm e) (a ⊓ b) = ↑(OrderIso.symm e) a ⊓ ↑(OrderIso.symm e) b) };\n {\n toLatticeHom :=\n {\n toSupHom :=\n { toFun := ↑(OrderIso.symm e),\n map_sup' :=\n (_ :\n ∀ (a b : ↑β.toBddDistLatCat.toDistLatCat),\n SupHom.toFun\n {\n toSupHom :=\n { toFun := ↑(OrderIso.symm e),\n map_sup' :=\n (_ :\n ∀ (a b : ↑β.toBddDistLatCat.toDistLatCat),\n ↑(OrderIso.symm e) (a ⊔ b) = ↑(OrderIso.symm e) a ⊔ ↑(OrderIso.symm e) b) },\n map_inf' :=\n (_ :\n ∀ (a b : ↑β.toBddDistLatCat.toDistLatCat),\n ↑(OrderIso.symm e) (a ⊓ b) = ↑(OrderIso.symm e) a ⊓ ↑(OrderIso.symm e) b) }.toSupHom\n (a ⊔ b) =\n SupHom.toFun\n {\n toSupHom :=\n { toFun := ↑(OrderIso.symm e),\n map_sup' :=\n (_ :\n ∀ (a b : ↑β.toBddDistLatCat.toDistLatCat),\n ↑(OrderIso.symm e) (a ⊔ b) = ↑(OrderIso.symm e) a ⊔ ↑(OrderIso.symm e) b) },\n map_inf' :=\n (_ :\n ∀ (a b : ↑β.toBddDistLatCat.toDistLatCat),\n ↑(OrderIso.symm e) (a ⊓ b) =\n ↑(OrderIso.symm e) a ⊓ ↑(OrderIso.symm e) b) }.toSupHom\n a ⊔\n SupHom.toFun\n {\n toSupHom :=\n { toFun := ↑(OrderIso.symm e),\n map_sup' :=\n (_ :\n ∀ (a b : ↑β.toBddDistLatCat.toDistLatCat),\n ↑(OrderIso.symm e) (a ⊔ b) = ↑(OrderIso.symm e) a ⊔ ↑(OrderIso.symm e) b) },\n map_inf' :=\n (_ :\n ∀ (a b : ↑β.toBddDistLatCat.toDistLatCat),\n ↑(OrderIso.symm e) (a ⊓ b) =\n ↑(OrderIso.symm e) a ⊓ ↑(OrderIso.symm e) b) }.toSupHom\n b) },\n map_inf' :=\n (_ :\n ∀ (a b : ↑β.toBddDistLatCat.toDistLatCat),\n SupHom.toFun src.toSupHom (a ⊓ b) = SupHom.toFun src.toSupHom a ⊓ SupHom.toFun src.toSupHom b) },\n map_top' := (_ : ↑(OrderIso.symm e) ⊤ = ⊤), map_bot' := (_ : ↑(OrderIso.symm e) ⊥ = ⊥) }) ≫\n let src :=\n {\n toSupHom :=\n { toFun := ↑e, map_sup' := (_ : ∀ (a b : ↑α.toBddDistLatCat.toDistLatCat), ↑e (a ⊔ b) = ↑e a ⊔ ↑e b) },\n map_inf' := (_ : ∀ (a b : ↑α.toBddDistLatCat.toDistLatCat), ↑e (a ⊓ b) = ↑e a ⊓ ↑e b) };\n {\n toLatticeHom :=\n {\n toSupHom :=\n { toFun := ↑e,\n map_sup' :=\n (_ :\n ∀ (a b : ↑α.toBddDistLatCat.toDistLatCat),\n SupHom.toFun\n {\n toSupHom :=\n { toFun := ↑e,\n map_sup' :=\n (_ : ∀ (a b : ↑α.toBddDistLatCat.toDistLatCat), ↑e (a ⊔ b) = ↑e a ⊔ ↑e b) },\n map_inf' :=\n (_ : ∀ (a b : ↑α.toBddDistLatCat.toDistLatCat), ↑e (a ⊓ b) = ↑e a ⊓ ↑e b) }.toSupHom\n (a ⊔ b) =\n SupHom.toFun\n {\n toSupHom :=\n { toFun := ↑e,\n map_sup' :=\n (_ : ∀ (a b : ↑α.toBddDistLatCat.toDistLatCat), ↑e (a ⊔ b) = ↑e a ⊔ ↑e b) },\n map_inf' :=\n (_ : ∀ (a b : ↑α.toBddDistLatCat.toDistLatCat), ↑e (a ⊓ b) = ↑e a ⊓ ↑e b) }.toSupHom\n a ⊔\n SupHom.toFun\n {\n toSupHom :=\n { toFun := ↑e,\n map_sup' :=\n (_ : ∀ (a b : ↑α.toBddDistLatCat.toDistLatCat), ↑e (a ⊔ b) = ↑e a ⊔ ↑e b) },\n map_inf' :=\n (_ : ∀ (a b : ↑α.toBddDistLatCat.toDistLatCat), ↑e (a ⊓ b) = ↑e a ⊓ ↑e b) }.toSupHom\n b) },\n map_inf' :=\n (_ :\n ∀ (a b : ↑α.toBddDistLatCat.toDistLatCat),\n SupHom.toFun src.toSupHom (a ⊓ b) = SupHom.toFun src.toSupHom a ⊓ SupHom.toFun src.toSupHom b) },\n map_top' := (_ : ↑e ⊤ = ⊤), map_bot' := (_ : ↑e ⊥ = ⊥) }) =\n 𝟙 β\n[PROOFSTEP]\next\n[GOAL]\ncase w\nα β : FinBddDistLatCat\ne : ↑α.toBddDistLatCat.toDistLatCat ≃o ↑β.toBddDistLatCat.toDistLatCat\nx✝ : (forget FinBddDistLatCat).obj β\n⊢ ↑((let src :=\n {\n toSupHom :=\n { toFun := ↑(OrderIso.symm e),\n map_sup' :=\n (_ :\n ∀ (a b : ↑β.toBddDistLatCat.toDistLatCat),\n ↑(OrderIso.symm e) (a ⊔ b) = ↑(OrderIso.symm e) a ⊔ ↑(OrderIso.symm e) b) },\n map_inf' :=\n (_ :\n ∀ (a b : ↑β.toBddDistLatCat.toDistLatCat),\n ↑(OrderIso.symm e) (a ⊓ b) = ↑(OrderIso.symm e) a ⊓ ↑(OrderIso.symm e) b) };\n {\n toLatticeHom :=\n {\n toSupHom :=\n { toFun := ↑(OrderIso.symm e),\n map_sup' :=\n (_ :\n ∀ (a b : ↑β.toBddDistLatCat.toDistLatCat),\n SupHom.toFun\n {\n toSupHom :=\n { toFun := ↑(OrderIso.symm e),\n map_sup' :=\n (_ :\n ∀ (a b : ↑β.toBddDistLatCat.toDistLatCat),\n ↑(OrderIso.symm e) (a ⊔ b) = ↑(OrderIso.symm e) a ⊔ ↑(OrderIso.symm e) b) },\n map_inf' :=\n (_ :\n ∀ (a b : ↑β.toBddDistLatCat.toDistLatCat),\n ↑(OrderIso.symm e) (a ⊓ b) =\n ↑(OrderIso.symm e) a ⊓ ↑(OrderIso.symm e) b) }.toSupHom\n (a ⊔ b) =\n SupHom.toFun\n {\n toSupHom :=\n { toFun := ↑(OrderIso.symm e),\n map_sup' :=\n (_ :\n ∀ (a b : ↑β.toBddDistLatCat.toDistLatCat),\n ↑(OrderIso.symm e) (a ⊔ b) =\n ↑(OrderIso.symm e) a ⊔ ↑(OrderIso.symm e) b) },\n map_inf' :=\n (_ :\n ∀ (a b : ↑β.toBddDistLatCat.toDistLatCat),\n ↑(OrderIso.symm e) (a ⊓ b) =\n ↑(OrderIso.symm e) a ⊓ ↑(OrderIso.symm e) b) }.toSupHom\n a ⊔\n SupHom.toFun\n {\n toSupHom :=\n { toFun := ↑(OrderIso.symm e),\n map_sup' :=\n (_ :\n ∀ (a b : ↑β.toBddDistLatCat.toDistLatCat),\n ↑(OrderIso.symm e) (a ⊔ b) =\n ↑(OrderIso.symm e) a ⊔ ↑(OrderIso.symm e) b) },\n map_inf' :=\n (_ :\n ∀ (a b : ↑β.toBddDistLatCat.toDistLatCat),\n ↑(OrderIso.symm e) (a ⊓ b) =\n ↑(OrderIso.symm e) a ⊓ ↑(OrderIso.symm e) b) }.toSupHom\n b) },\n map_inf' :=\n (_ :\n ∀ (a b : ↑β.toBddDistLatCat.toDistLatCat),\n SupHom.toFun src.toSupHom (a ⊓ b) = SupHom.toFun src.toSupHom a ⊓ SupHom.toFun src.toSupHom b) },\n map_top' := (_ : ↑(OrderIso.symm e) ⊤ = ⊤), map_bot' := (_ : ↑(OrderIso.symm e) ⊥ = ⊥) }) ≫\n let src :=\n {\n toSupHom :=\n { toFun := ↑e, map_sup' := (_ : ∀ (a b : ↑α.toBddDistLatCat.toDistLatCat), ↑e (a ⊔ b) = ↑e a ⊔ ↑e b) },\n map_inf' := (_ : ∀ (a b : ↑α.toBddDistLatCat.toDistLatCat), ↑e (a ⊓ b) = ↑e a ⊓ ↑e b) };\n {\n toLatticeHom :=\n {\n toSupHom :=\n { toFun := ↑e,\n map_sup' :=\n (_ :\n ∀ (a b : ↑α.toBddDistLatCat.toDistLatCat),\n SupHom.toFun\n {\n toSupHom :=\n { toFun := ↑e,\n map_sup' :=\n (_ : ∀ (a b : ↑α.toBddDistLatCat.toDistLatCat), ↑e (a ⊔ b) = ↑e a ⊔ ↑e b) },\n map_inf' :=\n (_ : ∀ (a b : ↑α.toBddDistLatCat.toDistLatCat), ↑e (a ⊓ b) = ↑e a ⊓ ↑e b) }.toSupHom\n (a ⊔ b) =\n SupHom.toFun\n {\n toSupHom :=\n { toFun := ↑e,\n map_sup' :=\n (_ : ∀ (a b : ↑α.toBddDistLatCat.toDistLatCat), ↑e (a ⊔ b) = ↑e a ⊔ ↑e b) },\n map_inf' :=\n (_ :\n ∀ (a b : ↑α.toBddDistLatCat.toDistLatCat), ↑e (a ⊓ b) = ↑e a ⊓ ↑e b) }.toSupHom\n a ⊔\n SupHom.toFun\n {\n toSupHom :=\n { toFun := ↑e,\n map_sup' :=\n (_ : ∀ (a b : ↑α.toBddDistLatCat.toDistLatCat), ↑e (a ⊔ b) = ↑e a ⊔ ↑e b) },\n map_inf' :=\n (_ :\n ∀ (a b : ↑α.toBddDistLatCat.toDistLatCat), ↑e (a ⊓ b) = ↑e a ⊓ ↑e b) }.toSupHom\n b) },\n map_inf' :=\n (_ :\n ∀ (a b : ↑α.toBddDistLatCat.toDistLatCat),\n SupHom.toFun src.toSupHom (a ⊓ b) = SupHom.toFun src.toSupHom a ⊓ SupHom.toFun src.toSupHom b) },\n map_top' := (_ : ↑e ⊤ = ⊤), map_bot' := (_ : ↑e ⊥ = ⊥) })\n x✝ =\n ↑(𝟙 β) x✝\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✝¹⁰ : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w₁\ninst✝⁹ : Category.{max v u, w₁} D\nE : Type w₂\ninst✝⁸ : Category.{max v u, w₂} E\nF : D ⥤ E\ninst✝⁷ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst✝⁶ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst✝⁵ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\ninst✝⁴ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ E\ninst✝³ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ F\ninst✝² : (X : C) → (W : Cover J X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\nP✝ P : Cᵒᵖ ⥤ D\ninst✝¹ : (F : D ⥤ E) → (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ F\ninst✝ :\n (F : D ⥤ E) →\n (X : C) → (W : Cover J X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\n⊢ (whiskeringLeft Cᵒᵖ D E).obj (sheafify J P) ≅ (whiskeringLeft Cᵒᵖ D E).obj P ⋙ sheafification J E\n[PROOFSTEP]\nrefine' J.plusFunctorWhiskerLeftIso _ ≪≫ _ ≪≫ Functor.associator _ _ _\n[GOAL]\nC : Type u\ninst✝¹⁰ : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w₁\ninst✝⁹ : Category.{max v u, w₁} D\nE : Type w₂\ninst✝⁸ : Category.{max v u, w₂} E\nF : D ⥤ E\ninst✝⁷ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst✝⁶ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst✝⁵ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\ninst✝⁴ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ E\ninst✝³ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ F\ninst✝² : (X : C) → (W : Cover J X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\nP✝ P : Cᵒᵖ ⥤ D\ninst✝¹ : (F : D ⥤ E) → (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ F\ninst✝ :\n (F : D ⥤ E) →\n (X : C) → (W : Cover J X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\n⊢ (whiskeringLeft Cᵒᵖ D E).obj (plusObj J P) ⋙ plusFunctor J E ≅\n ((whiskeringLeft Cᵒᵖ D E).obj P ⋙ plusFunctor J E) ⋙ plusFunctor J E\n[PROOFSTEP]\nrefine' isoWhiskerRight _ _\n[GOAL]\nC : Type u\ninst✝¹⁰ : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w₁\ninst✝⁹ : Category.{max v u, w₁} D\nE : Type w₂\ninst✝⁸ : Category.{max v u, w₂} E\nF : D ⥤ E\ninst✝⁷ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst✝⁶ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst✝⁵ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\ninst✝⁴ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ E\ninst✝³ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ F\ninst✝² : (X : C) → (W : Cover J X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\nP✝ P : Cᵒᵖ ⥤ D\ninst✝¹ : (F : D ⥤ E) → (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ F\ninst✝ :\n (F : D ⥤ E) →\n (X : C) → (W : Cover J X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\n⊢ (whiskeringLeft Cᵒᵖ D E).obj (plusObj J P) ≅ (whiskeringLeft Cᵒᵖ D E).obj P ⋙ plusFunctor J E\n[PROOFSTEP]\nrefine' J.plusFunctorWhiskerLeftIso _\n[GOAL]\nC : Type u\ninst✝¹⁰ : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w₁\ninst✝⁹ : Category.{max v u, w₁} D\nE : Type w₂\ninst✝⁸ : Category.{max v u, w₂} E\nF✝ : D ⥤ E\ninst✝⁷ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst✝⁶ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst✝⁵ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\ninst✝⁴ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ E\ninst✝³ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ F✝\ninst✝² : (X : C) → (W : Cover J X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F✝\nP✝ P : Cᵒᵖ ⥤ D\nF : D ⥤ E\ninst✝¹ : (F : D ⥤ E) → (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ F\ninst✝ :\n (F : D ⥤ E) →\n (X : C) → (W : Cover J X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\n⊢ NatTrans.app (sheafificationWhiskerLeftIso J P).hom F = (sheafifyCompIso J F P).hom\n[PROOFSTEP]\ndsimp [sheafificationWhiskerLeftIso, sheafifyCompIso]\n[GOAL]\nC : Type u\ninst✝¹⁰ : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w₁\ninst✝⁹ : Category.{max v u, w₁} D\nE : Type w₂\ninst✝⁸ : Category.{max v u, w₂} E\nF✝ : D ⥤ E\ninst✝⁷ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst✝⁶ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst✝⁵ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\ninst✝⁴ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ E\ninst✝³ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ F✝\ninst✝² : (X : C) → (W : Cover J X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F✝\nP✝ P : Cᵒᵖ ⥤ D\nF : D ⥤ E\ninst✝¹ : (F : D ⥤ E) → (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ F\ninst✝ :\n (F : D ⥤ E) →\n (X : C) → (W : Cover J X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\n⊢ (plusCompIso J F (plusObj J P)).hom ≫ plusMap J (plusCompIso J F P).hom ≫ 𝟙 (plusObj J (plusObj J (P ⋙ F))) =\n (plusCompIso J F (plusObj J P)).hom ≫ plusMap J (plusCompIso J F P).hom\n[PROOFSTEP]\nrw [Category.comp_id]\n[GOAL]\nC : Type u\ninst✝¹⁰ : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w₁\ninst✝⁹ : Category.{max v u, w₁} D\nE : Type w₂\ninst✝⁸ : Category.{max v u, w₂} E\nF✝ : D ⥤ E\ninst✝⁷ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst✝⁶ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst✝⁵ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\ninst✝⁴ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ E\ninst✝³ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ F✝\ninst✝² : (X : C) → (W : Cover J X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F✝\nP✝ P : Cᵒᵖ ⥤ D\nF : D ⥤ E\ninst✝¹ : (F : D ⥤ E) → (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ F\ninst✝ :\n (F : D ⥤ E) →\n (X : C) → (W : Cover J X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\n⊢ NatTrans.app (sheafificationWhiskerLeftIso J P).inv F = (sheafifyCompIso J F P).inv\n[PROOFSTEP]\ndsimp [sheafificationWhiskerLeftIso, sheafifyCompIso]\n[GOAL]\nC : Type u\ninst✝¹⁰ : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w₁\ninst✝⁹ : Category.{max v u, w₁} D\nE : Type w₂\ninst✝⁸ : Category.{max v u, w₂} E\nF✝ : D ⥤ E\ninst✝⁷ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst✝⁶ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst✝⁵ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\ninst✝⁴ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ E\ninst✝³ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ F✝\ninst✝² : (X : C) → (W : Cover J X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F✝\nP✝ P : Cᵒᵖ ⥤ D\nF : D ⥤ E\ninst✝¹ : (F : D ⥤ E) → (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ F\ninst✝ :\n (F : D ⥤ E) →\n (X : C) → (W : Cover J X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\n⊢ (𝟙 (plusObj J (plusObj J (P ⋙ F))) ≫ plusMap J (plusCompIso J F P).inv) ≫ (plusCompIso J F (plusObj J P)).inv =\n plusMap J (plusCompIso J F P).inv ≫ (plusCompIso J F (plusObj J P)).inv\n[PROOFSTEP]\nerw [Category.id_comp]\n[GOAL]\nC : Type u\ninst✝⁸ : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w₁\ninst✝⁷ : Category.{max v u, w₁} D\nE : Type w₂\ninst✝⁶ : Category.{max v u, w₂} E\nF : D ⥤ E\ninst✝⁵ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst✝⁴ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst✝³ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\ninst✝² : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ E\ninst✝¹ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ F\ninst✝ : (X : C) → (W : Cover J X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\nP : Cᵒᵖ ⥤ D\n⊢ sheafification J D ⋙ (whiskeringRight Cᵒᵖ D E).obj F ≅ (whiskeringRight Cᵒᵖ D E).obj F ⋙ sheafification J E\n[PROOFSTEP]\nrefine' Functor.associator _ _ _ ≪≫ _\n[GOAL]\nC : Type u\ninst✝⁸ : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w₁\ninst✝⁷ : Category.{max v u, w₁} D\nE : Type w₂\ninst✝⁶ : Category.{max v u, w₂} E\nF : D ⥤ E\ninst✝⁵ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst✝⁴ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst✝³ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\ninst✝² : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ E\ninst✝¹ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ F\ninst✝ : (X : C) → (W : Cover J X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\nP : Cᵒᵖ ⥤ D\n⊢ plusFunctor J D ⋙ plusFunctor J D ⋙ (whiskeringRight Cᵒᵖ D E).obj F ≅\n (whiskeringRight Cᵒᵖ D E).obj F ⋙ sheafification J E\n[PROOFSTEP]\nrefine' isoWhiskerLeft (J.plusFunctor D) (J.plusFunctorWhiskerRightIso _) ≪≫ _\n[GOAL]\nC : Type u\ninst✝⁸ : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w₁\ninst✝⁷ : Category.{max v u, w₁} D\nE : Type w₂\ninst✝⁶ : Category.{max v u, w₂} E\nF : D ⥤ E\ninst✝⁵ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst✝⁴ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst✝³ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\ninst✝² : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ E\ninst✝¹ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ F\ninst✝ : (X : C) → (W : Cover J X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\nP : Cᵒᵖ ⥤ D\n⊢ plusFunctor J D ⋙ (whiskeringRight Cᵒᵖ D E).obj F ⋙ plusFunctor J E ≅\n (whiskeringRight Cᵒᵖ D E).obj F ⋙ sheafification J E\n[PROOFSTEP]\nrefine' _ ≪≫ Functor.associator _ _ _\n[GOAL]\nC : Type u\ninst✝⁸ : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w₁\ninst✝⁷ : Category.{max v u, w₁} D\nE : Type w₂\ninst✝⁶ : Category.{max v u, w₂} E\nF : D ⥤ E\ninst✝⁵ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst✝⁴ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst✝³ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\ninst✝² : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ E\ninst✝¹ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ F\ninst✝ : (X : C) → (W : Cover J X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\nP : Cᵒᵖ ⥤ D\n⊢ plusFunctor J D ⋙ (whiskeringRight Cᵒᵖ D E).obj F ⋙ plusFunctor J E ≅\n ((whiskeringRight Cᵒᵖ D E).obj F ⋙ plusFunctor J E) ⋙ plusFunctor J E\n[PROOFSTEP]\nrefine' (Functor.associator _ _ _).symm ≪≫ _\n[GOAL]\nC : Type u\ninst✝⁸ : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w₁\ninst✝⁷ : Category.{max v u, w₁} D\nE : Type w₂\ninst✝⁶ : Category.{max v u, w₂} E\nF : D ⥤ E\ninst✝⁵ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst✝⁴ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst✝³ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\ninst✝² : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ E\ninst✝¹ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ F\ninst✝ : (X : C) → (W : Cover J X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\nP : Cᵒᵖ ⥤ D\n⊢ (plusFunctor J D ⋙ (whiskeringRight Cᵒᵖ D E).obj F) ⋙ plusFunctor J E ≅\n ((whiskeringRight Cᵒᵖ D E).obj F ⋙ plusFunctor J E) ⋙ plusFunctor J E\n[PROOFSTEP]\nexact isoWhiskerRight (J.plusFunctorWhiskerRightIso _) (J.plusFunctor E)\n[GOAL]\nC : Type u\ninst✝⁸ : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w₁\ninst✝⁷ : Category.{max v u, w₁} D\nE : Type w₂\ninst✝⁶ : Category.{max v u, w₂} E\nF : D ⥤ E\ninst✝⁵ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst✝⁴ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst✝³ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\ninst✝² : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ E\ninst✝¹ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ F\ninst✝ : (X : C) → (W : Cover J X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\nP : Cᵒᵖ ⥤ D\n⊢ NatTrans.app (sheafificationWhiskerRightIso J F).hom P = (sheafifyCompIso J F P).hom\n[PROOFSTEP]\ndsimp [sheafificationWhiskerRightIso, sheafifyCompIso]\n[GOAL]\nC : Type u\ninst✝⁸ : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w₁\ninst✝⁷ : Category.{max v u, w₁} D\nE : Type w₂\ninst✝⁶ : Category.{max v u, w₂} E\nF : D ⥤ E\ninst✝⁵ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst✝⁴ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst✝³ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\ninst✝² : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ E\ninst✝¹ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ F\ninst✝ : (X : C) → (W : Cover J X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\nP : Cᵒᵖ ⥤ D\n⊢ 𝟙 (plusObj J (plusObj J P) ⋙ F) ≫\n (plusCompIso J F (plusObj J P)).hom ≫\n (𝟙 (plusObj J (plusObj J P ⋙ F)) ≫ plusMap J (plusCompIso J F P).hom) ≫ 𝟙 (plusObj J (plusObj J (P ⋙ F))) =\n (plusCompIso J F (plusObj J P)).hom ≫ plusMap J (plusCompIso J F P).hom\n[PROOFSTEP]\nsimp only [Category.id_comp, Category.comp_id]\n[GOAL]\nC : Type u\ninst✝⁸ : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w₁\ninst✝⁷ : Category.{max v u, w₁} D\nE : Type w₂\ninst✝⁶ : Category.{max v u, w₂} E\nF : D ⥤ E\ninst✝⁵ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst✝⁴ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst✝³ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\ninst✝² : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ E\ninst✝¹ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ F\ninst✝ : (X : C) → (W : Cover J X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\nP : Cᵒᵖ ⥤ D\n⊢ 𝟙 (plusObj J (plusObj J P) ⋙ F) ≫ (plusCompIso J F (plusObj J P)).hom ≫ plusMap J (plusCompIso J F P).hom =\n (plusCompIso J F (plusObj J P)).hom ≫ plusMap J (plusCompIso J F P).hom\n[PROOFSTEP]\nerw [Category.id_comp]\n[GOAL]\nC : Type u\ninst✝⁸ : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w₁\ninst✝⁷ : Category.{max v u, w₁} D\nE : Type w₂\ninst✝⁶ : Category.{max v u, w₂} E\nF : D ⥤ E\ninst✝⁵ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst✝⁴ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst✝³ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\ninst✝² : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ E\ninst✝¹ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ F\ninst✝ : (X : C) → (W : Cover J X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\nP : Cᵒᵖ ⥤ D\n⊢ NatTrans.app (sheafificationWhiskerRightIso J F).inv P = (sheafifyCompIso J F P).inv\n[PROOFSTEP]\ndsimp [sheafificationWhiskerRightIso, sheafifyCompIso]\n[GOAL]\nC : Type u\ninst✝⁸ : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w₁\ninst✝⁷ : Category.{max v u, w₁} D\nE : Type w₂\ninst✝⁶ : Category.{max v u, w₂} E\nF : D ⥤ E\ninst✝⁵ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst✝⁴ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst✝³ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\ninst✝² : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ E\ninst✝¹ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ F\ninst✝ : (X : C) → (W : Cover J X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\nP : Cᵒᵖ ⥤ D\n⊢ ((𝟙 (plusObj J (plusObj J (P ⋙ F))) ≫ plusMap J (plusCompIso J F P).inv ≫ 𝟙 (plusObj J (plusObj J P ⋙ F))) ≫\n (plusCompIso J F (plusObj J P)).inv) ≫\n 𝟙 (plusObj J (plusObj J P) ⋙ F) =\n plusMap J (plusCompIso J F P).inv ≫ (plusCompIso J F (plusObj J P)).inv\n[PROOFSTEP]\nsimp only [Category.id_comp, Category.comp_id]\n[GOAL]\nC : Type u\ninst✝⁸ : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w₁\ninst✝⁷ : Category.{max v u, w₁} D\nE : Type w₂\ninst✝⁶ : Category.{max v u, w₂} E\nF : D ⥤ E\ninst✝⁵ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst✝⁴ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst✝³ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\ninst✝² : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ E\ninst✝¹ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ F\ninst✝ : (X : C) → (W : Cover J X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\nP : Cᵒᵖ ⥤ D\n⊢ (𝟙 (plusObj J (plusObj J (P ⋙ F))) ≫ plusMap J (plusCompIso J F P).inv) ≫ (plusCompIso J F (plusObj J P)).inv =\n plusMap J (plusCompIso J F P).inv ≫ (plusCompIso J F (plusObj J P)).inv\n[PROOFSTEP]\nerw [Category.id_comp]\n[GOAL]\nC : Type u\ninst✝⁸ : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w₁\ninst✝⁷ : Category.{max v u, w₁} D\nE : Type w₂\ninst✝⁶ : Category.{max v u, w₂} E\nF : D ⥤ E\ninst✝⁵ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst✝⁴ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst✝³ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\ninst✝² : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ E\ninst✝¹ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ F\ninst✝ : (X : C) → (W : Cover J X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\nP : Cᵒᵖ ⥤ D\n⊢ whiskerRight (toSheafify J P) F ≫ (sheafifyCompIso J F P).hom = toSheafify J (P ⋙ F)\n[PROOFSTEP]\ndsimp [sheafifyCompIso]\n[GOAL]\nC : Type u\ninst✝⁸ : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w₁\ninst✝⁷ : Category.{max v u, w₁} D\nE : Type w₂\ninst✝⁶ : Category.{max v u, w₂} E\nF : D ⥤ E\ninst✝⁵ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst✝⁴ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst✝³ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\ninst✝² : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ E\ninst✝¹ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ F\ninst✝ : (X : C) → (W : Cover J X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\nP : Cᵒᵖ ⥤ D\n⊢ whiskerRight (toSheafify J P) F ≫ (plusCompIso J F (plusObj J P)).hom ≫ plusMap J (plusCompIso J F P).hom =\n toSheafify J (P ⋙ F)\n[PROOFSTEP]\nerw [whiskerRight_comp, Category.assoc]\n[GOAL]\nC : Type u\ninst✝⁸ : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w₁\ninst✝⁷ : Category.{max v u, w₁} D\nE : Type w₂\ninst✝⁶ : Category.{max v u, w₂} E\nF : D ⥤ E\ninst✝⁵ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst✝⁴ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst✝³ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\ninst✝² : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ E\ninst✝¹ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ F\ninst✝ : (X : C) → (W : Cover J X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\nP : Cᵒᵖ ⥤ D\n⊢ whiskerRight (toPlus J P) F ≫\n whiskerRight (plusMap J (toPlus J P)) F ≫\n (plusCompIso J F (plusObj J P)).hom ≫ plusMap J (plusCompIso J F P).hom =\n toSheafify J (P ⋙ F)\n[PROOFSTEP]\nslice_lhs 2 3 => rw [plusCompIso_whiskerRight]\n[GOAL]\ncase a.a\nC : Type u\ninst✝⁸ : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w₁\ninst✝⁷ : Category.{max v u, w₁} D\nE : Type w₂\ninst✝⁶ : Category.{max v u, w₂} E\nF : D ⥤ E\ninst✝⁵ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst✝⁴ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst✝³ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\ninst✝² : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ E\ninst✝¹ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ F\ninst✝ : (X : C) → (W : Cover J X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\nP : Cᵒᵖ ⥤ D\n| whiskerRight (plusMap J (toPlus J P)) F ≫ (plusCompIso J F (plusObj J P)).hom\ncase a.a\nC : Type u\ninst✝⁸ : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w₁\ninst✝⁷ : Category.{max v u, w₁} D\nE : Type w₂\ninst✝⁶ : Category.{max v u, w₂} E\nF : D ⥤ E\ninst✝⁵ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst✝⁴ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst✝³ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\ninst✝² : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ E\ninst✝¹ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ F\ninst✝ : (X : C) → (W : Cover J X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\nP : Cᵒᵖ ⥤ D\n| plusMap J (plusCompIso J F P).hom\ncase a\nC : Type u\ninst✝⁸ : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w₁\ninst✝⁷ : Category.{max v u, w₁} D\nE : Type w₂\ninst✝⁶ : Category.{max v u, w₂} E\nF : D ⥤ E\ninst✝⁵ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst✝⁴ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst✝³ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\ninst✝² : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ E\ninst✝¹ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ F\ninst✝ : (X : C) → (W : Cover J X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\nP : Cᵒᵖ ⥤ D\n| whiskerRight (toPlus J P) F\n[PROOFSTEP]\nrw [plusCompIso_whiskerRight]\n[GOAL]\ncase a.a\nC : Type u\ninst✝⁸ : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w₁\ninst✝⁷ : Category.{max v u, w₁} D\nE : Type w₂\ninst✝⁶ : Category.{max v u, w₂} E\nF : D ⥤ E\ninst✝⁵ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst✝⁴ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst✝³ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\ninst✝² : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ E\ninst✝¹ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ F\ninst✝ : (X : C) → (W : Cover J X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\nP : Cᵒᵖ ⥤ D\n| whiskerRight (plusMap J (toPlus J P)) F ≫ (plusCompIso J F (plusObj J P)).hom\ncase a.a\nC : Type u\ninst✝⁸ : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w₁\ninst✝⁷ : Category.{max v u, w₁} D\nE : Type w₂\ninst✝⁶ : Category.{max v u, w₂} E\nF : D ⥤ E\ninst✝⁵ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst✝⁴ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst✝³ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\ninst✝² : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ E\ninst✝¹ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ F\ninst✝ : (X : C) → (W : Cover J X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\nP : Cᵒᵖ ⥤ D\n| plusMap J (plusCompIso J F P).hom\ncase a\nC : Type u\ninst✝⁸ : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w₁\ninst✝⁷ : Category.{max v u, w₁} D\nE : Type w₂\ninst✝⁶ : Category.{max v u, w₂} E\nF : D ⥤ E\ninst✝⁵ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst✝⁴ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst✝³ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\ninst✝² : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ E\ninst✝¹ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ F\ninst✝ : (X : C) → (W : Cover J X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\nP : Cᵒᵖ ⥤ D\n| whiskerRight (toPlus J P) F\n[PROOFSTEP]\nrw [plusCompIso_whiskerRight]\n[GOAL]\ncase a.a\nC : Type u\ninst✝⁸ : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w₁\ninst✝⁷ : Category.{max v u, w₁} D\nE : Type w₂\ninst✝⁶ : Category.{max v u, w₂} E\nF : D ⥤ E\ninst✝⁵ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst✝⁴ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst✝³ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\ninst✝² : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ E\ninst✝¹ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ F\ninst✝ : (X : C) → (W : Cover J X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\nP : Cᵒᵖ ⥤ D\n| whiskerRight (plusMap J (toPlus J P)) F ≫ (plusCompIso J F (plusObj J P)).hom\ncase a.a\nC : Type u\ninst✝⁸ : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w₁\ninst✝⁷ : Category.{max v u, w₁} D\nE : Type w₂\ninst✝⁶ : Category.{max v u, w₂} E\nF : D ⥤ E\ninst✝⁵ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst✝⁴ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst✝³ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\ninst✝² : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ E\ninst✝¹ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ F\ninst✝ : (X : C) → (W : Cover J X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\nP : Cᵒᵖ ⥤ D\n| plusMap J (plusCompIso J F P).hom\ncase a\nC : Type u\ninst✝⁸ : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w₁\ninst✝⁷ : Category.{max v u, w₁} D\nE : Type w₂\ninst✝⁶ : Category.{max v u, w₂} E\nF : D ⥤ E\ninst✝⁵ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst✝⁴ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst✝³ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\ninst✝² : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ E\ninst✝¹ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ F\ninst✝ : (X : C) → (W : Cover J X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\nP : Cᵒᵖ ⥤ D\n| whiskerRight (toPlus J P) F\n[PROOFSTEP]\nrw [plusCompIso_whiskerRight]\n[GOAL]\nC : Type u\ninst✝⁸ : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w₁\ninst✝⁷ : Category.{max v u, w₁} D\nE : Type w₂\ninst✝⁶ : Category.{max v u, w₂} E\nF : D ⥤ E\ninst✝⁵ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst✝⁴ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst✝³ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\ninst✝² : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ E\ninst✝¹ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ F\ninst✝ : (X : C) → (W : Cover J X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\nP : Cᵒᵖ ⥤ D\n⊢ whiskerRight (toPlus J P) F ≫\n ((plusCompIso J F P).hom ≫ plusMap J (whiskerRight (toPlus J P) F)) ≫ plusMap J (plusCompIso J F P).hom =\n toSheafify J (P ⋙ F)\n[PROOFSTEP]\nrw [Category.assoc, ← J.plusMap_comp, whiskerRight_toPlus_comp_plusCompIso_hom, ← Category.assoc,\n whiskerRight_toPlus_comp_plusCompIso_hom]\n[GOAL]\nC : Type u\ninst✝⁸ : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w₁\ninst✝⁷ : Category.{max v u, w₁} D\nE : Type w₂\ninst✝⁶ : Category.{max v u, w₂} E\nF : D ⥤ E\ninst✝⁵ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst✝⁴ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst✝³ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\ninst✝² : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ E\ninst✝¹ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ F\ninst✝ : (X : C) → (W : Cover J X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\nP : Cᵒᵖ ⥤ D\n⊢ toPlus J (P ⋙ F) ≫ plusMap J (toPlus J (P ⋙ F)) = toSheafify J (P ⋙ F)\n[PROOFSTEP]\nrfl\n[GOAL]\nC : Type u\ninst✝⁸ : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w₁\ninst✝⁷ : Category.{max v u, w₁} D\nE : Type w₂\ninst✝⁶ : Category.{max v u, w₂} E\nF : D ⥤ E\ninst✝⁵ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst✝⁴ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst✝³ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\ninst✝² : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ E\ninst✝¹ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ F\ninst✝ : (X : C) → (W : Cover J X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\nP : Cᵒᵖ ⥤ D\n⊢ toSheafify J (P ⋙ F) ≫ (sheafifyCompIso J F P).inv = whiskerRight (toSheafify J P) F\n[PROOFSTEP]\nrw [Iso.comp_inv_eq]\n[GOAL]\nC : Type u\ninst✝⁸ : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w₁\ninst✝⁷ : Category.{max v u, w₁} D\nE : Type w₂\ninst✝⁶ : Category.{max v u, w₂} E\nF : D ⥤ E\ninst✝⁵ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst✝⁴ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst✝³ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\ninst✝² : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ E\ninst✝¹ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ F\ninst✝ : (X : C) → (W : Cover J X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\nP : Cᵒᵖ ⥤ D\n⊢ toSheafify J (P ⋙ F) = whiskerRight (toSheafify J P) F ≫ (sheafifyCompIso J F P).hom\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\ninst✝¹² : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w₁\ninst✝¹¹ : Category.{max v u, w₁} D\nE : Type w₂\ninst✝¹⁰ : Category.{max v u, w₂} E\nF : D ⥤ E\ninst✝⁹ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst✝⁸ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst✝⁷ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\ninst✝⁶ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ E\ninst✝⁵ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ F\ninst✝⁴ : (X : C) → (W : Cover J X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\nP : Cᵒᵖ ⥤ D\ninst✝³ : ConcreteCategory D\ninst✝² : PreservesLimits (forget D)\ninst✝¹ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ (forget D)\ninst✝ : ReflectsIsomorphisms (forget D)\n⊢ (sheafifyCompIso J F P).inv =\n sheafifyLift J (whiskerRight (toSheafify J P) F) (_ : Presheaf.IsSheaf J (sheafify J P ⋙ F))\n[PROOFSTEP]\napply J.sheafifyLift_unique\n[GOAL]\ncase a\nC : Type u\ninst✝¹² : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w₁\ninst✝¹¹ : Category.{max v u, w₁} D\nE : Type w₂\ninst✝¹⁰ : Category.{max v u, w₂} E\nF : D ⥤ E\ninst✝⁹ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst✝⁸ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst✝⁷ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\ninst✝⁶ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ E\ninst✝⁵ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ F\ninst✝⁴ : (X : C) → (W : Cover J X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\nP : Cᵒᵖ ⥤ D\ninst✝³ : ConcreteCategory D\ninst✝² : PreservesLimits (forget D)\ninst✝¹ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ (forget D)\ninst✝ : ReflectsIsomorphisms (forget D)\n⊢ toSheafify J (P ⋙ F) ≫ (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✝¹² : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w₁\ninst✝¹¹ : Category.{max v u, w₁} D\nE : Type w₂\ninst✝¹⁰ : Category.{max v u, w₂} E\nF : D ⥤ E\ninst✝⁹ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst✝⁸ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst✝⁷ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\ninst✝⁶ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ E\ninst✝⁵ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ F\ninst✝⁴ : (X : C) → (W : Cover J X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\nP : Cᵒᵖ ⥤ D\ninst✝³ : ConcreteCategory D\ninst✝² : PreservesLimits (forget D)\ninst✝¹ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ (forget D)\ninst✝ : ReflectsIsomorphisms (forget D)\n⊢ toSheafify J (P ⋙ F) = whiskerRight (toSheafify J P) F ≫ (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 u₁\ninst✝² : Category.{v₁, u₁} C\nD : Type u₂\ninst✝¹ : Category.{v₂, u₂} D\nA : Type u₃\ninst✝ : Category.{v₃, u₃} A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nL : GrothendieckTopology A\nU✝ : C\nS✝ : Sieve U✝\nhS : S✝ ∈ GrothendieckTopology.sieves J U✝\n⊢ Sieve.functorPushforward (𝟭 C) S✝ ∈ GrothendieckTopology.sieves J ((𝟭 C).obj U✝)\n[PROOFSTEP]\nsimpa using hS\n[GOAL]\nC : Type u₁\ninst✝² : Category.{v₁, u₁} C\nD : Type u₂\ninst✝¹ : Category.{v₂, u₂} D\nA : Type u₃\ninst✝ : Category.{v₃, u₃} A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nL : GrothendieckTopology A\nF : C ⥤ D\nhF : CoverPreserving J K F\nG : D ⥤ A\nhG : CoverPreserving K L G\nU✝ : C\nS✝ : Sieve U✝\nhS : S✝ ∈ GrothendieckTopology.sieves J U✝\n⊢ Sieve.functorPushforward (F ⋙ G) S✝ ∈ GrothendieckTopology.sieves L ((F ⋙ G).obj U✝)\n[PROOFSTEP]\nrw [Sieve.functorPushforward_comp]\n[GOAL]\nC : Type u₁\ninst✝² : Category.{v₁, u₁} C\nD : Type u₂\ninst✝¹ : Category.{v₂, u₂} D\nA : Type u₃\ninst✝ : Category.{v₃, u₃} A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nL : GrothendieckTopology A\nF : C ⥤ D\nhF : CoverPreserving J K F\nG : D ⥤ A\nhG : CoverPreserving K L G\nU✝ : C\nS✝ : Sieve U✝\nhS : S✝ ∈ GrothendieckTopology.sieves J U✝\n⊢ Sieve.functorPushforward G (Sieve.functorPushforward F S✝) ∈ GrothendieckTopology.sieves L ((F ⋙ G).obj U✝)\n[PROOFSTEP]\nexact hG.cover_preserve (hF.cover_preserve hS)\n[GOAL]\nC : Type u₁\ninst✝² : Category.{v₁, u₁} C\nD : Type u₂\ninst✝¹ : Category.{v₂, u₂} D\nA : Type u₃\ninst✝ : Category.{v₃, u₃} A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nL : GrothendieckTopology A\nG : C ⥤ D\nhG : CompatiblePreserving K G\nℱ : SheafOfTypes K\nZ : C\nT : Presieve Z\nx : FamilyOfElements (G.op ⋙ ℱ.val) T\nh : Compatible x\n⊢ Compatible (FamilyOfElements.functorPushforward G x)\n[PROOFSTEP]\nrintro Z₁ Z₂ W g₁ g₂ f₁' f₂' H₁ H₂ eq\n[GOAL]\nC : Type u₁\ninst✝² : Category.{v₁, u₁} C\nD : Type u₂\ninst✝¹ : Category.{v₂, u₂} D\nA : Type u₃\ninst✝ : Category.{v₃, u₃} A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nL : GrothendieckTopology A\nG : C ⥤ D\nhG : CompatiblePreserving K G\nℱ : SheafOfTypes K\nZ : C\nT : Presieve Z\nx : FamilyOfElements (G.op ⋙ ℱ.val) T\nh : Compatible x\nZ₁ Z₂ W : D\ng₁ : W ⟶ Z₁\ng₂ : W ⟶ Z₂\nf₁' : Z₁ ⟶ G.obj Z\nf₂' : Z₂ ⟶ G.obj Z\nH₁ : Presieve.functorPushforward G T f₁'\nH₂ : Presieve.functorPushforward G T f₂'\neq : g₁ ≫ f₁' = g₂ ≫ f₂'\n⊢ ℱ.val.map g₁.op (FamilyOfElements.functorPushforward G x f₁' H₁) =\n ℱ.val.map g₂.op (FamilyOfElements.functorPushforward G x f₂' H₂)\n[PROOFSTEP]\nunfold FamilyOfElements.functorPushforward\n[GOAL]\nC : Type u₁\ninst✝² : Category.{v₁, u₁} C\nD : Type u₂\ninst✝¹ : Category.{v₂, u₂} D\nA : Type u₃\ninst✝ : Category.{v₃, u₃} A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nL : GrothendieckTopology A\nG : C ⥤ D\nhG : CompatiblePreserving K G\nℱ : SheafOfTypes K\nZ : C\nT : Presieve Z\nx : FamilyOfElements (G.op ⋙ ℱ.val) T\nh : Compatible x\nZ₁ Z₂ W : D\ng₁ : W ⟶ Z₁\ng₂ : W ⟶ Z₂\nf₁' : Z₁ ⟶ G.obj Z\nf₂' : Z₂ ⟶ G.obj Z\nH₁ : Presieve.functorPushforward G T f₁'\nH₂ : Presieve.functorPushforward G T f₂'\neq : g₁ ≫ f₁' = g₂ ≫ f₂'\n⊢ ℱ.val.map g₁.op\n (FunctorPushforwardStructure.casesOn (getFunctorPushforwardStructure H₁) fun Z_1 g h h₁ fac =>\n ℱ.val.map h.op (x g h₁)) =\n ℱ.val.map g₂.op\n (FunctorPushforwardStructure.casesOn (getFunctorPushforwardStructure H₂) fun Z_1 g h h₁ fac =>\n ℱ.val.map h.op (x g h₁))\n[PROOFSTEP]\nrcases getFunctorPushforwardStructure H₁ with ⟨X₁, f₁, h₁, hf₁, rfl⟩\n[GOAL]\ncase mk\nC : Type u₁\ninst✝² : Category.{v₁, u₁} C\nD : Type u₂\ninst✝¹ : Category.{v₂, u₂} D\nA : Type u₃\ninst✝ : Category.{v₃, u₃} A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nL : GrothendieckTopology A\nG : C ⥤ D\nhG : CompatiblePreserving K G\nℱ : SheafOfTypes K\nZ : C\nT : Presieve Z\nx : FamilyOfElements (G.op ⋙ ℱ.val) T\nh : Compatible x\nZ₁ Z₂ W : D\ng₁ : W ⟶ Z₁\ng₂ : W ⟶ Z₂\nf₂' : Z₂ ⟶ G.obj Z\nH₂ : Presieve.functorPushforward G T f₂'\nX₁ : C\nf₁ : X₁ ⟶ Z\nh₁ : Z₁ ⟶ G.obj X₁\nhf₁ : T f₁\nH₁ : Presieve.functorPushforward G T (h₁ ≫ G.map f₁)\neq : g₁ ≫ h₁ ≫ G.map f₁ = g₂ ≫ f₂'\n⊢ ℱ.val.map g₁.op\n (FunctorPushforwardStructure.casesOn\n { preobj := X₁, premap := f₁, lift := h₁, cover := hf₁, fac := (_ : h₁ ≫ G.map f₁ = h₁ ≫ G.map f₁) }\n fun Z_1 g h h₁_1 fac => ℱ.val.map h.op (x g h₁_1)) =\n ℱ.val.map g₂.op\n (FunctorPushforwardStructure.casesOn (getFunctorPushforwardStructure H₂) fun Z_1 g h h₁ fac =>\n ℱ.val.map h.op (x g h₁))\n[PROOFSTEP]\nrcases getFunctorPushforwardStructure H₂ with ⟨X₂, f₂, h₂, hf₂, rfl⟩\n[GOAL]\ncase mk.mk\nC : Type u₁\ninst✝² : Category.{v₁, u₁} C\nD : Type u₂\ninst✝¹ : Category.{v₂, u₂} D\nA : Type u₃\ninst✝ : Category.{v₃, u₃} A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nL : GrothendieckTopology A\nG : C ⥤ D\nhG : CompatiblePreserving K G\nℱ : SheafOfTypes K\nZ : C\nT : Presieve Z\nx : FamilyOfElements (G.op ⋙ ℱ.val) T\nh : Compatible x\nZ₁ Z₂ W : D\ng₁ : W ⟶ Z₁\ng₂ : W ⟶ Z₂\nX₁ : C\nf₁ : X₁ ⟶ Z\nh₁ : Z₁ ⟶ G.obj X₁\nhf₁ : T f₁\nH₁ : Presieve.functorPushforward G T (h₁ ≫ G.map f₁)\nX₂ : C\nf₂ : X₂ ⟶ Z\nh₂ : Z₂ ⟶ G.obj X₂\nhf₂ : T f₂\nH₂ : Presieve.functorPushforward G T (h₂ ≫ G.map f₂)\neq : g₁ ≫ h₁ ≫ G.map f₁ = g₂ ≫ h₂ ≫ G.map f₂\n⊢ ℱ.val.map g₁.op\n (FunctorPushforwardStructure.casesOn\n { preobj := X₁, premap := f₁, lift := h₁, cover := hf₁, fac := (_ : h₁ ≫ G.map f₁ = h₁ ≫ G.map f₁) }\n fun Z_1 g h h₁_1 fac => ℱ.val.map h.op (x g h₁_1)) =\n ℱ.val.map g₂.op\n (FunctorPushforwardStructure.casesOn\n { preobj := X₂, premap := f₂, lift := h₂, cover := hf₂, fac := (_ : h₂ ≫ G.map f₂ = h₂ ≫ G.map f₂) }\n fun Z_1 g h h₁ fac => ℱ.val.map h.op (x g h₁))\n[PROOFSTEP]\nsuffices ℱ.val.map (g₁ ≫ h₁).op (x f₁ hf₁) = ℱ.val.map (g₂ ≫ h₂).op (x f₂ hf₂) by simpa using this\n[GOAL]\nC : Type u₁\ninst✝² : Category.{v₁, u₁} C\nD : Type u₂\ninst✝¹ : Category.{v₂, u₂} D\nA : Type u₃\ninst✝ : Category.{v₃, u₃} A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nL : GrothendieckTopology A\nG : C ⥤ D\nhG : CompatiblePreserving K G\nℱ : SheafOfTypes K\nZ : C\nT : Presieve Z\nx : FamilyOfElements (G.op ⋙ ℱ.val) T\nh : Compatible x\nZ₁ Z₂ W : D\ng₁ : W ⟶ Z₁\ng₂ : W ⟶ Z₂\nX₁ : C\nf₁ : X₁ ⟶ Z\nh₁ : Z₁ ⟶ G.obj X₁\nhf₁ : T f₁\nH₁ : Presieve.functorPushforward G T (h₁ ≫ G.map f₁)\nX₂ : C\nf₂ : X₂ ⟶ Z\nh₂ : Z₂ ⟶ G.obj X₂\nhf₂ : T f₂\nH₂ : Presieve.functorPushforward G T (h₂ ≫ G.map f₂)\neq : g₁ ≫ h₁ ≫ G.map f₁ = g₂ ≫ h₂ ≫ G.map f₂\nthis : ℱ.val.map (g₁ ≫ h₁).op (x f₁ hf₁) = ℱ.val.map (g₂ ≫ h₂).op (x f₂ hf₂)\n⊢ ℱ.val.map g₁.op\n (FunctorPushforwardStructure.casesOn\n { preobj := X₁, premap := f₁, lift := h₁, cover := hf₁, fac := (_ : h₁ ≫ G.map f₁ = h₁ ≫ G.map f₁) }\n fun Z_1 g h h₁_1 fac => ℱ.val.map h.op (x g h₁_1)) =\n ℱ.val.map g₂.op\n (FunctorPushforwardStructure.casesOn\n { preobj := X₂, premap := f₂, lift := h₂, cover := hf₂, fac := (_ : h₂ ≫ G.map f₂ = h₂ ≫ G.map f₂) }\n fun Z_1 g h h₁ fac => ℱ.val.map h.op (x g h₁))\n[PROOFSTEP]\nsimpa using this\n[GOAL]\ncase mk.mk\nC : Type u₁\ninst✝² : Category.{v₁, u₁} C\nD : Type u₂\ninst✝¹ : Category.{v₂, u₂} D\nA : Type u₃\ninst✝ : Category.{v₃, u₃} A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nL : GrothendieckTopology A\nG : C ⥤ D\nhG : CompatiblePreserving K G\nℱ : SheafOfTypes K\nZ : C\nT : Presieve Z\nx : FamilyOfElements (G.op ⋙ ℱ.val) T\nh : Compatible x\nZ₁ Z₂ W : D\ng₁ : W ⟶ Z₁\ng₂ : W ⟶ Z₂\nX₁ : C\nf₁ : X₁ ⟶ Z\nh₁ : Z₁ ⟶ G.obj X₁\nhf₁ : T f₁\nH₁ : Presieve.functorPushforward G T (h₁ ≫ G.map f₁)\nX₂ : C\nf₂ : X₂ ⟶ Z\nh₂ : Z₂ ⟶ G.obj X₂\nhf₂ : T f₂\nH₂ : Presieve.functorPushforward G T (h₂ ≫ G.map f₂)\neq : g₁ ≫ h₁ ≫ G.map f₁ = g₂ ≫ h₂ ≫ G.map f₂\n⊢ ℱ.val.map (g₁ ≫ h₁).op (x f₁ hf₁) = ℱ.val.map (g₂ ≫ h₂).op (x f₂ hf₂)\n[PROOFSTEP]\napply hG.Compatible ℱ h _ _ hf₁ hf₂\n[GOAL]\ncase mk.mk\nC : Type u₁\ninst✝² : Category.{v₁, u₁} C\nD : Type u₂\ninst✝¹ : Category.{v₂, u₂} D\nA : Type u₃\ninst✝ : Category.{v₃, u₃} A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nL : GrothendieckTopology A\nG : C ⥤ D\nhG : CompatiblePreserving K G\nℱ : SheafOfTypes K\nZ : C\nT : Presieve Z\nx : FamilyOfElements (G.op ⋙ ℱ.val) T\nh : Compatible x\nZ₁ Z₂ W : D\ng₁ : W ⟶ Z₁\ng₂ : W ⟶ Z₂\nX₁ : C\nf₁ : X₁ ⟶ Z\nh₁ : Z₁ ⟶ G.obj X₁\nhf₁ : T f₁\nH₁ : Presieve.functorPushforward G T (h₁ ≫ G.map f₁)\nX₂ : C\nf₂ : X₂ ⟶ Z\nh₂ : Z₂ ⟶ G.obj X₂\nhf₂ : T f₂\nH₂ : Presieve.functorPushforward G T (h₂ ≫ G.map f₂)\neq : g₁ ≫ h₁ ≫ G.map f₁ = g₂ ≫ h₂ ≫ G.map f₂\n⊢ (g₁ ≫ h₁) ≫ G.map f₁ = (g₂ ≫ h₂) ≫ G.map f₂\n[PROOFSTEP]\nsimpa using eq\n[GOAL]\nC : Type u₁\ninst✝² : Category.{v₁, u₁} C\nD : Type u₂\ninst✝¹ : Category.{v₂, u₂} D\nA : Type u₃\ninst✝ : Category.{v₃, u₃} A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nL : GrothendieckTopology A\nG : C ⥤ D\nhG : CompatiblePreserving K G\nℱ : SheafOfTypes K\nZ : C\nT : Presieve Z\nx : FamilyOfElements (G.op ⋙ ℱ.val) T\nh : FamilyOfElements.Compatible x\nY : C\nf : Y ⟶ Z\nhf : T f\n⊢ 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 u₁\ninst✝² : Category.{v₁, u₁} C\nD : Type u₂\ninst✝¹ : Category.{v₂, u₂} D\nA : Type u₃\ninst✝ : Category.{v₃, u₃} A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nL : GrothendieckTopology A\nG : C ⥤ D\nhG : CompatiblePreserving K G\nℱ : SheafOfTypes K\nZ : C\nT : Presieve Z\nx : FamilyOfElements (G.op ⋙ ℱ.val) T\nh : FamilyOfElements.Compatible x\nY : C\nf : Y ⟶ Z\nhf : T f\n⊢ (FunctorPushforwardStructure.casesOn (getFunctorPushforwardStructure (_ : Presieve.functorPushforward G T (G.map f)))\n fun Z_1 g h h₁ fac => ℱ.val.map h.op (x g h₁)) =\n x f hf\n[PROOFSTEP]\nrcases e₁ : getFunctorPushforwardStructure (image_mem_functorPushforward G T hf) with ⟨X, g, f', hg, eq⟩\n[GOAL]\ncase mk\nC : Type u₁\ninst✝² : Category.{v₁, u₁} C\nD : Type u₂\ninst✝¹ : Category.{v₂, u₂} D\nA : Type u₃\ninst✝ : Category.{v₃, u₃} A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nL : GrothendieckTopology A\nG : C ⥤ D\nhG : CompatiblePreserving K G\nℱ : SheafOfTypes K\nZ : C\nT : Presieve Z\nx : FamilyOfElements (G.op ⋙ ℱ.val) T\nh : FamilyOfElements.Compatible x\nY : C\nf : Y ⟶ Z\nhf : T f\nX : C\ng : X ⟶ Z\nf' : G.obj Y ⟶ G.obj X\nhg : T g\neq : G.map f = f' ≫ G.map g\ne₁ :\n getFunctorPushforwardStructure (_ : Presieve.functorPushforward G T (G.map f)) =\n { preobj := X, premap := g, lift := f', cover := hg, fac := eq }\n⊢ (FunctorPushforwardStructure.casesOn { preobj := X, premap := g, lift := f', cover := hg, fac := eq }\n fun Z_1 g h h₁ fac => ℱ.val.map h.op (x g h₁)) =\n x f hf\n[PROOFSTEP]\nsimpa using hG.Compatible ℱ h f' (𝟙 _) hg hf (by simp [eq])\n[GOAL]\nC : Type u₁\ninst✝² : Category.{v₁, u₁} C\nD : Type u₂\ninst✝¹ : Category.{v₂, u₂} D\nA : Type u₃\ninst✝ : Category.{v₃, u₃} A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nL : GrothendieckTopology A\nG : C ⥤ D\nhG : CompatiblePreserving K G\nℱ : SheafOfTypes K\nZ : C\nT : Presieve Z\nx : FamilyOfElements (G.op ⋙ ℱ.val) T\nh : FamilyOfElements.Compatible x\nY : C\nf : Y ⟶ Z\nhf : T f\nX : C\ng : X ⟶ Z\nf' : G.obj Y ⟶ G.obj X\nhg : T g\neq : G.map f = f' ≫ G.map g\ne₁ :\n getFunctorPushforwardStructure (_ : Presieve.functorPushforward G T (G.map f)) =\n { preobj := X, premap := g, lift := f', cover := hg, fac := eq }\n⊢ f' ≫ G.map g = 𝟙 (G.obj Y) ≫ G.map f\n[PROOFSTEP]\nsimp [eq]\n[GOAL]\nC✝ : Type u₁\ninst✝⁵ : Category.{v₁, u₁} C✝\nD✝ : Type u₂\ninst✝⁴ : Category.{v₂, u₂} D✝\nA : Type u₃\ninst✝³ : Category.{v₃, u₃} A\nJ : GrothendieckTopology C✝\nK✝ : GrothendieckTopology D✝\nL : GrothendieckTopology A\nG✝ : C✝ ⥤ D✝\nhG : CompatiblePreserving K✝ G✝\nℱ : SheafOfTypes K✝\nZ : C✝\nT : Presieve Z\nx : FamilyOfElements (G✝.op ⋙ ℱ.val) T\nh : Compatible x\nC : Type u₁\ninst✝² : Category.{v₁, u₁} C\nD : Type u₁\ninst✝¹ : Category.{v₁, u₁} D\nK : GrothendieckTopology D\nG : C ⥤ D\ninst✝ : RepresentablyFlat G\n⊢ CompatiblePreserving K G\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase Compatible\nC✝ : Type u₁\ninst✝⁵ : Category.{v₁, u₁} C✝\nD✝ : Type u₂\ninst✝⁴ : Category.{v₂, u₂} D✝\nA : Type u₃\ninst✝³ : Category.{v₃, u₃} A\nJ : GrothendieckTopology C✝\nK✝ : GrothendieckTopology D✝\nL : GrothendieckTopology A\nG✝ : C✝ ⥤ D✝\nhG : CompatiblePreserving K✝ G✝\nℱ : SheafOfTypes K✝\nZ : C✝\nT : Presieve Z\nx : FamilyOfElements (G✝.op ⋙ ℱ.val) T\nh : Compatible x\nC : Type u₁\ninst✝² : Category.{v₁, u₁} C\nD : Type u₁\ninst✝¹ : Category.{v₁, u₁} D\nK : GrothendieckTopology D\nG : C ⥤ D\ninst✝ : RepresentablyFlat G\n⊢ ∀ (ℱ : SheafOfTypes K) {Z : C} {T : Presieve Z} {x : FamilyOfElements (G.op ⋙ ℱ.val) T},\n Compatible x →\n ∀ {Y₁ Y₂ : C} {X : D} (f₁ : X ⟶ G.obj Y₁) (f₂ : X ⟶ G.obj Y₂) {g₁ : Y₁ ⟶ Z} {g₂ : Y₂ ⟶ Z} (hg₁ : T g₁)\n (hg₂ : T g₂), f₁ ≫ G.map g₁ = f₂ ≫ G.map g₂ → ℱ.val.map f₁.op (x g₁ hg₁) = ℱ.val.map f₂.op (x g₂ hg₂)\n[PROOFSTEP]\nintro ℱ Z T x hx Y₁ Y₂ X f₁ f₂ g₁ g₂ hg₁ hg₂ e\n[GOAL]\ncase Compatible\nC✝ : Type u₁\ninst✝⁵ : Category.{v₁, u₁} C✝\nD✝ : Type u₂\ninst✝⁴ : Category.{v₂, u₂} D✝\nA : Type u₃\ninst✝³ : Category.{v₃, u₃} A\nJ : GrothendieckTopology C✝\nK✝ : GrothendieckTopology D✝\nL : GrothendieckTopology A\nG✝ : C✝ ⥤ D✝\nhG : CompatiblePreserving K✝ G✝\nℱ✝ : SheafOfTypes K✝\nZ✝ : C✝\nT✝ : Presieve Z✝\nx✝ : FamilyOfElements (G✝.op ⋙ ℱ✝.val) T✝\nh : Compatible x✝\nC : Type u₁\ninst✝² : Category.{v₁, u₁} C\nD : Type u₁\ninst✝¹ : Category.{v₁, u₁} D\nK : GrothendieckTopology D\nG : C ⥤ D\ninst✝ : RepresentablyFlat G\nℱ : SheafOfTypes K\nZ : C\nT : Presieve Z\nx : FamilyOfElements (G.op ⋙ ℱ.val) T\nhx : Compatible x\nY₁ Y₂ : C\nX : D\nf₁ : X ⟶ G.obj Y₁\nf₂ : X ⟶ G.obj Y₂\ng₁ : Y₁ ⟶ Z\ng₂ : Y₂ ⟶ Z\nhg₁ : T g₁\nhg₂ : T g₂\ne : f₁ ≫ G.map g₁ = f₂ ≫ G.map g₂\n⊢ ℱ.val.map f₁.op (x g₁ hg₁) = ℱ.val.map f₂.op (x g₂ hg₂)\n[PROOFSTEP]\nlet c : Cone (cospan g₁ g₂ ⋙ G) :=\n (Cones.postcompose (diagramIsoCospan (cospan g₁ g₂ ⋙ G)).inv).obj\n (PullbackCone.mk f₁ f₂ 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✝ : Type u₁\ninst✝⁵ : Category.{v₁, u₁} C✝\nD✝ : Type u₂\ninst✝⁴ : Category.{v₂, u₂} D✝\nA : Type u₃\ninst✝³ : Category.{v₃, u₃} A\nJ : GrothendieckTopology C✝\nK✝ : GrothendieckTopology D✝\nL : GrothendieckTopology A\nG✝ : C✝ ⥤ D✝\nhG : CompatiblePreserving K✝ G✝\nℱ✝ : SheafOfTypes K✝\nZ✝ : C✝\nT✝ : Presieve Z✝\nx✝ : FamilyOfElements (G✝.op ⋙ ℱ✝.val) T✝\nh : Compatible x✝\nC : Type u₁\ninst✝² : Category.{v₁, u₁} C\nD : Type u₁\ninst✝¹ : Category.{v₁, u₁} D\nK : GrothendieckTopology D\nG : C ⥤ D\ninst✝ : RepresentablyFlat G\nℱ : SheafOfTypes K\nZ : C\nT : Presieve Z\nx : FamilyOfElements (G.op ⋙ ℱ.val) T\nhx : Compatible x\nY₁ Y₂ : C\nX : D\nf₁ : X ⟶ G.obj Y₁\nf₂ : X ⟶ G.obj Y₂\ng₁ : Y₁ ⟶ Z\ng₂ : Y₂ ⟶ Z\nhg₁ : T g₁\nhg₂ : T g₂\ne : f₁ ≫ G.map g₁ = f₂ ≫ G.map g₂\nc : Cone (cospan g₁ g₂ ⋙ G) :=\n (Cones.postcompose (diagramIsoCospan (cospan g₁ g₂ ⋙ G)).inv).obj (PullbackCone.mk f₁ f₂ e)\n⊢ ℱ.val.map f₁.op (x g₁ hg₁) = ℱ.val.map f₂.op (x g₂ hg₂)\n[PROOFSTEP]\nlet c' := IsCofiltered.cone (c.toStructuredArrow ⋙ StructuredArrow.pre _ _ _)\n[GOAL]\ncase Compatible\nC✝ : Type u₁\ninst✝⁵ : Category.{v₁, u₁} C✝\nD✝ : Type u₂\ninst✝⁴ : Category.{v₂, u₂} D✝\nA : Type u₃\ninst✝³ : Category.{v₃, u₃} A\nJ : GrothendieckTopology C✝\nK✝ : GrothendieckTopology D✝\nL : GrothendieckTopology A\nG✝ : C✝ ⥤ D✝\nhG : CompatiblePreserving K✝ G✝\nℱ✝ : SheafOfTypes K✝\nZ✝ : C✝\nT✝ : Presieve Z✝\nx✝ : FamilyOfElements (G✝.op ⋙ ℱ✝.val) T✝\nh : Compatible x✝\nC : Type u₁\ninst✝² : Category.{v₁, u₁} C\nD : Type u₁\ninst✝¹ : Category.{v₁, u₁} D\nK : GrothendieckTopology D\nG : C ⥤ D\ninst✝ : RepresentablyFlat G\nℱ : SheafOfTypes K\nZ : C\nT : Presieve Z\nx : FamilyOfElements (G.op ⋙ ℱ.val) T\nhx : Compatible x\nY₁ Y₂ : C\nX : D\nf₁ : X ⟶ G.obj Y₁\nf₂ : X ⟶ G.obj Y₂\ng₁ : Y₁ ⟶ Z\ng₂ : Y₂ ⟶ Z\nhg₁ : T g₁\nhg₂ : T g₂\ne : f₁ ≫ G.map g₁ = f₂ ≫ G.map g₂\nc : Cone (cospan g₁ g₂ ⋙ G) :=\n (Cones.postcompose (diagramIsoCospan (cospan g₁ g₂ ⋙ G)).inv).obj (PullbackCone.mk f₁ f₂ e)\nc' : Cone (Cone.toStructuredArrow c ⋙ StructuredArrow.pre c.pt (cospan g₁ g₂) G) :=\n IsCofiltered.cone (Cone.toStructuredArrow c ⋙ StructuredArrow.pre c.pt (cospan g₁ g₂) G)\n⊢ ℱ.val.map f₁.op (x g₁ hg₁) = ℱ.val.map f₂.op (x g₂ hg₂)\n[PROOFSTEP]\nhave eq₁ : f₁ = (c'.pt.hom ≫ G.map (c'.π.app left).right) ≫ eqToHom (by simp) :=\n by\n erw [← (c'.π.app left).w]\n dsimp\n simp\n[GOAL]\nC✝ : Type u₁\ninst✝⁵ : Category.{v₁, u₁} C✝\nD✝ : Type u₂\ninst✝⁴ : Category.{v₂, u₂} D✝\nA : Type u₃\ninst✝³ : Category.{v₃, u₃} A\nJ : GrothendieckTopology C✝\nK✝ : GrothendieckTopology D✝\nL : GrothendieckTopology A\nG✝ : C✝ ⥤ D✝\nhG : CompatiblePreserving K✝ G✝\nℱ✝ : SheafOfTypes K✝\nZ✝ : C✝\nT✝ : Presieve Z✝\nx✝ : FamilyOfElements (G✝.op ⋙ ℱ✝.val) T✝\nh : Compatible x✝\nC : Type u₁\ninst✝² : Category.{v₁, u₁} C\nD : Type u₁\ninst✝¹ : Category.{v₁, u₁} D\nK : GrothendieckTopology D\nG : C ⥤ D\ninst✝ : RepresentablyFlat G\nℱ : SheafOfTypes K\nZ : C\nT : Presieve Z\nx : FamilyOfElements (G.op ⋙ ℱ.val) T\nhx : Compatible x\nY₁ Y₂ : C\nX : D\nf₁ : X ⟶ G.obj Y₁\nf₂ : X ⟶ G.obj Y₂\ng₁ : Y₁ ⟶ Z\ng₂ : Y₂ ⟶ Z\nhg₁ : T g₁\nhg₂ : T g₂\ne : f₁ ≫ G.map g₁ = f₂ ≫ G.map g₂\nc : Cone (cospan g₁ g₂ ⋙ G) :=\n (Cones.postcompose (diagramIsoCospan (cospan g₁ g₂ ⋙ G)).inv).obj (PullbackCone.mk f₁ f₂ e)\nc' : Cone (Cone.toStructuredArrow c ⋙ StructuredArrow.pre c.pt (cospan g₁ g₂) G) :=\n IsCofiltered.cone (Cone.toStructuredArrow c ⋙ StructuredArrow.pre c.pt (cospan g₁ g₂) G)\n⊢ G.obj ((Cone.toStructuredArrow c ⋙ StructuredArrow.pre c.pt (cospan g₁ g₂) G).obj left).right = G.obj Y₁\n[PROOFSTEP]\nsimp\n[GOAL]\nC✝ : Type u₁\ninst✝⁵ : Category.{v₁, u₁} C✝\nD✝ : Type u₂\ninst✝⁴ : Category.{v₂, u₂} D✝\nA : Type u₃\ninst✝³ : Category.{v₃, u₃} A\nJ : GrothendieckTopology C✝\nK✝ : GrothendieckTopology D✝\nL : GrothendieckTopology A\nG✝ : C✝ ⥤ D✝\nhG : CompatiblePreserving K✝ G✝\nℱ✝ : SheafOfTypes K✝\nZ✝ : C✝\nT✝ : Presieve Z✝\nx✝ : FamilyOfElements (G✝.op ⋙ ℱ✝.val) T✝\nh : Compatible x✝\nC : Type u₁\ninst✝² : Category.{v₁, u₁} C\nD : Type u₁\ninst✝¹ : Category.{v₁, u₁} D\nK : GrothendieckTopology D\nG : C ⥤ D\ninst✝ : RepresentablyFlat G\nℱ : SheafOfTypes K\nZ : C\nT : Presieve Z\nx : FamilyOfElements (G.op ⋙ ℱ.val) T\nhx : Compatible x\nY₁ Y₂ : C\nX : D\nf₁ : X ⟶ G.obj Y₁\nf₂ : X ⟶ G.obj Y₂\ng₁ : Y₁ ⟶ Z\ng₂ : Y₂ ⟶ Z\nhg₁ : T g₁\nhg₂ : T g₂\ne : f₁ ≫ G.map g₁ = f₂ ≫ G.map g₂\nc : Cone (cospan g₁ g₂ ⋙ G) :=\n (Cones.postcompose (diagramIsoCospan (cospan g₁ g₂ ⋙ G)).inv).obj (PullbackCone.mk f₁ f₂ e)\nc' : Cone (Cone.toStructuredArrow c ⋙ StructuredArrow.pre c.pt (cospan g₁ g₂) G) :=\n IsCofiltered.cone (Cone.toStructuredArrow c ⋙ StructuredArrow.pre c.pt (cospan g₁ g₂) G)\n⊢ f₁ =\n (c'.pt.hom ≫ G.map (NatTrans.app c'.π left).right) ≫\n eqToHom\n (_ :\n G.obj ((StructuredArrow.pre X (cospan g₁ g₂) G).obj (StructuredArrow.mk (f₁ ≫ 𝟙 (G.obj Y₁)))).right =\n G.obj Y₁)\n[PROOFSTEP]\nerw [← (c'.π.app left).w]\n[GOAL]\nC✝ : Type u₁\ninst✝⁵ : Category.{v₁, u₁} C✝\nD✝ : Type u₂\ninst✝⁴ : Category.{v₂, u₂} D✝\nA : Type u₃\ninst✝³ : Category.{v₃, u₃} A\nJ : GrothendieckTopology C✝\nK✝ : GrothendieckTopology D✝\nL : GrothendieckTopology A\nG✝ : C✝ ⥤ D✝\nhG : CompatiblePreserving K✝ G✝\nℱ✝ : SheafOfTypes K✝\nZ✝ : C✝\nT✝ : Presieve Z✝\nx✝ : FamilyOfElements (G✝.op ⋙ ℱ✝.val) T✝\nh : Compatible x✝\nC : Type u₁\ninst✝² : Category.{v₁, u₁} C\nD : Type u₁\ninst✝¹ : Category.{v₁, u₁} D\nK : GrothendieckTopology D\nG : C ⥤ D\ninst✝ : RepresentablyFlat G\nℱ : SheafOfTypes K\nZ : C\nT : Presieve Z\nx : FamilyOfElements (G.op ⋙ ℱ.val) T\nhx : Compatible x\nY₁ Y₂ : C\nX : D\nf₁ : X ⟶ G.obj Y₁\nf₂ : X ⟶ G.obj Y₂\ng₁ : Y₁ ⟶ Z\ng₂ : Y₂ ⟶ Z\nhg₁ : T g₁\nhg₂ : T g₂\ne : f₁ ≫ G.map g₁ = f₂ ≫ G.map g₂\nc : Cone (cospan g₁ g₂ ⋙ G) :=\n (Cones.postcompose (diagramIsoCospan (cospan g₁ g₂ ⋙ G)).inv).obj (PullbackCone.mk f₁ f₂ e)\nc' : Cone (Cone.toStructuredArrow c ⋙ StructuredArrow.pre c.pt (cospan g₁ g₂) G) :=\n IsCofiltered.cone (Cone.toStructuredArrow c ⋙ StructuredArrow.pre c.pt (cospan g₁ g₂) G)\n⊢ f₁ =\n ((Functor.fromPUnit c.pt).map (NatTrans.app c'.π left).left ≫\n ((Cone.toStructuredArrow c ⋙ StructuredArrow.pre c.pt (cospan g₁ g₂) G).obj left).hom) ≫\n eqToHom\n (_ :\n G.obj ((StructuredArrow.pre X (cospan g₁ g₂) G).obj (StructuredArrow.mk (f₁ ≫ 𝟙 (G.obj Y₁)))).right =\n G.obj Y₁)\n[PROOFSTEP]\ndsimp\n[GOAL]\nC✝ : Type u₁\ninst✝⁵ : Category.{v₁, u₁} C✝\nD✝ : Type u₂\ninst✝⁴ : Category.{v₂, u₂} D✝\nA : Type u₃\ninst✝³ : Category.{v₃, u₃} A\nJ : GrothendieckTopology C✝\nK✝ : GrothendieckTopology D✝\nL : GrothendieckTopology A\nG✝ : C✝ ⥤ D✝\nhG : CompatiblePreserving K✝ G✝\nℱ✝ : SheafOfTypes K✝\nZ✝ : C✝\nT✝ : Presieve Z✝\nx✝ : FamilyOfElements (G✝.op ⋙ ℱ✝.val) T✝\nh : Compatible x✝\nC : Type u₁\ninst✝² : Category.{v₁, u₁} C\nD : Type u₁\ninst✝¹ : Category.{v₁, u₁} D\nK : GrothendieckTopology D\nG : C ⥤ D\ninst✝ : RepresentablyFlat G\nℱ : SheafOfTypes K\nZ : C\nT : Presieve Z\nx : FamilyOfElements (G.op ⋙ ℱ.val) T\nhx : Compatible x\nY₁ Y₂ : C\nX : D\nf₁ : X ⟶ G.obj Y₁\nf₂ : X ⟶ G.obj Y₂\ng₁ : Y₁ ⟶ Z\ng₂ : Y₂ ⟶ Z\nhg₁ : T g₁\nhg₂ : T g₂\ne : f₁ ≫ G.map g₁ = f₂ ≫ G.map g₂\nc : Cone (cospan g₁ g₂ ⋙ G) :=\n (Cones.postcompose (diagramIsoCospan (cospan g₁ g₂ ⋙ G)).inv).obj (PullbackCone.mk f₁ f₂ e)\nc' : Cone (Cone.toStructuredArrow c ⋙ StructuredArrow.pre c.pt (cospan g₁ g₂) G) :=\n IsCofiltered.cone (Cone.toStructuredArrow c ⋙ StructuredArrow.pre c.pt (cospan g₁ g₂) G)\n⊢ f₁ = (𝟙 X ≫ f₁ ≫ 𝟙 (G.obj Y₁)) ≫ 𝟙 (G.obj Y₁)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase Compatible\nC✝ : Type u₁\ninst✝⁵ : Category.{v₁, u₁} C✝\nD✝ : Type u₂\ninst✝⁴ : Category.{v₂, u₂} D✝\nA : Type u₃\ninst✝³ : Category.{v₃, u₃} A\nJ : GrothendieckTopology C✝\nK✝ : GrothendieckTopology D✝\nL : GrothendieckTopology A\nG✝ : C✝ ⥤ D✝\nhG : CompatiblePreserving K✝ G✝\nℱ✝ : SheafOfTypes K✝\nZ✝ : C✝\nT✝ : Presieve Z✝\nx✝ : FamilyOfElements (G✝.op ⋙ ℱ✝.val) T✝\nh : Compatible x✝\nC : Type u₁\ninst✝² : Category.{v₁, u₁} C\nD : Type u₁\ninst✝¹ : Category.{v₁, u₁} D\nK : GrothendieckTopology D\nG : C ⥤ D\ninst✝ : RepresentablyFlat G\nℱ : SheafOfTypes K\nZ : C\nT : Presieve Z\nx : FamilyOfElements (G.op ⋙ ℱ.val) T\nhx : Compatible x\nY₁ Y₂ : C\nX : D\nf₁ : X ⟶ G.obj Y₁\nf₂ : X ⟶ G.obj Y₂\ng₁ : Y₁ ⟶ Z\ng₂ : Y₂ ⟶ Z\nhg₁ : T g₁\nhg₂ : T g₂\ne : f₁ ≫ G.map g₁ = f₂ ≫ G.map g₂\nc : Cone (cospan g₁ g₂ ⋙ G) :=\n (Cones.postcompose (diagramIsoCospan (cospan g₁ g₂ ⋙ G)).inv).obj (PullbackCone.mk f₁ f₂ e)\nc' : Cone (Cone.toStructuredArrow c ⋙ StructuredArrow.pre c.pt (cospan g₁ g₂) G) :=\n IsCofiltered.cone (Cone.toStructuredArrow c ⋙ StructuredArrow.pre c.pt (cospan g₁ g₂) G)\neq₁ :\n f₁ =\n (c'.pt.hom ≫ G.map (NatTrans.app c'.π left).right) ≫\n eqToHom\n (_ :\n G.obj ((StructuredArrow.pre X (cospan g₁ g₂) G).obj (StructuredArrow.mk (f₁ ≫ 𝟙 (G.obj Y₁)))).right =\n G.obj Y₁)\n⊢ ℱ.val.map f₁.op (x g₁ hg₁) = ℱ.val.map f₂.op (x g₂ hg₂)\n[PROOFSTEP]\nhave eq₂ : f₂ = (c'.pt.hom ≫ G.map (c'.π.app right).right) ≫ eqToHom (by simp) :=\n by\n erw [← (c'.π.app right).w]\n dsimp\n simp\n[GOAL]\nC✝ : Type u₁\ninst✝⁵ : Category.{v₁, u₁} C✝\nD✝ : Type u₂\ninst✝⁴ : Category.{v₂, u₂} D✝\nA : Type u₃\ninst✝³ : Category.{v₃, u₃} A\nJ : GrothendieckTopology C✝\nK✝ : GrothendieckTopology D✝\nL : GrothendieckTopology A\nG✝ : C✝ ⥤ D✝\nhG : CompatiblePreserving K✝ G✝\nℱ✝ : SheafOfTypes K✝\nZ✝ : C✝\nT✝ : Presieve Z✝\nx✝ : FamilyOfElements (G✝.op ⋙ ℱ✝.val) T✝\nh : Compatible x✝\nC : Type u₁\ninst✝² : Category.{v₁, u₁} C\nD : Type u₁\ninst✝¹ : Category.{v₁, u₁} D\nK : GrothendieckTopology D\nG : C ⥤ D\ninst✝ : RepresentablyFlat G\nℱ : SheafOfTypes K\nZ : C\nT : Presieve Z\nx : FamilyOfElements (G.op ⋙ ℱ.val) T\nhx : Compatible x\nY₁ Y₂ : C\nX : D\nf₁ : X ⟶ G.obj Y₁\nf₂ : X ⟶ G.obj Y₂\ng₁ : Y₁ ⟶ Z\ng₂ : Y₂ ⟶ Z\nhg₁ : T g₁\nhg₂ : T g₂\ne : f₁ ≫ G.map g₁ = f₂ ≫ G.map g₂\nc : Cone (cospan g₁ g₂ ⋙ G) :=\n (Cones.postcompose (diagramIsoCospan (cospan g₁ g₂ ⋙ G)).inv).obj (PullbackCone.mk f₁ f₂ e)\nc' : Cone (Cone.toStructuredArrow c ⋙ StructuredArrow.pre c.pt (cospan g₁ g₂) G) :=\n IsCofiltered.cone (Cone.toStructuredArrow c ⋙ StructuredArrow.pre c.pt (cospan g₁ g₂) G)\neq₁ :\n f₁ =\n (c'.pt.hom ≫ G.map (NatTrans.app c'.π left).right) ≫\n eqToHom\n (_ :\n G.obj ((StructuredArrow.pre X (cospan g₁ g₂) G).obj (StructuredArrow.mk (f₁ ≫ 𝟙 (G.obj Y₁)))).right =\n G.obj Y₁)\n⊢ G.obj ((Cone.toStructuredArrow c ⋙ StructuredArrow.pre c.pt (cospan g₁ g₂) G).obj right).right = G.obj Y₂\n[PROOFSTEP]\nsimp\n[GOAL]\nC✝ : Type u₁\ninst✝⁵ : Category.{v₁, u₁} C✝\nD✝ : Type u₂\ninst✝⁴ : Category.{v₂, u₂} D✝\nA : Type u₃\ninst✝³ : Category.{v₃, u₃} A\nJ : GrothendieckTopology C✝\nK✝ : GrothendieckTopology D✝\nL : GrothendieckTopology A\nG✝ : C✝ ⥤ D✝\nhG : CompatiblePreserving K✝ G✝\nℱ✝ : SheafOfTypes K✝\nZ✝ : C✝\nT✝ : Presieve Z✝\nx✝ : FamilyOfElements (G✝.op ⋙ ℱ✝.val) T✝\nh : Compatible x✝\nC : Type u₁\ninst✝² : Category.{v₁, u₁} C\nD : Type u₁\ninst✝¹ : Category.{v₁, u₁} D\nK : GrothendieckTopology D\nG : C ⥤ D\ninst✝ : RepresentablyFlat G\nℱ : SheafOfTypes K\nZ : C\nT : Presieve Z\nx : FamilyOfElements (G.op ⋙ ℱ.val) T\nhx : Compatible x\nY₁ Y₂ : C\nX : D\nf₁ : X ⟶ G.obj Y₁\nf₂ : X ⟶ G.obj Y₂\ng₁ : Y₁ ⟶ Z\ng₂ : Y₂ ⟶ Z\nhg₁ : T g₁\nhg₂ : T g₂\ne : f₁ ≫ G.map g₁ = f₂ ≫ G.map g₂\nc : Cone (cospan g₁ g₂ ⋙ G) :=\n (Cones.postcompose (diagramIsoCospan (cospan g₁ g₂ ⋙ G)).inv).obj (PullbackCone.mk f₁ f₂ e)\nc' : Cone (Cone.toStructuredArrow c ⋙ StructuredArrow.pre c.pt (cospan g₁ g₂) G) :=\n IsCofiltered.cone (Cone.toStructuredArrow c ⋙ StructuredArrow.pre c.pt (cospan g₁ g₂) G)\neq₁ :\n f₁ =\n (c'.pt.hom ≫ G.map (NatTrans.app c'.π left).right) ≫\n eqToHom\n (_ :\n G.obj ((StructuredArrow.pre X (cospan g₁ g₂) G).obj (StructuredArrow.mk (f₁ ≫ 𝟙 (G.obj Y₁)))).right =\n G.obj Y₁)\n⊢ f₂ =\n (c'.pt.hom ≫ G.map (NatTrans.app c'.π right).right) ≫\n eqToHom\n (_ :\n G.obj ((StructuredArrow.pre X (cospan g₁ g₂) G).obj (StructuredArrow.mk (f₂ ≫ 𝟙 (G.obj Y₂)))).right =\n G.obj Y₂)\n[PROOFSTEP]\nerw [← (c'.π.app right).w]\n[GOAL]\nC✝ : Type u₁\ninst✝⁵ : Category.{v₁, u₁} C✝\nD✝ : Type u₂\ninst✝⁴ : Category.{v₂, u₂} D✝\nA : Type u₃\ninst✝³ : Category.{v₃, u₃} A\nJ : GrothendieckTopology C✝\nK✝ : GrothendieckTopology D✝\nL : GrothendieckTopology A\nG✝ : C✝ ⥤ D✝\nhG : CompatiblePreserving K✝ G✝\nℱ✝ : SheafOfTypes K✝\nZ✝ : C✝\nT✝ : Presieve Z✝\nx✝ : FamilyOfElements (G✝.op ⋙ ℱ✝.val) T✝\nh : Compatible x✝\nC : Type u₁\ninst✝² : Category.{v₁, u₁} C\nD : Type u₁\ninst✝¹ : Category.{v₁, u₁} D\nK : GrothendieckTopology D\nG : C ⥤ D\ninst✝ : RepresentablyFlat G\nℱ : SheafOfTypes K\nZ : C\nT : Presieve Z\nx : FamilyOfElements (G.op ⋙ ℱ.val) T\nhx : Compatible x\nY₁ Y₂ : C\nX : D\nf₁ : X ⟶ G.obj Y₁\nf₂ : X ⟶ G.obj Y₂\ng₁ : Y₁ ⟶ Z\ng₂ : Y₂ ⟶ Z\nhg₁ : T g₁\nhg₂ : T g₂\ne : f₁ ≫ G.map g₁ = f₂ ≫ G.map g₂\nc : Cone (cospan g₁ g₂ ⋙ G) :=\n (Cones.postcompose (diagramIsoCospan (cospan g₁ g₂ ⋙ G)).inv).obj (PullbackCone.mk f₁ f₂ e)\nc' : Cone (Cone.toStructuredArrow c ⋙ StructuredArrow.pre c.pt (cospan g₁ g₂) G) :=\n IsCofiltered.cone (Cone.toStructuredArrow c ⋙ StructuredArrow.pre c.pt (cospan g₁ g₂) G)\neq₁ :\n f₁ =\n (c'.pt.hom ≫ G.map (NatTrans.app c'.π left).right) ≫\n eqToHom\n (_ :\n G.obj ((StructuredArrow.pre X (cospan g₁ g₂) G).obj (StructuredArrow.mk (f₁ ≫ 𝟙 (G.obj Y₁)))).right =\n G.obj Y₁)\n⊢ f₂ =\n ((Functor.fromPUnit c.pt).map (NatTrans.app c'.π right).left ≫\n ((Cone.toStructuredArrow c ⋙ StructuredArrow.pre c.pt (cospan g₁ g₂) G).obj right).hom) ≫\n eqToHom\n (_ :\n G.obj ((StructuredArrow.pre X (cospan g₁ g₂) G).obj (StructuredArrow.mk (f₂ ≫ 𝟙 (G.obj Y₂)))).right =\n G.obj Y₂)\n[PROOFSTEP]\ndsimp\n[GOAL]\nC✝ : Type u₁\ninst✝⁵ : Category.{v₁, u₁} C✝\nD✝ : Type u₂\ninst✝⁴ : Category.{v₂, u₂} D✝\nA : Type u₃\ninst✝³ : Category.{v₃, u₃} A\nJ : GrothendieckTopology C✝\nK✝ : GrothendieckTopology D✝\nL : GrothendieckTopology A\nG✝ : C✝ ⥤ D✝\nhG : CompatiblePreserving K✝ G✝\nℱ✝ : SheafOfTypes K✝\nZ✝ : C✝\nT✝ : Presieve Z✝\nx✝ : FamilyOfElements (G✝.op ⋙ ℱ✝.val) T✝\nh : Compatible x✝\nC : Type u₁\ninst✝² : Category.{v₁, u₁} C\nD : Type u₁\ninst✝¹ : Category.{v₁, u₁} D\nK : GrothendieckTopology D\nG : C ⥤ D\ninst✝ : RepresentablyFlat G\nℱ : SheafOfTypes K\nZ : C\nT : Presieve Z\nx : FamilyOfElements (G.op ⋙ ℱ.val) T\nhx : Compatible x\nY₁ Y₂ : C\nX : D\nf₁ : X ⟶ G.obj Y₁\nf₂ : X ⟶ G.obj Y₂\ng₁ : Y₁ ⟶ Z\ng₂ : Y₂ ⟶ Z\nhg₁ : T g₁\nhg₂ : T g₂\ne : f₁ ≫ G.map g₁ = f₂ ≫ G.map g₂\nc : Cone (cospan g₁ g₂ ⋙ G) :=\n (Cones.postcompose (diagramIsoCospan (cospan g₁ g₂ ⋙ G)).inv).obj (PullbackCone.mk f₁ f₂ e)\nc' : Cone (Cone.toStructuredArrow c ⋙ StructuredArrow.pre c.pt (cospan g₁ g₂) G) :=\n IsCofiltered.cone (Cone.toStructuredArrow c ⋙ StructuredArrow.pre c.pt (cospan g₁ g₂) G)\neq₁ :\n f₁ =\n (c'.pt.hom ≫ G.map (NatTrans.app c'.π left).right) ≫\n eqToHom\n (_ :\n G.obj ((StructuredArrow.pre X (cospan g₁ g₂) G).obj (StructuredArrow.mk (f₁ ≫ 𝟙 (G.obj Y₁)))).right =\n G.obj Y₁)\n⊢ f₂ = (𝟙 X ≫ f₂ ≫ 𝟙 (G.obj Y₂)) ≫ 𝟙 (G.obj Y₂)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase Compatible\nC✝ : Type u₁\ninst✝⁵ : Category.{v₁, u₁} C✝\nD✝ : Type u₂\ninst✝⁴ : Category.{v₂, u₂} D✝\nA : Type u₃\ninst✝³ : Category.{v₃, u₃} A\nJ : GrothendieckTopology C✝\nK✝ : GrothendieckTopology D✝\nL : GrothendieckTopology A\nG✝ : C✝ ⥤ D✝\nhG : CompatiblePreserving K✝ G✝\nℱ✝ : SheafOfTypes K✝\nZ✝ : C✝\nT✝ : Presieve Z✝\nx✝ : FamilyOfElements (G✝.op ⋙ ℱ✝.val) T✝\nh : Compatible x✝\nC : Type u₁\ninst✝² : Category.{v₁, u₁} C\nD : Type u₁\ninst✝¹ : Category.{v₁, u₁} D\nK : GrothendieckTopology D\nG : C ⥤ D\ninst✝ : RepresentablyFlat G\nℱ : SheafOfTypes K\nZ : C\nT : Presieve Z\nx : FamilyOfElements (G.op ⋙ ℱ.val) T\nhx : Compatible x\nY₁ Y₂ : C\nX : D\nf₁ : X ⟶ G.obj Y₁\nf₂ : X ⟶ G.obj Y₂\ng₁ : Y₁ ⟶ Z\ng₂ : Y₂ ⟶ Z\nhg₁ : T g₁\nhg₂ : T g₂\ne : f₁ ≫ G.map g₁ = f₂ ≫ G.map g₂\nc : Cone (cospan g₁ g₂ ⋙ G) :=\n (Cones.postcompose (diagramIsoCospan (cospan g₁ g₂ ⋙ G)).inv).obj (PullbackCone.mk f₁ f₂ e)\nc' : Cone (Cone.toStructuredArrow c ⋙ StructuredArrow.pre c.pt (cospan g₁ g₂) G) :=\n IsCofiltered.cone (Cone.toStructuredArrow c ⋙ StructuredArrow.pre c.pt (cospan g₁ g₂) G)\neq₁ :\n f₁ =\n (c'.pt.hom ≫ G.map (NatTrans.app c'.π left).right) ≫\n eqToHom\n (_ :\n G.obj ((StructuredArrow.pre X (cospan g₁ g₂) G).obj (StructuredArrow.mk (f₁ ≫ 𝟙 (G.obj Y₁)))).right =\n G.obj Y₁)\neq₂ :\n f₂ =\n (c'.pt.hom ≫ G.map (NatTrans.app c'.π right).right) ≫\n eqToHom\n (_ :\n G.obj ((StructuredArrow.pre X (cospan g₁ g₂) G).obj (StructuredArrow.mk (f₂ ≫ 𝟙 (G.obj Y₂)))).right =\n G.obj Y₂)\n⊢ ℱ.val.map f₁.op (x g₁ hg₁) = ℱ.val.map f₂.op (x g₂ hg₂)\n[PROOFSTEP]\nconv_lhs => rw [eq₁]\n[GOAL]\nC✝ : Type u₁\ninst✝⁵ : Category.{v₁, u₁} C✝\nD✝ : Type u₂\ninst✝⁴ : Category.{v₂, u₂} D✝\nA : Type u₃\ninst✝³ : Category.{v₃, u₃} A\nJ : GrothendieckTopology C✝\nK✝ : GrothendieckTopology D✝\nL : GrothendieckTopology A\nG✝ : C✝ ⥤ D✝\nhG : CompatiblePreserving K✝ G✝\nℱ✝ : SheafOfTypes K✝\nZ✝ : C✝\nT✝ : Presieve Z✝\nx✝ : FamilyOfElements (G✝.op ⋙ ℱ✝.val) T✝\nh : Compatible x✝\nC : Type u₁\ninst✝² : Category.{v₁, u₁} C\nD : Type u₁\ninst✝¹ : Category.{v₁, u₁} D\nK : GrothendieckTopology D\nG : C ⥤ D\ninst✝ : RepresentablyFlat G\nℱ : SheafOfTypes K\nZ : C\nT : Presieve Z\nx : FamilyOfElements (G.op ⋙ ℱ.val) T\nhx : Compatible x\nY₁ Y₂ : C\nX : D\nf₁ : X ⟶ G.obj Y₁\nf₂ : X ⟶ G.obj Y₂\ng₁ : Y₁ ⟶ Z\ng₂ : Y₂ ⟶ Z\nhg₁ : T g₁\nhg₂ : T g₂\ne : f₁ ≫ G.map g₁ = f₂ ≫ G.map g₂\nc : Cone (cospan g₁ g₂ ⋙ G) :=\n (Cones.postcompose (diagramIsoCospan (cospan g₁ g₂ ⋙ G)).inv).obj (PullbackCone.mk f₁ f₂ e)\nc' : Cone (Cone.toStructuredArrow c ⋙ StructuredArrow.pre c.pt (cospan g₁ g₂) G) :=\n IsCofiltered.cone (Cone.toStructuredArrow c ⋙ StructuredArrow.pre c.pt (cospan g₁ g₂) G)\neq₁ :\n f₁ =\n (c'.pt.hom ≫ G.map (NatTrans.app c'.π left).right) ≫\n eqToHom\n (_ :\n G.obj ((StructuredArrow.pre X (cospan g₁ g₂) G).obj (StructuredArrow.mk (f₁ ≫ 𝟙 (G.obj Y₁)))).right =\n G.obj Y₁)\neq₂ :\n f₂ =\n (c'.pt.hom ≫ G.map (NatTrans.app c'.π right).right) ≫\n eqToHom\n (_ :\n G.obj ((StructuredArrow.pre X (cospan g₁ g₂) G).obj (StructuredArrow.mk (f₂ ≫ 𝟙 (G.obj Y₂)))).right =\n G.obj Y₂)\n| ℱ.val.map f₁.op (x g₁ hg₁)\n[PROOFSTEP]\nrw [eq₁]\n[GOAL]\nC✝ : Type u₁\ninst✝⁵ : Category.{v₁, u₁} C✝\nD✝ : Type u₂\ninst✝⁴ : Category.{v₂, u₂} D✝\nA : Type u₃\ninst✝³ : Category.{v₃, u₃} A\nJ : GrothendieckTopology C✝\nK✝ : GrothendieckTopology D✝\nL : GrothendieckTopology A\nG✝ : C✝ ⥤ D✝\nhG : CompatiblePreserving K✝ G✝\nℱ✝ : SheafOfTypes K✝\nZ✝ : C✝\nT✝ : Presieve Z✝\nx✝ : FamilyOfElements (G✝.op ⋙ ℱ✝.val) T✝\nh : Compatible x✝\nC : Type u₁\ninst✝² : Category.{v₁, u₁} C\nD : Type u₁\ninst✝¹ : Category.{v₁, u₁} D\nK : GrothendieckTopology D\nG : C ⥤ D\ninst✝ : RepresentablyFlat G\nℱ : SheafOfTypes K\nZ : C\nT : Presieve Z\nx : FamilyOfElements (G.op ⋙ ℱ.val) T\nhx : Compatible x\nY₁ Y₂ : C\nX : D\nf₁ : X ⟶ G.obj Y₁\nf₂ : X ⟶ G.obj Y₂\ng₁ : Y₁ ⟶ Z\ng₂ : Y₂ ⟶ Z\nhg₁ : T g₁\nhg₂ : T g₂\ne : f₁ ≫ G.map g₁ = f₂ ≫ G.map g₂\nc : Cone (cospan g₁ g₂ ⋙ G) :=\n (Cones.postcompose (diagramIsoCospan (cospan g₁ g₂ ⋙ G)).inv).obj (PullbackCone.mk f₁ f₂ e)\nc' : Cone (Cone.toStructuredArrow c ⋙ StructuredArrow.pre c.pt (cospan g₁ g₂) G) :=\n IsCofiltered.cone (Cone.toStructuredArrow c ⋙ StructuredArrow.pre c.pt (cospan g₁ g₂) G)\neq₁ :\n f₁ =\n (c'.pt.hom ≫ G.map (NatTrans.app c'.π left).right) ≫\n eqToHom\n (_ :\n G.obj ((StructuredArrow.pre X (cospan g₁ g₂) G).obj (StructuredArrow.mk (f₁ ≫ 𝟙 (G.obj Y₁)))).right =\n G.obj Y₁)\neq₂ :\n f₂ =\n (c'.pt.hom ≫ G.map (NatTrans.app c'.π right).right) ≫\n eqToHom\n (_ :\n G.obj ((StructuredArrow.pre X (cospan g₁ g₂) G).obj (StructuredArrow.mk (f₂ ≫ 𝟙 (G.obj Y₂)))).right =\n G.obj Y₂)\n| ℱ.val.map f₁.op (x g₁ hg₁)\n[PROOFSTEP]\nrw [eq₁]\n[GOAL]\nC✝ : Type u₁\ninst✝⁵ : Category.{v₁, u₁} C✝\nD✝ : Type u₂\ninst✝⁴ : Category.{v₂, u₂} D✝\nA : Type u₃\ninst✝³ : Category.{v₃, u₃} A\nJ : GrothendieckTopology C✝\nK✝ : GrothendieckTopology D✝\nL : GrothendieckTopology A\nG✝ : C✝ ⥤ D✝\nhG : CompatiblePreserving K✝ G✝\nℱ✝ : SheafOfTypes K✝\nZ✝ : C✝\nT✝ : Presieve Z✝\nx✝ : FamilyOfElements (G✝.op ⋙ ℱ✝.val) T✝\nh : Compatible x✝\nC : Type u₁\ninst✝² : Category.{v₁, u₁} C\nD : Type u₁\ninst✝¹ : Category.{v₁, u₁} D\nK : GrothendieckTopology D\nG : C ⥤ D\ninst✝ : RepresentablyFlat G\nℱ : SheafOfTypes K\nZ : C\nT : Presieve Z\nx : FamilyOfElements (G.op ⋙ ℱ.val) T\nhx : Compatible x\nY₁ Y₂ : C\nX : D\nf₁ : X ⟶ G.obj Y₁\nf₂ : X ⟶ G.obj Y₂\ng₁ : Y₁ ⟶ Z\ng₂ : Y₂ ⟶ Z\nhg₁ : T g₁\nhg₂ : T g₂\ne : f₁ ≫ G.map g₁ = f₂ ≫ G.map g₂\nc : Cone (cospan g₁ g₂ ⋙ G) :=\n (Cones.postcompose (diagramIsoCospan (cospan g₁ g₂ ⋙ G)).inv).obj (PullbackCone.mk f₁ f₂ e)\nc' : Cone (Cone.toStructuredArrow c ⋙ StructuredArrow.pre c.pt (cospan g₁ g₂) G) :=\n IsCofiltered.cone (Cone.toStructuredArrow c ⋙ StructuredArrow.pre c.pt (cospan g₁ g₂) G)\neq₁ :\n f₁ =\n (c'.pt.hom ≫ G.map (NatTrans.app c'.π left).right) ≫\n eqToHom\n (_ :\n G.obj ((StructuredArrow.pre X (cospan g₁ g₂) G).obj (StructuredArrow.mk (f₁ ≫ 𝟙 (G.obj Y₁)))).right =\n G.obj Y₁)\neq₂ :\n f₂ =\n (c'.pt.hom ≫ G.map (NatTrans.app c'.π right).right) ≫\n eqToHom\n (_ :\n G.obj ((StructuredArrow.pre X (cospan g₁ g₂) G).obj (StructuredArrow.mk (f₂ ≫ 𝟙 (G.obj Y₂)))).right =\n G.obj Y₂)\n| ℱ.val.map f₁.op (x g₁ hg₁)\n[PROOFSTEP]\nrw [eq₁]\n[GOAL]\ncase Compatible\nC✝ : Type u₁\ninst✝⁵ : Category.{v₁, u₁} C✝\nD✝ : Type u₂\ninst✝⁴ : Category.{v₂, u₂} D✝\nA : Type u₃\ninst✝³ : Category.{v₃, u₃} A\nJ : GrothendieckTopology C✝\nK✝ : GrothendieckTopology D✝\nL : GrothendieckTopology A\nG✝ : C✝ ⥤ D✝\nhG : CompatiblePreserving K✝ G✝\nℱ✝ : SheafOfTypes K✝\nZ✝ : C✝\nT✝ : Presieve Z✝\nx✝ : FamilyOfElements (G✝.op ⋙ ℱ✝.val) T✝\nh : Compatible x✝\nC : Type u₁\ninst✝² : Category.{v₁, u₁} C\nD : Type u₁\ninst✝¹ : Category.{v₁, u₁} D\nK : GrothendieckTopology D\nG : C ⥤ D\ninst✝ : RepresentablyFlat G\nℱ : SheafOfTypes K\nZ : C\nT : Presieve Z\nx : FamilyOfElements (G.op ⋙ ℱ.val) T\nhx : Compatible x\nY₁ Y₂ : C\nX : D\nf₁ : X ⟶ G.obj Y₁\nf₂ : X ⟶ G.obj Y₂\ng₁ : Y₁ ⟶ Z\ng₂ : Y₂ ⟶ Z\nhg₁ : T g₁\nhg₂ : T g₂\ne : f₁ ≫ G.map g₁ = f₂ ≫ G.map g₂\nc : Cone (cospan g₁ g₂ ⋙ G) :=\n (Cones.postcompose (diagramIsoCospan (cospan g₁ g₂ ⋙ G)).inv).obj (PullbackCone.mk f₁ f₂ e)\nc' : Cone (Cone.toStructuredArrow c ⋙ StructuredArrow.pre c.pt (cospan g₁ g₂) G) :=\n IsCofiltered.cone (Cone.toStructuredArrow c ⋙ StructuredArrow.pre c.pt (cospan g₁ g₂) G)\neq₁ :\n f₁ =\n (c'.pt.hom ≫ G.map (NatTrans.app c'.π left).right) ≫\n eqToHom\n (_ :\n G.obj ((StructuredArrow.pre X (cospan g₁ g₂) G).obj (StructuredArrow.mk (f₁ ≫ 𝟙 (G.obj Y₁)))).right =\n G.obj Y₁)\neq₂ :\n f₂ =\n (c'.pt.hom ≫ G.map (NatTrans.app c'.π right).right) ≫\n eqToHom\n (_ :\n G.obj ((StructuredArrow.pre X (cospan g₁ g₂) G).obj (StructuredArrow.mk (f₂ ≫ 𝟙 (G.obj Y₂)))).right =\n G.obj Y₂)\n⊢ ℱ.val.map\n ((c'.pt.hom ≫ G.map (NatTrans.app c'.π left).right) ≫\n eqToHom\n (_ :\n G.obj ((StructuredArrow.pre X (cospan g₁ g₂) G).obj (StructuredArrow.mk (f₁ ≫ 𝟙 (G.obj Y₁)))).right =\n G.obj Y₁)).op\n (x g₁ hg₁) =\n ℱ.val.map f₂.op (x g₂ hg₂)\n[PROOFSTEP]\nconv_rhs => rw [eq₂]\n[GOAL]\nC✝ : Type u₁\ninst✝⁵ : Category.{v₁, u₁} C✝\nD✝ : Type u₂\ninst✝⁴ : Category.{v₂, u₂} D✝\nA : Type u₃\ninst✝³ : Category.{v₃, u₃} A\nJ : GrothendieckTopology C✝\nK✝ : GrothendieckTopology D✝\nL : GrothendieckTopology A\nG✝ : C✝ ⥤ D✝\nhG : CompatiblePreserving K✝ G✝\nℱ✝ : SheafOfTypes K✝\nZ✝ : C✝\nT✝ : Presieve Z✝\nx✝ : FamilyOfElements (G✝.op ⋙ ℱ✝.val) T✝\nh : Compatible x✝\nC : Type u₁\ninst✝² : Category.{v₁, u₁} C\nD : Type u₁\ninst✝¹ : Category.{v₁, u₁} D\nK : GrothendieckTopology D\nG : C ⥤ D\ninst✝ : RepresentablyFlat G\nℱ : SheafOfTypes K\nZ : C\nT : Presieve Z\nx : FamilyOfElements (G.op ⋙ ℱ.val) T\nhx : Compatible x\nY₁ Y₂ : C\nX : D\nf₁ : X ⟶ G.obj Y₁\nf₂ : X ⟶ G.obj Y₂\ng₁ : Y₁ ⟶ Z\ng₂ : Y₂ ⟶ Z\nhg₁ : T g₁\nhg₂ : T g₂\ne : f₁ ≫ G.map g₁ = f₂ ≫ G.map g₂\nc : Cone (cospan g₁ g₂ ⋙ G) :=\n (Cones.postcompose (diagramIsoCospan (cospan g₁ g₂ ⋙ G)).inv).obj (PullbackCone.mk f₁ f₂ e)\nc' : Cone (Cone.toStructuredArrow c ⋙ StructuredArrow.pre c.pt (cospan g₁ g₂) G) :=\n IsCofiltered.cone (Cone.toStructuredArrow c ⋙ StructuredArrow.pre c.pt (cospan g₁ g₂) G)\neq₁ :\n f₁ =\n (c'.pt.hom ≫ G.map (NatTrans.app c'.π left).right) ≫\n eqToHom\n (_ :\n G.obj ((StructuredArrow.pre X (cospan g₁ g₂) G).obj (StructuredArrow.mk (f₁ ≫ 𝟙 (G.obj Y₁)))).right =\n G.obj Y₁)\neq₂ :\n f₂ =\n (c'.pt.hom ≫ G.map (NatTrans.app c'.π right).right) ≫\n eqToHom\n (_ :\n G.obj ((StructuredArrow.pre X (cospan g₁ g₂) G).obj (StructuredArrow.mk (f₂ ≫ 𝟙 (G.obj Y₂)))).right =\n G.obj Y₂)\n| ℱ.val.map f₂.op (x g₂ hg₂)\n[PROOFSTEP]\nrw [eq₂]\n[GOAL]\nC✝ : Type u₁\ninst✝⁵ : Category.{v₁, u₁} C✝\nD✝ : Type u₂\ninst✝⁴ : Category.{v₂, u₂} D✝\nA : Type u₃\ninst✝³ : Category.{v₃, u₃} A\nJ : GrothendieckTopology C✝\nK✝ : GrothendieckTopology D✝\nL : GrothendieckTopology A\nG✝ : C✝ ⥤ D✝\nhG : CompatiblePreserving K✝ G✝\nℱ✝ : SheafOfTypes K✝\nZ✝ : C✝\nT✝ : Presieve Z✝\nx✝ : FamilyOfElements (G✝.op ⋙ ℱ✝.val) T✝\nh : Compatible x✝\nC : Type u₁\ninst✝² : Category.{v₁, u₁} C\nD : Type u₁\ninst✝¹ : Category.{v₁, u₁} D\nK : GrothendieckTopology D\nG : C ⥤ D\ninst✝ : RepresentablyFlat G\nℱ : SheafOfTypes K\nZ : C\nT : Presieve Z\nx : FamilyOfElements (G.op ⋙ ℱ.val) T\nhx : Compatible x\nY₁ Y₂ : C\nX : D\nf₁ : X ⟶ G.obj Y₁\nf₂ : X ⟶ G.obj Y₂\ng₁ : Y₁ ⟶ Z\ng₂ : Y₂ ⟶ Z\nhg₁ : T g₁\nhg₂ : T g₂\ne : f₁ ≫ G.map g₁ = f₂ ≫ G.map g₂\nc : Cone (cospan g₁ g₂ ⋙ G) :=\n (Cones.postcompose (diagramIsoCospan (cospan g₁ g₂ ⋙ G)).inv).obj (PullbackCone.mk f₁ f₂ e)\nc' : Cone (Cone.toStructuredArrow c ⋙ StructuredArrow.pre c.pt (cospan g₁ g₂) G) :=\n IsCofiltered.cone (Cone.toStructuredArrow c ⋙ StructuredArrow.pre c.pt (cospan g₁ g₂) G)\neq₁ :\n f₁ =\n (c'.pt.hom ≫ G.map (NatTrans.app c'.π left).right) ≫\n eqToHom\n (_ :\n G.obj ((StructuredArrow.pre X (cospan g₁ g₂) G).obj (StructuredArrow.mk (f₁ ≫ 𝟙 (G.obj Y₁)))).right =\n G.obj Y₁)\neq₂ :\n f₂ =\n (c'.pt.hom ≫ G.map (NatTrans.app c'.π right).right) ≫\n eqToHom\n (_ :\n G.obj ((StructuredArrow.pre X (cospan g₁ g₂) G).obj (StructuredArrow.mk (f₂ ≫ 𝟙 (G.obj Y₂)))).right =\n G.obj Y₂)\n| ℱ.val.map f₂.op (x g₂ hg₂)\n[PROOFSTEP]\nrw [eq₂]\n[GOAL]\nC✝ : Type u₁\ninst✝⁵ : Category.{v₁, u₁} C✝\nD✝ : Type u₂\ninst✝⁴ : Category.{v₂, u₂} D✝\nA : Type u₃\ninst✝³ : Category.{v₃, u₃} A\nJ : GrothendieckTopology C✝\nK✝ : GrothendieckTopology D✝\nL : GrothendieckTopology A\nG✝ : C✝ ⥤ D✝\nhG : CompatiblePreserving K✝ G✝\nℱ✝ : SheafOfTypes K✝\nZ✝ : C✝\nT✝ : Presieve Z✝\nx✝ : FamilyOfElements (G✝.op ⋙ ℱ✝.val) T✝\nh : Compatible x✝\nC : Type u₁\ninst✝² : Category.{v₁, u₁} C\nD : Type u₁\ninst✝¹ : Category.{v₁, u₁} D\nK : GrothendieckTopology D\nG : C ⥤ D\ninst✝ : RepresentablyFlat G\nℱ : SheafOfTypes K\nZ : C\nT : Presieve Z\nx : FamilyOfElements (G.op ⋙ ℱ.val) T\nhx : Compatible x\nY₁ Y₂ : C\nX : D\nf₁ : X ⟶ G.obj Y₁\nf₂ : X ⟶ G.obj Y₂\ng₁ : Y₁ ⟶ Z\ng₂ : Y₂ ⟶ Z\nhg₁ : T g₁\nhg₂ : T g₂\ne : f₁ ≫ G.map g₁ = f₂ ≫ G.map g₂\nc : Cone (cospan g₁ g₂ ⋙ G) :=\n (Cones.postcompose (diagramIsoCospan (cospan g₁ g₂ ⋙ G)).inv).obj (PullbackCone.mk f₁ f₂ e)\nc' : Cone (Cone.toStructuredArrow c ⋙ StructuredArrow.pre c.pt (cospan g₁ g₂) G) :=\n IsCofiltered.cone (Cone.toStructuredArrow c ⋙ StructuredArrow.pre c.pt (cospan g₁ g₂) G)\neq₁ :\n f₁ =\n (c'.pt.hom ≫ G.map (NatTrans.app c'.π left).right) ≫\n eqToHom\n (_ :\n G.obj ((StructuredArrow.pre X (cospan g₁ g₂) G).obj (StructuredArrow.mk (f₁ ≫ 𝟙 (G.obj Y₁)))).right =\n G.obj Y₁)\neq₂ :\n f₂ =\n (c'.pt.hom ≫ G.map (NatTrans.app c'.π right).right) ≫\n eqToHom\n (_ :\n G.obj ((StructuredArrow.pre X (cospan g₁ g₂) G).obj (StructuredArrow.mk (f₂ ≫ 𝟙 (G.obj Y₂)))).right =\n G.obj Y₂)\n| ℱ.val.map f₂.op (x g₂ hg₂)\n[PROOFSTEP]\nrw [eq₂]\n[GOAL]\ncase Compatible\nC✝ : Type u₁\ninst✝⁵ : Category.{v₁, u₁} C✝\nD✝ : Type u₂\ninst✝⁴ : Category.{v₂, u₂} D✝\nA : Type u₃\ninst✝³ : Category.{v₃, u₃} A\nJ : GrothendieckTopology C✝\nK✝ : GrothendieckTopology D✝\nL : GrothendieckTopology A\nG✝ : C✝ ⥤ D✝\nhG : CompatiblePreserving K✝ G✝\nℱ✝ : SheafOfTypes K✝\nZ✝ : C✝\nT✝ : Presieve Z✝\nx✝ : FamilyOfElements (G✝.op ⋙ ℱ✝.val) T✝\nh : Compatible x✝\nC : Type u₁\ninst✝² : Category.{v₁, u₁} C\nD : Type u₁\ninst✝¹ : Category.{v₁, u₁} D\nK : GrothendieckTopology D\nG : C ⥤ D\ninst✝ : RepresentablyFlat G\nℱ : SheafOfTypes K\nZ : C\nT : Presieve Z\nx : FamilyOfElements (G.op ⋙ ℱ.val) T\nhx : Compatible x\nY₁ Y₂ : C\nX : D\nf₁ : X ⟶ G.obj Y₁\nf₂ : X ⟶ G.obj Y₂\ng₁ : Y₁ ⟶ Z\ng₂ : Y₂ ⟶ Z\nhg₁ : T g₁\nhg₂ : T g₂\ne : f₁ ≫ G.map g₁ = f₂ ≫ G.map g₂\nc : Cone (cospan g₁ g₂ ⋙ G) :=\n (Cones.postcompose (diagramIsoCospan (cospan g₁ g₂ ⋙ G)).inv).obj (PullbackCone.mk f₁ f₂ e)\nc' : Cone (Cone.toStructuredArrow c ⋙ StructuredArrow.pre c.pt (cospan g₁ g₂) G) :=\n IsCofiltered.cone (Cone.toStructuredArrow c ⋙ StructuredArrow.pre c.pt (cospan g₁ g₂) G)\neq₁ :\n f₁ =\n (c'.pt.hom ≫ G.map (NatTrans.app c'.π left).right) ≫\n eqToHom\n (_ :\n G.obj ((StructuredArrow.pre X (cospan g₁ g₂) G).obj (StructuredArrow.mk (f₁ ≫ 𝟙 (G.obj Y₁)))).right =\n G.obj Y₁)\neq₂ :\n f₂ =\n (c'.pt.hom ≫ G.map (NatTrans.app c'.π right).right) ≫\n eqToHom\n (_ :\n G.obj ((StructuredArrow.pre X (cospan g₁ g₂) G).obj (StructuredArrow.mk (f₂ ≫ 𝟙 (G.obj Y₂)))).right =\n G.obj Y₂)\n⊢ ℱ.val.map\n ((c'.pt.hom ≫ G.map (NatTrans.app c'.π left).right) ≫\n eqToHom\n (_ :\n G.obj ((StructuredArrow.pre X (cospan g₁ g₂) G).obj (StructuredArrow.mk (f₁ ≫ 𝟙 (G.obj Y₁)))).right =\n G.obj Y₁)).op\n (x g₁ hg₁) =\n ℱ.val.map\n ((c'.pt.hom ≫ G.map (NatTrans.app c'.π right).right) ≫\n eqToHom\n (_ :\n G.obj ((StructuredArrow.pre X (cospan g₁ g₂) G).obj (StructuredArrow.mk (f₂ ≫ 𝟙 (G.obj Y₂)))).right =\n G.obj Y₂)).op\n (x g₂ hg₂)\n[PROOFSTEP]\nsimp only [op_comp, Functor.map_comp, types_comp_apply, eqToHom_op, eqToHom_map]\n[GOAL]\ncase Compatible\nC✝ : Type u₁\ninst✝⁵ : Category.{v₁, u₁} C✝\nD✝ : Type u₂\ninst✝⁴ : Category.{v₂, u₂} D✝\nA : Type u₃\ninst✝³ : Category.{v₃, u₃} A\nJ : GrothendieckTopology C✝\nK✝ : GrothendieckTopology D✝\nL : GrothendieckTopology A\nG✝ : C✝ ⥤ D✝\nhG : CompatiblePreserving K✝ G✝\nℱ✝ : SheafOfTypes K✝\nZ✝ : C✝\nT✝ : Presieve Z✝\nx✝ : FamilyOfElements (G✝.op ⋙ ℱ✝.val) T✝\nh : Compatible x✝\nC : Type u₁\ninst✝² : Category.{v₁, u₁} C\nD : Type u₁\ninst✝¹ : Category.{v₁, u₁} D\nK : GrothendieckTopology D\nG : C ⥤ D\ninst✝ : RepresentablyFlat G\nℱ : SheafOfTypes K\nZ : C\nT : Presieve Z\nx : FamilyOfElements (G.op ⋙ ℱ.val) T\nhx : Compatible x\nY₁ Y₂ : C\nX : D\nf₁ : X ⟶ G.obj Y₁\nf₂ : X ⟶ G.obj Y₂\ng₁ : Y₁ ⟶ Z\ng₂ : Y₂ ⟶ Z\nhg₁ : T g₁\nhg₂ : T g₂\ne : f₁ ≫ G.map g₁ = f₂ ≫ G.map g₂\nc : Cone (cospan g₁ g₂ ⋙ G) :=\n (Cones.postcompose (diagramIsoCospan (cospan g₁ g₂ ⋙ G)).inv).obj (PullbackCone.mk f₁ f₂ e)\nc' : Cone (Cone.toStructuredArrow c ⋙ StructuredArrow.pre c.pt (cospan g₁ g₂) G) :=\n IsCofiltered.cone (Cone.toStructuredArrow c ⋙ StructuredArrow.pre c.pt (cospan g₁ g₂) G)\neq₁ :\n f₁ =\n (c'.pt.hom ≫ G.map (NatTrans.app c'.π left).right) ≫\n eqToHom\n (_ :\n G.obj ((StructuredArrow.pre X (cospan g₁ g₂) G).obj (StructuredArrow.mk (f₁ ≫ 𝟙 (G.obj Y₁)))).right =\n G.obj Y₁)\neq₂ :\n f₂ =\n (c'.pt.hom ≫ G.map (NatTrans.app c'.π right).right) ≫\n eqToHom\n (_ :\n G.obj ((StructuredArrow.pre X (cospan g₁ g₂) G).obj (StructuredArrow.mk (f₂ ≫ 𝟙 (G.obj Y₂)))).right =\n G.obj Y₂)\n⊢ ℱ.val.map\n (IsCofiltered.cone\n (Cone.toStructuredArrow\n ((Cones.postcompose (diagramIsoCospan (cospan g₁ g₂ ⋙ G)).inv).obj (PullbackCone.mk f₁ f₂ e)) ⋙\n StructuredArrow.pre\n ((Cones.postcompose (diagramIsoCospan (cospan g₁ g₂ ⋙ G)).inv).obj (PullbackCone.mk f₁ f₂ e)).pt\n (cospan g₁ g₂) G)).pt.hom.op\n (ℱ.val.map\n (G.map\n (NatTrans.app\n (IsCofiltered.cone\n (Cone.toStructuredArrow\n ((Cones.postcompose (diagramIsoCospan (cospan g₁ g₂ ⋙ G)).inv).obj (PullbackCone.mk f₁ f₂ e)) ⋙\n StructuredArrow.pre\n ((Cones.postcompose (diagramIsoCospan (cospan g₁ g₂ ⋙ G)).inv).obj (PullbackCone.mk f₁ f₂ e)).pt\n (cospan g₁ g₂) G)).π\n left).right).op\n (eqToHom\n (_ :\n ℱ.val.obj (op (G.obj Y₁)) =\n ℱ.val.obj\n (op\n (G.obj\n ((Cone.toStructuredArrow\n ((Cones.postcompose (diagramIsoCospan (cospan g₁ g₂ ⋙ G)).inv).obj\n (PullbackCone.mk f₁ f₂ e)) ⋙\n StructuredArrow.pre\n ((Cones.postcompose (diagramIsoCospan (cospan g₁ g₂ ⋙ G)).inv).obj\n (PullbackCone.mk f₁ f₂ e)).pt\n (cospan g₁ g₂) G).obj\n left).right)))\n (x g₁ hg₁))) =\n ℱ.val.map\n (IsCofiltered.cone\n (Cone.toStructuredArrow\n ((Cones.postcompose (diagramIsoCospan (cospan g₁ g₂ ⋙ G)).inv).obj (PullbackCone.mk f₁ f₂ e)) ⋙\n StructuredArrow.pre\n ((Cones.postcompose (diagramIsoCospan (cospan g₁ g₂ ⋙ G)).inv).obj (PullbackCone.mk f₁ f₂ e)).pt\n (cospan g₁ g₂) G)).pt.hom.op\n (ℱ.val.map\n (G.map\n (NatTrans.app\n (IsCofiltered.cone\n (Cone.toStructuredArrow\n ((Cones.postcompose (diagramIsoCospan (cospan g₁ g₂ ⋙ G)).inv).obj (PullbackCone.mk f₁ f₂ e)) ⋙\n StructuredArrow.pre\n ((Cones.postcompose (diagramIsoCospan (cospan g₁ g₂ ⋙ G)).inv).obj (PullbackCone.mk f₁ f₂ e)).pt\n (cospan g₁ g₂) G)).π\n right).right).op\n (eqToHom\n (_ :\n ℱ.val.obj (op (G.obj Y₂)) =\n ℱ.val.obj\n (op\n (G.obj\n ((Cone.toStructuredArrow\n ((Cones.postcompose (diagramIsoCospan (cospan g₁ g₂ ⋙ G)).inv).obj\n (PullbackCone.mk f₁ f₂ e)) ⋙\n StructuredArrow.pre\n ((Cones.postcompose (diagramIsoCospan (cospan g₁ g₂ ⋙ G)).inv).obj\n (PullbackCone.mk f₁ f₂ e)).pt\n (cospan g₁ g₂) G).obj\n right).right)))\n (x g₂ hg₂)))\n[PROOFSTEP]\napply congr_arg\n[GOAL]\ncase Compatible.h\nC✝ : Type u₁\ninst✝⁵ : Category.{v₁, u₁} C✝\nD✝ : Type u₂\ninst✝⁴ : Category.{v₂, u₂} D✝\nA : Type u₃\ninst✝³ : Category.{v₃, u₃} A\nJ : GrothendieckTopology C✝\nK✝ : GrothendieckTopology D✝\nL : GrothendieckTopology A\nG✝ : C✝ ⥤ D✝\nhG : CompatiblePreserving K✝ G✝\nℱ✝ : SheafOfTypes K✝\nZ✝ : C✝\nT✝ : Presieve Z✝\nx✝ : FamilyOfElements (G✝.op ⋙ ℱ✝.val) T✝\nh : Compatible x✝\nC : Type u₁\ninst✝² : Category.{v₁, u₁} C\nD : Type u₁\ninst✝¹ : Category.{v₁, u₁} D\nK : GrothendieckTopology D\nG : C ⥤ D\ninst✝ : RepresentablyFlat G\nℱ : SheafOfTypes K\nZ : C\nT : Presieve Z\nx : FamilyOfElements (G.op ⋙ ℱ.val) T\nhx : Compatible x\nY₁ Y₂ : C\nX : D\nf₁ : X ⟶ G.obj Y₁\nf₂ : X ⟶ G.obj Y₂\ng₁ : Y₁ ⟶ Z\ng₂ : Y₂ ⟶ Z\nhg₁ : T g₁\nhg₂ : T g₂\ne : f₁ ≫ G.map g₁ = f₂ ≫ G.map g₂\nc : Cone (cospan g₁ g₂ ⋙ G) :=\n (Cones.postcompose (diagramIsoCospan (cospan g₁ g₂ ⋙ G)).inv).obj (PullbackCone.mk f₁ f₂ e)\nc' : Cone (Cone.toStructuredArrow c ⋙ StructuredArrow.pre c.pt (cospan g₁ g₂) G) :=\n IsCofiltered.cone (Cone.toStructuredArrow c ⋙ StructuredArrow.pre c.pt (cospan g₁ g₂) G)\neq₁ :\n f₁ =\n (c'.pt.hom ≫ G.map (NatTrans.app c'.π left).right) ≫\n eqToHom\n (_ :\n G.obj ((StructuredArrow.pre X (cospan g₁ g₂) G).obj (StructuredArrow.mk (f₁ ≫ 𝟙 (G.obj Y₁)))).right =\n G.obj Y₁)\neq₂ :\n f₂ =\n (c'.pt.hom ≫ G.map (NatTrans.app c'.π right).right) ≫\n eqToHom\n (_ :\n G.obj ((StructuredArrow.pre X (cospan g₁ g₂) G).obj (StructuredArrow.mk (f₂ ≫ 𝟙 (G.obj Y₂)))).right =\n G.obj Y₂)\n⊢ ℱ.val.map\n (G.map\n (NatTrans.app\n (IsCofiltered.cone\n (Cone.toStructuredArrow\n ((Cones.postcompose (diagramIsoCospan (cospan g₁ g₂ ⋙ G)).inv).obj (PullbackCone.mk f₁ f₂ e)) ⋙\n StructuredArrow.pre\n ((Cones.postcompose (diagramIsoCospan (cospan g₁ g₂ ⋙ G)).inv).obj (PullbackCone.mk f₁ f₂ e)).pt\n (cospan g₁ g₂) G)).π\n left).right).op\n (eqToHom\n (_ :\n ℱ.val.obj (op (G.obj Y₁)) =\n ℱ.val.obj\n (op\n (G.obj\n ((Cone.toStructuredArrow\n ((Cones.postcompose (diagramIsoCospan (cospan g₁ g₂ ⋙ G)).inv).obj\n (PullbackCone.mk f₁ f₂ e)) ⋙\n StructuredArrow.pre\n ((Cones.postcompose (diagramIsoCospan (cospan g₁ g₂ ⋙ G)).inv).obj\n (PullbackCone.mk f₁ f₂ e)).pt\n (cospan g₁ g₂) G).obj\n left).right)))\n (x g₁ hg₁)) =\n ℱ.val.map\n (G.map\n (NatTrans.app\n (IsCofiltered.cone\n (Cone.toStructuredArrow\n ((Cones.postcompose (diagramIsoCospan (cospan g₁ g₂ ⋙ G)).inv).obj (PullbackCone.mk f₁ f₂ e)) ⋙\n StructuredArrow.pre\n ((Cones.postcompose (diagramIsoCospan (cospan g₁ g₂ ⋙ G)).inv).obj (PullbackCone.mk f₁ f₂ e)).pt\n (cospan g₁ g₂) G)).π\n right).right).op\n (eqToHom\n (_ :\n ℱ.val.obj (op (G.obj Y₂)) =\n ℱ.val.obj\n (op\n (G.obj\n ((Cone.toStructuredArrow\n ((Cones.postcompose (diagramIsoCospan (cospan g₁ g₂ ⋙ G)).inv).obj\n (PullbackCone.mk f₁ f₂ e)) ⋙\n StructuredArrow.pre\n ((Cones.postcompose (diagramIsoCospan (cospan g₁ g₂ ⋙ G)).inv).obj\n (PullbackCone.mk f₁ f₂ e)).pt\n (cospan g₁ g₂) G).obj\n right).right)))\n (x g₂ hg₂))\n[PROOFSTEP]\ninjection c'.π.naturality WalkingCospan.Hom.inl with _ e₁\n[GOAL]\ncase Compatible.h\nC✝ : Type u₁\ninst✝⁵ : Category.{v₁, u₁} C✝\nD✝ : Type u₂\ninst✝⁴ : Category.{v₂, u₂} D✝\nA : Type u₃\ninst✝³ : Category.{v₃, u₃} A\nJ : GrothendieckTopology C✝\nK✝ : GrothendieckTopology D✝\nL : GrothendieckTopology A\nG✝ : C✝ ⥤ D✝\nhG : CompatiblePreserving K✝ G✝\nℱ✝ : SheafOfTypes K✝\nZ✝ : C✝\nT✝ : Presieve Z✝\nx✝ : FamilyOfElements (G✝.op ⋙ ℱ✝.val) T✝\nh : Compatible x✝\nC : Type u₁\ninst✝² : Category.{v₁, u₁} C\nD : Type u₁\ninst✝¹ : Category.{v₁, u₁} D\nK : GrothendieckTopology D\nG : C ⥤ D\ninst✝ : RepresentablyFlat G\nℱ : SheafOfTypes K\nZ : C\nT : Presieve Z\nx : FamilyOfElements (G.op ⋙ ℱ.val) T\nhx : Compatible x\nY₁ Y₂ : C\nX : D\nf₁ : X ⟶ G.obj Y₁\nf₂ : X ⟶ G.obj Y₂\ng₁ : Y₁ ⟶ Z\ng₂ : Y₂ ⟶ Z\nhg₁ : T g₁\nhg₂ : T g₂\ne : f₁ ≫ G.map g₁ = f₂ ≫ G.map g₂\nc : Cone (cospan g₁ g₂ ⋙ G) :=\n (Cones.postcompose (diagramIsoCospan (cospan g₁ g₂ ⋙ G)).inv).obj (PullbackCone.mk f₁ f₂ e)\nc' : Cone (Cone.toStructuredArrow c ⋙ StructuredArrow.pre c.pt (cospan g₁ g₂) G) :=\n IsCofiltered.cone (Cone.toStructuredArrow c ⋙ StructuredArrow.pre c.pt (cospan g₁ g₂) G)\neq₁ :\n f₁ =\n (c'.pt.hom ≫ G.map (NatTrans.app c'.π left).right) ≫\n eqToHom\n (_ :\n G.obj ((StructuredArrow.pre X (cospan g₁ g₂) G).obj (StructuredArrow.mk (f₁ ≫ 𝟙 (G.obj Y₁)))).right =\n G.obj Y₁)\neq₂ :\n f₂ =\n (c'.pt.hom ≫ G.map (NatTrans.app c'.π right).right) ≫\n eqToHom\n (_ :\n G.obj ((StructuredArrow.pre X (cospan g₁ g₂) G).obj (StructuredArrow.mk (f₂ ≫ 𝟙 (G.obj Y₂)))).right =\n G.obj Y₂)\nleft_eq✝ :\n (((Functor.const WalkingCospan).obj c'.pt).map Hom.inl).left ≫ (NatTrans.app c'.π one).left =\n (NatTrans.app c'.π left).left ≫\n ((Cone.toStructuredArrow c ⋙ StructuredArrow.pre c.pt (cospan g₁ g₂) G).map Hom.inl).left\ne₁ :\n (((Functor.const WalkingCospan).obj c'.pt).map Hom.inl).right ≫ (NatTrans.app c'.π one).right =\n (NatTrans.app c'.π left).right ≫\n ((Cone.toStructuredArrow c ⋙ StructuredArrow.pre c.pt (cospan g₁ g₂) G).map Hom.inl).right\n⊢ ℱ.val.map\n (G.map\n (NatTrans.app\n (IsCofiltered.cone\n (Cone.toStructuredArrow\n ((Cones.postcompose (diagramIsoCospan (cospan g₁ g₂ ⋙ G)).inv).obj (PullbackCone.mk f₁ f₂ e)) ⋙\n StructuredArrow.pre\n ((Cones.postcompose (diagramIsoCospan (cospan g₁ g₂ ⋙ G)).inv).obj (PullbackCone.mk f₁ f₂ e)).pt\n (cospan g₁ g₂) G)).π\n left).right).op\n (eqToHom\n (_ :\n ℱ.val.obj (op (G.obj Y₁)) =\n ℱ.val.obj\n (op\n (G.obj\n ((Cone.toStructuredArrow\n ((Cones.postcompose (diagramIsoCospan (cospan g₁ g₂ ⋙ G)).inv).obj\n (PullbackCone.mk f₁ f₂ e)) ⋙\n StructuredArrow.pre\n ((Cones.postcompose (diagramIsoCospan (cospan g₁ g₂ ⋙ G)).inv).obj\n (PullbackCone.mk f₁ f₂ e)).pt\n (cospan g₁ g₂) G).obj\n left).right)))\n (x g₁ hg₁)) =\n ℱ.val.map\n (G.map\n (NatTrans.app\n (IsCofiltered.cone\n (Cone.toStructuredArrow\n ((Cones.postcompose (diagramIsoCospan (cospan g₁ g₂ ⋙ G)).inv).obj (PullbackCone.mk f₁ f₂ e)) ⋙\n StructuredArrow.pre\n ((Cones.postcompose (diagramIsoCospan (cospan g₁ g₂ ⋙ G)).inv).obj (PullbackCone.mk f₁ f₂ e)).pt\n (cospan g₁ g₂) G)).π\n right).right).op\n (eqToHom\n (_ :\n ℱ.val.obj (op (G.obj Y₂)) =\n ℱ.val.obj\n (op\n (G.obj\n ((Cone.toStructuredArrow\n ((Cones.postcompose (diagramIsoCospan (cospan g₁ g₂ ⋙ G)).inv).obj\n (PullbackCone.mk f₁ f₂ e)) ⋙\n StructuredArrow.pre\n ((Cones.postcompose (diagramIsoCospan (cospan g₁ g₂ ⋙ G)).inv).obj\n (PullbackCone.mk f₁ f₂ e)).pt\n (cospan g₁ g₂) G).obj\n right).right)))\n (x g₂ hg₂))\n[PROOFSTEP]\ninjection c'.π.naturality WalkingCospan.Hom.inr with _ e₂\n[GOAL]\ncase Compatible.h\nC✝ : Type u₁\ninst✝⁵ : Category.{v₁, u₁} C✝\nD✝ : Type u₂\ninst✝⁴ : Category.{v₂, u₂} D✝\nA : Type u₃\ninst✝³ : Category.{v₃, u₃} A\nJ : GrothendieckTopology C✝\nK✝ : GrothendieckTopology D✝\nL : GrothendieckTopology A\nG✝ : C✝ ⥤ D✝\nhG : CompatiblePreserving K✝ G✝\nℱ✝ : SheafOfTypes K✝\nZ✝ : C✝\nT✝ : Presieve Z✝\nx✝ : FamilyOfElements (G✝.op ⋙ ℱ✝.val) T✝\nh : Compatible x✝\nC : Type u₁\ninst✝² : Category.{v₁, u₁} C\nD : Type u₁\ninst✝¹ : Category.{v₁, u₁} D\nK : GrothendieckTopology D\nG : C ⥤ D\ninst✝ : RepresentablyFlat G\nℱ : SheafOfTypes K\nZ : C\nT : Presieve Z\nx : FamilyOfElements (G.op ⋙ ℱ.val) T\nhx : Compatible x\nY₁ Y₂ : C\nX : D\nf₁ : X ⟶ G.obj Y₁\nf₂ : X ⟶ G.obj Y₂\ng₁ : Y₁ ⟶ Z\ng₂ : Y₂ ⟶ Z\nhg₁ : T g₁\nhg₂ : T g₂\ne : f₁ ≫ G.map g₁ = f₂ ≫ G.map g₂\nc : Cone (cospan g₁ g₂ ⋙ G) :=\n (Cones.postcompose (diagramIsoCospan (cospan g₁ g₂ ⋙ G)).inv).obj (PullbackCone.mk f₁ f₂ e)\nc' : Cone (Cone.toStructuredArrow c ⋙ StructuredArrow.pre c.pt (cospan g₁ g₂) G) :=\n IsCofiltered.cone (Cone.toStructuredArrow c ⋙ StructuredArrow.pre c.pt (cospan g₁ g₂) G)\neq₁ :\n f₁ =\n (c'.pt.hom ≫ G.map (NatTrans.app c'.π left).right) ≫\n eqToHom\n (_ :\n G.obj ((StructuredArrow.pre X (cospan g₁ g₂) G).obj (StructuredArrow.mk (f₁ ≫ 𝟙 (G.obj Y₁)))).right =\n G.obj Y₁)\neq₂ :\n f₂ =\n (c'.pt.hom ≫ G.map (NatTrans.app c'.π right).right) ≫\n eqToHom\n (_ :\n G.obj ((StructuredArrow.pre X (cospan g₁ g₂) G).obj (StructuredArrow.mk (f₂ ≫ 𝟙 (G.obj Y₂)))).right =\n G.obj Y₂)\nleft_eq✝¹ :\n (((Functor.const WalkingCospan).obj c'.pt).map Hom.inl).left ≫ (NatTrans.app c'.π one).left =\n (NatTrans.app c'.π left).left ≫\n ((Cone.toStructuredArrow c ⋙ StructuredArrow.pre c.pt (cospan g₁ g₂) G).map Hom.inl).left\ne₁ :\n (((Functor.const WalkingCospan).obj c'.pt).map Hom.inl).right ≫ (NatTrans.app c'.π one).right =\n (NatTrans.app c'.π left).right ≫\n ((Cone.toStructuredArrow c ⋙ StructuredArrow.pre c.pt (cospan g₁ g₂) G).map Hom.inl).right\nleft_eq✝ :\n (((Functor.const WalkingCospan).obj c'.pt).map Hom.inr).left ≫ (NatTrans.app c'.π one).left =\n (NatTrans.app c'.π right).left ≫\n ((Cone.toStructuredArrow c ⋙ StructuredArrow.pre c.pt (cospan g₁ g₂) G).map Hom.inr).left\ne₂ :\n (((Functor.const WalkingCospan).obj c'.pt).map Hom.inr).right ≫ (NatTrans.app c'.π one).right =\n (NatTrans.app c'.π right).right ≫\n ((Cone.toStructuredArrow c ⋙ StructuredArrow.pre c.pt (cospan g₁ g₂) G).map Hom.inr).right\n⊢ ℱ.val.map\n (G.map\n (NatTrans.app\n (IsCofiltered.cone\n (Cone.toStructuredArrow\n ((Cones.postcompose (diagramIsoCospan (cospan g₁ g₂ ⋙ G)).inv).obj (PullbackCone.mk f₁ f₂ e)) ⋙\n StructuredArrow.pre\n ((Cones.postcompose (diagramIsoCospan (cospan g₁ g₂ ⋙ G)).inv).obj (PullbackCone.mk f₁ f₂ e)).pt\n (cospan g₁ g₂) G)).π\n left).right).op\n (eqToHom\n (_ :\n ℱ.val.obj (op (G.obj Y₁)) =\n ℱ.val.obj\n (op\n (G.obj\n ((Cone.toStructuredArrow\n ((Cones.postcompose (diagramIsoCospan (cospan g₁ g₂ ⋙ G)).inv).obj\n (PullbackCone.mk f₁ f₂ e)) ⋙\n StructuredArrow.pre\n ((Cones.postcompose (diagramIsoCospan (cospan g₁ g₂ ⋙ G)).inv).obj\n (PullbackCone.mk f₁ f₂ e)).pt\n (cospan g₁ g₂) G).obj\n left).right)))\n (x g₁ hg₁)) =\n ℱ.val.map\n (G.map\n (NatTrans.app\n (IsCofiltered.cone\n (Cone.toStructuredArrow\n ((Cones.postcompose (diagramIsoCospan (cospan g₁ g₂ ⋙ G)).inv).obj (PullbackCone.mk f₁ f₂ e)) ⋙\n StructuredArrow.pre\n ((Cones.postcompose (diagramIsoCospan (cospan g₁ g₂ ⋙ G)).inv).obj (PullbackCone.mk f₁ f₂ e)).pt\n (cospan g₁ g₂) G)).π\n right).right).op\n (eqToHom\n (_ :\n ℱ.val.obj (op (G.obj Y₂)) =\n ℱ.val.obj\n (op\n (G.obj\n ((Cone.toStructuredArrow\n ((Cones.postcompose (diagramIsoCospan (cospan g₁ g₂ ⋙ G)).inv).obj\n (PullbackCone.mk f₁ f₂ e)) ⋙\n StructuredArrow.pre\n ((Cones.postcompose (diagramIsoCospan (cospan g₁ g₂ ⋙ G)).inv).obj\n (PullbackCone.mk f₁ f₂ e)).pt\n (cospan g₁ g₂) G).obj\n right).right)))\n (x g₂ hg₂))\n[PROOFSTEP]\nexact hx (c'.π.app left).right (c'.π.app right).right hg₁ hg₂ (e₁.symm.trans e₂)\n[GOAL]\nC : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝³ : Category.{v₂, u₂} D\nA : Type u₃\ninst✝² : Category.{v₃, u₃} A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nL : GrothendieckTopology A\nG : C ⥤ D\nhG : CompatiblePreserving K G\nℱ : SheafOfTypes K\nZ : C\nT : Presieve Z\nx : FamilyOfElements (G.op ⋙ ℱ.val) T\nh : Compatible x\nF : C ⥤ D\ninst✝¹ : Full F\ninst✝ : Faithful F\nhF : {c : C} → {d : D} → (d ⟶ F.obj c) → (c' : C) × (F.obj c' ≅ d)\n⊢ CompatiblePreserving K F\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase Compatible\nC : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝³ : Category.{v₂, u₂} D\nA : Type u₃\ninst✝² : Category.{v₃, u₃} A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nL : GrothendieckTopology A\nG : C ⥤ D\nhG : CompatiblePreserving K G\nℱ : SheafOfTypes K\nZ : C\nT : Presieve Z\nx : FamilyOfElements (G.op ⋙ ℱ.val) T\nh : Compatible x\nF : C ⥤ D\ninst✝¹ : Full F\ninst✝ : Faithful F\nhF : {c : C} → {d : D} → (d ⟶ F.obj c) → (c' : C) × (F.obj c' ≅ d)\n⊢ ∀ (ℱ : SheafOfTypes K) {Z : C} {T : Presieve Z} {x : FamilyOfElements (F.op ⋙ ℱ.val) T},\n Compatible x →\n ∀ {Y₁ Y₂ : C} {X : D} (f₁ : X ⟶ F.obj Y₁) (f₂ : X ⟶ F.obj Y₂) {g₁ : Y₁ ⟶ Z} {g₂ : Y₂ ⟶ Z} (hg₁ : T g₁)\n (hg₂ : T g₂), f₁ ≫ F.map g₁ = f₂ ≫ F.map g₂ → ℱ.val.map f₁.op (x g₁ hg₁) = ℱ.val.map f₂.op (x g₂ hg₂)\n[PROOFSTEP]\nintrov hx he\n[GOAL]\ncase Compatible\nC : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝³ : Category.{v₂, u₂} D\nA : Type u₃\ninst✝² : Category.{v₃, u₃} A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nL : GrothendieckTopology A\nG : C ⥤ D\nhG : CompatiblePreserving K G\nℱ✝ : SheafOfTypes K\nZ✝ : C\nT✝ : Presieve Z✝\nx✝ : FamilyOfElements (G.op ⋙ ℱ✝.val) T✝\nh : Compatible x✝\nF : C ⥤ D\ninst✝¹ : Full F\ninst✝ : Faithful F\nhF : {c : C} → {d : D} → (d ⟶ F.obj c) → (c' : C) × (F.obj c' ≅ d)\nℱ : SheafOfTypes K\nZ : C\nT : Presieve Z\nx : FamilyOfElements (F.op ⋙ ℱ.val) T\nhx : Compatible x\nY₁ Y₂ : C\nX : D\nf₁ : X ⟶ F.obj Y₁\nf₂ : X ⟶ F.obj Y₂\ng₁ : Y₁ ⟶ Z\ng₂ : Y₂ ⟶ Z\nhg₁ : T g₁\nhg₂ : T g₂\nhe : f₁ ≫ F.map g₁ = f₂ ≫ F.map g₂\n⊢ ℱ.val.map f₁.op (x g₁ hg₁) = ℱ.val.map f₂.op (x g₂ hg₂)\n[PROOFSTEP]\nobtain ⟨X', e⟩ := hF f₁\n[GOAL]\ncase Compatible.mk\nC : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝³ : Category.{v₂, u₂} D\nA : Type u₃\ninst✝² : Category.{v₃, u₃} A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nL : GrothendieckTopology A\nG : C ⥤ D\nhG : CompatiblePreserving K G\nℱ✝ : SheafOfTypes K\nZ✝ : C\nT✝ : Presieve Z✝\nx✝ : FamilyOfElements (G.op ⋙ ℱ✝.val) T✝\nh : Compatible x✝\nF : C ⥤ D\ninst✝¹ : Full F\ninst✝ : Faithful F\nhF : {c : C} → {d : D} → (d ⟶ F.obj c) → (c' : C) × (F.obj c' ≅ d)\nℱ : SheafOfTypes K\nZ : C\nT : Presieve Z\nx : FamilyOfElements (F.op ⋙ ℱ.val) T\nhx : Compatible x\nY₁ Y₂ : C\nX : D\nf₁ : X ⟶ F.obj Y₁\nf₂ : X ⟶ F.obj Y₂\ng₁ : Y₁ ⟶ Z\ng₂ : Y₂ ⟶ Z\nhg₁ : T g₁\nhg₂ : T g₂\nhe : f₁ ≫ F.map g₁ = f₂ ≫ F.map g₂\nX' : C\ne : F.obj X' ≅ X\n⊢ ℱ.val.map f₁.op (x g₁ hg₁) = ℱ.val.map f₂.op (x g₂ hg₂)\n[PROOFSTEP]\napply (ℱ.1.mapIso e.op).toEquiv.injective\n[GOAL]\ncase Compatible.mk.a\nC : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝³ : Category.{v₂, u₂} D\nA : Type u₃\ninst✝² : Category.{v₃, u₃} A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nL : GrothendieckTopology A\nG : C ⥤ D\nhG : CompatiblePreserving K G\nℱ✝ : SheafOfTypes K\nZ✝ : C\nT✝ : Presieve Z✝\nx✝ : FamilyOfElements (G.op ⋙ ℱ✝.val) T✝\nh : Compatible x✝\nF : C ⥤ D\ninst✝¹ : Full F\ninst✝ : Faithful F\nhF : {c : C} → {d : D} → (d ⟶ F.obj c) → (c' : C) × (F.obj c' ≅ d)\nℱ : SheafOfTypes K\nZ : C\nT : Presieve Z\nx : FamilyOfElements (F.op ⋙ ℱ.val) T\nhx : Compatible x\nY₁ Y₂ : C\nX : D\nf₁ : X ⟶ F.obj Y₁\nf₂ : X ⟶ F.obj Y₂\ng₁ : Y₁ ⟶ Z\ng₂ : Y₂ ⟶ Z\nhg₁ : T g₁\nhg₂ : T g₂\nhe : f₁ ≫ F.map g₁ = f₂ ≫ F.map g₂\nX' : C\ne : F.obj X' ≅ X\n⊢ ↑(ℱ.val.mapIso (Iso.op e)).toEquiv (ℱ.val.map f₁.op (x g₁ hg₁)) =\n ↑(ℱ.val.mapIso (Iso.op e)).toEquiv (ℱ.val.map f₂.op (x g₂ hg₂))\n[PROOFSTEP]\nsimp only [Iso.op_hom, Iso.toEquiv_fun, ℱ.1.mapIso_hom, ← FunctorToTypes.map_comp_apply]\n[GOAL]\ncase Compatible.mk.a\nC : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝³ : Category.{v₂, u₂} D\nA : Type u₃\ninst✝² : Category.{v₃, u₃} A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nL : GrothendieckTopology A\nG : C ⥤ D\nhG : CompatiblePreserving K G\nℱ✝ : SheafOfTypes K\nZ✝ : C\nT✝ : Presieve Z✝\nx✝ : FamilyOfElements (G.op ⋙ ℱ✝.val) T✝\nh : Compatible x✝\nF : C ⥤ D\ninst✝¹ : Full F\ninst✝ : Faithful F\nhF : {c : C} → {d : D} → (d ⟶ F.obj c) → (c' : C) × (F.obj c' ≅ d)\nℱ : SheafOfTypes K\nZ : C\nT : Presieve Z\nx : FamilyOfElements (F.op ⋙ ℱ.val) T\nhx : Compatible x\nY₁ Y₂ : C\nX : D\nf₁ : X ⟶ F.obj Y₁\nf₂ : X ⟶ F.obj Y₂\ng₁ : Y₁ ⟶ Z\ng₂ : Y₂ ⟶ Z\nhg₁ : T g₁\nhg₂ : T g₂\nhe : f₁ ≫ F.map g₁ = f₂ ≫ F.map g₂\nX' : C\ne : F.obj X' ≅ X\n⊢ ℱ.val.map (f₁.op ≫ e.hom.op) (x g₁ hg₁) = ℱ.val.map (f₂.op ≫ e.hom.op) (x g₂ hg₂)\n[PROOFSTEP]\nsimpa using hx (F.preimage <| e.hom ≫ f₁) (F.preimage <| e.hom ≫ f₂) hg₁ hg₂ (F.map_injective <| by simpa using he)\n[GOAL]\nC : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝³ : Category.{v₂, u₂} D\nA : Type u₃\ninst✝² : Category.{v₃, u₃} A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nL : GrothendieckTopology A\nG : C ⥤ D\nhG : CompatiblePreserving K G\nℱ✝ : SheafOfTypes K\nZ✝ : C\nT✝ : Presieve Z✝\nx✝ : FamilyOfElements (G.op ⋙ ℱ✝.val) T✝\nh : Compatible x✝\nF : C ⥤ D\ninst✝¹ : Full F\ninst✝ : Faithful F\nhF : {c : C} → {d : D} → (d ⟶ F.obj c) → (c' : C) × (F.obj c' ≅ d)\nℱ : SheafOfTypes K\nZ : C\nT : Presieve Z\nx : FamilyOfElements (F.op ⋙ ℱ.val) T\nhx : Compatible x\nY₁ Y₂ : C\nX : D\nf₁ : X ⟶ F.obj Y₁\nf₂ : X ⟶ F.obj Y₂\ng₁ : Y₁ ⟶ Z\ng₂ : Y₂ ⟶ Z\nhg₁ : T g₁\nhg₂ : T g₂\nhe : f₁ ≫ F.map g₁ = f₂ ≫ F.map g₂\nX' : C\ne : F.obj X' ≅ X\n⊢ F.map (F.preimage (e.hom ≫ f₁) ≫ g₁) = F.map (F.preimage (e.hom ≫ f₂) ≫ g₂)\n[PROOFSTEP]\nsimpa using he\n[GOAL]\nC : Type u₁\ninst✝² : Category.{v₁, u₁} C\nD : Type u₂\ninst✝¹ : Category.{v₂, u₂} D\nA : Type u₃\ninst✝ : Category.{v₃, u₃} A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nL : GrothendieckTopology A\nG✝ : C ⥤ D\nhG : CompatiblePreserving K G✝\nℱ✝ : SheafOfTypes K\nZ : C\nT : Presieve Z\nx : FamilyOfElements (G✝.op ⋙ ℱ✝.val) T\nh : Compatible x\nG : C ⥤ D\nhG₁ : CompatiblePreserving K G\nhG₂ : CoverPreserving J K G\nℱ : Sheaf K A\n⊢ Presheaf.IsSheaf J (G.op ⋙ ℱ.val)\n[PROOFSTEP]\nintro X U S hS x hx\n[GOAL]\nC : Type u₁\ninst✝² : Category.{v₁, u₁} C\nD : Type u₂\ninst✝¹ : Category.{v₂, u₂} D\nA : Type u₃\ninst✝ : Category.{v₃, u₃} A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nL : GrothendieckTopology A\nG✝ : C ⥤ D\nhG : CompatiblePreserving K G✝\nℱ✝ : SheafOfTypes K\nZ : C\nT : Presieve Z\nx✝ : FamilyOfElements (G✝.op ⋙ ℱ✝.val) T\nh : Compatible x✝\nG : C ⥤ D\nhG₁ : CompatiblePreserving K G\nhG₂ : CoverPreserving J K G\nℱ : Sheaf K A\nX : A\nU : C\nS : Sieve U\nhS : S ∈ GrothendieckTopology.sieves J U\nx : FamilyOfElements ((G.op ⋙ ℱ.val) ⋙ coyoneda.obj (op X)) S.arrows\nhx : Compatible x\n⊢ ∃! t, IsAmalgamation x t\n[PROOFSTEP]\nchange FamilyOfElements (G.op ⋙ ℱ.val ⋙ coyoneda.obj (op X)) _ at x \n[GOAL]\nC : Type u₁\ninst✝² : Category.{v₁, u₁} C\nD : Type u₂\ninst✝¹ : Category.{v₂, u₂} D\nA : Type u₃\ninst✝ : Category.{v₃, u₃} A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nL : GrothendieckTopology A\nG✝ : C ⥤ D\nhG : CompatiblePreserving K G✝\nℱ✝ : SheafOfTypes K\nZ : C\nT : Presieve Z\nx✝ : FamilyOfElements (G✝.op ⋙ ℱ✝.val) T\nh : Compatible x✝\nG : C ⥤ D\nhG₁ : CompatiblePreserving K G\nhG₂ : CoverPreserving J K G\nℱ : Sheaf K A\nX : A\nU : C\nS : Sieve U\nhS : S ∈ GrothendieckTopology.sieves J U\nx : FamilyOfElements (G.op ⋙ ℱ.val ⋙ coyoneda.obj (op X)) S.arrows\nhx : Compatible x\n⊢ ∃! t, IsAmalgamation x t\n[PROOFSTEP]\nlet H := ℱ.2 X _ (hG₂.cover_preserve hS)\n[GOAL]\nC : Type u₁\ninst✝² : Category.{v₁, u₁} C\nD : Type u₂\ninst✝¹ : Category.{v₂, u₂} D\nA : Type u₃\ninst✝ : Category.{v₃, u₃} A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nL : GrothendieckTopology A\nG✝ : C ⥤ D\nhG : CompatiblePreserving K G✝\nℱ✝ : SheafOfTypes K\nZ : C\nT : Presieve Z\nx✝ : FamilyOfElements (G✝.op ⋙ ℱ✝.val) T\nh : Compatible x✝\nG : C ⥤ D\nhG₁ : CompatiblePreserving K G\nhG₂ : CoverPreserving J K G\nℱ : Sheaf K A\nX : A\nU : C\nS : Sieve U\nhS : S ∈ GrothendieckTopology.sieves J U\nx : FamilyOfElements (G.op ⋙ ℱ.val ⋙ coyoneda.obj (op X)) S.arrows\nhx : Compatible x\nH : IsSheafFor (ℱ.val ⋙ coyoneda.obj (op X)) (Sieve.functorPushforward G S).arrows :=\n Sheaf.cond ℱ X (Sieve.functorPushforward G S) (CoverPreserving.cover_preserve hG₂ hS)\n⊢ ∃! t, IsAmalgamation x t\n[PROOFSTEP]\nlet hx' := hx.functorPushforward hG₁ (sheafOver ℱ X)\n[GOAL]\nC : Type u₁\ninst✝² : Category.{v₁, u₁} C\nD : Type u₂\ninst✝¹ : Category.{v₂, u₂} D\nA : Type u₃\ninst✝ : Category.{v₃, u₃} A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nL : GrothendieckTopology A\nG✝ : C ⥤ D\nhG : CompatiblePreserving K G✝\nℱ✝ : SheafOfTypes K\nZ : C\nT : Presieve Z\nx✝ : FamilyOfElements (G✝.op ⋙ ℱ✝.val) T\nh : Compatible x✝\nG : C ⥤ D\nhG₁ : CompatiblePreserving K G\nhG₂ : CoverPreserving J K G\nℱ : Sheaf K A\nX : A\nU : C\nS : Sieve U\nhS : S ∈ GrothendieckTopology.sieves J U\nx : FamilyOfElements (G.op ⋙ ℱ.val ⋙ coyoneda.obj (op X)) S.arrows\nhx : Compatible x\nH : IsSheafFor (ℱ.val ⋙ coyoneda.obj (op X)) (Sieve.functorPushforward G S).arrows :=\n Sheaf.cond ℱ X (Sieve.functorPushforward G S) (CoverPreserving.cover_preserve hG₂ hS)\nhx' : Compatible (FamilyOfElements.functorPushforward G x) := Compatible.functorPushforward hG₁ (sheafOver ℱ X) hx\n⊢ ∃! t, IsAmalgamation x t\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase h\nC : Type u₁\ninst✝² : Category.{v₁, u₁} C\nD : Type u₂\ninst✝¹ : Category.{v₂, u₂} D\nA : Type u₃\ninst✝ : Category.{v₃, u₃} A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nL : GrothendieckTopology A\nG✝ : C ⥤ D\nhG : CompatiblePreserving K G✝\nℱ✝ : SheafOfTypes K\nZ : C\nT : Presieve Z\nx✝ : FamilyOfElements (G✝.op ⋙ ℱ✝.val) T\nh : Compatible x✝\nG : C ⥤ D\nhG₁ : CompatiblePreserving K G\nhG₂ : CoverPreserving J K G\nℱ : Sheaf K A\nX : A\nU : C\nS : Sieve U\nhS : S ∈ GrothendieckTopology.sieves J U\nx : FamilyOfElements (G.op ⋙ ℱ.val ⋙ coyoneda.obj (op X)) S.arrows\nhx : Compatible x\nH : IsSheafFor (ℱ.val ⋙ coyoneda.obj (op X)) (Sieve.functorPushforward G S).arrows :=\n Sheaf.cond ℱ X (Sieve.functorPushforward G S) (CoverPreserving.cover_preserve hG₂ hS)\nhx' : Compatible (FamilyOfElements.functorPushforward G x) := Compatible.functorPushforward hG₁ (sheafOver ℱ X) hx\n⊢ (fun t => IsAmalgamation x t) ?w ∧\n ∀ (y : ((G.op ⋙ ℱ.val) ⋙ coyoneda.obj (op X)).obj (op U)), (fun t => IsAmalgamation x t) y → y = ?w\ncase w\nC : Type u₁\ninst✝² : Category.{v₁, u₁} C\nD : Type u₂\ninst✝¹ : Category.{v₂, u₂} D\nA : Type u₃\ninst✝ : Category.{v₃, u₃} A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nL : GrothendieckTopology A\nG✝ : C ⥤ D\nhG : CompatiblePreserving K G✝\nℱ✝ : SheafOfTypes K\nZ : C\nT : Presieve Z\nx✝ : FamilyOfElements (G✝.op ⋙ ℱ✝.val) T\nh : Compatible x✝\nG : C ⥤ D\nhG₁ : CompatiblePreserving K G\nhG₂ : CoverPreserving J K G\nℱ : Sheaf K A\nX : A\nU : C\nS : Sieve U\nhS : S ∈ GrothendieckTopology.sieves J U\nx : FamilyOfElements (G.op ⋙ ℱ.val ⋙ coyoneda.obj (op X)) S.arrows\nhx : Compatible x\nH : IsSheafFor (ℱ.val ⋙ coyoneda.obj (op X)) (Sieve.functorPushforward G S).arrows :=\n Sheaf.cond ℱ X (Sieve.functorPushforward G S) (CoverPreserving.cover_preserve hG₂ hS)\nhx' : Compatible (FamilyOfElements.functorPushforward G x) := Compatible.functorPushforward hG₁ (sheafOver ℱ X) hx\n⊢ ((G.op ⋙ ℱ.val) ⋙ coyoneda.obj (op X)).obj (op U)\n[PROOFSTEP]\nswap\n[GOAL]\ncase w\nC : Type u₁\ninst✝² : Category.{v₁, u₁} C\nD : Type u₂\ninst✝¹ : Category.{v₂, u₂} D\nA : Type u₃\ninst✝ : Category.{v₃, u₃} A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nL : GrothendieckTopology A\nG✝ : C ⥤ D\nhG : CompatiblePreserving K G✝\nℱ✝ : SheafOfTypes K\nZ : C\nT : Presieve Z\nx✝ : FamilyOfElements (G✝.op ⋙ ℱ✝.val) T\nh : Compatible x✝\nG : C ⥤ D\nhG₁ : CompatiblePreserving K G\nhG₂ : CoverPreserving J K G\nℱ : Sheaf K A\nX : A\nU : C\nS : Sieve U\nhS : S ∈ GrothendieckTopology.sieves J U\nx : FamilyOfElements (G.op ⋙ ℱ.val ⋙ coyoneda.obj (op X)) S.arrows\nhx : Compatible x\nH : IsSheafFor (ℱ.val ⋙ coyoneda.obj (op X)) (Sieve.functorPushforward G S).arrows :=\n Sheaf.cond ℱ X (Sieve.functorPushforward G S) (CoverPreserving.cover_preserve hG₂ hS)\nhx' : Compatible (FamilyOfElements.functorPushforward G x) := Compatible.functorPushforward hG₁ (sheafOver ℱ X) hx\n⊢ ((G.op ⋙ ℱ.val) ⋙ coyoneda.obj (op X)).obj (op U)\n[PROOFSTEP]\napply H.amalgamate (x.functorPushforward G)\n[GOAL]\ncase w\nC : Type u₁\ninst✝² : Category.{v₁, u₁} C\nD : Type u₂\ninst✝¹ : Category.{v₂, u₂} D\nA : Type u₃\ninst✝ : Category.{v₃, u₃} A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nL : GrothendieckTopology A\nG✝ : C ⥤ D\nhG : CompatiblePreserving K G✝\nℱ✝ : SheafOfTypes K\nZ : C\nT : Presieve Z\nx✝ : FamilyOfElements (G✝.op ⋙ ℱ✝.val) T\nh : Compatible x✝\nG : C ⥤ D\nhG₁ : CompatiblePreserving K G\nhG₂ : CoverPreserving J K G\nℱ : Sheaf K A\nX : A\nU : C\nS : Sieve U\nhS : S ∈ GrothendieckTopology.sieves J U\nx : FamilyOfElements (G.op ⋙ ℱ.val ⋙ coyoneda.obj (op X)) S.arrows\nhx : Compatible x\nH : IsSheafFor (ℱ.val ⋙ coyoneda.obj (op X)) (Sieve.functorPushforward G S).arrows :=\n Sheaf.cond ℱ X (Sieve.functorPushforward G S) (CoverPreserving.cover_preserve hG₂ hS)\nhx' : Compatible (FamilyOfElements.functorPushforward G x) := Compatible.functorPushforward hG₁ (sheafOver ℱ X) hx\n⊢ Compatible (FamilyOfElements.functorPushforward G x)\n[PROOFSTEP]\nexact hx'\n[GOAL]\ncase h\nC : Type u₁\ninst✝² : Category.{v₁, u₁} C\nD : Type u₂\ninst✝¹ : Category.{v₂, u₂} D\nA : Type u₃\ninst✝ : Category.{v₃, u₃} A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nL : GrothendieckTopology A\nG✝ : C ⥤ D\nhG : CompatiblePreserving K G✝\nℱ✝ : SheafOfTypes K\nZ : C\nT : Presieve Z\nx✝ : FamilyOfElements (G✝.op ⋙ ℱ✝.val) T\nh : Compatible x✝\nG : C ⥤ D\nhG₁ : CompatiblePreserving K G\nhG₂ : CoverPreserving J K G\nℱ : Sheaf K A\nX : A\nU : C\nS : Sieve U\nhS : S ∈ GrothendieckTopology.sieves J U\nx : FamilyOfElements (G.op ⋙ ℱ.val ⋙ coyoneda.obj (op X)) S.arrows\nhx : Compatible x\nH : IsSheafFor (ℱ.val ⋙ coyoneda.obj (op X)) (Sieve.functorPushforward G S).arrows :=\n Sheaf.cond ℱ X (Sieve.functorPushforward G S) (CoverPreserving.cover_preserve hG₂ hS)\nhx' : Compatible (FamilyOfElements.functorPushforward G x) := Compatible.functorPushforward hG₁ (sheafOver ℱ X) hx\n⊢ (fun t => IsAmalgamation x t) (IsSheafFor.amalgamate H (FamilyOfElements.functorPushforward G x) hx') ∧\n ∀ (y : ((G.op ⋙ ℱ.val) ⋙ coyoneda.obj (op X)).obj (op U)),\n (fun t => IsAmalgamation x t) y → y = IsSheafFor.amalgamate H (FamilyOfElements.functorPushforward G x) hx'\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase h.left\nC : Type u₁\ninst✝² : Category.{v₁, u₁} C\nD : Type u₂\ninst✝¹ : Category.{v₂, u₂} D\nA : Type u₃\ninst✝ : Category.{v₃, u₃} A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nL : GrothendieckTopology A\nG✝ : C ⥤ D\nhG : CompatiblePreserving K G✝\nℱ✝ : SheafOfTypes K\nZ : C\nT : Presieve Z\nx✝ : FamilyOfElements (G✝.op ⋙ ℱ✝.val) T\nh : Compatible x✝\nG : C ⥤ D\nhG₁ : CompatiblePreserving K G\nhG₂ : CoverPreserving J K G\nℱ : Sheaf K A\nX : A\nU : C\nS : Sieve U\nhS : S ∈ GrothendieckTopology.sieves J U\nx : FamilyOfElements (G.op ⋙ ℱ.val ⋙ coyoneda.obj (op X)) S.arrows\nhx : Compatible x\nH : IsSheafFor (ℱ.val ⋙ coyoneda.obj (op X)) (Sieve.functorPushforward G S).arrows :=\n Sheaf.cond ℱ X (Sieve.functorPushforward G S) (CoverPreserving.cover_preserve hG₂ hS)\nhx' : Compatible (FamilyOfElements.functorPushforward G x) := Compatible.functorPushforward hG₁ (sheafOver ℱ X) hx\n⊢ (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 u₁\ninst✝² : Category.{v₁, u₁} C\nD : Type u₂\ninst✝¹ : Category.{v₂, u₂} D\nA : Type u₃\ninst✝ : Category.{v₃, u₃} A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nL : GrothendieckTopology A\nG✝ : C ⥤ D\nhG : CompatiblePreserving K G✝\nℱ✝ : SheafOfTypes K\nZ : C\nT : Presieve Z\nx✝ : FamilyOfElements (G✝.op ⋙ ℱ✝.val) T\nh : Compatible x✝\nG : C ⥤ D\nhG₁ : CompatiblePreserving K G\nhG₂ : CoverPreserving J K G\nℱ : Sheaf K A\nX : A\nU : C\nS : Sieve U\nhS : S ∈ GrothendieckTopology.sieves J U\nx : FamilyOfElements (G.op ⋙ ℱ.val ⋙ coyoneda.obj (op X)) S.arrows\nhx : Compatible x\nH : IsSheafFor (ℱ.val ⋙ coyoneda.obj (op X)) (Sieve.functorPushforward G S).arrows :=\n Sheaf.cond ℱ X (Sieve.functorPushforward G S) (CoverPreserving.cover_preserve hG₂ hS)\nhx' : Compatible (FamilyOfElements.functorPushforward G x) := Compatible.functorPushforward hG₁ (sheafOver ℱ X) hx\nV : C\nf : V ⟶ U\nhf : S.arrows f\n⊢ ((G.op ⋙ ℱ.val) ⋙ 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 u₁\ninst✝² : Category.{v₁, u₁} C\nD : Type u₂\ninst✝¹ : Category.{v₂, u₂} D\nA : Type u₃\ninst✝ : Category.{v₃, u₃} A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nL : GrothendieckTopology A\nG✝ : C ⥤ D\nhG : CompatiblePreserving K G✝\nℱ✝ : SheafOfTypes K\nZ : C\nT : Presieve Z\nx✝ : FamilyOfElements (G✝.op ⋙ ℱ✝.val) T\nh : Compatible x✝\nG : C ⥤ D\nhG₁ : CompatiblePreserving K G\nhG₂ : CoverPreserving J K G\nℱ : Sheaf K A\nX : A\nU : C\nS : Sieve U\nhS : S ∈ GrothendieckTopology.sieves J U\nx : FamilyOfElements (G.op ⋙ ℱ.val ⋙ coyoneda.obj (op X)) S.arrows\nhx : Compatible x\nH : IsSheafFor (ℱ.val ⋙ coyoneda.obj (op X)) (Sieve.functorPushforward G S).arrows :=\n Sheaf.cond ℱ X (Sieve.functorPushforward G S) (CoverPreserving.cover_preserve hG₂ hS)\nhx' : Compatible (FamilyOfElements.functorPushforward G x) := Compatible.functorPushforward hG₁ (sheafOver ℱ X) hx\nV : C\nf : V ⟶ U\nhf : S.arrows f\ne_1✝ : ((G.op ⋙ ℱ.val) ⋙ coyoneda.obj (op X)).obj (op V) = (ℱ.val ⋙ coyoneda.obj (op X)).obj (op (G.obj V))\n⊢ x f hf = FamilyOfElements.functorPushforward G x (G.map f) (_ : Presieve.functorPushforward G S.arrows (G.map f))\n[PROOFSTEP]\nrw [hG₁.apply_map (sheafOver ℱ X) hx]\n[GOAL]\ncase h.right\nC : Type u₁\ninst✝² : Category.{v₁, u₁} C\nD : Type u₂\ninst✝¹ : Category.{v₂, u₂} D\nA : Type u₃\ninst✝ : Category.{v₃, u₃} A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nL : GrothendieckTopology A\nG✝ : C ⥤ D\nhG : CompatiblePreserving K G✝\nℱ✝ : SheafOfTypes K\nZ : C\nT : Presieve Z\nx✝ : FamilyOfElements (G✝.op ⋙ ℱ✝.val) T\nh : Compatible x✝\nG : C ⥤ D\nhG₁ : CompatiblePreserving K G\nhG₂ : CoverPreserving J K G\nℱ : Sheaf K A\nX : A\nU : C\nS : Sieve U\nhS : S ∈ GrothendieckTopology.sieves J U\nx : FamilyOfElements (G.op ⋙ ℱ.val ⋙ coyoneda.obj (op X)) S.arrows\nhx : Compatible x\nH : IsSheafFor (ℱ.val ⋙ coyoneda.obj (op X)) (Sieve.functorPushforward G S).arrows :=\n Sheaf.cond ℱ X (Sieve.functorPushforward G S) (CoverPreserving.cover_preserve hG₂ hS)\nhx' : Compatible (FamilyOfElements.functorPushforward G x) := Compatible.functorPushforward hG₁ (sheafOver ℱ X) hx\n⊢ ∀ (y : ((G.op ⋙ ℱ.val) ⋙ coyoneda.obj (op X)).obj (op U)),\n (fun t => IsAmalgamation x t) y → y = IsSheafFor.amalgamate H (FamilyOfElements.functorPushforward G x) hx'\n[PROOFSTEP]\nintro y hy\n[GOAL]\ncase h.right\nC : Type u₁\ninst✝² : Category.{v₁, u₁} C\nD : Type u₂\ninst✝¹ : Category.{v₂, u₂} D\nA : Type u₃\ninst✝ : Category.{v₃, u₃} A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nL : GrothendieckTopology A\nG✝ : C ⥤ D\nhG : CompatiblePreserving K G✝\nℱ✝ : SheafOfTypes K\nZ : C\nT : Presieve Z\nx✝ : FamilyOfElements (G✝.op ⋙ ℱ✝.val) T\nh : Compatible x✝\nG : C ⥤ D\nhG₁ : CompatiblePreserving K G\nhG₂ : CoverPreserving J K G\nℱ : Sheaf K A\nX : A\nU : C\nS : Sieve U\nhS : S ∈ GrothendieckTopology.sieves J U\nx : FamilyOfElements (G.op ⋙ ℱ.val ⋙ coyoneda.obj (op X)) S.arrows\nhx : Compatible x\nH : IsSheafFor (ℱ.val ⋙ coyoneda.obj (op X)) (Sieve.functorPushforward G S).arrows :=\n Sheaf.cond ℱ X (Sieve.functorPushforward G S) (CoverPreserving.cover_preserve hG₂ hS)\nhx' : Compatible (FamilyOfElements.functorPushforward G x) := Compatible.functorPushforward hG₁ (sheafOver ℱ X) hx\ny : ((G.op ⋙ ℱ.val) ⋙ coyoneda.obj (op X)).obj (op U)\nhy : IsAmalgamation x y\n⊢ y = IsSheafFor.amalgamate H (FamilyOfElements.functorPushforward G x) hx'\n[PROOFSTEP]\nrefine' H.isSeparatedFor _ y _ _ (H.isAmalgamation (hx.functorPushforward hG₁ (sheafOver ℱ X)))\n[GOAL]\ncase h.right\nC : Type u₁\ninst✝² : Category.{v₁, u₁} C\nD : Type u₂\ninst✝¹ : Category.{v₂, u₂} D\nA : Type u₃\ninst✝ : Category.{v₃, u₃} A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nL : GrothendieckTopology A\nG✝ : C ⥤ D\nhG : CompatiblePreserving K G✝\nℱ✝ : SheafOfTypes K\nZ : C\nT : Presieve Z\nx✝ : FamilyOfElements (G✝.op ⋙ ℱ✝.val) T\nh : Compatible x✝\nG : C ⥤ D\nhG₁ : CompatiblePreserving K G\nhG₂ : CoverPreserving J K G\nℱ : Sheaf K A\nX : A\nU : C\nS : Sieve U\nhS : S ∈ GrothendieckTopology.sieves J U\nx : FamilyOfElements (G.op ⋙ ℱ.val ⋙ coyoneda.obj (op X)) S.arrows\nhx : Compatible x\nH : IsSheafFor (ℱ.val ⋙ coyoneda.obj (op X)) (Sieve.functorPushforward G S).arrows :=\n Sheaf.cond ℱ X (Sieve.functorPushforward G S) (CoverPreserving.cover_preserve hG₂ hS)\nhx' : Compatible (FamilyOfElements.functorPushforward G x) := Compatible.functorPushforward hG₁ (sheafOver ℱ X) hx\ny : ((G.op ⋙ ℱ.val) ⋙ coyoneda.obj (op X)).obj (op U)\nhy : IsAmalgamation x y\n⊢ IsAmalgamation (FamilyOfElements.functorPushforward G x) y\n[PROOFSTEP]\nrintro V f\n ⟨Z, f', g', h, rfl⟩\n -- porting note: didn't need coercion (S : Presieve U) in Lean 3\n[GOAL]\ncase h.right.intro.intro.intro.intro\nC : Type u₁\ninst✝² : Category.{v₁, u₁} C\nD : Type u₂\ninst✝¹ : Category.{v₂, u₂} D\nA : Type u₃\ninst✝ : Category.{v₃, u₃} A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nL : GrothendieckTopology A\nG✝ : C ⥤ D\nhG : CompatiblePreserving K G✝\nℱ✝ : SheafOfTypes K\nZ✝ : C\nT : Presieve Z✝\nx✝ : FamilyOfElements (G✝.op ⋙ ℱ✝.val) T\nh✝ : Compatible x✝\nG : C ⥤ D\nhG₁ : CompatiblePreserving K G\nhG₂ : CoverPreserving J K G\nℱ : Sheaf K A\nX : A\nU : C\nS : Sieve U\nhS : S ∈ GrothendieckTopology.sieves J U\nx : FamilyOfElements (G.op ⋙ ℱ.val ⋙ coyoneda.obj (op X)) S.arrows\nhx : Compatible x\nH : IsSheafFor (ℱ.val ⋙ coyoneda.obj (op X)) (Sieve.functorPushforward G S).arrows :=\n Sheaf.cond ℱ X (Sieve.functorPushforward G S) (CoverPreserving.cover_preserve hG₂ hS)\nhx' : Compatible (FamilyOfElements.functorPushforward G x) := Compatible.functorPushforward hG₁ (sheafOver ℱ X) hx\ny : ((G.op ⋙ ℱ.val) ⋙ coyoneda.obj (op X)).obj (op U)\nhy : IsAmalgamation x y\nV : D\nZ : C\nf' : Z ⟶ U\ng' : V ⟶ G.obj Z\nh : S.arrows f'\n⊢ (ℱ.val ⋙ coyoneda.obj (op X)).map (g' ≫ G.map f').op y =\n FamilyOfElements.functorPushforward G x (g' ≫ G.map f') (_ : ∃ Z_1 g h, S.arrows g ∧ g' ≫ G.map f' = h ≫ 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 u₁\ninst✝² : Category.{v₁, u₁} C\nD : Type u₂\ninst✝¹ : Category.{v₂, u₂} D\nA : Type u₃\ninst✝ : Category.{v₃, u₃} A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nL : GrothendieckTopology A\nG✝ : C ⥤ D\nhG : CompatiblePreserving K G✝\nℱ✝ : SheafOfTypes K\nZ✝ : C\nT : Presieve Z✝\nx✝ : FamilyOfElements (G✝.op ⋙ ℱ✝.val) T\nh✝ : Compatible x✝\nG : C ⥤ D\nhG₁ : CompatiblePreserving K G\nhG₂ : CoverPreserving J K G\nℱ : Sheaf K A\nX : A\nU : C\nS : Sieve U\nhS : S ∈ GrothendieckTopology.sieves J U\nx : FamilyOfElements (G.op ⋙ ℱ.val ⋙ coyoneda.obj (op X)) S.arrows\nhx : Compatible x\nH : IsSheafFor (ℱ.val ⋙ coyoneda.obj (op X)) (Sieve.functorPushforward G S).arrows :=\n Sheaf.cond ℱ X (Sieve.functorPushforward G S) (CoverPreserving.cover_preserve hG₂ hS)\nhx' : Compatible (FamilyOfElements.functorPushforward G x) := Compatible.functorPushforward hG₁ (sheafOver ℱ X) hx\ny : ((G.op ⋙ ℱ.val) ⋙ coyoneda.obj (op X)).obj (op U)\nhy : IsAmalgamation x y\nV : D\nZ : C\nf' : Z ⟶ U\ng' : V ⟶ G.obj Z\nh : S.arrows f'\n⊢ (ℱ.val ⋙ coyoneda.obj (op X)).map (g' ≫ G.map f').op y =\n (sheafOver ℱ 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 u₁\ninst✝² : Category.{v₁, u₁} C\nD : Type u₂\ninst✝¹ : Category.{v₂, u₂} D\nA : Type u₃\ninst✝ : Category.{v₃, u₃} A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nL : GrothendieckTopology A\nG✝ : C ⥤ D\nhG : CompatiblePreserving K G✝\nℱ✝ : SheafOfTypes K\nZ✝ : C\nT : Presieve Z✝\nx✝ : FamilyOfElements (G✝.op ⋙ ℱ✝.val) T\nh✝ : Compatible x✝\nG : C ⥤ D\nhG₁ : CompatiblePreserving K G\nhG₂ : CoverPreserving J K G\nℱ : Sheaf K A\nX : A\nU : C\nS : Sieve U\nhS : S ∈ GrothendieckTopology.sieves J U\nx : FamilyOfElements (G.op ⋙ ℱ.val ⋙ coyoneda.obj (op X)) S.arrows\nhx : Compatible x\nH : IsSheafFor (ℱ.val ⋙ coyoneda.obj (op X)) (Sieve.functorPushforward G S).arrows :=\n Sheaf.cond ℱ X (Sieve.functorPushforward G S) (CoverPreserving.cover_preserve hG₂ hS)\nhx' : Compatible (FamilyOfElements.functorPushforward G x) := Compatible.functorPushforward hG₁ (sheafOver ℱ X) hx\ny : ((G.op ⋙ ℱ.val) ⋙ coyoneda.obj (op X)).obj (op U)\nhy : IsAmalgamation x y\nV : D\nZ : C\nf' : Z ⟶ U\ng' : V ⟶ G.obj Z\nh : S.arrows f'\n⊢ y ≫ ℱ.val.map ((G.map f').op ≫ g'.op) =\n FamilyOfElements.functorPushforward G x (G.map f') (_ : Presieve.functorPushforward G S.arrows (G.map f')) ≫\n ℱ.val.map g'.op\n[PROOFSTEP]\nsimp [hG₁.apply_map (sheafOver ℱ X) hx h, ← hy f' h]\n[GOAL]\nC : Type u₁\ninst✝² : Category.{v₁, u₁} C\nD : Type u₂\ninst✝¹ : Category.{v₂, u₂} D\nA : Type u₃\ninst✝ : Category.{v₃, u₃} A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nL : GrothendieckTopology A\nG✝ : C ⥤ D\nhG : CompatiblePreserving K G✝\nℱ✝ : SheafOfTypes K\nZ : C\nT : Presieve Z\nx : FamilyOfElements (G✝.op ⋙ ℱ✝.val) T\nh : Compatible x\nG : C ⥤ D\nhG₁ : CompatiblePreserving K G\nhG₂ : CoverPreserving J K G\nℱ : Sheaf K A\n⊢ { obj := fun ℱ => pullbackSheaf hG₁ hG₂ ℱ,\n map := fun {X Y} f => { val := ((whiskeringLeft Cᵒᵖ Dᵒᵖ A).obj G.op).map f.val } }.map\n (𝟙 ℱ) =\n 𝟙\n ({ obj := fun ℱ => pullbackSheaf hG₁ hG₂ ℱ,\n map := fun {X Y} f => { val := ((whiskeringLeft Cᵒᵖ Dᵒᵖ A).obj G.op).map f.val } }.obj\n ℱ)\n[PROOFSTEP]\next1\n[GOAL]\ncase h\nC : Type u₁\ninst✝² : Category.{v₁, u₁} C\nD : Type u₂\ninst✝¹ : Category.{v₂, u₂} D\nA : Type u₃\ninst✝ : Category.{v₃, u₃} A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nL : GrothendieckTopology A\nG✝ : C ⥤ D\nhG : CompatiblePreserving K G✝\nℱ✝ : SheafOfTypes K\nZ : C\nT : Presieve Z\nx : FamilyOfElements (G✝.op ⋙ ℱ✝.val) T\nh : Compatible x\nG : C ⥤ D\nhG₁ : CompatiblePreserving K G\nhG₂ : CoverPreserving J K G\nℱ : Sheaf K A\n⊢ ({ obj := fun ℱ => pullbackSheaf hG₁ hG₂ ℱ,\n map := fun {X Y} f => { val := ((whiskeringLeft Cᵒᵖ Dᵒᵖ A).obj G.op).map f.val } }.map\n (𝟙 ℱ)).val =\n (𝟙\n ({ obj := fun ℱ => pullbackSheaf hG₁ hG₂ ℱ,\n map := fun {X Y} f => { val := ((whiskeringLeft Cᵒᵖ Dᵒᵖ A).obj G.op).map f.val } }.obj\n ℱ)).val\n[PROOFSTEP]\napply ((whiskeringLeft _ _ _).obj G.op).map_id\n[GOAL]\nC : Type u₁\ninst✝² : Category.{v₁, u₁} C\nD : Type u₂\ninst✝¹ : Category.{v₂, u₂} D\nA : Type u₃\ninst✝ : Category.{v₃, u₃} A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nL : GrothendieckTopology A\nG✝ : C ⥤ D\nhG : CompatiblePreserving K G✝\nℱ : SheafOfTypes K\nZ : C\nT : Presieve Z\nx : FamilyOfElements (G✝.op ⋙ ℱ.val) T\nh : Compatible x\nG : C ⥤ D\nhG₁ : CompatiblePreserving K G\nhG₂ : CoverPreserving J K G\nX✝ Y✝ Z✝ : Sheaf K A\nf : X✝ ⟶ Y✝\ng : Y✝ ⟶ Z✝\n⊢ { obj := fun ℱ => pullbackSheaf hG₁ hG₂ ℱ,\n map := fun {X Y} f => { val := ((whiskeringLeft Cᵒᵖ Dᵒᵖ A).obj G.op).map f.val } }.map\n (f ≫ g) =\n { obj := fun ℱ => pullbackSheaf hG₁ hG₂ ℱ,\n map := fun {X Y} f => { val := ((whiskeringLeft Cᵒᵖ Dᵒᵖ A).obj G.op).map f.val } }.map\n f ≫\n { obj := fun ℱ => pullbackSheaf hG₁ hG₂ ℱ,\n map := fun {X Y} f => { val := ((whiskeringLeft Cᵒᵖ Dᵒᵖ A).obj G.op).map f.val } }.map\n g\n[PROOFSTEP]\next1\n[GOAL]\ncase h\nC : Type u₁\ninst✝² : Category.{v₁, u₁} C\nD : Type u₂\ninst✝¹ : Category.{v₂, u₂} D\nA : Type u₃\ninst✝ : Category.{v₃, u₃} A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nL : GrothendieckTopology A\nG✝ : C ⥤ D\nhG : CompatiblePreserving K G✝\nℱ : SheafOfTypes K\nZ : C\nT : Presieve Z\nx : FamilyOfElements (G✝.op ⋙ ℱ.val) T\nh : Compatible x\nG : C ⥤ D\nhG₁ : CompatiblePreserving K G\nhG₂ : CoverPreserving J K G\nX✝ Y✝ Z✝ : Sheaf K A\nf : X✝ ⟶ Y✝\ng : Y✝ ⟶ Z✝\n⊢ ({ obj := fun ℱ => pullbackSheaf hG₁ hG₂ ℱ,\n map := fun {X Y} f => { val := ((whiskeringLeft Cᵒᵖ Dᵒᵖ A).obj G.op).map f.val } }.map\n (f ≫ g)).val =\n ({ obj := fun ℱ => pullbackSheaf hG₁ hG₂ ℱ,\n map := fun {X Y} f => { val := ((whiskeringLeft Cᵒᵖ Dᵒᵖ A).obj G.op).map f.val } }.map\n f ≫\n { obj := fun ℱ => pullbackSheaf hG₁ hG₂ ℱ,\n map := fun {X Y} f => { val := ((whiskeringLeft Cᵒᵖ Dᵒᵖ 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]\nι : Type u_1\ninst✝ : Fintype ι\nI J : Box ι\nc c₁ c₂ : ℝ≥0\nr r₁ r₂ : (ι → ℝ) → ↑(Set.Ioi 0)\nπ π₁ π₂ : TaggedPrepartition I\nl l₁ l₂ : IntegrationParams\n⊢ Henstock ≤ Riemann\n[PROOFSTEP]\ntrivial\n[GOAL]\nι : Type u_1\ninst✝ : Fintype ι\nI J : Box ι\nc c₁ c₂ : ℝ≥0\nr r₁ r₂ : (ι → ℝ) → ↑(Set.Ioi 0)\nπ π₁ π₂ : TaggedPrepartition I\nl l₁ l₂ : IntegrationParams\n⊢ Henstock ≤ McShane\n[PROOFSTEP]\ntrivial\n[GOAL]\nι✝ : Type u_1\ninst✝ : Fintype ι✝\nI J : Box ι✝\nc c₁ c₂ : ℝ≥0\nr✝ r₁ r₂ : (ι✝ → ℝ) → ↑(Set.Ioi 0)\nπ π₁ π₂ : TaggedPrepartition I\nl✝ l₁ l₂ : IntegrationParams\nι : Type u_2\nl : IntegrationParams\nhl : l.bRiemann = false\nr : (ι → ℝ) → ↑(Set.Ioi 0)\n⊢ RCond l r\n[PROOFSTEP]\nsimp [RCond, hl]\n[GOAL]\nι : Type u_1\ninst✝ : Fintype ι\nI J : Box ι\nc c₁ c₂ : ℝ≥0\nr r₁ r₂ : (ι → ℝ) → ↑(Set.Ioi 0)\nπ π₁ π₂ : TaggedPrepartition I\nl l₁ l₂ : IntegrationParams\nh₁ : MemBaseSet l I c₁ r₁ π₁\nh₂ : MemBaseSet l I c₂ r₂ π₂\nhU : TaggedPrepartition.iUnion π₁ = TaggedPrepartition.iUnion π₂\n⊢ ∃ π,\n Prepartition.iUnion π = ↑I \\ TaggedPrepartition.iUnion π₁ ∧\n (l.bDistortion = true → Prepartition.distortion π ≤ c₁) ∧ (l.bDistortion = true → Prepartition.distortion π ≤ c₂)\n[PROOFSTEP]\nwlog hc : c₁ ≤ c₂ with H\n[GOAL]\ncase inr\nι : Type u_1\ninst✝ : Fintype ι\nI J : Box ι\nc c₁ c₂ : ℝ≥0\nr r₁ r₂ : (ι → ℝ) → ↑(Set.Ioi 0)\nπ π₁ π₂ : TaggedPrepartition I\nl l₁ l₂ : IntegrationParams\nh₁ : MemBaseSet l I c₁ r₁ π₁\nh₂ : MemBaseSet l I c₂ r₂ π₂\nhU : TaggedPrepartition.iUnion π₁ = TaggedPrepartition.iUnion π₂\nH :\n ∀ {ι : Type u_1} [inst : Fintype ι] {I : Box ι} {J : Box ι} {c : ℝ≥0} {c₁ c₂ : ℝ≥0} {r : (ι → ℝ) → ↑(Set.Ioi 0)}\n {r₁ r₂ : (ι → ℝ) → ↑(Set.Ioi 0)} {π : TaggedPrepartition I} {π₁ π₂ : TaggedPrepartition I} {l : IntegrationParams}\n {l₁ l₂ : IntegrationParams},\n MemBaseSet l I c₁ r₁ π₁ →\n MemBaseSet l I c₂ r₂ π₂ →\n TaggedPrepartition.iUnion π₁ = TaggedPrepartition.iUnion π₂ →\n c₁ ≤ c₂ →\n ∃ π,\n Prepartition.iUnion π = ↑I \\ TaggedPrepartition.iUnion π₁ ∧\n (l.bDistortion = true → Prepartition.distortion π ≤ c₁) ∧\n (l.bDistortion = true → Prepartition.distortion π ≤ c₂)\nhc : ¬c₁ ≤ c₂\n⊢ ∃ π,\n Prepartition.iUnion π = ↑I \\ TaggedPrepartition.iUnion π₁ ∧\n (l.bDistortion = true → Prepartition.distortion π ≤ c₁) ∧ (l.bDistortion = true → Prepartition.distortion π ≤ c₂)\n[PROOFSTEP]\nsimpa [hU, _root_.and_comm] using @H _ _ I J c c₂ c₁ r r₂ r₁ π π₂ π₁ _ l₂ l₁ h₂ h₁ hU.symm (le_of_not_le hc)\n[GOAL]\nι✝ : Type u_1\ninst✝¹ : Fintype ι✝\nI✝ : Box ι✝\nc₁✝ c₂✝ : ℝ≥0\nr₁✝ r₂✝ : (ι✝ → ℝ) → ↑(Set.Ioi 0)\nπ₁✝ π₂✝ : TaggedPrepartition I✝\nl✝ : IntegrationParams\nι : Type u_1\ninst✝ : Fintype ι\nI J : Box ι\nc c₁ c₂ : ℝ≥0\nr r₁ r₂ : (ι → ℝ) → ↑(Set.Ioi 0)\nπ π₁ π₂ : TaggedPrepartition I\nl l₁ l₂ : IntegrationParams\nh₁ : MemBaseSet l I c₁ r₁ π₁\nh₂ : MemBaseSet l I c₂ r₂ π₂\nhU : TaggedPrepartition.iUnion π₁ = TaggedPrepartition.iUnion π₂\nhc : c₁ ≤ c₂\n⊢ ∃ π,\n Prepartition.iUnion π = ↑I \\ TaggedPrepartition.iUnion π₁ ∧\n (l.bDistortion = true → Prepartition.distortion π ≤ c₁) ∧ (l.bDistortion = true → Prepartition.distortion π ≤ c₂)\n[PROOFSTEP]\nby_cases hD : (l.bDistortion : Prop)\n[GOAL]\ncase pos\nι✝ : Type u_1\ninst✝¹ : Fintype ι✝\nI✝ : Box ι✝\nc₁✝ c₂✝ : ℝ≥0\nr₁✝ r₂✝ : (ι✝ → ℝ) → ↑(Set.Ioi 0)\nπ₁✝ π₂✝ : TaggedPrepartition I✝\nl✝ : IntegrationParams\nι : Type u_1\ninst✝ : Fintype ι\nI J : Box ι\nc c₁ c₂ : ℝ≥0\nr r₁ r₂ : (ι → ℝ) → ↑(Set.Ioi 0)\nπ π₁ π₂ : TaggedPrepartition I\nl l₁ l₂ : IntegrationParams\nh₁ : MemBaseSet l I c₁ r₁ π₁\nh₂ : MemBaseSet l I c₂ r₂ π₂\nhU : TaggedPrepartition.iUnion π₁ = TaggedPrepartition.iUnion π₂\nhc : c₁ ≤ c₂\nhD : l.bDistortion = true\n⊢ ∃ π,\n Prepartition.iUnion π = ↑I \\ TaggedPrepartition.iUnion π₁ ∧\n (l.bDistortion = true → Prepartition.distortion π ≤ c₁) ∧ (l.bDistortion = true → Prepartition.distortion π ≤ c₂)\n[PROOFSTEP]\nrcases h₁.4 hD with ⟨π, hπU, hπc⟩\n[GOAL]\ncase pos.intro.intro\nι✝ : Type u_1\ninst✝¹ : Fintype ι✝\nI✝ : Box ι✝\nc₁✝ c₂✝ : ℝ≥0\nr₁✝ r₂✝ : (ι✝ → ℝ) → ↑(Set.Ioi 0)\nπ₁✝ π₂✝ : TaggedPrepartition I✝\nl✝ : IntegrationParams\nι : Type u_1\ninst✝ : Fintype ι\nI J : Box ι\nc c₁ c₂ : ℝ≥0\nr r₁ r₂ : (ι → ℝ) → ↑(Set.Ioi 0)\nπ✝ π₁ π₂ : TaggedPrepartition I\nl l₁ l₂ : IntegrationParams\nh₁ : MemBaseSet l I c₁ r₁ π₁\nh₂ : MemBaseSet l I c₂ r₂ π₂\nhU : TaggedPrepartition.iUnion π₁ = TaggedPrepartition.iUnion π₂\nhc : c₁ ≤ c₂\nhD : l.bDistortion = true\nπ : Prepartition I\nhπU : Prepartition.iUnion π = ↑I \\ TaggedPrepartition.iUnion π₁\nhπc : Prepartition.distortion π ≤ c₁\n⊢ ∃ π,\n Prepartition.iUnion π = ↑I \\ TaggedPrepartition.iUnion π₁ ∧\n (l.bDistortion = true → Prepartition.distortion π ≤ c₁) ∧ (l.bDistortion = true → Prepartition.distortion π ≤ c₂)\n[PROOFSTEP]\nexact ⟨π, hπU, fun _ => hπc, fun _ => hπc.trans hc⟩\n[GOAL]\ncase neg\nι✝ : Type u_1\ninst✝¹ : Fintype ι✝\nI✝ : Box ι✝\nc₁✝ c₂✝ : ℝ≥0\nr₁✝ r₂✝ : (ι✝ → ℝ) → ↑(Set.Ioi 0)\nπ₁✝ π₂✝ : TaggedPrepartition I✝\nl✝ : IntegrationParams\nι : Type u_1\ninst✝ : Fintype ι\nI J : Box ι\nc c₁ c₂ : ℝ≥0\nr r₁ r₂ : (ι → ℝ) → ↑(Set.Ioi 0)\nπ π₁ π₂ : TaggedPrepartition I\nl l₁ l₂ : IntegrationParams\nh₁ : MemBaseSet l I c₁ r₁ π₁\nh₂ : MemBaseSet l I c₂ r₂ π₂\nhU : TaggedPrepartition.iUnion π₁ = TaggedPrepartition.iUnion π₂\nhc : c₁ ≤ c₂\nhD : ¬l.bDistortion = true\n⊢ ∃ π,\n Prepartition.iUnion π = ↑I \\ TaggedPrepartition.iUnion π₁ ∧\n (l.bDistortion = true → Prepartition.distortion π ≤ c₁) ∧ (l.bDistortion = true → Prepartition.distortion π ≤ c₂)\n[PROOFSTEP]\nexact ⟨π₁.toPrepartition.compl, π₁.toPrepartition.iUnion_compl, fun h => (hD h).elim, fun h => (hD h).elim⟩\n[GOAL]\nι : Type u_1\ninst✝ : Fintype ι\nI J : Box ι\nc c₁ c₂ : ℝ≥0\nr r₁ r₂ : (ι → ℝ) → ↑(Set.Ioi 0)\nπ π₁ π₂✝ : TaggedPrepartition I\nl l₁ l₂ : IntegrationParams\nhπ₁ : MemBaseSet l I c r₁ π₁\nhle : ∀ (x : ι → ℝ), x ∈ ↑Box.Icc I → r₂ x ≤ r₁ x\nπ₂ : Prepartition I\nhU : Prepartition.iUnion π₂ = ↑I \\ TaggedPrepartition.iUnion π₁\nhc : l.bDistortion = true → Prepartition.distortion π₂ ≤ c\nx✝ : l.bDistortion = true\n⊢ Prepartition.iUnion ⊥ = ↑I \\ TaggedPrepartition.iUnion (unionComplToSubordinate π₁ π₂ hU r₂) ∧\n Prepartition.distortion ⊥ ≤ c\n[PROOFSTEP]\nsimp\n[GOAL]\nι : Type u_1\ninst✝ : Fintype ι\nI J : Box ι\nc c₁ c₂ : ℝ≥0\nr r₁ r₂ : (ι → ℝ) → ↑(Set.Ioi 0)\nπ π₁ π₂ : TaggedPrepartition I\nl l₁ l₂ : IntegrationParams\nhπ : MemBaseSet l I c r π\np : Box ι → Prop\n⊢ MemBaseSet l I c r (TaggedPrepartition.filter π p)\n[PROOFSTEP]\nrefine'\n ⟨fun J hJ => hπ.1 J (π.mem_filter.1 hJ).1, fun hH J hJ => hπ.2 hH J (π.mem_filter.1 hJ).1, fun hD =>\n (distortion_filter_le _ _).trans (hπ.3 hD), fun hD => _⟩\n[GOAL]\nι : Type u_1\ninst✝ : Fintype ι\nI J : Box ι\nc c₁ c₂ : ℝ≥0\nr r₁ r₂ : (ι → ℝ) → ↑(Set.Ioi 0)\nπ π₁ π₂ : TaggedPrepartition I\nl l₁ l₂ : IntegrationParams\nhπ : MemBaseSet l I c r π\np : Box ι → Prop\nhD : l.bDistortion = true\n⊢ ∃ π',\n Prepartition.iUnion π' = ↑I \\ TaggedPrepartition.iUnion (TaggedPrepartition.filter π p) ∧\n Prepartition.distortion π' ≤ c\n[PROOFSTEP]\nrcases hπ.4 hD with ⟨π₁, hπ₁U, hc⟩\n[GOAL]\ncase intro.intro\nι : Type u_1\ninst✝ : Fintype ι\nI J : Box ι\nc c₁ c₂ : ℝ≥0\nr r₁ r₂ : (ι → ℝ) → ↑(Set.Ioi 0)\nπ π₁✝ π₂ : TaggedPrepartition I\nl l₁ l₂ : IntegrationParams\nhπ : MemBaseSet l I c r π\np : Box ι → Prop\nhD : l.bDistortion = true\nπ₁ : Prepartition I\nhπ₁U : Prepartition.iUnion π₁ = ↑I \\ TaggedPrepartition.iUnion π\nhc : Prepartition.distortion π₁ ≤ c\n⊢ ∃ π',\n Prepartition.iUnion π' = ↑I \\ TaggedPrepartition.iUnion (TaggedPrepartition.filter π p) ∧\n Prepartition.distortion π' ≤ c\n[PROOFSTEP]\nset π₂ := π.filter fun J => ¬p J\n[GOAL]\ncase intro.intro\nι : Type u_1\ninst✝ : Fintype ι\nI J : Box ι\nc c₁ c₂ : ℝ≥0\nr r₁ r₂ : (ι → ℝ) → ↑(Set.Ioi 0)\nπ π₁✝ π₂✝ : TaggedPrepartition I\nl l₁ l₂ : IntegrationParams\nhπ : MemBaseSet l I c r π\np : Box ι → Prop\nhD : l.bDistortion = true\nπ₁ : Prepartition I\nhπ₁U : Prepartition.iUnion π₁ = ↑I \\ TaggedPrepartition.iUnion π\nhc : Prepartition.distortion π₁ ≤ c\nπ₂ : TaggedPrepartition I := TaggedPrepartition.filter π fun J => ¬p J\n⊢ ∃ π',\n Prepartition.iUnion π' = ↑I \\ TaggedPrepartition.iUnion (TaggedPrepartition.filter π p) ∧\n Prepartition.distortion π' ≤ c\n[PROOFSTEP]\nhave : Disjoint π₁.iUnion π₂.iUnion := by simpa [hπ₁U] using disjoint_sdiff_self_left.mono_right sdiff_le\n[GOAL]\nι : Type u_1\ninst✝ : Fintype ι\nI J : Box ι\nc c₁ c₂ : ℝ≥0\nr r₁ r₂ : (ι → ℝ) → ↑(Set.Ioi 0)\nπ π₁✝ π₂✝ : TaggedPrepartition I\nl l₁ l₂ : IntegrationParams\nhπ : MemBaseSet l I c r π\np : Box ι → Prop\nhD : l.bDistortion = true\nπ₁ : Prepartition I\nhπ₁U : Prepartition.iUnion π₁ = ↑I \\ TaggedPrepartition.iUnion π\nhc : Prepartition.distortion π₁ ≤ c\nπ₂ : TaggedPrepartition I := TaggedPrepartition.filter π fun J => ¬p J\n⊢ Disjoint (Prepartition.iUnion π₁) (TaggedPrepartition.iUnion π₂)\n[PROOFSTEP]\nsimpa [hπ₁U] using disjoint_sdiff_self_left.mono_right sdiff_le\n[GOAL]\ncase intro.intro\nι : Type u_1\ninst✝ : Fintype ι\nI J : Box ι\nc c₁ c₂ : ℝ≥0\nr r₁ r₂ : (ι → ℝ) → ↑(Set.Ioi 0)\nπ π₁✝ π₂✝ : TaggedPrepartition I\nl l₁ l₂ : IntegrationParams\nhπ : MemBaseSet l I c r π\np : Box ι → Prop\nhD : l.bDistortion = true\nπ₁ : Prepartition I\nhπ₁U : Prepartition.iUnion π₁ = ↑I \\ TaggedPrepartition.iUnion π\nhc : Prepartition.distortion π₁ ≤ c\nπ₂ : TaggedPrepartition I := TaggedPrepartition.filter π fun J => ¬p J\nthis : Disjoint (Prepartition.iUnion π₁) (TaggedPrepartition.iUnion π₂)\n⊢ ∃ π',\n Prepartition.iUnion π' = ↑I \\ TaggedPrepartition.iUnion (TaggedPrepartition.filter π p) ∧\n Prepartition.distortion π' ≤ c\n[PROOFSTEP]\nrefine' ⟨π₁.disjUnion π₂.toPrepartition this, _, _⟩\n[GOAL]\ncase intro.intro.refine'_1\nι : Type u_1\ninst✝ : Fintype ι\nI J : Box ι\nc c₁ c₂ : ℝ≥0\nr r₁ r₂ : (ι → ℝ) → ↑(Set.Ioi 0)\nπ π₁✝ π₂✝ : TaggedPrepartition I\nl l₁ l₂ : IntegrationParams\nhπ : MemBaseSet l I c r π\np : Box ι → Prop\nhD : l.bDistortion = true\nπ₁ : Prepartition I\nhπ₁U : Prepartition.iUnion π₁ = ↑I \\ TaggedPrepartition.iUnion π\nhc : Prepartition.distortion π₁ ≤ c\nπ₂ : TaggedPrepartition I := TaggedPrepartition.filter π fun J => ¬p J\nthis : Disjoint (Prepartition.iUnion π₁) (TaggedPrepartition.iUnion π₂)\n⊢ Prepartition.iUnion (Prepartition.disjUnion π₁ π₂.toPrepartition this) =\n ↑I \\ TaggedPrepartition.iUnion (TaggedPrepartition.filter π p)\n[PROOFSTEP]\nsuffices ↑I \\ π.iUnion ∪ π.iUnion \\ (π.filter p).iUnion = ↑I \\ (π.filter p).iUnion by simp [*]\n[GOAL]\nι : Type u_1\ninst✝ : Fintype ι\nI J : Box ι\nc c₁ c₂ : ℝ≥0\nr r₁ r₂ : (ι → ℝ) → ↑(Set.Ioi 0)\nπ π₁✝ π₂✝ : TaggedPrepartition I\nl l₁ l₂ : IntegrationParams\nhπ : MemBaseSet l I c r π\np : Box ι → Prop\nhD : l.bDistortion = true\nπ₁ : Prepartition I\nhπ₁U : Prepartition.iUnion π₁ = ↑I \\ TaggedPrepartition.iUnion π\nhc : Prepartition.distortion π₁ ≤ c\nπ₂ : TaggedPrepartition I := TaggedPrepartition.filter π fun J => ¬p J\nthis✝ : Disjoint (Prepartition.iUnion π₁) (TaggedPrepartition.iUnion π₂)\nthis :\n ↑I \\ TaggedPrepartition.iUnion π ∪\n TaggedPrepartition.iUnion π \\ TaggedPrepartition.iUnion (TaggedPrepartition.filter π p) =\n ↑I \\ TaggedPrepartition.iUnion (TaggedPrepartition.filter π p)\n⊢ Prepartition.iUnion (Prepartition.disjUnion π₁ π₂.toPrepartition this✝) =\n ↑I \\ TaggedPrepartition.iUnion (TaggedPrepartition.filter π p)\n[PROOFSTEP]\nsimp [*]\n[GOAL]\ncase intro.intro.refine'_1\nι : Type u_1\ninst✝ : Fintype ι\nI J : Box ι\nc c₁ c₂ : ℝ≥0\nr r₁ r₂ : (ι → ℝ) → ↑(Set.Ioi 0)\nπ π₁✝ π₂✝ : TaggedPrepartition I\nl l₁ l₂ : IntegrationParams\nhπ : MemBaseSet l I c r π\np : Box ι → Prop\nhD : l.bDistortion = true\nπ₁ : Prepartition I\nhπ₁U : Prepartition.iUnion π₁ = ↑I \\ TaggedPrepartition.iUnion π\nhc : Prepartition.distortion π₁ ≤ c\nπ₂ : TaggedPrepartition I := TaggedPrepartition.filter π fun J => ¬p J\nthis : Disjoint (Prepartition.iUnion π₁) (TaggedPrepartition.iUnion π₂)\n⊢ ↑I \\ TaggedPrepartition.iUnion π ∪\n TaggedPrepartition.iUnion π \\ TaggedPrepartition.iUnion (TaggedPrepartition.filter π p) =\n ↑I \\ TaggedPrepartition.iUnion (TaggedPrepartition.filter π p)\n[PROOFSTEP]\nhave h : (π.filter p).iUnion ⊆ π.iUnion := biUnion_subset_biUnion_left (Finset.filter_subset _ _)\n[GOAL]\ncase intro.intro.refine'_1\nι : Type u_1\ninst✝ : Fintype ι\nI J : Box ι\nc c₁ c₂ : ℝ≥0\nr r₁ r₂ : (ι → ℝ) → ↑(Set.Ioi 0)\nπ π₁✝ π₂✝ : TaggedPrepartition I\nl l₁ l₂ : IntegrationParams\nhπ : MemBaseSet l I c r π\np : Box ι → Prop\nhD : l.bDistortion = true\nπ₁ : Prepartition I\nhπ₁U : Prepartition.iUnion π₁ = ↑I \\ TaggedPrepartition.iUnion π\nhc : Prepartition.distortion π₁ ≤ c\nπ₂ : TaggedPrepartition I := TaggedPrepartition.filter π fun J => ¬p J\nthis : Disjoint (Prepartition.iUnion π₁) (TaggedPrepartition.iUnion π₂)\nh : TaggedPrepartition.iUnion (TaggedPrepartition.filter π p) ⊆ TaggedPrepartition.iUnion π\n⊢ ↑I \\ TaggedPrepartition.iUnion π ∪\n TaggedPrepartition.iUnion π \\ TaggedPrepartition.iUnion (TaggedPrepartition.filter π p) =\n ↑I \\ TaggedPrepartition.iUnion (TaggedPrepartition.filter π p)\n[PROOFSTEP]\next x\n[GOAL]\ncase intro.intro.refine'_1.h\nι : Type u_1\ninst✝ : Fintype ι\nI J : Box ι\nc c₁ c₂ : ℝ≥0\nr r₁ r₂ : (ι → ℝ) → ↑(Set.Ioi 0)\nπ π₁✝ π₂✝ : TaggedPrepartition I\nl l₁ l₂ : IntegrationParams\nhπ : MemBaseSet l I c r π\np : Box ι → Prop\nhD : l.bDistortion = true\nπ₁ : Prepartition I\nhπ₁U : Prepartition.iUnion π₁ = ↑I \\ TaggedPrepartition.iUnion π\nhc : Prepartition.distortion π₁ ≤ c\nπ₂ : TaggedPrepartition I := TaggedPrepartition.filter π fun J => ¬p J\nthis : Disjoint (Prepartition.iUnion π₁) (TaggedPrepartition.iUnion π₂)\nh : TaggedPrepartition.iUnion (TaggedPrepartition.filter π p) ⊆ TaggedPrepartition.iUnion π\nx : ι → ℝ\n⊢ x ∈\n ↑I \\ TaggedPrepartition.iUnion π ∪\n TaggedPrepartition.iUnion π \\ TaggedPrepartition.iUnion (TaggedPrepartition.filter π p) ↔\n x ∈ ↑I \\ TaggedPrepartition.iUnion (TaggedPrepartition.filter π p)\n[PROOFSTEP]\nfconstructor\n[GOAL]\ncase intro.intro.refine'_1.h.mp\nι : Type u_1\ninst✝ : Fintype ι\nI J : Box ι\nc c₁ c₂ : ℝ≥0\nr r₁ r₂ : (ι → ℝ) → ↑(Set.Ioi 0)\nπ π₁✝ π₂✝ : TaggedPrepartition I\nl l₁ l₂ : IntegrationParams\nhπ : MemBaseSet l I c r π\np : Box ι → Prop\nhD : l.bDistortion = true\nπ₁ : Prepartition I\nhπ₁U : Prepartition.iUnion π₁ = ↑I \\ TaggedPrepartition.iUnion π\nhc : Prepartition.distortion π₁ ≤ c\nπ₂ : TaggedPrepartition I := TaggedPrepartition.filter π fun J => ¬p J\nthis : Disjoint (Prepartition.iUnion π₁) (TaggedPrepartition.iUnion π₂)\nh : TaggedPrepartition.iUnion (TaggedPrepartition.filter π p) ⊆ TaggedPrepartition.iUnion π\nx : ι → ℝ\n⊢ x ∈\n ↑I \\ TaggedPrepartition.iUnion π ∪\n TaggedPrepartition.iUnion π \\ TaggedPrepartition.iUnion (TaggedPrepartition.filter π p) →\n x ∈ ↑I \\ TaggedPrepartition.iUnion (TaggedPrepartition.filter π p)\n[PROOFSTEP]\nrintro (⟨hxI, hxπ⟩ | ⟨hxπ, hxp⟩)\n[GOAL]\ncase intro.intro.refine'_1.h.mp.inl.intro\nι : Type u_1\ninst✝ : Fintype ι\nI J : Box ι\nc c₁ c₂ : ℝ≥0\nr r₁ r₂ : (ι → ℝ) → ↑(Set.Ioi 0)\nπ π₁✝ π₂✝ : TaggedPrepartition I\nl l₁ l₂ : IntegrationParams\nhπ : MemBaseSet l I c r π\np : Box ι → Prop\nhD : l.bDistortion = true\nπ₁ : Prepartition I\nhπ₁U : Prepartition.iUnion π₁ = ↑I \\ TaggedPrepartition.iUnion π\nhc : Prepartition.distortion π₁ ≤ c\nπ₂ : TaggedPrepartition I := TaggedPrepartition.filter π fun J => ¬p J\nthis : Disjoint (Prepartition.iUnion π₁) (TaggedPrepartition.iUnion π₂)\nh : TaggedPrepartition.iUnion (TaggedPrepartition.filter π p) ⊆ TaggedPrepartition.iUnion π\nx : ι → ℝ\nhxI : x ∈ ↑I\nhxπ : ¬x ∈ TaggedPrepartition.iUnion π\n⊢ x ∈ ↑I \\ TaggedPrepartition.iUnion (TaggedPrepartition.filter π p)\ncase intro.intro.refine'_1.h.mp.inr.intro\nι : Type u_1\ninst✝ : Fintype ι\nI J : Box ι\nc c₁ c₂ : ℝ≥0\nr r₁ r₂ : (ι → ℝ) → ↑(Set.Ioi 0)\nπ π₁✝ π₂✝ : TaggedPrepartition I\nl l₁ l₂ : IntegrationParams\nhπ : MemBaseSet l I c r π\np : Box ι → Prop\nhD : l.bDistortion = true\nπ₁ : Prepartition I\nhπ₁U : Prepartition.iUnion π₁ = ↑I \\ TaggedPrepartition.iUnion π\nhc : Prepartition.distortion π₁ ≤ c\nπ₂ : TaggedPrepartition I := TaggedPrepartition.filter π fun J => ¬p J\nthis : Disjoint (Prepartition.iUnion π₁) (TaggedPrepartition.iUnion π₂)\nh : TaggedPrepartition.iUnion (TaggedPrepartition.filter π p) ⊆ TaggedPrepartition.iUnion π\nx : ι → ℝ\nhxπ : x ∈ TaggedPrepartition.iUnion π\nhxp : ¬x ∈ TaggedPrepartition.iUnion (TaggedPrepartition.filter π p)\n⊢ x ∈ ↑I \\ TaggedPrepartition.iUnion (TaggedPrepartition.filter π p)\n[PROOFSTEP]\nexacts [⟨hxI, mt (@h x) hxπ⟩, ⟨π.iUnion_subset hxπ, hxp⟩]\n[GOAL]\ncase intro.intro.refine'_1.h.mpr\nι : Type u_1\ninst✝ : Fintype ι\nI J : Box ι\nc c₁ c₂ : ℝ≥0\nr r₁ r₂ : (ι → ℝ) → ↑(Set.Ioi 0)\nπ π₁✝ π₂✝ : TaggedPrepartition I\nl l₁ l₂ : IntegrationParams\nhπ : MemBaseSet l I c r π\np : Box ι → Prop\nhD : l.bDistortion = true\nπ₁ : Prepartition I\nhπ₁U : Prepartition.iUnion π₁ = ↑I \\ TaggedPrepartition.iUnion π\nhc : Prepartition.distortion π₁ ≤ c\nπ₂ : TaggedPrepartition I := TaggedPrepartition.filter π fun J => ¬p J\nthis : Disjoint (Prepartition.iUnion π₁) (TaggedPrepartition.iUnion π₂)\nh : TaggedPrepartition.iUnion (TaggedPrepartition.filter π p) ⊆ TaggedPrepartition.iUnion π\nx : ι → ℝ\n⊢ x ∈ ↑I \\ TaggedPrepartition.iUnion (TaggedPrepartition.filter π p) →\n x ∈\n ↑I \\ TaggedPrepartition.iUnion π ∪\n TaggedPrepartition.iUnion π \\ TaggedPrepartition.iUnion (TaggedPrepartition.filter π p)\n[PROOFSTEP]\nrintro ⟨hxI, hxp⟩\n[GOAL]\ncase intro.intro.refine'_1.h.mpr.intro\nι : Type u_1\ninst✝ : Fintype ι\nI J : Box ι\nc c₁ c₂ : ℝ≥0\nr r₁ r₂ : (ι → ℝ) → ↑(Set.Ioi 0)\nπ π₁✝ π₂✝ : TaggedPrepartition I\nl l₁ l₂ : IntegrationParams\nhπ : MemBaseSet l I c r π\np : Box ι → Prop\nhD : l.bDistortion = true\nπ₁ : Prepartition I\nhπ₁U : Prepartition.iUnion π₁ = ↑I \\ TaggedPrepartition.iUnion π\nhc : Prepartition.distortion π₁ ≤ c\nπ₂ : TaggedPrepartition I := TaggedPrepartition.filter π fun J => ¬p J\nthis : Disjoint (Prepartition.iUnion π₁) (TaggedPrepartition.iUnion π₂)\nh : TaggedPrepartition.iUnion (TaggedPrepartition.filter π p) ⊆ TaggedPrepartition.iUnion π\nx : ι → ℝ\nhxI : x ∈ ↑I\nhxp : ¬x ∈ TaggedPrepartition.iUnion (TaggedPrepartition.filter π p)\n⊢ x ∈\n ↑I \\ TaggedPrepartition.iUnion π ∪\n TaggedPrepartition.iUnion π \\ TaggedPrepartition.iUnion (TaggedPrepartition.filter π p)\n[PROOFSTEP]\nby_cases hxπ : x ∈ π.iUnion\n[GOAL]\ncase pos\nι : Type u_1\ninst✝ : Fintype ι\nI J : Box ι\nc c₁ c₂ : ℝ≥0\nr r₁ r₂ : (ι → ℝ) → ↑(Set.Ioi 0)\nπ π₁✝ π₂✝ : TaggedPrepartition I\nl l₁ l₂ : IntegrationParams\nhπ : MemBaseSet l I c r π\np : Box ι → Prop\nhD : l.bDistortion = true\nπ₁ : Prepartition I\nhπ₁U : Prepartition.iUnion π₁ = ↑I \\ TaggedPrepartition.iUnion π\nhc : Prepartition.distortion π₁ ≤ c\nπ₂ : TaggedPrepartition I := TaggedPrepartition.filter π fun J => ¬p J\nthis : Disjoint (Prepartition.iUnion π₁) (TaggedPrepartition.iUnion π₂)\nh : TaggedPrepartition.iUnion (TaggedPrepartition.filter π p) ⊆ TaggedPrepartition.iUnion π\nx : ι → ℝ\nhxI : x ∈ ↑I\nhxp : ¬x ∈ TaggedPrepartition.iUnion (TaggedPrepartition.filter π p)\nhxπ : x ∈ TaggedPrepartition.iUnion π\n⊢ x ∈\n ↑I \\ TaggedPrepartition.iUnion π ∪\n TaggedPrepartition.iUnion π \\ TaggedPrepartition.iUnion (TaggedPrepartition.filter π p)\ncase neg\nι : Type u_1\ninst✝ : Fintype ι\nI J : Box ι\nc c₁ c₂ : ℝ≥0\nr r₁ r₂ : (ι → ℝ) → ↑(Set.Ioi 0)\nπ π₁✝ π₂✝ : TaggedPrepartition I\nl l₁ l₂ : IntegrationParams\nhπ : MemBaseSet l I c r π\np : Box ι → Prop\nhD : l.bDistortion = true\nπ₁ : Prepartition I\nhπ₁U : Prepartition.iUnion π₁ = ↑I \\ TaggedPrepartition.iUnion π\nhc : Prepartition.distortion π₁ ≤ c\nπ₂ : TaggedPrepartition I := TaggedPrepartition.filter π fun J => ¬p J\nthis : Disjoint (Prepartition.iUnion π₁) (TaggedPrepartition.iUnion π₂)\nh : TaggedPrepartition.iUnion (TaggedPrepartition.filter π p) ⊆ TaggedPrepartition.iUnion π\nx : ι → ℝ\nhxI : x ∈ ↑I\nhxp : ¬x ∈ TaggedPrepartition.iUnion (TaggedPrepartition.filter π p)\nhxπ : ¬x ∈ TaggedPrepartition.iUnion π\n⊢ x ∈\n ↑I \\ TaggedPrepartition.iUnion π ∪\n TaggedPrepartition.iUnion π \\ TaggedPrepartition.iUnion (TaggedPrepartition.filter π p)\n[PROOFSTEP]\nexacts [Or.inr ⟨hxπ, hxp⟩, Or.inl ⟨hxI, hxπ⟩]\n[GOAL]\ncase intro.intro.refine'_2\nι : Type u_1\ninst✝ : Fintype ι\nI J : Box ι\nc c₁ c₂ : ℝ≥0\nr r₁ r₂ : (ι → ℝ) → ↑(Set.Ioi 0)\nπ π₁✝ π₂✝ : TaggedPrepartition I\nl l₁ l₂ : IntegrationParams\nhπ : MemBaseSet l I c r π\np : Box ι → Prop\nhD : l.bDistortion = true\nπ₁ : Prepartition I\nhπ₁U : Prepartition.iUnion π₁ = ↑I \\ TaggedPrepartition.iUnion π\nhc : Prepartition.distortion π₁ ≤ c\nπ₂ : TaggedPrepartition I := TaggedPrepartition.filter π fun J => ¬p J\nthis : Disjoint (Prepartition.iUnion π₁) (TaggedPrepartition.iUnion π₂)\n⊢ Prepartition.distortion (Prepartition.disjUnion π₁ π₂.toPrepartition this) ≤ c\n[PROOFSTEP]\nhave : (π.filter fun J => ¬p J).distortion ≤ c := (distortion_filter_le _ _).trans (hπ.3 hD)\n[GOAL]\ncase intro.intro.refine'_2\nι : Type u_1\ninst✝ : Fintype ι\nI J : Box ι\nc c₁ c₂ : ℝ≥0\nr r₁ r₂ : (ι → ℝ) → ↑(Set.Ioi 0)\nπ π₁✝ π₂✝ : TaggedPrepartition I\nl l₁ l₂ : IntegrationParams\nhπ : MemBaseSet l I c r π\np : Box ι → Prop\nhD : l.bDistortion = true\nπ₁ : Prepartition I\nhπ₁U : Prepartition.iUnion π₁ = ↑I \\ TaggedPrepartition.iUnion π\nhc : Prepartition.distortion π₁ ≤ c\nπ₂ : TaggedPrepartition I := TaggedPrepartition.filter π fun J => ¬p J\nthis✝ : Disjoint (Prepartition.iUnion π₁) (TaggedPrepartition.iUnion π₂)\nthis : distortion (TaggedPrepartition.filter π fun J => ¬p J) ≤ c\n⊢ Prepartition.distortion (Prepartition.disjUnion π₁ π₂.toPrepartition this✝) ≤ c\n[PROOFSTEP]\nsimpa [hc]\n[GOAL]\nι : Type u_1\ninst✝ : Fintype ι\nI J : Box ι\nc c₁ c₂ : ℝ≥0\nr r₁ r₂ : (ι → ℝ) → ↑(Set.Ioi 0)\nπ✝ π₁ π₂ : TaggedPrepartition I\nl l₁ l₂ : IntegrationParams\nπ : Prepartition I\nπi : (J : Box ι) → TaggedPrepartition J\nh : ∀ (J : Box ι), J ∈ π → MemBaseSet l J c r (πi J)\nhp : ∀ (J : Box ι), J ∈ π → IsPartition (πi J)\nhc : l.bDistortion = true → Prepartition.distortion (Prepartition.compl π) ≤ c\n⊢ MemBaseSet l I c r (Prepartition.biUnionTagged π πi)\n[PROOFSTEP]\nrefine'\n ⟨TaggedPrepartition.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 => _⟩\n[GOAL]\ncase refine'_1\nι : Type u_1\ninst✝ : Fintype ι\nI J : Box ι\nc c₁ c₂ : ℝ≥0\nr r₁ r₂ : (ι → ℝ) → ↑(Set.Ioi 0)\nπ✝ π₁ π₂ : TaggedPrepartition I\nl l₁ l₂ : IntegrationParams\nπ : Prepartition I\nπi : (J : Box ι) → TaggedPrepartition J\nh : ∀ (J : Box ι), J ∈ π → MemBaseSet l J c r (πi J)\nhp : ∀ (J : Box ι), J ∈ π → IsPartition (πi J)\nhc : l.bDistortion = true → Prepartition.distortion (Prepartition.compl π) ≤ c\nhD : l.bDistortion = true\n⊢ distortion (Prepartition.biUnionTagged π πi) ≤ c\n[PROOFSTEP]\nrw [Prepartition.distortion_biUnionTagged, Finset.sup_le_iff]\n[GOAL]\ncase refine'_1\nι : Type u_1\ninst✝ : Fintype ι\nI J : Box ι\nc c₁ c₂ : ℝ≥0\nr r₁ r₂ : (ι → ℝ) → ↑(Set.Ioi 0)\nπ✝ π₁ π₂ : TaggedPrepartition I\nl l₁ l₂ : IntegrationParams\nπ : Prepartition I\nπi : (J : Box ι) → TaggedPrepartition J\nh : ∀ (J : Box ι), J ∈ π → MemBaseSet l J c r (πi J)\nhp : ∀ (J : Box ι), J ∈ π → IsPartition (πi J)\nhc : l.bDistortion = true → Prepartition.distortion (Prepartition.compl π) ≤ c\nhD : l.bDistortion = true\n⊢ ∀ (b : Box ι), b ∈ π.boxes → distortion (πi b) ≤ c\n[PROOFSTEP]\nexact fun J hJ => (h J hJ).3 hD\n[GOAL]\ncase refine'_2\nι : Type u_1\ninst✝ : Fintype ι\nI J : Box ι\nc c₁ c₂ : ℝ≥0\nr r₁ r₂ : (ι → ℝ) → ↑(Set.Ioi 0)\nπ✝ π₁ π₂ : TaggedPrepartition I\nl l₁ l₂ : IntegrationParams\nπ : Prepartition I\nπi : (J : Box ι) → TaggedPrepartition J\nh : ∀ (J : Box ι), J ∈ π → MemBaseSet l J c r (πi J)\nhp : ∀ (J : Box ι), J ∈ π → IsPartition (πi J)\nhc : l.bDistortion = true → Prepartition.distortion (Prepartition.compl π) ≤ c\nhD : l.bDistortion = true\n⊢ ∃ π',\n Prepartition.iUnion π' = ↑I \\ TaggedPrepartition.iUnion (Prepartition.biUnionTagged π πi) ∧\n Prepartition.distortion π' ≤ c\n[PROOFSTEP]\nrefine' ⟨_, _, hc hD⟩\n[GOAL]\ncase refine'_2\nι : Type u_1\ninst✝ : Fintype ι\nI J : Box ι\nc c₁ c₂ : ℝ≥0\nr r₁ r₂ : (ι → ℝ) → ↑(Set.Ioi 0)\nπ✝ π₁ π₂ : TaggedPrepartition I\nl l₁ l₂ : IntegrationParams\nπ : Prepartition I\nπi : (J : Box ι) → TaggedPrepartition J\nh : ∀ (J : Box ι), J ∈ π → MemBaseSet l J c r (πi J)\nhp : ∀ (J : Box ι), J ∈ π → IsPartition (πi J)\nhc : l.bDistortion = true → Prepartition.distortion (Prepartition.compl π) ≤ c\nhD : l.bDistortion = true\n⊢ Prepartition.iUnion (Prepartition.compl π) = ↑I \\ TaggedPrepartition.iUnion (Prepartition.biUnionTagged π πi)\n[PROOFSTEP]\nrw [π.iUnion_compl, ← π.iUnion_biUnion_partition hp]\n[GOAL]\ncase refine'_2\nι : Type u_1\ninst✝ : Fintype ι\nI J : Box ι\nc c₁ c₂ : ℝ≥0\nr r₁ r₂ : (ι → ℝ) → ↑(Set.Ioi 0)\nπ✝ π₁ π₂ : TaggedPrepartition I\nl l₁ l₂ : IntegrationParams\nπ : Prepartition I\nπi : (J : Box ι) → TaggedPrepartition J\nh : ∀ (J : Box ι), J ∈ π → MemBaseSet l J c r (πi J)\nhp : ∀ (J : Box ι), J ∈ π → IsPartition (πi J)\nhc : l.bDistortion = true → Prepartition.distortion (Prepartition.compl π) ≤ c\nhD : l.bDistortion = true\n⊢ ↑I \\ Prepartition.iUnion (Prepartition.biUnion π fun J => (πi J).toPrepartition) =\n ↑I \\ TaggedPrepartition.iUnion (Prepartition.biUnionTagged π πi)\n[PROOFSTEP]\nrfl\n[GOAL]\nι : Type u_1\ninst✝ : Fintype ι\nI✝ J : Box ι\nc c₁ c₂ : ℝ≥0\nr r₁ r₂ : (ι → ℝ) → ↑(Set.Ioi 0)\nπ π₁✝ π₂✝ : TaggedPrepartition I✝\nl✝ l₁ l₂ : IntegrationParams\nI : Box ι\nl : IntegrationParams\nπ₁ π₂ : Prepartition I\nh : Prepartition.iUnion π₁ = Prepartition.iUnion π₂\n⊢ toFilteriUnion l I π₁ = toFilteriUnion l I π₂\n[PROOFSTEP]\nsimp only [toFilteriUnion, toFilterDistortioniUnion, h]\n[GOAL]\nι : Type u_1\ninst✝ : Fintype ι\nI✝ J : Box ι\nc c₁ c₂ : ℝ≥0\nr r₁ r₂ : (ι → ℝ) → ↑(Set.Ioi 0)\nπ π₁ π₂ : TaggedPrepartition I✝\nl✝ l₁ l₂ l : IntegrationParams\nI : Box ι\nπ₀ : Prepartition I\n⊢ HasBasis (toFilteriUnion l I π₀) (fun r => ∀ (c : ℝ≥0), RCond l (r c)) fun r =>\n {π | ∃ c, MemBaseSet l I c (r c) π ∧ TaggedPrepartition.iUnion π = Prepartition.iUnion π₀}\n[PROOFSTEP]\nhave := fun c => l.hasBasis_toFilterDistortioniUnion I c π₀\n[GOAL]\nι : Type u_1\ninst✝ : Fintype ι\nI✝ J : Box ι\nc c₁ c₂ : ℝ≥0\nr r₁ r₂ : (ι → ℝ) → ↑(Set.Ioi 0)\nπ π₁ π₂ : TaggedPrepartition I✝\nl✝ l₁ l₂ l : IntegrationParams\nI : Box ι\nπ₀ : Prepartition I\nthis :\n ∀ (c : ℝ≥0),\n HasBasis (toFilterDistortioniUnion l I c π₀) (RCond l) fun r =>\n {π | MemBaseSet l I c r π ∧ TaggedPrepartition.iUnion π = Prepartition.iUnion π₀}\n⊢ HasBasis (toFilteriUnion l I π₀) (fun r => ∀ (c : ℝ≥0), RCond l (r c)) fun r =>\n {π | ∃ c, MemBaseSet l I c (r c) π ∧ TaggedPrepartition.iUnion π = Prepartition.iUnion π₀}\n[PROOFSTEP]\nsimpa only [setOf_and, setOf_exists] using hasBasis_iSup this\n[GOAL]\nι : Type u_1\ninst✝ : Fintype ι\nI✝ J : Box ι\nc c₁ c₂ : ℝ≥0\nr r₁ r₂ : (ι → ℝ) → ↑(Set.Ioi 0)\nπ π₁ π₂ : TaggedPrepartition I✝\nl✝ l₁ l₂ l : IntegrationParams\nI : Box ι\n⊢ HasBasis (toFilteriUnion l I ⊤) (fun r => ∀ (c : ℝ≥0), RCond l (r c)) fun r =>\n {π | ∃ c, MemBaseSet l I c (r c) π ∧ IsPartition π}\n[PROOFSTEP]\nsimpa only [TaggedPrepartition.isPartition_iff_iUnion_eq, Prepartition.iUnion_top] using l.hasBasis_toFilteriUnion I ⊤\n[GOAL]\nι : Type u_1\ninst✝ : Fintype ι\nI✝ J : Box ι\nc c₁ c₂ : ℝ≥0\nr r₁ r₂ : (ι → ℝ) → ↑(Set.Ioi 0)\nπ π₁ π₂ : TaggedPrepartition I✝\nl✝ l₁ l₂ l : IntegrationParams\nI : Box ι\n⊢ HasBasis (toFilter l I) (fun r => ∀ (c : ℝ≥0), RCond l (r c)) fun r => {π | ∃ c, MemBaseSet l I c (r c) π}\n[PROOFSTEP]\nsimpa only [setOf_exists] using hasBasis_iSup (l.hasBasis_toFilterDistortion I)\n[GOAL]\nι : Type u_1\ninst✝ : Fintype ι\nI J : Box ι\nc c₁ c₂ : ℝ≥0\nr r₁ r₂ : (ι → ℝ) → ↑(Set.Ioi 0)\nπ π₁ π₂ : TaggedPrepartition I\nl✝ l₁ l₂ l : IntegrationParams\nh : I ≤ J\n⊢ Tendsto (↑(embedBox I J h)) (toFilteriUnion l I ⊤) (toFilteriUnion l J (Prepartition.single J I h))\n[PROOFSTEP]\nsimp only [toFilteriUnion, tendsto_iSup]\n[GOAL]\nι : Type u_1\ninst✝ : Fintype ι\nI J : Box ι\nc c₁ c₂ : ℝ≥0\nr r₁ r₂ : (ι → ℝ) → ↑(Set.Ioi 0)\nπ π₁ π₂ : TaggedPrepartition I\nl✝ l₁ l₂ l : IntegrationParams\nh : I ≤ J\n⊢ ∀ (i : ℝ≥0),\n Tendsto (↑(embedBox I J h)) (toFilterDistortioniUnion l I i ⊤)\n (⨆ (c : ℝ≥0), toFilterDistortioniUnion l J c (Prepartition.single J I h))\n[PROOFSTEP]\nintro c\n[GOAL]\nι : Type u_1\ninst✝ : Fintype ι\nI J : Box ι\nc✝ c₁ c₂ : ℝ≥0\nr r₁ r₂ : (ι → ℝ) → ↑(Set.Ioi 0)\nπ π₁ π₂ : TaggedPrepartition I\nl✝ l₁ l₂ l : IntegrationParams\nh : I ≤ J\nc : ℝ≥0\n⊢ Tendsto (↑(embedBox I J h)) (toFilterDistortioniUnion l I c ⊤)\n (⨆ (c : ℝ≥0), toFilterDistortioniUnion l J c (Prepartition.single J I h))\n[PROOFSTEP]\nset π₀ := Prepartition.single J I h\n[GOAL]\nι : Type u_1\ninst✝ : Fintype ι\nI J : Box ι\nc✝ c₁ c₂ : ℝ≥0\nr r₁ r₂ : (ι → ℝ) → ↑(Set.Ioi 0)\nπ π₁ π₂ : TaggedPrepartition I\nl✝ l₁ l₂ l : IntegrationParams\nh : I ≤ J\nc : ℝ≥0\nπ₀ : Prepartition J := Prepartition.single J I h\n⊢ Tendsto (↑(embedBox I J h)) (toFilterDistortioniUnion l I c ⊤) (⨆ (c : ℝ≥0), toFilterDistortioniUnion l J c π₀)\n[PROOFSTEP]\nrefine' le_iSup_of_le (max c π₀.compl.distortion) _\n[GOAL]\nι : Type u_1\ninst✝ : Fintype ι\nI J : Box ι\nc✝ c₁ c₂ : ℝ≥0\nr r₁ r₂ : (ι → ℝ) → ↑(Set.Ioi 0)\nπ π₁ π₂ : TaggedPrepartition I\nl✝ l₁ l₂ l : IntegrationParams\nh : I ≤ J\nc : ℝ≥0\nπ₀ : Prepartition J := Prepartition.single J I h\n⊢ Filter.map (↑(embedBox I J h)) (toFilterDistortioniUnion l I c ⊤) ≤\n toFilterDistortioniUnion l J (max c (Prepartition.distortion (Prepartition.compl π₀))) π₀\n[PROOFSTEP]\nrefine'\n ((l.hasBasis_toFilterDistortioniUnion I c ⊤).tendsto_iff (l.hasBasis_toFilterDistortioniUnion J _ _)).2 fun r hr => _\n[GOAL]\nι : Type u_1\ninst✝ : Fintype ι\nI J : Box ι\nc✝ c₁ c₂ : ℝ≥0\nr✝ r₁ r₂ : (ι → ℝ) → ↑(Set.Ioi 0)\nπ π₁ π₂ : TaggedPrepartition I\nl✝ l₁ l₂ l : IntegrationParams\nh : I ≤ J\nc : ℝ≥0\nπ₀ : Prepartition J := Prepartition.single J I h\nr : (ι → ℝ) → ↑(Set.Ioi 0)\nhr : RCond l r\n⊢ ∃ ia,\n RCond l ia ∧\n ∀ (x : TaggedPrepartition I),\n x ∈ {π | MemBaseSet l I c ia π ∧ TaggedPrepartition.iUnion π = Prepartition.iUnion ⊤} →\n ↑(embedBox I J h) x ∈\n {π |\n MemBaseSet l J (max c (Prepartition.distortion (Prepartition.compl π₀))) r π ∧\n TaggedPrepartition.iUnion π = Prepartition.iUnion π₀}\n[PROOFSTEP]\nrefine' ⟨r, hr, fun π hπ => _⟩\n[GOAL]\nι : Type u_1\ninst✝ : Fintype ι\nI J : Box ι\nc✝ c₁ c₂ : ℝ≥0\nr✝ r₁ r₂ : (ι → ℝ) → ↑(Set.Ioi 0)\nπ✝ π₁ π₂ : TaggedPrepartition I\nl✝ l₁ l₂ l : IntegrationParams\nh : I ≤ J\nc : ℝ≥0\nπ₀ : Prepartition J := Prepartition.single J I h\nr : (ι → ℝ) → ↑(Set.Ioi 0)\nhr : RCond l r\nπ : TaggedPrepartition I\nhπ : π ∈ {π | MemBaseSet l I c r π ∧ TaggedPrepartition.iUnion π = Prepartition.iUnion ⊤}\n⊢ ↑(embedBox I J h) π ∈\n {π |\n MemBaseSet l J (max c (Prepartition.distortion (Prepartition.compl π₀))) r π ∧\n TaggedPrepartition.iUnion π = Prepartition.iUnion π₀}\n[PROOFSTEP]\nrw [mem_setOf_eq, Prepartition.iUnion_top] at hπ \n[GOAL]\nι : Type u_1\ninst✝ : Fintype ι\nI J : Box ι\nc✝ c₁ c₂ : ℝ≥0\nr✝ r₁ r₂ : (ι → ℝ) → ↑(Set.Ioi 0)\nπ✝ π₁ π₂ : TaggedPrepartition I\nl✝ l₁ l₂ l : IntegrationParams\nh : I ≤ J\nc : ℝ≥0\nπ₀ : Prepartition J := Prepartition.single J I h\nr : (ι → ℝ) → ↑(Set.Ioi 0)\nhr : RCond l r\nπ : TaggedPrepartition I\nhπ : MemBaseSet l I c r π ∧ TaggedPrepartition.iUnion π = ↑I\n⊢ ↑(embedBox I J h) π ∈\n {π |\n MemBaseSet l J (max c (Prepartition.distortion (Prepartition.compl π₀))) r π ∧\n TaggedPrepartition.iUnion π = Prepartition.iUnion π₀}\n[PROOFSTEP]\nrefine' ⟨⟨hπ.1.1, hπ.1.2, fun hD => le_trans (hπ.1.3 hD) (le_max_left _ _), fun _ => _⟩, _⟩\n[GOAL]\ncase refine'_1\nι : Type u_1\ninst✝ : Fintype ι\nI J : Box ι\nc✝ c₁ c₂ : ℝ≥0\nr✝ r₁ r₂ : (ι → ℝ) → ↑(Set.Ioi 0)\nπ✝ π₁ π₂ : TaggedPrepartition I\nl✝ l₁ l₂ l : IntegrationParams\nh : I ≤ J\nc : ℝ≥0\nπ₀ : Prepartition J := Prepartition.single J I h\nr : (ι → ℝ) → ↑(Set.Ioi 0)\nhr : RCond l r\nπ : TaggedPrepartition I\nhπ : MemBaseSet l I c r π ∧ TaggedPrepartition.iUnion π = ↑I\nx✝ : l.bDistortion = true\n⊢ ∃ π',\n Prepartition.iUnion π' = ↑J \\ TaggedPrepartition.iUnion (↑(embedBox I J h) π) ∧\n Prepartition.distortion π' ≤ max c (Prepartition.distortion (Prepartition.compl π₀))\n[PROOFSTEP]\nrefine' ⟨_, π₀.iUnion_compl.trans _, le_max_right _ _⟩\n[GOAL]\ncase refine'_1\nι : Type u_1\ninst✝ : Fintype ι\nI J : Box ι\nc✝ c₁ c₂ : ℝ≥0\nr✝ r₁ r₂ : (ι → ℝ) → ↑(Set.Ioi 0)\nπ✝ π₁ π₂ : TaggedPrepartition I\nl✝ l₁ l₂ l : IntegrationParams\nh : I ≤ J\nc : ℝ≥0\nπ₀ : Prepartition J := Prepartition.single J I h\nr : (ι → ℝ) → ↑(Set.Ioi 0)\nhr : RCond l r\nπ : TaggedPrepartition I\nhπ : MemBaseSet l I c r π ∧ TaggedPrepartition.iUnion π = ↑I\nx✝ : l.bDistortion = true\n⊢ ↑J \\ Prepartition.iUnion π₀ = ↑J \\ TaggedPrepartition.iUnion (↑(embedBox I J h) π)\n[PROOFSTEP]\ncongr 1\n[GOAL]\ncase refine'_1.e_a\nι : Type u_1\ninst✝ : Fintype ι\nI J : Box ι\nc✝ c₁ c₂ : ℝ≥0\nr✝ r₁ r₂ : (ι → ℝ) → ↑(Set.Ioi 0)\nπ✝ π₁ π₂ : TaggedPrepartition I\nl✝ l₁ l₂ l : IntegrationParams\nh : I ≤ J\nc : ℝ≥0\nπ₀ : Prepartition J := Prepartition.single J I h\nr : (ι → ℝ) → ↑(Set.Ioi 0)\nhr : RCond l r\nπ : TaggedPrepartition I\nhπ : MemBaseSet l I c r π ∧ TaggedPrepartition.iUnion π = ↑I\nx✝ : l.bDistortion = true\n⊢ Prepartition.iUnion π₀ = TaggedPrepartition.iUnion (↑(embedBox I J h) π)\n[PROOFSTEP]\nexact (Prepartition.iUnion_single h).trans hπ.2.symm\n[GOAL]\ncase refine'_2\nι : Type u_1\ninst✝ : Fintype ι\nI J : Box ι\nc✝ c₁ c₂ : ℝ≥0\nr✝ r₁ r₂ : (ι → ℝ) → ↑(Set.Ioi 0)\nπ✝ π₁ π₂ : TaggedPrepartition I\nl✝ l₁ l₂ l : IntegrationParams\nh : I ≤ J\nc : ℝ≥0\nπ₀ : Prepartition J := Prepartition.single J I h\nr : (ι → ℝ) → ↑(Set.Ioi 0)\nhr : RCond l r\nπ : TaggedPrepartition I\nhπ : MemBaseSet l I c r π ∧ TaggedPrepartition.iUnion π = ↑I\n⊢ TaggedPrepartition.iUnion (↑(embedBox I J h) π) = Prepartition.iUnion π₀\n[PROOFSTEP]\nexact hπ.2.trans (Prepartition.iUnion_single _).symm\n[GOAL]\nι : Type u_1\ninst✝ : Fintype ι\nI J : Box ι\nc c₁ c₂ : ℝ≥0\nr✝ r₁ r₂ : (ι → ℝ) → ↑(Set.Ioi 0)\nπ π₁ π₂ : TaggedPrepartition I\nl✝ l₁ l₂ l : IntegrationParams\nπ₀ : Prepartition I\nhc₁ : Prepartition.distortion π₀ ≤ c\nhc₂ : Prepartition.distortion (Prepartition.compl π₀) ≤ c\nr : (ι → ℝ) → ↑(Set.Ioi 0)\n⊢ ∃ π, MemBaseSet l I c r π ∧ π.toPrepartition ≤ π₀ ∧ TaggedPrepartition.iUnion π = Prepartition.iUnion π₀\n[PROOFSTEP]\nrcases π₀.exists_tagged_le_isHenstock_isSubordinate_iUnion_eq r with ⟨π, hle, hH, hr, hd, hU⟩\n[GOAL]\ncase intro.intro.intro.intro.intro\nι : Type u_1\ninst✝ : Fintype ι\nI J : Box ι\nc c₁ c₂ : ℝ≥0\nr✝ r₁ r₂ : (ι → ℝ) → ↑(Set.Ioi 0)\nπ✝ π₁ π₂ : TaggedPrepartition I\nl✝ l₁ l₂ l : IntegrationParams\nπ₀ : Prepartition I\nhc₁ : Prepartition.distortion π₀ ≤ c\nhc₂ : Prepartition.distortion (Prepartition.compl π₀) ≤ c\nr : (ι → ℝ) → ↑(Set.Ioi 0)\nπ : TaggedPrepartition I\nhle : π.toPrepartition ≤ π₀\nhH : IsHenstock π\nhr : IsSubordinate π r\nhd : distortion π = Prepartition.distortion π₀\nhU : TaggedPrepartition.iUnion π = Prepartition.iUnion π₀\n⊢ ∃ π, MemBaseSet l I c r π ∧ π.toPrepartition ≤ π₀ ∧ TaggedPrepartition.iUnion π = Prepartition.iUnion π₀\n[PROOFSTEP]\nrefine' ⟨π, ⟨hr, fun _ => hH, fun _ => hd.trans_le hc₁, fun _ => ⟨π₀.compl, _, hc₂⟩⟩, ⟨hle, hU⟩⟩\n[GOAL]\ncase intro.intro.intro.intro.intro\nι : Type u_1\ninst✝ : Fintype ι\nI J : Box ι\nc c₁ c₂ : ℝ≥0\nr✝ r₁ r₂ : (ι → ℝ) → ↑(Set.Ioi 0)\nπ✝ π₁ π₂ : TaggedPrepartition I\nl✝ l₁ l₂ l : IntegrationParams\nπ₀ : Prepartition I\nhc₁ : Prepartition.distortion π₀ ≤ c\nhc₂ : Prepartition.distortion (Prepartition.compl π₀) ≤ c\nr : (ι → ℝ) → ↑(Set.Ioi 0)\nπ : TaggedPrepartition I\nhle : π.toPrepartition ≤ π₀\nhH : IsHenstock π\nhr : IsSubordinate π r\nhd : distortion π = Prepartition.distortion π₀\nhU : TaggedPrepartition.iUnion π = Prepartition.iUnion π₀\nx✝ : l.bDistortion = true\n⊢ Prepartition.iUnion (Prepartition.compl π₀) = ↑I \\ TaggedPrepartition.iUnion π\n[PROOFSTEP]\nexact Prepartition.compl_congr hU ▸ π.toPrepartition.iUnion_compl\n[GOAL]\nι : Type u_1\ninst✝ : Fintype ι\nI✝ J : Box ι\nc c₁ c₂ : ℝ≥0\nr✝ r₁ r₂ : (ι → ℝ) → ↑(Set.Ioi 0)\nπ π₁ π₂ : TaggedPrepartition I✝\nl✝ l₁ l₂ l : IntegrationParams\nI : Box ι\nhc : Box.distortion I ≤ c\nr : (ι → ℝ) → ↑(Set.Ioi 0)\n⊢ ∃ π, MemBaseSet l I c r π ∧ IsPartition π\n[PROOFSTEP]\nrw [← Prepartition.distortion_top] at hc \n[GOAL]\nι : Type u_1\ninst✝ : Fintype ι\nI✝ J : Box ι\nc c₁ c₂ : ℝ≥0\nr✝ r₁ r₂ : (ι → ℝ) → ↑(Set.Ioi 0)\nπ π₁ π₂ : TaggedPrepartition I✝\nl✝ l₁ l₂ l : IntegrationParams\nI : Box ι\nhc : Prepartition.distortion ⊤ ≤ c\nr : (ι → ℝ) → ↑(Set.Ioi 0)\n⊢ ∃ π, MemBaseSet l I c r π ∧ IsPartition π\n[PROOFSTEP]\nhave hc' : (⊤ : Prepartition I).compl.distortion ≤ c := by simp\n[GOAL]\nι : Type u_1\ninst✝ : Fintype ι\nI✝ J : Box ι\nc c₁ c₂ : ℝ≥0\nr✝ r₁ r₂ : (ι → ℝ) → ↑(Set.Ioi 0)\nπ π₁ π₂ : TaggedPrepartition I✝\nl✝ l₁ l₂ l : IntegrationParams\nI : Box ι\nhc : Prepartition.distortion ⊤ ≤ c\nr : (ι → ℝ) → ↑(Set.Ioi 0)\n⊢ Prepartition.distortion (Prepartition.compl ⊤) ≤ c\n[PROOFSTEP]\nsimp\n[GOAL]\nι : Type u_1\ninst✝ : Fintype ι\nI✝ J : Box ι\nc c₁ c₂ : ℝ≥0\nr✝ r₁ r₂ : (ι → ℝ) → ↑(Set.Ioi 0)\nπ π₁ π₂ : TaggedPrepartition I✝\nl✝ l₁ l₂ l : IntegrationParams\nI : Box ι\nhc : Prepartition.distortion ⊤ ≤ c\nr : (ι → ℝ) → ↑(Set.Ioi 0)\nhc' : Prepartition.distortion (Prepartition.compl ⊤) ≤ c\n⊢ ∃ π, MemBaseSet l I c r π ∧ IsPartition π\n[PROOFSTEP]\nsimpa [isPartition_iff_iUnion_eq] using l.exists_memBaseSet_le_iUnion_eq ⊤ hc hc' r\n[GOAL]\nι : Type u_1\ninst✝ : Fintype ι\nI✝ J : Box ι\nc c₁ c₂ : ℝ≥0\nr r₁ r₂ : (ι → ℝ) → ↑(Set.Ioi 0)\nπ π₁ π₂ : TaggedPrepartition I✝\nl✝ l₁ l₂ l : IntegrationParams\nI : Box ι\n⊢ NeBot (toFilterDistortion l I (Box.distortion I))\n[PROOFSTEP]\nsimpa using (l.toFilterDistortioniUnion_neBot' I ⊤).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}} |
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