awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.ConstantSheaf (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
J : GrothendieckTopology.{v_1, u_1} C
D : Type u_2
inst_1 : Category.{v_2, u_2} D
inst_2 : HasWeakSheafify.{v_1, v_2, u_1, u_2} J D
B : Type u_3
inst_3 : Category.{v_3, u_3} B
U : Functor.{v_2, v_3, u_2, u_3} D B
inst_4 : HasWeakSheafify.{v_1, v_3, u_1, u_3} J B
inst_5 : J.PreservesSheafification U
inst_6 : J.HasSheafCompose U
F : Sheaf.{v_1, v_2, u_1, u_2} J D
T : C
hT : IsTerminal.{v_1, u_1} T
⊢ Eq (α :=
    ObjectProperty.FullSubcategory.obj (P := Presheaf.IsSheaf J)
          (obj (constantSheaf J D) (F.obj (P := Presheaf.IsSheaf J).obj (Opposite.op T))) ⋙
        U ⟶
      F.obj (P := Presheaf.IsSheaf J) ⋙ U)
    (CategoryStruct.comp (Y :=
        ObjectProperty.FullSubcategory.obj (P := Presheaf.IsSheaf J)
          (obj (constantSheaf J B) (U.obj (F.obj (P := Presheaf.IsSheaf J).obj (Opposite.op T)))))
        (InducedCategory.Hom.hom
          (NatTrans.app (Iso.hom (constantCommuteCompose J U)) (F.obj (P := Presheaf.IsSheaf J).obj (Opposite.op T))))
        (InducedCategory.Hom.hom (NatTrans.app (counit (constantSheafAdj J B hT)) ((sheafCompose J U).obj F))) :
      ObjectProperty.FullSubcategory.obj (P := Presheaf.IsSheaf J)
            (obj (constantSheaf J D) (F.obj (P := Presheaf.IsSheaf J).obj (Opposite.op T))) ⋙
          U ⟶
        F.obj (P := Presheaf.IsSheaf J) ⋙ U)
    (whiskerRight (InducedCategory.Hom.hom (NatTrans.app (counit (constantSheafAdj J D hT)) F)) U :
      ObjectProperty.FullSubcategory.obj (P := Presheaf.IsSheaf J)
            (obj (constantSheaf J D) (F.obj (P := Presheaf.IsSheaf J).obj (Opposite.op T))) ⋙
          U ⟶
        F.obj (P := Presheaf.IsSheaf J) ⋙ U)
have h : Eq (α :=
  sheafify J (obj (const Cᵒᵖ) (F.obj (P := Presheaf.IsSheaf J).obj (Opposite.op T)) ⋙ U) ⟶
    F.obj (P := Presheaf.IsSheaf J) ⋙ U)
  (sheafifyMap J (Iso.hom (constComp Cᵒᵖ (F.obj (P := Presheaf.IsSheaf J).obj (Opposite.op T)) U)) ≫
    InducedCategory.Hom.hom (NatTrans.app (counit (constantSheafAdj J B hT)) ((sheafCompose J U).obj F)))
  (CategoryStruct.comp (Y :=
    ObjectProperty.FullSubcategory.obj (P := Presheaf.IsSheaf J)
        (obj (constantSheaf J D) (F.obj (P := Presheaf.IsSheaf J).obj (Opposite.op T))) ⋙
      U)
    (Iso.hom (sheafifyComposeIso J U (obj (const Cᵒᵖ) (F.obj (P := Presheaf.IsSheaf J).obj (Opposite.op T)))))
    (whiskerRight (InducedCategory.Hom.hom (NatTrans.app (counit (constantSheafAdj J D hT)) F)) U))
Tactic: rw [constantCommuteCompose_hom_app_hom, assoc, Iso.inv_comp_eq]
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.ConstantSheaf (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
J : GrothendieckTopology.{v_1, u_1} C
D : Type u_2
inst_1 : Category.{v_2, u_2} D
inst_2 : HasWeakSheafify.{v_1, v_2, u_1, u_2} J D
B : Type u_3
inst_3 : Category.{v_3, u_3} B
U : Functor.{v_2, v_3, u_2, u_3} D B
inst_4 : HasWeakSheafify.{v_1, v_3, u_1, u_3} J B
inst_5 : J.PreservesSheafification U
inst_6 : J.HasSheafCompose U
F : Sheaf.{v_1, v_2, u_1, u_2} J D
T : C
hT : IsTerminal.{v_1, u_1} T
⊢ Eq (α :=
    sheafify J (obj (const Cᵒᵖ) (F.obj (P := Presheaf.IsSheaf J).obj (Opposite.op T)) ⋙ U) ⟶
      F.obj (P := Presheaf.IsSheaf J) ⋙ U)
    (sheafifyMap J (Iso.hom (constComp Cᵒᵖ (F.obj (P := Presheaf.IsSheaf J).obj (Opposite.op T)) U)) ≫
      InducedCategory.Hom.hom (NatTrans.app (counit (constantSheafAdj J B hT)) ((sheafCompose J U).obj F)))
    (CategoryStruct.comp (Y :=
      ObjectProperty.FullSubcategory.obj (P := Presheaf.IsSheaf J)
          (obj (constantSheaf J D) (F.obj (P := Presheaf.IsSheaf J).obj (Opposite.op T))) ⋙
        U)
      (Iso.hom (sheafifyComposeIso J U (obj (const Cᵒᵖ) (F.obj (P := Presheaf.IsSheaf J).obj (Opposite.op T)))))
      (whiskerRight (InducedCategory.Hom.hom (NatTrans.app (counit (constantSheafAdj J D hT)) F)) U))
have h : CategoryStruct.comp (Y := sheafify J (obj (const Cᵒᵖ) (F.obj (P := Presheaf.IsSheaf J).obj (Opposite.op T)) ⋙ U))
    (toSheafify J (obj (const Cᵒᵖ) (F.obj (P := Presheaf.IsSheaf J).obj (Opposite.op T)) ⋙ U))
    (sheafifyMap J (Iso.hom (constComp Cᵒᵖ (F.obj (P := Presheaf.IsSheaf J).obj (Opposite.op T)) U)) ≫
      InducedCategory.Hom.hom (NatTrans.app (counit (constantSheafAdj J B hT)) ((sheafCompose J U).obj F))) =
  CategoryStruct.comp (Y := sheafify J (obj (const Cᵒᵖ) (F.obj (P := Presheaf.IsSheaf J).obj (Opposite.op T)) ⋙ U))
    (toSheafify J (obj (const Cᵒᵖ) (F.obj (P := Presheaf.IsSheaf J).obj (Opposite.op T)) ⋙ U))
    (CategoryStruct.comp (Y :=
      ObjectProperty.FullSubcategory.obj (P := Presheaf.IsSheaf J)
          (obj (constantSheaf J D) (F.obj (P := Presheaf.IsSheaf J).obj (Opposite.op T))) ⋙
        U)
      (Iso.hom (sheafifyComposeIso J U (obj (const Cᵒᵖ) (F.obj (P := Presheaf.IsSheaf J).obj (Opposite.op T)))))
      (whiskerRight (InducedCategory.Hom.hom (NatTrans.app (counit (constantSheafAdj J D hT)) F)) U))
Tactic: apply sheafify_hom_ext _ _ _ ((sheafCompose J U).obj F).property
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.ConstantSheaf (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
J : GrothendieckTopology.{v_1, u_1} C
D : Type u_2
inst_1 : Category.{v_2, u_2} D
inst_2 : HasWeakSheafify.{v_1, v_2, u_1, u_2} J D
B : Type u_3
inst_3 : Category.{v_3, u_3} B
U : Functor.{v_2, v_3, u_2, u_3} D B
inst_4 : HasWeakSheafify.{v_1, v_3, u_1, u_3} J B
inst_5 : J.PreservesSheafification U
inst_6 : J.HasSheafCompose U
F : Sheaf.{v_1, v_2, u_1, u_2} J D
T : C
hT : IsTerminal.{v_1, u_1} T
⊢ CategoryStruct.comp (Y := sheafify J (obj (const Cᵒᵖ) (F.obj (P := Presheaf.IsSheaf J).obj (Opposite.op T)) ⋙ U))
      (toSheafify J (obj (const Cᵒᵖ) (F.obj (P := Presheaf.IsSheaf J).obj (Opposite.op T)) ⋙ U))
      (sheafifyMap J (Iso.hom (constComp Cᵒᵖ (F.obj (P := Presheaf.IsSheaf J).obj (Opposite.op T)) U)) ≫
        InducedCategory.Hom.hom (NatTrans.app (counit (constantSheafAdj J B hT)) ((sheafCompose J U).obj F))) =
    CategoryStruct.comp (Y := sheafify J (obj (const Cᵒᵖ) (F.obj (P := Presheaf.IsSheaf J).obj (Opposite.op T)) ⋙ U))
      (toSheafify J (obj (const Cᵒᵖ) (F.obj (P := Presheaf.IsSheaf J).obj (Opposite.op T)) ⋙ U))
      (CategoryStruct.comp (Y :=
        ObjectProperty.FullSubcategory.obj (P := Presheaf.IsSheaf J)
            (obj (constantSheaf J D) (F.obj (P := Presheaf.IsSheaf J).obj (Opposite.op T))) ⋙
          U)
        (Iso.hom (sheafifyComposeIso J U (obj (const Cᵒᵖ) (F.obj (P := Presheaf.IsSheaf J).obj (Opposite.op T)))))
        (whiskerRight (InducedCategory.Hom.hom (NatTrans.app (counit (constantSheafAdj J D hT)) F)) U))
have h : ∀ (x : Cᵒᵖ),
  (CategoryStruct.comp (Y := sheafify J (obj (const Cᵒᵖ) (F.obj (P := Presheaf.IsSheaf J).obj (Opposite.op T)) ⋙ U))
          (toSheafify J (obj (const Cᵒᵖ) (F.obj (P := Presheaf.IsSheaf J).obj (Opposite.op T)) ⋙ U))
          (sheafifyMap J (Iso.hom (constComp Cᵒᵖ (F.obj (P := Presheaf.IsSheaf J).obj (Opposite.op T)) U)) ≫
            InducedCategory.Hom.hom (NatTrans.app (counit (constantSheafAdj J B hT)) ((sheafCompose J U).obj F)))).app
      x =
    (CategoryStruct.comp (Y := sheafify J (obj (const Cᵒᵖ) (F.obj (P := Presheaf.IsSheaf J).obj (Opposite.op T)) ⋙ U))
          (toSheafify J (obj (const Cᵒᵖ) (F.obj (P := Presheaf.IsSheaf J).obj (Opposite.op T)) ⋙ U))
          (CategoryStruct.comp (Y :=
            ObjectProperty.FullSubcategory.obj (P := Presheaf.IsSheaf J)
                (obj (constantSheaf J D) (F.obj (P := Presheaf.IsSheaf J).obj (Opposite.op T))) ⋙
              U)
            (Iso.hom (sheafifyComposeIso J U (obj (const Cᵒᵖ) (F.obj (P := Presheaf.IsSheaf J).obj (Opposite.op T)))))
            (whiskerRight (InducedCategory.Hom.hom (NatTrans.app (counit (constantSheafAdj J D hT)) F)) U))).app
      x
Tactic: ext x
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.ConstantSheaf (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
J : GrothendieckTopology.{v_1, u_1} C
D : Type u_2
inst_1 : Category.{v_2, u_2} D
inst_2 : HasWeakSheafify.{v_1, v_2, u_1, u_2} J D
B : Type u_3
inst_3 : Category.{v_3, u_3} B
U : Functor.{v_2, v_3, u_2, u_3} D B
inst_4 : HasWeakSheafify.{v_1, v_3, u_1, u_3} J B
inst_5 : J.PreservesSheafification U
inst_6 : J.HasSheafCompose U
F : Sheaf.{v_1, v_2, u_1, u_2} J D
T : C
hT : IsTerminal.{v_1, u_1} T
x : Cᵒᵖ
⊢ (CategoryStruct.comp (Y := sheafify J (obj (const Cᵒᵖ) (F.obj (P := Presheaf.IsSheaf J).obj (Opposite.op T)) ⋙ U))
          (toSheafify J (obj (const Cᵒᵖ) (F.obj (P := Presheaf.IsSheaf J).obj (Opposite.op T)) ⋙ U))
          (sheafifyMap J (Iso.hom (constComp Cᵒᵖ (F.obj (P := Presheaf.IsSheaf J).obj (Opposite.op T)) U)) ≫
            InducedCategory.Hom.hom (NatTrans.app (counit (constantSheafAdj J B hT)) ((sheafCompose J U).obj F)))).app
      x =
    (CategoryStruct.comp (Y := sheafify J (obj (const Cᵒᵖ) (F.obj (P := Presheaf.IsSheaf J).obj (Opposite.op T)) ⋙ U))
          (toSheafify J (obj (const Cᵒᵖ) (F.obj (P := Presheaf.IsSheaf J).obj (Opposite.op T)) ⋙ U))
          (CategoryStruct.comp (Y :=
            ObjectProperty.FullSubcategory.obj (P := Presheaf.IsSheaf J)
                (obj (constantSheaf J D) (F.obj (P := Presheaf.IsSheaf J).obj (Opposite.op T))) ⋙
              U)
            (Iso.hom (sheafifyComposeIso J U (obj (const Cᵒᵖ) (F.obj (P := Presheaf.IsSheaf J).obj (Opposite.op T)))))
            (whiskerRight (InducedCategory.Hom.hom (NatTrans.app (counit (constantSheafAdj J D hT)) F)) U))).app
      x
have h : U.map (F.obj (P := Presheaf.IsSheaf J).map (Quiver.Hom.op (hT.from (Opposite.unop x)))) =
  CategoryStruct.comp (Y :=
    U.obj
      (obj.{v_1, v_2, u_1, u_2} (sheafify J (obj (const Cᵒᵖ) (F.obj (P := Presheaf.IsSheaf J).obj (Opposite.op T)))) x))
    (U.map (NatTrans.app (toSheafify J (obj (const Cᵒᵖ) (F.obj (P := Presheaf.IsSheaf J).obj (Opposite.op T)))) x))
    (U.map
      (NatTrans.app
        (sheafifyLift J (NatTrans.app (counit (constantPresheafAdj D hT)) (F.obj (P := Presheaf.IsSheaf J))) sorry) x))
Tactic: simp [NatTrans.comp_app]
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.ConstantSheaf (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
J : GrothendieckTopology.{v_1, u_1} C
D : Type u_2
inst_1 : Category.{v_2, u_2} D
inst_2 : HasWeakSheafify.{v_1, v_2, u_1, u_2} J D
B : Type u_3
inst_3 : Category.{v_3, u_3} B
U : Functor.{v_2, v_3, u_2, u_3} D B
inst_4 : HasWeakSheafify.{v_1, v_3, u_1, u_3} J B
inst_5 : J.PreservesSheafification U
inst_6 : J.HasSheafCompose U
F : Sheaf.{v_1, v_2, u_1, u_2} J D
inst_7 : Faithful (constantSheaf J D)
inst_8 : Full (constantSheaf J D)
inst_9 : Faithful (constantSheaf J B)
inst_10 : Full (constantSheaf J B)
inst_11 : (sheafCompose J U).ReflectsIsomorphisms
inst_12 : IsConstant J ((sheafCompose J U).obj F)
T : C
hT : IsTerminal.{v_1, u_1} T
⊢ IsIso (X := (sheafCompose J U).obj (obj (obj (sheafSections J D) (Opposite.op T) ⋙ constantSheaf J D) F))
    ((sheafCompose J U).map (NatTrans.app (counit (constantSheafAdj J D hT)) F))
have rw : IsIso (X := (sheafCompose J U).obj (obj (obj (sheafSections J D) (Opposite.op T) ⋙ constantSheaf J D) F))
  (CategoryStruct.comp (Y := (U ⋙ constantSheaf J B).obj (F.obj (P := Presheaf.IsSheaf J).obj (Opposite.op T)))
      (NatTrans.app (Iso.hom (constantCommuteCompose J U)) (F.obj (P := Presheaf.IsSheaf J).obj (Opposite.op T)))
      (NatTrans.app (counit (constantSheafAdj J B hT)) ((sheafCompose J U).obj F)) :
    (constantSheaf J D ⋙ sheafCompose J U).obj (F.obj (P := Presheaf.IsSheaf J).obj (Opposite.op T)) ⟶
      obj (𝟭 (Sheaf.{v_1, v_3, u_1, u_3} J B)) ((sheafCompose J U).obj F))
Tactic: rw [← constantSheafAdj_counit_w]
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.ConstantSheaf (hypothesis)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
J : GrothendieckTopology.{v_1, u_1} C
D : Type u_2
inst_1 : Category.{v_2, u_2} D
inst_2 : HasWeakSheafify.{v_1, v_2, u_1, u_2} J D
B : Type u_3
inst_3 : Category.{v_3, u_3} B
U : Functor.{v_2, v_3, u_2, u_3} D B
inst_4 : HasWeakSheafify.{v_1, v_3, u_1, u_3} J B
inst_5 : J.PreservesSheafification U
inst_6 : J.HasSheafCompose U
F : Sheaf.{v_1, v_2, u_1, u_2} J D
inst_7 : Faithful (constantSheaf J D)
inst_8 : Full (constantSheaf J D)
inst_9 : Faithful (constantSheaf J B)
inst_10 : Full (constantSheaf J B)
inst_11 : (sheafCompose J U).ReflectsIsomorphisms
inst_12 : IsConstant J ((sheafCompose J U).obj F)
T : C
hT : IsTerminal.{v_1, u_1} T
⊢ IsConstant J F
have this : IsIso (X := (sheafCompose J U).obj (obj (obj (sheafSections J D) (Opposite.op T) ⋙ constantSheaf J D) F))
  ((sheafCompose J U).map (NatTrans.app (counit (constantSheafAdj J D hT)) F))
Tactic: have : IsIso ((sheafCompose J U).map ((constantSheafAdj J D hT).counit.app F)) := by
    rw [← constantSheafAdj_counit_w]
    infer_instance
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.ConstantSheaf (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
J : GrothendieckTopology.{v_1, u_1} C
D : Type u_2
inst_1 : Category.{v_2, u_2} D
inst_2 : HasWeakSheafify.{v_1, v_2, u_1, u_2} J D
B : Type u_3
inst_3 : Category.{v_3, u_3} B
U : Functor.{v_2, v_3, u_2, u_3} D B
inst_4 : HasWeakSheafify.{v_1, v_3, u_1, u_3} J B
inst_5 : J.PreservesSheafification U
inst_6 : J.HasSheafCompose U
F : Sheaf.{v_1, v_2, u_1, u_2} J D
inst_7 : Faithful (constantSheaf J D)
inst_8 : Full (constantSheaf J D)
inst_9 : Faithful (constantSheaf J B)
inst_10 : Full (constantSheaf J B)
inst_11 : (sheafCompose J U).ReflectsIsomorphisms
inst_12 : IsConstant J ((sheafCompose J U).obj F)
T : C
hT : IsTerminal.{v_1, u_1} T
this :
  IsIso (X := (sheafCompose J U).obj (obj (obj (sheafSections J D) (Opposite.op T) ⋙ constantSheaf J D) F))
    ((sheafCompose J U).map (NatTrans.app (counit (constantSheafAdj J D hT)) F))
⊢ IsConstant J F
have rw : IsIso (X := obj (obj (sheafSections J D) (Opposite.op T) ⋙ constantSheaf J D) F)
  (NatTrans.app (counit (constantSheafAdj J D hT)) F)
Tactic: rw [F.isConstant_iff_isIso_counit_app (hT := hT)]
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.ConstantSheaf (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
J : GrothendieckTopology.{v_1, u_1} C
D : Type u_2
inst_1 : Category.{v_2, u_2} D
inst_2 : HasWeakSheafify.{v_1, v_2, u_1, u_2} J D
B : Type u_3
inst_3 : Category.{v_3, u_3} B
U : Functor.{v_2, v_3, u_2, u_3} D B
inst_4 : HasWeakSheafify.{v_1, v_3, u_1, u_3} J B
inst_5 : J.PreservesSheafification U
inst_6 : J.HasSheafCompose U
F : Sheaf.{v_1, v_2, u_1, u_2} J D
h : IsConstant J F
⊢ IsConstant J ((sheafCompose J U).obj F)
have obtain : ∀ (Y : D) (i : obj (constantSheaf J D) Y ≅ F), IsConstant J ((sheafCompose J U).obj F)
Tactic: obtain ⟨Y, ⟨i⟩⟩ := h
Cleared hypotheses: h
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Continuous (goal)
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
D : Type u₂
inst_1 : Category.{v₂, u₂} D
X : C
E : PreOneHypercover.{u_1, v₁, u₁} X
F : Functor.{v₁, v₂, u₁, u₂} C D
W : C
i₁ i₂ : E.I₀
p₁ : W ⟶ E.X i₁
p₂ : W ⟶ E.X i₂
⊢ Sieve.functorPushforward F (E.sieve₁ p₁ p₂) ≤ (E.map F).sieve₁ (F.map p₁) (F.map p₂)
have rw : E.sieve₁ p₁ p₂ ≤ Sieve.functorPullback F ((E.map F).sieve₁ (F.map p₁) (F.map p₂))
Tactic: rw [Sieve.functorPushforward_le_iff_le_functorPullback]
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Continuous (goal)
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
D : Type u₂
inst_1 : Category.{v₂, u₂} D
X : C
E : PreOneHypercover.{u_1, v₁, u₁} X
F : Functor.{v₁, v₂, u₁, u₂} C D
W : C
i₁ i₂ : E.I₀
p₁ : W ⟶ E.X i₁
p₂ : W ⟶ E.X i₂
Y : C
f : Y ⟶ W
⊢ (E.sieve₁ p₁ p₂).arrows f → (Sieve.functorPullback F ((E.map F).sieve₁ (F.map p₁) (F.map p₂))).arrows f
have this : ∀ (k : E.I₁ i₁ i₂) (u : Y ⟶ E.Y k) (hf₁ : f ≫ p₁ = u ≫ E.p₁ k) (hf₂ : f ≫ p₂ = u ≫ E.p₂ k),
  (Sieve.functorPullback F ((E.map F).sieve₁ (F.map p₁) (F.map p₂))).arrows f
Tactic: ⟨k, u, hf₁, hf₂⟩
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Continuous (hypothesis)
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
D : Type u₂
inst_1 : Category.{v₂, u₂} D
X : C
E : PreOneHypercover.{u_1, v₁, u₁} X
F : Functor.{v₁, v₂, u₁, u₂} C D
i₁ i₂ : E.I₀
inst_2 : HasPullback (E.f i₁) (E.f i₂)
inst_3 : PreservesLimit (cospan (E.f i₁) (E.f i₂)) F
⊢ Eq (α := Sieve.{v₂, u₂} (F.obj (pullback (E.f i₁) (E.f i₂)))) (Sieve.functorPushforward F (E.sieve₁' i₁ i₂))
    ((E.map F).sieve₁ (F.map (pullback.fst (E.f i₁) (E.f i₂))) (F.map (pullback.snd (E.f i₁) (E.f i₂))))
have this : HasPullback ((E.map F).f i₁) ((E.map F).f i₂)
Tactic: have : HasPullback ((E.map F).f i₁) ((E.map F).f i₂) :=
    hasPullback_of_preservesPullback F (E.f i₁) (E.f i₂)
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Continuous (goal)
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
D : Type u₂
inst_1 : Category.{v₂, u₂} D
X : C
E : PreOneHypercover.{u_1, v₁, u₁} X
F : Functor.{v₁, v₂, u₁, u₂} C D
i₁ i₂ : E.I₀
inst_2 : HasPullback (E.f i₁) (E.f i₂)
inst_3 : PreservesLimit (cospan (E.f i₁) (E.f i₂)) F
this : HasPullback ((E.map F).f i₁) ((E.map F).f i₂)
⊢ Eq (α := Sieve.{v₂, u₂} (F.obj (pullback (E.f i₁) (E.f i₂)))) (Sieve.functorPushforward F (E.sieve₁' i₁ i₂))
    ((E.map F).sieve₁ (F.map (pullback.fst (E.f i₁) (E.f i₂))) (F.map (pullback.snd (E.f i₁) (E.f i₂))))
have refine_1 : LE.le (α := Sieve.{v₂, u₂} (F.obj (pullback (E.f i₁) (E.f i₂)))) (Sieve.functorPushforward F (E.sieve₁' i₁ i₂))
  ((E.map F).sieve₁ (F.map (pullback.fst (E.f i₁) (E.f i₂))) (F.map (pullback.snd (E.f i₁) (E.f i₂))))
Tactic: refine le_antisymm ?_ ?_
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Continuous (goal)
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
D : Type u₂
inst_1 : Category.{v₂, u₂} D
X : C
E : PreOneHypercover.{u_1, v₁, u₁} X
F : Functor.{v₁, v₂, u₁, u₂} C D
i₁ i₂ : E.I₀
inst_2 : HasPullback (E.f i₁) (E.f i₂)
inst_3 : PreservesLimit (cospan (E.f i₁) (E.f i₂)) F
this : HasPullback ((E.map F).f i₁) ((E.map F).f i₂)
refine_1 :
  LE.le (α := Sieve.{v₂, u₂} (F.obj (pullback (E.f i₁) (E.f i₂)))) (Sieve.functorPushforward F (E.sieve₁' i₁ i₂))
    ((E.map F).sieve₁ (F.map (pullback.fst (E.f i₁) (E.f i₂))) (F.map (pullback.snd (E.f i₁) (E.f i₂))))
⊢ Eq (α := Sieve.{v₂, u₂} (F.obj (pullback (E.f i₁) (E.f i₂)))) (Sieve.functorPushforward F (E.sieve₁' i₁ i₂))
    ((E.map F).sieve₁ (F.map (pullback.fst (E.f i₁) (E.f i₂))) (F.map (pullback.snd (E.f i₁) (E.f i₂))))
have refine_2 : LE.le (α := Sieve.{v₂, u₂} (F.obj (pullback (E.f i₁) (E.f i₂))))
  ((E.map F).sieve₁ (F.map (pullback.fst (E.f i₁) (E.f i₂))) (F.map (pullback.snd (E.f i₁) (E.f i₂))))
  (Sieve.functorPushforward F (E.sieve₁' i₁ i₂))
Tactic: refine le_antisymm ?_ ?_
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Continuous (goal)
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
D : Type u₂
inst_1 : Category.{v₂, u₂} D
X : C
E : PreOneHypercover.{u_1, v₁, u₁} X
F : Functor.{v₁, v₂, u₁, u₂} C D
i₁ i₂ : E.I₀
inst_2 : HasPullback (E.f i₁) (E.f i₂)
inst_3 : PreservesLimit (cospan (E.f i₁) (E.f i₂)) F
this : HasPullback ((E.map F).f i₁) ((E.map F).f i₂)
⊢ LE.le (α := Sieve.{v₂, u₂} (F.obj (pullback (E.f i₁) (E.f i₂)))) (Sieve.functorPushforward F (E.sieve₁' i₁ i₂))
    ((E.map F).sieve₁ (F.map (pullback.fst (E.f i₁) (E.f i₂))) (F.map (pullback.snd (E.f i₁) (E.f i₂))))
have refine_1 : LE.le (α := Sieve.{v₂, u₂} (F.obj (pullback (E.f i₁) (E.f i₂))))
  (Sieve.functorPushforward F (E.sieve₁ (pullback.fst (E.f i₁) (E.f i₂)) (pullback.snd (E.f i₁) (E.f i₂))))
  ((E.map F).sieve₁ (F.map (pullback.fst (E.f i₁) (E.f i₂))) (F.map (pullback.snd (E.f i₁) (E.f i₂))))
Tactic: rw [PreOneHypercover.sieve₁'_eq_sieve₁]
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Continuous (goal)
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
D : Type u₂
inst_1 : Category.{v₂, u₂} D
X : C
E : PreOneHypercover.{u_1, v₁, u₁} X
F : Functor.{v₁, v₂, u₁, u₂} C D
i₁ i₂ : E.I₀
inst_2 : HasPullback (E.f i₁) (E.f i₂)
inst_3 : PreservesLimit (cospan (E.f i₁) (E.f i₂)) F
this : HasPullback ((E.map F).f i₁) ((E.map F).f i₂)
⊢ LE.le (α := Sieve.{v₂, u₂} (F.obj (pullback (E.f i₁) (E.f i₂))))
    ((E.map F).sieve₁ (F.map (pullback.fst (E.f i₁) (E.f i₂))) (F.map (pullback.snd (E.f i₁) (E.f i₂))))
    (Sieve.functorPushforward F (E.sieve₁' i₁ i₂))
have refine_2 : LE.le (α := Sieve.{v₂, u₂} (F.obj (pullback (E.f i₁) (E.f i₂))))
  (Sieve.pullback (X := pullback ((E.map F).f i₁) ((E.map F).f i₂))
    (pullback.lift (F.map (pullback.fst (E.f i₁) (E.f i₂))) (F.map (pullback.snd (E.f i₁) (E.f i₂))) sorry)
    ((E.map F).sieve₁' i₁ i₂))
  (Sieve.functorPushforward F (E.sieve₁' i₁ i₂))
Tactic: rw [PreOneHypercover.sieve₁_eq_pullback_sieve₁' _ _ _
      (by simp [← Functor.map_comp, pullback.condition])]
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Continuous (goal)
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
D : Type u₂
inst_1 : Category.{v₂, u₂} D
X : C
E : PreOneHypercover.{u_1, v₁, u₁} X
F : Functor.{v₁, v₂, u₁, u₂} C D
i₁ i₂ : E.I₀
inst_2 : HasPullback (E.f i₁) (E.f i₂)
inst_3 : PreservesLimit (cospan (E.f i₁) (E.f i₂)) F
this : HasPullback ((E.map F).f i₁) ((E.map F).f i₂)
W : D
f : W ⟶ F.obj (pullback (E.f i₁) (E.f i₂))
Y : D
k : (E.map F).I₁ i₁ i₂
u : W ⟶ (E.map F).Y k
h :
  Eq (α := W ⟶ pullback ((E.map F).f i₁) ((E.map F).f i₂)) (u ≫ (E.map F).toPullback k)
    (f ≫ pullback.lift (F.map (pullback.fst (E.f i₁) (E.f i₂))) (F.map (pullback.snd (E.f i₁) (E.f i₂))) ⋯)
⊢ Sieve.arrows (Sieve.functorPushforward F (E.sieve₁' i₁ i₂)) f
have refine_2 : Sieve.arrows (X := pullback (E.f i₁) (E.f i₂)) (E.sieve₁' i₁ i₂) (pullback.lift (E.p₁ k) (E.p₂ k) sorry)
Tactic: refine ⟨E.Y k, pullback.lift (E.p₁ k) (E.p₂ k) (E.w _), u, ?_, ?_⟩
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Continuous (goal)
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
D : Type u₂
inst_1 : Category.{v₂, u₂} D
X : C
E : PreOneHypercover.{u_1, v₁, u₁} X
F : Functor.{v₁, v₂, u₁, u₂} C D
i₁ i₂ : E.I₀
inst_2 : HasPullback (E.f i₁) (E.f i₂)
inst_3 : PreservesLimit (cospan (E.f i₁) (E.f i₂)) F
this : HasPullback ((E.map F).f i₁) ((E.map F).f i₂)
W : D
f : W ⟶ F.obj (pullback (E.f i₁) (E.f i₂))
Y : D
k : (E.map F).I₁ i₁ i₂
u : W ⟶ (E.map F).Y k
h :
  Eq (α := W ⟶ pullback ((E.map F).f i₁) ((E.map F).f i₂)) (u ≫ (E.map F).toPullback k)
    (f ≫ pullback.lift (F.map (pullback.fst (E.f i₁) (E.f i₂))) (F.map (pullback.snd (E.f i₁) (E.f i₂))) ⋯)
refine_2 : Sieve.arrows (X := pullback (E.f i₁) (E.f i₂)) (E.sieve₁' i₁ i₂) (pullback.lift (E.p₁ k) (E.p₂ k) ⋯)
⊢ Sieve.arrows (Sieve.functorPushforward F (E.sieve₁' i₁ i₂)) f
have refine_2₁ : f = u ≫ F.map (pullback.lift (E.p₁ k) (E.p₂ k) sorry)
Tactic: refine ⟨E.Y k, pullback.lift (E.p₁ k) (E.p₂ k) (E.w _), u, ?_, ?_⟩
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Continuous (goal)
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
D : Type u₂
inst_1 : Category.{v₂, u₂} D
X : C
E : PreOneHypercover.{u_1, v₁, u₁} X
F : Functor.{v₁, v₂, u₁, u₂} C D
i₁ i₂ : E.I₀
inst_2 : HasPullback (E.f i₁) (E.f i₂)
inst_3 : PreservesLimit (cospan (E.f i₁) (E.f i₂)) F
this : HasPullback ((E.map F).f i₁) ((E.map F).f i₂)
W : D
f : W ⟶ F.obj (pullback (E.f i₁) (E.f i₂))
Y : D
k : (E.map F).I₁ i₁ i₂
u : W ⟶ (E.map F).Y k
h :
  Eq (α := W ⟶ pullback ((E.map F).f i₁) ((E.map F).f i₂)) (u ≫ (E.map F).toPullback k)
    (f ≫ pullback.lift (F.map (pullback.fst (E.f i₁) (E.f i₂))) (F.map (pullback.snd (E.f i₁) (E.f i₂))) ⋯)
⊢ Sieve.arrows (X := pullback (E.f i₁) (E.f i₂)) (E.sieve₁' i₁ i₂) (pullback.lift (E.p₁ k) (E.p₂ k) ⋯)
have right : Eq (α := E.Y k ⟶ pullback (E.f i₁) (E.f i₂)) (𝟙 (E.Y k) ≫ pullback.lift (E.p₁ k) (E.p₂ k) sorry)
  (pullback.lift (E.p₁ k) (E.p₂ k) sorry)
Tactic: use E.Y k, 𝟙 _, pullback.lift (E.p₁ k) (E.p₂ k) (E.w _), ⟨k⟩