awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Over (goal)
Goal:
C : Type u
inst : Category.{v, u} C
J_1 J : GrothendieckTopology.{v, u} C
X : C
Y : Over.{v, u} X
E : (J.over X).OneHypercover Y
i₁ i₂ : E.I₀
W : C
p₁ : W ⟶ Functor.obj (Over.forget X) (E.X i₁)
p₂ : W ⟶ Functor.obj (Over.forget X) (E.X i₂)
w : p₁ ≫ Functor.map (Over.forget X) (E.f i₁) = p₂ ≫ Functor.map (Over.forget X) (E.f i₂)
this_1 : p₁ ≫ (E.X i₁).hom = p₂ ≫ (E.X i₂).hom
this :
  Membership.mem (γ := Set (Sieve (Over.mk (p₁ ≫ (E.X i₁).hom))))
    ((J.over X : (a : Over X) → Set (Sieve a)) (Over.mk (p₁ ≫ (E.X i₁).hom)))
    (E.sieve₁ (i₁ := i₁) (i₂ := i₂) (Over.homMk p₁ ⋯) (Over.homMk p₂ ⋯))
refine_2 :
  ∀
    (this :
      Membership.mem (α := Sieve.{v, u} (Over.mk (p₁ ≫ (E.X i₁).hom)).left) (γ :=
        Set (Sieve.{v, u} (Over.mk (p₁ ≫ (E.X i₁).hom)).left))
        ((J : (a : C) → Set (Sieve a)) (Over.mk (p₁ ≫ (E.X i₁).hom)).left)
        ((Sieve.overEquiv (Over.mk (p₁ ≫ (E.X i₁).hom)) :
              (a : Sieve (Over.mk (p₁ ≫ (E.X i₁).hom))) → Sieve (Over.mk (p₁ ≫ (E.X i₁).hom)).left)
            (E.sieve₁ (i₁ := i₁) (i₂ := i₂) (Over.homMk p₁ ⋯) (Over.homMk p₂ ⋯)) :
          Sieve.{v, u} (Over.mk (p₁ ≫ (E.X i₁).hom)).left)),
    (E.map (Over.forget X)).sieve₁ p₁ p₂ ∈ (J : (a : C) → Set (Sieve a)) W
⊢ (E.map (Over.forget X)).sieve₁ p₁ p₂ ∈ (J : (a : C) → Set (Sieve a)) W
have refine_2₁ : ∀
  (this :
    Membership.mem (γ := Set (Sieve.{v, u} (Over.mk (p₁ ≫ (E.X i₁).hom)).left))
      ((J : (a : C) → Set (Sieve a)) (Over.mk (p₁ ≫ (E.X i₁).hom)).left)
      ((E.map (Over.forget X)).sieve₁ (i₁ := i₁) (i₂ := i₂) (Over.Hom.left (Over.homMk p₁ sorry))
        (Over.Hom.left (Over.homMk p₂ sorry)))),
  (E.map (Over.forget X)).sieve₁ p₁ p₂ ∈ (J : (a : C) → Set (Sieve a)) W
Tactic: rwa [GrothendieckTopology.mem_over_iff, Sieve.overEquiv_preOneHypercover_sieve₁] at this
Cleared hypotheses: this
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Over (goal)
Goal:
C : Type u
inst : Category.{v, u} C
J_1 : GrothendieckTopology.{v, u} C
D : Type u_1
inst_1 : Category.{v_1, u_1} D
J : GrothendieckTopology.{v, u} C
K : GrothendieckTopology.{v_1, u_1} D
F : Functor.{v, v_1, u, u_1} C D
X : C
inst_2 : F.PreservesOneHypercovers J K
Y : Over.{v, u} X
E : (J.over X).OneHypercover Y
E' : OneHypercover.{w, v_1, u_1} K (F.obj (Functor.obj (Over.forget X) Y)) := (E.map (Over.forget X) J).map F K
⊢ E.IsPreservedBy (Over.post F) (K.over (F.obj X))
have refine_1 : Membership.mem (γ := Set (Sieve (Functor.obj (Over.post F) Y)))
  ((K.over (F.obj X) : (a : Over (F.obj X)) → Set (Sieve a)) (Functor.obj (Over.post F) Y)) (E.map (Over.post F)).sieve₀
Tactic: refine ⟨?_, ?_⟩
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Over (goal)
Goal:
C : Type u
inst : Category.{v, u} C
J_1 : GrothendieckTopology.{v, u} C
D : Type u_1
inst_1 : Category.{v_1, u_1} D
J : GrothendieckTopology.{v, u} C
K : GrothendieckTopology.{v_1, u_1} D
F : Functor.{v, v_1, u, u_1} C D
X : C
inst_2 : F.PreservesOneHypercovers J K
Y : Over.{v, u} X
E : (J.over X).OneHypercover Y
E' : OneHypercover.{w, v_1, u_1} K (F.obj (Functor.obj (Over.forget X) Y)) := (E.map (Over.forget X) J).map F K
refine_1 :
  Membership.mem (γ := Set (Sieve (Functor.obj (Over.post F) Y)))
    ((K.over (F.obj X) : (a : Over (F.obj X)) → Set (Sieve a)) (Functor.obj (Over.post F) Y))
    (E.map (Over.post F)).sieve₀
⊢ E.IsPreservedBy (Over.post F) (K.over (F.obj X))
have refine_2 : ∀ (i₁ i₂ : E.I₀) ⦃W : Over.{v_1, u_1} (F.obj X)⦄ (p₁ : W ⟶ Functor.obj (Over.post F) (E.X i₁))
  (p₂ : W ⟶ Functor.obj (Over.post F) (E.X i₂))
  (w : p₁ ≫ Functor.map (Over.post F) (E.f i₁) = p₂ ≫ Functor.map (Over.post F) (E.f i₂)),
  (E.map (Over.post F)).sieve₁ p₁ p₂ ∈ (K.over (F.obj X) : (a : Over (F.obj X)) → Set (Sieve a)) W
Tactic: refine ⟨?_, ?_⟩
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Over (goal)
Goal:
C : Type u
inst : Category.{v, u} C
J_1 : GrothendieckTopology.{v, u} C
D : Type u_1
inst_1 : Category.{v_1, u_1} D
J : GrothendieckTopology.{v, u} C
K : GrothendieckTopology.{v_1, u_1} D
F : Functor.{v, v_1, u, u_1} C D
X : C
inst_2 : F.PreservesOneHypercovers J K
Y : Over.{v, u} X
E : (J.over X).OneHypercover Y
E' : OneHypercover.{w, v_1, u_1} K (F.obj (Functor.obj (Over.forget X) Y)) := (E.map (Over.forget X) J).map F K
⊢ Membership.mem (γ := Set (Sieve (Functor.obj (Over.post F) Y)))
    ((K.over (F.obj X) : (a : Over (F.obj X)) → Set (Sieve a)) (Functor.obj (Over.post F) Y))
    (PreZeroHypercover.sieve₀ (PreZeroHypercover.map (Over.post F) E.toPreZeroHypercover))
have refine_1 : Membership.mem (γ := Set (Sieve.{v_1, u_1} (Functor.obj (Over.post F) Y).left))
  ((K : (a : D) → Set (Sieve a)) (Functor.obj (Over.post F) Y).left)
  (Sieve.ofArrows (fun (i : E.I₀) ↦ (Functor.obj (Over.post F) (E.X i)).left) fun (i : E.I₀) ↦
    Over.Hom.left (Functor.map (Over.post F) (E.f i)))
Tactic: rw [PreZeroHypercover.sieve₀_map, GrothendieckTopology.mem_over_iff,
      Sieve.functorPushforward_ofArrows, Sieve.overEquiv_ofArrows]
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Over (goal)
Goal:
C : Type u
inst : Category.{v, u} C
J_1 : GrothendieckTopology.{v, u} C
D : Type u_1
inst_1 : Category.{v_1, u_1} D
J : GrothendieckTopology.{v, u} C
K : GrothendieckTopology.{v_1, u_1} D
F : Functor.{v, v_1, u, u_1} C D
X : C
inst_2 : F.PreservesOneHypercovers J K
Y : Over.{v, u} X
E : (J.over X).OneHypercover Y
E' : OneHypercover.{w, v_1, u_1} K (F.obj (Functor.obj (Over.forget X) Y)) := (E.map (Over.forget X) J).map F K
i₁ i₂ : E.I₀
W : Over.{v_1, u_1} (F.obj X)
p₁ : W ⟶ Functor.obj (Over.post F) (E.X i₁)
p₂ : W ⟶ Functor.obj (Over.post F) (E.X i₂)
w : p₁ ≫ Functor.map (Over.post F) (E.f i₁) = p₂ ≫ Functor.map (Over.post F) (E.f i₂)
⊢ (E.map (Over.post F)).sieve₁ p₁ p₂ ∈ (K.over (F.obj X) : (a : Over (F.obj X)) → Set (Sieve a)) W
have refine_2 : Membership.mem (γ := Set (Sieve.{v_1, u_1} W.left)) ((K : (a : D) → Set (Sieve a)) W.left)
  (((E.map (Over.forget X)).map F).sieve₁ (Over.Hom.left p₁) (Over.Hom.left p₂))
Tactic: simp_rw [GrothendieckTopology.mem_over_iff, Sieve.overEquiv_preOneHypercover_sieve₁,
      ← PreOneHypercover.map_comp, Over.post_forget_eq_forget_comp, PreOneHypercover.map_comp]
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Over (goal)
Goal:
C : Type u
inst : Category.{v, u} C
J : GrothendieckTopology.{v, u} C
inst_1 : HasPullbacks.{v, u} C
X Y : C
f : X ⟶ Y
⊢ CoverPreserving (J.over Y) (J.over X) (Over.pullback f)
have rw : (Over.map f).IsCocontinuous (J.over X) (J.over Y)
Tactic: rw [← (Over.mapPullbackAdj f).isCocontinuous_iff_coverPreserving]
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Over (goal)
Goal:
C : Type u'
inst : Category.{v_1, u'} C
inst_1 : HasBinaryProducts.{v_1, u'} C
J : GrothendieckTopology.{v_1, u'} C
X U : C
S : Sieve.{v_1, u'} U
hs : S ∈ (J : (a : C) → Set (Sieve a)) U
⊢ Membership.mem (γ := Set (Sieve (C := Over.{v_1, u'} X) (Functor.obj (Over.star X) U)))
    ((J.over X : (a : Over X) → Set (Sieve a)) (Functor.obj (Over.star X) U)) (Sieve.functorPushforward (Over.star X) S)
have refine : LE.le (α := Sieve.{v_1, u'} (Over.left.{v_1, u'} (Functor.obj (Over.star X) U))) (Sieve.pullback prod.snd S)
  ((Sieve.overEquiv (Functor.obj (Over.star X) U) :
      (a : Sieve ((Over.star X).obj U)) → Sieve ((Over.star X).obj U).left)
    (Sieve.functorPushforward (Over.star X) S))
Tactic: refine J.superset_covering ?_ (J.pullback_stable prod.snd hs)
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Over (goal)
Goal:
C : Type u'
inst : Category.{v_1, u'} C
inst_1 : HasBinaryProducts.{v_1, u'} C
J : GrothendieckTopology.{v_1, u'} C
X U : C
S : Sieve.{v_1, u'} U
hs : S ∈ (J : (a : C) → Set (Sieve a)) U
y : C
f : y ⟶ Over.left.{v_1, u'} (Functor.obj (Over.star X) U)
hf : Sieve.arrows (Sieve.pullback prod.snd S) f
⊢ Presieve.functorPushforward (Over.forget X) (X := Functor.obj (Over.star X) U)
    (Presieve.functorPushforward (Over.star X) S.arrows) (Y := y) f
have rw : Presieve.functorPushforward (Over.star X ⋙ Over.forget X) S.arrows f
Tactic: rw [← Presieve.functorPushforward_comp]
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Over (goal)
Goal:
C : Type u'
inst : Category.{v_1, u'} C
inst_1 : HasBinaryProducts.{v_1, u'} C
J : GrothendieckTopology.{v_1, u'} C
X U : C
S : Sieve.{v_1, u'} U
hs : S ∈ (J : (a : C) → Set (Sieve a)) U
y : C
f : y ⟶ Over.left.{v_1, u'} (Functor.obj (Over.star X) U)
hf : Sieve.arrows (Sieve.pullback prod.snd S) f
⊢ Presieve.functorPushforward (Over.star X ⋙ Over.forget X) S.arrows f
have refine_1 : f ≫ prod.fst =
  CategoryStruct.comp (Y := Limits.prod.{v_1, u'} X U)
    (CategoryStruct.comp (Y := Functor.obj.{v_1, v_1, u', u'} (Over.star X ⋙ Over.forget X) y)
      (prod.lift (f ≫ prod.fst) (𝟙 y)) (Functor.map (Over.star X ⋙ Over.forget X) (f ≫ prod.snd)))
    prod.fst
Tactic: refine ⟨_, _, prod.lift (f ≫ prod.fst) (𝟙 _), hf, Limits.prod.hom_ext ?_ ?_⟩
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Over (goal)
Goal:
C : Type u'
inst : Category.{v_1, u'} C
inst_1 : HasBinaryProducts.{v_1, u'} C
J : GrothendieckTopology.{v_1, u'} C
X U : C
S : Sieve.{v_1, u'} U
hs : S ∈ (J : (a : C) → Set (Sieve a)) U
y : C
f : y ⟶ Over.left.{v_1, u'} (Functor.obj (Over.star X) U)
hf : Sieve.arrows (Sieve.pullback prod.snd S) f
refine_1 :
  f ≫ prod.fst =
    CategoryStruct.comp (Y := Limits.prod.{v_1, u'} X U)
      (CategoryStruct.comp (Y := Functor.obj.{v_1, v_1, u', u'} (Over.star X ⋙ Over.forget X) y)
        (prod.lift (f ≫ prod.fst) (𝟙 y)) (Functor.map (Over.star X ⋙ Over.forget X) (f ≫ prod.snd)))
      prod.fst
⊢ Presieve.functorPushforward (Over.star X ⋙ Over.forget X) S.arrows f
have refine_2 : f ≫ prod.snd =
  CategoryStruct.comp (Y := Limits.prod.{v_1, u'} X U)
    (CategoryStruct.comp (Y := Functor.obj.{v_1, v_1, u', u'} (Over.star X ⋙ Over.forget X) y)
      (prod.lift (f ≫ prod.fst) (𝟙 y)) (Functor.map (Over.star X ⋙ Over.forget X) (f ≫ prod.snd)))
    prod.snd
Tactic: refine ⟨_, _, prod.lift (f ≫ prod.fst) (𝟙 _), hf, Limits.prod.hom_ext ?_ ?_⟩
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Over (goal)
Goal:
C : Type u
inst : Category.{v, u} C
J : GrothendieckTopology.{v, u} C
A : Type u'
inst_1 : Category.{v', u'} A
X Y : C
f g : X ⟶ Y
h : f = g
⊢ Eq (α := Sheaf (J.over Y) A ⥤ Sheaf (J.over X) A) (J.overMapPullback A f) (J.overMapPullback A g)
have subst : Eq (α := Sheaf (J.over Y) A ⥤ Sheaf (J.over X) A) (J.overMapPullback A f) (J.overMapPullback A f)
Tactic: subst h
Cleared hypotheses: g, h
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Over (goal)
Goal:
C : Type u
inst : Category.{v, u} C
J : GrothendieckTopology.{v, u} C
A : Type u'
inst_1 : Category.{v', u'} A
X Y : C
f : X ⟶ Y
⊢ Eq (α :=
    Quiver.Hom (V := Sheaf (J.over Y) A ⥤ Sheaf (J.over X) A) (J.overMapPullback A (f ≫ 𝟙 Y)) (J.overMapPullback A f))
    (Iso.inv (J.overMapPullbackComp A f (𝟙 Y)) ≫
      Functor.whiskerRight (Iso.hom (J.overMapPullbackId A Y)) (J.overMapPullback A f) ≫
        Iso.hom (Functor.leftUnitor (J.overMapPullback A f)))
    (Iso.hom (J.overMapPullbackCongr A ⋯))
have w : ∀ (x : Sheaf (J.over Y) A) (x_1 : (Over.{v, u} X)ᵒᵖ),
  (NatTrans.app
            (Iso.inv (J.overMapPullbackComp A f (𝟙 Y)) ≫
              Functor.whiskerRight (Iso.hom (J.overMapPullbackId A Y)) (J.overMapPullback A f) ≫
                Iso.hom (Functor.leftUnitor (J.overMapPullback A f)))
            x).hom.app
      x_1 =
    (NatTrans.app (Iso.hom (J.overMapPullbackCongr A sorry)) x).hom.app x_1
Tactic: ext
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Over (goal)
Goal:
C : Type u
inst : Category.{v, u} C
J : GrothendieckTopology.{v, u} C
A : Type u'
inst_1 : Category.{v', u'} A
X Y : C
f : X ⟶ Y
x : Sheaf (J.over Y) A
x_1 : (Over.{v, u} X)ᵒᵖ
⊢ (NatTrans.app
            (Iso.inv (J.overMapPullbackComp A f (𝟙 Y)) ≫
              Functor.whiskerRight (Iso.hom (J.overMapPullbackId A Y)) (J.overMapPullback A f) ≫
                Iso.hom (Functor.leftUnitor (J.overMapPullback A f)))
            x).hom.app
      x_1 =
    (NatTrans.app (Iso.hom (J.overMapPullbackCongr A ⋯)) x).hom.app x_1
have w : (NatTrans.app (Iso.inv (J.overMapPullbackComp A f (𝟙 Y))) x).hom.app x_1 ≫
    (NatTrans.app (Iso.hom (J.overMapPullbackId A Y)) x).hom.app (Opposite.op ((Over.map f).obj (Opposite.unop x_1))) ≫
      𝟙 (x.obj (P := Presheaf.IsSheaf (J.over Y)).obj (Opposite.op ((Over.map f).obj (Opposite.unop x_1)))) =
  x.obj (P := Presheaf.IsSheaf (J.over Y)).map
    (Quiver.Hom.op ((Over.mapCongr (f ≫ 𝟙 Y) f sorry).inv.app (Opposite.unop x_1)))
Tactic: dsimp
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Over (goal)
Goal:
C : Type u
inst : Category.{v, u} C
J : GrothendieckTopology.{v, u} C
A : Type u'
inst_1 : Category.{v', u'} A
X Y : C
f : X ⟶ Y
x : Sheaf (J.over Y) A
x_1 : (Over.{v, u} X)ᵒᵖ
⊢ (NatTrans.app (Iso.inv (J.overMapPullbackComp A f (𝟙 Y))) x).hom.app x_1 ≫
      (NatTrans.app (Iso.hom (J.overMapPullbackId A Y)) x).hom.app
          (Opposite.op ((Over.map f).obj (Opposite.unop x_1))) ≫
        𝟙 (x.obj (P := Presheaf.IsSheaf (J.over Y)).obj (Opposite.op ((Over.map f).obj (Opposite.unop x_1)))) =
    x.obj (P := Presheaf.IsSheaf (J.over Y)).map
      (Quiver.Hom.op ((Over.mapCongr (f ≫ 𝟙 Y) f ⋯).inv.app (Opposite.unop x_1)))
have w : x.obj (P := Presheaf.IsSheaf (J.over Y)).map
    (NatTrans.app (Iso.inv (Over.mapId Y)) ((Over.map f).obj (Opposite.unop x_1)) ≫
        (Over.mapComp f (𝟙 Y)).inv.app (Opposite.unop x_1)).op =
  x.obj (P := Presheaf.IsSheaf (J.over Y)).map
    (Quiver.Hom.op ((Over.mapCongr (f ≫ 𝟙 Y) f sorry).inv.app (Opposite.unop x_1)))
Tactic: simp only [overMapPullbackComp_inv_app_hom_app, overMapPullbackId_hom_app_hom_app,
    comp_id, ← Functor.map_comp, ← op_comp]
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Over (goal)
Goal:
C : Type u
inst : Category.{v, u} C
J : GrothendieckTopology.{v, u} C
A : Type u'
inst_1 : Category.{v', u'} A
X Y : C
f : X ⟶ Y
x : Sheaf (J.over Y) A
x_1 : (Over.{v, u} X)ᵒᵖ
⊢ x.obj (P := Presheaf.IsSheaf (J.over Y)).map
      (NatTrans.app (Iso.inv (Over.mapId Y)) ((Over.map f).obj (Opposite.unop x_1)) ≫
          (Over.mapComp f (𝟙 Y)).inv.app (Opposite.unop x_1)).op =
    x.obj (P := Presheaf.IsSheaf (J.over Y)).map
      (Quiver.Hom.op ((Over.mapCongr (f ≫ 𝟙 Y) f ⋯).inv.app (Opposite.unop x_1)))
have w : NatTrans.app (Iso.inv (Over.mapId Y)) ((Over.map f).obj (Opposite.unop x_1)) ≫
    (Over.mapComp f (𝟙 Y)).inv.app (Opposite.unop x_1) =
  (Over.mapCongr (f ≫ 𝟙 Y) f sorry).inv.app (Opposite.unop x_1)
Tactic: congr
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Over (goal)
Goal:
C : Type u
inst : Category.{v, u} C
J : GrothendieckTopology.{v, u} C
A : Type u'
inst_1 : Category.{v', u'} A
X Y : C
f : X ⟶ Y
⊢ Eq (α :=
    Quiver.Hom (V := Sheaf (J.over Y) A ⥤ Sheaf (J.over X) A) (J.overMapPullback A (𝟙 X ≫ f)) (J.overMapPullback A f))
    (Iso.inv (J.overMapPullbackComp A (𝟙 X) f) ≫
      Functor.whiskerLeft (J.overMapPullback A f) (Iso.hom (J.overMapPullbackId A X)) ≫
        Iso.hom (Functor.rightUnitor (J.overMapPullback A f)))
    (Iso.hom (J.overMapPullbackCongr A ⋯))
have w : ∀ (x : Sheaf (J.over Y) A) (x_1 : (Over.{v, u} X)ᵒᵖ),
  (NatTrans.app
            (Iso.inv (J.overMapPullbackComp A (𝟙 X) f) ≫
              Functor.whiskerLeft (J.overMapPullback A f) (Iso.hom (J.overMapPullbackId A X)) ≫
                Iso.hom (Functor.rightUnitor (J.overMapPullback A f)))
            x).hom.app
      x_1 =
    (NatTrans.app (Iso.hom (J.overMapPullbackCongr A sorry)) x).hom.app x_1
Tactic: ext
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Over (goal)
Goal:
C : Type u
inst : Category.{v, u} C
J : GrothendieckTopology.{v, u} C
A : Type u'
inst_1 : Category.{v', u'} A
X Y : C
f : X ⟶ Y
x : Sheaf (J.over Y) A
x_1 : (Over.{v, u} X)ᵒᵖ
⊢ (NatTrans.app
            (Iso.inv (J.overMapPullbackComp A (𝟙 X) f) ≫
              Functor.whiskerLeft (J.overMapPullback A f) (Iso.hom (J.overMapPullbackId A X)) ≫
                Iso.hom (Functor.rightUnitor (J.overMapPullback A f)))
            x).hom.app
      x_1 =
    (NatTrans.app (Iso.hom (J.overMapPullbackCongr A ⋯)) x).hom.app x_1
have w : (NatTrans.app (Iso.inv (J.overMapPullbackComp A (𝟙 X) f)) x).hom.app x_1 ≫
    (NatTrans.app (Iso.hom (J.overMapPullbackId A X)) (Functor.obj (J.overMapPullback A f) x)).hom.app x_1 ≫
      𝟙 (x.obj (P := Presheaf.IsSheaf (J.over Y)).obj (Opposite.op ((Over.map f).obj (Opposite.unop x_1)))) =
  x.obj (P := Presheaf.IsSheaf (J.over Y)).map
    (Quiver.Hom.op ((Over.mapCongr (𝟙 X ≫ f) f sorry).inv.app (Opposite.unop x_1)))
Tactic: dsimp
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Over (goal)
Goal:
C : Type u
inst : Category.{v, u} C
J : GrothendieckTopology.{v, u} C
A : Type u'
inst_1 : Category.{v', u'} A
X Y : C
f : X ⟶ Y
x : Sheaf (J.over Y) A
x_1 : (Over.{v, u} X)ᵒᵖ
⊢ (NatTrans.app (Iso.inv (J.overMapPullbackComp A (𝟙 X) f)) x).hom.app x_1 ≫
      (NatTrans.app (Iso.hom (J.overMapPullbackId A X)) (Functor.obj (J.overMapPullback A f) x)).hom.app x_1 ≫
        𝟙 (x.obj (P := Presheaf.IsSheaf (J.over Y)).obj (Opposite.op ((Over.map f).obj (Opposite.unop x_1)))) =
    x.obj (P := Presheaf.IsSheaf (J.over Y)).map
      (Quiver.Hom.op ((Over.mapCongr (𝟙 X ≫ f) f ⋯).inv.app (Opposite.unop x_1)))
have w : x.obj (P := Presheaf.IsSheaf (J.over Y)).map
    ((Over.map f).map (NatTrans.app (Iso.inv (Over.mapId X)) (Opposite.unop x_1)) ≫
        NatTrans.app (Iso.inv (Over.mapComp (𝟙 X) f)) (Opposite.unop x_1)).op =
  x.obj (P := Presheaf.IsSheaf (J.over Y)).map
    (Quiver.Hom.op ((Over.mapCongr (𝟙 X ≫ f) f sorry).inv.app (Opposite.unop x_1)))
Tactic: simp only [overMapPullbackComp_inv_app_hom_app, overMapPullbackId_hom_app_hom_app,
    Functor.sheafPushforwardContinuous_obj_obj_map, Quiver.Hom.unop_op,
    comp_id, ← Functor.map_comp, ← op_comp]
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Over (goal)
Goal:
C : Type u
inst : Category.{v, u} C
J : GrothendieckTopology.{v, u} C
A : Type u'
inst_1 : Category.{v', u'} A
X Y : C
f : X ⟶ Y
x : Sheaf (J.over Y) A
x_1 : (Over.{v, u} X)ᵒᵖ
⊢ x.obj (P := Presheaf.IsSheaf (J.over Y)).map
      ((Over.map f).map (NatTrans.app (Iso.inv (Over.mapId X)) (Opposite.unop x_1)) ≫
          NatTrans.app (Iso.inv (Over.mapComp (𝟙 X) f)) (Opposite.unop x_1)).op =
    x.obj (P := Presheaf.IsSheaf (J.over Y)).map
      (Quiver.Hom.op ((Over.mapCongr (𝟙 X ≫ f) f ⋯).inv.app (Opposite.unop x_1)))
have w : (Over.map f).map (NatTrans.app (Iso.inv (Over.mapId X)) (Opposite.unop x_1)) ≫
    NatTrans.app (Iso.inv (Over.mapComp (𝟙 X) f)) (Opposite.unop x_1) =
  (Over.mapCongr (𝟙 X ≫ f) f sorry).inv.app (Opposite.unop x_1)
Tactic: congr
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Over (goal)
Goal:
C : Type u
inst : Category.{v, u} C
J : GrothendieckTopology.{v, u} C
A : Type u'
inst_1 : Category.{v', u'} A
X Y Z T : C
f : X ⟶ Y
g : Y ⟶ Z
h : Z ⟶ T
⊢ Eq (α :=
    Quiver.Hom (V := Sheaf (J.over T) A ⥤ Sheaf (J.over X) A) (J.overMapPullback A (f ≫ g ≫ h))
      (J.overMapPullback A ((f ≫ g) ≫ h)))
    (Iso.inv (J.overMapPullbackComp A f (g ≫ h)) ≫
      Functor.whiskerRight (Iso.inv (J.overMapPullbackComp A g h)) (J.overMapPullback A f) ≫
        Iso.hom (Functor.associator (J.overMapPullback A h) (J.overMapPullback A g) (J.overMapPullback A f)) ≫
          Functor.whiskerLeft (J.overMapPullback A h) (Iso.hom (J.overMapPullbackComp A f g)) ≫
            Iso.hom (J.overMapPullbackComp A (f ≫ g) h))
    (Iso.hom (J.overMapPullbackCongr A ⋯))
have w : ∀ (x : Sheaf (J.over T) A) (x_1 : (Over.{v, u} X)ᵒᵖ),
  (NatTrans.app
            (Iso.inv (J.overMapPullbackComp A f (g ≫ h)) ≫
              Functor.whiskerRight (Iso.inv (J.overMapPullbackComp A g h)) (J.overMapPullback A f) ≫
                Iso.hom (Functor.associator (J.overMapPullback A h) (J.overMapPullback A g) (J.overMapPullback A f)) ≫
                  Functor.whiskerLeft (J.overMapPullback A h) (Iso.hom (J.overMapPullbackComp A f g)) ≫
                    Iso.hom (J.overMapPullbackComp A (f ≫ g) h))
            x).hom.app
      x_1 =
    (NatTrans.app (Iso.hom (J.overMapPullbackCongr A sorry)) x).hom.app x_1
Tactic: ext
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Over (goal)
Goal:
C : Type u
inst : Category.{v, u} C
J : GrothendieckTopology.{v, u} C
A : Type u'
inst_1 : Category.{v', u'} A
X Y Z T : C
f : X ⟶ Y
g : Y ⟶ Z
h : Z ⟶ T
x : Sheaf (J.over T) A
x_1 : (Over.{v, u} X)ᵒᵖ
⊢ (NatTrans.app
            (Iso.inv (J.overMapPullbackComp A f (g ≫ h)) ≫
              Functor.whiskerRight (Iso.inv (J.overMapPullbackComp A g h)) (J.overMapPullback A f) ≫
                Iso.hom (Functor.associator (J.overMapPullback A h) (J.overMapPullback A g) (J.overMapPullback A f)) ≫
                  Functor.whiskerLeft (J.overMapPullback A h) (Iso.hom (J.overMapPullbackComp A f g)) ≫
                    Iso.hom (J.overMapPullbackComp A (f ≫ g) h))
            x).hom.app
      x_1 =
    (NatTrans.app (Iso.hom (J.overMapPullbackCongr A ⋯)) x).hom.app x_1
have w : (NatTrans.app (Iso.inv (J.overMapPullbackComp A f (g ≫ h))) x).hom.app x_1 ≫
    (NatTrans.app (Iso.inv (J.overMapPullbackComp A g h)) x).hom.app
        (Opposite.op ((Over.map f).obj (Opposite.unop x_1))) ≫
      𝟙
          (x.obj (P := Presheaf.IsSheaf (J.over T)).obj
            (Opposite.op ((Over.map h).obj ((Over.map g).obj ((Over.map f).obj (Opposite.unop x_1)))))) ≫
        (NatTrans.app (Iso.hom (J.overMapPullbackComp A f g)) (Functor.obj (J.overMapPullback A h) x)).hom.app x_1 ≫
          (NatTrans.app (Iso.hom (J.overMapPullbackComp A (f ≫ g) h)) x).hom.app x_1 =
  x.obj (P := Presheaf.IsSheaf (J.over T)).map
    (Quiver.Hom.op ((Over.mapCongr (f ≫ g ≫ h) ((f ≫ g) ≫ h) sorry).inv.app (Opposite.unop x_1)))
Tactic: dsimp
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Over (goal)
Goal:
C : Type u
inst : Category.{v, u} C
J : GrothendieckTopology.{v, u} C
A : Type u'
inst_1 : Category.{v', u'} A
X Y Z T : C
f : X ⟶ Y
g : Y ⟶ Z
h : Z ⟶ T
x : Sheaf (J.over T) A
x_1 : (Over.{v, u} X)ᵒᵖ
⊢ (NatTrans.app (Iso.inv (J.overMapPullbackComp A f (g ≫ h))) x).hom.app x_1 ≫
      (NatTrans.app (Iso.inv (J.overMapPullbackComp A g h)) x).hom.app
          (Opposite.op ((Over.map f).obj (Opposite.unop x_1))) ≫
        𝟙
            (x.obj (P := Presheaf.IsSheaf (J.over T)).obj
              (Opposite.op ((Over.map h).obj ((Over.map g).obj ((Over.map f).obj (Opposite.unop x_1)))))) ≫
          (NatTrans.app (Iso.hom (J.overMapPullbackComp A f g)) (Functor.obj (J.overMapPullback A h) x)).hom.app x_1 ≫
            (NatTrans.app (Iso.hom (J.overMapPullbackComp A (f ≫ g) h)) x).hom.app x_1 =
    x.obj (P := Presheaf.IsSheaf (J.over T)).map
      (Quiver.Hom.op ((Over.mapCongr (f ≫ g ≫ h) ((f ≫ g) ≫ h) ⋯).inv.app (Opposite.unop x_1)))
have w : Eq (α :=
  x.obj (P := Presheaf.IsSheaf (J.over T)).obj (Opposite.op ((Over.map (f ≫ g ≫ h)).obj (Opposite.unop x_1))) ⟶
    x.obj (P := Presheaf.IsSheaf (J.over T)).obj (Opposite.op ((Over.map ((f ≫ g) ≫ h)).obj (Opposite.unop x_1))))
  (x.obj (P := Presheaf.IsSheaf (J.over T)).map
    (Quiver.Hom.op
      ((Over.mapComp (f ≫ g) h).hom.app (Opposite.unop x_1) ≫
        (Over.map h).map ((Over.mapComp f g).hom.app (Opposite.unop x_1)) ≫
          (Over.mapComp g h).inv.app ((Over.map f).obj (Opposite.unop x_1)) ≫
            (Over.mapComp f (g ≫ h)).inv.app (Opposite.unop x_1))))
  (x.obj (P := Presheaf.IsSheaf (J.over T)).map
    (Quiver.Hom.op ((Over.mapCongr (f ≫ g ≫ h) ((f ≫ g) ≫ h) sorry).inv.app (Opposite.unop x_1))))
Tactic: simp only [overMapPullbackComp_inv_app_hom_app,
    overMapPullbackComp_hom_app_hom_app, Functor.sheafPushforwardContinuous_obj_obj_map,
    Quiver.Hom.unop_op, ← Functor.map_comp, ← op_comp, id_comp, assoc]