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 : GrothendieckTopology.{v, u} C
X : C
Y₁ Y₂ : Over.{v, u} X
S₁ : Sieve Y₁
f : Y₂ ⟶ Y₁
h₁ :
  Membership.mem (γ := Set (Sieve Y₁))
    (Set.preimage (β := Sieve.{v, u} Y₁.left) (⇑(Sieve.overEquiv Y₁) : (a : Sieve Y₁) → Sieve Y₁.left)
      ((J : (a : C) → Set (Sieve a)) Y₁.left))
    S₁
⊢ Membership.mem (γ := Set (Sieve Y₂))
    (Set.preimage (β := Sieve.{v, u} Y₂.left) (⇑(Sieve.overEquiv Y₂) : (a : Sieve Y₂) → Sieve Y₂.left)
      ((J : (a : C) → Set (Sieve a)) Y₂.left))
    (Sieve.pullback f S₁)
have rw : Membership.mem (γ := Set (Sieve.{v, u} Y₂.left)) ((J : (a : C) → Set (Sieve a)) Y₂.left)
  (Sieve.pullback (Over.Hom.left f) ((Sieve.overEquiv Y₁ : (a : Sieve Y₁) → Sieve Y₁.left) S₁))
Tactic: rw [Set.mem_preimage, Sieve.overEquiv_pullback]
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Over (hypothesis)
Goal:
C : Type u
inst : Category.{v, u} C
J : GrothendieckTopology.{v, u} C
X : C
Y : Over.{v, u} X
S : Sieve Y
hS :
  Membership.mem (γ := Set (Sieve Y))
    (Set.preimage (β := Sieve.{v, u} Y.left) (⇑(Sieve.overEquiv Y) : (a : Sieve Y) → Sieve Y.left)
      ((J : (a : C) → Set (Sieve a)) Y.left))
    S
R : Sieve Y
hR :
  ∀ ⦃Y_1 : Over.{v, u} X⦄ ⦃f : Y_1 ⟶ Y⦄,
    S.arrows f →
      Membership.mem (γ := Set (Sieve Y_1))
        (Set.preimage (β := Sieve.{v, u} Y_1.left) (⇑(Sieve.overEquiv Y_1) : (a : Sieve Y_1) → Sieve Y_1.left)
          ((J : (a : C) → Set (Sieve a)) Y_1.left))
        (Sieve.pullback f R)
Z : C
f : Z ⟶ Y.left
hf : Sieve.arrows ((Sieve.overEquiv Y : (a : Sieve Y) → Sieve Y.left) S) f
⊢ Sieve.pullback f ((Sieve.overEquiv Y : (a : Sieve Y) → Sieve Y.left) R) ∈ (J : (a : C) → Set (Sieve a)) Z
have hR₁ : Membership.mem (γ := Set (Sieve (Over.mk (f ≫ Y.hom))))
  (Set.preimage (β := Sieve.{v, u} (Over.mk (f ≫ Y.hom)).left)
    (⇑(Sieve.overEquiv (Over.mk (f ≫ Y.hom))) : (a : Sieve (Over.mk (f ≫ Y.hom))) → Sieve (Over.mk (f ≫ Y.hom)).left)
    ((J : (a : C) → Set (Sieve a)) (Over.mk (f ≫ Y.hom)).left))
  (Sieve.pullback (Over.homMk f sorry) R)
Tactic: specialize hR ((Sieve.overEquiv_iff _ _).1 hf)
Cleared hypotheses: hR
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Over (goal)
Goal:
C : Type u
inst : Category.{v, u} C
J : GrothendieckTopology.{v, u} C
X : C
Y : Over.{v, u} X
S : Sieve Y
hS :
  Membership.mem (γ := Set (Sieve Y))
    (Set.preimage (β := Sieve.{v, u} Y.left) (⇑(Sieve.overEquiv Y) : (a : Sieve Y) → Sieve Y.left)
      ((J : (a : C) → Set (Sieve a)) Y.left))
    S
R : Sieve Y
Z : C
f : Z ⟶ Y.left
hf : Sieve.arrows ((Sieve.overEquiv Y : (a : Sieve Y) → Sieve Y.left) S) f
hR :
  Membership.mem (γ := Set (Sieve (Over.mk (f ≫ Y.hom))))
    (Set.preimage (β := Sieve.{v, u} (Over.mk (f ≫ Y.hom)).left)
      (⇑(Sieve.overEquiv (Over.mk (f ≫ Y.hom))) : (a : Sieve (Over.mk (f ≫ Y.hom))) → Sieve (Over.mk (f ≫ Y.hom)).left)
      ((J : (a : C) → Set (Sieve a)) (Over.mk (f ≫ Y.hom)).left))
    (Sieve.pullback (Over.homMk f ⋯) R)
⊢ Sieve.pullback f ((Sieve.overEquiv Y : (a : Sieve Y) → Sieve Y.left) R) ∈ (J : (a : C) → Set (Sieve a)) Z
have rwa : ∀
  (hR :
    Membership.mem (α := Sieve.{v, u} (Over.mk (f ≫ Y.hom)).left) (γ := Set (Sieve.{v, u} (Over.mk (f ≫ Y.hom)).left))
      ((J : (a : C) → Set (Sieve a)) (Over.mk (f ≫ Y.hom)).left)
      ((Sieve.overEquiv (Over.mk (f ≫ Y.hom)) : (a : Sieve (Over.mk (f ≫ Y.hom))) → Sieve (Over.mk (f ≫ Y.hom)).left)
          (Sieve.pullback (Over.homMk f sorry) R) :
        Sieve.{v, u} (Over.mk (f ≫ Y.hom)).left)),
  Sieve.pullback f ((Sieve.overEquiv Y : (a : Sieve Y) → Sieve Y.left) R) ∈ (J : (a : C) → Set (Sieve a)) Z
Tactic: rwa [Set.mem_preimage, Sieve.overEquiv_pullback] at hR
Cleared hypotheses: hR
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Over (goal)
Goal:
C : Type u
inst : Category.{v, u} C
J : GrothendieckTopology.{v, u} C
X : C
Y : Over.{v, u} X
S : Sieve Y
hS :
  Membership.mem (γ := Set (Sieve Y))
    (Set.preimage (β := Sieve.{v, u} Y.left) (⇑(Sieve.overEquiv Y) : (a : Sieve Y) → Sieve Y.left)
      ((J : (a : C) → Set (Sieve a)) Y.left))
    S
R : Sieve Y
Z : C
f : Z ⟶ Y.left
hf : Sieve.arrows ((Sieve.overEquiv Y : (a : Sieve Y) → Sieve Y.left) S) f
hR :
  Membership.mem (γ := Set (Sieve (Over.mk (f ≫ Y.hom))))
    (Set.preimage (β := Sieve.{v, u} (Over.mk (f ≫ Y.hom)).left)
      (⇑(Sieve.overEquiv (Over.mk (f ≫ Y.hom))) : (a : Sieve (Over.mk (f ≫ Y.hom))) → Sieve (Over.mk (f ≫ Y.hom)).left)
      ((J : (a : C) → Set (Sieve a)) (Over.mk (f ≫ Y.hom)).left))
    (Sieve.pullback (Over.homMk f ⋯) R)
rwa :
  ∀
    (hR :
      Membership.mem (α := Sieve.{v, u} (Over.mk (f ≫ Y.hom)).left) (γ := Set (Sieve.{v, u} (Over.mk (f ≫ Y.hom)).left))
        ((J : (a : C) → Set (Sieve a)) (Over.mk (f ≫ Y.hom)).left)
        ((Sieve.overEquiv (Over.mk (f ≫ Y.hom)) : (a : Sieve (Over.mk (f ≫ Y.hom))) → Sieve (Over.mk (f ≫ Y.hom)).left)
            (Sieve.pullback (Over.homMk f ⋯) R) :
          Sieve.{v, u} (Over.mk (f ≫ Y.hom)).left)),
    Sieve.pullback f ((Sieve.overEquiv Y : (a : Sieve Y) → Sieve Y.left) R) ∈ (J : (a : C) → Set (Sieve a)) Z
⊢ Sieve.pullback f ((Sieve.overEquiv Y : (a : Sieve Y) → Sieve Y.left) R) ∈ (J : (a : C) → Set (Sieve a)) Z
have rwa₁ : ∀
  (hR :
    Membership.mem (γ := Set (Sieve.{v, u} (Over.mk (f ≫ Y.hom)).left))
      ((J : (a : C) → Set (Sieve a)) (Over.mk (f ≫ Y.hom)).left)
      (Sieve.pullback (X := Y.left) (Over.Hom.left (Over.homMk f sorry))
        ((Sieve.overEquiv Y : (a : Sieve Y) → Sieve Y.left) R))),
  Sieve.pullback f ((Sieve.overEquiv Y : (a : Sieve Y) → Sieve Y.left) R) ∈ (J : (a : C) → Set (Sieve a)) Z
Tactic: rwa [Set.mem_preimage, Sieve.overEquiv_pullback] at hR
Cleared hypotheses: hR
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Over (goal)
Goal:
C : Type u
inst : Category.{v, u} C
J : GrothendieckTopology.{v, u} C
X : C
x : Sheaf J (Type u_1)
Z : Over.{v, u} X
x_1 : Presieve Z
x_2 : Presieve.FamilyOfElements (Functor.op (Over.forget X) ⋙ x.obj (P := Presheaf.IsSheaf J)) x_1
hx : x_2.Compatible (R := x_1)
Y₁ Y₂ : Over.{v, u} X
W : C
f₁ : W ⟶ Functor.obj (Over.forget X) Y₁
f₂ : W ⟶ Functor.obj (Over.forget X) Y₂
g₁ : Y₁ ⟶ Z
g₂ : Y₂ ⟶ Z
hg₁ : x_1 g₁
hg₂ : x_1 g₂
h : f₁ ≫ Functor.map (Over.forget X) g₁ = f₂ ≫ Functor.map (Over.forget X) g₂
W' : Over.{v, u} X := Over.mk (f₁ ≫ Y₁.hom)
g₁' : W' ⟶ Y₁ := Over.homMk f₁ ⋯
g₂' : W' ⟶ Y₂ := Over.homMk f₂ ⋯
⊢ g₁' ≫ g₁ = g₂' ≫ g₂
have h₁ : Over.Hom.left (g₁' ≫ g₁) = Over.Hom.left (g₂' ≫ g₂)
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
X Y : C
f : X ⟶ Y
U : Over.{v, u} X
S : Sieve U
hS : S ∈ (J.over X : (a : Over X) → Set (Sieve a)) U
⊢ Membership.mem (γ := Set (Sieve ((Over.map f).obj U)))
    ((J.over Y : (a : Over Y) → Set (Sieve a)) ((Over.map f).obj U)) (Sieve.functorPushforward (Over.map f) S)
have obtain : ∀ (S : Sieve.{v, u} U.left)
  (hS :
    Membership.mem (α := Sieve U) ((J.over X : (a : Over X) → Set (Sieve a)) U)
      (((Sieve.overEquiv U).symm : (a : Sieve U.left) → Sieve U) S : Sieve U)),
  Membership.mem (γ := Set (Sieve ((Over.map f).obj U)))
    ((J.over Y : (a : Over Y) → Set (Sieve a)) ((Over.map f).obj U))
    (Sieve.functorPushforward (Over.map f) (((Sieve.overEquiv U).symm : (a : Sieve U.left) → Sieve U) S))
Tactic: obtain ⟨S, rfl⟩ := (Sieve.overEquiv U).symm.surjective S
Cleared hypotheses: S, hS
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Over (goal)
Goal:
C : Type u
inst : Category.{v, u} C
J : GrothendieckTopology.{v, u} C
X Y : C
f : X ⟶ Y
U : Over.{v, u} X
S : Sieve.{v, u} U.left
hS :
  Membership.mem (α := Sieve U) ((J.over X : (a : Over X) → Set (Sieve a)) U)
    (((Sieve.overEquiv U).symm : (a : Sieve U.left) → Sieve U) S : Sieve U)
⊢ Membership.mem (γ := Set (Sieve ((Over.map f).obj U)))
    ((J.over Y : (a : Over Y) → Set (Sieve a)) ((Over.map f).obj U))
    (Sieve.functorPushforward (Over.map f) (((Sieve.overEquiv U).symm : (a : Sieve U.left) → Sieve U) S))
have rw : Membership.mem (α := Sieve ((Over.map f).obj U)) (γ := Set (Sieve ((Over.map f).obj U)))
  ((J.over Y : (a : Over Y) → Set (Sieve a)) ((Over.map f).obj U))
  (((Sieve.overEquiv ((Over.map f).obj U)).symm : (a : Sieve ((Over.map f).obj U).left) → Sieve ((Over.map f).obj U))
      S :
    Sieve ((Over.map f).obj U))
Tactic: rw [Sieve.functorPushforward_over_map]
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Over (goal)
Goal:
C : Type u
inst : Category.{v, u} C
J : GrothendieckTopology.{v, u} C
X Y : C
f : X ⟶ Y
U : Over.{v, u} X
S : Sieve.{v, u} U.left
hS :
  Membership.mem (α := Sieve U) ((J.over X : (a : Over X) → Set (Sieve a)) U)
    (((Sieve.overEquiv U).symm : (a : Sieve U.left) → Sieve U) S : Sieve U)
⊢ Membership.mem (α := Sieve ((Over.map f).obj U)) (γ := Set (Sieve ((Over.map f).obj U)))
    ((J.over Y : (a : Over Y) → Set (Sieve a)) ((Over.map f).obj U))
    (((Sieve.overEquiv ((Over.map f).obj U)).symm : (a : Sieve ((Over.map f).obj U).left) → Sieve ((Over.map f).obj U))
        S :
      Sieve ((Over.map f).obj U))
have hS₁ : Membership.mem (γ := Set (Sieve.{v, u} ((Over.map f).obj U).left))
  ((J : (a : C) → Set (Sieve a)) ((Over.map f).obj U).left) S
Tactic: apply overEquiv_symm_mem_over
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Over (goal)
Goal:
C : Type u
inst : Category.{v, u} C
J : GrothendieckTopology.{v, u} C
X Y : C
f : X ⟶ Y
F : Sheaf (J.over Y) (Type u_1)
Z : Over.{v, u} X
x_1 : Presieve Z
x : Presieve.FamilyOfElements ((Over.map f).op ⋙ F.obj (P := Presheaf.IsSheaf (J.over Y))) x_1
hx : x.Compatible (R := x_1)
Y₁ Y₂ : Over.{v, u} X
W : Over.{v, u} Y
f₁ : W ⟶ (Over.map f).obj Y₁
f₂ : W ⟶ (Over.map f).obj Y₂
g₁ : Y₁ ⟶ Z
g₂ : Y₂ ⟶ Z
hg₁ : x_1 g₁
hg₂ : x_1 g₂
h : f₁ ≫ (Over.map f).map g₁ = f₂ ≫ (Over.map f).map g₂
W' : Over.{v, u} X := Over.mk (Over.Hom.left f₁ ≫ Y₁.hom)
g₁' : W' ⟶ Y₁ := Over.homMk (Over.Hom.left f₁) ⋯
g₂' : W' ⟶ Y₂ := Over.homMk (Over.Hom.left f₂) ⋯
e : (Over.map f).obj W' ≅ W := Over.isoMk (Iso.refl ((Over.map f).obj W').left) ⋯
⊢ g₁' ≫ g₁ = g₂' ≫ g₂
have h₁ : Over.Hom.left (g₁' ≫ g₁) = Over.Hom.left (g₂' ≫ g₂)
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
X Y : C
f : X ⟶ Y
F : Sheaf (J.over Y) (Type u_1)
Z : Over.{v, u} X
x_1 : Presieve Z
x : Presieve.FamilyOfElements ((Over.map f).op ⋙ F.obj (P := Presheaf.IsSheaf (J.over Y))) x_1
hx : x.Compatible (R := x_1)
Y₁ Y₂ : Over.{v, u} X
W : Over.{v, u} Y
f₁ : W ⟶ (Over.map f).obj Y₁
f₂ : W ⟶ (Over.map f).obj Y₂
g₁ : Y₁ ⟶ Z
g₂ : Y₂ ⟶ Z
hg₁ : x_1 g₁
hg₂ : x_1 g₂
h : f₁ ≫ (Over.map f).map g₁ = f₂ ≫ (Over.map f).map g₂
W' : Over.{v, u} X := Over.mk (Over.Hom.left f₁ ≫ Y₁.hom)
g₁' : W' ⟶ Y₁ := Over.homMk (Over.Hom.left f₁) ⋯
g₂' : W' ⟶ Y₂ := Over.homMk (Over.Hom.left f₂) ⋯
e : (Over.map f).obj W' ≅ W := Over.isoMk (Iso.refl ((Over.map f).obj W').left) ⋯
⊢ Eq (α := (fun (X : Type u_1) ↦ X) (F.obj (P := Presheaf.IsSheaf (J.over Y)).obj (Opposite.op W)))
    ((ConcreteCategory.hom (FC := TypeCat.Fun) (F.obj (P := Presheaf.IsSheaf (J.over Y)).map f₁.op) :
          (a : (fun (X : Type u_1) ↦ X) (F.obj.obj (Opposite.op ((Over.map f).obj Y₁)))) →
            (fun (X : Type u_1) ↦ X) (F.obj.obj (Opposite.op W)))
        (x g₁ hg₁) :
      (fun (X : Type u_1) ↦ X) (F.obj (P := Presheaf.IsSheaf (J.over Y)).obj (Opposite.op W)))
    ((ConcreteCategory.hom (FC := TypeCat.Fun) (F.obj (P := Presheaf.IsSheaf (J.over Y)).map f₂.op) :
          (a : (fun (X : Type u_1) ↦ X) (F.obj.obj (Opposite.op ((Over.map f).obj Y₂)))) →
            (fun (X : Type u_1) ↦ X) (F.obj.obj (Opposite.op W)))
        (x g₂ hg₂) :
      (fun (X : Type u_1) ↦ X) (F.obj (P := Presheaf.IsSheaf (J.over Y)).obj (Opposite.op W)))
have h₁ : Eq (α := (fun (X : Type u_1) ↦ X) (F.obj (P := Presheaf.IsSheaf (J.over Y)).obj (Opposite.op W)))
  ((ConcreteCategory.hom (FC := TypeCat.Fun) (F.obj (P := Presheaf.IsSheaf (J.over Y)).map f₁.op) :
        (a : (fun (X : Type u_1) ↦ X) (F.obj.obj (Opposite.op ((Over.map f).obj Y₁)))) →
          (fun (X : Type u_1) ↦ X) (F.obj.obj (Opposite.op W)))
      (x g₁ hg₁) :
    (fun (X : Type u_1) ↦ X) (F.obj (P := Presheaf.IsSheaf (J.over Y)).obj (Opposite.op W)))
  ((ConcreteCategory.hom (FC := TypeCat.Fun) (F.obj (P := Presheaf.IsSheaf (J.over Y)).map e.inv.op) :
        (a : (fun (X : Type u_1) ↦ X) (F.obj.obj (Opposite.op ((Over.map f).obj W')))) →
          (fun (X : Type u_1) ↦ X) (F.obj.obj (Opposite.op W)))
      ((ConcreteCategory.hom (FC := TypeCat.Fun)
              (((Over.map f).op ⋙ F.obj (P := Presheaf.IsSheaf (J.over Y))).map g₁'.op) :
            (a : (fun (X : Type u_1) ↦ X) (((Over.map f).op ⋙ F.obj).obj (Opposite.op Y₁))) →
              (fun (X : Type u_1) ↦ X) (((Over.map f).op ⋙ F.obj).obj (Opposite.op W')))
          (x g₁ hg₁) :
        (fun (X : Type u_1) ↦ X) (((Over.map f).op ⋙ F.obj (P := Presheaf.IsSheaf (J.over Y))).obj (Opposite.op W'))) :
    (fun (X : Type u_1) ↦ X) (F.obj (P := Presheaf.IsSheaf (J.over Y)).obj (Opposite.op W)))
Tactic: convert!
      congr_arg (F.obj.map e.inv.op)
        (hx g₁' g₂' hg₁ hg₂ (by ext; exact (Over.forget _).congr_map h)) using 1
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Over (goal)
Goal:
C : Type u
inst : Category.{v, u} C
J : GrothendieckTopology.{v, u} C
X Y : C
f : X ⟶ Y
F : Sheaf (J.over Y) (Type u_1)
Z : Over.{v, u} X
x_1 : Presieve Z
x : Presieve.FamilyOfElements ((Over.map f).op ⋙ F.obj (P := Presheaf.IsSheaf (J.over Y))) x_1
hx : x.Compatible (R := x_1)
Y₁ Y₂ : Over.{v, u} X
W : Over.{v, u} Y
f₁ : W ⟶ (Over.map f).obj Y₁
f₂ : W ⟶ (Over.map f).obj Y₂
g₁ : Y₁ ⟶ Z
g₂ : Y₂ ⟶ Z
hg₁ : x_1 g₁
hg₂ : x_1 g₂
h : f₁ ≫ (Over.map f).map g₁ = f₂ ≫ (Over.map f).map g₂
W' : Over.{v, u} X := Over.mk (Over.Hom.left f₁ ≫ Y₁.hom)
g₁' : W' ⟶ Y₁ := Over.homMk (Over.Hom.left f₁) ⋯
g₂' : W' ⟶ Y₂ := Over.homMk (Over.Hom.left f₂) ⋯
e : (Over.map f).obj W' ≅ W := Over.isoMk (Iso.refl ((Over.map f).obj W').left) ⋯
⊢ Eq (α := F.obj (P := Presheaf.IsSheaf (J.over Y)).obj (Opposite.op W))
    ((ConcreteCategory.hom (FC := TypeCat.Fun) (F.obj (P := Presheaf.IsSheaf (J.over Y)).map f₁.op) :
          (a : F.obj.obj (Opposite.op ((Over.map f).obj Y₁))) → F.obj.obj (Opposite.op W))
        (x g₁ hg₁) :
      F.obj (P := Presheaf.IsSheaf (J.over Y)).obj (Opposite.op W))
    ((ConcreteCategory.hom (FC := TypeCat.Fun)
            (F.obj (P := Presheaf.IsSheaf (J.over Y)).map (Over.isoMk (Iso.refl W.left) ⋯).inv.op) :
          (a : F.obj.obj (Opposite.op ((Over.map f).obj (Over.mk (Over.Hom.left f₁ ≫ Y₁.hom))))) →
            F.obj.obj (Opposite.op W))
        ((ConcreteCategory.hom (FC := TypeCat.Fun)
                (F.obj (P := Presheaf.IsSheaf (J.over Y)).map ((Over.map f).map (Over.homMk (Over.Hom.left f₁) ⋯)).op) :
              (a : F.obj.obj (Opposite.op ((Over.map f).obj Y₁))) →
                F.obj.obj (Opposite.op ((Over.map f).obj (Over.mk (Over.Hom.left f₁ ≫ Y₁.hom)))))
            (x g₁ hg₁) :
          F.obj (P := Presheaf.IsSheaf (J.over Y)).obj
            (Opposite.op ((Over.map f).obj (Over.mk (Over.Hom.left f₁ ≫ Y₁.hom))))) :
      F.obj (P := Presheaf.IsSheaf (J.over Y)).obj (Opposite.op W))
have h₁ : Eq (α := F.obj (P := Presheaf.IsSheaf (J.over Y)).obj (Opposite.op W))
  ((ConcreteCategory.hom (FC := TypeCat.Fun) (F.obj (P := Presheaf.IsSheaf (J.over Y)).map f₁.op) :
        (a : F.obj.obj (Opposite.op ((Over.map f).obj Y₁))) → F.obj.obj (Opposite.op W))
      (x g₁ hg₁) :
    F.obj (P := Presheaf.IsSheaf (J.over Y)).obj (Opposite.op W))
  ((ConcreteCategory.hom (FC := TypeCat.Fun)
          (F.obj (P := Presheaf.IsSheaf (J.over Y)).map
            (((Over.map f).map (Over.homMk (Over.Hom.left f₁) sorry)).op ≫ (Over.isoMk (Iso.refl W.left) sorry).inv.op)) :
        (a : F.obj.obj (Opposite.op ((Over.map f).obj Y₁))) → F.obj.obj (Opposite.op W))
      (x g₁ hg₁) :
    F.obj (P := Presheaf.IsSheaf (J.over Y)).obj (Opposite.op W))
Tactic: rw [← Functor.map_comp_apply]
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Over (goal)
Goal:
C : Type u
inst : Category.{v, u} C
J : GrothendieckTopology.{v, u} C
X Y : C
f : X ⟶ Y
F : Sheaf (J.over Y) (Type u_1)
Z : Over.{v, u} X
x_1 : Presieve Z
x : Presieve.FamilyOfElements ((Over.map f).op ⋙ F.obj (P := Presheaf.IsSheaf (J.over Y))) x_1
hx : x.Compatible (R := x_1)
Y₁ Y₂ : Over.{v, u} X
W : Over.{v, u} Y
f₁ : W ⟶ (Over.map f).obj Y₁
f₂ : W ⟶ (Over.map f).obj Y₂
g₁ : Y₁ ⟶ Z
g₂ : Y₂ ⟶ Z
hg₁ : x_1 g₁
hg₂ : x_1 g₂
h : f₁ ≫ (Over.map f).map g₁ = f₂ ≫ (Over.map f).map g₂
W' : Over.{v, u} X := Over.mk (Over.Hom.left f₁ ≫ Y₁.hom)
g₁' : W' ⟶ Y₁ := Over.homMk (Over.Hom.left f₁) ⋯
g₂' : W' ⟶ Y₂ := Over.homMk (Over.Hom.left f₂) ⋯
e : (Over.map f).obj W' ≅ W := Over.isoMk (Iso.refl ((Over.map f).obj W').left) ⋯
⊢ Eq (α := F.obj (P := Presheaf.IsSheaf (J.over Y)).obj (Opposite.op W))
    ((ConcreteCategory.hom (FC := TypeCat.Fun) (F.obj (P := Presheaf.IsSheaf (J.over Y)).map f₁.op) :
          (a : F.obj.obj (Opposite.op ((Over.map f).obj Y₁))) → F.obj.obj (Opposite.op W))
        (x g₁ hg₁) :
      F.obj (P := Presheaf.IsSheaf (J.over Y)).obj (Opposite.op W))
    ((ConcreteCategory.hom (FC := TypeCat.Fun)
            (F.obj (P := Presheaf.IsSheaf (J.over Y)).map
              (((Over.map f).map (Over.homMk (Over.Hom.left f₁) ⋯)).op ≫ (Over.isoMk (Iso.refl W.left) ⋯).inv.op)) :
          (a : F.obj.obj (Opposite.op ((Over.map f).obj Y₁))) → F.obj.obj (Opposite.op W))
        (x g₁ hg₁) :
      F.obj (P := Presheaf.IsSheaf (J.over Y)).obj (Opposite.op W))
have h₁ : F.obj (P := Presheaf.IsSheaf (J.over Y)).map f₁.op =
  F.obj (P := Presheaf.IsSheaf (J.over Y)).map
    (((Over.map f).map (Over.homMk (Over.Hom.left f₁) sorry)).op ≫ (Over.isoMk (Iso.refl W.left) sorry).inv.op)
Tactic: apply ConcreteCategory.congr_hom
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Over (goal)
Goal:
C : Type u
inst : Category.{v, u} C
J : GrothendieckTopology.{v, u} C
X Y : C
f : X ⟶ Y
F : Sheaf (J.over Y) (Type u_1)
Z : Over.{v, u} X
x_1 : Presieve Z
x : Presieve.FamilyOfElements ((Over.map f).op ⋙ F.obj (P := Presheaf.IsSheaf (J.over Y))) x_1
hx : x.Compatible (R := x_1)
Y₁ Y₂ : Over.{v, u} X
W : Over.{v, u} Y
f₁ : W ⟶ (Over.map f).obj Y₁
f₂ : W ⟶ (Over.map f).obj Y₂
g₁ : Y₁ ⟶ Z
g₂ : Y₂ ⟶ Z
hg₁ : x_1 g₁
hg₂ : x_1 g₂
h : f₁ ≫ (Over.map f).map g₁ = f₂ ≫ (Over.map f).map g₂
W' : Over.{v, u} X := Over.mk (Over.Hom.left f₁ ≫ Y₁.hom)
g₁' : W' ⟶ Y₁ := Over.homMk (Over.Hom.left f₁) ⋯
g₂' : W' ⟶ Y₂ := Over.homMk (Over.Hom.left f₂) ⋯
e : (Over.map f).obj W' ≅ W := Over.isoMk (Iso.refl ((Over.map f).obj W').left) ⋯
⊢ F.obj (P := Presheaf.IsSheaf (J.over Y)).map f₁.op =
    F.obj (P := Presheaf.IsSheaf (J.over Y)).map
      (((Over.map f).map (Over.homMk (Over.Hom.left f₁) ⋯)).op ≫ (Over.isoMk (Iso.refl W.left) ⋯).inv.op)
have h₁ : f₁.op = ((Over.map f).map (Over.homMk (Over.Hom.left f₁) sorry)).op ≫ (Over.isoMk (Iso.refl W.left) sorry).inv.op
Tactic: congr 1
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Over (goal)
Goal:
C : Type u
inst : Category.{v, u} C
J : GrothendieckTopology.{v, u} C
X Y : C
f : X ⟶ Y
F : Sheaf (J.over Y) (Type u_1)
Z : Over.{v, u} X
x_1 : Presieve Z
x : Presieve.FamilyOfElements ((Over.map f).op ⋙ F.obj (P := Presheaf.IsSheaf (J.over Y))) x_1
hx : x.Compatible (R := x_1)
Y₁ Y₂ : Over.{v, u} X
W : Over.{v, u} Y
f₁ : W ⟶ (Over.map f).obj Y₁
f₂ : W ⟶ (Over.map f).obj Y₂
g₁ : Y₁ ⟶ Z
g₂ : Y₂ ⟶ Z
hg₁ : x_1 g₁
hg₂ : x_1 g₂
h : f₁ ≫ (Over.map f).map g₁ = f₂ ≫ (Over.map f).map g₂
W' : Over.{v, u} X := Over.mk (Over.Hom.left f₁ ≫ Y₁.hom)
g₁' : W' ⟶ Y₁ := Over.homMk (Over.Hom.left f₁) ⋯
g₂' : W' ⟶ Y₂ := Over.homMk (Over.Hom.left f₂) ⋯
e : (Over.map f).obj W' ≅ W := Over.isoMk (Iso.refl ((Over.map f).obj W').left) ⋯
⊢ f₁.op = ((Over.map f).map (Over.homMk (Over.Hom.left f₁) ⋯)).op ≫ (Over.isoMk (Iso.refl W.left) ⋯).inv.op
have h₁ : f₁.op = Quiver.Hom.op ((Over.isoMk (Iso.refl W.left) sorry).inv ≫ (Over.map f).map (Over.homMk (Over.Hom.left f₁) sorry))
Tactic: rw [← 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
X Y : C
f : X ⟶ Y
F : Sheaf (J.over Y) (Type u_1)
Z : Over.{v, u} X
x_1 : Presieve Z
x : Presieve.FamilyOfElements ((Over.map f).op ⋙ F.obj (P := Presheaf.IsSheaf (J.over Y))) x_1
hx : x.Compatible (R := x_1)
Y₁ Y₂ : Over.{v, u} X
W : Over.{v, u} Y
f₁ : W ⟶ (Over.map f).obj Y₁
f₂ : W ⟶ (Over.map f).obj Y₂
g₁ : Y₁ ⟶ Z
g₂ : Y₂ ⟶ Z
hg₁ : x_1 g₁
hg₂ : x_1 g₂
h : f₁ ≫ (Over.map f).map g₁ = f₂ ≫ (Over.map f).map g₂
W' : Over.{v, u} X := Over.mk (Over.Hom.left f₁ ≫ Y₁.hom)
g₁' : W' ⟶ Y₁ := Over.homMk (Over.Hom.left f₁) ⋯
g₂' : W' ⟶ Y₂ := Over.homMk (Over.Hom.left f₂) ⋯
e : (Over.map f).obj W' ≅ W := Over.isoMk (Iso.refl ((Over.map f).obj W').left) ⋯
⊢ f₁.op = Quiver.Hom.op ((Over.isoMk (Iso.refl W.left) ⋯).inv ≫ (Over.map f).map (Over.homMk (Over.Hom.left f₁) ⋯))
have h₁ : f₁ = (Over.isoMk (Iso.refl W.left) sorry).inv ≫ (Over.map f).map (Over.homMk (Over.Hom.left f₁) sorry)
Tactic: congr 1
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Over (goal)
Goal:
C : Type u
inst : Category.{v, u} C
J : GrothendieckTopology.{v, u} C
X Y : C
f : X ⟶ Y
F : Sheaf (J.over Y) (Type u_1)
Z : Over.{v, u} X
x_1 : Presieve Z
x : Presieve.FamilyOfElements ((Over.map f).op ⋙ F.obj (P := Presheaf.IsSheaf (J.over Y))) x_1
hx : x.Compatible (R := x_1)
Y₁ Y₂ : Over.{v, u} X
W : Over.{v, u} Y
f₁ : W ⟶ (Over.map f).obj Y₁
f₂ : W ⟶ (Over.map f).obj Y₂
g₁ : Y₁ ⟶ Z
g₂ : Y₂ ⟶ Z
hg₁ : x_1 g₁
hg₂ : x_1 g₂
h : f₁ ≫ (Over.map f).map g₁ = f₂ ≫ (Over.map f).map g₂
W' : Over.{v, u} X := Over.mk (Over.Hom.left f₁ ≫ Y₁.hom)
g₁' : W' ⟶ Y₁ := Over.homMk (Over.Hom.left f₁) ⋯
g₂' : W' ⟶ Y₂ := Over.homMk (Over.Hom.left f₂) ⋯
e : (Over.map f).obj W' ≅ W := Over.isoMk (Iso.refl ((Over.map f).obj W').left) ⋯
⊢ f₁ = (Over.isoMk (Iso.refl W.left) ⋯).inv ≫ (Over.map f).map (Over.homMk (Over.Hom.left f₁) ⋯)
have h₁ : Over.Hom.left f₁ =
  Over.Hom.left ((Over.isoMk (Iso.refl W.left) sorry).inv ≫ (Over.map f).map (Over.homMk (Over.Hom.left f₁) sorry))
Tactic: ext