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 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 f₂.op) :
          (a : F.obj.obj (Opposite.op ((Over.map f).obj Y₂))) → F.obj.obj (Opposite.op W))
        (x g₂ hg₂) :
      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 f₂) ⋯)).op) :
              (a : F.obj.obj (Opposite.op ((Over.map f).obj Y₂))) →
                F.obj.obj (Opposite.op ((Over.map f).obj (Over.mk (Over.Hom.left f₁ ≫ Y₁.hom)))))
            (x g₂ hg₂) :
          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 f₂.op) :
        (a : F.obj.obj (Opposite.op ((Over.map f).obj Y₂))) → F.obj.obj (Opposite.op W))
      (x g₂ hg₂) :
    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 f₂) sorry)).op ≫ (Over.isoMk (Iso.refl W.left) sorry).inv.op)) :
        (a : F.obj.obj (Opposite.op ((Over.map f).obj Y₂))) → F.obj.obj (Opposite.op W))
      (x g₂ hg₂) :
    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 f₂.op) :
          (a : F.obj.obj (Opposite.op ((Over.map f).obj Y₂))) → F.obj.obj (Opposite.op W))
        (x g₂ hg₂) :
      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 f₂) ⋯)).op ≫ (Over.isoMk (Iso.refl W.left) ⋯).inv.op)) :
          (a : F.obj.obj (Opposite.op ((Over.map f).obj Y₂))) → F.obj.obj (Opposite.op W))
        (x g₂ hg₂) :
      F.obj (P := Presheaf.IsSheaf (J.over Y)).obj (Opposite.op W))
have h₁ : F.obj (P := Presheaf.IsSheaf (J.over Y)).map f₂.op =
  F.obj (P := Presheaf.IsSheaf (J.over Y)).map
    (((Over.map f).map (Over.homMk (Over.Hom.left f₂) 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 f₂.op =
    F.obj (P := Presheaf.IsSheaf (J.over Y)).map
      (((Over.map f).map (Over.homMk (Over.Hom.left f₂) ⋯)).op ≫ (Over.isoMk (Iso.refl W.left) ⋯).inv.op)
have h₁ : f₂.op = ((Over.map f).map (Over.homMk (Over.Hom.left f₂) 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) ⋯
⊢ f₂.op = ((Over.map f).map (Over.homMk (Over.Hom.left f₂) ⋯)).op ≫ (Over.isoMk (Iso.refl W.left) ⋯).inv.op
have h₁ : f₂.op = Quiver.Hom.op ((Over.isoMk (Iso.refl W.left) sorry).inv ≫ (Over.map f).map (Over.homMk (Over.Hom.left f₂) 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) ⋯
⊢ f₂.op = Quiver.Hom.op ((Over.isoMk (Iso.refl W.left) ⋯).inv ≫ (Over.map f).map (Over.homMk (Over.Hom.left f₂) ⋯))
have h₁ : f₂ = (Over.isoMk (Iso.refl W.left) sorry).inv ≫ (Over.map f).map (Over.homMk (Over.Hom.left f₂) 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) ⋯
⊢ f₂ = (Over.isoMk (Iso.refl W.left) ⋯).inv ≫ (Over.map f).map (Over.homMk (Over.Hom.left f₂) ⋯)
have h₁ : Over.Hom.left f₂ =
  Over.Hom.left ((Over.isoMk (Iso.refl W.left) sorry).inv ≫ (Over.map f).map (Over.homMk (Over.Hom.left f₂) sorry))
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 ((Over.map f).obj U)
hS :
  Membership.mem (γ := Set (Sieve ((Over.map f).obj U)))
    ((J.over Y : (a : Over Y) → Set (Sieve a)) ((Over.map f).obj U)) S
⊢ Sieve.functorPullback (Over.map f) S ∈ (J.over X : (a : Over X) → Set (Sieve a)) U
have rw : ∀
  (hS :
    Membership.mem (α := Sieve.{v, u} ((Over.map f).obj U).left) (γ := Set (Sieve.{v, u} ((Over.map f).obj U).left))
      ((J : (a : C) → Set (Sieve a)) ((Over.map f).obj U).left)
      ((Sieve.overEquiv ((Over.map f).obj U) : (a : Sieve ((Over.map f).obj U)) → Sieve ((Over.map f).obj U).left) S :
        Sieve.{v, u} ((Over.map f).obj U).left)),
  Membership.mem (α := Sieve.{v, u} U.left) (γ := Set (Sieve.{v, u} U.left)) ((J : (a : C) → Set (Sieve a)) U.left)
    ((Sieve.overEquiv U : (a : Sieve U) → Sieve U.left) (Sieve.functorPullback (Over.map f) S) : Sieve.{v, u} U.left)
Tactic: rw [J.mem_over_iff] at hS ⊢
Cleared hypotheses: 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 ((Over.map f).obj U)
hS :
  Membership.mem (α := Sieve.{v, u} ((Over.map f).obj U).left) (γ := Set (Sieve.{v, u} ((Over.map f).obj U).left))
    ((J : (a : C) → Set (Sieve a)) ((Over.map f).obj U).left)
    ((Sieve.overEquiv ((Over.map f).obj U) : (a : Sieve ((Over.map f).obj U)) → Sieve ((Over.map f).obj U).left) S :
      Sieve.{v, u} ((Over.map f).obj U).left)
⊢ Membership.mem (α := Sieve.{v, u} U.left) (γ := Set (Sieve.{v, u} U.left)) ((J : (a : C) → Set (Sieve a)) U.left)
    ((Sieve.overEquiv U : (a : Sieve U) → Sieve U.left) (Sieve.functorPullback (Over.map f) S) : Sieve.{v, u} U.left)
have rwa : Membership.mem (α := Sieve.{v, u} U.left) (γ := Set (Sieve.{v, u} U.left)) ((J : (a : C) → Set (Sieve a)) U.left)
  ((Sieve.overEquiv ((Over.map f).obj U) : (a : Sieve ((Over.map f).obj U)) → Sieve ((Over.map f).obj U).left) S :
    Sieve.{v, u} ((Over.map f).obj U).left)
Tactic: rwa [Sieve.overEquiv_functorPullback_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
D : Type u_1
inst_1 : Category.{v_1, u_1} D
K : GrothendieckTopology.{v_1, u_1} D
F : Functor.{v, v_1, u, u_1} C D
X : C
inst_2 : F.IsCocontinuous J K
U : Over.{v, u} X
S : Sieve (Functor.obj (Over.post F) U)
hS :
  Membership.mem (γ := Set (Sieve (Functor.obj (Over.post F) U)))
    ((K.over (F.obj X) : (a : Over (F.obj X)) → Set (Sieve a)) (Functor.obj (Over.post F) U)) S
⊢ Sieve.functorPullback (Over.post F) S ∈ (J.over X : (a : Over X) → Set (Sieve a)) U
have rw : ∀
  (hS :
    Membership.mem (α := Sieve.{v_1, u_1} (Functor.obj (Over.post F) U).left) (γ :=
      Set (Sieve.{v_1, u_1} (Functor.obj (Over.post F) U).left))
      ((K : (a : D) → Set (Sieve a)) (Functor.obj (Over.post F) U).left)
      ((Sieve.overEquiv (Functor.obj (Over.post F) U) :
            (a : Sieve ((Over.post F).obj U)) → Sieve ((Over.post F).obj U).left)
          S :
        Sieve.{v_1, u_1} (Functor.obj (Over.post F) U).left)),
  Membership.mem (α := Sieve.{v, u} U.left) (γ := Set (Sieve.{v, u} U.left)) ((J : (a : C) → Set (Sieve a)) U.left)
    ((Sieve.overEquiv U : (a : Sieve U) → Sieve U.left) (Sieve.functorPullback (Over.post F) S) : Sieve.{v, u} U.left)
Tactic: rw [GrothendieckTopology.mem_over_iff] at hS ⊢
Cleared hypotheses: 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
D : Type u_1
inst_1 : Category.{v_1, u_1} D
K : GrothendieckTopology.{v_1, u_1} D
F : Functor.{v, v_1, u, u_1} C D
X : C
inst_2 : F.IsCocontinuous J K
U : Over.{v, u} X
S : Sieve (Functor.obj (Over.post F) U)
hS :
  Membership.mem (α := Sieve.{v_1, u_1} (Functor.obj (Over.post F) U).left) (γ :=
    Set (Sieve.{v_1, u_1} (Functor.obj (Over.post F) U).left))
    ((K : (a : D) → Set (Sieve a)) (Functor.obj (Over.post F) U).left)
    ((Sieve.overEquiv (Functor.obj (Over.post F) U) :
          (a : Sieve ((Over.post F).obj U)) → Sieve ((Over.post F).obj U).left)
        S :
      Sieve.{v_1, u_1} (Functor.obj (Over.post F) U).left)
⊢ Membership.mem (α := Sieve.{v, u} U.left) (γ := Set (Sieve.{v, u} U.left)) ((J : (a : C) → Set (Sieve a)) U.left)
    ((Sieve.overEquiv U : (a : Sieve U) → Sieve U.left) (Sieve.functorPullback (Over.post F) S) : Sieve.{v, u} U.left)
have rw : Membership.mem (γ := Set (Sieve.{v, u} U.left)) ((J : (a : C) → Set (Sieve a)) U.left)
  (Sieve.functorPullback F
    ((Sieve.overEquiv (Functor.obj (Over.post F) U) :
        (a : Sieve ((Over.post F).obj U)) → Sieve ((Over.post F).obj U).left)
      S))
Tactic: rw [Sieve.overEquiv_functorPullback_post]
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Over (goal)
Goal:
C : Type u
inst : Category.{v, u} C
J : GrothendieckTopology.{v, u} C
D : Type u_1
inst_1 : Category.{v_1, u_1} D
K : GrothendieckTopology.{v_1, u_1} D
F : Functor.{v, v_1, u, u_1} C D
X : C
h : CoverPreserving J K F
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 (Functor.obj (Over.post F) U)))
    ((K.over (F.obj X) : (a : Over (F.obj X)) → Set (Sieve a)) (Functor.obj (Over.post F) U))
    (Sieve.functorPushforward (Over.post F) S)
have rw : ∀
  (hS :
    Membership.mem (α := Sieve.{v, u} U.left) (γ := Set (Sieve.{v, u} U.left)) ((J : (a : C) → Set (Sieve a)) U.left)
      ((Sieve.overEquiv U : (a : Sieve U) → Sieve U.left) S : Sieve.{v, u} U.left)),
  Membership.mem (α := Sieve.{v_1, u_1} (Functor.obj (Over.post F) U).left) (γ :=
    Set (Sieve.{v_1, u_1} (Functor.obj (Over.post F) U).left))
    ((K : (a : D) → Set (Sieve a)) (Functor.obj (Over.post F) U).left)
    ((Sieve.overEquiv (Functor.obj (Over.post F) U) :
          (a : Sieve ((Over.post F).obj U)) → Sieve ((Over.post F).obj U).left)
        (Sieve.functorPushforward (Over.post F) S) :
      Sieve.{v_1, u_1} (Functor.obj (Over.post F) U).left)
Tactic: rw [GrothendieckTopology.mem_over_iff] at hS ⊢
Cleared hypotheses: 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
D : Type u_1
inst_1 : Category.{v_1, u_1} D
K : GrothendieckTopology.{v_1, u_1} D
F : Functor.{v, v_1, u, u_1} C D
X : C
h : CoverPreserving J K F
U : Over.{v, u} X
S : Sieve U
hS :
  Membership.mem (α := Sieve.{v, u} U.left) (γ := Set (Sieve.{v, u} U.left)) ((J : (a : C) → Set (Sieve a)) U.left)
    ((Sieve.overEquiv U : (a : Sieve U) → Sieve U.left) S : Sieve.{v, u} U.left)
⊢ Membership.mem (α := Sieve.{v_1, u_1} (Functor.obj (Over.post F) U).left) (γ :=
    Set (Sieve.{v_1, u_1} (Functor.obj (Over.post F) U).left))
    ((K : (a : D) → Set (Sieve a)) (Functor.obj (Over.post F) U).left)
    ((Sieve.overEquiv (Functor.obj (Over.post F) U) :
          (a : Sieve ((Over.post F).obj U)) → Sieve ((Over.post F).obj U).left)
        (Sieve.functorPushforward (Over.post F) S) :
      Sieve.{v_1, u_1} (Functor.obj (Over.post F) U).left)
have rw : Membership.mem (γ := Set (Sieve.{v_1, u_1} (Functor.obj (Over.post F) U).left))
  ((K : (a : D) → Set (Sieve a)) (Functor.obj (Over.post F) U).left)
  (Sieve.functorPushforward F ((Sieve.overEquiv U : (a : Sieve U) → Sieve U.left) S))
Tactic: rw [Sieve.overEquiv_functorPushforward_post]
-- 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
⊢ E.IsPreservedBy (Over.forget X) J
have refine_1 : Membership.mem (γ := Set (Sieve.{v, u} (Functor.obj (Over.forget X) Y)))
  ((J : (a : C) → Set (Sieve a)) (Functor.obj (Over.forget X) Y))
  (PreZeroHypercover.sieve₀.{u_1, v, u} (E.map (Over.forget X)).toPreZeroHypercover)
Tactic: refine ⟨?_, ?_⟩
-- 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
refine_1 :
  Membership.mem (γ := Set (Sieve.{v, u} (Functor.obj (Over.forget X) Y)))
    ((J : (a : C) → Set (Sieve a)) (Functor.obj (Over.forget X) Y))
    (PreZeroHypercover.sieve₀.{u_1, v, u} (E.map (Over.forget X)).toPreZeroHypercover)
⊢ E.IsPreservedBy (Over.forget X) J
have refine_2 : ∀ (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)),
  (E.map (Over.forget X)).sieve₁ p₁ p₂ ∈ (J : (a : C) → 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 J : GrothendieckTopology.{v, u} C
X : C
Y : Over.{v, u} X
E : (J.over X).OneHypercover Y
⊢ Membership.mem (γ := Set (Sieve.{v, u} (Functor.obj (Over.forget X) Y)))
    ((J : (a : C) → Set (Sieve a)) (Functor.obj (Over.forget X) Y))
    (PreZeroHypercover.sieve₀.{u_1, v, u} (E.map (Over.forget X)).toPreZeroHypercover)
have refine_1 : Membership.mem (γ := Set (Sieve.{v, u} Y.left)) ((J : (a : C) → Set (Sieve a)) Y.left)
  (PreZeroHypercover.sieve₀.{u_1, v, u} (PreZeroHypercover.map (Over.forget X) E.toPreZeroHypercover))
Tactic: dsimp
-- 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
⊢ Membership.mem (γ := Set (Sieve.{v, u} Y.left)) ((J : (a : C) → Set (Sieve a)) Y.left)
    (PreZeroHypercover.sieve₀.{u_1, v, u} (PreZeroHypercover.map (Over.forget X) E.toPreZeroHypercover))
have refine_1 : Membership.mem (γ := Set (Sieve.{v, u} Y.left)) ((J : (a : C) → Set (Sieve a)) Y.left)
  (Sieve.functorPushforward (Over.forget X) E.sieve₀)
Tactic: rw [dsimp% PreZeroHypercover.sieve₀_map (F := Over.forget X)]
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Over (hypothesis)
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)
⊢ (E.map (Over.forget X)).sieve₁ p₁ p₂ ∈ (J : (a : C) → Set (Sieve a)) W
have this : (p₁ ≫ Functor.map (Over.forget X) (E.f i₁)) ≫ Y.hom = (p₂ ≫ Functor.map (Over.forget X) (E.f i₂)) ≫ Y.hom
Tactic: have := w =≫ Over.hom _
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Over (hypothesis)
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 : (p₁ ≫ Functor.map (Over.forget X) (E.f i₁)) ≫ Y.hom = (p₂ ≫ Functor.map (Over.forget X) (E.f i₂)) ≫ Y.hom
⊢ (E.map (Over.forget X)).sieve₁ p₁ p₂ ∈ (J : (a : C) → Set (Sieve a)) W
have this₁ : p₁ ≫ (E.X i₁).hom = p₂ ≫ (E.X i₂).hom
Tactic: simp only [Over.forget_obj, Over.forget_map, Category.assoc, Over.w] 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 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 : p₁ ≫ (E.X i₁).hom = p₂ ≫ (E.X i₂).hom
⊢ Eq (α := Over.mk (p₁ ≫ (E.X i₁).hom) ⟶ Y) (Over.homMk p₁ ⋯ ≫ E.f i₁) (Over.homMk p₂ ⋯ ≫ E.f i₂)
have h : Eq (α := (Over.mk (p₁ ≫ (E.X i₁).hom)).left ⟶ Y.left) (Over.Hom.left (Over.homMk p₁ sorry ≫ E.f i₁))
  (Over.Hom.left (Over.homMk p₂ sorry ≫ E.f i₂))
Tactic: ext
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Over (hypothesis)
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 : p₁ ≫ (E.X i₁).hom = p₂ ≫ (E.X i₂).hom
⊢ (E.map (Over.forget X)).sieve₁ p₁ p₂ ∈ (J : (a : C) → Set (Sieve a)) W
have this₁ : 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 sorry) (Over.homMk p₂ sorry))
Tactic: have := E.mem₁ i₁ i₂ (Over.homMk (U := Over.mk (p₁ ≫ Over.hom _)) p₁)
      (Over.homMk (U := Over.mk (p₁ ≫ Over.hom _)) p₂ this.symm) (by ext; simpa)
-- 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 ⋯))
⊢ (E.map (Over.forget X)).sieve₁ p₁ p₂ ∈ (J : (a : C) → Set (Sieve a)) W
have 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 sorry) (Over.homMk p₂ sorry)) :
        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
Tactic: rwa [GrothendieckTopology.mem_over_iff, Sieve.overEquiv_preOneHypercover_sieve₁] at this
Cleared hypotheses: this