awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.DenseSubsite.OneHypercoverDense (goal)
Goal:
C₀ : Type u₀
C : Type u
inst : Category.{v₀, u₀} C₀
inst_1 : Category.{v, u} C
F : Functor.{v₀, v, u₀, u} C₀ C
J₀ : GrothendieckTopology.{v₀, u₀} C₀
J : GrothendieckTopology.{v, u} C
A : Type u'
inst_2 : Category.{v', u'} A
inst_3 : IsDenseSubsite J₀ J F
data : (X : C) → OneHypercoverDenseData.{w, v₀, v, u₀, u} F J₀ J X
inst_4 : HasLimitsOfSize.{w, w, v', u'} A
G₀ : Sheaf.{v₀, v', u₀, u'} J₀ A
X₀ Y₀ : C₀
f : F.obj Y₀ ⟶ F.obj X₀
W₀ : C₀
a : W₀ ⟶ Y₀
b : W₀ ⟶ X₀
fac₁ : F.map b = F.map a ≫ f
T₀ : C₀
c : T₀ ⟶ W₀
Y : C
i : PreOneHypercoverDenseData.I₀ (S := F.obj X₀) (toPreOneHypercoverDenseData (data (F.obj X₀)))
d : F.obj T₀ ⟶ F.obj (PreOneHypercoverDenseData.X (toPreOneHypercoverDenseData (data (F.obj X₀))) i)
fac₂ : d ≫ PreOneHypercoverDenseData.f (toPreOneHypercoverDenseData (data (F.obj X₀))) i = F.map (c ≫ b)
rw :
  CategoryStruct.comp (Y := presheafObj data G₀ (F.obj X₀))
      (inv data G₀ X₀ : G₀.obj (P := Presheaf.IsSheaf J₀).obj (op X₀) ⟶ (presheaf data G₀).obj (op (F.obj X₀)))
      (presheafObjπ data G₀ (F.obj X₀) i ≫ IsDenseSubsite.mapPreimage J F G₀ d) =
    IsDenseSubsite.mapPreimage J F G₀ f ≫ G₀.obj (P := Presheaf.IsSheaf J₀).map (c ≫ a).op
rw₁ :
  IsDenseSubsite.mapPreimage J F G₀ (PreOneHypercoverDenseData.f (toPreOneHypercoverDenseData (data (F.obj X₀))) i) ≫
      IsDenseSubsite.mapPreimage J F G₀ d =
    IsDenseSubsite.mapPreimage J F G₀ f ≫ G₀.obj (P := Presheaf.IsSheaf J₀).map (c ≫ a).op
rw₂ :
  IsDenseSubsite.mapPreimage J F G₀
      (d ≫ PreOneHypercoverDenseData.f (toPreOneHypercoverDenseData (data (F.obj X₀))) i) =
    IsDenseSubsite.mapPreimage J F G₀ f ≫ G₀.obj (P := Presheaf.IsSheaf J₀).map (c ≫ a).op
rw₃ :
  IsDenseSubsite.mapPreimage J F G₀ (F.map (c ≫ b)) =
    IsDenseSubsite.mapPreimage J F G₀ f ≫ G₀.obj (P := Presheaf.IsSheaf J₀).map (c ≫ a).op
rw₄ : IsDenseSubsite.mapPreimage J F G₀ (F.map (c ≫ b)) = IsDenseSubsite.mapPreimage J F G₀ (F.map (c ≫ a) ≫ f)
⊢ CategoryStruct.comp (Y := presheafObj data G₀ (F.obj X₀))
      (inv data G₀ X₀ : G₀.obj (P := Presheaf.IsSheaf J₀).obj (op X₀) ⟶ (presheaf data G₀).obj (op (F.obj X₀)))
      (restriction data G₀ f ≫ G₀.obj (P := Presheaf.IsSheaf J₀).map (c ≫ a).op) =
    IsDenseSubsite.mapPreimage J F G₀ f ≫ G₀.obj (P := Presheaf.IsSheaf J₀).map (c ≫ a).op
have rw₅ : IsDenseSubsite.mapPreimage J F G₀ (F.map c ≫ F.map b) = IsDenseSubsite.mapPreimage J F G₀ (F.map (c ≫ a) ≫ f)
Tactic: rw [restriction_map data G₀ f (c ≫ a) d
    (by rw [fac₂, map_comp, map_comp_assoc, fac₁]), inv_π_assoc,
    ← IsDenseSubsite.mapPreimage_comp, fac₂,
    IsDenseSubsite.mapPreimage_comp_map J F G₀, map_comp,
      map_comp_assoc, fac₁]
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.DenseSubsite.OneHypercoverDense (goal)
Goal:
C₀ : Type u₀
C : Type u
inst : Category.{v₀, u₀} C₀
inst_1 : Category.{v, u} C
F : Functor.{v₀, v, u₀, u} C₀ C
J₀ : GrothendieckTopology.{v₀, u₀} C₀
J : GrothendieckTopology.{v, u} C
A : Type u'
inst_2 : Category.{v', u'} A
inst_3 : IsDenseSubsite J₀ J F
data : (X : C) → OneHypercoverDenseData.{w, v₀, v, u₀, u} F J₀ J X
inst_4 : HasLimitsOfSize.{w, w, v', u'} A
G₀ : Sheaf.{v₀, v', u₀, u'} J₀ A
X₀ Y₀ : C₀
f : F.obj Y₀ ⟶ F.obj X₀
W₀ : C₀
a : W₀ ⟶ Y₀
b : W₀ ⟶ X₀
fac₁ : F.map b = F.map a ≫ f
T₀ : C₀
c : T₀ ⟶ W₀
Y : C
i : PreOneHypercoverDenseData.I₀ (S := F.obj X₀) (toPreOneHypercoverDenseData (data (F.obj X₀)))
d : F.obj T₀ ⟶ F.obj (PreOneHypercoverDenseData.X (toPreOneHypercoverDenseData (data (F.obj X₀))) i)
fac₂ : d ≫ PreOneHypercoverDenseData.f (toPreOneHypercoverDenseData (data (F.obj X₀))) i = F.map (c ≫ b)
rw :
  CategoryStruct.comp (Y := presheafObj data G₀ (F.obj X₀))
      (inv data G₀ X₀ : G₀.obj (P := Presheaf.IsSheaf J₀).obj (op X₀) ⟶ (presheaf data G₀).obj (op (F.obj X₀)))
      (presheafObjπ data G₀ (F.obj X₀) i ≫ IsDenseSubsite.mapPreimage J F G₀ d) =
    IsDenseSubsite.mapPreimage J F G₀ f ≫ G₀.obj (P := Presheaf.IsSheaf J₀).map (c ≫ a).op
rw₁ :
  IsDenseSubsite.mapPreimage J F G₀ (PreOneHypercoverDenseData.f (toPreOneHypercoverDenseData (data (F.obj X₀))) i) ≫
      IsDenseSubsite.mapPreimage J F G₀ d =
    IsDenseSubsite.mapPreimage J F G₀ f ≫ G₀.obj (P := Presheaf.IsSheaf J₀).map (c ≫ a).op
rw₂ :
  IsDenseSubsite.mapPreimage J F G₀
      (d ≫ PreOneHypercoverDenseData.f (toPreOneHypercoverDenseData (data (F.obj X₀))) i) =
    IsDenseSubsite.mapPreimage J F G₀ f ≫ G₀.obj (P := Presheaf.IsSheaf J₀).map (c ≫ a).op
rw₃ :
  IsDenseSubsite.mapPreimage J F G₀ (F.map (c ≫ b)) =
    IsDenseSubsite.mapPreimage J F G₀ f ≫ G₀.obj (P := Presheaf.IsSheaf J₀).map (c ≫ a).op
rw₄ : IsDenseSubsite.mapPreimage J F G₀ (F.map (c ≫ b)) = IsDenseSubsite.mapPreimage J F G₀ (F.map (c ≫ a) ≫ f)
rw₅ : IsDenseSubsite.mapPreimage J F G₀ (F.map c ≫ F.map b) = IsDenseSubsite.mapPreimage J F G₀ (F.map (c ≫ a) ≫ f)
⊢ CategoryStruct.comp (Y := presheafObj data G₀ (F.obj X₀))
      (inv data G₀ X₀ : G₀.obj (P := Presheaf.IsSheaf J₀).obj (op X₀) ⟶ (presheaf data G₀).obj (op (F.obj X₀)))
      (restriction data G₀ f ≫ G₀.obj (P := Presheaf.IsSheaf J₀).map (c ≫ a).op) =
    IsDenseSubsite.mapPreimage J F G₀ f ≫ G₀.obj (P := Presheaf.IsSheaf J₀).map (c ≫ a).op
have rw₆ : IsDenseSubsite.mapPreimage J F G₀ (F.map c ≫ F.map b) = IsDenseSubsite.mapPreimage J F G₀ (F.map c ≫ F.map a ≫ f)
Tactic: rw [restriction_map data G₀ f (c ≫ a) d
    (by rw [fac₂, map_comp, map_comp_assoc, fac₁]), inv_π_assoc,
    ← IsDenseSubsite.mapPreimage_comp, fac₂,
    IsDenseSubsite.mapPreimage_comp_map J F G₀, map_comp,
      map_comp_assoc, fac₁]
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.DenseSubsite.OneHypercoverDense (goal)
Goal:
C₀ : Type u₀
C : Type u
inst : Category.{v₀, u₀} C₀
inst_1 : Category.{v, u} C
F : Functor.{v₀, v, u₀, u} C₀ C
J₀ : GrothendieckTopology.{v₀, u₀} C₀
J : GrothendieckTopology.{v, u} C
A : Type u'
inst_2 : Category.{v', u'} A
inst_3 : IsDenseSubsite J₀ J F
data : (X : C) → OneHypercoverDenseData.{w, v₀, v, u₀, u} F J₀ J X
inst_4 : HasLimitsOfSize.{w, w, v', u'} A
G₀ : Sheaf.{v₀, v', u₀, u'} J₀ A
X₀ Y₀ : C₀
f : F.obj Y₀ ⟶ F.obj X₀
W₀ : C₀
a : W₀ ⟶ Y₀
b : W₀ ⟶ X₀
fac₁ : F.map b = F.map a ≫ f
T₀ : C₀
c : T₀ ⟶ W₀
Y : C
i : PreOneHypercoverDenseData.I₀ (S := F.obj X₀) (toPreOneHypercoverDenseData (data (F.obj X₀)))
d : F.obj T₀ ⟶ F.obj (PreOneHypercoverDenseData.X (toPreOneHypercoverDenseData (data (F.obj X₀))) i)
fac₂ : d ≫ PreOneHypercoverDenseData.f (toPreOneHypercoverDenseData (data (F.obj X₀))) i = F.map (c ≫ b)
rw :
  CategoryStruct.comp (Y := presheafObj data G₀ (F.obj X₀))
      (inv data G₀ X₀ : G₀.obj (P := Presheaf.IsSheaf J₀).obj (op X₀) ⟶ (presheaf data G₀).obj (op (F.obj X₀)))
      (presheafObjπ data G₀ (F.obj X₀) i ≫ IsDenseSubsite.mapPreimage J F G₀ d) =
    IsDenseSubsite.mapPreimage J F G₀ f ≫ G₀.obj (P := Presheaf.IsSheaf J₀).map (c ≫ a).op
rw₁ :
  IsDenseSubsite.mapPreimage J F G₀ (PreOneHypercoverDenseData.f (toPreOneHypercoverDenseData (data (F.obj X₀))) i) ≫
      IsDenseSubsite.mapPreimage J F G₀ d =
    IsDenseSubsite.mapPreimage J F G₀ f ≫ G₀.obj (P := Presheaf.IsSheaf J₀).map (c ≫ a).op
rw₂ :
  IsDenseSubsite.mapPreimage J F G₀
      (d ≫ PreOneHypercoverDenseData.f (toPreOneHypercoverDenseData (data (F.obj X₀))) i) =
    IsDenseSubsite.mapPreimage J F G₀ f ≫ G₀.obj (P := Presheaf.IsSheaf J₀).map (c ≫ a).op
rw₃ :
  IsDenseSubsite.mapPreimage J F G₀ (F.map (c ≫ b)) =
    IsDenseSubsite.mapPreimage J F G₀ f ≫ G₀.obj (P := Presheaf.IsSheaf J₀).map (c ≫ a).op
rw₄ : IsDenseSubsite.mapPreimage J F G₀ (F.map (c ≫ b)) = IsDenseSubsite.mapPreimage J F G₀ (F.map (c ≫ a) ≫ f)
rw₅ : IsDenseSubsite.mapPreimage J F G₀ (F.map c ≫ F.map b) = IsDenseSubsite.mapPreimage J F G₀ (F.map (c ≫ a) ≫ f)
rw₆ : IsDenseSubsite.mapPreimage J F G₀ (F.map c ≫ F.map b) = IsDenseSubsite.mapPreimage J F G₀ (F.map c ≫ F.map a ≫ f)
⊢ CategoryStruct.comp (Y := presheafObj data G₀ (F.obj X₀))
      (inv data G₀ X₀ : G₀.obj (P := Presheaf.IsSheaf J₀).obj (op X₀) ⟶ (presheaf data G₀).obj (op (F.obj X₀)))
      (restriction data G₀ f ≫ G₀.obj (P := Presheaf.IsSheaf J₀).map (c ≫ a).op) =
    IsDenseSubsite.mapPreimage J F G₀ f ≫ G₀.obj (P := Presheaf.IsSheaf J₀).map (c ≫ a).op
have rw₇ : IsDenseSubsite.mapPreimage J F G₀ (F.map c ≫ F.map a ≫ f) = IsDenseSubsite.mapPreimage J F G₀ (F.map c ≫ F.map a ≫ f)
Tactic: rw [restriction_map data G₀ f (c ≫ a) d
    (by rw [fac₂, map_comp, map_comp_assoc, fac₁]), inv_π_assoc,
    ← IsDenseSubsite.mapPreimage_comp, fac₂,
    IsDenseSubsite.mapPreimage_comp_map J F G₀, map_comp,
      map_comp_assoc, fac₁]
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.DenseSubsite.OneHypercoverDense (goal)
Goal:
C₀ : Type u₀
C : Type u
inst : Category.{v₀, u₀} C₀
inst_1 : Category.{v, u} C
F : Functor.{v₀, v, u₀, u} C₀ C
J₀ : GrothendieckTopology.{v₀, u₀} C₀
J : GrothendieckTopology.{v, u} C
A : Type u'
inst_2 : Category.{v', u'} A
inst_3 : IsDenseSubsite J₀ J F
data : (X : C) → OneHypercoverDenseData.{w, v₀, v, u₀, u} F J₀ J X
inst_4 : HasLimitsOfSize.{w, w, v', u'} A
G₀ : Sheaf.{v₀, v', u₀, u'} J₀ A
X₀ : C₀
i : PreOneHypercoverDenseData.I₀ (S := unop (op (F.obj X₀))) (toPreOneHypercoverDenseData (data (unop (op (F.obj X₀)))))
⊢ CategoryStruct.comp (Y := presheafObj data G₀ (unop (op (F.obj X₀))))
      (presheafObjObjIso.hom data G₀ X₀ ≫ presheafObjObjIso.inv data G₀ X₀ :
        (presheaf data G₀).obj (op (F.obj X₀)) ⟶ (presheaf data G₀).obj (op (F.obj X₀)))
      (presheafObjπ data G₀ (unop (op (F.obj X₀))) i) =
    𝟙 ((presheaf data G₀).obj (op (F.obj X₀))) ≫ presheafObjπ data G₀ (unop (op (F.obj X₀))) i
have rw : presheafObjObjIso.hom data G₀ X₀ ≫
    CategoryStruct.comp (Y := presheafObj data G₀ (unop (op (F.obj X₀))))
      (presheafObjObjIso.inv data G₀ X₀ :
        G₀.obj (P := Presheaf.IsSheaf J₀).obj (op X₀) ⟶ (presheaf data G₀).obj (op (F.obj X₀)))
      (presheafObjπ data G₀ (unop (op (F.obj X₀))) i) =
  𝟙 ((presheaf data G₀).obj (op (F.obj X₀))) ≫ presheafObjπ data G₀ (unop (op (F.obj X₀))) i
Tactic: rw [assoc, presheafObjObjIso.inv_π, id_comp,
      presheafObjObjIso.hom_mapPreimage data G₀ _ (𝟙 _) (fac := by simp),
      IsDenseSubsite.mapPreimage_id, comp_id]
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.DenseSubsite.OneHypercoverDense (goal)
Goal:
C₀ : Type u₀
C : Type u
inst : Category.{v₀, u₀} C₀
inst_1 : Category.{v, u} C
F : Functor.{v₀, v, u₀, u} C₀ C
J₀ : GrothendieckTopology.{v₀, u₀} C₀
J : GrothendieckTopology.{v, u} C
A : Type u'
inst_2 : Category.{v', u'} A
inst_3 : IsDenseSubsite J₀ J F
data : (X : C) → OneHypercoverDenseData.{w, v₀, v, u₀, u} F J₀ J X
inst_4 : HasLimitsOfSize.{w, w, v', u'} A
G₀ : Sheaf.{v₀, v', u₀, u'} J₀ A
X₀ : C₀
i : PreOneHypercoverDenseData.I₀ (S := unop (op (F.obj X₀))) (toPreOneHypercoverDenseData (data (unop (op (F.obj X₀)))))
rw :
  presheafObjObjIso.hom data G₀ X₀ ≫
      CategoryStruct.comp (Y := presheafObj data G₀ (unop (op (F.obj X₀))))
        (presheafObjObjIso.inv data G₀ X₀ :
          G₀.obj (P := Presheaf.IsSheaf J₀).obj (op X₀) ⟶ (presheaf data G₀).obj (op (F.obj X₀)))
        (presheafObjπ data G₀ (unop (op (F.obj X₀))) i) =
    𝟙 ((presheaf data G₀).obj (op (F.obj X₀))) ≫ presheafObjπ data G₀ (unop (op (F.obj X₀))) i
⊢ CategoryStruct.comp (Y := presheafObj data G₀ (unop (op (F.obj X₀))))
      (presheafObjObjIso.hom data G₀ X₀ ≫ presheafObjObjIso.inv data G₀ X₀ :
        (presheaf data G₀).obj (op (F.obj X₀)) ⟶ (presheaf data G₀).obj (op (F.obj X₀)))
      (presheafObjπ data G₀ (unop (op (F.obj X₀))) i) =
    𝟙 ((presheaf data G₀).obj (op (F.obj X₀))) ≫ presheafObjπ data G₀ (unop (op (F.obj X₀))) i
have rw₁ : presheafObjObjIso.hom data G₀ X₀ ≫
    IsDenseSubsite.mapPreimage J F G₀ (PreOneHypercoverDenseData.f (toPreOneHypercoverDenseData (data (F.obj X₀))) i) =
  𝟙 ((presheaf data G₀).obj (op (F.obj X₀))) ≫ presheafObjπ data G₀ (unop (op (F.obj X₀))) i
Tactic: rw [assoc, presheafObjObjIso.inv_π, id_comp,
      presheafObjObjIso.hom_mapPreimage data G₀ _ (𝟙 _) (fac := by simp),
      IsDenseSubsite.mapPreimage_id, comp_id]
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.DenseSubsite.OneHypercoverDense (goal)
Goal:
C₀ : Type u₀
C : Type u
inst : Category.{v₀, u₀} C₀
inst_1 : Category.{v, u} C
F : Functor.{v₀, v, u₀, u} C₀ C
J₀ : GrothendieckTopology.{v₀, u₀} C₀
J : GrothendieckTopology.{v, u} C
A : Type u'
inst_2 : Category.{v', u'} A
inst_3 : IsDenseSubsite J₀ J F
data : (X : C) → OneHypercoverDenseData.{w, v₀, v, u₀, u} F J₀ J X
inst_4 : HasLimitsOfSize.{w, w, v', u'} A
G₀ : Sheaf.{v₀, v', u₀, u'} J₀ A
X₀ : C₀
i : PreOneHypercoverDenseData.I₀ (S := unop (op (F.obj X₀))) (toPreOneHypercoverDenseData (data (unop (op (F.obj X₀)))))
rw :
  presheafObjObjIso.hom data G₀ X₀ ≫
      CategoryStruct.comp (Y := presheafObj data G₀ (unop (op (F.obj X₀))))
        (presheafObjObjIso.inv data G₀ X₀ :
          G₀.obj (P := Presheaf.IsSheaf J₀).obj (op X₀) ⟶ (presheaf data G₀).obj (op (F.obj X₀)))
        (presheafObjπ data G₀ (unop (op (F.obj X₀))) i) =
    𝟙 ((presheaf data G₀).obj (op (F.obj X₀))) ≫ presheafObjπ data G₀ (unop (op (F.obj X₀))) i
rw₁ :
  presheafObjObjIso.hom data G₀ X₀ ≫
      IsDenseSubsite.mapPreimage J F G₀
        (PreOneHypercoverDenseData.f (toPreOneHypercoverDenseData (data (F.obj X₀))) i) =
    𝟙 ((presheaf data G₀).obj (op (F.obj X₀))) ≫ presheafObjπ data G₀ (unop (op (F.obj X₀))) i
⊢ CategoryStruct.comp (Y := presheafObj data G₀ (unop (op (F.obj X₀))))
      (presheafObjObjIso.hom data G₀ X₀ ≫ presheafObjObjIso.inv data G₀ X₀ :
        (presheaf data G₀).obj (op (F.obj X₀)) ⟶ (presheaf data G₀).obj (op (F.obj X₀)))
      (presheafObjπ data G₀ (unop (op (F.obj X₀))) i) =
    𝟙 ((presheaf data G₀).obj (op (F.obj X₀))) ≫ presheafObjπ data G₀ (unop (op (F.obj X₀))) i
have rw₂ : Eq (α :=
  (presheaf data G₀).obj (op (F.obj X₀)) ⟶
    G₀.obj (P := Presheaf.IsSheaf J₀).obj
      (op (PreOneHypercoverDenseData.X (toPreOneHypercoverDenseData (data (unop (op (F.obj X₀))))) i)))
  (presheafObjObjIso.hom data G₀ X₀ ≫
    IsDenseSubsite.mapPreimage J F G₀ (PreOneHypercoverDenseData.f (toPreOneHypercoverDenseData (data (F.obj X₀))) i))
  (presheafObjπ data G₀ (unop (op (F.obj X₀))) i)
Tactic: rw [assoc, presheafObjObjIso.inv_π, id_comp,
      presheafObjObjIso.hom_mapPreimage data G₀ _ (𝟙 _) (fac := by simp),
      IsDenseSubsite.mapPreimage_id, comp_id]
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.DenseSubsite.OneHypercoverDense (goal)
Goal:
C₀ : Type u₀
C : Type u
inst : Category.{v₀, u₀} C₀
inst_1 : Category.{v, u} C
F : Functor.{v₀, v, u₀, u} C₀ C
J₀ : GrothendieckTopology.{v₀, u₀} C₀
J : GrothendieckTopology.{v, u} C
A : Type u'
inst_2 : Category.{v', u'} A
inst_3 : IsDenseSubsite J₀ J F
data : (X : C) → OneHypercoverDenseData.{w, v₀, v, u₀, u} F J₀ J X
inst_4 : HasLimitsOfSize.{w, w, v', u'} A
G₀ : Sheaf.{v₀, v', u₀, u'} J₀ A
X₀ : C₀
i : PreOneHypercoverDenseData.I₀ (S := unop (op (F.obj X₀))) (toPreOneHypercoverDenseData (data (unop (op (F.obj X₀)))))
rw :
  presheafObjObjIso.hom data G₀ X₀ ≫
      CategoryStruct.comp (Y := presheafObj data G₀ (unop (op (F.obj X₀))))
        (presheafObjObjIso.inv data G₀ X₀ :
          G₀.obj (P := Presheaf.IsSheaf J₀).obj (op X₀) ⟶ (presheaf data G₀).obj (op (F.obj X₀)))
        (presheafObjπ data G₀ (unop (op (F.obj X₀))) i) =
    𝟙 ((presheaf data G₀).obj (op (F.obj X₀))) ≫ presheafObjπ data G₀ (unop (op (F.obj X₀))) i
rw₁ :
  presheafObjObjIso.hom data G₀ X₀ ≫
      IsDenseSubsite.mapPreimage J F G₀
        (PreOneHypercoverDenseData.f (toPreOneHypercoverDenseData (data (F.obj X₀))) i) =
    𝟙 ((presheaf data G₀).obj (op (F.obj X₀))) ≫ presheafObjπ data G₀ (unop (op (F.obj X₀))) i
rw₂ :
  Eq (α :=
    (presheaf data G₀).obj (op (F.obj X₀)) ⟶
      G₀.obj (P := Presheaf.IsSheaf J₀).obj
        (op (PreOneHypercoverDenseData.X (toPreOneHypercoverDenseData (data (unop (op (F.obj X₀))))) i)))
    (presheafObjObjIso.hom data G₀ X₀ ≫
      IsDenseSubsite.mapPreimage J F G₀ (PreOneHypercoverDenseData.f (toPreOneHypercoverDenseData (data (F.obj X₀))) i))
    (presheafObjπ data G₀ (unop (op (F.obj X₀))) i)
⊢ CategoryStruct.comp (Y := presheafObj data G₀ (unop (op (F.obj X₀))))
      (presheafObjObjIso.hom data G₀ X₀ ≫ presheafObjObjIso.inv data G₀ X₀ :
        (presheaf data G₀).obj (op (F.obj X₀)) ⟶ (presheaf data G₀).obj (op (F.obj X₀)))
      (presheafObjπ data G₀ (unop (op (F.obj X₀))) i) =
    𝟙 ((presheaf data G₀).obj (op (F.obj X₀))) ≫ presheafObjπ data G₀ (unop (op (F.obj X₀))) i
have rw₃ : Eq (α :=
  (presheaf data G₀).obj (op (F.obj X₀)) ⟶
    G₀.obj (P := Presheaf.IsSheaf J₀).obj
      (op (PreOneHypercoverDenseData.X (toPreOneHypercoverDenseData (data (unop (op (F.obj X₀))))) i)))
  (presheafObjπ data G₀ (F.obj X₀) i ≫
      IsDenseSubsite.mapPreimage J F G₀
        (𝟙 (F.obj (PreOneHypercoverDenseData.X (toPreOneHypercoverDenseData (data (F.obj X₀))) i))) :
    presheafObj data G₀ (F.obj X₀) ⟶
      G₀.obj (P := Presheaf.IsSheaf J₀).obj
        (op (PreOneHypercoverDenseData.X (toPreOneHypercoverDenseData (data (F.obj X₀))) i)))
  (presheafObjπ data G₀ (unop (op (F.obj X₀))) i)
Tactic: rw [assoc, presheafObjObjIso.inv_π, id_comp,
      presheafObjObjIso.hom_mapPreimage data G₀ _ (𝟙 _) (fac := by simp),
      IsDenseSubsite.mapPreimage_id, comp_id]
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.DenseSubsite.OneHypercoverDense (goal)
Goal:
C₀ : Type u₀
C : Type u
inst : Category.{v₀, u₀} C₀
inst_1 : Category.{v, u} C
F : Functor.{v₀, v, u₀, u} C₀ C
J₀ : GrothendieckTopology.{v₀, u₀} C₀
J : GrothendieckTopology.{v, u} C
A : Type u'
inst_2 : Category.{v', u'} A
inst_3 : IsDenseSubsite J₀ J F
data : (X : C) → OneHypercoverDenseData.{w, v₀, v, u₀, u} F J₀ J X
inst_4 : HasLimitsOfSize.{w, w, v', u'} A
G₀ : Sheaf.{v₀, v', u₀, u'} J₀ A
X₀ : C₀
⊢ CategoryStruct.comp (Y := (presheaf data G₀).obj (op (F.obj X₀))) (presheafObjObjIso.inv data G₀ X₀)
      (presheafObjObjIso.hom data G₀ X₀) =
    𝟙 (G₀.obj (P := Presheaf.IsSheaf J₀).obj (op X₀))
have refine : ∀
  (I :
    GrothendieckTopology.Cover.Arrow (X := X₀)
      (Subtype.mk (p := fun (S : Sieve.{v₀, u₀} X₀) ↦ S ∈ (J₀ : (a : C₀) → Set (Sieve a)) X₀)
        (Sieve.functorPullback F
          (PreZeroHypercover.sieve₀.{w, v, u}
            (PreOneHypercover.toPreZeroHypercover
              (PreOneHypercoverDenseData.toPreOneHypercover (toPreOneHypercoverDenseData (data (F.obj X₀)))))))
        sorry)),
  CategoryStruct.comp (Y := (presheaf data G₀).obj (op (F.obj X₀))) (presheafObjObjIso.inv data G₀ X₀)
        (presheafObjObjIso.hom data G₀ X₀) ≫
      G₀.obj (P := Presheaf.IsSheaf J₀).map I.f.op =
    𝟙 (G₀.obj (P := Presheaf.IsSheaf J₀).obj (op X₀)) ≫ G₀.obj (P := Presheaf.IsSheaf J₀).map I.f.op
Tactic: refine Presheaf.IsSheaf.hom_ext G₀.property
      ⟨_, cover_lift F J₀ _ (data (F.obj X₀)).mem₀⟩ _ _ ?_
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.DenseSubsite.OneHypercoverDense (goal)
Goal:
C₀ : Type u₀
C : Type u
inst : Category.{v₀, u₀} C₀
inst_1 : Category.{v, u} C
F : Functor.{v₀, v, u₀, u} C₀ C
J₀ : GrothendieckTopology.{v₀, u₀} C₀
J : GrothendieckTopology.{v, u} C
A : Type u'
inst_2 : Category.{v', u'} A
inst_3 : IsDenseSubsite J₀ J F
data : (X : C) → OneHypercoverDenseData.{w, v₀, v, u₀, u} F J₀ J X
inst_4 : HasLimitsOfSize.{w, w, v', u'} A
G₀ : Sheaf.{v₀, v', u₀, u'} J₀ A
X₀ Y₀ : C₀
a : Y₀ ⟶ X₀
Y : C
i :
  PreZeroHypercover.I₀ (S := F.obj X₀)
    (PreOneHypercover.toPreZeroHypercover
      (PreOneHypercoverDenseData.toPreOneHypercover (toPreOneHypercoverDenseData (data (F.obj X₀)))))
b :
  F.obj Y₀ ⟶
    PreZeroHypercover.X
      (PreOneHypercover.toPreZeroHypercover
        (PreOneHypercoverDenseData.toPreOneHypercover (toPreOneHypercoverDenseData (data (F.obj X₀)))))
      i
fac :
  b ≫
      PreZeroHypercover.f
        (PreOneHypercover.toPreZeroHypercover
          (PreOneHypercoverDenseData.toPreOneHypercover (toPreOneHypercoverDenseData (data (F.obj X₀)))))
        i =
    F.map a
⊢ Eq (α :=
    G₀.obj (P := Presheaf.IsSheaf J₀).obj (op X₀) ⟶
      G₀.obj (P := Presheaf.IsSheaf J₀).obj
        (op
          (GrothendieckTopology.Cover.Arrow.Y (J := J₀) (S :=
            Subtype.mk (p := fun (S : Sieve.{v₀, u₀} X₀) ↦ S ∈ (J₀ : (a : C₀) → Set (Sieve a)) X₀)
              (Sieve.functorPullback F
                (PreZeroHypercover.sieve₀.{w, v, u}
                  (PreOneHypercover.toPreZeroHypercover
                    (PreOneHypercoverDenseData.toPreOneHypercover (toPreOneHypercoverDenseData (data (F.obj X₀)))))))
              ⋯)
            { Y := Y₀, f := a, hf := ⋯ })))
    (CategoryStruct.comp (Y := (presheaf data G₀).obj (op (F.obj X₀))) (presheafObjObjIso.inv data G₀ X₀)
        (presheafObjObjIso.hom data G₀ X₀) ≫
      G₀.obj (P := Presheaf.IsSheaf J₀).map
        (Quiver.Hom.op (GrothendieckTopology.Cover.Arrow.f { Y := Y₀, f := a, hf := ⋯ })))
    (𝟙 (G₀.obj (P := Presheaf.IsSheaf J₀).obj (op X₀)) ≫
      G₀.obj (P := Presheaf.IsSheaf J₀).map
        (Quiver.Hom.op (GrothendieckTopology.Cover.Arrow.f { Y := Y₀, f := a, hf := ⋯ })))
have dsimp : CategoryStruct.comp (Y := presheafObj data G₀ (F.obj X₀))
      (presheafObjObjIso.inv data G₀ X₀ :
        G₀.obj (P := Presheaf.IsSheaf J₀).obj (op X₀) ⟶ (presheaf data G₀).obj (op (F.obj X₀)))
      (presheafObjObjIso.hom data G₀ X₀) ≫
    G₀.obj (P := Presheaf.IsSheaf J₀).map a.op =
  𝟙 (G₀.obj (P := Presheaf.IsSheaf J₀).obj (op X₀)) ≫ G₀.obj (P := Presheaf.IsSheaf J₀).map a.op
Tactic: dsimp at i b fac ⊢
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.DenseSubsite.OneHypercoverDense (goal)
Goal:
C₀ : Type u₀
C : Type u
inst : Category.{v₀, u₀} C₀
inst_1 : Category.{v, u} C
F : Functor.{v₀, v, u₀, u} C₀ C
J₀ : GrothendieckTopology.{v₀, u₀} C₀
J : GrothendieckTopology.{v, u} C
A : Type u'
inst_2 : Category.{v', u'} A
inst_3 : IsDenseSubsite J₀ J F
data : (X : C) → OneHypercoverDenseData.{w, v₀, v, u₀, u} F J₀ J X
inst_4 : HasLimitsOfSize.{w, w, v', u'} A
G₀ : Sheaf.{v₀, v', u₀, u'} J₀ A
X : C
i : (data X).I₀
⊢ Eq (α := presheafObj data G₀ X ⟶ G₀.obj (P := Presheaf.IsSheaf J₀).obj (op ((data X).X i)))
    (presheafMap data G₀ ((data X).f i) ≫ Iso.hom (presheafObjObjIso data G₀ ((data X).X i))) (presheafObjπ data G₀ X i)
have rw : Eq (α := presheafObj data G₀ X ⟶ (presheaf data G₀).obj (op (F.obj ((data X).X i))))
  (presheafMap data G₀ ((data X).f i)) (presheafObjπ data G₀ X i ≫ Iso.inv (presheafObjObjIso data G₀ ((data X).X i)))
Tactic: rw [← cancel_mono (presheafObjObjIso data G₀ ((data X).X i)).inv, assoc, Iso.hom_inv_id,
    comp_id]
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.DenseSubsite.OneHypercoverDense (goal)
Goal:
C₀ : Type u₀
C : Type u
inst : Category.{v₀, u₀} C₀
inst_1 : Category.{v, u} C
F : Functor.{v₀, v, u₀, u} C₀ C
J₀ : GrothendieckTopology.{v₀, u₀} C₀
J : GrothendieckTopology.{v, u} C
A : Type u'
inst_2 : Category.{v', u'} A
inst_3 : IsDenseSubsite J₀ J F
data : (X : C) → OneHypercoverDenseData.{w, v₀, v, u₀, u} F J₀ J X
inst_4 : HasLimitsOfSize.{w, w, v', u'} A
G₀ : Sheaf.{v₀, v', u₀, u'} J₀ A
X : C
i : (data X).I₀
⊢ Eq (α := presheafObj data G₀ X ⟶ (presheaf data G₀).obj (op (F.obj ((data X).X i))))
    (presheafMap data G₀ ((data X).f i)) (presheafObjπ data G₀ X i ≫ Iso.inv (presheafObjObjIso data G₀ ((data X).X i)))
have h : ∀
  (i_1 :
    PreOneHypercoverDenseData.I₀ (S := F.obj ((data X).X i))
      (toPreOneHypercoverDenseData (data (F.obj ((data X).X i))))),
  Eq (α :=
    presheafObj data G₀ X ⟶
      G₀.obj (P := Presheaf.IsSheaf J₀).obj
        (op (PreOneHypercoverDenseData.X (toPreOneHypercoverDenseData (data (F.obj ((data X).X i)))) i_1)))
    (presheafMap data G₀ ((data X).f i) ≫ presheafObjπ data G₀ (F.obj ((data X).X i)) i_1)
    (CategoryStruct.comp (Y := presheafObj data G₀ (F.obj ((data X).X i)))
      (presheafObjπ data G₀ X i ≫ Iso.inv (presheafObjObjIso data G₀ ((data X).X i)) :
        presheafObj data G₀ X ⟶ (presheaf data G₀).obj (op (F.obj ((data X).X i))))
      (presheafObjπ data G₀ (F.obj ((data X).X i)) i_1))
Tactic: apply presheafObj_hom_ext