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
⊢ CategoryStruct.comp (Y := (presheaf data G₀).obj (op (F.obj X₀))) (inv data G₀ X₀) (restriction data G₀ f) =
    IsDenseSubsite.mapPreimage J F G₀ f
have refine : ∀
  (I :
    GrothendieckTopology.Cover.Arrow
      (Subtype.mk (p := fun (S : Sieve.{v₀, u₀} Y₀) ↦ S ∈ (J₀ : (a : C₀) → Set (Sieve a)) Y₀) (F.imageSieve f) sorry)),
  CategoryStruct.comp (Y := (presheaf data G₀).obj (op (F.obj X₀))) (inv data G₀ X₀) (restriction data G₀ f) ≫
      G₀.obj (P := Presheaf.IsSheaf J₀).map I.f.op =
    IsDenseSubsite.mapPreimage J F G₀ f ≫ G₀.obj (P := Presheaf.IsSheaf J₀).map I.f.op
Tactic: refine Presheaf.IsSheaf.hom_ext G₀.property
    ⟨_, IsDenseSubsite.imageSieve_mem J₀ J F f⟩ _ _ ?_
-- 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
⊢ 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} Y₀) ↦ S ∈ (J₀ : (a : C₀) → Set (Sieve a)) Y₀) (F.imageSieve f) ⋯)
            { Y := W₀, f := a, hf := ⋯ })))
    (CategoryStruct.comp (Y := (presheaf data G₀).obj (op (F.obj X₀))) (inv data G₀ X₀) (restriction data G₀ f) ≫
      G₀.obj (P := Presheaf.IsSheaf J₀).map
        (Quiver.Hom.op (GrothendieckTopology.Cover.Arrow.f { Y := W₀, f := a, hf := ⋯ })))
    (IsDenseSubsite.mapPreimage J F G₀ f ≫
      G₀.obj (P := Presheaf.IsSheaf J₀).map
        (Quiver.Hom.op (GrothendieckTopology.Cover.Arrow.f { Y := W₀, f := a, hf := ⋯ })))
have refine : ∀
  (I :
    GrothendieckTopology.Cover.Arrow
      (Subtype.mk (p := fun (S : Sieve.{v₀, u₀} W₀) ↦ S ∈ (J₀ : (a : C₀) → Set (Sieve a)) W₀)
        (Sieve.pullback b
          (Sieve.functorPullback F
            (PreZeroHypercover.sieve₀.{w, v, u}
              (PreOneHypercover.toPreZeroHypercover
                (PreOneHypercoverDenseData.toPreOneHypercover (toPreOneHypercoverDenseData (data (F.obj X₀))))))))
        sorry)),
  Eq (α := G₀.obj (P := Presheaf.IsSheaf J₀).obj (op X₀) ⟶ G₀.obj (P := Presheaf.IsSheaf J₀).obj (op I.Y))
    ((CategoryStruct.comp (Y := (presheaf data G₀).obj (op (F.obj X₀))) (inv data G₀ X₀) (restriction data G₀ f) ≫
        G₀.obj (P := Presheaf.IsSheaf J₀).map
          (Quiver.Hom.op (GrothendieckTopology.Cover.Arrow.f { Y := W₀, f := a, hf := sorry }))) ≫
      G₀.obj (P := Presheaf.IsSheaf J₀).map I.f.op)
    ((IsDenseSubsite.mapPreimage J F G₀ f ≫
        G₀.obj (P := Presheaf.IsSheaf J₀).map
          (Quiver.Hom.op (GrothendieckTopology.Cover.Arrow.f { Y := W₀, f := a, hf := sorry }))) ≫
      G₀.obj (P := Presheaf.IsSheaf J₀).map I.f.op)
Tactic: refine Presheaf.IsSheaf.hom_ext G₀.property
    ⟨_, J₀.pullback_stable b (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
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)
⊢ (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 a.op) ≫
      G₀.obj (P := Presheaf.IsSheaf J₀).map c.op =
    (IsDenseSubsite.mapPreimage J F G₀ f ≫ G₀.obj (P := Presheaf.IsSheaf J₀).map a.op) ≫
      G₀.obj (P := Presheaf.IsSheaf J₀).map c.op
have simp : 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
Tactic: simp only [assoc, ← Functor.map_comp, ← op_comp]
-- 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)
⊢ d ≫ PreOneHypercoverDenseData.f (toPreOneHypercoverDenseData (data (F.obj X₀))) i = F.map (c ≫ a) ≫ f
have rw : F.map (c ≫ b) = F.map (c ≫ a) ≫ f
Tactic: rw [fac₂, 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 : F.map (c ≫ b) = F.map (c ≫ a) ≫ f
⊢ d ≫ PreOneHypercoverDenseData.f (toPreOneHypercoverDenseData (data (F.obj X₀))) i = F.map (c ≫ a) ≫ f
have rw₁ : F.map c ≫ F.map b = F.map (c ≫ a) ≫ f
Tactic: rw [fac₂, 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 : F.map (c ≫ b) = F.map (c ≫ a) ≫ f
rw₁ : F.map c ≫ F.map b = F.map (c ≫ a) ≫ f
⊢ d ≫ PreOneHypercoverDenseData.f (toPreOneHypercoverDenseData (data (F.obj X₀))) i = F.map (c ≫ a) ≫ f
have rw₂ : F.map c ≫ F.map b = F.map c ≫ F.map a ≫ f
Tactic: rw [fac₂, 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 : F.map (c ≫ b) = F.map (c ≫ a) ≫ f
rw₁ : F.map c ≫ F.map b = F.map (c ≫ a) ≫ f
rw₂ : F.map c ≫ F.map b = F.map c ≫ F.map a ≫ f
⊢ d ≫ PreOneHypercoverDenseData.f (toPreOneHypercoverDenseData (data (F.obj X₀))) i = F.map (c ≫ a) ≫ f
have rw₃ : F.map c ≫ F.map a ≫ f = F.map c ≫ F.map a ≫ f
Tactic: rw [fac₂, 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)
⊢ 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 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
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
⊢ 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 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
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
⊢ 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 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
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
⊢ 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 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
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
⊢ 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 rw₄ : IsDenseSubsite.mapPreimage J F G₀ (F.map (c ≫ 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]