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
X : C
X₀ Y₀ : C₀
f : F.obj X₀ ⟶ X
g : Y₀ ⟶ X₀
h₁ h₂ : (data X).SieveStruct f g
Z₀ : C₀
a : Z₀ ⟶ Y₀
b₁ : Z₀ ⟶ (data X).X h₁.i₀
w₁ : F.map b₁ = F.map a ≫ h₁.q
b₂ : Z₀ ⟶ (data X).X h₂.i₀
w₂ : F.map b₂ = F.map a ≫ h₂.q
W₀ : C₀
c : W₀ ⟶ Z₀
hc :
  (↑(Subtype.mk (p := fun (S : Sieve.{v₀, u₀} Z₀) ↦ S ∈ (J₀ : (a : C₀) → Set (Sieve a)) Z₀) ((data X).sieve₁₀ b₁ b₂)
            ⋯) :
        Sieve Z₀).arrows
    c
⊢ Eq (α := presheafObj data G₀ X ⟶ G₀.obj (P := Presheaf.IsSheaf J₀).obj (op W₀))
    (((presheafObjπ data G₀ X h₁.i₀ ≫ IsDenseSubsite.mapPreimage J F G₀ h₁.q) ≫
        G₀.obj (P := Presheaf.IsSheaf J₀).map a.op) ≫
      G₀.obj (P := Presheaf.IsSheaf J₀).map c.op)
    (((presheafObjπ data G₀ X h₂.i₀ ≫ IsDenseSubsite.mapPreimage J F G₀ h₂.q) ≫
        G₀.obj (P := Presheaf.IsSheaf J₀).map a.op) ≫
      G₀.obj (P := Presheaf.IsSheaf J₀).map c.op)
have simp : Eq (α := presheafObj data G₀ X ⟶ G₀.obj (P := Presheaf.IsSheaf J₀).obj (op W₀))
  (presheafObjπ data G₀ X h₁.i₀ ≫ IsDenseSubsite.mapPreimage J F G₀ (F.map c ≫ F.map a ≫ h₁.q))
  (presheafObjπ data G₀ X h₂.i₀ ≫ IsDenseSubsite.mapPreimage J F G₀ (F.map c ≫ F.map a ≫ h₂.q))
Tactic: simp only [assoc, IsDenseSubsite.mapPreimage_comp_map]
-- 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
X₀ Y₀ : C₀
f : F.obj X₀ ⟶ X
g : Y₀ ⟶ X₀
h₁ h₂ : (data X).SieveStruct f g
Z₀ : C₀
a : Z₀ ⟶ Y₀
b₁ : Z₀ ⟶ (data X).X h₁.i₀
w₁ : F.map b₁ = F.map a ≫ h₁.q
b₂ : Z₀ ⟶ (data X).X h₂.i₀
w₂ : F.map b₂ = F.map a ≫ h₂.q
W₀ : C₀
c : W₀ ⟶ Z₀
hc :
  (↑(Subtype.mk (p := fun (S : Sieve.{v₀, u₀} Z₀) ↦ S ∈ (J₀ : (a : C₀) → Set (Sieve a)) Z₀) ((data X).sieve₁₀ b₁ b₂)
            ⋯) :
        Sieve Z₀).arrows
    c
⊢ Eq (α := presheafObj data G₀ X ⟶ G₀.obj (P := Presheaf.IsSheaf J₀).obj (op W₀))
    (presheafObjπ data G₀ X h₁.i₀ ≫ IsDenseSubsite.mapPreimage J F G₀ (F.map c ≫ F.map a ≫ h₁.q))
    (presheafObjπ data G₀ X h₂.i₀ ≫ IsDenseSubsite.mapPreimage J F G₀ (F.map c ≫ F.map a ≫ h₂.q))
have fac : (F.map c ≫ F.map a ≫ h₁.q) ≫ (data X).f h₁.i₀ = (F.map c ≫ F.map a ≫ h₂.q) ≫ (data X).f h₂.i₀
Tactic: apply presheafObj_mapPreimage_condition
-- 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
X₀ : C₀
f : F.obj X₀ ⟶ X
Z₁ : C₀
g₁ : Z₁ ⟶ X₀
h₁ : (data X).SieveStruct f g₁
Z₂ : C₀
g₂ : Z₂ ⟶ X₀
h₂ : (data X).SieveStruct f g₂
T₀ : C₀
p₁ : T₀ ⟶ Z₁
p₂ : T₀ ⟶ Z₂
w : p₁ ≫ g₁ = p₂ ≫ g₂
⊢ Eq (α := presheafObj data G₀ X ⟶ G₀.obj (P := Presheaf.IsSheaf J₀).obj (op T₀))
    (restriction.res data G₀ (X₀ := X₀) (f := f) (g := g₁) (Nonempty.some ⋯) ≫
      G₀.obj (P := Presheaf.IsSheaf J₀).map p₁.op)
    (restriction.res data G₀ (X₀ := X₀) (f := f) (g := g₂) (Nonempty.some ⋯) ≫
      G₀.obj (P := Presheaf.IsSheaf J₀).map p₂.op)
have rw : Eq (α := presheafObj data G₀ X ⟶ G₀.obj (P := Presheaf.IsSheaf J₀).obj (op T₀))
  (restriction.res data G₀ h₁ ≫ G₀.obj (P := Presheaf.IsSheaf J₀).map p₁.op)
  (restriction.res data G₀ h₂ ≫ G₀.obj (P := Presheaf.IsSheaf J₀).map p₂.op)
Tactic: rw [restriction.res_eq_res data G₀ _ h₁, restriction.res_eq_res data G₀ _ h₂]
-- 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
X₀ : C₀
f : F.obj X₀ ⟶ X
Z₁ : C₀
g₁ : Z₁ ⟶ X₀
h₁ : (data X).SieveStruct f g₁
Z₂ : C₀
g₂ : Z₂ ⟶ X₀
h₂ : (data X).SieveStruct f g₂
T₀ : C₀
p₁ : T₀ ⟶ Z₁
p₂ : T₀ ⟶ Z₂
w : p₁ ≫ g₁ = p₂ ≫ g₂
⊢ Eq (α := presheafObj data G₀ X ⟶ G₀.obj (P := Presheaf.IsSheaf J₀).obj (op T₀))
    (restriction.res data G₀ h₁ ≫ G₀.obj (P := Presheaf.IsSheaf J₀).map p₁.op)
    (restriction.res data G₀ h₂ ≫ G₀.obj (P := Presheaf.IsSheaf J₀).map p₂.op)
have refine : ∀
  (I :
    GrothendieckTopology.Cover.Arrow
      (Subtype.mk (p := fun (S : Sieve.{v₀, u₀} T₀) ↦ S ∈ (J₀ : (a : C₀) → Set (Sieve a)) T₀)
        (F.imageSieve (F.map p₁ ≫ h₁.q) ⊓ F.imageSieve (F.map p₂ ≫ h₂.q)) sorry)),
  Eq (α := presheafObj data G₀ X ⟶ G₀.obj (P := Presheaf.IsSheaf J₀).obj (op I.Y))
    ((restriction.res data G₀ h₁ ≫ G₀.obj (P := Presheaf.IsSheaf J₀).map p₁.op) ≫
      G₀.obj (P := Presheaf.IsSheaf J₀).map I.f.op)
    ((restriction.res data G₀ h₂ ≫ G₀.obj (P := Presheaf.IsSheaf J₀).map p₂.op) ≫
      G₀.obj (P := Presheaf.IsSheaf J₀).map I.f.op)
Tactic: refine Presheaf.IsSheaf.hom_ext G₀.property
        ⟨_, J₀.intersection_covering
          (IsDenseSubsite.imageSieve_mem J₀ J F (F.map p₁ ≫ h₁.q))
          (IsDenseSubsite.imageSieve_mem J₀ J F (F.map p₂ ≫ h₂.q))⟩ _ _ ?_
-- 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
X₀ : C₀
f : F.obj X₀ ⟶ X
Z₁ : C₀
g₁ : Z₁ ⟶ X₀
h₁ : (data X).SieveStruct f g₁
Z₂ : C₀
g₂ : Z₂ ⟶ X₀
h₂ : (data X).SieveStruct f g₂
T₀ : C₀
p₁ : T₀ ⟶ Z₁
p₂ : T₀ ⟶ Z₂
w : p₁ ≫ g₁ = p₂ ≫ g₂
W₀ : C₀
a : W₀ ⟶ T₀
q₁ : W₀ ⟶ (data X).X h₁.i₀
w₁ : F.map q₁ = F.map a ≫ F.map p₁ ≫ h₁.q
q₂ : W₀ ⟶ (data X).X h₂.i₀
w₂ : F.map q₂ = F.map a ≫ F.map p₂ ≫ h₂.q
⊢ Eq (α :=
    presheafObj data G₀ X ⟶
      G₀.obj (P := Presheaf.IsSheaf J₀).obj
        (op
          (GrothendieckTopology.Cover.Arrow.Y (J := J₀) (S :=
            Subtype.mk (p := fun (S : Sieve.{v₀, u₀} T₀) ↦ S ∈ (J₀ : (a : C₀) → Set (Sieve a)) T₀)
              (F.imageSieve (F.map p₁ ≫ h₁.q) ⊓ F.imageSieve (F.map p₂ ≫ h₂.q)) ⋯)
            { Y := W₀, f := a, hf := ⋯ })))
    ((restriction.res data G₀ h₁ ≫ G₀.obj (P := Presheaf.IsSheaf J₀).map p₁.op) ≫
      G₀.obj (P := Presheaf.IsSheaf J₀).map
        (Quiver.Hom.op (GrothendieckTopology.Cover.Arrow.f { Y := W₀, f := a, hf := ⋯ })))
    ((restriction.res data G₀ h₂ ≫ G₀.obj (P := Presheaf.IsSheaf J₀).map p₂.op) ≫
      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) ((data X).sieve₁₀ q₁ q₂)
        sorry)),
  Eq (α := presheafObj data G₀ X ⟶ G₀.obj (P := Presheaf.IsSheaf J₀).obj (op I.Y))
    (((restriction.res data G₀ h₁ ≫ G₀.obj (P := Presheaf.IsSheaf J₀).map p₁.op) ≫
        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)
    (((restriction.res data G₀ h₂ ≫ G₀.obj (P := Presheaf.IsSheaf J₀).map p₂.op) ≫
        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
        ⟨_, (data X).mem₁₀ h₁.i₀ h₂.i₀ q₁ q₂ (by
        simp only [w₁, w₂, assoc, h₁.fac, h₂.fac, ← Functor.map_comp_assoc, w])⟩ _ _ ?_
-- 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
X₀ : C₀
f : F.obj X₀ ⟶ X
Z₁ : C₀
g₁ : Z₁ ⟶ X₀
h₁ : (data X).SieveStruct f g₁
Z₂ : C₀
g₂ : Z₂ ⟶ X₀
h₂ : (data X).SieveStruct f g₂
T₀ : C₀
p₁ : T₀ ⟶ Z₁
p₂ : T₀ ⟶ Z₂
w : p₁ ≫ g₁ = p₂ ≫ g₂
W₀ : C₀
a : W₀ ⟶ T₀
q₁ : W₀ ⟶ (data X).X h₁.i₀
w₁ : F.map q₁ = F.map a ≫ F.map p₁ ≫ h₁.q
q₂ : W₀ ⟶ (data X).X h₂.i₀
w₂ : F.map q₂ = F.map a ≫ F.map p₂ ≫ h₂.q
U₀ : C₀
b : U₀ ⟶ W₀
hb :
  (↑(Subtype.mk (p := fun (S : Sieve.{v₀, u₀} W₀) ↦ S ∈ (J₀ : (a : C₀) → Set (Sieve a)) W₀) ((data X).sieve₁₀ q₁ q₂)
            ⋯) :
        Sieve W₀).arrows
    b
⊢ Eq (α :=
    presheafObj data G₀ X ⟶
      G₀.obj (P := Presheaf.IsSheaf J₀).obj
        (op
          { Y := U₀, f := b, hf := hb :
              GrothendieckTopology.Cover.Arrow
                (Subtype.mk (p := fun (S : Sieve.{v₀, u₀} W₀) ↦ S ∈ (J₀ : (a : C₀) → Set (Sieve a)) W₀)
                  ((data X).sieve₁₀ q₁ q₂) ⋯) }.Y))
    (((restriction.res data G₀ h₁ ≫ G₀.obj (P := Presheaf.IsSheaf J₀).map p₁.op) ≫
        G₀.obj (P := Presheaf.IsSheaf J₀).map
          (Quiver.Hom.op (GrothendieckTopology.Cover.Arrow.f { Y := W₀, f := a, hf := ⋯ }))) ≫
      G₀.obj (P := Presheaf.IsSheaf J₀).map
        { Y := U₀, f := b, hf := hb :
              GrothendieckTopology.Cover.Arrow
                (Subtype.mk (p := fun (S : Sieve.{v₀, u₀} W₀) ↦ S ∈ (J₀ : (a : C₀) → Set (Sieve a)) W₀)
                  ((data X).sieve₁₀ q₁ q₂) ⋯) }.f.op)
    (((restriction.res data G₀ h₂ ≫ G₀.obj (P := Presheaf.IsSheaf J₀).map p₂.op) ≫
        G₀.obj (P := Presheaf.IsSheaf J₀).map
          (Quiver.Hom.op (GrothendieckTopology.Cover.Arrow.f { Y := W₀, f := a, hf := ⋯ }))) ≫
      G₀.obj (P := Presheaf.IsSheaf J₀).map
        { Y := U₀, f := b, hf := hb :
              GrothendieckTopology.Cover.Arrow
                (Subtype.mk (p := fun (S : Sieve.{v₀, u₀} W₀) ↦ S ∈ (J₀ : (a : C₀) → Set (Sieve a)) W₀)
                  ((data X).sieve₁₀ q₁ q₂) ⋯) }.f.op)
have dsimp : Eq (α := presheafObj data G₀ X ⟶ G₀.obj (P := Presheaf.IsSheaf J₀).obj (op U₀))
  (((restriction.res data G₀ h₁ ≫ G₀.obj (P := Presheaf.IsSheaf J₀).map p₁.op) ≫
      G₀.obj (P := Presheaf.IsSheaf J₀).map a.op) ≫
    G₀.obj (P := Presheaf.IsSheaf J₀).map b.op)
  (((restriction.res data G₀ h₂ ≫ G₀.obj (P := Presheaf.IsSheaf J₀).map p₂.op) ≫
      G₀.obj (P := Presheaf.IsSheaf J₀).map a.op) ≫
    G₀.obj (P := Presheaf.IsSheaf J₀).map b.op)
Tactic: dsimp
-- 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
X₀ : C₀
f : F.obj X₀ ⟶ X
Z₁ : C₀
g₁ : Z₁ ⟶ X₀
h₁ : (data X).SieveStruct f g₁
Z₂ : C₀
g₂ : Z₂ ⟶ X₀
h₂ : (data X).SieveStruct f g₂
T₀ : C₀
p₁ : T₀ ⟶ Z₁
p₂ : T₀ ⟶ Z₂
w : p₁ ≫ g₁ = p₂ ≫ g₂
W₀ : C₀
a : W₀ ⟶ T₀
q₁ : W₀ ⟶ (data X).X h₁.i₀
w₁ : F.map q₁ = F.map a ≫ F.map p₁ ≫ h₁.q
q₂ : W₀ ⟶ (data X).X h₂.i₀
w₂ : F.map q₂ = F.map a ≫ F.map p₂ ≫ h₂.q
U₀ : C₀
b : U₀ ⟶ W₀
hb :
  (↑(Subtype.mk (p := fun (S : Sieve.{v₀, u₀} W₀) ↦ S ∈ (J₀ : (a : C₀) → Set (Sieve a)) W₀) ((data X).sieve₁₀ q₁ q₂)
            ⋯) :
        Sieve W₀).arrows
    b
⊢ Eq (α := presheafObj data G₀ X ⟶ G₀.obj (P := Presheaf.IsSheaf J₀).obj (op U₀))
    (((restriction.res data G₀ h₁ ≫ G₀.obj (P := Presheaf.IsSheaf J₀).map p₁.op) ≫
        G₀.obj (P := Presheaf.IsSheaf J₀).map a.op) ≫
      G₀.obj (P := Presheaf.IsSheaf J₀).map b.op)
    (((restriction.res data G₀ h₂ ≫ G₀.obj (P := Presheaf.IsSheaf J₀).map p₂.op) ≫
        G₀.obj (P := Presheaf.IsSheaf J₀).map a.op) ≫
      G₀.obj (P := Presheaf.IsSheaf J₀).map b.op)
have simp : Eq (α := presheafObj data G₀ X ⟶ G₀.obj (P := Presheaf.IsSheaf J₀).obj (op U₀))
  (presheafObjπ data G₀ X h₁.i₀ ≫ IsDenseSubsite.mapPreimage J F G₀ (F.map b ≫ F.map a ≫ F.map p₁ ≫ h₁.q))
  (presheafObjπ data G₀ X h₂.i₀ ≫ IsDenseSubsite.mapPreimage J F G₀ (F.map b ≫ F.map a ≫ F.map p₂ ≫ h₂.q))
Tactic: simp only [assoc, restriction.res, IsDenseSubsite.mapPreimage_comp_map]
-- 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
X₀ : C₀
f : F.obj X₀ ⟶ X
Z₁ : C₀
g₁ : Z₁ ⟶ X₀
h₁ : (data X).SieveStruct f g₁
Z₂ : C₀
g₂ : Z₂ ⟶ X₀
h₂ : (data X).SieveStruct f g₂
T₀ : C₀
p₁ : T₀ ⟶ Z₁
p₂ : T₀ ⟶ Z₂
w : p₁ ≫ g₁ = p₂ ≫ g₂
W₀ : C₀
a : W₀ ⟶ T₀
q₁ : W₀ ⟶ (data X).X h₁.i₀
w₁ : F.map q₁ = F.map a ≫ F.map p₁ ≫ h₁.q
q₂ : W₀ ⟶ (data X).X h₂.i₀
w₂ : F.map q₂ = F.map a ≫ F.map p₂ ≫ h₂.q
U₀ : C₀
b : U₀ ⟶ W₀
hb :
  (↑(Subtype.mk (p := fun (S : Sieve.{v₀, u₀} W₀) ↦ S ∈ (J₀ : (a : C₀) → Set (Sieve a)) W₀) ((data X).sieve₁₀ q₁ q₂)
            ⋯) :
        Sieve W₀).arrows
    b
⊢ Eq (α := presheafObj data G₀ X ⟶ G₀.obj (P := Presheaf.IsSheaf J₀).obj (op U₀))
    (presheafObjπ data G₀ X h₁.i₀ ≫ IsDenseSubsite.mapPreimage J F G₀ (F.map b ≫ F.map a ≫ F.map p₁ ≫ h₁.q))
    (presheafObjπ data G₀ X h₂.i₀ ≫ IsDenseSubsite.mapPreimage J F G₀ (F.map b ≫ F.map a ≫ F.map p₂ ≫ h₂.q))
have fac : (F.map b ≫ F.map a ≫ F.map p₁ ≫ h₁.q) ≫ (data X).f h₁.i₀ = (F.map b ≫ F.map a ≫ F.map p₂ ≫ h₂.q) ≫ (data X).f h₂.i₀
Tactic: apply presheafObj_mapPreimage_condition
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.DenseSubsite.OneHypercoverDense (hypothesis)
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
X₀ : C₀
f : F.obj X₀ ⟶ X
Y₀ : C₀
g : Y₀ ⟶ X₀
i : (data X).I₀
p : F.obj Y₀ ⟶ F.obj ((data X).X i)
fac : p ≫ (data X).f i = F.map g ≫ f
⊢ Eq (α := presheafObj data G₀ X ⟶ G₀.obj (P := Presheaf.IsSheaf J₀).obj (op Y₀))
    (restriction data G₀ f ≫ G₀.obj (P := Presheaf.IsSheaf J₀).map g.op)
    (presheafObjπ data G₀ X i ≫ IsDenseSubsite.mapPreimage J F G₀ p)
have hg : ((data X).sieve f).arrows g
Tactic: have hg : (data X).sieve f g := ⟨i, p, 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
X₀ : C₀
f : F.obj X₀ ⟶ X
Y₀ : C₀
g : Y₀ ⟶ X₀
i : (data X).I₀
p : F.obj Y₀ ⟶ F.obj ((data X).X i)
fac : p ≫ (data X).f i = F.map g ≫ f
hg : ((data X).sieve f).arrows g
⊢ Eq (α := presheafObj data G₀ X ⟶ G₀.obj (P := Presheaf.IsSheaf J₀).obj (op Y₀))
    (restriction data G₀ f ≫ G₀.obj (P := Presheaf.IsSheaf J₀).map g.op)
    (presheafObjπ data G₀ X i ≫ IsDenseSubsite.mapPreimage J F G₀ p)
have dsimp : Eq (α := presheafObj data G₀ X ⟶ G₀.obj (P := Presheaf.IsSheaf J₀).obj (op Y₀))
  (sorry.amalgamate
      (Subtype.mk (p := fun (S : Sieve.{v₀, u₀} X₀) ↦ S ∈ (J₀ : (a : C₀) → Set (Sieve a)) X₀) ((data X).sieve f) sorry)
      (fun
          (x :
            GrothendieckTopology.Cover.Arrow
              (Subtype.mk (p := fun (S : Sieve.{v₀, u₀} X₀) ↦ S ∈ (J₀ : (a : C₀) → Set (Sieve a)) X₀) ((data X).sieve f)
                sorry)) ↦
        (restriction.res data G₀ (Nonempty.some sorry) :
          presheafObj data G₀ X ⟶ G₀.obj (P := Presheaf.IsSheaf J₀).obj (op x.Y)))
      sorry ≫
    G₀.obj (P := Presheaf.IsSheaf J₀).map g.op)
  (presheafObjπ data G₀ X i ≫ IsDenseSubsite.mapPreimage J F G₀ p)
Tactic: dsimp only [restriction]
-- 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
X₀ : C₀
f : F.obj X₀ ⟶ X
Y₀ : C₀
g : Y₀ ⟶ X₀
i : (data X).I₀
p : F.obj Y₀ ⟶ F.obj ((data X).X i)
fac : p ≫ (data X).f i = F.map g ≫ f
hg : ((data X).sieve f).arrows g
⊢ Eq (α := presheafObj data G₀ X ⟶ G₀.obj (P := Presheaf.IsSheaf J₀).obj (op Y₀))
    (⋯.amalgamate
        (Subtype.mk (p := fun (S : Sieve.{v₀, u₀} X₀) ↦ S ∈ (J₀ : (a : C₀) → Set (Sieve a)) X₀) ((data X).sieve f) ⋯)
        (fun
            (x :
              GrothendieckTopology.Cover.Arrow
                (Subtype.mk (p := fun (S : Sieve.{v₀, u₀} X₀) ↦ S ∈ (J₀ : (a : C₀) → Set (Sieve a)) X₀)
                  ((data X).sieve f) ⋯)) ↦
          (restriction.res data G₀ (Nonempty.some ⋯) :
            presheafObj data G₀ X ⟶ G₀.obj (P := Presheaf.IsSheaf J₀).obj (op x.Y)))
        ⋯ ≫
      G₀.obj (P := Presheaf.IsSheaf J₀).map g.op)
    (presheafObjπ data G₀ X i ≫ IsDenseSubsite.mapPreimage J F G₀ p)
have rw : Eq (α := presheafObj data G₀ X ⟶ G₀.obj (P := Presheaf.IsSheaf J₀).obj (op Y₀))
  (restriction.res data G₀ (X₀ := X₀) (f := f) (g := GrothendieckTopology.Cover.Arrow.f { Y := Y₀, f := g, hf := hg })
    (Nonempty.some sorry))
  (presheafObjπ data G₀ X i ≫ IsDenseSubsite.mapPreimage J F G₀ p)
Tactic: rw [G₀.2.amalgamate_map _ _ _ ⟨_, g, hg⟩]
-- 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
X₀ : C₀
f : F.obj X₀ ⟶ X
Y₀ : C₀
g : Y₀ ⟶ X₀
i : (data X).I₀
p : F.obj Y₀ ⟶ F.obj ((data X).X i)
fac : p ≫ (data X).f i = F.map g ≫ f
hg : ((data X).sieve f).arrows g
⊢ Eq (α := presheafObj data G₀ X ⟶ G₀.obj (P := Presheaf.IsSheaf J₀).obj (op Y₀))
    (restriction.res data G₀ (X₀ := X₀) (f := f) (g := GrothendieckTopology.Cover.Arrow.f { Y := Y₀, f := g, hf := hg })
      (Nonempty.some ⋯))
    (presheafObjπ data G₀ X i ≫ IsDenseSubsite.mapPreimage J F G₀ p)
have fac₁ : CategoryStruct.comp (Y :=
    F.obj
      ((data X).X
        (SieveStruct.i₀ (X₀ := X₀) (f := f) (Y₀ :=
          GrothendieckTopology.Cover.Arrow.Y (J := J₀) (S :=
            Subtype.mk (p := fun (S : Sieve.{v₀, u₀} X₀) ↦ S ∈ (J₀ : (a : C₀) → Set (Sieve a)) X₀) ((data X).sieve f) sorry)
            { Y := Y₀, f := g, hf := hg })
          (g := GrothendieckTopology.Cover.Arrow.f { Y := Y₀, f := g, hf := hg }) (Nonempty.some sorry))))
    (SieveStruct.q (Nonempty.some sorry))
    ((data X).f
      (SieveStruct.i₀ (X₀ := X₀) (f := f) (Y₀ :=
        GrothendieckTopology.Cover.Arrow.Y (J := J₀) (S :=
          Subtype.mk (p := fun (S : Sieve.{v₀, u₀} X₀) ↦ S ∈ (J₀ : (a : C₀) → Set (Sieve a)) X₀) ((data X).sieve f) sorry)
          { Y := Y₀, f := g, hf := hg })
        (g := GrothendieckTopology.Cover.Arrow.f { Y := Y₀, f := g, hf := hg }) (Nonempty.some sorry))) =
  p ≫ (data X).f i
Tactic: apply presheafObj_mapPreimage_condition
-- 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
X₀ : C₀
f : F.obj X₀ ⟶ X
Y₀ : C₀
g : Y₀ ⟶ X₀
i : (data X).I₀
p : F.obj Y₀ ⟶ F.obj ((data X).X i)
fac : p ≫ (data X).f i = F.map g ≫ f
hg : ((data X).sieve f).arrows g
⊢ CategoryStruct.comp (Y :=
      F.obj
        ((data X).X
          (SieveStruct.i₀ (X₀ := X₀) (f := f) (Y₀ :=
            GrothendieckTopology.Cover.Arrow.Y (J := J₀) (S :=
              Subtype.mk (p := fun (S : Sieve.{v₀, u₀} X₀) ↦ S ∈ (J₀ : (a : C₀) → Set (Sieve a)) X₀) ((data X).sieve f)
                ⋯)
              { Y := Y₀, f := g, hf := hg })
            (g := GrothendieckTopology.Cover.Arrow.f { Y := Y₀, f := g, hf := hg }) (Nonempty.some ⋯))))
      (SieveStruct.q (Nonempty.some ⋯))
      ((data X).f
        (SieveStruct.i₀ (X₀ := X₀) (f := f) (Y₀ :=
          GrothendieckTopology.Cover.Arrow.Y (J := J₀) (S :=
            Subtype.mk (p := fun (S : Sieve.{v₀, u₀} X₀) ↦ S ∈ (J₀ : (a : C₀) → Set (Sieve a)) X₀) ((data X).sieve f) ⋯)
            { Y := Y₀, f := g, hf := hg })
          (g := GrothendieckTopology.Cover.Arrow.f { Y := Y₀, f := g, hf := hg }) (Nonempty.some ⋯))) =
    p ≫ (data X).f i
have fac₁ : F.map g ≫ f = p ≫ (data X).f i
Tactic: rw [hg.some.fac, fac]