awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.DenseSubsite.Basic (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
D : Type u_2
inst_1 : Category.{v_2, u_2} D
J : GrothendieckTopology.{v_1, u_1} C
K : GrothendieckTopology.{v_2, u_2} D
G : Functor.{v_1, v_2, u_1, u_2} C D
A : Type u_4
inst_2 : Category.{v_4, u_4} A
inst_3 : IsDenseSubsite J K G
F : Sheaf.{v_1, v_4, u_1, u_4} J A
X Y Z : C
f : G.obj X ⟶ G.obj Y
g : G.obj Y ⟶ G.obj Z
T₀ : C
a : T₀ ⟶ X
b : T₀ ⟶ Y
fac₁ : G.map b = G.map a ≫ f
U₀ : C
c : U₀ ⟶ T₀
d : U₀ ⟶ Z
fac₂ : G.map d = G.map (c ≫ b) ≫ g
⊢ Eq (α :=
    F.obj (P := Presheaf.IsSheaf J).obj (op Z) ⟶
      F.obj (P := Presheaf.IsSheaf J).obj
        (op
          (GrothendieckTopology.Cover.Arrow.Y (J := J) (S :=
            Subtype.mk (p := fun (S : Sieve.{v_1, u_1} T₀) ↦ S ∈ (J : (a : C) → Set (Sieve a)) T₀)
              (Sieve.pullback b (G.imageSieve g)) ⋯)
            { Y := U₀, f := c, hf := ⋯ })))
    (CategoryStruct.comp (Y := F.obj (P := Presheaf.IsSheaf J).obj (op T₀))
      (mapPreimage K G F (f ≫ g) ≫
          F.obj (P := Presheaf.IsSheaf J).map
            (Quiver.Hom.op (GrothendieckTopology.Cover.Arrow.f { Y := T₀, f := a, hf := ⋯ })) :
        F.obj (P := Presheaf.IsSheaf J).obj (op Z) ⟶
          F.obj (P := Presheaf.IsSheaf J).obj
            (op
              (GrothendieckTopology.Cover.Arrow.Y (J := J) (S :=
                Subtype.mk (p := fun (S : Sieve.{v_1, u_1} X) ↦ S ∈ (J : (a : C) → Set (Sieve a)) X) (G.imageSieve f) ⋯)
                { Y := T₀, f := a, hf := ⋯ })))
      (F.obj (P := Presheaf.IsSheaf J).map
        (Quiver.Hom.op (GrothendieckTopology.Cover.Arrow.f { Y := U₀, f := c, hf := ⋯ }))))
    (CategoryStruct.comp (Y := F.obj (P := Presheaf.IsSheaf J).obj (op T₀))
      ((mapPreimage K G F g ≫ mapPreimage K G F f) ≫
          F.obj (P := Presheaf.IsSheaf J).map
            (Quiver.Hom.op (GrothendieckTopology.Cover.Arrow.f { Y := T₀, f := a, hf := ⋯ })) :
        F.obj (P := Presheaf.IsSheaf J).obj (op Z) ⟶
          F.obj (P := Presheaf.IsSheaf J).obj
            (op
              (GrothendieckTopology.Cover.Arrow.Y (J := J) (S :=
                Subtype.mk (p := fun (S : Sieve.{v_1, u_1} X) ↦ S ∈ (J : (a : C) → Set (Sieve a)) X) (G.imageSieve f) ⋯)
                { Y := T₀, f := a, hf := ⋯ })))
      (F.obj (P := Presheaf.IsSheaf J).map
        (Quiver.Hom.op (GrothendieckTopology.Cover.Arrow.f { Y := U₀, f := c, hf := ⋯ }))))
have h : (mapPreimage K G F (f ≫ g) ≫ F.obj (P := Presheaf.IsSheaf J).map a.op) ≫ F.obj (P := Presheaf.IsSheaf J).map c.op =
  ((mapPreimage K G F g ≫ mapPreimage K G F f) ≫ F.obj (P := Presheaf.IsSheaf J).map a.op) ≫
    F.obj (P := Presheaf.IsSheaf J).map c.op
Tactic: dsimp
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.DenseSubsite.Basic (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
D : Type u_2
inst_1 : Category.{v_2, u_2} D
J : GrothendieckTopology.{v_1, u_1} C
K : GrothendieckTopology.{v_2, u_2} D
G : Functor.{v_1, v_2, u_1, u_2} C D
A : Type u_4
inst_2 : Category.{v_4, u_4} A
inst_3 : IsDenseSubsite J K G
F : Sheaf.{v_1, v_4, u_1, u_4} J A
X Y Z : C
f : G.obj X ⟶ G.obj Y
g : G.obj Y ⟶ G.obj Z
T₀ : C
a : T₀ ⟶ X
b : T₀ ⟶ Y
fac₁ : G.map b = G.map a ≫ f
U₀ : C
c : U₀ ⟶ T₀
d : U₀ ⟶ Z
fac₂ : G.map d = G.map (c ≫ b) ≫ g
⊢ (mapPreimage K G F (f ≫ g) ≫ F.obj (P := Presheaf.IsSheaf J).map a.op) ≫ F.obj (P := Presheaf.IsSheaf J).map c.op =
    ((mapPreimage K G F g ≫ mapPreimage K G F f) ≫ F.obj (P := Presheaf.IsSheaf J).map a.op) ≫
      F.obj (P := Presheaf.IsSheaf J).map c.op
have h : mapPreimage K G F (f ≫ g) ≫ F.obj (P := Presheaf.IsSheaf J).map (c ≫ a).op =
  mapPreimage K G F g ≫ mapPreimage K G F f ≫ F.obj (P := Presheaf.IsSheaf J).map (c ≫ a).op
Tactic: simp only [Category.assoc, ← Functor.map_comp, ← op_comp]
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.DenseSubsite.Basic (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
D : Type u_2
inst_1 : Category.{v_2, u_2} D
J : GrothendieckTopology.{v_1, u_1} C
K : GrothendieckTopology.{v_2, u_2} D
G : Functor.{v_1, v_2, u_1, u_2} C D
A : Type u_4
inst_2 : Category.{v_4, u_4} A
inst_3 : IsDenseSubsite J K G
F : Sheaf.{v_1, v_4, u_1, u_4} J A
X Y Z : C
f : G.obj X ⟶ G.obj Y
g : G.obj Y ⟶ G.obj Z
T₀ : C
a : T₀ ⟶ X
b : T₀ ⟶ Y
fac₁ : G.map b = G.map a ≫ f
U₀ : C
c : U₀ ⟶ T₀
d : U₀ ⟶ Z
fac₂ : G.map d = G.map (c ≫ b) ≫ g
⊢ mapPreimage K G F (f ≫ g) ≫ F.obj (P := Presheaf.IsSheaf J).map (c ≫ a).op =
    mapPreimage K G F g ≫ mapPreimage K G F f ≫ F.obj (P := Presheaf.IsSheaf J).map (c ≫ a).op
have h : G.map (c ≫ b) ≫ g = G.map d
Tactic: rw [mapPreimage_map_of_fac K G F (f ≫ g) (c ≫ a) d,
        mapPreimage_map_of_fac K G F f (c ≫ a) (c ≫ b),
        mapPreimage_map_of_fac K G F g (c ≫ b) d]
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.DenseSubsite.Basic (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
D : Type u_2
inst_1 : Category.{v_2, u_2} D
J : GrothendieckTopology.{v_1, u_1} C
K : GrothendieckTopology.{v_2, u_2} D
G : Functor.{v_1, v_2, u_1, u_2} C D
A : Type u_4
inst_2 : Category.{v_4, u_4} A
inst_3 : IsDenseSubsite J K G
F : Sheaf.{v_1, v_4, u_1, u_4} J A
X Y Z : C
f : G.obj X ⟶ G.obj Y
g : G.obj Y ⟶ G.obj Z
T₀ : C
a : T₀ ⟶ X
b : T₀ ⟶ Y
fac₁ : G.map b = G.map a ≫ f
U₀ : C
c : U₀ ⟶ T₀
d : U₀ ⟶ Z
fac₂ : G.map d = G.map (c ≫ b) ≫ g
h : G.map (c ≫ b) ≫ g = G.map d
⊢ mapPreimage K G F (f ≫ g) ≫ F.obj (P := Presheaf.IsSheaf J).map (c ≫ a).op =
    mapPreimage K G F g ≫ mapPreimage K G F f ≫ F.obj (P := Presheaf.IsSheaf J).map (c ≫ a).op
have h₁ : G.map (c ≫ a) ≫ f = G.map (c ≫ b)
Tactic: rw [mapPreimage_map_of_fac K G F (f ≫ g) (c ≫ a) d,
        mapPreimage_map_of_fac K G F f (c ≫ a) (c ≫ b),
        mapPreimage_map_of_fac K G F g (c ≫ b) d]
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.DenseSubsite.Basic (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
D : Type u_2
inst_1 : Category.{v_2, u_2} D
J : GrothendieckTopology.{v_1, u_1} C
K : GrothendieckTopology.{v_2, u_2} D
G : Functor.{v_1, v_2, u_1, u_2} C D
A : Type u_4
inst_2 : Category.{v_4, u_4} A
inst_3 : IsDenseSubsite J K G
F : Sheaf.{v_1, v_4, u_1, u_4} J A
X Y Z : C
f : G.obj X ⟶ G.obj Y
g : G.obj Y ⟶ G.obj Z
T₀ : C
a : T₀ ⟶ X
b : T₀ ⟶ Y
fac₁ : G.map b = G.map a ≫ f
U₀ : C
c : U₀ ⟶ T₀
d : U₀ ⟶ Z
fac₂ : G.map d = G.map (c ≫ b) ≫ g
h : G.map (c ≫ b) ≫ g = G.map d
h₁ : G.map (c ≫ a) ≫ f = G.map (c ≫ b)
⊢ mapPreimage K G F (f ≫ g) ≫ F.obj (P := Presheaf.IsSheaf J).map (c ≫ a).op =
    mapPreimage K G F g ≫ mapPreimage K G F f ≫ F.obj (P := Presheaf.IsSheaf J).map (c ≫ a).op
have h₂ : G.map (c ≫ a) ≫ f ≫ g = G.map d
Tactic: rw [mapPreimage_map_of_fac K G F (f ≫ g) (c ≫ a) d,
        mapPreimage_map_of_fac K G F f (c ≫ a) (c ≫ b),
        mapPreimage_map_of_fac K G F g (c ≫ b) d]
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.DenseSubsite.Basic (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
D : Type u_2
inst_1 : Category.{v_2, u_2} D
J : GrothendieckTopology.{v_1, u_1} C
K : GrothendieckTopology.{v_2, u_2} D
G : Functor.{v_1, v_2, u_1, u_2} C D
A : Type u_4
inst_2 : Category.{v_4, u_4} A
inst_3 : IsDenseSubsite J K G
F : Sheaf.{v_1, v_4, u_1, u_4} J A
X Y Z : C
f : G.obj X ⟶ G.obj Y
g : Z ⟶ X
⊢ mapPreimage K G F f ≫ F.obj (P := Presheaf.IsSheaf J).map g.op = mapPreimage K G F (G.map g ≫ f)
have rw : mapPreimage K G F f ≫ F.obj (P := Presheaf.IsSheaf J).map g.op = mapPreimage K G F f ≫ mapPreimage K G F (G.map g)
Tactic: rw [mapPreimage_comp, mapPreimage_map]
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.DenseSubsite.Basic (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
D : Type u_2
inst_1 : Category.{v_2, u_2} D
J : GrothendieckTopology.{v_1, u_1} C
K : GrothendieckTopology.{v_2, u_2} D
G : Functor.{v_1, v_2, u_1, u_2} C D
A : Type u_4
inst_2 : Category.{v_4, u_4} A
inst_3 : IsDenseSubsite J K G
F : Sheaf.{v_1, v_4, u_1, u_4} J A
X Y Z : C
f : G.obj X ⟶ G.obj Y
g : Z ⟶ X
rw : mapPreimage K G F f ≫ F.obj (P := Presheaf.IsSheaf J).map g.op = mapPreimage K G F f ≫ mapPreimage K G F (G.map g)
⊢ mapPreimage K G F f ≫ F.obj (P := Presheaf.IsSheaf J).map g.op = mapPreimage K G F (G.map g ≫ f)
have rw₁ : mapPreimage K G F f ≫ F.obj (P := Presheaf.IsSheaf J).map g.op =
  mapPreimage K G F f ≫ F.obj (P := Presheaf.IsSheaf J).map g.op
Tactic: rw [mapPreimage_comp, mapPreimage_map]
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.DenseSubsite.Basic (hypothesis)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
D : Type u_2
inst_1 : Category.{v_2, u_2} D
J : GrothendieckTopology.{v_1, u_1} C
K : GrothendieckTopology.{v_2, u_2} D
G : Functor.{v_1, v_2, u_1, u_2} C D
A : Type u_4
inst_2 : Category.{v_4, u_4} A
inst_3 : IsDenseSubsite J K G
inst_4 : IsEquivalence (C := Sheaf.{v_2, v_4, u_2, u_4} K A) (G.sheafPushforwardContinuous A J K)
inst_5 : HasWeakSheafify.{v_1, v_4, u_1, u_4} J A
P₁ P₂ : Functor.{v_2, v_4, u_2, u_4} Dᵒᵖ A
f : P₁ ⟶ P₂
Q : Sheaf.{v_2, v_4, u_2, u_4} K A
g : obj (sheafifyOfIsEquivalence J K G A) P₂ ⟶ Q
⊢ Eq (α := P₁ ⟶ Q.obj (P := Presheaf.IsSheaf K))
    ((sheafifyHomEquivOfIsEquivalence J K G : (a : (sheafifyOfIsEquivalence J K G A).obj P₁ ⟶ Q) → P₁ ⟶ Q.obj)
        (map (sheafifyOfIsEquivalence J K G A) f ≫ g) :
      P₁ ⟶ Q.obj (P := Presheaf.IsSheaf K))
    (f ≫ (sheafifyHomEquivOfIsEquivalence J K G : (a : (sheafifyOfIsEquivalence J K G A).obj P₂ ⟶ Q) → P₂ ⟶ Q.obj) g :
      P₁ ⟶ Q.obj (P := Presheaf.IsSheaf K))
have this : G.IsLocallyFull K
Tactic: have := IsDenseSubsite.isLocallyFull J K G
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.DenseSubsite.Basic (hypothesis)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
D : Type u_2
inst_1 : Category.{v_2, u_2} D
J : GrothendieckTopology.{v_1, u_1} C
K : GrothendieckTopology.{v_2, u_2} D
G : Functor.{v_1, v_2, u_1, u_2} C D
A : Type u_4
inst_2 : Category.{v_4, u_4} A
inst_3 : IsDenseSubsite J K G
inst_4 : IsEquivalence (C := Sheaf.{v_2, v_4, u_2, u_4} K A) (G.sheafPushforwardContinuous A J K)
inst_5 : HasWeakSheafify.{v_1, v_4, u_1, u_4} J A
P₁ P₂ : Functor.{v_2, v_4, u_2, u_4} Dᵒᵖ A
f : P₁ ⟶ P₂
Q : Sheaf.{v_2, v_4, u_2, u_4} K A
g : obj (sheafifyOfIsEquivalence J K G A) P₂ ⟶ Q
this : G.IsLocallyFull K
⊢ Eq (α := P₁ ⟶ Q.obj (P := Presheaf.IsSheaf K))
    ((sheafifyHomEquivOfIsEquivalence J K G : (a : (sheafifyOfIsEquivalence J K G A).obj P₁ ⟶ Q) → P₁ ⟶ Q.obj)
        (map (sheafifyOfIsEquivalence J K G A) f ≫ g) :
      P₁ ⟶ Q.obj (P := Presheaf.IsSheaf K))
    (f ≫ (sheafifyHomEquivOfIsEquivalence J K G : (a : (sheafifyOfIsEquivalence J K G A).obj P₂ ⟶ Q) → P₂ ⟶ Q.obj) g :
      P₁ ⟶ Q.obj (P := Presheaf.IsSheaf K))
have this₁ : G.IsCoverDense K
Tactic: have := IsDenseSubsite.isCoverDense J K G
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.DenseSubsite.Basic (hypothesis)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
D : Type u_2
inst_1 : Category.{v_2, u_2} D
J : GrothendieckTopology.{v_1, u_1} C
K : GrothendieckTopology.{v_2, u_2} D
G : Functor.{v_1, v_2, u_1, u_2} C D
A : Type u_4
inst_2 : Category.{v_4, u_4} A
inst_3 : IsDenseSubsite J K G
inst_4 : IsEquivalence (C := Sheaf.{v_2, v_4, u_2, u_4} K A) (G.sheafPushforwardContinuous A J K)
inst_5 : HasWeakSheafify.{v_1, v_4, u_1, u_4} J A
P : Functor.{v_2, v_4, u_2, u_4} Dᵒᵖ A
Q₁ Q₂ : Sheaf.{v_2, v_4, u_2, u_4} K A
f : obj (sheafifyOfIsEquivalence J K G A) P ⟶ Q₁
g : Q₁ ⟶ Q₂
⊢ Eq (α := P ⟶ Q₂.obj (P := Presheaf.IsSheaf K))
    ((sheafifyHomEquivOfIsEquivalence J K G : (a : (sheafifyOfIsEquivalence J K G A).obj P ⟶ Q₂) → P ⟶ Q₂.obj) (f ≫ g) :
      P ⟶ Q₂.obj (P := Presheaf.IsSheaf K))
    ((sheafifyHomEquivOfIsEquivalence J K G : (a : (sheafifyOfIsEquivalence J K G A).obj P ⟶ Q₁) → P ⟶ Q₁.obj) f ≫
        g.hom :
      P ⟶ Q₂.obj (P := Presheaf.IsSheaf K))
have this : G.IsLocallyFull K
Tactic: have := IsDenseSubsite.isLocallyFull J K G
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.DenseSubsite.Basic (hypothesis)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
D : Type u_2
inst_1 : Category.{v_2, u_2} D
J : GrothendieckTopology.{v_1, u_1} C
K : GrothendieckTopology.{v_2, u_2} D
G : Functor.{v_1, v_2, u_1, u_2} C D
A : Type u_4
inst_2 : Category.{v_4, u_4} A
inst_3 : IsDenseSubsite J K G
inst_4 : IsEquivalence (C := Sheaf.{v_2, v_4, u_2, u_4} K A) (G.sheafPushforwardContinuous A J K)
inst_5 : HasWeakSheafify.{v_1, v_4, u_1, u_4} J A
P : Functor.{v_2, v_4, u_2, u_4} Dᵒᵖ A
Q₁ Q₂ : Sheaf.{v_2, v_4, u_2, u_4} K A
f : obj (sheafifyOfIsEquivalence J K G A) P ⟶ Q₁
g : Q₁ ⟶ Q₂
this : G.IsLocallyFull K
⊢ Eq (α := P ⟶ Q₂.obj (P := Presheaf.IsSheaf K))
    ((sheafifyHomEquivOfIsEquivalence J K G : (a : (sheafifyOfIsEquivalence J K G A).obj P ⟶ Q₂) → P ⟶ Q₂.obj) (f ≫ g) :
      P ⟶ Q₂.obj (P := Presheaf.IsSheaf K))
    ((sheafifyHomEquivOfIsEquivalence J K G : (a : (sheafifyOfIsEquivalence J K G A).obj P ⟶ Q₁) → P ⟶ Q₁.obj) f ≫
        g.hom :
      P ⟶ Q₂.obj (P := Presheaf.IsSheaf K))
have this₁ : G.IsCoverDense K
Tactic: have := IsDenseSubsite.isCoverDense J K G
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.DenseSubsite.Basic (hypothesis)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
D : Type u_2
inst_1 : Category.{v_2, u_2} D
J : GrothendieckTopology.{v_1, u_1} C
K : GrothendieckTopology.{v_2, u_2} D
G : Functor.{v_1, v_2, u_1, u_2} C D
A : Type u_4
inst_2 : Category.{v_4, u_4} A
inst_3 : IsDenseSubsite J K G
inst_4 : IsEquivalence (C := Sheaf.{v_2, v_4, u_2, u_4} K A) (G.sheafPushforwardContinuous A J K)
inst_5 : HasSheafify.{v_1, v_4, u_1, u_4} J A
inst_6 : HasFiniteLimits.{v_4, u_4} A
⊢ HasSheafify.{v_2, v_4, u_2, u_4} K A
have this : PreservesFiniteLimits (C := Functor.{v_1, v_4, u_1, u_4} Cᵒᵖ A)
  (presheafToSheaf J A ⋙ inv (G.sheafPushforwardContinuous A J K))
Tactic: have : PreservesFiniteLimits (presheafToSheaf J A ⋙
    (G.sheafPushforwardContinuous A J K).inv) := by
    apply comp_preservesFiniteLimits
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.DenseSubsite.Basic (hypothesis)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
D : Type u_2
inst_1 : Category.{v_2, u_2} D
J : GrothendieckTopology.{v_1, u_1} C
K : GrothendieckTopology.{v_2, u_2} D
G : Functor.{v_1, v_2, u_1, u_2} C D
A : Type u_4
inst_2 : Category.{v_4, u_4} A
inst_3 : IsDenseSubsite J K G
inst_4 : IsEquivalence (C := Sheaf.{v_2, v_4, u_2, u_4} K A) (G.sheafPushforwardContinuous A J K)
inst_5 : HasSheafify.{v_1, v_4, u_1, u_4} J A
inst_6 : HasFiniteLimits.{v_4, u_4} A
this :
  PreservesFiniteLimits (C := Functor.{v_1, v_4, u_1, u_4} Cᵒᵖ A)
    (presheafToSheaf J A ⋙ inv (G.sheafPushforwardContinuous A J K))
⊢ HasSheafify.{v_2, v_4, u_2, u_4} K A
have this₁ : PreservesFiniteLimits (C := Functor.{v_2, v_4, u_2, u_4} Dᵒᵖ A) (sheafifyOfIsEquivalence J K G A)
Tactic: have : PreservesFiniteLimits (sheafifyOfIsEquivalence J K G A) := by
    apply comp_preservesFiniteLimits
-- 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
inst_2 : F.IsOneHypercoverDense J₀ J
inst_3 : F.IsLocallyFull J
inst_4 : F.IsLocallyFaithful J
h :
  ∀ {X₀ : C₀} {S₀ : Sieve.{v₀, u₀} X₀},
    Membership.mem (γ := Set (Sieve.{v, u} (F.obj X₀))) (J.sieves (F.obj X₀)) (Sieve.functorPushforward F S₀) ↔
      S₀ ∈ J₀.sieves X₀
X : C
⊢ Sieve.coverByImage F X ∈ (J : (a : C) → Set (Sieve a)) X
have refine : PreZeroHypercover.sieve₀.{w, v, u} (F.oneHypercoverDenseData J₀ J X).toPreOneHypercover.toPreZeroHypercover ≤
  Sieve.coverByImage F X
Tactic: refine J.superset_covering ?_ (F.oneHypercoverDenseData J₀ J X).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
inst_2 : F.IsOneHypercoverDense J₀ J
inst_3 : F.IsLocallyFull J
inst_4 : F.IsLocallyFaithful J
h_1 :
  ∀ {X₀ : C₀} {S₀ : Sieve.{v₀, u₀} X₀},
    Membership.mem (γ := Set (Sieve.{v, u} (F.obj X₀))) (J.sieves (F.obj X₀)) (Sieve.functorPushforward F S₀) ↔
      S₀ ∈ J₀.sieves X₀
X Y w : C
a : Y ⟶ w
w_1 : w ⟶ X
h :
  Presieve.ofArrows (F.oneHypercoverDenseData J₀ J X).toPreOneHypercover.X
    (F.oneHypercoverDenseData J₀ J X).toPreOneHypercover.f w_1
⊢ (Sieve.coverByImage F X).arrows (a ≫ w_1)
have mk : ∀ (i : (F.oneHypercoverDenseData J₀ J X).toPreOneHypercover.I₀)
  (a : Y ⟶ (F.oneHypercoverDenseData J₀ J X).toPreOneHypercover.X i),
  (Sieve.coverByImage F X).arrows (a ≫ (F.oneHypercoverDenseData J₀ J X).toPreOneHypercover.f i)
Tactic: cases h
Cleared hypotheses: w, a, w_1, h
-- 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
inst_2 : IsDenseSubsite J₀ J F
inst_3 : HasPullbacks.{v, u} C
inst_4 : F.Full
inst_5 : F.Faithful
hF :
  ∀ (S : C),
    ∃ (ι : Type w) (U : ι → C₀) (f : (i : ι) → F.obj (U i) ⟶ S),
      Sieve.ofArrows (fun (i : ι) ↦ F.obj (U i)) f ∈ (J : (a : C) → Set (Sieve a)) S
S : C
⊢ Nonempty (OneHypercoverDenseData.{w, v₀, v, u₀, u} F J₀ J S)
have choose : ∀ (ι : (S : C) → Type w) (U : (S : C) → ι S → C₀) (f : (S : C) → (i : ι S) → F.obj (U S i) ⟶ S)
  (hf : ∀ (S : C), Sieve.ofArrows (fun (i : ι S) ↦ F.obj (U S i)) (f S) ∈ (J : (a : C) → Set (Sieve a)) S),
  Nonempty (OneHypercoverDenseData.{w, v₀, v, u₀, u} F J₀ J S)
Tactic: choose ι U f hf using hF
Cleared hypotheses: hF
-- 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
inst_2 : IsDenseSubsite J₀ J F
inst_3 : HasPullbacks.{v, u} C
inst_4 : F.Full
inst_5 : F.Faithful
S : C
ι : (S : C) → Type w
U : (S : C) → ι S → C₀
f : (S : C) → (i : ι S) → F.obj (U S i) ⟶ S
hf : ∀ (S : C), Sieve.ofArrows (fun (i : ι S) ↦ F.obj (U S i)) (f S) ∈ (J : (a : C) → Set (Sieve a)) S
i j : ι S
W₀ : C₀
p₁ : W₀ ⟶ U S i
p₂ : W₀ ⟶ U S j
hp : F.map p₁ ≫ f S i = F.map p₂ ≫ f S j
⊢ { I₀ := ι S, X := U S, f := f S, I₁ := fun (i j : ι S) ↦ ι (pullback (f S i) (f S j)),
          Y := fun (i j : ι S) ↦ U (pullback (f S i) (f S j)),
          p₁ := fun (i j : ι S) (k : ι (pullback (f S i) (f S j))) ↦
            F.preimage (f (pullback (f S i) (f S j)) k ≫ pullback.fst (f S i) (f S j)),
          p₂ := fun (i j : ι S) (k : ι (pullback (f S i) (f S j))) ↦
            F.preimage (f (pullback (f S i) (f S j)) k ≫ pullback.snd (f S i) (f S j)),
          w := ⋯ : PreOneHypercoverDenseData.{w, v₀, v, u₀, u} F S }.sieve₁₀
      p₁ p₂ ∈
    (J₀ : (a : C₀) → Set (Sieve a)) W₀
have this : F.IsCoverDense J
Tactic: have := IsDenseSubsite.isCoverDense J₀ J 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
inst_2 : IsDenseSubsite J₀ J F
inst_3 : HasPullbacks.{v, u} C
inst_4 : F.Full
inst_5 : F.Faithful
S : C
ι : (S : C) → Type w
U : (S : C) → ι S → C₀
f : (S : C) → (i : ι S) → F.obj (U S i) ⟶ S
hf : ∀ (S : C), Sieve.ofArrows (fun (i : ι S) ↦ F.obj (U S i)) (f S) ∈ (J : (a : C) → Set (Sieve a)) S
i j : ι S
W₀ : C₀
p₁ : W₀ ⟶ U S i
p₂ : W₀ ⟶ U S j
hp : F.map p₁ ≫ f S i = F.map p₂ ≫ f S j
this : F.IsCoverDense J
⊢ { I₀ := ι S, X := U S, f := f S, I₁ := fun (i j : ι S) ↦ ι (pullback (f S i) (f S j)),
          Y := fun (i j : ι S) ↦ U (pullback (f S i) (f S j)),
          p₁ := fun (i j : ι S) (k : ι (pullback (f S i) (f S j))) ↦
            F.preimage (f (pullback (f S i) (f S j)) k ≫ pullback.fst (f S i) (f S j)),
          p₂ := fun (i j : ι S) (k : ι (pullback (f S i) (f S j))) ↦
            F.preimage (f (pullback (f S i) (f S j)) k ≫ pullback.snd (f S i) (f S j)),
          w := ⋯ : PreOneHypercoverDenseData.{w, v₀, v, u₀, u} F S }.sieve₁₀
      p₁ p₂ ∈
    (J₀ : (a : C₀) → Set (Sieve a)) W₀
have rw : Membership.mem (γ := Set (Sieve.{v, u} (F.obj W₀))) ((J : (a : C) → Set (Sieve a)) (F.obj W₀))
  (Sieve.functorPushforward F
    ({ I₀ := ι S, X := U S, f := f S, I₁ := fun (i j : ι S) ↦ ι (pullback (f S i) (f S j)),
          Y := fun (i j : ι S) ↦ U (pullback (f S i) (f S j)),
          p₁ := fun (i j : ι S) (k : ι (pullback (f S i) (f S j))) ↦
            F.preimage (f (pullback (f S i) (f S j)) k ≫ pullback.fst (f S i) (f S j)),
          p₂ := fun (i j : ι S) (k : ι (pullback (f S i) (f S j))) ↦
            F.preimage (f (pullback (f S i) (f S j)) k ≫ pullback.snd (f S i) (f S j)),
          w := sorry : PreOneHypercoverDenseData.{w, v₀, v, u₀, u} F S }.sieve₁₀
      p₁ p₂))
Tactic: rw [← functorPushforward_mem_iff J₀ J 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
inst_2 : IsDenseSubsite J₀ J F
inst_3 : HasPullbacks.{v, u} C
inst_4 : F.Full
inst_5 : F.Faithful
S : C
ι : (S : C) → Type w
U : (S : C) → ι S → C₀
f : (S : C) → (i : ι S) → F.obj (U S i) ⟶ S
hf : ∀ (S : C), Sieve.ofArrows (fun (i : ι S) ↦ F.obj (U S i)) (f S) ∈ (J : (a : C) → Set (Sieve a)) S
i j : ι S
W₀ : C₀
p₁ : W₀ ⟶ U S i
p₂ : W₀ ⟶ U S j
hp : F.map p₁ ≫ f S i = F.map p₂ ≫ f S j
this : F.IsCoverDense J
⊢ Membership.mem (γ := Set (Sieve.{v, u} (F.obj W₀))) ((J : (a : C) → Set (Sieve a)) (F.obj W₀))
    (Sieve.functorPushforward F
      ({ I₀ := ι S, X := U S, f := f S, I₁ := fun (i j : ι S) ↦ ι (pullback (f S i) (f S j)),
            Y := fun (i j : ι S) ↦ U (pullback (f S i) (f S j)),
            p₁ := fun (i j : ι S) (k : ι (pullback (f S i) (f S j))) ↦
              F.preimage (f (pullback (f S i) (f S j)) k ≫ pullback.fst (f S i) (f S j)),
            p₂ := fun (i j : ι S) (k : ι (pullback (f S i) (f S j))) ↦
              F.preimage (f (pullback (f S i) (f S j)) k ≫ pullback.snd (f S i) (f S j)),
            w := ⋯ : PreOneHypercoverDenseData.{w, v₀, v, u₀, u} F S }.sieve₁₀
        p₁ p₂))
have refine : Sieve.functorPushforward F
    (Sieve.functorPullback F
      (↑(Subtype.mk (p := fun (x : Sieve.{v, u} (F.obj W₀)) ↦
            Membership.mem (γ := Set (Sieve.{v, u} (F.obj W₀))) ((J : (a : C) → Set (Sieve a)) (F.obj W₀)) x)
            (Sieve.pullback (X := pullback (f S i) (f S j)) (pullback.lift (F.map p₁) (F.map p₂) hp)
              (Sieve.ofArrows (fun (i_1 : ι (pullback (f S i) (f S j))) ↦ F.obj (U (pullback (f S i) (f S j)) i_1))
                (f (pullback (f S i) (f S j)))))
            sorry) :
        Sieve (F.obj W₀))) ≤
  Sieve.functorPushforward F
    ({ I₀ := ι S, X := U S, f := f S, I₁ := fun (i j : ι S) ↦ ι (pullback (f S i) (f S j)),
          Y := fun (i j : ι S) ↦ U (pullback (f S i) (f S j)),
          p₁ := fun (i j : ι S) (k : ι (pullback (f S i) (f S j))) ↦
            F.preimage (f (pullback (f S i) (f S j)) k ≫ pullback.fst (f S i) (f S j)),
          p₂ := fun (i j : ι S) (k : ι (pullback (f S i) (f S j))) ↦
            F.preimage (f (pullback (f S i) (f S j)) k ≫ pullback.snd (f S i) (f S j)),
          w := sorry : PreOneHypercoverDenseData.{w, v₀, v, u₀, u} F S }.sieve₁₀
      p₁ p₂)
Tactic: refine J.superset_covering ?_
          (IsCoverDense.functorPullback_pushforward_covering
            ⟨_, J.pullback_stable (pullback.lift _ _ hp) (hf (pullback (f _ i) (f _ j)))⟩)