awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Hypercover.One (goal)
Goal:
C : Type u
inst : Category.{v, u} C
A : Type u_1
inst_1 : Category.{v_1, u_1} A
J : GrothendieckTopology.{v, u} C
S : C
E : OneHypercover.{u_2, v, u} J S
F : Sheaf.{v, v_1, u, u_1} J A
c : Multifork (E.multicospanIndex (F.obj (P := Presheaf.IsSheaf J)))
W : C
i₁ i₂ : E.I₀
p₁ : W ⟶ E.X i₁
p₂ : W ⟶ E.X i₂
w : p₁ ≫ E.f i₁ = p₂ ≫ E.f i₂
⊢ c.ι i₁ ≫ F.obj (P := Presheaf.IsSheaf J).map p₁.op = c.ι i₂ ≫ F.obj (P := Presheaf.IsSheaf J).map p₂.op
have h : ∀
  (I :
    Cover.Arrow (Subtype.mk (p := fun (S : Sieve.{v, u} W) ↦ S ∈ (J : (a : C) → Set (Sieve a)) W) (E.sieve₁ p₁ p₂) sorry)),
  (c.ι i₁ ≫ F.obj (P := Presheaf.IsSheaf J).map p₁.op) ≫ F.obj (P := Presheaf.IsSheaf J).map I.f.op =
    (c.ι i₂ ≫ F.obj (P := Presheaf.IsSheaf J).map p₂.op) ≫ F.obj (P := Presheaf.IsSheaf J).map I.f.op
Tactic: apply F.property.hom_ext ⟨_, E.mem₁ _ _ _ _ w⟩
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Hypercover.One (goal)
Goal:
C : Type u
inst : Category.{v, u} C
A : Type u_1
inst_1 : Category.{v_1, u_1} A
J : GrothendieckTopology.{v, u} C
S : C
E : OneHypercover.{u_2, v, u} J S
F : Sheaf.{v, v_1, u, u_1} J A
c : Multifork (E.multicospanIndex (F.obj (P := Presheaf.IsSheaf J)))
W : C
i₁ i₂ : E.I₀
p₁ : W ⟶ E.X i₁
p₂ : W ⟶ E.X i₂
w : p₁ ≫ E.f i₁ = p₂ ≫ E.f i₂
T : C
g : T ⟶ W
j : E.I₁ i₁ i₂
h : T ⟶ E.Y j
fac₁ : g ≫ p₁ = h ≫ E.p₁ j
fac₂ : g ≫ p₂ = h ≫ E.p₂ j
⊢ Eq (α :=
    c.pt ⟶
      F.obj (P := Presheaf.IsSheaf J).obj
        (Opposite.op
          (Cover.Arrow.Y (J := J) (S :=
            Subtype.mk (p := fun (S : Sieve.{v, u} W) ↦ S ∈ (J : (a : C) → Set (Sieve a)) W) (E.sieve₁ p₁ p₂) ⋯)
            { Y := T, f := g, hf := ⋯ })))
    ((c.ι i₁ ≫ F.obj (P := Presheaf.IsSheaf J).map p₁.op) ≫
      F.obj (P := Presheaf.IsSheaf J).map (Quiver.Hom.op (Cover.Arrow.f { Y := T, f := g, hf := ⋯ })))
    ((c.ι i₂ ≫ F.obj (P := Presheaf.IsSheaf J).map p₂.op) ≫
      F.obj (P := Presheaf.IsSheaf J).map (Quiver.Hom.op (Cover.Arrow.f { Y := T, f := g, hf := ⋯ })))
have h₁ : (c.ι i₁ ≫ F.obj (P := Presheaf.IsSheaf J).map p₁.op) ≫ F.obj (P := Presheaf.IsSheaf J).map g.op =
  (c.ι i₂ ≫ F.obj (P := Presheaf.IsSheaf J).map p₂.op) ≫ F.obj (P := Presheaf.IsSheaf J).map g.op
Tactic: dsimp
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Hypercover.One (goal)
Goal:
C : Type u
inst : Category.{v, u} C
A : Type u_1
inst_1 : Category.{v_1, u_1} A
J : GrothendieckTopology.{v, u} C
S : C
E : OneHypercover.{u_2, v, u} J S
F : Sheaf.{v, v_1, u, u_1} J A
c : Multifork (E.multicospanIndex (F.obj (P := Presheaf.IsSheaf J)))
W : C
i₁ i₂ : E.I₀
p₁ : W ⟶ E.X i₁
p₂ : W ⟶ E.X i₂
w : p₁ ≫ E.f i₁ = p₂ ≫ E.f i₂
T : C
g : T ⟶ W
j : E.I₁ i₁ i₂
h : T ⟶ E.Y j
fac₁ : g ≫ p₁ = h ≫ E.p₁ j
fac₂ : g ≫ p₂ = h ≫ E.p₂ j
⊢ (c.ι i₁ ≫ F.obj (P := Presheaf.IsSheaf J).map p₁.op) ≫ F.obj (P := Presheaf.IsSheaf J).map g.op =
    (c.ι i₂ ≫ F.obj (P := Presheaf.IsSheaf J).map p₂.op) ≫ F.obj (P := Presheaf.IsSheaf J).map g.op
have h₁ : c.ι i₁ ≫ F.obj (P := Presheaf.IsSheaf J).map (h ≫ E.p₁ j).op =
  c.ι i₂ ≫ F.obj (P := Presheaf.IsSheaf J).map (h ≫ E.p₂ j).op
Tactic: simp only [assoc, ← Functor.map_comp, ← op_comp, fac₁, fac₂]
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Hypercover.One (goal)
Goal:
C : Type u
inst : Category.{v, u} C
A : Type u_1
inst_1 : Category.{v_1, u_1} A
J : GrothendieckTopology.{v, u} C
S : C
E : OneHypercover.{u_2, v, u} J S
F : Sheaf.{v, v_1, u, u_1} J A
c : Multifork (E.multicospanIndex (F.obj (P := Presheaf.IsSheaf J)))
W : C
i₁ i₂ : E.I₀
p₁ : W ⟶ E.X i₁
p₂ : W ⟶ E.X i₂
w : p₁ ≫ E.f i₁ = p₂ ≫ E.f i₂
T : C
g : T ⟶ W
j : E.I₁ i₁ i₂
h : T ⟶ E.Y j
fac₁ : g ≫ p₁ = h ≫ E.p₁ j
fac₂ : g ≫ p₂ = h ≫ E.p₂ j
⊢ c.ι i₁ ≫ F.obj (P := Presheaf.IsSheaf J).map (h ≫ E.p₁ j).op =
    c.ι i₂ ≫ F.obj (P := Presheaf.IsSheaf J).map (h ≫ E.p₂ j).op
have h₁ : c.ι i₁ ≫ F.obj (P := Presheaf.IsSheaf J).map (E.p₁ j).op ≫ F.obj (P := Presheaf.IsSheaf J).map h.op =
  c.ι i₂ ≫ F.obj (P := Presheaf.IsSheaf J).map (E.p₂ j).op ≫ F.obj (P := Presheaf.IsSheaf J).map h.op
Tactic: simp only [op_comp, Functor.map_comp]
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Hypercover.One (goal)
Goal:
C : Type u
inst : Category.{v, u} C
A : Type u_1
inst_1 : Category.{v_1, u_1} A
J : GrothendieckTopology.{v, u} C
S : C
E : OneHypercover.{u_2, v, u} J S
F : Sheaf.{v, v_1, u, u_1} J A
c : Multifork (E.multicospanIndex (F.obj (P := Presheaf.IsSheaf J)))
m : c.pt ⟶ (E.multifork (F.obj (P := Presheaf.IsSheaf J))).pt
hm : ∀ (i : E.multicospanShape.L), m ≫ (E.multifork (F.obj (P := Presheaf.IsSheaf J))).ι i = c.ι i
⊢ m = (fun (c : Multifork (E.multicospanIndex (F.obj (P := Presheaf.IsSheaf J)))) ↦ multiforkLift c) c
have apply : ∀ (i : E.I₀),
  m ≫ F.obj (P := Presheaf.IsSheaf J).map (E.f i).op =
    (fun (c : Multifork (E.multicospanIndex (F.obj (P := Presheaf.IsSheaf J)))) ↦ multiforkLift c) c ≫
      F.obj (P := Presheaf.IsSheaf J).map (E.f i).op
Tactic: apply F.property.hom_ext_ofArrows _ E.mem₀
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Hypercover.One (goal)
Goal:
C : Type u
inst : Category.{v, u} C
A : Type u_1
inst_1 : Category.{v_1, u_1} A
J : GrothendieckTopology.{v, u} C
S : C
E : OneHypercover.{u_2, v, u} J S
F : Sheaf.{v, v_1, u, u_1} J A
c : Multifork (E.multicospanIndex (F.obj (P := Presheaf.IsSheaf J)))
m : c.pt ⟶ (E.multifork (F.obj (P := Presheaf.IsSheaf J))).pt
hm : ∀ (i : E.multicospanShape.L), m ≫ (E.multifork (F.obj (P := Presheaf.IsSheaf J))).ι i = c.ι i
i₀ : E.I₀
⊢ m ≫ F.obj (P := Presheaf.IsSheaf J).map (E.f i₀).op =
    multiforkLift c ≫ F.obj (P := Presheaf.IsSheaf J).map (E.f i₀).op
have rw : m ≫ F.obj (P := Presheaf.IsSheaf J).map (E.f i₀).op = c.ι i₀
Tactic: rw [multiforkLift_map]
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Hypercover.One (hypothesis)
Goal:
C : Type u
inst : Category.{v, u} C
A : Type u_1
inst_1 : Category.{v_1, u_1} A
J : GrothendieckTopology.{v, u} C
S_1 S : C
x x_1 : PreZeroHypercover.I₀.{w, v, u} (PreOneHypercover.toPreZeroHypercover (PreOneHypercover.trivial S))
x_2 : C
x_3 : x_2 ⟶ PreZeroHypercover.X (PreOneHypercover.toPreZeroHypercover (PreOneHypercover.trivial S)) x
x_4 : x_2 ⟶ PreZeroHypercover.X (PreOneHypercover.toPreZeroHypercover (PreOneHypercover.trivial S)) x_1
h :
  x_3 ≫ PreZeroHypercover.f (PreOneHypercover.toPreZeroHypercover (PreOneHypercover.trivial S)) x =
    x_4 ≫ PreZeroHypercover.f (PreOneHypercover.toPreZeroHypercover (PreOneHypercover.trivial S)) x_1
⊢ PreOneHypercover.sieve₁ (PreOneHypercover.trivial S) x_3 x_4 ∈ (J : (a : C) → Set (Sieve a)) x_2
have h₁ : x_3 = x_4
Tactic: simp only [PreOneHypercover.trivial_toPreZeroHypercover, PreZeroHypercover.singleton_X,
      PreZeroHypercover.singleton_f, Category.comp_id] at h
Cleared hypotheses: h
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Hypercover.One (goal)
Goal:
C : Type u
inst : Category.{v, u} C
A : Type u_1
inst_1 : Category.{v_1, u_1} A
J : GrothendieckTopology.{v, u} C
S_1 S : C
x x_1 : PreZeroHypercover.I₀.{w, v, u} (PreOneHypercover.toPreZeroHypercover (PreOneHypercover.trivial S))
x_2 : C
x_3 : x_2 ⟶ PreZeroHypercover.X (PreOneHypercover.toPreZeroHypercover (PreOneHypercover.trivial S)) x
x_4 : x_2 ⟶ PreZeroHypercover.X (PreOneHypercover.toPreZeroHypercover (PreOneHypercover.trivial S)) x_1
h : x_3 = x_4
⊢ PreOneHypercover.sieve₁ (PreOneHypercover.trivial S) x_3 x_4 ∈ (J : (a : C) → Set (Sieve a)) x_2
have subst : PreOneHypercover.sieve₁ (PreOneHypercover.trivial S) x_3 x_3 ∈ (J : (a : C) → Set (Sieve a)) x_2
Tactic: subst h
Cleared hypotheses: x_4, h
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Hypercover.One (goal)
Goal:
C : Type u
inst : Category.{v, u} C
A : Type u_1
inst_1 : Category.{v_1, u_1} A
J : GrothendieckTopology.{v, u} C
S : C
inst_2 : HasPullbacks.{v, u} C
E : OneHypercover.{u_2, v, u} J S
F : OneHypercover.{u_3, v, u} J S
inst_3 : ∀ (i : E.I₀) (j : F.I₀), HasPullback (E.f i) (F.f j)
inst_4 : ∀ (i j : E.I₀) (k : E.I₁ i j) (a b : F.I₀) (l : F.I₁ a b), HasPullback (E.p₁ k ≫ E.f i) (F.p₁ l ≫ F.f a)
i₁ i₂ : (PreOneHypercover.inter.{v, u, u_2, u_3} E.toPreOneHypercover F.toPreOneHypercover).I₀
W : C
p₁ : W ⟶ (PreOneHypercover.inter.{v, u, u_2, u_3} E.toPreOneHypercover F.toPreOneHypercover).X i₁
p₂ : W ⟶ (PreOneHypercover.inter.{v, u, u_2, u_3} E.toPreOneHypercover F.toPreOneHypercover).X i₂
h :
  p₁ ≫ (PreOneHypercover.inter.{v, u, u_2, u_3} E.toPreOneHypercover F.toPreOneHypercover).f i₁ =
    p₂ ≫ (PreOneHypercover.inter.{v, u, u_2, u_3} E.toPreOneHypercover F.toPreOneHypercover).f i₂
⊢ (PreOneHypercover.inter.{v, u, u_2, u_3} E.toPreOneHypercover F.toPreOneHypercover).sieve₁ p₁ p₂ ∈
    (J : (a : C) → Set (Sieve a)) W
have rw : (Sieve.bind (E.sieve₁ (p₁ ≫ pullback.fst (E.f i₁.1) (F.f i₁.2)) (p₂ ≫ pullback.fst (E.f i₂.1) (F.f i₂.2))).arrows
    fun (x : C) (f : x ⟶ W)
      (x_1 : (E.sieve₁ (p₁ ≫ pullback.fst (E.f i₁.1) (F.f i₁.2)) (p₂ ≫ pullback.fst (E.f i₂.1) (F.f i₂.2))).arrows f) ↦
    Sieve.pullback f (F.sieve₁ (p₁ ≫ pullback.snd (E.f i₁.1) (F.f i₁.2)) (p₂ ≫ pullback.snd (E.f i₂.1) (F.f i₂.2)))) ∈
  (J : (a : C) → Set (Sieve a)) W
Tactic: rw [PreOneHypercover.sieve₁_inter h]
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Hypercover.One (goal)
Goal:
C : Type u
inst : Category.{v, u} C
A : Type u_1
inst_1 : Category.{v_1, u_1} A
J : GrothendieckTopology.{v, u} C
S : C
inst_2 : HasPullbacks.{v, u} C
E : OneHypercover.{u_2, v, u} J S
F : OneHypercover.{u_3, v, u} J S
inst_3 : ∀ (i : E.I₀) (j : F.I₀), HasPullback (E.f i) (F.f j)
inst_4 : ∀ (i j : E.I₀) (k : E.I₁ i j) (a b : F.I₀) (l : F.I₁ a b), HasPullback (E.p₁ k ≫ E.f i) (F.p₁ l ≫ F.f a)
i₁ i₂ : (PreOneHypercover.inter.{v, u, u_2, u_3} E.toPreOneHypercover F.toPreOneHypercover).I₀
W : C
p₁ : W ⟶ (PreOneHypercover.inter.{v, u, u_2, u_3} E.toPreOneHypercover F.toPreOneHypercover).X i₁
p₂ : W ⟶ (PreOneHypercover.inter.{v, u, u_2, u_3} E.toPreOneHypercover F.toPreOneHypercover).X i₂
h :
  p₁ ≫ (PreOneHypercover.inter.{v, u, u_2, u_3} E.toPreOneHypercover F.toPreOneHypercover).f i₁ =
    p₂ ≫ (PreOneHypercover.inter.{v, u, u_2, u_3} E.toPreOneHypercover F.toPreOneHypercover).f i₂
⊢ (Sieve.bind (E.sieve₁ (p₁ ≫ pullback.fst (E.f i₁.1) (F.f i₁.2)) (p₂ ≫ pullback.fst (E.f i₂.1) (F.f i₂.2))).arrows
      fun (x : C) (f : x ⟶ W)
        (x_1 :
          (E.sieve₁ (p₁ ≫ pullback.fst (E.f i₁.1) (F.f i₁.2)) (p₂ ≫ pullback.fst (E.f i₂.1) (F.f i₂.2))).arrows f) ↦
      Sieve.pullback f (F.sieve₁ (p₁ ≫ pullback.snd (E.f i₁.1) (F.f i₁.2)) (p₂ ≫ pullback.snd (E.f i₂.1) (F.f i₂.2)))) ∈
    (J : (a : C) → Set (Sieve a)) W
have refine : ∀ (x : C) (x_1 : x ⟶ W),
  (E.sieve₁ (p₁ ≫ pullback.fst (E.f i₁.1) (F.f i₁.2)) (p₂ ≫ pullback.fst (E.f i₂.1) (F.f i₂.2))).arrows x_1 →
    Sieve.pullback x_1 (F.sieve₁ (p₁ ≫ pullback.snd (E.f i₁.1) (F.f i₁.2)) (p₂ ≫ pullback.snd (E.f i₂.1) (F.f i₂.2))) ∈
      (J : (a : C) → Set (Sieve a)) x
Tactic: refine J.bind_covering (E.mem₁ _ _ _ _ (by simpa using h)) fun _ _ _ ↦ ?_
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Hypercover.One (goal)
Goal:
C : Type u
inst : Category.{v, u} C
J : GrothendieckTopology.{v, u} C
X : C
S : J.Cover X
⊢ S.preOneHypercover.sieve₀ = (↑S : Sieve X)
have h : ∀ ⦃Y : C⦄ (f : Y ⟶ X), S.preOneHypercover.sieve₀.arrows f ↔ (↑S : Sieve X).arrows f
Tactic: ext Y f
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Hypercover.One (goal)
Goal:
C : Type u
inst : Category.{v, u} C
J : GrothendieckTopology.{v, u} C
X : C
S : J.Cover X
Y : C
f : Y ⟶ X
⊢ S.preOneHypercover.sieve₀.arrows f ↔ (↑S : Sieve X).arrows f
have h : S.preOneHypercover.sieve₀.arrows f → (↑S : Sieve X).arrows f
Tactic: constructor
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Hypercover.One (goal)
Goal:
C : Type u
inst : Category.{v, u} C
J : GrothendieckTopology.{v, u} C
X : C
S : J.Cover X
Y : C
f : Y ⟶ X
h : S.preOneHypercover.sieve₀.arrows f → (↑S : Sieve X).arrows f
⊢ S.preOneHypercover.sieve₀.arrows f ↔ (↑S : Sieve X).arrows f
have h₁ : (↑S : Sieve X).arrows f → S.preOneHypercover.sieve₀.arrows f
Tactic: constructor
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Hypercover.One (goal)
Goal:
C : Type u
inst : Category.{v, u} C
J : GrothendieckTopology.{v, u} C
X : C
S : J.Cover X
f₁ f₂ : S.Arrow
W : C
p₁ : W ⟶ f₁.Y
p₂ : W ⟶ f₂.Y
w : p₁ ≫ f₁.f = p₂ ≫ f₂.f
⊢ S.preOneHypercover.sieve₁ p₁ p₂ = ⊤
have h : ∀ ⦃Y : C⦄ (f : Y ⟶ W), (S.preOneHypercover.sieve₁ p₁ p₂).arrows f ↔ Sieve.arrows ⊤ f
Tactic: ext Y f
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Hypercover.One (goal)
Goal:
C : Type u
inst : Category.{v, u} C
J : GrothendieckTopology.{v, u} C
X : C
S : J.Cover X
f₁ f₂ : S.Arrow
W : C
p₁ : W ⟶ f₁.Y
p₂ : W ⟶ f₂.Y
w : p₁ ≫ f₁.f = p₂ ≫ f₂.f
Y : C
f : Y ⟶ W
⊢ (S.preOneHypercover.sieve₁ p₁ p₂).arrows f ↔ Sieve.arrows ⊤ f
have h : (S.preOneHypercover.sieve₁ p₁ p₂).arrows f
Tactic: simp only [Sieve.top_apply, iff_true]
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Hypercover.One (goal)
Goal:
C : Type u
inst : Category.{v, u} C
S : C
E : PreZeroHypercover.{u_2, v, u} S
inst_1 : E.HasPullbacks
i j : E.I₀
⊢ Eq (α := Sieve.{v, u} (pullback (E.toPreOneHypercover.f i) (E.toPreOneHypercover.f j)))
    (E.toPreOneHypercover.sieve₁' i j) ⊤
have rw : LE.le (α := Sieve.{v, u} (pullback (E.toPreOneHypercover.f i) (E.toPreOneHypercover.f j))) ⊤
  (E.toPreOneHypercover.sieve₁' i j)
Tactic: rw [eq_top_iff]
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Hypercover.One (goal)
Goal:
C : Type u
inst : Category.{v, u} C
S : C
E : PreZeroHypercover.{u_2, v, u} S
inst_1 : E.HasPullbacks
i j : E.I₀
Y : C
f : Y ⟶ pullback (E.toPreOneHypercover.f i) (E.toPreOneHypercover.f j)
h : Sieve.arrows ⊤ f
⊢ Sieve.arrows (E.toPreOneHypercover.sieve₁' i j) f
have refine : Presieve.ofArrows.{v, u, u_2} E.toPreOneHypercover.Y
  (fun (j_1 : E.toPreOneHypercover.I₁ i j) ↦ E.toPreOneHypercover.toPullback j_1) (𝟙 (pullback (E.f i) (E.f j)))
Tactic: refine ⟨pullback (E.f i) (E.f j), f, 𝟙 _, ?_, by simp⟩
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Hypercover.One (goal)
Goal:
C : Type u
inst : Category.{v, u} C
S : C
E : PreZeroHypercover.{u_2, v, u} S
inst_1 : E.HasPullbacks
i j : E.I₀
Y : C
f : Y ⟶ pullback (E.toPreOneHypercover.f i) (E.toPreOneHypercover.f j)
h : Sieve.arrows ⊤ f
⊢ Presieve.ofArrows.{v, u, u_2} E.toPreOneHypercover.Y
    (fun (j_1 : E.toPreOneHypercover.I₁ i j) ↦ E.toPreOneHypercover.toPullback j_1) (𝟙 (pullback (E.f i) (E.f j)))
have refine : 𝟙 (pullback (E.f i) (E.f j)) = eqToHom sorry ≫ E.toPreOneHypercover.toPullback PUnit.unit
Tactic: refine Presieve.ofArrows.mk' ⟨⟩ rfl ?_
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Hypercover.One (goal)
Goal:
C : Type u
inst : Category.{v, u} C
S : C
E : PreZeroHypercover.{u_2, v, u} S
inst_1 : E.HasPullbacks
i j : E.I₀
Y : C
f : Y ⟶ pullback (E.toPreOneHypercover.f i) (E.toPreOneHypercover.f j)
h : Sieve.arrows ⊤ f
⊢ 𝟙 (pullback (E.f i) (E.f j)) = eqToHom ⋯ ≫ E.toPreOneHypercover.toPullback PUnit.unit
have h₀ : 𝟙 (pullback (E.f i) (E.f j)) ≫ pullback.fst (E.f i) (E.f j) =
  CategoryStruct.comp (Y := pullback (E.f i) (E.f j)) (eqToHom sorry ≫ E.toPreOneHypercover.toPullback PUnit.unit)
    (pullback.fst (E.f i) (E.f j))
Tactic: apply pullback.hom_ext
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Hypercover.One (goal)
Goal:
C : Type u
inst : Category.{v, u} C
S : C
E : PreZeroHypercover.{u_2, v, u} S
inst_1 : E.HasPullbacks
i j : E.I₀
Y : C
f : Y ⟶ pullback (E.toPreOneHypercover.f i) (E.toPreOneHypercover.f j)
h : Sieve.arrows ⊤ f
h₀ :
  𝟙 (pullback (E.f i) (E.f j)) ≫ pullback.fst (E.f i) (E.f j) =
    CategoryStruct.comp (Y := pullback (E.f i) (E.f j)) (eqToHom ⋯ ≫ E.toPreOneHypercover.toPullback PUnit.unit)
      (pullback.fst (E.f i) (E.f j))
⊢ 𝟙 (pullback (E.f i) (E.f j)) = eqToHom ⋯ ≫ E.toPreOneHypercover.toPullback PUnit.unit
have h₁ : 𝟙 (pullback (E.f i) (E.f j)) ≫ pullback.snd (E.f i) (E.f j) =
  CategoryStruct.comp (Y := pullback (E.f i) (E.f j)) (eqToHom sorry ≫ E.toPreOneHypercover.toPullback PUnit.unit)
    (pullback.snd (E.f i) (E.f j))
Tactic: apply pullback.hom_ext
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Hypercover.One (goal)
Goal:
C : Type u
inst : Category.{v, u} C
X : C
E : PreZeroHypercover.{w, v, u} X
inst_1 : E.HasPullbacks
F : (i j : E.I₀) → PreZeroHypercover.{w, v, u} (pullback (E.f i) (E.f j))
i j : E.I₀
⊢ (Sieve.ofArrows (E.refineOneHypercover F).Y fun (j_1 : (E.refineOneHypercover F).I₁ i j) ↦
      (E.refineOneHypercover F).toPullback j_1) =
    sieve₀.{w, v, u} (F i j)
have e_f : (fun (j_1 : (E.refineOneHypercover F).I₁ i j) ↦ (E.refineOneHypercover F).toPullback j_1) = (F i j).f
Tactic: congr
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Hypercover.One (goal)
Goal:
C : Type u
inst : Category.{v, u} C
X : C
E : PreZeroHypercover.{w, v, u} X
inst_1 : E.HasPullbacks
F : (i j : E.I₀) → PreZeroHypercover.{w, v, u} (pullback (E.f i) (E.f j))
i j : E.I₀
⊢ (fun (j_1 : (E.refineOneHypercover F).I₁ i j) ↦ (E.refineOneHypercover F).toPullback j_1) = (F i j).f
have e_f : ∀ (x : (E.refineOneHypercover F).I₁ i j),
  CategoryStruct.comp (Y := pullback ((E.refineOneHypercover F).f i) ((E.refineOneHypercover F).f j))
      ((E.refineOneHypercover F).toPullback x)
      (pullback.fst ((E.refineOneHypercover F).f i) ((E.refineOneHypercover F).f j)) =
    (F i j).f x ≫ pullback.fst ((E.refineOneHypercover F).f i) ((E.refineOneHypercover F).f j)
Tactic: ext
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Hypercover.One (goal)
Goal:
C : Type u
inst : Category.{v, u} C
X : C
E : PreZeroHypercover.{w, v, u} X
inst_1 : E.HasPullbacks
F : (i j : E.I₀) → PreZeroHypercover.{w, v, u} (pullback (E.f i) (E.f j))
i j : E.I₀
e_f :
  ∀ (x : (E.refineOneHypercover F).I₁ i j),
    CategoryStruct.comp (Y := pullback ((E.refineOneHypercover F).f i) ((E.refineOneHypercover F).f j))
        ((E.refineOneHypercover F).toPullback x)
        (pullback.fst ((E.refineOneHypercover F).f i) ((E.refineOneHypercover F).f j)) =
      (F i j).f x ≫ pullback.fst ((E.refineOneHypercover F).f i) ((E.refineOneHypercover F).f j)
⊢ (fun (j_1 : (E.refineOneHypercover F).I₁ i j) ↦ (E.refineOneHypercover F).toPullback j_1) = (F i j).f
have e_f₁ : ∀ (x : (E.refineOneHypercover F).I₁ i j),
  CategoryStruct.comp (Y := pullback ((E.refineOneHypercover F).f i) ((E.refineOneHypercover F).f j))
      ((E.refineOneHypercover F).toPullback x)
      (pullback.snd ((E.refineOneHypercover F).f i) ((E.refineOneHypercover F).f j)) =
    (F i j).f x ≫ pullback.snd ((E.refineOneHypercover F).f i) ((E.refineOneHypercover F).f j)
Tactic: ext
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Hypercover.Saturate (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
S : C
E : PreZeroHypercover S
F : Functor.{v_1, u_3, u_1, u_3 + 1} Cᵒᵖ (Type u_3)
⊢ Nonempty (IsLimit (E.saturate.multifork F)) ↔ Presieve.IsSheafFor F E.presieve₀
have rw : Function.Bijective (E.saturate.multifork F).toSections ↔
  Function.Bijective (β := (E.saturate.multicospanIndex F).sections)
    ((⇑(E.sectionsSaturateEquiv F).symm :
        (a : Subtype (Presieve.Arrows.Compatible F E.f)) → (E.saturate.multicospanIndex F).sections) ∘
      Presieve.Arrows.toCompatible F E.f)
Tactic: rw [Multifork.isLimit_types_iff, Presieve.isSheafFor_ofArrows_iff_bijective_toCompabible,
    ← Function.Bijective.of_comp_iff' (E.sectionsSaturateEquiv F).symm.bijective]
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Hypercover.Saturate (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
A : Type u_2
inst_1 : Category.{v_2, u_2} A
S_1 : C
E_1 : PreZeroHypercover.{?u.18915, v_1, u_1} S_1
inst_2 : E_1.HasPullbacks
S_2 : C
E_2 : PreZeroHypercover.{?u.19098, v_1, u_1} S_2
inst_3 : E_2.HasPullbacks
S : C
E : PreZeroHypercover S
inst_4 : E.HasPullbacks
F : Functor.{v_1, v_2, u_1, u_2} Cᵒᵖ A
⊢ PreOneHypercover.Homotopy (PreOneHypercover.Hom.comp E.toSaturateOfHasPullbacks E.fromSaturateOfHasPullbacks)
    (PreOneHypercover.Hom.id E.toPreOneHypercover)
have rw : PreOneHypercover.Homotopy (PreOneHypercover.Hom.id E.toPreOneHypercover) (PreOneHypercover.Hom.id E.toPreOneHypercover)
Tactic: rw [toSaturateOfHasPullbacks_fromSaturateOfHasPullbacks]
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Hypercover.SheafOfTypes (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
S : C
E : PreZeroHypercover.{u_3, v_1, u_1} S
inst_1 : E.HasPullbacks
F : Functor.{v_1, u_2, u_1, u_2 + 1} Cᵒᵖ (Type u_2)
s : (E.toPreOneHypercover.multicospanIndex F).sections
i j : E.I₀
W : C
gi : W ⟶ E.X i
gj : W ⟶ E.X j
hgij : gi ≫ E.f i = gj ≫ E.f j
heq :
  Eq (α := F.obj (op (pullback (E.f i) (E.f j))))
    ((ConcreteCategory.hom (FC := TypeCat.Fun) (F.map (Quiver.Hom.op (pullback.fst (E.f i) (E.f j)))) :
          (a : F.obj (op (E.X i))) → F.obj (op (pullback (E.f i) (E.f j))))
        (s.val i) :
      F.obj (op (pullback (E.f i) (E.f j))))
    ((ConcreteCategory.hom (FC := TypeCat.Fun) (F.map (Quiver.Hom.op (pullback.snd (E.f i) (E.f j)))) :
          (a : F.obj (op (E.X j))) → F.obj (op (pullback (E.f i) (E.f j))))
        (s.val j) :
      F.obj (op (pullback (E.f i) (E.f j))))
⊢ Eq (α := (fun (X : Type u_2) ↦ X) (F.obj (op W)))
    ((ConcreteCategory.hom (FC := TypeCat.Fun) (F.map gi.op) :
          (a : (fun (X : Type u_2) ↦ X) (F.obj (op (E.X i)))) → (fun (X : Type u_2) ↦ X) (F.obj (op W)))
        (s.val i) :
      (fun (X : Type u_2) ↦ X) (F.obj (op W)))
    ((ConcreteCategory.hom (FC := TypeCat.Fun) (F.map gj.op) :
          (a : (fun (X : Type u_2) ↦ X) (F.obj (op (E.X j)))) → (fun (X : Type u_2) ↦ X) (F.obj (op W)))
        (s.val j) :
      (fun (X : Type u_2) ↦ X) (F.obj (op W)))
have rw : Eq (α := (fun (X : Type u_2) ↦ X) (F.obj (op W)))
  ((ConcreteCategory.hom (FC := TypeCat.Fun)
          (F.map
            (CategoryStruct.comp (Y := pullback (E.f i) (E.f j)) (pullback.lift gi gj hgij)
                (pullback.fst (E.f i) (E.f j))).op) :
        (a : (fun (X : Type u_2) ↦ X) (F.obj (op (E.X i)))) → (fun (X : Type u_2) ↦ X) (F.obj (op W)))
      (s.val i) :
    (fun (X : Type u_2) ↦ X) (F.obj (op W)))
  ((ConcreteCategory.hom (FC := TypeCat.Fun) (F.map gj.op) :
        (a : (fun (X : Type u_2) ↦ X) (F.obj (op (E.X j)))) → (fun (X : Type u_2) ↦ X) (F.obj (op W)))
      (s.val j) :
    (fun (X : Type u_2) ↦ X) (F.obj (op W)))
Tactic: rw [← pullback.lift_fst _ _ hgij]
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Hypercover.SheafOfTypes (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
S : C
E : PreZeroHypercover.{u_3, v_1, u_1} S
inst_1 : E.HasPullbacks
F : Functor.{v_1, u_2, u_1, u_2 + 1} Cᵒᵖ (Type u_2)
s : (E.toPreOneHypercover.multicospanIndex F).sections
i j : E.I₀
W : C
gi : W ⟶ E.X i
gj : W ⟶ E.X j
hgij : gi ≫ E.f i = gj ≫ E.f j
heq :
  Eq (α := F.obj (op (pullback (E.f i) (E.f j))))
    ((ConcreteCategory.hom (FC := TypeCat.Fun) (F.map (Quiver.Hom.op (pullback.fst (E.f i) (E.f j)))) :
          (a : F.obj (op (E.X i))) → F.obj (op (pullback (E.f i) (E.f j))))
        (s.val i) :
      F.obj (op (pullback (E.f i) (E.f j))))
    ((ConcreteCategory.hom (FC := TypeCat.Fun) (F.map (Quiver.Hom.op (pullback.snd (E.f i) (E.f j)))) :
          (a : F.obj (op (E.X j))) → F.obj (op (pullback (E.f i) (E.f j))))
        (s.val j) :
      F.obj (op (pullback (E.f i) (E.f j))))
| ((ConcreteCategory.hom (FC := TypeCat.Fun) (F.map gj.op) :
        (a : (fun (X : Type u_2) ↦ X) (F.obj (op (E.X j)))) → (fun (X : Type u_2) ↦ X) (F.obj (op W)))
      (s.val j) :
    (fun (X : Type u_2) ↦ X) (F.obj (op W)))
have rw : ((ConcreteCategory.hom (FC := TypeCat.Fun)
        (F.map
          (CategoryStruct.comp (Y := pullback (E.f i) (E.f j)) (pullback.lift gi gj hgij)
              (pullback.snd (E.f i) (E.f j))).op) :
      (a : (fun (X : Type u_2) ↦ X) (F.obj (op (E.X j)))) → (fun (X : Type u_2) ↦ X) (F.obj (op W)))
    (s.val j) :
  (fun (X : Type u_2) ↦ X) (F.obj (op W)))
Tactic: rw [← pullback.lift_snd _ _ hgij]