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
S : C
E : PreOneHypercover.{w, v, u} S
F : PreOneHypercover.{u_2, v, u} S
inst_1 : HasPullbacks.{v, u} C
i j : Prod.{w, u_2} E.I₀ F.I₀
W : C
p₁ : W ⟶ pullback (X := E.X i.1) (Y := F.X i.2) (E.f i.1) (F.f i.2)
p₂ : W ⟶ pullback (X := E.X j.1) (Y := F.X j.2) (E.f j.1) (F.f j.2)
w : p₁ ≫ pullback.fst (E.f i.1) (F.f i.2) ≫ E.f i.1 = p₂ ≫ pullback.fst (E.f j.1) (F.f j.2) ≫ E.f j.1
Y : C
f : Y ⟶ W
p : W ⟶ pullback ((E.inter F).f i) ((E.inter F).f j) := pullback.lift p₁ p₂ w
h :
  ((E.inter F).sieve₁ p₁ p₂).arrows f →
    ∀ (k : (E.inter F).I₁ i j) (a : Y ⟶ (E.inter F).Y k) (h₁ : f ≫ p₁ = a ≫ (E.inter F).p₁ k)
      (h₂ : f ≫ p₂ = a ≫ (E.inter F).p₂ k),
      (Sieve.bind (E.sieve₁ (p₁ ≫ pullback.fst (E.f i.1) (F.f i.2)) (p₂ ≫ pullback.fst (E.f j.1) (F.f j.2))).arrows
            fun (x : C) (f : x ⟶ W)
              (x_1 :
                (E.sieve₁ (p₁ ≫ pullback.fst (E.f i.1) (F.f i.2)) (p₂ ≫ pullback.fst (E.f j.1) (F.f j.2))).arrows f) ↦
            Sieve.pullback f
              (F.sieve₁ (p₁ ≫ pullback.snd (E.f i.1) (F.f i.2)) (p₂ ≫ pullback.snd (E.f j.1) (F.f j.2)))).arrows
        f
⊢ ((E.inter F).sieve₁ p₁ p₂).arrows f ↔
    (Sieve.bind (E.sieve₁ (p₁ ≫ pullback.fst (E.f i.1) (F.f i.2)) (p₂ ≫ pullback.fst (E.f j.1) (F.f j.2))).arrows
          fun (x : C) (f : x ⟶ W)
            (x_1 :
              (E.sieve₁ (p₁ ≫ pullback.fst (E.f i.1) (F.f i.2)) (p₂ ≫ pullback.fst (E.f j.1) (F.f j.2))).arrows f) ↦
          Sieve.pullback f
            (F.sieve₁ (p₁ ≫ pullback.snd (E.f i.1) (F.f i.2)) (p₂ ≫ pullback.snd (E.f j.1) (F.f j.2)))).arrows
      f
have h₁ : (Sieve.bind (E.sieve₁ (p₁ ≫ pullback.fst (E.f i.1) (F.f i.2)) (p₂ ≫ pullback.fst (E.f j.1) (F.f j.2))).arrows
        fun (x : C) (f : x ⟶ W)
          (x_1 : (E.sieve₁ (p₁ ≫ pullback.fst (E.f i.1) (F.f i.2)) (p₂ ≫ pullback.fst (E.f j.1) (F.f j.2))).arrows f) ↦
        Sieve.pullback f
          (F.sieve₁ (p₁ ≫ pullback.snd (E.f i.1) (F.f i.2)) (p₂ ≫ pullback.snd (E.f j.1) (F.f j.2)))).arrows
    f →
  ∀ (Z : C) (a : Y ⟶ Z) (b : Z ⟶ W) (k : E.I₁ i.1 j.1) (e : Z ⟶ E.Y k)
    (h₁ : b ≫ p₁ ≫ pullback.fst (E.f i.1) (F.f i.2) = e ≫ E.p₁ k)
    (h₂ : b ≫ p₂ ≫ pullback.fst (E.f j.1) (F.f j.2) = e ≫ E.p₂ k) (l : F.I₁ i.2 j.2) (u : Y ⟶ F.Y l)
    (u₁ : (a ≫ b) ≫ p₁ ≫ pullback.snd (E.f i.1) (F.f i.2) = u ≫ F.p₁ l)
    (u₂ : (a ≫ b) ≫ p₂ ≫ pullback.snd (E.f j.1) (F.f j.2) = u ≫ F.p₂ l) (hab : a ≫ b = f),
    ((E.inter F).sieve₁ p₁ p₂).arrows f
Tactic: refine ⟨fun ⟨k, a, h₁, h₂⟩ ↦ ?_, fun ⟨Z, a, b, ⟨k, e, h₁, h₂⟩, ⟨l, u, u₁, u₂⟩, hab⟩ ↦ ?_⟩
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Hypercover.One (goal)
Goal:
C : Type u
inst : Category.{v, u} C
S : C
E : PreOneHypercover.{w, v, u} S
F : PreOneHypercover.{u_2, v, u} S
inst_1 : HasPullbacks.{v, u} C
i j : Prod.{w, u_2} E.I₀ F.I₀
W : C
p₁ : W ⟶ pullback (X := E.X i.1) (Y := F.X i.2) (E.f i.1) (F.f i.2)
p₂ : W ⟶ pullback (X := E.X j.1) (Y := F.X j.2) (E.f j.1) (F.f j.2)
w : p₁ ≫ pullback.fst (E.f i.1) (F.f i.2) ≫ E.f i.1 = p₂ ≫ pullback.fst (E.f j.1) (F.f j.2) ≫ E.f j.1
Y : C
f : Y ⟶ W
p : W ⟶ pullback ((E.inter F).f i) ((E.inter F).f j) := pullback.lift p₁ p₂ w
h : ((E.inter F).sieve₁ p₁ p₂).arrows f
k : (E.inter F).I₁ i j
a : Y ⟶ (E.inter F).Y k
h₁ : f ≫ p₁ = a ≫ (E.inter F).p₁ k
h₂ : f ≫ p₂ = a ≫ (E.inter F).p₂ k
⊢ (Sieve.bind (E.sieve₁ (p₁ ≫ pullback.fst (E.f i.1) (F.f i.2)) (p₂ ≫ pullback.fst (E.f j.1) (F.f j.2))).arrows
        fun (x : C) (f : x ⟶ W)
          (x_1 : (E.sieve₁ (p₁ ≫ pullback.fst (E.f i.1) (F.f i.2)) (p₂ ≫ pullback.fst (E.f j.1) (F.f j.2))).arrows f) ↦
        Sieve.pullback f
          (F.sieve₁ (p₁ ≫ pullback.snd (E.f i.1) (F.f i.2)) (p₂ ≫ pullback.snd (E.f j.1) (F.f j.2)))).arrows
    f
have h : f ≫ p = a ≫ (E.inter F).toPullback k
Tactic: refine ⟨pullback p ((E.inter F).toPullback k), pullback.lift f a ?_,
        pullback.fst _ _, ?_, ?_, ?_⟩
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Hypercover.One (goal)
Goal:
C : Type u
inst : Category.{v, u} C
S : C
E : PreOneHypercover.{w, v, u} S
F : PreOneHypercover.{u_2, v, u} S
inst_1 : HasPullbacks.{v, u} C
i j : Prod.{w, u_2} E.I₀ F.I₀
W : C
p₁ : W ⟶ pullback (X := E.X i.1) (Y := F.X i.2) (E.f i.1) (F.f i.2)
p₂ : W ⟶ pullback (X := E.X j.1) (Y := F.X j.2) (E.f j.1) (F.f j.2)
w : p₁ ≫ pullback.fst (E.f i.1) (F.f i.2) ≫ E.f i.1 = p₂ ≫ pullback.fst (E.f j.1) (F.f j.2) ≫ E.f j.1
Y : C
f : Y ⟶ W
p : W ⟶ pullback ((E.inter F).f i) ((E.inter F).f j) := pullback.lift p₁ p₂ w
h_1 : ((E.inter F).sieve₁ p₁ p₂).arrows f
k : (E.inter F).I₁ i j
a : Y ⟶ (E.inter F).Y k
h₁ : f ≫ p₁ = a ≫ (E.inter F).p₁ k
h₂ : f ≫ p₂ = a ≫ (E.inter F).p₂ k
h : f ≫ p = a ≫ (E.inter F).toPullback k
⊢ (Sieve.bind (E.sieve₁ (p₁ ≫ pullback.fst (E.f i.1) (F.f i.2)) (p₂ ≫ pullback.fst (E.f j.1) (F.f j.2))).arrows
        fun (x : C) (f : x ⟶ W)
          (x_1 : (E.sieve₁ (p₁ ≫ pullback.fst (E.f i.1) (F.f i.2)) (p₂ ≫ pullback.fst (E.f j.1) (F.f j.2))).arrows f) ↦
        Sieve.pullback f
          (F.sieve₁ (p₁ ≫ pullback.snd (E.f i.1) (F.f i.2)) (p₂ ≫ pullback.snd (E.f j.1) (F.f j.2)))).arrows
    f
have h₃ : (E.sieve₁ (p₁ ≫ pullback.fst (E.f i.1) (F.f i.2)) (p₂ ≫ pullback.fst (E.f j.1) (F.f j.2))).arrows (Y :=
  pullback p ((E.inter F).toPullback k)) (pullback.fst p ((E.inter F).toPullback k))
Tactic: refine ⟨pullback p ((E.inter F).toPullback k), pullback.lift f a ?_,
        pullback.fst _ _, ?_, ?_, ?_⟩
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Hypercover.One (goal)
Goal:
C : Type u
inst : Category.{v, u} C
S : C
E : PreOneHypercover.{w, v, u} S
F : PreOneHypercover.{u_2, v, u} S
inst_1 : HasPullbacks.{v, u} C
i j : Prod.{w, u_2} E.I₀ F.I₀
W : C
p₁ : W ⟶ pullback (X := E.X i.1) (Y := F.X i.2) (E.f i.1) (F.f i.2)
p₂ : W ⟶ pullback (X := E.X j.1) (Y := F.X j.2) (E.f j.1) (F.f j.2)
w : p₁ ≫ pullback.fst (E.f i.1) (F.f i.2) ≫ E.f i.1 = p₂ ≫ pullback.fst (E.f j.1) (F.f j.2) ≫ E.f j.1
Y : C
f : Y ⟶ W
p : W ⟶ pullback ((E.inter F).f i) ((E.inter F).f j) := pullback.lift p₁ p₂ w
h_1 : ((E.inter F).sieve₁ p₁ p₂).arrows f
k : (E.inter F).I₁ i j
a : Y ⟶ (E.inter F).Y k
h₁ : f ≫ p₁ = a ≫ (E.inter F).p₁ k
h₂ : f ≫ p₂ = a ≫ (E.inter F).p₂ k
h : f ≫ p = a ≫ (E.inter F).toPullback k
h₃ :
  (E.sieve₁ (p₁ ≫ pullback.fst (E.f i.1) (F.f i.2)) (p₂ ≫ pullback.fst (E.f j.1) (F.f j.2))).arrows (Y :=
    pullback p ((E.inter F).toPullback k)) (pullback.fst p ((E.inter F).toPullback k))
⊢ (Sieve.bind (E.sieve₁ (p₁ ≫ pullback.fst (E.f i.1) (F.f i.2)) (p₂ ≫ pullback.fst (E.f j.1) (F.f j.2))).arrows
        fun (x : C) (f : x ⟶ W)
          (x_1 : (E.sieve₁ (p₁ ≫ pullback.fst (E.f i.1) (F.f i.2)) (p₂ ≫ pullback.fst (E.f j.1) (F.f j.2))).arrows f) ↦
        Sieve.pullback f
          (F.sieve₁ (p₁ ≫ pullback.snd (E.f i.1) (F.f i.2)) (p₂ ≫ pullback.snd (E.f j.1) (F.f j.2)))).arrows
    f
have h₄ : (fun (x : C) (x_1 : x ⟶ W)
      (h : (E.sieve₁ (p₁ ≫ pullback.fst (E.f i.1) (F.f i.2)) (p₂ ≫ pullback.fst (E.f j.1) (F.f j.2))).arrows x_1) ↦
    (Sieve.arrows
        ((fun (x : C) (f : x ⟶ W)
              (x_2 :
                (E.sieve₁ (p₁ ≫ pullback.fst (E.f i.1) (F.f i.2)) (p₂ ≫ pullback.fst (E.f j.1) (F.f j.2))).arrows f) ↦
            Sieve.pullback f (F.sieve₁ (p₁ ≫ pullback.snd (E.f i.1) (F.f i.2)) (p₂ ≫ pullback.snd (E.f j.1) (F.f j.2))))
          x x_1 h) :
      Presieve.{v, u} x))
  (pullback p ((E.inter F).toPullback k)) (pullback.fst p ((E.inter F).toPullback k)) sorry (Y := Y) (pullback.lift f a sorry)
Tactic: refine ⟨pullback p ((E.inter F).toPullback k), pullback.lift f a ?_,
        pullback.fst _ _, ?_, ?_, ?_⟩
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Hypercover.One (goal)
Goal:
C : Type u
inst : Category.{v, u} C
S : C
E : PreOneHypercover.{w, v, u} S
F : PreOneHypercover.{u_2, v, u} S
inst_1 : HasPullbacks.{v, u} C
i j : Prod.{w, u_2} E.I₀ F.I₀
W : C
p₁ : W ⟶ pullback (X := E.X i.1) (Y := F.X i.2) (E.f i.1) (F.f i.2)
p₂ : W ⟶ pullback (X := E.X j.1) (Y := F.X j.2) (E.f j.1) (F.f j.2)
w : p₁ ≫ pullback.fst (E.f i.1) (F.f i.2) ≫ E.f i.1 = p₂ ≫ pullback.fst (E.f j.1) (F.f j.2) ≫ E.f j.1
Y : C
f : Y ⟶ W
p : W ⟶ pullback ((E.inter F).f i) ((E.inter F).f j) := pullback.lift p₁ p₂ w
h_1 : ((E.inter F).sieve₁ p₁ p₂).arrows f
k : (E.inter F).I₁ i j
a : Y ⟶ (E.inter F).Y k
h₁ : f ≫ p₁ = a ≫ (E.inter F).p₁ k
h₂ : f ≫ p₂ = a ≫ (E.inter F).p₂ k
h : f ≫ p = a ≫ (E.inter F).toPullback k
h₃ :
  (E.sieve₁ (p₁ ≫ pullback.fst (E.f i.1) (F.f i.2)) (p₂ ≫ pullback.fst (E.f j.1) (F.f j.2))).arrows (Y :=
    pullback p ((E.inter F).toPullback k)) (pullback.fst p ((E.inter F).toPullback k))
h₄ :
  (fun (x : C) (x_1 : x ⟶ W)
        (h : (E.sieve₁ (p₁ ≫ pullback.fst (E.f i.1) (F.f i.2)) (p₂ ≫ pullback.fst (E.f j.1) (F.f j.2))).arrows x_1) ↦
      (Sieve.arrows
          ((fun (x : C) (f : x ⟶ W)
                (x_2 :
                  (E.sieve₁ (p₁ ≫ pullback.fst (E.f i.1) (F.f i.2)) (p₂ ≫ pullback.fst (E.f j.1) (F.f j.2))).arrows f) ↦
              Sieve.pullback f
                (F.sieve₁ (p₁ ≫ pullback.snd (E.f i.1) (F.f i.2)) (p₂ ≫ pullback.snd (E.f j.1) (F.f j.2))))
            x x_1 h) :
        Presieve.{v, u} x))
    (pullback p ((E.inter F).toPullback k)) (pullback.fst p ((E.inter F).toPullback k)) ⋯ (Y := Y) (pullback.lift f a ⋯)
⊢ (Sieve.bind (E.sieve₁ (p₁ ≫ pullback.fst (E.f i.1) (F.f i.2)) (p₂ ≫ pullback.fst (E.f j.1) (F.f j.2))).arrows
        fun (x : C) (f : x ⟶ W)
          (x_1 : (E.sieve₁ (p₁ ≫ pullback.fst (E.f i.1) (F.f i.2)) (p₂ ≫ pullback.fst (E.f j.1) (F.f j.2))).arrows f) ↦
        Sieve.pullback f
          (F.sieve₁ (p₁ ≫ pullback.snd (E.f i.1) (F.f i.2)) (p₂ ≫ pullback.snd (E.f j.1) (F.f j.2)))).arrows
    f
have h₅ : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)
Tactic: refine ⟨pullback p ((E.inter F).toPullback k), pullback.lift f a ?_,
        pullback.fst _ _, ?_, ?_, ?_⟩
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Hypercover.One (goal)
Goal:
C : Type u
inst : Category.{v, u} C
S : C
E : PreOneHypercover.{w, v, u} S
F : PreOneHypercover.{u_2, v, u} S
inst_1 : HasPullbacks.{v, u} C
i j : Prod.{w, u_2} E.I₀ F.I₀
W : C
p₁ : W ⟶ pullback (X := E.X i.1) (Y := F.X i.2) (E.f i.1) (F.f i.2)
p₂ : W ⟶ pullback (X := E.X j.1) (Y := F.X j.2) (E.f j.1) (F.f j.2)
w : p₁ ≫ pullback.fst (E.f i.1) (F.f i.2) ≫ E.f i.1 = p₂ ≫ pullback.fst (E.f j.1) (F.f j.2) ≫ E.f j.1
Y : C
f : Y ⟶ W
p : W ⟶ pullback ((E.inter F).f i) ((E.inter F).f j) := pullback.lift p₁ p₂ w
h : ((E.inter F).sieve₁ p₁ p₂).arrows f
k : (E.inter F).I₁ i j
a : Y ⟶ (E.inter F).Y k
h₁ : f ≫ p₁ = a ≫ (E.inter F).p₁ k
h₂ : f ≫ p₂ = a ≫ (E.inter F).p₂ k
⊢ f ≫ p = a ≫ (E.inter F).toPullback k
have h : (f ≫ p) ≫ pullback.fst ((E.inter F).f i) ((E.inter F).f j) =
  CategoryStruct.comp (Y := pullback ((E.inter F).f i) ((E.inter F).f j)) (a ≫ (E.inter F).toPullback k)
    (pullback.fst ((E.inter F).f i) ((E.inter F).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 : PreOneHypercover.{w, v, u} S
F : PreOneHypercover.{u_2, v, u} S
inst_1 : HasPullbacks.{v, u} C
i j : Prod.{w, u_2} E.I₀ F.I₀
W : C
p₁ : W ⟶ pullback (X := E.X i.1) (Y := F.X i.2) (E.f i.1) (F.f i.2)
p₂ : W ⟶ pullback (X := E.X j.1) (Y := F.X j.2) (E.f j.1) (F.f j.2)
w : p₁ ≫ pullback.fst (E.f i.1) (F.f i.2) ≫ E.f i.1 = p₂ ≫ pullback.fst (E.f j.1) (F.f j.2) ≫ E.f j.1
Y : C
f : Y ⟶ W
p : W ⟶ pullback ((E.inter F).f i) ((E.inter F).f j) := pullback.lift p₁ p₂ w
h_1 : ((E.inter F).sieve₁ p₁ p₂).arrows f
k : (E.inter F).I₁ i j
a : Y ⟶ (E.inter F).Y k
h₁ : f ≫ p₁ = a ≫ (E.inter F).p₁ k
h₂ : f ≫ p₂ = a ≫ (E.inter F).p₂ k
h :
  (f ≫ p) ≫ pullback.fst ((E.inter F).f i) ((E.inter F).f j) =
    CategoryStruct.comp (Y := pullback ((E.inter F).f i) ((E.inter F).f j)) (a ≫ (E.inter F).toPullback k)
      (pullback.fst ((E.inter F).f i) ((E.inter F).f j))
⊢ f ≫ p = a ≫ (E.inter F).toPullback k
have h₃ : (f ≫ p) ≫ pullback.snd ((E.inter F).f i) ((E.inter F).f j) =
  CategoryStruct.comp (Y := pullback ((E.inter F).f i) ((E.inter F).f j)) (a ≫ (E.inter F).toPullback k)
    (pullback.snd ((E.inter F).f i) ((E.inter F).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 : PreOneHypercover.{w, v, u} S
F : PreOneHypercover.{u_2, v, u} S
inst_1 : HasPullbacks.{v, u} C
i j : Prod.{w, u_2} E.I₀ F.I₀
W : C
p₁ : W ⟶ pullback (X := E.X i.1) (Y := F.X i.2) (E.f i.1) (F.f i.2)
p₂ : W ⟶ pullback (X := E.X j.1) (Y := F.X j.2) (E.f j.1) (F.f j.2)
w : p₁ ≫ pullback.fst (E.f i.1) (F.f i.2) ≫ E.f i.1 = p₂ ≫ pullback.fst (E.f j.1) (F.f j.2) ≫ E.f j.1
Y : C
f : Y ⟶ W
p : W ⟶ pullback ((E.inter F).f i) ((E.inter F).f j) := pullback.lift p₁ p₂ w
h : ((E.inter F).sieve₁ p₁ p₂).arrows f
k : (E.inter F).I₁ i j
a : Y ⟶ (E.inter F).Y k
h₁ : f ≫ p₁ = a ≫ (E.inter F).p₁ k
h₂ : f ≫ p₂ = a ≫ (E.inter F).p₂ k
⊢ (f ≫ p) ≫ pullback.fst ((E.inter F).f i) ((E.inter F).f j) =
    CategoryStruct.comp (Y := pullback ((E.inter F).f i) ((E.inter F).f j)) (a ≫ (E.inter F).toPullback k)
      (pullback.fst ((E.inter F).f i) ((E.inter F).f j))
have h : ((f ≫ p) ≫ pullback.fst ((E.inter F).f i) ((E.inter F).f j)) ≫
    pullback.fst
      (E.f
        (Sigma.fst (β := fun (i : E.I₀) ↦
          PreZeroHypercover.I₀.{u_2, v, u} (C := C) (S := E.X i)
            ((fun (i : E.I₀) ↦ PreZeroHypercover.pullback₁.{u_2, v, u} (E.f i) F.toPreZeroHypercover) i))
          ((Equiv.symm (Equiv.sigmaEquivProd E.I₀ F.toPreZeroHypercover.1) :
              (a : E.I₀ × F.toPreZeroHypercover.1) →
                (E.bind fun (i : E.I₀) ↦ PreZeroHypercover.pullback₁ (E.f i) F.toPreZeroHypercover).I₀)
            i)))
      (F.f
        (Sigma.snd (β := fun (i : E.I₀) ↦
          PreZeroHypercover.I₀.{u_2, v, u} (C := C) (S := E.X i)
            ((fun (i : E.I₀) ↦ PreZeroHypercover.pullback₁.{u_2, v, u} (E.f i) F.toPreZeroHypercover) i))
          ((Equiv.symm (Equiv.sigmaEquivProd E.I₀ F.toPreZeroHypercover.1) :
              (a : E.I₀ × F.toPreZeroHypercover.1) →
                (E.bind fun (i : E.I₀) ↦ PreZeroHypercover.pullback₁ (E.f i) F.toPreZeroHypercover).I₀)
            i))) =
  CategoryStruct.comp (Y := pullback ((E.inter F).f i) ((E.inter F).f j)) (a ≫ (E.inter F).toPullback k)
      (pullback.fst ((E.inter F).f i) ((E.inter F).f j)) ≫
    pullback.fst
      (E.f
        (Sigma.fst (β := fun (i : E.I₀) ↦
          PreZeroHypercover.I₀.{u_2, v, u} (C := C) (S := E.X i)
            ((fun (i : E.I₀) ↦ PreZeroHypercover.pullback₁.{u_2, v, u} (E.f i) F.toPreZeroHypercover) i))
          ((Equiv.symm (Equiv.sigmaEquivProd E.I₀ F.toPreZeroHypercover.1) :
              (a : E.I₀ × F.toPreZeroHypercover.1) →
                (E.bind fun (i : E.I₀) ↦ PreZeroHypercover.pullback₁ (E.f i) F.toPreZeroHypercover).I₀)
            i)))
      (F.f
        (Sigma.snd (β := fun (i : E.I₀) ↦
          PreZeroHypercover.I₀.{u_2, v, u} (C := C) (S := E.X i)
            ((fun (i : E.I₀) ↦ PreZeroHypercover.pullback₁.{u_2, v, u} (E.f i) F.toPreZeroHypercover) i))
          ((Equiv.symm (Equiv.sigmaEquivProd E.I₀ F.toPreZeroHypercover.1) :
              (a : E.I₀ × F.toPreZeroHypercover.1) →
                (E.bind fun (i : E.I₀) ↦ PreZeroHypercover.pullback₁ (E.f i) F.toPreZeroHypercover).I₀)
            i)))
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 : PreOneHypercover.{w, v, u} S
F : PreOneHypercover.{u_2, v, u} S
inst_1 : HasPullbacks.{v, u} C
i j : Prod.{w, u_2} E.I₀ F.I₀
W : C
p₁ : W ⟶ pullback (X := E.X i.1) (Y := F.X i.2) (E.f i.1) (F.f i.2)
p₂ : W ⟶ pullback (X := E.X j.1) (Y := F.X j.2) (E.f j.1) (F.f j.2)
w : p₁ ≫ pullback.fst (E.f i.1) (F.f i.2) ≫ E.f i.1 = p₂ ≫ pullback.fst (E.f j.1) (F.f j.2) ≫ E.f j.1
Y : C
f : Y ⟶ W
p : W ⟶ pullback ((E.inter F).f i) ((E.inter F).f j) := pullback.lift p₁ p₂ w
h : ((E.inter F).sieve₁ p₁ p₂).arrows f
k : (E.inter F).I₁ i j
a : Y ⟶ (E.inter F).Y k
h₁ : f ≫ p₁ = a ≫ (E.inter F).p₁ k
h₂ : f ≫ p₂ = a ≫ (E.inter F).p₂ k
⊢ (f ≫ p) ≫ pullback.snd ((E.inter F).f i) ((E.inter F).f j) =
    CategoryStruct.comp (Y := pullback ((E.inter F).f i) ((E.inter F).f j)) (a ≫ (E.inter F).toPullback k)
      (pullback.snd ((E.inter F).f i) ((E.inter F).f j))
have h : ((f ≫ p) ≫ pullback.snd ((E.inter F).f i) ((E.inter F).f j)) ≫
    pullback.fst
      (E.f
        (Sigma.fst (β := fun (i : E.I₀) ↦
          PreZeroHypercover.I₀.{u_2, v, u} (C := C) (S := E.X i)
            ((fun (i : E.I₀) ↦ PreZeroHypercover.pullback₁.{u_2, v, u} (E.f i) F.toPreZeroHypercover) i))
          ((Equiv.symm (Equiv.sigmaEquivProd E.I₀ F.toPreZeroHypercover.1) :
              (a : E.I₀ × F.toPreZeroHypercover.1) →
                (E.bind fun (i : E.I₀) ↦ PreZeroHypercover.pullback₁ (E.f i) F.toPreZeroHypercover).I₀)
            j)))
      (F.f
        (Sigma.snd (β := fun (i : E.I₀) ↦
          PreZeroHypercover.I₀.{u_2, v, u} (C := C) (S := E.X i)
            ((fun (i : E.I₀) ↦ PreZeroHypercover.pullback₁.{u_2, v, u} (E.f i) F.toPreZeroHypercover) i))
          ((Equiv.symm (Equiv.sigmaEquivProd E.I₀ F.toPreZeroHypercover.1) :
              (a : E.I₀ × F.toPreZeroHypercover.1) →
                (E.bind fun (i : E.I₀) ↦ PreZeroHypercover.pullback₁ (E.f i) F.toPreZeroHypercover).I₀)
            j))) =
  CategoryStruct.comp (Y := pullback ((E.inter F).f i) ((E.inter F).f j)) (a ≫ (E.inter F).toPullback k)
      (pullback.snd ((E.inter F).f i) ((E.inter F).f j)) ≫
    pullback.fst
      (E.f
        (Sigma.fst (β := fun (i : E.I₀) ↦
          PreZeroHypercover.I₀.{u_2, v, u} (C := C) (S := E.X i)
            ((fun (i : E.I₀) ↦ PreZeroHypercover.pullback₁.{u_2, v, u} (E.f i) F.toPreZeroHypercover) i))
          ((Equiv.symm (Equiv.sigmaEquivProd E.I₀ F.toPreZeroHypercover.1) :
              (a : E.I₀ × F.toPreZeroHypercover.1) →
                (E.bind fun (i : E.I₀) ↦ PreZeroHypercover.pullback₁ (E.f i) F.toPreZeroHypercover).I₀)
            j)))
      (F.f
        (Sigma.snd (β := fun (i : E.I₀) ↦
          PreZeroHypercover.I₀.{u_2, v, u} (C := C) (S := E.X i)
            ((fun (i : E.I₀) ↦ PreZeroHypercover.pullback₁.{u_2, v, u} (E.f i) F.toPreZeroHypercover) i))
          ((Equiv.symm (Equiv.sigmaEquivProd E.I₀ F.toPreZeroHypercover.1) :
              (a : E.I₀ × F.toPreZeroHypercover.1) →
                (E.bind fun (i : E.I₀) ↦ PreZeroHypercover.pullback₁ (E.f i) F.toPreZeroHypercover).I₀)
            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 : PreOneHypercover.{w, v, u} S
F : PreOneHypercover.{u_2, v, u} S
inst_1 : HasPullbacks.{v, u} C
i j : Prod.{w, u_2} E.I₀ F.I₀
W : C
p₁ : W ⟶ pullback (X := E.X i.1) (Y := F.X i.2) (E.f i.1) (F.f i.2)
p₂ : W ⟶ pullback (X := E.X j.1) (Y := F.X j.2) (E.f j.1) (F.f j.2)
w : p₁ ≫ pullback.fst (E.f i.1) (F.f i.2) ≫ E.f i.1 = p₂ ≫ pullback.fst (E.f j.1) (F.f j.2) ≫ E.f j.1
Y : C
f : Y ⟶ W
p : W ⟶ pullback ((E.inter F).f i) ((E.inter F).f j) := pullback.lift p₁ p₂ w
h : ((E.inter F).sieve₁ p₁ p₂).arrows f
k : (E.inter F).I₁ i j
a : Y ⟶ (E.inter F).Y k
h₁ : f ≫ p₁ = a ≫ (E.inter F).p₁ k
h₂ : f ≫ p₂ = a ≫ (E.inter F).p₂ k
⊢ (E.sieve₁ (p₁ ≫ pullback.fst (E.f i.1) (F.f i.2)) (p₂ ≫ pullback.fst (E.f j.1) (F.f j.2))).arrows (Y :=
    pullback p ((E.inter F).toPullback k)) (pullback.fst p ((E.inter F).toPullback k))
have h : Eq (α := pullback p ((E.inter F).toPullback k) ⟶ E.X i.1)
  (pullback.fst p ((E.inter F).toPullback k) ≫ p₁ ≫ pullback.fst (E.f i.1) (F.f i.2))
  ((pullback.snd p ((E.inter F).toPullback k) ≫ pullback.fst (E.p₁ k.1 ≫ E.f i.1) (F.p₁ k.2 ≫ F.f i.2)) ≫ E.p₁ k.1)
Tactic: refine ⟨k.1, pullback.snd _ _ ≫ pullback.fst _ _, ?_, ?_⟩
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Hypercover.One (goal)
Goal:
C : Type u
inst : Category.{v, u} C
S : C
E : PreOneHypercover.{w, v, u} S
F : PreOneHypercover.{u_2, v, u} S
inst_1 : HasPullbacks.{v, u} C
i j : Prod.{w, u_2} E.I₀ F.I₀
W : C
p₁ : W ⟶ pullback (X := E.X i.1) (Y := F.X i.2) (E.f i.1) (F.f i.2)
p₂ : W ⟶ pullback (X := E.X j.1) (Y := F.X j.2) (E.f j.1) (F.f j.2)
w : p₁ ≫ pullback.fst (E.f i.1) (F.f i.2) ≫ E.f i.1 = p₂ ≫ pullback.fst (E.f j.1) (F.f j.2) ≫ E.f j.1
Y : C
f : Y ⟶ W
p : W ⟶ pullback ((E.inter F).f i) ((E.inter F).f j) := pullback.lift p₁ p₂ w
h_1 : ((E.inter F).sieve₁ p₁ p₂).arrows f
k : (E.inter F).I₁ i j
a : Y ⟶ (E.inter F).Y k
h₁ : f ≫ p₁ = a ≫ (E.inter F).p₁ k
h₂ : f ≫ p₂ = a ≫ (E.inter F).p₂ k
h :
  Eq (α := pullback p ((E.inter F).toPullback k) ⟶ E.X i.1)
    (pullback.fst p ((E.inter F).toPullback k) ≫ p₁ ≫ pullback.fst (E.f i.1) (F.f i.2))
    ((pullback.snd p ((E.inter F).toPullback k) ≫ pullback.fst (E.p₁ k.1 ≫ E.f i.1) (F.p₁ k.2 ≫ F.f i.2)) ≫ E.p₁ k.1)
⊢ (E.sieve₁ (p₁ ≫ pullback.fst (E.f i.1) (F.f i.2)) (p₂ ≫ pullback.fst (E.f j.1) (F.f j.2))).arrows (Y :=
    pullback p ((E.inter F).toPullback k)) (pullback.fst p ((E.inter F).toPullback k))
have h₃ : Eq (α := pullback p ((E.inter F).toPullback k) ⟶ E.X j.1)
  (pullback.fst p ((E.inter F).toPullback k) ≫ p₂ ≫ pullback.fst (E.f j.1) (F.f j.2))
  ((pullback.snd p ((E.inter F).toPullback k) ≫ pullback.fst (E.p₁ k.1 ≫ E.f i.1) (F.p₁ k.2 ≫ F.f i.2)) ≫ E.p₂ k.1)
Tactic: refine ⟨k.1, pullback.snd _ _ ≫ pullback.fst _ _, ?_, ?_⟩
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Hypercover.One (hypothesis)
Goal:
C : Type u
inst : Category.{v, u} C
S : C
E : PreOneHypercover.{w, v, u} S
F : PreOneHypercover.{u_2, v, u} S
inst_1 : HasPullbacks.{v, u} C
i j : Prod.{w, u_2} E.I₀ F.I₀
W : C
p₁ : W ⟶ pullback (X := E.X i.1) (Y := F.X i.2) (E.f i.1) (F.f i.2)
p₂ : W ⟶ pullback (X := E.X j.1) (Y := F.X j.2) (E.f j.1) (F.f j.2)
w : p₁ ≫ pullback.fst (E.f i.1) (F.f i.2) ≫ E.f i.1 = p₂ ≫ pullback.fst (E.f j.1) (F.f j.2) ≫ E.f j.1
Y : C
f : Y ⟶ W
p : W ⟶ pullback ((E.inter F).f i) ((E.inter F).f j) := pullback.lift p₁ p₂ w
h : ((E.inter F).sieve₁ p₁ p₂).arrows f
k : (E.inter F).I₁ i j
a : Y ⟶ (E.inter F).Y k
h₁ : f ≫ p₁ = a ≫ (E.inter F).p₁ k
h₂ : f ≫ p₂ = a ≫ (E.inter F).p₂ k
⊢ Eq (α := pullback p ((E.inter F).toPullback k) ⟶ E.X i.1)
    (pullback.fst p ((E.inter F).toPullback k) ≫ p₁ ≫ pullback.fst (E.f i.1) (F.f i.2))
    ((pullback.snd p ((E.inter F).toPullback k) ≫ pullback.fst (E.p₁ k.1 ≫ E.f i.1) (F.p₁ k.2 ≫ F.f i.2)) ≫ E.p₁ k.1)
have this : p₁ ≫ pullback.fst (E.f i.1) (F.f i.2) =
  p ≫
    pullback.fst ((E.inter F).f i) ((E.inter F).f j) ≫
      pullback.fst
        (E.f
          (Sigma.fst (β := fun (i : E.I₀) ↦
            PreZeroHypercover.I₀.{u_2, v, u} (C := C) (S := E.X i)
              ((fun (i : E.I₀) ↦ PreZeroHypercover.pullback₁.{u_2, v, u} (E.f i) F.toPreZeroHypercover) i))
            ((Equiv.symm (Equiv.sigmaEquivProd E.I₀ F.toPreZeroHypercover.1) :
                (a : E.I₀ × F.toPreZeroHypercover.1) →
                  (E.bind fun (i : E.I₀) ↦ PreZeroHypercover.pullback₁ (E.f i) F.toPreZeroHypercover).I₀)
              i)))
        (F.f
          (Sigma.snd (β := fun (i : E.I₀) ↦
            PreZeroHypercover.I₀.{u_2, v, u} (C := C) (S := E.X i)
              ((fun (i : E.I₀) ↦ PreZeroHypercover.pullback₁.{u_2, v, u} (E.f i) F.toPreZeroHypercover) i))
            ((Equiv.symm (Equiv.sigmaEquivProd E.I₀ F.toPreZeroHypercover.1) :
                (a : E.I₀ × F.toPreZeroHypercover.1) →
                  (E.bind fun (i : E.I₀) ↦ PreZeroHypercover.pullback₁ (E.f i) F.toPreZeroHypercover).I₀)
              i)))
Tactic: have : p₁ ≫ pullback.fst (E.f i.1) (F.f i.2) = p ≫ pullback.fst _ _ ≫ pullback.fst _ _ := by
          simp [p]