awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Module: Mathlib.CategoryTheory.Sites.Hypercover.One
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 ⋯)
h₅ : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)
⊢ (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

Tactic:
refine ⟨pullback p ((E.inter F).toPullback k), pullback.lift f a ?_,
        pullback.fst _ _, ?_, ?_, ?_⟩
-- Module: Mathlib.CategoryTheory.Sites.Hypercover.One
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))
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))
⊢ f ≫ p = a ≫ (E.inter F).toPullback k

Tactic:
apply pullback.hom_ext

Premises used: CategoryTheory.Limits.pullback.hom_ext
-- Module: Mathlib.CategoryTheory.Sites.Hypercover.One
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)) ≫
      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:
simp [p, h₁, toPullback]

Premises used: CategoryTheory.PreOneHypercover.toPullback
-- Module: Mathlib.CategoryTheory.Sites.Hypercover.One
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)) ≫
      pullback.snd
        (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.snd
        (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:
simp [p, h₁, toPullback]

Premises used: CategoryTheory.PreOneHypercover.toPullback
-- Module: Mathlib.CategoryTheory.Sites.Hypercover.One
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)) ≫
      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:
simp [p, h₂, toPullback]

Premises used: CategoryTheory.PreOneHypercover.toPullback
-- Module: Mathlib.CategoryTheory.Sites.Hypercover.One
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)) ≫
      pullback.snd
        (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.snd
        (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:
simp [p, h₂, toPullback]

Premises used: CategoryTheory.PreOneHypercover.toPullback
-- Module: Mathlib.CategoryTheory.Sites.Hypercover.One
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))

Tactic:
⟨k.1, pullback.snd _ _ ≫ pullback.fst _ _, ?_, ?_⟩

Premises used: CategoryTheory.Limits.pullback.snd, Prod.fst, And.intro, CategoryTheory.Limits.pullback.fst, Exists.intro, CategoryTheory.CategoryStruct.comp
-- Module: Mathlib.CategoryTheory.Sites.Hypercover.One
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)
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)
⊢ (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 ⟨k.1, pullback.snd _ _ ≫ pullback.fst _ _, ?_, ?_⟩
-- Module: Mathlib.CategoryTheory.Sites.Hypercover.One
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
⊢ 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:
simp [p]
-- Module: Mathlib.CategoryTheory.Sites.Hypercover.One
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
⊢ 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]