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 : ((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
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)))
⊢ 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:
simp [this, pullback.condition_assoc, toPullback]

Premises used: CategoryTheory.Limits.pullback.condition_assoc, 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
⊢ p₂ ≫ pullback.fst (E.f j.1) (F.f j.2) =
    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)))

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
⊢ p₂ ≫ pullback.fst (E.f j.1) (F.f j.2) =
    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)))

Tactic:
have : p₂ ≫ pullback.fst (E.f j.1) (F.f j.2) = p ≫ pullback.snd _ _ ≫ pullback.fst _ _ := by
          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
this :
  p₂ ≫ pullback.fst (E.f j.1) (F.f j.2) =
    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)))
⊢ 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:
simp [this, pullback.condition_assoc, toPullback]

Premises used: CategoryTheory.Limits.pullback.condition_assoc, 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
⊢ failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)

Tactic:
simp [reassoc_of% h₁]
-- 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
⊢ failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)

Tactic:
simp [reassoc_of% h₂]
-- 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
⊢ (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 ⋯)

Tactic:
exact ⟨k.2, a ≫ pullback.snd _ _, by simp [reassoc_of% h₁], by simp [reassoc_of% h₂]⟩
-- 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
⊢ failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)

Tactic:
simp
-- 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 :
  (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
h :
  (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
      (a ≫ b) →
    ((E.inter F).sieve₁ p₁ p₂).arrows (a ≫ b)
⊢ ((E.inter F).sieve₁ p₁ p₂).arrows f

Tactic:
subst hab
-- 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
p : W ⟶ pullback ((E.inter F).f i) ((E.inter F).f j) := pullback.lift p₁ p₂ w
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
h :
  (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
    (a ≫ b)
⊢ ((E.inter F).sieve₁ p₁ p₂).arrows (a ≫ b)

Tactic:
⟨(k, l), pullback.lift (a ≫ e) u ?_, ?_, ?_⟩

Premises used: CategoryTheory.Limits.pullback.lift, Prod.mk, And.intro, 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
p : W ⟶ pullback ((E.inter F).f i) ((E.inter F).f j) := pullback.lift p₁ p₂ w
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
h_1 :
  (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
    (a ≫ b)
h : (a ≫ e) ≫ E.p₁ (k, l).1 ≫ E.f i.1 = u ≫ F.p₁ (k, l).2 ≫ F.f i.2
h₃ : (a ≫ b) ≫ p₁ = pullback.lift (a ≫ e) u ⋯ ≫ (E.inter F).p₁ (k, l)
h₄ : (a ≫ b) ≫ p₂ = pullback.lift (a ≫ e) u ⋯ ≫ (E.inter F).p₂ (k, l)
⊢ ((E.inter F).sieve₁ p₁ p₂).arrows (a ≫ b)

Tactic:
refine ⟨(k, l), pullback.lift (a ≫ e) u ?_, ?_, ?_⟩
-- 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
p : W ⟶ pullback ((E.inter F).f i) ((E.inter F).f j) := pullback.lift p₁ p₂ w
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
h :
  (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
    (a ≫ b)
⊢ a ≫ b ≫ p₁ ≫ pullback.snd (E.f i.1) (F.f i.2) = u ≫ F.p₁ l

Tactic:
simp only [Category.assoc] at u₁

Premises used: CategoryTheory.Category.assoc
-- 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
p : W ⟶ pullback ((E.inter F).f i) ((E.inter F).f j) := pullback.lift p₁ p₂ w
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
u₂ : (a ≫ b) ≫ p₂ ≫ pullback.snd (E.f j.1) (F.f j.2) = u ≫ F.p₂ l
h :
  (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
    (a ≫ b)
u₁ : a ≫ b ≫ p₁ ≫ pullback.snd (E.f i.1) (F.f i.2) = u ≫ F.p₁ l
⊢ (a ≫ e) ≫ E.p₁ (k, l).1 ≫ E.f i.1 = u ≫ F.p₁ (k, l).2 ≫ F.f i.2

Tactic:
simp [← reassoc_of% h₁, w, ← reassoc_of% u₁, ← pullback.condition]

Premises used: CategoryTheory.Limits.pullback.condition
-- Module: Mathlib.CategoryTheory.Sites.Hypercover.One
Goal:
C : Type u
inst : Category.{v, u} C
A : Type u_1
inst_1 : Category.{v_1, u_1} A
S : C
E_1 E : PreOneHypercover.{w, v, u} S
F : PreOneHypercover.{w', v, u} S
G : PreOneHypercover.{u_2, v, u} S
f : E.Hom F
g : F.Hom G
⊢ ∀ {i j : E.I₀} (k : E.I₁ i j), (f.h₁ k ≫ g.h₁ (f.s₁ k)) ≫ G.p₁ ((g.s₁ ∘ f.s₁) k) = E.p₁ k ≫ (f.comp g.toHom).h₀ i

Tactic:
simp [w₁₁, w₁₁_assoc]

Premises used: CategoryTheory.PreOneHypercover.Hom.w₁₁, CategoryTheory.PreOneHypercover.Hom.w₁₁_assoc
-- Module: Mathlib.CategoryTheory.Sites.Hypercover.One
Goal:
C : Type u
inst : Category.{v, u} C
A : Type u_1
inst_1 : Category.{v_1, u_1} A
S : C
E_1 E : PreOneHypercover.{w, v, u} S
F : PreOneHypercover.{w', v, u} S
G : PreOneHypercover.{u_2, v, u} S
f : E.Hom F
g : F.Hom G
⊢ ∀ {i j : E.I₀} (k : E.I₁ i j), (f.h₁ k ≫ g.h₁ (f.s₁ k)) ≫ G.p₂ ((g.s₁ ∘ f.s₁) k) = E.p₂ k ≫ (f.comp g.toHom).h₀ j

Tactic:
simp [w₁₂, w₁₂_assoc]

Premises used: CategoryTheory.PreOneHypercover.Hom.w₁₂_assoc, CategoryTheory.PreOneHypercover.Hom.w₁₂
-- Module: Mathlib.CategoryTheory.Sites.Hypercover.One
Goal:
C : Type u
inst : Category.{v, u} C
A : Type u_1
inst_1 : Category.{v_1, u_1} A
S : C
E_1 E : PreOneHypercover.{w, v, u} S
F : PreOneHypercover.{w', v, u} S
G : PreOneHypercover.{?u.219515, v, u} S
f : E.Hom F
P : Functor.{v, v_1, u, u_1} Cᵒᵖ A
c : Multifork (E.multicospanIndex P)
hc : IsLimit c
d : Multifork (F.multicospanIndex P)
k : E.I₁'
simp :
  d.ι (f.s₀ k.fst (β := fun (i : E.I₀ × E.I₀) ↦ E.I₁ i.1 i.2).1) ≫
      P.map (Quiver.Hom.op (f.h₀ k.fst (β := fun (i : E.I₀ × E.I₀) ↦ E.I₁ i.1 i.2).1)) ≫
        P.map (E.p₁ (k.snd (β := fun (i : E.I₀ × E.I₀) ↦ E.I₁ i.1 i.2))).op =
    d.ι (f.s₀ k.fst (β := fun (i : E.I₀ × E.I₀) ↦ E.I₁ i.1 i.2).2) ≫
      P.map (Quiver.Hom.op (f.h₀ k.fst (β := fun (i : E.I₀ × E.I₀) ↦ E.I₁ i.1 i.2).2)) ≫
        P.map (E.p₂ (k.snd (β := fun (i : E.I₀ × E.I₀) ↦ E.I₁ i.1 i.2))).op
⊢ Eq (α := d.pt ⟶ (E.multicospanIndex P).right k)
    ((fun (a : E.multicospanShape.L) ↦ d.ι (f.s₀ a) ≫ P.map (f.h₀ a).op) (E.multicospanShape.fst k) ≫
      (E.multicospanIndex P).fst k)
    ((fun (a : E.multicospanShape.L) ↦ d.ι (f.s₀ a) ≫ P.map (f.h₀ a).op) (E.multicospanShape.snd k) ≫
      (E.multicospanIndex P).snd k)

Tactic:
simp only [multicospanIndex_right, multicospanShape_fst, multicospanIndex_left,
      multicospanIndex_fst, assoc, multicospanShape_snd, multicospanIndex_snd]

Premises used: CategoryTheory.Category.assoc, CategoryTheory.PreOneHypercover.multicospanIndex_fst, CategoryTheory.PreOneHypercover.multicospanIndex_right, CategoryTheory.PreOneHypercover.multicospanShape_fst, CategoryTheory.PreOneHypercover.multicospanIndex_snd, CategoryTheory.PreOneHypercover.multicospanIndex_left, CategoryTheory.PreOneHypercover.multicospanShape_snd
-- Module: Mathlib.CategoryTheory.Sites.Hypercover.One
Goal:
C : Type u
inst : Category.{v, u} C
A : Type u_1
inst_1 : Category.{v_1, u_1} A
S : C
E_1 E : PreOneHypercover.{w, v, u} S
F : PreOneHypercover.{w', v, u} S
G : PreOneHypercover.{?u.219515, v, u} S
f : E.Hom F
P : Functor.{v, v_1, u, u_1} Cᵒᵖ A
c : Multifork (E.multicospanIndex P)
hc : IsLimit c
d : Multifork (F.multicospanIndex P)
k : E.I₁'
⊢ Eq (α := d.pt ⟶ (F.multicospanIndex P).right (f.s₁' k))
    (d.ι (F.multicospanShape.fst (f.s₁' k)) ≫ (F.multicospanIndex P).fst (f.s₁' k))
    (d.ι (F.multicospanShape.snd (f.s₁' k)) ≫ (F.multicospanIndex P).snd (f.s₁' k))

Tactic:
have heq := d.condition (f.s₁' k)