Module: Mathlib.CategoryTheory.Sites.Hypercover.One

Goal state before:
pp: case refine_1.refine_1.h₀.h₀
C : Type u
inst✝¹ : Category.{v, u} C
S : C
E : PreOneHypercover S
F : PreOneHypercover S
inst✝ : HasPullbacks C
i j : E.I₀ × F.I₀
W : C
p₁ : W ⟶ pullback (E.f i.1) (F.f i.2)
p₂ : W ⟶ pullback (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) := ⋯
x✝ : ((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 ((Equiv.sigmaEquivProd E.I₀ F.toPreZeroHypercover.1).symm i).fst)
        (F.f ((Equiv.sigmaEquivProd E.I₀ F.toPreZeroHypercover.1).symm i).snd) =
    ((a ≫ (E.inter F).toPullback k) ≫ pullback.fst ((E.inter F).f i) ((E.inter F).f j)) ≫
      pullback.fst (E.f ((Equiv.sigmaEquivProd E.I₀ F.toPreZeroHypercover.1).symm i).fst)
        (F.f ((Equiv.sigmaEquivProd E.I₀ F.toPreZeroHypercover.1).symm i).snd)

pp: case h₁
C : Type u
inst✝¹ : Category.{v, u} C
S : C
E : PreOneHypercover S
F : PreOneHypercover S
inst✝ : HasPullbacks C
i j : E.I₀ × F.I₀
W : C
p₁ : W ⟶ pullback (E.f i.1) (F.f i.2)
p₂ : W ⟶ pullback (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) := ⋯
x✝ : ((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 ((Equiv.sigmaEquivProd E.I₀ F.toPreZeroHypercover.1).symm i).fst)
        (F.f ((Equiv.sigmaEquivProd E.I₀ F.toPreZeroHypercover.1).symm i).snd) =
    ((a ≫ (E.inter F).toPullback k) ≫ pullback.fst ((E.inter F).f i) ((E.inter F).f j)) ≫
      pullback.snd (E.f ((Equiv.sigmaEquivProd E.I₀ F.toPreZeroHypercover.1).symm i).fst)
        (F.f ((Equiv.sigmaEquivProd E.I₀ F.toPreZeroHypercover.1).symm i).snd)

Tactic:
<;>

Goal state after:
case refine_1.refine_1.h₀.h₀
C : Type u
inst✝¹ : Category.{v, u} C
S : C
E : PreOneHypercover S
F : PreOneHypercover S
inst✝ : HasPullbacks C
i j : E.I₀ × F.I₀
W : C
p₁ : W ⟶ pullback (E.f i.1) (F.f i.2)
p₂ : W ⟶ pullback (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) := ⋯
x✝ : ((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 ((Equiv.sigmaEquivProd E.I₀ F.toPreZeroHypercover.1).symm i).fst)
        (F.f ((Equiv.sigmaEquivProd E.I₀ F.toPreZeroHypercover.1).symm i).snd) =
    ((a ≫ (E.inter F).toPullback k) ≫ pullback.fst ((E.inter F).f i) ((E.inter F).f j)) ≫
      pullback.fst (E.f ((Equiv.sigmaEquivProd E.I₀ F.toPreZeroHypercover.1).symm i).fst)
        (F.f ((Equiv.sigmaEquivProd E.I₀ F.toPreZeroHypercover.1).symm i).snd)

case h₁
C : Type u
inst✝¹ : Category.{v, u} C
S : C
E : PreOneHypercover S
F : PreOneHypercover S
inst✝ : HasPullbacks C
i j : E.I₀ × F.I₀
W : C
p₁ : W ⟶ pullback (E.f i.1) (F.f i.2)
p₂ : W ⟶ pullback (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) := ⋯
x✝ : ((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 ((Equiv.sigmaEquivProd E.I₀ F.toPreZeroHypercover.1).symm i).fst)
        (F.f ((Equiv.sigmaEquivProd E.I₀ F.toPreZeroHypercover.1).symm i).snd) =
    ((a ≫ (E.inter F).toPullback k) ≫ pullback.fst ((E.inter F).f i) ((E.inter F).f j)) ≫
      pullback.snd (E.f ((Equiv.sigmaEquivProd E.I₀ F.toPreZeroHypercover.1).symm i).fst)
        (F.f ((Equiv.sigmaEquivProd E.I₀ F.toPreZeroHypercover.1).symm i).snd)