Module: Mathlib.AlgebraicGeometry.Pullbacks

Goal state before:
pp: case h₀
X Y Z : Scheme
𝒰✝ : X.OpenCover
f✝ : X ⟶ Z
g : Y ⟶ Z
inst✝ : ∀ (i : 𝒰✝.I₀), HasPullback (𝒰✝.f i ≫ f✝) g
s : PullbackCone f✝ g
f : X ⟶ Y
𝒰 : Y.OpenCover
𝒱 : (i : (Precoverage.ZeroHypercover.pullback₁ f 𝒰).I₀) → ((Precoverage.ZeroHypercover.pullback₁ f 𝒰).X i).OpenCover
i : (openCoverOfBase 𝒰 f f).I₀
j : (𝒱 i).I₀
⊢ ((pullback.snd (pullback.diagonal f) ((diagonalCover f 𝒰 𝒱).f ⟨i, (j, j)⟩) ≫
          pullback.snd ((𝒱 ⟨i, (j, j)⟩.fst).f ⟨i, (j, j)⟩.snd.1 ≫ Cover.pullbackHom 𝒰 f ⟨i, (j, j)⟩.fst)
            ((𝒱 ⟨i, (j, j)⟩.fst).f ⟨i, (j, j)⟩.snd.2 ≫ Cover.pullbackHom 𝒰 f ⟨i, (j, j)⟩.fst)) ≫
        (𝒱 i).f ⟨i, (j, j)⟩.snd.2) ≫
      pullback.fst f (𝒰.f i) =
    ((pullback.snd (pullback.diagonal f) ((diagonalCover f 𝒰 𝒱).f ⟨i, (j, j)⟩) ≫
          pullback.fst ((𝒱 ⟨i, (j, j)⟩.fst).f ⟨i, (j, j)⟩.snd.1 ≫ Cover.pullbackHom 𝒰 f ⟨i, (j, j)⟩.fst)
            ((𝒱 ⟨i, (j, j)⟩.fst).f ⟨i, (j, j)⟩.snd.2 ≫ Cover.pullbackHom 𝒰 f ⟨i, (j, j)⟩.fst)) ≫
        (𝒱 i).f ⟨i, (j, j)⟩.snd.2) ≫
      pullback.fst f (𝒰.f i)

pp: case h₁
X Y Z : Scheme
𝒰✝ : X.OpenCover
f✝ : X ⟶ Z
g : Y ⟶ Z
inst✝ : ∀ (i : 𝒰✝.I₀), HasPullback (𝒰✝.f i ≫ f✝) g
s : PullbackCone f✝ g
f : X ⟶ Y
𝒰 : Y.OpenCover
𝒱 : (i : (Precoverage.ZeroHypercover.pullback₁ f 𝒰).I₀) → ((Precoverage.ZeroHypercover.pullback₁ f 𝒰).X i).OpenCover
i : (openCoverOfBase 𝒰 f f).I₀
j : (𝒱 i).I₀
⊢ ((pullback.snd (pullback.diagonal f) ((diagonalCover f 𝒰 𝒱).f ⟨i, (j, j)⟩) ≫
          pullback.snd ((𝒱 ⟨i, (j, j)⟩.fst).f ⟨i, (j, j)⟩.snd.1 ≫ Cover.pullbackHom 𝒰 f ⟨i, (j, j)⟩.fst)
            ((𝒱 ⟨i, (j, j)⟩.fst).f ⟨i, (j, j)⟩.snd.2 ≫ Cover.pullbackHom 𝒰 f ⟨i, (j, j)⟩.fst)) ≫
        (𝒱 i).f ⟨i, (j, j)⟩.snd.2) ≫
      pullback.snd f (𝒰.f i) =
    ((pullback.snd (pullback.diagonal f) ((diagonalCover f 𝒰 𝒱).f ⟨i, (j, j)⟩) ≫
          pullback.fst ((𝒱 ⟨i, (j, j)⟩.fst).f ⟨i, (j, j)⟩.snd.1 ≫ Cover.pullbackHom 𝒰 f ⟨i, (j, j)⟩.fst)
            ((𝒱 ⟨i, (j, j)⟩.fst).f ⟨i, (j, j)⟩.snd.2 ≫ Cover.pullbackHom 𝒰 f ⟨i, (j, j)⟩.fst)) ≫
        (𝒱 i).f ⟨i, (j, j)⟩.snd.2) ≫
      pullback.snd f (𝒰.f i)

Tactic:
· trans pullback.snd (pullback.diagonal f) ((diagonalCover f 𝒰 𝒱).f ⟨i, (j, j)⟩) ≫
        (diagonalCover f 𝒰 𝒱).f _ ≫ pullback.snd _ _
      · simp [diagonalCover_map]
      symm
      trans pullback.snd (pullback.diagonal f) ((diagonalCover f 𝒰 𝒱).f ⟨i, (j, j)⟩) ≫
        (diagonalCover f 𝒰 𝒱).f _ ≫ pullback.fst _ _
      · simp [diagonalCover_map]
      · rw [← pullback.condition_assoc, ← pullback.condition_assoc]
        simp

Goal state after:
case h₁
X Y Z : Scheme
𝒰✝ : X.OpenCover
f✝ : X ⟶ Z
g : Y ⟶ Z
inst✝ : ∀ (i : 𝒰✝.I₀), HasPullback (𝒰✝.f i ≫ f✝) g
s : PullbackCone f✝ g
f : X ⟶ Y
𝒰 : Y.OpenCover
𝒱 : (i : (Precoverage.ZeroHypercover.pullback₁ f 𝒰).I₀) → ((Precoverage.ZeroHypercover.pullback₁ f 𝒰).X i).OpenCover
i : (openCoverOfBase 𝒰 f f).I₀
j : (𝒱 i).I₀
⊢ ((pullback.snd (pullback.diagonal f) ((diagonalCover f 𝒰 𝒱).f ⟨i, (j, j)⟩) ≫
          pullback.snd ((𝒱 ⟨i, (j, j)⟩.fst).f ⟨i, (j, j)⟩.snd.1 ≫ Cover.pullbackHom 𝒰 f ⟨i, (j, j)⟩.fst)
            ((𝒱 ⟨i, (j, j)⟩.fst).f ⟨i, (j, j)⟩.snd.2 ≫ Cover.pullbackHom 𝒰 f ⟨i, (j, j)⟩.fst)) ≫
        (𝒱 i).f ⟨i, (j, j)⟩.snd.2) ≫
      pullback.snd f (𝒰.f i) =
    ((pullback.snd (pullback.diagonal f) ((diagonalCover f 𝒰 𝒱).f ⟨i, (j, j)⟩) ≫
          pullback.fst ((𝒱 ⟨i, (j, j)⟩.fst).f ⟨i, (j, j)⟩.snd.1 ≫ Cover.pullbackHom 𝒰 f ⟨i, (j, j)⟩.fst)
            ((𝒱 ⟨i, (j, j)⟩.fst).f ⟨i, (j, j)⟩.snd.2 ≫ Cover.pullbackHom 𝒰 f ⟨i, (j, j)⟩.fst)) ≫
        (𝒱 i).f ⟨i, (j, j)⟩.snd.2) ≫
      pullback.snd f (𝒰.f i)