Module: Mathlib.AlgebraicGeometry.AffineTransitionLimit

Goal state before:
pp: case h₀
I : Type u
inst✝³ : Category.{u, u} I
S X : Scheme
D : I ⥤ Scheme
t : D ⟶ (Functor.const I).obj S
f : X ⟶ S
inst✝² : ∀ (i : I), CompactSpace ↥(D.obj i)
inst✝¹ : IsCofiltered I
inst✝ : ∀ {i j : I} (f : i ⟶ j), IsAffineHom (D.map f)
A : ExistsHomHomCompEqCompAux D t f
W : (pullback.diagonalObj f).Opens := ⋯
h :
  ∀ (i' : I) (hii' : i' ⟶ A.i),
    (⇑(D.map hii' ≫ pullback.lift A.a A.b ⋯) ⁻¹' (↑(Scheme.Pullback.diagonalCoverDiagonalRange f A.𝒰S A.𝒰X))ᶜ).Nonempty
Z : (i' : I) → (i' ⟶ A.i) → Set ↥(D.obj i') := ⋯
hZ : ∀ (i' : I) (hii' : i' ⟶ A.i), IsClosed (Z i' hii')
s : ↥A.c.pt
hs : ∀ (i : I) (hij : i ⟶ A.i), (A.c.π.app i) s ∈ Z i hij
⊢ (A.c.π.app A.i ≫ A.a ≫ pullback.diagonal f) ≫ pullback.fst f f =
    (A.c.π.app A.i ≫ pullback.lift A.a A.b ⋯) ≫ pullback.fst f f

pp: case h₁
I : Type u
inst✝³ : Category.{u, u} I
S X : Scheme
D : I ⥤ Scheme
t : D ⟶ (Functor.const I).obj S
f : X ⟶ S
inst✝² : ∀ (i : I), CompactSpace ↥(D.obj i)
inst✝¹ : IsCofiltered I
inst✝ : ∀ {i j : I} (f : i ⟶ j), IsAffineHom (D.map f)
A : ExistsHomHomCompEqCompAux D t f
W : (pullback.diagonalObj f).Opens := ⋯
h :
  ∀ (i' : I) (hii' : i' ⟶ A.i),
    (⇑(D.map hii' ≫ pullback.lift A.a A.b ⋯) ⁻¹' (↑(Scheme.Pullback.diagonalCoverDiagonalRange f A.𝒰S A.𝒰X))ᶜ).Nonempty
Z : (i' : I) → (i' ⟶ A.i) → Set ↥(D.obj i') := ⋯
hZ : ∀ (i' : I) (hii' : i' ⟶ A.i), IsClosed (Z i' hii')
s : ↥A.c.pt
hs : ∀ (i : I) (hij : i ⟶ A.i), (A.c.π.app i) s ∈ Z i hij
⊢ (A.c.π.app A.i ≫ A.a ≫ pullback.diagonal f) ≫ pullback.snd f f =
    (A.c.π.app A.i ≫ pullback.lift A.a A.b ⋯) ≫ pullback.snd f f

Tactic:
<;>

Goal state after:
case h₀
I : Type u
inst✝³ : Category.{u, u} I
S X : Scheme
D : I ⥤ Scheme
t : D ⟶ (Functor.const I).obj S
f : X ⟶ S
inst✝² : ∀ (i : I), CompactSpace ↥(D.obj i)
inst✝¹ : IsCofiltered I
inst✝ : ∀ {i j : I} (f : i ⟶ j), IsAffineHom (D.map f)
A : ExistsHomHomCompEqCompAux D t f
W : (pullback.diagonalObj f).Opens := ⋯
h :
  ∀ (i' : I) (hii' : i' ⟶ A.i),
    (⇑(D.map hii' ≫ pullback.lift A.a A.b ⋯) ⁻¹' (↑(Scheme.Pullback.diagonalCoverDiagonalRange f A.𝒰S A.𝒰X))ᶜ).Nonempty
Z : (i' : I) → (i' ⟶ A.i) → Set ↥(D.obj i') := ⋯
hZ : ∀ (i' : I) (hii' : i' ⟶ A.i), IsClosed (Z i' hii')
s : ↥A.c.pt
hs : ∀ (i : I) (hij : i ⟶ A.i), (A.c.π.app i) s ∈ Z i hij
⊢ (A.c.π.app A.i ≫ A.a ≫ pullback.diagonal f) ≫ pullback.fst f f =
    (A.c.π.app A.i ≫ pullback.lift A.a A.b ⋯) ≫ pullback.fst f f

case h₁
I : Type u
inst✝³ : Category.{u, u} I
S X : Scheme
D : I ⥤ Scheme
t : D ⟶ (Functor.const I).obj S
f : X ⟶ S
inst✝² : ∀ (i : I), CompactSpace ↥(D.obj i)
inst✝¹ : IsCofiltered I
inst✝ : ∀ {i j : I} (f : i ⟶ j), IsAffineHom (D.map f)
A : ExistsHomHomCompEqCompAux D t f
W : (pullback.diagonalObj f).Opens := ⋯
h :
  ∀ (i' : I) (hii' : i' ⟶ A.i),
    (⇑(D.map hii' ≫ pullback.lift A.a A.b ⋯) ⁻¹' (↑(Scheme.Pullback.diagonalCoverDiagonalRange f A.𝒰S A.𝒰X))ᶜ).Nonempty
Z : (i' : I) → (i' ⟶ A.i) → Set ↥(D.obj i') := ⋯
hZ : ∀ (i' : I) (hii' : i' ⟶ A.i), IsClosed (Z i' hii')
s : ↥A.c.pt
hs : ∀ (i : I) (hij : i ⟶ A.i), (A.c.π.app i) s ∈ Z i hij
⊢ (A.c.π.app A.i ≫ A.a ≫ pullback.diagonal f) ≫ pullback.snd f f =
    (A.c.π.app A.i ≫ pullback.lift A.a A.b ⋯) ≫ pullback.snd f f