-- Mathlib/AlgebraicGeometry/Morphisms/Separated.lean
lemma Scheme.Pullback.diagonalCoverDiagonalRange_eq_top_of_injective
    (hf : Function.Injective f.base) :
    diagonalCoverDiagonalRange f 𝒰 𝒱 = ⊤

Proof state:
X Y : Scheme
f : X ⟶ Y
𝒰 : Y.OpenCover
𝒱 : (i : 𝒰.J) → (pullback f (𝒰.map i)).OpenCover
hf : Function.Injective ⇑(ConcreteCategory.hom f.base)
x : ↑↑(pullback.diagonalObj f).toPresheafedSpace
H : (ConcreteCategory.hom (pullback.fst f f).base) x = (ConcreteCategory.hom (pullback.snd f f).base) x
⊢ ∃ a b y, (ConcreteCategory.hom ((diagonalCover f 𝒰 𝒱).map ⟨a, (b, b)⟩).base) y = x

Tactic:
let i := 𝒰.f (f.base ((pullback.fst f f).base x))

Result:
X Y : Scheme
f : X ⟶ Y
𝒰 : Y.OpenCover
𝒱 : (i : 𝒰.J) → (pullback f (𝒰.map i)).OpenCover
hf : Function.Injective ⇑(ConcreteCategory.hom f.base)
x : ↑↑(pullback.diagonalObj f).toPresheafedSpace
H : (ConcreteCategory.hom (pullback.fst f f).base) x = (ConcreteCategory.hom (pullback.snd f f).base) x
i : 𝒰.J := 𝒰.f ((ConcreteCategory.hom f.base) ((ConcreteCategory.hom (pullback.fst f f).base) x))
⊢ ∃ a b y, (ConcreteCategory.hom ((diagonalCover f 𝒰 𝒱).map ⟨a, (b, b)⟩).base) y = x

Proof state:
case intro.intro.intro.intro
X Y : Scheme
f : X ⟶ Y
𝒰 : Y.OpenCover
𝒱 : (i : 𝒰.J) → (pullback f (𝒰.map i)).OpenCover
hf : Function.Injective ⇑(ConcreteCategory.hom f.base)
x : ↑↑(pullback.diagonalObj f).toPresheafedSpace
H : (ConcreteCategory.hom (pullback.fst f f).base) x = (ConcreteCategory.hom (pullback.snd f f).base) x
i : 𝒰.J := 𝒰.f ((ConcreteCategory.hom f.base) ((ConcreteCategory.hom (pullback.fst f f).base) x))
y : ↑↑(𝒰.obj i).toPresheafedSpace
hy✝ :
  (ConcreteCategory.hom (𝒰.map i).base) y =
    (ConcreteCategory.hom f.base) ((ConcreteCategory.hom (pullback.fst f f).base) x)
z : ↑↑(pullback f (𝒰.map i)).toPresheafedSpace
hz₁ : (ConcreteCategory.hom (pullback.fst f (𝒰.map i)).base) z = (ConcreteCategory.hom (pullback.fst f f).base) x
hz₂ : (ConcreteCategory.hom (pullback.snd f (𝒰.map i)).base) z = y
j : (𝒱 i).J := (𝒱 i).f z
w : ↑↑((𝒱 i).obj j).toPresheafedSpace
hy : (ConcreteCategory.hom ((𝒱 i).map j).base) w = z
⊢ x ∈
    ⇑(ConcreteCategory.hom (pullback.fst f f).base) ⁻¹'
        Set.range ⇑(ConcreteCategory.hom ((𝒱 i).map j ≫ pullback.fst f (𝒰.map i)).base) ∩
      ⇑(ConcreteCategory.hom (pullback.snd f f).base) ⁻¹'
        Set.range ⇑(ConcreteCategory.hom ((𝒱 i).map j ≫ pullback.fst f (𝒰.map i)).base)

Tactic:
simp [← H, ← hz₁, ← hy]

Result:
no goals

Proof state:
X Y : Scheme
f : X ⟶ Y
𝒰 : Y.OpenCover
𝒱 : (i : 𝒰.J) → (pullback f (𝒰.map i)).OpenCover
hf : Function.Injective ⇑(ConcreteCategory.hom f.base)
x : ↑↑(pullback.diagonalObj f).toPresheafedSpace
⊢ (ConcreteCategory.hom f.base) ((ConcreteCategory.hom (pullback.fst f f).base) x) =
    (ConcreteCategory.hom f.base) ((ConcreteCategory.hom (pullback.snd f f).base) x)

Tactic:
rw [← Scheme.comp_base_apply, ← Scheme.comp_base_apply, pullback.condition]

Result:
no goals

Proof state:
case intro.intro.intro
X Y : Scheme
f : X ⟶ Y
𝒰 : Y.OpenCover
𝒱 : (i : 𝒰.J) → (pullback f (𝒰.map i)).OpenCover
hf : Function.Injective ⇑(ConcreteCategory.hom f.base)
x : ↑↑(pullback.diagonalObj f).toPresheafedSpace
H : (ConcreteCategory.hom (pullback.fst f f).base) x = (ConcreteCategory.hom (pullback.snd f f).base) x
i : 𝒰.J := 𝒰.f ((ConcreteCategory.hom f.base) ((ConcreteCategory.hom (pullback.fst f f).base) x))
y : ↑↑(𝒰.obj i).toPresheafedSpace
hy :
  (ConcreteCategory.hom (𝒰.map i).base) y =
    (ConcreteCategory.hom f.base) ((ConcreteCategory.hom (pullback.fst f f).base) x)
z : ↑↑(pullback f (𝒰.map i)).toPresheafedSpace
hz₁ : (ConcreteCategory.hom (pullback.fst f (𝒰.map i)).base) z = (ConcreteCategory.hom (pullback.fst f f).base) x
hz₂ : (ConcreteCategory.hom (pullback.snd f (𝒰.map i)).base) z = y
j : (𝒱 i).J := (𝒱 i).f z
⊢ ∃ a b y, (ConcreteCategory.hom ((diagonalCover f 𝒰 𝒱).map ⟨a, (b, b)⟩).base) y = x

Tactic:
obtain ⟨w : (𝒱 i).obj j, hy : ((𝒱 i).map j).base w = z⟩ := (𝒱 i).covers z

Result:
case intro.intro.intro.intro
X Y : Scheme
f : X ⟶ Y
𝒰 : Y.OpenCover
𝒱 : (i : 𝒰.J) → (pullback f (𝒰.map i)).OpenCover
hf : Function.Injective ⇑(ConcreteCategory.hom f.base)
x : ↑↑(pullback.diagonalObj f).toPresheafedSpace
H : (ConcreteCategory.hom (pullback.fst f f).base) x = (ConcreteCategory.hom (pullback.snd f f).base) x
i : 𝒰.J := 𝒰.f ((ConcreteCategory.hom f.base) ((ConcreteCategory.hom (pullback.fst f f).base) x))
y : ↑↑(𝒰.obj i).toPresheafedSpace
hy✝ :
  (ConcreteCategory.hom (𝒰.map i).base) y =
    (ConcreteCategory.hom f.base) ((ConcreteCategory.hom (pullback.fst f f).base) x)
z : ↑↑(pullback f (𝒰.map i)).toPresheafedSpace
hz₁ : (ConcreteCategory.hom (pullback.fst f (𝒰.map i)).base) z = (ConcreteCategory.hom (pullback.fst f f).base) x
hz₂ : (ConcreteCategory.hom (pullback.snd f (𝒰.map i)).base) z = y
j : (𝒱 i).J := (𝒱 i).f z
w : ↑↑((𝒱 i).obj j).toPresheafedSpace
hy : (ConcreteCategory.hom ((𝒱 i).map j).base) w = z
⊢ ∃ a b y, (ConcreteCategory.hom ((diagonalCover f 𝒰 𝒱).map ⟨a, (b, b)⟩).base) y = x

Proof state:
case intro.intro.intro
X Y : Scheme
f : X ⟶ Y
𝒰 : Y.OpenCover
𝒱 : (i : 𝒰.J) → (pullback f (𝒰.map i)).OpenCover
hf : Function.Injective ⇑(ConcreteCategory.hom f.base)
x : ↑↑(pullback.diagonalObj f).toPresheafedSpace
H : (ConcreteCategory.hom (pullback.fst f f).base) x = (ConcreteCategory.hom (pullback.snd f f).base) x
i : 𝒰.J := 𝒰.f ((ConcreteCategory.hom f.base) ((ConcreteCategory.hom (pullback.fst f f).base) x))
y : ↑↑(𝒰.obj i).toPresheafedSpace
hy :
  (ConcreteCategory.hom (𝒰.map i).base) y =
    (ConcreteCategory.hom f.base) ((ConcreteCategory.hom (pullback.fst f f).base) x)
z : ↑↑(pullback f (𝒰.map i)).toPresheafedSpace
hz₁ : (ConcreteCategory.hom (pullback.fst f (𝒰.map i)).base) z = (ConcreteCategory.hom (pullback.fst f f).base) x
hz₂ : (ConcreteCategory.hom (pullback.snd f (𝒰.map i)).base) z = y
⊢ ∃ a b y, (ConcreteCategory.hom ((diagonalCover f 𝒰 𝒱).map ⟨a, (b, b)⟩).base) y = x

Tactic:
let j := (𝒱 i).f z

Result:
case intro.intro.intro
X Y : Scheme
f : X ⟶ Y
𝒰 : Y.OpenCover
𝒱 : (i : 𝒰.J) → (pullback f (𝒰.map i)).OpenCover
hf : Function.Injective ⇑(ConcreteCategory.hom f.base)
x : ↑↑(pullback.diagonalObj f).toPresheafedSpace
H : (ConcreteCategory.hom (pullback.fst f f).base) x = (ConcreteCategory.hom (pullback.snd f f).base) x
i : 𝒰.J := 𝒰.f ((ConcreteCategory.hom f.base) ((ConcreteCategory.hom (pullback.fst f f).base) x))
y : ↑↑(𝒰.obj i).toPresheafedSpace
hy :
  (ConcreteCategory.hom (𝒰.map i).base) y =
    (ConcreteCategory.hom f.base) ((ConcreteCategory.hom (pullback.fst f f).base) x)
z : ↑↑(pullback f (𝒰.map i)).toPresheafedSpace
hz₁ : (ConcreteCategory.hom (pullback.fst f (𝒰.map i)).base) z = (ConcreteCategory.hom (pullback.fst f f).base) x
hz₂ : (ConcreteCategory.hom (pullback.snd f (𝒰.map i)).base) z = y
j : (𝒱 i).J := (𝒱 i).f z
⊢ ∃ a b y, (ConcreteCategory.hom ((diagonalCover f 𝒰 𝒱).map ⟨a, (b, b)⟩).base) y = x