l3lab/ntp-mathlib huggingface dataset:
private theorem I_prop : innerProp' E (I : 𝕜) := by
  by_cases hI : (I : 𝕜) = 0
  · rw [hI, ← Nat.cast_zero]; exact nat_prop _
  intro x y
  have hI' : (-I : 𝕜) * I = 1 := by rw [← inv_I, inv_mul_cancel hI]
  rw [conj_I, inner_, inner_, mul_left_comm]
  congr 1
  rw [smul_smul, I_mul_I_of_nonzero hI, neg_one_smul]
  rw [mul_sub, mul_add, mul_sub, mul_assoc I (𝓚 ‖I • x - y‖), ← mul_assoc (-I) I, hI', one_mul,
    mul_assoc I (𝓚 ‖I • x + y‖), ← mul_assoc (-I) I, hI', one_mul]
  have h₁ : ‖-x - y‖ = ‖x + y‖ := by rw [← neg_add', norm_neg]
  have h₂ : ‖-x + y‖ = ‖x - y‖ := by rw [← neg_sub, norm_neg, sub_eq_neg_add]
  rw [h₁, h₂]
  

Proof state:
case neg.e_a
𝕜 : Type u_1
inst✝³ : IsROrC 𝕜
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
inst✝ : InnerProductSpaceable E
hI : ¬I = 0
x y : E
hI' : -I * I = 1
h₁ : ‖-x - y‖ = ‖x + y‖
h₂ : ‖-x + y‖ = ‖x - y‖
⊢ 𝓚 ‖I • x + y‖ * 𝓚 ‖I • x + y‖ - 𝓚 ‖I • x - y‖ * 𝓚 ‖I • x - y‖ + I * 𝓚 ‖x - y‖ * 𝓚 ‖x - y‖ -
      I * 𝓚 ‖x + y‖ * 𝓚 ‖x + y‖ =
    -I * (𝓚 ‖x + y‖ * 𝓚 ‖x + y‖) - -I * (𝓚 ‖x - y‖ * 𝓚 ‖x - y‖) + 𝓚 ‖I • x + y‖ * 𝓚 ‖I • x + y‖ -
      𝓚 ‖I • x - y‖ * 𝓚 ‖I • x - y‖

Next tactic:
simp only [sub_eq_add_neg, mul_assoc]
private theorem I_prop : innerProp' E (I : 𝕜) := by
  by_cases hI : (I : 𝕜) = 0
  · rw [hI, ← Nat.cast_zero]; exact nat_prop _
  intro x y
  have hI' : (-I : 𝕜) * I = 1 := by rw [← inv_I, inv_mul_cancel hI]
  rw [conj_I, inner_, inner_, mul_left_comm]
  congr 1
  rw [smul_smul, I_mul_I_of_nonzero hI, neg_one_smul]
  rw [mul_sub, mul_add, mul_sub, mul_assoc I (𝓚 ‖I • x - y‖), ← mul_assoc (-I) I, hI', one_mul,
    mul_assoc I (𝓚 ‖I • x + y‖), ← mul_assoc (-I) I, hI', one_mul]
  have h₁ : ‖-x - y‖ = ‖x + y‖ := by rw [← neg_add', norm_neg]
  have h₂ : ‖-x + y‖ = ‖x - y‖ := by rw [← neg_sub, norm_neg, sub_eq_neg_add]
  rw [h₁, h₂]
  simp only [sub_eq_add_neg, mul_assoc]
  

Proof state:
case neg.e_a
𝕜 : Type u_1
inst✝³ : IsROrC 𝕜
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
inst✝ : InnerProductSpaceable E
hI : ¬I = 0
x y : E
hI' : -I * I = 1
h₁ : ‖-x - y‖ = ‖x + y‖
h₂ : ‖-x + y‖ = ‖x - y‖
⊢ 𝓚 ‖I • x + y‖ * 𝓚 ‖I • x + y‖ + -(𝓚 ‖I • x + -y‖ * 𝓚 ‖I • x + -y‖) + I * (𝓚 ‖x + -y‖ * 𝓚 ‖x + -y‖) +
      -(I * (𝓚 ‖x + y‖ * 𝓚 ‖x + y‖)) =
    -I * (𝓚 ‖x + y‖ * 𝓚 ‖x + y‖) + -(-I * (𝓚 ‖x + -y‖ * 𝓚 ‖x + -y‖)) + 𝓚 ‖I • x + y‖ * 𝓚 ‖I • x + y‖ +
      -(𝓚 ‖I • x + -y‖ * 𝓚 ‖I • x + -y‖)

Next tactic:
rw [← neg_mul_eq_neg_mul, ← neg_mul_eq_neg_mul]
private theorem I_prop : innerProp' E (I : 𝕜) := by
  by_cases hI : (I : 𝕜) = 0
  · rw [hI, ← Nat.cast_zero]; exact nat_prop _
  intro x y
  have hI' : (-I : 𝕜) * I = 1 := by rw [← inv_I, inv_mul_cancel hI]
  rw [conj_I, inner_, inner_, mul_left_comm]
  congr 1
  rw [smul_smul, I_mul_I_of_nonzero hI, neg_one_smul]
  rw [mul_sub, mul_add, mul_sub, mul_assoc I (𝓚 ‖I • x - y‖), ← mul_assoc (-I) I, hI', one_mul,
    mul_assoc I (𝓚 ‖I • x + y‖), ← mul_assoc (-I) I, hI', one_mul]
  have h₁ : ‖-x - y‖ = ‖x + y‖ := by rw [← neg_add', norm_neg]
  have h₂ : ‖-x + y‖ = ‖x - y‖ := by rw [← neg_sub, norm_neg, sub_eq_neg_add]
  rw [h₁, h₂]
  simp only [sub_eq_add_neg, mul_assoc]
  rw [← neg_mul_eq_neg_mul, ← neg_mul_eq_neg_mul]
  

Proof state:
case neg.e_a
𝕜 : Type u_1
inst✝³ : IsROrC 𝕜
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
inst✝ : InnerProductSpaceable E
hI : ¬I = 0
x y : E
hI' : -I * I = 1
h₁ : ‖-x - y‖ = ‖x + y‖
h₂ : ‖-x + y‖ = ‖x - y‖
⊢ 𝓚 ‖I • x + y‖ * 𝓚 ‖I • x + y‖ + -(𝓚 ‖I • x + -y‖ * 𝓚 ‖I • x + -y‖) + I * (𝓚 ‖x + -y‖ * 𝓚 ‖x + -y‖) +
      -(I * (𝓚 ‖x + y‖ * 𝓚 ‖x + y‖)) =
    -(I * (𝓚 ‖x + y‖ * 𝓚 ‖x + y‖)) + - -(I * (𝓚 ‖x + -y‖ * 𝓚 ‖x + -y‖)) + 𝓚 ‖I • x + y‖ * 𝓚 ‖I • x + y‖ +
      -(𝓚 ‖I • x + -y‖ * 𝓚 ‖I • x + -y‖)

Next tactic:
abel
theorem innerProp (r : 𝕜) : innerProp' E r := by
  

Proof state:
𝕜 : Type u_1
inst✝³ : IsROrC 𝕜
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
inst✝ : InnerProductSpaceable E
r : 𝕜
⊢ InnerProductSpaceable.innerProp' E r

Next tactic:
intro x y
theorem innerProp (r : 𝕜) : innerProp' E r := by
  intro x y
  

Proof state:
𝕜 : Type u_1
inst✝³ : IsROrC 𝕜
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
inst✝ : InnerProductSpaceable E
r : 𝕜
x y : E
⊢ inner_ 𝕜 (r • x) y = (starRingEnd 𝕜) r * inner_ 𝕜 x y

Next tactic:
rw [← re_add_im r, add_smul, add_left, real_prop _ x, ← smul_smul, real_prop _ _ y, I_prop,
    map_add, map_mul, conj_ofReal, conj_ofReal, conj_I]
theorem innerProp (r : 𝕜) : innerProp' E r := by
  intro x y
  rw [← re_add_im r, add_smul, add_left, real_prop _ x, ← smul_smul, real_prop _ _ y, I_prop,
    map_add, map_mul, conj_ofReal, conj_ofReal, conj_I]
  

Proof state:
𝕜 : Type u_1
inst✝³ : IsROrC 𝕜
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
inst✝ : InnerProductSpaceable E
r : 𝕜
x y : E
⊢ ↑(re r) * inner_ 𝕜 x y + ↑(im r) * (-I * inner_ 𝕜 x y) = (↑(re r) + ↑(im r) * -I) * inner_ 𝕜 x y

Next tactic:
ring
instance (priority := 100) InnerProductSpaceable.to_uniformConvexSpace : UniformConvexSpace E := by
  

Proof state:
𝕜 : Type u_1
inst✝⁴ : IsROrC 𝕜
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : InnerProductSpaceable E
inst✝ : NormedSpace ℝ E
⊢ UniformConvexSpace E

Next tactic:
cases nonempty_innerProductSpace ℝ E
instance (priority := 100) InnerProductSpaceable.to_uniformConvexSpace : UniformConvexSpace E := by
  cases nonempty_innerProductSpace ℝ E; 

Proof state:
case intro
𝕜 : Type u_1
inst✝⁴ : IsROrC 𝕜
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : InnerProductSpaceable E
inst✝ : NormedSpace ℝ E
val✝ : InnerProductSpace ℝ E
⊢ UniformConvexSpace E

Next tactic:
infer_instance
/-- To show that a locally ringed space is a scheme, it suffices to show that it has a jointly
surjective family of open immersions from affine schemes. -/
protected def scheme (X : LocallyRingedSpace)
    (h :
      ∀ x : X,
        ∃ (R : CommRingCat) (f : Spec.toLocallyRingedSpace.obj (op R) ⟶ X),
          (x ∈ Set.range f.1.base : _) ∧ LocallyRingedSpace.IsOpenImmersion f) :
    Scheme where
  toLocallyRingedSpace := X
  local_affine := by
    

Proof state:
C : Type u
inst✝ : Category.{v, u} C
X : LocallyRingedSpace
h : ∀ (x : ↑(toTopCat X)), ∃ R f, x ∈ Set.range ⇑f.val.base ∧ IsOpenImmersion f
⊢ ∀ (x : ↑(toTopCat X)),
    ∃ U R, Nonempty (restrict X (_ : OpenEmbedding ⇑(Opens.inclusion U.obj)) ≅ Spec.toLocallyRingedSpace.obj (op R))

Next tactic:
intro x
/-- To show that a locally ringed space is a scheme, it suffices to show that it has a jointly
surjective family of open immersions from affine schemes. -/
protected def scheme (X : LocallyRingedSpace)
    (h :
      ∀ x : X,
        ∃ (R : CommRingCat) (f : Spec.toLocallyRingedSpace.obj (op R) ⟶ X),
          (x ∈ Set.range f.1.base : _) ∧ LocallyRingedSpace.IsOpenImmersion f) :
    Scheme where
  toLocallyRingedSpace := X
  local_affine := by
    intro x
    

Proof state:
C : Type u
inst✝ : Category.{v, u} C
X : LocallyRingedSpace
h : ∀ (x : ↑(toTopCat X)), ∃ R f, x ∈ Set.range ⇑f.val.base ∧ IsOpenImmersion f
x : ↑(toTopCat X)
⊢ ∃ U R, Nonempty (restrict X (_ : OpenEmbedding ⇑(Opens.inclusion U.obj)) ≅ Spec.toLocallyRingedSpace.obj (op R))

Next tactic:
obtain ⟨R, f, h₁, h₂⟩ := h x
/-- To show that a locally ringed space is a scheme, it suffices to show that it has a jointly
surjective family of open immersions from affine schemes. -/
protected def scheme (X : LocallyRingedSpace)
    (h :
      ∀ x : X,
        ∃ (R : CommRingCat) (f : Spec.toLocallyRingedSpace.obj (op R) ⟶ X),
          (x ∈ Set.range f.1.base : _) ∧ LocallyRingedSpace.IsOpenImmersion f) :
    Scheme where
  toLocallyRingedSpace := X
  local_affine := by
    intro x
    obtain ⟨R, f, h₁, h₂⟩ := h x
    

Proof state:
case intro.intro.intro
C : Type u
inst✝ : Category.{v, u} C
X : LocallyRingedSpace
h : ∀ (x : ↑(toTopCat X)), ∃ R f, x ∈ Set.range ⇑f.val.base ∧ IsOpenImmersion f
x : ↑(toTopCat X)
R : CommRingCat
f : Spec.toLocallyRingedSpace.obj (op R) ⟶ X
h₁ : x ∈ Set.range ⇑f.val.base
h₂ : IsOpenImmersion f
⊢ ∃ U R, Nonempty (restrict X (_ : OpenEmbedding ⇑(Opens.inclusion U.obj)) ≅ Spec.toLocallyRingedSpace.obj (op R))

Next tactic:
refine' ⟨⟨⟨_, h₂.base_open.open_range⟩, h₁⟩, R, ⟨_⟩⟩
/-- To show that a locally ringed space is a scheme, it suffices to show that it has a jointly
surjective family of open immersions from affine schemes. -/
protected def scheme (X : LocallyRingedSpace)
    (h :
      ∀ x : X,
        ∃ (R : CommRingCat) (f : Spec.toLocallyRingedSpace.obj (op R) ⟶ X),
          (x ∈ Set.range f.1.base : _) ∧ LocallyRingedSpace.IsOpenImmersion f) :
    Scheme where
  toLocallyRingedSpace := X
  local_affine := by
    intro x
    obtain ⟨R, f, h₁, h₂⟩ := h x
    refine' ⟨⟨⟨_, h₂.base_open.open_range⟩, h₁⟩, R, ⟨_⟩⟩
    

Proof state:
case intro.intro.intro
C : Type u
inst✝ : Category.{v, u} C
X : LocallyRingedSpace
h : ∀ (x : ↑(toTopCat X)), ∃ R f, x ∈ Set.range ⇑f.val.base ∧ IsOpenImmersion f
x : ↑(toTopCat X)
R : CommRingCat
f : Spec.toLocallyRingedSpace.obj (op R) ⟶ X
h₁ : x ∈ Set.range ⇑f.val.base
h₂ : IsOpenImmersion f
⊢ restrict X
      (_ :
        OpenEmbedding
          ⇑(Opens.inclusion
              { obj := { carrier := Set.range ⇑f.val.base, is_open' := (_ : IsOpen (Set.range ⇑f.val.base)) },
                  property := h₁ }.obj)) ≅
    Spec.toLocallyRingedSpace.obj (op R)

Next tactic:
apply LocallyRingedSpace.isoOfSheafedSpaceIso
/-- To show that a locally ringed space is a scheme, it suffices to show that it has a jointly
surjective family of open immersions from affine schemes. -/
protected def scheme (X : LocallyRingedSpace)
    (h :
      ∀ x : X,
        ∃ (R : CommRingCat) (f : Spec.toLocallyRingedSpace.obj (op R) ⟶ X),
          (x ∈ Set.range f.1.base : _) ∧ LocallyRingedSpace.IsOpenImmersion f) :
    Scheme where
  toLocallyRingedSpace := X
  local_affine := by
    intro x
    obtain ⟨R, f, h₁, h₂⟩ := h x
    refine' ⟨⟨⟨_, h₂.base_open.open_range⟩, h₁⟩, R, ⟨_⟩⟩
    apply LocallyRingedSpace.isoOfSheafedSpaceIso
    

Proof state:
case intro.intro.intro.f
C : Type u
inst✝ : Category.{v, u} C
X : LocallyRingedSpace
h : ∀ (x : ↑(toTopCat X)), ∃ R f, x ∈ Set.range ⇑f.val.base ∧ IsOpenImmersion f
x : ↑(toTopCat X)
R : CommRingCat
f : Spec.toLocallyRingedSpace.obj (op R) ⟶ X
h₁ : x ∈ Set.range ⇑f.val.base
h₂ : IsOpenImmersion f
⊢ (restrict X
        (_ :
          OpenEmbedding
            ⇑(Opens.inclusion
                { obj := { carrier := Set.range ⇑f.val.base, is_open' := (_ : IsOpen (Set.range ⇑f.val.base)) },
                    property := h₁ }.obj))).toSheafedSpace ≅
    (Spec.toLocallyRingedSpace.obj (op R)).toSheafedSpace

Next tactic:
refine' SheafedSpace.forgetToPresheafedSpace.preimageIso _
/-- To show that a locally ringed space is a scheme, it suffices to show that it has a jointly
surjective family of open immersions from affine schemes. -/
protected def scheme (X : LocallyRingedSpace)
    (h :
      ∀ x : X,
        ∃ (R : CommRingCat) (f : Spec.toLocallyRingedSpace.obj (op R) ⟶ X),
          (x ∈ Set.range f.1.base : _) ∧ LocallyRingedSpace.IsOpenImmersion f) :
    Scheme where
  toLocallyRingedSpace := X
  local_affine := by
    intro x
    obtain ⟨R, f, h₁, h₂⟩ := h x
    refine' ⟨⟨⟨_, h₂.base_open.open_range⟩, h₁⟩, R, ⟨_⟩⟩
    apply LocallyRingedSpace.isoOfSheafedSpaceIso
    refine' SheafedSpace.forgetToPresheafedSpace.preimageIso _
    

Proof state:
case intro.intro.intro.f
C : Type u
inst✝ : Category.{v, u} C
X : LocallyRingedSpace
h : ∀ (x : ↑(toTopCat X)), ∃ R f, x ∈ Set.range ⇑f.val.base ∧ IsOpenImmersion f
x : ↑(toTopCat X)
R : CommRingCat
f : Spec.toLocallyRingedSpace.obj (op R) ⟶ X
h₁ : x ∈ Set.range ⇑f.val.base
h₂ : IsOpenImmersion f
⊢ SheafedSpace.forgetToPresheafedSpace.obj
      (restrict X
          (_ :
            OpenEmbedding
              ⇑(Opens.inclusion
                  { obj := { carrier := Set.range ⇑f.val.base, is_open' := (_ : IsOpen (Set.range ⇑f.val.base)) },
                      property := h₁ }.obj))).toSheafedSpace ≅
    SheafedSpace.forgetToPresheafedSpace.obj (Spec.toLocallyRingedSpace.obj (op R)).toSheafedSpace

Next tactic:
skip
/-- To show that a locally ringed space is a scheme, it suffices to show that it has a jointly
surjective family of open immersions from affine schemes. -/
protected def scheme (X : LocallyRingedSpace)
    (h :
      ∀ x : X,
        ∃ (R : CommRingCat) (f : Spec.toLocallyRingedSpace.obj (op R) ⟶ X),
          (x ∈ Set.range f.1.base : _) ∧ LocallyRingedSpace.IsOpenImmersion f) :
    Scheme where
  toLocallyRingedSpace := X
  local_affine := by
    intro x
    obtain ⟨R, f, h₁, h₂⟩ := h x
    refine' ⟨⟨⟨_, h₂.base_open.open_range⟩, h₁⟩, R, ⟨_⟩⟩
    apply LocallyRingedSpace.isoOfSheafedSpaceIso
    refine' SheafedSpace.forgetToPresheafedSpace.preimageIso _
    skip
    

Proof state:
case intro.intro.intro.f
C : Type u
inst✝ : Category.{v, u} C
X : LocallyRingedSpace
h : ∀ (x : ↑(toTopCat X)), ∃ R f, x ∈ Set.range ⇑f.val.base ∧ IsOpenImmersion f
x : ↑(toTopCat X)
R : CommRingCat
f : Spec.toLocallyRingedSpace.obj (op R) ⟶ X
h₁ : x ∈ Set.range ⇑f.val.base
h₂ : IsOpenImmersion f
⊢ SheafedSpace.forgetToPresheafedSpace.obj
      (restrict X
          (_ :
            OpenEmbedding
              ⇑(Opens.inclusion
                  { obj := { carrier := Set.range ⇑f.val.base, is_open' := (_ : IsOpen (Set.range ⇑f.val.base)) },
                      property := h₁ }.obj))).toSheafedSpace ≅
    SheafedSpace.forgetToPresheafedSpace.obj (Spec.toLocallyRingedSpace.obj (op R)).toSheafedSpace

Next tactic:
apply PresheafedSpace.IsOpenImmersion.isoOfRangeEq (PresheafedSpace.ofRestrict _ _) f.1
/-- To show that a locally ringed space is a scheme, it suffices to show that it has a jointly
surjective family of open immersions from affine schemes. -/
protected def scheme (X : LocallyRingedSpace)
    (h :
      ∀ x : X,
        ∃ (R : CommRingCat) (f : Spec.toLocallyRingedSpace.obj (op R) ⟶ X),
          (x ∈ Set.range f.1.base : _) ∧ LocallyRingedSpace.IsOpenImmersion f) :
    Scheme where
  toLocallyRingedSpace := X
  local_affine := by
    intro x
    obtain ⟨R, f, h₁, h₂⟩ := h x
    refine' ⟨⟨⟨_, h₂.base_open.open_range⟩, h₁⟩, R, ⟨_⟩⟩
    apply LocallyRingedSpace.isoOfSheafedSpaceIso
    refine' SheafedSpace.forgetToPresheafedSpace.preimageIso _
    skip
    apply PresheafedSpace.IsOpenImmersion.isoOfRangeEq (PresheafedSpace.ofRestrict _ _) f.1
    · 

Proof state:
case intro.intro.intro.f
C : Type u
inst✝ : Category.{v, u} C
X : LocallyRingedSpace
h : ∀ (x : ↑(toTopCat X)), ∃ R f, x ∈ Set.range ⇑f.val.base ∧ IsOpenImmersion f
x : ↑(toTopCat X)
R : CommRingCat
f : Spec.toLocallyRingedSpace.obj (op R) ⟶ X
h₁ : x ∈ Set.range ⇑f.val.base
h₂ : IsOpenImmersion f
⊢ Set.range ⇑(PresheafedSpace.ofRestrict X.toPresheafedSpace ?m.3822).base = Set.range ⇑f.val.base

Next tactic:
exact Subtype.range_coe_subtype
/-- To show that a locally ringed space is a scheme, it suffices to show that it has a jointly
surjective family of open immersions from affine schemes. -/
protected def scheme (X : LocallyRingedSpace)
    (h :
      ∀ x : X,
        ∃ (R : CommRingCat) (f : Spec.toLocallyRingedSpace.obj (op R) ⟶ X),
          (x ∈ Set.range f.1.base : _) ∧ LocallyRingedSpace.IsOpenImmersion f) :
    Scheme where
  toLocallyRingedSpace := X
  local_affine := by
    intro x
    obtain ⟨R, f, h₁, h₂⟩ := h x
    refine' ⟨⟨⟨_, h₂.base_open.open_range⟩, h₁⟩, R, ⟨_⟩⟩
    apply LocallyRingedSpace.isoOfSheafedSpaceIso
    refine' SheafedSpace.forgetToPresheafedSpace.preimageIso _
    skip
    apply PresheafedSpace.IsOpenImmersion.isoOfRangeEq (PresheafedSpace.ofRestrict _ _) f.1
    · exact Subtype.range_coe_subtype
    · 

Proof state:
C : Type u
inst✝ : Category.{v, u} C
X : LocallyRingedSpace
h : ∀ (x : ↑(toTopCat X)), ∃ R f, x ∈ Set.range ⇑f.val.base ∧ IsOpenImmersion f
x : ↑(toTopCat X)
R : CommRingCat
f : Spec.toLocallyRingedSpace.obj (op R) ⟶ X
h₁ : x ∈ Set.range ⇑f.val.base
h₂ : IsOpenImmersion f
⊢ OpenEmbedding
    ⇑(Opens.inclusion
        { obj := { carrier := Set.range ⇑f.val.base, is_open' := (_ : IsOpen (Set.range ⇑f.val.base)) },
            property := h₁ }.obj)

Next tactic:
exact Opens.openEmbedding _
/-- The affine cover of a scheme. -/
def affineCover (X : Scheme) : OpenCover X where
  J := X.carrier
  obj x := Spec.obj <| Opposite.op (X.local_affine x).choose_spec.choose
  map x :=
    ((X.local_affine x).choose_spec.choose_spec.some.inv ≫ X.toLocallyRingedSpace.ofRestrict _ : _)
  f x := x
  IsOpen x := by
    apply (config := { allowSynthFailures := true }) PresheafedSpace.IsOpenImmersion.comp
    apply PresheafedSpace.IsOpenImmersion.ofRestrict
  Covers := by
    intro x
    erw [coe_comp]
    rw [Set.range_comp, Set.range_iff_surjective.mpr, Set.image_univ]
    erw [Subtype.range_coe_subtype]
    

Proof state:
C : Type u
inst✝¹ : Category.{v, u} C
X✝ Y Z : Scheme
𝒰 : OpenCover X✝
f : X✝ ⟶ Z
g : Y ⟶ Z
inst✝ : ∀ (x : 𝒰.J), HasPullback (OpenCover.map 𝒰 x ≫ f) g
X : Scheme
x : ↑↑X.toPresheafedSpace
⊢ x ∈
    {x_1 |
      x_1 ∈
        (Exists.choose
            (_ :
              ∃ U R,
                Nonempty
                  (LocallyRingedSpace.restrict X.toLocallyRingedSpace (_ : OpenEmbedding ⇑(Opens.inclusion U.obj)) ≅
                    Spec.toLocallyRingedSpace.obj (op R)))).obj}
C : Type u
inst✝¹ : Category.{v, u} C
X✝ Y Z : Scheme
𝒰 : OpenCover X✝
f : X✝ ⟶ Z
g : Y ⟶ Z
inst✝ : ∀ (x : 𝒰.J), HasPullback (OpenCover.map 𝒰 x ≫ f) g
X : Scheme
x : ↑↑X.toPresheafedSpace
⊢ Function.Surjective
    ⇑(Nonempty.some
              (_ :
                Nonempty
                  (LocallyRingedSpace.restrict X.toLocallyRingedSpace
                      (_ :
                        OpenEmbedding
                          ⇑(Opens.inclusion
                              (Exists.choose
                                  (_ :
                                    ∃ U R,
                                      Nonempty
                                        (LocallyRingedSpace.restrict X.toLocallyRingedSpace
                                            (_ : OpenEmbedding ⇑(Opens.inclusion U.obj)) ≅
                                          Spec.toLocallyRingedSpace.obj (op R)))).obj)) ≅
                    Spec.toLocallyRingedSpace.obj
                      (op
                        (Exists.choose
                          (_ :
                            ∃ R,
                              Nonempty
                                (LocallyRingedSpace.restrict X.toLocallyRingedSpace
                                    (_ :
                                      OpenEmbedding
                                        ⇑(Opens.inclusion
                                            (Exists.choose
                                                (_ :
                                                  ∃ U R,
                                                    Nonempty
                                                      (LocallyRingedSpace.restrict X.toLocallyRingedSpace
                                                          (_ : OpenEmbedding ⇑(Opens.inclusion U.obj)) ≅
                                                        Spec.toLocallyRingedSpace.obj (op R)))).obj)) ≅
                                  Spec.toLocallyRingedSpace.obj (op R)))))))).inv.val.base

Next tactic:
exact (X.local_affine x).choose.2
/-- The affine cover of a scheme. -/
def affineCover (X : Scheme) : OpenCover X where
  J := X.carrier
  obj x := Spec.obj <| Opposite.op (X.local_affine x).choose_spec.choose
  map x :=
    ((X.local_affine x).choose_spec.choose_spec.some.inv ≫ X.toLocallyRingedSpace.ofRestrict _ : _)
  f x := x
  IsOpen x := by
    apply (config := { allowSynthFailures := true }) PresheafedSpace.IsOpenImmersion.comp
    apply PresheafedSpace.IsOpenImmersion.ofRestrict
  Covers := by
    intro x
    erw [coe_comp]
    rw [Set.range_comp, Set.range_iff_surjective.mpr, Set.image_univ]
    erw [Subtype.range_coe_subtype]
    exact (X.local_affine x).choose.2
    

Proof state:
C : Type u
inst✝¹ : Category.{v, u} C
X✝ Y Z : Scheme
𝒰 : OpenCover X✝
f : X✝ ⟶ Z
g : Y ⟶ Z
inst✝ : ∀ (x : 𝒰.J), HasPullback (OpenCover.map 𝒰 x ≫ f) g
X : Scheme
x : ↑↑X.toPresheafedSpace
⊢ Function.Surjective
    ⇑(Nonempty.some
              (_ :
                Nonempty
                  (LocallyRingedSpace.restrict X.toLocallyRingedSpace
                      (_ :
                        OpenEmbedding
                          ⇑(Opens.inclusion
                              (Exists.choose
                                  (_ :
                                    ∃ U R,
                                      Nonempty
                                        (LocallyRingedSpace.restrict X.toLocallyRingedSpace
                                            (_ : OpenEmbedding ⇑(Opens.inclusion U.obj)) ≅
                                          Spec.toLocallyRingedSpace.obj (op R)))).obj)) ≅
                    Spec.toLocallyRingedSpace.obj
                      (op
                        (Exists.choose
                          (_ :
                            ∃ R,
                              Nonempty
                                (LocallyRingedSpace.restrict X.toLocallyRingedSpace
                                    (_ :
                                      OpenEmbedding
                                        ⇑(Opens.inclusion
                                            (Exists.choose
                                                (_ :
                                                  ∃ U R,
                                                    Nonempty
                                                      (LocallyRingedSpace.restrict X.toLocallyRingedSpace
                                                          (_ : OpenEmbedding ⇑(Opens.inclusion U.obj)) ≅
                                                        Spec.toLocallyRingedSpace.obj (op R)))).obj)) ≅
                                  Spec.toLocallyRingedSpace.obj (op R)))))))).inv.val.base

Next tactic:
rw [← TopCat.epi_iff_surjective]