mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.CategoryTheory.Limits.Types.ColimitTypeFiltered

Goal state before:
pp: case mpr
J : Type u
inst✝¹ : Category.{v, u} J
inst✝ : IsFiltered J
F : J ⥤ Type w₀
c : F.CoconeTypes
h :
  ∀ (j j' : J) (x : F.obj j) (x' : F.obj j'),
    c.ι j x = c.ι j' x' → ∃ k f f', (ConcreteCategory.hom (F.map f)) x = (ConcreteCategory.hom (F.map f')) x'
i : J
x : F.obj i
i' : J
x' : F.obj i'
eq✝ : c.ι i x = c.ι i' x'
k : J
f : i ⟶ k
f' : i' ⟶ k
eq : (ConcreteCategory.hom (F.map f)) x = (ConcreteCategory.hom (F.map f')) x'
⊢ F.ιColimitType i' x' = F.ιColimitType i' x'

Tactic:
]

Goal state after:
Module: Mathlib.AlgebraicGeometry.AffineSpace

Goal state before:
pp: case e'_7
n : Type u
S T : Scheme
f : S ⟶ T
e_3✝ : 𝔸(n; S) = pullback (terminal.from S) (terminal.from (Spec ℤ[n]))
⊢ toSpecMvPoly n S = toSpecMvPoly n S

Tactic:
]

Goal state after:
Module: Mathlib.CategoryTheory.Limits.Types.ColimitTypeFiltered

Goal state before:
pp: J : Type u
inst✝¹ : Category.{v, u} J
inst✝ : IsFiltered J
F : J ⥤ Type w₀
c : F.CoconeTypes
⊢ (∀ (j j' : J) (x : F.obj j) (x' : F.obj j'),
      c.ι j x = c.ι j' x' → ∃ k f f', (ConcreteCategory.hom (F.map f)) x = (ConcreteCategory.hom (F.map f')) x') ↔
    ∀ (j : J) (x x' : F.obj j),
      c.ι j x = c.ι j x' → ∃ k f, (ConcreteCategory.hom (F.map f)) x = (ConcreteCategory.hom (F.map f)) x'

Tactic:
]

Goal state after:
J : Type u
inst✝¹ : Category.{v, u} J
inst✝ : IsFiltered J
F : J ⥤ Type w₀
c : F.CoconeTypes
⊢ (∀ (j j' : J) (x : F.obj j) (x' : F.obj j'),
      c.ι j x = c.ι j' x' → ∃ k f f', (ConcreteCategory.hom (F.map f)) x = (ConcreteCategory.hom (F.map f')) x') ↔
    ∀ (j : J) (x x' : F.obj j),
      c.ι j x = c.ι j x' → ∃ k f, (ConcreteCategory.hom (F.map f)) x = (ConcreteCategory.hom (F.map f)) x'
Module: Mathlib.AlgebraicGeometry.AffineSpace

Goal state before:
pp: n : Type u
S : Scheme
this : IsIso (terminal.from (Spec (CommRingCat.of (ULift.{u, 0} ℤ))))
⊢ Surjective (Spec.map (CommRingCat.ofHom C))

Tactic:
]

Goal state after:
n : Type u
S : Scheme
this : IsIso (terminal.from (Spec (CommRingCat.of (ULift.{u, 0} ℤ))))
⊢ Surjective (Spec.map (CommRingCat.ofHom C))
Module: Mathlib.AlgebraicGeometry.AffineSpace

Goal state before:
pp: n : Type u
S : Scheme
inst✝ : Finite n
this : IsIso (terminal.from (Spec (CommRingCat.of (ULift.{u, 0} ℤ))))
⊢ Algebra.FinitePresentation ↑(CommRingCat.of (ULift.{u, 0} ℤ)) ↑ℤ[n]

Tactic:
]

Goal state after:
n : Type u
S : Scheme
inst✝ : Finite n
this : IsIso (terminal.from (Spec (CommRingCat.of (ULift.{u, 0} ℤ))))
⊢ Algebra.FinitePresentation ↑(CommRingCat.of (ULift.{u, 0} ℤ)) ↑ℤ[n]
Module: Mathlib.CategoryTheory.Limits.Types.ColimitTypeFiltered

Goal state before:
pp: case mp
J : Type u
inst✝¹ : Category.{v, u} J
inst✝ : IsFiltered J
F : J ⥤ Type w₀
c : F.CoconeTypes
h :
  ∀ (j j' : J) (x : F.obj j) (x' : F.obj j'),
    c.ι j x = c.ι j' x' → ∃ k f f', (ConcreteCategory.hom (F.map f)) x = (ConcreteCategory.hom (F.map f')) x'
j : J
x x' : F.obj j
eq✝ : c.ι j x = c.ι j x'
k : J
f f' : j ⟶ k
eq : (ConcreteCategory.hom (F.map f)) x = (ConcreteCategory.hom (F.map f')) x'
⊢ (ConcreteCategory.hom (F.map (coeqHom f f'))) ((ConcreteCategory.hom (F.map f')) x') =
    (ConcreteCategory.hom (F.map f ≫ F.map (coeqHom f f'))) x'

Tactic:
]

Goal state after:
case mp
J : Type u
inst✝¹ : Category.{v, u} J
inst✝ : IsFiltered J
F : J ⥤ Type w₀
c : F.CoconeTypes
h :
  ∀ (j j' : J) (x : F.obj j) (x' : F.obj j'),
    c.ι j x = c.ι j' x' → ∃ k f f', (ConcreteCategory.hom (F.map f)) x = (ConcreteCategory.hom (F.map f')) x'
j : J
x x' : F.obj j
eq✝ : c.ι j x = c.ι j x'
k : J
f f' : j ⟶ k
eq : (ConcreteCategory.hom (F.map f)) x = (ConcreteCategory.hom (F.map f')) x'
⊢ (ConcreteCategory.hom (F.map (coeqHom f f'))) ((ConcreteCategory.hom (F.map f')) x') =
    (ConcreteCategory.hom (F.map f ≫ F.map (coeqHom f f'))) x'
Module: Mathlib.CategoryTheory.Limits.Types.ColimitTypeFiltered

Goal state before:
pp: case mp
J : Type u
inst✝¹ : Category.{v, u} J
inst✝ : IsFiltered J
F : J ⥤ Type w₀
c : F.CoconeTypes
h :
  ∀ (j j' : J) (x : F.obj j) (x' : F.obj j'),
    c.ι j x = c.ι j' x' → ∃ k f f', (ConcreteCategory.hom (F.map f)) x = (ConcreteCategory.hom (F.map f')) x'
j : J
x x' : F.obj j
eq✝ : c.ι j x = c.ι j x'
k : J
f f' : j ⟶ k
eq : (ConcreteCategory.hom (F.map f)) x = (ConcreteCategory.hom (F.map f')) x'
⊢ (ConcreteCategory.hom (F.map (f' ≫ coeqHom f f'))) x' = (ConcreteCategory.hom (F.map (f' ≫ coeqHom f f'))) x'

Tactic:
]

Goal state after:
Module: Mathlib.AlgebraicGeometry.AffineSpace

Goal state before:
pp: case inr
n : Type u
S : Scheme
this✝ : ∀ {n : Type u} (S : Scheme), (∃ R, S = Spec R) → topologically (@IsOpenMap) (𝔸(n; S) ↘ S)
hS : ¬∃ R, S = Spec R
i : S.affineCover.toPreZeroHypercover.1
this : topologically (@IsOpenMap) (pullback.snd (𝔸(n; S) ↘ S) (S.affineCover.f i))
⊢ topologically (@IsOpenMap) (Scheme.Cover.pullbackHom S.affineCover (𝔸(n; S) ↘ S) i)

Tactic:
]

Goal state after:
case inr
n : Type u
S : Scheme
this✝ : ∀ {n : Type u} (S : Scheme), (∃ R, S = Spec R) → topologically (@IsOpenMap) (𝔸(n; S) ↘ S)
hS : ¬∃ R, S = Spec R
i : S.affineCover.toPreZeroHypercover.1
this : topologically (@IsOpenMap) (pullback.snd (𝔸(n; S) ↘ S) (S.affineCover.f i))
⊢ topologically (@IsOpenMap) (Scheme.Cover.pullbackHom S.affineCover (𝔸(n; S) ↘ S) i)
Module: Mathlib.AlgebraicGeometry.AffineSpace

Goal state before:
pp: n✝ : Type u
S : Scheme
n : Type u
R : CommRingCat
⊢ topologically (@IsOpenMap) (Spec.map (CommRingCat.ofHom C))

Tactic:
]

Goal state after:
n✝ : Type u
S : Scheme
n : Type u
R : CommRingCat
⊢ topologically (@IsOpenMap) (Spec.map (CommRingCat.ofHom C))
Module: Mathlib.AlgebraicGeometry.AffineSpace

Goal state before:
pp: n : Type u
S : Scheme
⊢ geometrically (fun x ↦ IrreducibleSpace ↥x) (𝔸(n; S) ↘ S)

Tactic:
]

Goal state after:
n : Type u
S : Scheme
⊢ geometrically (fun x ↦ IrreducibleSpace ↥x) (𝔸(n; S) ↘ S)
Module: Mathlib.AlgebraicGeometry.AffineSpace

Goal state before:
pp: n : Type u
S : Scheme
⊢ geometrically IsReduced (𝔸(n; S) ↘ S)

Tactic:
]

Goal state after:
n : Type u
S : Scheme
⊢ geometrically IsReduced (𝔸(n; S) ↘ S)
Module: Mathlib.AlgebraicGeometry.AffineSpace

Goal state before:
pp: case inr
n : Type u
S : Scheme
h : IsReduced S
this : ∀ {n : Type u} (S : Scheme) [h : IsReduced S], (∃ R, S = Spec R) → IsReduced 𝔸(n; S)
hS : ¬∃ R, S = Spec R
⊢ ∀ (i : (Precoverage.ZeroHypercover.pullback₁ (𝔸(n; S) ↘ S) S.affineCover).I₀),
    IsReduced ((Precoverage.ZeroHypercover.pullback₁ (𝔸(n; S) ↘ S) S.affineCover).X i)

Tactic:
]

Goal state after:
case inr
n : Type u
S : Scheme
h : IsReduced S
this : ∀ {n : Type u} (S : Scheme) [h : IsReduced S], (∃ R, S = Spec R) → IsReduced 𝔸(n; S)
hS : ¬∃ R, S = Spec R
⊢ ∀ (i : (Precoverage.ZeroHypercover.pullback₁ (𝔸(n; S) ↘ S) S.affineCover).I₀),
    IsReduced ((Precoverage.ZeroHypercover.pullback₁ (𝔸(n; S) ↘ S) S.affineCover).X i)
Module: Mathlib.AlgebraicGeometry.AffineSpace

Goal state before:
pp: n✝ : Type u
S : Scheme
n : Type u
R : CommRingCat
h : _root_.IsReduced ↑R
⊢ IsReduced 𝔸(n; Spec R)

Tactic:
]

Goal state after:
n✝ : Type u
S : Scheme
n : Type u
R : CommRingCat
h : _root_.IsReduced ↑R
⊢ IsReduced 𝔸(n; Spec R)
Module: Mathlib.AlgebraicGeometry.AffineSpace

Goal state before:
pp: n : Type u
S : Scheme
inst✝ : IsEmpty n
⊢ isomorphisms Scheme (Spec.map (CommRingCat.ofHom C) ≫ terminal.from (Spec (CommRingCat.of (ULift.{u, 0} ℤ))))

Tactic:
]

Goal state after:
n : Type u
S : Scheme
inst✝ : IsEmpty n
⊢ isomorphisms Scheme (Spec.map (CommRingCat.ofHom C) ≫ terminal.from (Spec (CommRingCat.of (ULift.{u, 0} ℤ))))
Module: Mathlib.RingTheory.Spectrum.Prime.ChevalleyComplexity

Goal state before:
pp: case right.refine_1
n : ℕ
P : (R : Type u) → [inst : CommRing R] → InductionObj R n → Prop
hP₁ : ∀ (R : Type u) [inst : CommRing R], P R { val := 0 }
hP₂ :
  ∀ (R : Type u) [inst : CommRing R] (e : InductionObj R n) (i : Fin n),
    (e.val i).Monic → (∀ (j : Fin n), j ≠ i → e.val j = 0) → P R e
hP₃ :
  ∀ (R : Type u) [inst : CommRing R] (e : InductionObj R n) (i j : Fin n),
    (e.val i).Monic →
      (e.val i).degree ≤ (e.val j).degree → i ≠ j → P R { val := Function.update e.val j (e.val j %ₘ e.val i) } → P R e
hP₄ :
  ∀ (R : Type u) [inst : CommRing R] (c : R) (i : Fin n) (e : InductionObj R n),
    c = (e.val i).leadingCoeff →
      c ≠ 0 →
        P (Away c)
            { val := Polynomial.C (IsLocalization.Away.invSelf c) • ⇑(mapRingHom (algebraMap R (Away c))) ∘ e.val } →
          P (R ⧸ Ideal.span {c}) { val := ⇑(mapRingHom (algebraMap R (R ⧸ Ideal.span {c}))) ∘ e.val } → P R e
v : DegreeType n
H_IH : ∀ y < v, ∀ {R : Type u} [inst : CommRing R] (e : InductionObj R n), y = e.degree → P R e
R : Type u
inst✝ : CommRing R
e : InductionObj R n
hv : v = e.degree
h✝ : Nontrivial R
he0 : ∃ x, ¬e.val x = 0
H : ¬∃ i, (e.val i).Monic ∧ ∀ (j : Fin n), e.val j ≠ 0 → (e.val i).degree ≤ (e.val j).degree
i : Fin n
hi : e.val i ≠ 0
i_min : ∀ (j : Fin n), e.val j ≠ 0 → (e.val i).degree ≤ (e.val j).degree
h_eq :
  ∀ (x : Fin n),
    (Polynomial.C (IsLocalization.Away.invSelf (e.val i).leadingCoeff) *
          Polynomial.map (algebraMap R (Away (e.val i).leadingCoeff)) (e.val x)).degree =
      (e.val x).degree
⊢ 1 = 1

Tactic:
]

Goal state after:
Module: Mathlib.AlgebraicGeometry.AffineSpace

Goal state before:
pp: case a
n : Type u
S : Scheme
inst✝ : IsEmpty n
⊢ IsAffine (Spec ℤ[n]) ∧ isomorphisms CommRingCat (CommRingCat.ofHom C)

Tactic:
]

Goal state after:
case a
n : Type u
S : Scheme
inst✝ : IsEmpty n
⊢ IsAffine (Spec ℤ[n]) ∧ isomorphisms CommRingCat (CommRingCat.ofHom C)
Module: Mathlib.AlgebraicGeometry.AffineSpace

Goal state before:
pp: case a
n : Type u
S : Scheme
inst✝ : IsEmpty n
⊢ isomorphisms Scheme (Spec.map (CommRingCat.ofHom C))

Tactic:
rw [HasAffineProperty.iff_of_isAffine (P := isomorphisms _), ← isomorphisms,
      ← arrow_mk_iso_iff (isomorphisms _) (arrowIsoΓSpecOfIsAffine _)]

Goal state after:
case a
n : Type u
S : Scheme
inst✝ : IsEmpty n
⊢ IsAffine (Spec ℤ[n]) ∧ isomorphisms CommRingCat (CommRingCat.ofHom C)
Module: Mathlib.AlgebraicGeometry.AffineSpace

Goal state before:
pp: n✝ : Type u
S : Scheme
n : Type u
R : CommRingCat
h :
  IsAffine (Spec (CommRingCat.of (MvPolynomial n ↑R))) ∧
    (CommRingCat.Hom.hom (Scheme.Hom.app (Spec.map (CommRingCat.ofHom C)) ⊤)).IsIntegral
h✝ : Nonempty ↥(Spec R)
this✝ : Nontrivial ↑R
i : n
this : (RingHom.toMorphismProperty fun {R S} [CommRing R] [CommRing S] f ↦ f.IsIntegral).RespectsIso
⊢ False

Tactic:
]

Goal state after:
n✝ : Type u
S : Scheme
n : Type u
R : CommRingCat
h :
  IsAffine (Spec (CommRingCat.of (MvPolynomial n ↑R))) ∧
    (CommRingCat.Hom.hom (Scheme.Hom.app (Spec.map (CommRingCat.ofHom C)) ⊤)).IsIntegral
h✝ : Nonempty ↥(Spec R)
this✝ : Nontrivial ↑R
i : n
this : (RingHom.toMorphismProperty fun {R S} [CommRing R] [CommRing S] f ↦ f.IsIntegral).RespectsIso
⊢ False
Module: Mathlib.AlgebraicGeometry.AffineSpace

Goal state before:
pp: n✝ : Type u
S : Scheme
n : Type u
R : CommRingCat
h :
  IsAffine (Spec (CommRingCat.of (MvPolynomial n ↑R))) ∧
    (CommRingCat.Hom.hom (Scheme.Hom.app (Spec.map (CommRingCat.ofHom C)) ⊤)).IsIntegral
h✝ : Nonempty ↥(Spec R)
this✝¹ : Nontrivial ↑R
i : n
this✝ : (RingHom.toMorphismProperty fun {R S} [CommRing R] [CommRing S] f ↦ f.IsIntegral).RespectsIso
p : Polynomial ↑R
hp : p.Monic
hp' : Polynomial.eval₂ (CommRingCat.Hom.hom (CommRingCat.ofHom C)) (X i) p = 0
this : ↑(uniqueAlgEquiv (↑R) PUnit.{u + 1}).symm p = 0
⊢ False

Tactic:
]

Goal state after:
n✝ : Type u
S : Scheme
n : Type u
R : CommRingCat
h :
  IsAffine (Spec (CommRingCat.of (MvPolynomial n ↑R))) ∧
    (CommRingCat.Hom.hom (Scheme.Hom.app (Spec.map (CommRingCat.ofHom C)) ⊤)).IsIntegral
h✝ : Nonempty ↥(Spec R)
this✝¹ : Nontrivial ↑R
i : n
this✝ : (RingHom.toMorphismProperty fun {R S} [CommRing R] [CommRing S] f ↦ f.IsIntegral).RespectsIso
p : Polynomial ↑R
hp : p.Monic
hp' : Polynomial.eval₂ (CommRingCat.Hom.hom (CommRingCat.ofHom C)) (X i) p = 0
this : ↑(uniqueAlgEquiv (↑R) PUnit.{u + 1}).symm p = 0
⊢ False
Module: Mathlib.AlgebraicGeometry.AlgClosed.Basic

Goal state before:
pp: X Y : Scheme
K : Type u
inst✝² : Field K
inst✝¹ : IsAlgClosed K
f : X ⟶ Spec (CommRingCat.of K)
inst✝ : LocallyOfFiniteType f
x : ↥X
hx : IsClosedImmersion (X.fromSpecResidueField x)
⊢ IsFinite (X.fromSpecResidueField x ≫ f)

Tactic:
]

Goal state after:
X Y : Scheme
K : Type u
inst✝² : Field K
inst✝¹ : IsAlgClosed K
f : X ⟶ Spec (CommRingCat.of K)
inst✝ : LocallyOfFiniteType f
x : ↥X
hx : IsClosedImmersion (X.fromSpecResidueField x)
⊢ IsFinite (X.fromSpecResidueField x ≫ f)
Module: Mathlib.AlgebraicGeometry.AlgClosed.Basic

Goal state before:
pp: X Y : Scheme
K : Type u
inst✝² : Field K
inst✝¹ : IsAlgClosed K
f : X ⟶ Spec (CommRingCat.of K)
inst✝ : LocallyOfFiniteType f
x : ↥X
hx : IsClosedImmersion (X.fromSpecResidueField x)
⊢ LocallyOfFiniteType (X.fromSpecResidueField x ≫ f)

Tactic:
]

Goal state after:
X Y : Scheme
K : Type u
inst✝² : Field K
inst✝¹ : IsAlgClosed K
f : X ⟶ Spec (CommRingCat.of K)
inst✝ : LocallyOfFiniteType f
x : ↥X
hx : IsClosedImmersion (X.fromSpecResidueField x)
⊢ LocallyOfFiniteType (X.fromSpecResidueField x ≫ f)
Module: Mathlib.AlgebraicGeometry.AlgClosed.Basic

Goal state before:
pp: X Y : Scheme
K : Type u
inst✝² : Field K
inst✝¹ : IsAlgClosed K
f : X ⟶ Spec (CommRingCat.of K)
inst✝ : LocallyOfFiniteType f
x : ↥X
hx : IsClosed {x}
this : IsFinite (X.fromSpecResidueField x ≫ f)
⊢ Function.Bijective ⇑(ConcreteCategory.hom (Spec.preimage (X.fromSpecResidueField x ≫ f)))

Tactic:
]

Goal state after:
X Y : Scheme
K : Type u
inst✝² : Field K
inst✝¹ : IsAlgClosed K
f : X ⟶ Spec (CommRingCat.of K)
inst✝ : LocallyOfFiniteType f
x : ↥X
hx : IsClosed {x}
this : IsFinite (X.fromSpecResidueField x ≫ f)
⊢ Function.Bijective ⇑(ConcreteCategory.hom (Spec.preimage (X.fromSpecResidueField x ≫ f)))
Module: Mathlib.AlgebraicGeometry.AlgClosed.Basic

Goal state before:
pp: X Y : Scheme
K : Type u
inst✝² : Field K
inst✝¹ : IsAlgClosed K
f : X ⟶ Spec (CommRingCat.of K)
inst✝ : LocallyOfFiniteType f
x : ↥X
hx : IsClosed {x}
this : IsFinite (X.fromSpecResidueField x ≫ f)
⊢ IsIntegralHom (X.fromSpecResidueField x ≫ f)

Tactic:
]

Goal state after:
X Y : Scheme
K : Type u
inst✝² : Field K
inst✝¹ : IsAlgClosed K
f : X ⟶ Spec (CommRingCat.of K)
inst✝ : LocallyOfFiniteType f
x : ↥X
hx : IsClosed {x}
this : IsFinite (X.fromSpecResidueField x ≫ f)
⊢ IsIntegralHom (X.fromSpecResidueField x ≫ f)
Module: Mathlib.AlgebraicGeometry.AlgClosed.Basic

Goal state before:
pp: X Y : Scheme
K : Type u
inst✝² : Field K
inst✝¹ : IsAlgClosed K
f : X ⟶ Spec (CommRingCat.of K)
inst✝ : LocallyOfFiniteType f
x : ↥X
hx : IsClosed {x}
p : { p // p ≫ f = 𝟙 (Spec (CommRingCat.of K)) }
this✝¹ : IsClosedImmersion ↑p
this✝ : IsClosed (Set.range ⇑↑p)
this : IsClosed {?m.94}
⊢ ↑p (IsLocalRing.closedPoint K) ∈ closedPoints ↥X

pp: X Y : Scheme
K : Type u
inst✝² : Field K
inst✝¹ : IsAlgClosed K
f : X ⟶ Spec (CommRingCat.of K)
inst✝ : LocallyOfFiniteType f
x : ↥X
hx : IsClosed {x}
p : { p // p ≫ f = 𝟙 (Spec (CommRingCat.of K)) }
this✝ : IsClosedImmersion ↑p
this : IsClosed (Set.range ⇑↑p)
⊢ ∀ (x : ↥(Spec (CommRingCat.of K))), ↑p x = ?m.94

pp: X Y : Scheme
K : Type u
inst✝² : Field K
inst✝¹ : IsAlgClosed K
f : X ⟶ Spec (CommRingCat.of K)
inst✝ : LocallyOfFiniteType f
x : ↥X
hx : IsClosed {x}
p : { p // p ≫ f = 𝟙 (Spec (CommRingCat.of K)) }
this✝ : IsClosedImmersion ↑p
this : IsClosed (Set.range ⇑↑p)
⊢ ↥X

Tactic:
]

Goal state after:
X Y : Scheme
K : Type u
inst✝² : Field K
inst✝¹ : IsAlgClosed K
f : X ⟶ Spec (CommRingCat.of K)
inst✝ : LocallyOfFiniteType f
x : ↥X
hx : IsClosed {x}
p : { p // p ≫ f = 𝟙 (Spec (CommRingCat.of K)) }
this✝¹ : IsClosedImmersion ↑p
this✝ : IsClosed (Set.range ⇑↑p)
this : IsClosed {?m.94}
⊢ ↑p (IsLocalRing.closedPoint K) ∈ closedPoints ↥X

X Y : Scheme
K : Type u
inst✝² : Field K
inst✝¹ : IsAlgClosed K
f : X ⟶ Spec (CommRingCat.of K)
inst✝ : LocallyOfFiniteType f
x : ↥X
hx : IsClosed {x}
p : { p // p ≫ f = 𝟙 (Spec (CommRingCat.of K)) }
this✝ : IsClosedImmersion ↑p
this : IsClosed (Set.range ⇑↑p)
⊢ ∀ (x : ↥(Spec (CommRingCat.of K))), ↑p x = ?m.94

X Y : Scheme
K : Type u
inst✝² : Field K
inst✝¹ : IsAlgClosed K
f : X ⟶ Spec (CommRingCat.of K)
inst✝ : LocallyOfFiniteType f
x : ↥X
hx : IsClosed {x}
p : { p // p ≫ f = 𝟙 (Spec (CommRingCat.of K)) }
this✝ : IsClosedImmersion ↑p
this : IsClosed (Set.range ⇑↑p)
⊢ ↥X
Module: Mathlib.AlgebraicGeometry.AlgClosed.Basic

Goal state before:
pp: X Y : Scheme
K : Type u
inst✝² : Field K
inst✝¹ : IsAlgClosed K
f : X ⟶ Spec (CommRingCat.of K)
inst✝ : LocallyOfFiniteType f
x : ↥X
hx : IsClosed {x}
p : { p // p ≫ f = 𝟙 (Spec (CommRingCat.of K)) }
⊢ ∃ (e :
    ⟨↑((fun p ↦ ⟨↑p (IsLocalRing.closedPoint K), ⋯⟩) p),
          (residueFieldIsoBase f ↑((fun p ↦ ⟨↑p (IsLocalRing.closedPoint K), ⋯⟩) p) ⋯).1⟩.fst =
      ((Scheme.SpecToEquivOfField K X) ↑p).fst),
    ⟨↑((fun p ↦ ⟨↑p (IsLocalRing.closedPoint K), ⋯⟩) p),
          (residueFieldIsoBase f ↑((fun p ↦ ⟨↑p (IsLocalRing.closedPoint K), ⋯⟩) p) ⋯).1⟩.snd =
      (Scheme.residueFieldCongr e).hom ≫ ((Scheme.SpecToEquivOfField K X) ↑p).snd

Tactic:
]

Goal state after:
X Y : Scheme
K : Type u
inst✝² : Field K
inst✝¹ : IsAlgClosed K
f : X ⟶ Spec (CommRingCat.of K)
inst✝ : LocallyOfFiniteType f
x : ↥X
hx : IsClosed {x}
p : { p // p ≫ f = 𝟙 (Spec (CommRingCat.of K)) }
⊢ ∃ (e :
    ⟨↑((fun p ↦ ⟨↑p (IsLocalRing.closedPoint K), ⋯⟩) p),
          (residueFieldIsoBase f ↑((fun p ↦ ⟨↑p (IsLocalRing.closedPoint K), ⋯⟩) p) ⋯).1⟩.fst =
      ((Scheme.SpecToEquivOfField K X) ↑p).fst),
    ⟨↑((fun p ↦ ⟨↑p (IsLocalRing.closedPoint K), ⋯⟩) p),
          (residueFieldIsoBase f ↑((fun p ↦ ⟨↑p (IsLocalRing.closedPoint K), ⋯⟩) p) ⋯).1⟩.snd =
      (Scheme.residueFieldCongr e).hom ≫ ((Scheme.SpecToEquivOfField K X) ↑p).snd
Module: Mathlib.AlgebraicGeometry.AlgClosed.Basic

Goal state before:
pp: X Y : Scheme
K : Type u
inst✝² : Field K
inst✝¹ : IsAlgClosed K
f : X ⟶ Spec (CommRingCat.of K)
inst✝ : LocallyOfFiniteType f
x : ↥X
hx : IsClosed {x}
p : { p // p ≫ f = 𝟙 (Spec (CommRingCat.of K)) }
pf✝¹ : IsLocalRing ↑(CommRingCat.of K)
h : IsClosed {↑p (IsLocalRing.closedPoint K)}
pf✝ : IsLocalHom (CommRingCat.Hom.hom (Scheme.stalkClosedPointTo ↑p))
⊢ Spec.map (𝟙 (CommRingCat.of K)) =
    Spec.map
      ((residueFieldIsoBase f (↑p (IsLocalRing.closedPoint K)) h).inv ≫
        Scheme.descResidueField (Scheme.stalkClosedPointTo ↑p))

Tactic:
]

Goal state after:
X Y : Scheme
K : Type u
inst✝² : Field K
inst✝¹ : IsAlgClosed K
f : X ⟶ Spec (CommRingCat.of K)
inst✝ : LocallyOfFiniteType f
x : ↥X
hx : IsClosed {x}
p : { p // p ≫ f = 𝟙 (Spec (CommRingCat.of K)) }
pf✝¹ : IsLocalRing ↑(CommRingCat.of K)
h : IsClosed {↑p (IsLocalRing.closedPoint K)}
pf✝ : IsLocalHom (CommRingCat.Hom.hom (Scheme.stalkClosedPointTo ↑p))
⊢ Spec.map (𝟙 (CommRingCat.of K)) =
    Spec.map
      ((residueFieldIsoBase f (↑p (IsLocalRing.closedPoint K)) h).inv ≫
        Scheme.descResidueField (Scheme.stalkClosedPointTo ↑p))
Module: Mathlib.AlgebraicGeometry.AlgClosed.Basic

Goal state before:
pp: X Y : Scheme
K : Type u
inst✝² : Field K
inst✝¹ : IsAlgClosed K
f : X ⟶ Spec (CommRingCat.of K)
inst✝ : LocallyOfFiniteType f
x : ↥X
hx : IsClosed {x}
p : { p // p ≫ f = 𝟙 (Spec (CommRingCat.of K)) }
pf✝¹ : IsLocalRing ↑(CommRingCat.of K)
h : IsClosed {↑p (IsLocalRing.closedPoint K)}
pf✝ : IsLocalHom (CommRingCat.Hom.hom (Scheme.stalkClosedPointTo ↑p))
⊢ 𝟙 (Spec (CommRingCat.of K)) = 𝟙 (Spec (CommRingCat.of K))

Tactic:
]

Goal state after:
Module: Mathlib.AlgebraicGeometry.AlgClosed.Basic

Goal state before:
pp: X Y : Scheme
K : Type u
inst✝² : Field K
inst✝¹ : IsAlgClosed K
f : X ⟶ Spec (CommRingCat.of K)
inst✝ : LocallyOfFiniteType f
x : ↥X
hx : IsClosed {x}
p : { p // p ≫ f = 𝟙 (Spec (CommRingCat.of K)) }
⊢ (fun x ↦ ⟨pointOfClosedPoint f ↑x ⋯, ⋯⟩) ((fun p ↦ ⟨↑p (IsLocalRing.closedPoint K), ⋯⟩) p) = p

Tactic:
ext
    refine ((Scheme.SpecToEquivOfField _ _).symm_apply_eq (x := ⟨_, _⟩)).mpr ?_
    rw [Scheme.SpecToEquivOfField_eq_iff]
    dsimp [Scheme.SpecToEquivOfField]
    simp only [Category.id_comp, exists_const]
    generalize_proofs _ h
    refine (Category.comp_id _).symm.trans (((residueFieldIsoBase f _ h).eq_inv_comp).mp ?_)
    rw [← Spec.map_injective.eq_iff]
    simp only [Spec.map_id, Spec.map_comp, SpecMap_residueFieldIsoBase_inv]
    rw [reassoc_of% Scheme.descResidueField_stalkClosedPointTo_fromSpecResidueField, p.2]

Goal state after:
Module: Mathlib.AlgebraicGeometry.AlgClosed.Basic

Goal state before:
pp: case refine_1
X Y : Scheme
K : Type u
inst✝⁵ : Field K
inst✝⁴ : IsAlgClosed K
f g : X ⟶ Y
i : Y ⟶ Spec (CommRingCat.of K)
inst✝³ : IsSeparated i
inst✝² : LocallyOfFiniteType i
inst✝¹ : IsReduced X
inst✝ : LocallyOfFiniteType (f ≫ i)
S : Set ↥X
hS : IsLocallyClosed S
hS' : Dense S
H : ∀ x ∈ S, IsClosed {x} → f x = g x
H' : f ≫ i = g ≫ i
this : JacobsonSpace ↥X
⊢ Dense S

Tactic:
]

Goal state after:
case refine_1
X Y : Scheme
K : Type u
inst✝⁵ : Field K
inst✝⁴ : IsAlgClosed K
f g : X ⟶ Y
i : Y ⟶ Spec (CommRingCat.of K)
inst✝³ : IsSeparated i
inst✝² : LocallyOfFiniteType i
inst✝¹ : IsReduced X
inst✝ : LocallyOfFiniteType (f ≫ i)
S : Set ↥X
hS : IsLocallyClosed S
hS' : Dense S
H : ∀ x ∈ S, IsClosed {x} → f x = g x
H' : f ≫ i = g ≫ i
this : JacobsonSpace ↥X
⊢ Dense S
Module: Mathlib.AlgebraicGeometry.AlgClosed.Basic

Goal state before:
pp: case refine_2
X Y : Scheme
K : Type u
inst✝⁵ : Field K
inst✝⁴ : IsAlgClosed K
f g : X ⟶ Y
i : Y ⟶ Spec (CommRingCat.of K)
inst✝³ : IsSeparated i
inst✝² : LocallyOfFiniteType i
inst✝¹ : IsReduced X
inst✝ : LocallyOfFiniteType (f ≫ i)
S : Set ↥X
hS : IsLocallyClosed S
hS' : Dense S
H : ∀ x ∈ S, IsClosed {x} → f x = g x
H' : f ≫ i = g ≫ i
this : JacobsonSpace ↥X
x : ↥X
hxS : x ∈ S
hx : x ∈ closedPoints ↥X
⊢ Spec.map (residueFieldIsoBase (f ≫ i) x hx).hom ≫ X.fromSpecResidueField x ≫ f =
    Spec.map (residueFieldIsoBase (f ≫ i) x hx).hom ≫ X.fromSpecResidueField x ≫ g

Tactic:
]

Goal state after:
case refine_2
X Y : Scheme
K : Type u
inst✝⁵ : Field K
inst✝⁴ : IsAlgClosed K
f g : X ⟶ Y
i : Y ⟶ Spec (CommRingCat.of K)
inst✝³ : IsSeparated i
inst✝² : LocallyOfFiniteType i
inst✝¹ : IsReduced X
inst✝ : LocallyOfFiniteType (f ≫ i)
S : Set ↥X
hS : IsLocallyClosed S
hS' : Dense S
H : ∀ x ∈ S, IsClosed {x} → f x = g x
H' : f ≫ i = g ≫ i
this : JacobsonSpace ↥X
x : ↥X
hxS : x ∈ S
hx : x ∈ closedPoints ↥X
⊢ Spec.map (residueFieldIsoBase (f ≫ i) x hx).hom ≫ X.fromSpecResidueField x ≫ f =
    Spec.map (residueFieldIsoBase (f ≫ i) x hx).hom ≫ X.fromSpecResidueField x ≫ g