mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.RingTheory.FinitePresentation

Goal state before:
pp: case h.right
R : Type w₁
A : Type w₂
B : Type w₃
inst✝⁵ : CommRing R
inst✝⁴ : CommRing A
inst✝³ : Algebra R A
inst✝² : CommRing B
inst✝¹ : Algebra R B
inst✝ : FinitePresentation R A
e : A ≃ₐ[R] B
n : ℕ
f : MvPolynomial (Fin n) R →ₐ[R] A
hf : Surjective ⇑f ∧ (RingHom.ker f.toRingHom).FG
hco : ((↑e).comp f).toRingHom = (↑e.toRingEquiv).comp f.toRingHom
⊢ Ideal.comap f.toRingHom ⊥ = Ideal.comap f.toRingHom ⊥

Tactic:
]

Goal state after:
Module: Mathlib.CategoryTheory.ObjectProperty.Small

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
D : Type u'
inst✝¹ : Category.{v', u'} D
P : ObjectProperty C
inst✝ : ObjectProperty.EssentiallySmall.{w, v, u} P
Q : ObjectProperty C
w✝ : ObjectProperty.Small.{w, v, u} Q
left✝ : Q ≤ P
right✝ : P ≤ Q.isoClosure
⊢ P ≤ Q.isoClosure

Tactic:
]

Goal state after:
C : Type u
inst✝² : Category.{v, u} C
D : Type u'
inst✝¹ : Category.{v', u'} D
P : ObjectProperty C
inst✝ : ObjectProperty.EssentiallySmall.{w, v, u} P
Q : ObjectProperty C
w✝ : ObjectProperty.Small.{w, v, u} Q
left✝ : Q ≤ P
right✝ : P ≤ Q.isoClosure
⊢ P ≤ Q.isoClosure
Module: Mathlib.CategoryTheory.ObjectProperty.Small

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
P : ObjectProperty C
inst✝¹ : P.IsClosedUnderIsomorphisms
inst✝ : ObjectProperty.EssentiallySmall.{w, v, u} P
Q : ObjectProperty C
w✝ : ObjectProperty.Small.{w, v, u} Q
hQ₁ : Q ≤ P
hQ₂ : P ≤ Q.isoClosure
⊢ Q ≤ P

Tactic:
]

Goal state after:
C : Type u
inst✝² : Category.{v, u} C
P : ObjectProperty C
inst✝¹ : P.IsClosedUnderIsomorphisms
inst✝ : ObjectProperty.EssentiallySmall.{w, v, u} P
Q : ObjectProperty C
w✝ : ObjectProperty.Small.{w, v, u} Q
hQ₁ : Q ≤ P
hQ₂ : P ≤ Q.isoClosure
⊢ Q ≤ P
Module: Mathlib.CategoryTheory.ObjectProperty.Small

Goal state before:
pp: C : Type u
inst✝ : Category.{v, u} C
P : ObjectProperty C
x✝ : ObjectProperty.EssentiallySmall.{w, v, u} P.op
Q : ObjectProperty Cᵒᵖ
h₁ : ObjectProperty.Small.{w, v, u} Q
left✝ : Q ≤ P.op
h₂ : P.op ≤ Q.isoClosure
⊢ P.op ≤ Q.isoClosure

Tactic:
]

Goal state after:
C : Type u
inst✝ : Category.{v, u} C
P : ObjectProperty C
x✝ : ObjectProperty.EssentiallySmall.{w, v, u} P.op
Q : ObjectProperty Cᵒᵖ
h₁ : ObjectProperty.Small.{w, v, u} Q
left✝ : Q ≤ P.op
h₂ : P.op ≤ Q.isoClosure
⊢ P.op ≤ Q.isoClosure
Module: Mathlib.CategoryTheory.ObjectProperty.Small

Goal state before:
pp: C : Type u
inst✝ : Category.{v, u} C
P : ObjectProperty C
x✝ : ObjectProperty.EssentiallySmall.{w, v, u} P
Q : ObjectProperty C
h₁ : ObjectProperty.Small.{w, v, u} Q
left✝ : Q ≤ P
h₂ : P ≤ Q.isoClosure
⊢ P ≤ Q.isoClosure

Tactic:
]

Goal state after:
C : Type u
inst✝ : Category.{v, u} C
P : ObjectProperty C
x✝ : ObjectProperty.EssentiallySmall.{w, v, u} P
Q : ObjectProperty C
h₁ : ObjectProperty.Small.{w, v, u} Q
left✝ : Q ≤ P
h₂ : P ≤ Q.isoClosure
⊢ P ≤ Q.isoClosure
Module: Mathlib.CategoryTheory.ObjectProperty.Small

Goal state before:
pp: C : Type u
inst✝ : Category.{v, u} C
P : ObjectProperty Cᵒᵖ
⊢ ObjectProperty.EssentiallySmall.{w, v, u} P ↔ ObjectProperty.EssentiallySmall.{w, v, u} P

Tactic:
]

Goal state after:
Module: Mathlib.CategoryTheory.ObjectProperty.Small

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
D : Type u'
inst✝² : Category.{v', u'} D
P : ObjectProperty C
inst✝¹ : LocallySmall.{w, v, u} C
inst✝ : ObjectProperty.EssentiallySmall.{w, v, u} P
Q : ObjectProperty C
w✝ : ObjectProperty.Small.{w, v, u} Q
h₁ : Q ≤ P
h₂ : P ≤ Q.isoClosure
this : (ιOfLE h₁).IsEquivalence
⊢ EssentiallySmall.{w, v, u} Q.FullSubcategory

Tactic:
]

Goal state after:
C : Type u
inst✝³ : Category.{v, u} C
D : Type u'
inst✝² : Category.{v', u'} D
P : ObjectProperty C
inst✝¹ : LocallySmall.{w, v, u} C
inst✝ : ObjectProperty.EssentiallySmall.{w, v, u} P
Q : ObjectProperty C
w✝ : ObjectProperty.Small.{w, v, u} Q
h₁ : Q ≤ P
h₂ : P ≤ Q.isoClosure
this : (ιOfLE h₁).IsEquivalence
⊢ EssentiallySmall.{w, v, u} Q.FullSubcategory
Module: Mathlib.CategoryTheory.MorphismProperty.Comma

Goal state before:
pp: A : Type u_1
inst✝³ : Category.{v_1, u_1} A
B : Type u_2
inst✝² : Category.{v_2, u_2} B
T : Type u_3
inst✝¹ : Category.{v_3, u_3} T
L : A ⥤ T
R : B ⥤ T
W : MorphismProperty T
X✝ : T
inst✝ : W.RespectsIso
X Y : Comma L R
e : X ≅ Y
h : commaObj L R W X
⊢ W X.hom

Tactic:
]

Goal state after:
A : Type u_1
inst✝³ : Category.{v_1, u_1} A
B : Type u_2
inst✝² : Category.{v_2, u_2} B
T : Type u_3
inst✝¹ : Category.{v_3, u_3} T
L : A ⥤ T
R : B ⥤ T
W : MorphismProperty T
X✝ : T
inst✝ : W.RespectsIso
X Y : Comma L R
e : X ≅ Y
h : commaObj L R W X
⊢ W X.hom
Module: Mathlib.CategoryTheory.ObjectProperty.Small

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
D : Type u'
inst✝¹ : Category.{v', u'} D
inst✝ : EssentiallySmall.{w, v, u} C
F : C ⥤ D
⊢ ObjectProperty.EssentiallySmall.{w, v', u'} (⊤.map F)

Tactic:
]

Goal state after:
C : Type u
inst✝² : Category.{v, u} C
D : Type u'
inst✝¹ : Category.{v', u'} D
inst✝ : EssentiallySmall.{w, v, u} C
F : C ⥤ D
⊢ ObjectProperty.EssentiallySmall.{w, v', u'} (⊤.map F)
Module: Mathlib.RingTheory.FinitePresentation

Goal state before:
pp: R : Type w₁
A : Type w₂
inst✝⁴ : CommRing R
inst✝³ : CommRing A
inst✝² : Algebra R A
inst✝¹ : FinitePresentation R A
ι : Type v
inst✝ : Finite ι
hfp : ∃ ι x f, Surjective ⇑f ∧ (RingHom.ker f.toRingHom).FG
⊢ ∃ ι_1 x f, Surjective ⇑f ∧ (RingHom.ker f.toRingHom).FG

Tactic:
]

Goal state after:
R : Type w₁
A : Type w₂
inst✝⁴ : CommRing R
inst✝³ : CommRing A
inst✝² : Algebra R A
inst✝¹ : FinitePresentation R A
ι : Type v
inst✝ : Finite ι
hfp : ∃ ι x f, Surjective ⇑f ∧ (RingHom.ker f.toRingHom).FG
⊢ ∃ ι_1 x f, Surjective ⇑f ∧ (RingHom.ker f.toRingHom).FG
Module: Mathlib.RingTheory.FinitePresentation

Goal state before:
pp: case intro.refine_1
R : Type w₁
A : Type w₂
inst✝⁴ : CommRing R
inst✝³ : CommRing A
inst✝² : Algebra R A
inst✝¹ : FinitePresentation R A
ι : Type v
inst✝ : Finite ι
ι' : Type v
w✝ : Fintype ι'
f : MvPolynomial ι' R →ₐ[R] A
hf_surj : Surjective ⇑f
hf_ker : (RingHom.ker f.toRingHom).FG
g : MvPolynomial (ι ⊕ ι') R →ₐ[R] MvPolynomial ι A := (MvPolynomial.mapAlgHom f).comp ↑(MvPolynomial.sumAlgEquiv R ι ι')
val✝ : Fintype (ι ⊕ ι')
⊢ ⊥.FG

Tactic:
]

Goal state after:
case intro.refine_1
R : Type w₁
A : Type w₂
inst✝⁴ : CommRing R
inst✝³ : CommRing A
inst✝² : Algebra R A
inst✝¹ : FinitePresentation R A
ι : Type v
inst✝ : Finite ι
ι' : Type v
w✝ : Fintype ι'
f : MvPolynomial ι' R →ₐ[R] A
hf_surj : Surjective ⇑f
hf_ker : (RingHom.ker f.toRingHom).FG
g : MvPolynomial (ι ⊕ ι') R →ₐ[R] MvPolynomial ι A := (MvPolynomial.mapAlgHom f).comp ↑(MvPolynomial.sumAlgEquiv R ι ι')
val✝ : Fintype (ι ⊕ ι')
⊢ ⊥.FG
Module: Mathlib.RingTheory.FinitePresentation

Goal state before:
pp: case intro.refine_2
R : Type w₁
A : Type w₂
inst✝⁴ : CommRing R
inst✝³ : CommRing A
inst✝² : Algebra R A
inst✝¹ : FinitePresentation R A
ι : Type v
inst✝ : Finite ι
ι' : Type v
w✝ : Fintype ι'
f : MvPolynomial ι' R →ₐ[R] A
hf_surj : Surjective ⇑f
hf_ker : (RingHom.ker f.toRingHom).FG
g : MvPolynomial (ι ⊕ ι') R →ₐ[R] MvPolynomial ι A := (MvPolynomial.mapAlgHom f).comp ↑(MvPolynomial.sumAlgEquiv R ι ι')
val✝ : Fintype (ι ⊕ ι')
⊢ (Ideal.map MvPolynomial.C (RingHom.ker ↑f)).FG

Tactic:
]

Goal state after:
case intro.refine_2
R : Type w₁
A : Type w₂
inst✝⁴ : CommRing R
inst✝³ : CommRing A
inst✝² : Algebra R A
inst✝¹ : FinitePresentation R A
ι : Type v
inst✝ : Finite ι
ι' : Type v
w✝ : Fintype ι'
f : MvPolynomial ι' R →ₐ[R] A
hf_surj : Surjective ⇑f
hf_ker : (RingHom.ker f.toRingHom).FG
g : MvPolynomial (ι ⊕ ι') R →ₐ[R] MvPolynomial ι A := (MvPolynomial.mapAlgHom f).comp ↑(MvPolynomial.sumAlgEquiv R ι ι')
val✝ : Fintype (ι ⊕ ι')
⊢ (Ideal.map MvPolynomial.C (RingHom.ker ↑f)).FG
Module: Mathlib.CategoryTheory.MorphismProperty.Comma

Goal state before:
pp: A : Type u_1
inst✝⁵ : Category.{v_1, u_1} A
B : Type u_2
inst✝⁴ : Category.{v_2, u_2} B
T : Type u_3
inst✝³ : Category.{v_3, u_3} T
L : A ⥤ T
R : B ⥤ T
P : MorphismProperty T
Q : MorphismProperty A
W : MorphismProperty B
inst✝² : Q.IsMultiplicative
inst✝¹ : W.IsMultiplicative
X Y : MorphismProperty.Comma L R P Q W
f : X ⟶ Y
inst✝ : IsIso f
⊢ 𝟙 X.toComma = 𝟙 X.toComma

Tactic:
]

Goal state after:
Module: Mathlib.CategoryTheory.ObjectProperty.Small

Goal state before:
pp: case refine_2
C : Type u
inst✝¹ : Category.{v, u} C
P : ObjectProperty C
inst✝ : LocallySmall.{w', v, u} C
x✝ : ObjectProperty.EssentiallySmall.{w, v, u} P
Q : ObjectProperty C
w✝ : ObjectProperty.Small.{w, v, u} Q
h₁ : Q ≤ P
h₂ : (ιOfLE h₁).IsEquivalence
⊢ ∃ J x, Nonempty (P.FullSubcategory ≌ J)

Tactic:
]

Goal state after:
case refine_2
C : Type u
inst✝¹ : Category.{v, u} C
P : ObjectProperty C
inst✝ : LocallySmall.{w', v, u} C
x✝ : ObjectProperty.EssentiallySmall.{w, v, u} P
Q : ObjectProperty C
w✝ : ObjectProperty.Small.{w, v, u} Q
h₁ : Q ≤ P
h₂ : (ιOfLE h₁).IsEquivalence
⊢ ∃ J x, Nonempty (P.FullSubcategory ≌ J)
Module: Mathlib.CategoryTheory.ObjectProperty.Small

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : LocallySmall.{w', v_1, u_1} C
⊢ (∃ J x, Nonempty (C ≌ J)) ↔ ∃ J x, Nonempty (⊤.FullSubcategory ≌ J)

Tactic:
]

Goal state after:
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : LocallySmall.{w', v_1, u_1} C
⊢ (∃ J x, Nonempty (C ≌ J)) ↔ ∃ J x, Nonempty (⊤.FullSubcategory ≌ J)
Module: Mathlib.RingTheory.FinitePresentation

Goal state before:
pp: case refine_1
R : Type w₁
A : Type w₂
B : Type w₃
inst✝⁸ : CommRing R
inst✝⁷ : CommRing A
inst✝⁶ : Algebra R A
inst✝⁵ : CommRing B
inst✝⁴ : Algebra R B
inst✝³ : Algebra A B
inst✝² : IsScalarTower R A B
inst✝¹ : FinitePresentation R B
inst✝ : FiniteType R A
n : ℕ
f : MvPolynomial (Fin n) R →ₐ[R] B
hf : Surjective ⇑f
s : Finset (MvPolynomial (Fin n) R)
hs : Ideal.span ↑s = RingHom.ker f.toRingHom
RX : Type (max w₁ 0) := MvPolynomial (Fin n) R
AX : Type (max w₂ 0) := MvPolynomial (Fin n) A
⊢ ∀ (x : B), x ∈ adjoin A ↑⊤

Tactic:
]

Goal state after:
case refine_1
R : Type w₁
A : Type w₂
B : Type w₃
inst✝⁸ : CommRing R
inst✝⁷ : CommRing A
inst✝⁶ : Algebra R A
inst✝⁵ : CommRing B
inst✝⁴ : Algebra R B
inst✝³ : Algebra A B
inst✝² : IsScalarTower R A B
inst✝¹ : FinitePresentation R B
inst✝ : FiniteType R A
n : ℕ
f : MvPolynomial (Fin n) R →ₐ[R] B
hf : Surjective ⇑f
s : Finset (MvPolynomial (Fin n) R)
hs : Ideal.span ↑s = RingHom.ker f.toRingHom
RX : Type (max w₁ 0) := MvPolynomial (Fin n) R
AX : Type (max w₂ 0) := MvPolynomial (Fin n) A
⊢ ∀ (x : B), x ∈ adjoin A ↑⊤
Module: Mathlib.CategoryTheory.ObjectProperty.Small

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : LocallySmall.{w, v, u} C
⊢ EssentiallySmall.{w, v, u} C ↔ ∃ J x, Nonempty (C ≌ J)

Tactic:
]

Goal state after:
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : LocallySmall.{w, v, u} C
⊢ EssentiallySmall.{w, v, u} C ↔ ∃ J x, Nonempty (C ≌ J)
Module: Mathlib.CategoryTheory.MorphismProperty.Comma

Goal state before:
pp: A : Type u_1
inst✝⁶ : Category.{v_1, u_1} A
B : Type u_2
inst✝⁵ : Category.{v_2, u_2} B
T : Type u_3
inst✝⁴ : Category.{v_3, u_3} T
L : A ⥤ T
R : B ⥤ T
P : MorphismProperty T
Q : MorphismProperty A
W : MorphismProperty B
inst✝³ : Q.IsMultiplicative
inst✝² : W.IsMultiplicative
inst✝¹ : Q.RespectsIso
inst✝ : W.RespectsIso
A✝ B✝ : MorphismProperty.Comma L R P Q W
f : A✝ ⟶ B✝
hf : IsIso (Hom.hom f)
⊢ IsIso (homFromCommaOfIsIso (Hom.hom f))

Tactic:
]

Goal state after:
A : Type u_1
inst✝⁶ : Category.{v_1, u_1} A
B : Type u_2
inst✝⁵ : Category.{v_2, u_2} B
T : Type u_3
inst✝⁴ : Category.{v_3, u_3} T
L : A ⥤ T
R : B ⥤ T
P : MorphismProperty T
Q : MorphismProperty A
W : MorphismProperty B
inst✝³ : Q.IsMultiplicative
inst✝² : W.IsMultiplicative
inst✝¹ : Q.RespectsIso
inst✝ : W.RespectsIso
A✝ B✝ : MorphismProperty.Comma L R P Q W
f : A✝ ⟶ B✝
hf : IsIso (Hom.hom f)
⊢ IsIso (homFromCommaOfIsIso (Hom.hom f))
Module: Mathlib.RingTheory.FinitePresentation

Goal state before:
pp: R : Type w₁
A : Type w₂
B : Type w₃
inst✝⁸ : CommRing R
inst✝⁷ : CommRing A
inst✝⁶ : Algebra R A
inst✝⁵ : CommRing B
inst✝⁴ : Algebra R B
inst✝³ : Algebra A B
inst✝² : IsScalarTower R A B
inst✝¹ : FinitePresentation R B
inst✝ : FiniteType R A
n : ℕ
f : MvPolynomial (Fin n) R →ₐ[R] B
hf : Surjective ⇑f
s : Finset (MvPolynomial (Fin n) R)
hs : Ideal.span ↑s = RingHom.ker f.toRingHom
RX : Type (max w₁ 0) := MvPolynomial (Fin n) R
AX : Type (max w₂ 0) := MvPolynomial (Fin n) A
t : Finset A
ht : adjoin R ↑t = ⊤
t' : ↥t → MvPolynomial (Fin n) R
ht' : ∀ (i : ↥t), f (t' i) = (algebraMap A B) ↑i
⊢ Subalgebra.restrictScalars R (adjoin (↥(adjoin R (⇑(algebraMap A AX) '' ↑t))) (Set.range MvPolynomial.X)) =
    Subalgebra.restrictScalars R ⊤

Tactic:
]

Goal state after:
R : Type w₁
A : Type w₂
B : Type w₃
inst✝⁸ : CommRing R
inst✝⁷ : CommRing A
inst✝⁶ : Algebra R A
inst✝⁵ : CommRing B
inst✝⁴ : Algebra R B
inst✝³ : Algebra A B
inst✝² : IsScalarTower R A B
inst✝¹ : FinitePresentation R B
inst✝ : FiniteType R A
n : ℕ
f : MvPolynomial (Fin n) R →ₐ[R] B
hf : Surjective ⇑f
s : Finset (MvPolynomial (Fin n) R)
hs : Ideal.span ↑s = RingHom.ker f.toRingHom
RX : Type (max w₁ 0) := MvPolynomial (Fin n) R
AX : Type (max w₂ 0) := MvPolynomial (Fin n) A
t : Finset A
ht : adjoin R ↑t = ⊤
t' : ↥t → MvPolynomial (Fin n) R
ht' : ∀ (i : ↥t), f (t' i) = (algebraMap A B) ↑i
⊢ Subalgebra.restrictScalars R (adjoin (↥(adjoin R (⇑(algebraMap A AX) '' ↑t))) (Set.range MvPolynomial.X)) =
    Subalgebra.restrictScalars R ⊤
Module: Mathlib.CategoryTheory.MorphismProperty.Comma

Goal state before:
pp: A : Type u_1
inst✝⁴ : Category.{v_1, u_1} A
B : Type u_2
inst✝³ : Category.{v_2, u_2} B
T : Type u_3
inst✝² : Category.{v_3, u_3} T
L : A ⥤ T
R : B ⥤ T
P : MorphismProperty T
Q : MorphismProperty A
W : MorphismProperty B
inst✝¹ : Q.IsMultiplicative
inst✝ : W.IsMultiplicative
X Y : MorphismProperty.Comma L R P Q W
h : X = Y
⊢ Y.left = Y.left

Tactic:
]

Goal state after:
Module: Mathlib.CategoryTheory.ObjectProperty.Small

Goal state before:
pp: C : Type u_1
D : Type u_2
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Category.{v_2, u_2} D
F : C ⥤ D
inst✝ : LocallySmall.{w, v_1, u_1} C
H₁ : ObjectProperty.EssentiallySmall.{w, v_2, u_2} F.essImage
H₂ : ∀ (Y : D), ObjectProperty.EssentiallySmall.{w, v_1, u_1} fun x ↦ Nonempty (F.obj x ≅ Y)
⊢ LocallySmall.{w, v_1, u_1} C ∧ ObjectProperty.EssentiallySmall.{w, v_1, u_1} ⊤

Tactic:
]

Goal state after:
C : Type u_1
D : Type u_2
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Category.{v_2, u_2} D
F : C ⥤ D
inst✝ : LocallySmall.{w, v_1, u_1} C
H₁ : ObjectProperty.EssentiallySmall.{w, v_2, u_2} F.essImage
H₂ : ∀ (Y : D), ObjectProperty.EssentiallySmall.{w, v_1, u_1} fun x ↦ Nonempty (F.obj x ≅ Y)
⊢ LocallySmall.{w, v_1, u_1} C ∧ ObjectProperty.EssentiallySmall.{w, v_1, u_1} ⊤
Module: Mathlib.RingTheory.FinitePresentation

Goal state before:
pp: R : Type w₁
A : Type w₂
B : Type w₃
inst✝⁸ : CommRing R
inst✝⁷ : CommRing A
inst✝⁶ : Algebra R A
inst✝⁵ : CommRing B
inst✝⁴ : Algebra R B
inst✝³ : Algebra A B
inst✝² : IsScalarTower R A B
inst✝¹ : FinitePresentation R B
inst✝ : FiniteType R A
n : ℕ
f : MvPolynomial (Fin n) R →ₐ[R] B
hf : Surjective ⇑f
s : Finset (MvPolynomial (Fin n) R)
hs : Ideal.span ↑s = RingHom.ker f.toRingHom
RX : Type (max w₁ 0) := MvPolynomial (Fin n) R
AX : Type (max w₂ 0) := MvPolynomial (Fin n) A
t : Finset A
ht : adjoin R ↑t = ⊤
t' : ↥t → MvPolynomial (Fin n) R
ht' : ∀ (i : ↥t), f (t' i) = (algebraMap A B) ↑i
⊢ adjoin (↥(Subalgebra.map (IsScalarTower.toAlgHom R A AX) ⊤)) (Set.range MvPolynomial.X) = ⊤

Tactic:
]

Goal state after:
R : Type w₁
A : Type w₂
B : Type w₃
inst✝⁸ : CommRing R
inst✝⁷ : CommRing A
inst✝⁶ : Algebra R A
inst✝⁵ : CommRing B
inst✝⁴ : Algebra R B
inst✝³ : Algebra A B
inst✝² : IsScalarTower R A B
inst✝¹ : FinitePresentation R B
inst✝ : FiniteType R A
n : ℕ
f : MvPolynomial (Fin n) R →ₐ[R] B
hf : Surjective ⇑f
s : Finset (MvPolynomial (Fin n) R)
hs : Ideal.span ↑s = RingHom.ker f.toRingHom
RX : Type (max w₁ 0) := MvPolynomial (Fin n) R
AX : Type (max w₂ 0) := MvPolynomial (Fin n) A
t : Finset A
ht : adjoin R ↑t = ⊤
t' : ↥t → MvPolynomial (Fin n) R
ht' : ∀ (i : ↥t), f (t' i) = (algebraMap A B) ↑i
⊢ adjoin (↥(Subalgebra.map (IsScalarTower.toAlgHom R A AX) ⊤)) (Set.range MvPolynomial.X) = ⊤
Module: Mathlib.CategoryTheory.MorphismProperty.Comma

Goal state before:
pp: A : Type u_1
inst✝⁴ : Category.{v_1, u_1} A
B : Type u_2
inst✝³ : Category.{v_2, u_2} B
T : Type u_3
inst✝² : Category.{v_3, u_3} T
L : A ⥤ T
R : B ⥤ T
P : MorphismProperty T
Q : MorphismProperty A
W : MorphismProperty B
inst✝¹ : Q.IsMultiplicative
inst✝ : W.IsMultiplicative
X Y : MorphismProperty.Comma L R P Q W
h : X = Y
⊢ Y.right = Y.right

Tactic:
]

Goal state after:
Module: Mathlib.RingTheory.FinitePresentation

Goal state before:
pp: R : Type w₁
A : Type w₂
B : Type w₃
inst✝⁸ : CommRing R
inst✝⁷ : CommRing A
inst✝⁶ : Algebra R A
inst✝⁵ : CommRing B
inst✝⁴ : Algebra R B
inst✝³ : Algebra A B
inst✝² : IsScalarTower R A B
inst✝¹ : FinitePresentation R B
inst✝ : FiniteType R A
n : ℕ
f : MvPolynomial (Fin n) R →ₐ[R] B
hf : Surjective ⇑f
s : Finset (MvPolynomial (Fin n) R)
hs : Ideal.span ↑s = RingHom.ker f.toRingHom
RX : Type (max w₁ 0) := MvPolynomial (Fin n) R
AX : Type (max w₂ 0) := MvPolynomial (Fin n) A
t : Finset A
ht : adjoin R ↑t = ⊤
t' : ↥t → MvPolynomial (Fin n) R
ht' : ∀ (i : ↥t), f (t' i) = (algebraMap A B) ↑i
⊢ ⊤ = ⊤

Tactic:
]

Goal state after:
Module: Mathlib.RingTheory.FinitePresentation

Goal state before:
pp: case refine_2
R : Type w₁
A : Type w₂
B : Type w₃
inst✝⁸ : CommRing R
inst✝⁷ : CommRing A
inst✝⁶ : Algebra R A
inst✝⁵ : CommRing B
inst✝⁴ : Algebra R B
inst✝³ : Algebra A B
inst✝² : IsScalarTower R A B
inst✝¹ : FinitePresentation R B
inst✝ : FiniteType R A
n : ℕ
f : MvPolynomial (Fin n) R →ₐ[R] B
hf : Surjective ⇑f
s : Finset (MvPolynomial (Fin n) R)
hs : Ideal.span ↑s = RingHom.ker f.toRingHom
RX : Type (max w₁ 0) := MvPolynomial (Fin n) R
AX : Type (max w₂ 0) := MvPolynomial (Fin n) A
t : Finset A
ht : adjoin R ↑t = ⊤
t' : ↥t → MvPolynomial (Fin n) R
ht' : ∀ (i : ↥t), f (t' i) = (algebraMap A B) ↑i
ht'' : adjoin R (⇑(algebraMap A AX) '' ↑t ∪ Set.range MvPolynomial.X) = ⊤
g : ↥t → AX := fun x ↦ MvPolynomial.C ↑x - (MvPolynomial.map (algebraMap R A)) (t' x)
⊢ Ideal.span (⇑(MvPolynomial.map (algebraMap R A)) '' ↑s ∪ Set.range g) =
    RingHom.ker (MvPolynomial.aeval (⇑f ∘ MvPolynomial.X)).toRingHom

Tactic:
]

Goal state after:
case refine_2
R : Type w₁
A : Type w₂
B : Type w₃
inst✝⁸ : CommRing R
inst✝⁷ : CommRing A
inst✝⁶ : Algebra R A
inst✝⁵ : CommRing B
inst✝⁴ : Algebra R B
inst✝³ : Algebra A B
inst✝² : IsScalarTower R A B
inst✝¹ : FinitePresentation R B
inst✝ : FiniteType R A
n : ℕ
f : MvPolynomial (Fin n) R →ₐ[R] B
hf : Surjective ⇑f
s : Finset (MvPolynomial (Fin n) R)
hs : Ideal.span ↑s = RingHom.ker f.toRingHom
RX : Type (max w₁ 0) := MvPolynomial (Fin n) R
AX : Type (max w₂ 0) := MvPolynomial (Fin n) A
t : Finset A
ht : adjoin R ↑t = ⊤
t' : ↥t → MvPolynomial (Fin n) R
ht' : ∀ (i : ↥t), f (t' i) = (algebraMap A B) ↑i
ht'' : adjoin R (⇑(algebraMap A AX) '' ↑t ∪ Set.range MvPolynomial.X) = ⊤
g : ↥t → AX := fun x ↦ MvPolynomial.C ↑x - (MvPolynomial.map (algebraMap R A)) (t' x)
⊢ Ideal.span (⇑(MvPolynomial.map (algebraMap R A)) '' ↑s ∪ Set.range g) =
    RingHom.ker (MvPolynomial.aeval (⇑f ∘ MvPolynomial.X)).toRingHom
Module: Mathlib.RingTheory.FinitePresentation

Goal state before:
pp: R : Type w₁
A : Type w₂
B : Type w₃
inst✝⁸ : CommRing R
inst✝⁷ : CommRing A
inst✝⁶ : Algebra R A
inst✝⁵ : CommRing B
inst✝⁴ : Algebra R B
inst✝³ : Algebra A B
inst✝² : IsScalarTower R A B
inst✝¹ : FinitePresentation R B
inst✝ : FiniteType R A
n : ℕ
f : MvPolynomial (Fin n) R →ₐ[R] B
hf : Surjective ⇑f
s : Finset (MvPolynomial (Fin n) R)
hs : Ideal.span ↑s = RingHom.ker f.toRingHom
RX : Type (max w₁ 0) := MvPolynomial (Fin n) R
AX : Type (max w₂ 0) := MvPolynomial (Fin n) A
t : Finset A
ht : adjoin R ↑t = ⊤
t' : ↥t → MvPolynomial (Fin n) R
ht' : ∀ (i : ↥t), f (t' i) = (algebraMap A B) ↑i
ht'' : adjoin R (⇑(algebraMap A AX) '' ↑t ∪ Set.range MvPolynomial.X) = ⊤
g : ↥t → AX := fun x ↦ MvPolynomial.C ↑x - (MvPolynomial.map (algebraMap R A)) (t' x)
s₀ : Set (MvPolynomial (Fin n) A) := ⇑(MvPolynomial.map (algebraMap R A)) '' ↑s ∪ Set.range g
I : Ideal (MvPolynomial (Fin n) A) := RingHom.ker (MvPolynomial.aeval (⇑f ∘ MvPolynomial.X))
⊢ ⇑(MvPolynomial.map (algebraMap R A)) '' ↑s ∪ Set.range g ⊆ ↑(RingHom.ker (MvPolynomial.aeval (⇑f ∘ MvPolynomial.X)))

Tactic:
]

Goal state after:
R : Type w₁
A : Type w₂
B : Type w₃
inst✝⁸ : CommRing R
inst✝⁷ : CommRing A
inst✝⁶ : Algebra R A
inst✝⁵ : CommRing B
inst✝⁴ : Algebra R B
inst✝³ : Algebra A B
inst✝² : IsScalarTower R A B
inst✝¹ : FinitePresentation R B
inst✝ : FiniteType R A
n : ℕ
f : MvPolynomial (Fin n) R →ₐ[R] B
hf : Surjective ⇑f
s : Finset (MvPolynomial (Fin n) R)
hs : Ideal.span ↑s = RingHom.ker f.toRingHom
RX : Type (max w₁ 0) := MvPolynomial (Fin n) R
AX : Type (max w₂ 0) := MvPolynomial (Fin n) A
t : Finset A
ht : adjoin R ↑t = ⊤
t' : ↥t → MvPolynomial (Fin n) R
ht' : ∀ (i : ↥t), f (t' i) = (algebraMap A B) ↑i
ht'' : adjoin R (⇑(algebraMap A AX) '' ↑t ∪ Set.range MvPolynomial.X) = ⊤
g : ↥t → AX := fun x ↦ MvPolynomial.C ↑x - (MvPolynomial.map (algebraMap R A)) (t' x)
s₀ : Set (MvPolynomial (Fin n) A) := ⇑(MvPolynomial.map (algebraMap R A)) '' ↑s ∪ Set.range g
I : Ideal (MvPolynomial (Fin n) A) := RingHom.ker (MvPolynomial.aeval (⇑f ∘ MvPolynomial.X))
⊢ ⇑(MvPolynomial.map (algebraMap R A)) '' ↑s ∪ Set.range g ⊆ ↑(RingHom.ker (MvPolynomial.aeval (⇑f ∘ MvPolynomial.X)))
Module: Mathlib.RingTheory.FinitePresentation

Goal state before:
pp: case inl
R : Type w₁
A : Type w₂
B : Type w₃
inst✝⁸ : CommRing R
inst✝⁷ : CommRing A
inst✝⁶ : Algebra R A
inst✝⁵ : CommRing B
inst✝⁴ : Algebra R B
inst✝³ : Algebra A B
inst✝² : IsScalarTower R A B
inst✝¹ : FinitePresentation R B
inst✝ : FiniteType R A
n : ℕ
f : MvPolynomial (Fin n) R →ₐ[R] B
hf : Surjective ⇑f
s : Finset (MvPolynomial (Fin n) R)
hs : Ideal.span ↑s = RingHom.ker f.toRingHom
RX : Type (max w₁ 0) := MvPolynomial (Fin n) R
AX : Type (max w₂ 0) := MvPolynomial (Fin n) A
t : Finset A
ht : adjoin R ↑t = ⊤
t' : ↥t → MvPolynomial (Fin n) R
ht' : ∀ (i : ↥t), f (t' i) = (algebraMap A B) ↑i
ht'' : adjoin R (⇑(algebraMap A AX) '' ↑t ∪ Set.range MvPolynomial.X) = ⊤
g : ↥t → AX := fun x ↦ MvPolynomial.C ↑x - (MvPolynomial.map (algebraMap R A)) (t' x)
s₀ : Set (MvPolynomial (Fin n) A) := ⇑(MvPolynomial.map (algebraMap R A)) '' ↑s ∪ Set.range g
I : Ideal (MvPolynomial (Fin n) A) := RingHom.ker (MvPolynomial.aeval (⇑f ∘ MvPolynomial.X))
x : MvPolynomial (Fin n) R
hx : x ∈ ↑s
⊢ (MvPolynomial.aeval (⇑f ∘ MvPolynomial.X)) ((MvPolynomial.map (algebraMap R A)) x) = 0

Tactic:
]

Goal state after:
case inl
R : Type w₁
A : Type w₂
B : Type w₃
inst✝⁸ : CommRing R
inst✝⁷ : CommRing A
inst✝⁶ : Algebra R A
inst✝⁵ : CommRing B
inst✝⁴ : Algebra R B
inst✝³ : Algebra A B
inst✝² : IsScalarTower R A B
inst✝¹ : FinitePresentation R B
inst✝ : FiniteType R A
n : ℕ
f : MvPolynomial (Fin n) R →ₐ[R] B
hf : Surjective ⇑f
s : Finset (MvPolynomial (Fin n) R)
hs : Ideal.span ↑s = RingHom.ker f.toRingHom
RX : Type (max w₁ 0) := MvPolynomial (Fin n) R
AX : Type (max w₂ 0) := MvPolynomial (Fin n) A
t : Finset A
ht : adjoin R ↑t = ⊤
t' : ↥t → MvPolynomial (Fin n) R
ht' : ∀ (i : ↥t), f (t' i) = (algebraMap A B) ↑i
ht'' : adjoin R (⇑(algebraMap A AX) '' ↑t ∪ Set.range MvPolynomial.X) = ⊤
g : ↥t → AX := fun x ↦ MvPolynomial.C ↑x - (MvPolynomial.map (algebraMap R A)) (t' x)
s₀ : Set (MvPolynomial (Fin n) A) := ⇑(MvPolynomial.map (algebraMap R A)) '' ↑s ∪ Set.range g
I : Ideal (MvPolynomial (Fin n) A) := RingHom.ker (MvPolynomial.aeval (⇑f ∘ MvPolynomial.X))
x : MvPolynomial (Fin n) R
hx : x ∈ ↑s
⊢ (MvPolynomial.aeval (⇑f ∘ MvPolynomial.X)) ((MvPolynomial.map (algebraMap R A)) x) = 0