l3lab/ntp-mathlib huggingface dataset:
theorem of_restrict_scalars_finitePresentation [Algebra A B] [IsScalarTower R A B]
    (hRB : FinitePresentation.{w₁, w₃} R B) [hRA : FiniteType R A] :
    FinitePresentation.{w₂, w₃} A B := by
  classical
  obtain ⟨n, f, hf, s, hs⟩ := hRB
  letI RX := MvPolynomial (Fin n) R
  letI AX := MvPolynomial (Fin n) A
  refine' ⟨n, MvPolynomial.aeval (f ∘ X), _, _⟩
  · rw [← Algebra.range_top_iff_surjective, ← Algebra.adjoin_range_eq_range_aeval,
      Set.range_comp f MvPolynomial.X, eq_top_iff, ← @adjoin_adjoin_of_tower R A B,
      adjoin_image, adjoin_range_X, Algebra.map_top, (Algebra.range_top_iff_surjective _).mpr hf]
    exact fun {x} => subset_adjoin ⟨⟩
  · obtain ⟨t, ht⟩ := hRA.out
    have := fun i : t => hf (algebraMap A B i)
    choose t' ht' using this
    have ht'' : Algebra.adjoin R (algebraMap A AX '' t ∪ Set.range (X : _ → AX)) = ⊤ := by
      rw [adjoin_union_eq_adjoin_adjoin, ← Subalgebra.restrictScalars_top R (A := AX)
        (S := { x // x ∈ adjoin R ((algebraMap A AX) '' t) })]
      refine congrArg (Subalgebra.restrictScalars R) ?_
      congr 1
      rw [adjoin_algebraMap, ht]
      apply Subalgebra.restrictScalars_injective R
      rw [← adjoin_restrictScalars, adjoin_range_X, Subalgebra.restrictScalars_top,
        Subalgebra.restrictScalars_top]
    letI g : t → AX := fun x => MvPolynomial.C (x : A) - map (algebraMap R A) (t' x)
    

Proof state:
case intro.intro.intro.intro.refine'_2.intro
R : Type w₁
A : Type w₂
B : Type w₃
inst✝⁶ : CommRing R
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R A
inst✝³ : CommRing B
inst✝² : Algebra R B
inst✝¹ : Algebra A B
inst✝ : IsScalarTower R A B
hRA : 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
RX : Type w₁ := MvPolynomial (Fin n) R
AX : Type w₂ := MvPolynomial (Fin n) A
t : Finset A
ht : adjoin R ↑t = ⊤
t' : { x // x ∈ t } → MvPolynomial (Fin n) R
ht' : ∀ (i : { x // x ∈ t }), f (t' i) = (algebraMap A B) ↑i
ht'' : adjoin R (⇑(algebraMap A AX) '' ↑t ∪ Set.range MvPolynomial.X) = ⊤
g : { x // x ∈ t } → AX := fun x => MvPolynomial.C ↑x - (MvPolynomial.map (algebraMap R A)) (t' x)
⊢ Ideal.FG (RingHom.ker ↑(MvPolynomial.aeval (⇑f ∘ MvPolynomial.X)))

Next tactic:
refine' ⟨s.image (map (algebraMap R A)) ∪ t.attach.image g, _⟩
theorem of_restrict_scalars_finitePresentation [Algebra A B] [IsScalarTower R A B]
    (hRB : FinitePresentation.{w₁, w₃} R B) [hRA : FiniteType R A] :
    FinitePresentation.{w₂, w₃} A B := by
  classical
  obtain ⟨n, f, hf, s, hs⟩ := hRB
  letI RX := MvPolynomial (Fin n) R
  letI AX := MvPolynomial (Fin n) A
  refine' ⟨n, MvPolynomial.aeval (f ∘ X), _, _⟩
  · rw [← Algebra.range_top_iff_surjective, ← Algebra.adjoin_range_eq_range_aeval,
      Set.range_comp f MvPolynomial.X, eq_top_iff, ← @adjoin_adjoin_of_tower R A B,
      adjoin_image, adjoin_range_X, Algebra.map_top, (Algebra.range_top_iff_surjective _).mpr hf]
    exact fun {x} => subset_adjoin ⟨⟩
  · obtain ⟨t, ht⟩ := hRA.out
    have := fun i : t => hf (algebraMap A B i)
    choose t' ht' using this
    have ht'' : Algebra.adjoin R (algebraMap A AX '' t ∪ Set.range (X : _ → AX)) = ⊤ := by
      rw [adjoin_union_eq_adjoin_adjoin, ← Subalgebra.restrictScalars_top R (A := AX)
        (S := { x // x ∈ adjoin R ((algebraMap A AX) '' t) })]
      refine congrArg (Subalgebra.restrictScalars R) ?_
      congr 1
      rw [adjoin_algebraMap, ht]
      apply Subalgebra.restrictScalars_injective R
      rw [← adjoin_restrictScalars, adjoin_range_X, Subalgebra.restrictScalars_top,
        Subalgebra.restrictScalars_top]
    letI g : t → AX := fun x => MvPolynomial.C (x : A) - map (algebraMap R A) (t' x)
    refine' ⟨s.image (map (algebraMap R A)) ∪ t.attach.image g, _⟩
    

Proof state:
case intro.intro.intro.intro.refine'_2.intro
R : Type w₁
A : Type w₂
B : Type w₃
inst✝⁶ : CommRing R
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R A
inst✝³ : CommRing B
inst✝² : Algebra R B
inst✝¹ : Algebra A B
inst✝ : IsScalarTower R A B
hRA : 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
RX : Type w₁ := MvPolynomial (Fin n) R
AX : Type w₂ := MvPolynomial (Fin n) A
t : Finset A
ht : adjoin R ↑t = ⊤
t' : { x // x ∈ t } → MvPolynomial (Fin n) R
ht' : ∀ (i : { x // x ∈ t }), f (t' i) = (algebraMap A B) ↑i
ht'' : adjoin R (⇑(algebraMap A AX) '' ↑t ∪ Set.range MvPolynomial.X) = ⊤
g : { x // x ∈ t } → AX := fun x => MvPolynomial.C ↑x - (MvPolynomial.map (algebraMap R A)) (t' x)
⊢ Ideal.span ↑(Finset.image (⇑(MvPolynomial.map (algebraMap R A))) s ∪ Finset.image g (Finset.attach t)) =
    RingHom.ker ↑(MvPolynomial.aeval (⇑f ∘ MvPolynomial.X))

Next tactic:
rw [Finset.coe_union, Finset.coe_image, Finset.coe_image, Finset.attach_eq_univ,
      Finset.coe_univ, Set.image_univ]
theorem of_restrict_scalars_finitePresentation [Algebra A B] [IsScalarTower R A B]
    (hRB : FinitePresentation.{w₁, w₃} R B) [hRA : FiniteType R A] :
    FinitePresentation.{w₂, w₃} A B := by
  classical
  obtain ⟨n, f, hf, s, hs⟩ := hRB
  letI RX := MvPolynomial (Fin n) R
  letI AX := MvPolynomial (Fin n) A
  refine' ⟨n, MvPolynomial.aeval (f ∘ X), _, _⟩
  · rw [← Algebra.range_top_iff_surjective, ← Algebra.adjoin_range_eq_range_aeval,
      Set.range_comp f MvPolynomial.X, eq_top_iff, ← @adjoin_adjoin_of_tower R A B,
      adjoin_image, adjoin_range_X, Algebra.map_top, (Algebra.range_top_iff_surjective _).mpr hf]
    exact fun {x} => subset_adjoin ⟨⟩
  · obtain ⟨t, ht⟩ := hRA.out
    have := fun i : t => hf (algebraMap A B i)
    choose t' ht' using this
    have ht'' : Algebra.adjoin R (algebraMap A AX '' t ∪ Set.range (X : _ → AX)) = ⊤ := by
      rw [adjoin_union_eq_adjoin_adjoin, ← Subalgebra.restrictScalars_top R (A := AX)
        (S := { x // x ∈ adjoin R ((algebraMap A AX) '' t) })]
      refine congrArg (Subalgebra.restrictScalars R) ?_
      congr 1
      rw [adjoin_algebraMap, ht]
      apply Subalgebra.restrictScalars_injective R
      rw [← adjoin_restrictScalars, adjoin_range_X, Subalgebra.restrictScalars_top,
        Subalgebra.restrictScalars_top]
    letI g : t → AX := fun x => MvPolynomial.C (x : A) - map (algebraMap R A) (t' x)
    refine' ⟨s.image (map (algebraMap R A)) ∪ t.attach.image g, _⟩
    rw [Finset.coe_union, Finset.coe_image, Finset.coe_image, Finset.attach_eq_univ,
      Finset.coe_univ, Set.image_univ]
    

Proof state:
case intro.intro.intro.intro.refine'_2.intro
R : Type w₁
A : Type w₂
B : Type w₃
inst✝⁶ : CommRing R
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R A
inst✝³ : CommRing B
inst✝² : Algebra R B
inst✝¹ : Algebra A B
inst✝ : IsScalarTower R A B
hRA : 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
RX : Type w₁ := MvPolynomial (Fin n) R
AX : Type w₂ := MvPolynomial (Fin n) A
t : Finset A
ht : adjoin R ↑t = ⊤
t' : { x // x ∈ t } → MvPolynomial (Fin n) R
ht' : ∀ (i : { x // x ∈ t }), f (t' i) = (algebraMap A B) ↑i
ht'' : adjoin R (⇑(algebraMap A AX) '' ↑t ∪ Set.range MvPolynomial.X) = ⊤
g : { x // x ∈ 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))

Next tactic:
let s₀ := (MvPolynomial.map (algebraMap R A)) '' s ∪ Set.range g
theorem of_restrict_scalars_finitePresentation [Algebra A B] [IsScalarTower R A B]
    (hRB : FinitePresentation.{w₁, w₃} R B) [hRA : FiniteType R A] :
    FinitePresentation.{w₂, w₃} A B := by
  classical
  obtain ⟨n, f, hf, s, hs⟩ := hRB
  letI RX := MvPolynomial (Fin n) R
  letI AX := MvPolynomial (Fin n) A
  refine' ⟨n, MvPolynomial.aeval (f ∘ X), _, _⟩
  · rw [← Algebra.range_top_iff_surjective, ← Algebra.adjoin_range_eq_range_aeval,
      Set.range_comp f MvPolynomial.X, eq_top_iff, ← @adjoin_adjoin_of_tower R A B,
      adjoin_image, adjoin_range_X, Algebra.map_top, (Algebra.range_top_iff_surjective _).mpr hf]
    exact fun {x} => subset_adjoin ⟨⟩
  · obtain ⟨t, ht⟩ := hRA.out
    have := fun i : t => hf (algebraMap A B i)
    choose t' ht' using this
    have ht'' : Algebra.adjoin R (algebraMap A AX '' t ∪ Set.range (X : _ → AX)) = ⊤ := by
      rw [adjoin_union_eq_adjoin_adjoin, ← Subalgebra.restrictScalars_top R (A := AX)
        (S := { x // x ∈ adjoin R ((algebraMap A AX) '' t) })]
      refine congrArg (Subalgebra.restrictScalars R) ?_
      congr 1
      rw [adjoin_algebraMap, ht]
      apply Subalgebra.restrictScalars_injective R
      rw [← adjoin_restrictScalars, adjoin_range_X, Subalgebra.restrictScalars_top,
        Subalgebra.restrictScalars_top]
    letI g : t → AX := fun x => MvPolynomial.C (x : A) - map (algebraMap R A) (t' x)
    refine' ⟨s.image (map (algebraMap R A)) ∪ t.attach.image g, _⟩
    rw [Finset.coe_union, Finset.coe_image, Finset.coe_image, Finset.attach_eq_univ,
      Finset.coe_univ, Set.image_univ]
    let s₀ := (MvPolynomial.map (algebraMap R A)) '' s ∪ Set.range g
    

Proof state:
case intro.intro.intro.intro.refine'_2.intro
R : Type w₁
A : Type w₂
B : Type w₃
inst✝⁶ : CommRing R
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R A
inst✝³ : CommRing B
inst✝² : Algebra R B
inst✝¹ : Algebra A B
inst✝ : IsScalarTower R A B
hRA : 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
RX : Type w₁ := MvPolynomial (Fin n) R
AX : Type w₂ := MvPolynomial (Fin n) A
t : Finset A
ht : adjoin R ↑t = ⊤
t' : { x // x ∈ t } → MvPolynomial (Fin n) R
ht' : ∀ (i : { x // x ∈ t }), f (t' i) = (algebraMap A B) ↑i
ht'' : adjoin R (⇑(algebraMap A AX) '' ↑t ∪ Set.range MvPolynomial.X) = ⊤
g : { x // x ∈ 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
⊢ Ideal.span (⇑(MvPolynomial.map (algebraMap R A)) '' ↑s ∪ Set.range g) =
    RingHom.ker ↑(MvPolynomial.aeval (⇑f ∘ MvPolynomial.X))

Next tactic:
let I := RingHom.ker (MvPolynomial.aeval (R := A) (f ∘ MvPolynomial.X))
theorem of_restrict_scalars_finitePresentation [Algebra A B] [IsScalarTower R A B]
    (hRB : FinitePresentation.{w₁, w₃} R B) [hRA : FiniteType R A] :
    FinitePresentation.{w₂, w₃} A B := by
  classical
  obtain ⟨n, f, hf, s, hs⟩ := hRB
  letI RX := MvPolynomial (Fin n) R
  letI AX := MvPolynomial (Fin n) A
  refine' ⟨n, MvPolynomial.aeval (f ∘ X), _, _⟩
  · rw [← Algebra.range_top_iff_surjective, ← Algebra.adjoin_range_eq_range_aeval,
      Set.range_comp f MvPolynomial.X, eq_top_iff, ← @adjoin_adjoin_of_tower R A B,
      adjoin_image, adjoin_range_X, Algebra.map_top, (Algebra.range_top_iff_surjective _).mpr hf]
    exact fun {x} => subset_adjoin ⟨⟩
  · obtain ⟨t, ht⟩ := hRA.out
    have := fun i : t => hf (algebraMap A B i)
    choose t' ht' using this
    have ht'' : Algebra.adjoin R (algebraMap A AX '' t ∪ Set.range (X : _ → AX)) = ⊤ := by
      rw [adjoin_union_eq_adjoin_adjoin, ← Subalgebra.restrictScalars_top R (A := AX)
        (S := { x // x ∈ adjoin R ((algebraMap A AX) '' t) })]
      refine congrArg (Subalgebra.restrictScalars R) ?_
      congr 1
      rw [adjoin_algebraMap, ht]
      apply Subalgebra.restrictScalars_injective R
      rw [← adjoin_restrictScalars, adjoin_range_X, Subalgebra.restrictScalars_top,
        Subalgebra.restrictScalars_top]
    letI g : t → AX := fun x => MvPolynomial.C (x : A) - map (algebraMap R A) (t' x)
    refine' ⟨s.image (map (algebraMap R A)) ∪ t.attach.image g, _⟩
    rw [Finset.coe_union, Finset.coe_image, Finset.coe_image, Finset.attach_eq_univ,
      Finset.coe_univ, Set.image_univ]
    let s₀ := (MvPolynomial.map (algebraMap R A)) '' s ∪ Set.range g
    let I := RingHom.ker (MvPolynomial.aeval (R := A) (f ∘ MvPolynomial.X))
    

Proof state:
case intro.intro.intro.intro.refine'_2.intro
R : Type w₁
A : Type w₂
B : Type w₃
inst✝⁶ : CommRing R
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R A
inst✝³ : CommRing B
inst✝² : Algebra R B
inst✝¹ : Algebra A B
inst✝ : IsScalarTower R A B
hRA : 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
RX : Type w₁ := MvPolynomial (Fin n) R
AX : Type w₂ := MvPolynomial (Fin n) A
t : Finset A
ht : adjoin R ↑t = ⊤
t' : { x // x ∈ t } → MvPolynomial (Fin n) R
ht' : ∀ (i : { x // x ∈ t }), f (t' i) = (algebraMap A B) ↑i
ht'' : adjoin R (⇑(algebraMap A AX) '' ↑t ∪ Set.range MvPolynomial.X) = ⊤
g : { x // x ∈ 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))
⊢ Ideal.span (⇑(MvPolynomial.map (algebraMap R A)) '' ↑s ∪ Set.range g) =
    RingHom.ker ↑(MvPolynomial.aeval (⇑f ∘ MvPolynomial.X))

Next tactic:
change Ideal.span s₀ = I
theorem of_restrict_scalars_finitePresentation [Algebra A B] [IsScalarTower R A B]
    (hRB : FinitePresentation.{w₁, w₃} R B) [hRA : FiniteType R A] :
    FinitePresentation.{w₂, w₃} A B := by
  classical
  obtain ⟨n, f, hf, s, hs⟩ := hRB
  letI RX := MvPolynomial (Fin n) R
  letI AX := MvPolynomial (Fin n) A
  refine' ⟨n, MvPolynomial.aeval (f ∘ X), _, _⟩
  · rw [← Algebra.range_top_iff_surjective, ← Algebra.adjoin_range_eq_range_aeval,
      Set.range_comp f MvPolynomial.X, eq_top_iff, ← @adjoin_adjoin_of_tower R A B,
      adjoin_image, adjoin_range_X, Algebra.map_top, (Algebra.range_top_iff_surjective _).mpr hf]
    exact fun {x} => subset_adjoin ⟨⟩
  · obtain ⟨t, ht⟩ := hRA.out
    have := fun i : t => hf (algebraMap A B i)
    choose t' ht' using this
    have ht'' : Algebra.adjoin R (algebraMap A AX '' t ∪ Set.range (X : _ → AX)) = ⊤ := by
      rw [adjoin_union_eq_adjoin_adjoin, ← Subalgebra.restrictScalars_top R (A := AX)
        (S := { x // x ∈ adjoin R ((algebraMap A AX) '' t) })]
      refine congrArg (Subalgebra.restrictScalars R) ?_
      congr 1
      rw [adjoin_algebraMap, ht]
      apply Subalgebra.restrictScalars_injective R
      rw [← adjoin_restrictScalars, adjoin_range_X, Subalgebra.restrictScalars_top,
        Subalgebra.restrictScalars_top]
    letI g : t → AX := fun x => MvPolynomial.C (x : A) - map (algebraMap R A) (t' x)
    refine' ⟨s.image (map (algebraMap R A)) ∪ t.attach.image g, _⟩
    rw [Finset.coe_union, Finset.coe_image, Finset.coe_image, Finset.attach_eq_univ,
      Finset.coe_univ, Set.image_univ]
    let s₀ := (MvPolynomial.map (algebraMap R A)) '' s ∪ Set.range g
    let I := RingHom.ker (MvPolynomial.aeval (R := A) (f ∘ MvPolynomial.X))
    change Ideal.span s₀ = I
    have leI : Ideal.span ((MvPolynomial.map (algebraMap R A)) '' s ∪ Set.range g) ≤
      RingHom.ker (MvPolynomial.aeval (R := A) (f ∘ MvPolynomial.X)) := by
      

Proof state:
R : Type w₁
A : Type w₂
B : Type w₃
inst✝⁶ : CommRing R
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R A
inst✝³ : CommRing B
inst✝² : Algebra R B
inst✝¹ : Algebra A B
inst✝ : IsScalarTower R A B
hRA : 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
RX : Type w₁ := MvPolynomial (Fin n) R
AX : Type w₂ := MvPolynomial (Fin n) A
t : Finset A
ht : adjoin R ↑t = ⊤
t' : { x // x ∈ t } → MvPolynomial (Fin n) R
ht' : ∀ (i : { x // x ∈ t }), f (t' i) = (algebraMap A B) ↑i
ht'' : adjoin R (⇑(algebraMap A AX) '' ↑t ∪ Set.range MvPolynomial.X) = ⊤
g : { x // x ∈ 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))
⊢ Ideal.span (⇑(MvPolynomial.map (algebraMap R A)) '' ↑s ∪ Set.range g) ≤
    RingHom.ker (MvPolynomial.aeval (⇑f ∘ MvPolynomial.X))

Next tactic:
rw [Ideal.span_le]
theorem of_restrict_scalars_finitePresentation [Algebra A B] [IsScalarTower R A B]
    (hRB : FinitePresentation.{w₁, w₃} R B) [hRA : FiniteType R A] :
    FinitePresentation.{w₂, w₃} A B := by
  classical
  obtain ⟨n, f, hf, s, hs⟩ := hRB
  letI RX := MvPolynomial (Fin n) R
  letI AX := MvPolynomial (Fin n) A
  refine' ⟨n, MvPolynomial.aeval (f ∘ X), _, _⟩
  · rw [← Algebra.range_top_iff_surjective, ← Algebra.adjoin_range_eq_range_aeval,
      Set.range_comp f MvPolynomial.X, eq_top_iff, ← @adjoin_adjoin_of_tower R A B,
      adjoin_image, adjoin_range_X, Algebra.map_top, (Algebra.range_top_iff_surjective _).mpr hf]
    exact fun {x} => subset_adjoin ⟨⟩
  · obtain ⟨t, ht⟩ := hRA.out
    have := fun i : t => hf (algebraMap A B i)
    choose t' ht' using this
    have ht'' : Algebra.adjoin R (algebraMap A AX '' t ∪ Set.range (X : _ → AX)) = ⊤ := by
      rw [adjoin_union_eq_adjoin_adjoin, ← Subalgebra.restrictScalars_top R (A := AX)
        (S := { x // x ∈ adjoin R ((algebraMap A AX) '' t) })]
      refine congrArg (Subalgebra.restrictScalars R) ?_
      congr 1
      rw [adjoin_algebraMap, ht]
      apply Subalgebra.restrictScalars_injective R
      rw [← adjoin_restrictScalars, adjoin_range_X, Subalgebra.restrictScalars_top,
        Subalgebra.restrictScalars_top]
    letI g : t → AX := fun x => MvPolynomial.C (x : A) - map (algebraMap R A) (t' x)
    refine' ⟨s.image (map (algebraMap R A)) ∪ t.attach.image g, _⟩
    rw [Finset.coe_union, Finset.coe_image, Finset.coe_image, Finset.attach_eq_univ,
      Finset.coe_univ, Set.image_univ]
    let s₀ := (MvPolynomial.map (algebraMap R A)) '' s ∪ Set.range g
    let I := RingHom.ker (MvPolynomial.aeval (R := A) (f ∘ MvPolynomial.X))
    change Ideal.span s₀ = I
    have leI : Ideal.span ((MvPolynomial.map (algebraMap R A)) '' s ∪ Set.range g) ≤
      RingHom.ker (MvPolynomial.aeval (R := A) (f ∘ MvPolynomial.X)) := by
      rw [Ideal.span_le]
      

Proof state:
R : Type w₁
A : Type w₂
B : Type w₃
inst✝⁶ : CommRing R
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R A
inst✝³ : CommRing B
inst✝² : Algebra R B
inst✝¹ : Algebra A B
inst✝ : IsScalarTower R A B
hRA : 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
RX : Type w₁ := MvPolynomial (Fin n) R
AX : Type w₂ := MvPolynomial (Fin n) A
t : Finset A
ht : adjoin R ↑t = ⊤
t' : { x // x ∈ t } → MvPolynomial (Fin n) R
ht' : ∀ (i : { x // x ∈ t }), f (t' i) = (algebraMap A B) ↑i
ht'' : adjoin R (⇑(algebraMap A AX) '' ↑t ∪ Set.range MvPolynomial.X) = ⊤
g : { x // x ∈ 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)))

Next tactic:
rintro _ (⟨x, hx, rfl⟩ | ⟨⟨x, hx⟩, rfl⟩)
theorem of_restrict_scalars_finitePresentation [Algebra A B] [IsScalarTower R A B]
    (hRB : FinitePresentation.{w₁, w₃} R B) [hRA : FiniteType R A] :
    FinitePresentation.{w₂, w₃} A B := by
  classical
  obtain ⟨n, f, hf, s, hs⟩ := hRB
  letI RX := MvPolynomial (Fin n) R
  letI AX := MvPolynomial (Fin n) A
  refine' ⟨n, MvPolynomial.aeval (f ∘ X), _, _⟩
  · rw [← Algebra.range_top_iff_surjective, ← Algebra.adjoin_range_eq_range_aeval,
      Set.range_comp f MvPolynomial.X, eq_top_iff, ← @adjoin_adjoin_of_tower R A B,
      adjoin_image, adjoin_range_X, Algebra.map_top, (Algebra.range_top_iff_surjective _).mpr hf]
    exact fun {x} => subset_adjoin ⟨⟩
  · obtain ⟨t, ht⟩ := hRA.out
    have := fun i : t => hf (algebraMap A B i)
    choose t' ht' using this
    have ht'' : Algebra.adjoin R (algebraMap A AX '' t ∪ Set.range (X : _ → AX)) = ⊤ := by
      rw [adjoin_union_eq_adjoin_adjoin, ← Subalgebra.restrictScalars_top R (A := AX)
        (S := { x // x ∈ adjoin R ((algebraMap A AX) '' t) })]
      refine congrArg (Subalgebra.restrictScalars R) ?_
      congr 1
      rw [adjoin_algebraMap, ht]
      apply Subalgebra.restrictScalars_injective R
      rw [← adjoin_restrictScalars, adjoin_range_X, Subalgebra.restrictScalars_top,
        Subalgebra.restrictScalars_top]
    letI g : t → AX := fun x => MvPolynomial.C (x : A) - map (algebraMap R A) (t' x)
    refine' ⟨s.image (map (algebraMap R A)) ∪ t.attach.image g, _⟩
    rw [Finset.coe_union, Finset.coe_image, Finset.coe_image, Finset.attach_eq_univ,
      Finset.coe_univ, Set.image_univ]
    let s₀ := (MvPolynomial.map (algebraMap R A)) '' s ∪ Set.range g
    let I := RingHom.ker (MvPolynomial.aeval (R := A) (f ∘ MvPolynomial.X))
    change Ideal.span s₀ = I
    have leI : Ideal.span ((MvPolynomial.map (algebraMap R A)) '' s ∪ Set.range g) ≤
      RingHom.ker (MvPolynomial.aeval (R := A) (f ∘ MvPolynomial.X)) := by
      rw [Ideal.span_le]
      rintro _ (⟨x, hx, rfl⟩ | ⟨⟨x, hx⟩, rfl⟩) <;>
      

Proof state:
case inl.intro.intro
R : Type w₁
A : Type w₂
B : Type w₃
inst✝⁶ : CommRing R
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R A
inst✝³ : CommRing B
inst✝² : Algebra R B
inst✝¹ : Algebra A B
inst✝ : IsScalarTower R A B
hRA : 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
RX : Type w₁ := MvPolynomial (Fin n) R
AX : Type w₂ := MvPolynomial (Fin n) A
t : Finset A
ht : adjoin R ↑t = ⊤
t' : { x // x ∈ t } → MvPolynomial (Fin n) R
ht' : ∀ (i : { x // x ∈ t }), f (t' i) = (algebraMap A B) ↑i
ht'' : adjoin R (⇑(algebraMap A AX) '' ↑t ∪ Set.range MvPolynomial.X) = ⊤
g : { x // x ∈ 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.map (algebraMap R A)) x ∈ ↑(RingHom.ker (MvPolynomial.aeval (⇑f ∘ MvPolynomial.X)))

Next tactic:
erw [RingHom.mem_ker]