-- Mathlib/RingTheory/RingHom/FinitePresentation.lean
theorem finitePresentation_ofLocalizationSpanTarget :
    OfLocalizationSpanTarget @FinitePresentation

Proof state:
case intro.intro.intro
R S : Type u_1
inst✝¹ : CommRing R
inst✝ : CommRing S
f✝ : R →+* S
s : Finset S
hs : Ideal.span ↑s = ⊤
this : Algebra R S := f✝.toAlgebra
H : ∀ (r : { x // x ∈ s }), Algebra.FinitePresentation R (Localization.Away ↑r)
hfintype : Algebra.FiniteType R S
n : ℕ
f : MvPolynomial (Fin n) R →ₐ[R] S
hf : Function.Surjective ⇑f
l : ↑↑s →₀ S
hl : (Finsupp.linearCombination S Subtype.val) l = 1
g' : { x // x ∈ s } → MvPolynomial (Fin n) R
hg' : ∀ (g : { x // x ∈ s }), f (g' g) = ↑g
h' : { x // x ∈ s } → MvPolynomial (Fin n) R
hh' : ∀ (g : { x // x ∈ s }), f (h' g) = l g
I : Ideal (MvPolynomial (Fin n) R) := Ideal.span {∑ g : { x // x ∈ s }, g' g * h' g - 1}
A : Type u_1 := MvPolynomial (Fin n) R ⧸ I
hfI : ∀ a ∈ I, f a = 0
f' : A →ₐ[R] S := Ideal.Quotient.liftₐ I f hfI
hf' : Function.Surjective ⇑f'
t : Finset A := Finset.image (fun g => (Ideal.Quotient.mk I) (g' g)) Finset.univ
⊢ Algebra.FinitePresentation R S

Tactic:
have ht : Ideal.span (t : Set A) = ⊤ := by
  rw [Ideal.eq_top_iff_one]
  have : ∑ g : { x // x ∈ s }, g' g * h' g = (1 : A) := by
    apply eq_of_sub_eq_zero
    rw [← map_one (Ideal.Quotient.mk I), ← map_sub, Ideal.Quotient.eq_zero_iff_mem]
    apply Ideal.subset_span
    simp
  simp_rw [← this, Finset.univ_eq_attach, map_sum, map_mul]
  refine Ideal.sum_mem _ (fun g _ ↦ Ideal.mul_mem_right _ _ <| Ideal.subset_span ?_)
  simp [t]

Result:
case intro.intro.intro
R S : Type u_1
inst✝¹ : CommRing R
inst✝ : CommRing S
f✝ : R →+* S
s : Finset S
hs : Ideal.span ↑s = ⊤
this : Algebra R S := f✝.toAlgebra
H : ∀ (r : { x // x ∈ s }), Algebra.FinitePresentation R (Localization.Away ↑r)
hfintype : Algebra.FiniteType R S
n : ℕ
f : MvPolynomial (Fin n) R →ₐ[R] S
hf : Function.Surjective ⇑f
l : ↑↑s →₀ S
hl : (Finsupp.linearCombination S Subtype.val) l = 1
g' : { x // x ∈ s } → MvPolynomial (Fin n) R
hg' : ∀ (g : { x // x ∈ s }), f (g' g) = ↑g
h' : { x // x ∈ s } → MvPolynomial (Fin n) R
hh' : ∀ (g : { x // x ∈ s }), f (h' g) = l g
I : Ideal (MvPolynomial (Fin n) R) := Ideal.span {∑ g : { x // x ∈ s }, g' g * h' g - 1}
A : Type u_1 := MvPolynomial (Fin n) R ⧸ I
hfI : ∀ a ∈ I, f a = 0
f' : A →ₐ[R] S := Ideal.Quotient.liftₐ I f hfI
hf' : Function.Surjective ⇑f'
t : Finset A := Finset.image (fun g => (Ideal.Quotient.mk I) (g' g)) Finset.univ
ht : Ideal.span ↑t = ⊤
⊢ Algebra.FinitePresentation R S

Proof state:
case intro
R S : Type u_1
inst✝¹ : CommRing R
inst✝ : CommRing S
f✝ : R →+* S
s : Finset S
hs : Ideal.span ↑s = ⊤
this : Algebra R S := f✝.toAlgebra
H : ∀ (r : { x // x ∈ s }), Algebra.FinitePresentation R (Localization.Away ↑r)
hfintype : Algebra.FiniteType R S
n : ℕ
f : MvPolynomial (Fin n) R →ₐ[R] S
hf : Function.Surjective ⇑f
l : ↑↑s →₀ S
hl : (Finsupp.linearCombination S Subtype.val) l = 1
g' : { x // x ∈ s } → MvPolynomial (Fin n) R
hg' : ∀ (g : { x // x ∈ s }), f (g' g) = ↑g
h' : { x // x ∈ s } → MvPolynomial (Fin n) R
hh' : ∀ (g : { x // x ∈ s }), f (h' g) = l g
I : Ideal (MvPolynomial (Fin n) R) := Ideal.span {∑ g : { x // x ∈ s }, g' g * h' g - 1}
A : Type u_1 := MvPolynomial (Fin n) R ⧸ I
q : MvPolynomial (Fin n) R
⊢ (∑ x ∈ s.attach, ↑x * l x - 1) * f q = 0

Tactic:
erw [Finsupp.linearCombination_apply_of_mem_supported S (s := s.attach)] at hl

Result:
case intro
R S : Type u_1
inst✝¹ : CommRing R
inst✝ : CommRing S
f✝ : R →+* S
s : Finset S
hs : Ideal.span ↑s = ⊤
this : Algebra R S := f✝.toAlgebra
H : ∀ (r : { x // x ∈ s }), Algebra.FinitePresentation R (Localization.Away ↑r)
hfintype : Algebra.FiniteType R S
n : ℕ
f : MvPolynomial (Fin n) R →ₐ[R] S
hf : Function.Surjective ⇑f
l : ↑↑s →₀ S
hl : ∑ i ∈ s.attach, l i • ↑i = 1
g' : { x // x ∈ s } → MvPolynomial (Fin n) R
hg' : ∀ (g : { x // x ∈ s }), f (g' g) = ↑g
h' : { x // x ∈ s } → MvPolynomial (Fin n) R
hh' : ∀ (g : { x // x ∈ s }), f (h' g) = l g
I : Ideal (MvPolynomial (Fin n) R) := Ideal.span {∑ g : { x // x ∈ s }, g' g * h' g - 1}
A : Type u_1 := MvPolynomial (Fin n) R ⧸ I
q : MvPolynomial (Fin n) R
⊢ (∑ x ∈ s.attach, ↑x * l x - 1) * f q = 0

case intro
R S : Type u_1
inst✝¹ : CommRing R
inst✝ : CommRing S
f✝ : R →+* S
s : Finset S
hs : Ideal.span ↑s = ⊤
this : Algebra R S := f✝.toAlgebra
H : ∀ (r : { x // x ∈ s }), Algebra.FinitePresentation R (Localization.Away ↑r)
hfintype : Algebra.FiniteType R S
n : ℕ
f : MvPolynomial (Fin n) R →ₐ[R] S
hf : Function.Surjective ⇑f
l : ↑↑s →₀ S
hl : (Finsupp.linearCombination S Subtype.val) l = 1
g' : { x // x ∈ s } → MvPolynomial (Fin n) R
hg' : ∀ (g : { x // x ∈ s }), f (g' g) = ↑g
h' : { x // x ∈ s } → MvPolynomial (Fin n) R
hh' : ∀ (g : { x // x ∈ s }), f (h' g) = l g
I : Ideal (MvPolynomial (Fin n) R) := Ideal.span {∑ g : { x // x ∈ s }, g' g * h' g - 1}
A : Type u_1 := MvPolynomial (Fin n) R ⧸ I
q : MvPolynomial (Fin n) R
⊢ l ∈ Finsupp.supported S S ↑s.attach

Proof state:
R S : Type u_1
inst✝¹ : CommRing R
inst✝ : CommRing S
f✝ : R →+* S
s : Finset S
hs : Ideal.span ↑s = ⊤
this : Algebra R S := f✝.toAlgebra
H : ∀ (r : { x // x ∈ s }), Algebra.FinitePresentation R (Localization.Away ↑r)
hfintype : Algebra.FiniteType R S
n : ℕ
f : MvPolynomial (Fin n) R →ₐ[R] S
hf : Function.Surjective ⇑f
l : ↑↑s →₀ S
hl : (Finsupp.linearCombination S Subtype.val) l = 1
g' : { x // x ∈ s } → MvPolynomial (Fin n) R
hg' : ∀ (g : { x // x ∈ s }), f (g' g) = ↑g
h' : { x // x ∈ s } → MvPolynomial (Fin n) R
hh' : ∀ (g : { x // x ∈ s }), f (h' g) = l g
I : Ideal (MvPolynomial (Fin n) R) := Ideal.span {∑ g : { x // x ∈ s }, g' g * h' g - 1}
A : Type u_1 := MvPolynomial (Fin n) R ⧸ I
p : MvPolynomial (Fin n) R
hp : ∑ g ∈ s.attach, g' g * h' g - 1 ∣ p
⊢ f p = 0

Tactic:
obtain ⟨q, rfl⟩ := hp

Result:
case intro
R S : Type u_1
inst✝¹ : CommRing R
inst✝ : CommRing S
f✝ : R →+* S
s : Finset S
hs : Ideal.span ↑s = ⊤
this : Algebra R S := f✝.toAlgebra
H : ∀ (r : { x // x ∈ s }), Algebra.FinitePresentation R (Localization.Away ↑r)
hfintype : Algebra.FiniteType R S
n : ℕ
f : MvPolynomial (Fin n) R →ₐ[R] S
hf : Function.Surjective ⇑f
l : ↑↑s →₀ S
hl : (Finsupp.linearCombination S Subtype.val) l = 1
g' : { x // x ∈ s } → MvPolynomial (Fin n) R
hg' : ∀ (g : { x // x ∈ s }), f (g' g) = ↑g
h' : { x // x ∈ s } → MvPolynomial (Fin n) R
hh' : ∀ (g : { x // x ∈ s }), f (h' g) = l g
I : Ideal (MvPolynomial (Fin n) R) := Ideal.span {∑ g : { x // x ∈ s }, g' g * h' g - 1}
A : Type u_1 := MvPolynomial (Fin n) R ⧸ I
q : MvPolynomial (Fin n) R
⊢ f ((∑ g ∈ s.attach, g' g * h' g - 1) * q) = 0