Theorem: RingHom.finiteType_ofLocalizationSpanTarget
File: Mathlib/RingTheory/RingHom/FiniteType.lean

tactic: rw [ofLocalizationSpanTarget_iff_finite]
state_before: ⊢ OfLocalizationSpanTarget @FiniteType
state_after: ⊢ OfLocalizationFiniteSpanTarget @FiniteType

tactic: introv R hs H
state_before: ⊢ OfLocalizationFiniteSpanTarget @FiniteType
state_after: R S : Type u_1
inst✝¹ : CommRing R
inst✝ : CommRing S
f : R →+* S
s : Finset S
hs : Ideal.span ↑s = ⊤
H : ∀ (r : { x // x ∈ s }), ((algebraMap S (Localization.Away ↑r)).comp f).FiniteType
⊢ f.FiniteType

tactic: letI := f.toAlgebra
state_before: R S : Type u_1
inst✝¹ : CommRing R
inst✝ : CommRing S
f : R →+* S
s : Finset S
hs : Ideal.span ↑s = ⊤
H : ∀ (r : { x // x ∈ s }), ((algebraMap S (Localization.Away ↑r)).comp f).FiniteType
⊢ f.FiniteType
state_after: R S : Type u_1
inst✝¹ : CommRing R
inst✝ : CommRing S
f : R →+* S
s : Finset S
hs : Ideal.span ↑s = ⊤
H : ∀ (r : { x // x ∈ s }), ((algebraMap S (Localization.Away ↑r)).comp f).FiniteType
this : Algebra R S := f.toAlgebra
⊢ f.FiniteType

tactic: replace H : ∀ r : s, Algebra.FiniteType R (Localization.Away (r : S)) := by
  intro r; simp_rw [RingHom.FiniteType] at H; convert H r; ext; simp_rw [Algebra.smul_def]; rfl
state_before: R S : Type u_1
inst✝¹ : CommRing R
inst✝ : CommRing S
f : R →+* S
s : Finset S
hs : Ideal.span ↑s = ⊤
H : ∀ (r : { x // x ∈ s }), ((algebraMap S (Localization.Away ↑r)).comp f).FiniteType
this : Algebra R S := f.toAlgebra
⊢ f.FiniteType
state_after: 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.FiniteType R (Localization.Away ↑r)
⊢ f.FiniteType

tactic: replace H := fun r => (H r).1
state_before: 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.FiniteType R (Localization.Away ↑r)
⊢ f.FiniteType
state_after: 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 }), ⊤.FG
⊢ f.FiniteType

tactic: constructor
state_before: 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 }), ⊤.FG
⊢ f.FiniteType
state_after: case out
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 }), ⊤.FG
⊢ ⊤.FG

tactic: choose t ht using H
state_before: case out
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 }), ⊤.FG
⊢ ⊤.FG
state_after: case out
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
t : (r : { x // x ∈ s }) → Finset (Localization.Away ↑r)
ht : ∀ (r : { x // x ∈ s }), Algebra.adjoin R ↑(t r) = ⊤
⊢ ⊤.FG

tactic: obtain ⟨l, hl⟩ :=
  (Finsupp.mem_span_iff_total S (s : Set S) 1).mp
    (show (1 : S) ∈ Ideal.span (s : Set S) by rw [hs]; trivial)
state_before: case out
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
t : (r : { x // x ∈ s }) → Finset (Localization.Away ↑r)
ht : ∀ (r : { x // x ∈ s }), Algebra.adjoin R ↑(t r) = ⊤
⊢ ⊤.FG
state_after: case out.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
t : (r : { x // x ∈ s }) → Finset (Localization.Away ↑r)
ht : ∀ (r : { x // x ∈ s }), Algebra.adjoin R ↑(t r) = ⊤
l : ↑↑s →₀ S
hl : (Finsupp.total (↑↑s) S S Subtype.val) l = 1
⊢ ⊤.FG

tactic: let sf := fun x : s => IsLocalization.finsetIntegerMultiple (Submonoid.powers (x : S)) (t x)
state_before: case out.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
t : (r : { x // x ∈ s }) → Finset (Localization.Away ↑r)
ht : ∀ (r : { x // x ∈ s }), Algebra.adjoin R ↑(t r) = ⊤
l : ↑↑s →₀ S
hl : (Finsupp.total (↑↑s) S S Subtype.val) l = 1
⊢ ⊤.FG
state_after: case out.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
t : (r : { x // x ∈ s }) → Finset (Localization.Away ↑r)
ht : ∀ (r : { x // x ∈ s }), Algebra.adjoin R ↑(t r) = ⊤
l : ↑↑s →₀ S
hl : (Finsupp.total (↑↑s) S S Subtype.val) l = 1
sf : { x // x ∈ s } → Finset S := fun x => IsLocalization.finsetIntegerMultiple (Submonoid.powers ↑x) (t x)
⊢ ⊤.FG

tactic: use s.attach.biUnion sf ∪ s ∪ l.support.image l
state_before: case out.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
t : (r : { x // x ∈ s }) → Finset (Localization.Away ↑r)
ht : ∀ (r : { x // x ∈ s }), Algebra.adjoin R ↑(t r) = ⊤
l : ↑↑s →₀ S
hl : (Finsupp.total (↑↑s) S S Subtype.val) l = 1
sf : { x // x ∈ s } → Finset S := fun x => IsLocalization.finsetIntegerMultiple (Submonoid.powers ↑x) (t x)
⊢ ⊤.FG
state_after: case h
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
t : (r : { x // x ∈ s }) → Finset (Localization.Away ↑r)
ht : ∀ (r : { x // x ∈ s }), Algebra.adjoin R ↑(t r) = ⊤
l : ↑↑s →₀ S
hl : (Finsupp.total (↑↑s) S S Subtype.val) l = 1
sf : { x // x ∈ s } → Finset S := fun x => IsLocalization.finsetIntegerMultiple (Submonoid.powers ↑x) (t x)
⊢ Algebra.adjoin R ↑(s.attach.biUnion sf ∪ s ∪ Finset.image (⇑l) l.support) = ⊤

tactic: rw [eq_top_iff]
state_before: case h
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
t : (r : { x // x ∈ s }) → Finset (Localization.Away ↑r)
ht : ∀ (r : { x // x ∈ s }), Algebra.adjoin R ↑(t r) = ⊤
l : ↑↑s →₀ S
hl : (Finsupp.total (↑↑s) S S Subtype.val) l = 1
sf : { x // x ∈ s } → Finset S := fun x => IsLocalization.finsetIntegerMultiple (Submonoid.powers ↑x) (t x)
⊢ Algebra.adjoin R ↑(s.attach.biUnion sf ∪ s ∪ Finset.image (⇑l) l.support) = ⊤
state_after: case h
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
t : (r : { x // x ∈ s }) → Finset (Localization.Away ↑r)
ht : ∀ (r : { x // x ∈ s }), Algebra.adjoin R ↑(t r) = ⊤
l : ↑↑s →₀ S
hl : (Finsupp.total (↑↑s) S S Subtype.val) l = 1
sf : { x // x ∈ s } → Finset S := fun x => IsLocalization.finsetIntegerMultiple (Submonoid.powers ↑x) (t x)
⊢ ⊤ ≤ Algebra.adjoin R ↑(s.attach.biUnion sf ∪ s ∪ Finset.image (⇑l) l.support)

tactic: rintro x -
state_before: case h
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
t : (r : { x // x ∈ s }) → Finset (Localization.Away ↑r)
ht : ∀ (r : { x // x ∈ s }), Algebra.adjoin R ↑(t r) = ⊤
l : ↑↑s →₀ S
hl : (Finsupp.total (↑↑s) S S Subtype.val) l = 1
sf : { x // x ∈ s } → Finset S := fun x => IsLocalization.finsetIntegerMultiple (Submonoid.powers ↑x) (t x)
⊢ ⊤ ≤ Algebra.adjoin R ↑(s.attach.biUnion sf ∪ s ∪ Finset.image (⇑l) l.support)
state_after: case h
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
t : (r : { x // x ∈ s }) → Finset (Localization.Away ↑r)
ht : ∀ (r : { x // x ∈ s }), Algebra.adjoin R ↑(t r) = ⊤
l : ↑↑s →₀ S
hl : (Finsupp.total (↑↑s) S S Subtype.val) l = 1
sf : { x // x ∈ s } → Finset S := fun x => IsLocalization.finsetIntegerMultiple (Submonoid.powers ↑x) (t x)
x : S
⊢ x ∈ Algebra.adjoin R ↑(s.attach.biUnion sf ∪ s ∪ Finset.image (⇑l) l.support)

tactic: apply Subalgebra.mem_of_span_eq_top_of_smul_pow_mem _ (s : Set S) l hl _ _ x _
state_before: case h
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
t : (r : { x // x ∈ s }) → Finset (Localization.Away ↑r)
ht : ∀ (r : { x // x ∈ s }), Algebra.adjoin R ↑(t r) = ⊤
l : ↑↑s →₀ S
hl : (Finsupp.total (↑↑s) S S Subtype.val) l = 1
sf : { x // x ∈ s } → Finset S := fun x => IsLocalization.finsetIntegerMultiple (Submonoid.powers ↑x) (t x)
x : S
⊢ x ∈ Algebra.adjoin R ↑(s.attach.biUnion sf ∪ s ∪ Finset.image (⇑l) l.support)
state_after: 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
t : (r : { x // x ∈ s }) → Finset (Localization.Away ↑r)
ht : ∀ (r : { x // x ∈ s }), Algebra.adjoin R ↑(t r) = ⊤
l : ↑↑s →₀ S
hl : (Finsupp.total (↑↑s) S S Subtype.val) l = 1
sf : { x // x ∈ s } → Finset S := fun x => IsLocalization.finsetIntegerMultiple (Submonoid.powers ↑x) (t x)
x : S
⊢ ↑s ⊆ ↑(Algebra.adjoin R ↑(s.attach.biUnion sf ∪ s ∪ Finset.image (⇑l) l.support))

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
t : (r : { x // x ∈ s }) → Finset (Localization.Away ↑r)
ht : ∀ (r : { x // x ∈ s }), Algebra.adjoin R ↑(t r) = ⊤
l : ↑↑s →₀ S
hl : (Finsupp.total (↑↑s) S S Subtype.val) l = 1
sf : { x // x ∈ s } → Finset S := fun x => IsLocalization.finsetIntegerMultiple (Submonoid.powers ↑x) (t x)
x : S
⊢ ∀ (i : ↑↑s), l i ∈ Algebra.adjoin R ↑(s.attach.biUnion sf ∪ s ∪ Finset.image (⇑l) l.support)

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
t : (r : { x // x ∈ s }) → Finset (Localization.Away ↑r)
ht : ∀ (r : { x // x ∈ s }), Algebra.adjoin R ↑(t r) = ⊤
l : ↑↑s →₀ S
hl : (Finsupp.total (↑↑s) S S Subtype.val) l = 1
sf : { x // x ∈ s } → Finset S := fun x => IsLocalization.finsetIntegerMultiple (Submonoid.powers ↑x) (t x)
x : S
⊢ ∀ (r : ↑↑s), ∃ n, ↑r ^ n • x ∈ Algebra.adjoin R ↑(s.attach.biUnion sf ∪ s ∪ Finset.image (⇑l) l.support)

tactic: intro r
state_before: R S : Type u_1
inst✝¹ : CommRing R
inst✝ : CommRing S
f : R →+* S
s : Finset S
hs : Ideal.span ↑s = ⊤
H : ∀ (r : { x // x ∈ s }), ((algebraMap S (Localization.Away ↑r)).comp f).FiniteType
this : Algebra R S := f.toAlgebra
⊢ ∀ (r : { x // x ∈ s }), Algebra.FiniteType R (Localization.Away ↑r)
state_after: R S : Type u_1
inst✝¹ : CommRing R
inst✝ : CommRing S
f : R →+* S
s : Finset S
hs : Ideal.span ↑s = ⊤
H : ∀ (r : { x // x ∈ s }), ((algebraMap S (Localization.Away ↑r)).comp f).FiniteType
this : Algebra R S := f.toAlgebra
r : { x // x ∈ s }
⊢ Algebra.FiniteType R (Localization.Away ↑r)

tactic: simp_rw [RingHom.FiniteType] at H
state_before: R S : Type u_1
inst✝¹ : CommRing R
inst✝ : CommRing S
f : R →+* S
s : Finset S
hs : Ideal.span ↑s = ⊤
H : ∀ (r : { x // x ∈ s }), ((algebraMap S (Localization.Away ↑r)).comp f).FiniteType
this : Algebra R S := f.toAlgebra
r : { x // x ∈ s }
⊢ Algebra.FiniteType R (Localization.Away ↑r)
state_after: R S : Type u_1
inst✝¹ : CommRing R
inst✝ : CommRing S
f : R →+* S
s : Finset S
hs : Ideal.span ↑s = ⊤
H : ∀ (r : { x // x ∈ s }), Algebra.FiniteType R (Localization.Away ↑r)
this : Algebra R S := f.toAlgebra
r : { x // x ∈ s }
⊢ Algebra.FiniteType R (Localization.Away ↑r)

tactic: convert H r
state_before: R S : Type u_1
inst✝¹ : CommRing R
inst✝ : CommRing S
f : R →+* S
s : Finset S
hs : Ideal.span ↑s = ⊤
H : ∀ (r : { x // x ∈ s }), Algebra.FiniteType R (Localization.Away ↑r)
this : Algebra R S := f.toAlgebra
r : { x // x ∈ s }
⊢ Algebra.FiniteType R (Localization.Away ↑r)
state_after: case h.e'_5
R S : Type u_1
inst✝¹ : CommRing R
inst✝ : CommRing S
f : R →+* S
s : Finset S
hs : Ideal.span ↑s = ⊤
H : ∀ (r : { x // x ∈ s }), Algebra.FiniteType R (Localization.Away ↑r)
this : Algebra R S := f.toAlgebra
r : { x // x ∈ s }
⊢ Localization.algebra = ((algebraMap S (Localization.Away ↑r)).comp f).toAlgebra

tactic: ext
state_before: case h.e'_5
R S : Type u_1
inst✝¹ : CommRing R
inst✝ : CommRing S
f : R →+* S
s : Finset S
hs : Ideal.span ↑s = ⊤
H : ∀ (r : { x // x ∈ s }), Algebra.FiniteType R (Localization.Away ↑r)
this : Algebra R S := f.toAlgebra
r : { x // x ∈ s }
⊢ Localization.algebra = ((algebraMap S (Localization.Away ↑r)).comp f).toAlgebra
state_after: case h.e'_5.h
R S : Type u_1
inst✝¹ : CommRing R
inst✝ : CommRing S
f : R →+* S
s : Finset S
hs : Ideal.span ↑s = ⊤
H : ∀ (r : { x // x ∈ s }), Algebra.FiniteType R (Localization.Away ↑r)
this : Algebra R S := f.toAlgebra
r : { x // x ∈ s }
r✝ : R
x✝ : Localization (Submonoid.powers ↑r)
⊢ (let_fun I := Localization.algebra;
    r✝ • x✝) =
    r✝ • x✝

tactic: simp_rw [Algebra.smul_def]
state_before: case h.e'_5.h
R S : Type u_1
inst✝¹ : CommRing R
inst✝ : CommRing S
f : R →+* S
s : Finset S
hs : Ideal.span ↑s = ⊤
H : ∀ (r : { x // x ∈ s }), Algebra.FiniteType R (Localization.Away ↑r)
this : Algebra R S := f.toAlgebra
r : { x // x ∈ s }
r✝ : R
x✝ : Localization (Submonoid.powers ↑r)
⊢ (let_fun I := Localization.algebra;
    r✝ • x✝) =
    r✝ • x✝
state_after: case h.e'_5.h
R S : Type u_1
inst✝¹ : CommRing R
inst✝ : CommRing S
f : R →+* S
s : Finset S
hs : Ideal.span ↑s = ⊤
H : ∀ (r : { x // x ∈ s }), Algebra.FiniteType R (Localization.Away ↑r)
this : Algebra R S := f.toAlgebra
r : { x // x ∈ s }
r✝ : R
x✝ : Localization (Submonoid.powers ↑r)
⊢ (algebraMap R (Localization (Submonoid.powers ↑r))) r✝ * x✝ =
    (algebraMap R (Localization (Submonoid.powers ↑r))) r✝ * x✝

tactic: rfl
state_before: case h.e'_5.h
R S : Type u_1
inst✝¹ : CommRing R
inst✝ : CommRing S
f : R →+* S
s : Finset S
hs : Ideal.span ↑s = ⊤
H : ∀ (r : { x // x ∈ s }), Algebra.FiniteType R (Localization.Away ↑r)
this : Algebra R S := f.toAlgebra
r : { x // x ∈ s }
r✝ : R
x✝ : Localization (Submonoid.powers ↑r)
⊢ (algebraMap R (Localization (Submonoid.powers ↑r))) r✝ * x✝ =
    (algebraMap R (Localization (Submonoid.powers ↑r))) r✝ * x✝
state_after: no goals

tactic: rw [hs]
state_before: 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
t : (r : { x // x ∈ s }) → Finset (Localization.Away ↑r)
ht : ∀ (r : { x // x ∈ s }), Algebra.adjoin R ↑(t r) = ⊤
⊢ 1 ∈ Ideal.span ↑s
state_after: 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
t : (r : { x // x ∈ s }) → Finset (Localization.Away ↑r)
ht : ∀ (r : { x // x ∈ s }), Algebra.adjoin R ↑(t r) = ⊤
⊢ 1 ∈ ⊤

tactic: trivial
state_before: 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
t : (r : { x // x ∈ s }) → Finset (Localization.Away ↑r)
ht : ∀ (r : { x // x ∈ s }), Algebra.adjoin R ↑(t r) = ⊤
⊢ 1 ∈ ⊤
state_after: no goals

tactic: intro x hx
state_before: 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
t : (r : { x // x ∈ s }) → Finset (Localization.Away ↑r)
ht : ∀ (r : { x // x ∈ s }), Algebra.adjoin R ↑(t r) = ⊤
l : ↑↑s →₀ S
hl : (Finsupp.total (↑↑s) S S Subtype.val) l = 1
sf : { x // x ∈ s } → Finset S := fun x => IsLocalization.finsetIntegerMultiple (Submonoid.powers ↑x) (t x)
x : S
⊢ ↑s ⊆ ↑(Algebra.adjoin R ↑(s.attach.biUnion sf ∪ s ∪ Finset.image (⇑l) l.support))
state_after: 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
t : (r : { x // x ∈ s }) → Finset (Localization.Away ↑r)
ht : ∀ (r : { x // x ∈ s }), Algebra.adjoin R ↑(t r) = ⊤
l : ↑↑s →₀ S
hl : (Finsupp.total (↑↑s) S S Subtype.val) l = 1
sf : { x // x ∈ s } → Finset S := fun x => IsLocalization.finsetIntegerMultiple (Submonoid.powers ↑x) (t x)
x✝ x : S
hx : x ∈ ↑s
⊢ x ∈ ↑(Algebra.adjoin R ↑(s.attach.biUnion sf ∪ s ∪ Finset.image (⇑l) l.support))

tactic: apply Algebra.subset_adjoin
state_before: 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
t : (r : { x // x ∈ s }) → Finset (Localization.Away ↑r)
ht : ∀ (r : { x // x ∈ s }), Algebra.adjoin R ↑(t r) = ⊤
l : ↑↑s →₀ S
hl : (Finsupp.total (↑↑s) S S Subtype.val) l = 1
sf : { x // x ∈ s } → Finset S := fun x => IsLocalization.finsetIntegerMultiple (Submonoid.powers ↑x) (t x)
x✝ x : S
hx : x ∈ ↑s
⊢ x ∈ ↑(Algebra.adjoin R ↑(s.attach.biUnion sf ∪ s ∪ Finset.image (⇑l) l.support))
state_after: case a
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
t : (r : { x // x ∈ s }) → Finset (Localization.Away ↑r)
ht : ∀ (r : { x // x ∈ s }), Algebra.adjoin R ↑(t r) = ⊤
l : ↑↑s →₀ S
hl : (Finsupp.total (↑↑s) S S Subtype.val) l = 1
sf : { x // x ∈ s } → Finset S := fun x => IsLocalization.finsetIntegerMultiple (Submonoid.powers ↑x) (t x)
x✝ x : S
hx : x ∈ ↑s
⊢ x ∈ ↑(s.attach.biUnion sf ∪ s ∪ Finset.image (⇑l) l.support)

tactic: rw [Finset.coe_union, Finset.coe_union]
state_before: case a
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
t : (r : { x // x ∈ s }) → Finset (Localization.Away ↑r)
ht : ∀ (r : { x // x ∈ s }), Algebra.adjoin R ↑(t r) = ⊤
l : ↑↑s →₀ S
hl : (Finsupp.total (↑↑s) S S Subtype.val) l = 1
sf : { x // x ∈ s } → Finset S := fun x => IsLocalization.finsetIntegerMultiple (Submonoid.powers ↑x) (t x)
x✝ x : S
hx : x ∈ ↑s
⊢ x ∈ ↑(s.attach.biUnion sf ∪ s ∪ Finset.image (⇑l) l.support)
state_after: case a
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
t : (r : { x // x ∈ s }) → Finset (Localization.Away ↑r)
ht : ∀ (r : { x // x ∈ s }), Algebra.adjoin R ↑(t r) = ⊤
l : ↑↑s →₀ S
hl : (Finsupp.total (↑↑s) S S Subtype.val) l = 1
sf : { x // x ∈ s } → Finset S := fun x => IsLocalization.finsetIntegerMultiple (Submonoid.powers ↑x) (t x)
x✝ x : S
hx : x ∈ ↑s
⊢ x ∈ ↑(s.attach.biUnion sf) ∪ ↑s ∪ ↑(Finset.image (⇑l) l.support)

tactic: exact Or.inl (Or.inr hx)
state_before: case a
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
t : (r : { x // x ∈ s }) → Finset (Localization.Away ↑r)
ht : ∀ (r : { x // x ∈ s }), Algebra.adjoin R ↑(t r) = ⊤
l : ↑↑s →₀ S
hl : (Finsupp.total (↑↑s) S S Subtype.val) l = 1
sf : { x // x ∈ s } → Finset S := fun x => IsLocalization.finsetIntegerMultiple (Submonoid.powers ↑x) (t x)
x✝ x : S
hx : x ∈ ↑s
⊢ x ∈ ↑(s.attach.biUnion sf) ∪ ↑s ∪ ↑(Finset.image (⇑l) l.support)
state_after: no goals

tactic: intro i
state_before: 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
t : (r : { x // x ∈ s }) → Finset (Localization.Away ↑r)
ht : ∀ (r : { x // x ∈ s }), Algebra.adjoin R ↑(t r) = ⊤
l : ↑↑s →₀ S
hl : (Finsupp.total (↑↑s) S S Subtype.val) l = 1
sf : { x // x ∈ s } → Finset S := fun x => IsLocalization.finsetIntegerMultiple (Submonoid.powers ↑x) (t x)
x : S
⊢ ∀ (i : ↑↑s), l i ∈ Algebra.adjoin R ↑(s.attach.biUnion sf ∪ s ∪ Finset.image (⇑l) l.support)
state_after: 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
t : (r : { x // x ∈ s }) → Finset (Localization.Away ↑r)
ht : ∀ (r : { x // x ∈ s }), Algebra.adjoin R ↑(t r) = ⊤
l : ↑↑s →₀ S
hl : (Finsupp.total (↑↑s) S S Subtype.val) l = 1
sf : { x // x ∈ s } → Finset S := fun x => IsLocalization.finsetIntegerMultiple (Submonoid.powers ↑x) (t x)
x : S
i : ↑↑s
⊢ l i ∈ Algebra.adjoin R ↑(s.attach.biUnion sf ∪ s ∪ Finset.image (⇑l) l.support)

tactic: by_cases h : l i = 0
state_before: 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
t : (r : { x // x ∈ s }) → Finset (Localization.Away ↑r)
ht : ∀ (r : { x // x ∈ s }), Algebra.adjoin R ↑(t r) = ⊤
l : ↑↑s →₀ S
hl : (Finsupp.total (↑↑s) S S Subtype.val) l = 1
sf : { x // x ∈ s } → Finset S := fun x => IsLocalization.finsetIntegerMultiple (Submonoid.powers ↑x) (t x)
x : S
i : ↑↑s
⊢ l i ∈ Algebra.adjoin R ↑(s.attach.biUnion sf ∪ s ∪ Finset.image (⇑l) l.support)
state_after: case pos
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
t : (r : { x // x ∈ s }) → Finset (Localization.Away ↑r)
ht : ∀ (r : { x // x ∈ s }), Algebra.adjoin R ↑(t r) = ⊤
l : ↑↑s →₀ S
hl : (Finsupp.total (↑↑s) S S Subtype.val) l = 1
sf : { x // x ∈ s } → Finset S := fun x => IsLocalization.finsetIntegerMultiple (Submonoid.powers ↑x) (t x)
x : S
i : ↑↑s
h : l i = 0
⊢ l i ∈ Algebra.adjoin R ↑(s.attach.biUnion sf ∪ s ∪ Finset.image (⇑l) l.support)

case neg
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
t : (r : { x // x ∈ s }) → Finset (Localization.Away ↑r)
ht : ∀ (r : { x // x ∈ s }), Algebra.adjoin R ↑(t r) = ⊤
l : ↑↑s →₀ S
hl : (Finsupp.total (↑↑s) S S Subtype.val) l = 1
sf : { x // x ∈ s } → Finset S := fun x => IsLocalization.finsetIntegerMultiple (Submonoid.powers ↑x) (t x)
x : S
i : ↑↑s
h : ¬l i = 0
⊢ l i ∈ Algebra.adjoin R ↑(s.attach.biUnion sf ∪ s ∪ Finset.image (⇑l) l.support)

tactic: apply Algebra.subset_adjoin
state_before: case neg
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
t : (r : { x // x ∈ s }) → Finset (Localization.Away ↑r)
ht : ∀ (r : { x // x ∈ s }), Algebra.adjoin R ↑(t r) = ⊤
l : ↑↑s →₀ S
hl : (Finsupp.total (↑↑s) S S Subtype.val) l = 1
sf : { x // x ∈ s } → Finset S := fun x => IsLocalization.finsetIntegerMultiple (Submonoid.powers ↑x) (t x)
x : S
i : ↑↑s
h : ¬l i = 0
⊢ l i ∈ Algebra.adjoin R ↑(s.attach.biUnion sf ∪ s ∪ Finset.image (⇑l) l.support)
state_after: case neg.a
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
t : (r : { x // x ∈ s }) → Finset (Localization.Away ↑r)
ht : ∀ (r : { x // x ∈ s }), Algebra.adjoin R ↑(t r) = ⊤
l : ↑↑s →₀ S
hl : (Finsupp.total (↑↑s) S S Subtype.val) l = 1
sf : { x // x ∈ s } → Finset S := fun x => IsLocalization.finsetIntegerMultiple (Submonoid.powers ↑x) (t x)
x : S
i : ↑↑s
h : ¬l i = 0
⊢ l i ∈ ↑(s.attach.biUnion sf ∪ s ∪ Finset.image (⇑l) l.support)

tactic: rw [Finset.coe_union, Finset.coe_image]
state_before: case neg.a
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
t : (r : { x // x ∈ s }) → Finset (Localization.Away ↑r)
ht : ∀ (r : { x // x ∈ s }), Algebra.adjoin R ↑(t r) = ⊤
l : ↑↑s →₀ S
hl : (Finsupp.total (↑↑s) S S Subtype.val) l = 1
sf : { x // x ∈ s } → Finset S := fun x => IsLocalization.finsetIntegerMultiple (Submonoid.powers ↑x) (t x)
x : S
i : ↑↑s
h : ¬l i = 0
⊢ l i ∈ ↑(s.attach.biUnion sf ∪ s ∪ Finset.image (⇑l) l.support)
state_after: case neg.a
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
t : (r : { x // x ∈ s }) → Finset (Localization.Away ↑r)
ht : ∀ (r : { x // x ∈ s }), Algebra.adjoin R ↑(t r) = ⊤
l : ↑↑s →₀ S
hl : (Finsupp.total (↑↑s) S S Subtype.val) l = 1
sf : { x // x ∈ s } → Finset S := fun x => IsLocalization.finsetIntegerMultiple (Submonoid.powers ↑x) (t x)
x : S
i : ↑↑s
h : ¬l i = 0
⊢ l i ∈ ↑(s.attach.biUnion sf ∪ s) ∪ ⇑l '' ↑l.support

tactic: exact Or.inr (Set.mem_image_of_mem _ (Finsupp.mem_support_iff.mpr h))
state_before: case neg.a
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
t : (r : { x // x ∈ s }) → Finset (Localization.Away ↑r)
ht : ∀ (r : { x // x ∈ s }), Algebra.adjoin R ↑(t r) = ⊤
l : ↑↑s →₀ S
hl : (Finsupp.total (↑↑s) S S Subtype.val) l = 1
sf : { x // x ∈ s } → Finset S := fun x => IsLocalization.finsetIntegerMultiple (Submonoid.powers ↑x) (t x)
x : S
i : ↑↑s
h : ¬l i = 0
⊢ l i ∈ ↑(s.attach.biUnion sf ∪ s) ∪ ⇑l '' ↑l.support
state_after: no goals

tactic: rw [h]
state_before: case pos
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
t : (r : { x // x ∈ s }) → Finset (Localization.Away ↑r)
ht : ∀ (r : { x // x ∈ s }), Algebra.adjoin R ↑(t r) = ⊤
l : ↑↑s →₀ S
hl : (Finsupp.total (↑↑s) S S Subtype.val) l = 1
sf : { x // x ∈ s } → Finset S := fun x => IsLocalization.finsetIntegerMultiple (Submonoid.powers ↑x) (t x)
x : S
i : ↑↑s
h : l i = 0
⊢ l i ∈ Algebra.adjoin R ↑(s.attach.biUnion sf ∪ s ∪ Finset.image (⇑l) l.support)
state_after: case pos
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
t : (r : { x // x ∈ s }) → Finset (Localization.Away ↑r)
ht : ∀ (r : { x // x ∈ s }), Algebra.adjoin R ↑(t r) = ⊤
l : ↑↑s →₀ S
hl : (Finsupp.total (↑↑s) S S Subtype.val) l = 1
sf : { x // x ∈ s } → Finset S := fun x => IsLocalization.finsetIntegerMultiple (Submonoid.powers ↑x) (t x)
x : S
i : ↑↑s
h : l i = 0
⊢ 0 ∈ Algebra.adjoin R ↑(s.attach.biUnion sf ∪ s ∪ Finset.image (⇑l) l.support)

tactic: exact zero_mem _
state_before: case pos
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
t : (r : { x // x ∈ s }) → Finset (Localization.Away ↑r)
ht : ∀ (r : { x // x ∈ s }), Algebra.adjoin R ↑(t r) = ⊤
l : ↑↑s →₀ S
hl : (Finsupp.total (↑↑s) S S Subtype.val) l = 1
sf : { x // x ∈ s } → Finset S := fun x => IsLocalization.finsetIntegerMultiple (Submonoid.powers ↑x) (t x)
x : S
i : ↑↑s
h : l i = 0
⊢ 0 ∈ Algebra.adjoin R ↑(s.attach.biUnion sf ∪ s ∪ Finset.image (⇑l) l.support)
state_after: no goals

tactic: intro r
state_before: 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
t : (r : { x // x ∈ s }) → Finset (Localization.Away ↑r)
ht : ∀ (r : { x // x ∈ s }), Algebra.adjoin R ↑(t r) = ⊤
l : ↑↑s →₀ S
hl : (Finsupp.total (↑↑s) S S Subtype.val) l = 1
sf : { x // x ∈ s } → Finset S := fun x => IsLocalization.finsetIntegerMultiple (Submonoid.powers ↑x) (t x)
x : S
⊢ ∀ (r : ↑↑s), ∃ n, ↑r ^ n • x ∈ Algebra.adjoin R ↑(s.attach.biUnion sf ∪ s ∪ Finset.image (⇑l) l.support)
state_after: 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
t : (r : { x // x ∈ s }) → Finset (Localization.Away ↑r)
ht : ∀ (r : { x // x ∈ s }), Algebra.adjoin R ↑(t r) = ⊤
l : ↑↑s →₀ S
hl : (Finsupp.total (↑↑s) S S Subtype.val) l = 1
sf : { x // x ∈ s } → Finset S := fun x => IsLocalization.finsetIntegerMultiple (Submonoid.powers ↑x) (t x)
x : S
r : ↑↑s
⊢ ∃ n, ↑r ^ n • x ∈ Algebra.adjoin R ↑(s.attach.biUnion sf ∪ s ∪ Finset.image (⇑l) l.support)

tactic: rw [Finset.coe_union, Finset.coe_union, Finset.coe_biUnion]
state_before: 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
t : (r : { x // x ∈ s }) → Finset (Localization.Away ↑r)
ht : ∀ (r : { x // x ∈ s }), Algebra.adjoin R ↑(t r) = ⊤
l : ↑↑s →₀ S
hl : (Finsupp.total (↑↑s) S S Subtype.val) l = 1
sf : { x // x ∈ s } → Finset S := fun x => IsLocalization.finsetIntegerMultiple (Submonoid.powers ↑x) (t x)
x : S
r : ↑↑s
⊢ ∃ n, ↑r ^ n • x ∈ Algebra.adjoin R ↑(s.attach.biUnion sf ∪ s ∪ Finset.image (⇑l) l.support)
state_after: 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
t : (r : { x // x ∈ s }) → Finset (Localization.Away ↑r)
ht : ∀ (r : { x // x ∈ s }), Algebra.adjoin R ↑(t r) = ⊤
l : ↑↑s →₀ S
hl : (Finsupp.total (↑↑s) S S Subtype.val) l = 1
sf : { x // x ∈ s } → Finset S := fun x => IsLocalization.finsetIntegerMultiple (Submonoid.powers ↑x) (t x)
x : S
r : ↑↑s
⊢ ∃ n, ↑r ^ n • x ∈ Algebra.adjoin R ((⋃ x ∈ ↑s.attach, ↑(sf x)) ∪ ↑s ∪ ↑(Finset.image (⇑l) l.support))

tactic: rw [show ∀ A : Set S, (∃ n, (r : S) ^ n • x ∈ Algebra.adjoin R A) ↔
  (∃ m : (Submonoid.powers (r : S)), (m : S) • x ∈ Algebra.adjoin R A) by
  { exact fun _ => by simp [Submonoid.mem_powers_iff] }]
state_before: 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
t : (r : { x // x ∈ s }) → Finset (Localization.Away ↑r)
ht : ∀ (r : { x // x ∈ s }), Algebra.adjoin R ↑(t r) = ⊤
l : ↑↑s →₀ S
hl : (Finsupp.total (↑↑s) S S Subtype.val) l = 1
sf : { x // x ∈ s } → Finset S := fun x => IsLocalization.finsetIntegerMultiple (Submonoid.powers ↑x) (t x)
x : S
r : ↑↑s
⊢ ∃ n, ↑r ^ n • x ∈ Algebra.adjoin R ((⋃ x ∈ ↑s.attach, ↑(sf x)) ∪ ↑s ∪ ↑(Finset.image (⇑l) l.support))
state_after: 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
t : (r : { x // x ∈ s }) → Finset (Localization.Away ↑r)
ht : ∀ (r : { x // x ∈ s }), Algebra.adjoin R ↑(t r) = ⊤
l : ↑↑s →₀ S
hl : (Finsupp.total (↑↑s) S S Subtype.val) l = 1
sf : { x // x ∈ s } → Finset S := fun x => IsLocalization.finsetIntegerMultiple (Submonoid.powers ↑x) (t x)
x : S
r : ↑↑s
⊢ ∃ m, ↑m • x ∈ Algebra.adjoin R ((⋃ x ∈ ↑s.attach, ↑(sf x)) ∪ ↑s ∪ ↑(Finset.image (⇑l) l.support))

tactic: refine IsLocalization.exists_smul_mem_of_mem_adjoin
  (Submonoid.powers (r : S)) x (t r) (Algebra.adjoin R _) ?_ ?_ ?_
state_before: 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
t : (r : { x // x ∈ s }) → Finset (Localization.Away ↑r)
ht : ∀ (r : { x // x ∈ s }), Algebra.adjoin R ↑(t r) = ⊤
l : ↑↑s →₀ S
hl : (Finsupp.total (↑↑s) S S Subtype.val) l = 1
sf : { x // x ∈ s } → Finset S := fun x => IsLocalization.finsetIntegerMultiple (Submonoid.powers ↑x) (t x)
x : S
r : ↑↑s
⊢ ∃ m, ↑m • x ∈ Algebra.adjoin R ((⋃ x ∈ ↑s.attach, ↑(sf x)) ∪ ↑s ∪ ↑(Finset.image (⇑l) l.support))
state_after: case refine_1
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
t : (r : { x // x ∈ s }) → Finset (Localization.Away ↑r)
ht : ∀ (r : { x // x ∈ s }), Algebra.adjoin R ↑(t r) = ⊤
l : ↑↑s →₀ S
hl : (Finsupp.total (↑↑s) S S Subtype.val) l = 1
sf : { x // x ∈ s } → Finset S := fun x => IsLocalization.finsetIntegerMultiple (Submonoid.powers ↑x) (t x)
x : S
r : ↑↑s
⊢ ↑(IsLocalization.finsetIntegerMultiple (Submonoid.powers ↑r) (t r)) ⊆
    ↑(Algebra.adjoin R ((⋃ x ∈ ↑s.attach, ↑(sf x)) ∪ ↑s ∪ ↑(Finset.image (⇑l) l.support)))

case refine_2
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
t : (r : { x // x ∈ s }) → Finset (Localization.Away ↑r)
ht : ∀ (r : { x // x ∈ s }), Algebra.adjoin R ↑(t r) = ⊤
l : ↑↑s →₀ S
hl : (Finsupp.total (↑↑s) S S Subtype.val) l = 1
sf : { x // x ∈ s } → Finset S := fun x => IsLocalization.finsetIntegerMultiple (Submonoid.powers ↑x) (t x)
x : S
r : ↑↑s
⊢ Submonoid.powers ↑r ≤
    (Algebra.adjoin R ((⋃ x ∈ ↑s.attach, ↑(sf x)) ∪ ↑s ∪ ↑(Finset.image (⇑l) l.support))).toSubmonoid

case refine_3
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
t : (r : { x // x ∈ s }) → Finset (Localization.Away ↑r)
ht : ∀ (r : { x // x ∈ s }), Algebra.adjoin R ↑(t r) = ⊤
l : ↑↑s →₀ S
hl : (Finsupp.total (↑↑s) S S Subtype.val) l = 1
sf : { x // x ∈ s } → Finset S := fun x => IsLocalization.finsetIntegerMultiple (Submonoid.powers ↑x) (t x)
x : S
r : ↑↑s
⊢ (algebraMap S (Localization.Away ↑r)) x ∈ Algebra.adjoin R ↑(t r)

tactic: exact fun _ => by simp [Submonoid.mem_powers_iff]
state_before: 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
t : (r : { x // x ∈ s }) → Finset (Localization.Away ↑r)
ht : ∀ (r : { x // x ∈ s }), Algebra.adjoin R ↑(t r) = ⊤
l : ↑↑s →₀ S
hl : (Finsupp.total (↑↑s) S S Subtype.val) l = 1
sf : { x // x ∈ s } → Finset S := fun x => IsLocalization.finsetIntegerMultiple (Submonoid.powers ↑x) (t x)
x : S
r : ↑↑s
⊢ ∀ (A : Set S), (∃ n, ↑r ^ n • x ∈ Algebra.adjoin R A) ↔ ∃ m, ↑m • x ∈ Algebra.adjoin R A
state_after: no goals

tactic: simp [Submonoid.mem_powers_iff]
state_before: 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
t : (r : { x // x ∈ s }) → Finset (Localization.Away ↑r)
ht : ∀ (r : { x // x ∈ s }), Algebra.adjoin R ↑(t r) = ⊤
l : ↑↑s →₀ S
hl : (Finsupp.total (↑↑s) S S Subtype.val) l = 1
sf : { x // x ∈ s } → Finset S := fun x => IsLocalization.finsetIntegerMultiple (Submonoid.powers ↑x) (t x)
x : S
r : ↑↑s
x✝ : Set S
⊢ (∃ n, ↑r ^ n • x ∈ Algebra.adjoin R x✝) ↔ ∃ m, ↑m • x ∈ Algebra.adjoin R x✝
state_after: no goals

tactic: intro x hx
state_before: case refine_1
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
t : (r : { x // x ∈ s }) → Finset (Localization.Away ↑r)
ht : ∀ (r : { x // x ∈ s }), Algebra.adjoin R ↑(t r) = ⊤
l : ↑↑s →₀ S
hl : (Finsupp.total (↑↑s) S S Subtype.val) l = 1
sf : { x // x ∈ s } → Finset S := fun x => IsLocalization.finsetIntegerMultiple (Submonoid.powers ↑x) (t x)
x : S
r : ↑↑s
⊢ ↑(IsLocalization.finsetIntegerMultiple (Submonoid.powers ↑r) (t r)) ⊆
    ↑(Algebra.adjoin R ((⋃ x ∈ ↑s.attach, ↑(sf x)) ∪ ↑s ∪ ↑(Finset.image (⇑l) l.support)))
state_after: case refine_1
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
t : (r : { x // x ∈ s }) → Finset (Localization.Away ↑r)
ht : ∀ (r : { x // x ∈ s }), Algebra.adjoin R ↑(t r) = ⊤
l : ↑↑s →₀ S
hl : (Finsupp.total (↑↑s) S S Subtype.val) l = 1
sf : { x // x ∈ s } → Finset S := fun x => IsLocalization.finsetIntegerMultiple (Submonoid.powers ↑x) (t x)
x✝ : S
r : ↑↑s
x : S
hx : x ∈ ↑(IsLocalization.finsetIntegerMultiple (Submonoid.powers ↑r) (t r))
⊢ x ∈ ↑(Algebra.adjoin R ((⋃ x ∈ ↑s.attach, ↑(sf x)) ∪ ↑s ∪ ↑(Finset.image (⇑l) l.support)))

tactic: apply Algebra.subset_adjoin
state_before: case refine_1
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
t : (r : { x // x ∈ s }) → Finset (Localization.Away ↑r)
ht : ∀ (r : { x // x ∈ s }), Algebra.adjoin R ↑(t r) = ⊤
l : ↑↑s →₀ S
hl : (Finsupp.total (↑↑s) S S Subtype.val) l = 1
sf : { x // x ∈ s } → Finset S := fun x => IsLocalization.finsetIntegerMultiple (Submonoid.powers ↑x) (t x)
x✝ : S
r : ↑↑s
x : S
hx : x ∈ ↑(IsLocalization.finsetIntegerMultiple (Submonoid.powers ↑r) (t r))
⊢ x ∈ ↑(Algebra.adjoin R ((⋃ x ∈ ↑s.attach, ↑(sf x)) ∪ ↑s ∪ ↑(Finset.image (⇑l) l.support)))
state_after: case refine_1.a
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
t : (r : { x // x ∈ s }) → Finset (Localization.Away ↑r)
ht : ∀ (r : { x // x ∈ s }), Algebra.adjoin R ↑(t r) = ⊤
l : ↑↑s →₀ S
hl : (Finsupp.total (↑↑s) S S Subtype.val) l = 1
sf : { x // x ∈ s } → Finset S := fun x => IsLocalization.finsetIntegerMultiple (Submonoid.powers ↑x) (t x)
x✝ : S
r : ↑↑s
x : S
hx : x ∈ ↑(IsLocalization.finsetIntegerMultiple (Submonoid.powers ↑r) (t r))
⊢ x ∈ (⋃ x ∈ ↑s.attach, ↑(sf x)) ∪ ↑s ∪ ↑(Finset.image (⇑l) l.support)

tactic: exact Or.inl (Or.inl ⟨_, ⟨r, rfl⟩, _, ⟨s.mem_attach r, rfl⟩, hx⟩)
state_before: case refine_1.a
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
t : (r : { x // x ∈ s }) → Finset (Localization.Away ↑r)
ht : ∀ (r : { x // x ∈ s }), Algebra.adjoin R ↑(t r) = ⊤
l : ↑↑s →₀ S
hl : (Finsupp.total (↑↑s) S S Subtype.val) l = 1
sf : { x // x ∈ s } → Finset S := fun x => IsLocalization.finsetIntegerMultiple (Submonoid.powers ↑x) (t x)
x✝ : S
r : ↑↑s
x : S
hx : x ∈ ↑(IsLocalization.finsetIntegerMultiple (Submonoid.powers ↑r) (t r))
⊢ x ∈ (⋃ x ∈ ↑s.attach, ↑(sf x)) ∪ ↑s ∪ ↑(Finset.image (⇑l) l.support)
state_after: no goals

tactic: rw [Submonoid.powers_eq_closure, Submonoid.closure_le, Set.singleton_subset_iff]
state_before: case refine_2
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
t : (r : { x // x ∈ s }) → Finset (Localization.Away ↑r)
ht : ∀ (r : { x // x ∈ s }), Algebra.adjoin R ↑(t r) = ⊤
l : ↑↑s →₀ S
hl : (Finsupp.total (↑↑s) S S Subtype.val) l = 1
sf : { x // x ∈ s } → Finset S := fun x => IsLocalization.finsetIntegerMultiple (Submonoid.powers ↑x) (t x)
x : S
r : ↑↑s
⊢ Submonoid.powers ↑r ≤
    (Algebra.adjoin R ((⋃ x ∈ ↑s.attach, ↑(sf x)) ∪ ↑s ∪ ↑(Finset.image (⇑l) l.support))).toSubmonoid
state_after: case refine_2
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
t : (r : { x // x ∈ s }) → Finset (Localization.Away ↑r)
ht : ∀ (r : { x // x ∈ s }), Algebra.adjoin R ↑(t r) = ⊤
l : ↑↑s →₀ S
hl : (Finsupp.total (↑↑s) S S Subtype.val) l = 1
sf : { x // x ∈ s } → Finset S := fun x => IsLocalization.finsetIntegerMultiple (Submonoid.powers ↑x) (t x)
x : S
r : ↑↑s
⊢ ↑r ∈ ↑(Algebra.adjoin R ((⋃ x ∈ ↑s.attach, ↑(sf x)) ∪ ↑s ∪ ↑(Finset.image (⇑l) l.support))).toSubmonoid

tactic: apply Algebra.subset_adjoin
state_before: case refine_2
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
t : (r : { x // x ∈ s }) → Finset (Localization.Away ↑r)
ht : ∀ (r : { x // x ∈ s }), Algebra.adjoin R ↑(t r) = ⊤
l : ↑↑s →₀ S
hl : (Finsupp.total (↑↑s) S S Subtype.val) l = 1
sf : { x // x ∈ s } → Finset S := fun x => IsLocalization.finsetIntegerMultiple (Submonoid.powers ↑x) (t x)
x : S
r : ↑↑s
⊢ ↑r ∈ ↑(Algebra.adjoin R ((⋃ x ∈ ↑s.attach, ↑(sf x)) ∪ ↑s ∪ ↑(Finset.image (⇑l) l.support))).toSubmonoid
state_after: case refine_2.a
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
t : (r : { x // x ∈ s }) → Finset (Localization.Away ↑r)
ht : ∀ (r : { x // x ∈ s }), Algebra.adjoin R ↑(t r) = ⊤
l : ↑↑s →₀ S
hl : (Finsupp.total (↑↑s) S S Subtype.val) l = 1
sf : { x // x ∈ s } → Finset S := fun x => IsLocalization.finsetIntegerMultiple (Submonoid.powers ↑x) (t x)
x : S
r : ↑↑s
⊢ ↑r ∈ (⋃ x ∈ ↑s.attach, ↑(sf x)) ∪ ↑s ∪ ↑(Finset.image (⇑l) l.support)

tactic: exact Or.inl (Or.inr r.2)
state_before: case refine_2.a
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
t : (r : { x // x ∈ s }) → Finset (Localization.Away ↑r)
ht : ∀ (r : { x // x ∈ s }), Algebra.adjoin R ↑(t r) = ⊤
l : ↑↑s →₀ S
hl : (Finsupp.total (↑↑s) S S Subtype.val) l = 1
sf : { x // x ∈ s } → Finset S := fun x => IsLocalization.finsetIntegerMultiple (Submonoid.powers ↑x) (t x)
x : S
r : ↑↑s
⊢ ↑r ∈ (⋃ x ∈ ↑s.attach, ↑(sf x)) ∪ ↑s ∪ ↑(Finset.image (⇑l) l.support)
state_after: no goals

tactic: rw [ht]
state_before: case refine_3
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
t : (r : { x // x ∈ s }) → Finset (Localization.Away ↑r)
ht : ∀ (r : { x // x ∈ s }), Algebra.adjoin R ↑(t r) = ⊤
l : ↑↑s →₀ S
hl : (Finsupp.total (↑↑s) S S Subtype.val) l = 1
sf : { x // x ∈ s } → Finset S := fun x => IsLocalization.finsetIntegerMultiple (Submonoid.powers ↑x) (t x)
x : S
r : ↑↑s
⊢ (algebraMap S (Localization.Away ↑r)) x ∈ Algebra.adjoin R ↑(t r)
state_after: case refine_3
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
t : (r : { x // x ∈ s }) → Finset (Localization.Away ↑r)
ht : ∀ (r : { x // x ∈ s }), Algebra.adjoin R ↑(t r) = ⊤
l : ↑↑s →₀ S
hl : (Finsupp.total (↑↑s) S S Subtype.val) l = 1
sf : { x // x ∈ s } → Finset S := fun x => IsLocalization.finsetIntegerMultiple (Submonoid.powers ↑x) (t x)
x : S
r : ↑↑s
⊢ (algebraMap S (Localization.Away ↑r)) x ∈ ⊤

tactic: trivial
state_before: case refine_3
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
t : (r : { x // x ∈ s }) → Finset (Localization.Away ↑r)
ht : ∀ (r : { x // x ∈ s }), Algebra.adjoin R ↑(t r) = ⊤
l : ↑↑s →₀ S
hl : (Finsupp.total (↑↑s) S S Subtype.val) l = 1
sf : { x // x ∈ s } → Finset S := fun x => IsLocalization.finsetIntegerMultiple (Submonoid.powers ↑x) (t x)
x : S
r : ↑↑s
⊢ (algebraMap S (Localization.Away ↑r)) x ∈ ⊤
state_after: no goals