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

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝⁶ : CommRing R
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : Flat R M
K : Type u_3
inst✝² : AddCommGroup K
inst✝¹ : Module R K
inst✝ : Module.Finite R K
K' K₁ K₂ : Submodule R K
hN₁✝ : K₁.FG
hN₂✝ : K₂.FG
ih₁ :
  ∀ {n : ℕ} {f : K →ₗ[R] Fin n →₀ R} {x : (Fin n →₀ R) →ₗ[R] M}, x ∘ₗ f = 0 → ∃ k a y, x = y ∘ₗ a ∧ K₁ ≤ (a ∘ₗ f).ker
ih₂ :
  ∀ {n : ℕ} {f : K →ₗ[R] Fin n →₀ R} {x : (Fin n →₀ R) →ₗ[R] M}, x ∘ₗ f = 0 → ∃ k a y, x = y ∘ₗ a ∧ K₂ ≤ (a ∘ₗ f).ker
n : ℕ
f : K →ₗ[R] Fin n →₀ R
k₁ : ℕ
a₁ : (Fin n →₀ R) →ₗ[R] Fin k₁ →₀ R
y₁ : (Fin k₁ →₀ R) →ₗ[R] M
ha₁ : K₁ ≤ (a₁ ∘ₗ f).ker
h : (y₁ ∘ₗ a₁) ∘ₗ f = 0
⊢ y₁ ∘ₗ a₁ ∘ₗ f = 0

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Support

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
f : R
H : Module.support R M ⊆ PrimeSpectrum.zeroLocus {f}
m : M
h : ¬(R ∙ m).annihilator = ⊤
⊢ f ∈ (R ∙ m).annihilator.radical

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Support

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
f : R
⊢ Subsingleton (LocalizedModule.Away f M) ↔ Module.support R M ⊆ PrimeSpectrum.zeroLocus {f}

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Support

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
⊢ support R M = ∅ ↔ Subsingleton M

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Support

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
⊢ (support R M).Nonempty ↔ Nontrivial M

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Support

Goal state before:
pp: R : Type u_1
inst✝² : CommRing R
A : Type u_3
inst✝¹ : Ring A
inst✝ : Algebra R A
⊢ support R A = PrimeSpectrum.zeroLocus ↑(RingHom.ker (algebraMap R A))

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Support

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝⁵ : CommRing R
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
inst✝² : IsDomain R
inst✝¹ : IsTorsionFree R M
inst✝ : Nontrivial M
⊢ support R M = Set.univ

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Localization.Finiteness

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝⁷ : CommSemiring R
inst✝⁶ : CommSemiring S
M : Submonoid R
S' : Type u_4
inst✝⁵ : CommSemiring S'
inst✝⁴ : Algebra S S'
inst✝³ : Algebra R S
inst✝² : Algebra R S'
inst✝¹ : IsScalarTower R S S'
inst✝ : IsLocalization (Submonoid.map (algebraMap R S) M) S'
x : S
s : Finset S'
hx : (algebraMap S S') x ∈ Submodule.span R ↑s
⊢ ∃ m, m • x ∈ Submodule.span R ↑(finsetIntegerMultiple (Submonoid.map (algebraMap R S) M) s)

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Support

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝⁴ : CommRing R
inst✝³ : AddCommGroup M
inst✝² : Module R M
N : Type u_3
inst✝¹ : AddCommGroup N
inst✝ : Module R N
f : M →ₗ[R] N
hf : Function.Injective ⇑f
x : PrimeSpectrum R
m : M
hm : ∀ r ∉ x.asIdeal, r • m ≠ 0
r : R
hr : r ∉ x.asIdeal
⊢ r • f m ≠ 0

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Support

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝⁴ : CommRing R
inst✝³ : AddCommGroup M
inst✝² : Module R M
N : Type u_3
inst✝¹ : AddCommGroup N
inst✝ : Module R N
f : M →ₗ[R] N
hf : Function.Injective ⇑f
⊢ support R M ⊆ support R N

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Support

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝⁴ : CommRing R
inst✝³ : AddCommGroup M
inst✝² : Module R M
N : Type u_3
inst✝¹ : AddCommGroup N
inst✝ : Module R N
f : M →ₗ[R] N
hf : Function.Surjective ⇑f
x : PrimeSpectrum R
m : M
hm : ∀ r ∉ x.asIdeal, r • f m ≠ 0
r : R
hr : r ∉ x.asIdeal
e : r • m = 0
⊢ r • f m = 0

Tactic:
simpa using congr(f $e)

Goal state after:
Module: Mathlib.RingTheory.Support

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝⁴ : CommRing R
inst✝³ : AddCommGroup M
inst✝² : Module R M
N : Type u_3
inst✝¹ : AddCommGroup N
inst✝ : Module R N
f : M →ₗ[R] N
hf : Function.Surjective ⇑f
x : PrimeSpectrum R
m : M
hm : ∀ r ∉ x.asIdeal, r • f m ≠ 0
r : R
hr : r ∉ x.asIdeal
e : r • m = 0
⊢ r • f m = 0

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Support

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝⁴ : CommRing R
inst✝³ : AddCommGroup M
inst✝² : Module R M
N : Type u_3
inst✝¹ : AddCommGroup N
inst✝ : Module R N
f : M →ₗ[R] N
hf : Function.Surjective ⇑f
⊢ support R N ⊆ support R M

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Localization.Finiteness

Goal state before:
pp: R : Type u_1
inst✝⁷ : CommSemiring R
M : Submonoid R
R' : Type u_3
inst✝⁶ : CommSemiring R'
inst✝⁵ : Algebra R R'
N : Type u_5
inst✝⁴ : AddCommMonoid N
inst✝³ : Module R N
inst✝² : Module R' N
inst✝¹ : IsScalarTower R R' N
inst✝ : IsLocalization M R'
s : Set N
x : N
hx : x ∈ Submodule.span R' s
⊢ ∃ t, t • x ∈ Submodule.span R s

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Localization.Finiteness

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝⁷ : CommSemiring R
inst✝⁶ : CommSemiring S
M : Submonoid R
R' : Type u_3
inst✝⁵ : CommSemiring R'
inst✝⁴ : Algebra R R'
inst✝³ : Algebra R' S
inst✝² : Algebra R S
inst✝¹ : IsScalarTower R R' S
inst✝ : IsLocalization M R'
s : Set S
x : S
hx : x ∈ Algebra.adjoin R' s
⊢ ∃ t, t • x ∈ Algebra.adjoin R s

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Localization.Finiteness

Goal state before:
pp: R : Type u_3
S : Type u_4
Rₚ : Type u_1
Sₚ : Type u_2
inst✝¹² : CommSemiring R
inst✝¹¹ : CommSemiring S
inst✝¹⁰ : CommSemiring Rₚ
inst✝⁹ : CommSemiring Sₚ
inst✝⁸ : Algebra R S
inst✝⁷ : Algebra R Rₚ
inst✝⁶ : Algebra R Sₚ
inst✝⁵ : Algebra S Sₚ
inst✝⁴ : Algebra Rₚ Sₚ
inst✝³ : IsScalarTower R S Sₚ
inst✝² : IsScalarTower R Rₚ Sₚ
M : Submonoid R
inst✝¹ : IsLocalization M Rₚ
inst✝ : IsLocalization (Algebra.algebraMapSubmonoid S M) Sₚ
hRS : Module.Finite R S
⊢ algebraMap Rₚ Sₚ = IsLocalization.map Sₚ (algebraMap R S) ⋯

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Flat.EquationalCriterion

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝⁶ : CommRing R
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : Flat R M
K : Type u_3
n : ℕ
inst✝² : AddCommGroup K
inst✝¹ : Module R K
inst✝ : Module.Finite R K
f : K →ₗ[R] Fin n →₀ R
x : (Fin n →₀ R) →ₗ[R] M
h : x ∘ₗ f = 0
K' : Submodule R K
hK' : K'.FG
⊢ ∃ k a y, x = y ∘ₗ a ∧ K' ≤ (a ∘ₗ f).ker

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Localization.Finiteness

Goal state before:
pp: R : Type u_3
S : Type u_4
Rₚ : Type u_1
Sₚ : Type u_2
inst✝¹² : CommSemiring R
inst✝¹¹ : CommSemiring S
inst✝¹⁰ : CommSemiring Rₚ
inst✝⁹ : CommSemiring Sₚ
inst✝⁸ : Algebra R S
inst✝⁷ : Algebra R Rₚ
inst✝⁶ : Algebra R Sₚ
inst✝⁵ : Algebra S Sₚ
inst✝⁴ : Algebra Rₚ Sₚ
inst✝³ : IsScalarTower R S Sₚ
inst✝² : IsScalarTower R Rₚ Sₚ
M : Submonoid R
inst✝¹ : IsLocalization M Rₚ
inst✝ : IsLocalization (Algebra.algebraMapSubmonoid S M) Sₚ
hRS : Module.Finite R S
⊢ Module.Finite Rₚ Sₚ

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Localization.Finiteness

Goal state before:
pp: R : Type u
inst✝¹¹ : CommSemiring R
S : Submonoid R
Rₚ : Type v
inst✝¹⁰ : CommSemiring Rₚ
inst✝⁹ : Algebra R Rₚ
inst✝⁸ : IsLocalization S Rₚ
M : Type w
inst✝⁷ : AddCommMonoid M
inst✝⁶ : Module R M
Mₚ : Type t
inst✝⁵ : AddCommMonoid Mₚ
inst✝⁴ : Module R Mₚ
inst✝³ : Module Rₚ Mₚ
inst✝² : IsScalarTower R Rₚ Mₚ
f : M →ₗ[R] Mₚ
inst✝¹ : IsLocalizedModule S f
inst✝ : Module.Finite R M
⊢ Module.Finite Rₚ Mₚ

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Flat.EquationalCriterion

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝⁶ : CommRing R
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : Flat R M
K : Type u_3
n : ℕ
inst✝² : AddCommGroup K
inst✝¹ : Module R K
inst✝ : Module.Finite R K
f : K →ₗ[R] Fin n →₀ R
x : (Fin n →₀ R) →ₗ[R] M
h : x ∘ₗ f = 0
⊢ ∃ k a y, x = y ∘ₗ a ∧ a ∘ₗ f = 0

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Flat.EquationalCriterion

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝¹⁰ : CommRing R
inst✝⁹ : AddCommGroup M
inst✝⁸ : Module R M
inst✝⁷ : Flat R M
K : Type u_3
N : Type u_4
inst✝⁶ : AddCommGroup K
inst✝⁵ : Module R K
inst✝⁴ : Module.Finite R K
inst✝³ : AddCommGroup N
inst✝² : Module R N
inst✝¹ : Free R N
inst✝ : Module.Finite R N
f : K →ₗ[R] N
x : N →ₗ[R] M
h : x ∘ₗ f = 0
e : (Fin (Fintype.card (Free.ChooseBasisIndex R N)) →₀ R) ≃ₗ[R] N
⊢ (x ∘ₗ ↑e) ∘ₗ ↑e.symm ∘ₗ f = 0

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Support

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝⁶ : CommRing R
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
N : Type u_3
P : Type u_4
inst✝³ : AddCommGroup N
inst✝² : Module R N
inst✝¹ : AddCommGroup P
inst✝ : Module R P
f : M →ₗ[R] N
g : N →ₗ[R] P
h : Function.Exact ⇑f ⇑g
hf : Function.Injective ⇑f
hg : Function.Surjective ⇑g
x : PrimeSpectrum R
H₁ : ∀ (m : M), ∃ r ∉ x.asIdeal, r • m = 0
H₂ : ∀ (m : P), ∃ r ∉ x.asIdeal, r • m = 0
m : N
r : R
hr : r ∉ x.asIdeal
m' : M
hm' : f m' = r • m
s : R
hs : s ∉ x.asIdeal
e₁ : s • m' = 0
⊢ (s * r) • m = 0

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Support

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝⁶ : CommRing R
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
N : Type u_3
P : Type u_4
inst✝³ : AddCommGroup N
inst✝² : Module R N
inst✝¹ : AddCommGroup P
inst✝ : Module R P
f : M →ₗ[R] N
g : N →ₗ[R] P
h : Function.Exact ⇑f ⇑g
hf : Function.Injective ⇑f
hg : Function.Surjective ⇑g
⊢ support R N = support R M ∪ support R P

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Flat.EquationalCriterion

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝¹⁰ : CommRing R
inst✝⁹ : AddCommGroup M
inst✝⁸ : Module R M
inst✝⁷ : Flat R M
K : Type u_3
N : Type u_4
inst✝⁶ : AddCommGroup K
inst✝⁵ : Module R K
inst✝⁴ : Module.Finite R K
inst✝³ : AddCommGroup N
inst✝² : Module R N
inst✝¹ : Free R N
inst✝ : Module.Finite R N
f : K →ₗ[R] N
x : N →ₗ[R] M
h : x ∘ₗ f = 0
e : (Fin (Fintype.card (Free.ChooseBasisIndex R N)) →₀ R) ≃ₗ[R] N
k : ℕ
a : (Fin (Fintype.card (Free.ChooseBasisIndex R N)) →₀ R) →ₗ[R] Fin k →₀ R
y : (Fin k →₀ R) →ₗ[R] M
hya : x ∘ₗ ↑e = y ∘ₗ a
haf : a ∘ₗ ↑e.symm ∘ₗ f = 0
⊢ x = y ∘ₗ a ∘ₗ ↑e.symm

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Localization.Finiteness

Goal state before:
pp: R : Type u
inst✝¹⁰ : CommSemiring R
M : Type w
inst✝⁹ : AddCommMonoid M
inst✝⁸ : Module R M
t : Finset R
ht : Ideal.span ↑t = ⊤
Mₚ : ↥t → Type u_1
inst✝⁷ : (g : ↥t) → AddCommMonoid (Mₚ g)
inst✝⁶ : (g : ↥t) → Module R (Mₚ g)
Rₚ : ↥t → Type u_2
inst✝⁵ : (g : ↥t) → CommSemiring (Rₚ g)
inst✝⁴ : (g : ↥t) → Algebra R (Rₚ g)
inst✝³ : ∀ (g : ↥t), IsLocalization.Away (↑g) (Rₚ g)
inst✝² : (g : ↥t) → Module (Rₚ g) (Mₚ g)
inst✝¹ : ∀ (g : ↥t), IsScalarTower R (Rₚ g) (Mₚ g)
f : (g : ↥t) → M →ₗ[R] Mₚ g
inst✝ : ∀ (g : ↥t), IsLocalizedModule.Away (↑g) (f g)
H : ∀ (g : ↥t), Module.Finite (Rₚ g) (Mₚ g)
s₁ : (g : ↥t) → Finset (Mₚ g)
s₂ : ∀ (g : ↥t), Submodule.span (Rₚ g) ↑(s₁ g) = ⊤
sf : ↥t → Finset M := fun x ↦ IsLocalizedModule.finsetIntegerMultiple (Submonoid.powers ↑x) (f x) (s₁ x)
x : M
r : ↑↑t
S : Submonoid R := Submonoid.powers ↑r
⊢ IsLocalizedModule.mk' (f r) x 1 ∈ Submodule.span (Rₚ r) ↑(s₁ r)

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Flat.EquationalCriterion

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝¹⁰ : CommRing R
inst✝⁹ : AddCommGroup M
inst✝⁸ : Module R M
inst✝⁷ : Flat R M
K : Type u_3
N : Type u_4
inst✝⁶ : AddCommGroup K
inst✝⁵ : Module R K
inst✝⁴ : Module.Finite R K
inst✝³ : AddCommGroup N
inst✝² : Module R N
inst✝¹ : Free R N
inst✝ : Module.Finite R N
f : K →ₗ[R] N
x : N →ₗ[R] M
h : x ∘ₗ f = 0
e : (Fin (Fintype.card (Free.ChooseBasisIndex R N)) →₀ R) ≃ₗ[R] N
k : ℕ
a : (Fin (Fintype.card (Free.ChooseBasisIndex R N)) →₀ R) →ₗ[R] Fin k →₀ R
y : (Fin k →₀ R) →ₗ[R] M
hya : x ∘ₗ ↑e = y ∘ₗ a
haf : a ∘ₗ ↑e.symm ∘ₗ f = 0
⊢ (a ∘ₗ ↑e.symm) ∘ₗ f = 0

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Support

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝³ : CommRing R
inst✝² : AddCommGroup M
inst✝¹ : Module R M
p : PrimeSpectrum R
inst✝ : Module.Finite R M
⊢ p ∈ support R M ↔ annihilator R M ≤ p.asIdeal

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Support

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝⁵ : CommRing R
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
inst✝² : Module.Finite R M
p : Ideal R
inst✝¹ : p.IsPrime
inst✝ : Subsingleton (LocalizedModule p.primeCompl M)
⊢ { asIdeal := p, isPrime := ⋯ } ∈ (Module.support R M)ᶜ

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Flat.EquationalCriterion

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝⁶ : CommRing R
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : Flat R M
P : Type u_3
inst✝² : AddCommGroup P
inst✝¹ : Module R P
inst✝ : FinitePresentation R P
h₁ : P →ₗ[R] M
w✝ : ℕ
K : Submodule R (Fin w✝ → R)
ϕ : P ≃ₗ[R] (Fin w✝ → R) ⧸ K
hK : K.FG
this : Module.Finite R ↥K
⊢ (h₁ ∘ₗ ↑ϕ.symm ∘ₗ K.mkQ) ∘ₗ K.subtype = 0

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Support

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝⁵ : CommRing R
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
inst✝² : Module.Finite R M
p : Ideal R
inst✝¹ : p.IsPrime
inst✝ : Subsingleton (LocalizedModule p.primeCompl M)
f : R
hf : f ∈ ↑(Module.annihilator R M)
hf' : { asIdeal := p, isPrime := ⋯ } ∈ (zeroLocus {f})ᶜ
⊢ f ∉ p

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Support

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝⁵ : CommRing R
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
inst✝² : Module.Finite R M
p : Ideal R
inst✝¹ : p.IsPrime
inst✝ : Subsingleton (LocalizedModule p.primeCompl M)
⊢ ∃ f ∉ p, Subsingleton (LocalizedModule.Away f M)

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Support

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝⁸ : CommRing R
inst✝⁷ : AddCommGroup M
inst✝⁶ : Module R M
inst✝⁵ : Module.Finite R M
M' : Type u_5
inst✝⁴ : AddCommMonoid M'
inst✝³ : Module R M'
l : M →ₗ[R] M'
p : Ideal R
inst✝² : p.IsPrime
inst✝¹ : IsLocalizedModule p.primeCompl l
inst✝ : Subsingleton M'
⊢ ∃ f ∉ p, Subsingleton (LocalizedModule.Away f M)

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Support

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝⁶ : CommRing R
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
N : Type u_3
inst✝³ : AddCommGroup N
inst✝² : Module R N
inst✝¹ : Module.Finite R M
p : Ideal R
inst✝ : p.IsPrime
φ : N →ₗ[R] M
hφ : Function.Surjective ⇑((LocalizedModule.map p.primeCompl) φ)
⊢ ∃ a ∉ p, Function.Surjective ⇑((LocalizedModule.map (Submonoid.powers a)) φ)

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Support

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝³ : CommRing R
inst✝² : AddCommGroup M
inst✝¹ : Module R M
inst✝ : Module.Finite R M
I : Ideal R
p : PrimeSpectrum R
hp₁ : Nontrivial (LocalizedModule p.asIdeal.primeCompl M)
hp₂ : p ∈ zeroLocus ↑I
Rₚ : Type u_1 := Localization.AtPrime p.asIdeal
Mₚ : Type (max u_1 u_2) := LocalizedModule p.asIdeal.primeCompl M
Mₚ' : Type (max u_1 u_2) := LocalizedModule p.asIdeal.primeCompl (M ⧸ I • ⊤)
Mₚ'' : Type (max u_1 u_2) := Mₚ ⧸ Ideal.map (algebraMap R Rₚ) I • ⊤
⊢ Submodule.localized p.asIdeal.primeCompl (I • ⊤) = Ideal.map (algebraMap R Rₚ) I • ⊤

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Flat.EquationalCriterion

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝⁶ : CommRing R
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : Flat R M
P : Type u_3
inst✝² : AddCommGroup P
inst✝¹ : Module R P
inst✝ : FinitePresentation R P
h₁ : P →ₗ[R] M
w✝ : ℕ
K : Submodule R (Fin w✝ → R)
ϕ : P ≃ₗ[R] (Fin w✝ → R) ⧸ K
hK : K.FG
this✝ : Module.Finite R ↥K
this : (h₁ ∘ₗ ↑ϕ.symm ∘ₗ K.mkQ) ∘ₗ K.subtype = 0
k : ℕ
a : (Fin w✝ → R) →ₗ[R] Fin k →₀ R
y : (Fin k →₀ R) →ₗ[R] M
hay : h₁ ∘ₗ ↑ϕ.symm ∘ₗ K.mkQ = y ∘ₗ a
ha : a ∘ₗ K.subtype = 0
⊢ K ≤ a.ker

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Support

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝³ : CommRing R
inst✝² : AddCommGroup M
inst✝¹ : Module R M
inst✝ : Module.Finite R M
I : Ideal R
p : PrimeSpectrum R
hp₁ : Nontrivial (LocalizedModule p.asIdeal.primeCompl M)
hp₂ : p ∈ zeroLocus ↑I
Rₚ : Type u_1 := Localization.AtPrime p.asIdeal
Mₚ : Type (max u_1 u_2) := LocalizedModule p.asIdeal.primeCompl M
Mₚ' : Type (max u_1 u_2) := LocalizedModule p.asIdeal.primeCompl (M ⧸ I • ⊤)
Mₚ'' : Type (max u_1 u_2) := Mₚ ⧸ Ideal.map (algebraMap R Rₚ) I • ⊤
e : Mₚ' ≃ₗ[Rₚ] Mₚ'' :=
  (localizedQuotientEquiv p.asIdeal.primeCompl (I • ⊤)).symm ≪≫ₗ
    (Submodule.localized p.asIdeal.primeCompl (I • ⊤)).quotEquivOfEq (Ideal.map (algebraMap R Rₚ) I • ⊤) ⋯
⊢ Nontrivial Mₚ''

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Support

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝³ : CommRing R
inst✝² : AddCommGroup M
inst✝¹ : Module R M
inst✝ : Module.Finite R M
I : Ideal R
⊢ support R (M ⧸ I • ⊤) = support R M ∩ zeroLocus ↑I

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Support

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝³ : CommRing R
inst✝² : AddCommGroup M
inst✝¹ : Module R M
inst✝ : Module.Finite R M
x : R
⊢ support R M ∩ zeroLocus ↑(Ideal.span {x}) = support R M ∩ zeroLocus {x}

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Flat.EquationalCriterion

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝⁶ : CommRing R
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : Flat R M
P : Type u_3
inst✝² : AddCommGroup P
inst✝¹ : Module R P
inst✝ : FinitePresentation R P
h₁ : P →ₗ[R] M
⊢ ∃ k h₂ h₃, h₁ = h₃ ∘ₗ h₂

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Module.StablyFree.Basic

Goal state before:
pp: R : Type u
inst✝⁵ : Ring R
M : Type v
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
N : Type w
inst✝² : AddCommGroup N
inst✝¹ : Module R N
e : M ≃ₗ[R] N
inst✝ : IsStablyFree R M
⊢ IsStablyFree R N

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Module.StablyFree.Basic

Goal state before:
pp: R : Type u
inst✝⁵ : Ring R
M : Type v
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
N : Type w
inst✝² : AddCommGroup N
inst✝¹ : Module R N
inst✝ : IsStablyFree R M
⊢ Projective R M

Tactic:
by

Goal state after:
Module: Mathlib.Data.Fin.Parity

Goal state before:
pp: n : ℕ
i : Fin (n + 1)
j : Fin n
⊢ Even (↑(i.succAbove j) + ↑(j.predAbove i)) ↔ Odd (↑i + ↑j)

Tactic:
by

Goal state after:
Module: Mathlib.Data.Fin.Parity

Goal state before:
pp: n : ℕ
R : Type u_1
inst✝¹ : Monoid R
inst✝ : HasDistribNeg R
i : Fin (n + 1)
j : Fin n
⊢ (-1) ^ (↑(i.succAbove j) + ↑(j.predAbove i)) = -(-1) ^ (↑i + ↑j)

Tactic:
by

Goal state after:
Module: Mathlib.Data.Fin.Parity

Goal state before:
pp: n : ℕ
k : Fin n
h : Even ↑k
⊢ Even k

Tactic:
by

Goal state after:
Module: Mathlib.Data.Fin.Parity

Goal state before:
pp: n : ℕ
k : Fin n
inst✝ : NeZero n
h : Odd ↑k
⊢ Odd k

Tactic:
by

Goal state after:
Module: Mathlib.Data.Fin.Parity

Goal state before:
pp: n : ℕ
hn : Odd n
k : Fin n
⊢ Even k

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Localization.Finiteness

Goal state before:
pp: R : Type u
inst✝¹⁰ : CommSemiring R
M : Type w
inst✝⁹ : AddCommMonoid M
inst✝⁸ : Module R M
t : Finset R
ht : Ideal.span ↑t = ⊤
Mₚ : ↥t → Type u_1
inst✝⁷ : (g : ↥t) → AddCommMonoid (Mₚ g)
inst✝⁶ : (g : ↥t) → Module R (Mₚ g)
Rₚ : ↥t → Type u_2
inst✝⁵ : (g : ↥t) → CommSemiring (Rₚ g)
inst✝⁴ : (g : ↥t) → Algebra R (Rₚ g)
inst✝³ : ∀ (g : ↥t), IsLocalization.Away (↑g) (Rₚ g)
inst✝² : (g : ↥t) → Module (Rₚ g) (Mₚ g)
inst✝¹ : ∀ (g : ↥t), IsScalarTower R (Rₚ g) (Mₚ g)
f : (g : ↥t) → M →ₗ[R] Mₚ g
inst✝ : ∀ (g : ↥t), IsLocalizedModule.Away (↑g) (f g)
H : ∀ (g : ↥t), Module.Finite (Rₚ g) (Mₚ g)
⊢ Module.Finite R M

Tactic:
by

Goal state after:
Module: Mathlib.Data.Fin.Parity

Goal state before:
pp: n : ℕ
inst✝ : NeZero n
hn : Odd n
k : Fin n
⊢ Odd k

Tactic:
by

Goal state after:
Module: Mathlib.Data.Fin.Parity

Goal state before:
pp: n : ℕ
k : Fin n
hn : Even n
⊢ Even k ↔ Even ↑k

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Localization.Finiteness

Goal state before:
pp: R : Type u
inst✝⁸ : CommSemiring R
M : Type w
inst✝⁷ : AddCommMonoid M
inst✝⁶ : Module R M
t : Set R
ht : Ideal.span t = ⊤
Mₚ : ↑t → Type u_1
inst✝⁵ : (g : ↑t) → AddCommMonoid (Mₚ g)
inst✝⁴ : (g : ↑t) → Module R (Mₚ g)
Rₚ : ↑t → Type u_2
inst✝³ : (g : ↑t) → CommSemiring (Rₚ g)
inst✝² : (g : ↑t) → Algebra R (Rₚ g)
h₁ : ∀ (g : ↑t), IsLocalization.Away (↑g) (Rₚ g)
inst✝¹ : (g : ↑t) → Module (Rₚ g) (Mₚ g)
inst✝ : ∀ (g : ↑t), IsScalarTower R (Rₚ g) (Mₚ g)
f : (g : ↑t) → M →ₗ[R] Mₚ g
h₂ : ∀ (g : ↑t), IsLocalizedModule.Away (↑g) (f g)
H : ∀ (g : ↑t), Module.Finite (Rₚ g) (Mₚ g)
⊢ Module.Finite R M

Tactic:
by

Goal state after:
Module: Mathlib.Data.Fin.Parity

Goal state before:
pp: n : ℕ
k : Fin n
inst✝ : NeZero n
hn : Even n
⊢ Odd k ↔ Odd ↑k

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Localization.Finiteness

Goal state before:
pp: R : Type u
inst✝¹⁰ : CommSemiring R
M : Type v
inst✝⁹ : AddCommMonoid M
inst✝⁸ : Module R M
N : Submodule R M
s : Set R
hs : Ideal.span s = ⊤
Mₚ : ↑s → Type u_1
inst✝⁷ : (g : ↑s) → AddCommMonoid (Mₚ g)
inst✝⁶ : (g : ↑s) → Module R (Mₚ g)
Rₚ : ↑s → Type u_2
inst✝⁵ : (g : ↑s) → CommSemiring (Rₚ g)
inst✝⁴ : (g : ↑s) → Algebra R (Rₚ g)
inst✝³ : ∀ (g : ↑s), IsLocalization.Away (↑g) (Rₚ g)
inst✝² : (g : ↑s) → Module (Rₚ g) (Mₚ g)
inst✝¹ : ∀ (g : ↑s), IsScalarTower R (Rₚ g) (Mₚ g)
ϕ : (g : ↑s) → M →ₗ[R] Mₚ g
inst✝ : ∀ (g : ↑s), IsLocalizedModule (Submonoid.powers ↑g) (ϕ g)
H : ∀ (g : ↑s), (localized' (Rₚ g) (Submonoid.powers ↑g) (ϕ g) N).FG
⊢ N.FG

Tactic:
by

Goal state after:
Module: Mathlib.Data.Fin.Parity

Goal state before:
pp: n : ℕ
k : Fin n
⊢ Even k ↔ Odd n ∨ Even ↑k

Tactic:
by

Goal state after:
Module: Mathlib.Data.Fin.Parity

Goal state before:
pp: n : ℕ
k : Fin n
⊢ Even k ↔ Even n → Even ↑k

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Localization.Finiteness

Goal state before:
pp: R : Type u
inst✝¹⁰ : CommSemiring R
M : Type v
inst✝⁹ : AddCommMonoid M
inst✝⁸ : Module R M
N : Submodule R M
R' : Type u_1
inst✝⁷ : CommSemiring R'
inst✝⁶ : Algebra R R'
M' : Type u_2
inst✝⁵ : AddCommMonoid M'
inst✝⁴ : Module R M'
inst✝³ : Module R' M'
inst✝² : IsScalarTower R R' M'
S : Submonoid R
inst✝¹ : IsLocalization S R'
f : M →ₗ[R] M'
inst✝ : IsLocalizedModule S f
s : Set M
hfin : s.Finite
hspan : span R s = N
⊢ span R' (⇑f '' s) = localized' R' S f N

Tactic:
by

Goal state after:
Module: Mathlib.Data.Fin.Parity

Goal state before:
pp: n : ℕ
k : Fin n
inst✝ : NeZero n
⊢ Odd k ↔ Odd n ∨ Odd ↑k

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Localization.Finiteness

Goal state before:
pp: R : Type u
inst✝¹⁰ : CommSemiring R
M : Type v
inst✝⁹ : AddCommMonoid M
inst✝⁸ : Module R M
N : Submodule R M
R' : Type u_1
inst✝⁷ : CommSemiring R'
inst✝⁶ : Algebra R R'
M' : Type u_2
inst✝⁵ : AddCommMonoid M'
inst✝⁴ : Module R M'
inst✝³ : Module R' M'
inst✝² : IsScalarTower R R' M'
S : Submonoid R
inst✝¹ : IsLocalization S R'
f : M →ₗ[R] M'
inst✝ : IsLocalizedModule S f
h : N.FG
⊢ (localized' R' S f N).FG

Tactic:
by

Goal state after:
Module: Mathlib.Data.Fin.Parity

Goal state before:
pp: n : ℕ
k : Fin n
inst✝ : NeZero n
⊢ Odd k ↔ Even n → Odd ↑k

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Localization.Finiteness

Goal state before:
pp: R : Type u
inst✝ : CommSemiring R
I : Ideal R
t : Set R
ht : span t = ⊤
H : ∀ (g : ↑t), (map (algebraMap R (Localization.Away ↑g)) I).FG
⊢ I.FG

Tactic:
by

Goal state after:
Module: Mathlib.Data.Fin.Parity

Goal state before:
pp: n : ℕ
k : Fin n
hn : Even n
⊢ Even k ↔ ↑k % 2 = 0

Tactic:
by

Goal state after:
Module: Mathlib.Data.Fin.Parity

Goal state before:
pp: n : ℕ
k : Fin n
inst✝ : NeZero n
hn : Even n
⊢ Odd k ↔ ↑k % 2 = 1

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Localization.Finiteness

Goal state before:
pp: R : Type u
inst✝¹ : CommSemiring R
S : Type v
inst✝ : CommSemiring S
f : R →+* S
t : Set R
ht : Ideal.span t = ⊤
H : ∀ (g : ↑t), (ker (Localization.awayMap f ↑g)).FG
⊢ (ker f).FG

Tactic:
by

Goal state after:
Module: Mathlib.Data.Fin.Parity

Goal state before:
pp: n : ℕ
k : Fin n
inst✝ : NeZero n
hn : Even n
⊢ ¬Odd k ↔ Even k

Tactic:
by

Goal state after: