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

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
this : annihilator R R = ⊥
⊢ ringKrullDim (R ⧸ annihilator R R) = ringKrullDim R

Tactic:
this

Goal state after:
R : Type u_1
inst✝ : CommRing R
this : annihilator R R = ⊥
⊢ ringKrullDim (R ⧸ ⊥) = ringKrullDim R
Module: Mathlib.Topology.Algebra.Category.ProfiniteGrp.Limits

Goal state before:
pp: P : ProfiniteGrp.{u}
U : Set ↑(limit P.diagram).toProfinite.toTop
s : Set ((j : OpenNormalSubgroup ↑P.toProfinite.toTop) → ↑(P.diagram.obj j).toProfinite.toTop)
hsO : IsOpen s
hsv : Subtype.val ⁻¹' s = U
spc : (j : OpenNormalSubgroup ↑P.toProfinite.toTop) → ↑(P.diagram.obj j).toProfinite.toTop
hspc : spc ∈ limitConePtAux P.diagram
uDefaultSpec : ⟨spc, hspc⟩ ∈ U
⊢ (U ∩ Set.range ⇑(Hom.hom P.toLimit)).Nonempty

Tactic:
[

Goal state after:
P : ProfiniteGrp.{u}
U : Set ↑(limit P.diagram).toProfinite.toTop
s : Set ((j : OpenNormalSubgroup ↑P.toProfinite.toTop) → ↑(P.diagram.obj j).toProfinite.toTop)
hsO : IsOpen s
hsv : Subtype.val ⁻¹' s = U
spc : (j : OpenNormalSubgroup ↑P.toProfinite.toTop) → ↑(P.diagram.obj j).toProfinite.toTop
hspc : spc ∈ limitConePtAux P.diagram
uDefaultSpec : ⟨spc, hspc⟩ ∈ U
⊢ (U ∩ Set.range ⇑(Hom.hom P.toLimit)).Nonempty
Module: Mathlib.RingTheory.KrullDimension.PID

Goal state before:
pp: R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsPrincipalIdealRing R
I : Ideal R
hI : I.IsPrime
hI' : I ∉ {x | Minimal Ideal.IsPrime x}
⊢ I.IsMaximal

Tactic:
Set.notMem_ofPred_iff,

Goal state after:
R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsPrincipalIdealRing R
I : Ideal R
hI : I.IsPrime
hI' : ¬Minimal Ideal.IsPrime I
⊢ I.IsMaximal
Module: Mathlib.RingTheory.KrullDimension.PID

Goal state before:
pp: R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : IsPrincipalIdealRing R
h : ¬IsField R
⊢ ringKrullDim R ≤ 1

pp: R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : IsPrincipalIdealRing R
h : ¬IsField R
h' : ringKrullDim R < 1
⊢ IsField R

Tactic:
· rw [← Nat.cast_one, ← Ring.krullDimLE_iff]
    infer_instance

Goal state after:
R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : IsPrincipalIdealRing R
h : ¬IsField R
h' : ringKrullDim R < 1
⊢ IsField R
Module: Mathlib.RingTheory.KrullDimension.Polynomial

Goal state before:
pp: R : Type u_1
inst✝³ : CommRing R
inst✝² : IsNoetherianRing R
p : Ideal R
P : Ideal R[X]
inst✝¹ : P.IsMaximal
inst✝ : P.LiesOver p
⊢ p.IsPrime

Tactic:
P.over_def p

Goal state after:
R : Type u_1
inst✝³ : CommRing R
inst✝² : IsNoetherianRing R
p : Ideal R
P : Ideal R[X]
inst✝¹ : P.IsMaximal
inst✝ : P.LiesOver p
⊢ (under R P).IsPrime
Module: Mathlib.RingTheory.Spectrum.Prime.LTSeries

Goal state before:
pp: case neg
R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsNoetherianRing R
inst✝ : IsLocalRing R
p₁ p₀ : PrimeSpectrum R
h₀ : p₀ < p₁
h₁ : p₁ < closedPoint R
x : R
hx : x ∈ 𝔪
hn : x ∉ p₀.asIdeal
e : ↑(zeroLocus ↑p₀.asIdeal) ≃o PrimeSpectrum (R ⧸ p₀.asIdeal) := p₀.asIdeal.primeSpectrumQuotientOrderIsoZeroLocus.symm
q✝ : Ideal R
hq : q✝ ∈ (p₀.asIdeal + span {x}).minimalPrimes
q : PrimeSpectrum R := { asIdeal := q✝, isPrime := ⋯ }
this : q.asIdeal.IsPrime
hxq : x ∈ q.asIdeal
hqm : q.asIdeal = 𝔪
h : (e ⟨q, ⋯⟩).asIdeal.height ≤ 1
⊢ False

Tactic:
[

Goal state after:
case neg
R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsNoetherianRing R
inst✝ : IsLocalRing R
p₁ p₀ : PrimeSpectrum R
h₀ : p₀ < p₁
h₁ : p₁ < closedPoint R
x : R
hx : x ∈ 𝔪
hn : x ∉ p₀.asIdeal
e : ↑(zeroLocus ↑p₀.asIdeal) ≃o PrimeSpectrum (R ⧸ p₀.asIdeal) := p₀.asIdeal.primeSpectrumQuotientOrderIsoZeroLocus.symm
q✝ : Ideal R
hq : q✝ ∈ (p₀.asIdeal + span {x}).minimalPrimes
q : PrimeSpectrum R := { asIdeal := q✝, isPrime := ⋯ }
this : q.asIdeal.IsPrime
hxq : x ∈ q.asIdeal
hqm : q.asIdeal = 𝔪
h : (e ⟨q, ⋯⟩).asIdeal.height ≤ 1
⊢ False
Module: Mathlib.RingTheory.Lasker

Goal state before:
pp: case refine_2.refine_1
R : Type u_1
M : Type u_2
inst✝² : CommSemiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
N : Submodule R M
s : Finset (Submodule R M)
hs : s.inf id = N
hs' : ∀ ⦃J : Submodule R M⦄, J ∈ s → J.IsPrimary
J : Submodule R M
hJ : J ∈ s
⊢ J ∈ {I ∈ s | (I.colon Set.univ).radical = (J.colon Set.univ).radical}

pp: case refine_2.refine_2
R : Type u_1
M : Type u_2
inst✝² : CommSemiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
N : Submodule R M
s : Finset (Submodule R M)
hs : s.inf id = N
hs' : ∀ ⦃J : Submodule R M⦄, J ∈ s → J.IsPrimary
J : Submodule R M
hJ : J ∈ s
⊢ ∀ ⦃y : Submodule R M⦄, y ∈ {I ∈ s | (I.colon Set.univ).radical = (J.colon Set.univ).radical} → (id y).IsPrimary

Tactic:
· simp [hJ]

Goal state after:
case refine_2.refine_2
R : Type u_1
M : Type u_2
inst✝² : CommSemiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
N : Submodule R M
s : Finset (Submodule R M)
hs : s.inf id = N
hs' : ∀ ⦃J : Submodule R M⦄, J ∈ s → J.IsPrimary
J : Submodule R M
hJ : J ∈ s
⊢ ∀ ⦃y : Submodule R M⦄, y ∈ {I ∈ s | (I.colon Set.univ).radical = (J.colon Set.univ).radical} → (id y).IsPrimary
Module: Mathlib.RingTheory.Lasker

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝² : CommSemiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
N : Submodule R M
s : Finset (Submodule R M)
hs : s.inf id = N
hs' : ∀ ⦃J : Submodule R M⦄, J ∈ s → J.IsPrimary
I J I' : Submodule R M
hI' : I' ∈ s
hI : {I ∈ s | (I.colon Set.univ).radical = (I'.colon Set.univ).radical}.inf id = I
J' : Submodule R M
hJ' : J' ∈ s
hJ : {I ∈ s | (I.colon Set.univ).radical = (J'.colon Set.univ).radical}.inf id = J
hIJ : (I.colon Set.univ).radical = (J.colon Set.univ).radical
this : (I'.colon Set.univ).radical = (J'.colon Set.univ).radical
⊢ {I ∈ s | (I.colon Set.univ).radical = (I'.colon Set.univ).radical}.inf id =
    {I ∈ s | (I.colon Set.univ).radical = (J'.colon Set.univ).radical}.inf id

Tactic:
this

Goal state after:
R : Type u_1
M : Type u_2
inst✝² : CommSemiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
N : Submodule R M
s : Finset (Submodule R M)
hs : s.inf id = N
hs' : ∀ ⦃J : Submodule R M⦄, J ∈ s → J.IsPrimary
I J I' : Submodule R M
hI' : I' ∈ s
hI : {I ∈ s | (I.colon Set.univ).radical = (I'.colon Set.univ).radical}.inf id = I
J' : Submodule R M
hJ' : J' ∈ s
hJ : {I ∈ s | (I.colon Set.univ).radical = (J'.colon Set.univ).radical}.inf id = J
hIJ : (I.colon Set.univ).radical = (J.colon Set.univ).radical
this : (I'.colon Set.univ).radical = (J'.colon Set.univ).radical
⊢ {I ∈ s | (I.colon Set.univ).radical = (J'.colon Set.univ).radical}.inf id =
    {I ∈ s | (I.colon Set.univ).radical = (J'.colon Set.univ).radical}.inf id
Module: Mathlib.RingTheory.Lasker

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝² : CommSemiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
N : Submodule R M
t : Finset (Submodule R M)
ht : N.IsMinimalPrimaryDecomposition t
x : M
⊢ (N.colon {x}).radical = {x_1 ∈ t | x ∉ x_1}.inf fun q ↦ (q.colon Set.univ).radical

Tactic:
[

Goal state after:
R : Type u_1
M : Type u_2
inst✝² : CommSemiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
N : Submodule R M
t : Finset (Submodule R M)
ht : N.IsMinimalPrimaryDecomposition t
x : M
⊢ (N.colon {x}).radical = {x_1 ∈ t | x ∉ x_1}.inf fun q ↦ (q.colon Set.univ).radical
Module: Mathlib.RingTheory.Lasker

Goal state before:
pp: case neg
R : Type u_3
M : Type u_4
inst✝² : CommRing R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
N : Submodule R M
s₀ : Finset ↑N.associatedPrimes
hs₀ : IsLowerSet ↑s₀
q : Submodule R M
hqp : q.IsPrimary
p : ↑N.associatedPrimes
S : Submonoid R := ⨅ q ∈ s₀, (↑q).primeCompl
f : M →ₗ[R] LocalizedModule S M := mkLinearMap S M
hp : p ∉ s₀
hq : ¬↑(q.colon Set.univ) ⊆ ⋃ r ∈ s₀, ↑↑r
y : R
hy1 : y ∈ ↑(q.colon Set.univ)
hy2 : y ∈ S
x : M
⊢ (mkLinearMap S M) x ∈ localized₀ S (mkLinearMap S M) q

Tactic:
[

Goal state after:
case neg
R : Type u_3
M : Type u_4
inst✝² : CommRing R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
N : Submodule R M
s₀ : Finset ↑N.associatedPrimes
hs₀ : IsLowerSet ↑s₀
q : Submodule R M
hqp : q.IsPrimary
p : ↑N.associatedPrimes
S : Submonoid R := ⨅ q ∈ s₀, (↑q).primeCompl
f : M →ₗ[R] LocalizedModule S M := mkLinearMap S M
hp : p ∉ s₀
hq : ¬↑(q.colon Set.univ) ⊆ ⋃ r ∈ s₀, ↑↑r
y : R
hy1 : y ∈ ↑(q.colon Set.univ)
hy2 : y ∈ S
x : M
⊢ (mkLinearMap S M) x ∈ localized₀ S (mkLinearMap S M) q
Module: Mathlib.RingTheory.Lasker

Goal state before:
pp: R : Type u_3
M : Type u_4
inst✝² : CommRing R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
N : Submodule R M
t : Finset (Submodule R M)
ht : N.IsMinimalPrimaryDecomposition t
s₀ : Finset ↑N.associatedPrimes
hs₀ : IsLowerSet ↑s₀
s : Finset (Submodule R M)
hs : s ⊆ t
hs' : Finset.image (fun q ↦ (q.colon Set.univ).radical) s = Finset.image Subtype.val s₀
S : Submonoid R := ⨅ q ∈ s₀, (↑q).primeCompl
f : M →ₗ[R] LocalizedModule S M := mkLinearMap S M
⊢ (t.inf fun i ↦ comap (mkLinearMap S M) ((⇑(localized₀FrameHom S (mkLinearMap S M)) ∘ id) i)) = ⨅ q ∈ s, q

Tactic:
[

Goal state after:
R : Type u_3
M : Type u_4
inst✝² : CommRing R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
N : Submodule R M
t : Finset (Submodule R M)
ht : N.IsMinimalPrimaryDecomposition t
s₀ : Finset ↑N.associatedPrimes
hs₀ : IsLowerSet ↑s₀
s : Finset (Submodule R M)
hs : s ⊆ t
hs' : Finset.image (fun q ↦ (q.colon Set.univ).radical) s = Finset.image Subtype.val s₀
S : Submonoid R := ⨅ q ∈ s₀, (↑q).primeCompl
f : M →ₗ[R] LocalizedModule S M := mkLinearMap S M
⊢ (t.inf fun i ↦ comap (mkLinearMap S M) ((⇑(localized₀FrameHom S (mkLinearMap S M)) ∘ id) i)) = ⨅ q ∈ s, q
Module: Mathlib.RingTheory.Lasker

Goal state before:
pp: R : Type u_3
M : Type u_4
inst✝² : CommRing R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
N : Submodule R M
t : Finset (Submodule R M)
ht : N.IsMinimalPrimaryDecomposition t
s₀ : Finset ↑N.associatedPrimes
hs₀ : IsLowerSet ↑s₀
s : Finset (Submodule R M)
hs : s ⊆ t
hs' : Finset.image (fun q ↦ (q.colon Set.univ).radical) s = Finset.image Subtype.val s₀
S : Submonoid R := ⨅ q ∈ s₀, (↑q).primeCompl
f : M →ₗ[R] LocalizedModule S M := mkLinearMap S M
this : ∀ q ∈ t, comap f (localized₀ S f q) = if q ∈ s then q else ⊤
⊢ (s.inf fun a ↦ a) = ⨅ q ∈ s, q

Tactic:
Finset.inf_eq_iInf

Goal state after:
R : Type u_3
M : Type u_4
inst✝² : CommRing R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
N : Submodule R M
t : Finset (Submodule R M)
ht : N.IsMinimalPrimaryDecomposition t
s₀ : Finset ↑N.associatedPrimes
hs₀ : IsLowerSet ↑s₀
s : Finset (Submodule R M)
hs : s ⊆ t
hs' : Finset.image (fun q ↦ (q.colon Set.univ).radical) s = Finset.image Subtype.val s₀
S : Submonoid R := ⨅ q ∈ s₀, (↑q).primeCompl
f : M →ₗ[R] LocalizedModule S M := mkLinearMap S M
this : ∀ q ∈ t, comap f (localized₀ S f q) = if q ∈ s then q else ⊤
⊢ ⨅ a ∈ s, a = ⨅ q ∈ s, q
Module: Mathlib.RingTheory.LittleWedderburn

Goal state before:
pp: case intro.h
n : ℕ
IH : ∀ m < n, ∀ (D : Type u_1) [inst : DivisionRing D] [Finite D] (val : Fintype D), card D = m → Subring.center D = ⊤
D : Type u_1
inst✝¹ : DivisionRing D
inst✝ : Finite D
val✝ : Fintype D
hn : card D = n
⊢ Subring.center D = ⊤

Tactic:
apply InductionHyp.center_eq_top

Goal state after:
case intro.h
n : ℕ
IH : ∀ m < n, ∀ (D : Type u_1) [inst : DivisionRing D] [Finite D] (val : Fintype D), card D = m → Subring.center D = ⊤
D : Type u_1
inst✝¹ : DivisionRing D
inst✝ : Finite D
val✝ : Fintype D
hn : card D = n
⊢ InductionHyp D
Module: Mathlib.RingTheory.LocalProperties.Injective

Goal state before:
pp: R : Type u
inst✝¹⁴ : CommRing R
M : Type v
inst✝¹³ : AddCommGroup M
inst✝¹² : Module R M
S : Submonoid R
inst✝¹¹ : Small.{v, u} R
inst✝¹⁰ : IsNoetherianRing R
Rₛ : Type u'
inst✝⁹ : Small.{v', u'} Rₛ
inst✝⁸ : CommRing Rₛ
inst✝⁷ : Algebra R Rₛ
Mₛ : Type v'
inst✝⁶ : AddCommGroup Mₛ
inst✝⁵ : Module R Mₛ
inst✝⁴ : Module Rₛ Mₛ
inst✝³ : IsScalarTower R Rₛ Mₛ
f : M →ₗ[R] Mₛ
inst✝² : IsLocalization S Rₛ
inst✝¹ : IsLocalizedModule S f
inst✝ : Injective R M
MB : ∀ (I : Ideal R), Function.Surjective ⇑(LinearMap.lcomp R M (Submodule.subtype I))
I : Submodule R R
g : ↥(Submodule.localized' Rₛ S (Algebra.linearMap R Rₛ) I) →ₗ[Rₛ] Mₛ
this : FinitePresentation R ↥I
a : ↥S
g' : R →ₗ[R] M
e :
  ↑a • g =
    (mapExtendScalars S (Submodule.toLocalized' Rₛ S (Algebra.linearMap R Rₛ) I) f Rₛ)
      ((LinearMap.lcomp R M I.subtype) g', a).1
⊢ ((algebraMap Rₛ (End Rₛ (↥(Submodule.localized' Rₛ S (Algebra.linearMap R Rₛ) I) →ₗ[Rₛ] Mₛ))) ((algebraMap R Rₛ) ↑a))
      ((LinearMap.lcomp Rₛ Mₛ (Submodule.localized' Rₛ S (Algebra.linearMap R Rₛ) I).subtype)
        (IsLocalization.mk' Rₛ 1 a • (mapExtendScalars S (Algebra.linearMap R Rₛ) f Rₛ) g')) =
    ((algebraMap Rₛ (End Rₛ (↥(Submodule.localized' Rₛ S (Algebra.linearMap R Rₛ) I) →ₗ[Rₛ] Mₛ)))
        ((algebraMap R Rₛ) ↑a))
      g

Tactic:
[

Goal state after:
R : Type u
inst✝¹⁴ : CommRing R
M : Type v
inst✝¹³ : AddCommGroup M
inst✝¹² : Module R M
S : Submonoid R
inst✝¹¹ : Small.{v, u} R
inst✝¹⁰ : IsNoetherianRing R
Rₛ : Type u'
inst✝⁹ : Small.{v', u'} Rₛ
inst✝⁸ : CommRing Rₛ
inst✝⁷ : Algebra R Rₛ
Mₛ : Type v'
inst✝⁶ : AddCommGroup Mₛ
inst✝⁵ : Module R Mₛ
inst✝⁴ : Module Rₛ Mₛ
inst✝³ : IsScalarTower R Rₛ Mₛ
f : M →ₗ[R] Mₛ
inst✝² : IsLocalization S Rₛ
inst✝¹ : IsLocalizedModule S f
inst✝ : Injective R M
MB : ∀ (I : Ideal R), Function.Surjective ⇑(LinearMap.lcomp R M (Submodule.subtype I))
I : Submodule R R
g : ↥(Submodule.localized' Rₛ S (Algebra.linearMap R Rₛ) I) →ₗ[Rₛ] Mₛ
this : FinitePresentation R ↥I
a : ↥S
g' : R →ₗ[R] M
e :
  ↑a • g =
    (mapExtendScalars S (Submodule.toLocalized' Rₛ S (Algebra.linearMap R Rₛ) I) f Rₛ)
      ((LinearMap.lcomp R M I.subtype) g', a).1
⊢ ((algebraMap Rₛ (End Rₛ (↥(Submodule.localized' Rₛ S (Algebra.linearMap R Rₛ) I) →ₗ[Rₛ] Mₛ))) ((algebraMap R Rₛ) ↑a))
      ((LinearMap.lcomp Rₛ Mₛ (Submodule.localized' Rₛ S (Algebra.linearMap R Rₛ) I).subtype)
        (IsLocalization.mk' Rₛ 1 a • (mapExtendScalars S (Algebra.linearMap R Rₛ) f Rₛ) g')) =
    ((algebraMap Rₛ (End Rₛ (↥(Submodule.localized' Rₛ S (Algebra.linearMap R Rₛ) I) →ₗ[Rₛ] Mₛ)))
        ((algebraMap R Rₛ) ↑a))
      g
Module: Mathlib.RingTheory.LocalIso

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝⁸ : CommSemiring R
inst✝⁷ : CommSemiring S
inst✝⁶ : Algebra R S
T : Type u_3
inst✝⁵ : CommSemiring T
inst✝⁴ : Algebra S T
inst✝³ : Algebra R T
inst✝² : IsScalarTower R S T
inst✝¹ : IsLocalIso R S
inst✝ : IsLocalIso S T
s : Set S := {g | IsStandardOpenImmersion R (Localization.Away g)}
T' : S → Type u_3 := fun g ↦ Localization.Away ((algebraMap S T) g)
this✝³ : (g : S) → Algebra (Localization.Away g) (T' g) := fun g ↦ localizationAlgebra (Submonoid.powers g) T
T'' : S → T → Type u_3 := fun g x ↦ Localization.Away ((algebraMap T (T' g)) x)
t : S → Set T := fun g ↦ {x | IsStandardOpenImmersion (Localization.Away g) (T'' g x)}
ι : Type (max u_3 u_2) := (i : ↑s) × ↑(t ↑i)
this✝² : ∀ (i : ι), IsStandardOpenImmersion (Localization.Away ↑i.fst) (T'' ↑i.fst ↑i.snd)
h1 : Ideal.span (⇑(algebraMap S T) '' {g | IsStandardOpenImmersion R (Localization.Away g)}) = ⊤
g : S
hg : IsStandardOpenImmersion R (Localization.Away g)
x : T
hx : IsStandardOpenImmersion S (Localization.Away x)
this✝¹ : Algebra (Localization.Away x) (T'' g x) := localizationAlgebra (Submonoid.powers x) (T' g)
this✝ : IsScalarTower S (Localization.Away x) (T'' g x)
this :
  algebraMapSubmonoid (Localization.Away x) (Submonoid.powers g) =
    algebraMapSubmonoid (Localization.Away x) (Submonoid.powers ((algebraMap S T) g))
⊢ IsLocalization (algebraMapSubmonoid (Localization.Away x) (Submonoid.powers g)) (T'' g x)

Tactic:
this

Goal state after:
R : Type u_1
S : Type u_2
inst✝⁸ : CommSemiring R
inst✝⁷ : CommSemiring S
inst✝⁶ : Algebra R S
T : Type u_3
inst✝⁵ : CommSemiring T
inst✝⁴ : Algebra S T
inst✝³ : Algebra R T
inst✝² : IsScalarTower R S T
inst✝¹ : IsLocalIso R S
inst✝ : IsLocalIso S T
s : Set S := {g | IsStandardOpenImmersion R (Localization.Away g)}
T' : S → Type u_3 := fun g ↦ Localization.Away ((algebraMap S T) g)
this✝³ : (g : S) → Algebra (Localization.Away g) (T' g) := fun g ↦ localizationAlgebra (Submonoid.powers g) T
T'' : S → T → Type u_3 := fun g x ↦ Localization.Away ((algebraMap T (T' g)) x)
t : S → Set T := fun g ↦ {x | IsStandardOpenImmersion (Localization.Away g) (T'' g x)}
ι : Type (max u_3 u_2) := (i : ↑s) × ↑(t ↑i)
this✝² : ∀ (i : ι), IsStandardOpenImmersion (Localization.Away ↑i.fst) (T'' ↑i.fst ↑i.snd)
h1 : Ideal.span (⇑(algebraMap S T) '' {g | IsStandardOpenImmersion R (Localization.Away g)}) = ⊤
g : S
hg : IsStandardOpenImmersion R (Localization.Away g)
x : T
hx : IsStandardOpenImmersion S (Localization.Away x)
this✝¹ : Algebra (Localization.Away x) (T'' g x) := localizationAlgebra (Submonoid.powers x) (T' g)
this✝ : IsScalarTower S (Localization.Away x) (T'' g x)
this :
  algebraMapSubmonoid (Localization.Away x) (Submonoid.powers g) =
    algebraMapSubmonoid (Localization.Away x) (Submonoid.powers ((algebraMap S T) g))
⊢ IsLocalization (algebraMapSubmonoid (Localization.Away x) (Submonoid.powers ((algebraMap S T) g))) (T'' g x)
Module: Mathlib.RingTheory.LocalProperties.Semilocal

Goal state before:
pp: R : Type u_1
inst✝¹¹ : CommSemiring R
inst✝¹⁰ : Finite (MaximalSpectrum R)
M : Type u_2
inst✝⁹ : AddCommMonoid M
inst✝⁸ : Module R M
Rₚ : (P : Ideal R) → [P.IsMaximal] → Type u_3
inst✝⁷ : (P : Ideal R) → [inst : P.IsMaximal] → CommSemiring (Rₚ P)
inst✝⁶ : (P : Ideal R) → [inst : P.IsMaximal] → Algebra R (Rₚ P)
inst✝⁵ : ∀ (P : Ideal R) [inst : P.IsMaximal], IsLocalization.AtPrime (Rₚ P) P
Mₚ : (P : Ideal R) → [P.IsMaximal] → Type u_4
inst✝⁴ : (P : Ideal R) → [inst : P.IsMaximal] → AddCommMonoid (Mₚ P)
inst✝³ : (P : Ideal R) → [inst : P.IsMaximal] → Module R (Mₚ P)
inst✝² : (P : Ideal R) → [inst : P.IsMaximal] → Module (Rₚ P) (Mₚ P)
inst✝¹ : ∀ (P : Ideal R) [inst : P.IsMaximal], IsScalarTower R (Rₚ P) (Mₚ P)
f : (P : Ideal R) → [inst : P.IsMaximal] → M →ₗ[R] Mₚ P
inst✝ : ∀ (P : Ideal R) [inst : P.IsMaximal], IsLocalizedModule P.primeCompl (f P)
N : Submodule R M
H : ∀ (P : Ideal R) [inst : P.IsMaximal], (localized' (Rₚ P) P.primeCompl (f P) N).FG
⊢ N.FG

Tactic:
[

Goal state after:
R : Type u_1
inst✝¹¹ : CommSemiring R
inst✝¹⁰ : Finite (MaximalSpectrum R)
M : Type u_2
inst✝⁹ : AddCommMonoid M
inst✝⁸ : Module R M
Rₚ : (P : Ideal R) → [P.IsMaximal] → Type u_3
inst✝⁷ : (P : Ideal R) → [inst : P.IsMaximal] → CommSemiring (Rₚ P)
inst✝⁶ : (P : Ideal R) → [inst : P.IsMaximal] → Algebra R (Rₚ P)
inst✝⁵ : ∀ (P : Ideal R) [inst : P.IsMaximal], IsLocalization.AtPrime (Rₚ P) P
Mₚ : (P : Ideal R) → [P.IsMaximal] → Type u_4
inst✝⁴ : (P : Ideal R) → [inst : P.IsMaximal] → AddCommMonoid (Mₚ P)
inst✝³ : (P : Ideal R) → [inst : P.IsMaximal] → Module R (Mₚ P)
inst✝² : (P : Ideal R) → [inst : P.IsMaximal] → Module (Rₚ P) (Mₚ P)
inst✝¹ : ∀ (P : Ideal R) [inst : P.IsMaximal], IsScalarTower R (Rₚ P) (Mₚ P)
f : (P : Ideal R) → [inst : P.IsMaximal] → M →ₗ[R] Mₚ P
inst✝ : ∀ (P : Ideal R) [inst : P.IsMaximal], IsLocalizedModule P.primeCompl (f P)
N : Submodule R M
H : ∀ (P : Ideal R) [inst : P.IsMaximal], (localized' (Rₚ P) P.primeCompl (f P) N).FG
⊢ N.FG
Module: Mathlib.RingTheory.Invariant.Profinite

Goal state before:
pp: case refine_2
A : Type u_1
B : Type u_2
inst✝¹⁵ : CommRing A
inst✝¹⁴ : CommRing B
inst✝¹³ : Algebra A B
G : Type u
inst✝¹² : Group G
inst✝¹¹ : MulSemiringAction G B
inst✝¹⁰ : SMulCommClass G A B
inst✝⁹ : TopologicalSpace G
inst✝⁸ : CompactSpace G
inst✝⁷ : TotallyDisconnectedSpace G
inst✝⁶ : IsTopologicalGroup G
inst✝⁵ : TopologicalSpace B
inst✝⁴ : DiscreteTopology B
inst✝³ : ContinuousSMul G B
P : Ideal A
Q : Ideal B
inst✝² : Q.IsPrime
inst✝¹ : Q.LiesOver P
inst✝ : Algebra.IsInvariant A B G
σ : (B ⧸ Q) ≃ₐ[A ⧸ P] B ⧸ Q
B' : (G : Type u) → [inst : Monoid G] → [inst_1 : MulSemiringAction G B] → [SMulCommClass G A B] → Subalgebra A B :=
  FixedPoints.subalgebra A B
s : (j : OpenNormalSubgroup G) → (stabilizerHomSurjectiveAuxFunctor P Q σ).obj j
hs : s ∈ (stabilizerHomSurjectiveAuxFunctor P Q σ).sections
a : ↑(ProfiniteGrp.of G).toProfinite.toTop :=
  (ProfiniteGrp.Hom.hom (ProfiniteGrp.of G).isoLimittoFiniteQuotientFunctor.inv) ⟨fun N ↦ ↑↑(s N), ⋯⟩
this : ∀ (N : OpenNormalSubgroup G), ↑a = ↑↑(s N)
N : OpenNormalSubgroup G
x : ↥(B' ↥↑N.toOpenSubgroup)
hN : ↑N ⊆ ↑(MulAction.stabilizer G ↑x)
⊢ (mk Q) ↑(↑a • x) = σ ((mk Q) ↑x)

Tactic:
this

Goal state after:
case refine_2
A : Type u_1
B : Type u_2
inst✝¹⁵ : CommRing A
inst✝¹⁴ : CommRing B
inst✝¹³ : Algebra A B
G : Type u
inst✝¹² : Group G
inst✝¹¹ : MulSemiringAction G B
inst✝¹⁰ : SMulCommClass G A B
inst✝⁹ : TopologicalSpace G
inst✝⁸ : CompactSpace G
inst✝⁷ : TotallyDisconnectedSpace G
inst✝⁶ : IsTopologicalGroup G
inst✝⁵ : TopologicalSpace B
inst✝⁴ : DiscreteTopology B
inst✝³ : ContinuousSMul G B
P : Ideal A
Q : Ideal B
inst✝² : Q.IsPrime
inst✝¹ : Q.LiesOver P
inst✝ : Algebra.IsInvariant A B G
σ : (B ⧸ Q) ≃ₐ[A ⧸ P] B ⧸ Q
B' : (G : Type u) → [inst : Monoid G] → [inst_1 : MulSemiringAction G B] → [SMulCommClass G A B] → Subalgebra A B :=
  FixedPoints.subalgebra A B
s : (j : OpenNormalSubgroup G) → (stabilizerHomSurjectiveAuxFunctor P Q σ).obj j
hs : s ∈ (stabilizerHomSurjectiveAuxFunctor P Q σ).sections
a : ↑(ProfiniteGrp.of G).toProfinite.toTop :=
  (ProfiniteGrp.Hom.hom (ProfiniteGrp.of G).isoLimittoFiniteQuotientFunctor.inv) ⟨fun N ↦ ↑↑(s N), ⋯⟩
this : ∀ (N : OpenNormalSubgroup G), ↑a = ↑↑(s N)
N : OpenNormalSubgroup G
x : ↥(B' ↥↑N.toOpenSubgroup)
hN : ↑N ⊆ ↑(MulAction.stabilizer G ↑x)
⊢ (mk Q) ↑(↑↑(s N) • x) = σ ((mk Q) ↑x)
Module: Mathlib.RingTheory.LocalRing.Pullback

Goal state before:
pp: case a
R : Type u_1
S : Type u_2
T : Type u_3
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : Semiring T
f : R →+* T
g : S →+* T
⊢ ker (f.pullbackFst g) ≤ Ideal.comap (f.pullbackSnd g) (ker g)

Tactic:
rintro ⟨⟨_, _⟩, h⟩

Goal state after:
case a
R : Type u_1
S : Type u_2
T : Type u_3
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : Semiring T
f : R →+* T
g : S →+* T
fst✝ : R
snd✝ : S
h : (fst✝, snd✝) ∈ f.pullback g
⊢ ⟨(fst✝, snd✝), h⟩ ∈ ker (f.pullbackFst g) → ⟨(fst✝, snd✝), h⟩ ∈ Ideal.comap (f.pullbackSnd g) (ker g)
Module: Mathlib.RingTheory.MvPolynomial.EulerIdentity

Goal state before:
pp: case refine_1
R : Type u_1
σ : Type u_2
M : Type u_3
inst✝¹ : CommSemiring R
φ : MvPolynomial σ R
inst✝ : AddCancelCommMonoid M
w : σ → M
n n' : M
i : σ
h : φ ∈ Submodule.span R ((fun m ↦ AddMonoidAlgebra.single m 1) '' {d | (weight w) d = n})
h' : n' + w i = n
m : σ →₀ ℕ
hm : m ∈ {d | (weight w) d = n}
⊢ IsWeightedHomogeneous w ((pderiv i) (AddMonoidAlgebra.single m 1)) n'

Tactic:
[

Goal state after:
case refine_1
R : Type u_1
σ : Type u_2
M : Type u_3
inst✝¹ : CommSemiring R
φ : MvPolynomial σ R
inst✝ : AddCancelCommMonoid M
w : σ → M
n n' : M
i : σ
h : φ ∈ Submodule.span R ((fun m ↦ AddMonoidAlgebra.single m 1) '' {d | (weight w) d = n})
h' : n' + w i = n
m : σ →₀ ℕ
hm : m ∈ {d | (weight w) d = n}
⊢ IsWeightedHomogeneous w ((pderiv i) (AddMonoidAlgebra.single m 1)) n'
Module: Mathlib.RingTheory.LaurentSeries

Goal state before:
pp: case neg
K : Type u_2
inst✝ : Field K
D : ℤ
f : K⸨X⸩
h_val_f : ∀ n < D, f.coeff n = 0
F : K⟦X⟧ := f.powerSeriesPart
ord_nonpos : HahnSeries.order f ≤ 0
s : ℕ
hs : HahnSeries.order f = -↑s
hDs : 0 < D + ↑s
d : ℕ
hd : D + ↑s = ↑d
⊢ Valued.v ((ofPowerSeries ℤ K) f.powerSeriesPart) ≤ exp (-↑d)

Tactic:
apply (intValuation_le_iff_coeff_lt_eq_zero K F).mpr

Goal state after:
case neg
K : Type u_2
inst✝ : Field K
D : ℤ
f : K⸨X⸩
h_val_f : ∀ n < D, f.coeff n = 0
F : K⟦X⟧ := ⋯
ord_nonpos : HahnSeries.order f ≤ 0
s : ℕ
hs : HahnSeries.order f = -↑s
hDs : 0 < D + ↑s
d : ℕ
hd : D + ↑s = ↑d
⊢ ∀ n < d, (PowerSeries.coeff n) F = 0
Module: Mathlib.RingTheory.LaurentSeries

Goal state before:
pp: case neg
K : Type u_2
inst✝ : Field K
D : ℤ
f : K⸨X⸩
h_val_f : ∀ n < D, f.coeff n = 0
F : K⟦X⟧ := f.powerSeriesPart
ord_nonpos : 0 < HahnSeries.order f
s : ℕ
hs : HahnSeries.order f = ↑s
hDs : 0 < D - ↑s
d : ℕ
hd : D - ↑s = ↑d
⊢ Valued.v ((ofPowerSeries ℤ K) f.powerSeriesPart) ≤ exp (-↑d)

Tactic:
apply (intValuation_le_iff_coeff_lt_eq_zero K F).mpr

Goal state after:
case neg
K : Type u_2
inst✝ : Field K
D : ℤ
f : K⸨X⸩
h_val_f : ∀ n < D, f.coeff n = 0
F : K⟦X⟧ := ⋯
ord_nonpos : 0 < HahnSeries.order f
s : ℕ
hs : HahnSeries.order f = ↑s
hDs : 0 < D - ↑s
d : ℕ
hd : D - ↑s = ↑d
⊢ ∀ n < d, (PowerSeries.coeff n) F = 0