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

Goal state before:
pp: case a
R✝ : Type u_1
inst✝¹⁷ : CommRing R✝
S✝ : Submonoid R✝
P : Type u_2
inst✝¹⁶ : CommRing P
inst✝¹⁵ : Algebra R✝ P
R₁ : Type u_3
inst✝¹⁴ : CommRing R₁
K✝ : Type u_4
inst✝¹³ : Field K✝
inst✝¹² : Algebra R₁ K✝
inst✝¹¹ : IsFractionRing R₁ K✝
inst✝¹⁰ : IsDomain R₁
R : Type u_5
S : Type u_6
K : Type u_7
L : Type u_8
inst✝⁹ : CommRing R
inst✝⁸ : IsDomain R
inst✝⁷ : CommRing S
inst✝⁶ : IsDomain S
inst✝⁵ : CommRing K
inst✝⁴ : CommRing L
inst✝³ : Algebra R K
inst✝² : Algebra S L
inst✝¹ : IsFractionRing R K
inst✝ : IsFractionRing S L
f : R ≃+* S
I J : FractionalIdeal R⁰ K
⊢ ∀ (m : L),
    (IsFractionRing.ringEquivOfRingEquiv f).symm m ∈ I →
      ∀ (n : L),
        (IsFractionRing.ringEquivOfRingEquiv f).symm n ∈ J →
          (IsFractionRing.ringEquivOfRingEquiv f).symm m * (IsFractionRing.ringEquivOfRingEquiv f).symm n ∈ ↑I * ↑J

Tactic:
·

Goal state after:
case a
R✝ : Type u_1
inst✝¹⁷ : CommRing R✝
S✝ : Submonoid R✝
P : Type u_2
inst✝¹⁶ : CommRing P
inst✝¹⁵ : Algebra R✝ P
R₁ : Type u_3
inst✝¹⁴ : CommRing R₁
K✝ : Type u_4
inst✝¹³ : Field K✝
inst✝¹² : Algebra R₁ K✝
inst✝¹¹ : IsFractionRing R₁ K✝
inst✝¹⁰ : IsDomain R₁
R : Type u_5
S : Type u_6
K : Type u_7
L : Type u_8
inst✝⁹ : CommRing R
inst✝⁸ : IsDomain R
inst✝⁷ : CommRing S
inst✝⁶ : IsDomain S
inst✝⁵ : CommRing K
inst✝⁴ : CommRing L
inst✝³ : Algebra R K
inst✝² : Algebra S L
inst✝¹ : IsFractionRing R K
inst✝ : IsFractionRing S L
f : R ≃+* S
I J : FractionalIdeal R⁰ K
⊢ ∀ (m : L),
    (IsFractionRing.ringEquivOfRingEquiv f).symm m ∈ I →
      ∀ (n : L),
        (IsFractionRing.ringEquivOfRingEquiv f).symm n ∈ J →
          (IsFractionRing.ringEquivOfRingEquiv f).symm m * (IsFractionRing.ringEquivOfRingEquiv f).symm n ∈ ↑I * ↑J
Module: Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas

Goal state before:
pp: A : Type u_2
inst✝¹ : CommRing A
inst✝ : IsDedekindDomain A
I J : Ideal A
this : NormalizedGCDMonoid (Ideal A) := UniqueFactorizationMonoid.toNormalizedGCDMonoid (Ideal A)
⊢ gcd I J ∣ I ⊔ J

pp: A : Type u_2
inst✝¹ : CommRing A
inst✝ : IsDedekindDomain A
I J : Ideal A
this : NormalizedGCDMonoid (Ideal A) := UniqueFactorizationMonoid.toNormalizedGCDMonoid (Ideal A)
⊢ I ⊔ J ∣ gcd I J

Tactic:
·

Goal state after:
A : Type u_2
inst✝¹ : CommRing A
inst✝ : IsDedekindDomain A
I J : Ideal A
this : NormalizedGCDMonoid (Ideal A) := UniqueFactorizationMonoid.toNormalizedGCDMonoid (Ideal A)
⊢ gcd I J ∣ I ⊔ J
Module: Mathlib.RingTheory.FractionalIdeal.Operations

Goal state before:
pp: case mp
R : Type u_5
S : Type u_6
K : Type u_7
L : Type u_8
inst✝⁹ : CommRing R
inst✝⁸ : IsDomain R
inst✝⁷ : CommRing S
inst✝⁶ : IsDomain S
inst✝⁵ : CommRing K
inst✝⁴ : CommRing L
inst✝³ : Algebra R K
inst✝² : Algebra S L
inst✝¹ : IsFractionRing R K
inst✝ : IsFractionRing S L
f : R ≃+* S
x : K
y : L
⊢ (∃ a, (semilinearEquivOfRingEquiv K L f) (a • x) = y) → ∃ a, a • (IsLocalization.map L ↑f ⋯) x = y

pp: case mpr
R : Type u_5
S : Type u_6
K : Type u_7
L : Type u_8
inst✝⁹ : CommRing R
inst✝⁸ : IsDomain R
inst✝⁷ : CommRing S
inst✝⁶ : IsDomain S
inst✝⁵ : CommRing K
inst✝⁴ : CommRing L
inst✝³ : Algebra R K
inst✝² : Algebra S L
inst✝¹ : IsFractionRing R K
inst✝ : IsFractionRing S L
f : R ≃+* S
x : K
y : L
⊢ (∃ a, a • (IsLocalization.map L ↑f ⋯) x = y) → ∃ a, (semilinearEquivOfRingEquiv K L f) (a • x) = y

Tactic:
·

Goal state after:
case mp
R : Type u_5
S : Type u_6
K : Type u_7
L : Type u_8
inst✝⁹ : CommRing R
inst✝⁸ : IsDomain R
inst✝⁷ : CommRing S
inst✝⁶ : IsDomain S
inst✝⁵ : CommRing K
inst✝⁴ : CommRing L
inst✝³ : Algebra R K
inst✝² : Algebra S L
inst✝¹ : IsFractionRing R K
inst✝ : IsFractionRing S L
f : R ≃+* S
x : K
y : L
⊢ (∃ a, (semilinearEquivOfRingEquiv K L f) (a • x) = y) → ∃ a, a • (IsLocalization.map L ↑f ⋯) x = y
Module: Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas

Goal state before:
pp: A : Type u_2
inst✝¹ : CommRing A
inst✝ : IsDedekindDomain A
I J : Ideal A
this : NormalizedGCDMonoid (Ideal A) := UniqueFactorizationMonoid.toNormalizedGCDMonoid (Ideal A)
⊢ I ⊔ J ∣ gcd I J

Tactic:
·

Goal state after:
A : Type u_2
inst✝¹ : CommRing A
inst✝ : IsDedekindDomain A
I J : Ideal A
this : NormalizedGCDMonoid (Ideal A) := UniqueFactorizationMonoid.toNormalizedGCDMonoid (Ideal A)
⊢ I ⊔ J ∣ gcd I J
Module: Mathlib.RingTheory.FractionalIdeal.Operations

Goal state before:
pp: case mpr
R : Type u_5
S : Type u_6
K : Type u_7
L : Type u_8
inst✝⁹ : CommRing R
inst✝⁸ : IsDomain R
inst✝⁷ : CommRing S
inst✝⁶ : IsDomain S
inst✝⁵ : CommRing K
inst✝⁴ : CommRing L
inst✝³ : Algebra R K
inst✝² : Algebra S L
inst✝¹ : IsFractionRing R K
inst✝ : IsFractionRing S L
f : R ≃+* S
x : K
y : L
⊢ (∃ a, a • (IsLocalization.map L ↑f ⋯) x = y) → ∃ a, (semilinearEquivOfRingEquiv K L f) (a • x) = y

Tactic:
·

Goal state after:
case mpr
R : Type u_5
S : Type u_6
K : Type u_7
L : Type u_8
inst✝⁹ : CommRing R
inst✝⁸ : IsDomain R
inst✝⁷ : CommRing S
inst✝⁶ : IsDomain S
inst✝⁵ : CommRing K
inst✝⁴ : CommRing L
inst✝³ : Algebra R K
inst✝² : Algebra S L
inst✝¹ : IsFractionRing R K
inst✝ : IsFractionRing S L
f : R ≃+* S
x : K
y : L
⊢ (∃ a, a • (IsLocalization.map L ↑f ⋯) x = y) → ∃ a, (semilinearEquivOfRingEquiv K L f) (a • x) = y
Module: Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas

Goal state before:
pp: A : Type u_2
inst✝¹ : CommRing A
inst✝ : IsDedekindDomain A
I J : Ideal A
this : NormalizedGCDMonoid (Ideal A) := UniqueFactorizationMonoid.toNormalizedGCDMonoid (Ideal A)
hgcd : gcd I J = I ⊔ J
⊢ lcm I J ∣ I ⊓ J

pp: A : Type u_2
inst✝¹ : CommRing A
inst✝ : IsDedekindDomain A
I J : Ideal A
this : NormalizedGCDMonoid (Ideal A) := UniqueFactorizationMonoid.toNormalizedGCDMonoid (Ideal A)
hgcd : gcd I J = I ⊔ J
⊢ I ⊓ J ∣ lcm I J

Tactic:
·

Goal state after:
A : Type u_2
inst✝¹ : CommRing A
inst✝ : IsDedekindDomain A
I J : Ideal A
this : NormalizedGCDMonoid (Ideal A) := UniqueFactorizationMonoid.toNormalizedGCDMonoid (Ideal A)
hgcd : gcd I J = I ⊔ J
⊢ lcm I J ∣ I ⊓ J
Module: Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas

Goal state before:
pp: A : Type u_2
inst✝¹ : CommRing A
inst✝ : IsDedekindDomain A
I J : Ideal A
this : NormalizedGCDMonoid (Ideal A) := UniqueFactorizationMonoid.toNormalizedGCDMonoid (Ideal A)
hgcd : gcd I J = I ⊔ J
⊢ I ⊓ J ∣ lcm I J

Tactic:
·

Goal state after:
A : Type u_2
inst✝¹ : CommRing A
inst✝ : IsDedekindDomain A
I J : Ideal A
this : NormalizedGCDMonoid (Ideal A) := UniqueFactorizationMonoid.toNormalizedGCDMonoid (Ideal A)
hgcd : gcd I J = I ⊔ J
⊢ I ⊓ J ∣ lcm I J
Module: Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas

Goal state before:
pp: A : Type u_2
inst✝¹ : CommRing A
inst✝ : IsDedekindDomain A
I J : Ideal A
this : NormalizedGCDMonoid (Ideal A) := UniqueFactorizationMonoid.toNormalizedGCDMonoid (Ideal A)
hgcd : gcd I J = I ⊔ J
hlcm : lcm I J = I ⊓ J
⊢ (I ⊔ J) * (I ⊓ J) = I * J

Tactic:
← hgcd,

Goal state after:
A : Type u_2
inst✝¹ : CommRing A
inst✝ : IsDedekindDomain A
I J : Ideal A
this : NormalizedGCDMonoid (Ideal A) := UniqueFactorizationMonoid.toNormalizedGCDMonoid (Ideal A)
hgcd : gcd I J = I ⊔ J
hlcm : lcm I J = I ⊓ J
⊢ gcd I J * (I ⊓ J) = I * J
Module: Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas

Goal state before:
pp: case inl
K : Type u_3
inst✝ : Field K
⊢ factors (span {0}) = Multiset.map (fun q ↦ span {q}) (factors 0)

pp: case inr
K : Type u_3
inst✝ : Field K
p : K[X]
hp : p ≠ 0
⊢ factors (span {p}) = Multiset.map (fun q ↦ span {q}) (factors p)

Tactic:
·

Goal state after:
case inl
K : Type u_3
inst✝ : Field K
⊢ factors (span {0}) = Multiset.map (fun q ↦ span {q}) (factors 0)
Module: Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas

Goal state before:
pp: case a
T : Type u_4
inst✝¹ : CommRing T
inst✝ : IsDedekindDomain T
I J : Ideal T
hI : I ≠ ⊥
hJ : J ≠ ⊥
this : (normalizedFactors I ∩ normalizedFactors J).prod ≠ 0
⊢ I ⊔ J ≤ (normalizedFactors I ∩ normalizedFactors J).prod

pp: case a
T : Type u_4
inst✝¹ : CommRing T
inst✝ : IsDedekindDomain T
I J : Ideal T
hI : I ≠ ⊥
hJ : J ≠ ⊥
this : (normalizedFactors I ∩ normalizedFactors J).prod ≠ 0
⊢ (normalizedFactors I ∩ normalizedFactors J).prod ≤ I ⊔ J

Tactic:
·

Goal state after:
case a
T : Type u_4
inst✝¹ : CommRing T
inst✝ : IsDedekindDomain T
I J : Ideal T
hI : I ≠ ⊥
hJ : J ≠ ⊥
this : (normalizedFactors I ∩ normalizedFactors J).prod ≠ 0
⊢ I ⊔ J ≤ (normalizedFactors I ∩ normalizedFactors J).prod
Module: Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas

Goal state before:
pp: case a
T : Type u_4
inst✝¹ : CommRing T
inst✝ : IsDedekindDomain T
I J : Ideal T
hI : I ≠ ⊥
hJ : J ≠ ⊥
this : (normalizedFactors I ∩ normalizedFactors J).prod ≠ 0
⊢ (normalizedFactors I ∩ normalizedFactors J).prod ≤ I ⊔ J

Tactic:
·

Goal state after:
case a
T : Type u_4
inst✝¹ : CommRing T
inst✝ : IsDedekindDomain T
I J : Ideal T
hI : I ≠ ⊥
hJ : J ≠ ⊥
this : (normalizedFactors I ∩ normalizedFactors J).prod ≠ 0
⊢ (normalizedFactors I ∩ normalizedFactors J).prod ≤ I ⊔ J
Module: Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas

Goal state before:
pp: case pos
T : Type u_4
inst✝¹ : CommRing T
inst✝ : IsDedekindDomain T
I J : Ideal T
hJ : Irreducible J
n : ℕ
hn : ↑n ≤ emultiplicity J I
hI : I = ⊥
⊢ J ^ n ⊔ I = J ^ n

pp: case neg
T : Type u_4
inst✝¹ : CommRing T
inst✝ : IsDedekindDomain T
I J : Ideal T
hJ : Irreducible J
n : ℕ
hn : ↑n ≤ emultiplicity J I
hI : ¬I = ⊥
⊢ J ^ n ⊔ I = J ^ n

Tactic:
·

Goal state after:
case pos
T : Type u_4
inst✝¹ : CommRing T
inst✝ : IsDedekindDomain T
I J : Ideal T
hJ : Irreducible J
n : ℕ
hn : ↑n ≤ emultiplicity J I
hI : I = ⊥
⊢ J ^ n ⊔ I = J ^ n
Module: Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas

Goal state before:
pp: T : Type u_4
inst✝¹ : CommRing T
inst✝ : IsDedekindDomain T
I J : Ideal T
hI : I ≠ ⊥
hJ : Irreducible J
n : ℕ
hn : emultiplicity J I ≤ ↑n
⊢ J ^ count J (normalizedFactors I) = J ^ multiplicity J I

pp: T : Type u_4
inst✝¹ : CommRing T
inst✝ : IsDedekindDomain T
I J : Ideal T
hI : I ≠ ⊥
hJ : Irreducible J
n : ℕ
hn : emultiplicity J I ≤ ↑n
⊢ count J (normalizedFactors I) ≤ n

Tactic:
·

Goal state after:
T : Type u_4
inst✝¹ : CommRing T
inst✝ : IsDedekindDomain T
I J : Ideal T
hI : I ≠ ⊥
hJ : Irreducible J
n : ℕ
hn : emultiplicity J I ≤ ↑n
⊢ J ^ count J (normalizedFactors I) = J ^ multiplicity J I
Module: Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas

Goal state before:
pp: T : Type u_4
inst✝¹ : CommRing T
inst✝ : IsDedekindDomain T
I J : Ideal T
hI : I ≠ ⊥
hJ : Irreducible J
n : ℕ
hn : emultiplicity J I ≤ ↑n
⊢ count J (normalizedFactors I) ≤ n

Tactic:
·

Goal state after:
T : Type u_4
inst✝¹ : CommRing T
inst✝ : IsDedekindDomain T
I J : Ideal T
hI : I ≠ ⊥
hJ : Irreducible J
n : ℕ
hn : emultiplicity J I ≤ ↑n
⊢ count J (normalizedFactors I) ≤ n
Module: Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas

Goal state before:
pp: case h.left
T : Type u_4
inst✝¹ : CommRing T
inst✝ : IsDedekindDomain T
I : Ideal T
hI : I ≠ ⊥
P : Ideal T
hpm : P.IsMaximal
⊢ P ⊔ (filter (fun x ↦ ¬P = x) (normalizedFactors I)).prod = ⊤

pp: case h.right
T : Type u_4
inst✝¹ : CommRing T
inst✝ : IsDedekindDomain T
I : Ideal T
hI : I ≠ ⊥
P : Ideal T
hpm : P.IsMaximal
⊢ I = P ^ count P (normalizedFactors I) * (filter (fun x ↦ ¬P = x) (normalizedFactors I)).prod

Tactic:
·

Goal state after:
case h.left
T : Type u_4
inst✝¹ : CommRing T
inst✝ : IsDedekindDomain T
I : Ideal T
hI : I ≠ ⊥
P : Ideal T
hpm : P.IsMaximal
⊢ P ⊔ (filter (fun x ↦ ¬P = x) (normalizedFactors I)).prod = ⊤
Module: Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas

Goal state before:
pp: case h.right
T : Type u_4
inst✝¹ : CommRing T
inst✝ : IsDedekindDomain T
I : Ideal T
hI : I ≠ ⊥
P : Ideal T
hpm : P.IsMaximal
⊢ I = P ^ count P (normalizedFactors I) * (filter (fun x ↦ ¬P = x) (normalizedFactors I)).prod

Tactic:
·

Goal state after:
case h.right
T : Type u_4
inst✝¹ : CommRing T
inst✝ : IsDedekindDomain T
I : Ideal T
hI : I ≠ ⊥
P : Ideal T
hpm : P.IsMaximal
⊢ I = P ^ count P (normalizedFactors I) * (filter (fun x ↦ ¬P = x) (normalizedFactors I)).prod
Module: Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas

Goal state before:
pp: case mp
R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsDedekindDomain R
hR : ¬IsField R
I : Ideal R
⊢ (∃ M, M.IsMaximal ∧ I ≤ M) → ∃ P, I ≤ P.asIdeal

pp: case mpr
R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsDedekindDomain R
hR : ¬IsField R
I : Ideal R
⊢ (∃ P, I ≤ P.asIdeal) → ∃ M, M.IsMaximal ∧ I ≤ M

Tactic:
·

Goal state after:
case mp
R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsDedekindDomain R
hR : ¬IsField R
I : Ideal R
⊢ (∃ M, M.IsMaximal ∧ I ≤ M) → ∃ P, I ≤ P.asIdeal
Module: Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas

Goal state before:
pp: case mpr
R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsDedekindDomain R
hR : ¬IsField R
I : Ideal R
⊢ (∃ P, I ≤ P.asIdeal) → ∃ M, M.IsMaximal ∧ I ≤ M

Tactic:
·

Goal state after:
case mpr
R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsDedekindDomain R
hR : ¬IsField R
I : Ideal R
⊢ (∃ P, I ≤ P.asIdeal) → ∃ M, M.IsMaximal ∧ I ≤ M
Module: Mathlib.Algebra.Module.FinitePresentation

Goal state before:
pp: case refine_1
R : Type u
M : Type u_1
inst✝² : Ring R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
fp : FinitePresentation R M
ι : Finset M
hι₁ : Submodule.span R ↑ι = ⊤
hι₂ : (linearCombination R Subtype.val).ker.FG
⊢ (linearCombination R Subtype.val ∘ₗ
        ↑(lcongr ι.equivFin (LinearEquiv.refl R R) ≪≫ₗ linearEquivFunOnFinite R R (Fin ι.card)).symm).range =
    ⊤

pp: case refine_2
R : Type u
M : Type u_1
inst✝² : Ring R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
fp : FinitePresentation R M
ι : Finset M
hι₁ : Submodule.span R ↑ι = ⊤
hι₂ : (linearCombination R Subtype.val).ker.FG
⊢ (linearCombination R Subtype.val ∘ₗ
        ↑(lcongr ι.equivFin (LinearEquiv.refl R R) ≪≫ₗ linearEquivFunOnFinite R R (Fin ι.card)).symm).ker.FG

Tactic:
·

Goal state after:
case refine_1
R : Type u
M : Type u_1
inst✝² : Ring R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
fp : FinitePresentation R M
ι : Finset M
hι₁ : Submodule.span R ↑ι = ⊤
hι₂ : (linearCombination R Subtype.val).ker.FG
⊢ (linearCombination R Subtype.val ∘ₗ
        ↑(lcongr ι.equivFin (LinearEquiv.refl R R) ≪≫ₗ linearEquivFunOnFinite R R (Fin ι.card)).symm).range =
    ⊤
Module: Mathlib.Algebra.Module.FinitePresentation

Goal state before:
pp: case refine_2
R : Type u
M : Type u_1
inst✝² : Ring R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
fp : FinitePresentation R M
ι : Finset M
hι₁ : Submodule.span R ↑ι = ⊤
hι₂ : (linearCombination R Subtype.val).ker.FG
⊢ (linearCombination R Subtype.val ∘ₗ
        ↑(lcongr ι.equivFin (LinearEquiv.refl R R) ≪≫ₗ linearEquivFunOnFinite R R (Fin ι.card)).symm).ker.FG

Tactic:
·

Goal state after:
case refine_2
R : Type u
M : Type u_1
inst✝² : Ring R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
fp : FinitePresentation R M
ι : Finset M
hι₁ : Submodule.span R ↑ι = ⊤
hι₂ : (linearCombination R Subtype.val).ker.FG
⊢ (linearCombination R Subtype.val ∘ₗ
        ↑(lcongr ι.equivFin (LinearEquiv.refl R R) ≪≫ₗ linearEquivFunOnFinite R R (Fin ι.card)).symm).ker.FG
Module: Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas

Goal state before:
pp: case pos
R : Type u_1
K : Type u_3
inst✝³ : CommRing R
inst✝² : Field K
inst✝¹ : IsDedekindDomain R
inst✝ : Algebra R K
hK : IsFractionRing R K
x : K
hR : IsField R
⊢ (∀ (i : HeightOneSpectrum R), x ∈ Localization.subalgebra.ofField K i.asIdeal.primeCompl ⋯) → x ∈ ⊥

pp: case neg
R : Type u_1
K : Type u_3
inst✝³ : CommRing R
inst✝² : Field K
inst✝¹ : IsDedekindDomain R
inst✝ : Algebra R K
hK : IsFractionRing R K
x : K
hR : ¬IsField R
⊢ (∀ (i : HeightOneSpectrum R), x ∈ Localization.subalgebra.ofField K i.asIdeal.primeCompl ⋯) → x ∈ ⊥

pp: case mpr
R : Type u_1
K : Type u_3
inst✝³ : CommRing R
inst✝² : Field K
inst✝¹ : IsDedekindDomain R
inst✝ : Algebra R K
hK : IsFractionRing R K
x : K
⊢ x ∈ ⊥ → ∀ (i : HeightOneSpectrum R), x ∈ Localization.subalgebra.ofField K i.asIdeal.primeCompl ⋯

Tactic:
·

Goal state after:
case pos
R : Type u_1
K : Type u_3
inst✝³ : CommRing R
inst✝² : Field K
inst✝¹ : IsDedekindDomain R
inst✝ : Algebra R K
hK : IsFractionRing R K
x : K
hR : IsField R
⊢ (∀ (i : HeightOneSpectrum R), x ∈ Localization.subalgebra.ofField K i.asIdeal.primeCompl ⋯) → x ∈ ⊥
Module: Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas

Goal state before:
pp: case neg
R : Type u_1
K : Type u_3
inst✝³ : CommRing R
inst✝² : Field K
inst✝¹ : IsDedekindDomain R
inst✝ : Algebra R K
hK : IsFractionRing R K
x : K
hR : ¬IsField R
⊢ (∀ (i : HeightOneSpectrum R), x ∈ Localization.subalgebra.ofField K i.asIdeal.primeCompl ⋯) →
    ∀ (i : MaximalSpectrum R), x ∈ Localization.subalgebra.ofField K i.asIdeal.primeCompl ⋯

pp: case mpr
R : Type u_1
K : Type u_3
inst✝³ : CommRing R
inst✝² : Field K
inst✝¹ : IsDedekindDomain R
inst✝ : Algebra R K
hK : IsFractionRing R K
x : K
⊢ (∀ (i : MaximalSpectrum R), x ∈ Localization.subalgebra.ofField K i.asIdeal.primeCompl ⋯) →
    ∀ (i : HeightOneSpectrum R), x ∈ Localization.subalgebra.ofField K i.asIdeal.primeCompl ⋯

Tactic:
·

Goal state after:
case neg
R : Type u_1
K : Type u_3
inst✝³ : CommRing R
inst✝² : Field K
inst✝¹ : IsDedekindDomain R
inst✝ : Algebra R K
hK : IsFractionRing R K
x : K
hR : ¬IsField R
⊢ (∀ (i : HeightOneSpectrum R), x ∈ Localization.subalgebra.ofField K i.asIdeal.primeCompl ⋯) →
    ∀ (i : MaximalSpectrum R), x ∈ Localization.subalgebra.ofField K i.asIdeal.primeCompl ⋯
Module: Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas

Goal state before:
pp: case mpr
R : Type u_1
K : Type u_3
inst✝³ : CommRing R
inst✝² : Field K
inst✝¹ : IsDedekindDomain R
inst✝ : Algebra R K
hK : IsFractionRing R K
x : K
⊢ (∀ (i : MaximalSpectrum R), x ∈ Localization.subalgebra.ofField K i.asIdeal.primeCompl ⋯) →
    ∀ (i : HeightOneSpectrum R), x ∈ Localization.subalgebra.ofField K i.asIdeal.primeCompl ⋯

Tactic:
·

Goal state after:
case mpr
R : Type u_1
K : Type u_3
inst✝³ : CommRing R
inst✝² : Field K
inst✝¹ : IsDedekindDomain R
inst✝ : Algebra R K
hK : IsFractionRing R K
x : K
⊢ (∀ (i : MaximalSpectrum R), x ∈ Localization.subalgebra.ofField K i.asIdeal.primeCompl ⋯) →
    ∀ (i : HeightOneSpectrum R), x ∈ Localization.subalgebra.ofField K i.asIdeal.primeCompl ⋯
Module: Mathlib.Algebra.Lie.Weights.IsSimple

Goal state before:
pp: case h_sub
K : Type u_1
L : Type u_2
inst✝⁷ : Field K
inst✝⁶ : CharZero K
inst✝⁵ : LieRing L
inst✝⁴ : LieAlgebra K L
inst✝³ : FiniteDimensional K L
inst✝² : IsKilling K L
H : LieSubalgebra K L
inst✝¹ : H.IsCartanSubalgebra
inst✝ : IsTriangularizable K (↥H) L
q : Submodule K (Dual K ↥H)
hq : ∀ (i : ↥LieSubalgebra.root), q ∈ End.invtSubmodule ↑((rootSystem H).reflection i)
χ : Weight K (↥H) L
x_χ m_α : L
hx_χ : x_χ ∈ genWeightSpace L ⇑χ
α : Weight K (↥H) L
hαq : Weight.toLinear K (↥H) L α ∈ q
hα₀ : α.IsNonZero
w_plus : Weight.toLinear K (↥H) L χ + Weight.toLinear K (↥H) L α ≠ 0
w_minus : Weight.toLinear K (↥H) L χ - Weight.toLinear K (↥H) L α ≠ 0
w_chi : Weight.toLinear K (↥H) L χ ≠ 0
m_pos m_neg : L
y : ↥H
hy : y ∈ corootSpace ⇑α
h_bracket_sum : ⁅x_χ, m_α⁆ = ⁅x_χ, m_pos⁆ + ⁅x_χ, m_neg⁆ + ⁅x_χ, ↑y⁆
h_pos_containment : ⁅x_χ, m_pos⁆ ∈ genWeightSpace L (⇑χ + ⇑α)
h_neg_containment : ⁅x_χ, m_neg⁆ ∈ genWeightSpace L (⇑χ - ⇑α)
h_chi_not_in_q : Weight.toLinear K (↥H) L χ ∉ q
S : RootPairing (↥LieSubalgebra.root) K (Dual K ↥H) ↥H := ⋯
exists_root_index : ∀ (γ : Weight K (↥H) L), γ.IsNonZero → ∃ i, S.root i = Weight.toLinear K (↥H) L γ
h_plus_bot : genWeightSpace L ⇑(Weight.toLinear K (↥H) L χ + Weight.toLinear K (↥H) L α) = ⊥
h_minus_bot : genWeightSpace L ⇑(Weight.toLinear K (↥H) L χ - Weight.toLinear K (↥H) L α) = ⊥
i : ↥LieSubalgebra.root
hi : S.root i = Weight.toLinear K (↥H) L χ
j : ↥LieSubalgebra.root
hj : S.root j = Weight.toLinear K (↥H) L α
⊢ S.root i - S.root j ∉ Set.range ⇑S.root

Tactic:
·

Goal state after:
case h_sub
K : Type u_1
L : Type u_2
inst✝⁷ : Field K
inst✝⁶ : CharZero K
inst✝⁵ : LieRing L
inst✝⁴ : LieAlgebra K L
inst✝³ : FiniteDimensional K L
inst✝² : IsKilling K L
H : LieSubalgebra K L
inst✝¹ : H.IsCartanSubalgebra
inst✝ : IsTriangularizable K (↥H) L
q : Submodule K (Dual K ↥H)
hq : ∀ (i : ↥LieSubalgebra.root), q ∈ End.invtSubmodule ↑((rootSystem H).reflection i)
χ : Weight K (↥H) L
x_χ m_α : L
hx_χ : x_χ ∈ genWeightSpace L ⇑χ
α : Weight K (↥H) L
hαq : Weight.toLinear K (↥H) L α ∈ q
hα₀ : α.IsNonZero
w_plus : Weight.toLinear K (↥H) L χ + Weight.toLinear K (↥H) L α ≠ 0
w_minus : Weight.toLinear K (↥H) L χ - Weight.toLinear K (↥H) L α ≠ 0
w_chi : Weight.toLinear K (↥H) L χ ≠ 0
m_pos m_neg : L
y : ↥H
hy : y ∈ corootSpace ⇑α
h_bracket_sum : ⁅x_χ, m_α⁆ = ⁅x_χ, m_pos⁆ + ⁅x_χ, m_neg⁆ + ⁅x_χ, ↑y⁆
h_pos_containment : ⁅x_χ, m_pos⁆ ∈ genWeightSpace L (⇑χ + ⇑α)
h_neg_containment : ⁅x_χ, m_neg⁆ ∈ genWeightSpace L (⇑χ - ⇑α)
h_chi_not_in_q : Weight.toLinear K (↥H) L χ ∉ q
S : RootPairing (↥LieSubalgebra.root) K (Dual K ↥H) ↥H := ⋯
exists_root_index : ∀ (γ : Weight K (↥H) L), γ.IsNonZero → ∃ i, S.root i = Weight.toLinear K (↥H) L γ
h_plus_bot : genWeightSpace L ⇑(Weight.toLinear K (↥H) L χ + Weight.toLinear K (↥H) L α) = ⊥
h_minus_bot : genWeightSpace L ⇑(Weight.toLinear K (↥H) L χ - Weight.toLinear K (↥H) L α) = ⊥
i : ↥LieSubalgebra.root
hi : S.root i = Weight.toLinear K (↥H) L χ
j : ↥LieSubalgebra.root
hj : S.root j = Weight.toLinear K (↥H) L α
⊢ S.root i - S.root j ∉ Set.range ⇑S.root
Module: Mathlib.Algebra.Module.FinitePresentation

Goal state before:
pp: case mp
R : Type u_1
M : Type u_2
N : Type u_3
inst✝⁶ : Ring R
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : AddCommGroup N
inst✝² : Module R N
inst✝¹ : Free R M
inst✝ : Module.Finite R M
l : M →ₗ[R] N
hl : Function.Surjective ⇑l
hl' : l.ker.FG
b : Basis (Free.ChooseBasisIndex R M) R M := ⋯
π : Free.ChooseBasisIndex R M → ↥⋯.toFinset := ⋯
σ : ↥⋯.toFinset → Free.ChooseBasisIndex R M
hσ : ∀ (b_1 : ↥⋯.toFinset), π (σ b_1) = b_1
hπ : Subtype.val ∘ π = ⇑l ∘ ⇑b
hσ₁ : π ∘ σ = id
hσ₂ : ⇑l ∘ ⇑b ∘ σ = Subtype.val
f : M →ₗ[R] ↥⋯.toFinset →₀ R := ⋯
x : ↥⋯.toFinset →₀ R
⊢ (linearCombination R Subtype.val) x = 0 → ∃ y, l y = 0 ∧ f y = x

pp: case mpr
R : Type u_1
M : Type u_2
N : Type u_3
inst✝⁶ : Ring R
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : AddCommGroup N
inst✝² : Module R N
inst✝¹ : Free R M
inst✝ : Module.Finite R M
l : M →ₗ[R] N
hl : Function.Surjective ⇑l
hl' : l.ker.FG
b : Basis (Free.ChooseBasisIndex R M) R M := ⋯
π : Free.ChooseBasisIndex R M → ↥⋯.toFinset := ⋯
σ : ↥⋯.toFinset → Free.ChooseBasisIndex R M
hσ : ∀ (b_1 : ↥⋯.toFinset), π (σ b_1) = b_1
hπ : Subtype.val ∘ π = ⇑l ∘ ⇑b
hσ₁ : π ∘ σ = id
hσ₂ : ⇑l ∘ ⇑b ∘ σ = Subtype.val
f : M →ₗ[R] ↥⋯.toFinset →₀ R := ⋯
x : ↥⋯.toFinset →₀ R
⊢ (∃ y, l y = 0 ∧ f y = x) → (linearCombination R Subtype.val) x = 0

Tactic:
·

Goal state after:
case mp
R : Type u_1
M : Type u_2
N : Type u_3
inst✝⁶ : Ring R
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : AddCommGroup N
inst✝² : Module R N
inst✝¹ : Free R M
inst✝ : Module.Finite R M
l : M →ₗ[R] N
hl : Function.Surjective ⇑l
hl' : l.ker.FG
b : Basis (Free.ChooseBasisIndex R M) R M := ⋯
π : Free.ChooseBasisIndex R M → ↥⋯.toFinset := ⋯
σ : ↥⋯.toFinset → Free.ChooseBasisIndex R M
hσ : ∀ (b_1 : ↥⋯.toFinset), π (σ b_1) = b_1
hπ : Subtype.val ∘ π = ⇑l ∘ ⇑b
hσ₁ : π ∘ σ = id
hσ₂ : ⇑l ∘ ⇑b ∘ σ = Subtype.val
f : M →ₗ[R] ↥⋯.toFinset →₀ R := ⋯
x : ↥⋯.toFinset →₀ R
⊢ (linearCombination R Subtype.val) x = 0 → ∃ y, l y = 0 ∧ f y = x