mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.NumberTheory.RamificationInertia.Galois

Goal state before:
pp: A : Type u_1
B : Type u_2
inst✝⁹ : CommRing A
inst✝⁸ : CommRing B
inst✝⁷ : Algebra A B
G : Type u_3
inst✝⁶ : Group G
inst✝⁵ : Finite G
inst✝⁴ : MulSemiringAction G B
inst✝³ : IsGaloisGroup G A B
inst✝² : Module.Finite A B
inst✝¹ : FaithfulSMul A B
p : Ideal A
inst✝ : p.IsPrime
⊢ p.inertiaDegIn B ≠ 0

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.NumberField.Discriminant.Basic

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
N : ℕ
hK : |discr K| ≤ ↑N
h_nz : N ≠ 0
h₂ : 1 < 3 * π / 4
h✝ : 1 < finrank ℚ K
h : Real.logb (3 * π / 4) (9 / 4 * ↑N) < ↑(finrank ℚ K)
⊢ 9 / 4 ≠ 0

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Invariant.Galois

Goal state before:
pp: A : Type u_1
B : Type u_2
inst✝¹⁹ : CommRing A
inst✝¹⁸ : CommRing B
inst✝¹⁷ : Algebra A B
G : Type u_3
inst✝¹⁶ : Finite G
inst✝¹⁵ : Group G
inst✝¹⁴ : MulSemiringAction G B
inst✝¹³ : Algebra.IsInvariant A B G
P : Ideal A
Q : Ideal B
inst✝¹² : Q.LiesOver P
inst✝¹¹ : P.IsPrime
inst✝¹⁰ : Q.IsPrime
K : Type u_4
L : Type u_5
inst✝⁹ : Field K
inst✝⁸ : Field L
inst✝⁷ : Algebra K L
inst✝⁶ : Algebra (A ⧸ P) K
inst✝⁵ : IsFractionRing (A ⧸ P) K
inst✝⁴ : Algebra (B ⧸ Q) L
inst✝³ : IsFractionRing (B ⧸ Q) L
inst✝² : Algebra (A ⧸ P) L
inst✝¹ : IsScalarTower (A ⧸ P) (B ⧸ Q) L
inst✝ : IsScalarTower (A ⧸ P) K L
this✝ : Algebra.IsIntegral A B
this : Algebra.IsAlgebraic K L
x y : B ⧸ Q
hy : y ∈ nonZeroDivisors (B ⧸ Q)
a : A
ha : (algebraMap A (A ⧸ P)) a ≠ 0
b : B
h : (algebraMap A (A ⧸ P)) a • x = y * (algebraMap B (B ⧸ Q)) b
val✝ : Fintype G
p : A[X]
hp : Polynomial.map (algebraMap A B) p = MulSemiringAction.charpoly G b
h_monic : p.Monic
h_eval : (aeval b) p = 0
q : A[X] := p.comp (C a * X)
d : L := (algebraMap (B ⧸ Q) L) x / (algebraMap (B ⧸ Q) L) y
comm₁ : (algebraMap K L).comp (algebraMap (A ⧸ P) K) = (algebraMap (B ⧸ Q) L).comp (algebraMap (A ⧸ P) (B ⧸ Q))
comm₂ : (algebraMap (A ⧸ P) (B ⧸ Q)).comp (algebraMap A (A ⧸ P)) = (algebraMap B (B ⧸ Q)).comp (algebraMap A B)
⊢ (aeval d) (Polynomial.map (algebraMap (A ⧸ P) K) (Polynomial.map (algebraMap A (A ⧸ P)) q)) = 0

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.IntegralClosure.IntegralRestrict

Goal state before:
pp: A : Type u_1
B : Type u_6
inst✝²⁵ : CommRing A
inst✝²⁴ : CommRing B
inst✝²³ : Algebra A B
Aₘ : Type u_9
Bₘ : Type u_10
inst✝²² : CommRing Aₘ
inst✝²¹ : CommRing Bₘ
inst✝²⁰ : Algebra Aₘ Bₘ
inst✝¹⁹ : Algebra A Aₘ
inst✝¹⁸ : Algebra B Bₘ
inst✝¹⁷ : Algebra A Bₘ
inst✝¹⁶ : IsScalarTower A Aₘ Bₘ
inst✝¹⁵ : IsScalarTower A B Bₘ
M : Submonoid A
inst✝¹⁴ : IsLocalization M Aₘ
inst✝¹³ : IsLocalization (algebraMapSubmonoid B M) Bₘ
inst✝¹² : IsIntegrallyClosed A
inst✝¹¹ : IsDomain A
inst✝¹⁰ : IsDomain B
inst✝⁹ : IsIntegrallyClosed B
inst✝⁸ : Algebra.IsIntegral A B
inst✝⁷ : IsTorsionFree A B
inst✝⁶ : IsDomain Aₘ
inst✝⁵ : IsIntegrallyClosed Aₘ
inst✝⁴ : IsDomain Bₘ
inst✝³ : IsIntegrallyClosed Bₘ
inst✝² : IsTorsionFree Aₘ Bₘ
inst✝¹ : Algebra.IsIntegral Aₘ Bₘ
inst✝ : FiniteDimensional (FractionRing A) (FractionRing B)
x : B
hM : M ≤ A⁰
K : Type u_1 := FractionRing A
L : Type u_6 := FractionRing B
f : Aₘ →+* K := IsLocalization.map K (RingHom.id A) hM
this✝⁵ : Algebra Aₘ K := f.toAlgebra
this✝⁴ : IsScalarTower A Aₘ K
this✝³ : IsFractionRing Aₘ K := IsFractionRing.isFractionRing_of_isDomain_of_isLocalization M Aₘ K
g : Bₘ →+* L := IsLocalization.map L (RingHom.id B) ⋯
this✝² : Algebra Bₘ L := g.toAlgebra
this✝¹ : IsScalarTower B Bₘ L
this✝ : Algebra Aₘ L := ((algebraMap K L).comp f).toAlgebra
this : IsScalarTower Aₘ K L
⊢ IsScalarTower Aₘ Bₘ L

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.RamificationInertia.Galois

Goal state before:
pp: A : Type u_1
B : Type u_2
inst✝¹⁹ : CommRing A
inst✝¹⁸ : CommRing B
inst✝¹⁷ : Algebra A B
p : Ideal A
P : Ideal B
hPp : P.IsPrime
hp : P.LiesOver p
G : Type u_3
inst✝¹⁶ : Group G
inst✝¹⁵ : Finite G
inst✝¹⁴ : MulSemiringAction G B
inst✝¹³ : IsGaloisGroup G A B
C : Type u_4
inst✝¹² : CommRing C
inst✝¹¹ : Algebra A C
inst✝¹⁰ : Algebra B C
inst✝⁹ : Nonempty ↑(P.primesOver C)
inst✝⁸ : IsScalarTower A B C
GAC : Type u_5
inst✝⁷ : Group GAC
inst✝⁶ : Finite GAC
inst✝⁵ : MulSemiringAction GAC C
inst✝⁴ : IsGaloisGroup GAC A C
GBC : Type u_6
inst✝³ : Group GBC
inst✝² : Finite GBC
inst✝¹ : MulSemiringAction GBC C
inst✝ : IsGaloisGroup GBC B C
⊢ p.inertiaDegIn B * P.inertiaDegIn C = p.inertiaDegIn C

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.NumberField.Discriminant.Basic

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
N : ℕ
hK : |discr K| ≤ ↑N
h_nz : N ≠ 0
h₂ : 1 < 3 * π / 4
h : 1 < finrank ℚ K
⊢ 0 < 4

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.NumberField.Discriminant.Basic

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
N : ℕ
hK : |discr K| ≤ ↑N
h_nz : N ≠ 0
h₂ : 1 < 3 * π / 4
h : 1 < finrank ℚ K
⊢ 0 < 9

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Invariant.Galois

Goal state before:
pp: A : Type u_1
B : Type u_2
inst✝¹⁹ : CommRing A
inst✝¹⁸ : CommRing B
inst✝¹⁷ : Algebra A B
G : Type u_3
inst✝¹⁶ : Finite G
inst✝¹⁵ : Group G
inst✝¹⁴ : MulSemiringAction G B
inst✝¹³ : Algebra.IsInvariant A B G
P : Ideal A
Q : Ideal B
inst✝¹² : Q.LiesOver P
inst✝¹¹ : P.IsPrime
inst✝¹⁰ : Q.IsPrime
K : Type u_4
L : Type u_5
inst✝⁹ : Field K
inst✝⁸ : Field L
inst✝⁷ : Algebra K L
inst✝⁶ : Algebra (A ⧸ P) K
inst✝⁵ : IsFractionRing (A ⧸ P) K
inst✝⁴ : Algebra (B ⧸ Q) L
inst✝³ : IsFractionRing (B ⧸ Q) L
inst✝² : Algebra (A ⧸ P) L
inst✝¹ : IsScalarTower (A ⧸ P) (B ⧸ Q) L
inst✝ : IsScalarTower (A ⧸ P) K L
this✝ : Algebra.IsIntegral A B
this : Algebra.IsAlgebraic K L
x y : B ⧸ Q
hy : y ∈ nonZeroDivisors (B ⧸ Q)
a : A
ha : (algebraMap A (A ⧸ P)) a ≠ 0
b : B
h : (algebraMap A (A ⧸ P)) a • x = y * (algebraMap B (B ⧸ Q)) b
val✝ : Fintype G
p : A[X]
hp : Polynomial.map (algebraMap A B) p = MulSemiringAction.charpoly G b
h_monic : p.Monic
q : A[X] := p.comp (C a * X)
d : L := (algebraMap (B ⧸ Q) L) x / (algebraMap (B ⧸ Q) L) y
comm₁ : (algebraMap K L).comp (algebraMap (A ⧸ P) K) = (algebraMap (B ⧸ Q) L).comp (algebraMap (A ⧸ P) (B ⧸ Q))
comm₂ : (algebraMap (A ⧸ P) (B ⧸ Q)).comp (algebraMap A (A ⧸ P)) = (algebraMap B (B ⧸ Q)).comp (algebraMap A B)
h_eval : (aeval d) (Polynomial.map (algebraMap (A ⧸ P) K) (Polynomial.map (algebraMap A (A ⧸ P)) q)) = 0
⊢ (Polynomial.map (algebraMap A B) p).Splits

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.RamificationInertia.Galois

Goal state before:
pp: A : Type u_1
B : Type u_2
inst✝²⁰ : CommRing A
inst✝¹⁹ : CommRing B
inst✝¹⁸ : Algebra A B
p : Ideal A
P : Ideal B
hPp : P.IsPrime
hp : P.LiesOver p
G : Type u_3
inst✝¹⁷ : Group G
inst✝¹⁶ : Finite G
inst✝¹⁵ : MulSemiringAction G B
inst✝¹⁴ : IsGaloisGroup G A B
C : Type u_4
inst✝¹³ : CommRing C
inst✝¹² : Algebra A C
inst✝¹¹ : Algebra B C
inst✝¹⁰ : Nonempty ↑(P.primesOver C)
inst✝⁹ : IsScalarTower A B C
GAC : Type u_5
inst✝⁸ : Group GAC
inst✝⁷ : Finite GAC
inst✝⁶ : MulSemiringAction GAC C
inst✝⁵ : IsGaloisGroup GAC A C
GBC : Type u_6
inst✝⁴ : Group GBC
inst✝³ : Finite GBC
inst✝² : MulSemiringAction GBC C
inst✝¹ : IsGaloisGroup GBC B C
inst✝ : Flat B C
⊢ p.ramificationIdxIn B * P.ramificationIdxIn C = p.ramificationIdxIn C

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.NumberField.Discriminant.Basic

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
N : ℕ
hK : |discr K| ≤ ↑N
h_nz : N ≠ 0
h₂ : 1 < 3 * π / 4
h : 1 < finrank ℚ K
⊢ 4 / 9 ≤ 1

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.IntegralClosure.IntegralRestrict

Goal state before:
pp: A : Type u_1
B : Type u_6
inst✝²⁵ : CommRing A
inst✝²⁴ : CommRing B
inst✝²³ : Algebra A B
Aₘ : Type u_9
Bₘ : Type u_10
inst✝²² : CommRing Aₘ
inst✝²¹ : CommRing Bₘ
inst✝²⁰ : Algebra Aₘ Bₘ
inst✝¹⁹ : Algebra A Aₘ
inst✝¹⁸ : Algebra B Bₘ
inst✝¹⁷ : Algebra A Bₘ
inst✝¹⁶ : IsScalarTower A Aₘ Bₘ
inst✝¹⁵ : IsScalarTower A B Bₘ
M : Submonoid A
inst✝¹⁴ : IsLocalization M Aₘ
inst✝¹³ : IsLocalization (algebraMapSubmonoid B M) Bₘ
inst✝¹² : IsIntegrallyClosed A
inst✝¹¹ : IsDomain A
inst✝¹⁰ : IsDomain B
inst✝⁹ : IsIntegrallyClosed B
inst✝⁸ : Algebra.IsIntegral A B
inst✝⁷ : IsTorsionFree A B
inst✝⁶ : IsDomain Aₘ
inst✝⁵ : IsIntegrallyClosed Aₘ
inst✝⁴ : IsDomain Bₘ
inst✝³ : IsIntegrallyClosed Bₘ
inst✝² : IsTorsionFree Aₘ Bₘ
inst✝¹ : Algebra.IsIntegral Aₘ Bₘ
inst✝ : FiniteDimensional (FractionRing A) (FractionRing B)
x : B
⊢ (algebraMap A Aₘ) ((intNorm A B) x) = (intNorm Aₘ Bₘ) ((algebraMap B Bₘ) x)

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.NumberField.Discriminant.Basic

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
N : ℕ
hK : |discr K| ≤ ↑N
⊢ finrank ℚ K ≤ rankOfDiscrBdd N

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.IntegralClosure.IntegralRestrict

Goal state before:
pp: A : Type u_1
B : Type u_6
inst✝⁹ : CommRing A
inst✝⁸ : CommRing B
inst✝⁷ : Algebra A B
inst✝⁶ : IsDomain A
inst✝⁵ : IsIntegrallyClosed A
inst✝⁴ : IsDomain B
inst✝³ : IsIntegrallyClosed B
inst✝² : Module.Finite A B
inst✝¹ : IsTorsionFree A B
inst✝ : IsGalois (FractionRing A) (FractionRing B)
x : B
⊢ (algebraMap A B) ((intNorm A B) x) = ∏ σ, σ x

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.IntegralClosure.IntegralRestrict

Goal state before:
pp: A : Type u_1
B : Type u_6
inst✝⁸ : CommRing A
inst✝⁷ : CommRing B
inst✝⁶ : Algebra A B
inst✝⁵ : IsDomain A
inst✝⁴ : IsIntegrallyClosed A
inst✝³ : IsDomain B
inst✝² : IsIntegrallyClosed B
inst✝¹ : Module.Finite A B
inst✝ : IsTorsionFree A B
x : B
this : FiniteDimensional (FractionRing A) (FractionRing B)
hx : x = 0
⊢ (algebraMap A B) ((intNorm A B) x) = x * 1

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.NumberField.Discriminant.Basic

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
N : ℕ
hK : |discr K| ≤ ↑N
⊢ boundOfDiscBdd N - 1 < boundOfDiscBdd N

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.RamificationInertia.Galois

Goal state before:
pp: A : Type u_1
inst✝¹¹ : CommRing A
inst✝¹⁰ : IsDomain A
p : Ideal A
inst✝⁹ : p.IsPrime
B : Type u_2
inst✝⁸ : CommRing B
inst✝⁷ : IsDomain B
inst✝⁶ : Algebra A B
inst✝⁵ : Module.Finite A B
inst✝⁴ : Flat A B
G : Type u_3
inst✝³ : Group G
inst✝² : Finite G
inst✝¹ : MulSemiringAction G B
inst✝ : IsGaloisGroup G A B
⊢ (p.primesOver B).ncard * (p.ramificationIdxIn B * p.inertiaDegIn B) = Nat.card G

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.IntegralClosure.IntegralRestrict

Goal state before:
pp: A : Type u_1
B : Type u_6
inst✝⁸ : CommRing A
inst✝⁷ : CommRing B
inst✝⁶ : Algebra A B
inst✝⁵ : IsDomain A
inst✝⁴ : IsIntegrallyClosed A
inst✝³ : IsDomain B
inst✝² : IsIntegrallyClosed B
inst✝¹ : Module.Finite A B
inst✝ : IsTorsionFree A B
x : B
this✝ : FiniteDimensional (FractionRing A) (FractionRing B)
hx : ¬x = 0
K : Type u_1 := FractionRing A
L : Type u_6 := FractionRing B
E : Type u_6 := AlgebraicClosure L
this : IsIntegral A (((algebraMap B L) x)⁻¹ * (algebraMap A L) ((intNorm A B) x))
⊢ x ∣ (algebraMap A B) ((intNorm A B) x)

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.RamificationInertia.Galois

Goal state before:
pp: A : Type u_1
B : Type u_2
inst✝²¹ : CommRing A
inst✝²⁰ : CommRing B
inst✝¹⁹ : Algebra A B
inst✝¹⁸ : FaithfulSMul A B
p : Ideal A
P : Ideal B
inst✝¹⁷ : P.IsPrime
inst✝¹⁶ : P.LiesOver p
G : Type u_3
inst✝¹⁵ : Group G
inst✝¹⁴ : Finite G
inst✝¹³ : MulSemiringAction G B
inst✝¹² : IsGaloisGroup G A B
C : Type u_4
inst✝¹¹ : CommRing C
inst✝¹⁰ : IsDomain C
inst✝⁹ : Algebra A C
inst✝⁸ : Algebra B C
inst✝⁷ : FaithfulSMul B C
inst✝⁶ : IsScalarTower A B C
GAC : Type u_5
inst✝⁵ : Group GAC
inst✝⁴ : Finite GAC
inst✝³ : MulSemiringAction GAC C
inst✝² : IsGaloisGroup GAC A C
this✝ : Algebra.IsIntegral A C
this : Algebra.IsIntegral B C
f : GAC →* G := restrictHom GAC G A B C
H : Subgroup GAC := Subgroup.comap f (stabilizer G P)
Q Q' : Ideal C
inst✝¹ : Q.LiesOver P
inst✝ : Q'.LiesOver P
g : GAC
hg : g • Q = Q'
⊢ g ∈ H

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.NumberField.Discriminant.Basic

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
N : ℕ
hK : |discr K| ≤ ↑N
this : boundOfDiscBdd N - 1 < boundOfDiscBdd N
⊢ 0 ≤ 2⁻¹

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.NumberField.Discriminant.Basic

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
N : ℕ
hK : |discr K| ≤ ↑N
this : boundOfDiscBdd N - 1 < boundOfDiscBdd N
⊢ 2⁻¹ ≤ 1

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.NumberField.Discriminant.Basic

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
N : ℕ
hK : |discr K| ≤ ↑N
⊢ minkowskiBound K 1 < ↑(boundOfDiscBdd N)

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.NumberField.Discriminant.Basic

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
N : ℕ
hK : |discr K| ≤ ↑N
a : 𝓞 K
h : ℚ⟮↑a⟯ = ⊤
⊢ (minpoly ℤ ↑a).natDegree ≤ rankOfDiscrBdd N

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.IntegralClosure.IntegralRestrict

Goal state before:
pp: A : Type u_1
B : Type u_6
inst✝⁸ : CommRing A
inst✝⁷ : CommRing B
inst✝⁶ : Algebra A B
inst✝⁵ : IsDomain A
inst✝⁴ : IsIntegrallyClosed A
inst✝³ : IsDomain B
inst✝² : IsIntegrallyClosed B
inst✝¹ : Module.Finite A B
inst✝ : IsTorsionFree A B
x : B
this✝¹ : FiniteDimensional (FractionRing A) (FractionRing B)
hx : ¬x = 0
K : Type u_1 := FractionRing A
L : Type u_6 := FractionRing B
E : Type u_6 := AlgebraicClosure L
this✝ : IsTorsionFree B E
a : E
ha : a ∈ (minpoly K ((algebraMap B L) x)).aroots E
this : (aeval a) (map (algebraMap A K) (minpoly A x)) = 0
⊢ eval₂ (algebraMap A E) a (minpoly A x) = 0

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.RamificationInertia.Galois

Goal state before:
pp: A : Type u_1
B : Type u_2
inst✝¹⁹ : CommRing A
inst✝¹⁸ : CommRing B
inst✝¹⁷ : Algebra A B
inst✝¹⁶ : FaithfulSMul A B
p : Ideal A
P : Ideal B
inst✝¹⁵ : P.IsPrime
inst✝¹⁴ : P.LiesOver p
G : Type u_3
inst✝¹³ : Group G
inst✝¹² : Finite G
inst✝¹¹ : MulSemiringAction G B
inst✝¹⁰ : IsGaloisGroup G A B
C : Type u_4
inst✝⁹ : CommRing C
inst✝⁸ : IsDomain C
inst✝⁷ : Algebra A C
inst✝⁶ : Algebra B C
inst✝⁵ : FaithfulSMul B C
inst✝⁴ : IsScalarTower A B C
GAC : Type u_5
inst✝³ : Group GAC
inst✝² : Finite GAC
inst✝¹ : MulSemiringAction GAC C
inst✝ : IsGaloisGroup GAC A C
this✝² : Algebra.IsIntegral A C
this✝¹ : Algebra.IsIntegral B C
f : GAC →* G := restrictHom GAC G A B C
H : Subgroup GAC := Subgroup.comap f (stabilizer G P)
key : ∀ (Q Q' : Ideal C) [Q.LiesOver P] [Q'.LiesOver P] (g : GAC), g • Q = Q' → g ∈ H
Q : Ideal C
left✝¹ : Q.IsPrime
right✝¹ : Q.LiesOver P
this✝ : Q.LiesOver p
Q' : Ideal C
left✝ : Q'.IsPrime
right✝ : Q'.LiesOver P
this : Q'.LiesOver p
g : GAC
hg : Q' = g • Q
⊢ (fun m ↦ m • Q) ⟨g, ⋯⟩ = Q'

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.IntegralClosure.IntegralRestrict

Goal state before:
pp: A : Type u_1
B : Type u_6
inst✝⁸ : CommRing A
inst✝⁷ : CommRing B
inst✝⁶ : Algebra A B
inst✝⁵ : IsDomain A
inst✝⁴ : IsIntegrallyClosed A
inst✝³ : IsDomain B
inst✝² : IsIntegrallyClosed B
inst✝¹ : Module.Finite A B
inst✝ : IsTorsionFree A B
x : B
⊢ x ∣ (algebraMap A B) ((intNorm A B) x)

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.RamificationInertia.Galois

Goal state before:
pp: A : Type u_1
B : Type u_2
inst✝¹⁹ : CommRing A
inst✝¹⁸ : CommRing B
inst✝¹⁷ : Algebra A B
inst✝¹⁶ : FaithfulSMul A B
p : Ideal A
P : Ideal B
inst✝¹⁵ : P.IsPrime
inst✝¹⁴ : P.LiesOver p
G : Type u_3
inst✝¹³ : Group G
inst✝¹² : Finite G
inst✝¹¹ : MulSemiringAction G B
inst✝¹⁰ : IsGaloisGroup G A B
C : Type u_4
inst✝⁹ : CommRing C
inst✝⁸ : IsDomain C
inst✝⁷ : Algebra A C
inst✝⁶ : Algebra B C
inst✝⁵ : FaithfulSMul B C
inst✝⁴ : IsScalarTower A B C
GAC : Type u_5
inst✝³ : Group GAC
inst✝² : Finite GAC
inst✝¹ : MulSemiringAction GAC C
inst✝ : IsGaloisGroup GAC A C
this✝¹ : Algebra.IsIntegral A C
this✝ : Algebra.IsIntegral B C
f : GAC →* G := restrictHom GAC G A B C
H : Subgroup GAC := Subgroup.comap f (stabilizer G P)
key : ∀ (Q Q' : Ideal C) [Q.LiesOver P] [Q'.LiesOver P] (g : GAC), g • Q = Q' → g ∈ H
Q : Ideal C
left✝ : Q.IsPrime
right✝ : Q.LiesOver P
this : Q.LiesOver p
⊢ orbit (↥H) Q = P.primesOver C

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.RamificationInertia.Galois

Goal state before:
pp: A : Type u_1
B : Type u_2
inst✝¹⁹ : CommRing A
inst✝¹⁸ : CommRing B
inst✝¹⁷ : Algebra A B
inst✝¹⁶ : FaithfulSMul A B
p : Ideal A
P : Ideal B
inst✝¹⁵ : P.IsPrime
inst✝¹⁴ : P.LiesOver p
G : Type u_3
inst✝¹³ : Group G
inst✝¹² : Finite G
inst✝¹¹ : MulSemiringAction G B
inst✝¹⁰ : IsGaloisGroup G A B
C : Type u_4
inst✝⁹ : CommRing C
inst✝⁸ : IsDomain C
inst✝⁷ : Algebra A C
inst✝⁶ : Algebra B C
inst✝⁵ : FaithfulSMul B C
inst✝⁴ : IsScalarTower A B C
GAC : Type u_5
inst✝³ : Group GAC
inst✝² : Finite GAC
inst✝¹ : MulSemiringAction GAC C
inst✝ : IsGaloisGroup GAC A C
this✝¹ : Algebra.IsIntegral A C
this✝ : Algebra.IsIntegral B C
f : GAC →* G := restrictHom GAC G A B C
H : Subgroup GAC := Subgroup.comap f (stabilizer G P)
key : ∀ (Q Q' : Ideal C) [Q.LiesOver P] [Q'.LiesOver P] (g : GAC), g • Q = Q' → g ∈ H
Q : Ideal C
left✝ : Q.IsPrime
right✝ : Q.LiesOver P
this : Q.LiesOver p
orbit_eq : orbit (↥H) Q = P.primesOver C
⊢ stabilizer (↥H) Q = (stabilizer GAC Q).subgroupOf H

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.NumberField.ClassNumber

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
C : ClassGroup (𝓞 K)
J : ↥(Ideal (𝓞 K))⁰
hJ : ClassGroup.mk0 J = C⁻¹
a : 𝓞 K
ha : a ∈ ↑↑J
h_nz : (Algebra.linearMap (𝓞 K) K) a ≠ 0
h_nm :
  ↑|(Algebra.norm ℚ) ((Algebra.linearMap (𝓞 K) K) a)| ≤
    ↑(FractionalIdeal.absNorm ↑((FractionalIdeal.mk0 K) J)) * (4 / π) ^ nrComplexPlaces K * ↑(finrank ℚ K)! /
        ↑(finrank ℚ K) ^ finrank ℚ K *
      √|↑(discr K)|
⊢ ?m.150 ∈ ?m.149

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.NumberField.ClassNumber

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
C : ClassGroup (𝓞 K)
J : ↥(Ideal (𝓞 K))⁰
hJ : ClassGroup.mk0 J = C⁻¹
a : 𝓞 K
ha : a ∈ ↑↑J
h_nz : (Algebra.linearMap (𝓞 K) K) a ≠ 0
h_nm :
  ↑|(Algebra.norm ℚ) ((Algebra.linearMap (𝓞 K) K) a)| ≤
    ↑(FractionalIdeal.absNorm ↑((FractionalIdeal.mk0 K) J)) * (4 / π) ^ nrComplexPlaces K * ↑(finrank ℚ K)! /
        ↑(finrank ℚ K) ^ finrank ℚ K *
      √|↑(discr K)|
I₀ : Ideal (𝓞 K)
hI : span {a} = ↑J * I₀
⊢ I₀ ≠ 0

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.RamificationInertia.Galois

Goal state before:
pp: A : Type u_1
B : Type u_2
inst✝¹⁹ : CommRing A
inst✝¹⁸ : CommRing B
inst✝¹⁷ : Algebra A B
inst✝¹⁶ : FaithfulSMul A B
p : Ideal A
P : Ideal B
inst✝¹⁵ : P.IsPrime
inst✝¹⁴ : P.LiesOver p
G : Type u_3
inst✝¹³ : Group G
inst✝¹² : Finite G
inst✝¹¹ : MulSemiringAction G B
inst✝¹⁰ : IsGaloisGroup G A B
C : Type u_4
inst✝⁹ : CommRing C
inst✝⁸ : IsDomain C
inst✝⁷ : Algebra A C
inst✝⁶ : Algebra B C
inst✝⁵ : FaithfulSMul B C
inst✝⁴ : IsScalarTower A B C
GAC : Type u_5
inst✝³ : Group GAC
inst✝² : Finite GAC
inst✝¹ : MulSemiringAction GAC C
inst✝ : IsGaloisGroup GAC A C
this✝¹ : Algebra.IsIntegral A C
this✝ : Algebra.IsIntegral B C
f : GAC →* G := restrictHom GAC G A B C
H : Subgroup GAC := Subgroup.comap f (stabilizer G P)
key : ∀ (Q Q' : Ideal C) [Q.LiesOver P] [Q'.LiesOver P] (g : GAC), g • Q = Q' → g ∈ H
Q : Ideal C
left✝ : Q.IsPrime
right✝ : Q.LiesOver P
this : Q.LiesOver p
orbit_eq : orbit (↥H) Q = P.primesOver C
stabilizer_eq : stabilizer (↥H) Q = (stabilizer GAC Q).subgroupOf H
⊢ (orbit G P).ncard * (P.primesOver C).ncard = (p.primesOver C).ncard

Tactic:
← index_stabilizer,

Goal state after:
A : Type u_1
B : Type u_2
inst✝¹⁹ : CommRing A
inst✝¹⁸ : CommRing B
inst✝¹⁷ : Algebra A B
inst✝¹⁶ : FaithfulSMul A B
p : Ideal A
P : Ideal B
inst✝¹⁵ : P.IsPrime
inst✝¹⁴ : P.LiesOver p
G : Type u_3
inst✝¹³ : Group G
inst✝¹² : Finite G
inst✝¹¹ : MulSemiringAction G B
inst✝¹⁰ : IsGaloisGroup G A B
C : Type u_4
inst✝⁹ : CommRing C
inst✝⁸ : IsDomain C
inst✝⁷ : Algebra A C
inst✝⁶ : Algebra B C
inst✝⁵ : FaithfulSMul B C
inst✝⁴ : IsScalarTower A B C
GAC : Type u_5
inst✝³ : Group GAC
inst✝² : Finite GAC
inst✝¹ : MulSemiringAction GAC C
inst✝ : IsGaloisGroup GAC A C
this✝¹ : Algebra.IsIntegral A C
this✝ : Algebra.IsIntegral B C
f : GAC →* G := restrictHom GAC G A B C
H : Subgroup GAC := Subgroup.comap f (stabilizer G P)
key : ∀ (Q Q' : Ideal C) [Q.LiesOver P] [Q'.LiesOver P] (g : GAC), g • Q = Q' → g ∈ H
Q : Ideal C
left✝ : Q.IsPrime
right✝ : Q.LiesOver P
this : Q.LiesOver p
orbit_eq : orbit (↥H) Q = P.primesOver C
stabilizer_eq : stabilizer (↥H) Q = (stabilizer GAC Q).subgroupOf H
⊢ (stabilizer G P).index * (P.primesOver C).ncard = (p.primesOver C).ncard
Module: Mathlib.NumberTheory.RamificationInertia.Galois

Goal state before:
pp: A : Type u_1
B : Type u_2
inst✝¹⁹ : CommRing A
inst✝¹⁸ : CommRing B
inst✝¹⁷ : Algebra A B
inst✝¹⁶ : FaithfulSMul A B
p : Ideal A
P : Ideal B
inst✝¹⁵ : P.IsPrime
inst✝¹⁴ : P.LiesOver p
G : Type u_3
inst✝¹³ : Group G
inst✝¹² : Finite G
inst✝¹¹ : MulSemiringAction G B
inst✝¹⁰ : IsGaloisGroup G A B
C : Type u_4
inst✝⁹ : CommRing C
inst✝⁸ : IsDomain C
inst✝⁷ : Algebra A C
inst✝⁶ : Algebra B C
inst✝⁵ : FaithfulSMul B C
inst✝⁴ : IsScalarTower A B C
GAC : Type u_5
inst✝³ : Group GAC
inst✝² : Finite GAC
inst✝¹ : MulSemiringAction GAC C
inst✝ : IsGaloisGroup GAC A C
this✝¹ : Algebra.IsIntegral A C
this✝ : Algebra.IsIntegral B C
f : GAC →* G := restrictHom GAC G A B C
H : Subgroup GAC := Subgroup.comap f (stabilizer G P)
key : ∀ (Q Q' : Ideal C) [Q.LiesOver P] [Q'.LiesOver P] (g : GAC), g • Q = Q' → g ∈ H
Q : Ideal C
left✝ : Q.IsPrime
right✝ : Q.LiesOver P
this : Q.LiesOver p
orbit_eq : orbit (↥H) Q = P.primesOver C
stabilizer_eq : stabilizer (↥H) Q = (stabilizer GAC Q).subgroupOf H
⊢ (stabilizer G P).index * (orbit (↥H) Q).ncard = (p.primesOver C).ncard

Tactic:
← index_stabilizer,

Goal state after:
A : Type u_1
B : Type u_2
inst✝¹⁹ : CommRing A
inst✝¹⁸ : CommRing B
inst✝¹⁷ : Algebra A B
inst✝¹⁶ : FaithfulSMul A B
p : Ideal A
P : Ideal B
inst✝¹⁵ : P.IsPrime
inst✝¹⁴ : P.LiesOver p
G : Type u_3
inst✝¹³ : Group G
inst✝¹² : Finite G
inst✝¹¹ : MulSemiringAction G B
inst✝¹⁰ : IsGaloisGroup G A B
C : Type u_4
inst✝⁹ : CommRing C
inst✝⁸ : IsDomain C
inst✝⁷ : Algebra A C
inst✝⁶ : Algebra B C
inst✝⁵ : FaithfulSMul B C
inst✝⁴ : IsScalarTower A B C
GAC : Type u_5
inst✝³ : Group GAC
inst✝² : Finite GAC
inst✝¹ : MulSemiringAction GAC C
inst✝ : IsGaloisGroup GAC A C
this✝¹ : Algebra.IsIntegral A C
this✝ : Algebra.IsIntegral B C
f : GAC →* G := restrictHom GAC G A B C
H : Subgroup GAC := Subgroup.comap f (stabilizer G P)
key : ∀ (Q Q' : Ideal C) [Q.LiesOver P] [Q'.LiesOver P] (g : GAC), g • Q = Q' → g ∈ H
Q : Ideal C
left✝ : Q.IsPrime
right✝ : Q.LiesOver P
this : Q.LiesOver p
orbit_eq : orbit (↥H) Q = P.primesOver C
stabilizer_eq : stabilizer (↥H) Q = (stabilizer GAC Q).subgroupOf H
⊢ (stabilizer G P).index * (stabilizer (↥H) Q).index = (p.primesOver C).ncard