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

Goal state before:
pp: A : Type u_1
K : Type u_2
L : Type u_3
L₂ : Type u_4
L₃ : Type u_5
B : Type u_6
B₂ : Type u_7
B₃ : Type u_8
inst✝⁴² : CommRing A
inst✝⁴¹ : CommRing B
inst✝⁴⁰ : CommRing B₂
inst✝³⁹ : CommRing B₃
inst✝³⁸ : Algebra A B
inst✝³⁷ : Algebra A B₂
inst✝³⁶ : Algebra A B₃
inst✝³⁵ : Field K
inst✝³⁴ : Field L
inst✝³³ : Field L₂
inst✝³² : Field L₃
inst✝³¹ : Algebra A K
inst✝³⁰ : IsFractionRing A K
inst✝²⁹ : Algebra K L
inst✝²⁸ : Algebra A L
inst✝²⁷ : IsScalarTower A K L
inst✝²⁶ : Algebra K L₂
inst✝²⁵ : Algebra A L₂
inst✝²⁴ : IsScalarTower A K L₂
inst✝²³ : Algebra K L₃
inst✝²² : Algebra A L₃
inst✝²¹ : IsScalarTower A K L₃
inst✝²⁰ : Algebra B L
inst✝¹⁹ : IsScalarTower A B L
inst✝¹⁸ : IsIntegralClosure B A L
inst✝¹⁷ : Algebra B₂ L₂
inst✝¹⁶ : IsScalarTower A B₂ L₂
inst✝¹⁵ : IsIntegralClosure B₂ A L₂
inst✝¹⁴ : Algebra B₃ L₃
inst✝¹³ : IsScalarTower A B₃ L₃
inst✝¹² : IsIntegralClosure B₃ A L₃
inst✝¹¹ : FiniteDimensional K L
Aₘ : Type ?u.109
Bₘ : Type ?u.111
inst✝¹⁰ : CommRing Aₘ
inst✝⁹ : CommRing Bₘ
inst✝⁸ : Algebra Aₘ Bₘ
inst✝⁷ : Algebra A Aₘ
inst✝⁶ : Algebra B Bₘ
inst✝⁵ : Algebra A Bₘ
inst✝⁴ : IsScalarTower A Aₘ Bₘ
inst✝³ : IsScalarTower A B Bₘ
M : Submonoid A
inst✝² : IsLocalization M Aₘ
inst✝¹ : IsLocalization (algebraMapSubmonoid B M) Bₘ
inst✝ : IsIntegrallyClosed A
⊢ IsIntegralClosure.mk' A ((norm K) ((algebraMap B L) 1)) ⋯ = 1

Tactic:
by

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

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
A : Set (mixedSpace K)
hA : ∀ (x : mixedSpace K), x ∈ A ↔ (fun w ↦ ‖x.1 w‖, x.2) ∈ A
inst✝ : NumberField K
⊢ ⋃ s, ⇑(negAt s) '' plusPart A =ᵐ[volume] A

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
h : 1 < finrank ℚ K
a : ℕ → ℝ := fun n ↦ ↑n ^ (n * 2) / ((4 / π) ^ n * ↑n.factorial ^ 2)
n m : ℕ
hmn✝ : 2 ≤ m
h_m : 4 / 9 * (3 * π / 4) ^ m ≤ a m
⊢ 3 = 1 + ↑(2 * m) * (1 / ↑m)

Tactic:
by

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

Goal state before:
pp: A : Type u_1
K : Type u_2
L : Type u_3
L₂ : Type u_4
L₃ : Type u_5
B : Type u_6
B₂ : Type u_7
B₃ : Type u_8
inst✝⁴² : CommRing A
inst✝⁴¹ : CommRing B
inst✝⁴⁰ : CommRing B₂
inst✝³⁹ : CommRing B₃
inst✝³⁸ : Algebra A B
inst✝³⁷ : Algebra A B₂
inst✝³⁶ : Algebra A B₃
inst✝³⁵ : Field K
inst✝³⁴ : Field L
inst✝³³ : Field L₂
inst✝³² : Field L₃
inst✝³¹ : Algebra A K
inst✝³⁰ : IsFractionRing A K
inst✝²⁹ : Algebra K L
inst✝²⁸ : Algebra A L
inst✝²⁷ : IsScalarTower A K L
inst✝²⁶ : Algebra K L₂
inst✝²⁵ : Algebra A L₂
inst✝²⁴ : IsScalarTower A K L₂
inst✝²³ : Algebra K L₃
inst✝²² : Algebra A L₃
inst✝²¹ : IsScalarTower A K L₃
inst✝²⁰ : Algebra B L
inst✝¹⁹ : IsScalarTower A B L
inst✝¹⁸ : IsIntegralClosure B A L
inst✝¹⁷ : Algebra B₂ L₂
inst✝¹⁶ : IsScalarTower A B₂ L₂
inst✝¹⁵ : IsIntegralClosure B₂ A L₂
inst✝¹⁴ : Algebra B₃ L₃
inst✝¹³ : IsScalarTower A B₃ L₃
inst✝¹² : IsIntegralClosure B₃ A L₃
inst✝¹¹ : FiniteDimensional K L
Aₘ : Type ?u.109
Bₘ : Type ?u.111
inst✝¹⁰ : CommRing Aₘ
inst✝⁹ : CommRing Bₘ
inst✝⁸ : Algebra Aₘ Bₘ
inst✝⁷ : Algebra A Aₘ
inst✝⁶ : Algebra B Bₘ
inst✝⁵ : Algebra A Bₘ
inst✝⁴ : IsScalarTower A Aₘ Bₘ
inst✝³ : IsScalarTower A B Bₘ
M : Submonoid A
inst✝² : IsLocalization M Aₘ
inst✝¹ : IsLocalization (algebraMapSubmonoid B M) Bₘ
inst✝ : IsIntegrallyClosed A
x y : B
⊢ IsIntegralClosure.mk' A ((norm K) ((algebraMap B L) (x * y))) ⋯ =
    IsIntegralClosure.mk' A ((norm K) ((algebraMap B L) x)) ⋯ *
      IsIntegralClosure.mk' A ((norm K) ((algebraMap B L) y)) ⋯

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
h : 1 < finrank ℚ K
a : ℕ → ℝ := fun n ↦ ↑n ^ (n * 2) / ((4 / π) ^ n * ↑n.factorial ^ 2)
n m : ℕ
hmn✝ : 2 ≤ m
h_m : 4 / 9 * (3 * π / 4) ^ m ≤ a m
⊢ -2 ≤ 0

Tactic:
by

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

Goal state before:
pp: A : Type u_1
K : Type u_2
L : Type u_3
B : Type u_6
inst✝¹³ : CommRing A
inst✝¹² : CommRing B
inst✝¹¹ : Algebra A B
inst✝¹⁰ : Field K
inst✝⁹ : Field L
inst✝⁸ : Algebra A K
inst✝⁷ : IsFractionRing A K
inst✝⁶ : Algebra K L
inst✝⁵ : Algebra A L
inst✝⁴ : IsScalarTower A K L
inst✝³ : Algebra B L
inst✝² : IsScalarTower A B L
inst✝¹ : IsIntegralClosure B A L
inst✝ : IsIntegrallyClosed A
x : B
⊢ (algebraMap A K) ((intNormAux A K L B) x) = (norm K) ((algebraMap B L) x)

Tactic:
by

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

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
A : Set (mixedSpace K)
inst✝ : NumberField K
hm : MeasurableSet A
x✝ : { w // w.IsReal }
⊢ Measurable fun x ↦ x.1 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
h : 1 < finrank ℚ K
a : ℕ → ℝ := fun n ↦ ↑n ^ (n * 2) / ((4 / π) ^ n * ↑n.factorial ^ 2)
n m : ℕ
hmn✝ : 2 ≤ m
h_m : 4 / 9 * (3 * π / 4) ^ m ≤ a m
⊢ 0 ≤ 1 / ↑m

Tactic:
by

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

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
A : Set (mixedSpace K)
inst✝ : NumberField K
hm : MeasurableSet A
⊢ MeasurableSet (plusPart A)

Tactic:
by

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

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
s : Set { w // w.IsReal }
A : Set (mixedSpace K)
inst✝ : NumberField K
hm : MeasurableSet A
⊢ volume (⇑(negAt s) '' plusPart A) = volume (plusPart A)

Tactic:
by

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

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
A : Set (mixedSpace K)
hA : ∀ (x : mixedSpace K), x ∈ A ↔ (fun w ↦ ‖x.1 w‖, x.2) ∈ A
inst✝ : NumberField K
hm : MeasurableSet A
⊢ volume A = 2 ^ nrRealPlaces K * volume (plusPart A)

Tactic:
by

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

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
w : InfinitePlace K
⊢ ∀ {a b : InfinitePlace K → ℝ}, a ∈ {x | x w = 0} → b ∈ {x | x w = 0} → a + b ∈ {x | x w = 0}

Tactic:
by

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

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
w : InfinitePlace K
⊢ ∀ (c : ℝ) {x : InfinitePlace K → ℝ},
    x ∈ { carrier := {x | x w = 0}, add_mem' := ⋯, zero_mem' := ⋯ }.carrier →
      c • x ∈ { carrier := {x | x w = 0}, add_mem' := ⋯, zero_mem' := ⋯ }.carrier

Tactic:
by

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

Goal state before:
pp: A : Type u_1
K : Type u_2
L : Type u_3
B : Type u_6
inst✝¹⁹ : CommRing A
inst✝¹⁸ : CommRing B
inst✝¹⁷ : Algebra A B
inst✝¹⁶ : Field K
inst✝¹⁵ : Field L
inst✝¹⁴ : Algebra A K
inst✝¹³ : IsFractionRing A K
inst✝¹² : Algebra K L
inst✝¹¹ : Algebra A L
inst✝¹⁰ : IsScalarTower A K L
inst✝⁹ : Algebra B L
inst✝⁸ : IsScalarTower A B L
inst✝⁷ : IsIntegralClosure B A L
inst✝⁶ : FiniteDimensional K L
inst✝⁵ : IsIntegrallyClosed A
inst✝⁴ : IsDomain A
inst✝³ : IsDomain B
inst✝² : IsIntegrallyClosed B
inst✝¹ : Algebra.IsIntegral A B
inst✝ : IsTorsionFree A B
x : B
⊢ (algebraMap A K) ((intNorm A B) x) = (norm K) ((algebraMap B L) x)

Tactic:
by

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

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
w : InfinitePlace K
⊢ volume {x | x w = 0} = 0

Tactic:
by

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

Goal state before:
pp: A : Type u_1
B : Type u_6
inst✝¹⁸ : CommRing A
inst✝¹⁷ : CommRing B
inst✝¹⁶ : Algebra A B
inst✝¹⁵ : IsIntegrallyClosed A
inst✝¹⁴ : IsDomain A
inst✝¹³ : IsDomain B
inst✝¹² : IsIntegrallyClosed B
inst✝¹¹ : Algebra.IsIntegral A B
inst✝¹⁰ : IsTorsionFree A B
C : Type u_9
inst✝⁹ : CommRing C
inst✝⁸ : IsDomain C
inst✝⁷ : IsIntegrallyClosed C
inst✝⁶ : Algebra A C
inst✝⁵ : Algebra B C
inst✝⁴ : IsScalarTower A B C
inst✝³ : Algebra.IsIntegral A C
inst✝² : Algebra.IsIntegral B C
inst✝¹ : IsTorsionFree A C
inst✝ : IsTorsionFree B C
x : C
⊢ (intNorm A B) ((intNorm B C) x) = (intNorm A C) x

Tactic:
by

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

Goal state before:
pp: K : Type u_1
inst✝ : Field K
⊢ Function.Injective ⇑mixedSpaceOfRealSpace

Tactic:
by

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

Goal state before:
pp: A : Type u_1
B : Type u_6
inst✝¹⁰ : CommRing A
inst✝⁹ : CommRing B
inst✝⁸ : Algebra A B
inst✝⁷ : IsIntegrallyClosed A
inst✝⁶ : IsDomain A
inst✝⁵ : IsDomain B
inst✝⁴ : IsIntegrallyClosed B
inst✝³ : Algebra.IsIntegral A B
inst✝² : IsTorsionFree A B
inst✝¹ : Free A B
inst✝ : Module.Finite A B
⊢ intNorm A B = norm A

Tactic:
by

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

Goal state before:
pp: K : Type u_1
inst✝ : Field K
x : realSpace K
w : InfinitePlace K
hx : 0 ≤ x w
⊢ (normAtPlace w) (mixedSpaceOfRealSpace x) = x w

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
h : 1 < finrank ℚ K
⊢ 4 / 9 * (3 * π / 4) ^ finrank ℚ K ≤ ↑|discr K|

Tactic:
by

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

Goal state before:
pp: K : Type u_1
inst✝ : Field K
x : mixedSpace K
w : { w // w.IsReal }
⊢ normAtComplexPlaces x ↑w = x.1 w

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 : IsIntegrallyClosed A
inst✝⁵ : IsDomain A
inst✝⁴ : IsDomain B
inst✝³ : IsIntegrallyClosed B
inst✝² : Algebra.IsIntegral A B
inst✝¹ : IsTorsionFree A B
inst✝ : FiniteDimensional (FractionRing A) (FractionRing B)
⊢ (intNorm A B) 0 = 0

Tactic:
by

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

Goal state before:
pp: K : Type u_1
inst✝ : Field K
x : mixedSpace K
w : { w // w.IsComplex }
⊢ normAtComplexPlaces x ↑w = ‖x.2 w‖

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
h : Nat.succ 1 ≤ finrank ℚ K
⊢ 2 < 4 / 9 * (3 * π / 4) ^ 2

Tactic:
by

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

Goal state before:
pp: A : Type u_1
B : Type u_6
B₂ : Type u_7
inst✝¹⁶ : CommRing A
inst✝¹⁵ : CommRing B
inst✝¹⁴ : CommRing B₂
inst✝¹³ : Algebra A B
inst✝¹² : Algebra A B₂
inst✝¹¹ : IsIntegrallyClosed A
inst✝¹⁰ : IsDomain A
inst✝⁹ : IsDomain B
inst✝⁸ : IsIntegrallyClosed B
inst✝⁷ : Algebra.IsIntegral A B
inst✝⁶ : IsTorsionFree A B
inst✝⁵ : IsDomain B₂
inst✝⁴ : IsIntegrallyClosed B₂
inst✝³ : Algebra.IsIntegral A B₂
inst✝² : IsTorsionFree A B₂
inst✝¹ : Algebra.IsAlgebraic (FractionRing A) (FractionRing B)
inst✝ : Algebra.IsAlgebraic (FractionRing A) (FractionRing B₂)
x : B
σ : B ≃ₐ[A] B₂
⊢ (intNorm A B₂) (σ x) = (intNorm A B) x

Tactic:
by

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

Goal state before:
pp: K : Type u_1
inst✝ : Field K
x : realSpace K
hx : ∀ (w : InfinitePlace K), w.IsComplex → 0 ≤ x w
⊢ normAtComplexPlaces (mixedSpaceOfRealSpace x) = x

Tactic:
by

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

Goal state before:
pp: K : Type u_1
inst✝ : Field K
⊢ Continuous normAtAllPlaces

Tactic:
by

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

Goal state before:
pp: K : Type u_1
inst✝ : Field K
x : realSpace K
hx : ∀ (w : InfinitePlace K), 0 ≤ x w
⊢ normAtAllPlaces (mixedSpaceOfRealSpace x) = 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
h : Nat.succ 1 ≤ finrank ℚ K
⊢ 4 / 9 * (3 * π / 4) ^ 2 ≤ 4 / 9 * (3 * π / 4) ^ finrank ℚ K

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 : IsIntegrallyClosed A
inst✝⁵ : IsDomain A
inst✝⁴ : IsDomain B
inst✝³ : IsIntegrallyClosed B
inst✝² : Algebra.IsIntegral A B
inst✝¹ : IsTorsionFree A B
inst✝ : FiniteDimensional (FractionRing A) (FractionRing B)
x : B
⊢ (intNorm A B) x = 0 ↔ x = 0

Tactic:
by

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

Goal state before:
pp: K : Type u_1
inst✝ : Field K
x : K
w : InfinitePlace K
⊢ normAtAllPlaces ((mixedEmbedding K) x) w = w x

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 : IsIntegrallyClosed A
inst✝⁵ : IsDomain A
inst✝⁴ : IsDomain B
inst✝³ : IsIntegrallyClosed B
inst✝² : Algebra.IsIntegral A B
inst✝¹ : IsTorsionFree A B
inst✝ : FiniteDimensional (FractionRing A) (FractionRing B)
x : B
⊢ (intNorm A B) x ≠ 0 ↔ x ≠ 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
h : 1 < finrank ℚ K
⊢ 2 < |discr K|

Tactic:
by

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

Goal state before:
pp: K : Type u_1
inst✝ : Field K
x : mixedSpace K
⊢ normAtAllPlaces (fun w ↦ ‖x.1 w‖, x.2) = normAtAllPlaces x

Tactic:
by

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

Goal state before:
pp: K : Type u_1
inst✝ : Field K
x y : mixedSpace K
h : normAtComplexPlaces x = normAtComplexPlaces y
⊢ normAtAllPlaces x = normAtAllPlaces y

Tactic:
by

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

Goal state before:
pp: K : Type u_1
inst✝ : Field K
s : Set (realSpace K)
hs : ∀ x ∈ s, ∀ (w : InfinitePlace K), 0 ≤ x w
⊢ normAtAllPlaces '' normAtAllPlaces ⁻¹' s = s

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 : Algebra Aₘ K := f.toAlgebra
⊢ algebraMap A K = (algebraMap Aₘ K).comp (algebraMap A Aₘ)

Tactic:
by

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

Goal state before:
pp: A : Type u_2
inst✝¹ : Field A
inst✝ : CharZero A
p : IntermediateField ℚ A → Prop
S : Set { F // p F }
T : Set A
hT : T.Finite
h : ∀ F ∈ S, ∃ x ∈ T, ↑F = ℚ⟮x⟯
⊢ S.Finite

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)
⊢ (algebraMap (B ⧸ Q) L) y ≠ 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
⊢ N ≠ 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
⊢ 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
⊢ 0 < 3

Tactic:
by

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

Goal state before:
pp: A : Type u_1
B : Type u_2
inst✝¹⁶ : CommRing A
inst✝¹⁵ : CommRing B
inst✝¹⁴ : Algebra A B
p : Ideal A
G : Type u_3
inst✝¹³ : Group G
inst✝¹² : MulSemiringAction G B
inst✝¹¹ : SMulCommClass G A B
K : Type u_4
L : Type u_5
inst✝¹⁰ : Field K
inst✝⁹ : Field L
inst✝⁸ : Algebra A K
inst✝⁷ : IsFractionRing A K
inst✝⁶ : Algebra B L
inst✝⁵ : Algebra K L
inst✝⁴ : Algebra A L
inst✝³ : IsScalarTower A B L
inst✝² : IsScalarTower A K L
inst✝¹ : IsIntegralClosure B A L
inst✝ : FiniteDimensional K L
σ τ : Gal(L/K)
Q : ↑(p.primesOver B)
⊢ (σ * τ) • Q = σ • τ • Q

Tactic:
by

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

Goal state before:
pp: A : Type u_1
B : Type u_2
inst✝¹⁶ : CommRing A
inst✝¹⁵ : CommRing B
inst✝¹⁴ : Algebra A B
p : Ideal A
G : Type u_3
inst✝¹³ : Group G
inst✝¹² : MulSemiringAction G B
inst✝¹¹ : SMulCommClass G A B
K : Type u_4
L : Type u_5
inst✝¹⁰ : Field K
inst✝⁹ : Field L
inst✝⁸ : Algebra A K
inst✝⁷ : IsFractionRing A K
inst✝⁶ : Algebra B L
inst✝⁵ : Algebra K L
inst✝⁴ : Algebra A L
inst✝³ : IsScalarTower A B L
inst✝² : IsScalarTower A K L
inst✝¹ : IsIntegralClosure B A L
inst✝ : FiniteDimensional K L
Q : ↑(p.primesOver B)
⊢ 1 • Q = Q

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
⊢ 1 < 3 * π / 4

Tactic:
by

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

Goal state before:
pp: A : Type u_1
B : Type u_2
inst✝⁷ : CommRing A
inst✝⁶ : CommRing B
inst✝⁵ : Algebra A B
p : Ideal A
P Q : Ideal B
hPp : P.IsPrime
hp : P.LiesOver p
hQp : Q.IsPrime
inst✝⁴ : Q.LiesOver p
G : Type u_3
inst✝³ : Group G
inst✝² : Finite G
inst✝¹ : MulSemiringAction G B
inst✝ : IsGaloisGroup G A B
⊢ ∃ σ, σ • P = Q

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✝² : Algebra Aₘ K := f.toAlgebra
this✝¹ : IsScalarTower A Aₘ K
this✝ : IsFractionRing Aₘ K := IsFractionRing.isFractionRing_of_isDomain_of_isLocalization M Aₘ K
g : Bₘ →+* L := IsLocalization.map L (RingHom.id B) ⋯
this : Algebra Bₘ L := g.toAlgebra
⊢ algebraMap B L = (algebraMap Bₘ L).comp (algebraMap B Bₘ)

Tactic:
by

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

Goal state before:
pp: A : Type u_1
B : Type u_2
inst✝⁷ : CommRing A
inst✝⁶ : CommRing B
inst✝⁵ : Algebra A B
p : Ideal A
P Q : Ideal B
hPp : P.IsPrime
hp : P.LiesOver p
hQp : Q.IsPrime
inst✝⁴ : Q.LiesOver p
G : Type u_3
inst✝³ : Group G
inst✝² : Finite G
inst✝¹ : MulSemiringAction G B
inst✝ : IsGaloisGroup G A B
⊢ ∀ (x y : ↑(p.primesOver B)), ∃ g, g • x = y

Tactic:
by

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

Goal state before:
pp: A : Type u_1
B : Type u_2
inst✝⁷ : CommRing A
inst✝⁶ : CommRing B
inst✝⁵ : Algebra A B
p : Ideal A
P Q : Ideal B
hPp : P.IsPrime
hp : P.LiesOver p
hQp : Q.IsPrime
inst✝⁴ : Q.LiesOver p
G : Type u_3
inst✝³ : Group G
inst✝² : Finite G
inst✝¹ : MulSemiringAction G B
inst✝ : IsGaloisGroup G A B
⊢ P.ramificationIdx A = Q.ramificationIdx A

Tactic:
by

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

Goal state before:
pp: A : Type u_1
B : Type u_2
inst✝⁷ : CommRing A
inst✝⁶ : CommRing B
inst✝⁵ : Algebra A B
p : Ideal A
P Q : Ideal B
hPp : P.IsPrime
hp : P.LiesOver p
hQp : Q.IsPrime
inst✝⁴ : Q.LiesOver p
G : Type u_3
inst✝³ : Group G
inst✝² : Finite G
inst✝¹ : MulSemiringAction G B
inst✝ : IsGaloisGroup G A B
⊢ P.inertiaDeg A = Q.inertiaDeg A

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)
⊢ 0 < 4 / 9

Tactic:
by

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

Goal state before:
pp: A : Type u_1
B : Type u_2
inst✝⁶ : CommRing A
inst✝⁵ : CommRing B
inst✝⁴ : Algebra A B
p : Ideal A
P : Ideal B
hPp : P.IsPrime
hp : P.LiesOver p
G : Type u_3
inst✝³ : Group G
inst✝² : Finite G
inst✝¹ : MulSemiringAction G B
inst✝ : IsGaloisGroup G A B
⊢ p.ramificationIdxIn B = P.ramificationIdx A

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)
⊢ 0 < 9 / 4 * ↑N

Tactic:
by

Goal state after:
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.ramificationIdxIn B ≠ 0

Tactic:
by

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

Goal state before:
pp: A : Type u_1
B : Type u_2
inst✝⁶ : CommRing A
inst✝⁵ : CommRing B
inst✝⁴ : Algebra A B
p : Ideal A
P : Ideal B
hPp : P.IsPrime
hp : P.LiesOver p
G : Type u_3
inst✝³ : Group G
inst✝² : Finite G
inst✝¹ : MulSemiringAction G B
inst✝ : IsGaloisGroup G A B
⊢ p.inertiaDegIn B = P.inertiaDeg A

Tactic:
by

Goal state after: