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
B : Type u_6
B₂ : Type u_7
inst✝²³ : CommRing A
inst✝²² : CommRing B
inst✝²¹ : CommRing B₂
inst✝²⁰ : Algebra A B
inst✝¹⁹ : Algebra A B₂
inst✝¹⁸ : Field K
inst✝¹⁷ : Field L
inst✝¹⁶ : Field L₂
inst✝¹⁵ : Algebra A K
inst✝¹⁴ : IsFractionRing A K
inst✝¹³ : Algebra K L
inst✝¹² : Algebra A L
inst✝¹¹ : IsScalarTower A K L
inst✝¹⁰ : Algebra K L₂
inst✝⁹ : Algebra A L₂
inst✝⁸ : IsScalarTower A K L₂
inst✝⁷ : Algebra B L
inst✝⁶ : IsScalarTower A B L
inst✝⁵ : IsIntegralClosure B A L
inst✝⁴ : Algebra B₂ L₂
inst✝³ : IsScalarTower A B₂ L₂
inst✝² : IsIntegralClosure B₂ A L₂
inst✝¹ : Algebra.IsAlgebraic K L
inst✝ : Algebra.IsAlgebraic K L₂
σ : B ≃ₐ[A] B₂
x : B
⊢ (galLiftEquiv K L L₂ σ) ((algebraMap B L) x) = (algebraMap B₂ L₂) (σ x)

Tactic:
by

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

Goal state before:
pp: K : Type u_1
inst✝ : Field K
s : Set { w // w.IsReal }
⊢ (negAt s).symm = negAt s

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✝¹⁵ : CommRing A
inst✝¹⁴ : CommRing B
inst✝¹³ : Algebra A B
inst✝¹² : Field K
inst✝¹¹ : Field L
inst✝¹⁰ : Algebra A K
inst✝⁹ : IsFractionRing A K
inst✝⁸ : Algebra K L
inst✝⁷ : Algebra A L
inst✝⁶ : IsScalarTower A K L
inst✝⁵ : Algebra B L
inst✝⁴ : IsScalarTower A B L
inst✝³ : IsIntegralClosure B A L
inst✝² : FiniteDimensional K L
inst✝¹ : IsGalois K L
inst✝ : IsIntegrallyClosed A
x : B
⊢ ∏ σ, ((galRestrict A K L B) σ) x =
    (algebraMap A B) (IsIntegralClosure.mk' A ((Algebra.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✝ : Field K
x : mixedSpace K
w : { w // w.IsReal }
⊢ ((negAt (signSet x)) x).1 w = ‖x.1 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
x : 𝓞 K
h_nz : x ≠ 0
h_nm : 1 ≤ ↑|(Algebra.norm ℚ) ↑x|
h_bd : ↑(finrank ℚ K) ^ finrank ℚ K / ((4 / π) ^ nrComplexPlaces K * ↑(finrank ℚ K).factorial) ≤ √|↑(discr K)|
⊢ 0 ≤ ↑(finrank ℚ K) ^ finrank ℚ K / ((4 / π) ^ nrComplexPlaces K * ↑(finrank ℚ K).factorial)

Tactic:
by

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

Goal state before:
pp: K : Type u_1
inst✝ : Field K
s : Set { w // w.IsReal }
A : Set (mixedSpace K)
⊢ ⇑(negAt s) ⁻¹' A = ⇑(negAt s) '' 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
x : 𝓞 K
h_nz : x ≠ 0
h_nm : 1 ≤ ↑|(Algebra.norm ℚ) ↑x|
h_bd : ↑(finrank ℚ K) ^ finrank ℚ K / ((4 / π) ^ nrComplexPlaces K * ↑(finrank ℚ K).factorial) ≤ √|↑(discr K)|
⊢ 0 ≤ |↑(discr K)|

Tactic:
by

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

Goal state before:
pp: K : Type u_1
inst✝ : Field K
s : Set { w // w.IsReal }
A : Set (mixedSpace K)
x : mixedSpace K
hx : x ∈ ⇑(negAt s) '' plusPart A
w : { w // w.IsReal }
hw : w ∈ s
⊢ x.1 w < 0

Tactic:
by

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

Goal state before:
pp: K : Type u_1
inst✝ : Field K
s : Set { w // w.IsReal }
A : Set (mixedSpace K)
x : mixedSpace K
hx : x ∈ ⇑(negAt s) '' plusPart A
w : { w // w.IsReal }
hw : w ∉ s
⊢ 0 < x.1 w

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 : 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
⊢ (algebraMap A K) ((intTrace A B) x) = (trace K L) ((algebraMap B L) 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
⊢ (algebraMap A (FractionRing A)) ((intTrace A B) x) =
    (trace (FractionRing A) (FractionRing B)) ((algebraMap B (FractionRing B)) x)

Tactic:
by

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

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
I : (FractionalIdeal (𝓞 K)⁰ K)ˣ
B : ℝ
h : minkowskiBound K I ≤ volume (convexBodySum K B)
⊢ ∃ a ∈ ↑I, a ≠ 0 ∧ ↑|(Algebra.norm ℚ) a| ≤ (B / ↑(finrank ℚ K)) ^ finrank ℚ K

Tactic:
by

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

Goal state before:
pp: K : Type u_1
inst✝ : Field K
A : Set (mixedSpace K)
⊢ Pairwise (Disjoint on fun s ↦ ⇑(negAt s) '' plusPart A)

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✝ : Free A B
⊢ intTrace A B = trace A B

Tactic:
by

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

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
B : ℝ
h : minkowskiBound K 1 ≤ volume (convexBodySum K B)
⊢ ∃ a, a ≠ 0 ∧ ↑|(Algebra.norm ℚ) ↑a| ≤ (B / ↑(finrank ℚ K)) ^ finrank ℚ K

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
x : 𝓞 K
h_nz : x ≠ 0
h_nm : 1 ≤ ↑|(Algebra.norm ℚ) ↑x|
h_bd : 1 ≤ (4 / π) ^ nrComplexPlaces K * ↑(finrank ℚ K).factorial / ↑(finrank ℚ K) ^ finrank ℚ K * √|↑(discr K)|
⊢ 0 < (4 / π) ^ nrComplexPlaces K * ↑(finrank ℚ K).factorial

Tactic:
by

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

Goal state before:
pp: A : Type u_1
K : Type u_2
L : Type u_3
B : Type u_4
inst✝¹⁵ : CommRing A
inst✝¹⁴ : CommRing B
inst✝¹³ : Field K
inst✝¹² : Field L
inst✝¹¹ : Algebra A K
inst✝¹⁰ : Algebra B L
inst✝⁹ : IsFractionRing A K
inst✝⁸ : IsFractionRing B L
inst✝⁷ : Algebra A B
inst✝⁶ : Algebra K L
inst✝⁵ : Algebra A L
inst✝⁴ : IsScalarTower A K L
inst✝³ : IsScalarTower A B L
inst✝² : IsIntegrallyClosed A
inst✝¹ : IsIntegralClosure B A L
inst✝ : Algebra.IsAlgebraic K L
this : MulSemiringAction Gal(L/K) B := IsIntegralClosure.MulSemiringAction A K L B
g : Gal(L/K)
b : B
l : L
⊢ g • b • l = (g • b) • g • l

Tactic:
by

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

Goal state before:
pp: A : Type u_1
K : Type u_2
L : Type u_3
B : Type u_4
inst✝¹⁵ : CommRing A
inst✝¹⁴ : CommRing B
inst✝¹³ : Field K
inst✝¹² : Field L
inst✝¹¹ : Algebra A K
inst✝¹⁰ : Algebra B L
inst✝⁹ : IsFractionRing A K
inst✝⁸ : IsFractionRing B L
inst✝⁷ : Algebra A B
inst✝⁶ : Algebra K L
inst✝⁵ : Algebra A L
inst✝⁴ : IsScalarTower A K L
inst✝³ : IsScalarTower A B L
inst✝² : IsIntegrallyClosed A
inst✝¹ : IsIntegralClosure B A L
inst✝ : FiniteDimensional K L
h✝ : IsGalois K L
h : IntermediateField.fixedField ⊤ = ⊥
this : MulSemiringAction Gal(L/K) B := IsIntegralClosure.MulSemiringAction A K L B
b : B
hb : ∀ (g : Gal(L/K)), g • b = b
⊢ (algebraMap B L) b ∈ IntermediateField.fixedField ⊤

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
⊢ ↑(finrank ℚ K) ^ (2 * finrank ℚ K) / ((4 / π) ^ (2 * nrComplexPlaces K) * ↑(finrank ℚ K).factorial ^ 2) ≤ ↑|discr K|

Tactic:
by

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

Goal state before:
pp: K : Type u_1
inst✝² : Field K
inst✝¹ : NumberField K
inst✝ : IsTotallyComplex K
⊢ ↑(finrank ℚ K) ^ (2 * finrank ℚ K) / ((4 / π) ^ finrank ℚ K * ↑(finrank ℚ K).factorial ^ 2) ≤ ↑|discr K|

Tactic:
by

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

Goal state before:
pp: K : Type u_1
inst✝² : Field K
inst✝¹ : NumberField K
inst✝ : IsTotallyComplex K
h : 0 < ↑(finrank ℚ K)
⊢ 0 ≤ ↑(finrank ℚ K) ^ 2 / (4 / π * ↑(finrank ℚ K).factorial ^ (2 * (↑(finrank ℚ K))⁻¹))

Tactic:
by

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

Goal state before:
pp: K : Type u_1
inst✝² : Field K
inst✝¹ : NumberField K
inst✝ : IsTotallyComplex K
h : 0 < ↑(finrank ℚ K)
⊢ 0 ≤ ↑|discr K| ^ (↑(finrank ℚ K))⁻¹

Tactic:
by

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

Goal state before:
pp: K : Type u_1
inst✝² : Field K
inst✝¹ : NumberField K
inst✝ : IsTotallyComplex K
h : 0 < ↑(finrank ℚ K)
⊢ 0 ≤ ↑(finrank ℚ K) ^ 2

Tactic:
by

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

Goal state before:
pp: K : Type u_1
inst✝² : Field K
inst✝¹ : NumberField K
inst✝ : IsTotallyComplex K
h : 0 < ↑(finrank ℚ K)
⊢ 0 ≤ 4 / π * ↑(finrank ℚ K).factorial ^ (2 * (↑(finrank ℚ K))⁻¹)

Tactic:
by

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

Goal state before:
pp: K : Type u_1
inst✝² : Field K
inst✝¹ : NumberField K
inst✝ : IsTotallyComplex K
h : 0 < ↑(finrank ℚ K)
⊢ 0 ≤ ↑|discr K|

Tactic:
by

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

Goal state before:
pp: K : Type u_1
inst✝² : Field K
inst✝¹ : NumberField K
inst✝ : IsTotallyComplex K
h : 0 < ↑(finrank ℚ K)
⊢ 0 ≤ 4 / π

Tactic:
by

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

Goal state before:
pp: K : Type u_1
inst✝² : Field K
inst✝¹ : NumberField K
inst✝ : IsTotallyComplex K
h : 0 < ↑(finrank ℚ K)
⊢ 0 ≤ ↑(finrank ℚ K).factorial ^ (2 * (↑(finrank ℚ K))⁻¹)

Tactic:
by

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

Goal state before:
pp: A : Type u_1
B : Type u_6
inst✝²⁴ : CommRing A
inst✝²³ : CommRing B
inst✝²² : Algebra A B
Aₘ : Type u_9
Bₘ : Type u_10
inst✝²¹ : CommRing Aₘ
inst✝²⁰ : CommRing Bₘ
inst✝¹⁹ : Algebra Aₘ Bₘ
inst✝¹⁸ : Algebra A Aₘ
inst✝¹⁷ : Algebra B Bₘ
inst✝¹⁶ : Algebra A Bₘ
inst✝¹⁵ : IsScalarTower A Aₘ Bₘ
inst✝¹⁴ : IsScalarTower A B Bₘ
M : Submonoid A
inst✝¹³ : IsLocalization M Aₘ
inst✝¹² : IsLocalization (algebraMapSubmonoid B M) Bₘ
inst✝¹¹ : IsDomain A
inst✝¹⁰ : IsIntegrallyClosed A
inst✝⁹ : IsDomain B
inst✝⁸ : IsIntegrallyClosed B
inst✝⁷ : Module.Finite A B
inst✝⁶ : IsTorsionFree A B
inst✝⁵ : IsDomain Aₘ
inst✝⁴ : IsIntegrallyClosed Aₘ
inst✝³ : IsDomain Bₘ
inst✝² : IsIntegrallyClosed Bₘ
inst✝¹ : IsTorsionFree Aₘ Bₘ
inst✝ : Module.Finite Aₘ Bₘ
x : B
hM : M ≤ A⁰
K : Type u_1 := FractionRing A
L : Type u_6 := FractionRing B
this✝² : IsIntegralClosure B A L
this✝¹ : Algebra.IsAlgebraic (FractionRing A) (FractionRing B)
this✝ : IsLocalization (algebraMapSubmonoid B A⁰) L
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: K : Type u_1
inst✝² : Field K
inst✝¹ : NumberField K
inst✝ : IsTotallyComplex K
h : 0 < ↑(finrank ℚ K)
⊢ 0 ≤ ↑(finrank ℚ K).factorial

Tactic:
by

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

Goal state before:
pp: A : Type u_1
K : Type u_2
L : Type u_3
B : Type u_4
inst✝¹⁵ : CommRing A
inst✝¹⁴ : CommRing B
inst✝¹³ : Field K
inst✝¹² : Field L
inst✝¹¹ : Algebra A K
inst✝¹⁰ : Algebra B L
inst✝⁹ : IsFractionRing A K
inst✝⁸ : IsFractionRing B L
inst✝⁷ : Algebra A B
inst✝⁶ : Algebra K L
inst✝⁵ : Algebra A L
inst✝⁴ : IsScalarTower A K L
inst✝³ : IsScalarTower A B L
inst✝² : IsIntegrallyClosed A
inst✝¹ : IsIntegralClosure B A L
inst✝ : FiniteDimensional K L
h : IsGalois K L
⊢ IsInvariant A B Gal(L/K)

Tactic:
by

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

Goal state before:
pp: K : Type u_1
inst✝² : Field K
inst✝¹ : NumberField K
inst✝ : IsTotallyComplex K
⊢ ↑(finrank ℚ K) ^ 2 / (4 / π * ↑(finrank ℚ K).factorial ^ (2 * (↑(finrank ℚ K))⁻¹)) ≤ ↑|discr K| ^ (↑(finrank ℚ K))⁻¹

Tactic:
by

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

Goal state before:
pp: A : Type u_1
B : Type u_6
inst✝²⁴ : CommRing A
inst✝²³ : CommRing B
inst✝²² : Algebra A B
Aₘ : Type u_9
Bₘ : Type u_10
inst✝²¹ : CommRing Aₘ
inst✝²⁰ : CommRing Bₘ
inst✝¹⁹ : Algebra Aₘ Bₘ
inst✝¹⁸ : Algebra A Aₘ
inst✝¹⁷ : Algebra B Bₘ
inst✝¹⁶ : Algebra A Bₘ
inst✝¹⁵ : IsScalarTower A Aₘ Bₘ
inst✝¹⁴ : IsScalarTower A B Bₘ
M : Submonoid A
inst✝¹³ : IsLocalization M Aₘ
inst✝¹² : IsLocalization (algebraMapSubmonoid B M) Bₘ
inst✝¹¹ : IsDomain A
inst✝¹⁰ : IsIntegrallyClosed A
inst✝⁹ : IsDomain B
inst✝⁸ : IsIntegrallyClosed B
inst✝⁷ : Module.Finite A B
inst✝⁶ : IsTorsionFree A B
inst✝⁵ : IsDomain Aₘ
inst✝⁴ : IsIntegrallyClosed Aₘ
inst✝³ : IsDomain Bₘ
inst✝² : IsIntegrallyClosed Bₘ
inst✝¹ : IsTorsionFree Aₘ Bₘ
inst✝ : Module.Finite Aₘ Bₘ
x : B
hM : M ≤ A⁰
K : Type u_1 := FractionRing A
L : Type u_6 := FractionRing B
this✝⁵ : IsIntegralClosure B A L
this✝⁴ : Algebra.IsAlgebraic (FractionRing A) (FractionRing B)
this✝³ : IsLocalization (algebraMapSubmonoid B A⁰) L
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.NumberField.CanonicalEmbedding.Basic

Goal state before:
pp: K : Type u_1
inst✝ : Field K
s : Set { w // w.IsReal }
A : Set (mixedSpace K)
x : mixedSpace K
hA : ∀ (x : mixedSpace K), x ∈ A ↔ (fun w ↦ ‖x.1 w‖, x.2) ∈ A
hx₁ : x ∈ A
hx₂ : ∀ (w : { w // w.IsReal }), x.1 w ≠ 0
⊢ x ∈ ⇑(negAt s) '' plusPart A ↔ (∀ w ∈ s, x.1 w < 0) ∧ ∀ w ∉ s, x.1 w > 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
a : ℕ → ℝ := fun n ↦ ↑n ^ (n * 2) / ((4 / π) ^ n * ↑n.factorial ^ 2)
this : ∀ (n : ℕ), 2 ≤ n → 4 / 9 * (3 * π / 4) ^ n ≤ a n
⊢ 4 / 9 * (3 * π / 4) ^ finrank ℚ K ≤
    ↑(finrank ℚ K) ^ (finrank ℚ K * 2) / ((4 / π) ^ (2 * nrComplexPlaces K) * ↑(finrank ℚ K).factorial ^ 2)

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 : ℕ
⊢ 4 / 9 * (3 * π / 4) ^ 2 = a 2

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
⊢ (aeval b) p = 0

Tactic:
by

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

Goal state before:
pp: A : Type u_1
B : Type u_6
inst✝²⁴ : CommRing A
inst✝²³ : CommRing B
inst✝²² : Algebra A B
Aₘ : Type u_9
Bₘ : Type u_10
inst✝²¹ : CommRing Aₘ
inst✝²⁰ : CommRing Bₘ
inst✝¹⁹ : Algebra Aₘ Bₘ
inst✝¹⁸ : Algebra A Aₘ
inst✝¹⁷ : Algebra B Bₘ
inst✝¹⁶ : Algebra A Bₘ
inst✝¹⁵ : IsScalarTower A Aₘ Bₘ
inst✝¹⁴ : IsScalarTower A B Bₘ
M : Submonoid A
inst✝¹³ : IsLocalization M Aₘ
inst✝¹² : IsLocalization (algebraMapSubmonoid B M) Bₘ
inst✝¹¹ : IsDomain A
inst✝¹⁰ : IsIntegrallyClosed A
inst✝⁹ : IsDomain B
inst✝⁸ : IsIntegrallyClosed B
inst✝⁷ : Module.Finite A B
inst✝⁶ : IsTorsionFree A B
inst✝⁵ : IsDomain Aₘ
inst✝⁴ : IsIntegrallyClosed Aₘ
inst✝³ : IsDomain Bₘ
inst✝² : IsIntegrallyClosed Bₘ
inst✝¹ : IsTorsionFree Aₘ Bₘ
inst✝ : Module.Finite Aₘ Bₘ
x : B
hM : M ≤ A⁰
K : Type u_1 := FractionRing A
L : Type u_6 := FractionRing B
this✝⁸ : IsIntegralClosure B A L
this✝⁷ : Algebra.IsAlgebraic (FractionRing A) (FractionRing B)
this✝⁶ : IsLocalization (algebraMapSubmonoid B A⁰) L
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.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
⊢ (algebraMap K L).comp (algebraMap (A ⧸ P) K) = (algebraMap (B ⧸ Q) L).comp (algebraMap (A ⧸ P) (B ⧸ 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
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
this : 3 ≤ (1 + 1 / ↑m) ^ (2 * m)
⊢ 0 < 4 / π

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))
⊢ (algebraMap (A ⧸ P) (B ⧸ Q)).comp (algebraMap A (A ⧸ P)) = (algebraMap B (B ⧸ Q)).comp (algebraMap A B)

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
this : 3 ≤ (1 + 1 / ↑m) ^ (2 * m)
⊢ 0 ≤ 3

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
this : 3 ≤ (1 + 1 / ↑m) ^ (2 * m)
⊢ 0 ≤ a m

Tactic:
by

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

Goal state before:
pp: K : Type u_1
inst✝ : Field K
A : Set (mixedSpace K)
hA : ∀ (x : mixedSpace K), x ∈ A ↔ (fun w ↦ ‖x.1 w‖, x.2) ∈ A
⊢ (⋃ s, ⇑(negAt s) '' plusPart A) ∪ A ∩ ⋃ w, {x | x.1 w = 0} = A

Tactic:
by

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

Goal state before:
pp: A : Type u_1
B : Type u_6
inst✝²⁴ : CommRing A
inst✝²³ : CommRing B
inst✝²² : Algebra A B
Aₘ : Type u_9
Bₘ : Type u_10
inst✝²¹ : CommRing Aₘ
inst✝²⁰ : CommRing Bₘ
inst✝¹⁹ : Algebra Aₘ Bₘ
inst✝¹⁸ : Algebra A Aₘ
inst✝¹⁷ : Algebra B Bₘ
inst✝¹⁶ : Algebra A Bₘ
inst✝¹⁵ : IsScalarTower A Aₘ Bₘ
inst✝¹⁴ : IsScalarTower A B Bₘ
M : Submonoid A
inst✝¹³ : IsLocalization M Aₘ
inst✝¹² : IsLocalization (algebraMapSubmonoid B M) Bₘ
inst✝¹¹ : IsDomain A
inst✝¹⁰ : IsIntegrallyClosed A
inst✝⁹ : IsDomain B
inst✝⁸ : IsIntegrallyClosed B
inst✝⁷ : Module.Finite A B
inst✝⁶ : IsTorsionFree A B
inst✝⁵ : IsDomain Aₘ
inst✝⁴ : IsIntegrallyClosed Aₘ
inst✝³ : IsDomain Bₘ
inst✝² : IsIntegrallyClosed Bₘ
inst✝¹ : IsTorsionFree Aₘ Bₘ
inst✝ : Module.Finite Aₘ Bₘ
x : B
⊢ (algebraMap A Aₘ) ((intTrace A B) x) = (intTrace 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
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
this : 3 ≤ (1 + 1 / ↑m) ^ (2 * m)
⊢ 4 / 9 * (3 * π / 4) ^ (m + 1) ≤ a (m + 1)

Tactic:
by

Goal state after: