mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.NumberTheory.NumberField.CanonicalEmbedding.Basic

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
I : (FractionalIdeal (𝓞 K)⁰ K)ˣ
e : ChooseBasisIndex ℤ (𝓞 K) ≃ ChooseBasisIndex ℤ ↥↑↑I
this :
  (latticeBasis K).det (⇑(mixedEmbedding K) ∘ ⇑(basisOfFractionalIdeal K I) ∘ ⇑e) =
    (algebraMap ℚ ℝ) ((integralBasis K).det (⇑(basisOfFractionalIdeal K I) ∘ ⇑e))
⊢ |(algebraMap ℚ ℝ) ((integralBasis K).det (⇑(basisOfFractionalIdeal K I) ∘ ⇑e))| = ↑(FractionalIdeal.absNorm ↑I)

Tactic:
eq_ratCast,

Goal state after:
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
I : (FractionalIdeal (𝓞 K)⁰ K)ˣ
e : ChooseBasisIndex ℤ (𝓞 K) ≃ ChooseBasisIndex ℤ ↥↑↑I
this :
  (latticeBasis K).det (⇑(mixedEmbedding K) ∘ ⇑(basisOfFractionalIdeal K I) ∘ ⇑e) =
    (algebraMap ℚ ℝ) ((integralBasis K).det (⇑(basisOfFractionalIdeal K I) ∘ ⇑e))
⊢ |↑((integralBasis K).det (⇑(basisOfFractionalIdeal K I) ∘ ⇑e))| = ↑(FractionalIdeal.absNorm ↑I)
Module: Mathlib.NumberTheory.NumberField.CanonicalEmbedding.Basic

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
I : (FractionalIdeal (𝓞 K)⁰ K)ˣ
x : mixedSpace K
⊢ Set.range ⇑(fractionalIdealLatticeBasis K I) =
    ⇑(mixedEmbedding K).toIntAlgHom.toLinearMap '' Set.range ⇑(basisOfFractionalIdeal K I)

Tactic:
← Set.range_comp

Goal state after:
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
I : (FractionalIdeal (𝓞 K)⁰ K)ˣ
x : mixedSpace K
⊢ Set.range ⇑(fractionalIdealLatticeBasis K I) =
    Set.range (⇑(mixedEmbedding K).toIntAlgHom.toLinearMap ∘ ⇑(basisOfFractionalIdeal K I))
Module: Mathlib.NumberTheory.NumberField.CanonicalEmbedding.Basic

Goal state before:
pp: K : Type u_1
inst✝ : Field K
s : Set { w // w.IsReal }
x : mixedSpace K
w : { w // w.IsReal }
hw : w ∈ s
⊢ ((negAt s) x).1 w = -x.1 w

Tactic:
by
  simp_rw [negAt, prodCongr_apply, piCongrRight_apply, if_pos hw,
    ContinuousLinearEquiv.neg_apply]

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
x : B
this✝¹ : IsIntegralClosure B A (FractionRing B)
this✝ : Algebra.IsAlgebraic (FractionRing A) (FractionRing B)
this : IsLocalization (algebraMapSubmonoid B A⁰) (FractionRing B)
⊢ (algebraMap A (FractionRing A)) ((intTrace A B) x) = (algebraMap A (FractionRing A)) ((trace A B) x)

Tactic:
Algebra.algebraMap_intTrace_fractionRing,

Goal state after:
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
x : B
this✝¹ : IsIntegralClosure B A (FractionRing B)
this✝ : Algebra.IsAlgebraic (FractionRing A) (FractionRing B)
this : IsLocalization (algebraMapSubmonoid B A⁰) (FractionRing B)
⊢ (trace (FractionRing A) (FractionRing B)) ((algebraMap B (FractionRing B)) x) =
    (algebraMap A (FractionRing A)) ((trace A B) x)
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 A K).comp (RingHom.id A)

Tactic:
RingHomCompTriple.comp_eq

Goal state after:
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 A K
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 B L).comp (RingHom.id B)

Tactic:
RingHomCompTriple.comp_eq

Goal state after:
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 B L
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 A K).comp (RingHom.id A)

Tactic:
RingHomCompTriple.comp_eq

Goal state after:
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 A K
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 B L).comp (RingHom.id B)

Tactic:
RingHomCompTriple.comp_eq

Goal state after:
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 B L
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
  have : FiniteDimensional (FractionRing A) (FractionRing B) := .of_isLocalization A B A⁰
  rw [← (galRestrict A (FractionRing A) (FractionRing B) B).toEquiv.prod_comp]
  simp only [MulEquiv.toEquiv_eq_coe, EquivLike.coe_coe]
  convert! (prod_galRestrict_eq_norm A (FractionRing A) (FractionRing B) B x).symm

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

Goal state before:
pp: case refine_2.refine_2
A : Type u_2
inst✝¹ : Field A
inst✝ : CharZero A
N : ℕ
D : ℕ := rankOfDiscrBdd N
B : ℝ≥0 := boundOfDiscBdd N
C : ℕ := ⌈max B 1 ^ D * ↑(D.choose (D / 2))⌉₊
x✝¹ : { F // FiniteDimensional ℚ ↥F }
K : IntermediateField ℚ A
hK₀ : FiniteDimensional ℚ ↥K
x✝ : ⟨K, hK₀⟩ ∈ {K | {w | w.IsReal}.Nonempty ∧ |discr ↥↑K| ≤ ↑N}
hK₂ : |discr ↥↑⟨K, hK₀⟩| ≤ ↑N
this✝¹ : CharZero ↥K
this✝ : NumberField ↥K
w₀ : InfinitePlace ↥↑⟨K, hK₀⟩
hw₀ : w₀ ∈ {w | w.IsReal}
this : minkowskiBound (↥K) 1 < ↑(convexBodyLTFactor ↥K) * ↑B
x : 𝓞 ↥K
hx₁ : ℚ⟮↑x⟯ = ⊤
hx₂ : ∀ (w : InfinitePlace ↥K), w ↑x < ↑(max B 1)
hx : IsIntegral ℤ ((algebraMap (𝓞 ↥K) ↥K) x)
i : ℕ
⊢ max (↑B) 1 ^ finrank ℚ ↥K * ↑((finrank ℚ ↥K).choose (finrank ℚ ↥K / 2)) ≤ ↑(max B 1 ^ D * ↑(D.choose (D / 2)))

Tactic:
val_eq_coe,

Goal state after:
case refine_2.refine_2
A : Type u_2
inst✝¹ : Field A
inst✝ : CharZero A
N : ℕ
D : ℕ := rankOfDiscrBdd N
B : ℝ≥0 := boundOfDiscBdd N
C : ℕ := ⌈max B 1 ^ D * ↑(D.choose (D / 2))⌉₊
x✝¹ : { F // FiniteDimensional ℚ ↥F }
K : IntermediateField ℚ A
hK₀ : FiniteDimensional ℚ ↥K
x✝ : ⟨K, hK₀⟩ ∈ {K | {w | w.IsReal}.Nonempty ∧ |discr ↥↑K| ≤ ↑N}
hK₂ : |discr ↥↑⟨K, hK₀⟩| ≤ ↑N
this✝¹ : CharZero ↥K
this✝ : NumberField ↥K
w₀ : InfinitePlace ↥↑⟨K, hK₀⟩
hw₀ : w₀ ∈ {w | w.IsReal}
this : minkowskiBound (↥K) 1 < ↑(convexBodyLTFactor ↥K) * ↑B
x : 𝓞 ↥K
hx₁ : ℚ⟮↑x⟯ = ⊤
hx₂ : ∀ (w : InfinitePlace ↥K), w ↑x < ↑(max B 1)
hx : IsIntegral ℤ ((algebraMap (𝓞 ↥K) ↥K) x)
i : ℕ
⊢ max (↑B) 1 ^ finrank ℚ ↥K * ↑((finrank ℚ ↥K).choose (finrank ℚ ↥K / 2)) ≤ ↑(max B 1 ^ D * ↑(D.choose (D / 2)))
Module: Mathlib.RingTheory.Ideal.Quotient.Index

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝⁴ : CommRing R
inst✝³ : AddCommGroup M
inst✝² : Module R M
I : Ideal R
N : Submodule R M
inst✝¹ : Finite (R ⧸ I)
inst✝ : Finite (M ⧸ N)
hN : N.FG
⊢ Finite (M ⧸ I • N)

Tactic:
suffices (I • N).toAddSubgroup.FiniteIndex by
    exact (I • N).toAddSubgroup.finite_quotient_of_finiteIndex

Goal state after:
R : Type u_1
M : Type u_2
inst✝⁴ : CommRing R
inst✝³ : AddCommGroup M
inst✝² : Module R M
I : Ideal R
N : Submodule R M
inst✝¹ : Finite (R ⧸ I)
inst✝ : Finite (M ⧸ N)
hN : N.FG
⊢ (I • N).toAddSubgroup.FiniteIndex
Module: Mathlib.RingTheory.FractionalIdeal.Extended

Goal state before:
pp: A : Type u_1
inst✝⁷ : CommRing A
B : Type u_2
inst✝⁶ : CommRing B
f : A →+* B
K : Type u_3
M : Submonoid A
inst✝⁵ : CommRing K
inst✝⁴ : Algebra A K
inst✝³ : IsLocalization M K
L : Type u_4
N : Submonoid B
inst✝² : CommRing L
inst✝¹ : Algebra B L
inst✝ : IsLocalization N L
hf : M ≤ Submonoid.comap f N
I✝ J I : FractionalIdeal M K
a : A
ha : a ∈ M
frac : ∀ b ∈ ↑I, IsInteger A (a • b)
b : L
hb : b ∈ span B (⇑(IsLocalization.map L f hf) '' ↑I)
k : K
kI : k ∈ ↑I
c : A
hc : (algebraMap A K) c = a • k
⊢ (algebraMap B L) (f c) = f a • (IsLocalization.map L f hf) k

Tactic:
by simp [← IsLocalization.map_smul, ← hc]

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

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
h : (4 / π) ^ nrComplexPlaces K * (↑(finrank ℚ K)! / ↑(finrank ℚ K) ^ finrank ℚ K * √|↑(discr K)|) < 2
this : 0 < finrank ℚ K
I : ↥(Ideal (𝓞 K))⁰
hI : ↑(absNorm ↑I) ≤ (4 / π) ^ nrComplexPlaces K * (↑(finrank ℚ K)! / ↑(finrank ℚ K) ^ finrank ℚ K * √|↑(discr K)|)
⊢ Submodule.IsPrincipal ⊤

Tactic:
exact top_isPrincipal

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

Goal state before:
pp: case refine_2.refine_2
A : Type u_2
inst✝¹ : Field A
inst✝ : CharZero A
N : ℕ
D : ℕ := rankOfDiscrBdd N
B : ℝ≥0 := boundOfDiscBdd N
C : ℕ := ⌈max (sqrt (1 + B ^ 2)) 1 ^ D * ↑(D.choose (D / 2))⌉₊
x✝¹ : { F // FiniteDimensional ℚ ↥F }
K : IntermediateField ℚ A
hK₀ : FiniteDimensional ℚ ↥K
x✝ : ⟨K, hK₀⟩ ∈ {K | {w | w.IsComplex}.Nonempty ∧ |discr ↥↑K| ≤ ↑N}
hK₂ : |discr ↥↑⟨K, hK₀⟩| ≤ ↑N
this✝¹ : CharZero ↥K
this✝ : NumberField ↥K
w₀ : InfinitePlace ↥↑⟨K, hK₀⟩
hw₀ : w₀ ∈ {w | w.IsComplex}
this : minkowskiBound (↥K) 1 < ↑(convexBodyLT'Factor ↥K) * ↑(boundOfDiscBdd N)
x : 𝓞 ↥K
hx₁ : ℚ⟮↑x⟯ = ⊤
hx₂ : ∀ (w : InfinitePlace ↥K), w ↑x < √(1 + ↑(boundOfDiscBdd N) ^ 2)
hx : IsIntegral ℤ ((algebraMap (𝓞 ↥K) ↥K) x)
i : ℕ
⊢ max (√(1 + ↑(boundOfDiscBdd N) ^ 2)) 1 ^ finrank ℚ ↥K * ↑((finrank ℚ ↥K).choose (finrank ℚ ↥K / 2)) ≤
    ↑(max (sqrt (1 + B ^ 2)) 1 ^ D * ↑(D.choose (D / 2)))

Tactic:
val_eq_coe,

Goal state after:
case refine_2.refine_2
A : Type u_2
inst✝¹ : Field A
inst✝ : CharZero A
N : ℕ
D : ℕ := rankOfDiscrBdd N
B : ℝ≥0 := boundOfDiscBdd N
C : ℕ := ⌈max (sqrt (1 + B ^ 2)) 1 ^ D * ↑(D.choose (D / 2))⌉₊
x✝¹ : { F // FiniteDimensional ℚ ↥F }
K : IntermediateField ℚ A
hK₀ : FiniteDimensional ℚ ↥K
x✝ : ⟨K, hK₀⟩ ∈ {K | {w | w.IsComplex}.Nonempty ∧ |discr ↥↑K| ≤ ↑N}
hK₂ : |discr ↥↑⟨K, hK₀⟩| ≤ ↑N
this✝¹ : CharZero ↥K
this✝ : NumberField ↥K
w₀ : InfinitePlace ↥↑⟨K, hK₀⟩
hw₀ : w₀ ∈ {w | w.IsComplex}
this : minkowskiBound (↥K) 1 < ↑(convexBodyLT'Factor ↥K) * ↑(boundOfDiscBdd N)
x : 𝓞 ↥K
hx₁ : ℚ⟮↑x⟯ = ⊤
hx₂ : ∀ (w : InfinitePlace ↥K), w ↑x < √(1 + ↑(boundOfDiscBdd N) ^ 2)
hx : IsIntegral ℤ ((algebraMap (𝓞 ↥K) ↥K) x)
i : ℕ
⊢ max (√(1 + ↑(boundOfDiscBdd N) ^ 2)) 1 ^ finrank ℚ ↥K * ↑((finrank ℚ ↥K).choose (finrank ℚ ↥K / 2)) ≤
    ↑(max (sqrt (1 + B ^ 2)) 1 ^ D * ↑(D.choose (D / 2)))
Module: Mathlib.RingTheory.DedekindDomain.Instances

Goal state before:
pp: R : Type u_1
S : Type u_2
T : Type u_3
inst✝⁸ : CommRing R
inst✝⁷ : CommRing S
inst✝⁶ : CommRing T
inst✝⁵ : IsDomain R
inst✝⁴ : IsDomain S
inst✝³ : IsDomain T
inst✝² : Algebra R S
P : Ideal R
inst✝¹ : P.IsPrime
inst✝ : FaithfulSMul R S
⊢ (algebraMap Rₚ L).comp (algebraMap R Rₚ) = (algebraMap S L).comp (algebraMap R S)

Tactic:
IsScalarTower.algebraMap_eq Rₚ K L,

Goal state after:
R : Type u_1
S : Type u_2
T : Type u_3
inst✝⁸ : CommRing R
inst✝⁷ : CommRing S
inst✝⁶ : CommRing T
inst✝⁵ : IsDomain R
inst✝⁴ : IsDomain S
inst✝³ : IsDomain T
inst✝² : Algebra R S
P : Ideal R
inst✝¹ : P.IsPrime
inst✝ : FaithfulSMul R S
⊢ ((algebraMap K L).comp (algebraMap Rₚ K)).comp (algebraMap R Rₚ) = (algebraMap S L).comp (algebraMap R S)
Module: Mathlib.RingTheory.Ideal.Norm.RelNorm

Goal state before:
pp: R : Type u_1
inst✝²⁴ : CommRing R
inst✝²³ : IsDomain R
S : Type u_3
inst✝²² : CommRing S
inst✝²¹ : IsDomain S
inst✝²⁰ : IsIntegrallyClosed R
inst✝¹⁹ : IsIntegrallyClosed S
inst✝¹⁸ : Algebra R S
inst✝¹⁷ : Module.Finite R S
inst✝¹⁶ : IsTorsionFree R S
I : Ideal S
M : Submonoid R
hM : M ≤ R⁰
Rₘ : Type u_4
Sₘ : Type u_5
inst✝¹⁵ : CommRing Rₘ
inst✝¹⁴ : Algebra R Rₘ
inst✝¹³ : CommRing Sₘ
inst✝¹² : Algebra S Sₘ
inst✝¹¹ : Algebra Rₘ Sₘ
inst✝¹⁰ : Algebra R Sₘ
inst✝⁹ : IsScalarTower R Rₘ Sₘ
inst✝⁸ : IsScalarTower R S Sₘ
inst✝⁷ : IsLocalization M Rₘ
inst✝⁶ : IsLocalization (Algebra.algebraMapSubmonoid S M) Sₘ
inst✝⁵ : IsIntegrallyClosed Rₘ
inst✝⁴ : IsDomain Rₘ
inst✝³ : IsDomain Sₘ
inst✝² : IsTorsionFree Rₘ Sₘ
inst✝¹ : Module.Finite Rₘ Sₘ
inst✝ : IsIntegrallyClosed Sₘ
K : Type u_1 := FractionRing R
f : Rₘ →+* K := IsLocalization.map K (RingHom.id R) hM
L : Type u_3 := FractionRing S
g : Sₘ →+* L := IsLocalization.map L (RingHom.id S) ⋯
algInst✝² : Algebra Rₘ K := f.toAlgebra
algInst✝¹ : Algebra Sₘ L := g.toAlgebra
algInst✝ : Algebra Rₘ L := ((algebraMap K L).comp f).toAlgebra
scalarTowerInst✝ : IsScalarTower Rₘ K L := IsScalarTower.of_algebraMap_eq' (Eq.refl (algebraMap Rₘ L))
⊢ algebraMap R K = (algebraMap R K).comp (RingHom.id R)

Tactic:
RingHomCompTriple.comp_eq

Goal state after:
R : Type u_1
inst✝²⁴ : CommRing R
inst✝²³ : IsDomain R
S : Type u_3
inst✝²² : CommRing S
inst✝²¹ : IsDomain S
inst✝²⁰ : IsIntegrallyClosed R
inst✝¹⁹ : IsIntegrallyClosed S
inst✝¹⁸ : Algebra R S
inst✝¹⁷ : Module.Finite R S
inst✝¹⁶ : IsTorsionFree R S
I : Ideal S
M : Submonoid R
hM : M ≤ R⁰
Rₘ : Type u_4
Sₘ : Type u_5
inst✝¹⁵ : CommRing Rₘ
inst✝¹⁴ : Algebra R Rₘ
inst✝¹³ : CommRing Sₘ
inst✝¹² : Algebra S Sₘ
inst✝¹¹ : Algebra Rₘ Sₘ
inst✝¹⁰ : Algebra R Sₘ
inst✝⁹ : IsScalarTower R Rₘ Sₘ
inst✝⁸ : IsScalarTower R S Sₘ
inst✝⁷ : IsLocalization M Rₘ
inst✝⁶ : IsLocalization (Algebra.algebraMapSubmonoid S M) Sₘ
inst✝⁵ : IsIntegrallyClosed Rₘ
inst✝⁴ : IsDomain Rₘ
inst✝³ : IsDomain Sₘ
inst✝² : IsTorsionFree Rₘ Sₘ
inst✝¹ : Module.Finite Rₘ Sₘ
inst✝ : IsIntegrallyClosed Sₘ
K : Type u_1 := FractionRing R
f : Rₘ →+* K := IsLocalization.map K (RingHom.id R) hM
L : Type u_3 := FractionRing S
g : Sₘ →+* L := IsLocalization.map L (RingHom.id S) ⋯
algInst✝² : Algebra Rₘ K := f.toAlgebra
algInst✝¹ : Algebra Sₘ L := g.toAlgebra
algInst✝ : Algebra Rₘ L := ((algebraMap K L).comp f).toAlgebra
scalarTowerInst✝ : IsScalarTower Rₘ K L := IsScalarTower.of_algebraMap_eq' (Eq.refl (algebraMap Rₘ L))
⊢ algebraMap R K = algebraMap R K