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

Goal state before:
pp: case neg
K : Type u_1
inst✝¹ : Field K
s : Set { w // w.IsReal }
inst✝ : NumberField K
w : { w // w.IsReal }
hw : w ∉ s
⊢ MeasurePreserving (↑↑(if w ∈ s then neg ℝ else ContinuousLinearEquiv.refl ℝ ℝ)).toAddHom.1 volume volume

Tactic:
if_neg hw

Goal state after:
case neg
K : Type u_1
inst✝¹ : Field K
s : Set { w // w.IsReal }
inst✝ : NumberField K
w : { w // w.IsReal }
hw : w ∉ s
⊢ MeasurePreserving (↑↑(ContinuousLinearEquiv.refl ℝ ℝ)).toAddHom.1 volume volume
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
  ext x
  have : IsIntegralClosure B A (FractionRing B) :=
    IsIntegralClosure.of_isIntegrallyClosed _ _ _
  -- TODO: How is this even supposed to fire? `R` and `S` cannot be inferred.
  have : Algebra.IsAlgebraic (FractionRing A) (FractionRing B) :=
    isAlgebraic_of_isFractionRing A B ..
  have : IsLocalization (algebraMapSubmonoid B A⁰) (FractionRing B) :=
    IsIntegralClosure.isLocalization _ (FractionRing A) _ _
  apply IsFractionRing.injective A (FractionRing A)
  rw [Algebra.algebraMap_intTrace_fractionRing, Algebra.trace_localization A A⁰]

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
⊢ (algebraMap K L).comp (f.comp (algebraMap A Aₘ)) = g.comp ((algebraMap Aₘ Bₘ).comp (algebraMap A Aₘ))

Tactic:
← IsScalarTower.algebraMap_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✝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
⊢ (algebraMap K L).comp (f.comp (algebraMap A Aₘ)) = g.comp (algebraMap A Bₘ)
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
⊢ (algebraMap K L).comp (algebraMap A K) = (algebraMap B L).comp (algebraMap A B)

Tactic:
← IsScalarTower.algebraMap_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✝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
⊢ algebraMap A L = (algebraMap B L).comp (algebraMap A B)
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
x✝ : InfinitePlace K
⊢ normAtAllPlaces (mixedSpaceOfRealSpace x) x✝ = x x✝

Tactic:
normAtAllPlaces_apply,

Goal state after:
K : Type u_1
inst✝ : Field K
x : realSpace K
hx : ∀ (w : InfinitePlace K), 0 ≤ x w
x✝ : InfinitePlace K
⊢ (normAtPlace x✝) (mixedSpaceOfRealSpace x) = x x✝
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:
normAtAllPlaces_apply,

Goal state after:
K : Type u_1
inst✝ : Field K
x : K
w : InfinitePlace K
⊢ (normAtPlace w) ((mixedEmbedding K) x) = w x
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
x : realSpace K
hx : x ∈ s
w : InfinitePlace K
⊢ normAtAllPlaces (mixedSpaceOfRealSpace x) w = x w

Tactic:
normAtAllPlaces_apply,

Goal state after:
K : Type u_1
inst✝ : Field K
s : Set (realSpace K)
hs : ∀ x ∈ s, ∀ (w : InfinitePlace K), 0 ≤ x w
x : realSpace K
hx : x ∈ s
w : InfinitePlace K
⊢ (normAtPlace w) (mixedSpaceOfRealSpace x) = x w
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
⊢ (algebraMap K L).comp (f.comp (algebraMap A Aₘ)) = g.comp ((algebraMap Aₘ Bₘ).comp (algebraMap A Aₘ))

Tactic:
← IsScalarTower.algebraMap_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✝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
⊢ (algebraMap K L).comp (f.comp (algebraMap A Aₘ)) = g.comp (algebraMap A Bₘ)
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
⊢ (algebraMap K L).comp (algebraMap A K) = (algebraMap B L).comp (algebraMap A B)

Tactic:
← IsScalarTower.algebraMap_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✝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
⊢ algebraMap A L = (algebraMap B L).comp (algebraMap A B)
Module: Mathlib.RingTheory.Ideal.Quotient.HasFiniteQuotients

Goal state before:
pp: R : Type u_1
inst✝¹ : CommRing R
inst✝ : HasFiniteQuotients R
⊢ IsNoetherianRing R

Tactic:
by
  refine (isNoetherianRing_iff_ideal_fg R).mpr fun I ↦ ?_
  by_cases hI : I = 0
  · exact hI ▸ Submodule.fg_bot
  obtain ⟨x, hx₁, hx₂⟩ := Submodule.exists_mem_ne_zero_of_ne_bot hI
  refine Submodule.fg_of_fg_map_of_fg_inf_ker (Submodule.mkQ (Ideal.span {x})) ?_ ?_
  · have := finiteQuotient (I := Ideal.span {x}) (by simp [hx₂])
    exact Submodule.FG.of_finite
  · rw [Submodule.ker_mkQ, inf_eq_right.mpr ((Ideal.span_singleton_le_iff_mem I).mpr hx₁)]
    exact Submodule.fg_span_singleton x

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

Goal state before:
pp: case neg
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
h :
  ∀ ⦃I : ↥(Ideal (𝓞 K))⁰⦄,
    (↑I).IsPrime →
      ↑(absNorm ↑I) ≤ (4 / π) ^ nrComplexPlaces K * (↑(finrank ℚ K)! / ↑(finrank ℚ K) ^ finrank ℚ K * √|↑(discr K)|) →
        Submodule.IsPrincipal ↑I
I : ↥(Ideal (𝓞 K))⁰
hI : ↑(absNorm ↑I) ≤ (4 / π) ^ nrComplexPlaces K * (↑(finrank ℚ K)! / ↑(finrank ℚ K) ^ finrank ℚ K * √|↑(discr K)|)
J : Ideal (𝓞 K)
hJ : J ∈ UniqueFactorizationMonoid.normalizedFactors ↑I
hJ0 : ¬J = 0
⊢ ↑(absNorm ↑⟨J, ⋯⟩) ≤ (4 / π) ^ nrComplexPlaces K * (↑(finrank ℚ K)! / ↑(finrank ℚ K) ^ finrank ℚ K * √|↑(discr K)|)

Tactic:
exact (cast_le.mpr <| le_of_dvd (absNorm_pos_of_nonZeroDivisors I) <|
    absNorm_dvd_absNorm_of_le <| le_of_dvd <|
      UniqueFactorizationMonoid.dvd_of_mem_normalizedFactors hJ).trans hI

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

Goal state before:
pp: case neg
R : Type u_1
S : Type u_2
T : Type u_3
inst✝¹³ : CommRing R
inst✝¹² : CommRing S
inst✝¹¹ : CommRing T
inst✝¹⁰ : Algebra R S
inst✝⁹ : Algebra S T
inst✝⁸ : Algebra R T
inst✝⁷ : IsScalarTower R S T
p : Ideal S
P : Ideal T
inst✝⁶ : P.LiesOver p
inst✝⁵ : p.IsPrime
inst✝⁴ : P.IsPrime
inst✝³ : IsUnramifiedAt R P
inst✝² : EssFiniteType R S
inst✝¹ : EssFiniteType R T
inst✝ : IsDedekindDomain S
hP₁ : P.primeCompl ≤ nonZeroDivisors T
hP₂ : p ≠ ⊥ → P ≠ ⊥
p₀ : Ideal R := Ideal.under R p
this✝⁵ : P.LiesOver p₀
this✝⁴ : Algebra (Localization.AtPrime p₀) (Localization.AtPrime p) := Localization.AtPrime.algebraOfLiesOver p₀ p
this✝³ : Algebra (Localization.AtPrime p) (Localization.AtPrime P) := Localization.AtPrime.algebraOfLiesOver p P
this✝² : Algebra (Localization.AtPrime p₀) (Localization.AtPrime P) := Localization.AtPrime.algebraOfLiesOver p₀ P
hp₀ : p₀ = Ideal.under R P
this✝¹ : EssFiniteType S T
this✝ : IsNoetherianRing T
h₁ : Algebra.IsSeparable p₀.ResidueField P.ResidueField
h₂ : Ideal.map (algebraMap R (Localization.AtPrime P)) p₀ = maximalIdeal (Localization.AtPrime P)
hp : ¬p = ⊥
this : ¬Ideal.map (algebraMap R T) p₀ ≤ P ^ 2
H :
  Ideal.map ((algebraMap S T).comp (algebraMap R S)) (Ideal.comap (algebraMap R S) p) ≤
    Ideal.map (algebraMap S T) (p ^ 2)
⊢ False

Tactic:
← IsScalarTower.algebraMap_eq,

Goal state after:
case neg
R : Type u_1
S : Type u_2
T : Type u_3
inst✝¹³ : CommRing R
inst✝¹² : CommRing S
inst✝¹¹ : CommRing T
inst✝¹⁰ : Algebra R S
inst✝⁹ : Algebra S T
inst✝⁸ : Algebra R T
inst✝⁷ : IsScalarTower R S T
p : Ideal S
P : Ideal T
inst✝⁶ : P.LiesOver p
inst✝⁵ : p.IsPrime
inst✝⁴ : P.IsPrime
inst✝³ : IsUnramifiedAt R P
inst✝² : EssFiniteType R S
inst✝¹ : EssFiniteType R T
inst✝ : IsDedekindDomain S
hP₁ : P.primeCompl ≤ nonZeroDivisors T
hP₂ : p ≠ ⊥ → P ≠ ⊥
p₀ : Ideal R := Ideal.under R p
this✝⁵ : P.LiesOver p₀
this✝⁴ : Algebra (Localization.AtPrime p₀) (Localization.AtPrime p) := Localization.AtPrime.algebraOfLiesOver p₀ p
this✝³ : Algebra (Localization.AtPrime p) (Localization.AtPrime P) := Localization.AtPrime.algebraOfLiesOver p P
this✝² : Algebra (Localization.AtPrime p₀) (Localization.AtPrime P) := Localization.AtPrime.algebraOfLiesOver p₀ P
hp₀ : p₀ = Ideal.under R P
this✝¹ : EssFiniteType S T
this✝ : IsNoetherianRing T
h₁ : Algebra.IsSeparable p₀.ResidueField P.ResidueField
h₂ : Ideal.map (algebraMap R (Localization.AtPrime P)) p₀ = maximalIdeal (Localization.AtPrime P)
hp : ¬p = ⊥
this : ¬Ideal.map (algebraMap R T) p₀ ≤ P ^ 2
H : Ideal.map (algebraMap R T) (Ideal.comap (algebraMap R S) p) ≤ Ideal.map (algebraMap S T) (p ^ 2)
⊢ False
Module: Mathlib.RingTheory.DedekindDomain.PID

Goal state before:
pp: case refine_2
R : Type u_2
A : Type u_3
inst✝³ : CommRing R
inst✝² : CommRing A
inst✝¹ : Algebra R A
S : Submonoid R
inst✝ : IsLocalization S A
I : (FractionalIdeal S A)ˣ
v : A
hv : v ∈ ↑I⁻¹
hinv : ↑I * ↑I⁻¹ = 1
J : Submodule R R := Submodule.comap (Algebra.linearMap R A) (↑↑I * (R ∙ v))
h : J = ⊤
hJ : IsLocalization.coeSubmodule A J = ↑↑I * (R ∙ v)
this : 1 ∈ ↑↑I * (R ∙ v)
w : A
hw : w ∈ ↑↑I
hvw : w * v = 1
⊢ 1 ∈ ↑I⁻¹ * spanSingleton S w

Tactic:
← hvw,

Goal state after:
case refine_2
R : Type u_2
A : Type u_3
inst✝³ : CommRing R
inst✝² : CommRing A
inst✝¹ : Algebra R A
S : Submonoid R
inst✝ : IsLocalization S A
I : (FractionalIdeal S A)ˣ
v : A
hv : v ∈ ↑I⁻¹
hinv : ↑I * ↑I⁻¹ = 1
J : Submodule R R := Submodule.comap (Algebra.linearMap R A) (↑↑I * (R ∙ v))
h : J = ⊤
hJ : IsLocalization.coeSubmodule A J = ↑↑I * (R ∙ v)
this : 1 ∈ ↑↑I * (R ∙ v)
w : A
hw : w ∈ ↑↑I
hvw : w * v = 1
⊢ w * v ∈ ↑I⁻¹ * spanSingleton S w
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 K L).comp ((algebraMap Rₚ K).comp (algebraMap R Rₚ)) = (algebraMap S L).comp (algebraMap R S)

Tactic:
← IsScalarTower.algebraMap_eq,

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 R K) = (algebraMap S L).comp (algebraMap R S)
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 K L).comp (algebraMap R K) = (algebraMap S L).comp (algebraMap R S)

Tactic:
← IsScalarTower.algebraMap_eq,

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 R L = (algebraMap S L).comp (algebraMap R S)