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

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
x : realMixedSpace K
⊢ (FDerivPolarCoordRealSymm K x).det = ∏ w, (x.2 w).1

Tactic:
by

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

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
⊢ ↑(polarCoordReal K).symm '' (polarCoordReal K).target =ᵐ[volume] Set.univ

Tactic:
by

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

Goal state before:
pp: K : Type u_1
inst✝³ : Field K
inst✝² : NumberField K
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : realMixedSpace K → E
⊢ ∫ (x : realMixedSpace K) in (polarCoordReal K).target, (∏ w, (x.2 w).1) • f (↑(polarCoordReal K).symm x) =
    ∫ (x : realMixedSpace K), f x

Tactic:
by

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

Goal state before:
pp: K : Type u_1
inst✝ : Field K
x✝² x✝¹ : (𝓞 K)ˣ
x✝ : mixedSpace K
⊢ (x✝² * x✝¹) • x✝ = x✝² • x✝¹ • x✝

Tactic:
by

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

Goal state before:
pp: K : Type u_1
inst✝ : Field K
x✝ : mixedSpace K
⊢ 1 • x✝ = x✝

Tactic:
by

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

Goal state before:
pp: K : Type u_1
inst✝ : Field K
x✝ : (𝓞 K)ˣ
⊢ x✝ • 0 = 0

Tactic:
by

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

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
f : realMixedSpace K → ℝ≥0∞
⊢ ∫⁻ (x : realMixedSpace K) in (polarCoordReal K).target,
      (∏ w, ENNReal.ofReal (x.2 w).1) * f (↑(polarCoordReal K).symm x) =
    ∫⁻ (x : realMixedSpace K), f x

Tactic:
by

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

Goal state before:
pp: K : Type u_1
inst✝ : Field K
u : (𝓞 K)ˣ
x : mixedSpace K
h : x = 0
⊢ u • x = 0

Tactic:
by

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

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
⊢ Measurable ↑(mixedEmbedding.polarCoord K).symm

Tactic:
by

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

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
x : realMixedSpace K
w : InfinitePlace K
hw : w.IsReal
⊢ (normAtPlace w) (↑(mixedEmbedding.polarCoord K).symm x) = ‖x.1 ⟨w, hw⟩‖

Tactic:
by

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

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
x : realMixedSpace K
w : InfinitePlace K
hw : w.IsComplex
⊢ (normAtPlace w) (↑(mixedEmbedding.polarCoord K).symm x) = ‖(x.2 ⟨w, hw⟩).1‖

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.NumberField.Units.Regulator

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
u : Fin (rank K) → (𝓞 K)ˣ
⊢ (Subgroup.closure (Set.range u) ⊔ torsion K).index ≠ 0 ↔
    Finite
      (↥(unitLattice K) ⧸ span ℤ (Set.range (⇑(logEmbeddingEquiv K) ∘ ⇑Additive.toMul.symm ∘ QuotientGroup.mk ∘ u)))

Tactic:
by

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

Goal state before:
pp: K : Type u_1
inst✝ : Field K
u : (𝓞 K)ˣ
x : mixedSpace K
h✝ : x ≠ 0
w : InfinitePlace K
h : (normAtPlace w) x ≠ 0
⊢ (normAtPlace w) ((mixedEmbedding K) ((algebraMap (𝓞 K) K) ↑u)) ≠ 0

Tactic:
by

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

Goal state before:
pp: K : Type u_1
inst✝ : Field K
u : (𝓞 K)ˣ
x : mixedSpace K
⊢ u • x = 0 ↔ x = 0

Tactic:
by

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

Goal state before:
pp: K : Type u_1
inst✝³ : Field K
inst✝² : NumberField K
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : mixedSpace K → E
⊢ ∫ (x : realMixedSpace K) in (mixedEmbedding.polarCoord K).target,
      (∏ w, (x.2 w).1) • f (↑(mixedEmbedding.polarCoord K).symm x) =
    ∫ (x : mixedSpace K), f x

Tactic:
by

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

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
x y : 𝓞 K
u : (𝓞 K)ˣ
⊢ u • (mixedEmbedding K) ↑x = (mixedEmbedding K) ↑y ↔ ↑u * x = y

Tactic:
by

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

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
u : (𝓞 K)ˣ
x : mixedSpace K
⊢ mixedEmbedding.norm (u • x) = mixedEmbedding.norm x

Tactic:
by

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

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
⊢ logMap 0 = 0

Tactic:
by

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

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
⊢ logMap 1 = 0

Tactic:
by

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

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
f : mixedSpace K → ℝ≥0∞
⊢ ∫⁻ (x : realMixedSpace K) in (mixedEmbedding.polarCoord K).target,
      (∏ w, ENNReal.ofReal (x.2 w).1) * f (↑(mixedEmbedding.polarCoord K).symm x) =
    ∫⁻ (x : mixedSpace K), f x

Tactic:
by

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

Goal state before:
pp: K : Type u_1
inst✝ : Field K
⊢ Continuous fun x w ↦ (x.2 w).2

Tactic:
by

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

Goal state before:
pp: K : Type u_1
inst✝ : Field K
⊢ Continuous (measurableEquivRealMixedSpacePolarSpace K).toFun

Tactic:
by

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

Goal state before:
pp: K : Type u_1
inst✝ : Field K
⊢ Continuous (measurableEquivRealMixedSpacePolarSpace K).invFun

Tactic:
by

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

Goal state before:
pp: K : Type u_1
inst✝ : Field K
x : realMixedSpace K
w : { w // w.IsReal }
⊢ ((homeoRealMixedSpacePolarSpace K) x).1 ↑w = x.1 w

Tactic:
by

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

Goal state before:
pp: K : Type u_1
inst✝ : Field K
x : realMixedSpace K
w : { w // w.IsComplex }
⊢ ((homeoRealMixedSpacePolarSpace K) x).1 ↑w = (x.2 w).1

Tactic:
by

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

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
⊢ Measurable ↑(polarSpaceCoord K).symm

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.NumberField.Units.Regulator

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
u : Fin (rank K) → (𝓞 K)ˣ
h₁ :
  (Subgroup.closure (Set.range u) ⊔ torsion K).index ≠ 0 ↔
    Finite
      (↥(unitLattice K) ⧸ span ℤ (Set.range (⇑(logEmbeddingEquiv K) ∘ ⇑Additive.toMul.symm ∘ QuotientGroup.mk ∘ u)))
⊢ DiscreteTopology ↥(span ℤ (Set.range fun i ↦ (logEmbedding K) (Additive.ofMul (u i))))

Tactic:
by

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

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
x y : mixedSpace K
hx : mixedEmbedding.norm x ≠ 0
hy : mixedEmbedding.norm y ≠ 0
⊢ logMap (x * y) = logMap x + logMap y

Tactic:
by

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

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
x : mixedSpace K
hx : mixedEmbedding.norm x = 1
w : { w // w ≠ w₀ }
⊢ logMap x w = ↑(↑w).mult * Real.log ((normAtPlace ↑w) x)

Tactic:
by

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

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
u : (𝓞 K)ˣ
⊢ logMap ((mixedEmbedding K) ((algebraMap (𝓞 K) K) ↑u)) = (logEmbedding K) (Additive.ofMul u)

Tactic:
by

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

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
x : mixedSpace K
u : (𝓞 K)ˣ
hx : mixedEmbedding.norm x ≠ 0
⊢ mixedEmbedding.norm ((mixedEmbedding K) ((algebraMap (𝓞 K) K) ↑u)) ≠ 0

Tactic:
by

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

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
⊢ (polarSpaceCoord K).target =
    (Set.univ.pi fun w ↦ if w.IsReal then Set.univ else Set.Ioi 0) ×ˢ Set.univ.pi fun x ↦ Set.Ioo (-π) π

Tactic:
by

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

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
x : mixedSpace K
u : (𝓞 K)ˣ
hx : mixedEmbedding.norm x ≠ 0
⊢ logMap (u • x) = (logEmbedding K) (Additive.ofMul u) + logMap x

Tactic:
by

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

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
x : mixedSpace K
ζ : (𝓞 K)ˣ
hζ : ζ ∈ torsion K
⊢ logMap (ζ • x) = logMap x

Tactic:
by

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

Goal state before:
pp: K : Type u_1
inst✝³ : Field K
inst✝² : NumberField K
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : mixedSpace K → E
⊢ ∫ (x : polarSpace K) in (polarSpaceCoord K).target, (∏ w, x.1 ↑w) • f (↑(polarSpaceCoord K).symm x) =
    ∫ (x : mixedSpace K), f x

Tactic:
by

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

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
c : ℝ
⊢ logMap (c • 1) = 0

Tactic:
by

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

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
x : mixedSpace K
hx : mixedEmbedding.norm x ≠ 0
c : ℝ
hc : c ≠ 0
⊢ mixedEmbedding.norm (c • 1) ≠ 0

Tactic:
by

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

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
f : mixedSpace K → ℝ≥0∞
⊢ ∫⁻ (x : polarSpace K) in (polarSpaceCoord K).target,
      (∏ w, ENNReal.ofReal (x.1 ↑w)) * f (↑(polarSpaceCoord K).symm x) =
    ∫⁻ (x : mixedSpace K), f x

Tactic:
by

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

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
x : polarSpace K
⊢ normAtComplexPlaces (↑(polarSpaceCoord K).symm x) = normAtComplexPlaces (mixedSpaceOfRealSpace x.1)

Tactic:
by

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

Goal state before:
pp: K : Type u_1
inst✝ : Field K
A : Set (mixedSpace K)
hA : normAtComplexPlaces ⁻¹' normAtComplexPlaces '' A = A
⊢ ∀ (x : mixedSpace K) (w : InfinitePlace K), w.IsComplex → 0 ≤ normAtComplexPlaces x w

Tactic:
by

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

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
x : mixedSpace K
hx : mixedEmbedding.norm x ≠ 0
c : ℝ
hc : c ≠ 0
⊢ logMap (c • x) = logMap x

Tactic:
by

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

Goal state before:
pp: K : Type u_1
inst✝ : Field K
A : Set (mixedSpace K)
hA : normAtComplexPlaces ⁻¹' normAtComplexPlaces '' A = A
h : ∀ (x : mixedSpace K) (w : InfinitePlace K), w.IsComplex → 0 ≤ normAtComplexPlaces x w
a : mixedSpace K
ha : a ∈ A
⊢ normAtComplexPlaces a ∈ Set.univ.pi fun w ↦ if w.IsReal then Set.univ else Set.Ici 0

Tactic:
by

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

Goal state before:
pp: K : Type u_1
inst✝ : Field K
A : Set (mixedSpace K)
hA : normAtComplexPlaces ⁻¹' normAtComplexPlaces '' A = A
h : ∀ (x : mixedSpace K) (w : InfinitePlace K), w.IsComplex → 0 ≤ normAtComplexPlaces x w
a : mixedSpace K
ha : a ∈ A
⊢ normAtComplexPlaces a ∈ Set.univ.pi fun w ↦ if w.IsReal then Set.univ else Set.Ici 0

Tactic:
by simpa using h a

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

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
x y : mixedSpace K
h : ∀ (w : InfinitePlace K), (normAtPlace w) x = (normAtPlace w) y
⊢ logMap x = logMap y

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.NumberField.Units.Regulator

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
u : Fin (rank K) → (𝓞 K)ˣ
⊢ IsMaxRank u ↔ (Subgroup.closure (Set.range u)).FiniteIndex

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.NumberField.Units.Regulator

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
u : Fin (rank K) → (𝓞 K)ˣ
hu : ¬IsMaxRank u
⊢ regOfFamily u = 0

Tactic:
by

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

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
⊢ MeasurableSet (fundamentalCone K)

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.NumberField.Units.Regulator

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
u : Fin (rank K) → (𝓞 K)ˣ
hu : IsMaxRank u
⊢ regOfFamily u = ZLattice.covolume (span ℤ (Set.range ⇑(basisOfIsMaxRank hu))) volume

Tactic:
by

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

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
x y : mixedSpace K
hx : x ∈ fundamentalCone K
hy : ∀ (w : InfinitePlace K), (normAtPlace w) y = (normAtPlace w) x
⊢ y ∉ {x | mixedEmbedding.norm x = 0}

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.NumberField.Units.Regulator

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
u : Fin (rank K) → (𝓞 K)ˣ
hu : IsMaxRank u
⊢ 0 < regOfFamily u

Tactic:
by

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

Goal state before:
pp: K : Type u_1
inst✝ : Field K
A : Set (mixedSpace K)
hA : normAtComplexPlaces ⁻¹' normAtComplexPlaces '' A = A
⊢ normAtComplexPlaces '' A = ⇑mixedSpaceOfRealSpace ⁻¹' A ∩ Set.univ.pi fun w ↦ if w.IsReal then Set.univ else Set.Ici 0

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.NumberField.Units.Regulator

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
u : Fin (rank K) → (𝓞 K)ˣ
⊢ regOfFamily u ≠ 0 → IsMaxRank u

Tactic:
by

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

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
x y : mixedSpace K
hx : x ∈ fundamentalCone K
hy : ∀ (w : InfinitePlace K), (normAtPlace w) y = (normAtPlace w) x
⊢ y ∈ fundamentalCone K

Tactic:
by

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

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
A : Set (mixedSpace K)
inst✝ : NumberField K
hA : normAtComplexPlaces ⁻¹' normAtComplexPlaces '' A = A
hm : MeasurableSet A
x : mixedSpace K
⊢ A.indicator 1 x = (normAtComplexPlaces '' A).indicator 1 (normAtComplexPlaces x)

Tactic:
by

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

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
x : mixedSpace K
c : ℝ
hx : x ∈ fundamentalCone K
hc : c ≠ 0
⊢ c • x ∈ fundamentalCone K

Tactic:
by

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

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
x : mixedSpace K
c : ℝ
hc : c ≠ 0
⊢ c • x ∈ fundamentalCone K ↔ x ∈ fundamentalCone K

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.NumberField.Units.Regulator

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
u : Fin (rank K) → (𝓞 K)ˣ
⊢ regOfFamily u = |(of fun i ↦ (logEmbedding K) (Additive.ofMul (u ((equivFinRank K).symm i)))).det|

Tactic:
by

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

Goal state before:
pp: K : Type u_1
inst✝³ : Field K
inst✝² : CharZero K
inst✝¹ : IsCMField K
inst✝ : NumberField K
⊢ Fintype.card (InfinitePlace ↥K⁺) = Fintype.card (InfinitePlace K)

Tactic:
by

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

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
x : mixedSpace K
hx : mixedEmbedding.norm x ≠ 0
B : Module.Basis (Fin (rank K)) ℝ (logSpace K) := Module.Basis.ofZLatticeBasis ℝ (unitLattice K) (basisUnitLattice K)
u : Additive (𝓞 K)ˣ
left✝ : u ∈ ↑⊤
hu : ↑⟨(logEmbedding K).toIntLinearMap u, ⋯⟩ + logMap x ∈ ZSpan.fundamentalDomain B
⊢ u • x ∈ logMap ⁻¹' ZSpan.fundamentalDomain (Module.Basis.ofZLatticeBasis ℝ (unitLattice K) (basisUnitLattice K))

Tactic:
by

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

Goal state before:
pp: K : Type u_1
inst✝³ : Field K
inst✝² : CharZero K
inst✝¹ : IsCMField K
inst✝ : NumberField K
w : InfinitePlace ↥K⁺
x : ↥K⁺
⊢ ((equivInfinitePlace K).symm w) ((algebraMap (↥K⁺) K) x) = w x

Tactic:
by

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

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
x : mixedSpace K
hx : mixedEmbedding.norm x ≠ 0
B : Module.Basis (Fin (rank K)) ℝ (logSpace K) := Module.Basis.ofZLatticeBasis ℝ (unitLattice K) (basisUnitLattice K)
u : Additive (𝓞 K)ˣ
left✝ : u ∈ ↑⊤
hu : ↑⟨(logEmbedding K).toIntLinearMap u, ⋯⟩ + logMap x ∈ ZSpan.fundamentalDomain B
⊢ u • x ∉ {x | mixedEmbedding.norm x = 0}

Tactic:
by

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

Goal state before:
pp: K : Type u_1
inst✝³ : Field K
inst✝² : CharZero K
inst✝¹ : IsCMField K
inst✝ : NumberField K
⊢ Units.rank ↥K⁺ = Units.rank K

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.NumberField.Units.Regulator

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
u : Fin (rank K) → (𝓞 K)ˣ
w₁ w₂ : InfinitePlace K
e₁ : { w // w ≠ w₁ } ≃ Fin (rank K)
e₂ : { w // w ≠ w₂ } ≃ Fin (rank K)
f : Fin (rank K + 1) ≃ InfinitePlace K := (finSuccEquiv (rank K)).trans ↑((Equiv.optionSubtype w₁).symm e₁.symm)
x✝ : InfinitePlace K
⊢ x✝ ≠ w₂ ↔ f.symm x✝ ≠ f.symm w₂

Tactic:
by

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

Goal state before:
pp: K : Type u_1
inst✝² : Field K
inst✝¹ : CharZero K
inst✝ : IsCMField K
φ ψ : K →+* ℂ
σ τ : Gal(K/↥K⁺)
hφ : ComplexEmbedding.IsConj φ σ
hψ : ComplexEmbedding.IsConj ψ τ
⊢ σ = τ

Tactic:
by

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

Goal state before:
pp: K : Type u_1
inst✝³ : Field K
inst✝² : CharZero K
inst✝¹ : IsCMField K
inst✝ : Algebra.IsIntegral ℚ K
φ : K →+* ℂ
⊢ ComplexEmbedding.IsConj φ (complexConj K)

Tactic:
by

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

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
x : mixedSpace K
hx : mixedEmbedding.norm x ≠ 0
B : Module.Basis (Fin (rank K)) ℝ (logSpace K) := Module.Basis.ofZLatticeBasis ℝ (unitLattice K) (basisUnitLattice K)
e : logSpace K
h₁ : e ∈ Submodule.span ℤ (Set.range ⇑B)
h₂ : ⟨e, h₁⟩ +ᵥ logMap x ∈ ZSpan.fundamentalDomain B
⊢ e ∈ unitLattice K

Tactic:
by

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

Goal state before:
pp: K : Type u_1
inst✝³ : Field K
inst✝² : CharZero K
inst✝¹ : IsCMField K
inst✝ : Algebra.IsIntegral ℚ K
φ : K →+* ℂ
x : K
⊢ φ ((complexConj K) x) = (starRingEnd ℂ) (φ x)

Tactic:
by

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

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
x : mixedSpace K
hx : mixedEmbedding.norm x ≠ 0
⊢ ∃ u, u • x ∈ fundamentalCone K

Tactic:
by

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

Goal state before:
pp: K : Type u_1
inst✝³ : Field K
inst✝² : CharZero K
inst✝¹ : IsCMField K
inst✝ : Algebra.IsIntegral ℚ K
w : InfinitePlace K
x : K
⊢ w ((complexConj K) x) = w x

Tactic:
by

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

Goal state before:
pp: K : Type u_1
inst✝³ : Field K
inst✝² : CharZero K
inst✝¹ : IsCMField K
inst✝ : Algebra.IsIntegral ℚ K
x : K
⊢ (complexConj K) ((complexConj K) x) = x

Tactic:
by

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

Goal state before:
pp: K : Type u_1
inst✝³ : Field K
inst✝² : CharZero K
inst✝¹ : IsCMField K
inst✝ : Algebra.IsIntegral ℚ K
⊢ complexConj K ^ 2 = 1

Tactic:
by

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

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
x : mixedSpace K
hx : x ∈ fundamentalCone K
ζ : (𝓞 K)ˣ
hζ : ζ ∈ torsion K
⊢ ζ • x ∈ fundamentalCone K

Tactic:
by

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

Goal state before:
pp: K : Type u_1
inst✝³ : Field K
inst✝² : CharZero K
inst✝¹ : IsCMField K
inst✝ : Algebra.IsIntegral ℚ K
⊢ Subgroup.zpowers (complexConj K) = ⊤

Tactic:
by

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

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
A : Set (mixedSpace K)
inst✝ : NumberField K
hA : normAtComplexPlaces ⁻¹' normAtComplexPlaces '' A = A
hm : MeasurableSet A
hA' : ∀ {x : mixedSpace K}, A.indicator 1 x = (normAtComplexPlaces '' A).indicator 1 (normAtComplexPlaces x)
⊢ MeasurableSet (⇑mixedSpaceOfRealSpace ⁻¹' normAtComplexPlaces ⁻¹' normAtComplexPlaces '' A)

Tactic:
by

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

Goal state before:
pp: K : Type u_1
inst✝³ : Field K
inst✝² : CharZero K
inst✝¹ : IsCMField K
inst✝ : Algebra.IsIntegral ℚ K
x : K
⊢ (complexConj K) x = x ↔ x ∈ K⁺

Tactic:
by

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

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
x : mixedSpace K
hx : x ∈ fundamentalCone K
u : (𝓞 K)ˣ
⊢ u • x ∈ fundamentalCone K ↔ u ∈ torsion K

Tactic:
by

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

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
a : mixedSpace K
⊢ a ∈ integerSet K ↔ a ∈ fundamentalCone K ∧ ∃ x, (mixedEmbedding K) ↑x = a

Tactic:
by

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

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
a : mixedSpace K
ha : a ∈ integerSet K
⊢ ∃! x, (mixedEmbedding K) ↑x = a

Tactic:
by

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

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
a : ↑(integerSet K)
⊢ ↑a ≠ 0

Tactic:
by

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

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
x y : mixedSpace K
c : ℝ
a : ↑(integerSet K)
⊢ ⋯.choose ≠ 0

Tactic:
by

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

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
A : Set (mixedSpace K)
inst✝ : NumberField K
hA : normAtComplexPlaces ⁻¹' normAtComplexPlaces '' A = A
hm : MeasurableSet A
hA' : ∀ {x : mixedSpace K}, A.indicator 1 x = (normAtComplexPlaces '' A).indicator 1 (normAtComplexPlaces x)
⊢ ⇑mixedSpaceOfRealSpace ⁻¹' A =ᵐ[Measure.pi fun x ↦ volume] ⇑mixedSpaceOfRealSpace ⁻¹' A

Tactic:
by

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

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
a : ↑(integerSet K)
⊢ (mixedEmbedding K) ↑↑(preimageOfMemIntegerSet a) = ↑a

Tactic:
by

Goal state after: