mathlib-initiative/mathlib-tactics huggingface dataset:
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)))
h₂ : DiscreteTopology ↥(span ℤ (Set.range fun i ↦ (logEmbedding K) (Additive.ofMul (u i))))
⊢ finrank ℤ ↥(span ℤ (Set.range fun i ↦ (logEmbedding K) (Additive.ofMul (u i)))) = Fintype.card (Fin (rank K)) ↔
    finrank ℤ
        ↥(span ℤ
            (Set.range fun x ↦
              (unitLattice K).subtype ((⇑(logEmbeddingEquiv K) ∘ ⇑Additive.toMul.symm ∘ QuotientGroup.mk ∘ u) x))) =
      rank K

Tactic:
]

Goal state after:
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)))
h₂ : DiscreteTopology ↥(span ℤ (Set.range fun i ↦ (logEmbedding K) (Additive.ofMul (u i))))
⊢ finrank ℤ ↥(span ℤ (Set.range fun i ↦ (logEmbedding K) (Additive.ofMul (u i)))) = Fintype.card (Fin (rank K)) ↔
    finrank ℤ
        ↥(span ℤ
            (Set.range fun x ↦
              (unitLattice K).subtype ((⇑(logEmbeddingEquiv K) ∘ ⇑Additive.toMul.symm ∘ QuotientGroup.mk ∘ u) x))) =
      rank K
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
hw : w.IsComplex
⊢ 0 ≤ ‖x.2 ⟨w, hw⟩‖

Tactic:
]

Goal state after:
K : Type u_1
inst✝ : Field K
A : Set (mixedSpace K)
hA : normAtComplexPlaces ⁻¹' normAtComplexPlaces '' A = A
x : mixedSpace K
w : InfinitePlace K
hw : w.IsComplex
⊢ 0 ≤ ‖x.2 ⟨w, hw⟩‖
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
this : mixedEmbedding.norm (c • 1) ≠ 0
⊢ logMap x = logMap x

Tactic:
]

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

Goal state before:
pp: case refine_1
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 ∈ normAtComplexPlaces '' A

Tactic:
]

Goal state after:
case refine_1
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 ∈ normAtComplexPlaces '' A
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 = 0

Tactic:
]

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

Goal state before:
pp: case refine_2
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
x : realSpace K
x✝ : x ∈ ⇑mixedSpaceOfRealSpace ⁻¹' A ∩ Set.univ.pi fun w ↦ if w.IsReal then Set.univ else Set.Ici 0
hx₁ : x ∈ normAtComplexPlaces '' A
hx₂ : x ∈ Set.univ.pi fun w ↦ if w.IsReal then Set.univ else Set.Ici 0
⊢ x ∈ normAtComplexPlaces '' A

pp: case refine_2
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
x : realSpace K
x✝ : x ∈ ⇑mixedSpaceOfRealSpace ⁻¹' A ∩ Set.univ.pi fun w ↦ if w.IsReal then Set.univ else Set.Ici 0
hx₁ : normAtComplexPlaces (mixedSpaceOfRealSpace x) ∈ normAtComplexPlaces '' A
hx₂ : x ∈ Set.univ.pi fun w ↦ if w.IsReal then Set.univ else Set.Ici 0
⊢ ∀ (w : InfinitePlace K), w.IsComplex → 0 ≤ x w

Tactic:
]

Goal state after:
case refine_2
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
x : realSpace K
x✝ : x ∈ ⇑mixedSpaceOfRealSpace ⁻¹' A ∩ Set.univ.pi fun w ↦ if w.IsReal then Set.univ else Set.Ici 0
hx₁ : x ∈ normAtComplexPlaces '' A
hx₂ : x ∈ Set.univ.pi fun w ↦ if w.IsReal then Set.univ else Set.Ici 0
⊢ x ∈ normAtComplexPlaces '' A

case refine_2
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
x : realSpace K
x✝ : x ∈ ⇑mixedSpaceOfRealSpace ⁻¹' A ∩ Set.univ.pi fun w ↦ if w.IsReal then Set.univ else Set.Ici 0
hx₁ : normAtComplexPlaces (mixedSpaceOfRealSpace x) ∈ normAtComplexPlaces '' A
hx₂ : x ∈ Set.univ.pi fun w ↦ if w.IsReal then Set.univ else Set.Ici 0
⊢ ∀ (w : InfinitePlace K), w.IsComplex → 0 ≤ x w
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
⊢ ZLattice.covolume (span ℤ (Set.range ⇑(basisOfIsMaxRank hu))) volume =
    ZLattice.covolume (span ℤ (Set.range ⇑(basisOfIsMaxRank hu))) volume

Tactic:
]

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 < ZLattice.covolume (span ℤ (Set.range ⇑(basisOfIsMaxRank hu))) volume

Tactic:
]

Goal state after:
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
u : Fin (rank K) → (𝓞 K)ˣ
hu : IsMaxRank u
⊢ 0 < ZLattice.covolume (span ℤ (Set.range ⇑(basisOfIsMaxRank hu))) volume
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
⊢ logMap x ∈ ZSpan.fundamentalDomain (Module.Basis.ofZLatticeBasis ℝ (unitLattice K) (basisUnitLattice K))

Tactic:
]

Goal state after:
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
⊢ logMap x ∈ ZSpan.fundamentalDomain (Module.Basis.ofZLatticeBasis ℝ (unitLattice K) (basisUnitLattice K))
Module: Mathlib.NumberTheory.NumberField.CanonicalEmbedding.FundamentalCone

Goal state before:
pp: case refine_1
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
x : mixedSpace K
c : ℝ
hx : x ∈ fundamentalCone K
hc : c ≠ 0
⊢ logMap x ∈ ZSpan.fundamentalDomain (Module.Basis.ofZLatticeBasis ℝ (unitLattice K) (basisUnitLattice K))

Tactic:
]

Goal state after:
case refine_1
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
x : mixedSpace K
c : ℝ
hx : x ∈ fundamentalCone K
hc : c ≠ 0
⊢ logMap x ∈ ZSpan.fundamentalDomain (Module.Basis.ofZLatticeBasis ℝ (unitLattice K) (basisUnitLattice K))
Module: Mathlib.NumberTheory.NumberField.Units.Regulator

Goal state before:
pp: case pos
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
u : Fin (rank K) → (𝓞 K)ˣ
hu : IsMaxRank u
⊢ |(of (Subtype.val ∘ ⇑(Basis.restrictScalars ℤ (basisOfIsMaxRank hu)) ∘ ⇑(equivFinRank K).symm)).det| =
    |(of fun i ↦ (logEmbedding K) (Additive.ofMul (u ((equivFinRank K).symm i)))).det|

Tactic:
]

Goal state after:
case pos
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
u : Fin (rank K) → (𝓞 K)ˣ
hu : IsMaxRank u
⊢ |(of (Subtype.val ∘ ⇑(Basis.restrictScalars ℤ (basisOfIsMaxRank hu)) ∘ ⇑(equivFinRank K).symm)).det| =
    |(of fun i ↦ (logEmbedding K) (Additive.ofMul (u ((equivFinRank K).symm i)))).det|
Module: Mathlib.NumberTheory.NumberField.CanonicalEmbedding.FundamentalCone

Goal state before:
pp: case refine_2
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
x : mixedSpace K
c : ℝ
hx : x ∈ fundamentalCone K
hc : c ≠ 0
⊢ ¬|c| ^ Module.finrank ℚ K = 0 ∧ ¬mixedEmbedding.norm x = 0

Tactic:
]

Goal state after:
case refine_2
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
x : mixedSpace K
c : ℝ
hx : x ∈ fundamentalCone K
hc : c ≠ 0
⊢ ¬|c| ^ Module.finrank ℚ K = 0 ∧ ¬mixedEmbedding.norm x = 0
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
h : c • x ∈ fundamentalCone K
⊢ c • x = c • x

Tactic:
]

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
⊢ (logEmbedding K) (Additive.ofMul u) + logMap x ∈
    ZSpan.fundamentalDomain (Module.Basis.ofZLatticeBasis ℝ (unitLattice K) (basisUnitLattice K))

Tactic:
]

Goal state after:
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
⊢ (logEmbedding K) (Additive.ofMul u) + logMap x ∈
    ZSpan.fundamentalDomain (Module.Basis.ofZLatticeBasis ℝ (unitLattice K) (basisUnitLattice K))
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)
⊢ ∫⁻ (x : InfinitePlace K → ℝ) in Set.univ.pi fun w ↦ if w.IsReal then Set.univ else Set.Ioi 0,
      ∫⁻ (y : { w // w.IsComplex } → ℝ) in Set.univ.pi fun x ↦ Set.Ioo (-π) π,
        (∏ w, ENNReal.ofReal ((x, y).1 ↑w)) * A.indicator 1 (↑(polarSpaceCoord K).symm (x, y)) =
    ENNReal.ofReal (2 * π) ^ nrComplexPlaces K *
      ∫⁻ (x : realSpace K) in normAtComplexPlaces '' A, ∏ w, ENNReal.ofReal (x ↑w)

pp: case hf
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)
⊢ AEMeasurable (fun x ↦ (∏ w, ENNReal.ofReal (x.1 ↑w)) * A.indicator 1 (↑(polarSpaceCoord K).symm x))
    ((volume.prod volume).restrict
      ((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:
]

Goal state after:
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)
⊢ ∫⁻ (x : InfinitePlace K → ℝ) in Set.univ.pi fun w ↦ if w.IsReal then Set.univ else Set.Ioi 0,
      ∫⁻ (y : { w // w.IsComplex } → ℝ) in Set.univ.pi fun x ↦ Set.Ioo (-π) π,
        (∏ w, ENNReal.ofReal ((x, y).1 ↑w)) * A.indicator 1 (↑(polarSpaceCoord K).symm (x, y)) =
    ENNReal.ofReal (2 * π) ^ nrComplexPlaces K *
      ∫⁻ (x : realSpace K) in normAtComplexPlaces '' A, ∏ w, ENNReal.ofReal (x ↑w)

case hf
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)
⊢ AEMeasurable (fun x ↦ (∏ w, ENNReal.ofReal (x.1 ↑w)) * A.indicator 1 (↑(polarSpaceCoord K).symm x))
    ((volume.prod volume).restrict
      ((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))
Module: Mathlib.NumberTheory.NumberField.Units.Regulator

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

pp: case neg
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
u : Fin (rank K) → (𝓞 K)ˣ
hu : ¬IsMaxRank u
⊢ ¬LinearIndependent ℝ fun i ↦ of (fun i ↦ (logEmbedding K) (Additive.ofMul (u ((equivFinRank K).symm i)))) i

Tactic:
]

Goal state after:
case neg
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
u : Fin (rank K) → (𝓞 K)ˣ
hu : ¬IsMaxRank u
⊢ ¬LinearIndependent ℝ fun i ↦ of (fun i ↦ (logEmbedding K) (Additive.ofMul (u ((equivFinRank K).symm i)))) i
Module: Mathlib.NumberTheory.NumberField.Units.Regulator

Goal state before:
pp: case neg
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
u : Fin (rank K) → (𝓞 K)ˣ
hu : ¬LinearIndependent ℝ ((fun i ↦ (logEmbedding K) (Additive.ofMul (u i))) ∘ ⇑(equivFinRank K).symm)
⊢ ¬LinearIndependent ℝ fun i ↦ of (fun i ↦ (logEmbedding K) (Additive.ofMul (u ((equivFinRank K).symm i)))) i

Tactic:
]

Goal state after:
case neg
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
u : Fin (rank K) → (𝓞 K)ˣ
hu : ¬LinearIndependent ℝ ((fun i ↦ (logEmbedding K) (Additive.ofMul (u i))) ∘ ⇑(equivFinRank K).symm)
⊢ ¬LinearIndependent ℝ fun i ↦ of (fun i ↦ (logEmbedding K) (Additive.ofMul (u ((equivFinRank K).symm i)))) i
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
⊢ Module.finrank ℚ ↥K⁺ * 2 = Module.finrank ℚ ↥K⁺ * 2

Tactic:
]

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⁺
⊢ w x = w x

Tactic:
]

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) - 1 = Fintype.card (InfinitePlace K) - 1

Tactic:
]

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
this : ∃! y, y ≠ 1
⊢ σ = τ

Tactic:
]

Goal state after:
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
this : ∃! y, y ≠ 1
⊢ σ = τ
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
σ : Gal(K/↥K⁺)
hσ : ComplexEmbedding.IsConj φ ⋯.choose
this : ComplexEmbedding.IsConj (Classical.choice ⋯) ⋯.choose
⊢ ComplexEmbedding.IsConj φ (complexConj K)

Tactic:
]

Goal state after:
K : Type u_1
inst✝³ : Field K
inst✝² : CharZero K
inst✝¹ : IsCMField K
inst✝ : Algebra.IsIntegral ℚ K
φ : K →+* ℂ
σ : Gal(K/↥K⁺)
hσ : ComplexEmbedding.IsConj φ ⋯.choose
this : ComplexEmbedding.IsConj (Classical.choice ⋯) ⋯.choose
⊢ ComplexEmbedding.IsConj φ (complexConj K)
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 ∈ Submodule.span ℤ (Set.range ⇑(Module.Basis.ofZLatticeBasis ℝ (unitLattice K) ?m.189))

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) := ⋯
e : logSpace K
h₁ : e ∈ Submodule.span ℤ (Set.range ⇑B)
h₂ : ⟨e, h₁⟩ +ᵥ logMap x ∈ ZSpan.fundamentalDomain B
⊢ Module.Basis ?m.187 ℤ ↥(unitLattice K)

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) := ⋯
e : logSpace K
h₁ : e ∈ Submodule.span ℤ (Set.range ⇑B)
h₂ : ⟨e, h₁⟩ +ᵥ logMap x ∈ ZSpan.fundamentalDomain B
⊢ Type ?u.59

Tactic:
]

Goal state after:
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 ∈ Submodule.span ℤ (Set.range ⇑(Module.Basis.ofZLatticeBasis ℝ (unitLattice K) ?m.189))

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) := ⋯
e : logSpace K
h₁ : e ∈ Submodule.span ℤ (Set.range ⇑B)
h₂ : ⟨e, h₁⟩ +ᵥ logMap x ∈ ZSpan.fundamentalDomain B
⊢ Module.Basis ?m.187 ℤ ↥(unitLattice K)

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) := ⋯
e : logSpace K
h₁ : e ∈ Submodule.span ℤ (Set.range ⇑B)
h₂ : ⟨e, h₁⟩ +ᵥ logMap x ∈ ZSpan.fundamentalDomain B
⊢ Type ?u.59
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
⊢ (starRingEnd ℂ) (φ x) = (starRingEnd ℂ) (φ x)

Tactic:
]

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 x = w x

Tactic:
]

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

Goal state before:
pp: case left
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
⊢ logMap x ∈ ZSpan.fundamentalDomain (Module.Basis.ofZLatticeBasis ℝ (unitLattice K) (basisUnitLattice K))

Tactic:
]

Goal state after:
case left
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
⊢ logMap x ∈ ZSpan.fundamentalDomain (Module.Basis.ofZLatticeBasis ℝ (unitLattice K) (basisUnitLattice K))
Module: Mathlib.NumberTheory.NumberField.CanonicalEmbedding.FundamentalCone

Goal state before:
pp: case right
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
⊢ ¬mixedEmbedding.norm x = 0

Tactic:
]

Goal state after:
case right
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
⊢ ¬mixedEmbedding.norm x = 0
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
h : u • x ∈ fundamentalCone K
⊢ (logEmbedding K) (Additive.ofMul u) = 0

Tactic:
]

Goal state after:
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
x : mixedSpace K
hx : x ∈ fundamentalCone K
u : (𝓞 K)ˣ
h : u • x ∈ fundamentalCone K
⊢ (logEmbedding K) (Additive.ofMul u) = 0
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
⊢ 2 = 2

Tactic:
]

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

Goal state before:
pp: case refine_1
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
x : mixedSpace K
hx : x ∈ fundamentalCone K
u : (𝓞 K)ˣ
h : u • x ∈ fundamentalCone K
B : Module.Basis (Fin (rank K)) ℝ (logSpace K) := Module.Basis.ofZLatticeBasis ℝ (unitLattice K) (basisUnitLattice K)
⊢ (logEmbedding K) (Additive.ofMul u) ∈ unitLattice K

Tactic:
]

Goal state after:
case refine_1
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
x : mixedSpace K
hx : x ∈ fundamentalCone K
u : (𝓞 K)ˣ
h : u • x ∈ fundamentalCone K
B : Module.Basis (Fin (rank K)) ℝ (logSpace K) := Module.Basis.ofZLatticeBasis ℝ (unitLattice K) (basisUnitLattice K)
⊢ (logEmbedding K) (Additive.ofMul u) ∈ unitLattice K
Module: Mathlib.NumberTheory.NumberField.CMField

Goal state before:
pp: case e'_1
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 ↔ ∀ (m : ℤ), (complexConj K ^ m) x = x

Tactic:
]

Goal state after:
case e'_1
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 ↔ ∀ (m : ℤ), (complexConj K ^ m) x = x
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)
⊢ (ENNReal.ofReal (2 * π) ^ Fintype.card { w // w.IsComplex } *
      ∫⁻ (a : InfinitePlace K → ℝ) in Set.univ.pi fun w ↦ if w.IsReal then Set.univ else Set.Ioi 0,
        (⇑mixedSpaceOfRealSpace ⁻¹' normAtComplexPlaces ⁻¹' normAtComplexPlaces '' A).indicator
          (fun x ↦ ∏ x, ENNReal.ofReal (a ↑x)) a ∂Measure.pi fun x ↦ volume) =
    ENNReal.ofReal (2 * π) ^ nrComplexPlaces K *
      ∫⁻ (x : realSpace K) in normAtComplexPlaces '' A, ∏ w, ENNReal.ofReal (x ↑w) ∂Measure.pi fun x ↦ volume

Tactic:
]

Goal state after:
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)
⊢ (ENNReal.ofReal (2 * π) ^ Fintype.card { w // w.IsComplex } *
      ∫⁻ (a : InfinitePlace K → ℝ) in Set.univ.pi fun w ↦ if w.IsReal then Set.univ else Set.Ioi 0,
        (⇑mixedSpaceOfRealSpace ⁻¹' normAtComplexPlaces ⁻¹' normAtComplexPlaces '' A).indicator
          (fun x ↦ ∏ x, ENNReal.ofReal (a ↑x)) a ∂Measure.pi fun x ↦ volume) =
    ENNReal.ofReal (2 * π) ^ nrComplexPlaces K *
      ∫⁻ (x : realSpace K) in normAtComplexPlaces '' A, ∏ w, ENNReal.ofReal (x ↑w) ∂Measure.pi fun x ↦ volume
Module: Mathlib.NumberTheory.NumberField.CMField

Goal state before:
pp: case e'_2
K : Type u_1
inst✝³ : Field K
inst✝² : CharZero K
inst✝¹ : IsCMField K
inst✝ : Algebra.IsIntegral ℚ K
x : K
⊢ x ∈ K⁺ ↔ x ∈ Set.range ⇑(algebraMap (↥K⁺) K)

Tactic:
]

Goal state after:
case e'_2
K : Type u_1
inst✝³ : Field K
inst✝² : CharZero K
inst✝¹ : IsCMField K
inst✝ : Algebra.IsIntegral ℚ K
x : K
⊢ x ∈ K⁺ ↔ x ∈ Set.range ⇑(algebraMap (↥K⁺) K)
Module: Mathlib.NumberTheory.NumberField.CanonicalEmbedding.FundamentalCone

Goal state before:
pp: case refine_2
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
x : mixedSpace K
hx : x ∈ fundamentalCone K
u : (𝓞 K)ˣ
h : u • x ∈ fundamentalCone K
B : Module.Basis (Fin (rank K)) ℝ (logSpace K) := Module.Basis.ofZLatticeBasis ℝ (unitLattice K) (basisUnitLattice K)
⊢ logMap (u • x) ∈ ZSpan.fundamentalDomain B

Tactic:
]

Goal state after:
case refine_2
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
x : mixedSpace K
hx : x ∈ fundamentalCone K
u : (𝓞 K)ˣ
h : u • x ∈ fundamentalCone K
B : Module.Basis (Fin (rank K)) ℝ (logSpace K) := Module.Basis.ofZLatticeBasis ℝ (unitLattice K) (basisUnitLattice K)
⊢ logMap (u • x) ∈ ZSpan.fundamentalDomain B