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
x : mixedSpace K
⊢ ‖x‖ = univ.sup' ⋯ fun w ↦ (normAtPlace w) x

Tactic:
by

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

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
⊢ ∏ w, (normAtPlace w) 0 ^ w.mult = 0

Tactic:
simp [mult]

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

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
⊢ ∏ w, (normAtPlace w) 0 ^ w.mult = 0

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.NumberField.InfinitePlace.Ramification

Goal state before:
pp: k : Type u_1
inst✝² : Field k
K : Type u_2
inst✝¹ : Field K
inst✝ : Algebra k K
w : InfinitePlace K
h : IsUnramified k w
⊢ Stab w = ⊥

Tactic:
by

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

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
⊢ ∏ w, (normAtPlace w) 1 ^ w.mult = 1

Tactic:
by

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

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
x✝¹ x✝ : mixedSpace K
⊢ ∏ w, (normAtPlace w) (x✝¹ * x✝) ^ w.mult = (∏ w, (normAtPlace w) x✝¹ ^ w.mult) * ∏ w, (normAtPlace w) x✝ ^ w.mult

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.NumberField.InfinitePlace.Ramification

Goal state before:
pp: k : Type u_1
inst✝² : Field k
K : Type u_2
inst✝¹ : Field K
inst✝ : Algebra k K
σ : Gal(K/k)
φ : K →+* ℂ
h : ComplexEmbedding.IsConj φ σ
⊢ ↑(Stab (mk φ)) = {1, σ}

Tactic:
by

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

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
x : mixedSpace K
⊢ mixedEmbedding.norm x = 0 ↔ ∃ w, (normAtPlace w) x = 0

Tactic:
by

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

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
f : Free.ChooseBasisIndex ℤ (𝓞 K) ≃ (K →+* ℂ) :=
  (canonicalEmbedding.latticeBasis K).indexEquiv (Pi.basisFun ℂ (K →+* ℂ))
e : index K ≃ Free.ChooseBasisIndex ℤ (𝓞 K) := (indexEquiv K).trans f.symm
M : Matrix (index K) (index K) ℝ := (mixedEmbedding.stdBasis K).toMatrix ⇑((latticeBasis K).reindex e.symm)
N : Matrix (K →+* ℂ) (K →+* ℂ) ℂ :=
  Algebra.embeddingsMatrixReindex ℚ ℂ (⇑(integralBasis K) ∘ ⇑f.symm) (RingHom.equivRatAlgHom K ℂ)
this : M.map ⇑ofRealHom = matrixToStdBasis K * ((reindex (indexEquiv K).symm (indexEquiv K).symm) N)ᵀ
⊢ volume (fundamentalDomain (latticeBasis K)) =
    ↑‖((mixedEmbedding.stdBasis K).toMatrix ⇑((latticeBasis K).reindex e.symm)).det‖₊

Tactic:
by

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

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
x : mixedSpace K
⊢ mixedEmbedding.norm x ≠ 0 ↔ ∀ (w : InfinitePlace K), (normAtPlace w) x ≠ 0

Tactic:
by

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

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
⊢ mixedEmbedding.norm x = mixedEmbedding.norm y

Tactic:
by

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

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
B : ℝ
hB : 0 < B
this : volume (convexBodySum K 1) = ↑(convexBodySumFactor K)
⊢ volume (convexBodySum K B) = ↑(convexBodySumFactor K) * ENNReal.ofReal B ^ finrank ℚ K

Tactic:
by

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

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
c : ℝ
x : mixedSpace K
⊢ mixedEmbedding.norm (c • x) = |c| ^ finrank ℚ K * mixedEmbedding.norm x

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.NumberField.InfinitePlace.Ramification

Goal state before:
pp: case neg
k : Type u_1
inst✝² : Field k
K : Type u_2
inst✝¹ : Field K
inst✝ : Algebra k K
w : InfinitePlace K
h : ∀ (σ : Gal(K/k)), ¬ComplexEmbedding.IsConj w.embedding σ
⊢ Nat.card ↑↑(Stab (mk w.embedding)) = 1 ∨ Nat.card ↑↑(Stab (mk w.embedding)) = 2

Tactic:
left
    trans Nat.card ({1} : Set Gal(K/k))
    · congr with x
      simp only [SetLike.mem_coe, mem_stabilizer_mk_iff, Set.mem_singleton_iff, or_iff_left_iff_imp,
        h x, IsEmpty.forall_iff]
    · simp

Goal state after:
Module: Mathlib.NumberTheory.NumberField.InfinitePlace.Ramification

Goal state before:
pp: k : Type u_1
inst✝² : Field k
K : Type u_2
inst✝¹ : Field K
inst✝ : Algebra k K
w : InfinitePlace K
⊢ Nat.card ↥(Stab w) = 1 ∨ Nat.card ↥(Stab w) = 2

Tactic:
by

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

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
B : ℝ
hB : 0 < B
this : ∫ (x : mixedSpace K), rexp (-convexBodySumFun x) = 2 ^ nrRealPlaces K * (π / 2) ^ nrComplexPlaces K
⊢ 0 ≤ 2 ^ nrRealPlaces K

Tactic:
by

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

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
c : ℝ
⊢ mixedEmbedding.norm (fun x ↦ c, fun x ↦ ↑c) = |c| ^ finrank ℚ K

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.NumberField.InfinitePlace.Ramification

Goal state before:
pp: k : Type u_1
inst✝³ : Field k
K : Type u_2
inst✝² : Field K
inst✝¹ : Algebra k K
w : InfinitePlace K
inst✝ : IsGalois k K
⊢ IsUnramified k w ↔ Stab w = ⊥

Tactic:
by

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

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
x : K
⊢ mixedEmbedding.norm ((mixedEmbedding K) x) = ↑|(Algebra.norm ℚ) x|

Tactic:
by

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

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
B : ℝ
hB : 0 < B
this : ∫ (x : mixedSpace K), rexp (-convexBodySumFun x) = 2 ^ nrRealPlaces K * (π / 2) ^ nrComplexPlaces K
⊢ 0 ≤ π / 2

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.NumberField.InfinitePlace.Ramification

Goal state before:
pp: k : Type u_1
inst✝³ : Field k
K : Type u_2
inst✝² : Field K
inst✝¹ : Algebra k K
w : InfinitePlace K
inst✝ : IsGalois k K
⊢ IsUnramified k w ↔ Nat.card ↥(Stab w) = 1

Tactic:
by

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

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

Tactic:
by

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

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
x : mixedSpace K
hx : x ∈ Set.range ⇑(mixedEmbedding K)
⊢ mixedEmbedding.norm x = 0 ↔ x = 0

Tactic:
by

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

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
⊢ Continuous ⇑mixedEmbedding.norm

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.NumberField.InfinitePlace.Ramification

Goal state before:
pp: k : Type u_1
inst✝³ : Field k
K : Type u_2
inst✝² : Field K
inst✝¹ : Algebra k K
w : InfinitePlace K
inst✝ : IsGalois k K
⊢ ¬IsUnramified k w ↔ Nat.card ↥(Stab w) = 2

Tactic:
by

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

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
⊢ fundamentalDomain (stdBasis K) =
    (Set.univ.pi fun x ↦ Set.Ico 0 1) ×ˢ Set.univ.pi fun x ↦ ⇑measurableEquivPi ⁻¹' Set.univ.pi fun x ↦ Set.Ico 0 1

Tactic:
by

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

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
f : Free.ChooseBasisIndex ℤ (𝓞 K) ≃ (K →+* ℂ) :=
  (canonicalEmbedding.latticeBasis K).indexEquiv (Pi.basisFun ℂ (K →+* ℂ))
e : index K ≃ Free.ChooseBasisIndex ℤ (𝓞 K) := (indexEquiv K).trans f.symm
M : Matrix (index K) (index K) ℝ := (mixedEmbedding.stdBasis K).toMatrix ⇑((latticeBasis K).reindex e.symm)
N : Matrix (K →+* ℂ) (K →+* ℂ) ℂ :=
  Algebra.embeddingsMatrixReindex ℚ ℂ (⇑(integralBasis K) ∘ ⇑f.symm) (RingHom.equivRatAlgHom K ℂ)
this : M.map ⇑ofRealHom = matrixToStdBasis K * ((reindex (indexEquiv K).symm (indexEquiv K).symm) N)ᵀ
⊢ ↑‖((mixedEmbedding.stdBasis K).toMatrix ⇑((latticeBasis K).reindex e.symm)).det‖₊ =
    ↑‖(matrixToStdBasis K).det * N.det‖₊

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.NumberField.InfinitePlace.Ramification

Goal state before:
pp: k : Type u_1
inst✝³ : Field k
K : Type u_2
inst✝² : Field K
inst✝¹ : Algebra k K
inst✝ : IsGalois k K
φ : K →+* ℂ
h : IsRamified k (mk φ)
⊢ ∃ σ, ComplexEmbedding.IsConj φ σ

Tactic:
by

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

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
B : ℝ
hB : 0 < B
this : ∫ (x : mixedSpace K), rexp (-convexBodySumFun x) = 2 ^ nrRealPlaces K * (π / 2) ^ nrComplexPlaces K
⊢ ENNReal.ofReal (∫ (x : mixedSpace K), rexp (-convexBodySumFun x)) / ↑(finrank ℚ K).factorial =
    ↑(convexBodySumFactor K)

Tactic:
by

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

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
f : Free.ChooseBasisIndex ℤ (𝓞 K) ≃ (K →+* ℂ) :=
  (canonicalEmbedding.latticeBasis K).indexEquiv (Pi.basisFun ℂ (K →+* ℂ))
e : index K ≃ Free.ChooseBasisIndex ℤ (𝓞 K) := (indexEquiv K).trans f.symm
M : Matrix (index K) (index K) ℝ := (mixedEmbedding.stdBasis K).toMatrix ⇑((latticeBasis K).reindex e.symm)
N : Matrix (K →+* ℂ) (K →+* ℂ) ℂ :=
  Algebra.embeddingsMatrixReindex ℚ ℂ (⇑(integralBasis K) ∘ ⇑f.symm) (RingHom.equivRatAlgHom K ℂ)
this : M.map ⇑ofRealHom = matrixToStdBasis K * ((reindex (indexEquiv K).symm (indexEquiv K).symm) N)ᵀ
⊢ ‖I‖₊ = 1

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.NumberField.InfinitePlace.Ramification

Goal state before:
pp: k : Type u_1
inst✝³ : Field k
K : Type u_2
inst✝² : Field K
inst✝¹ : Algebra k K
w : InfinitePlace K
inst✝ : IsGalois k K
⊢ Nat.card ↥(Stab w) = if IsUnramified k w then 1 else 2

Tactic:
by

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

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
⊢ volume (fundamentalDomain (stdBasis K)) = 1

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.NumberField.InfinitePlace.Ramification

Goal state before:
pp: k : Type u_1
inst✝³ : Field k
K : Type u_2
inst✝² : Field K
inst✝¹ : Algebra k K
w : InfinitePlace K
inst✝ : IsGalois k K
hw : ¬IsUnramified k w
⊢ Even (Nat.card Gal(K/k))

Tactic:
by

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

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
φ : K →+* ℂ
hφ : ComplexEmbedding.IsReal φ
⊢ (fun c ↦
        Sum.casesOn (motive := fun t ↦ c = t → K →+* ℂ) c (fun w h ↦ (↑w).embedding)
          (fun wj h ↦
            Prod.casesOn wj fun w j ↦ if j = 0 then (↑w).embedding else ComplexEmbedding.conjugate (↑w).embedding)
          ⋯)
      (Sum.inl (mkReal ⟨φ, hφ⟩)) =
    φ

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.NumberField.InfinitePlace.Ramification

Goal state before:
pp: k : Type u_1
inst✝³ : Field k
K : Type u_2
inst✝² : Field K
inst✝¹ : Algebra k K
w : InfinitePlace K
inst✝ : IsGalois k K
hw : ¬IsUnramified k w
⊢ Even (finrank k K)

Tactic:
by

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

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
φ : K →+* ℂ
hφ : ¬ComplexEmbedding.IsReal φ
hw : (InfinitePlace.mk φ).embedding = φ
⊢ (fun c ↦
        Sum.casesOn (motive := fun t ↦ c = t → K →+* ℂ) c (fun w h ↦ (↑w).embedding)
          (fun wj h ↦
            Prod.casesOn wj fun w j ↦ if j = 0 then (↑w).embedding else ComplexEmbedding.conjugate (↑w).embedding)
          ⋯)
      (Sum.inr (mkComplex ⟨φ, hφ⟩, 0)) =
    φ

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.NumberField.InfinitePlace.Ramification

Goal state before:
pp: k : Type u_1
inst✝² : Field k
K : Type u_2
inst✝¹ : Field K
inst✝ : Algebra k K
σ : Gal(K/k)
w : InfinitePlace K
⊢ IsUnramified k (σ • w) ↔ IsUnramified k w

Tactic:
by

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

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
φ : K →+* ℂ
hφ : ¬ComplexEmbedding.IsReal φ
hw : ¬(InfinitePlace.mk φ).embedding = φ
⊢ (fun c ↦
        Sum.casesOn (motive := fun t ↦ c = t → K →+* ℂ) c (fun w h ↦ (↑w).embedding)
          (fun wj h ↦
            Prod.casesOn wj fun w j ↦ if j = 0 then (↑w).embedding else ComplexEmbedding.conjugate (↑w).embedding)
          ⋯)
      (Sum.inr (mkComplex ⟨φ, hφ⟩, 1)) =
    φ

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.NumberField.InfinitePlace.Ramification

Goal state before:
pp: k : Type u_1
inst✝³ : Field k
K : Type u_2
inst✝² : Field K
inst✝¹ : Algebra k K
inst✝ : IsGalois k K
w : InfinitePlace K
⊢ IsUnramifiedIn K (w.comap (algebraMap k K)) ↔ IsUnramified k w

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.NumberField.InfinitePlace.Ramification

Goal state before:
pp: k : Type u_1
inst✝³ : Field k
K : Type u_2
inst✝² : Field K
inst✝¹ : Algebra k K
inst✝ : IsGalois k K
w : InfinitePlace k
hw : ¬IsUnramifiedIn K w
⊢ Even (Nat.card Gal(K/k))

Tactic:
by

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

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
⊢ index K ≃ (K →+* ℂ)

Tactic:
by

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

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
B : ℝ
hB : 0 < B
⊢ ∫ (x : mixedSpace K), rexp (-convexBodySumFun x) =
    (∫ (x : { w // w.IsReal } → ℝ), ∏ w, rexp (-‖x w‖)) * ∫ (x : { w // w.IsComplex } → ℂ), ∏ w, rexp (-2 * ‖x w‖)

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.NumberField.InfinitePlace.Ramification

Goal state before:
pp: k : Type u_1
inst✝³ : Field k
K : Type u_2
inst✝² : Field K
inst✝¹ : Algebra k K
inst✝ : IsGalois k K
w : InfinitePlace k
hw : ¬IsUnramifiedIn K w
⊢ Even (finrank k K)

Tactic:
by

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

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
⊢ (matrixToStdBasis K).det = ∏ _k, (2⁻¹ • !![1, 1; -I, I]).det

Tactic:
by

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

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
B : ℝ
hB : 0 < B
⊢ (∫ (x : { w // w.IsReal } → ℝ), ∏ w, rexp (-‖x w‖)) * ∫ (x : { w // w.IsComplex } → ℂ), ∏ w, rexp (-2 * ‖x w‖) =
    (∫ (x : ℝ), rexp (-|x|)) ^ nrRealPlaces K * (∫ (x : ℂ), rexp (-2 * ‖x‖)) ^ nrComplexPlaces K

Tactic:
by

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

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
⊢ ∏ _k, (2⁻¹ • !![1, 1; -I, I]).det = ∏ _k, 2⁻¹ * I

Tactic:
by

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

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
⊢ ∏ _k, 2⁻¹ * I = (2⁻¹ * I) ^ Fintype.card { w // w.IsComplex }

Tactic:
by

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

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
B : ℝ
hB : 0 < B
⊢ (∫ (x : ℝ), rexp (-|x|)) ^ nrRealPlaces K * (∫ (x : ℂ), rexp (-2 * ‖x‖)) ^ nrComplexPlaces K =
    (2 * Gamma (1 / 1 + 1)) ^ nrRealPlaces K * (π * 2 ^ (-2 / 1) * Gamma (2 / 1 + 1)) ^ nrComplexPlaces K

Tactic:
by

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

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
f : Free.ChooseBasisIndex ℤ (𝓞 K) ≃ (K →+* ℂ) :=
  (canonicalEmbedding.latticeBasis K).indexEquiv (Pi.basisFun ℂ (K →+* ℂ))
e : index K ≃ Free.ChooseBasisIndex ℤ (𝓞 K) := (indexEquiv K).trans f.symm
M : Matrix (index K) (index K) ℝ := (mixedEmbedding.stdBasis K).toMatrix ⇑((latticeBasis K).reindex e.symm)
N : Matrix (K →+* ℂ) (K →+* ℂ) ℂ :=
  Algebra.embeddingsMatrixReindex ℚ ℂ (⇑(integralBasis K) ∘ ⇑f.symm) (RingHom.equivRatAlgHom K ℂ)
this✝ : M.map ⇑ofRealHom = matrixToStdBasis K * ((reindex (indexEquiv K).symm (indexEquiv K).symm) N)ᵀ
this : ‖I‖₊ = 1
⊢ ↑(Real.toNNReal 2)⁻¹ ^ nrComplexPlaces K * ↑‖N.det‖₊ = 2⁻¹ ^ Fintype.card { w // w.IsComplex } * ↑(sqrt ‖N.det ^ 2‖₊)

Tactic:
Real.toNNReal_ofNat,

Goal state after:
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
f : Free.ChooseBasisIndex ℤ (𝓞 K) ≃ (K →+* ℂ) :=
  (canonicalEmbedding.latticeBasis K).indexEquiv (Pi.basisFun ℂ (K →+* ℂ))
e : index K ≃ Free.ChooseBasisIndex ℤ (𝓞 K) := (indexEquiv K).trans f.symm
M : Matrix (index K) (index K) ℝ := (mixedEmbedding.stdBasis K).toMatrix ⇑((latticeBasis K).reindex e.symm)
N : Matrix (K →+* ℂ) (K →+* ℂ) ℂ :=
  Algebra.embeddingsMatrixReindex ℚ ℂ (⇑(integralBasis K) ∘ ⇑f.symm) (RingHom.equivRatAlgHom K ℂ)
this✝ : M.map ⇑ofRealHom = matrixToStdBasis K * ((reindex (indexEquiv K).symm (indexEquiv K).symm) N)ᵀ
this : ‖I‖₊ = 1
⊢ ↑2⁻¹ ^ nrComplexPlaces K * ↑‖N.det‖₊ = 2⁻¹ ^ Fintype.card { w // w.IsComplex } * ↑(sqrt ‖N.det ^ 2‖₊)
Module: Mathlib.NumberTheory.NumberField.CanonicalEmbedding.ConvexBody

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
B : ℝ
hB : 0 < B
⊢ (2 * Gamma (1 / 1 + 1)) ^ nrRealPlaces K * (π * 2 ^ (-2 / 1) * Gamma (2 / 1 + 1)) ^ nrComplexPlaces K =
    2 ^ nrRealPlaces K * (π / 2) ^ nrComplexPlaces K

Tactic:
by

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

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
B : ℝ
⊢ volume (convexBodySum K B) = ↑(convexBodySumFactor K) * ENNReal.ofReal B ^ finrank ℚ K

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.NumberField.InfinitePlace.Ramification

Goal state before:
pp: k : Type u_1
inst✝⁵ : Field k
K : Type u_2
inst✝⁴ : Field K
inst✝³ : Algebra k K
inst✝² : NumberField K
inst✝¹ : NumberField k
inst✝ : IsGalois k K
⊢ #{w | IsUnramified k w} = #{w | IsUnramifiedIn K w} * finrank k K

Tactic:
by

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

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
I : (FractionalIdeal (𝓞 K)⁰ K)ˣ
⊢ Free.ChooseBasisIndex ℤ ↥↑↑I ≃ Free.ChooseBasisIndex ℤ (𝓞 K)

Tactic:
by

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

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
f : Free.ChooseBasisIndex ℤ (𝓞 K) ≃ (K →+* ℂ) :=
  (canonicalEmbedding.latticeBasis K).indexEquiv (Pi.basisFun ℂ (K →+* ℂ))
e : index K ≃ Free.ChooseBasisIndex ℤ (𝓞 K) := (indexEquiv K).trans f.symm
M : Matrix (index K) (index K) ℝ := (mixedEmbedding.stdBasis K).toMatrix ⇑((latticeBasis K).reindex e.symm)
N : Matrix (K →+* ℂ) (K →+* ℂ) ℂ :=
  Algebra.embeddingsMatrixReindex ℚ ℂ (⇑(integralBasis K) ∘ ⇑f.symm) (RingHom.equivRatAlgHom K ℂ)
this : M.map ⇑ofRealHom = matrixToStdBasis K * ((reindex (indexEquiv K).symm (indexEquiv K).symm) N)ᵀ
⊢ ↑‖(matrixToStdBasis K).det * N.det‖₊ = 2⁻¹ ^ Fintype.card { w // w.IsComplex } * ↑(sqrt ‖N.det ^ 2‖₊)

Tactic:
by

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

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
I : (FractionalIdeal (𝓞 K)⁰ K)ˣ
⊢ volume (fundamentalDomain (fractionalIdealLatticeBasis K I)) =
    ENNReal.ofReal ↑(FractionalIdeal.absNorm ↑I) * volume (fundamentalDomain (latticeBasis K))

Tactic:
by

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

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
I : (FractionalIdeal (𝓞 K)⁰ K)ˣ
⊢ minkowskiBound K I < ∞

Tactic:
by

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

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
f : Free.ChooseBasisIndex ℤ (𝓞 K) ≃ (K →+* ℂ) :=
  (canonicalEmbedding.latticeBasis K).indexEquiv (Pi.basisFun ℂ (K →+* ℂ))
e : index K ≃ Free.ChooseBasisIndex ℤ (𝓞 K) := (indexEquiv K).trans f.symm
M : Matrix (index K) (index K) ℝ := (mixedEmbedding.stdBasis K).toMatrix ⇑((latticeBasis K).reindex e.symm)
N : Matrix (K →+* ℂ) (K →+* ℂ) ℂ :=
  Algebra.embeddingsMatrixReindex ℚ ℂ (⇑(integralBasis K) ∘ ⇑f.symm) (RingHom.equivRatAlgHom K ℂ)
this : M.map ⇑ofRealHom = matrixToStdBasis K * ((reindex (indexEquiv K).symm (indexEquiv K).symm) N)ᵀ
⊢ 2⁻¹ ^ Fintype.card { w // w.IsComplex } * ↑(sqrt ‖N.det ^ 2‖₊) =
    2⁻¹ ^ Fintype.card { w // w.IsComplex } * ↑(sqrt ‖discr K‖₊)

Tactic:
by

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

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
I : (FractionalIdeal (𝓞 K)⁰ K)ˣ
⊢ volume (fundamentalDomain (fractionalIdealLatticeBasis K I)) ≠ 0

Tactic:
by

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

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
f : Free.ChooseBasisIndex ℤ (𝓞 K) ≃ (K →+* ℂ) :=
  (canonicalEmbedding.latticeBasis K).indexEquiv (Pi.basisFun ℂ (K →+* ℂ))
e : index K ≃ Free.ChooseBasisIndex ℤ (𝓞 K) := (indexEquiv K).trans f.symm
M : Matrix (index K) (index K) ℝ := (mixedEmbedding.stdBasis K).toMatrix ⇑((latticeBasis K).reindex e.symm)
N : Matrix (K →+* ℂ) (K →+* ℂ) ℂ :=
  Algebra.embeddingsMatrixReindex ℚ ℂ (⇑(integralBasis K) ∘ ⇑f.symm) (RingHom.equivRatAlgHom K ℂ)
this : M.map ⇑ofRealHom = matrixToStdBasis K * ((reindex (indexEquiv K).symm (indexEquiv K).symm) N)ᵀ
⊢ volume (fundamentalDomain (latticeBasis K)) = 2⁻¹ ^ nrComplexPlaces K * ↑(sqrt ‖discr K‖₊)

Tactic:
by

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

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
f : InfinitePlace K → ℝ≥0
I : (FractionalIdeal (𝓞 K)⁰ K)ˣ
h : minkowskiBound K I < volume (convexBodyLT K f)
h_fund :
  IsAddFundamentalDomain (↥(span ℤ (Set.range ⇑(fractionalIdealLatticeBasis K I))).toAddSubgroup)
    (fundamentalDomain (fractionalIdealLatticeBasis K I)) volume
⊢ Countable ↥(span ℤ (Set.range ⇑(fractionalIdealLatticeBasis K I))).toAddSubgroup

Tactic:
by

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

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
f : Free.ChooseBasisIndex ℤ (𝓞 K) ≃ (K →+* ℂ) :=
  (canonicalEmbedding.latticeBasis K).indexEquiv (Pi.basisFun ℂ (K →+* ℂ))
e : index K ≃ Free.ChooseBasisIndex ℤ (𝓞 K) := (indexEquiv K).trans f.symm
M : Matrix (index K) (index K) ℝ := (mixedEmbedding.stdBasis K).toMatrix ⇑((latticeBasis K).reindex e.symm)
N : Matrix (K →+* ℂ) (K →+* ℂ) ℂ :=
  Algebra.embeddingsMatrixReindex ℚ ℂ (⇑(integralBasis K) ∘ ⇑f.symm) (RingHom.equivRatAlgHom K ℂ)
i✝ j✝ : index K
⊢ ofRealHom (((mixedEmbedding.stdBasis K).repr (((latticeBasis K).reindex e.symm) j✝)) i✝) =
    (matrixToStdBasis K * ((reindex (indexEquiv K).symm (indexEquiv K).symm) N)ᵀ) i✝ j✝

Tactic:
Basis.coe_reindex,

Goal state after:
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
f : Free.ChooseBasisIndex ℤ (𝓞 K) ≃ (K →+* ℂ) :=
  (canonicalEmbedding.latticeBasis K).indexEquiv (Pi.basisFun ℂ (K →+* ℂ))
e : index K ≃ Free.ChooseBasisIndex ℤ (𝓞 K) := (indexEquiv K).trans f.symm
M : Matrix (index K) (index K) ℝ := (mixedEmbedding.stdBasis K).toMatrix ⇑((latticeBasis K).reindex e.symm)
N : Matrix (K →+* ℂ) (K →+* ℂ) ℂ :=
  Algebra.embeddingsMatrixReindex ℚ ℂ (⇑(integralBasis K) ∘ ⇑f.symm) (RingHom.equivRatAlgHom K ℂ)
i✝ j✝ : index K
⊢ ofRealHom (((mixedEmbedding.stdBasis K).repr ((⇑(latticeBasis K) ∘ ⇑e.symm.symm) j✝)) i✝) =
    (matrixToStdBasis K * ((reindex (indexEquiv K).symm (indexEquiv K).symm) N)ᵀ) i✝ j✝