Module: Mathlib.NumberTheory.NumberField.Ideal.Asymptotics

Goal state before:
pp: case e'_3
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
C : ClassGroup (𝓞 K)
h₁ :
  ∀ (s : ℝ),
    {x | x ∈ ⇑(toMixed K) ⁻¹' fundamentalCone K ∧ mixedEmbedding.norm ((toMixed K) x) ≤ s} =
      ⇑(toMixed K) ⁻¹' {x | x ∈ fundamentalCone K ∧ mixedEmbedding.norm x ≤ s}
h₂ : {x | x ∈ fundamentalCone K ∧ mixedEmbedding.norm x ≤ 1} = normLeOne K
J : ↥(Ideal (𝓞 K))⁰
hJ : ClassGroup.mk0 J = C⁻¹
h₃ : ↑(absNorm ↑J) ≠ 0
s : ℝ
⊢ ↑(Nat.card { I // ↑(absNorm ↑I) ≤ s ∧ ClassGroup.mk0 I = C }) / s =
    ((fun x ↦
          ↑(Nat.card
                  ↑({x_1 | x_1 ∈ ⇑(toMixed K) ⁻¹' fundamentalCone K ∧ mixedEmbedding.norm ((toMixed K) x_1) ≤ x} ∩
                      ↑(ZLattice.comap ℝ (idealLattice K ((FractionalIdeal.mk0 K) J)) ↑↑(toMixed K)))) /
              x *
            (↑(absNorm ↑J) * (↑(torsionOrder K))⁻¹)) ∘
        fun x ↦ id x * ↑(absNorm ↑J))
      s

pp: case e'_5
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
C : ClassGroup (𝓞 K)
h₁ :
  ∀ (s : ℝ),
    {x | x ∈ ⇑(toMixed K) ⁻¹' fundamentalCone K ∧ mixedEmbedding.norm ((toMixed K) x) ≤ s} =
      ⇑(toMixed K) ⁻¹' {x | x ∈ fundamentalCone K ∧ mixedEmbedding.norm x ≤ s}
h₂ : {x | x ∈ fundamentalCone K ∧ mixedEmbedding.norm x ≤ 1} = normLeOne K
J : ↥(Ideal (𝓞 K))⁰
hJ : ClassGroup.mk0 J = C⁻¹
h₃ : ↑(absNorm ↑J) ≠ 0
⊢ 2 ^ nrRealPlaces K * (2 * π) ^ nrComplexPlaces K * regulator K / (↑(torsionOrder K) * √|↑(discr K)|) =
    MeasureTheory.volume.real {x | x ∈ ⇑(toMixed K) ⁻¹' fundamentalCone K ∧ mixedEmbedding.norm ((toMixed K) x) ≤ 1} /
        ZLattice.covolume (ZLattice.comap ℝ (idealLattice K ((FractionalIdeal.mk0 K) J)) ↑↑(toMixed K))
          MeasureTheory.volume *
      (↑(absNorm ↑J) * (↑(torsionOrder K))⁻¹)

pp: case convert_1
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
C : ClassGroup (𝓞 K)
h₁ :
  ∀ (s : ℝ),
    {x | x ∈ ⇑(toMixed K) ⁻¹' fundamentalCone K ∧ mixedEmbedding.norm ((toMixed K) x) ≤ s} =
      ⇑(toMixed K) ⁻¹' {x | x ∈ fundamentalCone K ∧ mixedEmbedding.norm x ≤ s}
h₂ : {x | x ∈ fundamentalCone K ∧ mixedEmbedding.norm x ≤ 1} = normLeOne K
J : ↥(Ideal (𝓞 K))⁰
hJ : ClassGroup.mk0 J = C⁻¹
h₃ : ↑(absNorm ↑J) ≠ 0
x✝² : euclidean.mixedSpace K
x✝¹ : ℝ
x✝ : x✝² ∈ ⇑(toMixed K) ⁻¹' fundamentalCone K
h : 0 < x✝¹
⊢ x✝¹ • x✝² ∈ ⇑(toMixed K) ⁻¹' fundamentalCone K

pp: case convert_2
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
C : ClassGroup (𝓞 K)
h₁ :
  ∀ (s : ℝ),
    {x | x ∈ ⇑(toMixed K) ⁻¹' fundamentalCone K ∧ mixedEmbedding.norm ((toMixed K) x) ≤ s} =
      ⇑(toMixed K) ⁻¹' {x | x ∈ fundamentalCone K ∧ mixedEmbedding.norm x ≤ s}
h₂ : {x | x ∈ fundamentalCone K ∧ mixedEmbedding.norm x ≤ 1} = normLeOne K
J : ↥(Ideal (𝓞 K))⁰
hJ : ClassGroup.mk0 J = C⁻¹
h₃ : ↑(absNorm ↑J) ≠ 0
x✝¹ : euclidean.mixedSpace K
x✝ : ℝ
h : 0 ≤ x✝
⊢ mixedEmbedding.norm ((toMixed K) (x✝ • x✝¹)) =
    x✝ ^ Module.finrank ℝ (euclidean.mixedSpace K) * mixedEmbedding.norm ((toMixed K) x✝¹)

pp: case convert_3
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
C : ClassGroup (𝓞 K)
h₁ :
  ∀ (s : ℝ),
    {x | x ∈ ⇑(toMixed K) ⁻¹' fundamentalCone K ∧ mixedEmbedding.norm ((toMixed K) x) ≤ s} =
      ⇑(toMixed K) ⁻¹' {x | x ∈ fundamentalCone K ∧ mixedEmbedding.norm x ≤ s}
h₂ : {x | x ∈ fundamentalCone K ∧ mixedEmbedding.norm x ≤ 1} = normLeOne K
J : ↥(Ideal (𝓞 K))⁰
hJ : ClassGroup.mk0 J = C⁻¹
h₃ : ↑(absNorm ↑J) ≠ 0
⊢ MeasurableSet {x | x ∈ ⇑(toMixed K) ⁻¹' fundamentalCone K ∧ mixedEmbedding.norm ((toMixed K) x) ≤ 1}

pp: case convert_4
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
C : ClassGroup (𝓞 K)
h₁ :
  ∀ (s : ℝ),
    {x | x ∈ ⇑(toMixed K) ⁻¹' fundamentalCone K ∧ mixedEmbedding.norm ((toMixed K) x) ≤ s} =
      ⇑(toMixed K) ⁻¹' {x | x ∈ fundamentalCone K ∧ mixedEmbedding.norm x ≤ s}
h₂ : {x | x ∈ fundamentalCone K ∧ mixedEmbedding.norm x ≤ 1} = normLeOne K
J : ↥(Ideal (𝓞 K))⁰
hJ : ClassGroup.mk0 J = C⁻¹
h₃ : ↑(absNorm ↑J) ≠ 0
⊢ MeasureTheory.volume
      (frontier {x | x ∈ ⇑(toMixed K) ⁻¹' fundamentalCone K ∧ mixedEmbedding.norm ((toMixed K) x) ≤ 1}) =
    0

Tactic:
· simp_rw [Ideal.tendsto_norm_le_and_mk_eq_div_atTop_aux₁ K hJ, id_eq,
      Nat.card_congr (Ideal.tendsto_norm_le_and_mk_eq_div_atTop_aux₂ K),
      ← card_isPrincipal_dvd_norm_le, Function.comp_def, Nat.cast_mul, div_eq_mul_inv, mul_inv,
      ← mul_assoc, mul_comm _ (torsionOrder K : ℝ)⁻¹, mul_comm _ (torsionOrder K : ℝ), mul_assoc]
    rw [inv_mul_cancel_left₀ (Nat.cast_ne_zero.mpr (torsionOrder_ne_zero K)), inv_mul_cancel₀ h₃,
      mul_one]

Goal state after:
case e'_5
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
C : ClassGroup (𝓞 K)
h₁ :
  ∀ (s : ℝ),
    {x | x ∈ ⇑(toMixed K) ⁻¹' fundamentalCone K ∧ mixedEmbedding.norm ((toMixed K) x) ≤ s} =
      ⇑(toMixed K) ⁻¹' {x | x ∈ fundamentalCone K ∧ mixedEmbedding.norm x ≤ s}
h₂ : {x | x ∈ fundamentalCone K ∧ mixedEmbedding.norm x ≤ 1} = normLeOne K
J : ↥(Ideal (𝓞 K))⁰
hJ : ClassGroup.mk0 J = C⁻¹
h₃ : ↑(absNorm ↑J) ≠ 0
⊢ 2 ^ nrRealPlaces K * (2 * π) ^ nrComplexPlaces K * regulator K / (↑(torsionOrder K) * √|↑(discr K)|) =
    MeasureTheory.volume.real {x | x ∈ ⇑(toMixed K) ⁻¹' fundamentalCone K ∧ mixedEmbedding.norm ((toMixed K) x) ≤ 1} /
        ZLattice.covolume (ZLattice.comap ℝ (idealLattice K ((FractionalIdeal.mk0 K) J)) ↑↑(toMixed K))
          MeasureTheory.volume *
      (↑(absNorm ↑J) * (↑(torsionOrder K))⁻¹)

case convert_1
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
C : ClassGroup (𝓞 K)
h₁ :
  ∀ (s : ℝ),
    {x | x ∈ ⇑(toMixed K) ⁻¹' fundamentalCone K ∧ mixedEmbedding.norm ((toMixed K) x) ≤ s} =
      ⇑(toMixed K) ⁻¹' {x | x ∈ fundamentalCone K ∧ mixedEmbedding.norm x ≤ s}
h₂ : {x | x ∈ fundamentalCone K ∧ mixedEmbedding.norm x ≤ 1} = normLeOne K
J : ↥(Ideal (𝓞 K))⁰
hJ : ClassGroup.mk0 J = C⁻¹
h₃ : ↑(absNorm ↑J) ≠ 0
x✝² : euclidean.mixedSpace K
x✝¹ : ℝ
x✝ : x✝² ∈ ⇑(toMixed K) ⁻¹' fundamentalCone K
h : 0 < x✝¹
⊢ x✝¹ • x✝² ∈ ⇑(toMixed K) ⁻¹' fundamentalCone K

case convert_2
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
C : ClassGroup (𝓞 K)
h₁ :
  ∀ (s : ℝ),
    {x | x ∈ ⇑(toMixed K) ⁻¹' fundamentalCone K ∧ mixedEmbedding.norm ((toMixed K) x) ≤ s} =
      ⇑(toMixed K) ⁻¹' {x | x ∈ fundamentalCone K ∧ mixedEmbedding.norm x ≤ s}
h₂ : {x | x ∈ fundamentalCone K ∧ mixedEmbedding.norm x ≤ 1} = normLeOne K
J : ↥(Ideal (𝓞 K))⁰
hJ : ClassGroup.mk0 J = C⁻¹
h₃ : ↑(absNorm ↑J) ≠ 0
x✝¹ : euclidean.mixedSpace K
x✝ : ℝ
h : 0 ≤ x✝
⊢ mixedEmbedding.norm ((toMixed K) (x✝ • x✝¹)) =
    x✝ ^ Module.finrank ℝ (euclidean.mixedSpace K) * mixedEmbedding.norm ((toMixed K) x✝¹)

case convert_3
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
C : ClassGroup (𝓞 K)
h₁ :
  ∀ (s : ℝ),
    {x | x ∈ ⇑(toMixed K) ⁻¹' fundamentalCone K ∧ mixedEmbedding.norm ((toMixed K) x) ≤ s} =
      ⇑(toMixed K) ⁻¹' {x | x ∈ fundamentalCone K ∧ mixedEmbedding.norm x ≤ s}
h₂ : {x | x ∈ fundamentalCone K ∧ mixedEmbedding.norm x ≤ 1} = normLeOne K
J : ↥(Ideal (𝓞 K))⁰
hJ : ClassGroup.mk0 J = C⁻¹
h₃ : ↑(absNorm ↑J) ≠ 0
⊢ MeasurableSet {x | x ∈ ⇑(toMixed K) ⁻¹' fundamentalCone K ∧ mixedEmbedding.norm ((toMixed K) x) ≤ 1}

case convert_4
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
C : ClassGroup (𝓞 K)
h₁ :
  ∀ (s : ℝ),
    {x | x ∈ ⇑(toMixed K) ⁻¹' fundamentalCone K ∧ mixedEmbedding.norm ((toMixed K) x) ≤ s} =
      ⇑(toMixed K) ⁻¹' {x | x ∈ fundamentalCone K ∧ mixedEmbedding.norm x ≤ s}
h₂ : {x | x ∈ fundamentalCone K ∧ mixedEmbedding.norm x ≤ 1} = normLeOne K
J : ↥(Ideal (𝓞 K))⁰
hJ : ClassGroup.mk0 J = C⁻¹
h₃ : ↑(absNorm ↑J) ≠ 0
⊢ MeasureTheory.volume
      (frontier {x | x ∈ ⇑(toMixed K) ⁻¹' fundamentalCone K ∧ mixedEmbedding.norm ((toMixed K) x) ≤ 1}) =
    0