-- Module: Mathlib.NumberTheory.NumberField.Ideal.Asymptotics
Goal:
K : Type u_1
inst : Field K
inst_1 : NumberField K
C : ClassGroup (𝓞 K)
h₁ :
  ∀ (s : ℝ),
    Eq (α := Set (euclidean.mixedSpace K))
      {x : euclidean.mixedSpace K |
        Membership.mem (γ := Set (euclidean.mixedSpace K))
            (Set.preimage (β := mixedSpace K) (⇑(toMixed K) : (a : euclidean.mixedSpace K) → mixedSpace K)
              (fundamentalCone K))
            x ∧
          (mixedEmbedding.norm : (a : mixedSpace K) → ℝ)
              ((toMixed K : (a : euclidean.mixedSpace K) → mixedSpace K) x : mixedSpace K) ≤
            s}
      (Set.preimage (β := mixedSpace K) (⇑(toMixed K) : (a : euclidean.mixedSpace K) → mixedSpace K)
        {x : mixedSpace K |
          Membership.mem (γ := Set (mixedSpace K)) (fundamentalCone K) x ∧
            (mixedEmbedding.norm : (a : mixedSpace K) → ℝ) x ≤ s})
h₂ :
  Eq (α := Set (mixedSpace K))
    {x : mixedSpace K |
      Membership.mem (γ := Set (mixedSpace K)) (fundamentalCone K) x ∧
        LE.le (α := ℝ) ((mixedEmbedding.norm : (a : mixedSpace K) → ℝ) x : ℝ) (1 : ℝ)}
    (normLeOne K)
J : ↥(Ideal (𝓞 K))⁰
hJ : (ClassGroup.mk0 : (a : ↥(Ideal (𝓞 K))⁰) → ClassGroup (𝓞 K)) J = C⁻¹
h₃ : Ne (α := ℝ) (↑((absNorm : (a : Ideal (𝓞 K)) → ℕ) (↑J : Ideal (𝓞 K))) : ℝ) 0
h :
  HPow.hPow (α := ℝ) 2 (nrRealPlaces K) * (2 * π) ^ nrComplexPlaces K * regulator K *
      ((↑(torsionOrder K) : ℝ)⁻¹ * (√|(↑(discr K) : ℝ)|)⁻¹) =
    HPow.hPow (α := ℝ) 2 (nrRealPlaces K) * π ^ nrComplexPlaces K * regulator K *
        ((↑((absNorm : (a : Ideal (𝓞 K)) → ℕ) (↑J : Ideal (𝓞 K))) : ℝ)⁻¹ * HPow.hPow (α := ℝ) 2 (nrComplexPlaces K) *
          (√|(↑(discr K) : ℝ)|)⁻¹) *
      HMul.hMul (α := ℝ) (↑((absNorm : (a : Ideal (𝓞 K)) → ℕ) (↑J : Ideal (𝓞 K))) : ℝ) (↑(torsionOrder K) : ℝ)⁻¹
⊢ HPow.hPow (α := ℝ) 2 (nrRealPlaces K) * (2 * π) ^ nrComplexPlaces K * regulator K /
      ((↑(torsionOrder K) : ℝ) * √|(↑(discr K) : ℝ)|) =
    MeasureTheory.Measure.real.{u_1} (α := euclidean.mixedSpace K) (m := MeasureTheory.MeasureSpace.toMeasurableSpace)
          MeasureTheory.volume
          {x : euclidean.mixedSpace K |
            Membership.mem (γ := Set (euclidean.mixedSpace K))
                (Set.preimage (β := mixedSpace K) (⇑(toMixed K) : (a : euclidean.mixedSpace K) → mixedSpace K)
                  (fundamentalCone K))
                x ∧
              LE.le (α := ℝ)
                ((mixedEmbedding.norm : (a : mixedSpace K) → ℝ)
                    ((toMixed K : (a : euclidean.mixedSpace K) → mixedSpace K) x : mixedSpace K) :
                  ℝ)
                (1 : ℝ)} /
        ZLattice.covolume.{u_1} (E := euclidean.mixedSpace K)
          (ZLattice.comap ℝ (E := mixedSpace K)
            (idealLattice K ((FractionalIdeal.mk0 K : (a : ↥(Ideal (𝓞 K))⁰) → (FractionalIdeal (𝓞 K)⁰ K)ˣ) J))
            (↑(↑(toMixed K) : euclidean.mixedSpace K ≃ₗ[ℝ] mixedSpace K) : euclidean.mixedSpace K →ₗ[ℝ] mixedSpace K))
          (MeasureTheory.volume : MeasureTheory.Measure (euclidean.mixedSpace K)) *
      HMul.hMul (α := ℝ) (↑((absNorm : (a : Ideal (𝓞 K)) → ℕ) (↑J : Ideal (𝓞 K))) : ℝ) (↑(torsionOrder K) : ℝ)⁻¹

Tactic:
rw [h₁, h₂, MeasureTheory.measureReal_def, (volumePreserving_toMixed K).measure_preimage
      (measurableSet_normLeOne K).nullMeasurableSet, volume_normLeOne, ZLattice.covolume_comap
      _ _ _ (volumePreserving_toMixed K), covolume_idealLattice, ENNReal.toReal_mul,
      ENNReal.toReal_mul, ENNReal.toReal_pow, ENNReal.toReal_pow, ENNReal.toReal_ofNat,
      ENNReal.coe_toReal, NNReal.coe_real_pi, ENNReal.toReal_ofReal (regulator_pos K).le,
      FractionalIdeal.coe_mk0, FractionalIdeal.coeIdeal_absNorm, Rat.cast_natCast, div_eq_mul_inv,
      div_eq_mul_inv, mul_inv, mul_inv, mul_inv, inv_pow, inv_inv]

Premises used: ENNReal.coe_toReal, mul_inv, MeasurableSet.nullMeasurableSet, NumberField.mixedEmbedding.euclidean.volumePreserving_toMixed, LT.lt.le, NumberField.mixedEmbedding.covolume_idealLattice, inv_pow, ENNReal.toReal_pow, div_eq_mul_inv, NNReal.coe_real_pi, ENNReal.toReal_ofReal, Rat.cast_natCast, NumberField.Units.regulator_pos, inv_inv, ENNReal.toReal_mul, ENNReal.toReal_ofNat, MeasureTheory.measureReal_def, ZLattice.covolume_comap, NumberField.mixedEmbedding.fundamentalCone.volume_normLeOne, FractionalIdeal.coeIdeal_absNorm, NumberField.mixedEmbedding.fundamentalCone.measurableSet_normLeOne, FractionalIdeal.coe_mk0, MeasureTheory.MeasurePreserving.measure_preimage