-- 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
s : ℝ
h :
  (↑(Nat.card.{u_1}
            { I : ↥(Ideal (𝓞 K))⁰ //
              (↑J : Ideal (𝓞 K)) ∣ (↑I : Ideal (𝓞 K)) ∧
                IsPrincipal (↑I : Ideal (𝓞 K)) ∧
                  (↑((absNorm : (a : Ideal (𝓞 K)) → ℕ) (↑I : Ideal (𝓞 K))) : ℝ) ≤
                    s * (↑((absNorm : (a : Ideal (𝓞 K)) → ℕ) (↑J : Ideal (𝓞 K))) : ℝ) }) :
        ℝ) *
      s⁻¹ =
    (↑(torsionOrder K) : ℝ)⁻¹ *
      ((↑(torsionOrder K) : ℝ) *
        ((↑(Nat.card.{u_1}
                { I : ↥(Ideal (𝓞 K))⁰ //
                  (↑J : Ideal (𝓞 K)) ∣ (↑I : Ideal (𝓞 K)) ∧
                    IsPrincipal (↑I : Ideal (𝓞 K)) ∧
                      (↑((absNorm : (a : Ideal (𝓞 K)) → ℕ) (↑I : Ideal (𝓞 K))) : ℝ) ≤
                        s * (↑((absNorm : (a : Ideal (𝓞 K)) → ℕ) (↑J : Ideal (𝓞 K))) : ℝ) }) :
            ℝ) *
          (s⁻¹ *
            HMul.hMul (α := ℝ) (↑((absNorm : (a : Ideal (𝓞 K)) → ℕ) (↑J : Ideal (𝓞 K))) : ℝ)⁻¹
              (↑((absNorm : (a : Ideal (𝓞 K)) → ℕ) (↑J : Ideal (𝓞 K))) : ℝ))))
⊢ (↑(Nat.card.{u_1}
            { I : ↥(Ideal (𝓞 K))⁰ //
              (↑((absNorm : (a : Ideal (𝓞 K)) → ℕ) (↑I : Ideal (𝓞 K))) : ℝ) ≤ s ∧
                (ClassGroup.mk0 : (a : ↥(Ideal (𝓞 K))⁰) → ClassGroup (𝓞 K)) I = C }) :
        ℝ) /
      s =
    ((fun (x : ℝ) ↦
          (↑(Nat.card.{u_1}
                    (↑({x_1 : 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_1 ∧
                              (mixedEmbedding.norm : (a : mixedSpace K) → ℝ)
                                  ((toMixed K : (a : euclidean.mixedSpace K) → mixedSpace K) x_1 : mixedSpace K) ≤
                                x} ∩
                          (↑(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)) :
                            Set (euclidean.mixedSpace K))) :
                      Type u_1)) :
                ℝ) /
              x *
            HMul.hMul (α := ℝ) (↑((absNorm : (a : Ideal (𝓞 K)) → ℕ) (↑J : Ideal (𝓞 K))) : ℝ)
              (↑(torsionOrder K) : ℝ)⁻¹) ∘
        fun (x : ℝ) ↦ id x * (↑((absNorm : (a : Ideal (𝓞 K)) → ℕ) (↑J : Ideal (𝓞 K))) : ℝ))
      s

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]

Premises used: NumberField.mixedEmbedding.fundamentalCone.card_isPrincipal_dvd_norm_le, mul_inv, mul_assoc, NumberField.Units.torsionOrder, _private.0.NumberField.Ideal.tendsto_norm_le_and_mk_eq_div_atTop_aux₂, div_eq_mul_inv, Nat.cast, Nat.card_congr, Function.comp_def, _private.0.NumberField.Ideal.tendsto_norm_le_and_mk_eq_div_atTop_aux₁, Inv.inv, Nat.cast_mul, mul_comm, id_eq