awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- 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
x : euclidean.mixedSpace K
x_1 : ℝ
h_1 :
  Membership.mem (γ := Set (euclidean.mixedSpace K))
    (Set.preimage (β := mixedSpace K) (⇑(toMixed K) : (a : euclidean.mixedSpace K) → mixedSpace K) (fundamentalCone K))
    x
h : 0 < x_1
convert_1 :
  Membership.mem (α := mixedSpace K) (γ := Set (mixedSpace K)) (fundamentalCone K)
    ((toMixed K : (a : euclidean.mixedSpace K) → mixedSpace K) (x_1 • x) : mixedSpace K)
convert_1₁ :
  Membership.mem (γ := Set (mixedSpace K)) (fundamentalCone K)
    (HSMul.hSMul (β := mixedSpace K) x_1 ((toMixed K : (a : euclidean.mixedSpace K) → mixedSpace K) x : mixedSpace K))
convert_1₂ convert_1₃ convert_1₄ convert_1₅ :
  Membership.mem (α := mixedSpace K) (γ := Set (mixedSpace K)) (fundamentalCone K)
    ((toMixed K : (a : euclidean.mixedSpace K) → mixedSpace K) x : 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 • x)

Tactic:
rwa [Set.mem_preimage, map_smul, smul_mem_iff_mem h.ne']

Premises used: LT.lt.ne', NumberField.mixedEmbedding.fundamentalCone.smul_mem_iff_mem, Set.mem_preimage, map_smul
-- 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
⊢ MeasurableSet.{u_1} (α := 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 ∧
        LE.le (α := ℝ)
          ((fun (x : euclidean.mixedSpace K) ↦
              ((mixedEmbedding.norm : (a : mixedSpace K) → ℝ)
                  ((toMixed K : (a : euclidean.mixedSpace K) → mixedSpace K) x : mixedSpace K) :
                ℝ))
            x)
          1}

Tactic:
(toMixed K).continuous.measurable (measurableSet_normLeOne K)

Premises used: Set.Mem, Set.diff, setOf, Set.inter, NumberField.mixedEmbedding.fundamentalCone, Set.preimage, ContinuousLinearEquiv.continuous, Continuous.measurable, NumberField.mixedEmbedding.euclidean.toMixed, NumberField.mixedEmbedding.fundamentalCone.measurableSet_normLeOne
-- 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
⊢ MeasurableSet.{u_1} (α := 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 ∧
        LE.le (α := ℝ)
          ((fun (x : euclidean.mixedSpace K) ↦
              ((mixedEmbedding.norm : (a : mixedSpace K) → ℝ)
                  ((toMixed K : (a : euclidean.mixedSpace K) → mixedSpace K) x : mixedSpace K) :
                ℝ))
            x)
          1}

Tactic:
exact (toMixed K).continuous.measurable (measurableSet_normLeOne K)

Premises used: ContinuousLinearEquiv.continuous, Continuous.measurable, NumberField.mixedEmbedding.euclidean.toMixed, NumberField.mixedEmbedding.fundamentalCone.measurableSet_normLeOne
-- Module: Mathlib.NumberTheory.NumberField.Ideal.Asymptotics
Goal:
K : Type u_1
inst : Field K
inst_1 : NumberField K
⊢ Tendsto (β := ℝ)
    (fun (s : ℝ) ↦
      (↑(Nat.card.{u_1} { I : ↥(Ideal (𝓞 K))⁰ // (↑((absNorm : (a : Ideal (𝓞 K)) → ℕ) (↑I : Ideal (𝓞 K))) : ℝ) ≤ s }) :
          ℝ) /
        s)
    atTop
    (𝓝
      (∑ c : ClassGroup (𝓞 K),
        HPow.hPow (α := ℝ) 2 (nrRealPlaces K) * (2 * π) ^ nrComplexPlaces K * regulator K /
          ((↑(torsionOrder K) : ℝ) * √|(↑(discr K) : ℝ)|)))

Tactic:
Filter.Tendsto.congr' ?_
      (tendsto_finsetSum Finset.univ (fun C _ ↦ tendsto_norm_le_and_mk_eq_div_atTop K C))

Premises used: Finset.univ, tendsto_finsetSum, NumberField.Ideal.tendsto_norm_le_and_mk_eq_div_atTop, Filter.Tendsto.congr'
-- Module: Mathlib.NumberTheory.NumberField.Ideal.Asymptotics
Goal:
K : Type u_1
inst : Field K
inst_1 : NumberField K
h :
  HPow.hPow (α := ℝ) 2 (nrRealPlaces K) * (2 * π) ^ nrComplexPlaces K * regulator K * (↑(classNumber K) : ℝ) /
      ((↑(torsionOrder K) : ℝ) * √|(↑(discr K) : ℝ)|) =
    ∑ c : ClassGroup (𝓞 K),
      HPow.hPow (α := ℝ) 2 (nrRealPlaces K) * (2 * π) ^ nrComplexPlaces K * regulator K /
        ((↑(torsionOrder K) : ℝ) * √|(↑(discr K) : ℝ)|)
convert_2 :
  EventuallyEq (β := ℝ) atTop
    (fun (b : ℝ) ↦
      ∑ c : ClassGroup (𝓞 K),
        (↑(Nat.card.{u_1}
                { I : ↥(Ideal (𝓞 K))⁰ //
                  (↑((absNorm : (a : Ideal (𝓞 K)) → ℕ) (↑I : Ideal (𝓞 K))) : ℝ) ≤ b ∧
                    (ClassGroup.mk0 : (a : ↥(Ideal (𝓞 K))⁰) → ClassGroup (𝓞 K)) I = c }) :
            ℝ) /
          b)
    fun (s : ℝ) ↦
    (↑(Nat.card.{u_1} { I : ↥(Ideal (𝓞 K))⁰ // (↑((absNorm : (a : Ideal (𝓞 K)) → ℕ) (↑I : Ideal (𝓞 K))) : ℝ) ≤ s }) :
        ℝ) /
      s
⊢ Tendsto
    (fun (s : ℝ) ↦
      (↑(Nat.card.{u_1} { I : ↥(Ideal (𝓞 K))⁰ // (↑((absNorm : (a : Ideal (𝓞 K)) → ℕ) (↑I : Ideal (𝓞 K))) : ℝ) ≤ s }) :
          ℝ) /
        s)
    atTop
    (𝓝
      (HPow.hPow (α := ℝ) 2 (nrRealPlaces K) * (2 * π) ^ nrComplexPlaces K * regulator K * (↑(classNumber K) : ℝ) /
        ((↑(torsionOrder K) : ℝ) * √|(↑(discr K) : ℝ)|)))

Tactic:
convert!
    Filter.Tendsto.congr' ?_
      (tendsto_finsetSum Finset.univ (fun C _ ↦ tendsto_norm_le_and_mk_eq_div_atTop K C))

Premises used: Finset.univ, tendsto_finsetSum, NumberField.Ideal.tendsto_norm_le_and_mk_eq_div_atTop, Filter.Tendsto.congr'
-- Module: Mathlib.NumberTheory.NumberField.Ideal.Asymptotics
Goal:
K : Type u_1
inst : Field K
inst_1 : NumberField K
h :
  HPow.hPow (α := ℝ) 2 (nrRealPlaces K) * (2 * π) ^ nrComplexPlaces K * regulator K *
        (↑(Fintype.card.{u_1} (ClassGroup (𝓞 K))) : ℝ) /
      ((↑(torsionOrder K) : ℝ) * √|(↑(discr K) : ℝ)|) =
    (↑(Fintype.card.{u_1} (ClassGroup (𝓞 K))) : ℝ) *
      (HPow.hPow (α := ℝ) 2 (nrRealPlaces K) * (2 * π) ^ nrComplexPlaces K * regulator K /
        ((↑(torsionOrder K) : ℝ) * √|(↑(discr K) : ℝ)|))
⊢ HPow.hPow (α := ℝ) 2 (nrRealPlaces K) * (2 * π) ^ nrComplexPlaces K * regulator K * (↑(classNumber K) : ℝ) /
      ((↑(torsionOrder K) : ℝ) * √|(↑(discr K) : ℝ)|) =
    ∑ c : ClassGroup (𝓞 K),
      HPow.hPow (α := ℝ) 2 (nrRealPlaces K) * (2 * π) ^ nrComplexPlaces K * regulator K /
        ((↑(torsionOrder K) : ℝ) * √|(↑(discr K) : ℝ)|)

Tactic:
rw [Finset.sum_const, Finset.card_univ, nsmul_eq_mul, classNumber]

Premises used: Finset.card_univ, nsmul_eq_mul, NumberField.classNumber, Finset.sum_const
-- Module: Mathlib.NumberTheory.NumberField.Ideal.Asymptotics
Goal:
K : Type u_1
inst : Field K
inst_1 : NumberField K
⊢ HPow.hPow (α := ℝ) 2 (nrRealPlaces K) * (2 * π) ^ nrComplexPlaces K * regulator K *
        (↑(Fintype.card.{u_1} (ClassGroup (𝓞 K))) : ℝ) /
      ((↑(torsionOrder K) : ℝ) * √|(↑(discr K) : ℝ)|) =
    (↑(Fintype.card.{u_1} (ClassGroup (𝓞 K))) : ℝ) *
      (HPow.hPow (α := ℝ) 2 (nrRealPlaces K) * (2 * π) ^ nrComplexPlaces K * regulator K /
        ((↑(torsionOrder K) : ℝ) * √|(↑(discr K) : ℝ)|))

Tactic:
ring
-- Module: Mathlib.NumberTheory.NumberField.Ideal.Asymptotics
Goal:
K : Type u_1
inst : Field K
inst_1 : NumberField K
h :
  ∀ (s : ℝ) (hs : 0 ≤ s),
    ∑ c : ClassGroup (𝓞 K),
        (↑(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 =
      (↑(Nat.card.{u_1} { I : ↥(Ideal (𝓞 K))⁰ // (↑((absNorm : (a : Ideal (𝓞 K)) → ℕ) (↑I : Ideal (𝓞 K))) : ℝ) ≤ s }) :
          ℝ) /
        s
⊢ EventuallyEq (β := ℝ) atTop
    (fun (b : ℝ) ↦
      ∑ c : ClassGroup (𝓞 K),
        (↑(Nat.card.{u_1}
                { I : ↥(Ideal (𝓞 K))⁰ //
                  (↑((absNorm : (a : Ideal (𝓞 K)) → ℕ) (↑I : Ideal (𝓞 K))) : ℝ) ≤ b ∧
                    (ClassGroup.mk0 : (a : ↥(Ideal (𝓞 K))⁰) → ClassGroup (𝓞 K)) I = c }) :
            ℝ) /
          b)
    fun (s : ℝ) ↦
    (↑(Nat.card.{u_1} { I : ↥(Ideal (𝓞 K))⁰ // (↑((absNorm : (a : Ideal (𝓞 K)) → ℕ) (↑I : Ideal (𝓞 K))) : ℝ) ≤ s }) :
        ℝ) /
      s

Tactic:
filter_upwards [eventually_ge_atTop 0] with s hs

Premises used: Filter.eventually_ge_atTop, OfNat.ofNat
-- Module: Mathlib.NumberTheory.NumberField.Ideal.Asymptotics
Goal:
K : Type u_1
inst : Field K
inst_1 : NumberField K
s : ℝ
hs : 0 ≤ s
simp_rw : Fintype { I : ↥(Ideal (𝓞 K))⁰ // (absNorm : (a : Ideal (𝓞 K)) → ℕ) (↑I : Ideal (𝓞 K)) ≤ ⌊s⌋₊ }
⊢ Fintype { I : ↥(Ideal (𝓞 K))⁰ // (↑((absNorm : (a : Ideal (𝓞 K)) → ℕ) (↑I : Ideal (𝓞 K))) : ℝ) ≤ s }

Tactic:
simp_rw [← Nat.le_floor_iff hs]

Premises used: Nat.le_floor_iff
-- Module: Mathlib.NumberTheory.NumberField.Ideal.Asymptotics
Goal:
K : Type u_1
inst : Field K
inst_1 : NumberField K
s : ℝ
hs : 0 ≤ s
⊢ Fintype { I : ↥(Ideal (𝓞 K))⁰ // (absNorm : (a : Ideal (𝓞 K)) → ℕ) (↑I : Ideal (𝓞 K)) ≤ ⌊s⌋₊ }

Tactic:
refine @Fintype.ofFinite _ (finite_setOf_absNorm_le₀ ⌊s⌋₊)

Premises used: Fintype.ofFinite
-- Module: Mathlib.NumberTheory.NumberField.Ideal.Asymptotics
Goal:
K : Type u_1
inst : Field K
inst_1 : NumberField K
s : ℝ
hs : 0 ≤ s
h :
  ∀ (this : Fintype { I : ↥(Ideal (𝓞 K))⁰ // (↑((absNorm : (a : Ideal (𝓞 K)) → ℕ) (↑I : Ideal (𝓞 K))) : ℝ) ≤ s }),
    ∑ c : ClassGroup (𝓞 K),
        (↑(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 =
      (↑(Nat.card.{u_1} { I : ↥(Ideal (𝓞 K))⁰ // (↑((absNorm : (a : Ideal (𝓞 K)) → ℕ) (↑I : Ideal (𝓞 K))) : ℝ) ≤ s }) :
          ℝ) /
        s
⊢ ∑ c : ClassGroup (𝓞 K),
      (↑(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 =
    (↑(Nat.card.{u_1} { I : ↥(Ideal (𝓞 K))⁰ // (↑((absNorm : (a : Ideal (𝓞 K)) → ℕ) (↑I : Ideal (𝓞 K))) : ℝ) ≤ s }) :
        ℝ) /
      s

Tactic:
have : Fintype {I : (Ideal (𝓞 K))⁰ // absNorm (I : Ideal (𝓞 K)) ≤ s} := by
      simp_rw [← Nat.le_floor_iff hs]
      refine @Fintype.ofFinite _ (finite_setOf_absNorm_le₀ ⌊s⌋₊)
-- Module: Mathlib.NumberTheory.NumberField.Ideal.Asymptotics
Goal:
K : Type u_1
inst : Field K
inst_1 : NumberField K
s : ℝ
hs : 0 ≤ s
this : Fintype { I : ↥(Ideal (𝓞 K))⁰ // (↑((absNorm : (a : Ideal (𝓞 K)) → ℕ) (↑I : Ideal (𝓞 K))) : ℝ) ≤ s }
e : (C : ClassGroup (𝓞 K)) →
  { x : { 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)) (↑x : ↥(Ideal (𝓞 K))⁰) = C } ≃
    { x : ↥(Ideal (𝓞 K))⁰ //
      (↑((absNorm : (a : Ideal (𝓞 K)) → ℕ) (↑x : Ideal (𝓞 K))) : ℝ) ≤ s ∧
        (ClassGroup.mk0 : (a : ↥(Ideal (𝓞 K))⁰) → ClassGroup (𝓞 K)) x = C } :=
  fun (C : ClassGroup (𝓞 K)) ↦
    Equiv.subtypeSubtypeEquivSubtypeInter
      (fun (I : ↥(Ideal (𝓞 K))⁰) ↦ (↑((absNorm : (a : Ideal (𝓞 K)) → ℕ) (↑I : Ideal (𝓞 K))) : ℝ) ≤ s)
      fun (I : ↥(Ideal (𝓞 K))⁰) ↦ (ClassGroup.mk0 : (a : ↥(Ideal (𝓞 K))⁰) → ClassGroup (𝓞 K)) I = C
h :
  ∑ x : ClassGroup (𝓞 K),
      (↑(Finset.card.{u_1} (α :=
              { I : ↥(Ideal (𝓞 K))⁰ // (↑((absNorm : (a : Ideal (𝓞 K)) → ℕ) (↑I : Ideal (𝓞 K))) : ℝ) ≤ s })
              {x_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)) (↑x_1 : ↥(Ideal (𝓞 K))⁰) = x}) :
          ℝ) /
        s =
    (↑(Fintype.card.{u_1}
            { I : ↥(Ideal (𝓞 K))⁰ // (↑((absNorm : (a : Ideal (𝓞 K)) → ℕ) (↑I : Ideal (𝓞 K))) : ℝ) ≤ s }) :
        ℝ) /
      s
⊢ ∑ c : ClassGroup (𝓞 K),
      (↑(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 =
    (↑(Nat.card.{u_1} { I : ↥(Ideal (𝓞 K))⁰ // (↑((absNorm : (a : Ideal (𝓞 K)) → ℕ) (↑I : Ideal (𝓞 K))) : ℝ) ≤ s }) :
        ℝ) /
      s

Tactic:
simp_rw [← Nat.card_congr (e _), Nat.card_eq_fintype_card, Fintype.subtype_card]

Premises used: Fintype.subtype_card, Nat.card_congr, Nat.card_eq_fintype_card
-- Module: Mathlib.NumberTheory.NumberField.Ideal.Asymptotics
Goal:
K : Type u_1
inst : Field K
inst_1 : NumberField K
h :
  HPow.hPow (α := ℝ) 2 (nrRealPlaces K) * (2 * π) ^ nrComplexPlaces K * regulator K * (↑(classNumber K) : ℝ) /
      ((↑(torsionOrder K) : ℝ) * √|(↑(discr K) : ℝ)|) =
    ∑ c : ClassGroup (𝓞 K),
      HPow.hPow (α := ℝ) 2 (nrRealPlaces K) * (2 * π) ^ nrComplexPlaces K * regulator K /
        ((↑(torsionOrder K) : ℝ) * √|(↑(discr K) : ℝ)|)
convert_2 convert_2₁ :
  EventuallyEq (β := ℝ) atTop
    (fun (b : ℝ) ↦
      ∑ c : ClassGroup (𝓞 K),
        (↑(Nat.card.{u_1}
                { I : ↥(Ideal (𝓞 K))⁰ //
                  (↑((absNorm : (a : Ideal (𝓞 K)) → ℕ) (↑I : Ideal (𝓞 K))) : ℝ) ≤ b ∧
                    (ClassGroup.mk0 : (a : ↥(Ideal (𝓞 K))⁰) → ClassGroup (𝓞 K)) I = c }) :
            ℝ) /
          b)
    fun (s : ℝ) ↦
    (↑(Nat.card.{u_1} { I : ↥(Ideal (𝓞 K))⁰ // (↑((absNorm : (a : Ideal (𝓞 K)) → ℕ) (↑I : Ideal (𝓞 K))) : ℝ) ≤ s }) :
        ℝ) /
      s
⊢ Tendsto
    (fun (s : ℝ) ↦
      (↑(Nat.card.{u_1} { I : ↥(Ideal (𝓞 K))⁰ // (↑((absNorm : (a : Ideal (𝓞 K)) → ℕ) (↑I : Ideal (𝓞 K))) : ℝ) ≤ s }) :
          ℝ) /
        s)
    atTop
    (𝓝
      (HPow.hPow (α := ℝ) 2 (nrRealPlaces K) * (2 * π) ^ nrComplexPlaces K * regulator K * (↑(classNumber K) : ℝ) /
        ((↑(torsionOrder K) : ℝ) * √|(↑(discr K) : ℝ)|)))

Tactic:
classical
  convert!
    Filter.Tendsto.congr' ?_
      (tendsto_finsetSum Finset.univ (fun C _ ↦ tendsto_norm_le_and_mk_eq_div_atTop K C))
  · rw [Finset.sum_const, Finset.card_univ, nsmul_eq_mul, classNumber]
    ring
  · filter_upwards [eventually_ge_atTop 0] with s hs
    have : Fintype {I : (Ideal (𝓞 K))⁰ // absNorm (I : Ideal (𝓞 K)) ≤ s} := by
      simp_rw [← Nat.le_floor_iff hs]
      refine @Fintype.ofFinite _ (finite_setOf_absNorm_le₀ ⌊s⌋₊)
    let e := fun C : ClassGroup (𝓞 K) ↦ Equiv.subtypeSubtypeEquivSubtypeInter
      (fun I : (Ideal (𝓞 K))⁰ ↦ absNorm I.1 ≤ s) (fun I ↦ ClassGroup.mk0 I = C)
    simp_rw [← Nat.card_congr (e _), Nat.card_eq_fintype_card, Fintype.subtype_card]
    rw [Fintype.card, Finset.card_eq_sum_card_fiberwise (f := fun I ↦ ClassGroup.mk0 I.1)
      (t := Finset.univ) (fun _ _ ↦ Finset.mem_univ _), Nat.cast_sum, Finset.sum_div]

Premises used: Finset.card_univ, Finset.univ, Fintype.subtype_card, Fintype.card, Finset.sum_div, Nat.cast_sum, tendsto_finsetSum, nsmul_eq_mul, Filter.eventually_ge_atTop, NumberField.Ideal.tendsto_norm_le_and_mk_eq_div_atTop, Filter.Tendsto.congr', NumberField.classNumber, OfNat.ofNat, Nat.card_eq_fintype_card, Finset.sum_const
-- Module: Mathlib.NumberTheory.NumberField.Ideal.Asymptotics
Goal:
K : Type u_1
inst : Field K
inst_1 : NumberField K
⊢ Tendsto
    (fun (x : ℝ) ↦
      (↑(Nat.card.{u_1} { I : ↥(Ideal (𝓞 K))⁰ // (↑((absNorm : (a : Ideal (𝓞 K)) → ℕ) (↑I : Ideal (𝓞 K))) : ℝ) ≤ x }) :
            ℝ) /
          x +
        x⁻¹)
    atTop
    (𝓝
      (HPow.hPow (α := ℝ) 2 (nrRealPlaces K) * (2 * π) ^ nrComplexPlaces K * regulator K * (↑(classNumber K) : ℝ) /
          ((↑(torsionOrder K) : ℝ) * √|(↑(discr K) : ℝ)|) +
        0))

Tactic:
have := (tendsto_norm_le_div_atTop₀ K).add tendsto_inv_atTop_zero
-- Module: Mathlib.NumberTheory.NumberField.Ideal.Asymptotics
Goal:
K : Type u_1
inst : Field K
inst_1 : NumberField K
this :
  Tendsto
    (fun (x : ℝ) ↦
      (↑(Nat.card.{u_1} { I : ↥(Ideal (𝓞 K))⁰ // (↑((absNorm : (a : Ideal (𝓞 K)) → ℕ) (↑I : Ideal (𝓞 K))) : ℝ) ≤ x }) :
            ℝ) /
          x +
        x⁻¹)
    atTop
    (𝓝
      (HPow.hPow (α := ℝ) 2 (nrRealPlaces K) * (2 * π) ^ nrComplexPlaces K * regulator K * (↑(classNumber K) : ℝ) /
          ((↑(torsionOrder K) : ℝ) * √|(↑(discr K) : ℝ)|) +
        0))
⊢ Tendsto
    (fun (x : ℝ) ↦
      (↑(Nat.card.{u_1} { I : ↥(Ideal (𝓞 K))⁰ // (↑((absNorm : (a : Ideal (𝓞 K)) → ℕ) (↑I : Ideal (𝓞 K))) : ℝ) ≤ x }) :
            ℝ) /
          x +
        x⁻¹)
    atTop
    (𝓝
      (HPow.hPow (α := ℝ) 2 (nrRealPlaces K) * (2 * π) ^ nrComplexPlaces K * regulator K * (↑(classNumber K) : ℝ) /
        ((↑(torsionOrder K) : ℝ) * √|(↑(discr K) : ℝ)|)))

Tactic:
rw [add_zero] at this

Premises used: add_zero
-- Module: Mathlib.NumberTheory.NumberField.Ideal.Asymptotics
Goal:
K : Type u_1
inst : Field K
inst_1 : NumberField K
this :
  Tendsto
    (fun (x : ℝ) ↦
      (↑(Nat.card.{u_1} { I : ↥(Ideal (𝓞 K))⁰ // (↑((absNorm : (a : Ideal (𝓞 K)) → ℕ) (↑I : Ideal (𝓞 K))) : ℝ) ≤ x }) :
            ℝ) /
          x +
        x⁻¹)
    atTop
    (𝓝
      (HPow.hPow (α := ℝ) 2 (nrRealPlaces K) * (2 * π) ^ nrComplexPlaces K * regulator K * (↑(classNumber K) : ℝ) /
        ((↑(torsionOrder K) : ℝ) * √|(↑(discr K) : ℝ)|)))
apply :
  EventuallyEq (β := ℝ) atTop
    (fun (x : ℝ) ↦
      (↑(Nat.card.{u_1} { I : ↥(Ideal (𝓞 K))⁰ // (↑((absNorm : (a : Ideal (𝓞 K)) → ℕ) (↑I : Ideal (𝓞 K))) : ℝ) ≤ x }) :
            ℝ) /
          x +
        x⁻¹)
    fun (s : ℝ) ↦ (↑(Nat.card.{u_1} { I : Ideal (𝓞 K) // (↑((absNorm : (a : Ideal (𝓞 K)) → ℕ) I) : ℝ) ≤ s }) : ℝ) / s
⊢ Tendsto
    (fun (s : ℝ) ↦ (↑(Nat.card.{u_1} { I : Ideal (𝓞 K) // (↑((absNorm : (a : Ideal (𝓞 K)) → ℕ) I) : ℝ) ≤ s }) : ℝ) / s)
    atTop
    (𝓝
      (HPow.hPow (α := ℝ) 2 (nrRealPlaces K) * (2 * π) ^ nrComplexPlaces K * regulator K * (↑(classNumber K) : ℝ) /
        ((↑(torsionOrder K) : ℝ) * √|(↑(discr K) : ℝ)|)))

Tactic:
apply this.congr'

Premises used: Filter.Tendsto.congr'
-- Module: Mathlib.NumberTheory.NumberField.Ideal.Asymptotics
Goal:
K : Type u_1
inst : Field K
inst_1 : NumberField K
this :
  Tendsto
    (fun (x : ℝ) ↦
      (↑(Nat.card.{u_1} { I : ↥(Ideal (𝓞 K))⁰ // (↑((absNorm : (a : Ideal (𝓞 K)) → ℕ) (↑I : Ideal (𝓞 K))) : ℝ) ≤ x }) :
            ℝ) /
          x +
        x⁻¹)
    atTop
    (𝓝
      (HPow.hPow (α := ℝ) 2 (nrRealPlaces K) * (2 * π) ^ nrComplexPlaces K * regulator K * (↑(classNumber K) : ℝ) /
        ((↑(torsionOrder K) : ℝ) * √|(↑(discr K) : ℝ)|)))
h :
  ∀ (s : ℝ) (hs : 0 ≤ s),
    (↑(Nat.card.{u_1} { I : ↥(Ideal (𝓞 K))⁰ // (↑((absNorm : (a : Ideal (𝓞 K)) → ℕ) (↑I : Ideal (𝓞 K))) : ℝ) ≤ s }) :
            ℝ) /
          s +
        s⁻¹ =
      (↑(Nat.card.{u_1} { I : Ideal (𝓞 K) // (↑((absNorm : (a : Ideal (𝓞 K)) → ℕ) I) : ℝ) ≤ s }) : ℝ) / s
⊢ EventuallyEq (β := ℝ) atTop
    (fun (x : ℝ) ↦
      (↑(Nat.card.{u_1} { I : ↥(Ideal (𝓞 K))⁰ // (↑((absNorm : (a : Ideal (𝓞 K)) → ℕ) (↑I : Ideal (𝓞 K))) : ℝ) ≤ x }) :
            ℝ) /
          x +
        x⁻¹)
    fun (s : ℝ) ↦ (↑(Nat.card.{u_1} { I : Ideal (𝓞 K) // (↑((absNorm : (a : Ideal (𝓞 K)) → ℕ) I) : ℝ) ≤ s }) : ℝ) / s

Tactic:
filter_upwards [eventually_ge_atTop 0] with s hs

Premises used: Filter.eventually_ge_atTop, OfNat.ofNat
-- Module: Mathlib.NumberTheory.NumberField.Ideal.Asymptotics
Goal:
K : Type u_1
inst : Field K
inst_1 : NumberField K
this :
  Tendsto
    (fun (x : ℝ) ↦
      (↑(Nat.card.{u_1} { I : ↥(Ideal (𝓞 K))⁰ // (↑((absNorm : (a : Ideal (𝓞 K)) → ℕ) (↑I : Ideal (𝓞 K))) : ℝ) ≤ x }) :
            ℝ) /
          x +
        x⁻¹)
    atTop
    (𝓝
      (HPow.hPow (α := ℝ) 2 (nrRealPlaces K) * (2 * π) ^ nrComplexPlaces K * regulator K * (↑(classNumber K) : ℝ) /
        ((↑(torsionOrder K) : ℝ) * √|(↑(discr K) : ℝ)|)))
s : ℝ
hs : 0 ≤ s
h :
  (↑(Nat.card.{u_1} { I : ↥(Ideal (𝓞 K))⁰ // (absNorm : (a : Ideal (𝓞 K)) → ℕ) (↑I : Ideal (𝓞 K)) ≤ ⌊s⌋₊ }) : ℝ) / s +
      s⁻¹ =
    (↑(Nat.card.{u_1} { I : Ideal (𝓞 K) // (absNorm : (a : Ideal (𝓞 K)) → ℕ) I ≤ ⌊s⌋₊ }) : ℝ) / s
⊢ (↑(Nat.card.{u_1} { I : ↥(Ideal (𝓞 K))⁰ // (↑((absNorm : (a : Ideal (𝓞 K)) → ℕ) (↑I : Ideal (𝓞 K))) : ℝ) ≤ s }) : ℝ) /
        s +
      s⁻¹ =
    (↑(Nat.card.{u_1} { I : Ideal (𝓞 K) // (↑((absNorm : (a : Ideal (𝓞 K)) → ℕ) I) : ℝ) ≤ s }) : ℝ) / s

Tactic:
simp_rw [← Nat.le_floor_iff hs]

Premises used: Nat.le_floor_iff