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
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))⁰ // (absNorm : (a : Ideal (𝓞 K)) → ℕ) (↑I : Ideal (𝓞 K)) ≤ ⌊s⌋₊ } + 1) : ℝ) /
      s
h₁ :
  (↑(Nat.card.{u_1} { I : ↥(Ideal (𝓞 K))⁰ // (absNorm : (a : Ideal (𝓞 K)) → ℕ) (↑I : Ideal (𝓞 K)) ≤ ⌊s⌋₊ }) : ℝ) / s +
      s⁻¹ =
    HAdd.hAdd (α := ℝ)
        (↑(Nat.card.{u_1} { I : ↥(Ideal (𝓞 K))⁰ // (absNorm : (a : Ideal (𝓞 K)) → ℕ) (↑I : Ideal (𝓞 K)) ≤ ⌊s⌋₊ }) : ℝ)
        (↑1 : ℝ) /
      s
h₂ :
  (↑(Nat.card.{u_1} { I : ↥(Ideal (𝓞 K))⁰ // (absNorm : (a : Ideal (𝓞 K)) → ℕ) (↑I : Ideal (𝓞 K)) ≤ ⌊s⌋₊ }) : ℝ) / s +
      s⁻¹ =
    HAdd.hAdd (α := ℝ)
        (↑(Nat.card.{u_1} { I : ↥(Ideal (𝓞 K))⁰ // (absNorm : (a : Ideal (𝓞 K)) → ℕ) (↑I : Ideal (𝓞 K)) ≤ ⌊s⌋₊ }) : ℝ)
        1 /
      s
h₃ :
  (↑(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))⁰ // (absNorm : (a : Ideal (𝓞 K)) → ℕ) (↑I : Ideal (𝓞 K)) ≤ ⌊s⌋₊ }) : ℝ) / s +
      1 / s
h₄ h₅ :
  (↑(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))⁰ // (absNorm : (a : Ideal (𝓞 K)) → ℕ) (↑I : Ideal (𝓞 K)) ≤ ⌊s⌋₊ }) : ℝ) / s +
      s⁻¹
⊢ (↑(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

Tactic:
rw [Ideal.card_norm_le_eq_card_norm_le_add_one, Nat.cast_add, Nat.cast_one, add_div, one_div]

Premises used: Ideal.card_norm_le_eq_card_norm_le_add_one, Nat.cast_add, one_div, Nat.cast_one, add_div
-- Module: Mathlib.NumberTheory.NumberField.Ideal.Basic
Goal:
K : Type u_1
inst : Field K
I : Ideal (𝓞 K)
inst_1 : NumberField K
hI : Ne (α := ℕ) ((absNorm : (a : Ideal (𝓞 K)) → ℕ) I : ℕ) (1 : ℕ)
n : ℕ
ζ : K
hn : 2 ≤ n
hζ : IsPrimitiveRoot ζ n
h : Eq (α := 𝓞 K ⧸ I) ((Ideal.Quotient.mk I : (a : 𝓞 K) → 𝓞 K ⧸ I) hζ.toInteger : 𝓞 K ⧸ I) (1 : 𝓞 K ⧸ I)
h₁ : Nat.Coprime ((absNorm : (a : Ideal (𝓞 K)) → ℕ) I) n
⊢ hζ.toInteger - 1 ∈ I

Tactic:
rw [← map_one (Ideal.Quotient.mk I), Ideal.Quotient.eq] at h

Premises used: Ideal.Quotient.mk, Ideal.Quotient.eq, map_one
-- Module: Mathlib.NumberTheory.NumberField.Ideal.Basic
Goal:
K : Type u_1
inst : Field K
I : Ideal (𝓞 K)
inst_1 : NumberField K
hI : Ne (α := ℕ) ((absNorm : (a : Ideal (𝓞 K)) → ℕ) I : ℕ) (1 : ℕ)
n : ℕ
ζ : K
hn : 2 ≤ n
hζ : IsPrimitiveRoot ζ n
h : Eq (α := 𝓞 K ⧸ I) ((Ideal.Quotient.mk I : (a : 𝓞 K) → 𝓞 K ⧸ I) hζ.toInteger : 𝓞 K ⧸ I) (1 : 𝓞 K ⧸ I)
h₁ : Nat.Coprime ((absNorm : (a : Ideal (𝓞 K)) → ℕ) I) n
h₂ : hζ.toInteger - 1 ∈ I
⊢ Nat.Coprime ((absNorm : (a : Ideal (𝓞 K)) → ℕ) I) n

Tactic:
rw [← map_one (Ideal.Quotient.mk I), Ideal.Quotient.eq] at h

Premises used: Ideal.Quotient.mk, Ideal.Quotient.eq, map_one
-- Module: Mathlib.NumberTheory.NumberField.Ideal.Basic
Goal:
K : Type u_1
inst : Field K
I : Ideal (𝓞 K)
inst_1 : NumberField K
hI : Ne (α := ℕ) ((absNorm : (a : Ideal (𝓞 K)) → ℕ) I : ℕ) (1 : ℕ)
n : ℕ
ζ : K
hn : 2 ≤ n
hζ : IsPrimitiveRoot ζ n
h : hζ.toInteger - 1 ∈ I
h₁ : Nat.Coprime ((absNorm : (a : Ideal (𝓞 K)) → ℕ) I) n
obtain : ∀ (p : ℕ) (hp : Nat.Prime p) (h₂ : p ∣ (absNorm : (a : Ideal (𝓞 K)) → ℕ) I), False
⊢ False

Tactic:
obtain ⟨p, hp, h₂⟩ := Nat.exists_prime_and_dvd hI

Premises used: Nat.exists_prime_and_dvd
-- Module: Mathlib.NumberTheory.NumberField.Ideal.Basic
Goal:
K : Type u_1
inst : Field K
I : Ideal (𝓞 K)
inst_1 : NumberField K
hI : Ne (α := ℕ) ((absNorm : (a : Ideal (𝓞 K)) → ℕ) I : ℕ) (1 : ℕ)
n : ℕ
ζ : K
hn : 2 ≤ n
hζ : IsPrimitiveRoot ζ n
h : hζ.toInteger - 1 ∈ I
h₁ : Nat.Coprime ((absNorm : (a : Ideal (𝓞 K)) → ℕ) I) n
p : ℕ
hp : Nat.Prime p
h₂ : p ∣ (absNorm : (a : Ideal (𝓞 K)) → ℕ) I
⊢ Fact (Nat.Prime p)

Tactic:
have : Fact (p.Prime) := ⟨hp⟩
-- Module: Mathlib.NumberTheory.NumberField.Ideal.Basic
Goal:
K : Type u_1
inst : Field K
I : Ideal (𝓞 K)
inst_1 : NumberField K
hI : Ne (α := ℕ) ((absNorm : (a : Ideal (𝓞 K)) → ℕ) I : ℕ) (1 : ℕ)
n : ℕ
ζ : K
hn : 2 ≤ n
hζ : IsPrimitiveRoot ζ n
h : hζ.toInteger - 1 ∈ I
h₁ : Nat.Coprime ((absNorm : (a : Ideal (𝓞 K)) → ℕ) I) n
p : ℕ
hp : Nat.Prime p
h₂ : p ∣ (absNorm : (a : Ideal (𝓞 K)) → ℕ) I
this : Fact (Nat.Prime p)
⊢ False

Tactic:
hp.not_dvd_one <| h₁ ▸ Nat.dvd_gcd h₂ ?_

Premises used: Eq.rec, Nat.dvd_gcd, Nat.Prime.not_dvd_one
-- Module: Mathlib.NumberTheory.NumberField.Ideal.Basic
Goal:
K : Type u_1
inst : Field K
I : Ideal (𝓞 K)
inst_1 : NumberField K
hI : Ne (α := ℕ) ((absNorm : (a : Ideal (𝓞 K)) → ℕ) I : ℕ) (1 : ℕ)
n : ℕ
ζ : K
hn : 2 ≤ n
hζ : IsPrimitiveRoot ζ n
h : hζ.toInteger - 1 ∈ I
h₁ : Nat.Coprime ((absNorm : (a : Ideal (𝓞 K)) → ℕ) I) n
p : ℕ
hp : Nat.Prime p
h₂ : p ∣ (absNorm : (a : Ideal (𝓞 K)) → ℕ) I
this : Fact (Nat.Prime p)
refine : p ∣ n
⊢ False

Tactic:
refine hp.not_dvd_one <| h₁ ▸ Nat.dvd_gcd h₂ ?_

Premises used: Nat.Prime.not_dvd_one
-- Module: Mathlib.NumberTheory.NumberField.Ideal.Basic
Goal:
K : Type u_1
inst : Field K
I : Ideal (𝓞 K)
inst_1 : NumberField K
hI : Ne (α := ℕ) ((absNorm : (a : Ideal (𝓞 K)) → ℕ) I : ℕ) (1 : ℕ)
n : ℕ
ζ : K
hn : 2 ≤ n
hζ : IsPrimitiveRoot ζ n
h : hζ.toInteger - 1 ∈ I
h₁ : Nat.Coprime ((absNorm : (a : Ideal (𝓞 K)) → ℕ) I) n
p : ℕ
hp : Nat.Prime p
h₂ : p ∣ (absNorm : (a : Ideal (𝓞 K)) → ℕ) I
this : Fact (Nat.Prime p)
⊢ p ∣ n

Tactic:
hζ.prime_dvd_of_dvd_norm_sub_one hn <|
    Int.dvd_trans (Int.natCast_dvd_natCast.mpr h₂) (absNorm_dvd_norm_of_mem h)

Premises used: Iff.mpr, Int.dvd_trans, IsPrimitiveRoot.prime_dvd_of_dvd_norm_sub_one, Int.natCast_dvd_natCast, Ideal.absNorm_dvd_norm_of_mem
-- Module: Mathlib.NumberTheory.NumberField.Ideal.Basic
Goal:
K : Type u_1
inst : Field K
I : Ideal (𝓞 K)
inst_1 : NumberField K
hI : Ne (α := ℕ) ((absNorm : (a : Ideal (𝓞 K)) → ℕ) I : ℕ) (1 : ℕ)
n : ℕ
ζ : K
hn : 2 ≤ n
hζ : IsPrimitiveRoot ζ n
h : hζ.toInteger - 1 ∈ I
h₁ : Nat.Coprime ((absNorm : (a : Ideal (𝓞 K)) → ℕ) I) n
p : ℕ
hp : Nat.Prime p
h₂ : p ∣ (absNorm : (a : Ideal (𝓞 K)) → ℕ) I
this : Fact (Nat.Prime p)
⊢ p ∣ n

Tactic:
exact hζ.prime_dvd_of_dvd_norm_sub_one hn <|
    Int.dvd_trans (Int.natCast_dvd_natCast.mpr h₂) (absNorm_dvd_norm_of_mem h)

Premises used: Iff.mpr, Int.dvd_trans, IsPrimitiveRoot.prime_dvd_of_dvd_norm_sub_one, Int.natCast_dvd_natCast, Ideal.absNorm_dvd_norm_of_mem
-- Module: Mathlib.NumberTheory.NumberField.Ideal.Basic
Goal:
K : Type u_1
inst : Field K
I : Ideal (𝓞 K)
n : ℕ
x : (𝓞 K)ˣ
hx : Membership.mem (γ := Subgroup (𝓞 K)ˣ) (rootsOfUnity n (𝓞 K)) x
⊢ Eq (α := 𝓞 K ⧸ I)
    ((↑((I.rootsOfUnityMapQuot n : (a : ↥(rootsOfUnity n (𝓞 K))) → (𝓞 K ⧸ I)ˣ)
            (Subtype.mk (p := fun (x : (𝓞 K)ˣ) ↦ Membership.mem (γ := Subgroup (𝓞 K)ˣ) (rootsOfUnity n (𝓞 K)) x) x
              hx)) :
        𝓞 K ⧸ I) :
      𝓞 K ⧸ I)
    ((Quotient.mk I : (a : 𝓞 K) → 𝓞 K ⧸ I) (↑x : 𝓞 K) : 𝓞 K ⧸ I)

Tactic:
rfl

Premises used: rfl
-- Module: Mathlib.NumberTheory.NumberField.Ideal.Basic
Goal:
K : Type u_1
inst : Field K
I : Ideal (𝓞 K)
x : (𝓞 K)ˣ
hx : Membership.mem (γ := Subgroup (𝓞 K)ˣ) (torsion K) x
⊢ Eq (α := 𝓞 K ⧸ I)
    ((↑((I.torsionMapQuot : (a : ↥(torsion K)) → (𝓞 K ⧸ I)ˣ)
            (Subtype.mk (p := fun (x : (𝓞 K)ˣ) ↦ Membership.mem (γ := Subgroup (𝓞 K)ˣ) (torsion K) x) x hx)) :
        𝓞 K ⧸ I) :
      𝓞 K ⧸ I)
    ((Quotient.mk I : (a : 𝓞 K) → 𝓞 K ⧸ I) (↑x : 𝓞 K) : 𝓞 K ⧸ I)

Tactic:
rfl

Premises used: rfl
-- Module: Mathlib.NumberTheory.NumberField.Ideal.Basic
Goal:
K : Type u_1
inst : Field K
I : Ideal (𝓞 K)
inst_1 : NumberField K
n : ℕ
inst_2 : NeZero n
hI₁ : Ne (α := ℕ) ((absNorm : (a : Ideal (𝓞 K)) → ℕ) I : ℕ) (1 : ℕ)
hI₂ : Nat.Coprime ((absNorm : (a : Ideal (𝓞 K)) → ℕ) I) n
⊢ Function.Injective (α := ↥(rootsOfUnity n (𝓞 K))) (β := (𝓞 K ⧸ I)ˣ)
    (⇑(I.rootsOfUnityMapQuot n) : (a : ↥(rootsOfUnity n (𝓞 K))) → (𝓞 K ⧸ I)ˣ)

Tactic:
(injective_iff_map_eq_one _).mpr fun ⟨ζ, hζ⟩ h ↦ ?_

Premises used: Iff.mpr, OfNat.ofNat, _private.0.Ideal.rootsOfUnityMapQuot_injective.match_1, Eq, injective_iff_map_eq_one
-- Module: Mathlib.NumberTheory.NumberField.Ideal.Basic
Goal:
K : Type u_1
inst : Field K
I : Ideal (𝓞 K)
inst_1 : NumberField K
n : ℕ
inst_2 : NeZero n
hI₁ : Ne (α := ℕ) ((absNorm : (a : Ideal (𝓞 K)) → ℕ) I : ℕ) (1 : ℕ)
hI₂ : Nat.Coprime ((absNorm : (a : Ideal (𝓞 K)) → ℕ) I) n
refine :
  ∀ (x : ↥(rootsOfUnity n (𝓞 K))) (ζ : (𝓞 K)ˣ) (hζ : Membership.mem (γ := Subgroup (𝓞 K)ˣ) (rootsOfUnity n (𝓞 K)) ζ)
    (h :
      Eq (α := (𝓞 K ⧸ I)ˣ)
        ((I.rootsOfUnityMapQuot n : (a : ↥(rootsOfUnity n (𝓞 K))) → (𝓞 K ⧸ I)ˣ)
            (Subtype.mk (p := fun (x : (𝓞 K)ˣ) ↦ Membership.mem (γ := Subgroup (𝓞 K)ˣ) (rootsOfUnity n (𝓞 K)) x) ζ hζ) :
          (𝓞 K ⧸ I)ˣ)
        (1 : (𝓞 K ⧸ I)ˣ)),
    Subtype.mk (p := fun (x : (𝓞 K)ˣ) ↦ Membership.mem (γ := Subgroup (𝓞 K)ˣ) (rootsOfUnity n (𝓞 K)) x) ζ hζ = 1
⊢ Function.Injective (α := ↥(rootsOfUnity n (𝓞 K))) (β := (𝓞 K ⧸ I)ˣ)
    (⇑(I.rootsOfUnityMapQuot n) : (a : ↥(rootsOfUnity n (𝓞 K))) → (𝓞 K ⧸ I)ˣ)

Tactic:
refine (injective_iff_map_eq_one _).mpr fun ⟨ζ, hζ⟩ h ↦ ?_

Premises used: Iff.mpr, injective_iff_map_eq_one
-- Module: Mathlib.NumberTheory.NumberField.Ideal.Basic
Goal:
K : Type u_1
inst : Field K
I : Ideal (𝓞 K)
inst_1 : NumberField K
n : ℕ
inst_2 : NeZero n
hI₁ : Ne (α := ℕ) ((absNorm : (a : Ideal (𝓞 K)) → ℕ) I : ℕ) (1 : ℕ)
hI₂ : Nat.Coprime ((absNorm : (a : Ideal (𝓞 K)) → ℕ) I) n
x : ↥(rootsOfUnity n (𝓞 K))
ζ : (𝓞 K)ˣ
hζ : Membership.mem (γ := Subgroup (𝓞 K)ˣ) (rootsOfUnity n (𝓞 K)) ζ
h :
  Eq (α := (𝓞 K ⧸ I)ˣ)
    ((I.rootsOfUnityMapQuot n : (a : ↥(rootsOfUnity n (𝓞 K))) → (𝓞 K ⧸ I)ˣ)
        (Subtype.mk (p := fun (x : (𝓞 K)ˣ) ↦ Membership.mem (γ := Subgroup (𝓞 K)ˣ) (rootsOfUnity n (𝓞 K)) x) ζ hζ) :
      (𝓞 K ⧸ I)ˣ)
    (1 : (𝓞 K ⧸ I)ˣ)
obtain :
  ∀ (t : ℕ) (ht₀ : t ≠ 0) (ht : t ∣ n) (hζ_1 : IsPrimitiveRoot ζ t),
    Subtype.mk (p := fun (x : (𝓞 K)ˣ) ↦ Membership.mem (γ := Subgroup (𝓞 K)ˣ) (rootsOfUnity n (𝓞 K)) x) ζ hζ = 1
⊢ Subtype.mk (p := fun (x : (𝓞 K)ˣ) ↦ Membership.mem (γ := Subgroup (𝓞 K)ˣ) (rootsOfUnity n (𝓞 K)) x) ζ hζ = 1

Tactic:
obtain ⟨t, ht₀, ht, hζ⟩ := isPrimitiveRoot_of_mem_rootsOfUnity hζ

Premises used: isPrimitiveRoot_of_mem_rootsOfUnity
-- Module: Mathlib.NumberTheory.NumberField.Ideal.Basic
Goal:
K : Type u_1
inst : Field K
I : Ideal (𝓞 K)
inst_1 : NumberField K
n : ℕ
inst_2 : NeZero n
hI₁ : Ne (α := ℕ) ((absNorm : (a : Ideal (𝓞 K)) → ℕ) I : ℕ) (1 : ℕ)
hI₂ : Nat.Coprime ((absNorm : (a : Ideal (𝓞 K)) → ℕ) I) n
x : ↥(rootsOfUnity n (𝓞 K))
ζ : (𝓞 K)ˣ
hζ_1 : Membership.mem (γ := Subgroup (𝓞 K)ˣ) (rootsOfUnity n (𝓞 K)) ζ
h :
  Eq (α := (𝓞 K ⧸ I)ˣ)
    ((I.rootsOfUnityMapQuot n : (a : ↥(rootsOfUnity n (𝓞 K))) → (𝓞 K ⧸ I)ˣ)
        (Subtype.mk (p := fun (x : (𝓞 K)ˣ) ↦ Membership.mem (γ := Subgroup (𝓞 K)ˣ) (rootsOfUnity n (𝓞 K)) x) ζ hζ_1) :
      (𝓞 K ⧸ I)ˣ)
    (1 : (𝓞 K ⧸ I)ˣ)
t : ℕ
ht₀ : t ≠ 0
ht : t ∣ n
hζ : IsPrimitiveRoot ζ t
this : ¬2 ≤ t
⊢ t = 1

Tactic:
grind
-- Module: Mathlib.NumberTheory.NumberField.Ideal.Basic
Goal:
K : Type u_1
inst : Field K
I : Ideal (𝓞 K)
inst_1 : NumberField K
n : ℕ
inst_2 : NeZero n
hI₁ : Ne (α := ℕ) ((absNorm : (a : Ideal (𝓞 K)) → ℕ) I : ℕ) (1 : ℕ)
hI₂ : Nat.Coprime ((absNorm : (a : Ideal (𝓞 K)) → ℕ) I) n
x : ↥(rootsOfUnity n (𝓞 K))
ζ : (𝓞 K)ˣ
hζ_1 : Membership.mem (γ := Subgroup (𝓞 K)ˣ) (rootsOfUnity n (𝓞 K)) ζ
h :
  Eq (α := (𝓞 K ⧸ I)ˣ)
    ((I.rootsOfUnityMapQuot n : (a : ↥(rootsOfUnity n (𝓞 K))) → (𝓞 K ⧸ I)ˣ)
        (Subtype.mk (p := fun (x : (𝓞 K)ˣ) ↦ Membership.mem (γ := Subgroup (𝓞 K)ˣ) (rootsOfUnity n (𝓞 K)) x) ζ hζ_1) :
      (𝓞 K ⧸ I)ˣ)
    (1 : (𝓞 K ⧸ I)ˣ)
t : ℕ
ht₀ : t ≠ 0
ht : t ∣ n
hζ : IsPrimitiveRoot ζ t
this : ¬2 ≤ t
⊢ Subtype.mk (p := fun (x : (𝓞 K)ˣ) ↦ Membership.mem (γ := Subgroup (𝓞 K)ˣ) (rootsOfUnity n (𝓞 K)) x) ζ hζ_1 = 1

Tactic:
simpa [show t = 1 by grind] using hζ
-- Module: Mathlib.NumberTheory.NumberField.Ideal.Basic
Goal:
K : Type u_1
inst : Field K
I : Ideal (𝓞 K)
inst_1 : NumberField K
n : ℕ
inst_2 : NeZero n
hI₁ : Ne (α := ℕ) ((absNorm : (a : Ideal (𝓞 K)) → ℕ) I : ℕ) (1 : ℕ)
hI₂ : Nat.Coprime ((absNorm : (a : Ideal (𝓞 K)) → ℕ) I) n
x : ↥(rootsOfUnity n (𝓞 K))
ζ : (𝓞 K)ˣ
hζ_1 : Membership.mem (γ := Subgroup (𝓞 K)ˣ) (rootsOfUnity n (𝓞 K)) ζ
h :
  Eq (α := (𝓞 K ⧸ I)ˣ)
    ((I.rootsOfUnityMapQuot n : (a : ↥(rootsOfUnity n (𝓞 K))) → (𝓞 K ⧸ I)ˣ)
        (Subtype.mk (p := fun (x : (𝓞 K)ˣ) ↦ Membership.mem (γ := Subgroup (𝓞 K)ˣ) (rootsOfUnity n (𝓞 K)) x) ζ hζ_1) :
      (𝓞 K ⧸ I)ˣ)
    (1 : (𝓞 K ⧸ I)ˣ)
t : ℕ
ht₀ : t ≠ 0
ht : t ∣ n
hζ : IsPrimitiveRoot ζ t
suffices : ¬2 ≤ t
⊢ Subtype.mk (p := fun (x : (𝓞 K)ˣ) ↦ Membership.mem (γ := Subgroup (𝓞 K)ˣ) (rootsOfUnity n (𝓞 K)) x) ζ hζ_1 = 1

Tactic:
suffices ¬ (2 ≤ t) by
    simpa [show t = 1 by grind] using hζ
-- Module: Mathlib.NumberTheory.NumberField.Ideal.Basic
Goal:
K : Type u_1
inst : Field K
I : Ideal (𝓞 K)
inst_1 : NumberField K
n : ℕ
inst_2 : NeZero n
hI₁ : Ne (α := ℕ) ((absNorm : (a : Ideal (𝓞 K)) → ℕ) I : ℕ) (1 : ℕ)
hI₂ : Nat.Coprime ((absNorm : (a : Ideal (𝓞 K)) → ℕ) I) n
x : ↥(rootsOfUnity n (𝓞 K))
ζ : (𝓞 K)ˣ
hζ_1 : Membership.mem (γ := Subgroup (𝓞 K)ˣ) (rootsOfUnity n (𝓞 K)) ζ
h :
  Eq (α := (𝓞 K ⧸ I)ˣ)
    ((I.rootsOfUnityMapQuot n : (a : ↥(rootsOfUnity n (𝓞 K))) → (𝓞 K ⧸ I)ˣ)
        (Subtype.mk (p := fun (x : (𝓞 K)ˣ) ↦ Membership.mem (γ := Subgroup (𝓞 K)ˣ) (rootsOfUnity n (𝓞 K)) x) ζ hζ_1) :
      (𝓞 K ⧸ I)ˣ)
    (1 : (𝓞 K ⧸ I)ˣ)
t : ℕ
ht₀ : t ≠ 0
ht : t ∣ n
hζ : IsPrimitiveRoot ζ t
ht' : 2 ≤ t
μ : K := (↑(↑ζ : 𝓞 K) : K)
⊢ IsPrimitiveRoot μ t

Tactic:
have hμ : IsPrimitiveRoot μ t :=
    (IsPrimitiveRoot.coe_units_iff.mpr hζ).map_of_injective RingOfIntegers.coe_injective
-- Module: Mathlib.NumberTheory.NumberField.Ideal.Basic
Goal:
K : Type u_1
inst : Field K
I : Ideal (𝓞 K)
inst_1 : NumberField K
n : ℕ
inst_2 : NeZero n
hI₁ : Ne (α := ℕ) ((absNorm : (a : Ideal (𝓞 K)) → ℕ) I : ℕ) (1 : ℕ)
hI₂ : Nat.Coprime ((absNorm : (a : Ideal (𝓞 K)) → ℕ) I) n
x : ↥(rootsOfUnity n (𝓞 K))
ζ : (𝓞 K)ˣ
hζ_1 : Membership.mem (γ := Subgroup (𝓞 K)ˣ) (rootsOfUnity n (𝓞 K)) ζ
h :
  Eq (α := (𝓞 K ⧸ I)ˣ)
    ((I.rootsOfUnityMapQuot n : (a : ↥(rootsOfUnity n (𝓞 K))) → (𝓞 K ⧸ I)ˣ)
        (Subtype.mk (p := fun (x : (𝓞 K)ˣ) ↦ Membership.mem (γ := Subgroup (𝓞 K)ˣ) (rootsOfUnity n (𝓞 K)) x) ζ hζ_1) :
      (𝓞 K ⧸ I)ˣ)
    (1 : (𝓞 K ⧸ I)ˣ)
t : ℕ
ht₀ : t ≠ 0
ht : t ∣ n
hζ : IsPrimitiveRoot ζ t
ht' : 2 ≤ t
μ : K := (↑(↑ζ : 𝓞 K) : K)
hμ : IsPrimitiveRoot μ t
rw :
  ∀ (h : Eq (α := 𝓞 K ⧸ I) ((Quotient.mk I : (a : 𝓞 K) → 𝓞 K ⧸ I) (↑ζ : 𝓞 K) : 𝓞 K ⧸ I) (1 : 𝓞 K ⧸ I)) (t : ℕ)
    (ht₀ : t ≠ 0) (ht : t ∣ n) (hζ : IsPrimitiveRoot ζ t) (ht' : 2 ≤ t),
    let μ := (↑(↑ζ : 𝓞 K) : K);
    ∀ (hμ : IsPrimitiveRoot μ t), False
⊢ False

Tactic:
rw [Units.ext_iff, rootsOfUnityMapQuot_apply, Units.val_one] at h

Premises used: Units.ext_iff, Ideal.rootsOfUnityMapQuot_apply, Units.val_one
-- Module: Mathlib.NumberTheory.NumberField.Ideal.Basic
Goal:
K : Type u_1
inst : Field K
I : Ideal (𝓞 K)
inst_1 : NumberField K
n : ℕ
inst_2 : NeZero n
hI₁ : Ne (α := ℕ) ((absNorm : (a : Ideal (𝓞 K)) → ℕ) I : ℕ) (1 : ℕ)
hI₂ : Nat.Coprime ((absNorm : (a : Ideal (𝓞 K)) → ℕ) I) n
x : ↥(rootsOfUnity n (𝓞 K))
ζ : (𝓞 K)ˣ
hζ_1 : Membership.mem (γ := Subgroup (𝓞 K)ˣ) (rootsOfUnity n (𝓞 K)) ζ
h : Eq (α := 𝓞 K ⧸ I) ((Quotient.mk I : (a : 𝓞 K) → 𝓞 K ⧸ I) (↑ζ : 𝓞 K) : 𝓞 K ⧸ I) (1 : 𝓞 K ⧸ I)
t : ℕ
ht₀ : t ≠ 0
ht : t ∣ n
hζ : IsPrimitiveRoot ζ t
ht' : 2 ≤ t
μ : K := (↑(↑ζ : 𝓞 K) : K)
hμ : IsPrimitiveRoot μ t
⊢ False

Tactic:
hμ.not_coprime_norm_of_mk_eq_one hI₁ ht' h ?_

Premises used: IsPrimitiveRoot.not_coprime_norm_of_mk_eq_one
-- Module: Mathlib.NumberTheory.NumberField.Ideal.Basic
Goal:
K : Type u_1
inst : Field K
I : Ideal (𝓞 K)
inst_1 : NumberField K
n : ℕ
inst_2 : NeZero n
hI₁ : Ne (α := ℕ) ((absNorm : (a : Ideal (𝓞 K)) → ℕ) I : ℕ) (1 : ℕ)
hI₂ : Nat.Coprime ((absNorm : (a : Ideal (𝓞 K)) → ℕ) I) n
x : ↥(rootsOfUnity n (𝓞 K))
ζ : (𝓞 K)ˣ
hζ_1 : Membership.mem (γ := Subgroup (𝓞 K)ˣ) (rootsOfUnity n (𝓞 K)) ζ
h : Eq (α := 𝓞 K ⧸ I) ((Quotient.mk I : (a : 𝓞 K) → 𝓞 K ⧸ I) (↑ζ : 𝓞 K) : 𝓞 K ⧸ I) (1 : 𝓞 K ⧸ I)
t : ℕ
ht₀ : t ≠ 0
ht : t ∣ n
hζ : IsPrimitiveRoot ζ t
ht' : 2 ≤ t
μ : K := (↑(↑ζ : 𝓞 K) : K)
hμ : IsPrimitiveRoot μ t
refine : Nat.Coprime ((absNorm : (a : Ideal (𝓞 K)) → ℕ) I) t
⊢ False

Tactic:
refine hμ.not_coprime_norm_of_mk_eq_one hI₁ ht' h ?_

Premises used: IsPrimitiveRoot.not_coprime_norm_of_mk_eq_one
-- Module: Mathlib.NumberTheory.NumberField.Ideal.Basic
Goal:
K : Type u_1
inst : Field K
I : Ideal (𝓞 K)
inst_1 : NumberField K
n : ℕ
inst_2 : NeZero n
hI₁ : Ne (α := ℕ) ((absNorm : (a : Ideal (𝓞 K)) → ℕ) I : ℕ) (1 : ℕ)
hI₂ : Nat.Coprime ((absNorm : (a : Ideal (𝓞 K)) → ℕ) I) n
x : ↥(rootsOfUnity n (𝓞 K))
ζ : (𝓞 K)ˣ
hζ_1 : Membership.mem (γ := Subgroup (𝓞 K)ˣ) (rootsOfUnity n (𝓞 K)) ζ
h : Eq (α := 𝓞 K ⧸ I) ((Quotient.mk I : (a : 𝓞 K) → 𝓞 K ⧸ I) (↑ζ : 𝓞 K) : 𝓞 K ⧸ I) (1 : 𝓞 K ⧸ I)
t : ℕ
ht₀ : t ≠ 0
ht : t ∣ n
hζ : IsPrimitiveRoot ζ t
ht' : 2 ≤ t
μ : K := (↑(↑ζ : 𝓞 K) : K)
hμ : IsPrimitiveRoot μ t
⊢ Nat.Coprime ((absNorm : (a : Ideal (𝓞 K)) → ℕ) I) t

Tactic:
Nat.dvd_one.mp (hI₂ ▸ Nat.gcd_dvd_gcd_of_dvd_right (absNorm I) ht)

Premises used: Ideal.absNorm, Nat.dvd_one, Eq.rec, Iff.mp, DFunLike.coe, Nat.gcd_dvd_gcd_of_dvd_right
-- Module: Mathlib.NumberTheory.NumberField.Ideal.Basic
Goal:
K : Type u_1
inst : Field K
I : Ideal (𝓞 K)
inst_1 : NumberField K
n : ℕ
inst_2 : NeZero n
hI₁ : Ne (α := ℕ) ((absNorm : (a : Ideal (𝓞 K)) → ℕ) I : ℕ) (1 : ℕ)
hI₂ : Nat.Coprime ((absNorm : (a : Ideal (𝓞 K)) → ℕ) I) n
x : ↥(rootsOfUnity n (𝓞 K))
ζ : (𝓞 K)ˣ
hζ_1 : Membership.mem (γ := Subgroup (𝓞 K)ˣ) (rootsOfUnity n (𝓞 K)) ζ
h : Eq (α := 𝓞 K ⧸ I) ((Quotient.mk I : (a : 𝓞 K) → 𝓞 K ⧸ I) (↑ζ : 𝓞 K) : 𝓞 K ⧸ I) (1 : 𝓞 K ⧸ I)
t : ℕ
ht₀ : t ≠ 0
ht : t ∣ n
hζ : IsPrimitiveRoot ζ t
ht' : 2 ≤ t
μ : K := (↑(↑ζ : 𝓞 K) : K)
hμ : IsPrimitiveRoot μ t
⊢ Nat.Coprime ((absNorm : (a : Ideal (𝓞 K)) → ℕ) I) t

Tactic:
exact Nat.dvd_one.mp (hI₂ ▸ Nat.gcd_dvd_gcd_of_dvd_right (absNorm I) ht)

Premises used: Nat.dvd_one, Iff.mp
-- Module: Mathlib.NumberTheory.NumberField.Ideal.Basic
Goal:
K : Type u_1
inst : Field K
I : Ideal (𝓞 K)
inst_1 : NumberField K
n : ℕ
inst_2 : NeZero n
ζ : 𝓞 K
hζ : IsPrimitiveRoot ζ n
hI₁ : Ne (α := ℕ) ((absNorm : (a : Ideal (𝓞 K)) → ℕ) I : ℕ) (1 : ℕ)
hI₂ : Nat.Coprime ((absNorm : (a : Ideal (𝓞 K)) → ℕ) I) n
⊢ IsPrimitiveRoot hζ.toRootsOfUnity n

Tactic:
have h : IsPrimitiveRoot hζ.toRootsOfUnity n :=
    IsPrimitiveRoot.coe_submonoidClass_iff.mp <| IsPrimitiveRoot.coe_units_iff.mp hζ
-- Module: Mathlib.NumberTheory.NumberField.Ideal.Basic
Goal:
K : Type u_1
inst : Field K
I : Ideal (𝓞 K)
inst_1 : NumberField K
n : ℕ
inst_2 : NeZero n
ζ : 𝓞 K
hζ : IsPrimitiveRoot ζ n
hI₁ : Ne (α := ℕ) ((absNorm : (a : Ideal (𝓞 K)) → ℕ) I : ℕ) (1 : ℕ)
hI₂ : Nat.Coprime ((absNorm : (a : Ideal (𝓞 K)) → ℕ) I) n
h : IsPrimitiveRoot hζ.toRootsOfUnity n
⊢ IsPrimitiveRoot (M := 𝓞 K ⧸ I) ((Ideal.Quotient.mk I : (a : 𝓞 K) → 𝓞 K ⧸ I) ζ : 𝓞 K ⧸ I) n

Tactic:
IsPrimitiveRoot.coe_units_iff.mpr <|
    h.map_of_injective <| Ideal.rootsOfUnityMapQuot_injective n hI₁ hI₂

Premises used: Iff.mpr, Ideal.rootsOfUnityMapQuot_injective, IsPrimitiveRoot.coe_units_iff, IsPrimitiveRoot.map_of_injective
-- Module: Mathlib.NumberTheory.NumberField.Ideal.Basic
Goal:
K : Type u_1
inst : Field K
I : Ideal (𝓞 K)
inst_1 : NumberField K
n : ℕ
inst_2 : NeZero n
ζ : 𝓞 K
hζ : IsPrimitiveRoot ζ n
hI₁ : Ne (α := ℕ) ((absNorm : (a : Ideal (𝓞 K)) → ℕ) I : ℕ) (1 : ℕ)
hI₂ : Nat.Coprime ((absNorm : (a : Ideal (𝓞 K)) → ℕ) I) n
h : IsPrimitiveRoot hζ.toRootsOfUnity n
⊢ IsPrimitiveRoot (M := 𝓞 K ⧸ I) ((Ideal.Quotient.mk I : (a : 𝓞 K) → 𝓞 K ⧸ I) ζ : 𝓞 K ⧸ I) n

Tactic:
exact IsPrimitiveRoot.coe_units_iff.mpr <|
    h.map_of_injective <| Ideal.rootsOfUnityMapQuot_injective n hI₁ hI₂

Premises used: Iff.mpr, Ideal.rootsOfUnityMapQuot_injective, IsPrimitiveRoot.coe_units_iff, IsPrimitiveRoot.map_of_injective
-- Module: Mathlib.NumberTheory.NumberField.Ideal.Basic
Goal:
K : Type u_1
inst : Field K
I : Ideal (𝓞 K)
inst_1 : NumberField K
hI₁ : Ne (α := ℕ) ((absNorm : (a : Ideal (𝓞 K)) → ℕ) I : ℕ) (1 : ℕ)
hI₂ : Nat.Coprime ((absNorm : (a : Ideal (𝓞 K)) → ℕ) I) (torsionOrder K)
x : (𝓞 K)ˣ
hx : Membership.mem (γ := Subgroup (𝓞 K)ˣ) (torsion K) x
this :
  ∀ (y : (𝓞 K)ˣ) (hy : Membership.mem (γ := Subgroup (𝓞 K)ˣ) (torsion K) y),
    Eq (α := (𝓞 K ⧸ I)ˣ)
        ((I.torsionMapQuot : (a : ↥(torsion K)) → (𝓞 K ⧸ I)ˣ)
            (Subtype.mk (p := fun (x : (𝓞 K)ˣ) ↦ Membership.mem (γ := Subgroup (𝓞 K)ˣ) (torsion K) x) x hx) :
          (𝓞 K ⧸ I)ˣ)
        ((I.torsionMapQuot : (a : ↥(torsion K)) → (𝓞 K ⧸ I)ˣ)
            (Subtype.mk (p := fun (x : (𝓞 K)ˣ) ↦ Membership.mem (γ := Subgroup (𝓞 K)ˣ) (torsion K) x) y hy) :
          (𝓞 K ⧸ I)ˣ) →
      Subtype.mk (p := fun (x : (𝓞 K)ˣ) ↦ Membership.mem (γ := Subgroup (𝓞 K)ˣ) (torsion K) x) x hx =
        Subtype.mk (p := fun (x : (𝓞 K)ˣ) ↦ Membership.mem (γ := Subgroup (𝓞 K)ˣ) (torsion K) x) y hy
⊢ ∀ ⦃a₂ : ↥(torsion K)⦄,
    Eq (α := (𝓞 K ⧸ I)ˣ)
        ((I.torsionMapQuot : (a : ↥(torsion K)) → (𝓞 K ⧸ I)ˣ)
            (Subtype.mk (p := fun (x : (𝓞 K)ˣ) ↦ Membership.mem (γ := Subgroup (𝓞 K)ˣ) (torsion K) x) x hx) :
          (𝓞 K ⧸ I)ˣ)
        ((I.torsionMapQuot : (a : ↥(torsion K)) → (𝓞 K ⧸ I)ˣ) a₂ : (𝓞 K ⧸ I)ˣ) →
      Subtype.mk (p := fun (x : (𝓞 K)ˣ) ↦ Membership.mem (γ := Subgroup (𝓞 K)ˣ) (torsion K) x) x hx = a₂

Tactic:
⟨y, hy⟩
-- Module: Mathlib.NumberTheory.NumberField.Ideal.Basic
Goal:
K : Type u_1
inst : Field K
I : Ideal (𝓞 K)
inst_1 : NumberField K
hI₁ : Ne (α := ℕ) ((absNorm : (a : Ideal (𝓞 K)) → ℕ) I : ℕ) (1 : ℕ)
hI₂ : Nat.Coprime ((absNorm : (a : Ideal (𝓞 K)) → ℕ) I) (torsionOrder K)
x : (𝓞 K)ˣ
hx : Membership.mem (γ := Subgroup (𝓞 K)ˣ) (torsion K) x
y : (𝓞 K)ˣ
hy : Membership.mem (γ := Subgroup (𝓞 K)ˣ) (torsion K) y
h :
  Eq (α := (𝓞 K ⧸ I)ˣ)
    ((I.torsionMapQuot : (a : ↥(torsion K)) → (𝓞 K ⧸ I)ˣ)
        (Subtype.mk (p := fun (x : (𝓞 K)ˣ) ↦ Membership.mem (γ := Subgroup (𝓞 K)ˣ) (torsion K) x) x hx) :
      (𝓞 K ⧸ I)ˣ)
    ((I.torsionMapQuot : (a : ↥(torsion K)) → (𝓞 K ⧸ I)ˣ)
        (Subtype.mk (p := fun (x : (𝓞 K)ˣ) ↦ Membership.mem (γ := Subgroup (𝓞 K)ˣ) (torsion K) x) y hy) :
      (𝓞 K ⧸ I)ˣ)
rw :
  ∀ (hx_1 : Membership.mem (γ := Subgroup (𝓞 K)ˣ) (rootsOfUnity (torsionOrder K) (𝓞 K)) x) (y : (𝓞 K)ˣ)
    (hy : Membership.mem (γ := Subgroup (𝓞 K)ˣ) (torsion K) y)
    (hy_1 : Membership.mem (γ := Subgroup (𝓞 K)ˣ) (rootsOfUnity (torsionOrder K) (𝓞 K)) y)
    (h :
      Eq (α := (𝓞 K ⧸ I)ˣ)
        ((I.torsionMapQuot : (a : ↥(torsion K)) → (𝓞 K ⧸ I)ˣ)
            (Subtype.mk (p := fun (x : (𝓞 K)ˣ) ↦ Membership.mem (γ := Subgroup (𝓞 K)ˣ) (torsion K) x) x hx) :
          (𝓞 K ⧸ I)ˣ)
        ((I.torsionMapQuot : (a : ↥(torsion K)) → (𝓞 K ⧸ I)ˣ)
            (Subtype.mk (p := fun (x : (𝓞 K)ˣ) ↦ Membership.mem (γ := Subgroup (𝓞 K)ˣ) (torsion K) x) y hy) :
          (𝓞 K ⧸ I)ˣ)),
    Subtype.mk (p := fun (x : (𝓞 K)ˣ) ↦ Membership.mem (γ := Subgroup (𝓞 K)ˣ) (torsion K) x) x hx =
      Subtype.mk (p := fun (x : (𝓞 K)ˣ) ↦ Membership.mem (γ := Subgroup (𝓞 K)ˣ) (torsion K) x) y hy
⊢ Subtype.mk (p := fun (x : (𝓞 K)ˣ) ↦ Membership.mem (γ := Subgroup (𝓞 K)ˣ) (torsion K) x) x hx =
    Subtype.mk (p := fun (x : (𝓞 K)ˣ) ↦ Membership.mem (γ := Subgroup (𝓞 K)ˣ) (torsion K) x) y hy

Tactic:
rw [← rootsOfUnity_eq_torsion] at hx hy

Premises used: NumberField.Units.rootsOfUnity_eq_torsion