awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.NumberTheory.MulChar.Lemmas (goal)
Goal:
F : Type u_1
inst : Field F
inst_1 : Fintype F
R : Type u_2
inst_2 : CommRing R
n : ℕ
h : n ∣ Fintype.card F - 1
ζ : R
hζ : IsPrimitiveRoot ζ n
have : 0 < n
⊢ ∃ (χ : MulChar F R), orderOf χ = n
have have₁ : ∀ (hn₀ : 0 < n), ∃ (χ : MulChar F R), orderOf χ = n
Tactic: classical
  have hn₀ : 0 < n := by
    refine Nat.pos_of_ne_zero fun hn ↦ ?_
    simp only [hn, zero_dvd_iff, Nat.sub_eq_zero_iff_le] at h
    exact (Fintype.one_lt_card.trans_le h).false
  let e := MulChar.equiv_rootsOfUnity F R
  let ζ' : Rˣ := (hζ.isUnit hn₀.ne').unit
  have h' : ζ' ^ (Fintype.card Fˣ : ℕ) = 1 :=
    Units.ext_iff.mpr <| (hζ.pow_eq_one_iff_dvd _).mpr <| Fintype.card_units (α := F) ▸ h
  use e.symm ⟨ζ', (mem_rootsOfUnity (Fintype.card Fˣ) ζ').mpr h'⟩
  rw [e.symm.orderOf_eq, orderOf_eq_iff hn₀]
  refine ⟨?_, fun m hm hm₀ h ↦ ?_⟩
  · ext
    push_cast
    exact hζ.pow_eq_one
  · rw [Subtype.ext_iff, Units.ext_iff] at h
    push_cast at h
    exact ((Nat.le_of_dvd hm₀ <| hζ.dvd_of_pow_eq_one _ h).trans_lt hm).false
-- Lean 4 proof step from Mathlib.NumberTheory.MulChar.Lemmas (goal)
Goal:
F : Type u_1
inst : Field F
inst_1 : Fintype F
R : Type u_2
inst_2 : CommRing R
n : ℕ
h : n ∣ Fintype.card F - 1
ζ : R
hζ : IsPrimitiveRoot ζ n
have : 0 < n
have₁ : ∀ (hn₀ : 0 < n), ∃ (χ : MulChar F R), orderOf χ = n
have₂ : 0 < n
have₃ have₄ have₅ have₆ : ∀ (hn₀ : 0 < n), ∃ (χ : MulChar F R), orderOf χ = n
⊢ ∃ (χ : MulChar F R), orderOf χ = n
have have₇ : ∀ (hn₀ : 0 < n),
  let ζ' := sorry.unit;
  ∀ (h' : ζ' ^ Fintype.card.{u_1} Fˣ = 1), ∃ (χ : MulChar F R), orderOf χ = n
Tactic: classical
  have hn₀ : 0 < n := by
    refine Nat.pos_of_ne_zero fun hn ↦ ?_
    simp only [hn, zero_dvd_iff, Nat.sub_eq_zero_iff_le] at h
    exact (Fintype.one_lt_card.trans_le h).false
  let e := MulChar.equiv_rootsOfUnity F R
  let ζ' : Rˣ := (hζ.isUnit hn₀.ne').unit
  have h' : ζ' ^ (Fintype.card Fˣ : ℕ) = 1 :=
    Units.ext_iff.mpr <| (hζ.pow_eq_one_iff_dvd _).mpr <| Fintype.card_units (α := F) ▸ h
  use e.symm ⟨ζ', (mem_rootsOfUnity (Fintype.card Fˣ) ζ').mpr h'⟩
  rw [e.symm.orderOf_eq, orderOf_eq_iff hn₀]
  refine ⟨?_, fun m hm hm₀ h ↦ ?_⟩
  · ext
    push_cast
    exact hζ.pow_eq_one
  · rw [Subtype.ext_iff, Units.ext_iff] at h
    push_cast at h
    exact ((Nat.le_of_dvd hm₀ <| hζ.dvd_of_pow_eq_one _ h).trans_lt hm).false
-- Lean 4 proof step from Mathlib.NumberTheory.MulChar.Lemmas (goal)
Goal:
F : Type u_1
inst : Field F
inst_1 : Fintype F
R : Type u_2
inst_2 : CommRing R
n : ℕ
h : n ∣ Fintype.card F - 1
ζ : R
hζ : IsPrimitiveRoot ζ n
have : 0 < n
have₁ : ∀ (hn₀ : 0 < n), ∃ (χ : MulChar F R), orderOf χ = n
have₂ : 0 < n
have₃ have₄ have₅ have₆ : ∀ (hn₀ : 0 < n), ∃ (χ : MulChar F R), orderOf χ = n
have₇ :
  ∀ (hn₀ : 0 < n),
    let ζ' := ⋯.unit;
    ∀ (h' : ζ' ^ Fintype.card.{u_1} Fˣ = 1), ∃ (χ : MulChar F R), orderOf χ = n
⊢ ∃ (χ : MulChar F R), orderOf χ = n
have h₁ : ∀ (hn₀ : 0 < n),
  let e : MulChar F R ≃* ↥(rootsOfUnity (Fintype.card.{u_1} Fˣ) R) := equiv_rootsOfUnity F R;
  let ζ' := sorry.unit;
  ∀ (h' : ζ' ^ Fintype.card.{u_1} Fˣ = 1),
    orderOf (G := MulChar F R)
        ((e.symm : (a : ↥(rootsOfUnity (Fintype.card Fˣ) R)) → MulChar F R)
            (Subtype.mk (p := fun (x : Rˣ) ↦
              Membership.mem (γ := Subgroup Rˣ) (rootsOfUnity (Fintype.card.{u_1} Fˣ) R) x) ζ' sorry) :
          MulChar F R) =
      n
Tactic: classical
  have hn₀ : 0 < n := by
    refine Nat.pos_of_ne_zero fun hn ↦ ?_
    simp only [hn, zero_dvd_iff, Nat.sub_eq_zero_iff_le] at h
    exact (Fintype.one_lt_card.trans_le h).false
  let e := MulChar.equiv_rootsOfUnity F R
  let ζ' : Rˣ := (hζ.isUnit hn₀.ne').unit
  have h' : ζ' ^ (Fintype.card Fˣ : ℕ) = 1 :=
    Units.ext_iff.mpr <| (hζ.pow_eq_one_iff_dvd _).mpr <| Fintype.card_units (α := F) ▸ h
  use e.symm ⟨ζ', (mem_rootsOfUnity (Fintype.card Fˣ) ζ').mpr h'⟩
  rw [e.symm.orderOf_eq, orderOf_eq_iff hn₀]
  refine ⟨?_, fun m hm hm₀ h ↦ ?_⟩
  · ext
    push_cast
    exact hζ.pow_eq_one
  · rw [Subtype.ext_iff, Units.ext_iff] at h
    push_cast at h
    exact ((Nat.le_of_dvd hm₀ <| hζ.dvd_of_pow_eq_one _ h).trans_lt hm).false
-- Lean 4 proof step from Mathlib.NumberTheory.MulChar.Lemmas (goal)
Goal:
F : Type u_1
inst : Field F
inst_1 : Fintype F
R : Type u_2
inst_2 : CommRing R
n : ℕ
h : n ∣ Fintype.card F - 1
ζ : R
hζ : IsPrimitiveRoot ζ n
have : 0 < n
have₁ : ∀ (hn₀ : 0 < n), ∃ (χ : MulChar F R), orderOf χ = n
have₂ : 0 < n
have₃ have₄ have₅ have₆ : ∀ (hn₀ : 0 < n), ∃ (χ : MulChar F R), orderOf χ = n
have₇ :
  ∀ (hn₀ : 0 < n),
    let ζ' := ⋯.unit;
    ∀ (h' : ζ' ^ Fintype.card.{u_1} Fˣ = 1), ∃ (χ : MulChar F R), orderOf χ = n
h₁ :
  ∀ (hn₀ : 0 < n),
    let e : MulChar F R ≃* ↥(rootsOfUnity (Fintype.card.{u_1} Fˣ) R) := equiv_rootsOfUnity F R;
    let ζ' := ⋯.unit;
    ∀ (h' : ζ' ^ Fintype.card.{u_1} Fˣ = 1),
      orderOf (G := MulChar F R)
          ((e.symm : (a : ↥(rootsOfUnity (Fintype.card Fˣ) R)) → MulChar F R)
              (Subtype.mk (p := fun (x : Rˣ) ↦
                Membership.mem (γ := Subgroup Rˣ) (rootsOfUnity (Fintype.card.{u_1} Fˣ) R) x) ζ' ⋯) :
            MulChar F R) =
        n
⊢ ∃ (χ : MulChar F R), orderOf χ = n
have h₂ : ∀ (hn₀ : 0 < n),
  let ζ' := sorry.unit;
  ∀ (h' : ζ' ^ Fintype.card.{u_1} Fˣ = 1),
    Subtype.mk (p := fun (x : Rˣ) ↦ Membership.mem (γ := Subgroup Rˣ) (rootsOfUnity (Fintype.card.{u_1} Fˣ) R) x) ζ' sorry ^
          n =
        1 ∧
      ∀ m < n,
        0 < m →
          Subtype.mk (p := fun (x : Rˣ) ↦ Membership.mem (γ := Subgroup Rˣ) (rootsOfUnity (Fintype.card.{u_1} Fˣ) R) x)
                ζ' sorry ^
              m ≠
            1
Tactic: classical
  have hn₀ : 0 < n := by
    refine Nat.pos_of_ne_zero fun hn ↦ ?_
    simp only [hn, zero_dvd_iff, Nat.sub_eq_zero_iff_le] at h
    exact (Fintype.one_lt_card.trans_le h).false
  let e := MulChar.equiv_rootsOfUnity F R
  let ζ' : Rˣ := (hζ.isUnit hn₀.ne').unit
  have h' : ζ' ^ (Fintype.card Fˣ : ℕ) = 1 :=
    Units.ext_iff.mpr <| (hζ.pow_eq_one_iff_dvd _).mpr <| Fintype.card_units (α := F) ▸ h
  use e.symm ⟨ζ', (mem_rootsOfUnity (Fintype.card Fˣ) ζ').mpr h'⟩
  rw [e.symm.orderOf_eq, orderOf_eq_iff hn₀]
  refine ⟨?_, fun m hm hm₀ h ↦ ?_⟩
  · ext
    push_cast
    exact hζ.pow_eq_one
  · rw [Subtype.ext_iff, Units.ext_iff] at h
    push_cast at h
    exact ((Nat.le_of_dvd hm₀ <| hζ.dvd_of_pow_eq_one _ h).trans_lt hm).false
-- Lean 4 proof step from Mathlib.NumberTheory.MulChar.Lemmas (goal)
Goal:
F : Type u_1
inst : Field F
inst_1 : Fintype F
R : Type u_2
inst_2 : CommRing R
n : ℕ
h : n ∣ Fintype.card F - 1
ζ : R
hζ : IsPrimitiveRoot ζ n
have : 0 < n
have₁ : ∀ (hn₀ : 0 < n), ∃ (χ : MulChar F R), orderOf χ = n
have₂ : 0 < n
have₃ have₄ have₅ have₆ : ∀ (hn₀ : 0 < n), ∃ (χ : MulChar F R), orderOf χ = n
have₇ :
  ∀ (hn₀ : 0 < n),
    let ζ' := ⋯.unit;
    ∀ (h' : ζ' ^ Fintype.card.{u_1} Fˣ = 1), ∃ (χ : MulChar F R), orderOf χ = n
h₁ :
  ∀ (hn₀ : 0 < n),
    let e : MulChar F R ≃* ↥(rootsOfUnity (Fintype.card.{u_1} Fˣ) R) := equiv_rootsOfUnity F R;
    let ζ' := ⋯.unit;
    ∀ (h' : ζ' ^ Fintype.card.{u_1} Fˣ = 1),
      orderOf (G := MulChar F R)
          ((e.symm : (a : ↥(rootsOfUnity (Fintype.card Fˣ) R)) → MulChar F R)
              (Subtype.mk (p := fun (x : Rˣ) ↦
                Membership.mem (γ := Subgroup Rˣ) (rootsOfUnity (Fintype.card.{u_1} Fˣ) R) x) ζ' ⋯) :
            MulChar F R) =
        n
h₂ :
  ∀ (hn₀ : 0 < n),
    let ζ' := ⋯.unit;
    ∀ (h' : ζ' ^ Fintype.card.{u_1} Fˣ = 1),
      Subtype.mk (p := fun (x : Rˣ) ↦ Membership.mem (γ := Subgroup Rˣ) (rootsOfUnity (Fintype.card.{u_1} Fˣ) R) x) ζ'
              ⋯ ^
            n =
          1 ∧
        ∀ m < n,
          0 < m →
            Subtype.mk (p := fun (x : Rˣ) ↦
                  Membership.mem (γ := Subgroup Rˣ) (rootsOfUnity (Fintype.card.{u_1} Fˣ) R) x) ζ' ⋯ ^
                m ≠
              1
⊢ ∃ (χ : MulChar F R), orderOf χ = n
have h₃ : ∀ (hn₀ : 0 < n),
  let ζ' := sorry.unit;
  ∀ (h' : ζ' ^ Fintype.card.{u_1} Fˣ = 1),
    Subtype.mk (p := fun (x : Rˣ) ↦ Membership.mem (γ := Subgroup Rˣ) (rootsOfUnity (Fintype.card.{u_1} Fˣ) R) x) ζ' sorry ^
        n =
      1
Tactic: classical
  have hn₀ : 0 < n := by
    refine Nat.pos_of_ne_zero fun hn ↦ ?_
    simp only [hn, zero_dvd_iff, Nat.sub_eq_zero_iff_le] at h
    exact (Fintype.one_lt_card.trans_le h).false
  let e := MulChar.equiv_rootsOfUnity F R
  let ζ' : Rˣ := (hζ.isUnit hn₀.ne').unit
  have h' : ζ' ^ (Fintype.card Fˣ : ℕ) = 1 :=
    Units.ext_iff.mpr <| (hζ.pow_eq_one_iff_dvd _).mpr <| Fintype.card_units (α := F) ▸ h
  use e.symm ⟨ζ', (mem_rootsOfUnity (Fintype.card Fˣ) ζ').mpr h'⟩
  rw [e.symm.orderOf_eq, orderOf_eq_iff hn₀]
  refine ⟨?_, fun m hm hm₀ h ↦ ?_⟩
  · ext
    push_cast
    exact hζ.pow_eq_one
  · rw [Subtype.ext_iff, Units.ext_iff] at h
    push_cast at h
    exact ((Nat.le_of_dvd hm₀ <| hζ.dvd_of_pow_eq_one _ h).trans_lt hm).false
-- Lean 4 proof step from Mathlib.NumberTheory.MulChar.Lemmas (goal)
Goal:
F : Type u_1
inst : Field F
inst_1 : Fintype F
R : Type u_2
inst_2 : CommRing R
n : ℕ
h : n ∣ Fintype.card F - 1
ζ : R
hζ : IsPrimitiveRoot ζ n
have : 0 < n
have₁ : ∀ (hn₀ : 0 < n), ∃ (χ : MulChar F R), orderOf χ = n
have₂ : 0 < n
have₃ have₄ have₅ have₆ : ∀ (hn₀ : 0 < n), ∃ (χ : MulChar F R), orderOf χ = n
have₇ :
  ∀ (hn₀ : 0 < n),
    let ζ' := ⋯.unit;
    ∀ (h' : ζ' ^ Fintype.card.{u_1} Fˣ = 1), ∃ (χ : MulChar F R), orderOf χ = n
h₁ :
  ∀ (hn₀ : 0 < n),
    let e : MulChar F R ≃* ↥(rootsOfUnity (Fintype.card.{u_1} Fˣ) R) := equiv_rootsOfUnity F R;
    let ζ' := ⋯.unit;
    ∀ (h' : ζ' ^ Fintype.card.{u_1} Fˣ = 1),
      orderOf (G := MulChar F R)
          ((e.symm : (a : ↥(rootsOfUnity (Fintype.card Fˣ) R)) → MulChar F R)
              (Subtype.mk (p := fun (x : Rˣ) ↦
                Membership.mem (γ := Subgroup Rˣ) (rootsOfUnity (Fintype.card.{u_1} Fˣ) R) x) ζ' ⋯) :
            MulChar F R) =
        n
h₂ :
  ∀ (hn₀ : 0 < n),
    let ζ' := ⋯.unit;
    ∀ (h' : ζ' ^ Fintype.card.{u_1} Fˣ = 1),
      Subtype.mk (p := fun (x : Rˣ) ↦ Membership.mem (γ := Subgroup Rˣ) (rootsOfUnity (Fintype.card.{u_1} Fˣ) R) x) ζ'
              ⋯ ^
            n =
          1 ∧
        ∀ m < n,
          0 < m →
            Subtype.mk (p := fun (x : Rˣ) ↦
                  Membership.mem (γ := Subgroup Rˣ) (rootsOfUnity (Fintype.card.{u_1} Fˣ) R) x) ζ' ⋯ ^
                m ≠
              1
h₃ :
  ∀ (hn₀ : 0 < n),
    let ζ' := ⋯.unit;
    ∀ (h' : ζ' ^ Fintype.card.{u_1} Fˣ = 1),
      Subtype.mk (p := fun (x : Rˣ) ↦ Membership.mem (γ := Subgroup Rˣ) (rootsOfUnity (Fintype.card.{u_1} Fˣ) R) x) ζ'
            ⋯ ^
          n =
        1
⊢ ∃ (χ : MulChar F R), orderOf χ = n
have h₄ : ∀ (hn₀ : 0 < n),
  let ζ' := sorry.unit;
  ∀ (h' : ζ' ^ Fintype.card.{u_1} Fˣ = 1) (m : ℕ) (hm : m < n) (hm₀ : 0 < m)
    (h :
      Subtype.mk (p := fun (x : Rˣ) ↦ Membership.mem (γ := Subgroup Rˣ) (rootsOfUnity (Fintype.card.{u_1} Fˣ) R) x) ζ'
            sorry ^
          m =
        1),
    False
Tactic: classical
  have hn₀ : 0 < n := by
    refine Nat.pos_of_ne_zero fun hn ↦ ?_
    simp only [hn, zero_dvd_iff, Nat.sub_eq_zero_iff_le] at h
    exact (Fintype.one_lt_card.trans_le h).false
  let e := MulChar.equiv_rootsOfUnity F R
  let ζ' : Rˣ := (hζ.isUnit hn₀.ne').unit
  have h' : ζ' ^ (Fintype.card Fˣ : ℕ) = 1 :=
    Units.ext_iff.mpr <| (hζ.pow_eq_one_iff_dvd _).mpr <| Fintype.card_units (α := F) ▸ h
  use e.symm ⟨ζ', (mem_rootsOfUnity (Fintype.card Fˣ) ζ').mpr h'⟩
  rw [e.symm.orderOf_eq, orderOf_eq_iff hn₀]
  refine ⟨?_, fun m hm hm₀ h ↦ ?_⟩
  · ext
    push_cast
    exact hζ.pow_eq_one
  · rw [Subtype.ext_iff, Units.ext_iff] at h
    push_cast at h
    exact ((Nat.le_of_dvd hm₀ <| hζ.dvd_of_pow_eq_one _ h).trans_lt hm).false
-- Lean 4 proof step from Mathlib.NumberTheory.MulChar.Lemmas (goal)
Goal:
F : Type u_1
inst : Field F
inst_1 : Fintype F
R : Type u_2
inst_2 : CommRing R
χ : MulChar F R
⊢ orderOf χ ∣ Fintype.card F - 1
have rw : orderOf χ ∣ Fintype.card.{u_1} Fˣ
Tactic: rw [← Fintype.card_units]
-- Lean 4 proof step from Mathlib.NumberTheory.MulChar.Lemmas (goal)
Goal:
F : Type u_1
inst : Field F
inst_1 : Fintype F
R : Type u_2
inst_2 : CommRing R
χ : MulChar F R
⊢ orderOf χ ∣ Fintype.card F - 1
have rw : orderOf χ ∣ Fintype.card.{u_1} Fˣ
Tactic: classical
  rw [← Fintype.card_units]
  exact orderOf_dvd_of_pow_eq_one χ.pow_card_eq_one
-- Lean 4 proof step from Mathlib.NumberTheory.MulChar.Lemmas (goal)
Goal:
F : Type u_1
inst : Field F
inst_1 : Fintype F
R : Type u_2
inst_2 : CommRing R
inst_3 : DecidableEq F
ζ : R
hζ : IsPrimitiveRoot ζ (Fintype.card.{u_1} Fˣ)
⊢ Fintype.card.{u_1} Fˣ ∣ Fintype.card F - 1
have rw : Fintype.card F - 1 ∣ Fintype.card F - 1
Tactic: rw [Fintype.card_units]
-- Lean 4 proof step from Mathlib.NumberTheory.MulChar.Lemmas (hypothesis)
Goal:
F : Type u_1
inst : Field F
inst_1 : Finite F
R : Type u_2
inst_2 : CommRing R
χ : MulChar F R
a : F
ha : a ≠ 0
⊢ ∃ (ζ : Rˣ), Membership.mem (γ := Subgroup Rˣ) (rootsOfUnity (orderOf χ) R) ζ ∧ (↑ζ : R) = (χ : (a : F) → R) a
have hu : IsUnit.{u_2} (M := R) ((χ : (a : F) → R) a : R)
Tactic: have hu : IsUnit (χ a) := ha.isUnit.map χ
-- Lean 4 proof step from Mathlib.NumberTheory.MulChar.Lemmas (goal)
Goal:
F : Type u_1
inst : Field F
inst_1 : Finite F
R : Type u_2
inst_2 : CommRing R
χ : MulChar F R
a : F
ha : a ≠ 0
hu : IsUnit.{u_2} (M := R) ((χ : (a : F) → R) a : R)
⊢ ∃ (ζ : Rˣ), Membership.mem (γ := Subgroup Rˣ) (rootsOfUnity (orderOf χ) R) ζ ∧ (↑ζ : R) = (χ : (a : F) → R) a
have refine : Membership.mem (γ := Subgroup Rˣ) (rootsOfUnity (orderOf χ) R) hu.unit
Tactic: refine ⟨hu.unit, ?_, hu.unit_spec⟩
-- Lean 4 proof step from Mathlib.NumberTheory.MulChar.Lemmas (goal)
Goal:
F : Type u_1
inst : Field F
inst_1 : Finite F
R : Type u_2
inst_2 : CommRing R
χ : MulChar F R
a : F
ha : a ≠ 0
hu : IsUnit.{u_2} (M := R) ((χ : (a : F) → R) a : R)
⊢ Membership.mem (γ := Subgroup Rˣ) (rootsOfUnity (orderOf χ) R) hu.unit
have rw : hu.unit ^ orderOf χ = 1
Tactic: rw [mem_rootsOfUnity, Units.ext_iff, Units.val_pow_eq_pow_val, Units.val_one,
    IsUnit.unit_spec, ← χ.pow_apply' χ.orderOf_pos.ne', pow_orderOf_eq_one,
    show a = (isUnit_iff_ne_zero.mpr ha).unit by simp only [IsUnit.unit_spec],
    MulChar.one_apply_coe]
-- Lean 4 proof step from Mathlib.NumberTheory.MulChar.Lemmas (goal)
Goal:
F : Type u_1
inst : Field F
inst_1 : Finite F
R : Type u_2
inst_2 : CommRing R
χ : MulChar F R
a : F
ha : a ≠ 0
hu : IsUnit.{u_2} (M := R) ((χ : (a : F) → R) a : R)
rw : hu.unit ^ orderOf χ = 1
⊢ Membership.mem (γ := Subgroup Rˣ) (rootsOfUnity (orderOf χ) R) hu.unit
have rw₁ : (↑(hu.unit ^ orderOf χ) : R) = (↑1 : R)
Tactic: rw [mem_rootsOfUnity, Units.ext_iff, Units.val_pow_eq_pow_val, Units.val_one,
    IsUnit.unit_spec, ← χ.pow_apply' χ.orderOf_pos.ne', pow_orderOf_eq_one,
    show a = (isUnit_iff_ne_zero.mpr ha).unit by simp only [IsUnit.unit_spec],
    MulChar.one_apply_coe]
-- Lean 4 proof step from Mathlib.NumberTheory.MulChar.Lemmas (goal)
Goal:
F : Type u_1
inst : Field F
inst_1 : Finite F
R : Type u_2
inst_2 : CommRing R
χ : MulChar F R
a : F
ha : a ≠ 0
hu : IsUnit.{u_2} (M := R) ((χ : (a : F) → R) a : R)
rw : hu.unit ^ orderOf χ = 1
rw₁ : (↑(hu.unit ^ orderOf χ) : R) = (↑1 : R)
⊢ Membership.mem (γ := Subgroup Rˣ) (rootsOfUnity (orderOf χ) R) hu.unit
have rw₂ : (↑hu.unit : R) ^ orderOf χ = (↑1 : R)
Tactic: rw [mem_rootsOfUnity, Units.ext_iff, Units.val_pow_eq_pow_val, Units.val_one,
    IsUnit.unit_spec, ← χ.pow_apply' χ.orderOf_pos.ne', pow_orderOf_eq_one,
    show a = (isUnit_iff_ne_zero.mpr ha).unit by simp only [IsUnit.unit_spec],
    MulChar.one_apply_coe]
-- Lean 4 proof step from Mathlib.NumberTheory.MulChar.Lemmas (goal)
Goal:
F : Type u_1
inst : Field F
inst_1 : Finite F
R : Type u_2
inst_2 : CommRing R
χ : MulChar F R
a : F
ha : a ≠ 0
hu : IsUnit.{u_2} (M := R) ((χ : (a : F) → R) a : R)
rw : hu.unit ^ orderOf χ = 1
rw₁ : (↑(hu.unit ^ orderOf χ) : R) = (↑1 : R)
rw₂ : (↑hu.unit : R) ^ orderOf χ = (↑1 : R)
⊢ Membership.mem (γ := Subgroup Rˣ) (rootsOfUnity (orderOf χ) R) hu.unit
have rw₃ : (↑hu.unit : R) ^ orderOf χ = 1
Tactic: rw [mem_rootsOfUnity, Units.ext_iff, Units.val_pow_eq_pow_val, Units.val_one,
    IsUnit.unit_spec, ← χ.pow_apply' χ.orderOf_pos.ne', pow_orderOf_eq_one,
    show a = (isUnit_iff_ne_zero.mpr ha).unit by simp only [IsUnit.unit_spec],
    MulChar.one_apply_coe]
-- Lean 4 proof step from Mathlib.NumberTheory.MulChar.Lemmas (goal)
Goal:
F : Type u_1
inst : Field F
inst_1 : Finite F
R : Type u_2
inst_2 : CommRing R
χ : MulChar F R
a : F
ha : a ≠ 0
hu : IsUnit.{u_2} (M := R) ((χ : (a : F) → R) a : R)
rw : hu.unit ^ orderOf χ = 1
rw₁ : (↑(hu.unit ^ orderOf χ) : R) = (↑1 : R)
rw₂ : (↑hu.unit : R) ^ orderOf χ = (↑1 : R)
rw₃ : (↑hu.unit : R) ^ orderOf χ = 1
⊢ Membership.mem (γ := Subgroup Rˣ) (rootsOfUnity (orderOf χ) R) hu.unit
have rw₄ : HPow.hPow (α := R) ((χ : (a : F) → R) a : R) (orderOf χ) = 1
Tactic: rw [mem_rootsOfUnity, Units.ext_iff, Units.val_pow_eq_pow_val, Units.val_one,
    IsUnit.unit_spec, ← χ.pow_apply' χ.orderOf_pos.ne', pow_orderOf_eq_one,
    show a = (isUnit_iff_ne_zero.mpr ha).unit by simp only [IsUnit.unit_spec],
    MulChar.one_apply_coe]
-- Lean 4 proof step from Mathlib.NumberTheory.MulChar.Lemmas (goal)
Goal:
F : Type u_1
inst : Field F
inst_1 : Finite F
R : Type u_2
inst_2 : CommRing R
χ : MulChar F R
a : F
ha : a ≠ 0
hu : IsUnit.{u_2} (M := R) ((χ : (a : F) → R) a : R)
rw : hu.unit ^ orderOf χ = 1
rw₁ : (↑(hu.unit ^ orderOf χ) : R) = (↑1 : R)
rw₂ : (↑hu.unit : R) ^ orderOf χ = (↑1 : R)
rw₃ : (↑hu.unit : R) ^ orderOf χ = 1
rw₄ : HPow.hPow (α := R) ((χ : (a : F) → R) a : R) (orderOf χ) = 1
⊢ Membership.mem (γ := Subgroup Rˣ) (rootsOfUnity (orderOf χ) R) hu.unit
have rw₅ : Eq.{u_2 + 1} (α := R) ((χ ^ orderOf χ : (a : F) → R) a : R) (1 : R)
Tactic: rw [mem_rootsOfUnity, Units.ext_iff, Units.val_pow_eq_pow_val, Units.val_one,
    IsUnit.unit_spec, ← χ.pow_apply' χ.orderOf_pos.ne', pow_orderOf_eq_one,
    show a = (isUnit_iff_ne_zero.mpr ha).unit by simp only [IsUnit.unit_spec],
    MulChar.one_apply_coe]
-- Lean 4 proof step from Mathlib.NumberTheory.MulChar.Lemmas (goal)
Goal:
F : Type u_1
inst : Field F
inst_1 : Finite F
R : Type u_2
inst_2 : CommRing R
χ : MulChar F R
a : F
ha : a ≠ 0
hu : IsUnit.{u_2} (M := R) ((χ : (a : F) → R) a : R)
rw : hu.unit ^ orderOf χ = 1
rw₁ : (↑(hu.unit ^ orderOf χ) : R) = (↑1 : R)
rw₂ : (↑hu.unit : R) ^ orderOf χ = (↑1 : R)
rw₃ : (↑hu.unit : R) ^ orderOf χ = 1
rw₄ : HPow.hPow (α := R) ((χ : (a : F) → R) a : R) (orderOf χ) = 1
rw₅ : Eq.{u_2 + 1} (α := R) ((χ ^ orderOf χ : (a : F) → R) a : R) (1 : R)
⊢ Membership.mem (γ := Subgroup Rˣ) (rootsOfUnity (orderOf χ) R) hu.unit
have rw₆ : Eq.{u_2 + 1} (α := R) ((1 : (a : F) → R) a : R) (1 : R)
Tactic: rw [mem_rootsOfUnity, Units.ext_iff, Units.val_pow_eq_pow_val, Units.val_one,
    IsUnit.unit_spec, ← χ.pow_apply' χ.orderOf_pos.ne', pow_orderOf_eq_one,
    show a = (isUnit_iff_ne_zero.mpr ha).unit by simp only [IsUnit.unit_spec],
    MulChar.one_apply_coe]
-- Lean 4 proof step from Mathlib.NumberTheory.MulChar.Lemmas (goal)
Goal:
F : Type u_1
inst : Field F
inst_1 : Finite F
R : Type u_2
inst_2 : CommRing R
χ : MulChar F R
a : F
ha : a ≠ 0
hu : IsUnit.{u_2} (M := R) ((χ : (a : F) → R) a : R)
rw : hu.unit ^ orderOf χ = 1
rw₁ : (↑(hu.unit ^ orderOf χ) : R) = (↑1 : R)
rw₂ : (↑hu.unit : R) ^ orderOf χ = (↑1 : R)
rw₃ : (↑hu.unit : R) ^ orderOf χ = 1
rw₄ : HPow.hPow (α := R) ((χ : (a : F) → R) a : R) (orderOf χ) = 1
rw₅ : Eq.{u_2 + 1} (α := R) ((χ ^ orderOf χ : (a : F) → R) a : R) (1 : R)
rw₆ : Eq.{u_2 + 1} (α := R) ((1 : (a : F) → R) a : R) (1 : R)
⊢ Membership.mem (γ := Subgroup Rˣ) (rootsOfUnity (orderOf χ) R) hu.unit
have rw₇ : Eq.{u_2 + 1} (α := R) ((1 : (a : F) → R) (↑(IsUnit.unit (a := a) sorry) : F) : R) (1 : R)
Tactic: rw [mem_rootsOfUnity, Units.ext_iff, Units.val_pow_eq_pow_val, Units.val_one,
    IsUnit.unit_spec, ← χ.pow_apply' χ.orderOf_pos.ne', pow_orderOf_eq_one,
    show a = (isUnit_iff_ne_zero.mpr ha).unit by simp only [IsUnit.unit_spec],
    MulChar.one_apply_coe]
-- Lean 4 proof step from Mathlib.NumberTheory.MulChar.Lemmas (goal)
Goal:
F : Type u_1
inst : Field F
inst_1 : Finite F
R : Type u_2
inst_2 : CommRing R
χ : MulChar F R
a : F
ha : a ≠ 0
hu : IsUnit.{u_2} (M := R) ((χ : (a : F) → R) a : R)
rw : hu.unit ^ orderOf χ = 1
rw₁ : (↑(hu.unit ^ orderOf χ) : R) = (↑1 : R)
rw₂ : (↑hu.unit : R) ^ orderOf χ = (↑1 : R)
rw₃ : (↑hu.unit : R) ^ orderOf χ = 1
rw₄ : HPow.hPow (α := R) ((χ : (a : F) → R) a : R) (orderOf χ) = 1
rw₅ : Eq.{u_2 + 1} (α := R) ((χ ^ orderOf χ : (a : F) → R) a : R) (1 : R)
rw₆ : Eq.{u_2 + 1} (α := R) ((1 : (a : F) → R) a : R) (1 : R)
rw₇ : Eq.{u_2 + 1} (α := R) ((1 : (a : F) → R) (↑(IsUnit.unit (a := a) ⋯) : F) : R) (1 : R)
⊢ Membership.mem (γ := Subgroup Rˣ) (rootsOfUnity (orderOf χ) R) hu.unit
have rw₈ : Eq.{u_2 + 1} (α := R) 1 1
Tactic: rw [mem_rootsOfUnity, Units.ext_iff, Units.val_pow_eq_pow_val, Units.val_one,
    IsUnit.unit_spec, ← χ.pow_apply' χ.orderOf_pos.ne', pow_orderOf_eq_one,
    show a = (isUnit_iff_ne_zero.mpr ha).unit by simp only [IsUnit.unit_spec],
    MulChar.one_apply_coe]
-- Lean 4 proof step from Mathlib.NumberTheory.MulChar.Lemmas (goal)
Goal:
F : Type u_1
inst : Field F
inst_1 : Finite F
R : Type u_2
inst_2 : CommRing R
χ : MulChar F R
n : ℕ
hχ : χ ^ n = 1
a : F
ha : a ≠ 0
⊢ ∃ (ζ : Rˣ), Membership.mem (γ := Subgroup Rˣ) (rootsOfUnity n R) ζ ∧ (↑ζ : R) = (χ : (a : F) → R) a
have obtain : ∀ (μ : Rˣ) (hμ₁ : Membership.mem (γ := Subgroup Rˣ) (rootsOfUnity (orderOf χ) R) μ)
  (hμ₂ : (↑μ : R) = (χ : (a : F) → R) a),
  ∃ (ζ : Rˣ), Membership.mem (γ := Subgroup Rˣ) (rootsOfUnity n R) ζ ∧ (↑ζ : R) = (χ : (a : F) → R) a
Tactic: obtain ⟨μ, hμ₁, hμ₂⟩ := χ.apply_mem_rootsOfUnity_orderOf ha
-- Lean 4 proof step from Mathlib.NumberTheory.MulChar.Lemmas (goal)
Goal:
F : Type u_1
inst : Field F
inst_1 : Finite F
R : Type u_2
inst_2 : CommRing R
inst_3 : IsDomain R
χ : MulChar F R
n : ℕ
inst_4 : NeZero n
hχ : χ ^ n = 1
μ : R
hμ : IsPrimitiveRoot μ n
a : F
ha : a ≠ 0
⊢ ∃ k < n, (χ : (a : F) → R) a = μ ^ k
have obtain : ∀ (ζ : Rˣ) (hζ₁ : Membership.mem (γ := Subgroup Rˣ) (rootsOfUnity n R) ζ) (hζ₂ : (↑ζ : R) = (χ : (a : F) → R) a),
  ∃ k < n, (χ : (a : F) → R) a = μ ^ k
Tactic: obtain ⟨ζ, hζ₁, hζ₂⟩ := apply_mem_rootsOfUnity_of_pow_eq_one hχ ha
-- Lean 4 proof step from Mathlib.NumberTheory.MulChar.Lemmas (hypothesis)
Goal:
F : Type u_1
inst : Field F
inst_1 : Finite F
R : Type u_2
inst_2 : CommRing R
inst_3 : IsDomain R
χ : MulChar F R
n : ℕ
inst_4 : NeZero n
hχ : χ ^ n = 1
μ : R
hμ : IsPrimitiveRoot μ n
a : F
ha : a ≠ 0
ζ : Rˣ
hζ₁ : Membership.mem (γ := Subgroup Rˣ) (rootsOfUnity n R) ζ
hζ₂ : (↑ζ : R) = (χ : (a : F) → R) a
⊢ ∃ k < n, (χ : (a : F) → R) a = μ ^ k
have hζ' : (↑ζ : R) ^ n = 1
Tactic: have hζ' : ζ.val ^ n = 1 := (mem_rootsOfUnity' n ↑ζ).mp hζ₁
-- Lean 4 proof step from Mathlib.NumberTheory.MulChar.Lemmas (goal)
Goal:
F : Type u_1
inst : Field F
inst_1 : Finite F
R : Type u_2
inst_2 : CommRing R
inst_3 : IsDomain R
χ : MulChar F R
n : ℕ
inst_4 : NeZero n
hχ : χ ^ n = 1
μ : R
hμ : IsPrimitiveRoot μ n
a : F
ha : a ≠ 0
ζ : Rˣ
hζ₁ : Membership.mem (γ := Subgroup Rˣ) (rootsOfUnity n R) ζ
hζ₂ : (↑ζ : R) = (χ : (a : F) → R) a
hζ' : (↑ζ : R) ^ n = 1
⊢ ∃ k < n, (χ : (a : F) → R) a = μ ^ k
have obtain : ∀ (k : ℕ) (hk₁ : k < n) (hk₂ : μ ^ k = (↑ζ : R)), ∃ k < n, (χ : (a : F) → R) a = μ ^ k
Tactic: obtain ⟨k, hk₁, hk₂⟩ := hμ.eq_pow_of_pow_eq_one hζ'
-- Lean 4 proof step from Mathlib.NumberTheory.MulChar.Lemmas (goal)
Goal:
F : Type u_1
inst : Field F
inst_1 : Finite F
R : Type u_2
inst_2 : CommRing R
inst_3 : IsDomain R
χ : MulChar F R
n : ℕ
inst_4 : NeZero n
hχ : χ ^ n = 1
μ : R
hμ : IsPrimitiveRoot μ n
a : F
⊢ Membership.mem (α := R) (γ := Subalgebra ℤ R) ℤ[μ] ((χ : (a : F) → R) a : R)
have inl : Membership.mem (α := R) (γ := Subalgebra ℤ R) ℤ[μ] ((χ : (a : F) → R) 0 : R)
Tactic: rcases eq_or_ne a 0 with rfl | h
Cleared hypotheses: a
-- Lean 4 proof step from Mathlib.NumberTheory.MulChar.Lemmas (goal)
Goal:
F : Type u_1
inst : Field F
inst_1 : Finite F
R : Type u_2
inst_2 : CommRing R
inst_3 : IsDomain R
χ : MulChar F R
n : ℕ
inst_4 : NeZero n
hχ : χ ^ n = 1
μ : R
hμ : IsPrimitiveRoot μ n
a : F
inl : Membership.mem (α := R) (γ := Subalgebra ℤ R) ℤ[μ] ((χ : (a : F) → R) 0 : R)
⊢ Membership.mem (α := R) (γ := Subalgebra ℤ R) ℤ[μ] ((χ : (a : F) → R) a : R)
have inr : ∀ (h : a ≠ 0), Membership.mem (α := R) (γ := Subalgebra ℤ R) ℤ[μ] ((χ : (a : F) → R) a : R)
Tactic: rcases eq_or_ne a 0 with rfl | h