awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.NumberTheory.MulChar.Lemmas (goal)
Goal:
R : Type u_1
inst : CommMonoid R
R' : Type u_3
inst_1 : CommMonoidWithZero R'
f : MonoidHom.{u_1, u_3} Rˣ R'ˣ
S : Submonoid R
x : (↥S)ˣ
h :
  Eq (α := R')
    ((((Equiv.symm equivToUnitHom : (a : Rˣ →* R'ˣ) → MulChar R R') f : MulChar R R') : (a : R) → R')
        (↑(↑((S.unitsEquivUnitsType.symm : (a : (↥S)ˣ) → ↥S.units) x) : Rˣ) : R) :
      R')
    ((↑((f : (a : Rˣ) → R'ˣ) (↑((S.unitsEquivUnitsType.symm : (a : (↥S)ˣ) → ↥S.units) x) : Rˣ)) : R') : R')
⊢ Eq (α := R')
    ((((Equiv.symm equivToUnitHom : (a : Rˣ →* R'ˣ) → MulChar R R') f : MulChar R R') : (a : R) → R') (↑(↑x : ↥S) : R) :
      R')
    ((↑((f : (a : Rˣ) → R'ˣ) (↑((S.unitsEquivUnitsType.symm : (a : (↥S)ˣ) → ↥S.units) x) : Rˣ)) : R') : R')
have h₁ : Eq (α := R') ((↑((f : (a : Rˣ) → R'ˣ) (↑((S.unitsEquivUnitsType.symm : (a : (↥S)ˣ) → ↥S.units) x) : Rˣ)) : R') : R')
  ((↑((f : (a : Rˣ) → R'ˣ) (↑((S.unitsEquivUnitsType.symm : (a : (↥S)ˣ) → ↥S.units) x) : Rˣ)) : R') : R')
Tactic: rw [← Submonoid.val_unitsEquivUnitsType_symm_apply_coe S x, equivToUnitHom_symm_coe]
-- Lean 4 proof step from Mathlib.NumberTheory.MulChar.Lemmas (goal)
Goal:
R : Type u_1
inst : CommMonoid R
S : Type u_2
inst_1 : SetLike S R
inst_2 : SubmonoidClass S R
T : S
R' : Type u_3
inst_3 : CommMonoidWithZero R'
⊢ Eq (α := MulChar.{u_1, u_3} (↥T) R') (restrict T 1) 1
have h : ∀ (x : (↥T)ˣ),
  Eq.{u_3 + 1} (α := R') ((restrict T 1 : (a : ↥T) → R') (↑x : ↥T) : R') ((1 : (a : ↥T) → R') (↑x : ↥T) : R')
Tactic: ext x
-- Lean 4 proof step from Mathlib.NumberTheory.MulChar.Lemmas (goal)
Goal:
R : Type u_1
inst : CommMonoid R
S : Type u_2
inst_1 : SetLike S R
inst_2 : SubmonoidClass S R
T : S
R' : Type u_3
inst_3 : CommMonoidWithZero R'
x : (↥T)ˣ
⊢ Eq.{u_3 + 1} (α := R') ((restrict T 1 : (a : ↥T) → R') (↑x : ↥T) : R') ((1 : (a : ↥T) → R') (↑x : ↥T) : R')
have h : Eq.{u_3 + 1} (α := R') (if IsUnit (↑x : ↥T) then (1 : (a : R) → R') (↑(↑x : ↥T) : R) else 0 : R')
  ((1 : (a : ↥T) → R') (↑x : ↥T) : R')
Tactic: rw [restrict_apply, if_pos x.isUnit, MulChar.one_apply x.isUnit.coe, one_apply_coe]
-- Lean 4 proof step from Mathlib.NumberTheory.MulChar.Lemmas (goal)
Goal:
R : Type u_1
inst : CommMonoid R
S : Type u_2
inst_1 : SetLike S R
inst_2 : SubmonoidClass S R
T : S
R' : Type u_3
inst_3 : CommMonoidWithZero R'
x : (↥T)ˣ
h :
  Eq.{u_3 + 1} (α := R') (if IsUnit (↑x : ↥T) then (1 : (a : R) → R') (↑(↑x : ↥T) : R) else 0 : R')
    ((1 : (a : ↥T) → R') (↑x : ↥T) : R')
⊢ Eq.{u_3 + 1} (α := R') ((restrict T 1 : (a : ↥T) → R') (↑x : ↥T) : R') ((1 : (a : ↥T) → R') (↑x : ↥T) : R')
have h₁ : Eq.{u_3 + 1} (α := R') ((1 : (a : R) → R') (↑(↑x : ↥T) : R) : R') ((1 : (a : ↥T) → R') (↑x : ↥T) : R')
Tactic: rw [restrict_apply, if_pos x.isUnit, MulChar.one_apply x.isUnit.coe, one_apply_coe]
-- Lean 4 proof step from Mathlib.NumberTheory.MulChar.Lemmas (goal)
Goal:
R : Type u_1
inst : CommMonoid R
S : Type u_2
inst_1 : SetLike S R
inst_2 : SubmonoidClass S R
T : S
R' : Type u_3
inst_3 : CommMonoidWithZero R'
x : (↥T)ˣ
h :
  Eq.{u_3 + 1} (α := R') (if IsUnit (↑x : ↥T) then (1 : (a : R) → R') (↑(↑x : ↥T) : R) else 0 : R')
    ((1 : (a : ↥T) → R') (↑x : ↥T) : R')
h₁ : Eq.{u_3 + 1} (α := R') ((1 : (a : R) → R') (↑(↑x : ↥T) : R) : R') ((1 : (a : ↥T) → R') (↑x : ↥T) : R')
⊢ Eq.{u_3 + 1} (α := R') ((restrict T 1 : (a : ↥T) → R') (↑x : ↥T) : R') ((1 : (a : ↥T) → R') (↑x : ↥T) : R')
have h₂ : Eq.{u_3 + 1} (α := R') (1 : R') ((1 : (a : ↥T) → R') (↑x : ↥T) : R')
Tactic: rw [restrict_apply, if_pos x.isUnit, MulChar.one_apply x.isUnit.coe, one_apply_coe]
-- Lean 4 proof step from Mathlib.NumberTheory.MulChar.Lemmas (goal)
Goal:
R : Type u_1
inst : CommMonoid R
S : Type u_2
inst_1 : SetLike S R
inst_2 : SubmonoidClass S R
T : S
R' : Type u_3
inst_3 : CommMonoidWithZero R'
x : (↥T)ˣ
h :
  Eq.{u_3 + 1} (α := R') (if IsUnit (↑x : ↥T) then (1 : (a : R) → R') (↑(↑x : ↥T) : R) else 0 : R')
    ((1 : (a : ↥T) → R') (↑x : ↥T) : R')
h₁ : Eq.{u_3 + 1} (α := R') ((1 : (a : R) → R') (↑(↑x : ↥T) : R) : R') ((1 : (a : ↥T) → R') (↑x : ↥T) : R')
h₂ : Eq.{u_3 + 1} (α := R') (1 : R') ((1 : (a : ↥T) → R') (↑x : ↥T) : R')
⊢ Eq.{u_3 + 1} (α := R') ((restrict T 1 : (a : ↥T) → R') (↑x : ↥T) : R') ((1 : (a : ↥T) → R') (↑x : ↥T) : R')
have h₃ : Eq.{u_3 + 1} (α := R') 1 1
Tactic: rw [restrict_apply, if_pos x.isUnit, MulChar.one_apply x.isUnit.coe, one_apply_coe]
-- Lean 4 proof step from Mathlib.NumberTheory.MulChar.Lemmas (goal)
Goal:
R : Type u_1
inst : CommMonoid R
S : Type u_2
inst_1 : SetLike S R
inst_2 : SubmonoidClass S R
T : S
R' : Type u_3
inst_3 : CommMonoidWithZero R'
x x_1 : MulChar R R'
⊢ restrict T (x * x_1) = HMul.hMul (α := MulChar.{u_1, u_3} (↥T) R') (restrict T x) (restrict T x_1)
have h : ∀ (a : (↥T)ˣ),
  Eq.{u_3 + 1} (α := R') ((restrict T (x * x_1) : (a : ↥T) → R') (↑a : ↥T) : R')
    ((HMul.hMul (α := MulChar.{u_1, u_3} (↥T) R') (restrict T x) (restrict T x_1) : (a : ↥T) → R') (↑a : ↥T) : R')
Tactic: ext
-- Lean 4 proof step from Mathlib.NumberTheory.MulChar.Lemmas (goal)
Goal:
R : Type u_1
R' : Type u_2
inst : CommRing R
inst_1 : CommRing R'
inst_2 : StarRing R'
χ : MulChar R R'
⊢ χ.starComp.starComp = χ
have h : ∀ (a : Rˣ),
  Eq.{u_2 + 1} (α := R') ((χ.starComp.starComp : (a : R) → R') (↑a : R) : R') ((χ : (a : R) → R') (↑a : R) : R')
Tactic: ext1
-- Lean 4 proof step from Mathlib.NumberTheory.MulChar.Lemmas (goal)
Goal:
R : Type u_1
R' : Type u_2
inst : CommRing R
inst_1 : CommRing R'
inst_2 : StarRing R'
χ χ' : MulChar R R'
⊢ (χ * χ').starComp = χ'.starComp * χ.starComp
have h : ∀ (a : Rˣ),
  Eq.{u_2 + 1} (α := R') (((χ * χ').starComp : (a : R) → R') (↑a : R) : R')
    ((χ'.starComp * χ.starComp : (a : R) → R') (↑a : R) : R')
Tactic: ext1
-- Lean 4 proof step from Mathlib.NumberTheory.MulChar.Lemmas (goal)
Goal:
R : Type u_1
R' : Type u_2
inst : CommRing R
inst_1 : CommRing R'
inst_2 : Fintype Rˣ
a : Rˣ
χ : MulChar R R'
⊢ Membership.mem (α := R'ˣ) (γ := Subgroup R'ˣ) (rootsOfUnity (Fintype.card.{u_1} Rˣ) R')
    ((((equivToUnitHom : (a : MulChar R R') → Rˣ →* R'ˣ) χ : MonoidHom.{u_1, u_2} Rˣ R'ˣ) : (a : Rˣ) → R'ˣ) a : R'ˣ)
have rw : HPow.hPow (α := R'ˣ)
    ((((equivToUnitHom : (a : MulChar R R') → Rˣ →* R'ˣ) χ : MonoidHom.{u_1, u_2} Rˣ R'ˣ) : (a : Rˣ) → R'ˣ) a : R'ˣ)
    (Fintype.card.{u_1} Rˣ) =
  1
Tactic: rw [mem_rootsOfUnity, ← map_pow, ← (equivToUnitHom χ).map_one, pow_card_eq_one]
-- Lean 4 proof step from Mathlib.NumberTheory.MulChar.Lemmas (goal)
Goal:
R : Type u_1
R' : Type u_2
inst : CommRing R
inst_1 : CommRing R'
inst_2 : Fintype Rˣ
a : Rˣ
χ : MulChar R R'
rw :
  HPow.hPow (α := R'ˣ)
      ((((equivToUnitHom : (a : MulChar R R') → Rˣ →* R'ˣ) χ : MonoidHom.{u_1, u_2} Rˣ R'ˣ) : (a : Rˣ) → R'ˣ) a : R'ˣ)
      (Fintype.card.{u_1} Rˣ) =
    1
⊢ Membership.mem (α := R'ˣ) (γ := Subgroup R'ˣ) (rootsOfUnity (Fintype.card.{u_1} Rˣ) R')
    ((((equivToUnitHom : (a : MulChar R R') → Rˣ →* R'ˣ) χ : MonoidHom.{u_1, u_2} Rˣ R'ˣ) : (a : Rˣ) → R'ˣ) a : R'ˣ)
have rw₁ : Eq.{u_2 + 1} (α := R'ˣ)
  ((((equivToUnitHom : (a : MulChar R R') → Rˣ →* R'ˣ) χ : MonoidHom.{u_1, u_2} Rˣ R'ˣ) : (a : Rˣ) → R'ˣ)
      (a ^ Fintype.card.{u_1} Rˣ) :
    R'ˣ)
  (1 : R'ˣ)
Tactic: rw [mem_rootsOfUnity, ← map_pow, ← (equivToUnitHom χ).map_one, pow_card_eq_one]
-- Lean 4 proof step from Mathlib.NumberTheory.MulChar.Lemmas (goal)
Goal:
R : Type u_1
R' : Type u_2
inst : CommRing R
inst_1 : CommRing R'
inst_2 : Fintype Rˣ
a : Rˣ
χ : MulChar R R'
rw :
  HPow.hPow (α := R'ˣ)
      ((((equivToUnitHom : (a : MulChar R R') → Rˣ →* R'ˣ) χ : MonoidHom.{u_1, u_2} Rˣ R'ˣ) : (a : Rˣ) → R'ˣ) a : R'ˣ)
      (Fintype.card.{u_1} Rˣ) =
    1
rw₁ :
  Eq.{u_2 + 1} (α := R'ˣ)
    ((((equivToUnitHom : (a : MulChar R R') → Rˣ →* R'ˣ) χ : MonoidHom.{u_1, u_2} Rˣ R'ˣ) : (a : Rˣ) → R'ˣ)
        (a ^ Fintype.card.{u_1} Rˣ) :
      R'ˣ)
    (1 : R'ˣ)
⊢ Membership.mem (α := R'ˣ) (γ := Subgroup R'ˣ) (rootsOfUnity (Fintype.card.{u_1} Rˣ) R')
    ((((equivToUnitHom : (a : MulChar R R') → Rˣ →* R'ˣ) χ : MonoidHom.{u_1, u_2} Rˣ R'ˣ) : (a : Rˣ) → R'ˣ) a : R'ˣ)
have rw₂ : Eq.{u_2 + 1} (α := R'ˣ)
  ((((equivToUnitHom : (a : MulChar R R') → Rˣ →* R'ˣ) χ : MonoidHom.{u_1, u_2} Rˣ R'ˣ) : (a : Rˣ) → R'ˣ)
      (a ^ Fintype.card.{u_1} Rˣ) :
    R'ˣ)
  ((((equivToUnitHom : (a : MulChar R R') → Rˣ →* R'ˣ) χ : MonoidHom.{u_1, u_2} Rˣ R'ˣ) : (a : Rˣ) → R'ˣ) 1 : R'ˣ)
Tactic: rw [mem_rootsOfUnity, ← map_pow, ← (equivToUnitHom χ).map_one, pow_card_eq_one]
-- Lean 4 proof step from Mathlib.NumberTheory.MulChar.Lemmas (goal)
Goal:
R : Type u_1
R' : Type u_2
inst : CommRing R
inst_1 : CommRing R'
inst_2 : Fintype Rˣ
a : Rˣ
χ : MulChar R R'
rw :
  HPow.hPow (α := R'ˣ)
      ((((equivToUnitHom : (a : MulChar R R') → Rˣ →* R'ˣ) χ : MonoidHom.{u_1, u_2} Rˣ R'ˣ) : (a : Rˣ) → R'ˣ) a : R'ˣ)
      (Fintype.card.{u_1} Rˣ) =
    1
rw₁ :
  Eq.{u_2 + 1} (α := R'ˣ)
    ((((equivToUnitHom : (a : MulChar R R') → Rˣ →* R'ˣ) χ : MonoidHom.{u_1, u_2} Rˣ R'ˣ) : (a : Rˣ) → R'ˣ)
        (a ^ Fintype.card.{u_1} Rˣ) :
      R'ˣ)
    (1 : R'ˣ)
rw₂ :
  Eq.{u_2 + 1} (α := R'ˣ)
    ((((equivToUnitHom : (a : MulChar R R') → Rˣ →* R'ˣ) χ : MonoidHom.{u_1, u_2} Rˣ R'ˣ) : (a : Rˣ) → R'ˣ)
        (a ^ Fintype.card.{u_1} Rˣ) :
      R'ˣ)
    ((((equivToUnitHom : (a : MulChar R R') → Rˣ →* R'ˣ) χ : MonoidHom.{u_1, u_2} Rˣ R'ˣ) : (a : Rˣ) → R'ˣ) 1 : R'ˣ)
⊢ Membership.mem (α := R'ˣ) (γ := Subgroup R'ˣ) (rootsOfUnity (Fintype.card.{u_1} Rˣ) R')
    ((((equivToUnitHom : (a : MulChar R R') → Rˣ →* R'ˣ) χ : MonoidHom.{u_1, u_2} Rˣ R'ˣ) : (a : Rˣ) → R'ˣ) a : R'ˣ)
have rw₃ : Eq.{u_2 + 1} (α := R'ˣ)
  ((((equivToUnitHom : (a : MulChar R R') → Rˣ →* R'ˣ) χ : MonoidHom.{u_1, u_2} Rˣ R'ˣ) : (a : Rˣ) → R'ˣ) 1)
  ((((equivToUnitHom : (a : MulChar R R') → Rˣ →* R'ˣ) χ : MonoidHom.{u_1, u_2} Rˣ R'ˣ) : (a : Rˣ) → R'ˣ) 1)
Tactic: rw [mem_rootsOfUnity, ← map_pow, ← (equivToUnitHom χ).map_one, pow_card_eq_one]
-- Lean 4 proof step from Mathlib.NumberTheory.MulChar.Lemmas (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : Finite.{u_1 + 1} Rˣ
χ : MulChar R ℂ
⊢ star χ = χ⁻¹
have intro : ∀ (val : Fintype Rˣ), star χ = χ⁻¹
Tactic: cases nonempty_fintype Rˣ
-- Lean 4 proof step from Mathlib.NumberTheory.MulChar.Lemmas (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : Finite.{u_1 + 1} Rˣ
χ : MulChar R ℂ
val : Fintype Rˣ
⊢ star χ = χ⁻¹
have intro : ∀ (a : Rˣ), Eq (α := ℂ) ((star χ : (a : R) → ℂ) (↑a : R) : ℂ) ((χ⁻¹ : (a : R) → ℂ) (↑a : R) : ℂ)
Tactic: ext1 a
-- Lean 4 proof step from Mathlib.NumberTheory.MulChar.Lemmas (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : Finite.{u_1 + 1} Rˣ
χ : MulChar R ℂ
val : Fintype Rˣ
a : Rˣ
⊢ Eq (α := ℂ) ((star χ : (a : R) → ℂ) (↑a : R) : ℂ) ((χ⁻¹ : (a : R) → ℂ) (↑a : R) : ℂ)
have intro : Eq (α := ℂ) ((star χ : (a : R) → ℂ) (↑a : R) : ℂ) (((χ : (a : R) → ℂ) (↑a : R))⁻¹ : ℂ)
Tactic: simp only [inv_apply_eq_inv']
-- Lean 4 proof step from Mathlib.NumberTheory.MulChar.Lemmas (hypothesis)
Goal:
M : Type u_1
inst : CommMonoid M
inst_1 : Fintype M
inst_2 : DecidableEq M
R : Type u_2
inst_3 : CommMonoidWithZero R
ζ : Rˣ
hζ : Membership.mem (γ := Subgroup Rˣ) (rootsOfUnity (Fintype.card.{u_1} Mˣ) R) ζ
g : Mˣ
hg : ∀ (x : Mˣ), Membership.mem (γ := Subgroup Mˣ) (Subgroup.zpowers g) x
⊢ MulChar M R
have this : orderOf ζ ∣ Fintype.card.{u_1} Mˣ
Tactic: have : orderOf ζ ∣ Fintype.card Mˣ :=
    orderOf_dvd_iff_pow_eq_one.mpr <| (mem_rootsOfUnity _ ζ).mp hζ
-- Lean 4 proof step from Mathlib.NumberTheory.MulChar.Lemmas (goal)
Goal:
M : Type u_1
inst : CommMonoid M
inst_1 : Fintype M
inst_2 : DecidableEq M
R : Type u_2
inst_3 : CommMonoidWithZero R
ζ : Rˣ
hζ : Membership.mem (γ := Subgroup Rˣ) (rootsOfUnity (Fintype.card.{u_1} Mˣ) R) ζ
g : Mˣ
hg : ∀ (x : Mˣ), Membership.mem (γ := Subgroup Mˣ) (Subgroup.zpowers g) x
this : orderOf ζ ∣ Fintype.card.{u_1} Mˣ
⊢ MulChar M R
have refine : Fintype.card.{u_1} Mˣ = orderOf g
Tactic: refine ofUnitHom <| monoidHomOfForallMemZpowers hg <| this.trans <| dvd_of_eq ?_
-- Lean 4 proof step from Mathlib.NumberTheory.MulChar.Lemmas (goal)
Goal:
M : Type u_1
inst : CommMonoid M
inst_1 : Fintype M
inst_2 : DecidableEq M
R : Type u_2
inst_3 : CommMonoidWithZero R
ζ : Rˣ
hζ : Membership.mem (γ := Subgroup Rˣ) (rootsOfUnity (Fintype.card.{u_1} Mˣ) R) ζ
g : Mˣ
hg : ∀ (x : Mˣ), Membership.mem (γ := Subgroup Mˣ) (Subgroup.zpowers g) x
this : orderOf ζ ∣ Fintype.card.{u_1} Mˣ
⊢ Fintype.card.{u_1} Mˣ = orderOf g
have rw : Fintype.card.{u_1} Mˣ = Nat.card.{u_1} Mˣ
Tactic: rw [orderOf_eq_card_of_forall_mem_zpowers hg, Nat.card_eq_fintype_card]
-- Lean 4 proof step from Mathlib.NumberTheory.MulChar.Lemmas (goal)
Goal:
M : Type u_1
inst : CommMonoid M
inst_1 : Fintype M
inst_2 : DecidableEq M
R : Type u_2
inst_3 : CommMonoidWithZero R
ζ : Rˣ
hζ : Membership.mem (γ := Subgroup Rˣ) (rootsOfUnity (Fintype.card.{u_1} Mˣ) R) ζ
g : Mˣ
hg : ∀ (x : Mˣ), Membership.mem (γ := Subgroup Mˣ) (Subgroup.zpowers g) x
this : orderOf ζ ∣ Fintype.card.{u_1} Mˣ
rw : Fintype.card.{u_1} Mˣ = Nat.card.{u_1} Mˣ
⊢ Fintype.card.{u_1} Mˣ = orderOf g
have rw₁ : Fintype.card.{u_1} Mˣ = Fintype.card.{u_1} Mˣ
Tactic: rw [orderOf_eq_card_of_forall_mem_zpowers hg, Nat.card_eq_fintype_card]
-- Lean 4 proof step from Mathlib.NumberTheory.MulChar.Lemmas (goal)
Goal:
M : Type u_1
inst : CommMonoid M
inst_1 : Fintype M
inst_2 : DecidableEq M
R : Type u_2
inst_3 : CommMonoidWithZero R
inst_cyc : IsCyclic.{u_1} Mˣ
ζ : ↥(rootsOfUnity (Fintype.card.{u_1} Mˣ) R)
⊢ (fun (χ : MulChar M R) ↦
        (Subtype.mk (p := fun (x : Rˣ) ↦ Membership.mem (γ := Subgroup Rˣ) (rootsOfUnity (Fintype.card.{u_1} Mˣ) R) x)
            ((χ.toUnitHom : (a : Mˣ) → Rˣ)
              (Classical.choose (p := fun (x : Mˣ) ↦
                ∀ (x_1 : Mˣ), Membership.mem (γ := Subgroup Mˣ) (Subgroup.zpowers x) x_1) ⋯))
            ⋯ :
          ↥(rootsOfUnity (Fintype.card.{u_1} Mˣ) R)))
      ((fun (ζ : ↥(rootsOfUnity (Fintype.card.{u_1} Mˣ) R)) ↦
          ofRootOfUnity ⋯ (g :=
            Classical.choose (p := fun (x : Mˣ) ↦
              ∀ (x_1 : Mˣ), Membership.mem (γ := Subgroup Mˣ) (Subgroup.zpowers x) x_1) ⋯)
            ⋯)
        ζ) =
    ζ
have a : (↑(↑((fun (χ : MulChar M R) ↦
              (Subtype.mk (p := fun (x : Rˣ) ↦
                  Membership.mem (γ := Subgroup Rˣ) (rootsOfUnity (Fintype.card.{u_1} Mˣ) R) x)
                  ((χ.toUnitHom : (a : Mˣ) → Rˣ)
                    (Classical.choose (p := fun (x : Mˣ) ↦
                      ∀ (x_1 : Mˣ), Membership.mem (γ := Subgroup Mˣ) (Subgroup.zpowers x) x_1) sorry))
                  sorry :
                ↥(rootsOfUnity (Fintype.card.{u_1} Mˣ) R)))
            ((fun (ζ : ↥(rootsOfUnity (Fintype.card.{u_1} Mˣ) R)) ↦
                ofRootOfUnity sorry (g :=
                  Classical.choose (p := fun (x : Mˣ) ↦
                    ∀ (x_1 : Mˣ), Membership.mem (γ := Subgroup Mˣ) (Subgroup.zpowers x) x_1) sorry)
                  sorry)
              ζ)) :
        Rˣ) :
    R) =
  (↑(↑ζ : Rˣ) : R)
Tactic: ext
-- 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
⊢ 0 < n
have refine : ∀ (hn : n = 0), False
Tactic: refine Nat.pos_of_ne_zero fun hn ↦ ?_
-- Lean 4 proof step from Mathlib.NumberTheory.MulChar.Lemmas (hypothesis)
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
hn : n = 0
⊢ False
have h₁ : Fintype.card F ≤ 1
Tactic: simp only [hn, zero_dvd_iff, Nat.sub_eq_zero_iff_le] at h
Cleared hypotheses: h
-- Lean 4 proof step from Mathlib.NumberTheory.MulChar.Lemmas (hypothesis)
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
⊢ ∃ (χ : MulChar F R), orderOf χ = n
have hn₀ : 0 < n
Tactic: 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
-- Lean 4 proof step from Mathlib.NumberTheory.MulChar.Lemmas (hypothesis)
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
hn₀ : 0 < n
e : MulChar F R ≃* ↥(rootsOfUnity (Fintype.card.{u_1} Fˣ) R) := equiv_rootsOfUnity F R
ζ' : Rˣ := ⋯.unit
⊢ ∃ (χ : MulChar F R), orderOf χ = n
have h' : ζ' ^ Fintype.card.{u_1} Fˣ = 1
Tactic: 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
-- 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
hn₀ : 0 < n
e : MulChar F R ≃* ↥(rootsOfUnity (Fintype.card.{u_1} Fˣ) R) := equiv_rootsOfUnity F R
ζ' : Rˣ := ⋯.unit
h' : ζ' ^ Fintype.card.{u_1} Fˣ = 1
⊢ ∃ (χ : MulChar F R), orderOf χ = n
have h₁ : 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: use e.symm ⟨ζ', (mem_rootsOfUnity (Fintype.card Fˣ) ζ').mpr h'⟩
-- 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
hn₀ : 0 < n
e : MulChar F R ≃* ↥(rootsOfUnity (Fintype.card.{u_1} Fˣ) R) := equiv_rootsOfUnity F R
ζ' : Rˣ := ⋯.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
have h₁ : 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: rw [e.symm.orderOf_eq, orderOf_eq_iff hn₀]
-- 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
hn₀ : 0 < n
e : MulChar F R ≃* ↥(rootsOfUnity (Fintype.card.{u_1} Fˣ) R) := equiv_rootsOfUnity F R
ζ' : Rˣ := ⋯.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
have h₁ : Subtype.mk (p := fun (x : Rˣ) ↦ Membership.mem (γ := Subgroup Rˣ) (rootsOfUnity (Fintype.card.{u_1} Fˣ) R) x) ζ' sorry ^ n =
  1
Tactic: refine ⟨?_, fun m hm hm₀ h ↦ ?_⟩
-- 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
hn₀ : 0 < n
e : MulChar F R ≃* ↥(rootsOfUnity (Fintype.card.{u_1} Fˣ) R) := equiv_rootsOfUnity F R
ζ' : Rˣ := ⋯.unit
h' : ζ' ^ Fintype.card.{u_1} Fˣ = 1
h₁ :
  Subtype.mk (p := fun (x : Rˣ) ↦ Membership.mem (γ := Subgroup Rˣ) (rootsOfUnity (Fintype.card.{u_1} Fˣ) R) x) ζ' ⋯ ^
      n =
    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
have h₂ : ∀ (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: refine ⟨?_, fun m hm hm₀ h ↦ ?_⟩
-- 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
hn₀ : 0 < n
e : MulChar F R ≃* ↥(rootsOfUnity (Fintype.card.{u_1} Fˣ) R) := equiv_rootsOfUnity F R
ζ' : Rˣ := ⋯.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
have h₁ : (↑(↑(Subtype.mk (p := fun (x : Rˣ) ↦ Membership.mem (γ := Subgroup Rˣ) (rootsOfUnity (Fintype.card.{u_1} Fˣ) R) x) ζ'
              sorry ^
            n) :
        Rˣ) :
    R) =
  (↑(↑1 : Rˣ) : R)
Tactic: ext
-- 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
hn₀ : 0 < n
e : MulChar F R ≃* ↥(rootsOfUnity (Fintype.card.{u_1} Fˣ) R) := equiv_rootsOfUnity F R
ζ' : Rˣ := ⋯.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) :
          Rˣ) :
      R) =
    (↑(↑1 : Rˣ) : R)
have h₁ : (↑ζ' : R) ^ n = 1
Tactic: push_cast
-- Lean 4 proof step from Mathlib.NumberTheory.MulChar.Lemmas (hypothesis)
Goal:
F : Type u_1
inst : Field F
inst_1 : Fintype F
R : Type u_2
inst_2 : CommRing R
n : ℕ
h_1 : n ∣ Fintype.card F - 1
ζ : R
hζ : IsPrimitiveRoot ζ n
hn₀ : 0 < n
e : MulChar F R ≃* ↥(rootsOfUnity (Fintype.card.{u_1} Fˣ) R) := equiv_rootsOfUnity F R
ζ' : Rˣ := ⋯.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' ⋯ ^
      m =
    1
⊢ False
have h₁ : (↑(↑(Subtype.mk (p := fun (x : Rˣ) ↦ Membership.mem (γ := Subgroup Rˣ) (rootsOfUnity (Fintype.card.{u_1} Fˣ) R) x) ζ'
              sorry ^
            m) :
        Rˣ) :
    R) =
  (↑(↑1 : Rˣ) : R)
Tactic: rw [Subtype.ext_iff, Units.ext_iff] at h
Cleared hypotheses: h
-- 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
⊢ ∃ (χ : MulChar F R), orderOf χ = n
have have : 0 < 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