awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.Complex (goal)
Goal:
ζ : ℂˣ
n : ℕ
inst : NeZero n
hζ : Membership.mem (γ := Subgroup ℂˣ) (rootsOfUnity n ℂ) ζ
rw : HMul.hMul (α := ℂ) ((starRingEnd ℂ : (a : ℂ) → ℂ) (↑ζ : ℂ) : ℂ) (↑ζ : ℂ) = 1
rw₁ : HPow.hPow (β := ℕ) (↑‖(↑ζ : ℂ)‖ : ℂ) 2 = 1
rw₂ : HPow.hPow (β := ℕ) (↑1 : ℂ) 2 = 1
⊢ (starRingEnd ℂ : (a : ℂ) → ℂ) (↑ζ : ℂ) = (↑ζ⁻¹ : ℂ)
have rw₃ : HPow.hPow (α := ℂ) (β := ℕ) 1 2 = 1
Tactic: rw [← Units.mul_eq_one_iff_eq_inv, conj_mul', norm_eq_one_of_mem_rootsOfUnity hζ, ofReal_one,
    one_pow]
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.Complex (goal)
Goal:
ζ : ℂˣ
n : ℕ
inst : NeZero n
hζ : Membership.mem (γ := Subgroup ℂˣ) (rootsOfUnity n ℂ) ζ
rw : HMul.hMul (α := ℂ) ((starRingEnd ℂ : (a : ℂ) → ℂ) (↑ζ : ℂ) : ℂ) (↑ζ : ℂ) = 1
rw₁ : HPow.hPow (β := ℕ) (↑‖(↑ζ : ℂ)‖ : ℂ) 2 = 1
rw₂ : HPow.hPow (β := ℕ) (↑1 : ℂ) 2 = 1
rw₃ : HPow.hPow (α := ℂ) (β := ℕ) 1 2 = 1
⊢ (starRingEnd ℂ : (a : ℂ) → ℂ) (↑ζ : ℂ) = (↑ζ⁻¹ : ℂ)
have rw₄ : Eq (α := ℂ) 1 1
Tactic: rw [← Units.mul_eq_one_iff_eq_inv, conj_mul', norm_eq_one_of_mem_rootsOfUnity hζ, ofReal_one,
    one_pow]
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.CyclotomicUnits (goal)
Goal:
n j : ℕ
A : Type u_1
ζ : A
inst : CommRing A
inst_1 : IsDomain A
hζ : IsPrimitiveRoot ζ n
hj : j.Coprime n
⊢ Associated (ζ - 1) (ζ ^ j - 1)
have refine : ζ ^ j - 1 ∣ ζ - 1
Tactic: refine associated_of_dvd_dvd ⟨∑ i ∈ range j, ζ ^ i, (mul_geom_sum _ _).symm⟩ ?_
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.CyclotomicUnits (goal)
Goal:
n_1 j : ℕ
A : Type u_1
ζ : A
inst : CommRing A
inst_1 : IsDomain A
n : ℕ
hζ : IsPrimitiveRoot ζ (n + 2)
hj : j.Coprime (n + 2)
⊢ ζ ^ j - 1 ∣ ζ - 1
have obtain : ∀ (m : ℕ) (hm : j * m % (n + 2) = 1), ζ ^ j - 1 ∣ ζ - 1
Tactic: obtain ⟨m, -, hm⟩ := exists_mul_mod_eq_one_of_coprime hj (by lia)
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.CyclotomicUnits (goal)
Goal:
n_1 j : ℕ
A : Type u_1
ζ : A
inst : CommRing A
inst_1 : IsDomain A
n : ℕ
hζ : IsPrimitiveRoot ζ (n + 2)
hj : j.Coprime (n + 2)
m : ℕ
hm : j * m % (n + 2) = 1
⊢ ζ ^ j - 1 ∣ ζ - 1
have h : ζ - 1 = (ζ ^ j - 1) * ∑ i ∈ range m, (ζ ^ j) ^ i
Tactic: use ∑ i ∈ range m, (ζ ^ j) ^ i
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.CyclotomicUnits (goal)
Goal:
n_1 j : ℕ
A : Type u_1
ζ : A
inst : CommRing A
inst_1 : IsDomain A
n : ℕ
hζ : IsPrimitiveRoot ζ (n + 2)
hj : j.Coprime (n + 2)
m : ℕ
hm : j * m % (n + 2) = 1
⊢ ζ - 1 = (ζ ^ j - 1) * ∑ i ∈ range m, (ζ ^ j) ^ i
have h : ζ - 1 = (ζ ^ j) ^ m - 1
Tactic: rw [mul_geom_sum, ← pow_mul, ← pow_mod_orderOf, ← hζ.eq_orderOf, hm, pow_one]
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.CyclotomicUnits (goal)
Goal:
n_1 j : ℕ
A : Type u_1
ζ : A
inst : CommRing A
inst_1 : IsDomain A
n : ℕ
hζ : IsPrimitiveRoot ζ (n + 2)
hj : j.Coprime (n + 2)
m : ℕ
hm : j * m % (n + 2) = 1
h : ζ - 1 = (ζ ^ j) ^ m - 1
⊢ ζ - 1 = (ζ ^ j - 1) * ∑ i ∈ range m, (ζ ^ j) ^ i
have h₁ : ζ - 1 = ζ ^ (j * m) - 1
Tactic: rw [mul_geom_sum, ← pow_mul, ← pow_mod_orderOf, ← hζ.eq_orderOf, hm, pow_one]
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.CyclotomicUnits (goal)
Goal:
n_1 j : ℕ
A : Type u_1
ζ : A
inst : CommRing A
inst_1 : IsDomain A
n : ℕ
hζ : IsPrimitiveRoot ζ (n + 2)
hj : j.Coprime (n + 2)
m : ℕ
hm : j * m % (n + 2) = 1
h : ζ - 1 = (ζ ^ j) ^ m - 1
h₁ : ζ - 1 = ζ ^ (j * m) - 1
⊢ ζ - 1 = (ζ ^ j - 1) * ∑ i ∈ range m, (ζ ^ j) ^ i
have h₂ : ζ - 1 = ζ ^ (j * m % orderOf ζ) - 1
Tactic: rw [mul_geom_sum, ← pow_mul, ← pow_mod_orderOf, ← hζ.eq_orderOf, hm, pow_one]
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.CyclotomicUnits (goal)
Goal:
n_1 j : ℕ
A : Type u_1
ζ : A
inst : CommRing A
inst_1 : IsDomain A
n : ℕ
hζ : IsPrimitiveRoot ζ (n + 2)
hj : j.Coprime (n + 2)
m : ℕ
hm : j * m % (n + 2) = 1
h : ζ - 1 = (ζ ^ j) ^ m - 1
h₁ : ζ - 1 = ζ ^ (j * m) - 1
h₂ : ζ - 1 = ζ ^ (j * m % orderOf ζ) - 1
⊢ ζ - 1 = (ζ ^ j - 1) * ∑ i ∈ range m, (ζ ^ j) ^ i
have h₃ : ζ - 1 = ζ ^ (j * m % (n + 2)) - 1
Tactic: rw [mul_geom_sum, ← pow_mul, ← pow_mod_orderOf, ← hζ.eq_orderOf, hm, pow_one]
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.CyclotomicUnits (goal)
Goal:
n_1 j : ℕ
A : Type u_1
ζ : A
inst : CommRing A
inst_1 : IsDomain A
n : ℕ
hζ : IsPrimitiveRoot ζ (n + 2)
hj : j.Coprime (n + 2)
m : ℕ
hm : j * m % (n + 2) = 1
h : ζ - 1 = (ζ ^ j) ^ m - 1
h₁ : ζ - 1 = ζ ^ (j * m) - 1
h₂ : ζ - 1 = ζ ^ (j * m % orderOf ζ) - 1
h₃ : ζ - 1 = ζ ^ (j * m % (n + 2)) - 1
⊢ ζ - 1 = (ζ ^ j - 1) * ∑ i ∈ range m, (ζ ^ j) ^ i
have h₄ : ζ - 1 = HPow.hPow (β := ℕ) ζ 1 - 1
Tactic: rw [mul_geom_sum, ← pow_mul, ← pow_mod_orderOf, ← hζ.eq_orderOf, hm, pow_one]
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.CyclotomicUnits (goal)
Goal:
n_1 j : ℕ
A : Type u_1
ζ : A
inst : CommRing A
inst_1 : IsDomain A
n : ℕ
hζ : IsPrimitiveRoot ζ (n + 2)
hj : j.Coprime (n + 2)
m : ℕ
hm : j * m % (n + 2) = 1
h : ζ - 1 = (ζ ^ j) ^ m - 1
h₁ : ζ - 1 = ζ ^ (j * m) - 1
h₂ : ζ - 1 = ζ ^ (j * m % orderOf ζ) - 1
h₃ : ζ - 1 = ζ ^ (j * m % (n + 2)) - 1
h₄ : ζ - 1 = HPow.hPow (β := ℕ) ζ 1 - 1
⊢ ζ - 1 = (ζ ^ j - 1) * ∑ i ∈ range m, (ζ ^ j) ^ i
have h₅ : ζ - 1 = ζ - 1
Tactic: rw [mul_geom_sum, ← pow_mul, ← pow_mod_orderOf, ← hζ.eq_orderOf, hm, pow_one]
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.CyclotomicUnits (goal)
Goal:
n i j : ℕ
A : Type u_1
ζ : A
inst : CommRing A
inst_1 : IsDomain A
hζ : IsPrimitiveRoot ζ n
hi : i.Coprime n
hj : j.Coprime n
⊢ Associated (ζ ^ j - 1) (ζ ^ i - 1)
have suffices : ∀ {j : ℕ}, j.Coprime n → Associated (ζ - 1) (ζ ^ j - 1)
Tactic: suffices ∀ {j}, j.Coprime n → Associated (ζ - 1) (ζ ^ j - 1) by
    grind [Associated.trans, Associated.symm]
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.CyclotomicUnits (goal)
Goal:
A : Type u_1
ζ : A
inst : CommRing A
inst_1 : IsDomain A
R : Type u_3
inst_2 : CommRing R
inst_3 : Algebra R A
n : ℕ
inst_4 : NeZero n
hζ : IsPrimitiveRoot ζ n
σ : A ≃ₐ[R] A
⊢ Associated (ζ - 1) ((σ : (a : A) → A) (ζ - 1))
have rw : Associated (ζ - 1)
  (ζ ^ ZMod.val (n := n) (↑(((autToPow R hζ : (A ≃ₐ[R] A) →* (ZMod n)ˣ) : (a : A ≃ₐ[R] A) → (ZMod n)ˣ) σ) : ZMod n) - 1)
Tactic: rw [map_sub, map_one, ← hζ.autToPow_spec R σ]
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.CyclotomicUnits (goal)
Goal:
A : Type u_1
ζ : A
inst : CommRing A
inst_1 : IsDomain A
R : Type u_3
inst_2 : CommRing R
inst_3 : Algebra R A
n : ℕ
inst_4 : NeZero n
hζ : IsPrimitiveRoot ζ n
σ : A ≃ₐ[R] A
⊢ Associated (ζ - 1)
    (ζ ^ ZMod.val (n := n) (↑(((autToPow R hζ : (A ≃ₐ[R] A) →* (ZMod n)ˣ) : (a : A ≃ₐ[R] A) → (ZMod n)ˣ) σ) : ZMod n) -
      1)
have apply : (ZMod.val (n := n) (↑(((autToPow R hζ : (A ≃ₐ[R] A) →* (ZMod n)ˣ) : (a : A ≃ₐ[R] A) → (ZMod n)ˣ) σ) : ZMod n)).Coprime n
Tactic: apply hζ.associated_sub_one_pow_sub_one_of_coprime
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.CyclotomicUnits (goal)
Goal:
A : Type u_1
ζ : A
inst : CommRing A
inst_1 : IsDomain A
R : Type u_3
inst_2 : CommRing R
inst_3 : Algebra R A
n : ℕ
inst_4 : NeZero n
hζ : IsPrimitiveRoot ζ n
σ τ : A ≃ₐ[R] A
⊢ Associated (M := A) ((σ : (a : A) → A) (ζ - 1) : A) ((τ : (a : A) → A) (ζ - 1) : A)
have rw : Associated
  (ζ ^ ZMod.val (n := n) (↑(((autToPow R hζ : (A ≃ₐ[R] A) →* (ZMod n)ˣ) : (a : A ≃ₐ[R] A) → (ZMod n)ˣ) σ) : ZMod n) - 1)
  (ζ ^ ZMod.val (n := n) (↑(((autToPow R hζ : (A ≃ₐ[R] A) →* (ZMod n)ˣ) : (a : A ≃ₐ[R] A) → (ZMod n)ˣ) τ) : ZMod n) - 1)
Tactic: rw [map_sub, map_sub, map_one, map_one, ← hζ.autToPow_spec R σ, ← hζ.autToPow_spec R τ]
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.CyclotomicUnits (goal)
Goal:
A : Type u_1
ζ : A
inst : CommRing A
inst_1 : IsDomain A
R : Type u_3
inst_2 : CommRing R
inst_3 : Algebra R A
n : ℕ
inst_4 : NeZero n
hζ : IsPrimitiveRoot ζ n
σ τ : A ≃ₐ[R] A
⊢ Associated
    (ζ ^ ZMod.val (n := n) (↑(((autToPow R hζ : (A ≃ₐ[R] A) →* (ZMod n)ˣ) : (a : A ≃ₐ[R] A) → (ZMod n)ˣ) σ) : ZMod n) -
      1)
    (ζ ^ ZMod.val (n := n) (↑(((autToPow R hζ : (A ≃ₐ[R] A) →* (ZMod n)ˣ) : (a : A ≃ₐ[R] A) → (ZMod n)ˣ) τ) : ZMod n) -
      1)
have hi : (ZMod.val (n := n) (↑(((autToPow R hζ : (A ≃ₐ[R] A) →* (ZMod n)ˣ) : (a : A ≃ₐ[R] A) → (ZMod n)ˣ) τ) : ZMod n)).Coprime n
Tactic: apply hζ.associated_pow_sub_one_pow_of_coprime
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.CyclotomicUnits (goal)
Goal:
A : Type u_1
ζ : A
inst : CommRing A
inst_1 : IsDomain A
R : Type u_3
inst_2 : CommRing R
inst_3 : Algebra R A
n : ℕ
inst_4 : NeZero n
hζ : IsPrimitiveRoot ζ n
σ τ : A ≃ₐ[R] A
hi :
  (ZMod.val (n := n) (↑(((autToPow R hζ : (A ≃ₐ[R] A) →* (ZMod n)ˣ) : (a : A ≃ₐ[R] A) → (ZMod n)ˣ) τ) : ZMod n)).Coprime
    n
⊢ Associated
    (ζ ^ ZMod.val (n := n) (↑(((autToPow R hζ : (A ≃ₐ[R] A) →* (ZMod n)ˣ) : (a : A ≃ₐ[R] A) → (ZMod n)ˣ) σ) : ZMod n) -
      1)
    (ζ ^ ZMod.val (n := n) (↑(((autToPow R hζ : (A ≃ₐ[R] A) →* (ZMod n)ˣ) : (a : A ≃ₐ[R] A) → (ZMod n)ˣ) τ) : ZMod n) -
      1)
have hj : (ZMod.val (n := n) (↑(((autToPow R hζ : (A ≃ₐ[R] A) →* (ZMod n)ˣ) : (a : A ≃ₐ[R] A) → (ZMod n)ˣ) σ) : ZMod n)).Coprime n
Tactic: apply hζ.associated_pow_sub_one_pow_of_coprime
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.CyclotomicUnits (goal)
Goal:
n j : ℕ
A : Type u_1
ζ : A
inst : CommRing A
inst_1 : IsDomain A
hζ : IsPrimitiveRoot ζ n
hn : 2 ≤ n
hj : j.Coprime n
⊢ IsUnit.{u_1} (M := A) (∑ i ∈ range j, ζ ^ i)
have obtain : ∀ (u : Aˣ) (hu : (HPow.hPow (β := ℕ) ζ 1 - 1) * (↑u : A) = ζ ^ j - 1), IsUnit.{u_1} (M := A) (∑ i ∈ range j, ζ ^ i)
Tactic: obtain ⟨u, hu⟩ := hζ.associated_pow_sub_one_pow_of_coprime hj (coprime_one_left n)
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.CyclotomicUnits (goal)
Goal:
n j : ℕ
A : Type u_1
ζ : A
inst : CommRing A
inst_1 : IsDomain A
hζ : IsPrimitiveRoot ζ n
hn : 2 ≤ n
hj : j.Coprime n
u : Aˣ
hu : (HPow.hPow (β := ℕ) ζ 1 - 1) * (↑u : A) = ζ ^ j - 1
⊢ IsUnit.{u_1} (M := A) (∑ i ∈ range j, ζ ^ i)
have h : ∑ i ∈ range j, ζ ^ i = (↑u : A)
Tactic: convert! u.isUnit
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.CyclotomicUnits (goal)
Goal:
n j : ℕ
A : Type u_1
ζ : A
inst : CommRing A
inst_1 : IsDomain A
hζ : IsPrimitiveRoot ζ n
hn : 2 ≤ n
hj : j.Coprime n
u : Aˣ
hu : (HPow.hPow (β := ℕ) ζ 1 - 1) * (↑u : A) = ζ ^ j - 1
⊢ ∑ i ∈ range j, ζ ^ i = (↑u : A)
have h : Eq.{u_1 + 1} (α := A) ((fun (x : A) ↦ (1 - ζ) * x) (∑ i ∈ range j, ζ ^ i)) ((fun (x : A) ↦ (1 - ζ) * x) (↑u : A))
Tactic: apply mul_right_injective₀ (show 1 - ζ ≠ 0 by grind [sub_one_ne_zero])
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.CyclotomicUnits (goal)
Goal:
n_1 j : ℕ
A : Type u_1
ζ : A
inst : CommRing A
inst_1 : IsDomain A
hj_Unit : IsUnit.{u_1} (M := A) (↑j : A)
n : ℕ
hζ : IsPrimitiveRoot ζ (n + 2)
hj : j.Coprime (n + 2)
⊢ 2 ≤ n + 2
have linarith : n + 2 < 2 → -1 + HSub.hSub (α := ℤ) 0 (↑n : ℤ) + (HAdd.hAdd (α := ℤ) (↑n : ℤ) 2 + 1 - 2) = 0
Tactic: linarith
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.CyclotomicUnits (goal)
Goal:
n j : ℕ
A : Type u_1
ζ : A
inst : CommRing A
inst_1 : IsDomain A
hζ : IsPrimitiveRoot ζ n
hn : 2 ≤ n
i : ℕ
hjn : j.Coprime n
⊢ Associated (ζ - 1) (ζ ^ (i + j) - ζ ^ i)
have h : (ζ - 1) * (↑(sorry.unit ^ i * sorry.unit) : A) = ζ ^ (i + j) - ζ ^ i
Tactic: use (hζ.isUnit (by lia)).unit ^ i * (hζ.geom_sum_isUnit hn hjn).unit
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.CyclotomicUnits (goal)
Goal:
n j : ℕ
A : Type u_1
ζ : A
inst : CommRing A
inst_1 : IsDomain A
hζ : IsPrimitiveRoot ζ n
hn : 2 ≤ n
i : ℕ
hjn : j.Coprime n
⊢ (ζ - 1) * (↑(⋯.unit ^ i * ⋯.unit) : A) = ζ ^ (i + j) - ζ ^ i
have h : (ζ - 1) * ζ ^ i * ∑ i ∈ range j, ζ ^ i = ζ ^ (i + j) - ζ ^ i
Tactic: suffices (ζ - 1) * ζ ^ i * ∑ i ∈ range j, ζ ^ i = (ζ ^ (i + j) - ζ ^ i) by
    simp [← this, mul_assoc]
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.CyclotomicUnits (hypothesis)
Goal:
p : ℕ
A : Type u_1
ζ : A
inst : CommRing A
inst_1 : IsDomain A
hζ : IsPrimitiveRoot ζ p
hp : Nat.Prime p
η₁ : A
hη₁ : η₁ ∈ (↑(nthRootsFinset p 1) : Set A)
η₂ : A
hη₂ : η₂ ∈ (↑(nthRootsFinset p 1) : Set A)
e : η₁ ≠ η₂
⊢ Associated (ζ - 1) (η₁ - η₂)
have this : NeZero p
Tactic: have : NeZero p := ⟨hp.ne_zero⟩
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.CyclotomicUnits (goal)
Goal:
p : ℕ
A : Type u_1
ζ : A
inst : CommRing A
inst_1 : IsDomain A
hζ : IsPrimitiveRoot ζ p
hp : Nat.Prime p
η₁ : A
hη₁ : η₁ ∈ (↑(nthRootsFinset p 1) : Set A)
η₂ : A
hη₂ : η₂ ∈ (↑(nthRootsFinset p 1) : Set A)
e : η₁ ≠ η₂
this : NeZero p
⊢ Associated (ζ - 1) (η₁ - η₂)
have obtain : ∀ (i : ℕ) (hi : i < p) (hη₁ : ζ ^ i ∈ (↑(nthRootsFinset p 1) : Set A)) (e : ζ ^ i ≠ η₂), Associated (ζ - 1) (ζ ^ i - η₂)
Tactic: obtain ⟨i, hi, rfl⟩ :=
    hζ.eq_pow_of_pow_eq_one ((Polynomial.mem_nthRootsFinset hp.pos 1).1 hη₁)
Cleared hypotheses: η₁, hη₁, e
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.CyclotomicUnits (goal)
Goal:
p : ℕ
A : Type u_1
ζ : A
inst : CommRing A
inst_1 : IsDomain A
hζ : IsPrimitiveRoot ζ p
hp : Nat.Prime p
η₂ : A
hη₂ : η₂ ∈ (↑(nthRootsFinset p 1) : Set A)
this : NeZero p
i : ℕ
hi : i < p
hη₁ : ζ ^ i ∈ (↑(nthRootsFinset p 1) : Set A)
e : ζ ^ i ≠ η₂
⊢ Associated (ζ - 1) (ζ ^ i - η₂)
have obtain : ∀ (j : ℕ) (hj : j < p) (hη₂ : ζ ^ j ∈ (↑(nthRootsFinset p 1) : Set A)) (e : ζ ^ i ≠ ζ ^ j),
  Associated (ζ - 1) (ζ ^ i - ζ ^ j)
Tactic: obtain ⟨j, hj, rfl⟩ :=
    hζ.eq_pow_of_pow_eq_one ((Polynomial.mem_nthRootsFinset hp.pos 1).1 hη₂)
Cleared hypotheses: η₂, hη₂, e
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.CyclotomicUnits (goal)
Goal:
p : ℕ
A : Type u_1
ζ : A
inst : CommRing A
inst_1 : IsDomain A
hζ : IsPrimitiveRoot ζ p
hp : Nat.Prime p
this : NeZero p
i : ℕ
hi : i < p
hη₁ : ζ ^ i ∈ (↑(nthRootsFinset p 1) : Set A)
j : ℕ
hj : j < p
hη₂ : ζ ^ j ∈ (↑(nthRootsFinset p 1) : Set A)
e : ζ ^ i ≠ ζ ^ j
⊢ Associated (ζ - 1) (ζ ^ i - ζ ^ j)
have inr : ∀
  (this :
    ∀ {p : ℕ} {A : Type u_1} {ζ : A} [inst : CommRing A] [inst_2 : IsDomain A] (hζ : IsPrimitiveRoot ζ p)
      (hp : Nat.Prime p) (this : NeZero p) (i : ℕ) (hi : i < p) (hη₁ : ζ ^ i ∈ (↑(nthRootsFinset p 1) : Set A)) (j : ℕ)
      (hj : j < p) (hη₂ : ζ ^ j ∈ (↑(nthRootsFinset p 1) : Set A)) (e : ζ ^ i ≠ ζ ^ j) (hij : j ≤ i),
      Associated (ζ - 1) (ζ ^ i - ζ ^ j))
  (hij : ¬j ≤ i), Associated (ζ - 1) (ζ ^ i - ζ ^ j)
Tactic: wlog hij : j ≤ i
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.CyclotomicUnits (goal)
Goal:
p : ℕ
A : Type u_1
ζ : A
inst : CommRing A
inst_1 : IsDomain A
hζ : IsPrimitiveRoot ζ p
hp : Nat.Prime p
this : NeZero p
i : ℕ
hi : i < p
hη₁ : ζ ^ i ∈ (↑(nthRootsFinset p 1) : Set A)
j : ℕ
hj : j < p
hη₂ : ζ ^ j ∈ (↑(nthRootsFinset p 1) : Set A)
e : ζ ^ i ≠ ζ ^ j
inr :
  ∀
    (this :
      ∀ {p : ℕ} {A : Type u_1} {ζ : A} [inst : CommRing A] [inst_2 : IsDomain A] (hζ : IsPrimitiveRoot ζ p)
        (hp : Nat.Prime p) (this : NeZero p) (i : ℕ) (hi : i < p) (hη₁ : ζ ^ i ∈ (↑(nthRootsFinset p 1) : Set A))
        (j : ℕ) (hj : j < p) (hη₂ : ζ ^ j ∈ (↑(nthRootsFinset p 1) : Set A)) (e : ζ ^ i ≠ ζ ^ j) (hij : j ≤ i),
        Associated (ζ - 1) (ζ ^ i - ζ ^ j))
    (hij : ¬j ≤ i), Associated (ζ - 1) (ζ ^ i - ζ ^ j)
⊢ Associated (ζ - 1) (ζ ^ i - ζ ^ j)
have wlog : ∀ {p : ℕ} {A : Type u_1} {ζ : A} [inst : CommRing A] [inst_2 : IsDomain A] (hζ : IsPrimitiveRoot ζ p) (hp : Nat.Prime p)
  (this : NeZero p) (i : ℕ) (hi : i < p) (hη₁ : ζ ^ i ∈ (↑(nthRootsFinset p 1) : Set A)) (j : ℕ) (hj : j < p)
  (hη₂ : ζ ^ j ∈ (↑(nthRootsFinset p 1) : Set A)) (e : ζ ^ i ≠ ζ ^ j) (hij : j ≤ i), Associated (ζ - 1) (ζ ^ i - ζ ^ j)
Tactic: wlog hij : j ≤ i
Cleared hypotheses: hζ, hp, this, i, hi, hη₁, j, hj, hη₂, e
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.CyclotomicUnits (hypothesis)
Goal:
p_1 : ℕ
A_1 : Type u_1
ζ_1 : A_1
inst : CommRing A_1
inst_1 : IsDomain A_1
p : ℕ
A : Type u_1
ζ : A
inst_2 : CommRing A
inst_3 : IsDomain A
hζ : IsPrimitiveRoot ζ p
hp : Nat.Prime p
this : NeZero p
i : ℕ
hi : i < p
hη₁ : ζ ^ i ∈ (↑(nthRootsFinset p 1) : Set A)
j : ℕ
hj : j < p
hη₂ : ζ ^ j ∈ (↑(nthRootsFinset p 1) : Set A)
e : ζ ^ i ≠ ζ ^ j
hij : j ≤ i
⊢ Associated (ζ - 1) (ζ ^ i - ζ ^ j)
have H : (i - j).Coprime p
Tactic: have H : (i - j).Coprime p := (coprime_of_lt_prime (by grind) (by grind) hp).symm
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.CyclotomicUnits (goal)
Goal:
p_1 : ℕ
A_1 : Type u_1
ζ_1 : A_1
inst : CommRing A_1
inst_1 : IsDomain A_1
p : ℕ
A : Type u_1
ζ : A
inst_2 : CommRing A
inst_3 : IsDomain A
hζ : IsPrimitiveRoot ζ p
hp : Nat.Prime p
this : NeZero p
i : ℕ
hi : i < p
hη₁ : ζ ^ i ∈ (↑(nthRootsFinset p 1) : Set A)
j : ℕ
hj : j < p
hη₂ : ζ ^ j ∈ (↑(nthRootsFinset p 1) : Set A)
e : ζ ^ i ≠ ζ ^ j
hij : j ≤ i
H : (i - j).Coprime p
⊢ Associated (ζ - 1) (ζ ^ i - ζ ^ j)
have obtain : ∀ (u : Aˣ) (h : (ζ - 1) * (↑u : A) = ζ ^ (j + (i - j)) - ζ ^ j), Associated (ζ - 1) (ζ ^ i - ζ ^ j)
Tactic: obtain ⟨u, h⟩ := hζ.associated_pow_add_sub_sub_one hp.two_le j H
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.CyclotomicUnits (hypothesis)
Goal:
p_1 : ℕ
A_1 : Type u_1
ζ_1 : A_1
inst : CommRing A_1
inst_1 : IsDomain A_1
p : ℕ
A : Type u_1
ζ : A
inst_2 : CommRing A
inst_3 : IsDomain A
hζ : IsPrimitiveRoot ζ p
hp : Nat.Prime p
this : NeZero p
i : ℕ
hi : i < p
hη₁ : ζ ^ i ∈ (↑(nthRootsFinset p 1) : Set A)
j : ℕ
hj : j < p
hη₂ : ζ ^ j ∈ (↑(nthRootsFinset p 1) : Set A)
e : ζ ^ i ≠ ζ ^ j
hij : j ≤ i
H : (i - j).Coprime p
u : Aˣ
h : (ζ - 1) * (↑u : A) = ζ ^ (j + (i - j)) - ζ ^ j
⊢ Associated (ζ - 1) (ζ ^ i - ζ ^ j)
have h₁ : (ζ - 1) * (↑u : A) = ζ ^ i - ζ ^ j
Tactic: simp only [hij, add_tsub_cancel_of_le] at h
Cleared hypotheses: h
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.CyclotomicUnits (goal)
Goal:
p_1 : ℕ
A_1 : Type u_1
ζ_1 : A_1
inst : CommRing A_1
inst_1 : IsDomain A_1
p : ℕ
A : Type u_1
ζ : A
inst_2 : CommRing A
inst_3 : IsDomain A
hζ : IsPrimitiveRoot ζ p
hp : Nat.Prime p
this : NeZero p
i : ℕ
hi : i < p
hη₁ : ζ ^ i ∈ (↑(nthRootsFinset p 1) : Set A)
j : ℕ
hj : j < p
hη₂ : ζ ^ j ∈ (↑(nthRootsFinset p 1) : Set A)
e : ζ ^ i ≠ ζ ^ j
hij : j ≤ i
H : (i - j).Coprime p
u : Aˣ
h : (ζ - 1) * (↑u : A) = ζ ^ i - ζ ^ j
⊢ Associated (ζ - 1) (ζ ^ i - ζ ^ j)
have rw : Associated (ζ - 1) ((ζ - 1) * (↑u : A))
Tactic: rw [← h, associated_mul_unit_right_iff]
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.CyclotomicUnits (goal)
Goal:
p_1 : ℕ
A_1 : Type u_1
ζ_1 : A_1
inst : CommRing A_1
inst_1 : IsDomain A_1
p : ℕ
A : Type u_1
ζ : A
inst_2 : CommRing A
inst_3 : IsDomain A
hζ : IsPrimitiveRoot ζ p
hp : Nat.Prime p
this : NeZero p
i : ℕ
hi : i < p
hη₁ : ζ ^ i ∈ (↑(nthRootsFinset p 1) : Set A)
j : ℕ
hj : j < p
hη₂ : ζ ^ j ∈ (↑(nthRootsFinset p 1) : Set A)
e : ζ ^ i ≠ ζ ^ j
hij : j ≤ i
H : (i - j).Coprime p
u : Aˣ
h : (ζ - 1) * (↑u : A) = ζ ^ i - ζ ^ j
rw : Associated (ζ - 1) ((ζ - 1) * (↑u : A))
⊢ Associated (ζ - 1) (ζ ^ i - ζ ^ j)
have rw₁ : Associated (ζ - 1) (ζ - 1)
Tactic: rw [← h, associated_mul_unit_right_iff]
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.EnoughRootsOfUnity (hypothesis)
Goal:
M : Type u_1
inst : CommMonoid M
n : ℕ
inst_1 : NeZero n
inst_2 : HasEnoughRootsOfUnity M n
⊢ Finite.{u_1 + 1} ↥(rootsOfUnity n M)
have this : IsCyclic.{u_1} ↥(rootsOfUnity n M)
Tactic: have := rootsOfUnity_isCyclic M n
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.EnoughRootsOfUnity (goal)
Goal:
M : Type u_1
inst : CommMonoid M
n : ℕ
inst_1 : NeZero n
inst_2 : HasEnoughRootsOfUnity M n
this : IsCyclic.{u_1} ↥(rootsOfUnity n M)
⊢ Finite.{u_1 + 1} ↥(rootsOfUnity n M)
have obtain : ∀ (g : ↥(rootsOfUnity n M))
  (hg : ∀ (x : ↥(rootsOfUnity n M)), Membership.mem (γ := Subgroup ↥(rootsOfUnity n M)) (Subgroup.zpowers g) x),
  Finite.{u_1 + 1} ↥(rootsOfUnity n M)
Tactic: obtain ⟨g, hg⟩ := IsCyclic.exists_generator (α := rootsOfUnity n M)
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.EnoughRootsOfUnity (hypothesis)
Goal:
M : Type u_1
inst : CommMonoid M
n : ℕ
inst_1 : NeZero n
inst_2 : HasEnoughRootsOfUnity M n
this : IsCyclic.{u_1} ↥(rootsOfUnity n M)
g : ↥(rootsOfUnity n M)
hg : ∀ (x : ↥(rootsOfUnity n M)), Membership.mem (γ := Subgroup ↥(rootsOfUnity n M)) (Subgroup.zpowers g) x
⊢ Finite.{u_1 + 1} ↥(rootsOfUnity n M)
have hg' : g ^ n = 1
Tactic: have hg' : g ^ n = 1 := OneMemClass.coe_eq_one.mp g.prop
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.EnoughRootsOfUnity (goal)
Goal:
M : Type u_1
inst : CommMonoid M
n : ℕ
inst_1 : NeZero n
inst_2 : HasEnoughRootsOfUnity M n
this : IsCyclic.{u_1} ↥(rootsOfUnity n M)
g : ↥(rootsOfUnity n M)
hg : ∀ (x : ↥(rootsOfUnity n M)), Membership.mem (γ := Subgroup ↥(rootsOfUnity n M)) (Subgroup.zpowers g) x
hg' : g ^ n = 1
f : (j : ZMod n) → ↥(rootsOfUnity n M) := fun (j : ZMod n) ↦ HPow.hPow (β := ℤ) g (↑j.val : ℤ)
⊢ Finite.{u_1 + 1} ↥(rootsOfUnity n M)
have refine : ∀ (x : ↥(rootsOfUnity n M)), ∃ (a : ZMod n), f a = x
Tactic: refine Finite.of_surjective f fun x ↦ ?_
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.EnoughRootsOfUnity (goal)
Goal:
M : Type u_1
inst : CommMonoid M
n : ℕ
inst_1 : NeZero n
inst_2 : HasEnoughRootsOfUnity M n
this : IsCyclic.{u_1} ↥(rootsOfUnity n M)
g : ↥(rootsOfUnity n M)
hg : ∀ (x : ↥(rootsOfUnity n M)), Membership.mem (γ := Subgroup ↥(rootsOfUnity n M)) (Subgroup.zpowers g) x
hg' : g ^ n = 1
f : (j : ZMod n) → ↥(rootsOfUnity n M) := fun (j : ZMod n) ↦ HPow.hPow (β := ℤ) g (↑j.val : ℤ)
x : ↥(rootsOfUnity n M)
⊢ ∃ (a : ZMod n), f a = x
have obtain : ∀ (k : ℤ) (hk : g ^ k = x), ∃ (a : ZMod n), f a = x
Tactic: obtain ⟨k, hk⟩ := Subgroup.mem_zpowers_iff.mp <| hg x
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.EnoughRootsOfUnity (goal)
Goal:
M : Type u_1
inst : CommMonoid M
n : ℕ
inst_1 : NeZero n
inst_2 : HasEnoughRootsOfUnity M n
this : IsCyclic.{u_1} ↥(rootsOfUnity n M)
g : ↥(rootsOfUnity n M)
hg : ∀ (x : ↥(rootsOfUnity n M)), Membership.mem (γ := Subgroup ↥(rootsOfUnity n M)) (Subgroup.zpowers g) x
hg' : g ^ n = 1
f : (j : ZMod n) → ↥(rootsOfUnity n M) := fun (j : ZMod n) ↦ HPow.hPow (β := ℤ) g (↑j.val : ℤ)
x : ↥(rootsOfUnity n M)
k : ℤ
hk : g ^ k = x
⊢ ∃ (a : ZMod n), f a = x
have refine : f (↑k : ZMod n) = x
Tactic: refine ⟨k, ?_⟩
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.EnoughRootsOfUnity (goal)
Goal:
M : Type u_1
inst : CommMonoid M
n : ℕ
inst_1 : NeZero n
inst_2 : HasEnoughRootsOfUnity M n
⊢ Nat.card.{u_1} ↥(rootsOfUnity n M) = n
have obtain : ∀ (ζ : M) (h : IsPrimitiveRoot ζ n), Nat.card.{u_1} ↥(rootsOfUnity n M) = n
Tactic: obtain ⟨ζ, h⟩ := exists_primitiveRoot M n
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.EnoughRootsOfUnity (goal)
Goal:
M : Type u_1
inst : CommMonoid M
n : ℕ
inst_1 : NeZero n
inst_2 : HasEnoughRootsOfUnity M n
ζ : M
h : IsPrimitiveRoot ζ n
⊢ Nat.card.{u_1} ↥(rootsOfUnity n M) = n
have rw : Monoid.exponent.{u_1} ↥(rootsOfUnity n M) = n
Tactic: rw [← IsCyclic.exponent_eq_card]
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.EnoughRootsOfUnity (goal)
Goal:
M : Type u_1
inst : CommMonoid M
n : ℕ
inst_1 : NeZero n
inst_2 : HasEnoughRootsOfUnity M n
ζ : M
h : IsPrimitiveRoot ζ n
⊢ Monoid.exponent.{u_1} ↥(rootsOfUnity n M) = n
have refine_1 : Monoid.exponent.{u_1} ↥(rootsOfUnity n M) ∣ n
Tactic: refine dvd_antisymm ?_ ?_
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.EnoughRootsOfUnity (goal)
Goal:
M : Type u_1
inst : CommMonoid M
n : ℕ
inst_1 : NeZero n
inst_2 : HasEnoughRootsOfUnity M n
ζ : M
h : IsPrimitiveRoot ζ n
refine_1 : Monoid.exponent.{u_1} ↥(rootsOfUnity n M) ∣ n
⊢ Monoid.exponent.{u_1} ↥(rootsOfUnity n M) = n
have refine_2 : n ∣ Monoid.exponent.{u_1} ↥(rootsOfUnity n M)
Tactic: refine dvd_antisymm ?_ ?_
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.EnoughRootsOfUnity (goal)
Goal:
M : Type u_1
inst : CommMonoid M
n : ℕ
inst_1 : NeZero n
inst_2 : HasEnoughRootsOfUnity M n
ζ : M
h : IsPrimitiveRoot ζ n
⊢ n ∣ Monoid.exponent.{u_1} ↥(rootsOfUnity n M)
have refine_2 : orderOf ζ ∣ Monoid.exponent.{u_1} ↥(rootsOfUnity n M)
Tactic: nth_rewrite 1 [h.eq_orderOf]
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.EnoughRootsOfUnity (goal)
Goal:
M : Type u_1
inst : CommMonoid M
n : ℕ
inst_1 : NeZero n
inst_2 : HasEnoughRootsOfUnity M n
ζ : M
h : IsPrimitiveRoot ζ n
⊢ orderOf ζ ∣ Monoid.exponent.{u_1} ↥(rootsOfUnity n M)
have refine_2 : orderOf sorry.unit ∣ Monoid.exponent.{u_1} ↥(rootsOfUnity n M)
Tactic: rw [← (h.isUnit NeZero.out).unit_spec, orderOf_units]