awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.Minpoly (goal)
Goal:
n : ℕ
K : Type u_1
inst : CommRing K
μ : K
h : IsPrimitiveRoot μ n
inst_1 : IsDomain K
inst_2 : CharZero K
inst_3 : DecidableEq K
hn : ¬n = 0
this : NeZero n
hpos : 0 < n
m : ℕ
left : m < n
hcop : m.Coprime n
hx : μ ^ m ∈ primitiveRoots n K
⊢ Membership.mem (γ := Multiset K) (roots (map (Int.castRingHom K) (minpoly ℤ μ))) (μ ^ m)
have a : Membership.mem (γ := Multiset K) (roots (map (Int.castRingHom K) (minpoly ℤ μ))) (μ ^ m) ↔
  IsRoot (map (Int.castRingHom K) (minpoly ℤ μ)) (μ ^ m)
Tactic: convert! pow_isRoot_minpoly h hcop using 0
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.Minpoly (goal)
Goal:
n : ℕ
K : Type u_1
inst : CommRing K
μ : K
h : IsPrimitiveRoot μ n
inst_1 : IsDomain K
inst_2 : CharZero K
inst_3 : DecidableEq K
hn : ¬n = 0
this : NeZero n
hpos : 0 < n
m : ℕ
left : m < n
hcop : m.Coprime n
hx : μ ^ m ∈ primitiveRoots n K
⊢ Membership.mem (γ := Multiset K) (roots (map (Int.castRingHom K) (minpoly ℤ μ))) (μ ^ m) ↔
    IsRoot (map (Int.castRingHom K) (minpoly ℤ μ)) (μ ^ m)
have a : Ne.{u_1 + 1} (α := K[X]) (map (Int.castRingHom K) (minpoly ℤ μ)) 0
Tactic: rw [← mem_roots]
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots (goal)
Goal:
R : Type u_4
k : ℕ
inst : CommRing R
inst_1 : IsDomain R
ζ : R
h0 : 0 < k
⊢ ζ ∈ primitiveRoots k R ↔ IsPrimitiveRoot ζ k
have rw : IsPrimitiveRoot ζ k → ζ ^ k = 1
Tactic: rw [primitiveRoots, mem_filter, Multiset.mem_toFinset, mem_nthRoots h0, and_iff_right_iff_imp]
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots (goal)
Goal:
R : Type u_4
k : ℕ
inst : CommRing R
inst_1 : IsDomain R
ζ : R
h0 : 0 < k
⊢ ζ ∈ primitiveRoots k R ↔ IsPrimitiveRoot ζ k
have rw : Membership.mem (γ := Finset R) ({ζ ∈ Multiset.toFinset (nthRoots k 1) | IsPrimitiveRoot ζ k}) ζ ↔ IsPrimitiveRoot ζ k
Tactic: classical
  rw [primitiveRoots, mem_filter, Multiset.mem_toFinset, mem_nthRoots h0, and_iff_right_iff_imp]
  exact IsPrimitiveRoot.pow_eq_one
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots (goal)
Goal:
R : Type u_4
k : ℕ
inst : CommRing R
inst_1 : IsDomain R
ζ : R
h0 : 0 < k
rw :
  Membership.mem (γ := Finset R) ({ζ ∈ Multiset.toFinset (nthRoots k 1) | IsPrimitiveRoot ζ k}) ζ ↔ IsPrimitiveRoot ζ k
⊢ ζ ∈ primitiveRoots k R ↔ IsPrimitiveRoot ζ k
have rw₁ : Membership.mem (γ := Finset R) (Multiset.toFinset (nthRoots k 1)) ζ ∧ IsPrimitiveRoot ζ k ↔ IsPrimitiveRoot ζ k
Tactic: classical
  rw [primitiveRoots, mem_filter, Multiset.mem_toFinset, mem_nthRoots h0, and_iff_right_iff_imp]
  exact IsPrimitiveRoot.pow_eq_one
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots (goal)
Goal:
R : Type u_4
k : ℕ
inst : CommRing R
inst_1 : IsDomain R
ζ : R
h0 : 0 < k
rw :
  Membership.mem (γ := Finset R) ({ζ ∈ Multiset.toFinset (nthRoots k 1) | IsPrimitiveRoot ζ k}) ζ ↔ IsPrimitiveRoot ζ k
rw₁ : Membership.mem (γ := Finset R) (Multiset.toFinset (nthRoots k 1)) ζ ∧ IsPrimitiveRoot ζ k ↔ IsPrimitiveRoot ζ k
⊢ ζ ∈ primitiveRoots k R ↔ IsPrimitiveRoot ζ k
have rw₂ : Membership.mem (γ := Multiset R) (nthRoots k 1) ζ ∧ IsPrimitiveRoot ζ k ↔ IsPrimitiveRoot ζ k
Tactic: classical
  rw [primitiveRoots, mem_filter, Multiset.mem_toFinset, mem_nthRoots h0, and_iff_right_iff_imp]
  exact IsPrimitiveRoot.pow_eq_one
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots (goal)
Goal:
R : Type u_4
k : ℕ
inst : CommRing R
inst_1 : IsDomain R
ζ : R
h0 : 0 < k
rw :
  Membership.mem (γ := Finset R) ({ζ ∈ Multiset.toFinset (nthRoots k 1) | IsPrimitiveRoot ζ k}) ζ ↔ IsPrimitiveRoot ζ k
rw₁ : Membership.mem (γ := Finset R) (Multiset.toFinset (nthRoots k 1)) ζ ∧ IsPrimitiveRoot ζ k ↔ IsPrimitiveRoot ζ k
rw₂ : Membership.mem (γ := Multiset R) (nthRoots k 1) ζ ∧ IsPrimitiveRoot ζ k ↔ IsPrimitiveRoot ζ k
⊢ ζ ∈ primitiveRoots k R ↔ IsPrimitiveRoot ζ k
have rw₃ : ζ ^ k = 1 ∧ IsPrimitiveRoot ζ k ↔ IsPrimitiveRoot ζ k
Tactic: classical
  rw [primitiveRoots, mem_filter, Multiset.mem_toFinset, mem_nthRoots h0, and_iff_right_iff_imp]
  exact IsPrimitiveRoot.pow_eq_one
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots (goal)
Goal:
R : Type u_4
k : ℕ
inst : CommRing R
inst_1 : IsDomain R
ζ : R
h0 : 0 < k
rw :
  Membership.mem (γ := Finset R) ({ζ ∈ Multiset.toFinset (nthRoots k 1) | IsPrimitiveRoot ζ k}) ζ ↔ IsPrimitiveRoot ζ k
rw₁ : Membership.mem (γ := Finset R) (Multiset.toFinset (nthRoots k 1)) ζ ∧ IsPrimitiveRoot ζ k ↔ IsPrimitiveRoot ζ k
rw₂ : Membership.mem (γ := Multiset R) (nthRoots k 1) ζ ∧ IsPrimitiveRoot ζ k ↔ IsPrimitiveRoot ζ k
rw₃ : ζ ^ k = 1 ∧ IsPrimitiveRoot ζ k ↔ IsPrimitiveRoot ζ k
⊢ ζ ∈ primitiveRoots k R ↔ IsPrimitiveRoot ζ k
have rw₄ : IsPrimitiveRoot ζ k → ζ ^ k = 1
Tactic: classical
  rw [primitiveRoots, mem_filter, Multiset.mem_toFinset, mem_nthRoots h0, and_iff_right_iff_imp]
  exact IsPrimitiveRoot.pow_eq_one
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots (goal)
Goal:
R : Type u_4
inst : CommRing R
inst_1 : IsDomain R
⊢ primitiveRoots 0 R = ∅
have rw : Eq.{u_4 + 1} (α := Finset R) ({ζ ∈ Multiset.toFinset (nthRoots 0 1) | IsPrimitiveRoot ζ 0}) ∅
Tactic: rw [primitiveRoots, nthRoots_zero, Multiset.toFinset_zero, Finset.filter_empty]
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots (goal)
Goal:
R : Type u_4
inst : CommRing R
inst_1 : IsDomain R
rw : Eq.{u_4 + 1} (α := Finset R) ({ζ ∈ Multiset.toFinset (nthRoots 0 1) | IsPrimitiveRoot ζ 0}) ∅
⊢ primitiveRoots 0 R = ∅
have rw₁ : Eq.{u_4 + 1} (α := Finset R) ({ζ ∈ Multiset.toFinset 0 | IsPrimitiveRoot ζ 0}) ∅
Tactic: rw [primitiveRoots, nthRoots_zero, Multiset.toFinset_zero, Finset.filter_empty]
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots (goal)
Goal:
R : Type u_4
inst : CommRing R
inst_1 : IsDomain R
rw : Eq.{u_4 + 1} (α := Finset R) ({ζ ∈ Multiset.toFinset (nthRoots 0 1) | IsPrimitiveRoot ζ 0}) ∅
rw₁ : Eq.{u_4 + 1} (α := Finset R) ({ζ ∈ Multiset.toFinset 0 | IsPrimitiveRoot ζ 0}) ∅
⊢ primitiveRoots 0 R = ∅
have rw₂ : Eq.{u_4 + 1} (α := Finset R) ({ζ ∈ ∅ | IsPrimitiveRoot ζ 0}) ∅
Tactic: rw [primitiveRoots, nthRoots_zero, Multiset.toFinset_zero, Finset.filter_empty]
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots (goal)
Goal:
R : Type u_4
inst : CommRing R
inst_1 : IsDomain R
rw : Eq.{u_4 + 1} (α := Finset R) ({ζ ∈ Multiset.toFinset (nthRoots 0 1) | IsPrimitiveRoot ζ 0}) ∅
rw₁ : Eq.{u_4 + 1} (α := Finset R) ({ζ ∈ Multiset.toFinset 0 | IsPrimitiveRoot ζ 0}) ∅
rw₂ : Eq.{u_4 + 1} (α := Finset R) ({ζ ∈ ∅ | IsPrimitiveRoot ζ 0}) ∅
⊢ primitiveRoots 0 R = ∅
have rw₃ : Eq.{u_4 + 1} (α := Finset R) ∅ ∅
Tactic: rw [primitiveRoots, nthRoots_zero, Multiset.toFinset_zero, Finset.filter_empty]
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots (goal)
Goal:
R : Type u_4
inst : CommRing R
inst_1 : IsDomain R
⊢ primitiveRoots 0 R = ∅
have rw : Eq.{u_4 + 1} (α := Finset R) ({ζ ∈ Multiset.toFinset (nthRoots 0 1) | IsPrimitiveRoot ζ 0}) ∅
Tactic: classical
  rw [primitiveRoots, nthRoots_zero, Multiset.toFinset_zero, Finset.filter_empty]
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots (goal)
Goal:
R : Type u_4
inst : CommRing R
inst_1 : IsDomain R
rw : Eq.{u_4 + 1} (α := Finset R) ({ζ ∈ Multiset.toFinset (nthRoots 0 1) | IsPrimitiveRoot ζ 0}) ∅
⊢ primitiveRoots 0 R = ∅
have rw₁ : Eq.{u_4 + 1} (α := Finset R) ({ζ ∈ Multiset.toFinset 0 | IsPrimitiveRoot ζ 0}) ∅
Tactic: classical
  rw [primitiveRoots, nthRoots_zero, Multiset.toFinset_zero, Finset.filter_empty]
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots (goal)
Goal:
R : Type u_4
inst : CommRing R
inst_1 : IsDomain R
rw : Eq.{u_4 + 1} (α := Finset R) ({ζ ∈ Multiset.toFinset (nthRoots 0 1) | IsPrimitiveRoot ζ 0}) ∅
rw₁ : Eq.{u_4 + 1} (α := Finset R) ({ζ ∈ Multiset.toFinset 0 | IsPrimitiveRoot ζ 0}) ∅
⊢ primitiveRoots 0 R = ∅
have rw₂ : Eq.{u_4 + 1} (α := Finset R) ({ζ ∈ ∅ | IsPrimitiveRoot ζ 0}) ∅
Tactic: classical
  rw [primitiveRoots, nthRoots_zero, Multiset.toFinset_zero, Finset.filter_empty]
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots (goal)
Goal:
R : Type u_4
inst : CommRing R
inst_1 : IsDomain R
rw : Eq.{u_4 + 1} (α := Finset R) ({ζ ∈ Multiset.toFinset (nthRoots 0 1) | IsPrimitiveRoot ζ 0}) ∅
rw₁ : Eq.{u_4 + 1} (α := Finset R) ({ζ ∈ Multiset.toFinset 0 | IsPrimitiveRoot ζ 0}) ∅
rw₂ : Eq.{u_4 + 1} (α := Finset R) ({ζ ∈ ∅ | IsPrimitiveRoot ζ 0}) ∅
⊢ primitiveRoots 0 R = ∅
have rw₃ : Eq.{u_4 + 1} (α := Finset R) ∅ ∅
Tactic: classical
  rw [primitiveRoots, nthRoots_zero, Multiset.toFinset_zero, Finset.filter_empty]
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots (goal)
Goal:
M : Type u_1
inst : CommMonoid M
k : ℕ
ζ : M
hk : 0 < k
h1 : ζ ^ k = 1
h : ∀ (l : ℕ), 0 < l → l < k → ζ ^ l ≠ 1
⊢ IsPrimitiveRoot ζ k
have refine : ∀ (l : ℕ) (hl : ζ ^ l = 1), k ∣ l
Tactic: refine ⟨h1, fun l hl ↦ ?_⟩
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots (goal)
Goal:
M : Type u_1
inst : CommMonoid M
k : ℕ
ζ : M
hk : 0 < k
h1 : ζ ^ k = 1
h : ∀ (l : ℕ), 0 < l → l < k → ζ ^ l ≠ 1
l : ℕ
hl : ζ ^ l = 1
⊢ k ∣ l
have suffices : k.gcd l = k
Tactic: suffices k.gcd l = k from this ▸ k.gcd_dvd_right l
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots (goal)
Goal:
M : Type u_1
inst : CommMonoid M
k : ℕ
ζ : M
hk : 0 < k
h1 : ζ ^ k = 1
h : ∀ (l : ℕ), 0 < l → l < k → ζ ^ l ≠ 1
l : ℕ
hl : ζ ^ l = 1
⊢ k.gcd l = k
have refine : ∀ (h' : k.gcd l < k), False
Tactic: refine (Nat.le_of_dvd hk (k.gcd_dvd_left l)).eq_of_not_lt fun h' ↦ ?_
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots (goal)
Goal:
M : Type u_1
inst : CommMonoid M
k : ℕ
ζ : M
h : IsPrimitiveRoot ζ k
h0 : k ≠ 0
⊢ ζ * ζ ^ (k - 1) = 1
have rw : ζ ^ (k - 1 + 1) = 1
Tactic: rw [← pow_succ', Nat.sub_one_add_one h0, h.pow_eq_one]
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots (goal)
Goal:
M : Type u_1
inst : CommMonoid M
k : ℕ
ζ : M
h : IsPrimitiveRoot ζ k
h0 : k ≠ 0
rw : ζ ^ (k - 1 + 1) = 1
⊢ ζ * ζ ^ (k - 1) = 1
have rw₁ : ζ ^ k = 1
Tactic: rw [← pow_succ', Nat.sub_one_add_one h0, h.pow_eq_one]
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots (goal)
Goal:
M : Type u_1
inst : CommMonoid M
k : ℕ
ζ : M
h : IsPrimitiveRoot ζ k
h0 : k ≠ 0
rw : ζ ^ (k - 1 + 1) = 1
rw₁ : ζ ^ k = 1
⊢ ζ * ζ ^ (k - 1) = 1
have rw₂ : Eq.{u_1 + 1} (α := M) 1 1
Tactic: rw [← pow_succ', Nat.sub_one_add_one h0, h.pow_eq_one]
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots (goal)
Goal:
M : Type u_1
inst : CommMonoid M
k : ℕ
ζ : M
h : IsPrimitiveRoot ζ k
i j : ℕ
hi : i < k
hj : j < k
H : ζ ^ i = ζ ^ j
⊢ i = j
have inr : ∀ (this : ∀ ⦃i j : ℕ⦄ (hi : i < k) (hj : j < k) (H : ζ ^ i = ζ ^ j) (hij : i ≤ j), i = j) (hij : ¬i ≤ j), i = j
Tactic: wlog hij : i ≤ j generalizing i j
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots (goal)
Goal:
M : Type u_1
inst : CommMonoid M
k : ℕ
ζ : M
h : IsPrimitiveRoot ζ k
i j : ℕ
hi : i < k
hj : j < k
H : ζ ^ i = ζ ^ j
inr : ∀ (this : ∀ ⦃i j : ℕ⦄ (hi : i < k) (hj : j < k) (H : ζ ^ i = ζ ^ j) (hij : i ≤ j), i = j) (hij : ¬i ≤ j), i = j
⊢ i = j
have wlog : ∀ ⦃i j : ℕ⦄ (hi : i < k) (hj : j < k) (H : ζ ^ i = ζ ^ j) (hij : i ≤ j), i = j
Tactic: wlog hij : i ≤ j generalizing i j
Cleared hypotheses: i, j, hi, hj, H
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots (goal)
Goal:
M : Type u_1
inst : CommMonoid M
k : ℕ
ζ : M
h : IsPrimitiveRoot ζ k
i j : ℕ
hi : i < k
hj : j < k
H : ζ ^ i = ζ ^ j
hij : i ≤ j
⊢ i = j
have apply : j ≤ i
Tactic: apply le_antisymm hij
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots (goal)
Goal:
M : Type u_1
inst : CommMonoid M
k : ℕ
ζ : M
h : IsPrimitiveRoot ζ k
i j : ℕ
hi : i < k
hj : j < k
H : ζ ^ i = ζ ^ j
hij : i ≤ j
⊢ j ≤ i
have rw : j - i = 0
Tactic: rw [← tsub_eq_zero_iff_le]
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots (goal)
Goal:
M : Type u_1
inst : CommMonoid M
k : ℕ
ζ : M
h : IsPrimitiveRoot ζ k
i j : ℕ
hi : i < k
hj : j < k
H : ζ ^ i = ζ ^ j
hij : i ≤ j
⊢ j - i = 0
have apply : k ∣ j - i
Tactic: apply Nat.eq_zero_of_dvd_of_lt _ (lt_of_le_of_lt tsub_le_self hj)
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots (goal)
Goal:
M : Type u_1
inst : CommMonoid M
k : ℕ
ζ : M
h : IsPrimitiveRoot ζ k
i j : ℕ
hi : i < k
hj : j < k
H : ζ ^ i = ζ ^ j
hij : i ≤ j
⊢ k ∣ j - i
have a : ζ ^ (j - i) = 1
Tactic: apply h.dvd_of_pow_eq_one
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots (goal)
Goal:
M : Type u_1
inst : CommMonoid M
k : ℕ
ζ : M
h : IsPrimitiveRoot ζ k
i j : ℕ
hi : i < k
hj : j < k
H : ζ ^ i = ζ ^ j
hij : i ≤ j
⊢ ζ ^ (j - i) = 1
have a : ζ ^ (j - i) * ζ ^ i = 1 * ζ ^ i
Tactic: rw [← ((h.isUnit (Nat.ne_zero_of_lt hi)).pow i).mul_left_inj, ← pow_add,
    tsub_add_cancel_of_le hij, H, one_mul]
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots (goal)
Goal:
M : Type u_1
inst : CommMonoid M
k : ℕ
ζ : M
h : IsPrimitiveRoot ζ k
i j : ℕ
hi : i < k
hj : j < k
H : ζ ^ i = ζ ^ j
hij : i ≤ j
a : ζ ^ (j - i) * ζ ^ i = 1 * ζ ^ i
⊢ ζ ^ (j - i) = 1
have a₁ : ζ ^ (j - i + i) = 1 * ζ ^ i
Tactic: rw [← ((h.isUnit (Nat.ne_zero_of_lt hi)).pow i).mul_left_inj, ← pow_add,
    tsub_add_cancel_of_le hij, H, one_mul]
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots (goal)
Goal:
M : Type u_1
inst : CommMonoid M
k : ℕ
ζ : M
h : IsPrimitiveRoot ζ k
i j : ℕ
hi : i < k
hj : j < k
H : ζ ^ i = ζ ^ j
hij : i ≤ j
a : ζ ^ (j - i) * ζ ^ i = 1 * ζ ^ i
a₁ : ζ ^ (j - i + i) = 1 * ζ ^ i
⊢ ζ ^ (j - i) = 1
have a₂ : ζ ^ j = 1 * ζ ^ i
Tactic: rw [← ((h.isUnit (Nat.ne_zero_of_lt hi)).pow i).mul_left_inj, ← pow_add,
    tsub_add_cancel_of_le hij, H, one_mul]
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots (goal)
Goal:
M : Type u_1
inst : CommMonoid M
k : ℕ
ζ : M
h : IsPrimitiveRoot ζ k
i j : ℕ
hi : i < k
hj : j < k
H : ζ ^ i = ζ ^ j
hij : i ≤ j
a : ζ ^ (j - i) * ζ ^ i = 1 * ζ ^ i
a₁ : ζ ^ (j - i + i) = 1 * ζ ^ i
a₂ : ζ ^ j = 1 * ζ ^ i
⊢ ζ ^ (j - i) = 1
have a₃ : ζ ^ j = 1 * ζ ^ j
Tactic: rw [← ((h.isUnit (Nat.ne_zero_of_lt hi)).pow i).mul_left_inj, ← pow_add,
    tsub_add_cancel_of_le hij, H, one_mul]
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots (goal)
Goal:
M : Type u_1
inst : CommMonoid M
k : ℕ
ζ : M
h : IsPrimitiveRoot ζ k
i j : ℕ
hi : i < k
hj : j < k
H : ζ ^ i = ζ ^ j
hij : i ≤ j
a : ζ ^ (j - i) * ζ ^ i = 1 * ζ ^ i
a₁ : ζ ^ (j - i + i) = 1 * ζ ^ i
a₂ : ζ ^ j = 1 * ζ ^ i
a₃ : ζ ^ j = 1 * ζ ^ j
⊢ ζ ^ (j - i) = 1
have a₄ : ζ ^ j = ζ ^ j
Tactic: rw [← ((h.isUnit (Nat.ne_zero_of_lt hi)).pow i).mul_left_inj, ← pow_add,
    tsub_add_cancel_of_le hij, H, one_mul]
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots (goal)
Goal:
M : Type u_1
inst : CommMonoid M
ζ : M
⊢ IsPrimitiveRoot ζ 1 ↔ ζ = 1
have mp : IsPrimitiveRoot ζ 1 → ζ = 1
Tactic: constructor
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots (goal)
Goal:
M : Type u_1
inst : CommMonoid M
ζ : M
mp : IsPrimitiveRoot ζ 1 → ζ = 1
⊢ IsPrimitiveRoot ζ 1 ↔ ζ = 1
have mpr : ζ = 1 → IsPrimitiveRoot ζ 1
Tactic: constructor
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots (goal)
Goal:
M : Type u_1
inst : CommMonoid M
ζ : M
h : IsPrimitiveRoot ζ 1
⊢ ζ = 1
have mp : HPow.hPow (β := ℕ) ζ 1 = 1
Tactic: rw [← pow_one ζ, h.pow_eq_one]
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots (goal)
Goal:
M : Type u_1
inst : CommMonoid M
ζ : M
h : IsPrimitiveRoot ζ 1
mp : HPow.hPow (β := ℕ) ζ 1 = 1
⊢ ζ = 1
have mp₁ : Eq.{u_1 + 1} (α := M) 1 1
Tactic: rw [← pow_one ζ, h.pow_eq_one]
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots (goal)
Goal:
k : ℕ
M : Type u_7
B : Type u_8
inst : CommMonoid M
inst_1 : SetLike B M
inst_2 : SubmonoidClass B M
N : B
ζ : ↥N
⊢ IsPrimitiveRoot (↑ζ : M) k ↔ IsPrimitiveRoot ζ k
have simp_rw : ((↑ζ : M) ^ k = 1 ∧ ∀ (l : ℕ), (↑ζ : M) ^ l = 1 → k ∣ l) ↔ ζ ^ k = 1 ∧ ∀ (l : ℕ), ζ ^ l = 1 → k ∣ l
Tactic: simp_rw [iff_def]
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots (goal)
Goal:
k : ℕ
M : Type u_7
B : Type u_8
inst : CommMonoid M
inst_1 : SetLike B M
inst_2 : SubmonoidClass B M
N : B
ζ : ↥N
⊢ ((↑ζ : M) ^ k = 1 ∧ ∀ (l : ℕ), (↑ζ : M) ^ l = 1 → k ∣ l) ↔ ζ ^ k = 1 ∧ ∀ (l : ℕ), ζ ^ l = 1 → k ∣ l
have norm_cast : (ζ ^ k = 1 ∧ ∀ (l : ℕ), ζ ^ l = 1 → k ∣ l) ↔ ζ ^ k = 1 ∧ ∀ (l : ℕ), ζ ^ l = 1 → k ∣ l
Tactic: norm_cast
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots (goal)
Goal:
M : Type u_1
inst : CommMonoid M
k : ℕ
ζ : M
h : IsPrimitiveRoot ζ k
i : ℕ
hi : i.Coprime k
⊢ IsPrimitiveRoot (ζ ^ i) k
have pos : ∀ (h0 : k = 0), IsPrimitiveRoot (ζ ^ i) k
Tactic: by_cases h0 : k = 0
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots (goal)
Goal:
M : Type u_1
inst : CommMonoid M
k : ℕ
ζ : M
h : IsPrimitiveRoot ζ k
i : ℕ
hi : i.Coprime k
pos : ∀ (h0 : k = 0), IsPrimitiveRoot (ζ ^ i) k
⊢ IsPrimitiveRoot (ζ ^ i) k
have neg : ∀ (h0 : ¬k = 0), IsPrimitiveRoot (ζ ^ i) k
Tactic: by_cases h0 : k = 0
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots (goal)
Goal:
M : Type u_1
inst : CommMonoid M
k : ℕ
ζ : M
h : IsPrimitiveRoot ζ k
i : ℕ
hi : i.Coprime k
h0 : k = 0
⊢ IsPrimitiveRoot (ζ ^ i) k
have pos : ∀ (h : IsPrimitiveRoot ζ 0) (hi : i.Coprime 0), IsPrimitiveRoot (ζ ^ i) 0
Tactic: subst k
Cleared hypotheses: k, h, hi, h0
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots (goal)
Goal:
M : Type u_1
inst : CommMonoid M
k : ℕ
ζ : M
h : IsPrimitiveRoot ζ k
i : ℕ
hi : i.Coprime k
h0 : ¬k = 0
⊢ IsPrimitiveRoot (ζ ^ i) k
have neg : ∀ (ζ : Mˣ) (h : IsPrimitiveRoot (↑ζ : M) k), IsPrimitiveRoot ((↑ζ : M) ^ i) k
Tactic: rcases h.isUnit h0 with ⟨ζ, rfl⟩
Cleared hypotheses: ζ, h
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots (goal)
Goal:
M : Type u_1
inst : CommMonoid M
k i : ℕ
hi : i.Coprime k
h0 : ¬k = 0
ζ : Mˣ
h : IsPrimitiveRoot (↑ζ : M) k
⊢ IsPrimitiveRoot ((↑ζ : M) ^ i) k
have neg : IsPrimitiveRoot (↑(ζ ^ i) : M) k
Tactic: rw [← Units.val_pow_eq_pow_val]
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots (goal)
Goal:
M : Type u_1
inst : CommMonoid M
k i : ℕ
hi : i.Coprime k
h0 : ¬k = 0
ζ : Mˣ
h : IsPrimitiveRoot (↑ζ : M) k
⊢ IsPrimitiveRoot (↑(ζ ^ i) : M) k
have neg : ∀ (h : IsPrimitiveRoot ζ k), IsPrimitiveRoot (ζ ^ i) k
Tactic: rw [coe_units_iff] at h ⊢
Cleared hypotheses: h
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots (goal)
Goal:
M : Type u_1
inst : CommMonoid M
k i : ℕ
hi : i.Coprime k
h0 : ¬k = 0
ζ : Mˣ
h : IsPrimitiveRoot ζ k
⊢ (ζ ^ i) ^ k = 1
have rw : ζ ^ (k * i) = 1
Tactic: rw [← pow_mul', pow_mul, h.pow_eq_one, one_pow]
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots (goal)
Goal:
M : Type u_1
inst : CommMonoid M
k i : ℕ
hi : i.Coprime k
h0 : ¬k = 0
ζ : Mˣ
h : IsPrimitiveRoot ζ k
rw : ζ ^ (k * i) = 1
⊢ (ζ ^ i) ^ k = 1
have rw₁ : (ζ ^ k) ^ i = 1
Tactic: rw [← pow_mul', pow_mul, h.pow_eq_one, one_pow]
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots (goal)
Goal:
M : Type u_1
inst : CommMonoid M
k i : ℕ
hi : i.Coprime k
h0 : ¬k = 0
ζ : Mˣ
h : IsPrimitiveRoot ζ k
rw : ζ ^ (k * i) = 1
rw₁ : (ζ ^ k) ^ i = 1
⊢ (ζ ^ i) ^ k = 1
have rw₂ : HPow.hPow (α := Mˣ) 1 i = 1
Tactic: rw [← pow_mul', pow_mul, h.pow_eq_one, one_pow]
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots (goal)
Goal:
M : Type u_1
inst : CommMonoid M
k i : ℕ
hi : i.Coprime k
h0 : ¬k = 0
ζ : Mˣ
h : IsPrimitiveRoot ζ k
rw : ζ ^ (k * i) = 1
rw₁ : (ζ ^ k) ^ i = 1
rw₂ : HPow.hPow (α := Mˣ) 1 i = 1
⊢ (ζ ^ i) ^ k = 1
have rw₃ : Eq.{u_1 + 1} (α := Mˣ) 1 1
Tactic: rw [← pow_mul', pow_mul, h.pow_eq_one, one_pow]
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots (goal)
Goal:
M : Type u_1
inst : CommMonoid M
k i : ℕ
hi : i.Coprime k
h0 : ¬k = 0
ζ : Mˣ
h : IsPrimitiveRoot ζ k
⊢ IsPrimitiveRoot (ζ ^ i) k
have neg : ∀ (l : ℕ) (hl : (ζ ^ i) ^ l = 1), ζ ^ l = 1
Tactic: refine
    { pow_eq_one := by rw [← pow_mul', pow_mul, h.pow_eq_one, one_pow]
      dvd_of_pow_eq_one := fun l hl ↦ h.dvd_of_pow_eq_one l ?_ }
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots (goal)
Goal:
M : Type u_1
inst : CommMonoid M
k i : ℕ
hi : i.Coprime k
h0 : ¬k = 0
ζ : Mˣ
h : IsPrimitiveRoot ζ k
l : ℕ
hl : (ζ ^ i) ^ l = 1
⊢ ζ ^ l = 1
have neg : ζ ^ (HMul.hMul (α := ℤ) (↑i : ℤ) (↑l : ℤ) * i.gcdA k) * (ζ ^ ((↑k : ℤ) * i.gcdB k)) ^ l = 1
Tactic: rw [← pow_one ζ, ← zpow_natCast ζ, ← hi.gcd_eq_one, Nat.gcd_eq_gcd_ab, zpow_add, mul_pow,
    ← zpow_natCast, ← zpow_mul, mul_right_comm]
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots (goal)
Goal:
M : Type u_1
inst : CommMonoid M
k : ℕ
ζ : M
h : IsPrimitiveRoot ζ k
h0 : 0 < k
i : ℕ
⊢ IsPrimitiveRoot (ζ ^ i) k ↔ i.Coprime k
have refine : ∀ (hi : IsPrimitiveRoot (ζ ^ i) k), i.Coprime k
Tactic: refine ⟨fun hi ↦ ?_, h.pow_of_coprime i⟩
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots (goal)
Goal:
M : Type u_1
inst : CommMonoid M
k : ℕ
ζ : M
h : IsPrimitiveRoot ζ k
h0 : 0 < k
i : ℕ
hi : IsPrimitiveRoot (ζ ^ i) k
⊢ i.Coprime k
have obtain : ∀ (a : ℕ) (ha : i = i.gcd k * a), i.Coprime k
Tactic: obtain ⟨a, ha⟩ := i.gcd_dvd_left k
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots (goal)
Goal:
M : Type u_1
inst : CommMonoid M
k : ℕ
ζ : M
h : IsPrimitiveRoot ζ k
h0 : 0 < k
i : ℕ
hi : IsPrimitiveRoot (ζ ^ i) k
a : ℕ
ha : i = i.gcd k * a
⊢ i.Coprime k
have obtain : ∀ (b : ℕ) (hb : k = i.gcd k * b), i.Coprime k
Tactic: obtain ⟨b, hb⟩ := i.gcd_dvd_right k
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots (goal)
Goal:
M : Type u_1
inst : CommMonoid M
k : ℕ
ζ : M
h : IsPrimitiveRoot ζ k
h0 : 0 < k
i : ℕ
hi : IsPrimitiveRoot (ζ ^ i) k
a : ℕ
ha : i = i.gcd k * a
b : ℕ
hb : k = i.gcd k * b
this : b = k
⊢ i.Coprime k
have rwa : ∀ (hb : k = i.gcd k * k) (this : b = k), i.Coprime k
Tactic: rwa [this, eq_comm, Nat.mul_eq_right h0.ne', ← Nat.coprime_iff_gcd_eq_one] at hb
Cleared hypotheses: hb, this
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots (goal)
Goal:
M : Type u_1
inst : CommMonoid M
k : ℕ
ζ : M
h : IsPrimitiveRoot ζ k
h0 : 0 < k
i : ℕ
hi : IsPrimitiveRoot (ζ ^ i) k
a : ℕ
ha : i = i.gcd k * a
b : ℕ
hb : k = i.gcd k * b
this : b = k
rwa : ∀ (hb : k = i.gcd k * k) (this : b = k), i.Coprime k
⊢ i.Coprime k
have rwa₁ : ∀ (hb : i.gcd k * k = k) (this : b = k), i.Coprime k
Tactic: rwa [this, eq_comm, Nat.mul_eq_right h0.ne', ← Nat.coprime_iff_gcd_eq_one] at hb
Cleared hypotheses: hb, this
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots (goal)
Goal:
M : Type u_1
inst : CommMonoid M
k : ℕ
ζ : M
h : IsPrimitiveRoot ζ k
h0 : 0 < k
i : ℕ
hi : IsPrimitiveRoot (ζ ^ i) k
a : ℕ
ha : i = i.gcd k * a
b : ℕ
hb : k = i.gcd k * b
this : b = k
rwa : ∀ (hb : k = i.gcd k * k) (this : b = k), i.Coprime k
rwa₁ : ∀ (hb : i.gcd k * k = k) (this : b = k), i.Coprime k
⊢ i.Coprime k
have rwa₂ : ∀ (hb : i.gcd k = 1) (this : b = k), i.Coprime k
Tactic: rwa [this, eq_comm, Nat.mul_eq_right h0.ne', ← Nat.coprime_iff_gcd_eq_one] at hb
Cleared hypotheses: hb, this
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots (goal)
Goal:
M : Type u_1
inst : CommMonoid M
k : ℕ
ζ : M
h : IsPrimitiveRoot ζ k
h0 : 0 < k
i : ℕ
hi : IsPrimitiveRoot (ζ ^ i) k
a : ℕ
ha : i = i.gcd k * a
b : ℕ
hb : k = i.gcd k * b
this : b = k
rwa : ∀ (hb : k = i.gcd k * k) (this : b = k), i.Coprime k
rwa₁ : ∀ (hb : i.gcd k * k = k) (this : b = k), i.Coprime k
rwa₂ : ∀ (hb : i.gcd k = 1) (this : b = k), i.Coprime k
⊢ i.Coprime k
have rwa₃ : ∀ (hb : i.Coprime k) (this : b = k), i.Coprime k
Tactic: rwa [this, eq_comm, Nat.mul_eq_right h0.ne', ← Nat.coprime_iff_gcd_eq_one] at hb
Cleared hypotheses: hb, this
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots (goal)
Goal:
M : Type u_1
inst : CommMonoid M
k : ℕ
ζ : M
h : IsPrimitiveRoot ζ k
h0 : 0 < k
i : ℕ
hi : IsPrimitiveRoot (ζ ^ i) k
a : ℕ
ha : i = i.gcd k * a
b : ℕ
hb : k = i.gcd k * b
⊢ i.Coprime k
have suffices : b = k
Tactic: suffices b = k by
    rwa [this, eq_comm, Nat.mul_eq_right h0.ne', ← Nat.coprime_iff_gcd_eq_one] at hb