awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots (goal)
Goal:
G : Type u_3
inst : DivisionCommMonoid G
k : ℕ
ζ : G
h : IsPrimitiveRoot ζ k
i : ℤ
hi : i.gcd (↑k : ℤ) = 1
h0 : ¬0 ≤ i
i' : ℕ
hi' : (↑i' : ℤ) = -i
⊢ IsPrimitiveRoot (ζ ^ i) k
have neg : IsPrimitiveRoot (ζ ^ i') k
Tactic: rw [← inv_iff, ← zpow_neg, ← hi', zpow_natCast]
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots (goal)
Goal:
G : Type u_3
inst : DivisionCommMonoid G
k : ℕ
ζ : G
h : IsPrimitiveRoot ζ k
i : ℤ
hi : i.gcd (↑k : ℤ) = 1
h0 : ¬0 ≤ i
i' : ℕ
hi' : (↑i' : ℤ) = -i
⊢ IsPrimitiveRoot (ζ ^ i') k
have neg : i'.Coprime k
Tactic: apply h.pow_of_coprime
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots (goal)
Goal:
G : Type u_3
inst : DivisionCommMonoid G
k : ℕ
ζ : G
h : IsPrimitiveRoot ζ k
i : ℤ
hi : i.gcd (↑k : ℤ) = 1
h0 : ¬0 ≤ i
i' : ℕ
hi' : (↑i' : ℤ) = -i
⊢ i'.Coprime k
have neg : ∀ (hi : i.natAbs.gcd (Int.natAbs (↑k : ℤ)) = 1) (h0 : ¬0 ≤ i) (i' : ℕ) (hi' : (↑i' : ℤ) = -i), i'.Coprime k
Tactic: rwa [Int.gcd, ← Int.natAbs_neg, ← hi'] at hi
Cleared hypotheses: hi, h0, i', hi'
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots (goal)
Goal:
G : Type u_3
inst : DivisionCommMonoid G
k : ℕ
ζ : G
h : IsPrimitiveRoot ζ k
i : ℤ
hi : i.gcd (↑k : ℤ) = 1
h0 : ¬0 ≤ i
i' : ℕ
hi' : (↑i' : ℤ) = -i
neg : ∀ (hi : i.natAbs.gcd (Int.natAbs (↑k : ℤ)) = 1) (h0 : ¬0 ≤ i) (i' : ℕ) (hi' : (↑i' : ℤ) = -i), i'.Coprime k
⊢ i'.Coprime k
have neg₁ : ∀ (hi : (-i).natAbs.gcd (Int.natAbs (↑k : ℤ)) = 1) (h0 : ¬0 ≤ i) (i' : ℕ) (hi' : (↑i' : ℤ) = -i), i'.Coprime k
Tactic: rwa [Int.gcd, ← Int.natAbs_neg, ← hi'] at hi
Cleared hypotheses: hi, h0, i', hi'
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots (goal)
Goal:
G : Type u_3
inst : DivisionCommMonoid G
k : ℕ
ζ : G
h : IsPrimitiveRoot ζ k
i : ℤ
hi : i.gcd (↑k : ℤ) = 1
h0 : ¬0 ≤ i
i' : ℕ
hi' : (↑i' : ℤ) = -i
neg : ∀ (hi : i.natAbs.gcd (Int.natAbs (↑k : ℤ)) = 1) (h0 : ¬0 ≤ i) (i' : ℕ) (hi' : (↑i' : ℤ) = -i), i'.Coprime k
neg₁ : ∀ (hi : (-i).natAbs.gcd (Int.natAbs (↑k : ℤ)) = 1) (h0 : ¬0 ≤ i) (i' : ℕ) (hi' : (↑i' : ℤ) = -i), i'.Coprime k
⊢ i'.Coprime k
have neg₂ : ∀ (h0 : ¬0 ≤ i) (i' : ℕ) (hi : (Int.natAbs (↑i' : ℤ)).gcd (Int.natAbs (↑k : ℤ)) = 1) (hi' : (↑i' : ℤ) = -i),
  i'.Coprime k
Tactic: rwa [Int.gcd, ← Int.natAbs_neg, ← hi'] at hi
Cleared hypotheses: hi, h0, i', hi'
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots (goal)
Goal:
R : Type u_4
inst : CommRing R
inst_1 : IsDomain R
⊢ primitiveRoots 1 R = {1}
have refine_1 : 1 ∈ primitiveRoots 1 R
Tactic: refine Finset.eq_singleton_iff_unique_mem.2 ⟨?_, fun x hx ↦ ?_⟩
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots (goal)
Goal:
R : Type u_4
inst : CommRing R
inst_1 : IsDomain R
refine_1 : 1 ∈ primitiveRoots 1 R
⊢ primitiveRoots 1 R = {1}
have refine_2 : ∀ (x : R) (hx : x ∈ primitiveRoots 1 R), x = 1
Tactic: refine Finset.eq_singleton_iff_unique_mem.2 ⟨?_, fun x hx ↦ ?_⟩
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots (goal)
Goal:
R : Type u_4
inst : CommRing R
inst_1 : IsDomain R
x : R
hx : x ∈ primitiveRoots 1 R
⊢ x = 1
have refine_2 : ∀ (hx : IsPrimitiveRoot x 1), x = 1
Tactic: rwa [mem_primitiveRoots zero_lt_one, IsPrimitiveRoot.one_right_iff] at hx
Cleared hypotheses: hx
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots (goal)
Goal:
R : Type u_4
inst : CommRing R
inst_1 : IsDomain R
x : R
hx : x ∈ primitiveRoots 1 R
refine_2 : ∀ (hx : IsPrimitiveRoot x 1), x = 1
⊢ x = 1
have refine_2₁ : ∀ (hx : x = 1), x = 1
Tactic: rwa [mem_primitiveRoots zero_lt_one, IsPrimitiveRoot.one_right_iff] at hx
Cleared hypotheses: hx
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots (goal)
Goal:
R : Type u_4
ζ : R
inst : CommRing R
inst_1 : IsDomain R
n : ℕ
inst_2 : NeZero n
hζ : IsPrimitiveRoot ζ n
p : ℕ := ringChar R
⊢ NeZero.{u_4} (R := R) (↑n : R)
have refine : ∀ (hpn : p ∣ n), False
Tactic: refine .of_not_dvd R (p := p) fun hpn ↦ ?_
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots (goal)
Goal:
R : Type u_4
ζ : R
inst : CommRing R
inst_1 : IsDomain R
n : ℕ
inst_2 : NeZero n
hζ : IsPrimitiveRoot ζ n
p : ℕ := ringChar R
hpn : p ∣ n
⊢ False
have obtain : ∀ (n : ℕ) [NeZero (p * n)] (hζ : IsPrimitiveRoot ζ (p * n)), False
Tactic: obtain ⟨n, rfl⟩ := hpn
Cleared hypotheses: n, inst_2, hζ, hpn
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots (hypothesis)
Goal:
R : Type u_4
ζ : R
inst : CommRing R
inst_1 : IsDomain R
p : ℕ := ringChar R
n : ℕ
inst_2 : NeZero (p * n)
hζ : IsPrimitiveRoot ζ (p * n)
⊢ False
have h : p ≠ 0 ∧ n ≠ 0
Tactic: have h : p ≠ 0 ∧ n ≠ 0 := by aesop
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots (hypothesis)
Goal:
R : Type u_4
ζ : R
inst : CommRing R
inst_1 : IsDomain R
p : ℕ := ringChar R
n : ℕ
inst_2 : NeZero (p * n)
hζ : IsPrimitiveRoot ζ (p * n)
h : p ≠ 0 ∧ n ≠ 0
⊢ False
have this : NeZero p
Tactic: have : NeZero p := .mk h.1
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots (hypothesis)
Goal:
R : Type u_4
ζ : R
inst : CommRing R
inst_1 : IsDomain R
p : ℕ := ringChar R
n : ℕ
inst_2 : NeZero (p * n)
hζ : IsPrimitiveRoot ζ (p * n)
h : p ≠ 0 ∧ n ≠ 0
this : NeZero p
⊢ False
have hp : Fact (Nat.Prime p)
Tactic: have hp : Fact p.Prime := CharP.char_is_prime_of_pos R p
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots (goal)
Goal:
R : Type u_4
ζ : R
inst : CommRing R
inst_1 : IsDomain R
p : ℕ := ringChar R
n : ℕ
inst_2 : NeZero (p * n)
hζ : IsPrimitiveRoot ζ (p * n)
h : p ≠ 0 ∧ n ≠ 0
this : NeZero p
hp : Fact (Nat.Prime p)
⊢ False
have refine : Eq (α := R) ((frobenius R p : (a : R) → R) (ζ ^ n) : R) ((frobenius R p : (a : R) → R) 1 : R)
Tactic: refine (hζ.pow_ne_one_of_pos_of_lt h.2 (lt_mul_of_one_lt_left (by grind) hp.1.one_lt) <|
    frobenius_inj R p ?_).elim
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots (goal)
Goal:
R : Type u_4
ζ : R
inst : CommRing R
inst_1 : IsDomain R
p : ℕ := ringChar R
n : ℕ
inst_2 : NeZero (p * n)
hζ : IsPrimitiveRoot ζ (p * n)
h : p ≠ 0 ∧ n ≠ 0
this : NeZero p
hp : Fact (Nat.Prime p)
⊢ Eq (α := R) ((frobenius R p : (a : R) → R) (ζ ^ n) : R) ((frobenius R p : (a : R) → R) 1 : R)
have rw : (ζ ^ n) ^ p = (frobenius R p : (a : R) → R) 1
Tactic: rw [frobenius_def, ← pow_mul', hζ.1, map_one]
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots (goal)
Goal:
R : Type u_4
ζ : R
inst : CommRing R
inst_1 : IsDomain R
p : ℕ := ringChar R
n : ℕ
inst_2 : NeZero (p * n)
hζ : IsPrimitiveRoot ζ (p * n)
h : p ≠ 0 ∧ n ≠ 0
this : NeZero p
hp : Fact (Nat.Prime p)
rw : (ζ ^ n) ^ p = (frobenius R p : (a : R) → R) 1
⊢ Eq (α := R) ((frobenius R p : (a : R) → R) (ζ ^ n) : R) ((frobenius R p : (a : R) → R) 1 : R)
have rw₁ : ζ ^ (p * n) = (frobenius R p : (a : R) → R) 1
Tactic: rw [frobenius_def, ← pow_mul', hζ.1, map_one]
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots (goal)
Goal:
R : Type u_4
ζ : R
inst : CommRing R
inst_1 : IsDomain R
p : ℕ := ringChar R
n : ℕ
inst_2 : NeZero (p * n)
hζ : IsPrimitiveRoot ζ (p * n)
h : p ≠ 0 ∧ n ≠ 0
this : NeZero p
hp : Fact (Nat.Prime p)
rw : (ζ ^ n) ^ p = (frobenius R p : (a : R) → R) 1
rw₁ : ζ ^ (p * n) = (frobenius R p : (a : R) → R) 1
⊢ Eq (α := R) ((frobenius R p : (a : R) → R) (ζ ^ n) : R) ((frobenius R p : (a : R) → R) 1 : R)
have rw₂ : Eq (α := R) (1 : R) ((frobenius R p : (a : R) → R) 1 : R)
Tactic: rw [frobenius_def, ← pow_mul', hζ.1, map_one]
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots (goal)
Goal:
R : Type u_4
ζ : R
inst : CommRing R
inst_1 : IsDomain R
p : ℕ := ringChar R
n : ℕ
inst_2 : NeZero (p * n)
hζ : IsPrimitiveRoot ζ (p * n)
h : p ≠ 0 ∧ n ≠ 0
this : NeZero p
hp : Fact (Nat.Prime p)
rw : (ζ ^ n) ^ p = (frobenius R p : (a : R) → R) 1
rw₁ : ζ ^ (p * n) = (frobenius R p : (a : R) → R) 1
rw₂ : Eq (α := R) (1 : R) ((frobenius R p : (a : R) → R) 1 : R)
⊢ Eq (α := R) ((frobenius R p : (a : R) → R) (ζ ^ n) : R) ((frobenius R p : (a : R) → R) 1 : R)
have rw₃ : Eq.{u_4 + 1} (α := R) 1 1
Tactic: rw [frobenius_def, ← pow_mul', hζ.1, map_one]
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots (goal)
Goal:
R : Type u_4
inst : CommRing R
p : ℕ
inst_1 : Nontrivial R
h : CharP R p
hp : p ≠ 2
⊢ IsPrimitiveRoot.{u_4} (M := R) (-1) 2
have h₁ : 2 = orderOf.{u_4} (G := R) (-1)
Tactic: convert! IsPrimitiveRoot.orderOf (-1 : R)
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots (goal)
Goal:
R : Type u_4
inst : CommRing R
p : ℕ
inst_1 : Nontrivial R
h : CharP R p
hp : p ≠ 2
⊢ ¬ringChar R = 2
have rwa : ¬p = 2
Tactic: rwa [ringChar.eq_iff.mpr h]
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots (goal)
Goal:
R : Type u_4
inst : CommRing R
p : ℕ
inst_1 : Nontrivial R
h : CharP R p
hp : p ≠ 2
⊢ 2 = orderOf.{u_4} (G := R) (-1)
have h₁ : Eq (α := ℕ) 2 (if ringChar R = 2 then 1 else 2)
Tactic: rw [orderOf_neg_one, if_neg <| by rwa [ringChar.eq_iff.mpr h]]
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots (goal)
Goal:
R : Type u_4
inst : CommRing R
p : ℕ
inst_1 : Nontrivial R
h : CharP R p
hp : p ≠ 2
h₁ : Eq (α := ℕ) 2 (if ringChar R = 2 then 1 else 2)
⊢ 2 = orderOf.{u_4} (G := R) (-1)
have h₂ : Eq (α := ℕ) 2 2
Tactic: rw [orderOf_neg_one, if_neg <| by rwa [ringChar.eq_iff.mpr h]]
-- 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
hζ : IsPrimitiveRoot ζ k
hk : 1 < k
⊢ Eq.{u_4 + 1} (α := R) (∑ i ∈ range k, ζ ^ i) 0
have refine : (1 - ζ) * ∑ i ∈ range k, ζ ^ i = 0
Tactic: refine eq_zero_of_ne_zero_of_mul_left_eq_zero (sub_ne_zero_of_ne (hζ.ne_one hk).symm) ?_
-- 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
hζ : IsPrimitiveRoot ζ k
hk : 1 < k
⊢ (1 - ζ) * ∑ i ∈ range k, ζ ^ i = 0
have rw : 1 - ζ ^ k = 0
Tactic: rw [mul_neg_geom_sum, hζ.pow_eq_one, sub_self]
-- 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
hζ : IsPrimitiveRoot ζ k
hk : 1 < k
rw : 1 - ζ ^ k = 0
⊢ (1 - ζ) * ∑ i ∈ range k, ζ ^ i = 0
have rw₁ : HSub.hSub (α := R) 1 1 = 0
Tactic: rw [mul_neg_geom_sum, hζ.pow_eq_one, sub_self]
-- 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
hζ : IsPrimitiveRoot ζ k
hk : 1 < k
rw : 1 - ζ ^ k = 0
rw₁ : HSub.hSub (α := R) 1 1 = 0
⊢ (1 - ζ) * ∑ i ∈ range k, ζ ^ i = 0
have rw₂ : Eq.{u_4 + 1} (α := R) 0 0
Tactic: rw [mul_neg_geom_sum, hζ.pow_eq_one, sub_self]
-- 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
hζ : IsPrimitiveRoot ζ k
hk : 1 < k
⊢ ζ ^ k.pred = -∑ i ∈ range k.pred, ζ ^ i
have rw : ζ ^ k.pred + ∑ i ∈ range k.pred, ζ ^ i = 0
Tactic: rw [eq_neg_iff_add_eq_zero, add_comm, ← sum_range_succ, ← Nat.succ_eq_add_one,
    Nat.succ_pred_eq_of_pos (pos_of_gt hk), hζ.geom_sum_eq_zero hk]
-- 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
hζ : IsPrimitiveRoot ζ k
hk : 1 < k
rw : ζ ^ k.pred + ∑ i ∈ range k.pred, ζ ^ i = 0
⊢ ζ ^ k.pred = -∑ i ∈ range k.pred, ζ ^ i
have rw₁ : ∑ i ∈ range k.pred, ζ ^ i + ζ ^ k.pred = 0
Tactic: rw [eq_neg_iff_add_eq_zero, add_comm, ← sum_range_succ, ← Nat.succ_eq_add_one,
    Nat.succ_pred_eq_of_pos (pos_of_gt hk), hζ.geom_sum_eq_zero hk]
-- 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
hζ : IsPrimitiveRoot ζ k
hk : 1 < k
rw : ζ ^ k.pred + ∑ i ∈ range k.pred, ζ ^ i = 0
rw₁ : ∑ i ∈ range k.pred, ζ ^ i + ζ ^ k.pred = 0
⊢ ζ ^ k.pred = -∑ i ∈ range k.pred, ζ ^ i
have rw₂ : Eq.{u_4 + 1} (α := R) (∑ x ∈ range (k.pred + 1), ζ ^ x) 0
Tactic: rw [eq_neg_iff_add_eq_zero, add_comm, ← sum_range_succ, ← Nat.succ_eq_add_one,
    Nat.succ_pred_eq_of_pos (pos_of_gt hk), hζ.geom_sum_eq_zero hk]
-- 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
hζ : IsPrimitiveRoot ζ k
hk : 1 < k
rw : ζ ^ k.pred + ∑ i ∈ range k.pred, ζ ^ i = 0
rw₁ : ∑ i ∈ range k.pred, ζ ^ i + ζ ^ k.pred = 0
rw₂ : Eq.{u_4 + 1} (α := R) (∑ x ∈ range (k.pred + 1), ζ ^ x) 0
rw₃ : Eq.{u_4 + 1} (α := R) (∑ x ∈ range k.pred.succ, ζ ^ x) 0
⊢ ζ ^ k.pred = -∑ i ∈ range k.pred, ζ ^ i
have rw₄ : Eq.{u_4 + 1} (α := R) (∑ x ∈ range k, ζ ^ x) 0
Tactic: rw [eq_neg_iff_add_eq_zero, add_comm, ← sum_range_succ, ← Nat.succ_eq_add_one,
    Nat.succ_pred_eq_of_pos (pos_of_gt hk), hζ.geom_sum_eq_zero hk]
-- 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
hζ : IsPrimitiveRoot ζ k
hk : 1 < k
rw : ζ ^ k.pred + ∑ i ∈ range k.pred, ζ ^ i = 0
rw₁ : ∑ i ∈ range k.pred, ζ ^ i + ζ ^ k.pred = 0
rw₂ : Eq.{u_4 + 1} (α := R) (∑ x ∈ range (k.pred + 1), ζ ^ x) 0
rw₃ : Eq.{u_4 + 1} (α := R) (∑ x ∈ range k.pred.succ, ζ ^ x) 0
rw₄ : Eq.{u_4 + 1} (α := R) (∑ x ∈ range k, ζ ^ x) 0
⊢ ζ ^ k.pred = -∑ i ∈ range k.pred, ζ ^ i
have rw₅ : Eq.{u_4 + 1} (α := R) 0 0
Tactic: rw [eq_neg_iff_add_eq_zero, add_comm, ← sum_range_succ, ← Nat.succ_eq_add_one,
    Nat.succ_pred_eq_of_pos (pos_of_gt hk), hζ.geom_sum_eq_zero hk]
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots (goal)
Goal:
M : Type u_1
N : Type u_2
G : Type u_3
R : Type u_4
S : Type u_5
F : Type u_6
inst : CommMonoid M
inst_1 : CommMonoid N
inst_2 : DivisionCommMonoid G
k l : ℕ
inst_3 : CommRing R
ζ : Rˣ
h_1 h : IsPrimitiveRoot ζ k
⊢ Eq (α := Additive ↥(Subgroup.zpowers ζ))
    ((Additive.ofMul : (a : ↥(Subgroup.zpowers ζ)) → Additive ↥(Subgroup.zpowers ζ))
        (Subtype.mk (p := fun (x : Rˣ) ↦ Membership.mem (γ := Subgroup Rˣ) (Subgroup.zpowers ζ) x)
          ((fun (x : ℤ) ↦ ζ ^ x) 0) ⋯) :
      Additive ↥(Subgroup.zpowers ζ))
    (0 : Additive ↥(Subgroup.zpowers ζ))
have simp : Eq (α := Additive ↥(Subgroup.zpowers ζ))
  ((Additive.ofMul : (a : ↥(Subgroup.zpowers ζ)) → Additive ↥(Subgroup.zpowers ζ))
      (Subtype.mk (p := fun (x : Rˣ) ↦ Membership.mem (γ := Subgroup Rˣ) (Subgroup.zpowers ζ) x) 1 sorry) :
    Additive ↥(Subgroup.zpowers ζ))
  (0 : Additive ↥(Subgroup.zpowers ζ))
Tactic: simp only [zpow_zero]
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots (goal)
Goal:
M : Type u_1
N : Type u_2
G : Type u_3
R : Type u_4
S : Type u_5
F : Type u_6
inst : CommMonoid M
inst_1 : CommMonoid N
inst_2 : DivisionCommMonoid G
k l : ℕ
inst_3 : CommRing R
ζ : Rˣ
h_1 h : IsPrimitiveRoot ζ k
i j : ℤ
⊢ (Additive.ofMul : (a : ↥(Subgroup.zpowers ζ)) → Additive ↥(Subgroup.zpowers ζ))
      (Subtype.mk (p := fun (x : Rˣ) ↦ Membership.mem (γ := Subgroup Rˣ) (Subgroup.zpowers ζ) x)
        ((fun (x : ℤ) ↦ ζ ^ x) (i + j)) ⋯) =
    HAdd.hAdd (α := Additive ↥(Subgroup.zpowers ζ))
      ((Additive.ofMul : (a : ↥(Subgroup.zpowers ζ)) → Additive ↥(Subgroup.zpowers ζ))
        (Subtype.mk (p := fun (x : Rˣ) ↦ Membership.mem (γ := Subgroup Rˣ) (Subgroup.zpowers ζ) x)
          ((fun (x : ℤ) ↦ ζ ^ x) i) ⋯))
      ((Additive.ofMul : (a : ↥(Subgroup.zpowers ζ)) → Additive ↥(Subgroup.zpowers ζ))
        (Subtype.mk (p := fun (x : Rˣ) ↦ Membership.mem (γ := Subgroup Rˣ) (Subgroup.zpowers ζ) x)
          ((fun (x : ℤ) ↦ ζ ^ x) j) ⋯))
have simp : (Additive.ofMul : (a : ↥(Subgroup.zpowers ζ)) → Additive ↥(Subgroup.zpowers ζ))
    (Subtype.mk (p := fun (x : Rˣ) ↦ Membership.mem (γ := Subgroup Rˣ) (Subgroup.zpowers ζ) x) (ζ ^ i * ζ ^ j) sorry) =
  HAdd.hAdd (α := Additive ↥(Subgroup.zpowers ζ)) (β := Additive ↥(Subgroup.zpowers ζ))
    ((Additive.ofMul : (a : ↥(Subgroup.zpowers ζ)) → Additive ↥(Subgroup.zpowers ζ))
        (Subtype.mk (p := fun (x : Rˣ) ↦ Membership.mem (γ := Subgroup Rˣ) (Subgroup.zpowers ζ) x) (ζ ^ i) sorry) :
      Additive ↥(Subgroup.zpowers ζ))
    ((Additive.ofMul : (a : ↥(Subgroup.zpowers ζ)) → Additive ↥(Subgroup.zpowers ζ))
        (Subtype.mk (p := fun (x : Rˣ) ↦ Membership.mem (γ := Subgroup Rˣ) (Subgroup.zpowers ζ) x) (ζ ^ j) sorry) :
      Additive ↥(Subgroup.zpowers ζ))
Tactic: simp only [zpow_add]
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots (hypothesis)
Goal:
M : Type u_1
N : Type u_2
G : Type u_3
R : Type u_4
S : Type u_5
F : Type u_6
inst : CommMonoid M
inst_1 : CommMonoid N
inst_2 : DivisionCommMonoid G
k l : ℕ
inst_3 : CommRing R
ζ : Rˣ
h_1 h : IsPrimitiveRoot ζ k
i : ℤ
hi : i ∈ AddMonoidHom.ker (M := ZMod k) (Int.castAddHom (ZMod k))
⊢ Membership.mem (γ := AddSubgroup ℤ)
    (AddMonoidHom.ker.{0, u_4} (M := Additive ↥(Subgroup.zpowers ζ))
      {
        toFun := fun (i : ℤ) ↦
          (Additive.ofMul : (a : ↥(Subgroup.zpowers ζ)) → Additive ↥(Subgroup.zpowers ζ))
            (Subtype.mk (p := fun (x : Rˣ) ↦ Membership.mem (γ := Subgroup Rˣ) (Subgroup.zpowers ζ) x)
              ((fun (x : ℤ) ↦ ζ ^ x) i) ⋯),
        map_zero' := ⋯, map_add' := ⋯ })
    i
have hi₁ : (↑k : ℤ) ∣ i
Tactic: simp only [AddMonoidHom.mem_ker, CharP.intCast_eq_zero_iff (ZMod k) k, AddMonoidHom.coe_mk,
          Int.coe_castAddHom] at hi ⊢
Cleared hypotheses: hi
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots (goal)
Goal:
M : Type u_1
N : Type u_2
G : Type u_3
R : Type u_4
S : Type u_5
F : Type u_6
inst : CommMonoid M
inst_1 : CommMonoid N
inst_2 : DivisionCommMonoid G
k l : ℕ
inst_3 : CommRing R
ζ : Rˣ
h_1 h : IsPrimitiveRoot ζ k
i : ℤ
hi : (↑k : ℤ) ∣ i
⊢ Eq.{u_4 + 1} (α := Additive ↥(Subgroup.zpowers ζ))
    (({
            toFun := fun (i : ℤ) ↦
              (Additive.ofMul : (a : ↥(Subgroup.zpowers ζ)) → Additive ↥(Subgroup.zpowers ζ))
                (Subtype.mk (p := fun (x : Rˣ) ↦ Membership.mem (γ := Subgroup Rˣ) (Subgroup.zpowers ζ) x) (ζ ^ i) ⋯),
            map_zero' := ⋯ } :
          (a : ℤ) → Additive ↥(Subgroup.zpowers ζ))
        i :
      Additive ↥(Subgroup.zpowers ζ))
    (0 : Additive ↥(Subgroup.zpowers ζ))
have obtain : ∀ (i : ℤ),
  Eq.{u_4 + 1} (α := Additive ↥(Subgroup.zpowers ζ))
    (({
            toFun := fun (i : ℤ) ↦
              (Additive.ofMul : (a : ↥(Subgroup.zpowers ζ)) → Additive ↥(Subgroup.zpowers ζ))
                (Subtype.mk (p := fun (x : Rˣ) ↦ Membership.mem (γ := Subgroup Rˣ) (Subgroup.zpowers ζ) x) (ζ ^ i) sorry),
            map_zero' := sorry } :
          (a : ℤ) → Additive ↥(Subgroup.zpowers ζ))
        ((↑k : ℤ) * i) :
      Additive ↥(Subgroup.zpowers ζ))
    (0 : Additive ↥(Subgroup.zpowers ζ))
Tactic: obtain ⟨i, rfl⟩ := hi
Cleared hypotheses: i, hi
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots (goal)
Goal:
M : Type u_1
N : Type u_2
G : Type u_3
R : Type u_4
S : Type u_5
F : Type u_6
inst : CommMonoid M
inst_1 : CommMonoid N
inst_2 : DivisionCommMonoid G
k l : ℕ
inst_3 : CommRing R
ζ : Rˣ
h_1 h : IsPrimitiveRoot ζ k
⊢ Function.Bijective.{1, u_4 + 1} (α := ZMod k) (β := Additive ↥(Subgroup.zpowers ζ))
    (⇑((AddMonoidHom.liftOfRightInverse (Int.castAddHom (ZMod k)) ZMod.cast ⋯ :
              (a : { g : ℤ →+ Additive ↥(Subgroup.zpowers ζ) // (Int.castAddHom (ZMod k)).ker ≤ g.ker }) →
                ZMod k →+ Additive ↥(Subgroup.zpowers ζ))
            (Subtype.mk (p := fun (g : ℤ →+ Additive ↥(Subgroup.zpowers ζ)) ↦
              AddMonoidHom.ker (M := ZMod k) (Int.castAddHom (ZMod k)) ≤ g.ker)
              {
                toFun := fun (i : ℤ) ↦
                  (Additive.ofMul : (a : ↥(Subgroup.zpowers ζ)) → Additive ↥(Subgroup.zpowers ζ))
                    (Subtype.mk (p := fun (x : Rˣ) ↦ Membership.mem (γ := Subgroup Rˣ) (Subgroup.zpowers ζ) x)
                      ((fun (x : ℤ) ↦ ζ ^ x) i) ⋯),
                map_zero' := ⋯, map_add' := ⋯ }
              ⋯) :
          ZMod k →+ Additive ↥(Subgroup.zpowers ζ)) :
      (a : ZMod k) → Additive ↥(Subgroup.zpowers ζ))
have left : Function.Injective.{1, u_4 + 1} (α := ZMod k) (β := Additive ↥(Subgroup.zpowers ζ))
  (⇑((AddMonoidHom.liftOfRightInverse (Int.castAddHom (ZMod k)) ZMod.cast sorry :
            (a : { g : ℤ →+ Additive ↥(Subgroup.zpowers ζ) // (Int.castAddHom (ZMod k)).ker ≤ g.ker }) →
              ZMod k →+ Additive ↥(Subgroup.zpowers ζ))
          (Subtype.mk (p := fun (g : ℤ →+ Additive ↥(Subgroup.zpowers ζ)) ↦
            AddMonoidHom.ker (M := ZMod k) (Int.castAddHom (ZMod k)) ≤ g.ker)
            {
              toFun := fun (i : ℤ) ↦
                (Additive.ofMul : (a : ↥(Subgroup.zpowers ζ)) → Additive ↥(Subgroup.zpowers ζ))
                  (Subtype.mk (p := fun (x : Rˣ) ↦ Membership.mem (γ := Subgroup Rˣ) (Subgroup.zpowers ζ) x)
                    ((fun (x : ℤ) ↦ ζ ^ x) i) sorry),
              map_zero' := sorry, map_add' := sorry }
            sorry) :
        ZMod k →+ Additive ↥(Subgroup.zpowers ζ)) :
    (a : ZMod k) → Additive ↥(Subgroup.zpowers ζ))
Tactic: constructor
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots (goal)
Goal:
M : Type u_1
N : Type u_2
G : Type u_3
R : Type u_4
S : Type u_5
F : Type u_6
inst : CommMonoid M
inst_1 : CommMonoid N
inst_2 : DivisionCommMonoid G
k l : ℕ
inst_3 : CommRing R
ζ : Rˣ
h_1 h : IsPrimitiveRoot ζ k
⊢ Function.Injective.{1, u_4 + 1} (α := ZMod k) (β := Additive ↥(Subgroup.zpowers ζ))
    (⇑((AddMonoidHom.liftOfRightInverse (Int.castAddHom (ZMod k)) ZMod.cast ⋯ :
              (a : { g : ℤ →+ Additive ↥(Subgroup.zpowers ζ) // (Int.castAddHom (ZMod k)).ker ≤ g.ker }) →
                ZMod k →+ Additive ↥(Subgroup.zpowers ζ))
            (Subtype.mk (p := fun (g : ℤ →+ Additive ↥(Subgroup.zpowers ζ)) ↦
              AddMonoidHom.ker (M := ZMod k) (Int.castAddHom (ZMod k)) ≤ g.ker)
              {
                toFun := fun (i : ℤ) ↦
                  (Additive.ofMul : (a : ↥(Subgroup.zpowers ζ)) → Additive ↥(Subgroup.zpowers ζ))
                    (Subtype.mk (p := fun (x : Rˣ) ↦ Membership.mem (γ := Subgroup Rˣ) (Subgroup.zpowers ζ) x)
                      ((fun (x : ℤ) ↦ ζ ^ x) i) ⋯),
                map_zero' := ⋯, map_add' := ⋯ }
              ⋯) :
          ZMod k →+ Additive ↥(Subgroup.zpowers ζ)) :
      (a : ZMod k) → Additive ↥(Subgroup.zpowers ζ))
have left : ∀ (a : ZMod k),
  Eq.{u_4 + 1} (α := Additive ↥(Subgroup.zpowers ζ))
      ((((AddMonoidHom.liftOfRightInverse (Int.castAddHom (ZMod k)) ZMod.cast sorry :
                  (a : { g : ℤ →+ Additive ↥(Subgroup.zpowers ζ) // (Int.castAddHom (ZMod k)).ker ≤ g.ker }) →
                    ZMod k →+ Additive ↥(Subgroup.zpowers ζ))
                (Subtype.mk (p := fun (g : ℤ →+ Additive ↥(Subgroup.zpowers ζ)) ↦
                  AddMonoidHom.ker (M := ZMod k) (Int.castAddHom (ZMod k)) ≤ g.ker)
                  {
                    toFun := fun (i : ℤ) ↦
                      (Additive.ofMul : (a : ↥(Subgroup.zpowers ζ)) → Additive ↥(Subgroup.zpowers ζ))
                        (Subtype.mk (p := fun (x : Rˣ) ↦ Membership.mem (γ := Subgroup Rˣ) (Subgroup.zpowers ζ) x)
                          ((fun (x : ℤ) ↦ ζ ^ x) i) sorry),
                    map_zero' := sorry, map_add' := sorry }
                  sorry) :
              ZMod k →+ Additive ↥(Subgroup.zpowers ζ)) :
            (a : ZMod k) → Additive ↥(Subgroup.zpowers ζ))
          a :
        Additive ↥(Subgroup.zpowers ζ))
      (0 : Additive ↥(Subgroup.zpowers ζ)) →
    a = 0
Tactic: rw [injective_iff_map_eq_zero]