awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots (goal)
Goal:
R : Type u_4
inst : CommRing R
inst_1 : IsDomain R
ζ : R
n : ℕ
h : IsPrimitiveRoot ζ n
⊢ card.{u_4} (α := R) (nthRootsFinset n 1) = n
have rw : card.{u_4} (α := R) (Multiset.toFinset (nthRoots n 1)) = n
Tactic: rw [nthRootsFinset, ← Multiset.toFinset_eq (nthRoots_one_nodup h), card_mk, h.card_nthRoots_one]
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots (goal)
Goal:
R : Type u_4
inst : CommRing R
inst_1 : IsDomain R
ζ : R
n : ℕ
h : IsPrimitiveRoot ζ n
rw : card.{u_4} (α := R) (Multiset.toFinset (nthRoots n 1)) = n
⊢ card.{u_4} (α := R) (nthRootsFinset n 1) = n
have rw₁ : #{ val := nthRoots n 1, nodup := sorry : Finset R } = n
Tactic: rw [nthRootsFinset, ← Multiset.toFinset_eq (nthRoots_one_nodup h), card_mk, h.card_nthRoots_one]
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots (goal)
Goal:
R : Type u_4
inst : CommRing R
inst_1 : IsDomain R
ζ : R
n : ℕ
h : IsPrimitiveRoot ζ n
rw : card.{u_4} (α := R) (Multiset.toFinset (nthRoots n 1)) = n
rw₁ : #{ val := nthRoots n 1, nodup := ⋯ : Finset R } = n
⊢ card.{u_4} (α := R) (nthRootsFinset n 1) = n
have rw₂ : Multiset.card.{u_4} (α := R) (nthRoots n 1) = n
Tactic: rw [nthRootsFinset, ← Multiset.toFinset_eq (nthRoots_one_nodup h), card_mk, h.card_nthRoots_one]
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots (goal)
Goal:
R : Type u_4
inst : CommRing R
inst_1 : IsDomain R
ζ : R
n : ℕ
h : IsPrimitiveRoot ζ n
rw : card.{u_4} (α := R) (Multiset.toFinset (nthRoots n 1)) = n
rw₁ : #{ val := nthRoots n 1, nodup := ⋯ : Finset R } = n
rw₂ : Multiset.card.{u_4} (α := R) (nthRoots n 1) = n
⊢ card.{u_4} (α := R) (nthRootsFinset n 1) = n
have rw₃ : n = n
Tactic: rw [nthRootsFinset, ← Multiset.toFinset_eq (nthRoots_one_nodup h), card_mk, h.card_nthRoots_one]
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots (goal)
Goal:
R : Type u_4
inst : CommRing R
inst_1 : IsDomain R
ζ : R
n : ℕ
h : IsPrimitiveRoot ζ n
⊢ card.{u_4} (α := R) (nthRootsFinset n 1) = n
have rw : card.{u_4} (α := R) (Multiset.toFinset (nthRoots n 1)) = n
Tactic: classical
  rw [nthRootsFinset, ← Multiset.toFinset_eq (nthRoots_one_nodup h), card_mk, h.card_nthRoots_one]
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots (goal)
Goal:
R : Type u_4
inst : CommRing R
inst_1 : IsDomain R
ζ : R
n : ℕ
h : IsPrimitiveRoot ζ n
rw : card.{u_4} (α := R) (Multiset.toFinset (nthRoots n 1)) = n
⊢ card.{u_4} (α := R) (nthRootsFinset n 1) = n
have rw₁ : #{ val := nthRoots n 1, nodup := sorry : Finset R } = n
Tactic: classical
  rw [nthRootsFinset, ← Multiset.toFinset_eq (nthRoots_one_nodup h), card_mk, h.card_nthRoots_one]
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots (goal)
Goal:
R : Type u_4
inst : CommRing R
inst_1 : IsDomain R
ζ : R
n : ℕ
h : IsPrimitiveRoot ζ n
rw : card.{u_4} (α := R) (Multiset.toFinset (nthRoots n 1)) = n
rw₁ : #{ val := nthRoots n 1, nodup := ⋯ : Finset R } = n
⊢ card.{u_4} (α := R) (nthRootsFinset n 1) = n
have rw₂ : Multiset.card.{u_4} (α := R) (nthRoots n 1) = n
Tactic: classical
  rw [nthRootsFinset, ← Multiset.toFinset_eq (nthRoots_one_nodup h), card_mk, h.card_nthRoots_one]
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots (goal)
Goal:
R : Type u_4
inst : CommRing R
inst_1 : IsDomain R
ζ : R
n : ℕ
h : IsPrimitiveRoot ζ n
rw : card.{u_4} (α := R) (Multiset.toFinset (nthRoots n 1)) = n
rw₁ : #{ val := nthRoots n 1, nodup := ⋯ : Finset R } = n
rw₂ : Multiset.card.{u_4} (α := R) (nthRoots n 1) = n
⊢ card.{u_4} (α := R) (nthRootsFinset n 1) = n
have rw₃ : n = n
Tactic: classical
  rw [nthRootsFinset, ← Multiset.toFinset_eq (nthRoots_one_nodup h), card_mk, h.card_nthRoots_one]
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots (goal)
Goal:
R : Type u_4
inst : CommRing R
inst_1 : IsDomain R
ζ : R
k : ℕ
h : IsPrimitiveRoot ζ k
⊢ #(primitiveRoots k R) = φ k
have pos : ∀ (h0 : k = 0), #(primitiveRoots k R) = φ k
Tactic: by_cases h0 : k = 0
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots (goal)
Goal:
R : Type u_4
inst : CommRing R
inst_1 : IsDomain R
ζ : R
k : ℕ
h : IsPrimitiveRoot ζ k
pos : ∀ (h0 : k = 0), #(primitiveRoots k R) = φ k
⊢ #(primitiveRoots k R) = φ k
have neg : ∀ (h0 : ¬k = 0), #(primitiveRoots k R) = φ k
Tactic: by_cases h0 : k = 0
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots (hypothesis)
Goal:
R : Type u_4
inst : CommRing R
inst_1 : IsDomain R
ζ : R
k : ℕ
h : IsPrimitiveRoot ζ k
h0 : ¬k = 0
⊢ #(primitiveRoots k R) = φ k
have this : NeZero k
Tactic: have : NeZero k := ⟨h0⟩
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots (goal)
Goal:
R : Type u_4
inst : CommRing R
inst_1 : IsDomain R
ζ : R
k : ℕ
h : IsPrimitiveRoot ζ k
h0 : ¬k = 0
this : NeZero k
⊢ #(primitiveRoots k R) = φ k
have neg : φ k = #(primitiveRoots k R)
Tactic: symm
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots (goal)
Goal:
R : Type u_4
inst : CommRing R
inst_1 : IsDomain R
ζ : R
k : ℕ
h : IsPrimitiveRoot ζ k
h0 : ¬k = 0
this : NeZero k
⊢ φ k = #(primitiveRoots k R)
have neg : ∀ (a : ℕ) (ha : a ∈ {a ∈ range k | k.Coprime a}),
  (fun (i : ℕ) (x : i ∈ {a ∈ range k | k.Coprime a}) ↦ ζ ^ i) a ha ∈ primitiveRoots k R
Tactic: refine Finset.card_bij (fun i _ ↦ ζ ^ i) ?_ ?_ ?_
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots (goal)
Goal:
R : Type u_4
inst : CommRing R
inst_1 : IsDomain R
ζ : R
k : ℕ
h : IsPrimitiveRoot ζ k
h0 : ¬k = 0
this : NeZero k
neg :
  ∀ (a : ℕ) (ha : a ∈ {a ∈ range k | k.Coprime a}),
    (fun (i : ℕ) (x : i ∈ {a ∈ range k | k.Coprime a}) ↦ ζ ^ i) a ha ∈ primitiveRoots k R
⊢ φ k = #(primitiveRoots k R)
have neg₁ : ∀ (a₁ : ℕ) (ha₁ : a₁ ∈ {a ∈ range k | k.Coprime a}) (a₂ : ℕ) (ha₂ : a₂ ∈ {a ∈ range k | k.Coprime a}),
  Eq.{u_4 + 1} (α := R) ((fun (i : ℕ) (x : i ∈ {a ∈ range k | k.Coprime a}) ↦ ζ ^ i) a₁ ha₁)
      ((fun (i : ℕ) (x : i ∈ {a ∈ range k | k.Coprime a}) ↦ ζ ^ i) a₂ ha₂) →
    a₁ = a₂
Tactic: refine Finset.card_bij (fun i _ ↦ ζ ^ i) ?_ ?_ ?_
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots (goal)
Goal:
R : Type u_4
inst : CommRing R
inst_1 : IsDomain R
ζ : R
k : ℕ
h : IsPrimitiveRoot ζ k
h0 : ¬k = 0
this : NeZero k
neg :
  ∀ (a : ℕ) (ha : a ∈ {a ∈ range k | k.Coprime a}),
    (fun (i : ℕ) (x : i ∈ {a ∈ range k | k.Coprime a}) ↦ ζ ^ i) a ha ∈ primitiveRoots k R
neg₁ :
  ∀ (a₁ : ℕ) (ha₁ : a₁ ∈ {a ∈ range k | k.Coprime a}) (a₂ : ℕ) (ha₂ : a₂ ∈ {a ∈ range k | k.Coprime a}),
    Eq.{u_4 + 1} (α := R) ((fun (i : ℕ) (x : i ∈ {a ∈ range k | k.Coprime a}) ↦ ζ ^ i) a₁ ha₁)
        ((fun (i : ℕ) (x : i ∈ {a ∈ range k | k.Coprime a}) ↦ ζ ^ i) a₂ ha₂) →
      a₁ = a₂
⊢ φ k = #(primitiveRoots k R)
have neg₂ : ∀ b ∈ primitiveRoots k R,
  ∃ (a : ℕ) (ha : a ∈ {a ∈ range k | k.Coprime a}), (fun (i : ℕ) (x : i ∈ {a ∈ range k | k.Coprime a}) ↦ ζ ^ i) a ha = b
Tactic: refine Finset.card_bij (fun i _ ↦ ζ ^ i) ?_ ?_ ?_
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots (goal)
Goal:
R : Type u_4
inst : CommRing R
inst_1 : IsDomain R
ζ : R
k : ℕ
h : IsPrimitiveRoot ζ k
h0 : ¬k = 0
this : NeZero k
⊢ ∀ (a : ℕ) (ha : a ∈ {a ∈ range k | k.Coprime a}),
    (fun (i : ℕ) (x : i ∈ {a ∈ range k | k.Coprime a}) ↦ ζ ^ i) a ha ∈ primitiveRoots k R
have neg : ∀ a < k, k.Coprime a → ζ ^ a ∈ primitiveRoots k R
Tactic: simp only [and_imp, mem_filter, mem_range]
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots (goal)
Goal:
R : Type u_4
inst : CommRing R
inst_1 : IsDomain R
ζ : R
k : ℕ
h : IsPrimitiveRoot ζ k
h0 : ¬k = 0
this : NeZero k
i : ℕ
hi : k.Coprime i
⊢ ζ ^ i ∈ primitiveRoots k R
have neg : IsPrimitiveRoot (ζ ^ i) k
Tactic: rw [mem_primitiveRoots (Nat.pos_of_ne_zero h0)]
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots (goal)
Goal:
R : Type u_4
inst : CommRing R
inst_1 : IsDomain R
ζ : R
k : ℕ
h : IsPrimitiveRoot ζ k
h0 : ¬k = 0
this : NeZero k
⊢ ∀ (a₁ : ℕ) (ha₁ : a₁ ∈ {a ∈ range k | k.Coprime a}) (a₂ : ℕ) (ha₂ : a₂ ∈ {a ∈ range k | k.Coprime a}),
    Eq.{u_4 + 1} (α := R) ((fun (i : ℕ) (x : i ∈ {a ∈ range k | k.Coprime a}) ↦ ζ ^ i) a₁ ha₁)
        ((fun (i : ℕ) (x : i ∈ {a ∈ range k | k.Coprime a}) ↦ ζ ^ i) a₂ ha₂) →
      a₁ = a₂
have neg : ∀ a₁ < k, k.Coprime a₁ → ∀ a₂ < k, k.Coprime a₂ → ζ ^ a₁ = ζ ^ a₂ → a₁ = a₂
Tactic: simp only [and_imp, mem_filter, mem_range]
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots (goal)
Goal:
R : Type u_4
inst : CommRing R
inst_1 : IsDomain R
ζ : R
k : ℕ
h : IsPrimitiveRoot ζ k
h0 : ¬k = 0
this : NeZero k
⊢ ∀ b ∈ primitiveRoots k R,
    ∃ (a : ℕ) (ha : a ∈ {a ∈ range k | k.Coprime a}),
      (fun (i : ℕ) (x : i ∈ {a ∈ range k | k.Coprime a}) ↦ ζ ^ i) a ha = b
have neg : ∀ b ∈ primitiveRoots k R, ∃ (a : ℕ), (a < k ∧ k.Coprime a) ∧ ζ ^ a = b
Tactic: simp only [exists_prop, mem_filter, mem_range]
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots (hypothesis)
Goal:
R : Type u_4
inst : CommRing R
inst_1 : IsDomain R
ζ : R
k : ℕ
h : IsPrimitiveRoot ζ k
h0 : ¬k = 0
this : NeZero k
ξ : R
hξ : ξ ∈ primitiveRoots k R
⊢ ∃ (a : ℕ), (a < k ∧ k.Coprime a) ∧ ζ ^ a = ξ
have hξ₁ : ∃ i < k, i.Coprime k ∧ ζ ^ i = ξ
Tactic: rw [mem_primitiveRoots (Nat.pos_of_ne_zero h0), h.isPrimitiveRoot_iff] at hξ
Cleared hypotheses: hξ
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots (goal)
Goal:
R : Type u_4
inst : CommRing R
inst_1 : IsDomain R
ζ : R
k : ℕ
h : IsPrimitiveRoot ζ k
h0 : ¬k = 0
this : NeZero k
ξ : R
hξ : ∃ i < k, i.Coprime k ∧ ζ ^ i = ξ
⊢ ∃ (a : ℕ), (a < k ∧ k.Coprime a) ∧ ζ ^ a = ξ
have neg : ∀ (i : ℕ) (hin : i < k) (hi : i.Coprime k) (H : ζ ^ i = ξ), ∃ (a : ℕ), (a < k ∧ k.Coprime a) ∧ ζ ^ a = ξ
Tactic: rcases hξ with ⟨i, hin, hi, H⟩
Cleared hypotheses: hξ
-- 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 : IsPrimitiveRoot ζ k
inst_4 : IsDomain R
a b n : ℕ
h : a * b ≡ 1 [MOD n]
x : ↥(primitiveRoots n R)
⊢ 0 < n
have by_contra! : ∀ (h : n ≤ 0), False
Tactic: by_contra! h
-- 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 : IsPrimitiveRoot ζ k
inst_4 : IsDomain R
a b n : ℕ
h_2 : a * b ≡ 1 [MOD n]
x : ↥(primitiveRoots n R)
h : n ≤ 0
⊢ False
have mk : ∀ val ∈ primitiveRoots n R, False
Tactic: cases x
Cleared hypotheses: x
-- 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 : IsPrimitiveRoot ζ k
inst_4 : IsDomain R
a b n : ℕ
h : a * b ≡ 1 [MOD n]
x : ↥(primitiveRoots n R)
⊢ (fun (x : ↥(primitiveRoots n R)) ↦
        (Subtype.mk (p := fun (x : R) ↦ x ∈ primitiveRoots n R) ((↑x : R) ^ b) ⋯ : ↥(primitiveRoots n R)))
      ((fun (x : ↥(primitiveRoots n R)) ↦
          (Subtype.mk (p := fun (x : R) ↦ x ∈ primitiveRoots n R) ((↑x : R) ^ a) ⋯ : ↥(primitiveRoots n R)))
        x) =
    x
have a₁ : (↑((fun (x : ↥(primitiveRoots n R)) ↦
          (Subtype.mk (p := fun (x : R) ↦ x ∈ primitiveRoots n R) ((↑x : R) ^ b) sorry : ↥(primitiveRoots n R)))
        ((fun (x : ↥(primitiveRoots n R)) ↦
            (Subtype.mk (p := fun (x : R) ↦ x ∈ primitiveRoots n R) ((↑x : R) ^ a) sorry : ↥(primitiveRoots n R)))
          x)) :
    R) =
  (↑x : R)
Tactic: ext
-- 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 : IsPrimitiveRoot ζ k
inst_4 : IsDomain R
a b n : ℕ
h : a * b ≡ 1 [MOD n]
x : ↥(primitiveRoots n R)
⊢ (fun (x : ↥(primitiveRoots n R)) ↦
        (Subtype.mk (p := fun (x : R) ↦ x ∈ primitiveRoots n R) ((↑x : R) ^ a) ⋯ : ↥(primitiveRoots n R)))
      ((fun (x : ↥(primitiveRoots n R)) ↦
          (Subtype.mk (p := fun (x : R) ↦ x ∈ primitiveRoots n R) ((↑x : R) ^ b) ⋯ : ↥(primitiveRoots n R)))
        x) =
    x
have a₁ : (↑((fun (x : ↥(primitiveRoots n R)) ↦
          (Subtype.mk (p := fun (x : R) ↦ x ∈ primitiveRoots n R) ((↑x : R) ^ a) sorry : ↥(primitiveRoots n R)))
        ((fun (x : ↥(primitiveRoots n R)) ↦
            (Subtype.mk (p := fun (x : R) ↦ x ∈ primitiveRoots n R) ((↑x : R) ^ b) sorry : ↥(primitiveRoots n R)))
          x)) :
    R) =
  (↑x : R)
Tactic: ext
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots (goal)
Goal:
R : Type u_4
inst : CommRing R
inst_1 : IsDomain R
inst_2 : DecidableEq R
n : ℕ
inst_3 : NeZero n
⊢ Eq.{u_4 + 1} (α := Finset R) (nthRootsFinset n 1) (n.divisors.biUnion fun (i : ℕ) ↦ primitiveRoots i R)
have h : ∀ (x : R),
  Membership.mem (γ := Finset R) (nthRootsFinset n 1) x ↔
    Membership.mem (γ := Finset R) (n.divisors.biUnion fun (i : ℕ) ↦ primitiveRoots i R) x
Tactic: ext x
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots (goal)
Goal:
R : Type u_4
inst : CommRing R
inst_1 : IsDomain R
inst_2 : DecidableEq R
n : ℕ
inst_3 : NeZero n
x : R
⊢ Membership.mem (γ := Finset R) (nthRootsFinset n 1) x ↔
    Membership.mem (γ := Finset R) (n.divisors.biUnion fun (i : ℕ) ↦ primitiveRoots i R) x
have h : x ^ n = 1 ↔ ∃ (a : ℕ), a ∣ n ∧ x ∈ primitiveRoots a R
Tactic: suffices x ^ n = 1 ↔ ∃ a, a ∣ n ∧ x ∈ primitiveRoots a R by
    simpa [Polynomial.mem_nthRootsFinset (NeZero.pos n), (NeZero.ne n)]
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots (goal)
Goal:
R : Type u_4
inst : CommRing R
inst_1 : IsDomain R
inst_2 : DecidableEq R
n : ℕ
inst_3 : NeZero n
x : R
⊢ x ^ n = 1 ↔ ∃ (a : ℕ), a ∣ n ∧ x ∈ primitiveRoots a R
have h : x ^ n = 1 → ∃ (a : ℕ), a ∣ n ∧ x ∈ primitiveRoots a R
Tactic: constructor
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots (goal)
Goal:
R : Type u_4
inst : CommRing R
inst_1 : IsDomain R
inst_2 : DecidableEq R
n : ℕ
inst_3 : NeZero n
x : R
h : x ^ n = 1 → ∃ (a : ℕ), a ∣ n ∧ x ∈ primitiveRoots a R
⊢ x ^ n = 1 ↔ ∃ (a : ℕ), a ∣ n ∧ x ∈ primitiveRoots a R
have h₁ : (∃ (a : ℕ), a ∣ n ∧ x ∈ primitiveRoots a R) → x ^ n = 1
Tactic: constructor
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots (goal)
Goal:
R : Type u_4
inst : CommRing R
inst_1 : IsDomain R
inst_2 : DecidableEq R
n : ℕ
inst_3 : NeZero n
x : R
H : x ^ n = 1
⊢ ∃ (a : ℕ), a ∣ n ∧ x ∈ primitiveRoots a R
have h : ∀ (k : ℕ) (hk : k > 0) (hx : IsPrimitiveRoot x k), ∃ (a : ℕ), a ∣ n ∧ x ∈ primitiveRoots a R
Tactic: obtain ⟨k, hk, hx⟩ := exists_pos H (NeZero.ne n)
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots (hypothesis)
Goal:
R : Type u_4
inst : CommRing R
inst_1 : IsDomain R
inst_2 : DecidableEq R
n : ℕ
inst_3 : NeZero n
x : R
a : ℕ
ha : x ∈ primitiveRoots a R
d : ℕ
hd : n = a * d
⊢ x ^ n = 1
have hazero : 0 < a
Tactic: have hazero : 0 < a := Nat.pos_of_ne_zero fun ha₀ ↦ by simp_all
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots (goal)
Goal:
R : Type u_4
inst : CommRing R
inst_1 : IsDomain R
inst_2 : DecidableEq R
n : ℕ
inst_3 : NeZero n
x : R
a : ℕ
ha : x ∈ primitiveRoots a R
d : ℕ
hd : n = a * d
hazero : 0 < a
⊢ x ^ n = 1
have h : ∀ (ha : IsPrimitiveRoot x a) (d : ℕ) (hd : n = a * d) (hazero : 0 < a), x ^ n = 1
Tactic: rw [mem_primitiveRoots hazero] at ha
Cleared hypotheses: ha, d, hd, hazero
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots (goal)
Goal:
R : Type u_4
inst : CommRing R
inst_1 : IsDomain R
inst_2 : DecidableEq R
n : ℕ
inst_3 : NeZero n
x : R
a : ℕ
ha : IsPrimitiveRoot x a
d : ℕ
hd : n = a * d
hazero : 0 < a
⊢ x ^ n = 1
have h : x ^ (a * d) = 1
Tactic: rw [hd, pow_mul, ha.pow_eq_one, one_pow]
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots (goal)
Goal:
R : Type u_4
inst : CommRing R
inst_1 : IsDomain R
inst_2 : DecidableEq R
n : ℕ
inst_3 : NeZero n
x : R
a : ℕ
ha : IsPrimitiveRoot x a
d : ℕ
hd : n = a * d
hazero : 0 < a
h : x ^ (a * d) = 1
⊢ x ^ n = 1
have h₁ : (x ^ a) ^ d = 1
Tactic: rw [hd, pow_mul, ha.pow_eq_one, one_pow]
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots (goal)
Goal:
R : Type u_4
inst : CommRing R
inst_1 : IsDomain R
inst_2 : DecidableEq R
n : ℕ
inst_3 : NeZero n
x : R
a : ℕ
ha : IsPrimitiveRoot x a
d : ℕ
hd : n = a * d
hazero : 0 < a
h : x ^ (a * d) = 1
h₁ : (x ^ a) ^ d = 1
⊢ x ^ n = 1
have h₂ : HPow.hPow (α := R) 1 d = 1
Tactic: rw [hd, pow_mul, ha.pow_eq_one, one_pow]
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots (goal)
Goal:
R : Type u_4
inst : CommRing R
inst_1 : IsDomain R
inst_2 : DecidableEq R
n : ℕ
inst_3 : NeZero n
x : R
a : ℕ
ha : IsPrimitiveRoot x a
d : ℕ
hd : n = a * d
hazero : 0 < a
h : x ^ (a * d) = 1
h₁ : (x ^ a) ^ d = 1
h₂ : HPow.hPow (α := R) 1 d = 1
⊢ x ^ n = 1
have h₃ : Eq.{u_4 + 1} (α := R) 1 1
Tactic: rw [hd, pow_mul, ha.pow_eq_one, one_pow]
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots (goal)
Goal:
R : Type u_4
inst : CommRing R
inst_1 : IsDomain R
inst_2 : DecidableEq R
n : ℕ
⊢ Eq.{u_4 + 1} (α := Finset R) (nthRootsFinset n 1) (n.divisors.biUnion fun (i : ℕ) ↦ primitiveRoots i R)
have pos : ∀ (hn : n = 0), Eq.{u_4 + 1} (α := Finset R) (nthRootsFinset n 1) (n.divisors.biUnion fun (i : ℕ) ↦ primitiveRoots i R)
Tactic: by_cases hn : n = 0
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots (goal)
Goal:
R : Type u_4
inst : CommRing R
inst_1 : IsDomain R
inst_2 : DecidableEq R
n : ℕ
pos :
  ∀ (hn : n = 0),
    Eq.{u_4 + 1} (α := Finset R) (nthRootsFinset n 1) (n.divisors.biUnion fun (i : ℕ) ↦ primitiveRoots i R)
⊢ Eq.{u_4 + 1} (α := Finset R) (nthRootsFinset n 1) (n.divisors.biUnion fun (i : ℕ) ↦ primitiveRoots i R)
have neg : ∀ (hn : ¬n = 0), Eq.{u_4 + 1} (α := Finset R) (nthRootsFinset n 1) (n.divisors.biUnion fun (i : ℕ) ↦ primitiveRoots i R)
Tactic: by_cases hn : n = 0
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots (hypothesis)
Goal:
R : Type u_4
inst : CommRing R
inst_1 : IsDomain R
inst_2 : DecidableEq R
n : ℕ
hn : ¬n = 0
⊢ Eq.{u_4 + 1} (α := Finset R) (nthRootsFinset n 1) (n.divisors.biUnion fun (i : ℕ) ↦ primitiveRoots i R)
have this : NeZero n
Tactic: have : NeZero n := ⟨hn⟩
-- 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 S
inst_4 : IsDomain S
μ : S
n : ℕ
hμ : IsPrimitiveRoot μ n
inst_5 : CommRing R
inst_6 : Algebra R S
inst_7 : NeZero n
μ' : ↥(rootsOfUnity n S) := hμ.toRootsOfUnity
⊢ orderOf μ' = n
have refine : orderOf μ' = orderOf μ
Tactic: refine Eq.trans ?_ hμ.eq_orderOf.symm
-- 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 S
inst_4 : IsDomain S
μ : S
n : ℕ
hμ : IsPrimitiveRoot μ n
inst_5 : CommRing R
inst_6 : Algebra R S
inst_7 : NeZero n
μ' : ↥(rootsOfUnity n S) := hμ.toRootsOfUnity
⊢ orderOf μ' = orderOf μ
have rw : orderOf μ' = orderOf (↑(↑hμ.toRootsOfUnity : Sˣ) : S)
Tactic: rw [← hμ.val_toRootsOfUnity_coe, orderOf_units, Subgroup.orderOf_coe]
-- 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 S
inst_4 : IsDomain S
μ : S
n : ℕ
hμ : IsPrimitiveRoot μ n
inst_5 : CommRing R
inst_6 : Algebra R S
inst_7 : NeZero n
μ' : ↥(rootsOfUnity n S) := hμ.toRootsOfUnity
rw : orderOf μ' = orderOf (↑(↑hμ.toRootsOfUnity : Sˣ) : S)
⊢ orderOf μ' = orderOf μ
have rw₁ : orderOf μ' = orderOf (↑hμ.toRootsOfUnity : Sˣ)
Tactic: rw [← hμ.val_toRootsOfUnity_coe, orderOf_units, Subgroup.orderOf_coe]
-- 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 S
inst_4 : IsDomain S
μ : S
n : ℕ
hμ : IsPrimitiveRoot μ n
inst_5 : CommRing R
inst_6 : Algebra R S
inst_7 : NeZero n
μ' : ↥(rootsOfUnity n S) := hμ.toRootsOfUnity
rw : orderOf μ' = orderOf (↑(↑hμ.toRootsOfUnity : Sˣ) : S)
rw₁ : orderOf μ' = orderOf (↑hμ.toRootsOfUnity : Sˣ)
⊢ orderOf μ' = orderOf μ
have rw₂ : orderOf μ' = orderOf hμ.toRootsOfUnity
Tactic: rw [← hμ.val_toRootsOfUnity_coe, orderOf_units, Subgroup.orderOf_coe]
-- 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 S
inst_4 : IsDomain S
μ : S
n : ℕ
hμ : IsPrimitiveRoot μ n
inst_5 : CommRing R
inst_6 : Algebra R S
inst_7 : NeZero n
μ' : ↥(rootsOfUnity n S) := hμ.toRootsOfUnity
ho : orderOf μ' = n
⊢ Eq (α := ZMod n)
    (↑(Exists.choose (p := fun (m : ℕ) ↦ ((↑1 : S →ₐ[R] S) : (a : S) → S) (↑(↑μ' : Sˣ) : S) = (↑(↑μ' : Sˣ) : S) ^ m)
          ⋯) :
      ZMod n)
    1
have h1 : ∃ (m : ℕ), ((↑1 : S →ₐ[R] S) : (a : S) → S) (↑(↑μ' : Sˣ) : S) = (↑(↑μ' : Sˣ) : S) ^ m
Tactic: generalize_proofs h1
-- 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 S
inst_4 : IsDomain S
μ : S
n : ℕ
hμ : IsPrimitiveRoot μ n
inst_5 : CommRing R
inst_6 : Algebra R S
inst_7 : NeZero n
μ' : ↥(rootsOfUnity n S) := hμ.toRootsOfUnity
ho : orderOf μ' = n
h1 : ∃ (m : ℕ), ((↑1 : S →ₐ[R] S) : (a : S) → S) (↑(↑μ' : Sˣ) : S) = (↑(↑μ' : Sˣ) : S) ^ m
⊢ Eq (α := ZMod n)
    (↑(h1.choose (p := fun (m : ℕ) ↦ ((↑1 : S →ₐ[R] S) : (a : S) → S) (↑(↑μ' : Sˣ) : S) = (↑(↑μ' : Sˣ) : S) ^ m)) :
      ZMod n)
    1
have h : ((↑1 : S →ₐ[R] S) : (a : S) → S) (↑(↑μ' : Sˣ) : S) =
  (↑(↑μ' : Sˣ) : S) ^
    h1.choose (p := fun (m : ℕ) ↦ ((↑1 : S →ₐ[R] S) : (a : S) → S) (↑(↑μ' : Sˣ) : S) = (↑(↑μ' : Sˣ) : S) ^ m)
Tactic: have h := h1.choose_spec
-- 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 S
inst_4 : IsDomain S
μ : S
n : ℕ
hμ : IsPrimitiveRoot μ n
inst_5 : CommRing R
inst_6 : Algebra R S
inst_7 : NeZero n
μ' : ↥(rootsOfUnity n S) := hμ.toRootsOfUnity
ho : orderOf μ' = n
h1 : ∃ (m : ℕ), ((↑1 : S →ₐ[R] S) : (a : S) → S) (↑(↑μ' : Sˣ) : S) = (↑(↑μ' : Sˣ) : S) ^ m
h :
  ((↑1 : S →ₐ[R] S) : (a : S) → S) (↑(↑μ' : Sˣ) : S) =
    (↑(↑μ' : Sˣ) : S) ^
      h1.choose (p := fun (m : ℕ) ↦ ((↑1 : S →ₐ[R] S) : (a : S) → S) (↑(↑μ' : Sˣ) : S) = (↑(↑μ' : Sˣ) : S) ^ m)
⊢ Eq (α := ZMod n)
    (↑(h1.choose (p := fun (m : ℕ) ↦ ((↑1 : S →ₐ[R] S) : (a : S) → S) (↑(↑μ' : Sˣ) : S) = (↑(↑μ' : Sˣ) : S) ^ m)) :
      ZMod n)
    1
have h₁ : μ' = μ' ^ h1.choose (p := fun (m : ℕ) ↦ ((↑1 : S →ₐ[R] S) : (a : S) → S) (↑(↑μ' : Sˣ) : S) = (↑(↑μ' : Sˣ) : S) ^ m)
Tactic: replace h : μ' = μ' ^ h1.choose :=
          rootsOfUnity.coe_injective (by simpa only [rootsOfUnity.coe_pow] using h)
Cleared hypotheses: h