awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- 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_2 : orderOf ⋯.unit ∣ Monoid.exponent.{u_1} ↥(rootsOfUnity n M)
⊢ orderOf ⋯.unit ∣ Monoid.exponent.{u_1} ↥(rootsOfUnity n M)
have refine_2₁ : Membership.mem (γ := Subgroup Mˣ) (rootsOfUnity n M) sorry.unit
Tactic: let ζ' : rootsOfUnity n M := ⟨(h.isUnit NeZero.out).unit, ?_⟩
-- 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
ζ' : ↥(rootsOfUnity n M) :=
  Subtype.mk (p := fun (x : Mˣ) ↦ Membership.mem (γ := Subgroup Mˣ) (rootsOfUnity n M) x) ⋯.unit ⋯
⊢ orderOf ⋯.unit ∣ Monoid.exponent.{u_1} ↥(rootsOfUnity n M)
have refine_2 : orderOf (Subtype.mk (p := fun (x : Mˣ) ↦ Membership.mem (γ := Subgroup Mˣ) (rootsOfUnity n M) x) sorry.unit sorry) ∣
  Monoid.exponent.{u_1} ↥(rootsOfUnity n M)
Tactic: rw [← Subgroup.orderOf_mk]
-- 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
ζ' : ↥(rootsOfUnity n M) :=
  Subtype.mk (p := fun (x : Mˣ) ↦ Membership.mem (γ := Subgroup Mˣ) (rootsOfUnity n M) x) ⋯.unit ⋯
refine_2 :
  orderOf (Subtype.mk (p := fun (x : Mˣ) ↦ Membership.mem (γ := Subgroup Mˣ) (rootsOfUnity n M) x) ⋯.unit ⋯) ∣
    Monoid.exponent.{u_1} ↥(rootsOfUnity n M)
⊢ orderOf ⋯.unit ∣ Monoid.exponent.{u_1} ↥(rootsOfUnity n M)
have refine_2₁ : Membership.mem (γ := Subgroup Mˣ) (rootsOfUnity n M) sorry.unit
Tactic: rw [← Subgroup.orderOf_mk]
-- 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
ζ' : ↥(rootsOfUnity n M) :=
  Subtype.mk (p := fun (x : Mˣ) ↦ Membership.mem (γ := Subgroup Mˣ) (rootsOfUnity n M) x) ⋯.unit ⋯
refine_2 :
  orderOf (Subtype.mk (p := fun (x : Mˣ) ↦ Membership.mem (γ := Subgroup Mˣ) (rootsOfUnity n M) x) ⋯.unit ⋯) ∣
    Monoid.exponent.{u_1} ↥(rootsOfUnity n M)
refine_2₁ : Membership.mem (γ := Subgroup Mˣ) (rootsOfUnity n M) ⋯.unit
⊢ orderOf ⋯.unit ∣ Monoid.exponent.{u_1} ↥(rootsOfUnity n M)
have rw : Subgroup Mˣ
Tactic: rw [← Subgroup.orderOf_mk]
-- 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
⊢ ⋯.unit ^ n = 1
have refine_2 : (↑(sorry.unit ^ n) : M) = (↑1 : M)
Tactic: rw [← Units.val_inj, Units.val_pow_eq_pow_val, IsUnit.unit_spec, h.pow_eq_one, Units.val_one]
-- 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_2 : (↑(⋯.unit ^ n) : M) = (↑1 : M)
⊢ ⋯.unit ^ n = 1
have refine_2₁ : (↑sorry.unit : M) ^ n = (↑1 : M)
Tactic: rw [← Units.val_inj, Units.val_pow_eq_pow_val, IsUnit.unit_spec, h.pow_eq_one, Units.val_one]
-- 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_2 : (↑(⋯.unit ^ n) : M) = (↑1 : M)
refine_2₁ : (↑⋯.unit : M) ^ n = (↑1 : M)
⊢ ⋯.unit ^ n = 1
have refine_2₂ : ζ ^ n = (↑1 : M)
Tactic: rw [← Units.val_inj, Units.val_pow_eq_pow_val, IsUnit.unit_spec, h.pow_eq_one, Units.val_one]
-- 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_2 : (↑(⋯.unit ^ n) : M) = (↑1 : M)
refine_2₁ : (↑⋯.unit : M) ^ n = (↑1 : M)
refine_2₂ : ζ ^ n = (↑1 : M)
⊢ ⋯.unit ^ n = 1
have refine_2₃ : Eq.{u_1 + 1} (α := M) 1 (↑1 : M)
Tactic: rw [← Units.val_inj, Units.val_pow_eq_pow_val, IsUnit.unit_spec, h.pow_eq_one, Units.val_one]
-- 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_2 : (↑(⋯.unit ^ n) : M) = (↑1 : M)
refine_2₁ : (↑⋯.unit : M) ^ n = (↑1 : M)
refine_2₂ : ζ ^ n = (↑1 : M)
refine_2₃ : Eq.{u_1 + 1} (α := M) 1 (↑1 : M)
⊢ ⋯.unit ^ n = 1
have refine_2₄ : Eq.{u_1 + 1} (α := M) 1 1
Tactic: rw [← Units.val_inj, Units.val_pow_eq_pow_val, IsUnit.unit_spec, h.pow_eq_one, Units.val_one]
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.EnoughRootsOfUnity (goal)
Goal:
n : ℕ
inst : NeZero n
M : Type u_1
N : Type u_2
inst_1 : CommMonoid M
inst_2 : CommMonoid N
hm : HasEnoughRootsOfUnity M n
e : MulEquiv.{u_1, u_2} ↥(rootsOfUnity n M) ↥(rootsOfUnity n N)
⊢ ∃ (m : N), IsPrimitiveRoot m n
have obtain : ∀ (m : M) (hm : IsPrimitiveRoot m n), ∃ (m : N), IsPrimitiveRoot m n
Tactic: obtain ⟨m, hm⟩ := hm.prim
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.EnoughRootsOfUnity (goal)
Goal:
n : ℕ
inst : NeZero n
M : Type u_1
N : Type u_2
inst_1 : CommMonoid M
inst_2 : CommMonoid N
hm_1 : HasEnoughRootsOfUnity M n
e : MulEquiv.{u_1, u_2} ↥(rootsOfUnity n M) ↥(rootsOfUnity n N)
m : M
hm : IsPrimitiveRoot m n
⊢ ∃ (m : N), IsPrimitiveRoot m n
have h : IsPrimitiveRoot.{u_2} (M := N) (↑(↑((e : (a : ↥(rootsOfUnity n M)) → ↥(rootsOfUnity n N)) hm.toRootsOfUnity) : Nˣ) : N)
  n
Tactic: use (e hm.toRootsOfUnity).val.val
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.EnoughRootsOfUnity (goal)
Goal:
n : ℕ
inst : NeZero n
M : Type u_1
N : Type u_2
inst_1 : CommMonoid M
inst_2 : CommMonoid N
hm_1 : HasEnoughRootsOfUnity M n
e : MulEquiv.{u_1, u_2} ↥(rootsOfUnity n M) ↥(rootsOfUnity n N)
m : M
hm : IsPrimitiveRoot m n
⊢ IsPrimitiveRoot.{u_2} (M := N)
    (↑(↑((e : (a : ↥(rootsOfUnity n M)) → ↥(rootsOfUnity n N)) hm.toRootsOfUnity) : Nˣ) : N) n
have h : IsPrimitiveRoot.{u_2} (M := ↥(rootsOfUnity n N))
  ((e : (a : ↥(rootsOfUnity n M)) → ↥(rootsOfUnity n N)) hm.toRootsOfUnity : ↥(rootsOfUnity n N)) n
Tactic: rw [IsPrimitiveRoot.coe_units_iff, IsPrimitiveRoot.coe_submonoidClass_iff]
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.EnoughRootsOfUnity (goal)
Goal:
n : ℕ
inst : NeZero n
M : Type u_1
N : Type u_2
inst_1 : CommMonoid M
inst_2 : CommMonoid N
hm_1 : HasEnoughRootsOfUnity M n
e : MulEquiv.{u_1, u_2} ↥(rootsOfUnity n M) ↥(rootsOfUnity n N)
m : M
hm : IsPrimitiveRoot m n
⊢ IsPrimitiveRoot.{u_2} (M := ↥(rootsOfUnity n N))
    ((e : (a : ↥(rootsOfUnity n M)) → ↥(rootsOfUnity n N)) hm.toRootsOfUnity : ↥(rootsOfUnity n N)) n
have h : IsPrimitiveRoot hm.toRootsOfUnity n
Tactic: refine .map_of_injective ?_ e.injective
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.EnoughRootsOfUnity (goal)
Goal:
n : ℕ
inst : NeZero n
M : Type u_1
N : Type u_2
inst_1 : CommMonoid M
inst_2 : CommMonoid N
hm_1 : HasEnoughRootsOfUnity M n
e : MulEquiv.{u_1, u_2} ↥(rootsOfUnity n M) ↥(rootsOfUnity n N)
m : M
hm : IsPrimitiveRoot m n
⊢ IsPrimitiveRoot hm.toRootsOfUnity n
have h : IsPrimitiveRoot (↑hm.toRootsOfUnity : Mˣ) n
Tactic: rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, ← IsPrimitiveRoot.coe_units_iff]
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.EnoughRootsOfUnity (goal)
Goal:
n : ℕ
inst : NeZero n
M : Type u_1
N : Type u_2
inst_1 : CommMonoid M
inst_2 : CommMonoid N
hm_1 : HasEnoughRootsOfUnity M n
e : MulEquiv.{u_1, u_2} ↥(rootsOfUnity n M) ↥(rootsOfUnity n N)
m : M
hm : IsPrimitiveRoot m n
h : IsPrimitiveRoot (↑hm.toRootsOfUnity : Mˣ) n
⊢ IsPrimitiveRoot hm.toRootsOfUnity n
have h₁ : IsPrimitiveRoot (↑(↑hm.toRootsOfUnity : Mˣ) : M) n
Tactic: rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, ← IsPrimitiveRoot.coe_units_iff]
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.EnoughRootsOfUnity (hypothesis)
Goal:
G : Type u_1
M : Type u_2
inst : CommGroup G
inst_1 : Finite G
inst_2 : IsCyclic G
inst_3 : CommMonoid M
inst_4 : HasEnoughRootsOfUnity M (Nat.card G)
⊢ Nonempty (MonoidHom.{u_1, u_2} G Mˣ ≃* G)
have this : NeZero (Nat.card G)
Tactic: have : NeZero (Nat.card G) := ⟨Nat.card_pos.ne'⟩
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.EnoughRootsOfUnity (hypothesis)
Goal:
G : Type u_1
M : Type u_2
inst : CommGroup G
inst_1 : Finite G
inst_2 : IsCyclic G
inst_3 : CommMonoid M
inst_4 : HasEnoughRootsOfUnity M (Nat.card G)
this : NeZero (Nat.card G)
⊢ Nonempty (MonoidHom.{u_1, u_2} G Mˣ ≃* G)
have hord : Nat.card.{u_2} ↥(rootsOfUnity (Nat.card G) M) = Nat.card G
Tactic: have hord := HasEnoughRootsOfUnity.natCard_rootsOfUnity M (Nat.card G)
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.EnoughRootsOfUnity (hypothesis)
Goal:
G : Type u_1
M : Type u_2
inst : Group G
inst_1 : Finite G
inst_2 : CommMonoid M
inst_3 : HasEnoughRootsOfUnity M (Monoid.exponent G)
S : Subgroup Mˣ := rootsOfUnity (Monoid.exponent G) M
⊢ Finite (MonoidHom.{u_1, u_2} G Mˣ)
have this : Finite (MonoidHom.{u_1, u_2} G ↥S)
Tactic: have : Finite (G →* S) := .of_injective _ DFunLike.coe_injective
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.EnoughRootsOfUnity (goal)
Goal:
G : Type u_1
M : Type u_2
inst : Group G
inst_1 : Finite G
inst_2 : CommMonoid M
inst_3 : HasEnoughRootsOfUnity M (Monoid.exponent G)
S : Subgroup Mˣ := rootsOfUnity (Monoid.exponent G) M
this : Finite (MonoidHom.{u_1, u_2} G ↥S)
⊢ Finite (MonoidHom.{u_1, u_2} G Mˣ)
have refine : ∀ (f : MonoidHom.{u_1, u_2} G Mˣ), ∃ (a : MonoidHom.{u_1, u_2} G ↥S), S.subtype.comp a = f
Tactic: refine .of_surjective S.subtype.comp fun f ↦ ?_
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.EnoughRootsOfUnity (goal)
Goal:
G : Type u_1
M : Type u_2
inst : Group G
inst_1 : Finite G
inst_2 : CommMonoid M
inst_3 : HasEnoughRootsOfUnity M (Monoid.exponent G)
S : Subgroup Mˣ := rootsOfUnity (Monoid.exponent G) M
this : Finite (MonoidHom.{u_1, u_2} G ↥S)
f : MonoidHom.{u_1, u_2} G Mˣ
a : G
⊢ Membership.mem (α := Mˣ) S ((f : (a : G) → Mˣ) a : Mˣ)
have rw : HPow.hPow (α := Mˣ) ((f : (a : G) → Mˣ) a : Mˣ) (Monoid.exponent G) = 1
Tactic: rw [mem_rootsOfUnity, ← map_pow, Monoid.pow_exponent_eq_one, map_one]
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.EnoughRootsOfUnity (goal)
Goal:
G : Type u_1
M : Type u_2
inst : Group G
inst_1 : Finite G
inst_2 : CommMonoid M
inst_3 : HasEnoughRootsOfUnity M (Monoid.exponent G)
S : Subgroup Mˣ := rootsOfUnity (Monoid.exponent G) M
this : Finite (MonoidHom.{u_1, u_2} G ↥S)
f : MonoidHom.{u_1, u_2} G Mˣ
a : G
rw : HPow.hPow (α := Mˣ) ((f : (a : G) → Mˣ) a : Mˣ) (Monoid.exponent G) = 1
⊢ Membership.mem (α := Mˣ) S ((f : (a : G) → Mˣ) a : Mˣ)
have rw₁ : Eq.{u_2 + 1} (α := Mˣ) ((f : (a : G) → Mˣ) (a ^ Monoid.exponent G) : Mˣ) (1 : Mˣ)
Tactic: rw [mem_rootsOfUnity, ← map_pow, Monoid.pow_exponent_eq_one, map_one]
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.EnoughRootsOfUnity (goal)
Goal:
G : Type u_1
M : Type u_2
inst : Group G
inst_1 : Finite G
inst_2 : CommMonoid M
inst_3 : HasEnoughRootsOfUnity M (Monoid.exponent G)
S : Subgroup Mˣ := rootsOfUnity (Monoid.exponent G) M
this : Finite (MonoidHom.{u_1, u_2} G ↥S)
f : MonoidHom.{u_1, u_2} G Mˣ
a : G
rw : HPow.hPow (α := Mˣ) ((f : (a : G) → Mˣ) a : Mˣ) (Monoid.exponent G) = 1
rw₁ : Eq.{u_2 + 1} (α := Mˣ) ((f : (a : G) → Mˣ) (a ^ Monoid.exponent G) : Mˣ) (1 : Mˣ)
⊢ Membership.mem (α := Mˣ) S ((f : (a : G) → Mˣ) a : Mˣ)
have rw₂ : Eq.{u_2 + 1} (α := Mˣ) ((f : (a : G) → Mˣ) 1) 1
Tactic: rw [mem_rootsOfUnity, ← map_pow, Monoid.pow_exponent_eq_one, map_one]
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.EnoughRootsOfUnity (goal)
Goal:
G : Type u_1
M : Type u_2
inst : Group G
inst_1 : Finite G
inst_2 : CommMonoid M
inst_3 : HasEnoughRootsOfUnity M (Monoid.exponent G)
S : Subgroup Mˣ := rootsOfUnity (Monoid.exponent G) M
this : Finite (MonoidHom.{u_1, u_2} G ↥S)
f : MonoidHom.{u_1, u_2} G Mˣ
a : G
rw : HPow.hPow (α := Mˣ) ((f : (a : G) → Mˣ) a : Mˣ) (Monoid.exponent G) = 1
rw₁ : Eq.{u_2 + 1} (α := Mˣ) ((f : (a : G) → Mˣ) (a ^ Monoid.exponent G) : Mˣ) (1 : Mˣ)
rw₂ : Eq.{u_2 + 1} (α := Mˣ) ((f : (a : G) → Mˣ) 1) 1
⊢ Membership.mem (α := Mˣ) S ((f : (a : G) → Mˣ) a : Mˣ)
have rw₃ : Eq.{u_2 + 1} (α := Mˣ) 1 1
Tactic: rw [mem_rootsOfUnity, ← map_pow, Monoid.pow_exponent_eq_one, map_one]
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.EnoughRootsOfUnity (hypothesis)
Goal:
G : Type u_1
M : Type u_2
inst : Group G
inst_1 : Finite G
inst_2 : CommMonoid M
inst_3 : HasEnoughRootsOfUnity M (Monoid.exponent G)
S : Subgroup Mˣ := rootsOfUnity (Monoid.exponent G) M
this : Finite (MonoidHom.{u_1, u_2} G ↥S)
f : MonoidHom.{u_1, u_2} G Mˣ
⊢ ∃ (a : MonoidHom.{u_1, u_2} G ↥S), S.subtype.comp a = f
have H : ∀ (a : G), Membership.mem (α := Mˣ) S ((f : (a : G) → Mˣ) a : Mˣ)
Tactic: have H a : f a ∈ S := by
    rw [mem_rootsOfUnity, ← map_pow, Monoid.pow_exponent_eq_one, map_one]
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.Lemmas (hypothesis)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDomain R
n : ℕ
μ : R
hμ : IsPrimitiveRoot μ (n + 1)
⊢ ∏ k ∈ range n, (1 - μ ^ (k + 1)) = HAdd.hAdd (α := R) (↑n : R) 1
have this : HPow.hPow (α := R[X]) X (n + 1) - (C : (a : R) → R[X]) 1 =
  ∏ i ∈ range (n + 1), HSub.hSub (α := R[X]) (β := R[X]) X ((C : (a : R) → R[X]) (μ ^ i * 1) : R[X])
Tactic: have := X_pow_sub_C_eq_prod hμ n.zero_lt_succ (one_pow (n + 1))
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.Lemmas (hypothesis)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDomain R
n : ℕ
μ : R
hμ : IsPrimitiveRoot μ (n + 1)
this :
  HPow.hPow (α := R[X]) X (n + 1) - (C : (a : R) → R[X]) 1 =
    ∏ i ∈ range (n + 1), HSub.hSub (α := R[X]) (β := R[X]) X ((C : (a : R) → R[X]) (μ ^ i * 1) : R[X])
⊢ ∏ k ∈ range n, (1 - μ ^ (k + 1)) = HAdd.hAdd (α := R) (↑n : R) 1
have this₁ : (∏ k ∈ range n, HSub.hSub (α := R[X]) (β := R[X]) X ((C : (a : R) → R[X]) (μ ^ (k + 1) * 1) : R[X])) *
    HSub.hSub (α := R[X]) X ((C : (a : R) → R[X]) 1) =
  (∑ i ∈ range (n + 1), HPow.hPow (α := R[X]) X i) * HSub.hSub (α := R[X]) X 1
Tactic: rw [C_1, ← mul_geom_sum, prod_range_succ', pow_zero, mul_one, mul_comm, eq_comm] at this
Cleared hypotheses: this
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.Lemmas (hypothesis)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDomain R
n : ℕ
μ : R
hμ : IsPrimitiveRoot μ (n + 1)
this :
  (∏ k ∈ range n, HSub.hSub (α := R[X]) (β := R[X]) X ((C : (a : R) → R[X]) (μ ^ (k + 1) * 1) : R[X])) *
      HSub.hSub (α := R[X]) X ((C : (a : R) → R[X]) 1) =
    (∑ i ∈ range (n + 1), HPow.hPow (α := R[X]) X i) * HSub.hSub (α := R[X]) X 1
⊢ ∏ k ∈ range n, (1 - μ ^ (k + 1)) = HAdd.hAdd (α := R) (↑n : R) 1
have this₁ : Eq.{u_1 + 1} (α := R[X])
  (∏ k ∈ range n, HSub.hSub (α := R[X]) (β := R[X]) X ((C : (a : R) → R[X]) (μ ^ (k + 1) * 1) : R[X]))
  (∑ i ∈ range (n + 1), HPow.hPow (α := R[X]) X i)
Tactic: replace this := mul_right_cancel₀ (Polynomial.X_sub_C_ne_zero 1) this
Cleared hypotheses: this
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.Lemmas (hypothesis)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDomain R
n : ℕ
μ : R
hμ : IsPrimitiveRoot μ (n + 1)
this :
  Eq.{u_1 + 1} (α := R[X])
    (∏ k ∈ range n, HSub.hSub (α := R[X]) (β := R[X]) X ((C : (a : R) → R[X]) (μ ^ (k + 1) * 1) : R[X]))
    (∑ i ∈ range (n + 1), HPow.hPow (α := R[X]) X i)
⊢ ∏ k ∈ range n, (1 - μ ^ (k + 1)) = HAdd.hAdd (α := R) (↑n : R) 1
have this₁ : Eq.{u_1 + 1} (α := R)
  (eval 1 (∏ k ∈ range n, HSub.hSub (α := R[X]) (β := R[X]) X ((C : (a : R) → R[X]) (μ ^ (k + 1) * 1) : R[X])))
  (eval 1 (∑ i ∈ range (n + 1), HPow.hPow (α := R[X]) X i))
Tactic: apply_fun Polynomial.eval 1 at this
Cleared hypotheses: this
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.Lemmas (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDomain R
n : ℕ
μ : R
hμ : IsPrimitiveRoot μ (n + 1)
⊢ HPow.hPow (α := R) (-1) n = ∏ k ∈ range n, -1
have rw : HPow.hPow (α := R) (-1) n = HPow.hPow (α := R) (-1) #(range n)
Tactic: rw [prod_const, card_range]
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.Lemmas (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDomain R
n : ℕ
μ : R
hμ : IsPrimitiveRoot μ (n + 1)
rw : HPow.hPow (α := R) (-1) n = HPow.hPow (α := R) (-1) #(range n)
⊢ HPow.hPow (α := R) (-1) n = ∏ k ∈ range n, -1
have rw₁ : HPow.hPow (α := R) (-1) n = HPow.hPow (α := R) (-1) n
Tactic: rw [prod_const, card_range]
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.Lemmas (hypothesis)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDomain R
n : ℕ
μ : R
hμ : IsPrimitiveRoot μ (n + 1)
⊢ HPow.hPow (α := R) (-1) n * ∏ k ∈ range n, (μ ^ (k + 1) - 1) = HAdd.hAdd (α := R) (↑n : R) 1
have this : HPow.hPow (α := R) (-1) n = ∏ k ∈ range n, -1
Tactic: have : (-1 : R) ^ n = ∏ k ∈ range n, -1 := by rw [prod_const, card_range]
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.Lemmas (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDomain R
k n : ℕ
hn : k < n
μ : R
hμ : IsPrimitiveRoot μ n
⊢ ∃ (z : R), Membership.mem (γ := Subalgebra ℤ R) ℤ[μ] z ∧ (↑n : R) = z * (μ - 1) ^ k
have let : ∀ (n' : ℕ) (hn : k < n' + 1) (hμ : IsPrimitiveRoot μ (n' + 1)),
  ∃ (z : R), Membership.mem (γ := Subalgebra ℤ R) ℤ[μ] z ∧ (↑(n' + 1) : R) = z * (μ - 1) ^ k
Tactic: let n' + 1 := n
Cleared hypotheses: hn, hμ
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.Lemmas (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDomain R
k n : ℕ
μ : R
n' : ℕ
hn : k < n' + 1
hμ : IsPrimitiveRoot μ (n' + 1)
⊢ ∃ (z : R), Membership.mem (γ := Subalgebra ℤ R) ℤ[μ] z ∧ (↑(n' + 1) : R) = z * (μ - 1) ^ k
have obtain : ∀ (m : ℕ) (hn : k < m + k + 1) (hμ : IsPrimitiveRoot μ (m + k + 1)),
  ∃ (z : R), Membership.mem (γ := Subalgebra ℤ R) ℤ[μ] z ∧ (↑(m + k + 1) : R) = z * (μ - 1) ^ k
Tactic: obtain ⟨m, rfl⟩ := Nat.exists_eq_add_of_le' (Nat.le_of_lt_succ hn)
Cleared hypotheses: n', hn, hμ
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.Lemmas (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDomain R
k_1 n : ℕ
μ : R
m : ℕ
hn : k_1 < m + k_1 + 1
hμ : IsPrimitiveRoot μ (m + k_1 + 1)
k : ℕ
⊢ ∃ (z : R), Membership.mem (γ := Subalgebra ℤ R) ℤ[μ] z ∧ μ ^ k - 1 = z * (μ - 1)
have refine : Membership.mem (γ := Subalgebra ℤ R) ℤ[μ] (∑ x ∈ range k, μ ^ x)
Tactic: refine ⟨(Finset.range k).sum (μ ^ ·), ?_, (geom_sum_mul μ k).symm⟩
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.Lemmas (hypothesis)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDomain R
k n : ℕ
μ : R
m : ℕ
hn : k < m + k + 1
hμ : IsPrimitiveRoot μ (m + k + 1)
⊢ ∃ (z : R), Membership.mem (γ := Subalgebra ℤ R) ℤ[μ] z ∧ (↑(m + k + 1) : R) = z * (μ - 1) ^ k
have hdvd : ∀ (k : ℕ), ∃ (z : R), Membership.mem (γ := Subalgebra ℤ R) ℤ[μ] z ∧ μ ^ k - 1 = z * (μ - 1)
Tactic: have hdvd k : ∃ z ∈ ℤ[μ], μ ^ k - 1 = z * (μ - 1) := by
    refine ⟨(Finset.range k).sum (μ ^ ·), ?_, (geom_sum_mul μ k).symm⟩
    exact Subalgebra.sum_mem _ fun m _ ↦ Subalgebra.pow_mem _ (self_mem_adjoin_singleton _ μ) _
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.Lemmas (hypothesis)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDomain R
k n : ℕ
μ : R
m : ℕ
hn : k < m + k + 1
hμ : IsPrimitiveRoot μ (m + k + 1)
hdvd : ∀ (k : ℕ), ∃ (z : R), Membership.mem (γ := Subalgebra ℤ R) ℤ[μ] z ∧ μ ^ k - 1 = z * (μ - 1)
Z : (k : ℕ) → R :=
  fun (k : ℕ) ↦
    Classical.choose (p := fun (x : R) ↦ Membership.mem (γ := Subalgebra ℤ R) ℤ[μ] x ∧ μ ^ k - 1 = x * (μ - 1)) ⋯
⊢ ∃ (z : R), Membership.mem (γ := Subalgebra ℤ R) ℤ[μ] z ∧ (↑(m + k + 1) : R) = z * (μ - 1) ^ k
have Zdef : ∀ (k : ℕ), Membership.mem (γ := Subalgebra ℤ R) ℤ[μ] (Z k) ∧ μ ^ k - 1 = Z k * (μ - 1)
Tactic: have Zdef k : Z k ∈ ℤ[μ] ∧ μ ^ k - 1 = Z k * (μ - 1) :=
    Classical.choose_spec <| hdvd k
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.Lemmas (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDomain R
k n : ℕ
μ : R
m : ℕ
hn : k < m + k + 1
hμ : IsPrimitiveRoot μ (m + k + 1)
hdvd : ∀ (k : ℕ), ∃ (z : R), Membership.mem (γ := Subalgebra ℤ R) ℤ[μ] z ∧ μ ^ k - 1 = z * (μ - 1)
Z : (k : ℕ) → R :=
  fun (k : ℕ) ↦
    Classical.choose (p := fun (x : R) ↦ Membership.mem (γ := Subalgebra ℤ R) ℤ[μ] x ∧ μ ^ k - 1 = x * (μ - 1)) ⋯
Zdef : ∀ (k : ℕ), Membership.mem (γ := Subalgebra ℤ R) ℤ[μ] (Z k) ∧ μ ^ k - 1 = Z k * (μ - 1)
⊢ ∃ (z : R), Membership.mem (γ := Subalgebra ℤ R) ℤ[μ] z ∧ (↑(m + k + 1) : R) = z * (μ - 1) ^ k
have refine_1 : Membership.mem (γ := Subalgebra ℤ R) ℤ[μ]
  ((HPow.hPow (α := R) (-1) (m + k) * ∏ j ∈ range k, Z (j + 1)) * ∏ j ∈ Ico k (m + k), (μ ^ (j + 1) - 1))
Tactic: refine ⟨(-1) ^ (m + k) * (∏ j ∈ range k, Z (j + 1)) * ∏ j ∈ Ico k (m + k), (μ ^ (j + 1) - 1),
    ?_, ?_⟩
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.Lemmas (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDomain R
k n : ℕ
μ : R
m : ℕ
hn : k < m + k + 1
hμ : IsPrimitiveRoot μ (m + k + 1)
hdvd : ∀ (k : ℕ), ∃ (z : R), Membership.mem (γ := Subalgebra ℤ R) ℤ[μ] z ∧ μ ^ k - 1 = z * (μ - 1)
Z : (k : ℕ) → R :=
  fun (k : ℕ) ↦
    Classical.choose (p := fun (x : R) ↦ Membership.mem (γ := Subalgebra ℤ R) ℤ[μ] x ∧ μ ^ k - 1 = x * (μ - 1)) ⋯
Zdef : ∀ (k : ℕ), Membership.mem (γ := Subalgebra ℤ R) ℤ[μ] (Z k) ∧ μ ^ k - 1 = Z k * (μ - 1)
refine_1 :
  Membership.mem (γ := Subalgebra ℤ R) ℤ[μ]
    ((HPow.hPow (α := R) (-1) (m + k) * ∏ j ∈ range k, Z (j + 1)) * ∏ j ∈ Ico k (m + k), (μ ^ (j + 1) - 1))
⊢ ∃ (z : R), Membership.mem (γ := Subalgebra ℤ R) ℤ[μ] z ∧ (↑(m + k + 1) : R) = z * (μ - 1) ^ k
have refine_2 : (↑(m + k + 1) : R) =
  ((HPow.hPow (α := R) (-1) (m + k) * ∏ j ∈ range k, Z (j + 1)) * ∏ j ∈ Ico k (m + k), (μ ^ (j + 1) - 1)) * (μ - 1) ^ k
Tactic: refine ⟨(-1) ^ (m + k) * (∏ j ∈ range k, Z (j + 1)) * ∏ j ∈ Ico k (m + k), (μ ^ (j + 1) - 1),
    ?_, ?_⟩
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.Lemmas (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDomain R
k n : ℕ
μ : R
m : ℕ
hn : k < m + k + 1
hμ : IsPrimitiveRoot μ (m + k + 1)
hdvd : ∀ (k : ℕ), ∃ (z : R), Membership.mem (γ := Subalgebra ℤ R) ℤ[μ] z ∧ μ ^ k - 1 = z * (μ - 1)
Z : (k : ℕ) → R :=
  fun (k : ℕ) ↦
    Classical.choose (p := fun (x : R) ↦ Membership.mem (γ := Subalgebra ℤ R) ℤ[μ] x ∧ μ ^ k - 1 = x * (μ - 1)) ⋯
Zdef : ∀ (k : ℕ), Membership.mem (γ := Subalgebra ℤ R) ℤ[μ] (Z k) ∧ μ ^ k - 1 = Z k * (μ - 1)
⊢ Membership.mem (γ := Subalgebra ℤ R) ℤ[μ]
    ((HPow.hPow (α := R) (-1) (m + k) * ∏ j ∈ range k, Z (j + 1)) * ∏ j ∈ Ico k (m + k), (μ ^ (j + 1) - 1))
have refine_1 : Membership.mem (γ := Subalgebra ℤ R) ℤ[μ] (HPow.hPow (α := R) (-1) (m + k) * ∏ j ∈ range k, Z (j + 1))
Tactic: apply Subalgebra.mul_mem
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.Lemmas (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDomain R
k n : ℕ
μ : R
m : ℕ
hn : k < m + k + 1
hμ : IsPrimitiveRoot μ (m + k + 1)
hdvd : ∀ (k : ℕ), ∃ (z : R), Membership.mem (γ := Subalgebra ℤ R) ℤ[μ] z ∧ μ ^ k - 1 = z * (μ - 1)
Z : (k : ℕ) → R :=
  fun (k : ℕ) ↦
    Classical.choose (p := fun (x : R) ↦ Membership.mem (γ := Subalgebra ℤ R) ℤ[μ] x ∧ μ ^ k - 1 = x * (μ - 1)) ⋯
Zdef : ∀ (k : ℕ), Membership.mem (γ := Subalgebra ℤ R) ℤ[μ] (Z k) ∧ μ ^ k - 1 = Z k * (μ - 1)
refine_1 : Membership.mem (γ := Subalgebra ℤ R) ℤ[μ] (HPow.hPow (α := R) (-1) (m + k) * ∏ j ∈ range k, Z (j + 1))
⊢ Membership.mem (γ := Subalgebra ℤ R) ℤ[μ]
    ((HPow.hPow (α := R) (-1) (m + k) * ∏ j ∈ range k, Z (j + 1)) * ∏ j ∈ Ico k (m + k), (μ ^ (j + 1) - 1))
have refine_1₁ : Membership.mem (γ := Subalgebra ℤ R) ℤ[μ] (∏ j ∈ Ico k (m + k), (μ ^ (j + 1) - 1))
Tactic: apply Subalgebra.mul_mem
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.Lemmas (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDomain R
k n : ℕ
μ : R
m : ℕ
hn : k < m + k + 1
hμ : IsPrimitiveRoot μ (m + k + 1)
hdvd : ∀ (k : ℕ), ∃ (z : R), Membership.mem (γ := Subalgebra ℤ R) ℤ[μ] z ∧ μ ^ k - 1 = z * (μ - 1)
Z : (k : ℕ) → R :=
  fun (k : ℕ) ↦
    Classical.choose (p := fun (x : R) ↦ Membership.mem (γ := Subalgebra ℤ R) ℤ[μ] x ∧ μ ^ k - 1 = x * (μ - 1)) ⋯
Zdef : ∀ (k : ℕ), Membership.mem (γ := Subalgebra ℤ R) ℤ[μ] (Z k) ∧ μ ^ k - 1 = Z k * (μ - 1)
⊢ Membership.mem (γ := Subalgebra ℤ R) ℤ[μ] (HPow.hPow (α := R) (-1) (m + k) * ∏ j ∈ range k, Z (j + 1))
have refine_1 : Membership.mem (γ := Subalgebra ℤ R) ℤ[μ] (HPow.hPow (α := R) (-1) (m + k))
Tactic: apply Subalgebra.mul_mem