awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.RingTheory.RingInvo (goal)
Goal:
F : Type u_1
R : Type u_2
inst : Semiring R
inst_1 : EquivLike F R Rᵐᵒᵖ
f : RingInvo R
tE : R ≃ Rᵐᵒᵖ
map_mul' : ∀ (x y : R), tE.toFun (x * y) = tE.toFun x * tE.toFun y
map_add' : ∀ (x y : R), tE.toFun (x + y) = tE.toFun x + tE.toFun y
involution' :
  ∀ (x : R),
    MulOpposite.unop
        ({ toEquiv := tE, map_mul' := map_mul', map_add' := map_add' : R ≃+* Rᵐᵒᵖ }.toFun
          (MulOpposite.unop ({ toEquiv := tE, map_mul' := map_mul', map_add' := map_add' : R ≃+* Rᵐᵒᵖ }.toFun x))) =
      x
h₁ :
  { toEquiv := tE, map_mul' := map_mul', map_add' := map_add', involution' := involution' : RingInvo R }.toFun = f.toFun
h₂ :
  { toEquiv := tE, map_mul' := map_mul', map_add' := map_add', involution' := involution' : RingInvo R }.invFun =
    f.invFun
⊢ { toEquiv := tE, map_mul' := map_mul', map_add' := map_add', involution' := involution' : RingInvo R } = f
have rcases : ∀ (tF : R ≃ Rᵐᵒᵖ) (map_mul'_1 : ∀ (x y : R), tF.toFun (x * y) = tF.toFun x * tF.toFun y)
  (map_add'_1 : ∀ (x y : R), tF.toFun (x + y) = tF.toFun x + tF.toFun y)
  (involution'_1 :
    ∀ (x : R),
      MulOpposite.unop
          ({ toEquiv := tF, map_mul' := map_mul'_1, map_add' := map_add'_1 : R ≃+* Rᵐᵒᵖ }.toFun
            (MulOpposite.unop
              ({ toEquiv := tF, map_mul' := map_mul'_1, map_add' := map_add'_1 : R ≃+* Rᵐᵒᵖ }.toFun x))) =
        x)
  (h₁ :
    { toEquiv := tE, map_mul' := map_mul', map_add' := map_add', involution' := involution' : RingInvo R }.toFun =
      { toEquiv := tF, map_mul' := map_mul'_1, map_add' := map_add'_1, involution' := involution'_1 :
          RingInvo R }.toFun)
  (h₂ :
    { toEquiv := tE, map_mul' := map_mul', map_add' := map_add', involution' := involution' : RingInvo R }.invFun =
      { toEquiv := tF, map_mul' := map_mul'_1, map_add' := map_add'_1, involution' := involution'_1 :
          RingInvo R }.invFun),
  { toEquiv := tE, map_mul' := map_mul', map_add' := map_add', involution' := involution' : RingInvo R } =
    { toEquiv := tF, map_mul' := map_mul'_1, map_add' := map_add'_1, involution' := involution'_1 }
Tactic: rcases f with ⟨⟨tF, _⟩, _⟩
Cleared hypotheses: f, h₁, h₂
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.AlgebraicallyClosed (hypothesis)
Goal:
F : Type u_1
inst : Field F
n k : ℕ
inst_1 : NeZero.{u_1} (R := F) (↑n : F)
inst_2 : IsSepClosed F
⊢ ∃ (m : F), IsPrimitiveRoot m n
have this : NeZero n
Tactic: have : NeZero n := .of_neZero_natCast F
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.AlgebraicallyClosed (hypothesis)
Goal:
F : Type u_1
inst : Field F
n k : ℕ
inst_1 : NeZero.{u_1} (R := F) (↑n : F)
inst_2 : IsSepClosed F
this : NeZero n
⊢ ∃ (m : F), IsPrimitiveRoot m n
have this₁ : IsCyclotomicExtension {n} F F
Tactic: have := isCyclotomicExtension {n} F fun _ h _ ↦ Set.mem_singleton_iff.mp h ▸ ‹NeZero (n : F)›
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.Basic (goal)
Goal:
M : Type u_1
inst : CommMonoid M
k : ℤ
⊢ Eq (α := Subgroup Mˣ) (MonoidHom.ker (zpowGroupHom k)) (rootsOfUnity k.natAbs M)
have h : ∀ (x : Mˣ),
  Membership.mem (γ := Subgroup Mˣ) (MonoidHom.ker (zpowGroupHom k)) x ↔
    Membership.mem (γ := Subgroup Mˣ) (rootsOfUnity k.natAbs M) x
Tactic: ext
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.Basic (goal)
Goal:
M : Type u_1
inst : CommMonoid M
k : ℕ
ζ : Mˣ
⊢ ζ ^ k = 1 ↔ (↑ζ : M) ^ k = 1
have norm_cast : ζ ^ k = 1 ↔ ζ ^ k = 1
Tactic: norm_cast
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.Basic (goal)
Goal:
M : Type u_7
inst : CommMonoid M
⊢ Eq (α := Subgroup Mˣ) (rootsOfUnity 1 M) ⊥
have h : ∀ (x : Mˣ), Membership.mem (γ := Subgroup Mˣ) (rootsOfUnity 1 M) x ↔ Membership.mem (γ := Subgroup Mˣ) ⊥ x
Tactic: ext1
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.Basic (goal)
Goal:
M : Type u_7
inst : CommMonoid M
⊢ Eq (α := Subgroup Mˣ) (rootsOfUnity 0 M) ⊤
have h : ∀ (x : Mˣ), Membership.mem (γ := Subgroup Mˣ) (rootsOfUnity 0 M) x ↔ Membership.mem (γ := Subgroup Mˣ) ⊤ x
Tactic: ext1
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.Basic (goal)
Goal:
M : Type u_1
inst : CommMonoid M
k l : ℕ
h : k ∣ l
⊢ LE.le (α := Subgroup Mˣ) (rootsOfUnity k M) (rootsOfUnity l M)
have obtain : ∀ (d : ℕ), LE.le (α := Subgroup Mˣ) (rootsOfUnity k M) (rootsOfUnity (k * d) M)
Tactic: obtain ⟨d, rfl⟩ := h
Cleared hypotheses: l, h
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.Basic (goal)
Goal:
M : Type u_1
inst : CommMonoid M
m n : ℕ
⊢ Eq (α := Subgroup Mˣ) (rootsOfUnity m M ⊓ rootsOfUnity n M) (rootsOfUnity (m.gcd n) M)
have h : ∀ (x : Mˣ),
  Membership.mem (γ := Subgroup Mˣ) (rootsOfUnity m M ⊓ rootsOfUnity n M) x ↔
    Membership.mem (γ := Subgroup Mˣ) (rootsOfUnity (m.gcd n) M) x
Tactic: ext
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.Basic (goal)
Goal:
R : Type u_4
k : ℕ
inst : CommMonoid R
ζ : ↥(rootsOfUnity k R)
m : ℕ
⊢ (↑(↑(ζ ^ m) : Rˣ) : R) = (↑(↑ζ : Rˣ) : R) ^ m
have rw : (↑((↑ζ : Rˣ) ^ m) : R) = (↑(↑ζ : Rˣ) : R) ^ m
Tactic: rw [Subgroup.coe_pow, Units.val_pow_eq_pow_val]
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.Basic (goal)
Goal:
R : Type u_4
k : ℕ
inst : CommMonoid R
ζ : ↥(rootsOfUnity k R)
m : ℕ
rw : (↑((↑ζ : Rˣ) ^ m) : R) = (↑(↑ζ : Rˣ) : R) ^ m
⊢ (↑(↑(ζ ^ m) : Rˣ) : R) = (↑(↑ζ : Rˣ) : R) ^ m
have rw₁ : (↑(↑ζ : Rˣ) : R) ^ m = (↑(↑ζ : Rˣ) : R) ^ m
Tactic: rw [Subgroup.coe_pow, Units.val_pow_eq_pow_val]
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.Basic (goal)
Goal:
M_1 : 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_1
inst_1 : CommMonoid N
inst_2 : DivisionCommMonoid G
k l : ℕ
M : Type u_7
inst_3 : CommMonoid M
n : ℕ
ζ : ↥(rootsOfUnity n M)
⊢ Membership.mem (α := Mˣˣ) (γ := Subgroup Mˣˣ) (rootsOfUnity n Mˣ) ((toUnits : (a : Mˣ) → Mˣˣ) (↑ζ : Mˣ) : Mˣˣ)
have simp : (↑ζ : Mˣ) ^ n = 1
Tactic: simp only [mem_rootsOfUnity, ← map_pow, EmbeddingLike.map_eq_one_iff]
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.Basic (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 : CommMonoid R
inst_4 : CommMonoid S
inst_5 : FunLike F R S
inst_6 : MonoidHomClass F R S
σ : F
n : ℕ
ξ : ↥(rootsOfUnity n R)
⊢ Membership.mem (α := Sˣ) (γ := Subgroup Sˣ) (rootsOfUnity n S)
    (((Units.map (↑σ : R →* S) : MonoidHom.{u_4, u_5} Rˣ Sˣ) : (a : Rˣ) → Sˣ) (↑ξ : Rˣ) : Sˣ)
have rw : Eq.{u_5 + 1} (α := S) (((↑σ : R →* S) : (a : R) → S) (↑1 : R) : S) ((↑1 : S) : S)
Tactic: rw [mem_rootsOfUnity, ← map_pow, Units.ext_iff, Units.coe_map, ξ.prop]
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.Basic (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 : CommMonoid R
inst_4 : CommMonoid S
inst_5 : FunLike F R S
inst_6 : MonoidHomClass F R S
σ : F
n : ℕ
⊢ Eq (α := ↥(rootsOfUnity n S))
    (Subtype.mk (p := fun (x : Sˣ) ↦ Membership.mem (γ := Subgroup Sˣ) (rootsOfUnity n S) x)
      (((Units.map (↑σ : R →* S) : MonoidHom.{u_4, u_5} Rˣ Sˣ) : (a : Rˣ) → Sˣ) (↑1 : Rˣ)) ⋯)
    1
have a : Eq (α := Sˣ)
  (↑(Subtype.mk (p := fun (x : Sˣ) ↦ Membership.mem (γ := Subgroup Sˣ) (rootsOfUnity n S) x)
        (((Units.map (↑σ : R →* S) : MonoidHom.{u_4, u_5} Rˣ Sˣ) : (a : Rˣ) → Sˣ) (↑1 : Rˣ)) sorry) :
    Sˣ)
  (↑1 : Sˣ)
Tactic: ext1
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.Basic (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 : CommMonoid R
inst_4 : CommMonoid S
inst_5 : FunLike F R S
inst_6 : MonoidHomClass F R S
σ : F
n : ℕ
ξ₁ ξ₂ : ↥(rootsOfUnity n R)
⊢ Subtype.mk (p := fun (x : Sˣ) ↦ Membership.mem (γ := Subgroup Sˣ) (rootsOfUnity n S) x)
      (((Units.map (↑σ : R →* S) : MonoidHom.{u_4, u_5} Rˣ Sˣ) : (a : Rˣ) → Sˣ) (↑(ξ₁ * ξ₂) : Rˣ)) ⋯ =
    HMul.hMul (α := ↥(rootsOfUnity n S)) (β := ↥(rootsOfUnity n S))
      (Subtype.mk (p := fun (x : Sˣ) ↦ Membership.mem (γ := Subgroup Sˣ) (rootsOfUnity n S) x)
          (((Units.map (↑σ : R →* S) : MonoidHom.{u_4, u_5} Rˣ Sˣ) : (a : Rˣ) → Sˣ) (↑ξ₁ : Rˣ)) ⋯ :
        ↥(rootsOfUnity n S))
      (Subtype.mk (p := fun (x : Sˣ) ↦ Membership.mem (γ := Subgroup Sˣ) (rootsOfUnity n S) x)
          (((Units.map (↑σ : R →* S) : MonoidHom.{u_4, u_5} Rˣ Sˣ) : (a : Rˣ) → Sˣ) (↑ξ₂ : Rˣ)) ⋯ :
        ↥(rootsOfUnity n S))
have a : (↑(Subtype.mk (p := fun (x : Sˣ) ↦ Membership.mem (γ := Subgroup Sˣ) (rootsOfUnity n S) x)
        (((Units.map (↑σ : R →* S) : MonoidHom.{u_4, u_5} Rˣ Sˣ) : (a : Rˣ) → Sˣ) (↑(ξ₁ * ξ₂) : Rˣ)) sorry) :
    Sˣ) =
  (↑(HMul.hMul (α := ↥(rootsOfUnity n S)) (β := ↥(rootsOfUnity n S))
        (Subtype.mk (p := fun (x : Sˣ) ↦ Membership.mem (γ := Subgroup Sˣ) (rootsOfUnity n S) x)
            (((Units.map (↑σ : R →* S) : MonoidHom.{u_4, u_5} Rˣ Sˣ) : (a : Rˣ) → Sˣ) (↑ξ₁ : Rˣ)) sorry :
          ↥(rootsOfUnity n S))
        (Subtype.mk (p := fun (x : Sˣ) ↦ Membership.mem (γ := Subgroup Sˣ) (rootsOfUnity n S) x)
            (((Units.map (↑σ : R →* S) : MonoidHom.{u_4, u_5} Rˣ Sˣ) : (a : Rˣ) → Sˣ) (↑ξ₂ : Rˣ)) sorry :
          ↥(rootsOfUnity n S))) :
    Sˣ)
Tactic: ext1
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.Basic (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 : CommMonoid R
inst_4 : CommMonoid S
inst_5 : FunLike F R S
σ : R ≃* S
n : ℕ
ξ : ↥(rootsOfUnity n R)
⊢ ((_root_.restrictRootsOfUnity σ.symm n : MonoidHom.{u_5, u_4} ↥(rootsOfUnity n S) ↥(rootsOfUnity n R)) :
        (a : ↥(rootsOfUnity n S)) → ↥(rootsOfUnity n R))
      (((_root_.restrictRootsOfUnity σ n : MonoidHom.{u_4, u_5} ↥(rootsOfUnity n R) ↥(rootsOfUnity n S)) :
          (a : ↥(rootsOfUnity n R)) → ↥(rootsOfUnity n S))
        ξ) =
    ξ
have a : (↑(↑(((_root_.restrictRootsOfUnity σ.symm n : MonoidHom.{u_5, u_4} ↥(rootsOfUnity n S) ↥(rootsOfUnity n R)) :
              (a : ↥(rootsOfUnity n S)) → ↥(rootsOfUnity n R))
            (((_root_.restrictRootsOfUnity σ n : MonoidHom.{u_4, u_5} ↥(rootsOfUnity n R) ↥(rootsOfUnity n S)) :
                (a : ↥(rootsOfUnity n R)) → ↥(rootsOfUnity n S))
              ξ)) :
        Rˣ) :
    R) =
  (↑(↑ξ : Rˣ) : R)
Tactic: ext
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.Basic (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 : CommMonoid R
inst_4 : CommMonoid S
inst_5 : FunLike F R S
σ : R ≃* S
n : ℕ
ξ : ↥(rootsOfUnity n S)
⊢ ((_root_.restrictRootsOfUnity σ n : MonoidHom.{u_4, u_5} ↥(rootsOfUnity n R) ↥(rootsOfUnity n S)) :
        (a : ↥(rootsOfUnity n R)) → ↥(rootsOfUnity n S))
      (((_root_.restrictRootsOfUnity σ.symm n : MonoidHom.{u_5, u_4} ↥(rootsOfUnity n S) ↥(rootsOfUnity n R)) :
          (a : ↥(rootsOfUnity n S)) → ↥(rootsOfUnity n R))
        ξ) =
    ξ
have a : (↑(↑(((_root_.restrictRootsOfUnity σ n : MonoidHom.{u_4, u_5} ↥(rootsOfUnity n R) ↥(rootsOfUnity n S)) :
              (a : ↥(rootsOfUnity n R)) → ↥(rootsOfUnity n S))
            (((_root_.restrictRootsOfUnity σ.symm n : MonoidHom.{u_5, u_4} ↥(rootsOfUnity n S) ↥(rootsOfUnity n R)) :
                (a : ↥(rootsOfUnity n S)) → ↥(rootsOfUnity n R))
              ξ)) :
        Sˣ) :
    S) =
  (↑(↑ξ : Sˣ) : S)
Tactic: ext
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.Basic (goal)
Goal:
R : Type u_4
k : ℕ
inst : CommMonoid R
inst_1 : NeZero k
⊢ val '' (↑(rootsOfUnity k R) : Set Rˣ) = {z : R | z ^ k = 1}
have h : ∀ (x : R), x ∈ val '' (↑(rootsOfUnity k R) : Set Rˣ) ↔ Membership.mem (γ := Set R) {z : R | z ^ k = 1} x
Tactic: ext x
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.Basic (goal)
Goal:
R : Type u_4
k : ℕ
inst : CommMonoid R
inst_1 : NeZero k
x : R
h : x ∈ val '' (↑(rootsOfUnity k R) : Set Rˣ)
y : Rˣ
hy1 : y ∈ (↑(rootsOfUnity k R) : Set Rˣ)
hy2 : (↑y : R) = x
⊢ Membership.mem (γ := Set R) {z : R | z ^ k = 1} x
have rw : Membership.mem (γ := Set R) {z : R | z ^ k = 1} (↑y : R)
Tactic: rw [← hy2]
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.Basic (goal)
Goal:
R : Type u_4
inst : CommRing R
inst_1 : NoZeroDivisors R
⊢ val '' (↑(rootsOfUnity 2 R) : Set Rˣ) = {1, -1}
have h : ∀ (x : R), x ∈ val '' (↑(rootsOfUnity 2 R) : Set Rˣ) ↔ Membership.mem (γ := Set R) {1, -1} x
Tactic: ext x
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.Basic (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 : NeZero k
inst_4 : CommRing R
inst_5 : IsDomain R
x : { x : R // Membership.mem (γ := Multiset R) (nthRoots k 1) x }
⊢ ↥(rootsOfUnity k R)
have refine_1 : (↑x : R) * (↑x : R) ^ (k - 1) = 1
Tactic: refine ⟨⟨x, ↑x ^ (k - 1 : ℕ), ?_, ?_⟩, ?_⟩
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.Basic (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 : NeZero k
inst_4 : CommRing R
inst_5 : IsDomain R
x : { x : R // Membership.mem (γ := Multiset R) (nthRoots k 1) x }
refine_1 : (↑x : R) * (↑x : R) ^ (k - 1) = 1
⊢ ↥(rootsOfUnity k R)
have refine_2 : (↑x : R) ^ (k - 1) * (↑x : R) = 1
Tactic: refine ⟨⟨x, ↑x ^ (k - 1 : ℕ), ?_, ?_⟩, ?_⟩
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.Basic (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 : NeZero k
inst_4 : CommRing R
inst_5 : IsDomain R
x : { x : R // Membership.mem (γ := Multiset R) (nthRoots k 1) x }
refine_1 : (↑x : R) * (↑x : R) ^ (k - 1) = 1
refine_2 : (↑x : R) ^ (k - 1) * (↑x : R) = 1
⊢ ↥(rootsOfUnity k R)
have refine_3 : Membership.mem (γ := Subgroup Rˣ) (rootsOfUnity k R)
  { val := (↑x : R), inv := (↑x : R) ^ (k - 1), val_inv := sorry, inv_val := sorry }
Tactic: refine ⟨⟨x, ↑x ^ (k - 1 : ℕ), ?_, ?_⟩, ?_⟩
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.Basic (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 : NeZero k
inst_4 : CommRing R
inst_5 : IsDomain R
x : { x : R // Membership.mem (γ := Multiset R) (nthRoots k 1) x }
⊢ (↑x : R) * (↑x : R) ^ (k - 1) = 1
have refine_1 : ∀ (x : R) (hx : Membership.mem (γ := Multiset R) (nthRoots k 1) x),
  (↑(Subtype.mk (p := fun (x : R) ↦ Membership.mem (γ := Multiset R) (nthRoots k 1) x) x hx) : R) *
      (↑(Subtype.mk (p := fun (x : R) ↦ Membership.mem (γ := Multiset R) (nthRoots k 1) x) x hx) : R) ^ (k - 1) =
    1
Tactic: rcases x with ⟨x, hx⟩
Cleared hypotheses: x
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.Basic (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 : NeZero k
inst_4 : CommRing R
inst_5 : IsDomain R
x : R
hx : Membership.mem (γ := Multiset R) (nthRoots k 1) x
⊢ (↑(Subtype.mk (p := fun (x : R) ↦ Membership.mem (γ := Multiset R) (nthRoots k 1) x) x hx) : R) *
      (↑(Subtype.mk (p := fun (x : R) ↦ Membership.mem (γ := Multiset R) (nthRoots k 1) x) x hx) : R) ^ (k - 1) =
    1
have hx₁ : x ^ k = 1
Tactic: rw [mem_nthRoots <| NeZero.pos k] at hx
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.Basic (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 : NeZero k
inst_4 : CommRing R
inst_5 : IsDomain R
x : { x : R // Membership.mem (γ := Multiset R) (nthRoots k 1) x }
⊢ (↑x : R) ^ (k - 1) * (↑x : R) = 1
have refine_2 : ∀ (x : R) (hx : Membership.mem (γ := Multiset R) (nthRoots k 1) x),
  (↑(Subtype.mk (p := fun (x : R) ↦ Membership.mem (γ := Multiset R) (nthRoots k 1) x) x hx) : R) ^ (k - 1) *
      (↑(Subtype.mk (p := fun (x : R) ↦ Membership.mem (γ := Multiset R) (nthRoots k 1) x) x hx) : R) =
    1
Tactic: rcases x with ⟨x, hx⟩
Cleared hypotheses: x
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.Basic (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 : NeZero k
inst_4 : CommRing R
inst_5 : IsDomain R
x : R
hx : Membership.mem (γ := Multiset R) (nthRoots k 1) x
⊢ (↑(Subtype.mk (p := fun (x : R) ↦ Membership.mem (γ := Multiset R) (nthRoots k 1) x) x hx) : R) ^ (k - 1) *
      (↑(Subtype.mk (p := fun (x : R) ↦ Membership.mem (γ := Multiset R) (nthRoots k 1) x) x hx) : R) =
    1
have hx₁ : x ^ k = 1
Tactic: rw [mem_nthRoots <| NeZero.pos k] at hx
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.Basic (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 : NeZero k
inst_4 : CommRing R
inst_5 : IsDomain R
x : { x : R // Membership.mem (γ := Multiset R) (nthRoots k 1) x }
⊢ Membership.mem (γ := Subgroup Rˣ) (rootsOfUnity k R)
    { val := (↑x : R), inv := (↑x : R) ^ (k - 1), val_inv := ⋯, inv_val := ⋯ }
have refine_3 : ∀ (x : R) (hx : Membership.mem (γ := Multiset R) (nthRoots k 1) x),
  Membership.mem (γ := Subgroup Rˣ) (rootsOfUnity k R)
    { val := (↑(Subtype.mk (p := fun (x : R) ↦ Membership.mem (γ := Multiset R) (nthRoots k 1) x) x hx) : R),
      inv := (↑(Subtype.mk (p := fun (x : R) ↦ Membership.mem (γ := Multiset R) (nthRoots k 1) x) x hx) : R) ^ (k - 1),
      val_inv := sorry, inv_val := sorry }
Tactic: rcases x with ⟨x, hx⟩
Cleared hypotheses: x
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.Basic (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 : NeZero k
inst_4 : CommRing R
inst_5 : IsDomain R
x : R
hx : Membership.mem (γ := Multiset R) (nthRoots k 1) x
⊢ Membership.mem (γ := Subgroup Rˣ) (rootsOfUnity k R)
    { val := (↑(Subtype.mk (p := fun (x : R) ↦ Membership.mem (γ := Multiset R) (nthRoots k 1) x) x hx) : R),
      inv := (↑(Subtype.mk (p := fun (x : R) ↦ Membership.mem (γ := Multiset R) (nthRoots k 1) x) x hx) : R) ^ (k - 1),
      val_inv := ⋯, inv_val := ⋯ }
have hx₁ : x ^ k = 1
Tactic: rw [mem_nthRoots <| NeZero.pos k] at hx
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.Basic (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 : NeZero k
inst_4 : CommRing R
inst_5 : IsDomain R
x : R
hx_1 : Membership.mem (γ := Multiset R) (nthRoots k 1) x
hx : x ^ k = 1
⊢ Membership.mem (γ := Subgroup Rˣ) (rootsOfUnity k R)
    { val := (↑(Subtype.mk (p := fun (x : R) ↦ Membership.mem (γ := Multiset R) (nthRoots k 1) x) x hx_1) : R),
      inv :=
        (↑(Subtype.mk (p := fun (x : R) ↦ Membership.mem (γ := Multiset R) (nthRoots k 1) x) x hx_1) : R) ^ (k - 1),
      val_inv := ⋯, inv_val := ⋯ }
have refine_3 : Membership.mem (γ := Subgroup Rˣ) (rootsOfUnity k R) { val := x, inv := x ^ (k - 1), val_inv := sorry, inv_val := sorry }
Tactic: simp only [← pow_succ, ← pow_succ', hx, tsub_add_cancel_of_le NeZero.one_le]
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.Basic (goal)
Goal:
R : Type u_4
F : Type u_6
k : ℕ
inst : NeZero k
inst_1 : CommRing R
inst_2 : IsDomain R
inst_3 : FunLike F R R
inst_4 : MonoidHomClass F R R
σ : F
ζ : ↥(rootsOfUnity k R)
⊢ ∃ (m : ℕ), (σ : (a : R) → R) (↑(↑ζ : Rˣ) : R) = (↑(↑ζ : Rˣ) : R) ^ m
have obtain : ∀ (m : ℤ)
  (hm :
    ∀ (g : ↥(rootsOfUnity k R)),
      (restrictRootsOfUnity σ k : (a : ↥(rootsOfUnity k R)) → ↥(rootsOfUnity k R)) g = g ^ m),
  ∃ (m : ℕ), (σ : (a : R) → R) (↑(↑ζ : Rˣ) : R) = (↑(↑ζ : Rˣ) : R) ^ m
Tactic: obtain ⟨m, hm⟩ := MonoidHom.map_cyclic (restrictRootsOfUnity σ k)
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.Basic (goal)
Goal:
R : Type u_4
F : Type u_6
k : ℕ
inst : NeZero k
inst_1 : CommRing R
inst_2 : IsDomain R
inst_3 : FunLike F R R
inst_4 : MonoidHomClass F R R
σ : F
ζ : ↥(rootsOfUnity k R)
m : ℤ
hm : ∀ (g : ↥(rootsOfUnity k R)), (restrictRootsOfUnity σ k : (a : ↥(rootsOfUnity k R)) → ↥(rootsOfUnity k R)) g = g ^ m
⊢ ∃ (m : ℕ), (σ : (a : R) → R) (↑(↑ζ : Rˣ) : R) = (↑(↑ζ : Rˣ) : R) ^ m
have rw : ∃ (m_1 : ℕ), (↑(↑ζ : Rˣ) : R) ^ (m % (↑(orderOf ζ) : ℤ)).toNat = (↑(↑ζ : Rˣ) : R) ^ m_1
Tactic: rw [← restrictRootsOfUnity_coe_apply, hm, ← zpow_mod_orderOf, ← Int.toNat_of_nonneg
      (m.emod_nonneg (Int.natCast_ne_zero.mpr (pos_iff_ne_zero.mp (orderOf_pos ζ)))),
    zpow_natCast, rootsOfUnity.coe_pow]
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.Basic (hypothesis)
Goal:
M : Type u_1
N : Type u_2
G : Type u_3
R : Type u_4
S_1 : Type u_5
F : Type u_6
inst : CommMonoid M
inst_1 : CommMonoid N
inst_2 : DivisionCommMonoid G
k l : ℕ
inst_3 : NeZero k
inst_4 : CommRing R
inst_5 : IsDomain R
L : Type u_7
inst_6 : LeftCancelMonoid L
inst_7 : Finite L
S : Subgroup Rˣ := rootsOfUnity (Monoid.exponent L) R
⊢ Finite (MonoidHom.{u_7, u_4} L Rˣ)
have this : Finite (MonoidHom.{u_7, u_4} L ↥S)
Tactic: have : Finite (L →* S) := .of_injective _ DFunLike.coe_injective
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.Basic (goal)
Goal:
M : Type u_1
N : Type u_2
G : Type u_3
R : Type u_4
S_1 : Type u_5
F : Type u_6
inst : CommMonoid M
inst_1 : CommMonoid N
inst_2 : DivisionCommMonoid G
k l : ℕ
inst_3 : NeZero k
inst_4 : CommRing R
inst_5 : IsDomain R
L : Type u_7
inst_6 : LeftCancelMonoid L
inst_7 : Finite L
S : Subgroup Rˣ := rootsOfUnity (Monoid.exponent L) R
this : Finite (MonoidHom.{u_7, u_4} L ↥S)
⊢ Finite (MonoidHom.{u_7, u_4} L Rˣ)
have refine : ∀ (f : MonoidHom.{u_7, u_4} L Rˣ),
  ∃ (a : MonoidHom.{u_7, u_4} L ↥S), (fun (f : MonoidHom.{u_7, u_4} L ↥S) ↦ S.subtype.comp f) a = f
Tactic: refine .of_surjective (fun f : L →* S ↦ (Subgroup.subtype _).comp f) fun f ↦ ?_
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.Basic (goal)
Goal:
M : Type u_1
N : Type u_2
G : Type u_3
R : Type u_4
S_1 : Type u_5
F : Type u_6
inst : CommMonoid M
inst_1 : CommMonoid N
inst_2 : DivisionCommMonoid G
k l : ℕ
inst_3 : NeZero k
inst_4 : CommRing R
inst_5 : IsDomain R
L : Type u_7
inst_6 : LeftCancelMonoid L
inst_7 : Finite L
S : Subgroup Rˣ := rootsOfUnity (Monoid.exponent L) R
this : Finite (MonoidHom.{u_7, u_4} L ↥S)
f : MonoidHom.{u_7, u_4} L Rˣ
a : L
⊢ Membership.mem (α := Rˣ) S ((f : (a : L) → Rˣ) a : Rˣ)
have rw : HPow.hPow (α := Rˣ) ((f : (a : L) → Rˣ) a : Rˣ) (Monoid.exponent L) = 1
Tactic: rw [mem_rootsOfUnity, ← map_pow, Monoid.pow_exponent_eq_one, map_one]
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.Basic (goal)
Goal:
M : Type u_1
N : Type u_2
G : Type u_3
R : Type u_4
S_1 : Type u_5
F : Type u_6
inst : CommMonoid M
inst_1 : CommMonoid N
inst_2 : DivisionCommMonoid G
k l : ℕ
inst_3 : NeZero k
inst_4 : CommRing R
inst_5 : IsDomain R
L : Type u_7
inst_6 : LeftCancelMonoid L
inst_7 : Finite L
S : Subgroup Rˣ := rootsOfUnity (Monoid.exponent L) R
this : Finite (MonoidHom.{u_7, u_4} L ↥S)
f : MonoidHom.{u_7, u_4} L Rˣ
a : L
rw : HPow.hPow (α := Rˣ) ((f : (a : L) → Rˣ) a : Rˣ) (Monoid.exponent L) = 1
⊢ Membership.mem (α := Rˣ) S ((f : (a : L) → Rˣ) a : Rˣ)
have rw₁ : Eq.{u_4 + 1} (α := Rˣ) ((f : (a : L) → Rˣ) (a ^ Monoid.exponent L) : Rˣ) (1 : Rˣ)
Tactic: rw [mem_rootsOfUnity, ← map_pow, Monoid.pow_exponent_eq_one, map_one]