awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- 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 : ZMod k
hi :
  Eq.{u_4 + 1} (α := 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))
        i :
      Additive ↥(Subgroup.zpowers ζ))
    (0 : Additive ↥(Subgroup.zpowers ζ))
⊢ i = 0
have hi₁ : Eq.{u_4 + 1} (α := Rˣ)
  (↑((((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))
        i) :
    Rˣ)
  (↑0 : Rˣ)
Tactic: rw [Subtype.ext_iff] at hi
Cleared hypotheses: hi
-- 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 : ZMod k
hi :
  Eq.{u_4 + 1} (α := Rˣ)
    (↑((((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))
          i) :
      Rˣ)
    (↑0 : Rˣ)
⊢ i = 0
have this : Dvd.dvd (α := ℤ) (↑k : ℤ) i.cast
Tactic: have := (h.zpow_eq_one_iff_dvd _).mp 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 : ZMod k
hi :
  Eq.{u_4 + 1} (α := Rˣ)
    (↑((((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))
          i) :
      Rˣ)
    (↑0 : Rˣ)
this : Dvd.dvd (α := ℤ) (↑k : ℤ) i.cast
⊢ i = 0
have left : (↑i.cast : ZMod k) = i
Tactic: rw [← (CharP.intCast_eq_zero_iff (ZMod k) k _).mpr this, eq_comm]
-- 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
⊢ ∃ (a : ZMod k),
    (((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))
        a =
      Subtype.mk (p := fun (x : Rˣ) ↦ Membership.mem (γ := Subgroup Rˣ) (Subgroup.zpowers ζ) x)
        ((fun (x : ℤ) ↦ ζ ^ x) i) ⋯
have right : (((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))
    ((Int.castAddHom (ZMod k) : (a : ℤ) → ZMod k) i : ZMod k) =
  Subtype.mk (p := fun (x : Rˣ) ↦ Membership.mem (γ := Subgroup Rˣ) (Subgroup.zpowers ζ) x) ((fun (x : ℤ) ↦ ζ ^ x) i) sorry
Tactic: refine ⟨Int.castAddHom (ZMod k) i, ?_⟩
-- 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
⊢ (((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))
      ((Int.castAddHom (ZMod k) : (a : ℤ) → ZMod k) i : ZMod k) =
    Subtype.mk (p := fun (x : Rˣ) ↦ Membership.mem (γ := Subgroup Rˣ) (Subgroup.zpowers ζ) x) ((fun (x : ℤ) ↦ ζ ^ x) i)
      ⋯
have right : ((↑(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) :
        ℤ →+ Additive ↥(Subgroup.zpowers ζ)) :
      (a : ℤ) → Additive ↥(Subgroup.zpowers ζ))
    i =
  Subtype.mk (p := fun (x : Rˣ) ↦ Membership.mem (γ := Subgroup Rˣ) (Subgroup.zpowers ζ) x) ((fun (x : ℤ) ↦ ζ ^ x) i) sorry
Tactic: rw [AddMonoidHom.liftOfRightInverse_comp_apply]
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots (goal)
Goal:
R : Type u_4
k : ℕ
inst : CommRing R
ζ : Rˣ
h : IsPrimitiveRoot ζ k
i : ℕ
⊢ Eq (α := ZMod k) (↑i : ZMod k) (↑(↑i : ℤ) : ZMod k)
have norm_cast : Eq (α := ZMod k) (↑i : ZMod k) (↑i : ZMod k)
Tactic: norm_cast
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots (hypothesis)
Goal:
R : Type u_4
k : ℕ
inst : CommRing R
ζ : Rˣ
h : IsPrimitiveRoot ζ k
i : ℕ
⊢ Eq (α := Additive ↥(Subgroup.zpowers ζ))
    ((h.zmodEquivZPowers : (a : ZMod k) → Additive ↥(Subgroup.zpowers ζ)) (↑i : ZMod k) :
      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) ⋯) :
      Additive ↥(Subgroup.zpowers ζ))
have this : Eq (α := ZMod k) (↑i : ZMod k) (↑(↑i : ℤ) : ZMod k)
Tactic: have : (i : ZMod k) = (i : ℤ) := by norm_cast
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots (goal)
Goal:
R : Type u_4
k : ℕ
inst : CommRing R
ζ : Rˣ
h : IsPrimitiveRoot ζ k
i : ℤ
⊢ Eq (α := ZMod k)
    ((h.zmodEquivZPowers.symm : (a : Additive ↥(Subgroup.zpowers ζ)) → ZMod k)
        ((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) ⋯)) :
      ZMod k)
    ((↑i : ZMod k) : ZMod k)
have rw : Eq (α := ZMod k)
  ((h.zmodEquivZPowers.symm : (a : Additive ↥(Subgroup.zpowers ζ)) → ZMod k)
      ((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)) :
    ZMod k)
  ((h.zmodEquivZPowers.symm : (a : Additive ↥(Subgroup.zpowers ζ)) → ZMod k)
      ((h.zmodEquivZPowers : (a : ZMod k) → Additive ↥(Subgroup.zpowers ζ)) (↑i : ZMod k)) :
    ZMod k)
Tactic: rw [← h.zmodEquivZPowers.symm_apply_apply i, zmodEquivZPowers_apply_coe_int]
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots (goal)
Goal:
R : Type u_4
k : ℕ
inst : CommRing R
ζ : Rˣ
h : IsPrimitiveRoot ζ k
i : ℤ
rw :
  Eq (α := ZMod k)
    ((h.zmodEquivZPowers.symm : (a : Additive ↥(Subgroup.zpowers ζ)) → ZMod k)
        ((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) ⋯)) :
      ZMod k)
    ((h.zmodEquivZPowers.symm : (a : Additive ↥(Subgroup.zpowers ζ)) → ZMod k)
        ((h.zmodEquivZPowers : (a : ZMod k) → Additive ↥(Subgroup.zpowers ζ)) (↑i : ZMod k)) :
      ZMod k)
⊢ Eq (α := ZMod k)
    ((h.zmodEquivZPowers.symm : (a : Additive ↥(Subgroup.zpowers ζ)) → ZMod k)
        ((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) ⋯)) :
      ZMod k)
    ((↑i : ZMod k) : ZMod k)
have rw₁ : Eq (α := ZMod k)
  ((h.zmodEquivZPowers.symm : (a : Additive ↥(Subgroup.zpowers ζ)) → ZMod k)
      ((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)) :
    ZMod k)
  ((h.zmodEquivZPowers.symm : (a : Additive ↥(Subgroup.zpowers ζ)) → ZMod k)
      ((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)) :
    ZMod k)
Tactic: rw [← h.zmodEquivZPowers.symm_apply_apply i, zmodEquivZPowers_apply_coe_int]
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots (goal)
Goal:
R : Type u_4
k : ℕ
inst : CommRing R
ζ : Rˣ
h : IsPrimitiveRoot ζ k
i : ℕ
⊢ Eq (α := ZMod k)
    ((h.zmodEquivZPowers.symm : (a : Additive ↥(Subgroup.zpowers ζ)) → ZMod k)
        ((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) ⋯)) :
      ZMod k)
    ((↑i : ZMod k) : ZMod k)
have rw : Eq (α := ZMod k)
  ((h.zmodEquivZPowers.symm : (a : Additive ↥(Subgroup.zpowers ζ)) → ZMod k)
      ((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)) :
    ZMod k)
  ((h.zmodEquivZPowers.symm : (a : Additive ↥(Subgroup.zpowers ζ)) → ZMod k)
      ((h.zmodEquivZPowers : (a : ZMod k) → Additive ↥(Subgroup.zpowers ζ)) (↑i : ZMod k)) :
    ZMod k)
Tactic: rw [← h.zmodEquivZPowers.symm_apply_apply i, zmodEquivZPowers_apply_coe_nat]
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots (goal)
Goal:
R : Type u_4
k : ℕ
inst : CommRing R
ζ : Rˣ
h : IsPrimitiveRoot ζ k
i : ℕ
rw :
  Eq (α := ZMod k)
    ((h.zmodEquivZPowers.symm : (a : Additive ↥(Subgroup.zpowers ζ)) → ZMod k)
        ((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) ⋯)) :
      ZMod k)
    ((h.zmodEquivZPowers.symm : (a : Additive ↥(Subgroup.zpowers ζ)) → ZMod k)
        ((h.zmodEquivZPowers : (a : ZMod k) → Additive ↥(Subgroup.zpowers ζ)) (↑i : ZMod k)) :
      ZMod k)
⊢ Eq (α := ZMod k)
    ((h.zmodEquivZPowers.symm : (a : Additive ↥(Subgroup.zpowers ζ)) → ZMod k)
        ((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) ⋯)) :
      ZMod k)
    ((↑i : ZMod k) : ZMod k)
have rw₁ : Eq (α := ZMod k)
  ((h.zmodEquivZPowers.symm : (a : Additive ↥(Subgroup.zpowers ζ)) → ZMod k)
      ((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)) :
    ZMod k)
  ((h.zmodEquivZPowers.symm : (a : Additive ↥(Subgroup.zpowers ζ)) → ZMod k)
      ((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)) :
    ZMod k)
Tactic: rw [← h.zmodEquivZPowers.symm_apply_apply i, zmodEquivZPowers_apply_coe_nat]
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots (goal)
Goal:
R : Type u_4
inst : CommRing R
inst_1 : IsDomain R
k : ℕ
inst_2 : NeZero k
ζ : Rˣ
h : IsPrimitiveRoot ζ k
⊢ Eq (α := Subgroup Rˣ) (Subgroup.zpowers ζ) (rootsOfUnity k R)
have a : (↑(Subgroup.zpowers ζ) : Set Rˣ) = (↑(rootsOfUnity k R : Subgroup Rˣ) : Set Rˣ)
Tactic: apply SetLike.coe_injective
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots (goal)
Goal:
R : Type u_4
inst : CommRing R
inst_1 : IsDomain R
k : ℕ
inst_2 : NeZero k
ζ : Rˣ
h : IsPrimitiveRoot ζ k
⊢ (↑(Subgroup.zpowers ζ) : Set Rˣ) = (↑(rootsOfUnity k R : Subgroup Rˣ) : Set Rˣ)
have a : ∀ (F : Fintype ↥(Subgroup.zpowers ζ)), (↑(Subgroup.zpowers ζ) : Set Rˣ) = (↑(rootsOfUnity k R : Subgroup Rˣ) : Set Rˣ)
Tactic: have F : Fintype (Subgroup.zpowers ζ) := Fintype.ofEquiv _ h.zmodEquivZPowers.toEquiv
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots (goal)
Goal:
R : Type u_4
inst : CommRing R
inst_1 : IsDomain R
k : ℕ
inst_2 : NeZero k
ζ : Rˣ
h : IsPrimitiveRoot ζ k
F : Fintype ↥(Subgroup.zpowers ζ)
⊢ (↑(Subgroup.zpowers ζ) : Set Rˣ) = (↑(rootsOfUnity k R : Subgroup Rˣ) : Set Rˣ)
have a : Fintype.card.{u_4} (↑(↑(rootsOfUnity k R : Subgroup Rˣ) : Set Rˣ) : Type u_4) ≤
  Fintype.card.{u_4} (↑(↑(Subgroup.zpowers ζ) : Set Rˣ) : Type u_4)
Tactic: refine
    @Set.eq_of_subset_of_card_le Rˣ _ _ F (rootsOfUnity.fintype R k)
      (Subgroup.zpowers_le_of_mem <| show ζ ∈ rootsOfUnity k R from h.pow_eq_one) ?_
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots (hypothesis)
Goal:
R : Type u_4
inst : CommRing R
inst_1 : IsDomain R
S : Type u_7
F : Type u_8
inst_2 : CommRing S
inst_3 : IsDomain S
inst_4 : FunLike F R S
inst_5 : MonoidHomClass F R S
ζ : R
n : ℕ
inst_6 : NeZero n
hζ : IsPrimitiveRoot ζ n
f : F
hf : Function.Injective.{u_4 + 1, u_7 + 1} (α := R) (β := S) (⇑f : (a : R) → S)
this : CommMonoid Sˣ := inferInstance
⊢ Eq (α := Subgroup Sˣ) (Subgroup.map (G := Rˣ) (Units.map (↑f : R →* S)) (rootsOfUnity n R)) (rootsOfUnity n S)
have hζ₁ : IsPrimitiveRoot sorry.unit n
Tactic: replace hζ := hζ.isUnit_unit NeZero.out
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots (goal)
Goal:
R : Type u_4
inst : CommRing R
inst_1 : IsDomain R
S : Type u_7
F : Type u_8
inst_2 : CommRing S
inst_3 : IsDomain S
inst_4 : FunLike F R S
inst_5 : MonoidHomClass F R S
ζ : R
n : ℕ
inst_6 : NeZero n
hζ_1 : IsPrimitiveRoot ζ n
f : F
hf : Function.Injective.{u_4 + 1, u_7 + 1} (α := R) (β := S) (⇑f : (a : R) → S)
this : CommMonoid Sˣ := inferInstance
hζ : IsPrimitiveRoot ⋯.unit n
⊢ Eq (α := Subgroup Sˣ) (Subgroup.map (G := Rˣ) (Units.map (↑f : R →* S)) (rootsOfUnity n R)) (rootsOfUnity n S)
have rw : Eq (α := Subgroup Sˣ) (Subgroup.map (Units.map (↑f : R →* S)) (Subgroup.zpowers sorry.unit)) (rootsOfUnity n S)
Tactic: rw [← hζ.zpowers_eq,
    ← (hζ.map_of_injective (Units.map_injective (f := (f : R →* S)) hf)).zpowers_eq,
    MonoidHom.map_zpowers]
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots (goal)
Goal:
R : Type u_4
inst : CommRing R
inst_1 : IsDomain R
S : Type u_7
F : Type u_8
inst_2 : CommRing S
inst_3 : IsDomain S
inst_4 : FunLike F R S
inst_5 : MonoidHomClass F R S
ζ : R
n : ℕ
inst_6 : NeZero n
hζ_1 : IsPrimitiveRoot ζ n
f : F
hf : Function.Injective.{u_4 + 1, u_7 + 1} (α := R) (β := S) (⇑f : (a : R) → S)
this : CommMonoid Sˣ := inferInstance
hζ : IsPrimitiveRoot ⋯.unit n
rw : Eq (α := Subgroup Sˣ) (Subgroup.map (Units.map (↑f : R →* S)) (Subgroup.zpowers ⋯.unit)) (rootsOfUnity n S)
⊢ Eq (α := Subgroup Sˣ) (Subgroup.map (G := Rˣ) (Units.map (↑f : R →* S)) (rootsOfUnity n R)) (rootsOfUnity n S)
have rw₁ : Eq (α := Subgroup Sˣ) (Subgroup.map (Units.map (↑f : R →* S)) (Subgroup.zpowers sorry.unit))
  (Subgroup.zpowers (((Units.map (↑f : R →* S) : MonoidHom.{u_4, u_7} Rˣ Sˣ) : (a : Rˣ) → Sˣ) sorry.unit))
Tactic: rw [← hζ.zpowers_eq,
    ← (hζ.map_of_injective (Units.map_injective (f := (f : R →* S)) hf)).zpowers_eq,
    MonoidHom.map_zpowers]
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots (goal)
Goal:
R : Type u_4
inst : CommRing R
inst_1 : IsDomain R
S : Type u_7
F : Type u_8
inst_2 : CommRing S
inst_3 : IsDomain S
inst_4 : FunLike F R S
inst_5 : MonoidHomClass F R S
ζ : R
n : ℕ
inst_6 : NeZero n
hζ_1 : IsPrimitiveRoot ζ n
f : F
hf : Function.Injective.{u_4 + 1, u_7 + 1} (α := R) (β := S) (⇑f : (a : R) → S)
this : CommMonoid Sˣ := inferInstance
hζ : IsPrimitiveRoot ⋯.unit n
rw : Eq (α := Subgroup Sˣ) (Subgroup.map (Units.map (↑f : R →* S)) (Subgroup.zpowers ⋯.unit)) (rootsOfUnity n S)
rw₁ :
  Eq (α := Subgroup Sˣ) (Subgroup.map (Units.map (↑f : R →* S)) (Subgroup.zpowers ⋯.unit))
    (Subgroup.zpowers (((Units.map (↑f : R →* S) : MonoidHom.{u_4, u_7} Rˣ Sˣ) : (a : Rˣ) → Sˣ) ⋯.unit))
⊢ Eq (α := Subgroup Sˣ) (Subgroup.map (G := Rˣ) (Units.map (↑f : R →* S)) (rootsOfUnity n R)) (rootsOfUnity n S)
have rw₂ : Eq.{u_7 + 1} (α := Subgroup Sˣ)
  (Subgroup.zpowers (((Units.map (↑f : R →* S) : MonoidHom.{u_4, u_7} Rˣ Sˣ) : (a : Rˣ) → Sˣ) sorry.unit) : Subgroup Sˣ)
  (Subgroup.zpowers (((Units.map (↑f : R →* S) : MonoidHom.{u_4, u_7} Rˣ Sˣ) : (a : Rˣ) → Sˣ) sorry.unit) : Subgroup Sˣ)
Tactic: rw [← hζ.zpowers_eq,
    ← (hζ.map_of_injective (Units.map_injective (f := (f : R →* S)) hf)).zpowers_eq,
    MonoidHom.map_zpowers]
-- Lean 4 proof step from Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots (goal)
Goal:
R : Type u_4
inst : CommRing R
inst_1 : IsDomain R
S : Type u_7
F : Type u_8
inst_2 : CommRing S
inst_3 : IsDomain S
inst_4 : FunLike F R S
inst_5 : MonoidHomClass F R S
n : ℕ
inst_6 : NeZero n
f : F
hf : Function.Injective.{u_4 + 1, u_7 + 1} (α := R) (β := S) (⇑f : (a : R) → S)
hζ : (primitiveRoots n R).Nonempty
η : ↥(rootsOfUnity n S)
⊢ (f : (a : R) → S)
      (↑(↑(((MulEquiv.symm (rootsOfUnityEquivOfPrimitiveRoots hf hζ) :
                    MulEquiv.{u_7, u_4} ↥(rootsOfUnity n S) ↥(rootsOfUnity n R)) :
                  (a : ↥(rootsOfUnity n S)) → ↥(rootsOfUnity n R))
                η) :
            Rˣ) :
        R) =
    (↑(↑η : Sˣ) : S)
have obtain : ∀ (ε : ↥(rootsOfUnity n R)),
  Eq.{u_7 + 1} (α := S)
    ((f : (a : R) → S)
        (↑(↑(((MulEquiv.symm (rootsOfUnityEquivOfPrimitiveRoots hf hζ) :
                      MulEquiv.{u_7, u_4} ↥(rootsOfUnity n S) ↥(rootsOfUnity n R)) :
                    (a : ↥(rootsOfUnity n S)) → ↥(rootsOfUnity n R))
                  (((rootsOfUnityEquivOfPrimitiveRoots hf hζ :
                        MulEquiv.{u_4, u_7} ↥(rootsOfUnity n R) ↥(rootsOfUnity n S)) :
                      (a : ↥(rootsOfUnity n R)) → ↥(rootsOfUnity n S))
                    ε)) :
              Rˣ) :
          R) :
      S)
    ((↑(↑(((rootsOfUnityEquivOfPrimitiveRoots hf hζ : MulEquiv.{u_4, u_7} ↥(rootsOfUnity n R) ↥(rootsOfUnity n S)) :
                  (a : ↥(rootsOfUnity n R)) → ↥(rootsOfUnity n S))
                ε) :
            Sˣ) :
        S) :
      S)
Tactic: obtain ⟨ε, rfl⟩ := (rootsOfUnityEquivOfPrimitiveRoots hf hζ).surjective η
Cleared hypotheses: η