awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.NumberTheory.NumberField.Cyclotomic.Three (hypothesis)
Goal:
K : Type u_1
inst : Field K
ζ : K
hζ : IsPrimitiveRoot ζ 3
u : (𝓞 K)ˣ
inst_1 : NumberField K
inst_2 : IsCyclotomicExtension {3} ℚ K
hrank : rank K = 0
x : ↥(torsion K)
e : Fin (rank K) → ℤ
hxu : u = (↑(x, e).1 : (𝓞 K)ˣ) * ∏ i : Fin (rank K), fundSystem K i ^ (x, e).2 i
⊢ u ∈
    [1, -1, IsUnit.unit (a := hζ.toInteger) ⋯, -IsUnit.unit (a := hζ.toInteger) ⋯,
      HPow.hPow (β := ℕ) (IsUnit.unit (a := hζ.toInteger) ⋯) 2,
      -HPow.hPow (β := ℕ) (IsUnit.unit (a := hζ.toInteger) ⋯) 2]
have hxu₁ : u = (↑x : (𝓞 K)ˣ)
Tactic: replace hxu : u = x := by
    rw [← mul_one x.1, hxu]
    apply congr_arg
    rw [← Finset.prod_empty]
    congr
    rw [Finset.univ_eq_empty_iff, hrank]
    infer_instance
Cleared hypotheses: hxu
-- Lean 4 proof step from Mathlib.NumberTheory.NumberField.Cyclotomic.Three (goal)
Goal:
K : Type u_1
inst : Field K
ζ : K
hζ : IsPrimitiveRoot ζ 3
u : (𝓞 K)ˣ
inst_1 : NumberField K
inst_2 : IsCyclotomicExtension {3} ℚ K
hrank : rank K = 0
x : ↥(torsion K)
e : Fin (rank K) → ℤ
hxu : u = (↑x : (𝓞 K)ˣ)
⊢ u ∈
    [1, -1, IsUnit.unit (a := hζ.toInteger) ⋯, -IsUnit.unit (a := hζ.toInteger) ⋯,
      HPow.hPow (β := ℕ) (IsUnit.unit (a := hζ.toInteger) ⋯) 2,
      -HPow.hPow (β := ℕ) (IsUnit.unit (a := hζ.toInteger) ⋯) 2]
have obtain : ∀ (n : ℕ) (hnpos : 0 < n) (hn : (↑x : (𝓞 K)ˣ) ^ n = 1),
  u ∈
    [1, -1, IsUnit.unit (a := hζ.toInteger) sorry, -IsUnit.unit (a := hζ.toInteger) sorry,
      HPow.hPow (β := ℕ) (IsUnit.unit (a := hζ.toInteger) sorry) 2,
      -HPow.hPow (β := ℕ) (IsUnit.unit (a := hζ.toInteger) sorry) 2]
Tactic: obtain ⟨n, hnpos, hn⟩ := isOfFinOrder_iff_pow_eq_one.1 <| (CommGroup.mem_torsion _ _).1 x.2
-- Lean 4 proof step from Mathlib.NumberTheory.NumberField.Cyclotomic.Three (goal)
Goal:
K : Type u_1
inst : Field K
ζ : K
hζ : IsPrimitiveRoot ζ 3
u : (𝓞 K)ˣ
inst_1 : NumberField K
inst_2 : IsCyclotomicExtension {3} ℚ K
hrank : rank K = 0
x : ↥(torsion K)
e : Fin (rank K) → ℤ
hxu : u = (↑x : (𝓞 K)ˣ)
n : ℕ
hnpos : 0 < n
hn : (↑x : (𝓞 K)ˣ) ^ n = 1
⊢ HPow.hPow (α := K) ((algebraMap (𝓞 K) K : (a : 𝓞 K) → K) (↑u : 𝓞 K) : K)
      (↑(Subtype.mk (p := fun (n : ℕ) ↦ 0 < n) n hnpos) : ℕ) =
    1
have rw : Eq (α := K)
  ((algebraMap (𝓞 K) K : (a : 𝓞 K) → K) ((↑u : 𝓞 K) ^ (↑(Subtype.mk (p := fun (n : ℕ) ↦ 0 < n) n hnpos) : ℕ)) : K)
  (1 : K)
Tactic: rw [← map_pow]
-- Lean 4 proof step from Mathlib.NumberTheory.NumberField.Cyclotomic.Three (goal)
Goal:
K : Type u_1
inst : Field K
ζ : K
hζ : IsPrimitiveRoot ζ 3
u : (𝓞 K)ˣ
inst_1 : NumberField K
inst_2 : IsCyclotomicExtension {3} ℚ K
hrank : rank K = 0
x : ↥(torsion K)
e : Fin (rank K) → ℤ
hxu : u = (↑x : (𝓞 K)ˣ)
n : ℕ
hnpos : 0 < n
hn : (↑x : (𝓞 K)ˣ) ^ n = 1
⊢ Eq (α := K)
    ((algebraMap (𝓞 K) K : (a : 𝓞 K) → K) ((↑u : 𝓞 K) ^ (↑(Subtype.mk (p := fun (n : ℕ) ↦ 0 < n) n hnpos) : ℕ)) : K)
    (1 : K)
have h : (↑u : 𝓞 K) ^ (↑(Subtype.mk (p := fun (n : ℕ) ↦ 0 < n) n hnpos) : ℕ) = 1
Tactic: convert! map_one (algebraMap (𝓞 K) K)
-- Lean 4 proof step from Mathlib.NumberTheory.NumberField.Cyclotomic.Three (goal)
Goal:
K : Type u_1
inst : Field K
ζ : K
hζ : IsPrimitiveRoot ζ 3
u : (𝓞 K)ˣ
inst_1 : NumberField K
inst_2 : IsCyclotomicExtension {3} ℚ K
hrank : rank K = 0
x : ↥(torsion K)
e : Fin (rank K) → ℤ
hxu : u = (↑x : (𝓞 K)ˣ)
n : ℕ
hnpos : 0 < n
hn : (↑x : (𝓞 K)ˣ) ^ n = 1
⊢ (↑u : 𝓞 K) ^ (↑(Subtype.mk (p := fun (n : ℕ) ↦ 0 < n) n hnpos) : ℕ) = 1
have h : ∀ (hn : x ^ n = 1), Eq.{u_1 + 1} (α := 𝓞 K) (↑(↑1 : (𝓞 K)ˣ) : 𝓞 K) 1
Tactic: rw_mod_cast [hxu, hn]
Cleared hypotheses: hn
-- Lean 4 proof step from Mathlib.NumberTheory.NumberField.Cyclotomic.Three (hypothesis)
Goal:
K : Type u_1
inst : Field K
ζ : K
hζ : IsPrimitiveRoot ζ 3
u : (𝓞 K)ˣ
inst_1 : NumberField K
inst_2 : IsCyclotomicExtension {3} ℚ K
hrank : rank K = 0
x : ↥(torsion K)
e : Fin (rank K) → ℤ
hxu : u = (↑x : (𝓞 K)ˣ)
n : ℕ
hnpos : 0 < n
hn : (↑x : (𝓞 K)ˣ) ^ n = 1
⊢ u ∈
    [1, -1, IsUnit.unit (a := hζ.toInteger) ⋯, -IsUnit.unit (a := hζ.toInteger) ⋯,
      HPow.hPow (β := ℕ) (IsUnit.unit (a := hζ.toInteger) ⋯) 2,
      -HPow.hPow (β := ℕ) (IsUnit.unit (a := hζ.toInteger) ⋯) 2]
have hn₁ : HPow.hPow (α := K) ((algebraMap (𝓞 K) K : (a : 𝓞 K) → K) (↑u : 𝓞 K) : K)
    (↑(Subtype.mk (p := fun (n : ℕ) ↦ 0 < n) n hnpos) : ℕ) =
  1
Tactic: replace hn : (↑u : K) ^ ((⟨n, hnpos⟩ : ℕ+) : ℕ) = 1 := by
    rw [← map_pow]
    convert! map_one (algebraMap (𝓞 K) K)
    rw_mod_cast [hxu, hn]
    simp
Cleared hypotheses: hn
-- Lean 4 proof step from Mathlib.NumberTheory.NumberField.Cyclotomic.Three (goal)
Goal:
K : Type u_1
inst : Field K
ζ : K
hζ : IsPrimitiveRoot ζ 3
u : (𝓞 K)ˣ
inst_1 : NumberField K
inst_2 : IsCyclotomicExtension {3} ℚ K
hrank : rank K = 0
x : ↥(torsion K)
e : Fin (rank K) → ℤ
hxu : u = (↑x : (𝓞 K)ˣ)
n : ℕ
hnpos : 0 < n
hn :
  HPow.hPow (α := K) ((algebraMap (𝓞 K) K : (a : 𝓞 K) → K) (↑u : 𝓞 K) : K)
      (↑(Subtype.mk (p := fun (n : ℕ) ↦ 0 < n) n hnpos) : ℕ) =
    1
⊢ u ∈
    [1, -1, IsUnit.unit (a := hζ.toInteger) ⋯, -IsUnit.unit (a := hζ.toInteger) ⋯,
      HPow.hPow (β := ℕ) (IsUnit.unit (a := hζ.toInteger) ⋯) 2,
      -HPow.hPow (β := ℕ) (IsUnit.unit (a := hζ.toInteger) ⋯) 2]
have obtain : ∀ (r : ℕ) (hr3 : r < 3)
  (hru :
    (algebraMap (𝓞 K) K : (a : 𝓞 K) → K) (↑u : 𝓞 K) = ζ ^ r ∨ (algebraMap (𝓞 K) K : (a : 𝓞 K) → K) (↑u : 𝓞 K) = -ζ ^ r),
  u ∈
    [1, -1, IsUnit.unit (a := hζ.toInteger) sorry, -IsUnit.unit (a := hζ.toInteger) sorry,
      HPow.hPow (β := ℕ) (IsUnit.unit (a := hζ.toInteger) sorry) 2,
      -HPow.hPow (β := ℕ) (IsUnit.unit (a := hζ.toInteger) sorry) 2]
Tactic: obtain ⟨r, hr3, hru⟩ := hζ.exists_pow_or_neg_mul_pow_of_isOfFinOrder (by decide)
    (isOfFinOrder_iff_pow_eq_one.2 ⟨n, hnpos, hn⟩)
-- Lean 4 proof step from Mathlib.NumberTheory.NumberField.Cyclotomic.Three (hypothesis)
Goal:
K : Type u_1
inst : Field K
ζ : K
hζ : IsPrimitiveRoot ζ 3
u : (𝓞 K)ˣ
inst_1 : NumberField K
inst_2 : IsCyclotomicExtension {3} ℚ K
hrank : rank K = 0
x : ↥(torsion K)
e : Fin (rank K) → ℤ
hxu : u = (↑x : (𝓞 K)ˣ)
n : ℕ
hnpos : 0 < n
hn :
  HPow.hPow (α := K) ((algebraMap (𝓞 K) K : (a : 𝓞 K) → K) (↑u : 𝓞 K) : K)
      (↑(Subtype.mk (p := fun (n : ℕ) ↦ 0 < n) n hnpos) : ℕ) =
    1
r : ℕ
hr3 : r < 3
hru : (algebraMap (𝓞 K) K : (a : 𝓞 K) → K) (↑u : 𝓞 K) = ζ ^ r ∨ (algebraMap (𝓞 K) K : (a : 𝓞 K) → K) (↑u : 𝓞 K) = -ζ ^ r
⊢ u ∈
    [1, -1, IsUnit.unit (a := hζ.toInteger) ⋯, -IsUnit.unit (a := hζ.toInteger) ⋯,
      HPow.hPow (β := ℕ) (IsUnit.unit (a := hζ.toInteger) ⋯) 2,
      -HPow.hPow (β := ℕ) (IsUnit.unit (a := hζ.toInteger) ⋯) 2]
have hr : Membership.mem (γ := Finset ℕ) (Finset.Ico 0 3) r
Tactic: replace hr : r ∈ Finset.Ico 0 3 := Finset.mem_Ico.2 ⟨by simp, hr3⟩
-- Lean 4 proof step from Mathlib.NumberTheory.NumberField.Cyclotomic.Three (goal)
Goal:
K : Type u_1
inst : Field K
ζ : K
hζ : IsPrimitiveRoot ζ 3
u : (𝓞 K)ˣ
inst_1 : NumberField K
inst_2 : IsCyclotomicExtension {3} ℚ K
hrank : rank K = 0
x : ↥(torsion K)
e : Fin (rank K) → ℤ
hxu : u = (↑x : (𝓞 K)ˣ)
n : ℕ
hnpos : 0 < n
hn :
  HPow.hPow (α := K) ((algebraMap (𝓞 K) K : (a : 𝓞 K) → K) (↑u : 𝓞 K) : K)
      (↑(Subtype.mk (p := fun (n : ℕ) ↦ 0 < n) n hnpos) : ℕ) =
    1
r : ℕ
hr3 : r < 3
hru : (algebraMap (𝓞 K) K : (a : 𝓞 K) → K) (↑u : 𝓞 K) = ζ ^ r ∨ (algebraMap (𝓞 K) K : (a : 𝓞 K) → K) (↑u : 𝓞 K) = -ζ ^ r
hr : Membership.mem (γ := Finset ℕ) (Finset.Ico 0 3) r
⊢ u = η ^ r ∨ u = -η ^ r
have inl : ∀ (h : (algebraMap (𝓞 K) K : (a : 𝓞 K) → K) (↑u : 𝓞 K) = ζ ^ r), u = η ^ r ∨ u = -η ^ r
Tactic: rcases hru with h | h
Cleared hypotheses: hru
-- Lean 4 proof step from Mathlib.NumberTheory.NumberField.Cyclotomic.Three (goal)
Goal:
K : Type u_1
inst : Field K
ζ : K
hζ : IsPrimitiveRoot ζ 3
u : (𝓞 K)ˣ
inst_1 : NumberField K
inst_2 : IsCyclotomicExtension {3} ℚ K
hrank : rank K = 0
x : ↥(torsion K)
e : Fin (rank K) → ℤ
hxu : u = (↑x : (𝓞 K)ˣ)
n : ℕ
hnpos : 0 < n
hn :
  HPow.hPow (α := K) ((algebraMap (𝓞 K) K : (a : 𝓞 K) → K) (↑u : 𝓞 K) : K)
      (↑(Subtype.mk (p := fun (n : ℕ) ↦ 0 < n) n hnpos) : ℕ) =
    1
r : ℕ
hr3 : r < 3
hru : (algebraMap (𝓞 K) K : (a : 𝓞 K) → K) (↑u : 𝓞 K) = ζ ^ r ∨ (algebraMap (𝓞 K) K : (a : 𝓞 K) → K) (↑u : 𝓞 K) = -ζ ^ r
hr : Membership.mem (γ := Finset ℕ) (Finset.Ico 0 3) r
inl : ∀ (h : (algebraMap (𝓞 K) K : (a : 𝓞 K) → K) (↑u : 𝓞 K) = ζ ^ r), u = η ^ r ∨ u = -η ^ r
⊢ u = η ^ r ∨ u = -η ^ r
have inr : ∀ (h : (algebraMap (𝓞 K) K : (a : 𝓞 K) → K) (↑u : 𝓞 K) = -ζ ^ r), u = η ^ r ∨ u = -η ^ r
Tactic: rcases hru with h | h
Cleared hypotheses: hru
-- Lean 4 proof step from Mathlib.NumberTheory.NumberField.Cyclotomic.Three (goal)
Goal:
K : Type u_1
inst : Field K
ζ : K
hζ : IsPrimitiveRoot ζ 3
u : (𝓞 K)ˣ
inst_1 : NumberField K
inst_2 : IsCyclotomicExtension {3} ℚ K
hrank : rank K = 0
x : ↥(torsion K)
e : Fin (rank K) → ℤ
hxu : u = (↑x : (𝓞 K)ˣ)
n : ℕ
hnpos : 0 < n
hn :
  HPow.hPow (α := K) ((algebraMap (𝓞 K) K : (a : 𝓞 K) → K) (↑u : 𝓞 K) : K)
      (↑(Subtype.mk (p := fun (n : ℕ) ↦ 0 < n) n hnpos) : ℕ) =
    1
r : ℕ
hr3 : r < 3
hr : Membership.mem (γ := Finset ℕ) (Finset.Ico 0 3) r
h : (algebraMap (𝓞 K) K : (a : 𝓞 K) → K) (↑u : 𝓞 K) = ζ ^ r
⊢ u = η ^ r ∨ u = -η ^ r
have inl : u = η ^ r
Tactic: left
-- Lean 4 proof step from Mathlib.NumberTheory.NumberField.Cyclotomic.Three (goal)
Goal:
K : Type u_1
inst : Field K
ζ : K
hζ : IsPrimitiveRoot ζ 3
u : (𝓞 K)ˣ
inst_1 : NumberField K
inst_2 : IsCyclotomicExtension {3} ℚ K
hrank : rank K = 0
x : ↥(torsion K)
e : Fin (rank K) → ℤ
hxu : u = (↑x : (𝓞 K)ˣ)
n : ℕ
hnpos : 0 < n
hn :
  HPow.hPow (α := K) ((algebraMap (𝓞 K) K : (a : 𝓞 K) → K) (↑u : 𝓞 K) : K)
      (↑(Subtype.mk (p := fun (n : ℕ) ↦ 0 < n) n hnpos) : ℕ) =
    1
r : ℕ
hr3 : r < 3
hr : Membership.mem (γ := Finset ℕ) (Finset.Ico 0 3) r
h : (algebraMap (𝓞 K) K : (a : 𝓞 K) → K) (↑u : 𝓞 K) = ζ ^ r
⊢ u = η ^ r
have inl : (↑(↑u : 𝓞 K) : K) = (↑(↑(η ^ r) : 𝓞 K) : K)
Tactic: ext
-- Lean 4 proof step from Mathlib.NumberTheory.NumberField.Cyclotomic.Three (goal)
Goal:
K : Type u_1
inst : Field K
ζ : K
hζ : IsPrimitiveRoot ζ 3
u : (𝓞 K)ˣ
inst_1 : NumberField K
inst_2 : IsCyclotomicExtension {3} ℚ K
hrank : rank K = 0
x : ↥(torsion K)
e : Fin (rank K) → ℤ
hxu : u = (↑x : (𝓞 K)ˣ)
n : ℕ
hnpos : 0 < n
hn :
  HPow.hPow (α := K) ((algebraMap (𝓞 K) K : (a : 𝓞 K) → K) (↑u : 𝓞 K) : K)
      (↑(Subtype.mk (p := fun (n : ℕ) ↦ 0 < n) n hnpos) : ℕ) =
    1
r : ℕ
hr3 : r < 3
hr : Membership.mem (γ := Finset ℕ) (Finset.Ico 0 3) r
h : (algebraMap (𝓞 K) K : (a : 𝓞 K) → K) (↑u : 𝓞 K) = -ζ ^ r
⊢ u = η ^ r ∨ u = -η ^ r
have inr : u = -η ^ r
Tactic: right
-- Lean 4 proof step from Mathlib.NumberTheory.NumberField.Cyclotomic.Three (goal)
Goal:
K : Type u_1
inst : Field K
ζ : K
hζ : IsPrimitiveRoot ζ 3
u : (𝓞 K)ˣ
inst_1 : NumberField K
inst_2 : IsCyclotomicExtension {3} ℚ K
hrank : rank K = 0
x : ↥(torsion K)
e : Fin (rank K) → ℤ
hxu : u = (↑x : (𝓞 K)ˣ)
n : ℕ
hnpos : 0 < n
hn :
  HPow.hPow (α := K) ((algebraMap (𝓞 K) K : (a : 𝓞 K) → K) (↑u : 𝓞 K) : K)
      (↑(Subtype.mk (p := fun (n : ℕ) ↦ 0 < n) n hnpos) : ℕ) =
    1
r : ℕ
hr3 : r < 3
hr : Membership.mem (γ := Finset ℕ) (Finset.Ico 0 3) r
h : (algebraMap (𝓞 K) K : (a : 𝓞 K) → K) (↑u : 𝓞 K) = -ζ ^ r
⊢ u = -η ^ r
have inr : (↑(↑u : 𝓞 K) : K) = (↑(↑(-η ^ r) : 𝓞 K) : K)
Tactic: ext
-- Lean 4 proof step from Mathlib.NumberTheory.NumberField.Cyclotomic.Three (hypothesis)
Goal:
K : Type u_1
inst : Field K
ζ : K
hζ : IsPrimitiveRoot ζ 3
u : (𝓞 K)ˣ
inst_1 : NumberField K
inst_2 : IsCyclotomicExtension {3} ℚ K
hrank : rank K = 0
x : ↥(torsion K)
e : Fin (rank K) → ℤ
hxu : u = (↑x : (𝓞 K)ˣ)
n : ℕ
hnpos : 0 < n
hn :
  HPow.hPow (α := K) ((algebraMap (𝓞 K) K : (a : 𝓞 K) → K) (↑u : 𝓞 K) : K)
      (↑(Subtype.mk (p := fun (n : ℕ) ↦ 0 < n) n hnpos) : ℕ) =
    1
r : ℕ
hr3 : r < 3
hru : (algebraMap (𝓞 K) K : (a : 𝓞 K) → K) (↑u : 𝓞 K) = ζ ^ r ∨ (algebraMap (𝓞 K) K : (a : 𝓞 K) → K) (↑u : 𝓞 K) = -ζ ^ r
hr : Membership.mem (γ := Finset ℕ) (Finset.Ico 0 3) r
⊢ u ∈
    [1, -1, IsUnit.unit (a := hζ.toInteger) ⋯, -IsUnit.unit (a := hζ.toInteger) ⋯,
      HPow.hPow (β := ℕ) (IsUnit.unit (a := hζ.toInteger) ⋯) 2,
      -HPow.hPow (β := ℕ) (IsUnit.unit (a := hζ.toInteger) ⋯) 2]
have hru₁ : u = η ^ r ∨ u = -η ^ r
Tactic: replace hru : ↑u = η ^ r ∨ ↑u = -η ^ r := by
    rcases hru with h | h
    · left; ext; exact h
    · right; ext; exact h
Cleared hypotheses: hru
-- Lean 4 proof step from Mathlib.NumberTheory.NumberField.Cyclotomic.Three (goal)
Goal:
K : Type u_1
inst : Field K
ζ : K
hζ : IsPrimitiveRoot ζ 3
u : (𝓞 K)ˣ
inst_1 : NumberField K
inst_2 : IsCyclotomicExtension {3} ℚ K
hrank : rank K = 0
x : ↥(torsion K)
e : Fin (rank K) → ℤ
hxu : u = (↑x : (𝓞 K)ˣ)
n : ℕ
hnpos : 0 < n
hn :
  HPow.hPow (α := K) ((algebraMap (𝓞 K) K : (a : 𝓞 K) → K) (↑u : 𝓞 K) : K)
      (↑(Subtype.mk (p := fun (n : ℕ) ↦ 0 < n) n hnpos) : ℕ) =
    1
r : ℕ
hr3 : r < 3
hr : Membership.mem (γ := Finset ℕ) (Finset.Ico 0 3) r
hru : u = η ^ r ∨ u = -η ^ r
⊢ u ∈
    [1, -1, IsUnit.unit (a := hζ.toInteger) ⋯, -IsUnit.unit (a := hζ.toInteger) ⋯,
      HPow.hPow (β := ℕ) (IsUnit.unit (a := hζ.toInteger) ⋯) 2,
      -HPow.hPow (β := ℕ) (IsUnit.unit (a := hζ.toInteger) ⋯) 2]
have 0 : ∀ (hr3 : LT.lt (α := ℕ) 0 3) (hru : u = HPow.hPow (β := ℕ) η 0 ∨ u = -HPow.hPow (β := ℕ) η 0),
  u ∈
    [1, -1, IsUnit.unit (a := hζ.toInteger) sorry, -IsUnit.unit (a := hζ.toInteger) sorry,
      HPow.hPow (β := ℕ) (IsUnit.unit (a := hζ.toInteger) sorry) 2,
      -HPow.hPow (β := ℕ) (IsUnit.unit (a := hζ.toInteger) sorry) 2]
Tactic: fin_cases hr
Cleared hypotheses: r, hr3, hr, hru
-- Lean 4 proof step from Mathlib.NumberTheory.NumberField.Cyclotomic.Three (goal)
Goal:
K : Type u_1
inst : Field K
ζ : K
hζ : IsPrimitiveRoot ζ 3
u : (𝓞 K)ˣ
inst_1 : NumberField K
inst_2 : IsCyclotomicExtension {3} ℚ K
hrank : rank K = 0
x : ↥(torsion K)
e : Fin (rank K) → ℤ
hxu : u = (↑x : (𝓞 K)ˣ)
n : ℕ
hnpos : 0 < n
hn :
  HPow.hPow (α := K) ((algebraMap (𝓞 K) K : (a : 𝓞 K) → K) (↑u : 𝓞 K) : K)
      (↑(Subtype.mk (p := fun (n : ℕ) ↦ 0 < n) n hnpos) : ℕ) =
    1
r : ℕ
hr3 : r < 3
hr : Membership.mem (γ := Finset ℕ) (Finset.Ico 0 3) r
hru : u = η ^ r ∨ u = -η ^ r
0 :
  ∀ (hr3 : LT.lt (α := ℕ) 0 3) (hru : u = HPow.hPow (β := ℕ) η 0 ∨ u = -HPow.hPow (β := ℕ) η 0),
    u ∈
      [1, -1, IsUnit.unit (a := hζ.toInteger) ⋯, -IsUnit.unit (a := hζ.toInteger) ⋯,
        HPow.hPow (β := ℕ) (IsUnit.unit (a := hζ.toInteger) ⋯) 2,
        -HPow.hPow (β := ℕ) (IsUnit.unit (a := hζ.toInteger) ⋯) 2]
⊢ u ∈
    [1, -1, IsUnit.unit (a := hζ.toInteger) ⋯, -IsUnit.unit (a := hζ.toInteger) ⋯,
      HPow.hPow (β := ℕ) (IsUnit.unit (a := hζ.toInteger) ⋯) 2,
      -HPow.hPow (β := ℕ) (IsUnit.unit (a := hζ.toInteger) ⋯) 2]
have 1 : ∀ (hr3 : HAdd.hAdd (α := ℕ) 0 1 < 3) (hru : u = η ^ HAdd.hAdd (α := ℕ) 0 1 ∨ u = -η ^ HAdd.hAdd (α := ℕ) 0 1),
  u ∈
    [1, -1, IsUnit.unit (a := hζ.toInteger) sorry, -IsUnit.unit (a := hζ.toInteger) sorry,
      HPow.hPow (β := ℕ) (IsUnit.unit (a := hζ.toInteger) sorry) 2,
      -HPow.hPow (β := ℕ) (IsUnit.unit (a := hζ.toInteger) sorry) 2]
Tactic: fin_cases hr
Cleared hypotheses: r, hr3, hr, hru
-- Lean 4 proof step from Mathlib.NumberTheory.NumberField.Cyclotomic.Three (goal)
Goal:
K : Type u_1
inst : Field K
ζ : K
hζ : IsPrimitiveRoot ζ 3
u : (𝓞 K)ˣ
inst_1 : NumberField K
inst_2 : IsCyclotomicExtension {3} ℚ K
hrank : rank K = 0
x : ↥(torsion K)
e : Fin (rank K) → ℤ
hxu : u = (↑x : (𝓞 K)ˣ)
n : ℕ
hnpos : 0 < n
hn :
  HPow.hPow (α := K) ((algebraMap (𝓞 K) K : (a : 𝓞 K) → K) (↑u : 𝓞 K) : K)
      (↑(Subtype.mk (p := fun (n : ℕ) ↦ 0 < n) n hnpos) : ℕ) =
    1
r : ℕ
hr3 : r < 3
hr : Membership.mem (γ := Finset ℕ) (Finset.Ico 0 3) r
hru : u = η ^ r ∨ u = -η ^ r
0 :
  ∀ (hr3 : LT.lt (α := ℕ) 0 3) (hru : u = HPow.hPow (β := ℕ) η 0 ∨ u = -HPow.hPow (β := ℕ) η 0),
    u ∈
      [1, -1, IsUnit.unit (a := hζ.toInteger) ⋯, -IsUnit.unit (a := hζ.toInteger) ⋯,
        HPow.hPow (β := ℕ) (IsUnit.unit (a := hζ.toInteger) ⋯) 2,
        -HPow.hPow (β := ℕ) (IsUnit.unit (a := hζ.toInteger) ⋯) 2]
1 :
  ∀ (hr3 : HAdd.hAdd (α := ℕ) 0 1 < 3) (hru : u = η ^ HAdd.hAdd (α := ℕ) 0 1 ∨ u = -η ^ HAdd.hAdd (α := ℕ) 0 1),
    u ∈
      [1, -1, IsUnit.unit (a := hζ.toInteger) ⋯, -IsUnit.unit (a := hζ.toInteger) ⋯,
        HPow.hPow (β := ℕ) (IsUnit.unit (a := hζ.toInteger) ⋯) 2,
        -HPow.hPow (β := ℕ) (IsUnit.unit (a := hζ.toInteger) ⋯) 2]
⊢ u ∈
    [1, -1, IsUnit.unit (a := hζ.toInteger) ⋯, -IsUnit.unit (a := hζ.toInteger) ⋯,
      HPow.hPow (β := ℕ) (IsUnit.unit (a := hζ.toInteger) ⋯) 2,
      -HPow.hPow (β := ℕ) (IsUnit.unit (a := hζ.toInteger) ⋯) 2]
have 2 : ∀ (hr3 : HAdd.hAdd (α := ℕ) 0 1 + 1 < 3)
  (hru : u = η ^ (HAdd.hAdd (α := ℕ) 0 1 + 1) ∨ u = -η ^ (HAdd.hAdd (α := ℕ) 0 1 + 1)),
  u ∈
    [1, -1, IsUnit.unit (a := hζ.toInteger) sorry, -IsUnit.unit (a := hζ.toInteger) sorry,
      HPow.hPow (β := ℕ) (IsUnit.unit (a := hζ.toInteger) sorry) 2,
      -HPow.hPow (β := ℕ) (IsUnit.unit (a := hζ.toInteger) sorry) 2]
Tactic: fin_cases hr
Cleared hypotheses: r, hr3, hr, hru
-- Lean 4 proof step from Mathlib.NumberTheory.NumberField.Cyclotomic.Three (goal)
Goal:
K : Type u_1
inst : Field K
ζ : K
hζ : IsPrimitiveRoot ζ 3
u : (𝓞 K)ˣ
inst_1 : NumberField K
inst_2 : IsCyclotomicExtension {3} ℚ K
hrank : rank K = 0
x : ↥(torsion K)
e : Fin (rank K) → ℤ
hxu : u = (↑x : (𝓞 K)ˣ)
n : ℕ
hnpos : 0 < n
hn :
  HPow.hPow (α := K) ((algebraMap (𝓞 K) K : (a : 𝓞 K) → K) (↑u : 𝓞 K) : K)
      (↑(Subtype.mk (p := fun (n : ℕ) ↦ 0 < n) n hnpos) : ℕ) =
    1
hr3 : LT.lt (α := ℕ) 0 3
hru : u = HPow.hPow (β := ℕ) η 0 ∨ u = -HPow.hPow (β := ℕ) η 0
⊢ u ∈
    [1, -1, IsUnit.unit (a := hζ.toInteger) ⋯, -IsUnit.unit (a := hζ.toInteger) ⋯,
      HPow.hPow (β := ℕ) (IsUnit.unit (a := hζ.toInteger) ⋯) 2,
      -HPow.hPow (β := ℕ) (IsUnit.unit (a := hζ.toInteger) ⋯) 2]
have 0 : ∀ (h : u = HPow.hPow (β := ℕ) η 0),
  u ∈
    [1, -1, IsUnit.unit (a := hζ.toInteger) sorry, -IsUnit.unit (a := hζ.toInteger) sorry,
      HPow.hPow (β := ℕ) (IsUnit.unit (a := hζ.toInteger) sorry) 2,
      -HPow.hPow (β := ℕ) (IsUnit.unit (a := hζ.toInteger) sorry) 2]
Tactic: rcases hru with h | h
Cleared hypotheses: hru
-- Lean 4 proof step from Mathlib.NumberTheory.NumberField.Cyclotomic.Three (goal)
Goal:
K : Type u_1
inst : Field K
ζ : K
hζ : IsPrimitiveRoot ζ 3
u : (𝓞 K)ˣ
inst_1 : NumberField K
inst_2 : IsCyclotomicExtension {3} ℚ K
hrank : rank K = 0
x : ↥(torsion K)
e : Fin (rank K) → ℤ
hxu : u = (↑x : (𝓞 K)ˣ)
n : ℕ
hnpos : 0 < n
hn :
  HPow.hPow (α := K) ((algebraMap (𝓞 K) K : (a : 𝓞 K) → K) (↑u : 𝓞 K) : K)
      (↑(Subtype.mk (p := fun (n : ℕ) ↦ 0 < n) n hnpos) : ℕ) =
    1
hr3 : LT.lt (α := ℕ) 0 3
hru : u = HPow.hPow (β := ℕ) η 0 ∨ u = -HPow.hPow (β := ℕ) η 0
0 :
  ∀ (h : u = HPow.hPow (β := ℕ) η 0),
    u ∈
      [1, -1, IsUnit.unit (a := hζ.toInteger) ⋯, -IsUnit.unit (a := hζ.toInteger) ⋯,
        HPow.hPow (β := ℕ) (IsUnit.unit (a := hζ.toInteger) ⋯) 2,
        -HPow.hPow (β := ℕ) (IsUnit.unit (a := hζ.toInteger) ⋯) 2]
⊢ u ∈
    [1, -1, IsUnit.unit (a := hζ.toInteger) ⋯, -IsUnit.unit (a := hζ.toInteger) ⋯,
      HPow.hPow (β := ℕ) (IsUnit.unit (a := hζ.toInteger) ⋯) 2,
      -HPow.hPow (β := ℕ) (IsUnit.unit (a := hζ.toInteger) ⋯) 2]
have 0₁ : ∀ (h : u = -HPow.hPow (β := ℕ) η 0),
  u ∈
    [1, -1, IsUnit.unit (a := hζ.toInteger) sorry, -IsUnit.unit (a := hζ.toInteger) sorry,
      HPow.hPow (β := ℕ) (IsUnit.unit (a := hζ.toInteger) sorry) 2,
      -HPow.hPow (β := ℕ) (IsUnit.unit (a := hζ.toInteger) sorry) 2]
Tactic: rcases hru with h | h
Cleared hypotheses: hru
-- Lean 4 proof step from Mathlib.NumberTheory.NumberField.Cyclotomic.Three (goal)
Goal:
K : Type u_1
inst : Field K
ζ : K
hζ : IsPrimitiveRoot ζ 3
u : (𝓞 K)ˣ
inst_1 : NumberField K
inst_2 : IsCyclotomicExtension {3} ℚ K
hrank : rank K = 0
x : ↥(torsion K)
e : Fin (rank K) → ℤ
hxu : u = (↑x : (𝓞 K)ˣ)
n : ℕ
hnpos : 0 < n
hn :
  HPow.hPow (α := K) ((algebraMap (𝓞 K) K : (a : 𝓞 K) → K) (↑u : 𝓞 K) : K)
      (↑(Subtype.mk (p := fun (n : ℕ) ↦ 0 < n) n hnpos) : ℕ) =
    1
hr3 : HAdd.hAdd (α := ℕ) 0 1 < 3
hru : u = η ^ HAdd.hAdd (α := ℕ) 0 1 ∨ u = -η ^ HAdd.hAdd (α := ℕ) 0 1
⊢ u ∈
    [1, -1, IsUnit.unit (a := hζ.toInteger) ⋯, -IsUnit.unit (a := hζ.toInteger) ⋯,
      HPow.hPow (β := ℕ) (IsUnit.unit (a := hζ.toInteger) ⋯) 2,
      -HPow.hPow (β := ℕ) (IsUnit.unit (a := hζ.toInteger) ⋯) 2]
have 1 : ∀ (h : u = η ^ HAdd.hAdd (α := ℕ) 0 1),
  u ∈
    [1, -1, IsUnit.unit (a := hζ.toInteger) sorry, -IsUnit.unit (a := hζ.toInteger) sorry,
      HPow.hPow (β := ℕ) (IsUnit.unit (a := hζ.toInteger) sorry) 2,
      -HPow.hPow (β := ℕ) (IsUnit.unit (a := hζ.toInteger) sorry) 2]
Tactic: rcases hru with h | h
Cleared hypotheses: hru
-- Lean 4 proof step from Mathlib.NumberTheory.NumberField.Cyclotomic.Three (goal)
Goal:
K : Type u_1
inst : Field K
ζ : K
hζ : IsPrimitiveRoot ζ 3
u : (𝓞 K)ˣ
inst_1 : NumberField K
inst_2 : IsCyclotomicExtension {3} ℚ K
hrank : rank K = 0
x : ↥(torsion K)
e : Fin (rank K) → ℤ
hxu : u = (↑x : (𝓞 K)ˣ)
n : ℕ
hnpos : 0 < n
hn :
  HPow.hPow (α := K) ((algebraMap (𝓞 K) K : (a : 𝓞 K) → K) (↑u : 𝓞 K) : K)
      (↑(Subtype.mk (p := fun (n : ℕ) ↦ 0 < n) n hnpos) : ℕ) =
    1
hr3 : HAdd.hAdd (α := ℕ) 0 1 < 3
hru : u = η ^ HAdd.hAdd (α := ℕ) 0 1 ∨ u = -η ^ HAdd.hAdd (α := ℕ) 0 1
1 :
  ∀ (h : u = η ^ HAdd.hAdd (α := ℕ) 0 1),
    u ∈
      [1, -1, IsUnit.unit (a := hζ.toInteger) ⋯, -IsUnit.unit (a := hζ.toInteger) ⋯,
        HPow.hPow (β := ℕ) (IsUnit.unit (a := hζ.toInteger) ⋯) 2,
        -HPow.hPow (β := ℕ) (IsUnit.unit (a := hζ.toInteger) ⋯) 2]
⊢ u ∈
    [1, -1, IsUnit.unit (a := hζ.toInteger) ⋯, -IsUnit.unit (a := hζ.toInteger) ⋯,
      HPow.hPow (β := ℕ) (IsUnit.unit (a := hζ.toInteger) ⋯) 2,
      -HPow.hPow (β := ℕ) (IsUnit.unit (a := hζ.toInteger) ⋯) 2]
have 1₁ : ∀ (h : u = -η ^ HAdd.hAdd (α := ℕ) 0 1),
  u ∈
    [1, -1, IsUnit.unit (a := hζ.toInteger) sorry, -IsUnit.unit (a := hζ.toInteger) sorry,
      HPow.hPow (β := ℕ) (IsUnit.unit (a := hζ.toInteger) sorry) 2,
      -HPow.hPow (β := ℕ) (IsUnit.unit (a := hζ.toInteger) sorry) 2]
Tactic: rcases hru with h | h
Cleared hypotheses: hru
-- Lean 4 proof step from Mathlib.NumberTheory.NumberField.Cyclotomic.Three (goal)
Goal:
K : Type u_1
inst : Field K
ζ : K
hζ : IsPrimitiveRoot ζ 3
u : (𝓞 K)ˣ
inst_1 : NumberField K
inst_2 : IsCyclotomicExtension {3} ℚ K
hrank : rank K = 0
x : ↥(torsion K)
e : Fin (rank K) → ℤ
hxu : u = (↑x : (𝓞 K)ˣ)
n : ℕ
hnpos : 0 < n
hn :
  HPow.hPow (α := K) ((algebraMap (𝓞 K) K : (a : 𝓞 K) → K) (↑u : 𝓞 K) : K)
      (↑(Subtype.mk (p := fun (n : ℕ) ↦ 0 < n) n hnpos) : ℕ) =
    1
hr3 : HAdd.hAdd (α := ℕ) 0 1 + 1 < 3
hru : u = η ^ (HAdd.hAdd (α := ℕ) 0 1 + 1) ∨ u = -η ^ (HAdd.hAdd (α := ℕ) 0 1 + 1)
⊢ u ∈
    [1, -1, IsUnit.unit (a := hζ.toInteger) ⋯, -IsUnit.unit (a := hζ.toInteger) ⋯,
      HPow.hPow (β := ℕ) (IsUnit.unit (a := hζ.toInteger) ⋯) 2,
      -HPow.hPow (β := ℕ) (IsUnit.unit (a := hζ.toInteger) ⋯) 2]
have 2 : ∀ (h : u = η ^ (HAdd.hAdd (α := ℕ) 0 1 + 1)),
  u ∈
    [1, -1, IsUnit.unit (a := hζ.toInteger) sorry, -IsUnit.unit (a := hζ.toInteger) sorry,
      HPow.hPow (β := ℕ) (IsUnit.unit (a := hζ.toInteger) sorry) 2,
      -HPow.hPow (β := ℕ) (IsUnit.unit (a := hζ.toInteger) sorry) 2]
Tactic: rcases hru with h | h
Cleared hypotheses: hru
-- Lean 4 proof step from Mathlib.NumberTheory.NumberField.Cyclotomic.Three (goal)
Goal:
K : Type u_1
inst : Field K
ζ : K
hζ : IsPrimitiveRoot ζ 3
u : (𝓞 K)ˣ
inst_1 : NumberField K
inst_2 : IsCyclotomicExtension {3} ℚ K
hrank : rank K = 0
x : ↥(torsion K)
e : Fin (rank K) → ℤ
hxu : u = (↑x : (𝓞 K)ˣ)
n : ℕ
hnpos : 0 < n
hn :
  HPow.hPow (α := K) ((algebraMap (𝓞 K) K : (a : 𝓞 K) → K) (↑u : 𝓞 K) : K)
      (↑(Subtype.mk (p := fun (n : ℕ) ↦ 0 < n) n hnpos) : ℕ) =
    1
hr3 : HAdd.hAdd (α := ℕ) 0 1 + 1 < 3
hru : u = η ^ (HAdd.hAdd (α := ℕ) 0 1 + 1) ∨ u = -η ^ (HAdd.hAdd (α := ℕ) 0 1 + 1)
2 :
  ∀ (h : u = η ^ (HAdd.hAdd (α := ℕ) 0 1 + 1)),
    u ∈
      [1, -1, IsUnit.unit (a := hζ.toInteger) ⋯, -IsUnit.unit (a := hζ.toInteger) ⋯,
        HPow.hPow (β := ℕ) (IsUnit.unit (a := hζ.toInteger) ⋯) 2,
        -HPow.hPow (β := ℕ) (IsUnit.unit (a := hζ.toInteger) ⋯) 2]
⊢ u ∈
    [1, -1, IsUnit.unit (a := hζ.toInteger) ⋯, -IsUnit.unit (a := hζ.toInteger) ⋯,
      HPow.hPow (β := ℕ) (IsUnit.unit (a := hζ.toInteger) ⋯) 2,
      -HPow.hPow (β := ℕ) (IsUnit.unit (a := hζ.toInteger) ⋯) 2]
have 2₁ : ∀ (h : u = -η ^ (HAdd.hAdd (α := ℕ) 0 1 + 1)),
  u ∈
    [1, -1, IsUnit.unit (a := hζ.toInteger) sorry, -IsUnit.unit (a := hζ.toInteger) sorry,
      HPow.hPow (β := ℕ) (IsUnit.unit (a := hζ.toInteger) sorry) 2,
      -HPow.hPow (β := ℕ) (IsUnit.unit (a := hζ.toInteger) sorry) 2]
Tactic: rcases hru with h | h
Cleared hypotheses: hru