awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- 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
⊢ HPow.hPow (β := ℕ) λ 2 = -3 * (↑η : 𝓞 K)
have h : (↑(HPow.hPow (β := ℕ) λ 2) : K) = (↑(-3 * (↑η : 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
⊢ HPow.hPow (β := ℕ) ((↑hζ.toInteger : K) - 1) 2 =
    HAdd.hAdd (β := K) (HPow.hPow (α := K) (β := ℕ) ((algebraMap (𝓞 K) K : (a : 𝓞 K) → K) (↑η : 𝓞 K) : K) 2)
          ((algebraMap (𝓞 K) K : (a : 𝓞 K) → K) (↑η : 𝓞 K) : K) +
        1 -
      HMul.hMul (α := K) (β := K) (3 : K) ((algebraMap (𝓞 K) K : (a : 𝓞 K) → K) (↑η : 𝓞 K) : K)
have simp : HPow.hPow (β := ℕ) (ζ - 1) 2 = HPow.hPow (β := ℕ) ζ 2 + ζ + 1 - 3 * ζ
Tactic: simp only [RingOfIntegers.map_mk, IsUnit.unit_spec]
-- 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
⊢ (↑(HPow.hPow (β := ℕ) λ 2) : K) = (↑(-3 * (↑η : 𝓞 K)) : K)
have calc : HPow.hPow (β := ℕ) ((↑hζ.toInteger : K) - 1) 2 =
  HAdd.hAdd (β := K) (HPow.hPow (α := K) (β := ℕ) ((algebraMap (𝓞 K) K : (a : 𝓞 K) → K) (↑η : 𝓞 K) : K) 2)
        ((algebraMap (𝓞 K) K : (a : 𝓞 K) → K) (↑η : 𝓞 K) : K) +
      1 -
    HMul.hMul (α := K) (β := K) (3 : K) ((algebraMap (𝓞 K) K : (a : 𝓞 K) → K) (↑η : 𝓞 K) : K)
Tactic: calc (λ ^ 2 : K) = η ^ 2 + η + 1 - 3 * η := by
        simp only [RingOfIntegers.map_mk, IsUnit.unit_spec]; ring
  _ = 0 - 3 * η := by simpa using hζ.isRoot_cyclotomic (by decide)
  _ = -3 * η := by ring
-- 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
calc :
  HPow.hPow (β := ℕ) ((↑hζ.toInteger : K) - 1) 2 =
    HAdd.hAdd (β := K) (HPow.hPow (α := K) (β := ℕ) ((algebraMap (𝓞 K) K : (a : 𝓞 K) → K) (↑η : 𝓞 K) : K) 2)
          ((algebraMap (𝓞 K) K : (a : 𝓞 K) → K) (↑η : 𝓞 K) : K) +
        1 -
      HMul.hMul (α := K) (β := K) (3 : K) ((algebraMap (𝓞 K) K : (a : 𝓞 K) → K) (↑η : 𝓞 K) : K)
⊢ (↑(HPow.hPow (β := ℕ) λ 2) : K) = (↑(-3 * (↑η : 𝓞 K)) : K)
have calc₁ : HAdd.hAdd (β := K) (HPow.hPow (α := K) (β := ℕ) ((algebraMap (𝓞 K) K : (a : 𝓞 K) → K) (↑η : 𝓞 K) : K) 2)
        ((algebraMap (𝓞 K) K : (a : 𝓞 K) → K) (↑η : 𝓞 K) : K) +
      1 -
    HMul.hMul (α := K) (β := K) (3 : K) ((algebraMap (𝓞 K) K : (a : 𝓞 K) → K) (↑η : 𝓞 K) : K) =
  0 - HMul.hMul (α := K) (β := K) (3 : K) ((algebraMap (𝓞 K) K : (a : 𝓞 K) → K) (↑η : 𝓞 K) : K)
Tactic: calc (λ ^ 2 : K) = η ^ 2 + η + 1 - 3 * η := by
        simp only [RingOfIntegers.map_mk, IsUnit.unit_spec]; ring
  _ = 0 - 3 * η := by simpa using hζ.isRoot_cyclotomic (by decide)
  _ = -3 * η := by ring
-- 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
calc :
  HPow.hPow (β := ℕ) ((↑hζ.toInteger : K) - 1) 2 =
    HAdd.hAdd (β := K) (HPow.hPow (α := K) (β := ℕ) ((algebraMap (𝓞 K) K : (a : 𝓞 K) → K) (↑η : 𝓞 K) : K) 2)
          ((algebraMap (𝓞 K) K : (a : 𝓞 K) → K) (↑η : 𝓞 K) : K) +
        1 -
      HMul.hMul (α := K) (β := K) (3 : K) ((algebraMap (𝓞 K) K : (a : 𝓞 K) → K) (↑η : 𝓞 K) : K)
calc₁ :
  HAdd.hAdd (β := K) (HPow.hPow (α := K) (β := ℕ) ((algebraMap (𝓞 K) K : (a : 𝓞 K) → K) (↑η : 𝓞 K) : K) 2)
          ((algebraMap (𝓞 K) K : (a : 𝓞 K) → K) (↑η : 𝓞 K) : K) +
        1 -
      HMul.hMul (α := K) (β := K) (3 : K) ((algebraMap (𝓞 K) K : (a : 𝓞 K) → K) (↑η : 𝓞 K) : K) =
    0 - HMul.hMul (α := K) (β := K) (3 : K) ((algebraMap (𝓞 K) K : (a : 𝓞 K) → K) (↑η : 𝓞 K) : K)
⊢ (↑(HPow.hPow (β := ℕ) λ 2) : K) = (↑(-3 * (↑η : 𝓞 K)) : K)
have calc₂ : 0 - HMul.hMul (α := K) (β := K) (3 : K) ((algebraMap (𝓞 K) K : (a : 𝓞 K) → K) (↑η : 𝓞 K) : K) =
  HMul.hMul (α := K) (β := K) (-3 : K) ((algebraMap (𝓞 K) K : (a : 𝓞 K) → K) (↑η : 𝓞 K) : K)
Tactic: calc (λ ^ 2 : K) = η ^ 2 + η + 1 - 3 * η := by
        simp only [RingOfIntegers.map_mk, IsUnit.unit_spec]; ring
  _ = 0 - 3 * η := by simpa using hζ.isRoot_cyclotomic (by decide)
  _ = -3 * η := by ring
-- 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
⊢ HPow.hPow (β := ℕ) (↑(IsUnit.unit (a := hζ.toInteger) ⋯) : 𝓞 K) 2 = -(↑(IsUnit.unit (a := hζ.toInteger) ⋯) : 𝓞 K) - 1
have rw : HPow.hPow (β := ℕ) (↑(IsUnit.unit (a := hζ.toInteger) sorry) : 𝓞 K) 2 + (↑(IsUnit.unit (a := hζ.toInteger) sorry) : 𝓞 K) + 1 = 0
Tactic: rw [← neg_add', ← add_eq_zero_iff_eq_neg, ← add_assoc]
-- 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
⊢ HPow.hPow (β := ℕ) (↑(IsUnit.unit (a := hζ.toInteger) ⋯) : 𝓞 K) 2 + (↑(IsUnit.unit (a := hζ.toInteger) ⋯) : 𝓞 K) + 1 =
    0
have h : (↑(HPow.hPow (β := ℕ) (↑(IsUnit.unit (a := hζ.toInteger) sorry) : 𝓞 K) 2 + (↑(IsUnit.unit (a := hζ.toInteger) sorry) : 𝓞 K) +
        1) :
    K) =
  (↑0 : 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
hcong : ∃ (n : ℤ), HPow.hPow (β := ℕ) λ 2 ∣ (↑u : 𝓞 K) - (↑n : 𝓞 K)
⊢ ∃ (n : ℤ), 3 ∣ (↑u : 𝓞 K) - (↑n : 𝓞 K)
have obtain : ∀ (n : ℤ) (x : 𝓞 K) (hx : (↑u : 𝓞 K) - (↑n : 𝓞 K) = HPow.hPow (β := ℕ) λ 2 * x), ∃ (n : ℤ), 3 ∣ (↑u : 𝓞 K) - (↑n : 𝓞 K)
Tactic: obtain ⟨n, x, hx⟩ := hcong
Cleared hypotheses: hcong
-- 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
n : ℤ
x : 𝓞 K
hx : (↑u : 𝓞 K) - (↑n : 𝓞 K) = HPow.hPow (β := ℕ) λ 2 * x
⊢ (↑u : 𝓞 K) - (↑n : 𝓞 K) = 3 * (-(↑η : 𝓞 K) * x)
have rw : (↑u : 𝓞 K) - (↑n : 𝓞 K) = 3 * -(↑η : 𝓞 K) * x
Tactic: rw [← mul_assoc, mul_neg, ← neg_mul, ← lambda_sq, hx]
-- 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
n : ℤ
x : 𝓞 K
hx : (↑u : 𝓞 K) - (↑n : 𝓞 K) = HPow.hPow (β := ℕ) λ 2 * x
rw : (↑u : 𝓞 K) - (↑n : 𝓞 K) = 3 * -(↑η : 𝓞 K) * x
⊢ (↑u : 𝓞 K) - (↑n : 𝓞 K) = 3 * (-(↑η : 𝓞 K) * x)
have rw₁ : (↑u : 𝓞 K) - (↑n : 𝓞 K) = -(3 * (↑η : 𝓞 K)) * x
Tactic: rw [← mul_assoc, mul_neg, ← neg_mul, ← lambda_sq, hx]
-- 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
n : ℤ
x : 𝓞 K
hx : (↑u : 𝓞 K) - (↑n : 𝓞 K) = HPow.hPow (β := ℕ) λ 2 * x
rw : (↑u : 𝓞 K) - (↑n : 𝓞 K) = 3 * -(↑η : 𝓞 K) * x
rw₁ : (↑u : 𝓞 K) - (↑n : 𝓞 K) = -(3 * (↑η : 𝓞 K)) * x
⊢ (↑u : 𝓞 K) - (↑n : 𝓞 K) = 3 * (-(↑η : 𝓞 K) * x)
have rw₂ : (↑u : 𝓞 K) - (↑n : 𝓞 K) = -3 * (↑η : 𝓞 K) * x
Tactic: rw [← mul_assoc, mul_neg, ← neg_mul, ← lambda_sq, hx]
-- 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
n : ℤ
x : 𝓞 K
hx : (↑u : 𝓞 K) - (↑n : 𝓞 K) = HPow.hPow (β := ℕ) λ 2 * x
rw : (↑u : 𝓞 K) - (↑n : 𝓞 K) = 3 * -(↑η : 𝓞 K) * x
rw₁ : (↑u : 𝓞 K) - (↑n : 𝓞 K) = -(3 * (↑η : 𝓞 K)) * x
rw₂ : (↑u : 𝓞 K) - (↑n : 𝓞 K) = -3 * (↑η : 𝓞 K) * x
rw₃ : (↑u : 𝓞 K) - (↑n : 𝓞 K) = HPow.hPow (β := ℕ) λ 2 * x
⊢ (↑u : 𝓞 K) - (↑n : 𝓞 K) = 3 * (-(↑η : 𝓞 K) * x)
have rw₄ : HPow.hPow (β := ℕ) λ 2 * x = HPow.hPow (β := ℕ) λ 2 * x
Tactic: rw [← mul_assoc, mul_neg, ← neg_mul, ← lambda_sq, hx]
-- 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
hcong : ∃ (n : ℤ), HPow.hPow (β := ℕ) λ 2 ∣ (↑u : 𝓞 K) - (↑n : 𝓞 K)
⊢ u = 1 ∨ u = -1
have hcong₁ : ∃ (n : ℤ), 3 ∣ (↑u : 𝓞 K) - (↑n : 𝓞 K)
Tactic: replace hcong : ∃ n : ℤ, (3 : 𝓞 K) ∣ (↑u - n : 𝓞 K) := by
    obtain ⟨n, x, hx⟩ := hcong
    exact ⟨n, -η * x, by rw [← mul_assoc, mul_neg, ← neg_mul, ← lambda_sq, hx]⟩
Cleared hypotheses: hcong
-- 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
hcong : ∃ (n : ℤ), 3 ∣ (↑u : 𝓞 K) - (↑n : 𝓞 K)
⊢ u = 1 ∨ u = -1
have this : 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: have := Units.mem hζ u
-- 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
hcong : ∃ (n : ℤ), 3 ∣ (↑u : 𝓞 K) - (↑n : 𝓞 K)
this :
  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 ∨ u = -1
have 0 : ∀ (hcong : ∃ (n : ℤ), 3 ∣ HSub.hSub (α := 𝓞 K) (↑1 : 𝓞 K) (↑n : 𝓞 K)),
  Eq.{u_1 + 1} (α := (𝓞 K)ˣ) 1 1 ∨ Eq.{u_1 + 1} (α := (𝓞 K)ˣ) 1 (-1)
Tactic: fin_cases this
Cleared hypotheses: u, hcong, this
-- 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
hcong : ∃ (n : ℤ), 3 ∣ (↑u : 𝓞 K) - (↑n : 𝓞 K)
this :
  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]
0 :
  ∀ (hcong : ∃ (n : ℤ), 3 ∣ HSub.hSub (α := 𝓞 K) (↑1 : 𝓞 K) (↑n : 𝓞 K)),
    Eq.{u_1 + 1} (α := (𝓞 K)ˣ) 1 1 ∨ Eq.{u_1 + 1} (α := (𝓞 K)ˣ) 1 (-1)
⊢ u = 1 ∨ u = -1
have 1 : ∀ (hcong : ∃ (n : ℤ), 3 ∣ HSub.hSub (α := 𝓞 K) (↑(-1) : 𝓞 K) (↑n : 𝓞 K)),
  Eq.{u_1 + 1} (α := (𝓞 K)ˣ) (-1) 1 ∨ Eq.{u_1 + 1} (α := (𝓞 K)ˣ) (-1) (-1)
Tactic: fin_cases this
Cleared hypotheses: u, hcong, this
-- 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
hcong : ∃ (n : ℤ), 3 ∣ (↑u : 𝓞 K) - (↑n : 𝓞 K)
this :
  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]
0 :
  ∀ (hcong : ∃ (n : ℤ), 3 ∣ HSub.hSub (α := 𝓞 K) (↑1 : 𝓞 K) (↑n : 𝓞 K)),
    Eq.{u_1 + 1} (α := (𝓞 K)ˣ) 1 1 ∨ Eq.{u_1 + 1} (α := (𝓞 K)ˣ) 1 (-1)
1 :
  ∀ (hcong : ∃ (n : ℤ), 3 ∣ HSub.hSub (α := 𝓞 K) (↑(-1) : 𝓞 K) (↑n : 𝓞 K)),
    Eq.{u_1 + 1} (α := (𝓞 K)ˣ) (-1) 1 ∨ Eq.{u_1 + 1} (α := (𝓞 K)ˣ) (-1) (-1)
⊢ u = 1 ∨ u = -1
have 2 : ∀ (hcong : ∃ (n : ℤ), 3 ∣ (↑(IsUnit.unit (a := hζ.toInteger) sorry) : 𝓞 K) - (↑n : 𝓞 K)),
  IsUnit.unit (a := hζ.toInteger) sorry = 1 ∨ IsUnit.unit (a := hζ.toInteger) sorry = -1
Tactic: fin_cases this
Cleared hypotheses: u, hcong, this
-- 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
hcong : ∃ (n : ℤ), 3 ∣ (↑u : 𝓞 K) - (↑n : 𝓞 K)
this :
  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]
0 :
  ∀ (hcong : ∃ (n : ℤ), 3 ∣ HSub.hSub (α := 𝓞 K) (↑1 : 𝓞 K) (↑n : 𝓞 K)),
    Eq.{u_1 + 1} (α := (𝓞 K)ˣ) 1 1 ∨ Eq.{u_1 + 1} (α := (𝓞 K)ˣ) 1 (-1)
1 :
  ∀ (hcong : ∃ (n : ℤ), 3 ∣ HSub.hSub (α := 𝓞 K) (↑(-1) : 𝓞 K) (↑n : 𝓞 K)),
    Eq.{u_1 + 1} (α := (𝓞 K)ˣ) (-1) 1 ∨ Eq.{u_1 + 1} (α := (𝓞 K)ˣ) (-1) (-1)
2 :
  ∀ (hcong : ∃ (n : ℤ), 3 ∣ (↑(IsUnit.unit (a := hζ.toInteger) ⋯) : 𝓞 K) - (↑n : 𝓞 K)),
    IsUnit.unit (a := hζ.toInteger) ⋯ = 1 ∨ IsUnit.unit (a := hζ.toInteger) ⋯ = -1
⊢ u = 1 ∨ u = -1
have 3 : ∀ (hcong : ∃ (n : ℤ), 3 ∣ (↑(-IsUnit.unit (a := hζ.toInteger) sorry) : 𝓞 K) - (↑n : 𝓞 K)),
  -IsUnit.unit (a := hζ.toInteger) sorry = 1 ∨ -IsUnit.unit (a := hζ.toInteger) sorry = -1
Tactic: fin_cases this
Cleared hypotheses: u, hcong, this
-- 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
hcong : ∃ (n : ℤ), 3 ∣ (↑u : 𝓞 K) - (↑n : 𝓞 K)
this :
  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]
0 :
  ∀ (hcong : ∃ (n : ℤ), 3 ∣ HSub.hSub (α := 𝓞 K) (↑1 : 𝓞 K) (↑n : 𝓞 K)),
    Eq.{u_1 + 1} (α := (𝓞 K)ˣ) 1 1 ∨ Eq.{u_1 + 1} (α := (𝓞 K)ˣ) 1 (-1)
1 :
  ∀ (hcong : ∃ (n : ℤ), 3 ∣ HSub.hSub (α := 𝓞 K) (↑(-1) : 𝓞 K) (↑n : 𝓞 K)),
    Eq.{u_1 + 1} (α := (𝓞 K)ˣ) (-1) 1 ∨ Eq.{u_1 + 1} (α := (𝓞 K)ˣ) (-1) (-1)
2 :
  ∀ (hcong : ∃ (n : ℤ), 3 ∣ (↑(IsUnit.unit (a := hζ.toInteger) ⋯) : 𝓞 K) - (↑n : 𝓞 K)),
    IsUnit.unit (a := hζ.toInteger) ⋯ = 1 ∨ IsUnit.unit (a := hζ.toInteger) ⋯ = -1
3 :
  ∀ (hcong : ∃ (n : ℤ), 3 ∣ (↑(-IsUnit.unit (a := hζ.toInteger) ⋯) : 𝓞 K) - (↑n : 𝓞 K)),
    -IsUnit.unit (a := hζ.toInteger) ⋯ = 1 ∨ -IsUnit.unit (a := hζ.toInteger) ⋯ = -1
⊢ u = 1 ∨ u = -1
have 4 : ∀ (hcong : ∃ (n : ℤ), 3 ∣ (↑(HPow.hPow (β := ℕ) (IsUnit.unit (a := hζ.toInteger) sorry) 2) : 𝓞 K) - (↑n : 𝓞 K)),
  HPow.hPow (β := ℕ) (IsUnit.unit (a := hζ.toInteger) sorry) 2 = 1 ∨
    HPow.hPow (β := ℕ) (IsUnit.unit (a := hζ.toInteger) sorry) 2 = -1
Tactic: fin_cases this
Cleared hypotheses: u, hcong, this
-- 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
hcong : ∃ (n : ℤ), 3 ∣ (↑u : 𝓞 K) - (↑n : 𝓞 K)
this :
  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]
0 :
  ∀ (hcong : ∃ (n : ℤ), 3 ∣ HSub.hSub (α := 𝓞 K) (↑1 : 𝓞 K) (↑n : 𝓞 K)),
    Eq.{u_1 + 1} (α := (𝓞 K)ˣ) 1 1 ∨ Eq.{u_1 + 1} (α := (𝓞 K)ˣ) 1 (-1)
1 :
  ∀ (hcong : ∃ (n : ℤ), 3 ∣ HSub.hSub (α := 𝓞 K) (↑(-1) : 𝓞 K) (↑n : 𝓞 K)),
    Eq.{u_1 + 1} (α := (𝓞 K)ˣ) (-1) 1 ∨ Eq.{u_1 + 1} (α := (𝓞 K)ˣ) (-1) (-1)
2 :
  ∀ (hcong : ∃ (n : ℤ), 3 ∣ (↑(IsUnit.unit (a := hζ.toInteger) ⋯) : 𝓞 K) - (↑n : 𝓞 K)),
    IsUnit.unit (a := hζ.toInteger) ⋯ = 1 ∨ IsUnit.unit (a := hζ.toInteger) ⋯ = -1
3 :
  ∀ (hcong : ∃ (n : ℤ), 3 ∣ (↑(-IsUnit.unit (a := hζ.toInteger) ⋯) : 𝓞 K) - (↑n : 𝓞 K)),
    -IsUnit.unit (a := hζ.toInteger) ⋯ = 1 ∨ -IsUnit.unit (a := hζ.toInteger) ⋯ = -1
4 :
  ∀ (hcong : ∃ (n : ℤ), 3 ∣ (↑(HPow.hPow (β := ℕ) (IsUnit.unit (a := hζ.toInteger) ⋯) 2) : 𝓞 K) - (↑n : 𝓞 K)),
    HPow.hPow (β := ℕ) (IsUnit.unit (a := hζ.toInteger) ⋯) 2 = 1 ∨
      HPow.hPow (β := ℕ) (IsUnit.unit (a := hζ.toInteger) ⋯) 2 = -1
⊢ u = 1 ∨ u = -1
have 5 : ∀ (hcong : ∃ (n : ℤ), 3 ∣ (↑(-HPow.hPow (β := ℕ) (IsUnit.unit (a := hζ.toInteger) sorry) 2) : 𝓞 K) - (↑n : 𝓞 K)),
  -HPow.hPow (β := ℕ) (IsUnit.unit (a := hζ.toInteger) sorry) 2 = 1 ∨
    -HPow.hPow (β := ℕ) (IsUnit.unit (a := hζ.toInteger) sorry) 2 = -1
Tactic: fin_cases this
Cleared hypotheses: u, hcong, this
-- 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
inst_1 : NumberField K
inst_2 : IsCyclotomicExtension {3} ℚ K
hcong : ∃ (n : ℤ), 3 ∣ HSub.hSub (α := 𝓞 K) (↑1 : 𝓞 K) (↑n : 𝓞 K)
⊢ Eq.{u_1 + 1} (α := (𝓞 K)ˣ) 1 1 ∨ Eq.{u_1 + 1} (α := (𝓞 K)ˣ) 1 (-1)
have 0 : Eq.{u_1 + 1} (α := (𝓞 K)ˣ) 1 1
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
inst_1 : NumberField K
inst_2 : IsCyclotomicExtension {3} ℚ K
hcong : ∃ (n : ℤ), 3 ∣ HSub.hSub (α := 𝓞 K) (↑(-1) : 𝓞 K) (↑n : 𝓞 K)
⊢ Eq.{u_1 + 1} (α := (𝓞 K)ˣ) (-1) 1 ∨ Eq.{u_1 + 1} (α := (𝓞 K)ˣ) (-1) (-1)
have 1 : Eq.{u_1 + 1} (α := (𝓞 K)ˣ) (-1) (-1)
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
inst_1 : NumberField K
inst_2 : IsCyclotomicExtension {3} ℚ K
hcong : ∃ (n : ℤ), 3 ∣ (↑(IsUnit.unit (a := hζ.toInteger) ⋯) : 𝓞 K) - (↑n : 𝓞 K)
⊢ IsUnit.unit (a := hζ.toInteger) ⋯ = 1 ∨ IsUnit.unit (a := hζ.toInteger) ⋯ = -1
have 2 : False
Tactic: exfalso
-- 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
inst_1 : NumberField K
inst_2 : IsCyclotomicExtension {3} ℚ K
hcong : ∃ (n : ℤ), 3 ∣ (↑(-IsUnit.unit (a := hζ.toInteger) ⋯) : 𝓞 K) - (↑n : 𝓞 K)
⊢ -IsUnit.unit (a := hζ.toInteger) ⋯ = 1 ∨ -IsUnit.unit (a := hζ.toInteger) ⋯ = -1
have 3 : False
Tactic: exfalso
-- 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
inst_1 : NumberField K
inst_2 : IsCyclotomicExtension {3} ℚ K
hcong : ∃ (n : ℤ), 3 ∣ (↑(HPow.hPow (β := ℕ) (IsUnit.unit (a := hζ.toInteger) ⋯) 2) : 𝓞 K) - (↑n : 𝓞 K)
⊢ HPow.hPow (β := ℕ) (IsUnit.unit (a := hζ.toInteger) ⋯) 2 = 1 ∨
    HPow.hPow (β := ℕ) (IsUnit.unit (a := hζ.toInteger) ⋯) 2 = -1
have 4 : False
Tactic: exfalso
-- 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
inst_1 : NumberField K
inst_2 : IsCyclotomicExtension {3} ℚ K
hcong : ∃ (n : ℤ), 3 ∣ (↑(-HPow.hPow (β := ℕ) (IsUnit.unit (a := hζ.toInteger) ⋯) 2) : 𝓞 K) - (↑n : 𝓞 K)
⊢ -HPow.hPow (β := ℕ) (IsUnit.unit (a := hζ.toInteger) ⋯) 2 = 1 ∨
    -HPow.hPow (β := ℕ) (IsUnit.unit (a := hζ.toInteger) ⋯) 2 = -1
have 5 : False
Tactic: exfalso
-- 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
inst_1 : NumberField K
inst_2 : IsCyclotomicExtension {3} ℚ K
hcong : ∃ (n : ℤ), 3 ∣ (↑(-IsUnit.unit (a := hζ.toInteger) ⋯) : 𝓞 K) - (↑n : 𝓞 K)
⊢ False
have 3 : ∃ (n : ℤ), (↑3 : 𝓞 K) ∣ hζ.toInteger - (↑n : 𝓞 K)
Tactic: apply hζ.not_exists_int_prime_dvd_sub_of_prime_ne_two' (by decide)
-- 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
inst_1 : NumberField K
inst_2 : IsCyclotomicExtension {3} ℚ K
hcong : ∃ (n : ℤ), 3 ∣ (↑(-IsUnit.unit (a := hζ.toInteger) ⋯) : 𝓞 K) - (↑n : 𝓞 K)
⊢ ∃ (n : ℤ), (↑3 : 𝓞 K) ∣ hζ.toInteger - (↑n : 𝓞 K)
have 3 : ∀ (n : ℤ) (x : 𝓞 K) (hx : (↑(-IsUnit.unit (a := hζ.toInteger) sorry) : 𝓞 K) - (↑n : 𝓞 K) = 3 * x),
  ∃ (n : ℤ), (↑3 : 𝓞 K) ∣ hζ.toInteger - (↑n : 𝓞 K)
Tactic: obtain ⟨n, x, hx⟩ := hcong
Cleared hypotheses: hcong
-- 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
inst_1 : NumberField K
inst_2 : IsCyclotomicExtension {3} ℚ K
n : ℤ
x : 𝓞 K
hx : (↑(-IsUnit.unit (a := hζ.toInteger) ⋯) : 𝓞 K) - (↑n : 𝓞 K) = 3 * x
⊢ ∃ (n : ℤ), (↑3 : 𝓞 K) ∣ hζ.toInteger - (↑n : 𝓞 K)
have hx₁ : (↑(-IsUnit.unit (a := hζ.toInteger) sorry) : 𝓞 K) = 3 * x + (↑n : 𝓞 K)
Tactic: rw [sub_eq_iff_eq_add] at hx
Cleared hypotheses: hx
-- 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
inst_1 : NumberField K
inst_2 : IsCyclotomicExtension {3} ℚ K
n : ℤ
x : 𝓞 K
hx : (↑(-IsUnit.unit (a := hζ.toInteger) ⋯) : 𝓞 K) = 3 * x + (↑n : 𝓞 K)
⊢ ∃ (n : ℤ), (↑3 : 𝓞 K) ∣ hζ.toInteger - (↑n : 𝓞 K)
have 3 : hζ.toInteger = (↑3 : 𝓞 K) * -x + (↑(-n) : 𝓞 K)
Tactic: refine ⟨-n, -x, sub_eq_iff_eq_add.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
inst_1 : NumberField K
inst_2 : IsCyclotomicExtension {3} ℚ K
hcong : ∃ (n : ℤ), 3 ∣ (↑(-HPow.hPow (β := ℕ) (IsUnit.unit (a := hζ.toInteger) ⋯) 2) : 𝓞 K) - (↑n : 𝓞 K)
⊢ False
have 5 : ∃ (n : ℤ), (↑3 : 𝓞 K) ∣ sorry.toInteger - (↑n : 𝓞 K)
Tactic: apply (hζ.pow_of_coprime 2 (by decide)).not_exists_int_prime_dvd_sub_of_prime_ne_two'
      (by decide)
-- 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
inst_1 : NumberField K
inst_2 : IsCyclotomicExtension {3} ℚ K
hcong : ∃ (n : ℤ), 3 ∣ (↑(-HPow.hPow (β := ℕ) (IsUnit.unit (a := hζ.toInteger) ⋯) 2) : 𝓞 K) - (↑n : 𝓞 K)
⊢ ∃ (n : ℤ), (↑3 : 𝓞 K) ∣ ⋯.toInteger - (↑n : 𝓞 K)
have 5 : ∀ (n : ℤ) (x : 𝓞 K) (hx : (↑(-HPow.hPow (β := ℕ) (IsUnit.unit (a := hζ.toInteger) sorry) 2) : 𝓞 K) - (↑n : 𝓞 K) = 3 * x),
  ∃ (n : ℤ), (↑3 : 𝓞 K) ∣ sorry.toInteger - (↑n : 𝓞 K)
Tactic: obtain ⟨n, x, hx⟩ := hcong
Cleared hypotheses: hcong
-- 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
inst_1 : NumberField K
inst_2 : IsCyclotomicExtension {3} ℚ K
n : ℤ
x : 𝓞 K
hx : (↑(-HPow.hPow (β := ℕ) (IsUnit.unit (a := hζ.toInteger) ⋯) 2) : 𝓞 K) - (↑n : 𝓞 K) = 3 * x
⊢ ∃ (n : ℤ), (↑3 : 𝓞 K) ∣ ⋯.toInteger - (↑n : 𝓞 K)
have 5 : sorry.toInteger = (↑3 : 𝓞 K) * -x + (↑(-n) : 𝓞 K)
Tactic: refine ⟨-n, -x, sub_eq_iff_eq_add.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
inst_1 : NumberField K
inst_2 : IsCyclotomicExtension {3} ℚ K
n : ℤ
x : 𝓞 K
hx : (↑(-HPow.hPow (β := ℕ) (IsUnit.unit (a := hζ.toInteger) ⋯) 2) : 𝓞 K) - (↑n : 𝓞 K) = 3 * x
⊢ ⋯.toInteger = HPow.hPow (β := ℕ) hζ.toInteger 2
have h : (↑sorry.toInteger : K) = (↑(HPow.hPow (β := ℕ) hζ.toInteger 2) : 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
inst_1 : NumberField K
inst_2 : IsCyclotomicExtension {3} ℚ K
n : ℤ
x : 𝓞 K
hx : (↑(-HPow.hPow (β := ℕ) (IsUnit.unit (a := hζ.toInteger) ⋯) 2) : 𝓞 K) - (↑n : 𝓞 K) = 3 * x
⊢ ⋯.toInteger = (↑3 : 𝓞 K) * -x + (↑(-n) : 𝓞 K)
have this : sorry.toInteger = HPow.hPow (β := ℕ) hζ.toInteger 2
Tactic: have : (hζ.pow_of_coprime 2 (by decide)).toInteger = hζ.toInteger ^ 2 := by ext; simp
-- 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
x : 𝓞 K
inst_1 : NumberField K
inst_2 : IsCyclotomicExtension {3} ℚ K
⊢ λ ∣ x ∨ λ ∣ x - 1 ∨ λ ∣ x + 1
have this : Finite.{u_1 + 1} (HasQuotient.Quotient (𝓞 K) (B := Ideal (𝓞 K)) (Ideal.span {λ}))
Tactic: have := hζ.finite_quotient_toInteger_sub_one (by decide)