awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.NumberTheory.FLT.Three (hypothesis)
Goal:
K : Type u_1
inst : Field K
ζ : K
hζ : IsPrimitiveRoot ζ 3
S : Solution hζ
inst_1 : NumberField K
inst_2 : IsCyclotomicExtension {3} ℚ K
this : HPow.hPow (β := ℕ) λ 2 ∣ HPow.hPow (β := ℕ) λ 4
X : 𝓞 K
hX : (↑S.u₅ : 𝓞 K) * HPow.hPow (β := ℕ) (λ ^ (S.multiplicity - 1) * S.X) 3 = HPow.hPow (β := ℕ) λ 2 * X
HZ : HPow.hPow (β := ℕ) λ 4 ∣ HPow.hPow (β := ℕ) S.Z 3 - 1
HY : HPow.hPow (β := ℕ) λ 2 ∣ HPow.hPow (β := ℕ) S.Y 3 + 1
⊢ ∃ (n : ℤ), HPow.hPow (β := ℕ) λ 2 ∣ (↑S.u₄ : 𝓞 K) - (↑n : 𝓞 K)
have HZ₁ : HPow.hPow (β := ℕ) λ 2 ∣ HPow.hPow (β := ℕ) S.Z 3 - 1
Tactic: replace HZ :=
      this.trans HZ
Cleared hypotheses: HZ
-- Lean 4 proof step from Mathlib.NumberTheory.FLT.Three (hypothesis)
Goal:
K : Type u_1
inst : Field K
ζ : K
hζ : IsPrimitiveRoot ζ 3
S : Solution hζ
inst_1 : NumberField K
inst_2 : IsCyclotomicExtension {3} ℚ K
this : HPow.hPow (β := ℕ) λ 2 ∣ HPow.hPow (β := ℕ) λ 4
X : 𝓞 K
hX : (↑S.u₅ : 𝓞 K) * HPow.hPow (β := ℕ) (λ ^ (S.multiplicity - 1) * S.X) 3 = HPow.hPow (β := ℕ) λ 2 * X
HZ : HPow.hPow (β := ℕ) λ 4 ∣ HPow.hPow (β := ℕ) S.Z 3 + 1
HY : HPow.hPow (β := ℕ) λ 2 ∣ HPow.hPow (β := ℕ) S.Y 3 + 1
⊢ ∃ (n : ℤ), HPow.hPow (β := ℕ) λ 2 ∣ (↑S.u₄ : 𝓞 K) - (↑n : 𝓞 K)
have HZ₁ : HPow.hPow (β := ℕ) λ 2 ∣ HPow.hPow (β := ℕ) S.Z 3 + 1
Tactic: replace HZ :=
      this.trans HZ
Cleared hypotheses: HZ
-- Lean 4 proof step from Mathlib.NumberTheory.FLT.Three (goal)
Goal:
K : Type u_1
inst : Field K
ζ : K
hζ : IsPrimitiveRoot ζ 3
S : Solution hζ
inst_1 : NumberField K
inst_2 : IsCyclotomicExtension {3} ℚ K
this : HPow.hPow (β := ℕ) λ 2 ∣ HPow.hPow (β := ℕ) λ 4
X : 𝓞 K
hX : (↑S.u₅ : 𝓞 K) * HPow.hPow (β := ℕ) (λ ^ (S.multiplicity - 1) * S.X) 3 = HPow.hPow (β := ℕ) λ 2 * X
HY : HPow.hPow (β := ℕ) λ 2 ∣ HPow.hPow (β := ℕ) S.Y 3 - 1
HZ : HPow.hPow (β := ℕ) λ 2 ∣ HPow.hPow (β := ℕ) S.Z 3 - 1
⊢ ∃ (n : ℤ), HPow.hPow (β := ℕ) λ 2 ∣ (↑S.u₄ : 𝓞 K) - (↑n : 𝓞 K)
have hcong : ∀ (Y : 𝓞 K) (hY : HPow.hPow (β := ℕ) S.Y 3 - 1 = HPow.hPow (β := ℕ) λ 2 * Y),
  ∃ (n : ℤ), HPow.hPow (β := ℕ) λ 2 ∣ (↑S.u₄ : 𝓞 K) - (↑n : 𝓞 K)
Tactic: rcases HY with ⟨Y, hY⟩
Cleared hypotheses: HY
-- Lean 4 proof step from Mathlib.NumberTheory.FLT.Three (goal)
Goal:
K : Type u_1
inst : Field K
ζ : K
hζ : IsPrimitiveRoot ζ 3
S : Solution hζ
inst_1 : NumberField K
inst_2 : IsCyclotomicExtension {3} ℚ K
this : HPow.hPow (β := ℕ) λ 2 ∣ HPow.hPow (β := ℕ) λ 4
X : 𝓞 K
hX : (↑S.u₅ : 𝓞 K) * HPow.hPow (β := ℕ) (λ ^ (S.multiplicity - 1) * S.X) 3 = HPow.hPow (β := ℕ) λ 2 * X
HY : HPow.hPow (β := ℕ) λ 2 ∣ HPow.hPow (β := ℕ) S.Y 3 - 1
HZ : HPow.hPow (β := ℕ) λ 2 ∣ HPow.hPow (β := ℕ) S.Z 3 + 1
⊢ ∃ (n : ℤ), HPow.hPow (β := ℕ) λ 2 ∣ (↑S.u₄ : 𝓞 K) - (↑n : 𝓞 K)
have hcong : ∀ (Y : 𝓞 K) (hY : HPow.hPow (β := ℕ) S.Y 3 - 1 = HPow.hPow (β := ℕ) λ 2 * Y),
  ∃ (n : ℤ), HPow.hPow (β := ℕ) λ 2 ∣ (↑S.u₄ : 𝓞 K) - (↑n : 𝓞 K)
Tactic: rcases HY with ⟨Y, hY⟩
Cleared hypotheses: HY
-- Lean 4 proof step from Mathlib.NumberTheory.FLT.Three (goal)
Goal:
K : Type u_1
inst : Field K
ζ : K
hζ : IsPrimitiveRoot ζ 3
S : Solution hζ
inst_1 : NumberField K
inst_2 : IsCyclotomicExtension {3} ℚ K
this : HPow.hPow (β := ℕ) λ 2 ∣ HPow.hPow (β := ℕ) λ 4
X : 𝓞 K
hX : (↑S.u₅ : 𝓞 K) * HPow.hPow (β := ℕ) (λ ^ (S.multiplicity - 1) * S.X) 3 = HPow.hPow (β := ℕ) λ 2 * X
HY : HPow.hPow (β := ℕ) λ 2 ∣ HPow.hPow (β := ℕ) S.Y 3 + 1
HZ : HPow.hPow (β := ℕ) λ 2 ∣ HPow.hPow (β := ℕ) S.Z 3 - 1
⊢ ∃ (n : ℤ), HPow.hPow (β := ℕ) λ 2 ∣ (↑S.u₄ : 𝓞 K) - (↑n : 𝓞 K)
have hcong : ∀ (Y : 𝓞 K) (hY : HPow.hPow (β := ℕ) S.Y 3 + 1 = HPow.hPow (β := ℕ) λ 2 * Y),
  ∃ (n : ℤ), HPow.hPow (β := ℕ) λ 2 ∣ (↑S.u₄ : 𝓞 K) - (↑n : 𝓞 K)
Tactic: rcases HY with ⟨Y, hY⟩
Cleared hypotheses: HY
-- Lean 4 proof step from Mathlib.NumberTheory.FLT.Three (goal)
Goal:
K : Type u_1
inst : Field K
ζ : K
hζ : IsPrimitiveRoot ζ 3
S : Solution hζ
inst_1 : NumberField K
inst_2 : IsCyclotomicExtension {3} ℚ K
this : HPow.hPow (β := ℕ) λ 2 ∣ HPow.hPow (β := ℕ) λ 4
X : 𝓞 K
hX : (↑S.u₅ : 𝓞 K) * HPow.hPow (β := ℕ) (λ ^ (S.multiplicity - 1) * S.X) 3 = HPow.hPow (β := ℕ) λ 2 * X
HY : HPow.hPow (β := ℕ) λ 2 ∣ HPow.hPow (β := ℕ) S.Y 3 + 1
HZ : HPow.hPow (β := ℕ) λ 2 ∣ HPow.hPow (β := ℕ) S.Z 3 + 1
⊢ ∃ (n : ℤ), HPow.hPow (β := ℕ) λ 2 ∣ (↑S.u₄ : 𝓞 K) - (↑n : 𝓞 K)
have hcong : ∀ (Y : 𝓞 K) (hY : HPow.hPow (β := ℕ) S.Y 3 + 1 = HPow.hPow (β := ℕ) λ 2 * Y),
  ∃ (n : ℤ), HPow.hPow (β := ℕ) λ 2 ∣ (↑S.u₄ : 𝓞 K) - (↑n : 𝓞 K)
Tactic: rcases HY with ⟨Y, hY⟩
Cleared hypotheses: HY
-- Lean 4 proof step from Mathlib.NumberTheory.FLT.Three (goal)
Goal:
K : Type u_1
inst : Field K
ζ : K
hζ : IsPrimitiveRoot ζ 3
S : Solution hζ
inst_1 : NumberField K
inst_2 : IsCyclotomicExtension {3} ℚ K
this : HPow.hPow (β := ℕ) λ 2 ∣ HPow.hPow (β := ℕ) λ 4
X : 𝓞 K
hX : (↑S.u₅ : 𝓞 K) * HPow.hPow (β := ℕ) (λ ^ (S.multiplicity - 1) * S.X) 3 = HPow.hPow (β := ℕ) λ 2 * X
HZ : HPow.hPow (β := ℕ) λ 2 ∣ HPow.hPow (β := ℕ) S.Z 3 - 1
Y : 𝓞 K
hY : HPow.hPow (β := ℕ) S.Y 3 - 1 = HPow.hPow (β := ℕ) λ 2 * Y
⊢ ∃ (n : ℤ), HPow.hPow (β := ℕ) λ 2 ∣ (↑S.u₄ : 𝓞 K) - (↑n : 𝓞 K)
have hcong : ∀ (Z : 𝓞 K) (hZ : HPow.hPow (β := ℕ) S.Z 3 - 1 = HPow.hPow (β := ℕ) λ 2 * Z),
  ∃ (n : ℤ), HPow.hPow (β := ℕ) λ 2 ∣ (↑S.u₄ : 𝓞 K) - (↑n : 𝓞 K)
Tactic: rcases HZ with ⟨Z, hZ⟩
Cleared hypotheses: HZ
-- Lean 4 proof step from Mathlib.NumberTheory.FLT.Three (goal)
Goal:
K : Type u_1
inst : Field K
ζ : K
hζ : IsPrimitiveRoot ζ 3
S : Solution hζ
inst_1 : NumberField K
inst_2 : IsCyclotomicExtension {3} ℚ K
this : HPow.hPow (β := ℕ) λ 2 ∣ HPow.hPow (β := ℕ) λ 4
X : 𝓞 K
hX : (↑S.u₅ : 𝓞 K) * HPow.hPow (β := ℕ) (λ ^ (S.multiplicity - 1) * S.X) 3 = HPow.hPow (β := ℕ) λ 2 * X
HZ : HPow.hPow (β := ℕ) λ 2 ∣ HPow.hPow (β := ℕ) S.Z 3 + 1
Y : 𝓞 K
hY : HPow.hPow (β := ℕ) S.Y 3 - 1 = HPow.hPow (β := ℕ) λ 2 * Y
⊢ ∃ (n : ℤ), HPow.hPow (β := ℕ) λ 2 ∣ (↑S.u₄ : 𝓞 K) - (↑n : 𝓞 K)
have hcong : ∀ (Z : 𝓞 K) (hZ : HPow.hPow (β := ℕ) S.Z 3 + 1 = HPow.hPow (β := ℕ) λ 2 * Z),
  ∃ (n : ℤ), HPow.hPow (β := ℕ) λ 2 ∣ (↑S.u₄ : 𝓞 K) - (↑n : 𝓞 K)
Tactic: rcases HZ with ⟨Z, hZ⟩
Cleared hypotheses: HZ
-- Lean 4 proof step from Mathlib.NumberTheory.FLT.Three (goal)
Goal:
K : Type u_1
inst : Field K
ζ : K
hζ : IsPrimitiveRoot ζ 3
S : Solution hζ
inst_1 : NumberField K
inst_2 : IsCyclotomicExtension {3} ℚ K
this : HPow.hPow (β := ℕ) λ 2 ∣ HPow.hPow (β := ℕ) λ 4
X : 𝓞 K
hX : (↑S.u₅ : 𝓞 K) * HPow.hPow (β := ℕ) (λ ^ (S.multiplicity - 1) * S.X) 3 = HPow.hPow (β := ℕ) λ 2 * X
HZ : HPow.hPow (β := ℕ) λ 2 ∣ HPow.hPow (β := ℕ) S.Z 3 - 1
Y : 𝓞 K
hY : HPow.hPow (β := ℕ) S.Y 3 + 1 = HPow.hPow (β := ℕ) λ 2 * Y
⊢ ∃ (n : ℤ), HPow.hPow (β := ℕ) λ 2 ∣ (↑S.u₄ : 𝓞 K) - (↑n : 𝓞 K)
have hcong : ∀ (Z : 𝓞 K) (hZ : HPow.hPow (β := ℕ) S.Z 3 - 1 = HPow.hPow (β := ℕ) λ 2 * Z),
  ∃ (n : ℤ), HPow.hPow (β := ℕ) λ 2 ∣ (↑S.u₄ : 𝓞 K) - (↑n : 𝓞 K)
Tactic: rcases HZ with ⟨Z, hZ⟩
Cleared hypotheses: HZ
-- Lean 4 proof step from Mathlib.NumberTheory.FLT.Three (goal)
Goal:
K : Type u_1
inst : Field K
ζ : K
hζ : IsPrimitiveRoot ζ 3
S : Solution hζ
inst_1 : NumberField K
inst_2 : IsCyclotomicExtension {3} ℚ K
this : HPow.hPow (β := ℕ) λ 2 ∣ HPow.hPow (β := ℕ) λ 4
X : 𝓞 K
hX : (↑S.u₅ : 𝓞 K) * HPow.hPow (β := ℕ) (λ ^ (S.multiplicity - 1) * S.X) 3 = HPow.hPow (β := ℕ) λ 2 * X
HZ : HPow.hPow (β := ℕ) λ 2 ∣ HPow.hPow (β := ℕ) S.Z 3 + 1
Y : 𝓞 K
hY : HPow.hPow (β := ℕ) S.Y 3 + 1 = HPow.hPow (β := ℕ) λ 2 * Y
⊢ ∃ (n : ℤ), HPow.hPow (β := ℕ) λ 2 ∣ (↑S.u₄ : 𝓞 K) - (↑n : 𝓞 K)
have hcong : ∀ (Z : 𝓞 K) (hZ : HPow.hPow (β := ℕ) S.Z 3 + 1 = HPow.hPow (β := ℕ) λ 2 * Z),
  ∃ (n : ℤ), HPow.hPow (β := ℕ) λ 2 ∣ (↑S.u₄ : 𝓞 K) - (↑n : 𝓞 K)
Tactic: rcases HZ with ⟨Z, hZ⟩
Cleared hypotheses: HZ
-- Lean 4 proof step from Mathlib.NumberTheory.FLT.Three (goal)
Goal:
K : Type u_1
inst : Field K
ζ : K
hζ : IsPrimitiveRoot ζ 3
S : Solution hζ
inst_1 : NumberField K
inst_2 : IsCyclotomicExtension {3} ℚ K
this : HPow.hPow (β := ℕ) λ 2 ∣ HPow.hPow (β := ℕ) λ 4
X : 𝓞 K
hX : (↑S.u₅ : 𝓞 K) * HPow.hPow (β := ℕ) (λ ^ (S.multiplicity - 1) * S.X) 3 = HPow.hPow (β := ℕ) λ 2 * X
Y : 𝓞 K
hY : HPow.hPow (β := ℕ) S.Y 3 - 1 = HPow.hPow (β := ℕ) λ 2 * Y
Z : 𝓞 K
hZ : HPow.hPow (β := ℕ) S.Z 3 - 1 = HPow.hPow (β := ℕ) λ 2 * Z
⊢ ∃ (n : ℤ), HPow.hPow (β := ℕ) λ 2 ∣ (↑S.u₄ : 𝓞 K) - (↑n : 𝓞 K)
have hcong : (↑S.u₄ : 𝓞 K) - (↑(-1) : 𝓞 K) = HPow.hPow (β := ℕ) λ 2 * (X - Y - (↑S.u₄ : 𝓞 K) * Z)
Tactic: refine ⟨-1, X - Y - S.u₄ * Z, ?_⟩
-- Lean 4 proof step from Mathlib.NumberTheory.FLT.Three (goal)
Goal:
K : Type u_1
inst : Field K
ζ : K
hζ : IsPrimitiveRoot ζ 3
S : Solution hζ
inst_1 : NumberField K
inst_2 : IsCyclotomicExtension {3} ℚ K
this : HPow.hPow (β := ℕ) λ 2 ∣ HPow.hPow (β := ℕ) λ 4
X : 𝓞 K
hX : (↑S.u₅ : 𝓞 K) * HPow.hPow (β := ℕ) (λ ^ (S.multiplicity - 1) * S.X) 3 = HPow.hPow (β := ℕ) λ 2 * X
Y : 𝓞 K
hY : HPow.hPow (β := ℕ) S.Y 3 - 1 = HPow.hPow (β := ℕ) λ 2 * Y
Z : 𝓞 K
hZ : HPow.hPow (β := ℕ) S.Z 3 - 1 = HPow.hPow (β := ℕ) λ 2 * Z
⊢ (↑S.u₄ : 𝓞 K) - (↑(-1) : 𝓞 K) = HPow.hPow (β := ℕ) λ 2 * (X - Y - (↑S.u₄ : 𝓞 K) * Z)
have hcong : (↑S.u₄ : 𝓞 K) - (↑(-1) : 𝓞 K) =
  HPow.hPow (β := ℕ) S.Y 3 + (↑S.u₄ : 𝓞 K) * HPow.hPow (β := ℕ) S.Z 3 - (HPow.hPow (β := ℕ) S.Y 3 - 1) -
    (↑S.u₄ : 𝓞 K) * (HPow.hPow (β := ℕ) S.Z 3 - 1)
Tactic: rw [show λ ^ 2 * (X - Y - S.u₄ * Z) = λ ^ 2 * X - λ ^ 2 * Y - S.u₄ * (λ ^ 2 * Z) by ring,
      ← hX, ← hY, ← hZ, ← formula2]
-- Lean 4 proof step from Mathlib.NumberTheory.FLT.Three (goal)
Goal:
K : Type u_1
inst : Field K
ζ : K
hζ : IsPrimitiveRoot ζ 3
S : Solution hζ
inst_1 : NumberField K
inst_2 : IsCyclotomicExtension {3} ℚ K
this : HPow.hPow (β := ℕ) λ 2 ∣ HPow.hPow (β := ℕ) λ 4
X : 𝓞 K
hX : (↑S.u₅ : 𝓞 K) * HPow.hPow (β := ℕ) (λ ^ (S.multiplicity - 1) * S.X) 3 = HPow.hPow (β := ℕ) λ 2 * X
Y : 𝓞 K
hY : HPow.hPow (β := ℕ) S.Y 3 - 1 = HPow.hPow (β := ℕ) λ 2 * Y
Z : 𝓞 K
hZ : HPow.hPow (β := ℕ) S.Z 3 + 1 = HPow.hPow (β := ℕ) λ 2 * Z
⊢ ∃ (n : ℤ), HPow.hPow (β := ℕ) λ 2 ∣ (↑S.u₄ : 𝓞 K) - (↑n : 𝓞 K)
have hcong : (↑S.u₄ : 𝓞 K) - (↑1 : 𝓞 K) = HPow.hPow (β := ℕ) λ 2 * (-X + Y + (↑S.u₄ : 𝓞 K) * Z)
Tactic: refine ⟨1, -X + Y + S.u₄ * Z, ?_⟩
-- Lean 4 proof step from Mathlib.NumberTheory.FLT.Three (goal)
Goal:
K : Type u_1
inst : Field K
ζ : K
hζ : IsPrimitiveRoot ζ 3
S : Solution hζ
inst_1 : NumberField K
inst_2 : IsCyclotomicExtension {3} ℚ K
this : HPow.hPow (β := ℕ) λ 2 ∣ HPow.hPow (β := ℕ) λ 4
X : 𝓞 K
hX : (↑S.u₅ : 𝓞 K) * HPow.hPow (β := ℕ) (λ ^ (S.multiplicity - 1) * S.X) 3 = HPow.hPow (β := ℕ) λ 2 * X
Y : 𝓞 K
hY : HPow.hPow (β := ℕ) S.Y 3 - 1 = HPow.hPow (β := ℕ) λ 2 * Y
Z : 𝓞 K
hZ : HPow.hPow (β := ℕ) S.Z 3 + 1 = HPow.hPow (β := ℕ) λ 2 * Z
⊢ (↑S.u₄ : 𝓞 K) - (↑1 : 𝓞 K) = HPow.hPow (β := ℕ) λ 2 * (-X + Y + (↑S.u₄ : 𝓞 K) * Z)
have hcong : (↑S.u₄ : 𝓞 K) - (↑1 : 𝓞 K) =
  -(HPow.hPow (β := ℕ) S.Y 3 + (↑S.u₄ : 𝓞 K) * HPow.hPow (β := ℕ) S.Z 3 - (HPow.hPow (β := ℕ) S.Y 3 - 1) -
      (↑S.u₄ : 𝓞 K) * (HPow.hPow (β := ℕ) S.Z 3 + 1))
Tactic: rw [show λ ^ 2 * (-X + Y + S.u₄ * Z) = -(λ ^ 2 * X - λ ^ 2 * Y - S.u₄ * (λ ^ 2 * Z)) by ring,
      ← hX, ← hY, ← hZ, ← formula2]
-- Lean 4 proof step from Mathlib.NumberTheory.FLT.Three (goal)
Goal:
K : Type u_1
inst : Field K
ζ : K
hζ : IsPrimitiveRoot ζ 3
S : Solution hζ
inst_1 : NumberField K
inst_2 : IsCyclotomicExtension {3} ℚ K
this : HPow.hPow (β := ℕ) λ 2 ∣ HPow.hPow (β := ℕ) λ 4
X : 𝓞 K
hX : (↑S.u₅ : 𝓞 K) * HPow.hPow (β := ℕ) (λ ^ (S.multiplicity - 1) * S.X) 3 = HPow.hPow (β := ℕ) λ 2 * X
Y : 𝓞 K
hY : HPow.hPow (β := ℕ) S.Y 3 + 1 = HPow.hPow (β := ℕ) λ 2 * Y
Z : 𝓞 K
hZ : HPow.hPow (β := ℕ) S.Z 3 - 1 = HPow.hPow (β := ℕ) λ 2 * Z
⊢ ∃ (n : ℤ), HPow.hPow (β := ℕ) λ 2 ∣ (↑S.u₄ : 𝓞 K) - (↑n : 𝓞 K)
have hcong : (↑S.u₄ : 𝓞 K) - (↑1 : 𝓞 K) = HPow.hPow (β := ℕ) λ 2 * (X - Y - (↑S.u₄ : 𝓞 K) * Z)
Tactic: refine ⟨1, X - Y - S.u₄ * Z, ?_⟩
-- Lean 4 proof step from Mathlib.NumberTheory.FLT.Three (goal)
Goal:
K : Type u_1
inst : Field K
ζ : K
hζ : IsPrimitiveRoot ζ 3
S : Solution hζ
inst_1 : NumberField K
inst_2 : IsCyclotomicExtension {3} ℚ K
this : HPow.hPow (β := ℕ) λ 2 ∣ HPow.hPow (β := ℕ) λ 4
X : 𝓞 K
hX : (↑S.u₅ : 𝓞 K) * HPow.hPow (β := ℕ) (λ ^ (S.multiplicity - 1) * S.X) 3 = HPow.hPow (β := ℕ) λ 2 * X
Y : 𝓞 K
hY : HPow.hPow (β := ℕ) S.Y 3 + 1 = HPow.hPow (β := ℕ) λ 2 * Y
Z : 𝓞 K
hZ : HPow.hPow (β := ℕ) S.Z 3 - 1 = HPow.hPow (β := ℕ) λ 2 * Z
⊢ (↑S.u₄ : 𝓞 K) - (↑1 : 𝓞 K) = HPow.hPow (β := ℕ) λ 2 * (X - Y - (↑S.u₄ : 𝓞 K) * Z)
have hcong : (↑S.u₄ : 𝓞 K) - (↑1 : 𝓞 K) =
  HPow.hPow (β := ℕ) S.Y 3 + (↑S.u₄ : 𝓞 K) * HPow.hPow (β := ℕ) S.Z 3 - (HPow.hPow (β := ℕ) S.Y 3 + 1) -
    (↑S.u₄ : 𝓞 K) * (HPow.hPow (β := ℕ) S.Z 3 - 1)
Tactic: rw [show λ ^ 2 * (X - Y - S.u₄ * Z) = λ ^ 2 * X - λ ^ 2 * Y - S.u₄ * (λ ^ 2 * Z) by ring,
      ← hX, ← hY, ← hZ, ← formula2]
-- Lean 4 proof step from Mathlib.NumberTheory.FLT.Three (goal)
Goal:
K : Type u_1
inst : Field K
ζ : K
hζ : IsPrimitiveRoot ζ 3
S : Solution hζ
inst_1 : NumberField K
inst_2 : IsCyclotomicExtension {3} ℚ K
this : HPow.hPow (β := ℕ) λ 2 ∣ HPow.hPow (β := ℕ) λ 4
X : 𝓞 K
hX : (↑S.u₅ : 𝓞 K) * HPow.hPow (β := ℕ) (λ ^ (S.multiplicity - 1) * S.X) 3 = HPow.hPow (β := ℕ) λ 2 * X
Y : 𝓞 K
hY : HPow.hPow (β := ℕ) S.Y 3 + 1 = HPow.hPow (β := ℕ) λ 2 * Y
Z : 𝓞 K
hZ : HPow.hPow (β := ℕ) S.Z 3 + 1 = HPow.hPow (β := ℕ) λ 2 * Z
⊢ ∃ (n : ℤ), HPow.hPow (β := ℕ) λ 2 ∣ (↑S.u₄ : 𝓞 K) - (↑n : 𝓞 K)
have hcong : (↑S.u₄ : 𝓞 K) - (↑(-1) : 𝓞 K) = HPow.hPow (β := ℕ) λ 2 * (-X + Y + (↑S.u₄ : 𝓞 K) * Z)
Tactic: refine ⟨-1, -X + Y + S.u₄ * Z, ?_⟩
-- Lean 4 proof step from Mathlib.NumberTheory.FLT.Three (goal)
Goal:
K : Type u_1
inst : Field K
ζ : K
hζ : IsPrimitiveRoot ζ 3
S : Solution hζ
inst_1 : NumberField K
inst_2 : IsCyclotomicExtension {3} ℚ K
this : HPow.hPow (β := ℕ) λ 2 ∣ HPow.hPow (β := ℕ) λ 4
X : 𝓞 K
hX : (↑S.u₅ : 𝓞 K) * HPow.hPow (β := ℕ) (λ ^ (S.multiplicity - 1) * S.X) 3 = HPow.hPow (β := ℕ) λ 2 * X
Y : 𝓞 K
hY : HPow.hPow (β := ℕ) S.Y 3 + 1 = HPow.hPow (β := ℕ) λ 2 * Y
Z : 𝓞 K
hZ : HPow.hPow (β := ℕ) S.Z 3 + 1 = HPow.hPow (β := ℕ) λ 2 * Z
⊢ (↑S.u₄ : 𝓞 K) - (↑(-1) : 𝓞 K) = HPow.hPow (β := ℕ) λ 2 * (-X + Y + (↑S.u₄ : 𝓞 K) * Z)
have hcong : (↑S.u₄ : 𝓞 K) - (↑(-1) : 𝓞 K) =
  -(HPow.hPow (β := ℕ) S.Y 3 + (↑S.u₄ : 𝓞 K) * HPow.hPow (β := ℕ) S.Z 3 - (HPow.hPow (β := ℕ) S.Y 3 + 1) -
      (↑S.u₄ : 𝓞 K) * (HPow.hPow (β := ℕ) S.Z 3 + 1))
Tactic: rw [show λ ^ 2 * (-X + Y + S.u₄ * Z) = -(λ ^ 2 * X - λ ^ 2 * Y - S.u₄ * (λ ^ 2 * Z)) by ring,
      ← hX, ← hY, ← hZ, ← formula2]
-- Lean 4 proof step from Mathlib.NumberTheory.FLT.Three (goal)
Goal:
K : Type u_1
inst : Field K
ζ : K
hζ : IsPrimitiveRoot ζ 3
S : Solution hζ
inst_1 : NumberField K
inst_2 : IsCyclotomicExtension {3} ℚ K
⊢ HPow.hPow (β := ℕ) S.u₄ 2 = 1
have inl : ∀ (h : S.u₄ = 1), HPow.hPow (β := ℕ) S.u₄ 2 = 1
Tactic: rcases S.u₄_eq_one_or_neg_one with h | h
-- Lean 4 proof step from Mathlib.NumberTheory.FLT.Three (goal)
Goal:
K : Type u_1
inst : Field K
ζ : K
hζ : IsPrimitiveRoot ζ 3
S : Solution hζ
inst_1 : NumberField K
inst_2 : IsCyclotomicExtension {3} ℚ K
inl : ∀ (h : S.u₄ = 1), HPow.hPow (β := ℕ) S.u₄ 2 = 1
⊢ HPow.hPow (β := ℕ) S.u₄ 2 = 1
have inr : ∀ (h : S.u₄ = -1), HPow.hPow (β := ℕ) S.u₄ 2 = 1
Tactic: rcases S.u₄_eq_one_or_neg_one with h | h
-- Lean 4 proof step from Mathlib.NumberTheory.FLT.Three (goal)
Goal:
K : Type u_1
inst : Field K
ζ : K
hζ : IsPrimitiveRoot ζ 3
S : Solution hζ
S' : Solution' hζ
inst_1 : NumberField K
inst_2 : IsCyclotomicExtension {3} ℚ K
⊢ λ ∣ λ ^ (S.multiplicity - 1) * S.X
have refine : ∀ (h : S.multiplicity - 1 = 0), False
Tactic: refine dvd_mul_of_dvd_left (dvd_pow_self _ (fun h ↦ ?_)) _
-- Lean 4 proof step from Mathlib.NumberTheory.FLT.Three (hypothesis)
Goal:
K : Type u_1
inst : Field K
ζ : K
hζ : IsPrimitiveRoot ζ 3
S : Solution hζ
S' : Solution' hζ
inst_1 : NumberField K
inst_2 : IsCyclotomicExtension {3} ℚ K
h : S.multiplicity - 1 = 0
⊢ False
have h₁ : S.multiplicity = 1
Tactic: rw [Nat.sub_eq_iff_eq_add (le_trans (by simp) S.two_le_multiplicity), zero_add] at h
Cleared hypotheses: h
-- Lean 4 proof step from Mathlib.NumberTheory.FLT.Three (goal)
Goal:
K : Type u_1
inst : Field K
ζ : K
hζ : IsPrimitiveRoot ζ 3
S : Solution hζ
inst_1 : NumberField K
inst_2 : IsCyclotomicExtension {3} ℚ K
⊢ S.Solution'_descent.multiplicity = S.multiplicity - 1
have refine : ∀ (h : λ ^ (S.multiplicity - 1 + 1) ∣ S.Solution'_descent.c), λ ∣ S.X
Tactic: refine multiplicity_eq_of_dvd_of_not_dvd
    (by simp [Solution'_descent]) (fun h ↦ S.lambda_not_dvd_X ?_)
-- Lean 4 proof step from Mathlib.NumberTheory.FLT.Three (goal)
Goal:
K : Type u_1
inst : Field K
ζ : K
hζ : IsPrimitiveRoot ζ 3
S : Solution hζ
inst_1 : NumberField K
inst_2 : IsCyclotomicExtension {3} ℚ K
h : λ ^ (S.multiplicity - 1 + 1) ∣ S.Solution'_descent.c
⊢ λ ∣ S.X
have obtain : ∀ (k : 𝓞 K) (hk : λ ^ (S.multiplicity - 1) * S.X = λ ^ (S.multiplicity - 1 + 1) * k), λ ∣ S.X
Tactic: obtain ⟨k, hk : λ ^ (S.multiplicity - 1) * S.X = λ ^ (S.multiplicity - 1 + 1) * k⟩ := h
Cleared hypotheses: h
-- Lean 4 proof step from Mathlib.NumberTheory.FLT.Three (hypothesis)
Goal:
K : Type u_1
inst : Field K
ζ : K
hζ : IsPrimitiveRoot ζ 3
S : Solution hζ
inst_1 : NumberField K
inst_2 : IsCyclotomicExtension {3} ℚ K
k : 𝓞 K
hk : λ ^ (S.multiplicity - 1) * S.X = λ ^ (S.multiplicity - 1 + 1) * k
⊢ λ ∣ S.X
have hk₁ : λ ^ (S.multiplicity - 1) * S.X = λ ^ (S.multiplicity - 1) * (λ * k)
Tactic: rw [pow_succ, mul_assoc] at hk
Cleared hypotheses: hk
-- Lean 4 proof step from Mathlib.NumberTheory.FLT.Three (hypothesis)
Goal:
K : Type u_1
inst : Field K
ζ : K
hζ : IsPrimitiveRoot ζ 3
S : Solution hζ
inst_1 : NumberField K
inst_2 : IsCyclotomicExtension {3} ℚ K
k : 𝓞 K
hk : λ ^ (S.multiplicity - 1) * S.X = λ ^ (S.multiplicity - 1) * (λ * k)
⊢ λ ∣ S.X
have hk₁ : S.X = λ * k
Tactic: simp only [mul_eq_mul_left_iff, pow_eq_zero_iff', hζ.zeta_sub_one_prime'.ne_zero, ne_eq,
    false_and, or_false] at hk
Cleared hypotheses: hk
-- Lean 4 proof step from Mathlib.NumberTheory.FLT.Three (goal)
Goal:
K : Type u_1
inst : Field K
ζ : K
hζ : IsPrimitiveRoot ζ 3
S : Solution hζ
inst_1 : NumberField K
inst_2 : IsCyclotomicExtension {3} ℚ K
⊢ S.Solution'_descent.multiplicity < S.multiplicity
have rw : S.multiplicity.pred < S.multiplicity
Tactic: rw [Solution'_descent_multiplicity S, Nat.sub_one]
-- Lean 4 proof step from Mathlib.NumberTheory.FLT.Three (hypothesis)
Goal:
K : Type u_1
inst : Field K
ζ : K
hζ : IsPrimitiveRoot ζ 3
S : Solution hζ
inst_1 : NumberField K
inst_2 : IsCyclotomicExtension {3} ℚ K
⊢ S.multiplicity ≠ 0
have this : 2 ≤ S.multiplicity
Tactic: have := S.two_le_multiplicity
-- Lean 4 proof step from Mathlib.NumberTheory.FLT.Three (goal)
Goal:
K : Type u_1
inst : Field K
ζ : K
hζ : IsPrimitiveRoot ζ 3
S : Solution hζ
inst_1 : NumberField K
inst_2 : IsCyclotomicExtension {3} ℚ K
⊢ ∃ (S₁ : Solution hζ), S₁.multiplicity < S.multiplicity
have obtain : ∀ (S' : Solution hζ) (hS' : S'.multiplicity = S.Solution'_descent.multiplicity),
  ∃ (S₁ : Solution hζ), S₁.multiplicity < S.multiplicity
Tactic: obtain ⟨S', hS'⟩ := exists_Solution_of_Solution' (Solution'_descent S)
-- Lean 4 proof step from Mathlib.NumberTheory.FLT.Three (goal)
Goal:
K : Type u_1
inst : Field K
ζ : K
hζ : IsPrimitiveRoot ζ 3
S : Solution hζ
inst_1 : NumberField K
inst_2 : IsCyclotomicExtension {3} ℚ K
⊢ ∃ (S₁ : Solution hζ), S₁.multiplicity < S.multiplicity
have obtain : ∀ (S' : Solution hζ) (hS' : S'.multiplicity = S.Solution'_descent.multiplicity),
  ∃ (S₁ : Solution hζ), S₁.multiplicity < S.multiplicity
Tactic: classical
  obtain ⟨S', hS'⟩ := exists_Solution_of_Solution' (Solution'_descent S)
  exact ⟨S', hS' ▸ Solution'_descent_multiplicity_lt S⟩
-- Lean 4 proof step from Mathlib.NumberTheory.FLT.Three (hypothesis)
Goal:
K : Type := CyclotomicField 3 ℚ
hζ : IsPrimitiveRoot (IsCyclotomicExtension.zeta 3 ℚ K) 3 := ⋯
⊢ FermatLastTheoremFor 3
have this : NumberField K
Tactic: have : NumberField K := IsCyclotomicExtension.numberField {3} ℚ _
-- Lean 4 proof step from Mathlib.NumberTheory.FLT.Three (goal)
Goal:
K : Type := CyclotomicField 3 ℚ
hζ : IsPrimitiveRoot (IsCyclotomicExtension.zeta 3 ℚ K) 3 := ⋯
this : NumberField K
⊢ FermatLastTheoremFor 3
have apply : FermatLastTheoremForThreeGen hζ
Tactic: apply FermatLastTheoremForThree_of_FermatLastTheoremThreeGen hζ
-- Lean 4 proof step from Mathlib.NumberTheory.FLT.Three (goal)
Goal:
K : Type := CyclotomicField 3 ℚ
hζ : IsPrimitiveRoot (IsCyclotomicExtension.zeta 3 ℚ K) 3 := ⋯
this : NumberField K
a b c : NumberField.RingOfIntegers K
u : (NumberField.RingOfIntegers K)ˣ
hc : c ≠ 0
ha : ¬hζ.toInteger - 1 ∣ a
hb : ¬hζ.toInteger - 1 ∣ b
hcdvd : hζ.toInteger - 1 ∣ c
coprime : IsCoprime a b
H : HPow.hPow (β := ℕ) a 3 + HPow.hPow (β := ℕ) b 3 = (↑u : NumberField.RingOfIntegers K) * HPow.hPow (β := ℕ) c 3
S' : FermatLastTheoremForThreeGen.Solution' hζ :=
  { a := a, b := b, c := c, u := u, ha := ha, hb := hb, hc := hc, coprime := coprime, hcdvd := hcdvd, H := H }
⊢ False
have obtain : ∀ (S : FermatLastTheoremForThreeGen.Solution hζ), False
Tactic: obtain ⟨S, -⟩ := FermatLastTheoremForThreeGen.exists_Solution_of_Solution' S'
-- Lean 4 proof step from Mathlib.NumberTheory.FLT.Three (goal)
Goal:
K : Type := CyclotomicField 3 ℚ
hζ : IsPrimitiveRoot (IsCyclotomicExtension.zeta 3 ℚ K) 3 := ⋯
this : NumberField K
a b c : NumberField.RingOfIntegers K
u : (NumberField.RingOfIntegers K)ˣ
hc : c ≠ 0
ha : ¬hζ.toInteger - 1 ∣ a
hb : ¬hζ.toInteger - 1 ∣ b
hcdvd : hζ.toInteger - 1 ∣ c
coprime : IsCoprime a b
H : HPow.hPow (β := ℕ) a 3 + HPow.hPow (β := ℕ) b 3 = (↑u : NumberField.RingOfIntegers K) * HPow.hPow (β := ℕ) c 3
S' : FermatLastTheoremForThreeGen.Solution' hζ :=
  { a := a, b := b, c := c, u := u, ha := ha, hb := hb, hc := hc, coprime := coprime, hcdvd := hcdvd, H := H }
S : FermatLastTheoremForThreeGen.Solution hζ
⊢ False
have obtain : ∀ (Smin : FermatLastTheoremForThreeGen.Solution hζ) (hSmin : Smin.isMinimal), False
Tactic: obtain ⟨Smin, hSmin⟩ := S.exists_minimal
-- Lean 4 proof step from Mathlib.NumberTheory.FLT.Three (goal)
Goal:
K : Type := CyclotomicField 3 ℚ
hζ : IsPrimitiveRoot (IsCyclotomicExtension.zeta 3 ℚ K) 3 := ⋯
this : NumberField K
a b c : NumberField.RingOfIntegers K
u : (NumberField.RingOfIntegers K)ˣ
hc : c ≠ 0
ha : ¬hζ.toInteger - 1 ∣ a
hb : ¬hζ.toInteger - 1 ∣ b
hcdvd : hζ.toInteger - 1 ∣ c
coprime : IsCoprime a b
H : HPow.hPow (β := ℕ) a 3 + HPow.hPow (β := ℕ) b 3 = (↑u : NumberField.RingOfIntegers K) * HPow.hPow (β := ℕ) c 3
S' : FermatLastTheoremForThreeGen.Solution' hζ :=
  { a := a, b := b, c := c, u := u, ha := ha, hb := hb, hc := hc, coprime := coprime, hcdvd := hcdvd, H := H }
S Smin : FermatLastTheoremForThreeGen.Solution hζ
hSmin : Smin.isMinimal
⊢ False
have obtain : ∀ (Sfin : FermatLastTheoremForThreeGen.Solution hζ) (hSfin : Sfin.multiplicity < Smin.multiplicity), False
Tactic: obtain ⟨Sfin, hSfin⟩ := Smin.exists_Solution_multiplicity_lt
-- Lean 4 proof step from Mathlib.NumberTheory.FLT.Three (goal)
Goal:
K : Type := CyclotomicField 3 ℚ
hζ : IsPrimitiveRoot (IsCyclotomicExtension.zeta 3 ℚ K) 3 := ⋯
this : NumberField K
a b c : NumberField.RingOfIntegers K
u : (NumberField.RingOfIntegers K)ˣ
hc : c ≠ 0
ha : ¬hζ.toInteger - 1 ∣ a
hb : ¬hζ.toInteger - 1 ∣ b
hcdvd : hζ.toInteger - 1 ∣ c
coprime : IsCoprime a b
H : HPow.hPow (β := ℕ) a 3 + HPow.hPow (β := ℕ) b 3 = (↑u : NumberField.RingOfIntegers K) * HPow.hPow (β := ℕ) c 3
S' : FermatLastTheoremForThreeGen.Solution' hζ :=
  { a := a, b := b, c := c, u := u, ha := ha, hb := hb, hc := hc, coprime := coprime, hcdvd := hcdvd, H := H }
S Smin : FermatLastTheoremForThreeGen.Solution hζ
hSmin : Smin.isMinimal
Sfin : FermatLastTheoremForThreeGen.Solution hζ
hSfin : Sfin.multiplicity < Smin.multiplicity
⊢ False
have linarith : -1 + (HAdd.hAdd (α := ℤ) (↑Sfin.multiplicity : ℤ) 1 - (↑Smin.multiplicity : ℤ)) +
    HSub.hSub (α := ℤ) (↑Smin.multiplicity : ℤ) (↑Sfin.multiplicity : ℤ) =
  0
Tactic: linarith [hSmin Sfin]