awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- 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
p : 𝓞 K
hp : Prime p
hpab : p ∣ 1 * S.a + 1 * S.b
hpaηb : p ∣ 1 * S.a + (↑η : 𝓞 K) * S.b
this : p ∣ (1 * (↑η : 𝓞 K) - HMul.hMul (α := 𝓞 K) 1 1) * @gcd _ CommSemiring.toCommMonoidWithZero _ S.a S.b
rwa : ∀ (this : p ∣ ((↑η : 𝓞 K) - HMul.hMul (α := 𝓞 K) 1 1) * @gcd _ CommSemiring.toCommMonoidWithZero _ S.a S.b), p ∣ λ
⊢ p ∣ λ
have rwa₁ : ∀ (this : p ∣ ((↑η : 𝓞 K) - 1) * @gcd _ CommSemiring.toCommMonoidWithZero _ S.a S.b), p ∣ λ
Tactic: rwa [one_mul, one_mul, coe_eta, IsUnit.dvd_mul_right <| (gcd_isUnit_iff _ _).2 S.coprime] at this
Cleared hypotheses: this
-- 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
p : 𝓞 K
hp : Prime p
hpab : p ∣ 1 * S.a + 1 * S.b
hpaηb : p ∣ 1 * S.a + (↑η : 𝓞 K) * S.b
this : p ∣ (1 * (↑η : 𝓞 K) - HMul.hMul (α := 𝓞 K) 1 1) * @gcd _ CommSemiring.toCommMonoidWithZero _ S.a S.b
rwa : ∀ (this : p ∣ ((↑η : 𝓞 K) - HMul.hMul (α := 𝓞 K) 1 1) * @gcd _ CommSemiring.toCommMonoidWithZero _ S.a S.b), p ∣ λ
rwa₁ : ∀ (this : p ∣ ((↑η : 𝓞 K) - 1) * @gcd _ CommSemiring.toCommMonoidWithZero _ S.a S.b), p ∣ λ
⊢ p ∣ λ
have rwa₂ : ∀ (this : p ∣ λ * @gcd _ CommSemiring.toCommMonoidWithZero _ S.a S.b), p ∣ λ
Tactic: rwa [one_mul, one_mul, coe_eta, IsUnit.dvd_mul_right <| (gcd_isUnit_iff _ _).2 S.coprime] at this
Cleared hypotheses: this
-- 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
p : 𝓞 K
hp : Prime p
hpab : p ∣ 1 * S.a + 1 * S.b
hpaηb : p ∣ 1 * S.a + (↑η : 𝓞 K) * S.b
this : p ∣ (1 * (↑η : 𝓞 K) - HMul.hMul (α := 𝓞 K) 1 1) * @gcd _ CommSemiring.toCommMonoidWithZero _ S.a S.b
rwa : ∀ (this : p ∣ ((↑η : 𝓞 K) - HMul.hMul (α := 𝓞 K) 1 1) * @gcd _ CommSemiring.toCommMonoidWithZero _ S.a S.b), p ∣ λ
rwa₁ : ∀ (this : p ∣ ((↑η : 𝓞 K) - 1) * @gcd _ CommSemiring.toCommMonoidWithZero _ S.a S.b), p ∣ λ
rwa₂ : ∀ (this : p ∣ λ * @gcd _ CommSemiring.toCommMonoidWithZero _ S.a S.b), p ∣ λ
⊢ p ∣ λ
have rwa₃ : ∀ (this : p ∣ λ), p ∣ λ
Tactic: rwa [one_mul, one_mul, coe_eta, IsUnit.dvd_mul_right <| (gcd_isUnit_iff _ _).2 S.coprime] at this
Cleared hypotheses: this
-- 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
p : 𝓞 K
hp : Prime p
hpab : p ∣ S.a + S.b
hpaηsqb : p ∣ S.a + HPow.hPow (β := ℕ) (↑η : 𝓞 K) 2 * S.b
⊢ Associated p λ
have suffices : p ∣ λ
Tactic: suffices p_lam : p ∣ λ from hp.associated_of_dvd hζ.zeta_sub_one_prime' p_lam
-- 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
p : 𝓞 K
hp : Prime p
hpab : p ∣ S.a + S.b
hpaηsqb : p ∣ S.a + HPow.hPow (β := ℕ) (↑η : 𝓞 K) 2 * S.b
⊢ p ∣ λ
have hpab₁ : p ∣ 1 * S.a + 1 * S.b
Tactic: rw [← one_mul S.a, ← one_mul S.b] at hpab
Cleared hypotheses: hpab, hpaηsqb
-- 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
p : 𝓞 K
hp : Prime p
hpab : p ∣ 1 * S.a + 1 * S.b
hpaηsqb : p ∣ S.a + HPow.hPow (β := ℕ) (↑η : 𝓞 K) 2 * S.b
⊢ p ∣ λ
have hpaηsqb₁ : p ∣ 1 * S.a + HPow.hPow (β := ℕ) (↑η : 𝓞 K) 2 * S.b
Tactic: rw [← one_mul S.a] at hpaηsqb
Cleared hypotheses: hpaηsqb
-- 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
p : 𝓞 K
hp : Prime p
hpab : p ∣ 1 * S.a + 1 * S.b
hpaηsqb : p ∣ 1 * S.a + HPow.hPow (β := ℕ) (↑η : 𝓞 K) 2 * S.b
⊢ p ∣ λ
have this : p ∣
  (1 * HPow.hPow (β := ℕ) (↑η : 𝓞 K) 2 - HMul.hMul (α := 𝓞 K) 1 1) * @gcd _ CommSemiring.toCommMonoidWithZero _ S.a S.b
Tactic: have := dvd_mul_sub_mul_mul_gcd_of_dvd hpab hpaηsqb
-- 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
p : 𝓞 K
hp : Prime p
hpab : p ∣ 1 * S.a + 1 * S.b
hpaηsqb : p ∣ 1 * S.a + HPow.hPow (β := ℕ) (↑η : 𝓞 K) 2 * S.b
this :
  p ∣
    (1 * HPow.hPow (β := ℕ) (↑η : 𝓞 K) 2 - HMul.hMul (α := 𝓞 K) 1 1) *
      @gcd _ CommSemiring.toCommMonoidWithZero _ S.a S.b
⊢ p ∣ λ
have this₁ : p ∣ -(HPow.hPow (β := ℕ) (↑η : 𝓞 K) 2 - 1)
Tactic: rw [one_mul, mul_one, IsUnit.dvd_mul_right <| (gcd_isUnit_iff _ _).2 S.coprime, ← dvd_neg] at this
Cleared hypotheses: this
-- 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
p : 𝓞 K
hp : Prime p
hpab : p ∣ 1 * S.a + 1 * S.b
hpaηsqb : p ∣ 1 * S.a + HPow.hPow (β := ℕ) (↑η : 𝓞 K) 2 * S.b
this : p ∣ -(HPow.hPow (β := ℕ) (↑η : 𝓞 K) 2 - 1)
⊢ p ∣ λ
have h : λ = -(HPow.hPow (β := ℕ) (↑η : 𝓞 K) 2 - 1) * (↑η : 𝓞 K)
Tactic: convert! dvd_mul_of_dvd_left this η using 1
-- 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
p : 𝓞 K
hp : Prime p
hpab : p ∣ 1 * S.a + 1 * S.b
hpaηsqb : p ∣ 1 * S.a + HPow.hPow (β := ℕ) (↑η : 𝓞 K) 2 * S.b
this : p ∣ -(HPow.hPow (β := ℕ) (↑η : 𝓞 K) 2 - 1)
⊢ λ = -(HPow.hPow (β := ℕ) (↑η : 𝓞 K) 2 - 1) * (↑η : 𝓞 K)
have h : λ = 1 * hζ.toInteger - (-(-hζ.toInteger - 1) - 1 * hζ.toInteger)
Tactic: rw [eta_sq, neg_sub, sub_mul, sub_mul, neg_mul, ← pow_two, eta_sq, coe_eta]
-- 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
p : 𝓞 K
hp : Prime p
hpaηb : p ∣ S.a + (↑η : 𝓞 K) * S.b
hpaηsqb : p ∣ S.a + HPow.hPow (β := ℕ) (↑η : 𝓞 K) 2 * S.b
⊢ Associated p λ
have suffices : p ∣ λ
Tactic: suffices p_lam : p ∣ λ from hp.associated_of_dvd hζ.zeta_sub_one_prime' p_lam
-- 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
p : 𝓞 K
hp : Prime p
hpaηb : p ∣ S.a + (↑η : 𝓞 K) * S.b
hpaηsqb : p ∣ S.a + HPow.hPow (β := ℕ) (↑η : 𝓞 K) 2 * S.b
⊢ p ∣ λ
have hpaηb₁ : p ∣ 1 * S.a + (↑η : 𝓞 K) * S.b
Tactic: rw [← one_mul S.a] at hpaηb
Cleared hypotheses: hpaηb, hpaηsqb
-- 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
p : 𝓞 K
hp : Prime p
hpaηb : p ∣ 1 * S.a + (↑η : 𝓞 K) * S.b
hpaηsqb : p ∣ S.a + HPow.hPow (β := ℕ) (↑η : 𝓞 K) 2 * S.b
⊢ p ∣ λ
have hpaηsqb₁ : p ∣ 1 * S.a + HPow.hPow (β := ℕ) (↑η : 𝓞 K) 2 * S.b
Tactic: rw [← one_mul S.a] at hpaηsqb
Cleared hypotheses: hpaηsqb
-- 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
p : 𝓞 K
hp : Prime p
hpaηb : p ∣ 1 * S.a + (↑η : 𝓞 K) * S.b
hpaηsqb : p ∣ 1 * S.a + HPow.hPow (β := ℕ) (↑η : 𝓞 K) 2 * S.b
⊢ p ∣ λ
have this : p ∣ (1 * HPow.hPow (β := ℕ) (↑η : 𝓞 K) 2 - (↑η : 𝓞 K) * 1) * @gcd _ CommSemiring.toCommMonoidWithZero _ S.a S.b
Tactic: have := dvd_mul_sub_mul_mul_gcd_of_dvd hpaηb hpaηsqb
-- 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
p : 𝓞 K
hp : Prime p
hpaηb : p ∣ 1 * S.a + (↑η : 𝓞 K) * S.b
hpaηsqb : p ∣ 1 * S.a + HPow.hPow (β := ℕ) (↑η : 𝓞 K) 2 * S.b
this : p ∣ (1 * HPow.hPow (β := ℕ) (↑η : 𝓞 K) 2 - (↑η : 𝓞 K) * 1) * @gcd _ CommSemiring.toCommMonoidWithZero _ S.a S.b
⊢ p ∣ λ
have this₁ : p ∣ HPow.hPow (β := ℕ) (↑η : 𝓞 K) 2 - (↑η : 𝓞 K)
Tactic: rw [one_mul, mul_one, IsUnit.dvd_mul_right <| (gcd_isUnit_iff _ _).2 S.coprime] at this
Cleared hypotheses: this
-- 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
p : 𝓞 K
hp : Prime p
hpaηb : p ∣ 1 * S.a + (↑η : 𝓞 K) * S.b
hpaηsqb : p ∣ 1 * S.a + HPow.hPow (β := ℕ) (↑η : 𝓞 K) 2 * S.b
this : p ∣ HPow.hPow (β := ℕ) (↑η : 𝓞 K) 2 - (↑η : 𝓞 K)
⊢ p ∣ λ
have h : λ = (HPow.hPow (β := ℕ) (↑η : 𝓞 K) 2 - (↑η : 𝓞 K)) * (↑η : 𝓞 K) * (↑η : 𝓞 K)
Tactic: convert! (dvd_mul_of_dvd_left (dvd_mul_of_dvd_left this η) η) using 1
-- 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
p : 𝓞 K
hp : Prime p
hpaηb : p ∣ 1 * S.a + (↑η : 𝓞 K) * S.b
hpaηsqb : p ∣ 1 * S.a + HPow.hPow (β := ℕ) (↑η : 𝓞 K) 2 * S.b
this : p ∣ HPow.hPow (β := ℕ) (↑η : 𝓞 K) 2 - (↑η : 𝓞 K)
⊢ λ = (HPow.hPow (β := ℕ) (↑η : 𝓞 K) 2 - (↑η : 𝓞 K)) * (↑η : 𝓞 K) * (↑η : 𝓞 K)
have h : (HPow.hPow (β := ℕ) (↑η : 𝓞 K) 2 - (↑η : 𝓞 K)) * (↑η : 𝓞 K) * (↑η : 𝓞 K) = λ
Tactic: symm
-- 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
p : 𝓞 K
hp : Prime p
hpaηb : p ∣ 1 * S.a + (↑η : 𝓞 K) * S.b
hpaηsqb : p ∣ 1 * S.a + HPow.hPow (β := ℕ) (↑η : 𝓞 K) 2 * S.b
this : p ∣ HPow.hPow (β := ℕ) (↑η : 𝓞 K) 2 - (↑η : 𝓞 K)
⊢ (HPow.hPow (β := ℕ) (↑η : 𝓞 K) 2 - (↑η : 𝓞 K)) * (↑η : 𝓞 K) * (↑η : 𝓞 K) =
    (-(↑η : 𝓞 K) - 1 - (↑η : 𝓞 K)) * (-(↑η : 𝓞 K) - 1)
have rw : (-(↑(IsUnit.unit (a := hζ.toInteger) sorry) : 𝓞 K) - 1 - (↑η : 𝓞 K)) * (↑η : 𝓞 K) * (↑η : 𝓞 K) =
  (-(↑η : 𝓞 K) - 1 - (↑η : 𝓞 K)) * (-(↑η : 𝓞 K) - 1)
Tactic: rw [eta_sq, mul_assoc, ← pow_two, eta_sq]
-- 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
p : 𝓞 K
hp : Prime p
hpaηb : p ∣ 1 * S.a + (↑η : 𝓞 K) * S.b
hpaηsqb : p ∣ 1 * S.a + HPow.hPow (β := ℕ) (↑η : 𝓞 K) 2 * S.b
this : p ∣ HPow.hPow (β := ℕ) (↑η : 𝓞 K) 2 - (↑η : 𝓞 K)
rw :
  (-(↑(IsUnit.unit (a := hζ.toInteger) ⋯) : 𝓞 K) - 1 - (↑η : 𝓞 K)) * (↑η : 𝓞 K) * (↑η : 𝓞 K) =
    (-(↑η : 𝓞 K) - 1 - (↑η : 𝓞 K)) * (-(↑η : 𝓞 K) - 1)
⊢ (HPow.hPow (β := ℕ) (↑η : 𝓞 K) 2 - (↑η : 𝓞 K)) * (↑η : 𝓞 K) * (↑η : 𝓞 K) =
    (-(↑η : 𝓞 K) - 1 - (↑η : 𝓞 K)) * (-(↑η : 𝓞 K) - 1)
have rw₁ : (-(↑(IsUnit.unit (a := hζ.toInteger) sorry) : 𝓞 K) - 1 - (↑η : 𝓞 K)) * ((↑η : 𝓞 K) * (↑η : 𝓞 K)) =
  (-(↑η : 𝓞 K) - 1 - (↑η : 𝓞 K)) * (-(↑η : 𝓞 K) - 1)
Tactic: rw [eta_sq, mul_assoc, ← pow_two, eta_sq]
-- 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
p : 𝓞 K
hp : Prime p
hpaηb : p ∣ 1 * S.a + (↑η : 𝓞 K) * S.b
hpaηsqb : p ∣ 1 * S.a + HPow.hPow (β := ℕ) (↑η : 𝓞 K) 2 * S.b
this : p ∣ HPow.hPow (β := ℕ) (↑η : 𝓞 K) 2 - (↑η : 𝓞 K)
rw :
  (-(↑(IsUnit.unit (a := hζ.toInteger) ⋯) : 𝓞 K) - 1 - (↑η : 𝓞 K)) * (↑η : 𝓞 K) * (↑η : 𝓞 K) =
    (-(↑η : 𝓞 K) - 1 - (↑η : 𝓞 K)) * (-(↑η : 𝓞 K) - 1)
rw₁ :
  (-(↑(IsUnit.unit (a := hζ.toInteger) ⋯) : 𝓞 K) - 1 - (↑η : 𝓞 K)) * ((↑η : 𝓞 K) * (↑η : 𝓞 K)) =
    (-(↑η : 𝓞 K) - 1 - (↑η : 𝓞 K)) * (-(↑η : 𝓞 K) - 1)
⊢ (HPow.hPow (β := ℕ) (↑η : 𝓞 K) 2 - (↑η : 𝓞 K)) * (↑η : 𝓞 K) * (↑η : 𝓞 K) =
    (-(↑η : 𝓞 K) - 1 - (↑η : 𝓞 K)) * (-(↑η : 𝓞 K) - 1)
have rw₂ : (-(↑(IsUnit.unit (a := hζ.toInteger) sorry) : 𝓞 K) - 1 - (↑η : 𝓞 K)) * HPow.hPow (β := ℕ) (↑η : 𝓞 K) 2 =
  (-(↑η : 𝓞 K) - 1 - (↑η : 𝓞 K)) * (-(↑η : 𝓞 K) - 1)
Tactic: rw [eta_sq, mul_assoc, ← pow_two, eta_sq]
-- 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
p : 𝓞 K
hp : Prime p
hpaηb : p ∣ 1 * S.a + (↑η : 𝓞 K) * S.b
hpaηsqb : p ∣ 1 * S.a + HPow.hPow (β := ℕ) (↑η : 𝓞 K) 2 * S.b
this : p ∣ HPow.hPow (β := ℕ) (↑η : 𝓞 K) 2 - (↑η : 𝓞 K)
rw :
  (-(↑(IsUnit.unit (a := hζ.toInteger) ⋯) : 𝓞 K) - 1 - (↑η : 𝓞 K)) * (↑η : 𝓞 K) * (↑η : 𝓞 K) =
    (-(↑η : 𝓞 K) - 1 - (↑η : 𝓞 K)) * (-(↑η : 𝓞 K) - 1)
rw₁ :
  (-(↑(IsUnit.unit (a := hζ.toInteger) ⋯) : 𝓞 K) - 1 - (↑η : 𝓞 K)) * ((↑η : 𝓞 K) * (↑η : 𝓞 K)) =
    (-(↑η : 𝓞 K) - 1 - (↑η : 𝓞 K)) * (-(↑η : 𝓞 K) - 1)
rw₂ :
  (-(↑(IsUnit.unit (a := hζ.toInteger) ⋯) : 𝓞 K) - 1 - (↑η : 𝓞 K)) * HPow.hPow (β := ℕ) (↑η : 𝓞 K) 2 =
    (-(↑η : 𝓞 K) - 1 - (↑η : 𝓞 K)) * (-(↑η : 𝓞 K) - 1)
⊢ (HPow.hPow (β := ℕ) (↑η : 𝓞 K) 2 - (↑η : 𝓞 K)) * (↑η : 𝓞 K) * (↑η : 𝓞 K) =
    (-(↑η : 𝓞 K) - 1 - (↑η : 𝓞 K)) * (-(↑η : 𝓞 K) - 1)
have rw₃ : (-(↑(IsUnit.unit (a := hζ.toInteger) sorry) : 𝓞 K) - 1 - (↑η : 𝓞 K)) * (-(↑(IsUnit.unit (a := hζ.toInteger) sorry) : 𝓞 K) - 1) =
  (-(↑η : 𝓞 K) - 1 - (↑η : 𝓞 K)) * (-(↑η : 𝓞 K) - 1)
Tactic: rw [eta_sq, mul_assoc, ← pow_two, eta_sq]
-- 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
p : 𝓞 K
hp : Prime p
hpaηb : p ∣ 1 * S.a + (↑η : 𝓞 K) * S.b
hpaηsqb : p ∣ 1 * S.a + HPow.hPow (β := ℕ) (↑η : 𝓞 K) 2 * S.b
this : p ∣ HPow.hPow (β := ℕ) (↑η : 𝓞 K) 2 - (↑η : 𝓞 K)
⊢ 2 * HPow.hPow (β := ℕ) (↑η : 𝓞 K) 2 + 3 * (↑η : 𝓞 K) + 1 = λ
have rw : 2 * (-hζ.toInteger - 1) + 3 * hζ.toInteger + 1 = λ
Tactic: rw [eta_sq, coe_eta]
-- 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
p : 𝓞 K
hp : Prime p
hpaηb : p ∣ 1 * S.a + (↑η : 𝓞 K) * S.b
hpaηsqb : p ∣ 1 * S.a + HPow.hPow (β := ℕ) (↑η : 𝓞 K) 2 * S.b
this : p ∣ HPow.hPow (β := ℕ) (↑η : 𝓞 K) 2 - (↑η : 𝓞 K)
⊢ (HPow.hPow (β := ℕ) (↑η : 𝓞 K) 2 - (↑η : 𝓞 K)) * (↑η : 𝓞 K) * (↑η : 𝓞 K) = λ
have calc : (HPow.hPow (β := ℕ) (↑η : 𝓞 K) 2 - (↑η : 𝓞 K)) * (↑η : 𝓞 K) * (↑η : 𝓞 K) =
  (-(↑η : 𝓞 K) - 1 - (↑η : 𝓞 K)) * (-(↑η : 𝓞 K) - 1)
Tactic: calc _ = (-η.1 - 1 - η) * (-η - 1) := by rw [eta_sq, mul_assoc, ← pow_two, eta_sq]
  _ = 2 * η.1 ^ 2 + 3 * η + 1 := by ring
  _ = λ := by rw [eta_sq, coe_eta]; ring
-- 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
p : 𝓞 K
hp : Prime p
hpaηb : p ∣ 1 * S.a + (↑η : 𝓞 K) * S.b
hpaηsqb : p ∣ 1 * S.a + HPow.hPow (β := ℕ) (↑η : 𝓞 K) 2 * S.b
this : p ∣ HPow.hPow (β := ℕ) (↑η : 𝓞 K) 2 - (↑η : 𝓞 K)
calc :
  (HPow.hPow (β := ℕ) (↑η : 𝓞 K) 2 - (↑η : 𝓞 K)) * (↑η : 𝓞 K) * (↑η : 𝓞 K) =
    (-(↑η : 𝓞 K) - 1 - (↑η : 𝓞 K)) * (-(↑η : 𝓞 K) - 1)
⊢ (HPow.hPow (β := ℕ) (↑η : 𝓞 K) 2 - (↑η : 𝓞 K)) * (↑η : 𝓞 K) * (↑η : 𝓞 K) = λ
have calc₁ : (-(↑η : 𝓞 K) - 1 - (↑η : 𝓞 K)) * (-(↑η : 𝓞 K) - 1) = 2 * HPow.hPow (β := ℕ) (↑η : 𝓞 K) 2 + 3 * (↑η : 𝓞 K) + 1
Tactic: calc _ = (-η.1 - 1 - η) * (-η - 1) := by rw [eta_sq, mul_assoc, ← pow_two, eta_sq]
  _ = 2 * η.1 ^ 2 + 3 * η + 1 := by ring
  _ = λ := by rw [eta_sq, coe_eta]; ring
-- 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
p : 𝓞 K
hp : Prime p
hpaηb : p ∣ 1 * S.a + (↑η : 𝓞 K) * S.b
hpaηsqb : p ∣ 1 * S.a + HPow.hPow (β := ℕ) (↑η : 𝓞 K) 2 * S.b
this : p ∣ HPow.hPow (β := ℕ) (↑η : 𝓞 K) 2 - (↑η : 𝓞 K)
calc :
  (HPow.hPow (β := ℕ) (↑η : 𝓞 K) 2 - (↑η : 𝓞 K)) * (↑η : 𝓞 K) * (↑η : 𝓞 K) =
    (-(↑η : 𝓞 K) - 1 - (↑η : 𝓞 K)) * (-(↑η : 𝓞 K) - 1)
calc₁ : (-(↑η : 𝓞 K) - 1 - (↑η : 𝓞 K)) * (-(↑η : 𝓞 K) - 1) = 2 * HPow.hPow (β := ℕ) (↑η : 𝓞 K) 2 + 3 * (↑η : 𝓞 K) + 1
⊢ (HPow.hPow (β := ℕ) (↑η : 𝓞 K) 2 - (↑η : 𝓞 K)) * (↑η : 𝓞 K) * (↑η : 𝓞 K) = λ
have calc₂ : 2 * HPow.hPow (β := ℕ) (↑η : 𝓞 K) 2 + 3 * (↑η : 𝓞 K) + 1 = λ
Tactic: calc _ = (-η.1 - 1 - η) * (-η - 1) := by rw [eta_sq, mul_assoc, ← pow_two, eta_sq]
  _ = 2 * η.1 ^ 2 + 3 * η + 1 := by ring
  _ = λ := by rw [eta_sq, coe_eta]; ring
-- 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
h : λ ∣ S.y
⊢ False
have h₁ : ((↑η : 𝓞 K) - 1) * λ ∣ ((↑η : 𝓞 K) - 1) * S.y
Tactic: replace h := mul_dvd_mul_left ((η : 𝓞 K) - 1) 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
h : ((↑η : 𝓞 K) - 1) * λ ∣ ((↑η : 𝓞 K) - 1) * S.y
⊢ False
have h₁ : HPow.hPow (β := ℕ) λ 2 ∣ S.a + (↑η : 𝓞 K) * S.b
Tactic: rw [coe_eta, ← y_spec, ← pow_two] at 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
h : λ ∣ S.z
⊢ False
have h₁ : ((↑η : 𝓞 K) - 1) * λ ∣ ((↑η : 𝓞 K) - 1) * S.z
Tactic: replace h := mul_dvd_mul_left ((η : 𝓞 K) - 1) 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
h : ((↑η : 𝓞 K) - 1) * λ ∣ ((↑η : 𝓞 K) - 1) * S.z
⊢ False
have h₁ : HPow.hPow (β := ℕ) λ 2 ∣ S.a + HPow.hPow (β := ℕ) (↑η : 𝓞 K) 2 * S.b
Tactic: rw [coe_eta, ← z_spec, ← pow_two] at 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
⊢ λ ^ (3 * S.multiplicity - 2) ∣ S.a + S.b
have h : λ ^ S.multiplicity ∣ S.c
Tactic: have h : λ ^ S.multiplicity ∣ S.c := pow_multiplicity_dvd _ _
-- 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
h : λ ^ S.multiplicity ∣ S.c
⊢ λ ^ (3 * S.multiplicity - 2) ∣ S.a + S.b
have h₁ : HPow.hPow (β := ℕ) (λ ^ S.multiplicity) 3 ∣ (↑S.u : 𝓞 K) * HPow.hPow (β := ℕ) S.c 3
Tactic: replace h : (λ ^ multiplicity S) ^ 3 ∣ S.u * S.c ^ 3 := by simp [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
h : HPow.hPow (β := ℕ) (λ ^ S.multiplicity) 3 ∣ (↑S.u : 𝓞 K) * HPow.hPow (β := ℕ) S.c 3
⊢ λ ^ (3 * S.multiplicity - 2) ∣ S.a + S.b
have h₁ : λ ^ (3 * S.multiplicity) ∣ (S.a + S.b) * (λ * S.y) * (λ * S.z)
Tactic: rw [← S.H, a_cube_add_b_cube_eq_mul, ← pow_mul, mul_comm, y_spec, z_spec] 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
h : λ ^ (3 * S.multiplicity) ∣ (S.a + S.b) * (λ * S.y) * (λ * S.z)
⊢ λ ^ (3 * S.multiplicity - 2) ∣ S.a + S.b
have apply : λ ^ (3 * S.multiplicity - 2) ∣ S.z * (S.a + S.b)
Tactic: apply hζ.zeta_sub_one_prime'.pow_dvd_of_dvd_mul_left _ S.lambda_not_dvd_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
h : λ ^ (3 * S.multiplicity) ∣ (S.a + S.b) * (λ * S.y) * (λ * S.z)
⊢ λ ^ (3 * S.multiplicity - 2) ∣ S.z * (S.a + S.b)
have apply : λ ^ (3 * S.multiplicity - 2) ∣ S.y * (S.z * (S.a + S.b))
Tactic: apply hζ.zeta_sub_one_prime'.pow_dvd_of_dvd_mul_left _ S.lambda_not_dvd_y
-- 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
h : λ ^ (3 * S.multiplicity) ∣ (S.a + S.b) * (λ * S.y) * (λ * S.z)
⊢ λ ^ (3 * S.multiplicity - 2) ∣ S.y * (S.z * (S.a + S.b))
have this : 2 ≤ S.multiplicity
Tactic: have := S.two_le_multiplicity
-- 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
h : λ ^ (3 * S.multiplicity) ∣ (S.a + S.b) * (λ * S.y) * (λ * S.z)
this : 2 ≤ S.multiplicity
⊢ λ ^ (3 * S.multiplicity - 2) ∣ S.y * (S.z * (S.a + S.b))
have h₁ : λ ^ (3 * S.multiplicity - 2) * λ * λ ∣ (S.a + S.b) * S.y * S.z * λ * λ
Tactic: rw [show 3 * multiplicity S = 3 * multiplicity S - 2 + 1 + 1 by lia, pow_succ, pow_succ,
    show (S.a + S.b) * (λ * y S) * (λ * z S) = (S.a + S.b) * y S * z S * λ * λ by ring] at h
Cleared hypotheses: h, this
-- 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
h : λ ^ (3 * S.multiplicity - 2) * λ * λ ∣ (S.a + S.b) * S.y * S.z * λ * λ
this : 2 ≤ S.multiplicity
⊢ λ ^ (3 * S.multiplicity - 2) ∣ S.y * (S.z * (S.a + S.b))
have h₁ : λ ^ (3 * S.multiplicity - 2) ∣ (S.a + S.b) * S.y * S.z
Tactic: simp only [mul_dvd_mul_iff_right hζ.zeta_sub_one_prime'.ne_zero] 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
this : 2 ≤ S.multiplicity
h : λ ^ (3 * S.multiplicity - 2) ∣ (S.a + S.b) * S.y * S.z
⊢ λ ^ (3 * S.multiplicity - 2) ∣ S.y * (S.z * (S.a + S.b))
have rwa : ∀ (h : λ ^ (3 * S.multiplicity - 2) ∣ S.y * (S.z * (S.a + S.b))),
  λ ^ (3 * S.multiplicity - 2) ∣ S.y * (S.z * (S.a + S.b))
Tactic: rwa [show (S.a + S.b) * y S * z S = y S * (z S * (S.a + S.b)) by ring] 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
h : λ ∣ S.w
⊢ False
have refine : λ ^ (S.multiplicity + 1) ∣ S.c
Tactic: refine S.toSolution'.multiplicity_lambda_c_finite.not_pow_dvd_of_multiplicity_lt
    (lt_add_one S.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
h : λ ∣ S.w
⊢ λ ^ (S.multiplicity + 1) ∣ S.c
have rw : λ ^ S.multiplicity * λ ∣ S.c
Tactic: rw [pow_succ', mul_comm]
-- 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
h : λ ∣ S.x
⊢ False
have h₁ : λ ^ (3 * S.multiplicity - 2) * λ ∣ λ ^ (3 * S.multiplicity - 2) * S.x
Tactic: replace h := mul_dvd_mul_left (λ ^ (3 * S.multiplicity - 2)) 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
h : λ ^ (3 * S.multiplicity - 2) * λ ∣ λ ^ (3 * S.multiplicity - 2) * S.x
⊢ False
have h₁ : λ * λ ^ (3 * S.multiplicity - 2) ∣ S.a + S.b
Tactic: rw [mul_comm, ← x_spec] at h
Cleared hypotheses: h