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
h : λ * λ ^ (3 * S.multiplicity - 2) ∣ S.a + S.b
⊢ False
have h₁ : λ * λ ^ (3 * S.multiplicity - 2) * λ * λ ∣
  (S.a + S.b) * (S.a + (↑η : 𝓞 K) * S.b) * (S.a + HPow.hPow (β := ℕ) (↑η : 𝓞 K) 2 * S.b)
Tactic: replace h :=
    mul_dvd_mul (mul_dvd_mul h S.lambda_dvd_a_add_eta_mul_b) S.lambda_dvd_a_add_eta_sq_mul_b
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 * λ ^ (3 * S.multiplicity - 2) * λ * λ ∣
    (S.a + S.b) * (S.a + (↑η : 𝓞 K) * S.b) * (S.a + HPow.hPow (β := ℕ) (↑η : 𝓞 K) 2 * S.b)
⊢ False
have h₁ : λ * λ ^ (3 * S.multiplicity - 2) * λ * λ ∣ HPow.hPow (β := ℕ) (λ ^ S.multiplicity * S.w) 3
Tactic: simp only [← a_cube_add_b_cube_eq_mul, S.H, w_spec, Units.isUnit, IsUnit.dvd_mul_left] 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 * λ ^ (3 * S.multiplicity - 2) * λ * λ ∣ HPow.hPow (β := ℕ) (λ ^ S.multiplicity * S.w) 3
⊢ False
have h₁ : λ ^ (3 * S.multiplicity - 2 + 1 + 1) * λ ∣ HPow.hPow (β := ℕ) (λ ^ S.multiplicity * S.w) 3
Tactic: rw [← pow_succ', mul_comm, ← mul_assoc, ← pow_succ'] 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 ^ (3 * S.multiplicity - 2 + 1 + 1) * λ ∣ HPow.hPow (β := ℕ) (λ ^ S.multiplicity * S.w) 3
⊢ False
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
h : λ ^ (3 * S.multiplicity - 2 + 1 + 1) * λ ∣ HPow.hPow (β := ℕ) (λ ^ S.multiplicity * S.w) 3
this : 2 ≤ S.multiplicity
⊢ False
have rw : ∀ (h : λ ∣ HPow.hPow (β := ℕ) S.w 3) (this : 2 ≤ S.multiplicity), False
Tactic: rw [show 3 * multiplicity S - 2 + 1 + 1 = 3 * multiplicity S by lia, mul_pow, ← pow_mul,
    mul_comm _ 3, mul_dvd_mul_iff_left _] at h
Cleared hypotheses: h, 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
h : λ ^ (3 * S.multiplicity - 2 + 1 + 1) * λ ∣ HPow.hPow (β := ℕ) (λ ^ S.multiplicity * S.w) 3
this : 2 ≤ S.multiplicity
rw : ∀ (h : λ ∣ HPow.hPow (β := ℕ) S.w 3) (this : 2 ≤ S.multiplicity), False
⊢ False
have rw₁ : ∀ (h : λ ^ (3 * S.multiplicity) * λ ∣ λ ^ (3 * S.multiplicity) * HPow.hPow (β := ℕ) S.w 3) (this : 2 ≤ S.multiplicity),
  λ ^ (3 * S.multiplicity) ≠ 0
Tactic: rw [show 3 * multiplicity S - 2 + 1 + 1 = 3 * multiplicity S by lia, mul_pow, ← pow_mul,
    mul_comm _ 3, mul_dvd_mul_iff_left _] at h
Cleared hypotheses: h, 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
inst_1 : NumberField K
inst_2 : IsCyclotomicExtension {3} ℚ K
r s t w : 𝓞 K
hr : ¬λ ∣ r
hs : ¬λ ∣ s
Hp : ∀ {p : 𝓞 K}, Prime p → p ∣ t → p ∣ w → Associated p λ
H₁ : ∀ {q : 𝓞 K}, q ∣ r → q ∣ t
H₂ : ∀ {q : 𝓞 K}, q ∣ s → q ∣ w
⊢ IsCoprime r s
have refine : ∀ (p : 𝓞 K) (hp : Prime p) (p_dvd_r : p ∣ r) (p_dvd_s : p ∣ s), λ ∣ r
Tactic: refine isCoprime_of_prime_dvd (not_and.2 (fun _ hz ↦ hs (by simp [hz])))
    (fun p hp p_dvd_r p_dvd_s ↦ hr ?_)
-- 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
hh :
  λ ^ (3 * S.multiplicity - 2) * S.x * λ * S.y * λ * S.z =
    (↑S.u : 𝓞 K) * λ ^ (3 * S.multiplicity) * HPow.hPow (β := ℕ) S.w 3
⊢ S.x * S.y * S.z = (↑S.u : 𝓞 K) * HPow.hPow (β := ℕ) S.w 3
have hh₁ : λ ^ (3 * S.multiplicity - 2) * λ * λ * S.x * S.y * S.z =
  (↑S.u : 𝓞 K) * λ ^ (3 * S.multiplicity) * HPow.hPow (β := ℕ) S.w 3
Tactic: rw [show λ ^ (3 * multiplicity S - 2) * x S * λ * y S * λ * z S =
      λ ^ (3 * multiplicity S - 2) * λ * λ * x S * y S * z S by ring] at hh
Cleared hypotheses: hh
-- 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
hh :
  λ ^ (3 * S.multiplicity - 2) * λ * λ * S.x * S.y * S.z =
    (↑S.u : 𝓞 K) * λ ^ (3 * S.multiplicity) * HPow.hPow (β := ℕ) S.w 3
⊢ S.x * S.y * S.z = (↑S.u : 𝓞 K) * HPow.hPow (β := ℕ) S.w 3
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
hh :
  λ ^ (3 * S.multiplicity - 2) * λ * λ * S.x * S.y * S.z =
    (↑S.u : 𝓞 K) * λ ^ (3 * S.multiplicity) * HPow.hPow (β := ℕ) S.w 3
this : 2 ≤ S.multiplicity
⊢ S.x * S.y * S.z = (↑S.u : 𝓞 K) * HPow.hPow (β := ℕ) S.w 3
have hh₁ : λ ^ (3 * S.multiplicity) * (S.x * (S.y * S.z)) = λ ^ (3 * S.multiplicity) * ((↑S.u : 𝓞 K) * HPow.hPow (β := ℕ) S.w 3)
Tactic: rw [mul_comm _ (λ ^ (3 * multiplicity S)), ← pow_succ, ← pow_succ,
      show 3 * multiplicity S - 2 + 1 + 1 = 3 * multiplicity S by lia, mul_assoc, mul_assoc,
      mul_assoc] at hh
Cleared hypotheses: hh, 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
hh :
  λ ^ (3 * S.multiplicity) * (S.x * (S.y * S.z)) = λ ^ (3 * S.multiplicity) * ((↑S.u : 𝓞 K) * HPow.hPow (β := ℕ) S.w 3)
this : 2 ≤ S.multiplicity
⊢ S.x * S.y * S.z = (↑S.u : 𝓞 K) * HPow.hPow (β := ℕ) S.w 3
have hh₁ : S.x * (S.y * S.z) = (↑S.u : 𝓞 K) * HPow.hPow (β := ℕ) S.w 3
Tactic: simp only [mul_eq_mul_left_iff, pow_eq_zero_iff', hζ.zeta_sub_one_prime'.ne_zero, ne_eq,
      mul_eq_zero, OfNat.ofNat_ne_zero, false_or, false_and, or_false] at hh
Cleared hypotheses: hh
-- 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
hh : S.x * (S.y * S.z) = (↑S.u : 𝓞 K) * HPow.hPow (β := ℕ) S.w 3
⊢ S.x * S.y * S.z = (↑S.u : 𝓞 K) * HPow.hPow (β := ℕ) S.w 3
have h : S.x * S.y * S.z = S.x * (S.y * S.z)
Tactic: convert! hh 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
⊢ S.x * S.y * S.z = (↑S.u : 𝓞 K) * HPow.hPow (β := ℕ) S.w 3
have suffices : λ ^ (3 * S.multiplicity - 2) * S.x * λ * S.y * λ * S.z =
  (↑S.u : 𝓞 K) * λ ^ (3 * S.multiplicity) * HPow.hPow (β := ℕ) S.w 3
Tactic: suffices hh : λ ^ (3 * S.multiplicity - 2) * S.x * λ * S.y * λ * S.z =
      S.u * λ ^ (3 * S.multiplicity) * S.w ^ 3 by
    rw [show λ ^ (3 * multiplicity S - 2) * x S * λ * y S * λ * z S =
      λ ^ (3 * multiplicity S - 2) * λ * λ * x S * y S * z S by ring] at hh
    have := S.two_le_multiplicity
    rw [mul_comm _ (λ ^ (3 * multiplicity S)), ← pow_succ, ← pow_succ,
      show 3 * multiplicity S - 2 + 1 + 1 = 3 * multiplicity S by lia, mul_assoc, mul_assoc,
      mul_assoc] at hh
    simp only [mul_eq_mul_left_iff, pow_eq_zero_iff', hζ.zeta_sub_one_prime'.ne_zero, ne_eq,
      mul_eq_zero, OfNat.ofNat_ne_zero, false_or, false_and, or_false] at hh
    convert! hh using 1
    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
⊢ λ ^ (3 * S.multiplicity - 2) * S.x * λ * S.y * λ * S.z =
    (↑S.u : 𝓞 K) * λ ^ (3 * S.multiplicity) * HPow.hPow (β := ℕ) S.w 3
have simp : (S.a + S.b) * ((S.a + (↑η : 𝓞 K) * S.b) * (S.a + HPow.hPow (β := ℕ) (↑η : 𝓞 K) 2 * S.b)) =
  (↑S.u : 𝓞 K) * (λ ^ (3 * S.multiplicity) * HPow.hPow (β := ℕ) S.w 3)
Tactic: simp only [← x_spec, mul_assoc, ← y_spec, ← z_spec]
-- 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.a + S.b) * ((S.a + (↑η : 𝓞 K) * S.b) * (S.a + HPow.hPow (β := ℕ) (↑η : 𝓞 K) 2 * S.b)) =
    (↑S.u : 𝓞 K) * (λ ^ (3 * S.multiplicity) * HPow.hPow (β := ℕ) S.w 3)
have rw : (S.a + S.b) * ((S.a + (↑η : 𝓞 K) * S.b) * (S.a + HPow.hPow (β := ℕ) (↑η : 𝓞 K) 2 * S.b)) =
  (S.a + S.b) * (S.a + (↑η : 𝓞 K) * S.b) * (S.a + HPow.hPow (β := ℕ) (↑η : 𝓞 K) 2 * S.b)
Tactic: rw [mul_comm 3, pow_mul, ← mul_pow, ← w_spec, ← S.H, a_cube_add_b_cube_eq_mul]
-- 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
⊢ (∃ (X : 𝓞 K), Associated (HPow.hPow (β := ℕ) X 3) S.x) ∧
    (∃ (Y : 𝓞 K), Associated (HPow.hPow (β := ℕ) Y 3) S.y) ∧ ∃ (Z : 𝓞 K), Associated (HPow.hPow (β := ℕ) Z 3) S.z
have h₁ : IsCoprime (S.x * S.y) S.z
Tactic: have h₁ := S.isCoprime_x_z.mul_left S.isCoprime_y_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
h₁ : IsCoprime (S.x * S.y) S.z
⊢ HPow.hPow (β := ℕ) S.w 3 * (↑S.u : 𝓞 K) = S.x * S.y * S.z
have rw : HPow.hPow (β := ℕ) S.w 3 * (↑S.u : 𝓞 K) = (↑S.u : 𝓞 K) * HPow.hPow (β := ℕ) S.w 3
Tactic: rw [x_mul_y_mul_z_eq_u_mul_w_cube S, mul_comm]
-- 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
h₁ : IsCoprime (S.x * S.y) S.z
rw : HPow.hPow (β := ℕ) S.w 3 * (↑S.u : 𝓞 K) = (↑S.u : 𝓞 K) * HPow.hPow (β := ℕ) S.w 3
⊢ HPow.hPow (β := ℕ) S.w 3 * (↑S.u : 𝓞 K) = S.x * S.y * S.z
have rw₁ : (↑S.u : 𝓞 K) * HPow.hPow (β := ℕ) S.w 3 = (↑S.u : 𝓞 K) * HPow.hPow (β := ℕ) S.w 3
Tactic: rw [x_mul_y_mul_z_eq_u_mul_w_cube S, 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
h₁ : IsCoprime (S.x * S.y) S.z
⊢ (∃ (X : 𝓞 K), Associated (HPow.hPow (β := ℕ) X 3) S.x) ∧
    (∃ (Y : 𝓞 K), Associated (HPow.hPow (β := ℕ) Y 3) S.y) ∧ ∃ (Z : 𝓞 K), Associated (HPow.hPow (β := ℕ) Z 3) S.z
have h₂ : Associated (HPow.hPow (β := ℕ) S.w 3) (S.x * S.y * S.z)
Tactic: have h₂ : Associated (S.w ^ 3) (S.x * S.y * S.z) :=
    ⟨S.u, by rw [x_mul_y_mul_z_eq_u_mul_w_cube S, mul_comm]⟩
-- 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
h₁ : IsCoprime (S.x * S.y) S.z
h₂ : Associated (HPow.hPow (β := ℕ) S.w 3) (S.x * S.y * S.z)
⊢ (∃ (X : 𝓞 K), Associated (HPow.hPow (β := ℕ) X 3) S.x) ∧
    (∃ (Y : 𝓞 K), Associated (HPow.hPow (β := ℕ) Y 3) S.y) ∧ ∃ (Z : 𝓞 K), Associated (HPow.hPow (β := ℕ) Z 3) S.z
have obtain : ∀ (T : 𝓞 K) (h₃ : Associated (HPow.hPow (β := ℕ) T 3) (S.x * S.y)),
  (∃ (X : 𝓞 K), Associated (HPow.hPow (β := ℕ) X 3) S.x) ∧
    (∃ (Y : 𝓞 K), Associated (HPow.hPow (β := ℕ) Y 3) S.y) ∧ ∃ (Z : 𝓞 K), Associated (HPow.hPow (β := ℕ) Z 3) S.z
Tactic: obtain ⟨T, h₃⟩ := exists_associated_pow_of_associated_pow_mul h₁ h₂
-- 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
⊢ (∃ (X : 𝓞 K), Associated (HPow.hPow (β := ℕ) X 3) S.x) ∧
    (∃ (Y : 𝓞 K), Associated (HPow.hPow (β := ℕ) Y 3) S.y) ∧ ∃ (Z : 𝓞 K), Associated (HPow.hPow (β := ℕ) Z 3) S.z
have have : ∀ (h₁ : IsCoprime (S.x * S.y) S.z),
  (∃ (X : 𝓞 K), Associated (HPow.hPow (β := ℕ) X 3) S.x) ∧
    (∃ (Y : 𝓞 K), Associated (HPow.hPow (β := ℕ) Y 3) S.y) ∧ ∃ (Z : 𝓞 K), Associated (HPow.hPow (β := ℕ) Z 3) S.z
Tactic: classical
  have h₁ := S.isCoprime_x_z.mul_left S.isCoprime_y_z
  have h₂ : Associated (S.w ^ 3) (S.x * S.y * S.z) :=
    ⟨S.u, by rw [x_mul_y_mul_z_eq_u_mul_w_cube S, mul_comm]⟩
  obtain ⟨T, h₃⟩ := exists_associated_pow_of_associated_pow_mul h₁ h₂
  exact ⟨exists_associated_pow_of_associated_pow_mul S.isCoprime_x_y h₃,
    exists_associated_pow_of_associated_pow_mul S.isCoprime_x_y.symm (mul_comm _ S.x ▸ h₃),
    exists_associated_pow_of_associated_pow_mul h₁.symm (mul_comm _ S.z ▸ h₂)⟩
-- 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
have have₁ have₂ :
  ∀ (h₁ : IsCoprime (S.x * S.y) S.z),
    (∃ (X : 𝓞 K), Associated (HPow.hPow (β := ℕ) X 3) S.x) ∧
      (∃ (Y : 𝓞 K), Associated (HPow.hPow (β := ℕ) Y 3) S.y) ∧ ∃ (Z : 𝓞 K), Associated (HPow.hPow (β := ℕ) Z 3) S.z
⊢ (∃ (X : 𝓞 K), Associated (HPow.hPow (β := ℕ) X 3) S.x) ∧
    (∃ (Y : 𝓞 K), Associated (HPow.hPow (β := ℕ) Y 3) S.y) ∧ ∃ (Z : 𝓞 K), Associated (HPow.hPow (β := ℕ) Z 3) S.z
have have₃ : ∀ (h₁ : IsCoprime (S.x * S.y) S.z) (h₂ : Associated (HPow.hPow (β := ℕ) S.w 3) (S.x * S.y * S.z)),
  (∃ (X : 𝓞 K), Associated (HPow.hPow (β := ℕ) X 3) S.x) ∧
    (∃ (Y : 𝓞 K), Associated (HPow.hPow (β := ℕ) Y 3) S.y) ∧ ∃ (Z : 𝓞 K), Associated (HPow.hPow (β := ℕ) Z 3) S.z
Tactic: classical
  have h₁ := S.isCoprime_x_z.mul_left S.isCoprime_y_z
  have h₂ : Associated (S.w ^ 3) (S.x * S.y * S.z) :=
    ⟨S.u, by rw [x_mul_y_mul_z_eq_u_mul_w_cube S, mul_comm]⟩
  obtain ⟨T, h₃⟩ := exists_associated_pow_of_associated_pow_mul h₁ h₂
  exact ⟨exists_associated_pow_of_associated_pow_mul S.isCoprime_x_y h₃,
    exists_associated_pow_of_associated_pow_mul S.isCoprime_x_y.symm (mul_comm _ S.x ▸ h₃),
    exists_associated_pow_of_associated_pow_mul h₁.symm (mul_comm _ S.z ▸ h₂)⟩
-- 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
have have₁ have₂ :
  ∀ (h₁ : IsCoprime (S.x * S.y) S.z),
    (∃ (X : 𝓞 K), Associated (HPow.hPow (β := ℕ) X 3) S.x) ∧
      (∃ (Y : 𝓞 K), Associated (HPow.hPow (β := ℕ) Y 3) S.y) ∧ ∃ (Z : 𝓞 K), Associated (HPow.hPow (β := ℕ) Z 3) S.z
have₃ :
  ∀ (h₁ : IsCoprime (S.x * S.y) S.z) (h₂ : Associated (HPow.hPow (β := ℕ) S.w 3) (S.x * S.y * S.z)),
    (∃ (X : 𝓞 K), Associated (HPow.hPow (β := ℕ) X 3) S.x) ∧
      (∃ (Y : 𝓞 K), Associated (HPow.hPow (β := ℕ) Y 3) S.y) ∧ ∃ (Z : 𝓞 K), Associated (HPow.hPow (β := ℕ) Z 3) S.z
⊢ (∃ (X : 𝓞 K), Associated (HPow.hPow (β := ℕ) X 3) S.x) ∧
    (∃ (Y : 𝓞 K), Associated (HPow.hPow (β := ℕ) Y 3) S.y) ∧ ∃ (Z : 𝓞 K), Associated (HPow.hPow (β := ℕ) Z 3) S.z
have have₄ : ∀ (h₁ : IsCoprime (S.x * S.y) S.z) (h₂ : Associated (HPow.hPow (β := ℕ) S.w 3) (S.x * S.y * S.z)) (T : 𝓞 K)
  (h₃ : Associated (HPow.hPow (β := ℕ) T 3) (S.x * S.y)),
  (∃ (X : 𝓞 K), Associated (HPow.hPow (β := ℕ) X 3) S.x) ∧
    (∃ (Y : 𝓞 K), Associated (HPow.hPow (β := ℕ) Y 3) S.y) ∧ ∃ (Z : 𝓞 K), Associated (HPow.hPow (β := ℕ) Z 3) S.z
Tactic: classical
  have h₁ := S.isCoprime_x_z.mul_left S.isCoprime_y_z
  have h₂ : Associated (S.w ^ 3) (S.x * S.y * S.z) :=
    ⟨S.u, by rw [x_mul_y_mul_z_eq_u_mul_w_cube S, mul_comm]⟩
  obtain ⟨T, h₃⟩ := exists_associated_pow_of_associated_pow_mul h₁ h₂
  exact ⟨exists_associated_pow_of_associated_pow_mul S.isCoprime_x_y h₃,
    exists_associated_pow_of_associated_pow_mul S.isCoprime_x_y.symm (mul_comm _ S.x ▸ h₃),
    exists_associated_pow_of_associated_pow_mul h₁.symm (mul_comm _ S.z ▸ h₂)⟩
-- 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
⊢ IsCoprime S.Y S.Z
have rw : IsCoprime S.y S.z
Tactic: rw [← IsCoprime.pow_iff (m := 3) (n := 3) (by decide) (by decide),
    ← isCoprime_mul_unit_right_left S.u₂.isUnit, ← isCoprime_mul_unit_right_right S.u₃.isUnit,
    Y_u₂_spec, Z_u₃_spec]
-- 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.X 3 * (↑S.u₁ : 𝓞 K) * λ ^ (3 * S.multiplicity - 2) +
        HPow.hPow (β := ℕ) S.Y 3 * (↑S.u₂ : 𝓞 K) * λ * (↑η : 𝓞 K) +
      HPow.hPow (β := ℕ) S.Z 3 * (↑S.u₃ : 𝓞 K) * λ * HPow.hPow (β := ℕ) (↑η : 𝓞 K) 2 =
    0
have rw : S.a + S.b + (S.a + (↑η : 𝓞 K) * S.b) * (↑η : 𝓞 K) +
    (S.a + (-(↑(IsUnit.unit (a := hζ.toInteger) sorry) : 𝓞 K) - 1) * S.b) *
      (-(↑(IsUnit.unit (a := hζ.toInteger) sorry) : 𝓞 K) - 1) =
  0
Tactic: rw [X_u₁_spec, Y_u₂_spec, Z_u₃_spec, mul_comm S.x, ← x_spec, mul_comm S.y, ← y_spec, mul_comm S.z,
    ← z_spec, 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
⊢ S.a + S.b + HPow.hPow (β := ℕ) (↑η : 𝓞 K) 2 * S.b - S.a + HPow.hPow (β := ℕ) (↑η : 𝓞 K) 2 * S.b +
        2 * (↑η : 𝓞 K) * S.b +
      S.b =
    0
have rw : S.a + S.b + (-(↑(IsUnit.unit (a := hζ.toInteger) sorry) : 𝓞 K) - 1) * S.b - S.a +
        (-(↑(IsUnit.unit (a := hζ.toInteger) sorry) : 𝓞 K) - 1) * S.b +
      2 * (↑η : 𝓞 K) * S.b +
    S.b =
  0
Tactic: rw [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
⊢ S.a + S.b + (S.a + (↑η : 𝓞 K) * S.b) * (↑η : 𝓞 K) +
      (S.a + (-(↑(IsUnit.unit (a := hζ.toInteger) ⋯) : 𝓞 K) - 1) * S.b) *
        (-(↑(IsUnit.unit (a := hζ.toInteger) ⋯) : 𝓞 K) - 1) =
    0
have calc : S.a + S.b + (S.a + (↑η : 𝓞 K) * S.b) * (↑η : 𝓞 K) +
    (S.a + (-(↑(IsUnit.unit (a := hζ.toInteger) sorry) : 𝓞 K) - 1) * S.b) *
      (-(↑(IsUnit.unit (a := hζ.toInteger) sorry) : 𝓞 K) - 1) =
  S.a + S.b + HPow.hPow (β := ℕ) (↑η : 𝓞 K) 2 * S.b - S.a + HPow.hPow (β := ℕ) (↑η : 𝓞 K) 2 * S.b +
      2 * (↑η : 𝓞 K) * S.b +
    S.b
Tactic: calc _ = S.a+S.b+η^2*S.b-S.a+η^2*S.b+2*η*S.b+S.b := by ring
  _ = 0 := by rw [eta_sq]; 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
calc :
  S.a + S.b + (S.a + (↑η : 𝓞 K) * S.b) * (↑η : 𝓞 K) +
      (S.a + (-(↑(IsUnit.unit (a := hζ.toInteger) ⋯) : 𝓞 K) - 1) * S.b) *
        (-(↑(IsUnit.unit (a := hζ.toInteger) ⋯) : 𝓞 K) - 1) =
    S.a + S.b + HPow.hPow (β := ℕ) (↑η : 𝓞 K) 2 * S.b - S.a + HPow.hPow (β := ℕ) (↑η : 𝓞 K) 2 * S.b +
        2 * (↑η : 𝓞 K) * S.b +
      S.b
⊢ S.a + S.b + (S.a + (↑η : 𝓞 K) * S.b) * (↑η : 𝓞 K) +
      (S.a + (-(↑(IsUnit.unit (a := hζ.toInteger) ⋯) : 𝓞 K) - 1) * S.b) *
        (-(↑(IsUnit.unit (a := hζ.toInteger) ⋯) : 𝓞 K) - 1) =
    0
have calc₁ : S.a + S.b + HPow.hPow (β := ℕ) (↑η : 𝓞 K) 2 * S.b - S.a + HPow.hPow (β := ℕ) (↑η : 𝓞 K) 2 * S.b + 2 * (↑η : 𝓞 K) * S.b +
    S.b =
  0
Tactic: calc _ = S.a+S.b+η^2*S.b-S.a+η^2*S.b+2*η*S.b+S.b := by ring
  _ = 0 := by rw [eta_sq]; 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
⊢ HPow.hPow (β := ℕ) S.Y 3 + (↑S.u₄ : 𝓞 K) * HPow.hPow (β := ℕ) S.Z 3 =
    (↑S.u₅ : 𝓞 K) * HPow.hPow (β := ℕ) (λ ^ (S.multiplicity - 1) * S.X) 3
have rw : (↑(HPow.hPow (β := ℕ) η 2 * S.u₁ * S.u₂⁻¹) : 𝓞 K) * HPow.hPow (β := ℕ) (λ ^ (S.multiplicity - 1) * S.X) 3 +
      HPow.hPow (β := ℕ) S.Y 3 +
    (↑S.u₄ : 𝓞 K) * HPow.hPow (β := ℕ) S.Z 3 =
  0
Tactic: rw [u₅_def, neg_mul, neg_mul, Units.val_neg, neg_mul, eq_neg_iff_add_eq_zero, add_assoc,
    add_comm (S.u₄ * S.Z ^ 3), ← add_assoc, add_comm (S.Y ^ 3)]
-- 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) η 2 * S.u₁ * S.u₂⁻¹) : 𝓞 K) * HPow.hPow (β := ℕ) (λ ^ (S.multiplicity - 1) * S.X) 3 +
        HPow.hPow (β := ℕ) S.Y 3 +
      (↑S.u₄ : 𝓞 K) * HPow.hPow (β := ℕ) S.Z 3 =
    0
have apply : ((↑(HPow.hPow (β := ℕ) η 2 * S.u₁ * S.u₂⁻¹) : 𝓞 K) * HPow.hPow (β := ℕ) (λ ^ (S.multiplicity - 1) * S.X) 3 +
        HPow.hPow (β := ℕ) S.Y 3 +
      (↑S.u₄ : 𝓞 K) * HPow.hPow (β := ℕ) S.Z 3) *
    (λ * (↑S.u₂ : 𝓞 K) * (↑η : 𝓞 K)) =
  0 * (λ * (↑S.u₂ : 𝓞 K) * (↑η : 𝓞 K))
Tactic: apply mul_right_cancel₀ <| mul_ne_zero
    (mul_ne_zero hζ.zeta_sub_one_prime'.ne_zero S.u₂.isUnit.ne_zero) (Units.isUnit η).ne_zero
-- 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) η 2 * S.u₁ * S.u₂⁻¹) : 𝓞 K) * HPow.hPow (β := ℕ) (λ ^ (S.multiplicity - 1) * S.X) 3 +
          HPow.hPow (β := ℕ) S.Y 3 +
        (↑S.u₄ : 𝓞 K) * HPow.hPow (β := ℕ) S.Z 3) *
      (λ * (↑S.u₂ : 𝓞 K) * (↑η : 𝓞 K)) =
    0 * (λ * (↑S.u₂ : 𝓞 K) * (↑η : 𝓞 K))
have simp : (↑(HPow.hPow (β := ℕ) η 2 * S.u₁ * S.u₂⁻¹) : 𝓞 K) * HPow.hPow (β := ℕ) (λ ^ (S.multiplicity - 1) * S.X) 3 *
        (λ * (↑S.u₂ : 𝓞 K) * (↑η : 𝓞 K)) +
      HPow.hPow (β := ℕ) S.Y 3 * (λ * (↑S.u₂ : 𝓞 K) * (↑η : 𝓞 K)) +
    (↑S.u₄ : 𝓞 K) * HPow.hPow (β := ℕ) S.Z 3 * (λ * (↑S.u₂ : 𝓞 K) * (↑η : 𝓞 K)) =
  0
Tactic: simp only [zero_mul, add_mul]
-- 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) η 2 * S.u₁ * S.u₂⁻¹) : 𝓞 K) * HPow.hPow (β := ℕ) (λ ^ (S.multiplicity - 1) * S.X) 3 *
          (λ * (↑S.u₂ : 𝓞 K) * (↑η : 𝓞 K)) +
        HPow.hPow (β := ℕ) S.Y 3 * (λ * (↑S.u₂ : 𝓞 K) * (↑η : 𝓞 K)) +
      (↑S.u₄ : 𝓞 K) * HPow.hPow (β := ℕ) S.Z 3 * (λ * (↑S.u₂ : 𝓞 K) * (↑η : 𝓞 K)) =
    0
have rw : (↑(HPow.hPow (β := ℕ) η 2 * S.u₁ * S.u₂⁻¹) : 𝓞 K) * HPow.hPow (β := ℕ) (λ ^ (S.multiplicity - 1) * S.X) 3 *
        (λ * (↑S.u₂ : 𝓞 K) * (↑η : 𝓞 K)) +
      HPow.hPow (β := ℕ) S.Y 3 * (λ * (↑S.u₂ : 𝓞 K) * (↑η : 𝓞 K)) +
    (↑S.u₄ : 𝓞 K) * HPow.hPow (β := ℕ) S.Z 3 * (λ * (↑S.u₂ : 𝓞 K) * (↑η : 𝓞 K)) =
  HPow.hPow (β := ℕ) S.X 3 * (↑S.u₁ : 𝓞 K) * λ ^ (3 * S.multiplicity - 2) +
      HPow.hPow (β := ℕ) S.Y 3 * (↑S.u₂ : 𝓞 K) * λ * (↑η : 𝓞 K) +
    HPow.hPow (β := ℕ) S.Z 3 * (↑S.u₃ : 𝓞 K) * λ * HPow.hPow (β := ℕ) (↑η : 𝓞 K) 2
Tactic: rw [← formula1 S]