awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Module: Mathlib.NumberTheory.FLT.Three
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
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
⊢ S.x * S.y * S.z = (↑S.u : 𝓞 K) * 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
-- Module: Mathlib.NumberTheory.FLT.Three
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
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)
⊢ 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:
simp only [← x_spec, mul_assoc, ← y_spec, ← z_spec]

Premises used: _private.0.FermatLastTheoremForThreeGen.Solution.z_spec, mul_assoc, _private.0.FermatLastTheoremForThreeGen.Solution.x_spec, _private.0.FermatLastTheoremForThreeGen.Solution.y_spec
-- Module: Mathlib.NumberTheory.FLT.Three
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
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)
⊢ (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:
rw [mul_comm 3, pow_mul, ← mul_pow, ← w_spec, ← S.H, a_cube_add_b_cube_eq_mul]

Premises used: _private.0.FermatLastTheoremForThreeGen.Solution'.H, pow_mul, _private.0.FermatLastTheoremForThreeGen.Solution.w_spec, mul_comm, _private.0.FermatLastTheoremForThreeGen.a_cube_add_b_cube_eq_mul, OfNat.ofNat, mul_pow, _private.0.FermatLastTheoremForThreeGen.Solution.toSolution'
-- Module: Mathlib.NumberTheory.FLT.Three
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.a + S.b) * (S.a + (↑η : 𝓞 K) * S.b) * (S.a + HPow.hPow (β := ℕ) (↑η : 𝓞 K) 2 * S.b)

Tactic:
ring
-- Module: Mathlib.NumberTheory.FLT.Three
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.x * S.y) S.z

Tactic:
have h₁ := S.isCoprime_x_z.mul_left S.isCoprime_y_z
-- Module: Mathlib.NumberTheory.FLT.Three
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
rw₁ rw₂ : (↑S.u : 𝓞 K) * HPow.hPow (β := ℕ) S.w 3 = (↑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

Tactic:
rw [x_mul_y_mul_z_eq_u_mul_w_cube S, mul_comm]

Premises used: _private.0.FermatLastTheoremForThreeGen.Solution.x_mul_y_mul_z_eq_u_mul_w_cube, mul_comm
-- Module: Mathlib.NumberTheory.FLT.Three
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
⊢ 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]⟩
-- Module: Mathlib.NumberTheory.FLT.Three
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)
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
⊢ (∃ (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

Premises used: exists_associated_pow_of_associated_pow_mul
-- Module: Mathlib.NumberTheory.FLT.Three
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)
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:
⟨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)⟩

Premises used: exists_associated_pow_of_associated_pow_mul, IsCoprime.symm, And.intro, _private.0.FermatLastTheoremForThreeGen.Solution.isCoprime_x_y
-- Module: Mathlib.NumberTheory.FLT.Three
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)
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:
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)⟩
-- Module: Mathlib.NumberTheory.FLT.Three
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
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
⊢ (∃ (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)⟩
-- Module: Mathlib.NumberTheory.FLT.Three
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) = S.x

Tactic:
(exists_cube_associated S).1.choose_spec.choose_spec

Premises used: Exists.choose_spec, _private.0.FermatLastTheoremForThreeGen.Solution.X, _private.0.FermatLastTheoremForThreeGen.Solution.exists_cube_associated, And.left
-- Module: Mathlib.NumberTheory.FLT.Three
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.u₂ : 𝓞 K) = S.y

Tactic:
(exists_cube_associated S).2.1.choose_spec.choose_spec

Premises used: Exists.choose_spec, _private.0.FermatLastTheoremForThreeGen.Solution.exists_cube_associated, _private.0.FermatLastTheoremForThreeGen.Solution.Y, And.right, And.left
-- Module: Mathlib.NumberTheory.FLT.Three
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.Z 3 * (↑S.u₃ : 𝓞 K) = S.z

Tactic:
(exists_cube_associated S).2.2.choose_spec.choose_spec

Premises used: Exists.choose_spec, _private.0.FermatLastTheoremForThreeGen.Solution.exists_cube_associated, _private.0.FermatLastTheoremForThreeGen.Solution.Z, And.right
-- Module: Mathlib.NumberTheory.FLT.Three
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 : S.X = 0
⊢ λ ∣ S.x

Tactic:
simp [← X_u₁_spec, h]

Premises used: _private.0.FermatLastTheoremForThreeGen.Solution.X_u₁_spec
-- Module: Mathlib.NumberTheory.FLT.Three
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
⊢ Ne (α := ℕ) 3 0

Tactic:
decide
-- Module: Mathlib.NumberTheory.FLT.Three
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
⊢ Ne (α := ℕ) 3 0

Tactic:
decide
-- Module: Mathlib.NumberTheory.FLT.Three
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
⊢ Ne (α := ℕ) 3 0

Tactic:
decide
-- Module: Mathlib.NumberTheory.FLT.Three
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
⊢ LT.lt (α := ℕ) 0 3

Tactic:
decide
-- Module: Mathlib.NumberTheory.FLT.Three
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
rw : IsCoprime S.y S.z
⊢ 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]

Premises used: isCoprime_mul_unit_right_right, _private.0.FermatLastTheoremForThreeGen.Solution.Z_u₃_spec, isCoprime_mul_unit_right_left, _private.0.FermatLastTheoremForThreeGen.Solution.u₃, IsCoprime.pow_iff, Units.isUnit, _private.0.FermatLastTheoremForThreeGen.Solution.Y_u₂_spec, _private.0.FermatLastTheoremForThreeGen.Solution.u₂
-- Module: Mathlib.NumberTheory.FLT.Three
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

Tactic:
isCoprime_y_z S

Premises used: _private.0.FermatLastTheoremForThreeGen.Solution.isCoprime_y_z
-- Module: Mathlib.NumberTheory.FLT.Three
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

Tactic:
exact isCoprime_y_z S

Premises used: _private.0.FermatLastTheoremForThreeGen.Solution.isCoprime_y_z
-- Module: Mathlib.NumberTheory.FLT.Three
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
rw :
  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
⊢ 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

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]

Premises used: _private.0.FermatLastTheoremForThreeGen.Solution.z_spec, _private.0.FermatLastTheoremForThreeGen.Solution.y, _private.0.FermatLastTheoremForThreeGen.Solution.Z_u₃_spec, IsCyclotomicExtension.Rat.Three.eta_sq, _private.0.FermatLastTheoremForThreeGen.Solution.x, mul_comm, _private.0.FermatLastTheoremForThreeGen.Solution.X_u₁_spec, _private.0.FermatLastTheoremForThreeGen.Solution.z, _private.0.FermatLastTheoremForThreeGen.Solution.Y_u₂_spec, _private.0.FermatLastTheoremForThreeGen.Solution.x_spec, _private.0.FermatLastTheoremForThreeGen.Solution.y_spec
-- Module: Mathlib.NumberTheory.FLT.Three
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) =
    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:
ring
-- Module: Mathlib.NumberTheory.FLT.Three
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
rw :
  S.a + S.b + (-(↑(IsUnit.unit (a := hζ.toInteger) ⋯) : 𝓞 K) - 1) * S.b - S.a +
          (-(↑(IsUnit.unit (a := hζ.toInteger) ⋯) : 𝓞 K) - 1) * S.b +
        2 * (↑η : 𝓞 K) * S.b +
      S.b =
    0
⊢ 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:
rw [eta_sq]

Premises used: IsCyclotomicExtension.Rat.Three.eta_sq
-- Module: Mathlib.NumberTheory.FLT.Three
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 + (-(↑(IsUnit.unit (a := hζ.toInteger) ⋯) : 𝓞 K) - 1) * S.b - S.a +
          (-(↑(IsUnit.unit (a := hζ.toInteger) ⋯) : 𝓞 K) - 1) * S.b +
        2 * (↑η : 𝓞 K) * S.b +
      S.b =
    0

Tactic:
ring
-- Module: Mathlib.NumberTheory.FLT.Three
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
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
⊢ 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

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

Premises used: _private.0.FermatLastTheoremForThreeGen.Solution'.a, _private.0.FermatLastTheoremForThreeGen.Solution'.b, _private.0.FermatLastTheoremForThreeGen.Solution.toSolution'
-- Module: Mathlib.NumberTheory.FLT.Three
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.u₄ = IsUnit.unit (a := hζ.toInteger) ⋯ * S.u₃ * S.u₂⁻¹

Tactic:
rfl

Premises used: rfl, _private.0.FermatLastTheoremForThreeGen.Solution.u₄
-- Module: Mathlib.NumberTheory.FLT.Three
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.u₅ = -HPow.hPow (β := ℕ) (IsUnit.unit (a := hζ.toInteger) ⋯) 2 * S.u₁ * S.u₂⁻¹

Tactic:
rfl

Premises used: _private.0.FermatLastTheoremForThreeGen.Solution.u₅, rfl
-- Module: Mathlib.NumberTheory.FLT.Three
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
a b : 𝓞 K
ha : a ≠ 0
hb : b ≠ 0
⊢ a * b ≠ 0

Tactic:
mul_ne_zero ha hb

Premises used: mul_ne_zero
-- Module: Mathlib.NumberTheory.FLT.Three
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
a b : 𝓞 K
ha : a ≠ 0
hb : b ≠ 0
⊢ a * b ≠ 0

Tactic:
exact mul_ne_zero ha hb

Premises used: mul_ne_zero
-- Module: Mathlib.NumberTheory.FLT.Three
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
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
⊢ 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

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)]

Premises used: Units.val, HMul.hMul, add_assoc, _private.0.FermatLastTheoremForThreeGen.Solution.Y, _private.0.FermatLastTheoremForThreeGen.Solution.u₅_def, add_comm, _private.0.FermatLastTheoremForThreeGen.Solution.Z, Units.val_neg, HPow.hPow, neg_mul, eq_neg_iff_add_eq_zero, OfNat.ofNat, _private.0.FermatLastTheoremForThreeGen.Solution.u₄
-- Module: Mathlib.NumberTheory.FLT.Three
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
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))
⊢ (↑(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:
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

Premises used: mul_right_cancel₀
-- Module: Mathlib.NumberTheory.FLT.Three
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
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
⊢ ((↑(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:
simp only [zero_mul, add_mul]

Premises used: add_mul, MulZeroClass.zero_mul
-- Module: Mathlib.NumberTheory.FLT.Three
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
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
⊢ (↑(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:
rw [← formula1 S]

Premises used: _private.0.FermatLastTheoremForThreeGen.Solution.formula1
-- Module: Mathlib.NumberTheory.FLT.Three
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
refine_1 :
  (↑(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.X 3 * (↑S.u₁ : 𝓞 K) * λ ^ (3 * S.multiplicity - 2)
refine_2 :
  HPow.hPow (β := ℕ) S.Y 3 * (λ * (↑S.u₂ : 𝓞 K) * (↑η : 𝓞 K)) =
    HPow.hPow (β := ℕ) S.Y 3 * (↑S.u₂ : 𝓞 K) * λ * (↑η : 𝓞 K)
refine_3 :
  (↑S.u₄ : 𝓞 K) * HPow.hPow (β := ℕ) S.Z 3 * (λ * (↑S.u₂ : 𝓞 K) * (↑η : 𝓞 K)) =
    HPow.hPow (β := ℕ) S.Z 3 * (↑S.u₃ : 𝓞 K) * λ * HPow.hPow (β := ℕ) (↑η : 𝓞 K) 2
⊢ (↑(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:
congrm ?_ + ?_ + ?_
-- Module: Mathlib.NumberTheory.FLT.Three
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
⊢ 2 ≤ S.multiplicity

Tactic:
have := S.two_le_multiplicity
-- Module: Mathlib.NumberTheory.FLT.Three
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
⊢ (S.multiplicity - 1) * 3 + 1 = 3 * S.multiplicity - 2

Tactic:
lia
-- Module: Mathlib.NumberTheory.FLT.Three
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 - 1) * 3 + 1 = 3 * S.multiplicity - 2

Tactic:
have : (S.multiplicity-1)*3+1 = 3*S.multiplicity-2 := by have := S.two_le_multiplicity; lia