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
h : HPow.hPow (β := ℕ) (λ ^ S.multiplicity) 3 ∣ (↑S.u : 𝓞 K) * HPow.hPow (β := ℕ) S.c 3
⊢ 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

Premises used: _private.0.FermatLastTheoremForThreeGen.Solution.z_spec, _private.0.FermatLastTheoremForThreeGen.Solution'.H, pow_mul, mul_comm, _private.0.FermatLastTheoremForThreeGen.a_cube_add_b_cube_eq_mul, _private.0.FermatLastTheoremForThreeGen.Solution.toSolution', _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
h : λ ^ (3 * S.multiplicity) ∣ (S.a + S.b) * (λ * S.y) * (λ * S.z)
apply : λ ^ (3 * S.multiplicity - 2) ∣ S.z * (S.a + S.b)
⊢ λ ^ (3 * S.multiplicity - 2) ∣ S.a + S.b

Tactic:
apply hζ.zeta_sub_one_prime'.pow_dvd_of_dvd_mul_left _ S.lambda_not_dvd_z

Premises used: _private.0.FermatLastTheoremForThreeGen.Solution.lambda_not_dvd_z, Prime.pow_dvd_of_dvd_mul_left, IsPrimitiveRoot.zeta_sub_one_prime'
-- 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 ^ (3 * S.multiplicity) ∣ (S.a + S.b) * (λ * S.y) * (λ * S.z)
apply : λ ^ (3 * S.multiplicity - 2) ∣ S.y * (S.z * (S.a + S.b))
⊢ 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_y

Premises used: _private.0.FermatLastTheoremForThreeGen.Solution.lambda_not_dvd_y, Prime.pow_dvd_of_dvd_mul_left, IsPrimitiveRoot.zeta_sub_one_prime'
-- 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 ^ (3 * S.multiplicity) ∣ (S.a + S.b) * (λ * S.y) * (λ * S.z)
⊢ 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
h : λ ^ (3 * S.multiplicity) ∣ (S.a + S.b) * (λ * S.y) * (λ * S.z)
this : 2 ≤ S.multiplicity
⊢ 3 * S.multiplicity = 3 * S.multiplicity - 2 + 1 + 1

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
h : λ ^ (3 * S.multiplicity - 2) * λ * λ ∣ (S.a + S.b) * (λ * S.y) * (λ * S.z)
this : 2 ≤ S.multiplicity
⊢ (S.a + S.b) * (λ * S.y) * (λ * S.z) = (S.a + S.b) * S.y * S.z * λ * λ

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
h : λ ^ (3 * S.multiplicity) ∣ (S.a + S.b) * (λ * S.y) * (λ * S.z)
this : 2 ≤ S.multiplicity
⊢ 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

Premises used: pow_succ
-- 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 ^ (3 * S.multiplicity) ∣ (S.a + S.b) * (λ * S.y) * (λ * S.z)
this : 2 ≤ S.multiplicity
h₁ : λ ^ (3 * S.multiplicity - 2) * λ * λ ∣ (S.a + S.b) * S.y * S.z * λ * λ
⊢ 2 ≤ S.multiplicity

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

Premises used: pow_succ
-- 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 ^ (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.a + S.b) * S.y * S.z

Tactic:
simp only [mul_dvd_mul_iff_right hζ.zeta_sub_one_prime'.ne_zero] at h

Premises used: mul_dvd_mul_iff_right, Prime.ne_zero, IsPrimitiveRoot.zeta_sub_one_prime'
-- 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
h : λ ^ (3 * S.multiplicity - 2) ∣ (S.a + S.b) * S.y * S.z
⊢ (S.a + S.b) * S.y * S.z = S.y * (S.z * (S.a + 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
this : 2 ≤ S.multiplicity
h : λ ^ (3 * S.multiplicity - 2) ∣ (S.a + S.b) * S.y * S.z
rwa rwa₁ rwa₂ rwa₃ :
  ∀ (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))
⊢ 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
-- 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 = (hζ.toInteger - 1) ^ (3 * S.multiplicity - 2) * S.x

Tactic:
(lambda_pow_dvd_a_add_b S).choose_spec

Premises used: Exists.choose_spec, _private.0.FermatLastTheoremForThreeGen.Solution.x, _private.0.FermatLastTheoremForThreeGen.Solution.lambda_pow_dvd_a_add_b
-- Module: Mathlib.NumberTheory.FLT.Three
Goal:
K : Type u_1
inst : Field K
ζ : K
hζ : IsPrimitiveRoot ζ 3
S : Solution hζ
⊢ S.c = (hζ.toInteger - 1) ^ S.multiplicity * S.w

Tactic:
(pow_multiplicity_dvd (hζ.toInteger - 1) S.c).choose_spec

Premises used: pow_multiplicity_dvd, Exists.choose_spec, _private.0.FermatLastTheoremForThreeGen.Solution.multiplicity, _private.0.FermatLastTheoremForThreeGen.Solution.w, HSub.hSub, _private.0.FermatLastTheoremForThreeGen.Solution'.multiplicity, IsPrimitiveRoot.toInteger, OfNat.ofNat, _private.0.FermatLastTheoremForThreeGen.Solution'.c, _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
h : λ ∣ S.w
⊢ False

Tactic:
S.toSolution'.multiplicity_lambda_c_finite.not_pow_dvd_of_multiplicity_lt
    (lt_add_one S.multiplicity) ?_

Premises used: _private.0.FermatLastTheoremForThreeGen.Solution.multiplicity, lt_add_one, FiniteMultiplicity.not_pow_dvd_of_multiplicity_lt, _private.0.FermatLastTheoremForThreeGen.Solution'.multiplicity_lambda_c_finite, _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
h : λ ∣ S.w
refine : λ ^ (S.multiplicity + 1) ∣ S.c
⊢ False

Tactic:
refine S.toSolution'.multiplicity_lambda_c_finite.not_pow_dvd_of_multiplicity_lt
    (lt_add_one S.multiplicity) ?_

Premises used: _private.0.FermatLastTheoremForThreeGen.Solution.multiplicity, lt_add_one, FiniteMultiplicity.not_pow_dvd_of_multiplicity_lt, _private.0.FermatLastTheoremForThreeGen.Solution'.multiplicity_lambda_c_finite, _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
h : λ ∣ S.w
rw : λ ^ S.multiplicity * λ ∣ S.c
⊢ λ ^ (S.multiplicity + 1) ∣ S.c

Tactic:
rw [pow_succ', mul_comm]

Premises used: mul_comm, pow_succ'
-- 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.w
⊢ λ ^ S.multiplicity * λ ∣ S.c

Tactic:
S.w_spec ▸ (mul_dvd_mul_left (λ ^ S.multiplicity) h)

Premises used: _private.0.FermatLastTheoremForThreeGen.Solution.multiplicity, HSub.hSub, Eq.rec, mul_dvd_mul_left, _private.0.FermatLastTheoremForThreeGen.Solution.w_spec, HPow.hPow, IsPrimitiveRoot.toInteger, OfNat.ofNat, Eq.symm
-- 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.w
⊢ λ ^ S.multiplicity * λ ∣ S.c

Tactic:
exact S.w_spec ▸ (mul_dvd_mul_left (λ ^ S.multiplicity) h)
-- 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
⊢ 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
-- 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 ^ (3 * S.multiplicity - 2) * λ ∣ λ ^ (3 * S.multiplicity - 2) * S.x
⊢ λ * λ ^ (3 * S.multiplicity - 2) ∣ S.a + S.b

Tactic:
rw [mul_comm, ← x_spec] at h

Premises used: mul_comm, _private.0.FermatLastTheoremForThreeGen.Solution.x_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 * λ ^ (3 * S.multiplicity - 2) ∣ S.a + S.b
⊢ 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
-- 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 * 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)
⊢ 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

Premises used: IsUnit.dvd_mul_left, _private.0.FermatLastTheoremForThreeGen.Solution'.H, _private.0.FermatLastTheoremForThreeGen.Solution.w_spec, Units.isUnit, _private.0.FermatLastTheoremForThreeGen.a_cube_add_b_cube_eq_mul, _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
h : λ * λ ^ (3 * S.multiplicity - 2) * λ * λ ∣ HPow.hPow (β := ℕ) (λ ^ S.multiplicity * S.w) 3
⊢ 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

Premises used: mul_assoc, mul_comm, pow_succ'
-- 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 ^ (3 * S.multiplicity - 2 + 1 + 1) * λ ∣ HPow.hPow (β := ℕ) (λ ^ S.multiplicity * S.w) 3
⊢ 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
h : λ ^ (3 * S.multiplicity - 2 + 1 + 1) * λ ∣ HPow.hPow (β := ℕ) (λ ^ S.multiplicity * S.w) 3
this : 2 ≤ S.multiplicity
⊢ 3 * S.multiplicity - 2 + 1 + 1 = 3 * S.multiplicity

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
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
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
⊢ 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

Premises used: pow_mul, mul_dvd_mul_iff_left, mul_comm, OfNat.ofNat, mul_pow
-- 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 ∣ HPow.hPow (β := ℕ) S.w 3
this : 2 ≤ S.multiplicity
⊢ False

Tactic:
lambda_not_dvd_w _ <| hζ.zeta_sub_one_prime'.dvd_of_dvd_pow h

Premises used: _private.0.FermatLastTheoremForThreeGen.Solution.lambda_not_dvd_w, Prime.dvd_of_dvd_pow, IsPrimitiveRoot.zeta_sub_one_prime'
-- 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 ∣ HPow.hPow (β := ℕ) S.w 3
this : 2 ≤ S.multiplicity
⊢ False

Tactic:
exact lambda_not_dvd_w _ <| hζ.zeta_sub_one_prime'.dvd_of_dvd_pow h

Premises used: _private.0.FermatLastTheoremForThreeGen.Solution.lambda_not_dvd_w, Prime.dvd_of_dvd_pow, IsPrimitiveRoot.zeta_sub_one_prime'
-- 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 ^ (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:
simp [hζ.zeta_sub_one_prime'.ne_zero]

Premises used: Prime.ne_zero, IsPrimitiveRoot.zeta_sub_one_prime'
-- Module: Mathlib.NumberTheory.FLT.Three
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
h : r = 0
hz : s = 0
⊢ λ ∣ s

Tactic:
simp [hz]
-- Module: Mathlib.NumberTheory.FLT.Three
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
refine : ∀ (p : 𝓞 K) (hp : Prime p) (p_dvd_r : p ∣ r) (p_dvd_s : p ∣ s), λ ∣ r
⊢ IsCoprime r s

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 ?_)

Premises used: Iff.mpr, isCoprime_of_prime_dvd, not_and
-- Module: Mathlib.NumberTheory.FLT.Three
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
p : 𝓞 K
hp : Prime p
p_dvd_r : p ∣ r
p_dvd_s : p ∣ s
rwa rwa₁ rwa₂ rwa₃ : p ∣ r
⊢ λ ∣ r

Tactic:
rwa [← Associated.dvd_iff_dvd_left <| Hp hp (H₁ p_dvd_r) (H₂ p_dvd_s)]

Premises used: Associated.dvd_iff_dvd_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
⊢ IsCoprime S.x S.y

Tactic:
isCoprime_helper (lambda_not_dvd_x S) (lambda_not_dvd_y S)
    (associated_of_dvd_a_add_b_of_dvd_a_add_eta_mul_b S) (fun hq ↦ x_spec S ▸ hq.mul_left _)
      (fun hq ↦ y_spec S ▸ hq.mul_left _)

Premises used: _private.0.FermatLastTheoremForThreeGen.Solution.multiplicity, HMul.hMul, HSub.hSub, _private.0.FermatLastTheoremForThreeGen.Solution.lambda_not_dvd_x, Eq.rec, Dvd.dvd.mul_left, _private.0.FermatLastTheoremForThreeGen.Solution.lambda_not_dvd_y, HPow.hPow, _private.0.FermatLastTheoremForThreeGen.Solution.associated_of_dvd_a_add_b_of_dvd_a_add_eta_mul_b, _private.0.FermatLastTheoremForThreeGen.Solution.isCoprime_helper, IsPrimitiveRoot.toInteger, OfNat.ofNat, Eq.symm, _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
⊢ IsCoprime S.x S.z

Tactic:
isCoprime_helper (lambda_not_dvd_x S) (lambda_not_dvd_z S)
    (associated_of_dvd_a_add_b_of_dvd_a_add_eta_sq_mul_b S) (fun hq ↦ x_spec S ▸ hq.mul_left _)
      (fun hq ↦ z_spec S ▸ hq.mul_left _)

Premises used: _private.0.FermatLastTheoremForThreeGen.Solution.multiplicity, HMul.hMul, _private.0.FermatLastTheoremForThreeGen.Solution.associated_of_dvd_a_add_b_of_dvd_a_add_eta_sq_mul_b, _private.0.FermatLastTheoremForThreeGen.Solution.z_spec, HSub.hSub, _private.0.FermatLastTheoremForThreeGen.Solution.lambda_not_dvd_x, Eq.rec, Dvd.dvd.mul_left, HPow.hPow, _private.0.FermatLastTheoremForThreeGen.Solution.isCoprime_helper, IsPrimitiveRoot.toInteger, OfNat.ofNat, Eq.symm, _private.0.FermatLastTheoremForThreeGen.Solution.lambda_not_dvd_z, _private.0.FermatLastTheoremForThreeGen.Solution.x_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
⊢ IsCoprime S.y S.z

Tactic:
isCoprime_helper (lambda_not_dvd_y S) (lambda_not_dvd_z S)
    (associated_of_dvd_a_add_eta_mul_b_of_dvd_a_add_eta_sq_mul_b S)
    (fun hq ↦ y_spec S ▸ hq.mul_left _) (fun hq ↦ z_spec S ▸ hq.mul_left _)

Premises used: _private.0.FermatLastTheoremForThreeGen.Solution.z_spec, HSub.hSub, Eq.rec, Dvd.dvd.mul_left, _private.0.FermatLastTheoremForThreeGen.Solution.lambda_not_dvd_y, _private.0.FermatLastTheoremForThreeGen.Solution.isCoprime_helper, IsPrimitiveRoot.toInteger, OfNat.ofNat, Eq.symm, _private.0.FermatLastTheoremForThreeGen.Solution.associated_of_dvd_a_add_eta_mul_b_of_dvd_a_add_eta_sq_mul_b, _private.0.FermatLastTheoremForThreeGen.Solution.lambda_not_dvd_z, _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
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
⊢ λ ^ (3 * S.multiplicity - 2) * S.x * λ * S.y * λ * S.z = λ ^ (3 * S.multiplicity - 2) * λ * λ * S.x * S.y * S.z

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
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
⊢ 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
-- 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
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
⊢ 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
hh :
  λ ^ (3 * S.multiplicity - 2 + 1 + 1) * 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
⊢ 3 * S.multiplicity - 2 + 1 + 1 = 3 * S.multiplicity

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
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
⊢ 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

Premises used: pow_succ, mul_assoc
-- 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
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
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)
⊢ 2 ≤ S.multiplicity

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

Premises used: pow_succ, mul_assoc
-- 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
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

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

Premises used: OfNat.ofNat_ne_zero, mul_eq_mul_left_iff, false_and, pow_eq_zero_iff', false_or, mul_eq_zero, ne_eq, Prime.ne_zero, or_false, IsPrimitiveRoot.zeta_sub_one_prime'
-- 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
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

Tactic:
hh
-- 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
hh : S.x * (S.y * S.z) = (↑S.u : 𝓞 K) * HPow.hPow (β := ℕ) S.w 3
h : S.x * S.y * S.z = S.x * (S.y * S.z)
⊢ S.x * S.y * S.z = (↑S.u : 𝓞 K) * HPow.hPow (β := ℕ) S.w 3

Tactic:
convert! hh using 1
-- 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
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.x * (S.y * S.z)

Tactic:
ring