ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- FermatLastTheoremForThreeGen.exists_Solution_of_Solution'
lemma exists_Solution_of_Solution' : ∃ (S₁ : Solution hζ), S₁.multiplicity = S'.multiplicity 
Arguments: {K : Type u_1} [Field K] [NumberField K] [IsCyclotomicExtension {3} ℚ K] {ζ : K} {hζ : IsPrimitiveRoot ζ ↑3} (S' : FermatLastTheoremForThreeGen.Solution' hζ) [DecidableRel fun a b => a ∣ b]
Goal: ∃ S₁, S₁.multiplicity = S'.multiplicity
-- FermatLastTheoremForThreeGen.Solution.a_add_eta_mul_b
lemma a_add_eta_mul_b : S.a + η * S.b = (S.a + S.b) + λ * S.b 
Arguments: {K : Type u_1} [Field K] {ζ : K} {hζ : IsPrimitiveRoot ζ ↑3} (S : FermatLastTheoremForThreeGen.Solution hζ)
Goal: S.a + ↑⋯.unit * S.b = S.a + S.b + (hζ.toInteger - 1) * S.b
-- FermatLastTheoremForThreeGen.Solution.lambda_dvd_a_add_eta_mul_b
lemma lambda_dvd_a_add_eta_mul_b : λ ∣ (S.a + η * S.b) 
Arguments: {K : Type u_1} [Field K] {ζ : K} {hζ : IsPrimitiveRoot ζ ↑3} (S : FermatLastTheoremForThreeGen.Solution hζ)
Goal: hζ.toInteger - 1 ∣ S.a + ↑⋯.unit * S.b
-- FermatLastTheoremForThreeGen.Solution.lambda_dvd_a_add_eta_sq_mul_b
lemma lambda_dvd_a_add_eta_sq_mul_b : λ ∣ (S.a + η ^ 2 * S.b) 
Arguments: {K : Type u_1} [Field K] {ζ : K} {hζ : IsPrimitiveRoot ζ ↑3} (S : FermatLastTheoremForThreeGen.Solution hζ)
Goal: hζ.toInteger - 1 ∣ S.a + ↑⋯.unit ^ 2 * S.b
-- FermatLastTheoremForThreeGen.Solution.lambda_sq_not_dvd_a_add_eta_mul_b
lemma lambda_sq_not_dvd_a_add_eta_mul_b : ¬ λ ^ 2 ∣ (S.a + η * S.b) 
Arguments: {K : Type u_1} [Field K] [NumberField K] [IsCyclotomicExtension {3} ℚ K] {ζ : K} {hζ : IsPrimitiveRoot ζ ↑3} (S : FermatLastTheoremForThreeGen.Solution hζ)
Goal: ¬(hζ.toInteger - 1) ^ 2 ∣ S.a + ↑⋯.unit * S.b
-- FermatLastTheoremForThreeGen.Solution.lambda_sq_not_dvd_a_add_eta_sq_mul_b
lemma lambda_sq_not_dvd_a_add_eta_sq_mul_b : ¬ λ ^ 2 ∣ (S.a + η ^ 2 * S.b) 
Arguments: {K : Type u_1} [Field K] [NumberField K] [IsCyclotomicExtension {3} ℚ K] {ζ : K} {hζ : IsPrimitiveRoot ζ ↑3} (S : FermatLastTheoremForThreeGen.Solution hζ)
Goal: ¬(hζ.toInteger - 1) ^ 2 ∣ S.a + ↑⋯.unit ^ 2 * S.b
-- FermatLastTheoremForThreeGen.Solution.associated_of_dvd_a_add_b_of_dvd_a_add_eta_mul_b
lemma associated_of_dvd_a_add_b_of_dvd_a_add_eta_mul_b {p : 𝓞 K} (hp : Prime p)
    (hpab : p ∣ S.a + S.b) (hpaηb : p ∣ S.a + η * S.b) : Associated p λ 
Arguments: {K : Type u_1} [Field K] [NumberField K] [IsCyclotomicExtension {3} ℚ K] {ζ : K} {hζ : IsPrimitiveRoot ζ ↑3} (S : FermatLastTheoremForThreeGen.Solution hζ) {p : NumberField.RingOfIntegers K} (hp : Prime p) (hpab : p ∣ S.a + S.b) (hpaηb : p ∣ S.a + ↑⋯.unit * S.b)
Goal: Associated p (hζ.toInteger - 1)
-- FermatLastTheoremForThreeGen.Solution.associated_of_dvd_a_add_b_of_dvd_a_add_eta_sq_mul_b
lemma associated_of_dvd_a_add_b_of_dvd_a_add_eta_sq_mul_b {p : 𝓞 K} (hp : Prime p)
    (hpab : p ∣ (S.a + S.b)) (hpaηsqb : p ∣ (S.a + η ^ 2 * S.b)) : Associated p λ 
Arguments: {K : Type u_1} [Field K] [NumberField K] [IsCyclotomicExtension {3} ℚ K] {ζ : K} {hζ : IsPrimitiveRoot ζ ↑3} (S : FermatLastTheoremForThreeGen.Solution hζ) {p : NumberField.RingOfIntegers K} (hp : Prime p) (hpab : p ∣ S.a + S.b) (hpaηsqb : p ∣ S.a + ↑⋯.unit ^ 2 * S.b)
Goal: Associated p (hζ.toInteger - 1)
-- FermatLastTheoremForThreeGen.Solution.associated_of_dvd_a_add_eta_mul_b_of_dvd_a_add_eta_sq_mul_b
lemma associated_of_dvd_a_add_eta_mul_b_of_dvd_a_add_eta_sq_mul_b {p : 𝓞 K} (hp : Prime p)
    (hpaηb : p ∣ S.a + η * S.b) (hpaηsqb : p ∣ S.a + η ^ 2 * S.b) : Associated p λ 
Arguments: {K : Type u_1} [Field K] [NumberField K] [IsCyclotomicExtension {3} ℚ K] {ζ : K} {hζ : IsPrimitiveRoot ζ ↑3} (S : FermatLastTheoremForThreeGen.Solution hζ) {p : NumberField.RingOfIntegers K} (hp : Prime p) (hpaηb : p ∣ S.a + ↑⋯.unit * S.b) (hpaηsqb : p ∣ S.a + ↑⋯.unit ^ 2 * S.b)
Goal: Associated p (hζ.toInteger - 1)
-- _private.Mathlib.NumberTheory.FLT.Three.0.FermatLastTheoremForThreeGen.Solution.y_spec
private lemma y_spec : S.a + η * S.b = λ * S.y 
Arguments: {K : Type u_1} [Field K] {ζ : K} {hζ : IsPrimitiveRoot ζ ↑3} (S : FermatLastTheoremForThreeGen.Solution hζ)
Goal: S.a + ↑⋯.unit * S.b = (hζ.toInteger - 1) * FermatLastTheoremForThreeGen.Solution.y S
-- _private.Mathlib.NumberTheory.FLT.Three.0.FermatLastTheoremForThreeGen.Solution.z_spec
private lemma z_spec : S.a + η ^ 2 * S.b = λ * S.z 
Arguments: {K : Type u_1} [Field K] {ζ : K} {hζ : IsPrimitiveRoot ζ ↑3} (S : FermatLastTheoremForThreeGen.Solution hζ)
Goal: S.a + ↑⋯.unit ^ 2 * S.b = (hζ.toInteger - 1) * FermatLastTheoremForThreeGen.Solution.z S
-- _private.Mathlib.NumberTheory.FLT.Three.0.FermatLastTheoremForThreeGen.Solution.lambda_not_dvd_y
private lemma lambda_not_dvd_y : ¬ λ ∣ S.y 
Arguments: {K : Type u_1} [Field K] [NumberField K] [IsCyclotomicExtension {3} ℚ K] {ζ : K} {hζ : IsPrimitiveRoot ζ ↑3} (S : FermatLastTheoremForThreeGen.Solution hζ)
Goal: ¬hζ.toInteger - 1 ∣ FermatLastTheoremForThreeGen.Solution.y S
-- _private.Mathlib.NumberTheory.FLT.Three.0.FermatLastTheoremForThreeGen.Solution.lambda_not_dvd_z
private lemma lambda_not_dvd_z : ¬ λ ∣ S.z 
Arguments: {K : Type u_1} [Field K] [NumberField K] [IsCyclotomicExtension {3} ℚ K] {ζ : K} {hζ : IsPrimitiveRoot ζ ↑3} (S : FermatLastTheoremForThreeGen.Solution hζ)
Goal: ¬hζ.toInteger - 1 ∣ FermatLastTheoremForThreeGen.Solution.z S
-- _private.Mathlib.NumberTheory.FLT.Three.0.FermatLastTheoremForThreeGen.Solution.lambda_pow_dvd_a_add_b
private lemma lambda_pow_dvd_a_add_b : λ ^ (3 * S.multiplicity - 2) ∣ S.a + S.b 
Arguments: {K : Type u_1} [Field K] [NumberField K] [IsCyclotomicExtension {3} ℚ K] {ζ : K} {hζ : IsPrimitiveRoot ζ ↑3} (S : FermatLastTheoremForThreeGen.Solution hζ) [DecidableRel fun a b => a ∣ b]
Goal: (hζ.toInteger - 1) ^ (3 * S.multiplicity - 2) ∣ S.a + S.b
-- _private.Mathlib.NumberTheory.FLT.Three.0.FermatLastTheoremForThreeGen.Solution.x_spec
private lemma x_spec : S.a + S.b = λ ^ (3 * S.multiplicity - 2) * S.x 
Arguments: {K : Type u_1} [Field K] [NumberField K] [IsCyclotomicExtension {3} ℚ K] {ζ : K} {hζ : IsPrimitiveRoot ζ ↑3} (S : FermatLastTheoremForThreeGen.Solution hζ) [DecidableRel fun a b => a ∣ b]
Goal: S.a + S.b = (hζ.toInteger - 1) ^ (3 * S.multiplicity - 2) * FermatLastTheoremForThreeGen.Solution.x S
-- _private.Mathlib.NumberTheory.FLT.Three.0.FermatLastTheoremForThreeGen.Solution.w_spec
private lemma w_spec : S.c = λ ^ S.multiplicity * S.w 
Arguments: {K : Type u_1} [Field K] [NumberField K] [IsCyclotomicExtension {3} ℚ K] {ζ : K} {hζ : IsPrimitiveRoot ζ ↑3} (S : FermatLastTheoremForThreeGen.Solution hζ) [DecidableRel fun a b => a ∣ b]
Goal: S.c = (hζ.toInteger - 1) ^ S.multiplicity * FermatLastTheoremForThreeGen.Solution.w S
-- _private.Mathlib.NumberTheory.FLT.Three.0.FermatLastTheoremForThreeGen.Solution.lambda_not_dvd_w
private lemma lambda_not_dvd_w : ¬ λ ∣ S.w 
Arguments: {K : Type u_1} [Field K] [NumberField K] [IsCyclotomicExtension {3} ℚ K] {ζ : K} {hζ : IsPrimitiveRoot ζ ↑3} (S : FermatLastTheoremForThreeGen.Solution hζ) [DecidableRel fun a b => a ∣ b]
Goal: ¬hζ.toInteger - 1 ∣ FermatLastTheoremForThreeGen.Solution.w S
-- _private.Mathlib.NumberTheory.FLT.Three.0.FermatLastTheoremForThreeGen.Solution.lambda_not_dvd_x
private lemma lambda_not_dvd_x : ¬ λ ∣ S.x 
Arguments: {K : Type u_1} [Field K] [NumberField K] [IsCyclotomicExtension {3} ℚ K] {ζ : K} {hζ : IsPrimitiveRoot ζ ↑3} (S : FermatLastTheoremForThreeGen.Solution hζ) [DecidableRel fun a b => a ∣ b]
Goal: ¬hζ.toInteger - 1 ∣ FermatLastTheoremForThreeGen.Solution.x S
-- _private.Mathlib.NumberTheory.FLT.Three.0.FermatLastTheoremForThreeGen.Solution.isCoprime_helper
private lemma isCoprime_helper {r s t w : 𝓞 K} (hr : ¬ λ ∣ r) (hs : ¬ λ ∣ s)
    (Hp : ∀ {p}, Prime p → p ∣ t → p ∣ w → Associated p λ) (H₁ : ∀ {q}, q ∣ r → q ∣ t)
    (H₂ : ∀ {q}, q ∣ s → q ∣ w) : IsCoprime r s 
Arguments: {K : Type u_1} [Field K] [NumberField K] [IsCyclotomicExtension {3} ℚ K] {ζ : K} {hζ : IsPrimitiveRoot ζ ↑3} {r s t w : NumberField.RingOfIntegers K} (hr : ¬hζ.toInteger - 1 ∣ r) (hs : ¬hζ.toInteger - 1 ∣ s) (Hp : ∀ {p : NumberField.RingOfIntegers K}, Prime p → p ∣ t → p ∣ w → Associated p (hζ.toInteger - 1)) (H₁ : ∀ {q : NumberField.RingOfIntegers K}, q ∣ r → q ∣ t) (H₂ : ∀ {q : NumberField.RingOfIntegers K}, q ∣ s → q ∣ w)
Goal: IsCoprime r s
-- _private.Mathlib.NumberTheory.FLT.Three.0.FermatLastTheoremForThreeGen.Solution.isCoprime_x_y
private lemma isCoprime_x_y : IsCoprime S.x S.y 
Arguments: {K : Type u_1} [Field K] [NumberField K] [IsCyclotomicExtension {3} ℚ K] {ζ : K} {hζ : IsPrimitiveRoot ζ ↑3} (S : FermatLastTheoremForThreeGen.Solution hζ) [DecidableRel fun a b => a ∣ b]
Goal: IsCoprime (FermatLastTheoremForThreeGen.Solution.x S) (FermatLastTheoremForThreeGen.Solution.y S)
-- _private.Mathlib.NumberTheory.FLT.Three.0.FermatLastTheoremForThreeGen.Solution.isCoprime_x_z
private lemma isCoprime_x_z : IsCoprime S.x S.z 
Arguments: {K : Type u_1} [Field K] [NumberField K] [IsCyclotomicExtension {3} ℚ K] {ζ : K} {hζ : IsPrimitiveRoot ζ ↑3} (S : FermatLastTheoremForThreeGen.Solution hζ) [DecidableRel fun a b => a ∣ b]
Goal: IsCoprime (FermatLastTheoremForThreeGen.Solution.x S) (FermatLastTheoremForThreeGen.Solution.z S)
-- _private.Mathlib.NumberTheory.FLT.Three.0.FermatLastTheoremForThreeGen.Solution.isCoprime_y_z
private lemma isCoprime_y_z : IsCoprime S.y S.z 
Arguments: {K : Type u_1} [Field K] [NumberField K] [IsCyclotomicExtension {3} ℚ K] {ζ : K} {hζ : IsPrimitiveRoot ζ ↑3} (S : FermatLastTheoremForThreeGen.Solution hζ)
Goal: IsCoprime (FermatLastTheoremForThreeGen.Solution.y S) (FermatLastTheoremForThreeGen.Solution.z S)
-- _private.Mathlib.NumberTheory.FLT.Three.0.FermatLastTheoremForThreeGen.Solution.x_mul_y_mul_z_eq_u_mul_w_cube
private lemma x_mul_y_mul_z_eq_u_mul_w_cube : S.x * S.y * S.z = S.u * S.w ^ 3 
Arguments: {K : Type u_1} [Field K] [NumberField K] [IsCyclotomicExtension {3} ℚ K] {ζ : K} {hζ : IsPrimitiveRoot ζ ↑3} (S : FermatLastTheoremForThreeGen.Solution hζ) [DecidableRel fun a b => a ∣ b]
Goal: FermatLastTheoremForThreeGen.Solution.x S * FermatLastTheoremForThreeGen.Solution.y S *
    FermatLastTheoremForThreeGen.Solution.z S =
  ↑S.u * FermatLastTheoremForThreeGen.Solution.w S ^ 3
-- _private.Mathlib.NumberTheory.FLT.Three.0.FermatLastTheoremForThreeGen.Solution.exists_cube_associated
private lemma exists_cube_associated :
    (∃ X, Associated (X ^ 3) S.x) ∧ (∃ Y, Associated (Y ^ 3) S.y) ∧
      ∃ Z, Associated (Z ^ 3) S.z 
Arguments: {K : Type u_1} [Field K] [NumberField K] [IsCyclotomicExtension {3} ℚ K] {ζ : K} {hζ : IsPrimitiveRoot ζ ↑3} (S : FermatLastTheoremForThreeGen.Solution hζ) [DecidableRel fun a b => a ∣ b]
Goal: (∃ X, Associated (X ^ 3) (FermatLastTheoremForThreeGen.Solution.x S)) ∧
  (∃ Y, Associated (Y ^ 3) (FermatLastTheoremForThreeGen.Solution.y S)) ∧
    ∃ Z, Associated (Z ^ 3) (FermatLastTheoremForThreeGen.Solution.z S)
-- _private.Mathlib.NumberTheory.FLT.Three.0.FermatLastTheoremForThreeGen.Solution.X_u₁_spec
private lemma X_u₁_spec : S.X ^ 3 * S.u₁ = S.x 
Arguments: {K : Type u_1} [Field K] [NumberField K] [IsCyclotomicExtension {3} ℚ K] {ζ : K} {hζ : IsPrimitiveRoot ζ ↑3} (S : FermatLastTheoremForThreeGen.Solution hζ) [DecidableRel fun a b => a ∣ b]
Goal: FermatLastTheoremForThreeGen.Solution.X S ^ 3 * ↑(FermatLastTheoremForThreeGen.Solution.u₁ S) =
  FermatLastTheoremForThreeGen.Solution.x S
-- _private.Mathlib.NumberTheory.FLT.Three.0.FermatLastTheoremForThreeGen.Solution.Y_u₂_spec
private lemma Y_u₂_spec : S.Y ^ 3 * S.u₂ = S.y 
Arguments: {K : Type u_1} [Field K] [NumberField K] [IsCyclotomicExtension {3} ℚ K] {ζ : K} {hζ : IsPrimitiveRoot ζ ↑3} (S : FermatLastTheoremForThreeGen.Solution hζ) [DecidableRel fun a b => a ∣ b]
Goal: FermatLastTheoremForThreeGen.Solution.Y S ^ 3 * ↑(FermatLastTheoremForThreeGen.Solution.u₂ S) =
  FermatLastTheoremForThreeGen.Solution.y S
-- _private.Mathlib.NumberTheory.FLT.Three.0.FermatLastTheoremForThreeGen.Solution.Z_u₃_spec
private lemma Z_u₃_spec : S.Z ^ 3 * S.u₃ = S.z 
Arguments: {K : Type u_1} [Field K] [NumberField K] [IsCyclotomicExtension {3} ℚ K] {ζ : K} {hζ : IsPrimitiveRoot ζ ↑3} (S : FermatLastTheoremForThreeGen.Solution hζ) [DecidableRel fun a b => a ∣ b]
Goal: FermatLastTheoremForThreeGen.Solution.Z S ^ 3 * ↑(FermatLastTheoremForThreeGen.Solution.u₃ S) =
  FermatLastTheoremForThreeGen.Solution.z S
-- _private.Mathlib.NumberTheory.FLT.Three.0.FermatLastTheoremForThreeGen.Solution.X_ne_zero
private lemma X_ne_zero : S.X ≠ 0 
Arguments: {K : Type u_1} [Field K] [NumberField K] [IsCyclotomicExtension {3} ℚ K] {ζ : K} {hζ : IsPrimitiveRoot ζ ↑3} (S : FermatLastTheoremForThreeGen.Solution hζ) [DecidableRel fun a b => a ∣ b]
Goal: FermatLastTheoremForThreeGen.Solution.X S ≠ 0
-- _private.Mathlib.NumberTheory.FLT.Three.0.FermatLastTheoremForThreeGen.Solution.lambda_not_dvd_X
private lemma lambda_not_dvd_X : ¬ λ ∣ S.X 
Arguments: {K : Type u_1} [Field K] [NumberField K] [IsCyclotomicExtension {3} ℚ K] {ζ : K} {hζ : IsPrimitiveRoot ζ ↑3} (S : FermatLastTheoremForThreeGen.Solution hζ) [DecidableRel fun a b => a ∣ b]
Goal: ¬hζ.toInteger - 1 ∣ FermatLastTheoremForThreeGen.Solution.X S
-- _private.Mathlib.NumberTheory.FLT.Three.0.FermatLastTheoremForThreeGen.Solution.lambda_not_dvd_Y
private lemma lambda_not_dvd_Y : ¬ λ ∣ S.Y 
Arguments: {K : Type u_1} [Field K] [NumberField K] [IsCyclotomicExtension {3} ℚ K] {ζ : K} {hζ : IsPrimitiveRoot ζ ↑3} (S : FermatLastTheoremForThreeGen.Solution hζ) [DecidableRel fun a b => a ∣ b]
Goal: ¬hζ.toInteger - 1 ∣ FermatLastTheoremForThreeGen.Solution.Y S
-- _private.Mathlib.NumberTheory.FLT.Three.0.FermatLastTheoremForThreeGen.Solution.lambda_not_dvd_Z
private lemma lambda_not_dvd_Z : ¬ λ ∣ S.Z 
Arguments: {K : Type u_1} [Field K] [NumberField K] [IsCyclotomicExtension {3} ℚ K] {ζ : K} {hζ : IsPrimitiveRoot ζ ↑3} (S : FermatLastTheoremForThreeGen.Solution hζ) [DecidableRel fun a b => a ∣ b]
Goal: ¬hζ.toInteger - 1 ∣ FermatLastTheoremForThreeGen.Solution.Z S
-- _private.Mathlib.NumberTheory.FLT.Three.0.FermatLastTheoremForThreeGen.Solution.isCoprime_Y_Z
private lemma isCoprime_Y_Z : IsCoprime S.Y S.Z 
Arguments: {K : Type u_1} [Field K] [NumberField K] [IsCyclotomicExtension {3} ℚ K] {ζ : K} {hζ : IsPrimitiveRoot ζ ↑3} (S : FermatLastTheoremForThreeGen.Solution hζ) [DecidableRel fun a b => a ∣ b]
Goal: IsCoprime (FermatLastTheoremForThreeGen.Solution.Y S) (FermatLastTheoremForThreeGen.Solution.Z S)
-- _private.Mathlib.NumberTheory.FLT.Three.0.FermatLastTheoremForThreeGen.Solution.formula1
private lemma formula1 : S.X^3*S.u₁*λ^(3*S.multiplicity-2)+S.Y^3*S.u₂*λ*η+S.Z^3*S.u₃*λ*η^2 = 0 
Arguments: {K : Type u_1} [Field K] [NumberField K] [IsCyclotomicExtension {3} ℚ K] {ζ : K} {hζ : IsPrimitiveRoot ζ ↑3} (S : FermatLastTheoremForThreeGen.Solution hζ) [DecidableRel fun a b => a ∣ b]
Goal: FermatLastTheoremForThreeGen.Solution.X S ^ 3 * ↑(FermatLastTheoremForThreeGen.Solution.u₁ S) *
        (hζ.toInteger - 1) ^ (3 * S.multiplicity - 2) +
      FermatLastTheoremForThreeGen.Solution.Y S ^ 3 * ↑(FermatLastTheoremForThreeGen.Solution.u₂ S) *
          (hζ.toInteger - 1) *
        ↑⋯.unit +
    FermatLastTheoremForThreeGen.Solution.Z S ^ 3 * ↑(FermatLastTheoremForThreeGen.Solution.u₃ S) * (hζ.toInteger - 1) *
      ↑⋯.unit ^ 2 =
  0
-- _private.Mathlib.NumberTheory.FLT.Three.0.FermatLastTheoremForThreeGen.Solution.u₄_def
private lemma u₄_def : S.u₄ = η * S.u₃ * S.u₂⁻¹ 
Arguments: {K : Type u_1} [Field K] [NumberField K] [IsCyclotomicExtension {3} ℚ K] {ζ : K} {hζ : IsPrimitiveRoot ζ ↑3} (S : FermatLastTheoremForThreeGen.Solution hζ) [DecidableRel fun a b => a ∣ b]
Goal: FermatLastTheoremForThreeGen.Solution.u₄ S =
  ⋯.unit * FermatLastTheoremForThreeGen.Solution.u₃ S * (FermatLastTheoremForThreeGen.Solution.u₂ S)⁻¹
-- _private.Mathlib.NumberTheory.FLT.Three.0.FermatLastTheoremForThreeGen.Solution.u₅_def
private lemma u₅_def : S.u₅ = -η ^ 2 * S.u₁ * S.u₂⁻¹ 
Arguments: {K : Type u_1} [Field K] [NumberField K] [IsCyclotomicExtension {3} ℚ K] {ζ : K} {hζ : IsPrimitiveRoot ζ ↑3} (S : FermatLastTheoremForThreeGen.Solution hζ) [DecidableRel fun a b => a ∣ b]
Goal: FermatLastTheoremForThreeGen.Solution.u₅ S =
  -⋯.unit ^ 2 * FermatLastTheoremForThreeGen.Solution.u₁ S * (FermatLastTheoremForThreeGen.Solution.u₂ S)⁻¹
-- _private.Mathlib.NumberTheory.FLT.Three.0.FermatLastTheoremForThreeGen.Solution.formula2
private lemma formula2 :
    S.Y ^ 3 + S.u₄ * S.Z ^ 3 = S.u₅ * (λ ^ (S.multiplicity - 1) * S.X) ^ 3 
Arguments: {K : Type u_1} [Field K] [NumberField K] [IsCyclotomicExtension {3} ℚ K] {ζ : K} {hζ : IsPrimitiveRoot ζ ↑3} (S : FermatLastTheoremForThreeGen.Solution hζ) [DecidableRel fun a b => a ∣ b]
Goal: FermatLastTheoremForThreeGen.Solution.Y S ^ 3 +
    ↑(FermatLastTheoremForThreeGen.Solution.u₄ S) * FermatLastTheoremForThreeGen.Solution.Z S ^ 3 =
  ↑(FermatLastTheoremForThreeGen.Solution.u₅ S) *
    ((hζ.toInteger - 1) ^ (S.multiplicity - 1) * FermatLastTheoremForThreeGen.Solution.X S) ^ 3
-- _private.Mathlib.NumberTheory.FLT.Three.0.FermatLastTheoremForThreeGen.Solution.lambda_sq_div_u₅_mul
private lemma lambda_sq_div_u₅_mul : λ ^ 2 ∣ S.u₅ * (λ ^ (S.multiplicity - 1) * S.X) ^ 3 
Arguments: {K : Type u_1} [Field K] [NumberField K] [IsCyclotomicExtension {3} ℚ K] {ζ : K} {hζ : IsPrimitiveRoot ζ ↑3} (S : FermatLastTheoremForThreeGen.Solution hζ) [DecidableRel fun a b => a ∣ b]
Goal: (hζ.toInteger - 1) ^ 2 ∣
  ↑(FermatLastTheoremForThreeGen.Solution.u₅ S) *
    ((hζ.toInteger - 1) ^ (S.multiplicity - 1) * FermatLastTheoremForThreeGen.Solution.X S) ^ 3
-- _private.Mathlib.NumberTheory.FLT.Three.0.FermatLastTheoremForThreeGen.Solution.u₄_eq_one_or_neg_one
private lemma u₄_eq_one_or_neg_one : S.u₄ = 1 ∨ S.u₄ = -1 
Arguments: {K : Type u_1} [Field K] [NumberField K] [IsCyclotomicExtension {3} ℚ K] {ζ : K} {hζ : IsPrimitiveRoot ζ ↑3} (S : FermatLastTheoremForThreeGen.Solution hζ) [DecidableRel fun a b => a ∣ b]
Goal: FermatLastTheoremForThreeGen.Solution.u₄ S = 1 ∨ FermatLastTheoremForThreeGen.Solution.u₄ S = -1
-- _private.Mathlib.NumberTheory.FLT.Three.0.FermatLastTheoremForThreeGen.Solution.u₄_sq
private lemma u₄_sq : S.u₄ ^ 2 = 1 
Arguments: {K : Type u_1} [Field K] [NumberField K] [IsCyclotomicExtension {3} ℚ K] {ζ : K} {hζ : IsPrimitiveRoot ζ ↑3} (S : FermatLastTheoremForThreeGen.Solution hζ) [DecidableRel fun a b => a ∣ b]
Goal: FermatLastTheoremForThreeGen.Solution.u₄ S ^ 2 = 1
-- _private.Mathlib.NumberTheory.FLT.Three.0.FermatLastTheoremForThreeGen.Solution.formula3
private lemma formula3 :
    S.Y ^ 3 + (S.u₄ * S.Z) ^ 3 = S.u₅ * (λ ^ (S.multiplicity - 1) * S.X) ^ 3 
Arguments: {K : Type u_1} [Field K] [NumberField K] [IsCyclotomicExtension {3} ℚ K] {ζ : K} {hζ : IsPrimitiveRoot ζ ↑3} (S : FermatLastTheoremForThreeGen.Solution hζ) [DecidableRel fun a b => a ∣ b]
Goal: FermatLastTheoremForThreeGen.Solution.Y S ^ 3 +
    (↑(FermatLastTheoremForThreeGen.Solution.u₄ S) * FermatLastTheoremForThreeGen.Solution.Z S) ^ 3 =
  ↑(FermatLastTheoremForThreeGen.Solution.u₅ S) *
    ((hζ.toInteger - 1) ^ (S.multiplicity - 1) * FermatLastTheoremForThreeGen.Solution.X S) ^ 3
-- FermatLastTheoremForThreeGen.Solution.Solution'_descent_multiplicity
lemma Solution'_descent_multiplicity : S.Solution'_descent.multiplicity = S.multiplicity - 1 
Arguments: {K : Type u_1} [Field K] [NumberField K] [IsCyclotomicExtension {3} ℚ K] {ζ : K} {hζ : IsPrimitiveRoot ζ ↑3} (S : FermatLastTheoremForThreeGen.Solution hζ) [DecidableRel fun a b => a ∣ b]
Goal: S.Solution'_descent.multiplicity = S.multiplicity - 1
-- FermatLastTheoremForThreeGen.Solution.Solution'_descent_multiplicity_lt
lemma Solution'_descent_multiplicity_lt :
    (Solution'_descent S).multiplicity < S.multiplicity 
Arguments: {K : Type u_1} [Field K] [NumberField K] [IsCyclotomicExtension {3} ℚ K] {ζ : K} {hζ : IsPrimitiveRoot ζ ↑3} (S : FermatLastTheoremForThreeGen.Solution hζ) [DecidableRel fun a b => a ∣ b]
Goal: S.Solution'_descent.multiplicity < S.multiplicity
-- FermatLastTheoremForThreeGen.Solution.exists_Solution_multiplicity_lt
theorem exists_Solution_multiplicity_lt :
    ∃ S₁ : Solution hζ, S₁.multiplicity < S.multiplicity 
Arguments: {K : Type u_1} [Field K] [NumberField K] [IsCyclotomicExtension {3} ℚ K] {ζ : K} {hζ : IsPrimitiveRoot ζ ↑3} (S : FermatLastTheoremForThreeGen.Solution hζ) [DecidableRel fun a b => a ∣ b]
Goal: ∃ S₁, S₁.multiplicity < S.multiplicity
-- fermatLastTheoremThree
theorem fermatLastTheoremThree : FermatLastTheoremFor 3 
Goal: FermatLastTheoremFor 3
-- FirstOrder.Field.realize_eqZero
@[simp] theorem realize_eqZero [CommRing K] [CompatibleRing K] (n : ℕ)
    (v : Empty → K) : (Formula.Realize (eqZero n) v) ↔ ((n : K) = 0) 
Arguments: {K : Type u_1} [CommRing K] [FirstOrder.Ring.CompatibleRing K] (n : ℕ) (v : Empty → K)
Goal: FirstOrder.Language.Formula.Realize (FirstOrder.Field.eqZero n) v ↔ ↑n = 0
-- FirstOrder.Field.charP_iff_model_fieldOfChar
theorem charP_iff_model_fieldOfChar [Field K] [CompatibleRing K] :
    (Theory.fieldOfChar p).Model K ↔ CharP K p 
Arguments: {p : ℕ} {K : Type u_1} [Field K] [FirstOrder.Ring.CompatibleRing K]
Goal: K ⊨ FirstOrder.Language.Theory.fieldOfChar p ↔ CharP K p
-- FirstOrder.Field.charP_of_model_fieldOfChar
theorem charP_of_model_fieldOfChar [Field K] [CompatibleRing K]
    [h : (Theory.fieldOfChar p).Model K] : CharP K p 
Arguments: (p : ℕ) (K : Type u_1) [Field K] [FirstOrder.Ring.CompatibleRing K] [h : K ⊨ FirstOrder.Language.Theory.fieldOfChar p]
Goal: CharP K p
-- FirstOrder.Language.Ultraproduct.funMap_cast
theorem funMap_cast {n : ℕ} (f : L.Functions n) (x : Fin n → ∀ a, M a) :
    (funMap f fun i => (x i : (u : Filter α).Product M)) =
      (fun a => funMap f fun i => x i a : (u : Filter α).Product M) 
Arguments: {α : Type u_1} {M : α → Type u_2} {u : Ultrafilter α} {L : FirstOrder.Language} [(a : α) → L.Structure (M a)] {n : ℕ} (f : L.Functions n) (x : Fin n → (a : α) → M a)
Goal: (FirstOrder.Language.Structure.funMap f fun i => Quotient.mk' (x i)) =
  Quotient.mk' fun a => FirstOrder.Language.Structure.funMap f fun i => x i a
-- FirstOrder.Language.Ultraproduct.term_realize_cast
theorem term_realize_cast {β : Type*} (x : β → ∀ a, M a) (t : L.Term β) :
    (t.realize fun i => (x i : (u : Filter α).Product M)) =
      (fun a => t.realize fun i => x i a : (u : Filter α).Product M) 
Arguments: {α : Type u_1} {M : α → Type u_2} {u : Ultrafilter α} {L : FirstOrder.Language} [(a : α) → L.Structure (M a)] {β : Type u_3} (x : β → (a : α) → M a) (t : L.Term β)
Goal: FirstOrder.Language.Term.realize (fun i => Quotient.mk' (x i)) t =
  Quotient.mk' fun a => FirstOrder.Language.Term.realize (fun i => x i a) t
-- FirstOrder.Language.Ultraproduct.boundedFormula_realize_cast
theorem boundedFormula_realize_cast {β : Type*} {n : ℕ} (φ : L.BoundedFormula β n)
    (x : β → ∀ a, M a) (v : Fin n → ∀ a, M a) :
    (φ.Realize (fun i : β => (x i : (u : Filter α).Product M))
        (fun i => (v i : (u : Filter α).Product M))) ↔
      ∀ᶠ a : α in u, φ.Realize (fun i : β => x i a) fun i => v i a 
Arguments: {α : Type u_1} {M : α → Type u_2} {u : Ultrafilter α} {L : FirstOrder.Language} [(a : α) → L.Structure (M a)] [∀ (a : α), Nonempty (M a)] {β : Type u_3} {n : ℕ} (φ : L.BoundedFormula β n) (x : β → (a : α) → M a) (v : Fin n → (a : α) → M a)
Goal: (φ.Realize (fun i => Quotient.mk' (x i)) fun i => Quotient.mk' (v i)) ↔
  ∀ᶠ (a : α) in ↑u, φ.Realize (fun i => x i a) fun i => v i a
-- FirstOrder.Language.Ultraproduct.realize_formula_cast
theorem realize_formula_cast {β : Type*} (φ : L.Formula β) (x : β → ∀ a, M a) :
    (φ.Realize fun i => (x i : (u : Filter α).Product M)) ↔
      ∀ᶠ a : α in u, φ.Realize fun i => x i a 
Arguments: {α : Type u_1} {M : α → Type u_2} {u : Ultrafilter α} {L : FirstOrder.Language} [(a : α) → L.Structure (M a)] [∀ (a : α), Nonempty (M a)] {β : Type u_3} (φ : L.Formula β) (x : β → (a : α) → M a)
Goal: (φ.Realize fun i => Quotient.mk' (x i)) ↔ ∀ᶠ (a : α) in ↑u, φ.Realize fun i => x i a
-- FirstOrder.Language.Ultraproduct.sentence_realize
theorem sentence_realize (φ : L.Sentence) :
    (u : Filter α).Product M ⊨ φ ↔ ∀ᶠ a : α in u, M a ⊨ φ 
Arguments: {α : Type u_1} {M : α → Type u_2} {u : Ultrafilter α} {L : FirstOrder.Language} [(a : α) → L.Structure (M a)] [∀ (a : α), Nonempty (M a)] (φ : L.Sentence)
Goal: (↑u).Product M ⊨ φ ↔ ∀ᶠ (a : α) in ↑u, M a ⊨ φ