awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.NumberTheory.FLT.Polynomial (goal)
Goal:
k : Type u_1
inst : Field k
n : ℕ
hn : 3 ≤ n
chn : Ne.{u_1 + 1} (α := k) (↑n : k) 0
⊢ FermatLastTheoremWith'.{u_1} k[X] n
have rw : ∀ (a b c : k[X]),
  a ≠ 0 →
    b ≠ 0 →
      c ≠ 0 →
        a ^ n + b ^ n = c ^ n →
          ∃ (d : k[X]) (a' : k[X]) (b' : k[X]) (c' : k[X]),
            (a = a' * d ∧ b = b' * d ∧ c = c' * d) ∧ IsUnit a' ∧ IsUnit b' ∧ IsUnit c'
Tactic: classical
  rw [FermatLastTheoremWith']
  intro a b c ha hb hc heq
  obtain ⟨a', eq_a⟩ := gcd_dvd_left a b
  obtain ⟨b', eq_b⟩ := gcd_dvd_right a b
  set d := gcd a b
  have hd : d ≠ 0 := gcd_ne_zero_of_left ha
  rw [eq_a, eq_b, mul_pow, mul_pow, ← mul_add] at heq
  have hdc : d ∣ c := by
    -- TODO: This is basically reproving `IsIntegrallyClosed.pow_dvd_pow_iff`
    have hn : 0 < n := by lia
    have hdncn : d ^ n ∣ c ^ n := ⟨_, heq.symm⟩
    simpa [dvd_iff_normalizedFactors_le_normalizedFactors, Multiset.le_iff_count, *] using hdncn
  obtain ⟨c', eq_c⟩ := hdc
  rw [eq_a, mul_ne_zero_iff] at ha
  rw [eq_b, mul_ne_zero_iff] at hb
  rw [eq_c, mul_ne_zero_iff] at hc
  rw [mul_comm] at eq_a eq_b eq_c
  refine ⟨d, a', b', c', ⟨eq_a, eq_b, eq_c⟩, ?_⟩
  rw [eq_c, mul_pow, mul_comm, mul_left_inj' (pow_ne_zero n hd)] at heq
  suffices goal : a'.natDegree = 0 ∧ b'.natDegree = 0 ∧ c'.natDegree = 0 by
    simp only [natDegree_eq_zero] at goal
    obtain ⟨⟨ca', ha'⟩, ⟨cb', hb'⟩, ⟨cc', hc'⟩⟩ := goal
    rw [← ha', ← hb', ← hc']
    rw [← ha', C_ne_zero] at ha
    rw [← hb', C_ne_zero] at hb
    rw [← hc', C_ne_zero] at hc
    exact ⟨ha.right.isUnit_C, hb.right.isUnit_C, hc.right.isUnit_C⟩
  apply flt hn chn ha.right hb.right hc.right _ heq
  convert! isCoprime_div_gcd_div_gcd _
  · exact EuclideanDomain.eq_div_of_mul_eq_left ha.left eq_a.symm
  · exact EuclideanDomain.eq_div_of_mul_eq_left ha.left eq_b.symm
  · rw [eq_b]
    exact mul_ne_zero hb.right hb.left
-- Lean 4 proof step from Mathlib.NumberTheory.FLT.Polynomial (goal)
Goal:
k : Type u_1
inst : Field k
n : ℕ
hn : 3 ≤ n
chn : Ne.{u_1 + 1} (α := k) (↑n : k) 0
rw rw₁ rw₂ rw₃ :
  ∀ (a b c : k[X]) (ha : a ≠ 0) (hb : b ≠ 0) (hc : c ≠ 0) (heq : a ^ n + b ^ n = c ^ n),
    ∃ (d : k[X]) (a' : k[X]) (b' : k[X]) (c' : k[X]),
      (a = a' * d ∧ b = b' * d ∧ c = c' * d) ∧ IsUnit a' ∧ IsUnit b' ∧ IsUnit c'
⊢ FermatLastTheoremWith'.{u_1} k[X] n
have rw₄ : ∀ (a b c : k[X]) (ha : a ≠ 0) (hb : b ≠ 0) (hc : c ≠ 0) (heq : a ^ n + b ^ n = c ^ n) (a' : k[X])
  (eq_a : a = @gcd _ CommSemiring.toCommMonoidWithZero _ a b * a'),
  ∃ (d : k[X]) (a' : k[X]) (b' : k[X]) (c' : k[X]),
    (a = a' * d ∧ b = b' * d ∧ c = c' * d) ∧ IsUnit a' ∧ IsUnit b' ∧ IsUnit c'
Tactic: classical
  rw [FermatLastTheoremWith']
  intro a b c ha hb hc heq
  obtain ⟨a', eq_a⟩ := gcd_dvd_left a b
  obtain ⟨b', eq_b⟩ := gcd_dvd_right a b
  set d := gcd a b
  have hd : d ≠ 0 := gcd_ne_zero_of_left ha
  rw [eq_a, eq_b, mul_pow, mul_pow, ← mul_add] at heq
  have hdc : d ∣ c := by
    -- TODO: This is basically reproving `IsIntegrallyClosed.pow_dvd_pow_iff`
    have hn : 0 < n := by lia
    have hdncn : d ^ n ∣ c ^ n := ⟨_, heq.symm⟩
    simpa [dvd_iff_normalizedFactors_le_normalizedFactors, Multiset.le_iff_count, *] using hdncn
  obtain ⟨c', eq_c⟩ := hdc
  rw [eq_a, mul_ne_zero_iff] at ha
  rw [eq_b, mul_ne_zero_iff] at hb
  rw [eq_c, mul_ne_zero_iff] at hc
  rw [mul_comm] at eq_a eq_b eq_c
  refine ⟨d, a', b', c', ⟨eq_a, eq_b, eq_c⟩, ?_⟩
  rw [eq_c, mul_pow, mul_comm, mul_left_inj' (pow_ne_zero n hd)] at heq
  suffices goal : a'.natDegree = 0 ∧ b'.natDegree = 0 ∧ c'.natDegree = 0 by
    simp only [natDegree_eq_zero] at goal
    obtain ⟨⟨ca', ha'⟩, ⟨cb', hb'⟩, ⟨cc', hc'⟩⟩ := goal
    rw [← ha', ← hb', ← hc']
    rw [← ha', C_ne_zero] at ha
    rw [← hb', C_ne_zero] at hb
    rw [← hc', C_ne_zero] at hc
    exact ⟨ha.right.isUnit_C, hb.right.isUnit_C, hc.right.isUnit_C⟩
  apply flt hn chn ha.right hb.right hc.right _ heq
  convert! isCoprime_div_gcd_div_gcd _
  · exact EuclideanDomain.eq_div_of_mul_eq_left ha.left eq_a.symm
  · exact EuclideanDomain.eq_div_of_mul_eq_left ha.left eq_b.symm
  · rw [eq_b]
    exact mul_ne_zero hb.right hb.left
-- Lean 4 proof step from Mathlib.NumberTheory.FLT.Polynomial (goal)
Goal:
k : Type u_1
inst : Field k
n : ℕ
hn : 3 ≤ n
chn : Ne.{u_1 + 1} (α := k) (↑n : k) 0
rw rw₁ rw₂ rw₃ :
  ∀ (a b c : k[X]) (ha : a ≠ 0) (hb : b ≠ 0) (hc : c ≠ 0) (heq : a ^ n + b ^ n = c ^ n),
    ∃ (d : k[X]) (a' : k[X]) (b' : k[X]) (c' : k[X]),
      (a = a' * d ∧ b = b' * d ∧ c = c' * d) ∧ IsUnit a' ∧ IsUnit b' ∧ IsUnit c'
rw₄ :
  ∀ (a b c : k[X]) (ha : a ≠ 0) (hb : b ≠ 0) (hc : c ≠ 0) (heq : a ^ n + b ^ n = c ^ n) (a' : k[X])
    (eq_a : a = @gcd _ CommSemiring.toCommMonoidWithZero _ a b * a'),
    ∃ (d : k[X]) (a' : k[X]) (b' : k[X]) (c' : k[X]),
      (a = a' * d ∧ b = b' * d ∧ c = c' * d) ∧ IsUnit a' ∧ IsUnit b' ∧ IsUnit c'
⊢ FermatLastTheoremWith'.{u_1} k[X] n
have rw₅ : ∀ (a b c : k[X]) (ha : a ≠ 0) (hb : b ≠ 0) (hc : c ≠ 0) (heq : a ^ n + b ^ n = c ^ n) (a' : k[X])
  (eq_a : a = @gcd _ CommSemiring.toCommMonoidWithZero _ a b * a') (b' : k[X])
  (eq_b : b = @gcd _ CommSemiring.toCommMonoidWithZero _ a b * b'),
  ∃ (d : k[X]) (a' : k[X]) (b' : k[X]) (c' : k[X]),
    (a = a' * d ∧ b = b' * d ∧ c = c' * d) ∧ IsUnit a' ∧ IsUnit b' ∧ IsUnit c'
Tactic: classical
  rw [FermatLastTheoremWith']
  intro a b c ha hb hc heq
  obtain ⟨a', eq_a⟩ := gcd_dvd_left a b
  obtain ⟨b', eq_b⟩ := gcd_dvd_right a b
  set d := gcd a b
  have hd : d ≠ 0 := gcd_ne_zero_of_left ha
  rw [eq_a, eq_b, mul_pow, mul_pow, ← mul_add] at heq
  have hdc : d ∣ c := by
    -- TODO: This is basically reproving `IsIntegrallyClosed.pow_dvd_pow_iff`
    have hn : 0 < n := by lia
    have hdncn : d ^ n ∣ c ^ n := ⟨_, heq.symm⟩
    simpa [dvd_iff_normalizedFactors_le_normalizedFactors, Multiset.le_iff_count, *] using hdncn
  obtain ⟨c', eq_c⟩ := hdc
  rw [eq_a, mul_ne_zero_iff] at ha
  rw [eq_b, mul_ne_zero_iff] at hb
  rw [eq_c, mul_ne_zero_iff] at hc
  rw [mul_comm] at eq_a eq_b eq_c
  refine ⟨d, a', b', c', ⟨eq_a, eq_b, eq_c⟩, ?_⟩
  rw [eq_c, mul_pow, mul_comm, mul_left_inj' (pow_ne_zero n hd)] at heq
  suffices goal : a'.natDegree = 0 ∧ b'.natDegree = 0 ∧ c'.natDegree = 0 by
    simp only [natDegree_eq_zero] at goal
    obtain ⟨⟨ca', ha'⟩, ⟨cb', hb'⟩, ⟨cc', hc'⟩⟩ := goal
    rw [← ha', ← hb', ← hc']
    rw [← ha', C_ne_zero] at ha
    rw [← hb', C_ne_zero] at hb
    rw [← hc', C_ne_zero] at hc
    exact ⟨ha.right.isUnit_C, hb.right.isUnit_C, hc.right.isUnit_C⟩
  apply flt hn chn ha.right hb.right hc.right _ heq
  convert! isCoprime_div_gcd_div_gcd _
  · exact EuclideanDomain.eq_div_of_mul_eq_left ha.left eq_a.symm
  · exact EuclideanDomain.eq_div_of_mul_eq_left ha.left eq_b.symm
  · rw [eq_b]
    exact mul_ne_zero hb.right hb.left
-- Lean 4 proof step from Mathlib.NumberTheory.FLT.Polynomial (goal)
Goal:
k : Type u_1
inst : Field k
n : ℕ
hn : 3 ≤ n
chn : Ne.{u_1 + 1} (α := k) (↑n : k) 0
rw rw₁ rw₂ rw₃ :
  ∀ (a b c : k[X]) (ha : a ≠ 0) (hb : b ≠ 0) (hc : c ≠ 0) (heq : a ^ n + b ^ n = c ^ n),
    ∃ (d : k[X]) (a' : k[X]) (b' : k[X]) (c' : k[X]),
      (a = a' * d ∧ b = b' * d ∧ c = c' * d) ∧ IsUnit a' ∧ IsUnit b' ∧ IsUnit c'
rw₄ :
  ∀ (a b c : k[X]) (ha : a ≠ 0) (hb : b ≠ 0) (hc : c ≠ 0) (heq : a ^ n + b ^ n = c ^ n) (a' : k[X])
    (eq_a : a = @gcd _ CommSemiring.toCommMonoidWithZero _ a b * a'),
    ∃ (d : k[X]) (a' : k[X]) (b' : k[X]) (c' : k[X]),
      (a = a' * d ∧ b = b' * d ∧ c = c' * d) ∧ IsUnit a' ∧ IsUnit b' ∧ IsUnit c'
rw₅ :
  ∀ (a b c : k[X]) (ha : a ≠ 0) (hb : b ≠ 0) (hc : c ≠ 0) (heq : a ^ n + b ^ n = c ^ n) (a' : k[X])
    (eq_a : a = @gcd _ CommSemiring.toCommMonoidWithZero _ a b * a') (b' : k[X])
    (eq_b : b = @gcd _ CommSemiring.toCommMonoidWithZero _ a b * b'),
    ∃ (d : k[X]) (a' : k[X]) (b' : k[X]) (c' : k[X]),
      (a = a' * d ∧ b = b' * d ∧ c = c' * d) ∧ IsUnit a' ∧ IsUnit b' ∧ IsUnit c'
⊢ FermatLastTheoremWith'.{u_1} k[X] n
have rw₆ : ∀ (a b c : k[X]) (ha : a ≠ 0) (hb : b ≠ 0) (hc : c ≠ 0) (heq : a ^ n + b ^ n = c ^ n) (a' b' : k[X]),
  let d := @gcd _ CommSemiring.toCommMonoidWithZero _ a b;
  ∀ (eq_a : a = d * a') (eq_b : b = d * b'),
    ∃ (d : k[X]) (a' : k[X]) (b' : k[X]) (c' : k[X]),
      (a = a' * d ∧ b = b' * d ∧ c = c' * d) ∧ IsUnit a' ∧ IsUnit b' ∧ IsUnit c'
Tactic: classical
  rw [FermatLastTheoremWith']
  intro a b c ha hb hc heq
  obtain ⟨a', eq_a⟩ := gcd_dvd_left a b
  obtain ⟨b', eq_b⟩ := gcd_dvd_right a b
  set d := gcd a b
  have hd : d ≠ 0 := gcd_ne_zero_of_left ha
  rw [eq_a, eq_b, mul_pow, mul_pow, ← mul_add] at heq
  have hdc : d ∣ c := by
    -- TODO: This is basically reproving `IsIntegrallyClosed.pow_dvd_pow_iff`
    have hn : 0 < n := by lia
    have hdncn : d ^ n ∣ c ^ n := ⟨_, heq.symm⟩
    simpa [dvd_iff_normalizedFactors_le_normalizedFactors, Multiset.le_iff_count, *] using hdncn
  obtain ⟨c', eq_c⟩ := hdc
  rw [eq_a, mul_ne_zero_iff] at ha
  rw [eq_b, mul_ne_zero_iff] at hb
  rw [eq_c, mul_ne_zero_iff] at hc
  rw [mul_comm] at eq_a eq_b eq_c
  refine ⟨d, a', b', c', ⟨eq_a, eq_b, eq_c⟩, ?_⟩
  rw [eq_c, mul_pow, mul_comm, mul_left_inj' (pow_ne_zero n hd)] at heq
  suffices goal : a'.natDegree = 0 ∧ b'.natDegree = 0 ∧ c'.natDegree = 0 by
    simp only [natDegree_eq_zero] at goal
    obtain ⟨⟨ca', ha'⟩, ⟨cb', hb'⟩, ⟨cc', hc'⟩⟩ := goal
    rw [← ha', ← hb', ← hc']
    rw [← ha', C_ne_zero] at ha
    rw [← hb', C_ne_zero] at hb
    rw [← hc', C_ne_zero] at hc
    exact ⟨ha.right.isUnit_C, hb.right.isUnit_C, hc.right.isUnit_C⟩
  apply flt hn chn ha.right hb.right hc.right _ heq
  convert! isCoprime_div_gcd_div_gcd _
  · exact EuclideanDomain.eq_div_of_mul_eq_left ha.left eq_a.symm
  · exact EuclideanDomain.eq_div_of_mul_eq_left ha.left eq_b.symm
  · rw [eq_b]
    exact mul_ne_zero hb.right hb.left
-- Lean 4 proof step from Mathlib.NumberTheory.FLT.Three (goal)
Goal:
n : ℤ
h : ¬3 ∣ n
⊢ HPow.hPow (α := ZMod 9) (β := ℕ) (↑n : ZMod 9) 3 = 1 ∨ HPow.hPow (α := ZMod 9) (β := ℕ) (↑n : ZMod 9) 3 = 8
have a : Ne (α := ZMod 3) ((castHom sorry (ZMod 3) : (a : ZMod 9) → ZMod 3) (↑n : ZMod 9)) 0
Tactic: apply cube_of_castHom_ne_zero
-- Lean 4 proof step from Mathlib.NumberTheory.FLT.Three (goal)
Goal:
n : ℤ
h : ¬3 ∣ n
⊢ Ne (α := ZMod 3) ((castHom ⋯ (ZMod 3) : (a : ZMod 9) → ZMod 3) (↑n : ZMod 9)) 0
have a : Ne (α := ZMod 3) (↑n : ZMod 3) 0
Tactic: rwa [map_intCast, Ne, ZMod.intCast_zmod_eq_zero_iff_dvd]
-- Lean 4 proof step from Mathlib.NumberTheory.FLT.Three (goal)
Goal:
n : ℤ
h : ¬3 ∣ n
a : Ne (α := ZMod 3) (↑n : ZMod 3) 0
⊢ Ne (α := ZMod 3) ((castHom ⋯ (ZMod 3) : (a : ZMod 9) → ZMod 3) (↑n : ZMod 9)) 0
have a₁ : ¬Eq (α := ZMod 3) (↑n : ZMod 3) 0
Tactic: rwa [map_intCast, Ne, ZMod.intCast_zmod_eq_zero_iff_dvd]
-- Lean 4 proof step from Mathlib.NumberTheory.FLT.Three (goal)
Goal:
n : ℤ
h : ¬3 ∣ n
a : Ne (α := ZMod 3) (↑n : ZMod 3) 0
a₁ : ¬Eq (α := ZMod 3) (↑n : ZMod 3) 0
⊢ Ne (α := ZMod 3) ((castHom ⋯ (ZMod 3) : (a : ZMod 9) → ZMod 3) (↑n : ZMod 9)) 0
have a₂ : ¬(↑3 : ℤ) ∣ n
Tactic: rwa [map_intCast, Ne, ZMod.intCast_zmod_eq_zero_iff_dvd]
-- Lean 4 proof step from Mathlib.NumberTheory.FLT.Three (hypothesis)
Goal:
a b c : ℤ
hdvd : ¬3 ∣ a * b * c
⊢ HPow.hPow (β := ℕ) a 3 + HPow.hPow (β := ℕ) b 3 ≠ HPow.hPow (β := ℕ) c 3
have hdvd₁ : (¬3 ∣ a ∧ ¬3 ∣ b) ∧ ¬3 ∣ c
Tactic: simp_rw [Int.prime_three.dvd_mul, not_or] at hdvd
Cleared hypotheses: hdvd
-- Lean 4 proof step from Mathlib.NumberTheory.FLT.Three (goal)
Goal:
a b c : ℤ
hdvd : (¬3 ∣ a ∧ ¬3 ∣ b) ∧ ¬3 ∣ c
⊢ HPow.hPow (β := ℕ) a 3 + HPow.hPow (β := ℕ) b 3 ≠ HPow.hPow (β := ℕ) c 3
have apply : ¬Eq (α := ZMod 9) (↑(HPow.hPow (β := ℕ) a 3 + HPow.hPow (β := ℕ) b 3) : ZMod 9) (↑(HPow.hPow (β := ℕ) c 3) : ZMod 9)
Tactic: apply mt (congrArg (Int.cast : ℤ → ZMod 9))
-- Lean 4 proof step from Mathlib.NumberTheory.FLT.Three (goal)
Goal:
a b c : ℤ
hdvd : (¬3 ∣ a ∧ ¬3 ∣ b) ∧ ¬3 ∣ c
⊢ ¬Eq (α := ZMod 9) (↑(HPow.hPow (β := ℕ) a 3 + HPow.hPow (β := ℕ) b 3) : ZMod 9) (↑(HPow.hPow (β := ℕ) c 3) : ZMod 9)
have simp_rw : ¬HPow.hPow (α := ZMod 9) (β := ℕ) (↑a : ZMod 9) 3 + HPow.hPow (α := ZMod 9) (β := ℕ) (↑b : ZMod 9) 3 =
    HPow.hPow (α := ZMod 9) (β := ℕ) (↑c : ZMod 9) 3
Tactic: simp_rw [Int.cast_add, Int.cast_pow]
-- Lean 4 proof step from Mathlib.NumberTheory.FLT.Three (goal)
Goal:
a b c : ℤ
hdvd : (¬3 ∣ a ∧ ¬3 ∣ b) ∧ ¬3 ∣ c
⊢ ¬HPow.hPow (α := ZMod 9) (β := ℕ) (↑a : ZMod 9) 3 + HPow.hPow (α := ZMod 9) (β := ℕ) (↑b : ZMod 9) 3 =
      HPow.hPow (α := ZMod 9) (β := ℕ) (↑c : ZMod 9) 3
have inl : ∀ (ha : HPow.hPow (α := ZMod 9) (β := ℕ) (↑a : ZMod 9) 3 = 1),
  ¬HPow.hPow (α := ZMod 9) (β := ℕ) (↑a : ZMod 9) 3 + HPow.hPow (α := ZMod 9) (β := ℕ) (↑b : ZMod 9) 3 =
      HPow.hPow (α := ZMod 9) (β := ℕ) (↑c : ZMod 9) 3
Tactic: rcases cube_of_not_dvd hdvd.1.1 with ha | ha
-- Lean 4 proof step from Mathlib.NumberTheory.FLT.Three (goal)
Goal:
a b c : ℤ
hdvd : (¬3 ∣ a ∧ ¬3 ∣ b) ∧ ¬3 ∣ c
inl :
  ∀ (ha : HPow.hPow (α := ZMod 9) (β := ℕ) (↑a : ZMod 9) 3 = 1),
    ¬HPow.hPow (α := ZMod 9) (β := ℕ) (↑a : ZMod 9) 3 + HPow.hPow (α := ZMod 9) (β := ℕ) (↑b : ZMod 9) 3 =
        HPow.hPow (α := ZMod 9) (β := ℕ) (↑c : ZMod 9) 3
⊢ ¬HPow.hPow (α := ZMod 9) (β := ℕ) (↑a : ZMod 9) 3 + HPow.hPow (α := ZMod 9) (β := ℕ) (↑b : ZMod 9) 3 =
      HPow.hPow (α := ZMod 9) (β := ℕ) (↑c : ZMod 9) 3
have inr : ∀ (ha : HPow.hPow (α := ZMod 9) (β := ℕ) (↑a : ZMod 9) 3 = 8),
  ¬HPow.hPow (α := ZMod 9) (β := ℕ) (↑a : ZMod 9) 3 + HPow.hPow (α := ZMod 9) (β := ℕ) (↑b : ZMod 9) 3 =
      HPow.hPow (α := ZMod 9) (β := ℕ) (↑c : ZMod 9) 3
Tactic: rcases cube_of_not_dvd hdvd.1.1 with ha | ha
-- Lean 4 proof step from Mathlib.NumberTheory.FLT.Three (goal)
Goal:
a b c : ℤ
hdvd : (¬3 ∣ a ∧ ¬3 ∣ b) ∧ ¬3 ∣ c
ha : HPow.hPow (α := ZMod 9) (β := ℕ) (↑a : ZMod 9) 3 = 1
⊢ ¬HPow.hPow (α := ZMod 9) (β := ℕ) (↑a : ZMod 9) 3 + HPow.hPow (α := ZMod 9) (β := ℕ) (↑b : ZMod 9) 3 =
      HPow.hPow (α := ZMod 9) (β := ℕ) (↑c : ZMod 9) 3
have inl : ∀ (hb : HPow.hPow (α := ZMod 9) (β := ℕ) (↑b : ZMod 9) 3 = 1),
  ¬HPow.hPow (α := ZMod 9) (β := ℕ) (↑a : ZMod 9) 3 + HPow.hPow (α := ZMod 9) (β := ℕ) (↑b : ZMod 9) 3 =
      HPow.hPow (α := ZMod 9) (β := ℕ) (↑c : ZMod 9) 3
Tactic: rcases cube_of_not_dvd hdvd.1.2 with hb | hb
-- Lean 4 proof step from Mathlib.NumberTheory.FLT.Three (goal)
Goal:
a b c : ℤ
hdvd : (¬3 ∣ a ∧ ¬3 ∣ b) ∧ ¬3 ∣ c
ha : HPow.hPow (α := ZMod 9) (β := ℕ) (↑a : ZMod 9) 3 = 1
inl :
  ∀ (hb : HPow.hPow (α := ZMod 9) (β := ℕ) (↑b : ZMod 9) 3 = 1),
    ¬HPow.hPow (α := ZMod 9) (β := ℕ) (↑a : ZMod 9) 3 + HPow.hPow (α := ZMod 9) (β := ℕ) (↑b : ZMod 9) 3 =
        HPow.hPow (α := ZMod 9) (β := ℕ) (↑c : ZMod 9) 3
⊢ ¬HPow.hPow (α := ZMod 9) (β := ℕ) (↑a : ZMod 9) 3 + HPow.hPow (α := ZMod 9) (β := ℕ) (↑b : ZMod 9) 3 =
      HPow.hPow (α := ZMod 9) (β := ℕ) (↑c : ZMod 9) 3
have inl₁ : ∀ (hb : HPow.hPow (α := ZMod 9) (β := ℕ) (↑b : ZMod 9) 3 = 8),
  ¬HPow.hPow (α := ZMod 9) (β := ℕ) (↑a : ZMod 9) 3 + HPow.hPow (α := ZMod 9) (β := ℕ) (↑b : ZMod 9) 3 =
      HPow.hPow (α := ZMod 9) (β := ℕ) (↑c : ZMod 9) 3
Tactic: rcases cube_of_not_dvd hdvd.1.2 with hb | hb
-- Lean 4 proof step from Mathlib.NumberTheory.FLT.Three (goal)
Goal:
a b c : ℤ
hdvd : (¬3 ∣ a ∧ ¬3 ∣ b) ∧ ¬3 ∣ c
ha : HPow.hPow (α := ZMod 9) (β := ℕ) (↑a : ZMod 9) 3 = 8
⊢ ¬HPow.hPow (α := ZMod 9) (β := ℕ) (↑a : ZMod 9) 3 + HPow.hPow (α := ZMod 9) (β := ℕ) (↑b : ZMod 9) 3 =
      HPow.hPow (α := ZMod 9) (β := ℕ) (↑c : ZMod 9) 3
have inr : ∀ (hb : HPow.hPow (α := ZMod 9) (β := ℕ) (↑b : ZMod 9) 3 = 1),
  ¬HPow.hPow (α := ZMod 9) (β := ℕ) (↑a : ZMod 9) 3 + HPow.hPow (α := ZMod 9) (β := ℕ) (↑b : ZMod 9) 3 =
      HPow.hPow (α := ZMod 9) (β := ℕ) (↑c : ZMod 9) 3
Tactic: rcases cube_of_not_dvd hdvd.1.2 with hb | hb
-- Lean 4 proof step from Mathlib.NumberTheory.FLT.Three (goal)
Goal:
a b c : ℤ
hdvd : (¬3 ∣ a ∧ ¬3 ∣ b) ∧ ¬3 ∣ c
ha : HPow.hPow (α := ZMod 9) (β := ℕ) (↑a : ZMod 9) 3 = 8
inr :
  ∀ (hb : HPow.hPow (α := ZMod 9) (β := ℕ) (↑b : ZMod 9) 3 = 1),
    ¬HPow.hPow (α := ZMod 9) (β := ℕ) (↑a : ZMod 9) 3 + HPow.hPow (α := ZMod 9) (β := ℕ) (↑b : ZMod 9) 3 =
        HPow.hPow (α := ZMod 9) (β := ℕ) (↑c : ZMod 9) 3
⊢ ¬HPow.hPow (α := ZMod 9) (β := ℕ) (↑a : ZMod 9) 3 + HPow.hPow (α := ZMod 9) (β := ℕ) (↑b : ZMod 9) 3 =
      HPow.hPow (α := ZMod 9) (β := ℕ) (↑c : ZMod 9) 3
have inr₁ : ∀ (hb : HPow.hPow (α := ZMod 9) (β := ℕ) (↑b : ZMod 9) 3 = 8),
  ¬HPow.hPow (α := ZMod 9) (β := ℕ) (↑a : ZMod 9) 3 + HPow.hPow (α := ZMod 9) (β := ℕ) (↑b : ZMod 9) 3 =
      HPow.hPow (α := ZMod 9) (β := ℕ) (↑c : ZMod 9) 3
Tactic: rcases cube_of_not_dvd hdvd.1.2 with hb | hb
-- Lean 4 proof step from Mathlib.NumberTheory.FLT.Three (goal)
Goal:
a b c : ℤ
hdvd : (¬3 ∣ a ∧ ¬3 ∣ b) ∧ ¬3 ∣ c
ha : HPow.hPow (α := ZMod 9) (β := ℕ) (↑a : ZMod 9) 3 = 1
hb : HPow.hPow (α := ZMod 9) (β := ℕ) (↑b : ZMod 9) 3 = 1
⊢ ¬HPow.hPow (α := ZMod 9) (β := ℕ) (↑a : ZMod 9) 3 + HPow.hPow (α := ZMod 9) (β := ℕ) (↑b : ZMod 9) 3 =
      HPow.hPow (α := ZMod 9) (β := ℕ) (↑c : ZMod 9) 3
have inl : ∀ (hc : HPow.hPow (α := ZMod 9) (β := ℕ) (↑c : ZMod 9) 3 = 1),
  ¬HPow.hPow (α := ZMod 9) (β := ℕ) (↑a : ZMod 9) 3 + HPow.hPow (α := ZMod 9) (β := ℕ) (↑b : ZMod 9) 3 =
      HPow.hPow (α := ZMod 9) (β := ℕ) (↑c : ZMod 9) 3
Tactic: rcases cube_of_not_dvd hdvd.2 with hc | hc
-- Lean 4 proof step from Mathlib.NumberTheory.FLT.Three (goal)
Goal:
a b c : ℤ
hdvd : (¬3 ∣ a ∧ ¬3 ∣ b) ∧ ¬3 ∣ c
ha : HPow.hPow (α := ZMod 9) (β := ℕ) (↑a : ZMod 9) 3 = 1
hb : HPow.hPow (α := ZMod 9) (β := ℕ) (↑b : ZMod 9) 3 = 1
inl :
  ∀ (hc : HPow.hPow (α := ZMod 9) (β := ℕ) (↑c : ZMod 9) 3 = 1),
    ¬HPow.hPow (α := ZMod 9) (β := ℕ) (↑a : ZMod 9) 3 + HPow.hPow (α := ZMod 9) (β := ℕ) (↑b : ZMod 9) 3 =
        HPow.hPow (α := ZMod 9) (β := ℕ) (↑c : ZMod 9) 3
⊢ ¬HPow.hPow (α := ZMod 9) (β := ℕ) (↑a : ZMod 9) 3 + HPow.hPow (α := ZMod 9) (β := ℕ) (↑b : ZMod 9) 3 =
      HPow.hPow (α := ZMod 9) (β := ℕ) (↑c : ZMod 9) 3
have inl₁ : ∀ (hc : HPow.hPow (α := ZMod 9) (β := ℕ) (↑c : ZMod 9) 3 = 8),
  ¬HPow.hPow (α := ZMod 9) (β := ℕ) (↑a : ZMod 9) 3 + HPow.hPow (α := ZMod 9) (β := ℕ) (↑b : ZMod 9) 3 =
      HPow.hPow (α := ZMod 9) (β := ℕ) (↑c : ZMod 9) 3
Tactic: rcases cube_of_not_dvd hdvd.2 with hc | hc
-- Lean 4 proof step from Mathlib.NumberTheory.FLT.Three (goal)
Goal:
a b c : ℤ
hdvd : (¬3 ∣ a ∧ ¬3 ∣ b) ∧ ¬3 ∣ c
ha : HPow.hPow (α := ZMod 9) (β := ℕ) (↑a : ZMod 9) 3 = 1
hb : HPow.hPow (α := ZMod 9) (β := ℕ) (↑b : ZMod 9) 3 = 8
⊢ ¬HPow.hPow (α := ZMod 9) (β := ℕ) (↑a : ZMod 9) 3 + HPow.hPow (α := ZMod 9) (β := ℕ) (↑b : ZMod 9) 3 =
      HPow.hPow (α := ZMod 9) (β := ℕ) (↑c : ZMod 9) 3
have inl : ∀ (hc : HPow.hPow (α := ZMod 9) (β := ℕ) (↑c : ZMod 9) 3 = 1),
  ¬HPow.hPow (α := ZMod 9) (β := ℕ) (↑a : ZMod 9) 3 + HPow.hPow (α := ZMod 9) (β := ℕ) (↑b : ZMod 9) 3 =
      HPow.hPow (α := ZMod 9) (β := ℕ) (↑c : ZMod 9) 3
Tactic: rcases cube_of_not_dvd hdvd.2 with hc | hc
-- Lean 4 proof step from Mathlib.NumberTheory.FLT.Three (goal)
Goal:
a b c : ℤ
hdvd : (¬3 ∣ a ∧ ¬3 ∣ b) ∧ ¬3 ∣ c
ha : HPow.hPow (α := ZMod 9) (β := ℕ) (↑a : ZMod 9) 3 = 1
hb : HPow.hPow (α := ZMod 9) (β := ℕ) (↑b : ZMod 9) 3 = 8
inl :
  ∀ (hc : HPow.hPow (α := ZMod 9) (β := ℕ) (↑c : ZMod 9) 3 = 1),
    ¬HPow.hPow (α := ZMod 9) (β := ℕ) (↑a : ZMod 9) 3 + HPow.hPow (α := ZMod 9) (β := ℕ) (↑b : ZMod 9) 3 =
        HPow.hPow (α := ZMod 9) (β := ℕ) (↑c : ZMod 9) 3
⊢ ¬HPow.hPow (α := ZMod 9) (β := ℕ) (↑a : ZMod 9) 3 + HPow.hPow (α := ZMod 9) (β := ℕ) (↑b : ZMod 9) 3 =
      HPow.hPow (α := ZMod 9) (β := ℕ) (↑c : ZMod 9) 3
have inl₁ : ∀ (hc : HPow.hPow (α := ZMod 9) (β := ℕ) (↑c : ZMod 9) 3 = 8),
  ¬HPow.hPow (α := ZMod 9) (β := ℕ) (↑a : ZMod 9) 3 + HPow.hPow (α := ZMod 9) (β := ℕ) (↑b : ZMod 9) 3 =
      HPow.hPow (α := ZMod 9) (β := ℕ) (↑c : ZMod 9) 3
Tactic: rcases cube_of_not_dvd hdvd.2 with hc | hc
-- Lean 4 proof step from Mathlib.NumberTheory.FLT.Three (goal)
Goal:
a b c : ℤ
hdvd : (¬3 ∣ a ∧ ¬3 ∣ b) ∧ ¬3 ∣ c
ha : HPow.hPow (α := ZMod 9) (β := ℕ) (↑a : ZMod 9) 3 = 8
hb : HPow.hPow (α := ZMod 9) (β := ℕ) (↑b : ZMod 9) 3 = 1
⊢ ¬HPow.hPow (α := ZMod 9) (β := ℕ) (↑a : ZMod 9) 3 + HPow.hPow (α := ZMod 9) (β := ℕ) (↑b : ZMod 9) 3 =
      HPow.hPow (α := ZMod 9) (β := ℕ) (↑c : ZMod 9) 3
have inr : ∀ (hc : HPow.hPow (α := ZMod 9) (β := ℕ) (↑c : ZMod 9) 3 = 1),
  ¬HPow.hPow (α := ZMod 9) (β := ℕ) (↑a : ZMod 9) 3 + HPow.hPow (α := ZMod 9) (β := ℕ) (↑b : ZMod 9) 3 =
      HPow.hPow (α := ZMod 9) (β := ℕ) (↑c : ZMod 9) 3
Tactic: rcases cube_of_not_dvd hdvd.2 with hc | hc
-- Lean 4 proof step from Mathlib.NumberTheory.FLT.Three (goal)
Goal:
a b c : ℤ
hdvd : (¬3 ∣ a ∧ ¬3 ∣ b) ∧ ¬3 ∣ c
ha : HPow.hPow (α := ZMod 9) (β := ℕ) (↑a : ZMod 9) 3 = 8
hb : HPow.hPow (α := ZMod 9) (β := ℕ) (↑b : ZMod 9) 3 = 1
inr :
  ∀ (hc : HPow.hPow (α := ZMod 9) (β := ℕ) (↑c : ZMod 9) 3 = 1),
    ¬HPow.hPow (α := ZMod 9) (β := ℕ) (↑a : ZMod 9) 3 + HPow.hPow (α := ZMod 9) (β := ℕ) (↑b : ZMod 9) 3 =
        HPow.hPow (α := ZMod 9) (β := ℕ) (↑c : ZMod 9) 3
⊢ ¬HPow.hPow (α := ZMod 9) (β := ℕ) (↑a : ZMod 9) 3 + HPow.hPow (α := ZMod 9) (β := ℕ) (↑b : ZMod 9) 3 =
      HPow.hPow (α := ZMod 9) (β := ℕ) (↑c : ZMod 9) 3
have inr₁ : ∀ (hc : HPow.hPow (α := ZMod 9) (β := ℕ) (↑c : ZMod 9) 3 = 8),
  ¬HPow.hPow (α := ZMod 9) (β := ℕ) (↑a : ZMod 9) 3 + HPow.hPow (α := ZMod 9) (β := ℕ) (↑b : ZMod 9) 3 =
      HPow.hPow (α := ZMod 9) (β := ℕ) (↑c : ZMod 9) 3
Tactic: rcases cube_of_not_dvd hdvd.2 with hc | hc
-- Lean 4 proof step from Mathlib.NumberTheory.FLT.Three (goal)
Goal:
a b c : ℤ
hdvd : (¬3 ∣ a ∧ ¬3 ∣ b) ∧ ¬3 ∣ c
ha : HPow.hPow (α := ZMod 9) (β := ℕ) (↑a : ZMod 9) 3 = 8
hb : HPow.hPow (α := ZMod 9) (β := ℕ) (↑b : ZMod 9) 3 = 8
⊢ ¬HPow.hPow (α := ZMod 9) (β := ℕ) (↑a : ZMod 9) 3 + HPow.hPow (α := ZMod 9) (β := ℕ) (↑b : ZMod 9) 3 =
      HPow.hPow (α := ZMod 9) (β := ℕ) (↑c : ZMod 9) 3
have inr : ∀ (hc : HPow.hPow (α := ZMod 9) (β := ℕ) (↑c : ZMod 9) 3 = 1),
  ¬HPow.hPow (α := ZMod 9) (β := ℕ) (↑a : ZMod 9) 3 + HPow.hPow (α := ZMod 9) (β := ℕ) (↑b : ZMod 9) 3 =
      HPow.hPow (α := ZMod 9) (β := ℕ) (↑c : ZMod 9) 3
Tactic: rcases cube_of_not_dvd hdvd.2 with hc | hc
-- Lean 4 proof step from Mathlib.NumberTheory.FLT.Three (goal)
Goal:
a b c : ℤ
hdvd : (¬3 ∣ a ∧ ¬3 ∣ b) ∧ ¬3 ∣ c
ha : HPow.hPow (α := ZMod 9) (β := ℕ) (↑a : ZMod 9) 3 = 8
hb : HPow.hPow (α := ZMod 9) (β := ℕ) (↑b : ZMod 9) 3 = 8
inr :
  ∀ (hc : HPow.hPow (α := ZMod 9) (β := ℕ) (↑c : ZMod 9) 3 = 1),
    ¬HPow.hPow (α := ZMod 9) (β := ℕ) (↑a : ZMod 9) 3 + HPow.hPow (α := ZMod 9) (β := ℕ) (↑b : ZMod 9) 3 =
        HPow.hPow (α := ZMod 9) (β := ℕ) (↑c : ZMod 9) 3
⊢ ¬HPow.hPow (α := ZMod 9) (β := ℕ) (↑a : ZMod 9) 3 + HPow.hPow (α := ZMod 9) (β := ℕ) (↑b : ZMod 9) 3 =
      HPow.hPow (α := ZMod 9) (β := ℕ) (↑c : ZMod 9) 3
have inr₁ : ∀ (hc : HPow.hPow (α := ZMod 9) (β := ℕ) (↑c : ZMod 9) 3 = 8),
  ¬HPow.hPow (α := ZMod 9) (β := ℕ) (↑a : ZMod 9) 3 + HPow.hPow (α := ZMod 9) (β := ℕ) (↑b : ZMod 9) 3 =
      HPow.hPow (α := ZMod 9) (β := ℕ) (↑c : ZMod 9) 3
Tactic: rcases cube_of_not_dvd hdvd.2 with hc | hc
-- Lean 4 proof step from Mathlib.NumberTheory.FLT.Three (goal)
Goal:
a b c : ℤ
hdvd : (¬3 ∣ a ∧ ¬3 ∣ b) ∧ ¬3 ∣ c
ha : HPow.hPow (α := ZMod 9) (β := ℕ) (↑a : ZMod 9) 3 = 1
hb : HPow.hPow (α := ZMod 9) (β := ℕ) (↑b : ZMod 9) 3 = 1
hc : HPow.hPow (α := ZMod 9) (β := ℕ) (↑c : ZMod 9) 3 = 1
⊢ ¬HPow.hPow (α := ZMod 9) (β := ℕ) (↑a : ZMod 9) 3 + HPow.hPow (α := ZMod 9) (β := ℕ) (↑b : ZMod 9) 3 =
      HPow.hPow (α := ZMod 9) (β := ℕ) (↑c : ZMod 9) 3
have inl : ¬HAdd.hAdd (α := ZMod 9) 1 1 = 1
Tactic: rw [ha, hb, hc]
-- Lean 4 proof step from Mathlib.NumberTheory.FLT.Three (goal)
Goal:
a b c : ℤ
hdvd : (¬3 ∣ a ∧ ¬3 ∣ b) ∧ ¬3 ∣ c
ha : HPow.hPow (α := ZMod 9) (β := ℕ) (↑a : ZMod 9) 3 = 1
hb : HPow.hPow (α := ZMod 9) (β := ℕ) (↑b : ZMod 9) 3 = 1
hc : HPow.hPow (α := ZMod 9) (β := ℕ) (↑c : ZMod 9) 3 = 8
⊢ ¬HPow.hPow (α := ZMod 9) (β := ℕ) (↑a : ZMod 9) 3 + HPow.hPow (α := ZMod 9) (β := ℕ) (↑b : ZMod 9) 3 =
      HPow.hPow (α := ZMod 9) (β := ℕ) (↑c : ZMod 9) 3
have inl : ¬HAdd.hAdd (α := ZMod 9) 1 1 = 8
Tactic: rw [ha, hb, hc]
-- Lean 4 proof step from Mathlib.NumberTheory.FLT.Three (goal)
Goal:
a b c : ℤ
hdvd : (¬3 ∣ a ∧ ¬3 ∣ b) ∧ ¬3 ∣ c
ha : HPow.hPow (α := ZMod 9) (β := ℕ) (↑a : ZMod 9) 3 = 1
hb : HPow.hPow (α := ZMod 9) (β := ℕ) (↑b : ZMod 9) 3 = 8
hc : HPow.hPow (α := ZMod 9) (β := ℕ) (↑c : ZMod 9) 3 = 1
⊢ ¬HPow.hPow (α := ZMod 9) (β := ℕ) (↑a : ZMod 9) 3 + HPow.hPow (α := ZMod 9) (β := ℕ) (↑b : ZMod 9) 3 =
      HPow.hPow (α := ZMod 9) (β := ℕ) (↑c : ZMod 9) 3
have inl : ¬HAdd.hAdd (α := ZMod 9) 1 8 = 1
Tactic: rw [ha, hb, hc]