awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Module: Mathlib.NumberTheory.FLT.Three
Goal:
H :
  ∀ (a b c : ℤ),
    c ≠ 0 →
      ¬3 ∣ a → ¬3 ∣ b → 3 ∣ c → IsCoprime a b → HPow.hPow (β := ℕ) a 3 + HPow.hPow (β := ℕ) b 3 ≠ HPow.hPow (β := ℕ) c 3
refine :
  ∀ (a b c : ℤ) (ha : a ≠ 0) (hb : b ≠ 0) (hc : c ≠ 0) (Hgcd : Eq (α := ℤ) (Finset.gcd {a, b, c} id) 1)
    (hF : HPow.hPow (β := ℕ) a 3 + HPow.hPow (β := ℕ) b 3 = HPow.hPow (β := ℕ) c 3), False
⊢ FermatLastTheoremWith ℤ 3

Tactic:
refine fermatLastTheoremWith_of_fermatLastTheoremWith_coprime (fun a b c ha hb hc Hgcd hF ↦ ?_)

Premises used: fermatLastTheoremWith_of_fermatLastTheoremWith_coprime
-- Module: Mathlib.NumberTheory.FLT.Three
Goal:
H :
  ∀ (a b c : ℤ),
    c ≠ 0 →
      ¬3 ∣ a → ¬3 ∣ b → 3 ∣ c → IsCoprime a b → HPow.hPow (β := ℕ) a 3 + HPow.hPow (β := ℕ) b 3 ≠ HPow.hPow (β := ℕ) c 3
a b c : ℤ
ha : a ≠ 0
hb : b ≠ 0
hc : c ≠ 0
Hgcd : Eq (α := ℤ) (Finset.gcd {a, b, c} id) 1
hF : HPow.hPow (β := ℕ) a 3 + HPow.hPow (β := ℕ) b 3 = HPow.hPow (β := ℕ) c 3
pos : ∀ (h1 : 3 ∣ a * b * c), False
neg : ∀ (h1 : ¬3 ∣ a * b * c), False
⊢ False

Tactic:
by_cases h1 : 3 ∣ a * b * c
-- Module: Mathlib.NumberTheory.FLT.Three
Goal:
H :
  ∀ (a b c : ℤ),
    c ≠ 0 →
      ¬3 ∣ a → ¬3 ∣ b → 3 ∣ c → IsCoprime a b → HPow.hPow (β := ℕ) a 3 + HPow.hPow (β := ℕ) b 3 ≠ HPow.hPow (β := ℕ) c 3
a b c : ℤ
ha : a ≠ 0
hb : b ≠ 0
hc : c ≠ 0
Hgcd : Eq (α := ℤ) (Finset.gcd {a, b, c} id) 1
hF : HPow.hPow (β := ℕ) a 3 + HPow.hPow (β := ℕ) b 3 = HPow.hPow (β := ℕ) c 3
h1 : ¬3 ∣ a * b * c
⊢ False

Tactic:
fermatLastTheoremThree_case_1 h1 hF

Premises used: _private.0.fermatLastTheoremThree_case_1
-- Module: Mathlib.NumberTheory.FLT.Three
Goal:
H :
  ∀ (a b c : ℤ),
    c ≠ 0 →
      ¬3 ∣ a → ¬3 ∣ b → 3 ∣ c → IsCoprime a b → HPow.hPow (β := ℕ) a 3 + HPow.hPow (β := ℕ) b 3 ≠ HPow.hPow (β := ℕ) c 3
a b c : ℤ
ha : a ≠ 0
hb : b ≠ 0
hc : c ≠ 0
Hgcd : Eq (α := ℤ) (Finset.gcd {a, b, c} id) 1
hF : HPow.hPow (β := ℕ) a 3 + HPow.hPow (β := ℕ) b 3 = HPow.hPow (β := ℕ) c 3
h1 : ¬3 ∣ a * b * c
⊢ False

Tactic:
exact fermatLastTheoremThree_case_1 h1 hF

Premises used: _private.0.fermatLastTheoremThree_case_1
-- Module: Mathlib.NumberTheory.FLT.Three
Goal:
H :
  ∀ (a b c : ℤ),
    c ≠ 0 →
      ¬3 ∣ a → ¬3 ∣ b → 3 ∣ c → IsCoprime a b → HPow.hPow (β := ℕ) a 3 + HPow.hPow (β := ℕ) b 3 ≠ HPow.hPow (β := ℕ) c 3
a b c : ℤ
ha : a ≠ 0
hb : b ≠ 0
hc : c ≠ 0
Hgcd : Eq (α := ℤ) (Finset.gcd {a, b, c} id) 1
hF : HPow.hPow (β := ℕ) a 3 + HPow.hPow (β := ℕ) b 3 = HPow.hPow (β := ℕ) c 3
h1 : 3 ∣ a * b * c
⊢ (3 ∣ a ∨ 3 ∣ b) ∨ 3 ∣ c

Tactic:
rw [prime_three.dvd_mul, prime_three.dvd_mul] at h1

Premises used: Prime.dvd_mul, Int.prime_three
-- Module: Mathlib.NumberTheory.FLT.Three
Goal:
H :
  ∀ (a b c : ℤ),
    c ≠ 0 →
      ¬3 ∣ a → ¬3 ∣ b → 3 ∣ c → IsCoprime a b → HPow.hPow (β := ℕ) a 3 + HPow.hPow (β := ℕ) b 3 ≠ HPow.hPow (β := ℕ) c 3
a b c : ℤ
ha : a ≠ 0
hb : b ≠ 0
hc : c ≠ 0
Hgcd : Eq (α := ℤ) (Finset.gcd {a, b, c} id) 1
hF : HPow.hPow (β := ℕ) a 3 + HPow.hPow (β := ℕ) b 3 + -HPow.hPow (β := ℕ) c 3 = 0
h1 : (3 ∣ a ∨ 3 ∣ b) ∨ 3 ∣ c
⊢ Odd (α := ℕ) 3

Tactic:
decide
-- Module: Mathlib.NumberTheory.FLT.Three
Goal:
H :
  ∀ (a b c : ℤ),
    c ≠ 0 →
      ¬3 ∣ a → ¬3 ∣ b → 3 ∣ c → IsCoprime a b → HPow.hPow (β := ℕ) a 3 + HPow.hPow (β := ℕ) b 3 ≠ HPow.hPow (β := ℕ) c 3
a b c : ℤ
ha : a ≠ 0
hb : b ≠ 0
hc : c ≠ 0
Hgcd : Eq (α := ℤ) (Finset.gcd {a, b, c} id) 1
hF : HPow.hPow (β := ℕ) a 3 + HPow.hPow (β := ℕ) b 3 = HPow.hPow (β := ℕ) c 3
h1 : (3 ∣ a ∨ 3 ∣ b) ∨ 3 ∣ c
⊢ HPow.hPow (β := ℕ) a 3 + HPow.hPow (β := ℕ) b 3 + HPow.hPow (β := ℕ) (-c) 3 = 0

Tactic:
rw [← sub_eq_zero, sub_eq_add_neg, ← (show Odd 3 by decide).neg_pow] at hF

Premises used: Odd.neg_pow, sub_eq_zero, sub_eq_add_neg
-- Module: Mathlib.NumberTheory.FLT.Three
Goal:
H :
  ∀ (a b c : ℤ),
    c ≠ 0 →
      ¬3 ∣ a → ¬3 ∣ b → 3 ∣ c → IsCoprime a b → HPow.hPow (β := ℕ) a 3 + HPow.hPow (β := ℕ) b 3 ≠ HPow.hPow (β := ℕ) c 3
a b c : ℤ
ha : a ≠ 0
hb : b ≠ 0
hc : c ≠ 0
Hgcd : Eq (α := ℤ) (Finset.gcd {a, b, c} id) 1
hF : HPow.hPow (β := ℕ) a 3 + HPow.hPow (β := ℕ) b 3 = HPow.hPow (β := ℕ) c 3
h1 : (3 ∣ a ∨ 3 ∣ b) ∨ 3 ∣ c
hF₁ : HPow.hPow (β := ℕ) a 3 + HPow.hPow (β := ℕ) b 3 + HPow.hPow (β := ℕ) (-c) 3 = 0
⊢ (3 ∣ a ∨ 3 ∣ b) ∨ 3 ∣ c

Tactic:
rw [← sub_eq_zero, sub_eq_add_neg, ← (show Odd 3 by decide).neg_pow] at hF

Premises used: Odd.neg_pow, sub_eq_zero, sub_eq_add_neg
-- Module: Mathlib.NumberTheory.FLT.Three
Goal:
H :
  ∀ (a b c : ℤ),
    c ≠ 0 →
      ¬3 ∣ a → ¬3 ∣ b → 3 ∣ c → IsCoprime a b → HPow.hPow (β := ℕ) a 3 + HPow.hPow (β := ℕ) b 3 ≠ HPow.hPow (β := ℕ) c 3
a b c : ℤ
ha : a ≠ 0
hb : b ≠ 0
hc : c ≠ 0
Hgcd : Eq (α := ℤ) (Finset.gcd {a, b, c} id) 1
hF : HPow.hPow (β := ℕ) a 3 + HPow.hPow (β := ℕ) b 3 + HPow.hPow (β := ℕ) (-c) 3 = 0
h1 : (3 ∣ a ∨ 3 ∣ b) ∨ 3 ∣ c
pos : ∀ (h3a : 3 ∣ a), False
pos₁ : ∀ (h3b : 3 ∣ b), False
pos₂ : ∀ (h3c : 3 ∣ c), False
⊢ False

Tactic:
rcases h1 with (h3a | h3b) | h3c
-- Module: Mathlib.NumberTheory.FLT.Three
Goal:
H :
  ∀ (a b c : ℤ),
    c ≠ 0 →
      ¬3 ∣ a → ¬3 ∣ b → 3 ∣ c → IsCoprime a b → HPow.hPow (β := ℕ) a 3 + HPow.hPow (β := ℕ) b 3 ≠ HPow.hPow (β := ℕ) c 3
a b c : ℤ
ha : a ≠ 0
hb : b ≠ 0
hc : c ≠ 0
Hgcd : Eq (α := ℤ) (Finset.gcd {a, b, c} id) 1
hF : HPow.hPow (β := ℕ) a 3 + HPow.hPow (β := ℕ) b 3 + HPow.hPow (β := ℕ) (-c) 3 = 0
h3a : 3 ∣ a
⊢ False

Tactic:
fermatLastTheoremThree_of_dvd_a_of_gcd_eq_one_of_case2 ha h3a ?_ H hF

Premises used: _private.0.fermatLastTheoremThree_of_dvd_a_of_gcd_eq_one_of_case2
-- Module: Mathlib.NumberTheory.FLT.Three
Goal:
H :
  ∀ (a b c : ℤ),
    c ≠ 0 →
      ¬3 ∣ a → ¬3 ∣ b → 3 ∣ c → IsCoprime a b → HPow.hPow (β := ℕ) a 3 + HPow.hPow (β := ℕ) b 3 ≠ HPow.hPow (β := ℕ) c 3
a b c : ℤ
ha : a ≠ 0
hb : b ≠ 0
hc : c ≠ 0
Hgcd : Eq (α := ℤ) (Finset.gcd {a, b, c} id) 1
hF : HPow.hPow (β := ℕ) a 3 + HPow.hPow (β := ℕ) b 3 + HPow.hPow (β := ℕ) (-c) 3 = 0
h3a : 3 ∣ a
pos : Eq (α := ℤ) (Finset.gcd {a, b, -c} id) 1
⊢ False

Tactic:
refine fermatLastTheoremThree_of_dvd_a_of_gcd_eq_one_of_case2 ha h3a ?_ H hF

Premises used: _private.0.fermatLastTheoremThree_of_dvd_a_of_gcd_eq_one_of_case2
-- Module: Mathlib.NumberTheory.FLT.Three
Goal:
H :
  ∀ (a b c : ℤ),
    c ≠ 0 →
      ¬3 ∣ a → ¬3 ∣ b → 3 ∣ c → IsCoprime a b → HPow.hPow (β := ℕ) a 3 + HPow.hPow (β := ℕ) b 3 ≠ HPow.hPow (β := ℕ) c 3
a b c : ℤ
ha : a ≠ 0
hb : b ≠ 0
hc : c ≠ 0
Hgcd : Eq (α := ℤ) (Finset.gcd {a, b, c} id) 1
hF : HPow.hPow (β := ℕ) a 3 + HPow.hPow (β := ℕ) b 3 + HPow.hPow (β := ℕ) (-c) 3 = 0
h3a : 3 ∣ a
⊢ Eq (α := ℤ) (Finset.gcd {a, b, -c} id) 1

Tactic:
simp only [← Hgcd, gcd_insert, gcd_singleton, id_eq, ← abs_eq_normalize, abs_neg]

Premises used: abs_neg, Finset.gcd_singleton, Finset.gcd_insert, Int.abs_eq_normalize, id_eq
-- Module: Mathlib.NumberTheory.FLT.Three
Goal:
H :
  ∀ (a b c : ℤ),
    c ≠ 0 →
      ¬3 ∣ a → ¬3 ∣ b → 3 ∣ c → IsCoprime a b → HPow.hPow (β := ℕ) a 3 + HPow.hPow (β := ℕ) b 3 ≠ HPow.hPow (β := ℕ) c 3
a b c : ℤ
ha : a ≠ 0
hb : b ≠ 0
hc : c ≠ 0
Hgcd : Eq (α := ℤ) (Finset.gcd {a, b, c} id) 1
hF : HPow.hPow (β := ℕ) a 3 + HPow.hPow (β := ℕ) b 3 + HPow.hPow (β := ℕ) (-c) 3 = 0
h3b : 3 ∣ b
⊢ HPow.hPow (β := ℕ) b 3 + HPow.hPow (β := ℕ) a 3 + HPow.hPow (β := ℕ) (-c) 3 = 0

Tactic:
rw [add_comm (a ^ 3)] at hF

Premises used: add_comm, HPow.hPow, OfNat.ofNat
-- Module: Mathlib.NumberTheory.FLT.Three
Goal:
H :
  ∀ (a b c : ℤ),
    c ≠ 0 →
      ¬3 ∣ a → ¬3 ∣ b → 3 ∣ c → IsCoprime a b → HPow.hPow (β := ℕ) a 3 + HPow.hPow (β := ℕ) b 3 ≠ HPow.hPow (β := ℕ) c 3
a b c : ℤ
ha : a ≠ 0
hb : b ≠ 0
hc : c ≠ 0
Hgcd : Eq (α := ℤ) (Finset.gcd {a, b, c} id) 1
hF : HPow.hPow (β := ℕ) a 3 + HPow.hPow (β := ℕ) b 3 + HPow.hPow (β := ℕ) (-c) 3 = 0
h3b : 3 ∣ b
hF₁ : HPow.hPow (β := ℕ) b 3 + HPow.hPow (β := ℕ) a 3 + HPow.hPow (β := ℕ) (-c) 3 = 0
⊢ 3 ∣ b

Tactic:
rw [add_comm (a ^ 3)] at hF

Premises used: add_comm, HPow.hPow, OfNat.ofNat
-- Module: Mathlib.NumberTheory.FLT.Three
Goal:
H :
  ∀ (a b c : ℤ),
    c ≠ 0 →
      ¬3 ∣ a → ¬3 ∣ b → 3 ∣ c → IsCoprime a b → HPow.hPow (β := ℕ) a 3 + HPow.hPow (β := ℕ) b 3 ≠ HPow.hPow (β := ℕ) c 3
a b c : ℤ
ha : a ≠ 0
hb : b ≠ 0
hc : c ≠ 0
Hgcd : Eq (α := ℤ) (Finset.gcd {a, b, c} id) 1
hF : HPow.hPow (β := ℕ) b 3 + HPow.hPow (β := ℕ) a 3 + HPow.hPow (β := ℕ) (-c) 3 = 0
h3b : 3 ∣ b
⊢ False

Tactic:
fermatLastTheoremThree_of_dvd_a_of_gcd_eq_one_of_case2 hb h3b ?_ H hF

Premises used: _private.0.fermatLastTheoremThree_of_dvd_a_of_gcd_eq_one_of_case2
-- Module: Mathlib.NumberTheory.FLT.Three
Goal:
H :
  ∀ (a b c : ℤ),
    c ≠ 0 →
      ¬3 ∣ a → ¬3 ∣ b → 3 ∣ c → IsCoprime a b → HPow.hPow (β := ℕ) a 3 + HPow.hPow (β := ℕ) b 3 ≠ HPow.hPow (β := ℕ) c 3
a b c : ℤ
ha : a ≠ 0
hb : b ≠ 0
hc : c ≠ 0
Hgcd : Eq (α := ℤ) (Finset.gcd {a, b, c} id) 1
hF : HPow.hPow (β := ℕ) b 3 + HPow.hPow (β := ℕ) a 3 + HPow.hPow (β := ℕ) (-c) 3 = 0
h3b : 3 ∣ b
pos : Eq (α := ℤ) (Finset.gcd {b, a, -c} id) 1
⊢ False

Tactic:
refine fermatLastTheoremThree_of_dvd_a_of_gcd_eq_one_of_case2 hb h3b ?_ H hF

Premises used: _private.0.fermatLastTheoremThree_of_dvd_a_of_gcd_eq_one_of_case2
-- Module: Mathlib.NumberTheory.FLT.Three
Goal:
H :
  ∀ (a b c : ℤ),
    c ≠ 0 →
      ¬3 ∣ a → ¬3 ∣ b → 3 ∣ c → IsCoprime a b → HPow.hPow (β := ℕ) a 3 + HPow.hPow (β := ℕ) b 3 ≠ HPow.hPow (β := ℕ) c 3
a b c : ℤ
ha : a ≠ 0
hb : b ≠ 0
hc : c ≠ 0
Hgcd : Eq (α := ℤ) (Finset.gcd {a, b, c} id) 1
hF : HPow.hPow (β := ℕ) b 3 + HPow.hPow (β := ℕ) a 3 + HPow.hPow (β := ℕ) (-c) 3 = 0
h3b : 3 ∣ b
⊢ Eq (α := ℤ) (Finset.gcd {b, a, -c} id) 1

Tactic:
simp only [← Hgcd, insert_comm, gcd_insert, gcd_singleton, id_eq, ← abs_eq_normalize, abs_neg]

Premises used: abs_neg, Finset.gcd_singleton, Finset.gcd_insert, Int.abs_eq_normalize, Finset.insert_comm, id_eq
-- Module: Mathlib.NumberTheory.FLT.Three
Goal:
H :
  ∀ (a b c : ℤ),
    c ≠ 0 →
      ¬3 ∣ a → ¬3 ∣ b → 3 ∣ c → IsCoprime a b → HPow.hPow (β := ℕ) a 3 + HPow.hPow (β := ℕ) b 3 ≠ HPow.hPow (β := ℕ) c 3
a b c : ℤ
ha : a ≠ 0
hb : b ≠ 0
hc : c ≠ 0
Hgcd : Eq (α := ℤ) (Finset.gcd {a, b, c} id) 1
hF : HPow.hPow (β := ℕ) a 3 + HPow.hPow (β := ℕ) b 3 + HPow.hPow (β := ℕ) (-c) 3 = 0
h3c : 3 ∣ c
⊢ HPow.hPow (β := ℕ) (-c) 3 + HPow.hPow (β := ℕ) a 3 + HPow.hPow (β := ℕ) b 3 = 0

Tactic:
rw [add_comm _ ((-c) ^ 3), ← add_assoc] at hF

Premises used: add_assoc, add_comm, HPow.hPow, OfNat.ofNat, Neg.neg
-- Module: Mathlib.NumberTheory.FLT.Three
Goal:
H :
  ∀ (a b c : ℤ),
    c ≠ 0 →
      ¬3 ∣ a → ¬3 ∣ b → 3 ∣ c → IsCoprime a b → HPow.hPow (β := ℕ) a 3 + HPow.hPow (β := ℕ) b 3 ≠ HPow.hPow (β := ℕ) c 3
a b c : ℤ
ha : a ≠ 0
hb : b ≠ 0
hc : c ≠ 0
Hgcd : Eq (α := ℤ) (Finset.gcd {a, b, c} id) 1
hF : HPow.hPow (β := ℕ) a 3 + HPow.hPow (β := ℕ) b 3 + HPow.hPow (β := ℕ) (-c) 3 = 0
h3c : 3 ∣ c
hF₁ : HPow.hPow (β := ℕ) (-c) 3 + HPow.hPow (β := ℕ) a 3 + HPow.hPow (β := ℕ) b 3 = 0
⊢ 3 ∣ c

Tactic:
rw [add_comm _ ((-c) ^ 3), ← add_assoc] at hF

Premises used: add_assoc, add_comm, HPow.hPow, OfNat.ofNat, Neg.neg
-- Module: Mathlib.NumberTheory.FLT.Three
Goal:
H :
  ∀ (a b c : ℤ),
    c ≠ 0 →
      ¬3 ∣ a → ¬3 ∣ b → 3 ∣ c → IsCoprime a b → HPow.hPow (β := ℕ) a 3 + HPow.hPow (β := ℕ) b 3 ≠ HPow.hPow (β := ℕ) c 3
a b c : ℤ
ha : a ≠ 0
hb : b ≠ 0
hc : c ≠ 0
Hgcd : Eq (α := ℤ) (Finset.gcd {a, b, c} id) 1
hF : HPow.hPow (β := ℕ) (-c) 3 + HPow.hPow (β := ℕ) a 3 + HPow.hPow (β := ℕ) b 3 = 0
h3c : 3 ∣ c
⊢ 3 ∣ -c

Tactic:
simp [h3c]
-- Module: Mathlib.NumberTheory.FLT.Three
Goal:
H :
  ∀ (a b c : ℤ),
    c ≠ 0 →
      ¬3 ∣ a → ¬3 ∣ b → 3 ∣ c → IsCoprime a b → HPow.hPow (β := ℕ) a 3 + HPow.hPow (β := ℕ) b 3 ≠ HPow.hPow (β := ℕ) c 3
a b c : ℤ
ha : a ≠ 0
hb : b ≠ 0
hc : c ≠ 0
Hgcd : Eq (α := ℤ) (Finset.gcd {a, b, c} id) 1
hF : HPow.hPow (β := ℕ) (-c) 3 + HPow.hPow (β := ℕ) a 3 + HPow.hPow (β := ℕ) b 3 = 0
h3c : 3 ∣ c
pos : Eq (α := ℤ) (Finset.gcd {-c, a, b} id) 1
⊢ False

Tactic:
refine fermatLastTheoremThree_of_dvd_a_of_gcd_eq_one_of_case2 (neg_ne_zero.2 hc) (by simp [h3c])
      ?_ H hF

Premises used: Iff.mpr, _private.0.fermatLastTheoremThree_of_dvd_a_of_gcd_eq_one_of_case2, neg_ne_zero
-- Module: Mathlib.NumberTheory.FLT.Three
Goal:
H :
  ∀ (a b c : ℤ),
    c ≠ 0 →
      ¬3 ∣ a → ¬3 ∣ b → 3 ∣ c → IsCoprime a b → HPow.hPow (β := ℕ) a 3 + HPow.hPow (β := ℕ) b 3 ≠ HPow.hPow (β := ℕ) c 3
a b c : ℤ
ha : a ≠ 0
hb : b ≠ 0
hc : c ≠ 0
Hgcd : Eq (α := ℤ) (Finset.gcd {a, b, c} id) 1
hF : HPow.hPow (β := ℕ) (-c) 3 + HPow.hPow (β := ℕ) a 3 + HPow.hPow (β := ℕ) b 3 = 0
h3c : 3 ∣ c
pos : Eq (α := ℤ) (Finset.gcd {a, b, -c} id) 1
⊢ Eq (α := ℤ) (Finset.gcd {-c, a, b} id) 1

Tactic:
rw [Finset.insert_comm (-c), Finset.pair_comm (-c) b]

Premises used: Finset.pair_comm, Finset.insert_comm, Neg.neg
-- Module: Mathlib.NumberTheory.FLT.Three
Goal:
H :
  ∀ (a b c : ℤ),
    c ≠ 0 →
      ¬3 ∣ a → ¬3 ∣ b → 3 ∣ c → IsCoprime a b → HPow.hPow (β := ℕ) a 3 + HPow.hPow (β := ℕ) b 3 ≠ HPow.hPow (β := ℕ) c 3
a b c : ℤ
ha : a ≠ 0
hb : b ≠ 0
hc : c ≠ 0
Hgcd : Eq (α := ℤ) (Finset.gcd {a, b, c} id) 1
hF : HPow.hPow (β := ℕ) (-c) 3 + HPow.hPow (β := ℕ) a 3 + HPow.hPow (β := ℕ) b 3 = 0
h3c : 3 ∣ c
⊢ Eq (α := ℤ) (Finset.gcd {a, b, -c} id) 1

Tactic:
simp only [← Hgcd, gcd_insert, gcd_singleton, id_eq, ← abs_eq_normalize, abs_neg]

Premises used: abs_neg, Finset.gcd_singleton, Finset.gcd_insert, Int.abs_eq_normalize, id_eq
-- Module: Mathlib.NumberTheory.FLT.Three
Goal:
K : Type u_1
inst : Field K
ζ : K
hζ : IsPrimitiveRoot ζ 3
⊢ 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
inst_1 : NumberField K
inst_2 : IsCyclotomicExtension {3} ℚ K
H : FermatLastTheoremForThreeGen hζ
⊢ FermatLastTheoremFor 3

Tactic:
fermatLastTheoremThree_of_three_dvd_only_c (fun a b c hc ha hb ⟨x, hx⟩ hcoprime h ↦ ?_)

Premises used: False, _private.0.fermatLastTheoremThree_of_three_dvd_only_c, _private.0.FermatLastTheoremForThree_of_FermatLastTheoremThreeGen.match_1
-- 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
H : FermatLastTheoremForThreeGen hζ
refine :
  ∀ (a b c : ℤ) (hc : c ≠ 0) (ha : ¬3 ∣ a) (hb : ¬3 ∣ b),
    3 ∣ c →
      ∀ (hcoprime : IsCoprime a b) (h : HPow.hPow (β := ℕ) a 3 + HPow.hPow (β := ℕ) b 3 = HPow.hPow (β := ℕ) c 3)
        (x : ℤ) (hx : c = 3 * x), False
⊢ FermatLastTheoremFor 3

Tactic:
refine fermatLastTheoremThree_of_three_dvd_only_c (fun a b c hc ha hb ⟨x, hx⟩ hcoprime h ↦ ?_)

Premises used: False, _private.0.fermatLastTheoremThree_of_three_dvd_only_c, _private.0.FermatLastTheoremForThree_of_FermatLastTheoremThreeGen.match_1
-- 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
H : FermatLastTheoremForThreeGen hζ
a b c : ℤ
hc : c ≠ 0
ha : ¬3 ∣ a
hb : ¬3 ∣ b
h_1 : 3 ∣ c
hcoprime : IsCoprime a b
h : HPow.hPow (β := ℕ) a 3 + HPow.hPow (β := ℕ) b 3 = HPow.hPow (β := ℕ) c 3
x : ℤ
hx : c = 3 * x
⊢ Ne.{u_1 + 1} (α := 𝓞 K) (↑c : 𝓞 K) 0

Tactic:
simp [hc]
-- 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
H : FermatLastTheoremForThreeGen hζ
a b c : ℤ
hc : c ≠ 0
ha : ¬3 ∣ a
hb : ¬3 ∣ b
h_1 : 3 ∣ c
hcoprime : IsCoprime a b
h : HPow.hPow (β := ℕ) a 3 + HPow.hPow (β := ℕ) b 3 = HPow.hPow (β := ℕ) c 3
x : ℤ
hx : c = 3 * x
refine_1 : ∀ (hdvd : λ ∣ (↑a : 𝓞 K)), 3 ∣ a
refine_2 : ∀ (hdvd : λ ∣ (↑b : 𝓞 K)), 3 ∣ b
refine_3 : λ ∣ (↑c : 𝓞 K)
refine_4 : IsCoprime.{u_1} (R := 𝓞 K) (↑a : 𝓞 K) (↑b : 𝓞 K)
refine_5 :
  HPow.hPow (α := 𝓞 K) (β := ℕ) (↑a : 𝓞 K) 3 + HPow.hPow (α := 𝓞 K) (β := ℕ) (↑b : 𝓞 K) 3 =
    (↑1 : 𝓞 K) * HPow.hPow (α := 𝓞 K) (β := ℕ) (↑c : 𝓞 K) 3
⊢ False

Tactic:
refine H a b c 1 (by simp [hc]) (fun hdvd ↦ ha ?_) (fun hdvd ↦ hb ?_) ?_ ?_ ?_

Premises used: Int.cast, OfNat.ofNat
-- 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
H : FermatLastTheoremForThreeGen hζ
a b c : ℤ
hc : c ≠ 0
ha : ¬3 ∣ a
hb : ¬3 ∣ b
h_1 : 3 ∣ c
hcoprime : IsCoprime a b
h : HPow.hPow (β := ℕ) a 3 + HPow.hPow (β := ℕ) b 3 = HPow.hPow (β := ℕ) c 3
x : ℤ
hx : c = 3 * x
hdvd : λ ∣ (↑a : 𝓞 K)
⊢ Ne (α := ℕ) 3 2

Tactic:
decide
-- 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
H : FermatLastTheoremForThreeGen hζ
a b c : ℤ
hc : c ≠ 0
ha : ¬3 ∣ a
hb : ¬3 ∣ b
h_1 : 3 ∣ c
hcoprime : IsCoprime a b
h : HPow.hPow (β := ℕ) a 3 + HPow.hPow (β := ℕ) b 3 = HPow.hPow (β := ℕ) c 3
x : ℤ
hx : c = 3 * x
hdvd : (Algebra.norm ℤ : (a : 𝓞 K) → ℤ) λ ∣ a
⊢ Ne (α := ℕ) 3 2

Tactic:
decide
-- 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
H : FermatLastTheoremForThreeGen hζ
a b c : ℤ
hc : c ≠ 0
ha : ¬3 ∣ a
hb : ¬3 ∣ b
h_1 : 3 ∣ c
hcoprime : IsCoprime a b
h : HPow.hPow (β := ℕ) a 3 + HPow.hPow (β := ℕ) b 3 = HPow.hPow (β := ℕ) c 3
x : ℤ
hx : c = 3 * x
hdvd : λ ∣ (↑a : 𝓞 K)
refine_1 : ∀ (hdvd : (Algebra.norm ℤ : (a : 𝓞 K) → ℤ) λ ∣ a), 3 ∣ a
refine_1₁ refine_1₂ refine_1₃ refine_1₄ : ∀ (hdvd : (↑3 : ℤ) ∣ a), 3 ∣ a
⊢ 3 ∣ a

Tactic:
rwa [← Ideal.norm_dvd_iff (hζ.prime_norm_toInteger_sub_one_of_prime_ne_two' (by decide)),
      hζ.norm_toInteger_sub_one_of_prime_ne_two' (by decide)] at hdvd

Premises used: Ideal.norm_dvd_iff, IsPrimitiveRoot.prime_norm_toInteger_sub_one_of_prime_ne_two', IsPrimitiveRoot.norm_toInteger_sub_one_of_prime_ne_two'
-- 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
H : FermatLastTheoremForThreeGen hζ
a b c : ℤ
hc : c ≠ 0
ha : ¬3 ∣ a
hb : ¬3 ∣ b
h_1 : 3 ∣ c
hcoprime : IsCoprime a b
h : HPow.hPow (β := ℕ) a 3 + HPow.hPow (β := ℕ) b 3 = HPow.hPow (β := ℕ) c 3
x : ℤ
hx : c = 3 * x
hdvd : λ ∣ (↑b : 𝓞 K)
⊢ Ne (α := ℕ) 3 2

Tactic:
decide
-- 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
H : FermatLastTheoremForThreeGen hζ
a b c : ℤ
hc : c ≠ 0
ha : ¬3 ∣ a
hb : ¬3 ∣ b
h_1 : 3 ∣ c
hcoprime : IsCoprime a b
h : HPow.hPow (β := ℕ) a 3 + HPow.hPow (β := ℕ) b 3 = HPow.hPow (β := ℕ) c 3
x : ℤ
hx : c = 3 * x
hdvd : (Algebra.norm ℤ : (a : 𝓞 K) → ℤ) λ ∣ b
⊢ Ne (α := ℕ) 3 2

Tactic:
decide
-- 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
H : FermatLastTheoremForThreeGen hζ
a b c : ℤ
hc : c ≠ 0
ha : ¬3 ∣ a
hb : ¬3 ∣ b
h_1 : 3 ∣ c
hcoprime : IsCoprime a b
h : HPow.hPow (β := ℕ) a 3 + HPow.hPow (β := ℕ) b 3 = HPow.hPow (β := ℕ) c 3
x : ℤ
hx : c = 3 * x
hdvd : λ ∣ (↑b : 𝓞 K)
refine_2 : ∀ (hdvd : (Algebra.norm ℤ : (a : 𝓞 K) → ℤ) λ ∣ b), 3 ∣ b
refine_2₁ refine_2₂ refine_2₃ refine_2₄ : ∀ (hdvd : (↑3 : ℤ) ∣ b), 3 ∣ b
⊢ 3 ∣ b

Tactic:
rwa [← Ideal.norm_dvd_iff (hζ.prime_norm_toInteger_sub_one_of_prime_ne_two' (by decide)),
      hζ.norm_toInteger_sub_one_of_prime_ne_two' (by decide)] at hdvd

Premises used: Ideal.norm_dvd_iff, IsPrimitiveRoot.prime_norm_toInteger_sub_one_of_prime_ne_two', IsPrimitiveRoot.norm_toInteger_sub_one_of_prime_ne_two'
-- 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
H : FermatLastTheoremForThreeGen hζ
a b c : ℤ
hc : c ≠ 0
ha : ¬3 ∣ a
hb : ¬3 ∣ b
h_1 : 3 ∣ c
hcoprime : IsCoprime a b
h : HPow.hPow (β := ℕ) a 3 + HPow.hPow (β := ℕ) b 3 = HPow.hPow (β := ℕ) c 3
x : ℤ
hx : c = 3 * x
⊢ (↑c : 𝓞 K) = HMul.hMul (α := 𝓞 K) (↑3 : 𝓞 K) (↑x : 𝓞 K)

Tactic:
simp [hx]
-- 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
H : FermatLastTheoremForThreeGen hζ
a b c : ℤ
hc : c ≠ 0
ha : ¬3 ∣ a
hb : ¬3 ∣ b
h_1 : 3 ∣ c
hcoprime : IsCoprime a b
h : HPow.hPow (β := ℕ) a 3 + HPow.hPow (β := ℕ) b 3 = HPow.hPow (β := ℕ) c 3
x : ℤ
hx : c = 3 * x
⊢ λ ∣ (↑c : 𝓞 K)

Tactic:
exact dvd_trans hζ.toInteger_sub_one_dvd_prime' ⟨x, by simp [hx]⟩

Premises used: Int.cast, dvd_trans, IsPrimitiveRoot.toInteger_sub_one_dvd_prime', Exists.intro
-- 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
H : FermatLastTheoremForThreeGen hζ
a b c : ℤ
hc : c ≠ 0
ha : ¬3 ∣ a
hb : ¬3 ∣ b
h_1 : 3 ∣ c
hcoprime : IsCoprime a b
h : HPow.hPow (β := ℕ) a 3 + HPow.hPow (β := ℕ) b 3 = HPow.hPow (β := ℕ) c 3
x : ℤ
hx : c = 3 * x
⊢ IsCoprime.{u_1} (R := 𝓞 K) (↑a : 𝓞 K) (↑b : 𝓞 K)

Tactic:
IsCoprime.intCast hcoprime

Premises used: IsCoprime.intCast
-- 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
H : FermatLastTheoremForThreeGen hζ
a b c : ℤ
hc : c ≠ 0
ha : ¬3 ∣ a
hb : ¬3 ∣ b
h_1 : 3 ∣ c
hcoprime : IsCoprime a b
h : HPow.hPow (β := ℕ) a 3 + HPow.hPow (β := ℕ) b 3 = HPow.hPow (β := ℕ) c 3
x : ℤ
hx : c = 3 * x
⊢ IsCoprime.{u_1} (R := 𝓞 K) (↑a : 𝓞 K) (↑b : 𝓞 K)

Tactic:
exact IsCoprime.intCast hcoprime

Premises used: IsCoprime.intCast
-- 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
H : FermatLastTheoremForThreeGen hζ
a b c : ℤ
hc : c ≠ 0
ha : ¬3 ∣ a
hb : ¬3 ∣ b
h_1 : 3 ∣ c
hcoprime : IsCoprime a b
h : HPow.hPow (β := ℕ) a 3 + HPow.hPow (β := ℕ) b 3 = HPow.hPow (β := ℕ) c 3
x : ℤ
hx : c = 3 * x
⊢ HPow.hPow (α := 𝓞 K) (β := ℕ) (↑a : 𝓞 K) 3 + HPow.hPow (α := 𝓞 K) (β := ℕ) (↑b : 𝓞 K) 3 =
    (↑1 : 𝓞 K) * HPow.hPow (α := 𝓞 K) (β := ℕ) (↑c : 𝓞 K) 3

Tactic:
simpa using mod_cast 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
⊢ FiniteMultiplicity (hζ.toInteger - 1) S'.c

Tactic:
.of_not_isUnit hζ.zeta_sub_one_prime'.not_unit S'.hc

Premises used: Prime.not_unit, FiniteMultiplicity.of_not_isUnit, _private.0.FermatLastTheoremForThreeGen.Solution'.hc, 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
T : Set ℕ := {n : ℕ | ∃ (S' : Solution hζ), S'.multiplicity = n}
rcases :
  ∀ (S₁ : Solution hζ) (hS₁ : S₁.multiplicity = Nat.find (p := Membership.mem T) ⋯), ∃ (S₁ : Solution hζ), S₁.isMinimal
⊢ ∃ (S₁ : Solution hζ), S₁.isMinimal

Tactic:
rcases Nat.find_spec (⟨S.multiplicity, ⟨S, rfl⟩⟩ : T.Nonempty) with ⟨S₁, hS₁⟩

Premises used: _private.0.FermatLastTheoremForThreeGen.Solution.multiplicity, Exists.intro, rfl, Nat.find_spec
-- Module: Mathlib.NumberTheory.FLT.Three
Goal:
K : Type u_1
inst : Field K
ζ : K
hζ : IsPrimitiveRoot ζ 3
S : Solution hζ
T : Set ℕ := {n : ℕ | ∃ (S' : Solution hζ), S'.multiplicity = n}
S₁ : Solution hζ
hS₁ : S₁.multiplicity = Nat.find (p := Membership.mem T) ⋯
⊢ ∃ (S₁ : Solution hζ), S₁.isMinimal

Tactic:
⟨S₁, fun S'' ↦ hS₁ ▸ Nat.find_min' _ ⟨S'', rfl⟩⟩

Premises used: _private.0.FermatLastTheoremForThreeGen.Solution.multiplicity, Eq.rec, Nat.find_min', Exists.intro, Eq.symm, rfl
-- Module: Mathlib.NumberTheory.FLT.Three
Goal:
K : Type u_1
inst : Field K
ζ : K
hζ : IsPrimitiveRoot ζ 3
S : Solution hζ
T : Set ℕ := {n : ℕ | ∃ (S' : Solution hζ), S'.multiplicity = n}
S₁ : Solution hζ
hS₁ : S₁.multiplicity = Nat.find (p := Membership.mem T) ⋯
⊢ ∃ (S₁ : Solution hζ), S₁.isMinimal

Tactic:
exact ⟨S₁, fun S'' ↦ hS₁ ▸ Nat.find_min' _ ⟨S'', rfl⟩⟩
-- Module: Mathlib.NumberTheory.FLT.Three
Goal:
K : Type u_1
inst : Field K
ζ : K
hζ : IsPrimitiveRoot ζ 3
S : Solution hζ
let let₁ let₂ : ∃ (S₁ : Solution hζ), S₁.isMinimal
let₃ :
  let T : Set ℕ := {n : ℕ | ∃ (S' : Solution hζ), S'.multiplicity = n};
  ∀ (S₁ : Solution hζ) (hS₁ : S₁.multiplicity = Nat.find (p := Membership.mem T) ⋯), ∃ (S₁ : Solution hζ), S₁.isMinimal
⊢ ∃ (S₁ : Solution hζ), S₁.isMinimal

Tactic:
classical
  let T := {n | ∃ (S' : Solution hζ), S'.multiplicity = n}
  rcases Nat.find_spec (⟨S.multiplicity, ⟨S, rfl⟩⟩ : T.Nonempty) with ⟨S₁, hS₁⟩
  exact ⟨S₁, fun S'' ↦ hS₁ ▸ Nat.find_min' _ ⟨S'', rfl⟩⟩