awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- 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 = 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 8 = 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 = 8
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 inr : ¬HAdd.hAdd (α := ZMod 9) 8 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 = 8
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 inr : ¬HAdd.hAdd (α := ZMod 9) 8 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 = 8
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 inr : ¬HAdd.hAdd (α := ZMod 9) 8 8 = 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 = 8
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 = 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 : ¬HAdd.hAdd (α := ZMod 9) 8 8 = 8
Tactic: rw [ha, hb, hc]
-- Lean 4 proof step from Mathlib.NumberTheory.FLT.Three (goal)
Goal:
a b c : ℤ
ha : a ≠ 0
Hgcd : Eq (α := ℤ) (Finset.gcd {a, b, c} id) 1
h3a : 3 ∣ a
HF : HPow.hPow (β := ℕ) a 3 + HPow.hPow (β := ℕ) b 3 + HPow.hPow (β := ℕ) c 3 = 0
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
⊢ IsCoprime (-b) (-c)
have refine : IsCoprime b c
Tactic: refine IsCoprime.neg_neg ?_
-- Lean 4 proof step from Mathlib.NumberTheory.FLT.Three (hypothesis)
Goal:
a b c : ℤ
ha : a ≠ 0
Hgcd : Eq (α := ℤ) (Finset.gcd {a, b, c} id) 1
h3a : 3 ∣ a
HF : HPow.hPow (β := ℕ) a 3 + HPow.hPow (β := ℕ) b 3 + HPow.hPow (β := ℕ) c 3 = 0
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
⊢ IsCoprime b c
have HF₁ : HPow.hPow (β := ℕ) b 3 + HPow.hPow (β := ℕ) c 3 + HPow.hPow (β := ℕ) a 3 = 0
Tactic: rw [add_comm (a ^ 3), add_assoc, add_comm (a ^ 3), ← add_assoc] at HF
Cleared hypotheses: HF, H
-- Lean 4 proof step from Mathlib.NumberTheory.FLT.Three (goal)
Goal:
a b c : ℤ
ha : a ≠ 0
Hgcd : Eq (α := ℤ) (Finset.gcd {a, b, c} id) 1
h3a : 3 ∣ a
HF : HPow.hPow (β := ℕ) b 3 + HPow.hPow (β := ℕ) c 3 + HPow.hPow (β := ℕ) a 3 = 0
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
⊢ IsCoprime b c
have refine : Eq (α := ℤ) (Finset.gcd {b, c, a} id) 1
Tactic: refine isCoprime_of_gcd_eq_one_of_FLT ?_ HF
-- Lean 4 proof step from Mathlib.NumberTheory.FLT.Three (goal)
Goal:
a b c : ℤ
ha : a ≠ 0
Hgcd : Eq (α := ℤ) (Finset.gcd {a, b, c} id) 1
h3a : 3 ∣ a
HF : HPow.hPow (β := ℕ) b 3 + HPow.hPow (β := ℕ) c 3 + HPow.hPow (β := ℕ) a 3 = 0
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
⊢ Eq (α := ℤ) (Finset.gcd {b, c, a} id) 1
have h : Eq (α := Finset ℤ) {b, c, a} {a, b, c}
Tactic: convert! Hgcd using 2
-- Lean 4 proof step from Mathlib.NumberTheory.FLT.Three (goal)
Goal:
a b c : ℤ
ha : a ≠ 0
Hgcd : Eq (α := ℤ) (Finset.gcd {a, b, c} id) 1
h3a : 3 ∣ a
HF : HPow.hPow (β := ℕ) b 3 + HPow.hPow (β := ℕ) c 3 + HPow.hPow (β := ℕ) a 3 = 0
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
⊢ Eq (α := Finset ℤ) {b, c, a} {a, b, c}
have h : Eq (α := Finset ℤ) {b, a, c} {a, b, c}
Tactic: rw [Finset.pair_comm, Finset.insert_comm]
-- Lean 4 proof step from Mathlib.NumberTheory.FLT.Three (goal)
Goal:
a b c : ℤ
ha : a ≠ 0
Hgcd : Eq (α := ℤ) (Finset.gcd {a, b, c} id) 1
h3a : 3 ∣ a
HF : HPow.hPow (β := ℕ) b 3 + HPow.hPow (β := ℕ) c 3 + HPow.hPow (β := ℕ) a 3 = 0
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
h : Eq (α := Finset ℤ) {b, a, c} {a, b, c}
⊢ Eq (α := Finset ℤ) {b, c, a} {a, b, c}
have h₁ : Eq (α := Finset ℤ) {a, b, c} {a, b, c}
Tactic: rw [Finset.pair_comm, Finset.insert_comm]
-- Lean 4 proof step from Mathlib.NumberTheory.FLT.Three (hypothesis)
Goal:
a b c : ℤ
ha : a ≠ 0
Hgcd : Eq (α := ℤ) (Finset.gcd {a, b, c} id) 1
h3a : 3 ∣ a
HF : HPow.hPow (β := ℕ) a 3 + HPow.hPow (β := ℕ) b 3 + HPow.hPow (β := ℕ) c 3 = 0
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
⊢ 3 ∣ b
have hbc : IsCoprime (-b) (-c)
Tactic: have hbc : IsCoprime (-b) (-c) := by
    refine IsCoprime.neg_neg ?_
    rw [add_comm (a ^ 3), add_assoc, add_comm (a ^ 3), ← add_assoc] at HF
    refine isCoprime_of_gcd_eq_one_of_FLT ?_ HF
    convert! Hgcd using 2
    rw [Finset.pair_comm, Finset.insert_comm]
-- Lean 4 proof step from Mathlib.NumberTheory.FLT.Three (goal)
Goal:
a b c : ℤ
ha : a ≠ 0
Hgcd : Eq (α := ℤ) (Finset.gcd {a, b, c} id) 1
h3a : 3 ∣ a
HF : HPow.hPow (β := ℕ) a 3 + HPow.hPow (β := ℕ) b 3 + HPow.hPow (β := ℕ) c 3 = 0
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
hbc : IsCoprime (-b) (-c)
⊢ 3 ∣ b
have by_contra! : ∀ (h3b : ¬3 ∣ b), False
Tactic: by_contra! h3b
-- Lean 4 proof step from Mathlib.NumberTheory.FLT.Three (goal)
Goal:
a b c : ℤ
ha : a ≠ 0
Hgcd : Eq (α := ℤ) (Finset.gcd {a, b, c} id) 1
h3a : 3 ∣ a
HF : HPow.hPow (β := ℕ) a 3 + HPow.hPow (β := ℕ) b 3 + HPow.hPow (β := ℕ) c 3 = 0
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
hbc : IsCoprime (-b) (-c)
h3b : ¬3 ∣ b
⊢ False
have pos : ∀ (h3c : 3 ∣ c), False
Tactic: by_cases h3c : 3 ∣ c
-- Lean 4 proof step from Mathlib.NumberTheory.FLT.Three (goal)
Goal:
a b c : ℤ
ha : a ≠ 0
Hgcd : Eq (α := ℤ) (Finset.gcd {a, b, c} id) 1
h3a : 3 ∣ a
HF : HPow.hPow (β := ℕ) a 3 + HPow.hPow (β := ℕ) b 3 + HPow.hPow (β := ℕ) c 3 = 0
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
hbc : IsCoprime (-b) (-c)
h3b : ¬3 ∣ b
pos : ∀ (h3c : 3 ∣ c), False
⊢ False
have neg : ∀ (h3c : ¬3 ∣ c), False
Tactic: by_cases h3c : 3 ∣ c
-- Lean 4 proof step from Mathlib.NumberTheory.FLT.Three (goal)
Goal:
a b c : ℤ
ha : a ≠ 0
Hgcd : Eq (α := ℤ) (Finset.gcd {a, b, c} id) 1
h3a : 3 ∣ a
HF : HPow.hPow (β := ℕ) a 3 + HPow.hPow (β := ℕ) b 3 + HPow.hPow (β := ℕ) c 3 = 0
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
hbc : IsCoprime (-b) (-c)
h3b : ¬3 ∣ b
h3c : 3 ∣ c
⊢ False
have pos : 3 ∣ b
Tactic: apply h3b
-- Lean 4 proof step from Mathlib.NumberTheory.FLT.Three (hypothesis)
Goal:
a b c : ℤ
ha : a ≠ 0
Hgcd : Eq (α := ℤ) (Finset.gcd {a, b, c} id) 1
h3a : 3 ∣ a
HF : HPow.hPow (β := ℕ) a 3 + HPow.hPow (β := ℕ) b 3 + HPow.hPow (β := ℕ) c 3 = 0
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
hbc : IsCoprime (-b) (-c)
h3b : ¬3 ∣ b
h3c : 3 ∣ c
⊢ 3 ∣ b
have HF₁ : HPow.hPow (β := ℕ) a 3 + HPow.hPow (β := ℕ) c 3 + HPow.hPow (β := ℕ) b 3 = 0
Tactic: rw [add_assoc, add_comm (b ^ 3), ← add_assoc] at HF
Cleared hypotheses: HF, H, hbc, h3b, h3c
-- Lean 4 proof step from Mathlib.NumberTheory.FLT.Three (goal)
Goal:
a b c : ℤ
ha : a ≠ 0
Hgcd : Eq (α := ℤ) (Finset.gcd {a, b, c} id) 1
h3a : 3 ∣ a
HF : HPow.hPow (β := ℕ) a 3 + HPow.hPow (β := ℕ) b 3 + HPow.hPow (β := ℕ) c 3 = 0
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
hbc : IsCoprime (-b) (-c)
h3b : ¬3 ∣ b
h3c : ¬3 ∣ c
⊢ False
have neg : HPow.hPow (β := ℕ) (-b) 3 + HPow.hPow (β := ℕ) (-c) 3 = HPow.hPow (β := ℕ) a 3
Tactic: refine H (-b) (-c) a ha (by simp [h3b]) (by simp [h3c]) h3a hbc ?_
-- Lean 4 proof step from Mathlib.NumberTheory.FLT.Three (hypothesis)
Goal:
a b c : ℤ
ha : a ≠ 0
Hgcd : Eq (α := ℤ) (Finset.gcd {a, b, c} id) 1
h3a : 3 ∣ a
HF : HPow.hPow (β := ℕ) a 3 + HPow.hPow (β := ℕ) b 3 + HPow.hPow (β := ℕ) c 3 = 0
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
hbc : IsCoprime (-b) (-c)
h3b : ¬3 ∣ b
h3c : ¬3 ∣ c
⊢ HPow.hPow (β := ℕ) (-b) 3 + HPow.hPow (β := ℕ) (-c) 3 = HPow.hPow (β := ℕ) a 3
have HF₁ : HPow.hPow (β := ℕ) a 3 + HPow.hPow (β := ℕ) b 3 = HPow.hPow (β := ℕ) (-c) 3
Tactic: rw [add_eq_zero_iff_eq_neg, ← (show Odd 3 by decide).neg_pow] at HF
Cleared hypotheses: HF, H, hbc, h3b, h3c
-- Lean 4 proof step from Mathlib.NumberTheory.FLT.Three (goal)
Goal:
a b c : ℤ
ha : a ≠ 0
Hgcd : Eq (α := ℤ) (Finset.gcd {a, b, c} id) 1
h3a : 3 ∣ a
HF : HPow.hPow (β := ℕ) a 3 + HPow.hPow (β := ℕ) b 3 = HPow.hPow (β := ℕ) (-c) 3
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
hbc : IsCoprime (-b) (-c)
h3b : ¬3 ∣ b
h3c : ¬3 ∣ c
⊢ HPow.hPow (β := ℕ) (-b) 3 + HPow.hPow (β := ℕ) (-c) 3 = HPow.hPow (β := ℕ) a 3
have neg : HPow.hPow (β := ℕ) (-b) 3 + (HPow.hPow (β := ℕ) a 3 + HPow.hPow (β := ℕ) b 3) = HPow.hPow (β := ℕ) a 3
Tactic: rw [← HF]
-- Lean 4 proof step from Mathlib.NumberTheory.FLT.Three (goal)
Goal:
a b c : ℤ
ha : a ≠ 0
h3a : 3 ∣ a
Hgcd : Eq (α := ℤ) (Finset.gcd {a, b, c} id) 1
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
HF : HPow.hPow (β := ℕ) a 3 + HPow.hPow (β := ℕ) b 3 + HPow.hPow (β := ℕ) c 3 = 0
⊢ False
have apply : Dvd.dvd (α := ℤ) 3 1
Tactic: apply (show ¬(3 ∣ (1 : ℤ)) by decide)
-- Lean 4 proof step from Mathlib.NumberTheory.FLT.Three (goal)
Goal:
a b c : ℤ
ha : a ≠ 0
h3a : 3 ∣ a
Hgcd : Eq (α := ℤ) (Finset.gcd {a, b, c} id) 1
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
HF : HPow.hPow (β := ℕ) a 3 + HPow.hPow (β := ℕ) b 3 + HPow.hPow (β := ℕ) c 3 = 0
⊢ Dvd.dvd (α := ℤ) 3 1
have rw : Dvd.dvd (α := ℤ) 3 (Finset.gcd {a, b, c} id)
Tactic: rw [← Hgcd]
-- Lean 4 proof step from Mathlib.NumberTheory.FLT.Three (goal)
Goal:
a b c : ℤ
ha : a ≠ 0
h3a : 3 ∣ a
Hgcd : Eq (α := ℤ) (Finset.gcd {a, b, c} id) 1
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
HF : HPow.hPow (β := ℕ) a 3 + HPow.hPow (β := ℕ) b 3 + HPow.hPow (β := ℕ) c 3 = 0
⊢ Dvd.dvd (α := ℤ) 3 (Finset.gcd {a, b, c} id)
have refine : ∀ (x : ℤ) (hx : Membership.mem (γ := Finset ℤ) {a, b, c} x), 3 ∣ id x
Tactic: refine dvd_gcd (fun x hx ↦ ?_)
-- Lean 4 proof step from Mathlib.NumberTheory.FLT.Three (hypothesis)
Goal:
a b c : ℤ
ha : a ≠ 0
h3a : 3 ∣ a
Hgcd : Eq (α := ℤ) (Finset.gcd {a, b, c} id) 1
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
HF : HPow.hPow (β := ℕ) a 3 + HPow.hPow (β := ℕ) b 3 + HPow.hPow (β := ℕ) c 3 = 0
x : ℤ
hx : Membership.mem (γ := Finset ℤ) {a, b, c} x
⊢ 3 ∣ id x
have hx₁ : x = a ∨ x = b ∨ x = c
Tactic: simp only [mem_insert, mem_singleton] at hx
Cleared hypotheses: hx
-- Lean 4 proof step from Mathlib.NumberTheory.FLT.Three (hypothesis)
Goal:
a b c : ℤ
ha : a ≠ 0
h3a : 3 ∣ a
Hgcd : Eq (α := ℤ) (Finset.gcd {a, b, c} id) 1
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
HF : HPow.hPow (β := ℕ) a 3 + HPow.hPow (β := ℕ) b 3 + HPow.hPow (β := ℕ) c 3 = 0
x : ℤ
hx : x = a ∨ x = b ∨ x = c
⊢ 3 ∣ id x
have h3b : 3 ∣ b
Tactic: have h3b : 3 ∣ b := by
    refine three_dvd_b_of_dvd_a_of_gcd_eq_one_of_case2 ha ?_ h3a HF H
    simp only [← Hgcd, gcd_insert, gcd_singleton, id_eq, ← Int.abs_eq_normalize]
-- Lean 4 proof step from Mathlib.NumberTheory.FLT.Three (goal)
Goal:
a b c : ℤ
ha : a ≠ 0
h3a : 3 ∣ a
Hgcd : Eq (α := ℤ) (Finset.gcd {a, b, c} id) 1
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
HF : HPow.hPow (β := ℕ) a 3 + HPow.hPow (β := ℕ) b 3 + HPow.hPow (β := ℕ) c 3 = 0
x : ℤ
hx : x = a ∨ x = b ∨ x = c
h3b : 3 ∣ b
⊢ 3 ∣ id x
have inl : ∀ (hx : x = a), 3 ∣ id x
Tactic: rcases hx with hx | hx | hx
Cleared hypotheses: hx
-- Lean 4 proof step from Mathlib.NumberTheory.FLT.Three (goal)
Goal:
a b c : ℤ
ha : a ≠ 0
h3a : 3 ∣ a
Hgcd : Eq (α := ℤ) (Finset.gcd {a, b, c} id) 1
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
HF : HPow.hPow (β := ℕ) a 3 + HPow.hPow (β := ℕ) b 3 + HPow.hPow (β := ℕ) c 3 = 0
x : ℤ
hx : x = a ∨ x = b ∨ x = c
h3b : 3 ∣ b
inl : ∀ (hx : x = a), 3 ∣ id x
⊢ 3 ∣ id x
have inr : ∀ (hx : x = b), 3 ∣ id x
Tactic: rcases hx with hx | hx | hx
Cleared hypotheses: hx
-- Lean 4 proof step from Mathlib.NumberTheory.FLT.Three (goal)
Goal:
a b c : ℤ
ha : a ≠ 0
h3a : 3 ∣ a
Hgcd : Eq (α := ℤ) (Finset.gcd {a, b, c} id) 1
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
HF : HPow.hPow (β := ℕ) a 3 + HPow.hPow (β := ℕ) b 3 + HPow.hPow (β := ℕ) c 3 = 0
x : ℤ
hx : x = a ∨ x = b ∨ x = c
h3b : 3 ∣ b
inl : ∀ (hx : x = a), 3 ∣ id x
inr : ∀ (hx : x = b), 3 ∣ id x
⊢ 3 ∣ id x
have inr₁ : ∀ (hx : x = c), 3 ∣ id x
Tactic: rcases hx with hx | hx | hx
Cleared hypotheses: hx
-- Lean 4 proof step from Mathlib.NumberTheory.FLT.Three (goal)
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
⊢ FermatLastTheoremFor 3
have rw : FermatLastTheoremWith ℤ 3
Tactic: rw [fermatLastTheoremFor_iff_int]
-- Lean 4 proof step from Mathlib.NumberTheory.FLT.Three (goal)
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
⊢ FermatLastTheoremWith ℤ 3
have 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
Tactic: refine fermatLastTheoremWith_of_fermatLastTheoremWith_coprime (fun a b c ha hb hc Hgcd hF ↦ ?_)
-- Lean 4 proof step from Mathlib.NumberTheory.FLT.Three (goal)
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
⊢ False
have pos : ∀ (h1 : 3 ∣ a * b * c), False
Tactic: by_cases h1 : 3 ∣ a * b * c
-- Lean 4 proof step from Mathlib.NumberTheory.FLT.Three (goal)
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
⊢ False
have neg : ∀ (h1 : ¬3 ∣ a * b * c), False
Tactic: by_cases h1 : 3 ∣ a * b * c
-- Lean 4 proof step from Mathlib.NumberTheory.FLT.Three (hypothesis)
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
have h1₁ : (3 ∣ a ∨ 3 ∣ b) ∨ 3 ∣ c
Tactic: rw [prime_three.dvd_mul, prime_three.dvd_mul] at h1
Cleared hypotheses: h1
-- Lean 4 proof step from Mathlib.NumberTheory.FLT.Three (hypothesis)
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
⊢ False
have hF₁ : 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
Cleared hypotheses: hF, h1
-- Lean 4 proof step from Mathlib.NumberTheory.FLT.Three (goal)
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
⊢ False
have pos : ∀ (h3a : 3 ∣ a), False
Tactic: rcases h1 with (h3a | h3b) | h3c
Cleared hypotheses: h1
-- Lean 4 proof step from Mathlib.NumberTheory.FLT.Three (goal)
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
⊢ False
have pos₁ : ∀ (h3b : 3 ∣ b), False
Tactic: rcases h1 with (h3a | h3b) | h3c
Cleared hypotheses: h1
-- Lean 4 proof step from Mathlib.NumberTheory.FLT.Three (goal)
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
⊢ False
have pos₂ : ∀ (h3c : 3 ∣ c), False
Tactic: rcases h1 with (h3a | h3b) | h3c
Cleared hypotheses: h1
-- Lean 4 proof step from Mathlib.NumberTheory.FLT.Three (goal)
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
have pos : Eq (α := ℤ) (Finset.gcd {a, b, -c} id) 1
Tactic: refine fermatLastTheoremThree_of_dvd_a_of_gcd_eq_one_of_case2 ha h3a ?_ H hF
-- Lean 4 proof step from Mathlib.NumberTheory.FLT.Three (hypothesis)
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
⊢ False
have hF₁ : 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
Cleared hypotheses: hF, h3b
-- Lean 4 proof step from Mathlib.NumberTheory.FLT.Three (goal)
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
have pos : Eq (α := ℤ) (Finset.gcd {b, a, -c} id) 1
Tactic: refine fermatLastTheoremThree_of_dvd_a_of_gcd_eq_one_of_case2 hb h3b ?_ H hF
-- Lean 4 proof step from Mathlib.NumberTheory.FLT.Three (hypothesis)
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
⊢ False
have hF₁ : 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
Cleared hypotheses: hF, h3c
-- Lean 4 proof step from Mathlib.NumberTheory.FLT.Three (goal)
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
⊢ False
have pos : Eq (α := ℤ) (Finset.gcd {-c, a, b} id) 1
Tactic: refine fermatLastTheoremThree_of_dvd_a_of_gcd_eq_one_of_case2 (neg_ne_zero.2 hc) (by simp [h3c])
      ?_ H hF
-- Lean 4 proof step from Mathlib.NumberTheory.FLT.Three (goal)
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 {-c, a, b} id) 1
have pos : Eq (α := ℤ) (Finset.gcd {a, b, -c} id) 1
Tactic: rw [Finset.insert_comm (-c), Finset.pair_comm (-c) b]
-- Lean 4 proof step from Mathlib.NumberTheory.FLT.Three (goal)
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
have 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
Tactic: refine fermatLastTheoremThree_of_three_dvd_only_c (fun a b c hc ha hb ⟨x, hx⟩ hcoprime h ↦ ?_)