l3lab/ntp-mathlib huggingface dataset:
theorem gcd_dvd : (Int.gcd x y : ℤ) ∣ z := by
  by_cases h0 : Int.gcd x y = 0
  · have hx : x = 0 := by
      apply Int.natAbs_eq_zero.mp
      apply Nat.eq_zero_of_gcd_eq_zero_left h0
    have hy : y = 0 := by
      apply Int.natAbs_eq_zero.mp
      apply Nat.eq_zero_of_gcd_eq_zero_right h0
    have hz : z = 0 := by
      simpa only [PythagoreanTriple, hx, hy, add_zero, zero_eq_mul, mul_zero,
        or_self_iff] using h
    simp only [hz, dvd_zero]
  obtain ⟨k, x0, y0, _, h2, rfl, rfl⟩ :
    ∃ (k : ℕ) (x0 y0 : _), 0 < k ∧ Int.gcd x0 y0 = 1 ∧ x = x0 * k ∧ y = y0 * k :=
    Int.exists_gcd_one' (Nat.pos_of_ne_zero h0)
  rw [Int.gcd_mul_right, h2, Int.natAbs_ofNat, one_mul]
  rw [← Int.pow_dvd_pow_iff zero_lt_two, sq z, ← h.eq]
  

Proof state:
case neg.intro.intro.intro.intro.intro.intro
z : ℤ
k : ℕ
x0 y0 : ℤ
left✝ : 0 < k
h2 : Int.gcd x0 y0 = 1
h : PythagoreanTriple (x0 * ↑k) (y0 * ↑k) z
h0 : ¬Int.gcd (x0 * ↑k) (y0 * ↑k) = 0
⊢ ↑k ^ 2 ∣ x0 * ↑k * (x0 * ↑k) + y0 * ↑k * (y0 * ↑k)

Next tactic:
rw [(by ring : x0 * k * (x0 * k) + y0 * k * (y0 * k) = (k : ℤ) ^ 2 * (x0 * x0 + y0 * y0))]
theorem gcd_dvd : (Int.gcd x y : ℤ) ∣ z := by
  by_cases h0 : Int.gcd x y = 0
  · have hx : x = 0 := by
      apply Int.natAbs_eq_zero.mp
      apply Nat.eq_zero_of_gcd_eq_zero_left h0
    have hy : y = 0 := by
      apply Int.natAbs_eq_zero.mp
      apply Nat.eq_zero_of_gcd_eq_zero_right h0
    have hz : z = 0 := by
      simpa only [PythagoreanTriple, hx, hy, add_zero, zero_eq_mul, mul_zero,
        or_self_iff] using h
    simp only [hz, dvd_zero]
  obtain ⟨k, x0, y0, _, h2, rfl, rfl⟩ :
    ∃ (k : ℕ) (x0 y0 : _), 0 < k ∧ Int.gcd x0 y0 = 1 ∧ x = x0 * k ∧ y = y0 * k :=
    Int.exists_gcd_one' (Nat.pos_of_ne_zero h0)
  rw [Int.gcd_mul_right, h2, Int.natAbs_ofNat, one_mul]
  rw [← Int.pow_dvd_pow_iff zero_lt_two, sq z, ← h.eq]
  rw [(by 

Proof state:
z : ℤ
k : ℕ
x0 y0 : ℤ
left✝ : 0 < k
h2 : Int.gcd x0 y0 = 1
h : PythagoreanTriple (x0 * ↑k) (y0 * ↑k) z
h0 : ¬Int.gcd (x0 * ↑k) (y0 * ↑k) = 0
⊢ x0 * ↑k * (x0 * ↑k) + y0 * ↑k * (y0 * ↑k) = ↑k ^ 2 * (x0 * x0 + y0 * y0)

Next tactic:
ring
theorem gcd_dvd : (Int.gcd x y : ℤ) ∣ z := by
  by_cases h0 : Int.gcd x y = 0
  · have hx : x = 0 := by
      apply Int.natAbs_eq_zero.mp
      apply Nat.eq_zero_of_gcd_eq_zero_left h0
    have hy : y = 0 := by
      apply Int.natAbs_eq_zero.mp
      apply Nat.eq_zero_of_gcd_eq_zero_right h0
    have hz : z = 0 := by
      simpa only [PythagoreanTriple, hx, hy, add_zero, zero_eq_mul, mul_zero,
        or_self_iff] using h
    simp only [hz, dvd_zero]
  obtain ⟨k, x0, y0, _, h2, rfl, rfl⟩ :
    ∃ (k : ℕ) (x0 y0 : _), 0 < k ∧ Int.gcd x0 y0 = 1 ∧ x = x0 * k ∧ y = y0 * k :=
    Int.exists_gcd_one' (Nat.pos_of_ne_zero h0)
  rw [Int.gcd_mul_right, h2, Int.natAbs_ofNat, one_mul]
  rw [← Int.pow_dvd_pow_iff zero_lt_two, sq z, ← h.eq]
  rw [(by ring : x0 * k * (x0 * k) + y0 * k * (y0 * k) = (k : ℤ) ^ 2 * (x0 * x0 + y0 * y0))]
  

Proof state:
case neg.intro.intro.intro.intro.intro.intro
z : ℤ
k : ℕ
x0 y0 : ℤ
left✝ : 0 < k
h2 : Int.gcd x0 y0 = 1
h : PythagoreanTriple (x0 * ↑k) (y0 * ↑k) z
h0 : ¬Int.gcd (x0 * ↑k) (y0 * ↑k) = 0
⊢ ↑k ^ 2 ∣ ↑k ^ 2 * (x0 * x0 + y0 * y0)

Next tactic:
exact dvd_mul_right _ _
theorem normalize : PythagoreanTriple (x / Int.gcd x y) (y / Int.gcd x y) (z / Int.gcd x y) := by
  

Proof state:
x y z : ℤ
h : PythagoreanTriple x y z
⊢ PythagoreanTriple (x / ↑(Int.gcd x y)) (y / ↑(Int.gcd x y)) (z / ↑(Int.gcd x y))

Next tactic:
by_cases h0 : Int.gcd x y = 0
theorem normalize : PythagoreanTriple (x / Int.gcd x y) (y / Int.gcd x y) (z / Int.gcd x y) := by
  by_cases h0 : Int.gcd x y = 0
  · 

Proof state:
case pos
x y z : ℤ
h : PythagoreanTriple x y z
h0 : Int.gcd x y = 0
⊢ PythagoreanTriple (x / ↑(Int.gcd x y)) (y / ↑(Int.gcd x y)) (z / ↑(Int.gcd x y))

Next tactic:
have hx : x = 0 := by
      apply Int.natAbs_eq_zero.mp
      apply Nat.eq_zero_of_gcd_eq_zero_left h0
theorem normalize : PythagoreanTriple (x / Int.gcd x y) (y / Int.gcd x y) (z / Int.gcd x y) := by
  by_cases h0 : Int.gcd x y = 0
  · have hx : x = 0 := by
      

Proof state:
x y z : ℤ
h : PythagoreanTriple x y z
h0 : Int.gcd x y = 0
⊢ x = 0

Next tactic:
apply Int.natAbs_eq_zero.mp
theorem normalize : PythagoreanTriple (x / Int.gcd x y) (y / Int.gcd x y) (z / Int.gcd x y) := by
  by_cases h0 : Int.gcd x y = 0
  · have hx : x = 0 := by
      apply Int.natAbs_eq_zero.mp
      

Proof state:
x y z : ℤ
h : PythagoreanTriple x y z
h0 : Int.gcd x y = 0
⊢ Int.natAbs x = 0

Next tactic:
apply Nat.eq_zero_of_gcd_eq_zero_left h0
theorem normalize : PythagoreanTriple (x / Int.gcd x y) (y / Int.gcd x y) (z / Int.gcd x y) := by
  by_cases h0 : Int.gcd x y = 0
  · have hx : x = 0 := by
      apply Int.natAbs_eq_zero.mp
      apply Nat.eq_zero_of_gcd_eq_zero_left h0
    

Proof state:
case pos
x y z : ℤ
h : PythagoreanTriple x y z
h0 : Int.gcd x y = 0
hx : x = 0
⊢ PythagoreanTriple (x / ↑(Int.gcd x y)) (y / ↑(Int.gcd x y)) (z / ↑(Int.gcd x y))

Next tactic:
have hy : y = 0 := by
      apply Int.natAbs_eq_zero.mp
      apply Nat.eq_zero_of_gcd_eq_zero_right h0
theorem normalize : PythagoreanTriple (x / Int.gcd x y) (y / Int.gcd x y) (z / Int.gcd x y) := by
  by_cases h0 : Int.gcd x y = 0
  · have hx : x = 0 := by
      apply Int.natAbs_eq_zero.mp
      apply Nat.eq_zero_of_gcd_eq_zero_left h0
    have hy : y = 0 := by
      

Proof state:
x y z : ℤ
h : PythagoreanTriple x y z
h0 : Int.gcd x y = 0
hx : x = 0
⊢ y = 0

Next tactic:
apply Int.natAbs_eq_zero.mp
theorem normalize : PythagoreanTriple (x / Int.gcd x y) (y / Int.gcd x y) (z / Int.gcd x y) := by
  by_cases h0 : Int.gcd x y = 0
  · have hx : x = 0 := by
      apply Int.natAbs_eq_zero.mp
      apply Nat.eq_zero_of_gcd_eq_zero_left h0
    have hy : y = 0 := by
      apply Int.natAbs_eq_zero.mp
      

Proof state:
x y z : ℤ
h : PythagoreanTriple x y z
h0 : Int.gcd x y = 0
hx : x = 0
⊢ Int.natAbs y = 0

Next tactic:
apply Nat.eq_zero_of_gcd_eq_zero_right h0
theorem normalize : PythagoreanTriple (x / Int.gcd x y) (y / Int.gcd x y) (z / Int.gcd x y) := by
  by_cases h0 : Int.gcd x y = 0
  · have hx : x = 0 := by
      apply Int.natAbs_eq_zero.mp
      apply Nat.eq_zero_of_gcd_eq_zero_left h0
    have hy : y = 0 := by
      apply Int.natAbs_eq_zero.mp
      apply Nat.eq_zero_of_gcd_eq_zero_right h0
    

Proof state:
case pos
x y z : ℤ
h : PythagoreanTriple x y z
h0 : Int.gcd x y = 0
hx : x = 0
hy : y = 0
⊢ PythagoreanTriple (x / ↑(Int.gcd x y)) (y / ↑(Int.gcd x y)) (z / ↑(Int.gcd x y))

Next tactic:
have hz : z = 0 := by
      simpa only [PythagoreanTriple, hx, hy, add_zero, zero_eq_mul, mul_zero,
        or_self_iff] using h
theorem normalize : PythagoreanTriple (x / Int.gcd x y) (y / Int.gcd x y) (z / Int.gcd x y) := by
  by_cases h0 : Int.gcd x y = 0
  · have hx : x = 0 := by
      apply Int.natAbs_eq_zero.mp
      apply Nat.eq_zero_of_gcd_eq_zero_left h0
    have hy : y = 0 := by
      apply Int.natAbs_eq_zero.mp
      apply Nat.eq_zero_of_gcd_eq_zero_right h0
    have hz : z = 0 := by
      

Proof state:
x y z : ℤ
h : PythagoreanTriple x y z
h0 : Int.gcd x y = 0
hx : x = 0
hy : y = 0
⊢ z = 0

Next tactic:
simpa only [PythagoreanTriple, hx, hy, add_zero, zero_eq_mul, mul_zero,
        or_self_iff] using h
theorem normalize : PythagoreanTriple (x / Int.gcd x y) (y / Int.gcd x y) (z / Int.gcd x y) := by
  by_cases h0 : Int.gcd x y = 0
  · have hx : x = 0 := by
      apply Int.natAbs_eq_zero.mp
      apply Nat.eq_zero_of_gcd_eq_zero_left h0
    have hy : y = 0 := by
      apply Int.natAbs_eq_zero.mp
      apply Nat.eq_zero_of_gcd_eq_zero_right h0
    have hz : z = 0 := by
      simpa only [PythagoreanTriple, hx, hy, add_zero, zero_eq_mul, mul_zero,
        or_self_iff] using h
    

Proof state:
case pos
x y z : ℤ
h : PythagoreanTriple x y z
h0 : Int.gcd x y = 0
hx : x = 0
hy : y = 0
hz : z = 0
⊢ PythagoreanTriple (x / ↑(Int.gcd x y)) (y / ↑(Int.gcd x y)) (z / ↑(Int.gcd x y))

Next tactic:
simp only [hx, hy, hz, Int.zero_div]
theorem normalize : PythagoreanTriple (x / Int.gcd x y) (y / Int.gcd x y) (z / Int.gcd x y) := by
  by_cases h0 : Int.gcd x y = 0
  · have hx : x = 0 := by
      apply Int.natAbs_eq_zero.mp
      apply Nat.eq_zero_of_gcd_eq_zero_left h0
    have hy : y = 0 := by
      apply Int.natAbs_eq_zero.mp
      apply Nat.eq_zero_of_gcd_eq_zero_right h0
    have hz : z = 0 := by
      simpa only [PythagoreanTriple, hx, hy, add_zero, zero_eq_mul, mul_zero,
        or_self_iff] using h
    simp only [hx, hy, hz, Int.zero_div]
    

Proof state:
case pos
x y z : ℤ
h : PythagoreanTriple x y z
h0 : Int.gcd x y = 0
hx : x = 0
hy : y = 0
hz : z = 0
⊢ PythagoreanTriple (0 / ↑(Int.gcd 0 0)) (0 / ↑(Int.gcd 0 0)) (0 / ↑(Int.gcd 0 0))

Next tactic:
exact zero
theorem normalize : PythagoreanTriple (x / Int.gcd x y) (y / Int.gcd x y) (z / Int.gcd x y) := by
  by_cases h0 : Int.gcd x y = 0
  · have hx : x = 0 := by
      apply Int.natAbs_eq_zero.mp
      apply Nat.eq_zero_of_gcd_eq_zero_left h0
    have hy : y = 0 := by
      apply Int.natAbs_eq_zero.mp
      apply Nat.eq_zero_of_gcd_eq_zero_right h0
    have hz : z = 0 := by
      simpa only [PythagoreanTriple, hx, hy, add_zero, zero_eq_mul, mul_zero,
        or_self_iff] using h
    simp only [hx, hy, hz, Int.zero_div]
    exact zero
  

Proof state:
case neg
x y z : ℤ
h : PythagoreanTriple x y z
h0 : ¬Int.gcd x y = 0
⊢ PythagoreanTriple (x / ↑(Int.gcd x y)) (y / ↑(Int.gcd x y)) (z / ↑(Int.gcd x y))

Next tactic:
rcases h.gcd_dvd with ⟨z0, rfl⟩
theorem normalize : PythagoreanTriple (x / Int.gcd x y) (y / Int.gcd x y) (z / Int.gcd x y) := by
  by_cases h0 : Int.gcd x y = 0
  · have hx : x = 0 := by
      apply Int.natAbs_eq_zero.mp
      apply Nat.eq_zero_of_gcd_eq_zero_left h0
    have hy : y = 0 := by
      apply Int.natAbs_eq_zero.mp
      apply Nat.eq_zero_of_gcd_eq_zero_right h0
    have hz : z = 0 := by
      simpa only [PythagoreanTriple, hx, hy, add_zero, zero_eq_mul, mul_zero,
        or_self_iff] using h
    simp only [hx, hy, hz, Int.zero_div]
    exact zero
  rcases h.gcd_dvd with ⟨z0, rfl⟩
  

Proof state:
case neg.intro
x y : ℤ
h0 : ¬Int.gcd x y = 0
z0 : ℤ
h : PythagoreanTriple x y (↑(Int.gcd x y) * z0)
⊢ PythagoreanTriple (x / ↑(Int.gcd x y)) (y / ↑(Int.gcd x y)) (↑(Int.gcd x y) * z0 / ↑(Int.gcd x y))

Next tactic:
obtain ⟨k, x0, y0, k0, h2, rfl, rfl⟩ :
    ∃ (k : ℕ) (x0 y0 : _), 0 < k ∧ Int.gcd x0 y0 = 1 ∧ x = x0 * k ∧ y = y0 * k :=
    Int.exists_gcd_one' (Nat.pos_of_ne_zero h0)
theorem normalize : PythagoreanTriple (x / Int.gcd x y) (y / Int.gcd x y) (z / Int.gcd x y) := by
  by_cases h0 : Int.gcd x y = 0
  · have hx : x = 0 := by
      apply Int.natAbs_eq_zero.mp
      apply Nat.eq_zero_of_gcd_eq_zero_left h0
    have hy : y = 0 := by
      apply Int.natAbs_eq_zero.mp
      apply Nat.eq_zero_of_gcd_eq_zero_right h0
    have hz : z = 0 := by
      simpa only [PythagoreanTriple, hx, hy, add_zero, zero_eq_mul, mul_zero,
        or_self_iff] using h
    simp only [hx, hy, hz, Int.zero_div]
    exact zero
  rcases h.gcd_dvd with ⟨z0, rfl⟩
  obtain ⟨k, x0, y0, k0, h2, rfl, rfl⟩ :
    ∃ (k : ℕ) (x0 y0 : _), 0 < k ∧ Int.gcd x0 y0 = 1 ∧ x = x0 * k ∧ y = y0 * k :=
    Int.exists_gcd_one' (Nat.pos_of_ne_zero h0)
  

Proof state:
case neg.intro.intro.intro.intro.intro.intro.intro
z0 : ℤ
k : ℕ
x0 y0 : ℤ
k0 : 0 < k
h2 : Int.gcd x0 y0 = 1
h0 : ¬Int.gcd (x0 * ↑k) (y0 * ↑k) = 0
h : PythagoreanTriple (x0 * ↑k) (y0 * ↑k) (↑(Int.gcd (x0 * ↑k) (y0 * ↑k)) * z0)
⊢ PythagoreanTriple (x0 * ↑k / ↑(Int.gcd (x0 * ↑k) (y0 * ↑k))) (y0 * ↑k / ↑(Int.gcd (x0 * ↑k) (y0 * ↑k)))
    (↑(Int.gcd (x0 * ↑k) (y0 * ↑k)) * z0 / ↑(Int.gcd (x0 * ↑k) (y0 * ↑k)))

Next tactic:
have hk : (k : ℤ) ≠ 0 := by
    norm_cast
    rwa [pos_iff_ne_zero] at k0
theorem normalize : PythagoreanTriple (x / Int.gcd x y) (y / Int.gcd x y) (z / Int.gcd x y) := by
  by_cases h0 : Int.gcd x y = 0
  · have hx : x = 0 := by
      apply Int.natAbs_eq_zero.mp
      apply Nat.eq_zero_of_gcd_eq_zero_left h0
    have hy : y = 0 := by
      apply Int.natAbs_eq_zero.mp
      apply Nat.eq_zero_of_gcd_eq_zero_right h0
    have hz : z = 0 := by
      simpa only [PythagoreanTriple, hx, hy, add_zero, zero_eq_mul, mul_zero,
        or_self_iff] using h
    simp only [hx, hy, hz, Int.zero_div]
    exact zero
  rcases h.gcd_dvd with ⟨z0, rfl⟩
  obtain ⟨k, x0, y0, k0, h2, rfl, rfl⟩ :
    ∃ (k : ℕ) (x0 y0 : _), 0 < k ∧ Int.gcd x0 y0 = 1 ∧ x = x0 * k ∧ y = y0 * k :=
    Int.exists_gcd_one' (Nat.pos_of_ne_zero h0)
  have hk : (k : ℤ) ≠ 0 := by
    

Proof state:
z0 : ℤ
k : ℕ
x0 y0 : ℤ
k0 : 0 < k
h2 : Int.gcd x0 y0 = 1
h0 : ¬Int.gcd (x0 * ↑k) (y0 * ↑k) = 0
h : PythagoreanTriple (x0 * ↑k) (y0 * ↑k) (↑(Int.gcd (x0 * ↑k) (y0 * ↑k)) * z0)
⊢ ↑k ≠ 0

Next tactic:
norm_cast
theorem normalize : PythagoreanTriple (x / Int.gcd x y) (y / Int.gcd x y) (z / Int.gcd x y) := by
  by_cases h0 : Int.gcd x y = 0
  · have hx : x = 0 := by
      apply Int.natAbs_eq_zero.mp
      apply Nat.eq_zero_of_gcd_eq_zero_left h0
    have hy : y = 0 := by
      apply Int.natAbs_eq_zero.mp
      apply Nat.eq_zero_of_gcd_eq_zero_right h0
    have hz : z = 0 := by
      simpa only [PythagoreanTriple, hx, hy, add_zero, zero_eq_mul, mul_zero,
        or_self_iff] using h
    simp only [hx, hy, hz, Int.zero_div]
    exact zero
  rcases h.gcd_dvd with ⟨z0, rfl⟩
  obtain ⟨k, x0, y0, k0, h2, rfl, rfl⟩ :
    ∃ (k : ℕ) (x0 y0 : _), 0 < k ∧ Int.gcd x0 y0 = 1 ∧ x = x0 * k ∧ y = y0 * k :=
    Int.exists_gcd_one' (Nat.pos_of_ne_zero h0)
  have hk : (k : ℤ) ≠ 0 := by
    norm_cast
    

Proof state:
z0 : ℤ
k : ℕ
x0 y0 : ℤ
k0 : 0 < k
h2 : Int.gcd x0 y0 = 1
h0 : ¬Int.gcd (x0 * ↑k) (y0 * ↑k) = 0
h : PythagoreanTriple (x0 * ↑k) (y0 * ↑k) (↑(Int.gcd (x0 * ↑k) (y0 * ↑k)) * z0)
⊢ ¬k = 0

Next tactic:
rwa [pos_iff_ne_zero] at k0
theorem normalize : PythagoreanTriple (x / Int.gcd x y) (y / Int.gcd x y) (z / Int.gcd x y) := by
  by_cases h0 : Int.gcd x y = 0
  · have hx : x = 0 := by
      apply Int.natAbs_eq_zero.mp
      apply Nat.eq_zero_of_gcd_eq_zero_left h0
    have hy : y = 0 := by
      apply Int.natAbs_eq_zero.mp
      apply Nat.eq_zero_of_gcd_eq_zero_right h0
    have hz : z = 0 := by
      simpa only [PythagoreanTriple, hx, hy, add_zero, zero_eq_mul, mul_zero,
        or_self_iff] using h
    simp only [hx, hy, hz, Int.zero_div]
    exact zero
  rcases h.gcd_dvd with ⟨z0, rfl⟩
  obtain ⟨k, x0, y0, k0, h2, rfl, rfl⟩ :
    ∃ (k : ℕ) (x0 y0 : _), 0 < k ∧ Int.gcd x0 y0 = 1 ∧ x = x0 * k ∧ y = y0 * k :=
    Int.exists_gcd_one' (Nat.pos_of_ne_zero h0)
  have hk : (k : ℤ) ≠ 0 := by
    norm_cast
    rwa [pos_iff_ne_zero] at k0
  

Proof state:
case neg.intro.intro.intro.intro.intro.intro.intro
z0 : ℤ
k : ℕ
x0 y0 : ℤ
k0 : 0 < k
h2 : Int.gcd x0 y0 = 1
h0 : ¬Int.gcd (x0 * ↑k) (y0 * ↑k) = 0
h : PythagoreanTriple (x0 * ↑k) (y0 * ↑k) (↑(Int.gcd (x0 * ↑k) (y0 * ↑k)) * z0)
hk : ↑k ≠ 0
⊢ PythagoreanTriple (x0 * ↑k / ↑(Int.gcd (x0 * ↑k) (y0 * ↑k))) (y0 * ↑k / ↑(Int.gcd (x0 * ↑k) (y0 * ↑k)))
    (↑(Int.gcd (x0 * ↑k) (y0 * ↑k)) * z0 / ↑(Int.gcd (x0 * ↑k) (y0 * ↑k)))

Next tactic:
rw [Int.gcd_mul_right, h2, Int.natAbs_ofNat, one_mul] at h ⊢
theorem normalize : PythagoreanTriple (x / Int.gcd x y) (y / Int.gcd x y) (z / Int.gcd x y) := by
  by_cases h0 : Int.gcd x y = 0
  · have hx : x = 0 := by
      apply Int.natAbs_eq_zero.mp
      apply Nat.eq_zero_of_gcd_eq_zero_left h0
    have hy : y = 0 := by
      apply Int.natAbs_eq_zero.mp
      apply Nat.eq_zero_of_gcd_eq_zero_right h0
    have hz : z = 0 := by
      simpa only [PythagoreanTriple, hx, hy, add_zero, zero_eq_mul, mul_zero,
        or_self_iff] using h
    simp only [hx, hy, hz, Int.zero_div]
    exact zero
  rcases h.gcd_dvd with ⟨z0, rfl⟩
  obtain ⟨k, x0, y0, k0, h2, rfl, rfl⟩ :
    ∃ (k : ℕ) (x0 y0 : _), 0 < k ∧ Int.gcd x0 y0 = 1 ∧ x = x0 * k ∧ y = y0 * k :=
    Int.exists_gcd_one' (Nat.pos_of_ne_zero h0)
  have hk : (k : ℤ) ≠ 0 := by
    norm_cast
    rwa [pos_iff_ne_zero] at k0
  rw [Int.gcd_mul_right, h2, Int.natAbs_ofNat, one_mul] at h ⊢
  

Proof state:
case neg.intro.intro.intro.intro.intro.intro.intro
z0 : ℤ
k : ℕ
x0 y0 : ℤ
k0 : 0 < k
h2 : Int.gcd x0 y0 = 1
h0 : ¬Int.gcd (x0 * ↑k) (y0 * ↑k) = 0
h : PythagoreanTriple (x0 * ↑k) (y0 * ↑k) (↑k * z0)
hk : ↑k ≠ 0
⊢ PythagoreanTriple (x0 * ↑k / ↑k) (y0 * ↑k / ↑k) (↑k * z0 / ↑k)

Next tactic:
rw [mul_comm x0, mul_comm y0, mul_iff k hk] at h
theorem normalize : PythagoreanTriple (x / Int.gcd x y) (y / Int.gcd x y) (z / Int.gcd x y) := by
  by_cases h0 : Int.gcd x y = 0
  · have hx : x = 0 := by
      apply Int.natAbs_eq_zero.mp
      apply Nat.eq_zero_of_gcd_eq_zero_left h0
    have hy : y = 0 := by
      apply Int.natAbs_eq_zero.mp
      apply Nat.eq_zero_of_gcd_eq_zero_right h0
    have hz : z = 0 := by
      simpa only [PythagoreanTriple, hx, hy, add_zero, zero_eq_mul, mul_zero,
        or_self_iff] using h
    simp only [hx, hy, hz, Int.zero_div]
    exact zero
  rcases h.gcd_dvd with ⟨z0, rfl⟩
  obtain ⟨k, x0, y0, k0, h2, rfl, rfl⟩ :
    ∃ (k : ℕ) (x0 y0 : _), 0 < k ∧ Int.gcd x0 y0 = 1 ∧ x = x0 * k ∧ y = y0 * k :=
    Int.exists_gcd_one' (Nat.pos_of_ne_zero h0)
  have hk : (k : ℤ) ≠ 0 := by
    norm_cast
    rwa [pos_iff_ne_zero] at k0
  rw [Int.gcd_mul_right, h2, Int.natAbs_ofNat, one_mul] at h ⊢
  rw [mul_comm x0, mul_comm y0, mul_iff k hk] at h
  

Proof state:
case neg.intro.intro.intro.intro.intro.intro.intro
z0 : ℤ
k : ℕ
x0 y0 : ℤ
k0 : 0 < k
h2 : Int.gcd x0 y0 = 1
h0 : ¬Int.gcd (x0 * ↑k) (y0 * ↑k) = 0
h : PythagoreanTriple x0 y0 z0
hk : ↑k ≠ 0
⊢ PythagoreanTriple (x0 * ↑k / ↑k) (y0 * ↑k / ↑k) (↑k * z0 / ↑k)

Next tactic:
rwa [Int.mul_ediv_cancel _ hk, Int.mul_ediv_cancel _ hk, Int.mul_ediv_cancel_left _ hk]
theorem isClassified_of_isPrimitiveClassified (hp : h.IsPrimitiveClassified) : h.IsClassified := by
  

Proof state:
x y z : ℤ
h : PythagoreanTriple x y z
hp : IsPrimitiveClassified h
⊢ IsClassified h

Next tactic:
obtain ⟨m, n, H⟩ := hp
theorem isClassified_of_isPrimitiveClassified (hp : h.IsPrimitiveClassified) : h.IsClassified := by
  obtain ⟨m, n, H⟩ := hp
  

Proof state:
case intro.intro
x y z : ℤ
h : PythagoreanTriple x y z
m n : ℤ
H :
  (x = m ^ 2 - n ^ 2 ∧ y = 2 * m * n ∨ x = 2 * m * n ∧ y = m ^ 2 - n ^ 2) ∧
    Int.gcd m n = 1 ∧ (m % 2 = 0 ∧ n % 2 = 1 ∨ m % 2 = 1 ∧ n % 2 = 0)
⊢ IsClassified h

Next tactic:
use 1, m, n
theorem isClassified_of_isPrimitiveClassified (hp : h.IsPrimitiveClassified) : h.IsClassified := by
  obtain ⟨m, n, H⟩ := hp
  use 1, m, n
  

Proof state:
case h
x y z : ℤ
h : PythagoreanTriple x y z
m n : ℤ
H :
  (x = m ^ 2 - n ^ 2 ∧ y = 2 * m * n ∨ x = 2 * m * n ∧ y = m ^ 2 - n ^ 2) ∧
    Int.gcd m n = 1 ∧ (m % 2 = 0 ∧ n % 2 = 1 ∨ m % 2 = 1 ∧ n % 2 = 0)
⊢ (x = 1 * (m ^ 2 - n ^ 2) ∧ y = 1 * (2 * m * n) ∨ x = 1 * (2 * m * n) ∧ y = 1 * (m ^ 2 - n ^ 2)) ∧ Int.gcd m n = 1

Next tactic:
rcases H with ⟨t, co, _⟩
theorem isClassified_of_isPrimitiveClassified (hp : h.IsPrimitiveClassified) : h.IsClassified := by
  obtain ⟨m, n, H⟩ := hp
  use 1, m, n
  rcases H with ⟨t, co, _⟩
  

Proof state:
case h.intro.intro
x y z : ℤ
h : PythagoreanTriple x y z
m n : ℤ
t : x = m ^ 2 - n ^ 2 ∧ y = 2 * m * n ∨ x = 2 * m * n ∧ y = m ^ 2 - n ^ 2
co : Int.gcd m n = 1
right✝ : m % 2 = 0 ∧ n % 2 = 1 ∨ m % 2 = 1 ∧ n % 2 = 0
⊢ (x = 1 * (m ^ 2 - n ^ 2) ∧ y = 1 * (2 * m * n) ∨ x = 1 * (2 * m * n) ∧ y = 1 * (m ^ 2 - n ^ 2)) ∧ Int.gcd m n = 1

Next tactic:
rw [one_mul, one_mul]
theorem isClassified_of_isPrimitiveClassified (hp : h.IsPrimitiveClassified) : h.IsClassified := by
  obtain ⟨m, n, H⟩ := hp
  use 1, m, n
  rcases H with ⟨t, co, _⟩
  rw [one_mul, one_mul]
  

Proof state:
case h.intro.intro
x y z : ℤ
h : PythagoreanTriple x y z
m n : ℤ
t : x = m ^ 2 - n ^ 2 ∧ y = 2 * m * n ∨ x = 2 * m * n ∧ y = m ^ 2 - n ^ 2
co : Int.gcd m n = 1
right✝ : m % 2 = 0 ∧ n % 2 = 1 ∨ m % 2 = 1 ∧ n % 2 = 0
⊢ (x = m ^ 2 - n ^ 2 ∧ y = 2 * m * n ∨ x = 2 * m * n ∧ y = m ^ 2 - n ^ 2) ∧ Int.gcd m n = 1

Next tactic:
exact ⟨t, co⟩
theorem isClassified_of_normalize_isPrimitiveClassified (hc : h.normalize.IsPrimitiveClassified) :
    h.IsClassified := by
  

Proof state:
x y z : ℤ
h : PythagoreanTriple x y z
hc : IsPrimitiveClassified (_ : PythagoreanTriple (x / ↑(Int.gcd x y)) (y / ↑(Int.gcd x y)) (z / ↑(Int.gcd x y)))
⊢ IsClassified h

Next tactic:
convert h.normalize.mul_isClassified (Int.gcd x y)
        (isClassified_of_isPrimitiveClassified h.normalize hc)
theorem isClassified_of_normalize_isPrimitiveClassified (hc : h.normalize.IsPrimitiveClassified) :
    h.IsClassified := by
  convert h.normalize.mul_isClassified (Int.gcd x y)
        (isClassified_of_isPrimitiveClassified h.normalize hc) <;>
    

Proof state:
case h.e'_1
x y z : ℤ
h : PythagoreanTriple x y z
hc : IsPrimitiveClassified (_ : PythagoreanTriple (x / ↑(Int.gcd x y)) (y / ↑(Int.gcd x y)) (z / ↑(Int.gcd x y)))
⊢ x = ↑(Int.gcd x y) * (x / ↑(Int.gcd x y))

Next tactic:
rw [Int.mul_ediv_cancel']
theorem isClassified_of_normalize_isPrimitiveClassified (hc : h.normalize.IsPrimitiveClassified) :
    h.IsClassified := by
  convert h.normalize.mul_isClassified (Int.gcd x y)
        (isClassified_of_isPrimitiveClassified h.normalize hc) <;>
    

Proof state:
case h.e'_2
x y z : ℤ
h : PythagoreanTriple x y z
hc : IsPrimitiveClassified (_ : PythagoreanTriple (x / ↑(Int.gcd x y)) (y / ↑(Int.gcd x y)) (z / ↑(Int.gcd x y)))
⊢ y = ↑(Int.gcd x y) * (y / ↑(Int.gcd x y))

Next tactic:
rw [Int.mul_ediv_cancel']
theorem isClassified_of_normalize_isPrimitiveClassified (hc : h.normalize.IsPrimitiveClassified) :
    h.IsClassified := by
  convert h.normalize.mul_isClassified (Int.gcd x y)
        (isClassified_of_isPrimitiveClassified h.normalize hc) <;>
    

Proof state:
case h.e'_3
x y z : ℤ
h : PythagoreanTriple x y z
hc : IsPrimitiveClassified (_ : PythagoreanTriple (x / ↑(Int.gcd x y)) (y / ↑(Int.gcd x y)) (z / ↑(Int.gcd x y)))
⊢ z = ↑(Int.gcd x y) * (z / ↑(Int.gcd x y))

Next tactic:
rw [Int.mul_ediv_cancel']
theorem isClassified_of_normalize_isPrimitiveClassified (hc : h.normalize.IsPrimitiveClassified) :
    h.IsClassified := by
  convert h.normalize.mul_isClassified (Int.gcd x y)
        (isClassified_of_isPrimitiveClassified h.normalize hc) <;>
    rw [Int.mul_ediv_cancel']
  · 

Proof state:
case h.e'_1
x y z : ℤ
h : PythagoreanTriple x y z
hc : IsPrimitiveClassified (_ : PythagoreanTriple (x / ↑(Int.gcd x y)) (y / ↑(Int.gcd x y)) (z / ↑(Int.gcd x y)))
⊢ ↑(Int.gcd x y) ∣ x

Next tactic:
exact Int.gcd_dvd_left x y
theorem isClassified_of_normalize_isPrimitiveClassified (hc : h.normalize.IsPrimitiveClassified) :
    h.IsClassified := by
  convert h.normalize.mul_isClassified (Int.gcd x y)
        (isClassified_of_isPrimitiveClassified h.normalize hc) <;>
    rw [Int.mul_ediv_cancel']
  · exact Int.gcd_dvd_left x y
  · 

Proof state:
case h.e'_2
x y z : ℤ
h : PythagoreanTriple x y z
hc : IsPrimitiveClassified (_ : PythagoreanTriple (x / ↑(Int.gcd x y)) (y / ↑(Int.gcd x y)) (z / ↑(Int.gcd x y)))
⊢ ↑(Int.gcd x y) ∣ y

Next tactic:
exact Int.gcd_dvd_right x y
theorem isClassified_of_normalize_isPrimitiveClassified (hc : h.normalize.IsPrimitiveClassified) :
    h.IsClassified := by
  convert h.normalize.mul_isClassified (Int.gcd x y)
        (isClassified_of_isPrimitiveClassified h.normalize hc) <;>
    rw [Int.mul_ediv_cancel']
  · exact Int.gcd_dvd_left x y
  · exact Int.gcd_dvd_right x y
  · 

Proof state:
case h.e'_3
x y z : ℤ
h : PythagoreanTriple x y z
hc : IsPrimitiveClassified (_ : PythagoreanTriple (x / ↑(Int.gcd x y)) (y / ↑(Int.gcd x y)) (z / ↑(Int.gcd x y)))
⊢ ↑(Int.gcd x y) ∣ z

Next tactic:
exact h.gcd_dvd
theorem ne_zero_of_coprime (hc : Int.gcd x y = 1) : z ≠ 0 := by
  

Proof state:
x y z : ℤ
h : PythagoreanTriple x y z
hc : Int.gcd x y = 1
⊢ z ≠ 0

Next tactic:
suffices 0 < z * z by
    rintro rfl
    norm_num at this
theorem ne_zero_of_coprime (hc : Int.gcd x y = 1) : z ≠ 0 := by
  suffices 0 < z * z by
    

Proof state:
x y z : ℤ
h : PythagoreanTriple x y z
hc : Int.gcd x y = 1
this : 0 < z * z
⊢ z ≠ 0

Next tactic:
rintro rfl
theorem ne_zero_of_coprime (hc : Int.gcd x y = 1) : z ≠ 0 := by
  suffices 0 < z * z by
    rintro rfl
    

Proof state:
x y : ℤ
hc : Int.gcd x y = 1
h : PythagoreanTriple x y 0
this : 0 < 0 * 0
⊢ False

Next tactic:
norm_num at this
theorem ne_zero_of_coprime (hc : Int.gcd x y = 1) : z ≠ 0 := by
  suffices 0 < z * z by
    rintro rfl
    norm_num at this
  

Proof state:
x y z : ℤ
h : PythagoreanTriple x y z
hc : Int.gcd x y = 1
⊢ 0 < z * z

Next tactic:
rw [← h.eq, ← sq, ← sq]