l3lab/ntp-mathlib huggingface dataset:
private theorem coprime_sq_sub_sq_sum_of_odd_odd {m n : ℤ} (h : Int.gcd m n = 1) (hm : m % 2 = 1)
    (hn : n % 2 = 1) :
    2 ∣ m ^ 2 + n ^ 2 ∧
      2 ∣ m ^ 2 - n ^ 2 ∧
        (m ^ 2 - n ^ 2) / 2 % 2 = 0 ∧ Int.gcd ((m ^ 2 - n ^ 2) / 2) ((m ^ 2 + n ^ 2) / 2) = 1 := by
  cases' exists_eq_mul_left_of_dvd (Int.dvd_sub_of_emod_eq hm) with m0 hm2
  cases' exists_eq_mul_left_of_dvd (Int.dvd_sub_of_emod_eq hn) with n0 hn2
  rw [sub_eq_iff_eq_add] at hm2 hn2
  subst m
  subst n
  have h1 : (m0 * 2 + 1) ^ 2 + (n0 * 2 + 1) ^ 2 = 2 * (2 * (m0 ^ 2 + n0 ^ 2 + m0 + n0) + 1) := by
    ring
  have h2 : (m0 * 2 + 1) ^ 2 - (n0 * 2 + 1) ^ 2 = 2 * (2 * (m0 ^ 2 - n0 ^ 2 + m0 - n0)) := by ring
  have h3 : ((m0 * 2 + 1) ^ 2 - (n0 * 2 + 1) ^ 2) / 2 % 2 = 0 := by
    rw [h2, Int.mul_ediv_cancel_left, Int.mul_emod_right]
    exact by decide
  refine' ⟨⟨_, h1⟩, ⟨_, h2⟩, h3, _⟩
  have h20 : (2 : ℤ) ≠ 0 := by decide
  rw [h1, h2, Int.mul_ediv_cancel_left _ h20, Int.mul_ediv_cancel_left _ h20]
  by_contra h4
  obtain ⟨p, hp, hp1, hp2⟩ := Nat.Prime.not_coprime_iff_dvd.mp h4
  apply hp.not_dvd_one
  rw [← h]
  rw [← Int.coe_nat_dvd_left] at hp1 hp2
  apply Nat.dvd_gcd
  · apply Int.Prime.dvd_natAbs_of_coe_dvd_sq hp
    convert dvd_add hp1 hp2
    ring
  · 

Proof state:
case intro.intro.intro.intro.intro.a
m0 n0 : ℤ
hm : (m0 * 2 + 1) % 2 = 1
hn : (n0 * 2 + 1) % 2 = 1
h : Int.gcd (m0 * 2 + 1) (n0 * 2 + 1) = 1
h1 : (m0 * 2 + 1) ^ 2 + (n0 * 2 + 1) ^ 2 = 2 * (2 * (m0 ^ 2 + n0 ^ 2 + m0 + n0) + 1)
h2 : (m0 * 2 + 1) ^ 2 - (n0 * 2 + 1) ^ 2 = 2 * (2 * (m0 ^ 2 - n0 ^ 2 + m0 - n0))
h3 : ((m0 * 2 + 1) ^ 2 - (n0 * 2 + 1) ^ 2) / 2 % 2 = 0
h20 : 2 ≠ 0
h4 : ¬Int.gcd (2 * (m0 ^ 2 - n0 ^ 2 + m0 - n0)) (2 * (m0 ^ 2 + n0 ^ 2 + m0 + n0) + 1) = 1
p : ℕ
hp : Nat.Prime p
hp1 : ↑p ∣ 2 * (m0 ^ 2 - n0 ^ 2 + m0 - n0)
hp2 : ↑p ∣ 2 * (m0 ^ 2 + n0 ^ 2 + m0 + n0) + 1
⊢ p ∣ Int.natAbs (n0 * 2 + 1)

Next tactic:
apply Int.Prime.dvd_natAbs_of_coe_dvd_sq hp
private theorem coprime_sq_sub_sq_sum_of_odd_odd {m n : ℤ} (h : Int.gcd m n = 1) (hm : m % 2 = 1)
    (hn : n % 2 = 1) :
    2 ∣ m ^ 2 + n ^ 2 ∧
      2 ∣ m ^ 2 - n ^ 2 ∧
        (m ^ 2 - n ^ 2) / 2 % 2 = 0 ∧ Int.gcd ((m ^ 2 - n ^ 2) / 2) ((m ^ 2 + n ^ 2) / 2) = 1 := by
  cases' exists_eq_mul_left_of_dvd (Int.dvd_sub_of_emod_eq hm) with m0 hm2
  cases' exists_eq_mul_left_of_dvd (Int.dvd_sub_of_emod_eq hn) with n0 hn2
  rw [sub_eq_iff_eq_add] at hm2 hn2
  subst m
  subst n
  have h1 : (m0 * 2 + 1) ^ 2 + (n0 * 2 + 1) ^ 2 = 2 * (2 * (m0 ^ 2 + n0 ^ 2 + m0 + n0) + 1) := by
    ring
  have h2 : (m0 * 2 + 1) ^ 2 - (n0 * 2 + 1) ^ 2 = 2 * (2 * (m0 ^ 2 - n0 ^ 2 + m0 - n0)) := by ring
  have h3 : ((m0 * 2 + 1) ^ 2 - (n0 * 2 + 1) ^ 2) / 2 % 2 = 0 := by
    rw [h2, Int.mul_ediv_cancel_left, Int.mul_emod_right]
    exact by decide
  refine' ⟨⟨_, h1⟩, ⟨_, h2⟩, h3, _⟩
  have h20 : (2 : ℤ) ≠ 0 := by decide
  rw [h1, h2, Int.mul_ediv_cancel_left _ h20, Int.mul_ediv_cancel_left _ h20]
  by_contra h4
  obtain ⟨p, hp, hp1, hp2⟩ := Nat.Prime.not_coprime_iff_dvd.mp h4
  apply hp.not_dvd_one
  rw [← h]
  rw [← Int.coe_nat_dvd_left] at hp1 hp2
  apply Nat.dvd_gcd
  · apply Int.Prime.dvd_natAbs_of_coe_dvd_sq hp
    convert dvd_add hp1 hp2
    ring
  · apply Int.Prime.dvd_natAbs_of_coe_dvd_sq hp
    

Proof state:
case intro.intro.intro.intro.intro.a.h
m0 n0 : ℤ
hm : (m0 * 2 + 1) % 2 = 1
hn : (n0 * 2 + 1) % 2 = 1
h : Int.gcd (m0 * 2 + 1) (n0 * 2 + 1) = 1
h1 : (m0 * 2 + 1) ^ 2 + (n0 * 2 + 1) ^ 2 = 2 * (2 * (m0 ^ 2 + n0 ^ 2 + m0 + n0) + 1)
h2 : (m0 * 2 + 1) ^ 2 - (n0 * 2 + 1) ^ 2 = 2 * (2 * (m0 ^ 2 - n0 ^ 2 + m0 - n0))
h3 : ((m0 * 2 + 1) ^ 2 - (n0 * 2 + 1) ^ 2) / 2 % 2 = 0
h20 : 2 ≠ 0
h4 : ¬Int.gcd (2 * (m0 ^ 2 - n0 ^ 2 + m0 - n0)) (2 * (m0 ^ 2 + n0 ^ 2 + m0 + n0) + 1) = 1
p : ℕ
hp : Nat.Prime p
hp1 : ↑p ∣ 2 * (m0 ^ 2 - n0 ^ 2 + m0 - n0)
hp2 : ↑p ∣ 2 * (m0 ^ 2 + n0 ^ 2 + m0 + n0) + 1
⊢ ↑p ∣ (n0 * 2 + 1) ^ 2

Next tactic:
convert dvd_sub hp2 hp1
private theorem coprime_sq_sub_sq_sum_of_odd_odd {m n : ℤ} (h : Int.gcd m n = 1) (hm : m % 2 = 1)
    (hn : n % 2 = 1) :
    2 ∣ m ^ 2 + n ^ 2 ∧
      2 ∣ m ^ 2 - n ^ 2 ∧
        (m ^ 2 - n ^ 2) / 2 % 2 = 0 ∧ Int.gcd ((m ^ 2 - n ^ 2) / 2) ((m ^ 2 + n ^ 2) / 2) = 1 := by
  cases' exists_eq_mul_left_of_dvd (Int.dvd_sub_of_emod_eq hm) with m0 hm2
  cases' exists_eq_mul_left_of_dvd (Int.dvd_sub_of_emod_eq hn) with n0 hn2
  rw [sub_eq_iff_eq_add] at hm2 hn2
  subst m
  subst n
  have h1 : (m0 * 2 + 1) ^ 2 + (n0 * 2 + 1) ^ 2 = 2 * (2 * (m0 ^ 2 + n0 ^ 2 + m0 + n0) + 1) := by
    ring
  have h2 : (m0 * 2 + 1) ^ 2 - (n0 * 2 + 1) ^ 2 = 2 * (2 * (m0 ^ 2 - n0 ^ 2 + m0 - n0)) := by ring
  have h3 : ((m0 * 2 + 1) ^ 2 - (n0 * 2 + 1) ^ 2) / 2 % 2 = 0 := by
    rw [h2, Int.mul_ediv_cancel_left, Int.mul_emod_right]
    exact by decide
  refine' ⟨⟨_, h1⟩, ⟨_, h2⟩, h3, _⟩
  have h20 : (2 : ℤ) ≠ 0 := by decide
  rw [h1, h2, Int.mul_ediv_cancel_left _ h20, Int.mul_ediv_cancel_left _ h20]
  by_contra h4
  obtain ⟨p, hp, hp1, hp2⟩ := Nat.Prime.not_coprime_iff_dvd.mp h4
  apply hp.not_dvd_one
  rw [← h]
  rw [← Int.coe_nat_dvd_left] at hp1 hp2
  apply Nat.dvd_gcd
  · apply Int.Prime.dvd_natAbs_of_coe_dvd_sq hp
    convert dvd_add hp1 hp2
    ring
  · apply Int.Prime.dvd_natAbs_of_coe_dvd_sq hp
    convert dvd_sub hp2 hp1
    

Proof state:
case h.e'_4
m0 n0 : ℤ
hm : (m0 * 2 + 1) % 2 = 1
hn : (n0 * 2 + 1) % 2 = 1
h : Int.gcd (m0 * 2 + 1) (n0 * 2 + 1) = 1
h1 : (m0 * 2 + 1) ^ 2 + (n0 * 2 + 1) ^ 2 = 2 * (2 * (m0 ^ 2 + n0 ^ 2 + m0 + n0) + 1)
h2 : (m0 * 2 + 1) ^ 2 - (n0 * 2 + 1) ^ 2 = 2 * (2 * (m0 ^ 2 - n0 ^ 2 + m0 - n0))
h3 : ((m0 * 2 + 1) ^ 2 - (n0 * 2 + 1) ^ 2) / 2 % 2 = 0
h20 : 2 ≠ 0
h4 : ¬Int.gcd (2 * (m0 ^ 2 - n0 ^ 2 + m0 - n0)) (2 * (m0 ^ 2 + n0 ^ 2 + m0 + n0) + 1) = 1
p : ℕ
hp : Nat.Prime p
hp1 : ↑p ∣ 2 * (m0 ^ 2 - n0 ^ 2 + m0 - n0)
hp2 : ↑p ∣ 2 * (m0 ^ 2 + n0 ^ 2 + m0 + n0) + 1
⊢ (n0 * 2 + 1) ^ 2 = 2 * (m0 ^ 2 + n0 ^ 2 + m0 + n0) + 1 - 2 * (m0 ^ 2 - n0 ^ 2 + m0 - n0)

Next tactic:
ring
theorem isPrimitiveClassified_aux (hc : x.gcd y = 1) (hzpos : 0 < z) {m n : ℤ}
    (hm2n2 : 0 < m ^ 2 + n ^ 2) (hv2 : (x : ℚ) / z = 2 * m * n / ((m : ℚ) ^ 2 + (n : ℚ) ^ 2))
    (hw2 : (y : ℚ) / z = ((m : ℚ) ^ 2 - (n : ℚ) ^ 2) / ((m : ℚ) ^ 2 + (n : ℚ) ^ 2))
    (H : Int.gcd (m ^ 2 - n ^ 2) (m ^ 2 + n ^ 2) = 1) (co : Int.gcd m n = 1)
    (pp : m % 2 = 0 ∧ n % 2 = 1 ∨ m % 2 = 1 ∧ n % 2 = 0) : h.IsPrimitiveClassified := by
  

Proof state:
x y z : ℤ
h : PythagoreanTriple x y z
hc : Int.gcd x y = 1
hzpos : 0 < z
m n : ℤ
hm2n2 : 0 < m ^ 2 + n ^ 2
hv2 : ↑x / ↑z = 2 * ↑m * ↑n / (↑m ^ 2 + ↑n ^ 2)
hw2 : ↑y / ↑z = (↑m ^ 2 - ↑n ^ 2) / (↑m ^ 2 + ↑n ^ 2)
H : Int.gcd (m ^ 2 - n ^ 2) (m ^ 2 + n ^ 2) = 1
co : Int.gcd m n = 1
pp : m % 2 = 0 ∧ n % 2 = 1 ∨ m % 2 = 1 ∧ n % 2 = 0
⊢ IsPrimitiveClassified h

Next tactic:
have hz : z ≠ 0
theorem isPrimitiveClassified_aux (hc : x.gcd y = 1) (hzpos : 0 < z) {m n : ℤ}
    (hm2n2 : 0 < m ^ 2 + n ^ 2) (hv2 : (x : ℚ) / z = 2 * m * n / ((m : ℚ) ^ 2 + (n : ℚ) ^ 2))
    (hw2 : (y : ℚ) / z = ((m : ℚ) ^ 2 - (n : ℚ) ^ 2) / ((m : ℚ) ^ 2 + (n : ℚ) ^ 2))
    (H : Int.gcd (m ^ 2 - n ^ 2) (m ^ 2 + n ^ 2) = 1) (co : Int.gcd m n = 1)
    (pp : m % 2 = 0 ∧ n % 2 = 1 ∨ m % 2 = 1 ∧ n % 2 = 0) : h.IsPrimitiveClassified := by
  have hz : z ≠ 0
  

Proof state:
case hz
x y z : ℤ
h : PythagoreanTriple x y z
hc : Int.gcd x y = 1
hzpos : 0 < z
m n : ℤ
hm2n2 : 0 < m ^ 2 + n ^ 2
hv2 : ↑x / ↑z = 2 * ↑m * ↑n / (↑m ^ 2 + ↑n ^ 2)
hw2 : ↑y / ↑z = (↑m ^ 2 - ↑n ^ 2) / (↑m ^ 2 + ↑n ^ 2)
H : Int.gcd (m ^ 2 - n ^ 2) (m ^ 2 + n ^ 2) = 1
co : Int.gcd m n = 1
pp : m % 2 = 0 ∧ n % 2 = 1 ∨ m % 2 = 1 ∧ n % 2 = 0
⊢ z ≠ 0
x y z : ℤ
h : PythagoreanTriple x y z
hc : Int.gcd x y = 1
hzpos : 0 < z
m n : ℤ
hm2n2 : 0 < m ^ 2 + n ^ 2
hv2 : ↑x / ↑z = 2 * ↑m * ↑n / (↑m ^ 2 + ↑n ^ 2)
hw2 : ↑y / ↑z = (↑m ^ 2 - ↑n ^ 2) / (↑m ^ 2 + ↑n ^ 2)
H : Int.gcd (m ^ 2 - n ^ 2) (m ^ 2 + n ^ 2) = 1
co : Int.gcd m n = 1
pp : m % 2 = 0 ∧ n % 2 = 1 ∨ m % 2 = 1 ∧ n % 2 = 0
hz : z ≠ 0
⊢ IsPrimitiveClassified h

Next tactic:
apply ne_of_gt hzpos
theorem isPrimitiveClassified_aux (hc : x.gcd y = 1) (hzpos : 0 < z) {m n : ℤ}
    (hm2n2 : 0 < m ^ 2 + n ^ 2) (hv2 : (x : ℚ) / z = 2 * m * n / ((m : ℚ) ^ 2 + (n : ℚ) ^ 2))
    (hw2 : (y : ℚ) / z = ((m : ℚ) ^ 2 - (n : ℚ) ^ 2) / ((m : ℚ) ^ 2 + (n : ℚ) ^ 2))
    (H : Int.gcd (m ^ 2 - n ^ 2) (m ^ 2 + n ^ 2) = 1) (co : Int.gcd m n = 1)
    (pp : m % 2 = 0 ∧ n % 2 = 1 ∨ m % 2 = 1 ∧ n % 2 = 0) : h.IsPrimitiveClassified := by
  have hz : z ≠ 0
  apply ne_of_gt hzpos
  

Proof state:
x y z : ℤ
h : PythagoreanTriple x y z
hc : Int.gcd x y = 1
hzpos : 0 < z
m n : ℤ
hm2n2 : 0 < m ^ 2 + n ^ 2
hv2 : ↑x / ↑z = 2 * ↑m * ↑n / (↑m ^ 2 + ↑n ^ 2)
hw2 : ↑y / ↑z = (↑m ^ 2 - ↑n ^ 2) / (↑m ^ 2 + ↑n ^ 2)
H : Int.gcd (m ^ 2 - n ^ 2) (m ^ 2 + n ^ 2) = 1
co : Int.gcd m n = 1
pp : m % 2 = 0 ∧ n % 2 = 1 ∨ m % 2 = 1 ∧ n % 2 = 0
hz : z ≠ 0
⊢ IsPrimitiveClassified h

Next tactic:
have h2 : y = m ^ 2 - n ^ 2 ∧ z = m ^ 2 + n ^ 2 := by
    apply Rat.div_int_inj hzpos hm2n2 (h.coprime_of_coprime hc) H
    rw [hw2]
    norm_cast
theorem isPrimitiveClassified_aux (hc : x.gcd y = 1) (hzpos : 0 < z) {m n : ℤ}
    (hm2n2 : 0 < m ^ 2 + n ^ 2) (hv2 : (x : ℚ) / z = 2 * m * n / ((m : ℚ) ^ 2 + (n : ℚ) ^ 2))
    (hw2 : (y : ℚ) / z = ((m : ℚ) ^ 2 - (n : ℚ) ^ 2) / ((m : ℚ) ^ 2 + (n : ℚ) ^ 2))
    (H : Int.gcd (m ^ 2 - n ^ 2) (m ^ 2 + n ^ 2) = 1) (co : Int.gcd m n = 1)
    (pp : m % 2 = 0 ∧ n % 2 = 1 ∨ m % 2 = 1 ∧ n % 2 = 0) : h.IsPrimitiveClassified := by
  have hz : z ≠ 0
  apply ne_of_gt hzpos
  have h2 : y = m ^ 2 - n ^ 2 ∧ z = m ^ 2 + n ^ 2 := by
    

Proof state:
x y z : ℤ
h : PythagoreanTriple x y z
hc : Int.gcd x y = 1
hzpos : 0 < z
m n : ℤ
hm2n2 : 0 < m ^ 2 + n ^ 2
hv2 : ↑x / ↑z = 2 * ↑m * ↑n / (↑m ^ 2 + ↑n ^ 2)
hw2 : ↑y / ↑z = (↑m ^ 2 - ↑n ^ 2) / (↑m ^ 2 + ↑n ^ 2)
H : Int.gcd (m ^ 2 - n ^ 2) (m ^ 2 + n ^ 2) = 1
co : Int.gcd m n = 1
pp : m % 2 = 0 ∧ n % 2 = 1 ∨ m % 2 = 1 ∧ n % 2 = 0
hz : z ≠ 0
⊢ y = m ^ 2 - n ^ 2 ∧ z = m ^ 2 + n ^ 2

Next tactic:
apply Rat.div_int_inj hzpos hm2n2 (h.coprime_of_coprime hc) H
theorem isPrimitiveClassified_aux (hc : x.gcd y = 1) (hzpos : 0 < z) {m n : ℤ}
    (hm2n2 : 0 < m ^ 2 + n ^ 2) (hv2 : (x : ℚ) / z = 2 * m * n / ((m : ℚ) ^ 2 + (n : ℚ) ^ 2))
    (hw2 : (y : ℚ) / z = ((m : ℚ) ^ 2 - (n : ℚ) ^ 2) / ((m : ℚ) ^ 2 + (n : ℚ) ^ 2))
    (H : Int.gcd (m ^ 2 - n ^ 2) (m ^ 2 + n ^ 2) = 1) (co : Int.gcd m n = 1)
    (pp : m % 2 = 0 ∧ n % 2 = 1 ∨ m % 2 = 1 ∧ n % 2 = 0) : h.IsPrimitiveClassified := by
  have hz : z ≠ 0
  apply ne_of_gt hzpos
  have h2 : y = m ^ 2 - n ^ 2 ∧ z = m ^ 2 + n ^ 2 := by
    apply Rat.div_int_inj hzpos hm2n2 (h.coprime_of_coprime hc) H
    

Proof state:
x y z : ℤ
h : PythagoreanTriple x y z
hc : Int.gcd x y = 1
hzpos : 0 < z
m n : ℤ
hm2n2 : 0 < m ^ 2 + n ^ 2
hv2 : ↑x / ↑z = 2 * ↑m * ↑n / (↑m ^ 2 + ↑n ^ 2)
hw2 : ↑y / ↑z = (↑m ^ 2 - ↑n ^ 2) / (↑m ^ 2 + ↑n ^ 2)
H : Int.gcd (m ^ 2 - n ^ 2) (m ^ 2 + n ^ 2) = 1
co : Int.gcd m n = 1
pp : m % 2 = 0 ∧ n % 2 = 1 ∨ m % 2 = 1 ∧ n % 2 = 0
hz : z ≠ 0
⊢ ↑y / ↑z = ↑(m ^ 2 - n ^ 2) / ↑(m ^ 2 + n ^ 2)

Next tactic:
rw [hw2]
theorem isPrimitiveClassified_aux (hc : x.gcd y = 1) (hzpos : 0 < z) {m n : ℤ}
    (hm2n2 : 0 < m ^ 2 + n ^ 2) (hv2 : (x : ℚ) / z = 2 * m * n / ((m : ℚ) ^ 2 + (n : ℚ) ^ 2))
    (hw2 : (y : ℚ) / z = ((m : ℚ) ^ 2 - (n : ℚ) ^ 2) / ((m : ℚ) ^ 2 + (n : ℚ) ^ 2))
    (H : Int.gcd (m ^ 2 - n ^ 2) (m ^ 2 + n ^ 2) = 1) (co : Int.gcd m n = 1)
    (pp : m % 2 = 0 ∧ n % 2 = 1 ∨ m % 2 = 1 ∧ n % 2 = 0) : h.IsPrimitiveClassified := by
  have hz : z ≠ 0
  apply ne_of_gt hzpos
  have h2 : y = m ^ 2 - n ^ 2 ∧ z = m ^ 2 + n ^ 2 := by
    apply Rat.div_int_inj hzpos hm2n2 (h.coprime_of_coprime hc) H
    rw [hw2]
    

Proof state:
x y z : ℤ
h : PythagoreanTriple x y z
hc : Int.gcd x y = 1
hzpos : 0 < z
m n : ℤ
hm2n2 : 0 < m ^ 2 + n ^ 2
hv2 : ↑x / ↑z = 2 * ↑m * ↑n / (↑m ^ 2 + ↑n ^ 2)
hw2 : ↑y / ↑z = (↑m ^ 2 - ↑n ^ 2) / (↑m ^ 2 + ↑n ^ 2)
H : Int.gcd (m ^ 2 - n ^ 2) (m ^ 2 + n ^ 2) = 1
co : Int.gcd m n = 1
pp : m % 2 = 0 ∧ n % 2 = 1 ∨ m % 2 = 1 ∧ n % 2 = 0
hz : z ≠ 0
⊢ (↑m ^ 2 - ↑n ^ 2) / (↑m ^ 2 + ↑n ^ 2) = ↑(m ^ 2 - n ^ 2) / ↑(m ^ 2 + n ^ 2)

Next tactic:
norm_cast
theorem isPrimitiveClassified_aux (hc : x.gcd y = 1) (hzpos : 0 < z) {m n : ℤ}
    (hm2n2 : 0 < m ^ 2 + n ^ 2) (hv2 : (x : ℚ) / z = 2 * m * n / ((m : ℚ) ^ 2 + (n : ℚ) ^ 2))
    (hw2 : (y : ℚ) / z = ((m : ℚ) ^ 2 - (n : ℚ) ^ 2) / ((m : ℚ) ^ 2 + (n : ℚ) ^ 2))
    (H : Int.gcd (m ^ 2 - n ^ 2) (m ^ 2 + n ^ 2) = 1) (co : Int.gcd m n = 1)
    (pp : m % 2 = 0 ∧ n % 2 = 1 ∨ m % 2 = 1 ∧ n % 2 = 0) : h.IsPrimitiveClassified := by
  have hz : z ≠ 0
  apply ne_of_gt hzpos
  have h2 : y = m ^ 2 - n ^ 2 ∧ z = m ^ 2 + n ^ 2 := by
    apply Rat.div_int_inj hzpos hm2n2 (h.coprime_of_coprime hc) H
    rw [hw2]
    norm_cast
  

Proof state:
x y z : ℤ
h : PythagoreanTriple x y z
hc : Int.gcd x y = 1
hzpos : 0 < z
m n : ℤ
hm2n2 : 0 < m ^ 2 + n ^ 2
hv2 : ↑x / ↑z = 2 * ↑m * ↑n / (↑m ^ 2 + ↑n ^ 2)
hw2 : ↑y / ↑z = (↑m ^ 2 - ↑n ^ 2) / (↑m ^ 2 + ↑n ^ 2)
H : Int.gcd (m ^ 2 - n ^ 2) (m ^ 2 + n ^ 2) = 1
co : Int.gcd m n = 1
pp : m % 2 = 0 ∧ n % 2 = 1 ∨ m % 2 = 1 ∧ n % 2 = 0
hz : z ≠ 0
h2 : y = m ^ 2 - n ^ 2 ∧ z = m ^ 2 + n ^ 2
⊢ IsPrimitiveClassified h

Next tactic:
use m, n
theorem isPrimitiveClassified_aux (hc : x.gcd y = 1) (hzpos : 0 < z) {m n : ℤ}
    (hm2n2 : 0 < m ^ 2 + n ^ 2) (hv2 : (x : ℚ) / z = 2 * m * n / ((m : ℚ) ^ 2 + (n : ℚ) ^ 2))
    (hw2 : (y : ℚ) / z = ((m : ℚ) ^ 2 - (n : ℚ) ^ 2) / ((m : ℚ) ^ 2 + (n : ℚ) ^ 2))
    (H : Int.gcd (m ^ 2 - n ^ 2) (m ^ 2 + n ^ 2) = 1) (co : Int.gcd m n = 1)
    (pp : m % 2 = 0 ∧ n % 2 = 1 ∨ m % 2 = 1 ∧ n % 2 = 0) : h.IsPrimitiveClassified := by
  have hz : z ≠ 0
  apply ne_of_gt hzpos
  have h2 : y = m ^ 2 - n ^ 2 ∧ z = m ^ 2 + n ^ 2 := by
    apply Rat.div_int_inj hzpos hm2n2 (h.coprime_of_coprime hc) H
    rw [hw2]
    norm_cast
  use m, n
  

Proof state:
case h
x y z : ℤ
h : PythagoreanTriple x y z
hc : Int.gcd x y = 1
hzpos : 0 < z
m n : ℤ
hm2n2 : 0 < m ^ 2 + n ^ 2
hv2 : ↑x / ↑z = 2 * ↑m * ↑n / (↑m ^ 2 + ↑n ^ 2)
hw2 : ↑y / ↑z = (↑m ^ 2 - ↑n ^ 2) / (↑m ^ 2 + ↑n ^ 2)
H : Int.gcd (m ^ 2 - n ^ 2) (m ^ 2 + n ^ 2) = 1
co : Int.gcd m n = 1
pp : m % 2 = 0 ∧ n % 2 = 1 ∨ m % 2 = 1 ∧ n % 2 = 0
hz : z ≠ 0
h2 : y = m ^ 2 - n ^ 2 ∧ z = m ^ 2 + n ^ 2
⊢ (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)

Next tactic:
apply And.intro _ (And.intro co pp)
theorem isPrimitiveClassified_aux (hc : x.gcd y = 1) (hzpos : 0 < z) {m n : ℤ}
    (hm2n2 : 0 < m ^ 2 + n ^ 2) (hv2 : (x : ℚ) / z = 2 * m * n / ((m : ℚ) ^ 2 + (n : ℚ) ^ 2))
    (hw2 : (y : ℚ) / z = ((m : ℚ) ^ 2 - (n : ℚ) ^ 2) / ((m : ℚ) ^ 2 + (n : ℚ) ^ 2))
    (H : Int.gcd (m ^ 2 - n ^ 2) (m ^ 2 + n ^ 2) = 1) (co : Int.gcd m n = 1)
    (pp : m % 2 = 0 ∧ n % 2 = 1 ∨ m % 2 = 1 ∧ n % 2 = 0) : h.IsPrimitiveClassified := by
  have hz : z ≠ 0
  apply ne_of_gt hzpos
  have h2 : y = m ^ 2 - n ^ 2 ∧ z = m ^ 2 + n ^ 2 := by
    apply Rat.div_int_inj hzpos hm2n2 (h.coprime_of_coprime hc) H
    rw [hw2]
    norm_cast
  use m, n
  apply And.intro _ (And.intro co pp)
  

Proof state:
x y z : ℤ
h : PythagoreanTriple x y z
hc : Int.gcd x y = 1
hzpos : 0 < z
m n : ℤ
hm2n2 : 0 < m ^ 2 + n ^ 2
hv2 : ↑x / ↑z = 2 * ↑m * ↑n / (↑m ^ 2 + ↑n ^ 2)
hw2 : ↑y / ↑z = (↑m ^ 2 - ↑n ^ 2) / (↑m ^ 2 + ↑n ^ 2)
H : Int.gcd (m ^ 2 - n ^ 2) (m ^ 2 + n ^ 2) = 1
co : Int.gcd m n = 1
pp : m % 2 = 0 ∧ n % 2 = 1 ∨ m % 2 = 1 ∧ n % 2 = 0
hz : z ≠ 0
h2 : y = m ^ 2 - n ^ 2 ∧ z = m ^ 2 + n ^ 2
⊢ x = m ^ 2 - n ^ 2 ∧ y = 2 * m * n ∨ x = 2 * m * n ∧ y = m ^ 2 - n ^ 2

Next tactic:
right
theorem isPrimitiveClassified_aux (hc : x.gcd y = 1) (hzpos : 0 < z) {m n : ℤ}
    (hm2n2 : 0 < m ^ 2 + n ^ 2) (hv2 : (x : ℚ) / z = 2 * m * n / ((m : ℚ) ^ 2 + (n : ℚ) ^ 2))
    (hw2 : (y : ℚ) / z = ((m : ℚ) ^ 2 - (n : ℚ) ^ 2) / ((m : ℚ) ^ 2 + (n : ℚ) ^ 2))
    (H : Int.gcd (m ^ 2 - n ^ 2) (m ^ 2 + n ^ 2) = 1) (co : Int.gcd m n = 1)
    (pp : m % 2 = 0 ∧ n % 2 = 1 ∨ m % 2 = 1 ∧ n % 2 = 0) : h.IsPrimitiveClassified := by
  have hz : z ≠ 0
  apply ne_of_gt hzpos
  have h2 : y = m ^ 2 - n ^ 2 ∧ z = m ^ 2 + n ^ 2 := by
    apply Rat.div_int_inj hzpos hm2n2 (h.coprime_of_coprime hc) H
    rw [hw2]
    norm_cast
  use m, n
  apply And.intro _ (And.intro co pp)
  right
  

Proof state:
case h
x y z : ℤ
h : PythagoreanTriple x y z
hc : Int.gcd x y = 1
hzpos : 0 < z
m n : ℤ
hm2n2 : 0 < m ^ 2 + n ^ 2
hv2 : ↑x / ↑z = 2 * ↑m * ↑n / (↑m ^ 2 + ↑n ^ 2)
hw2 : ↑y / ↑z = (↑m ^ 2 - ↑n ^ 2) / (↑m ^ 2 + ↑n ^ 2)
H : Int.gcd (m ^ 2 - n ^ 2) (m ^ 2 + n ^ 2) = 1
co : Int.gcd m n = 1
pp : m % 2 = 0 ∧ n % 2 = 1 ∨ m % 2 = 1 ∧ n % 2 = 0
hz : z ≠ 0
h2 : y = m ^ 2 - n ^ 2 ∧ z = m ^ 2 + n ^ 2
⊢ x = 2 * m * n ∧ y = m ^ 2 - n ^ 2

Next tactic:
refine' ⟨_, h2.left⟩
theorem isPrimitiveClassified_aux (hc : x.gcd y = 1) (hzpos : 0 < z) {m n : ℤ}
    (hm2n2 : 0 < m ^ 2 + n ^ 2) (hv2 : (x : ℚ) / z = 2 * m * n / ((m : ℚ) ^ 2 + (n : ℚ) ^ 2))
    (hw2 : (y : ℚ) / z = ((m : ℚ) ^ 2 - (n : ℚ) ^ 2) / ((m : ℚ) ^ 2 + (n : ℚ) ^ 2))
    (H : Int.gcd (m ^ 2 - n ^ 2) (m ^ 2 + n ^ 2) = 1) (co : Int.gcd m n = 1)
    (pp : m % 2 = 0 ∧ n % 2 = 1 ∨ m % 2 = 1 ∧ n % 2 = 0) : h.IsPrimitiveClassified := by
  have hz : z ≠ 0
  apply ne_of_gt hzpos
  have h2 : y = m ^ 2 - n ^ 2 ∧ z = m ^ 2 + n ^ 2 := by
    apply Rat.div_int_inj hzpos hm2n2 (h.coprime_of_coprime hc) H
    rw [hw2]
    norm_cast
  use m, n
  apply And.intro _ (And.intro co pp)
  right
  refine' ⟨_, h2.left⟩
  

Proof state:
case h
x y z : ℤ
h : PythagoreanTriple x y z
hc : Int.gcd x y = 1
hzpos : 0 < z
m n : ℤ
hm2n2 : 0 < m ^ 2 + n ^ 2
hv2 : ↑x / ↑z = 2 * ↑m * ↑n / (↑m ^ 2 + ↑n ^ 2)
hw2 : ↑y / ↑z = (↑m ^ 2 - ↑n ^ 2) / (↑m ^ 2 + ↑n ^ 2)
H : Int.gcd (m ^ 2 - n ^ 2) (m ^ 2 + n ^ 2) = 1
co : Int.gcd m n = 1
pp : m % 2 = 0 ∧ n % 2 = 1 ∨ m % 2 = 1 ∧ n % 2 = 0
hz : z ≠ 0
h2 : y = m ^ 2 - n ^ 2 ∧ z = m ^ 2 + n ^ 2
⊢ x = 2 * m * n

Next tactic:
rw [← Rat.coe_int_inj _ _, ← div_left_inj' ((mt (Rat.coe_int_inj z 0).mp) hz), hv2, h2.right]
theorem isPrimitiveClassified_aux (hc : x.gcd y = 1) (hzpos : 0 < z) {m n : ℤ}
    (hm2n2 : 0 < m ^ 2 + n ^ 2) (hv2 : (x : ℚ) / z = 2 * m * n / ((m : ℚ) ^ 2 + (n : ℚ) ^ 2))
    (hw2 : (y : ℚ) / z = ((m : ℚ) ^ 2 - (n : ℚ) ^ 2) / ((m : ℚ) ^ 2 + (n : ℚ) ^ 2))
    (H : Int.gcd (m ^ 2 - n ^ 2) (m ^ 2 + n ^ 2) = 1) (co : Int.gcd m n = 1)
    (pp : m % 2 = 0 ∧ n % 2 = 1 ∨ m % 2 = 1 ∧ n % 2 = 0) : h.IsPrimitiveClassified := by
  have hz : z ≠ 0
  apply ne_of_gt hzpos
  have h2 : y = m ^ 2 - n ^ 2 ∧ z = m ^ 2 + n ^ 2 := by
    apply Rat.div_int_inj hzpos hm2n2 (h.coprime_of_coprime hc) H
    rw [hw2]
    norm_cast
  use m, n
  apply And.intro _ (And.intro co pp)
  right
  refine' ⟨_, h2.left⟩
  rw [← Rat.coe_int_inj _ _, ← div_left_inj' ((mt (Rat.coe_int_inj z 0).mp) hz), hv2, h2.right]
  

Proof state:
case h
x y z : ℤ
h : PythagoreanTriple x y z
hc : Int.gcd x y = 1
hzpos : 0 < z
m n : ℤ
hm2n2 : 0 < m ^ 2 + n ^ 2
hv2 : ↑x / ↑z = 2 * ↑m * ↑n / (↑m ^ 2 + ↑n ^ 2)
hw2 : ↑y / ↑z = (↑m ^ 2 - ↑n ^ 2) / (↑m ^ 2 + ↑n ^ 2)
H : Int.gcd (m ^ 2 - n ^ 2) (m ^ 2 + n ^ 2) = 1
co : Int.gcd m n = 1
pp : m % 2 = 0 ∧ n % 2 = 1 ∨ m % 2 = 1 ∧ n % 2 = 0
hz : z ≠ 0
h2 : y = m ^ 2 - n ^ 2 ∧ z = m ^ 2 + n ^ 2
⊢ 2 * ↑m * ↑n / (↑m ^ 2 + ↑n ^ 2) = ↑(2 * m * n) / ↑(m ^ 2 + n ^ 2)

Next tactic:
norm_cast
theorem isPrimitiveClassified_of_coprime_of_odd_of_pos (hc : Int.gcd x y = 1) (hyo : y % 2 = 1)
    (hzpos : 0 < z) : h.IsPrimitiveClassified := by
  

Proof state:
x y z : ℤ
h : PythagoreanTriple x y z
hc : Int.gcd x y = 1
hyo : y % 2 = 1
hzpos : 0 < z
⊢ IsPrimitiveClassified h

Next tactic:
by_cases h0 : x = 0
theorem isPrimitiveClassified_of_coprime_of_odd_of_pos (hc : Int.gcd x y = 1) (hyo : y % 2 = 1)
    (hzpos : 0 < z) : h.IsPrimitiveClassified := by
  by_cases h0 : x = 0
  · 

Proof state:
case pos
x y z : ℤ
h : PythagoreanTriple x y z
hc : Int.gcd x y = 1
hyo : y % 2 = 1
hzpos : 0 < z
h0 : x = 0
⊢ IsPrimitiveClassified h

Next tactic:
exact h.isPrimitiveClassified_of_coprime_of_zero_left hc h0
theorem isPrimitiveClassified_of_coprime_of_odd_of_pos (hc : Int.gcd x y = 1) (hyo : y % 2 = 1)
    (hzpos : 0 < z) : h.IsPrimitiveClassified := by
  by_cases h0 : x = 0
  · exact h.isPrimitiveClassified_of_coprime_of_zero_left hc h0
  

Proof state:
case neg
x y z : ℤ
h : PythagoreanTriple x y z
hc : Int.gcd x y = 1
hyo : y % 2 = 1
hzpos : 0 < z
h0 : ¬x = 0
⊢ IsPrimitiveClassified h

Next tactic:
let v := (x : ℚ) / z
theorem isPrimitiveClassified_of_coprime_of_odd_of_pos (hc : Int.gcd x y = 1) (hyo : y % 2 = 1)
    (hzpos : 0 < z) : h.IsPrimitiveClassified := by
  by_cases h0 : x = 0
  · exact h.isPrimitiveClassified_of_coprime_of_zero_left hc h0
  let v := (x : ℚ) / z
  

Proof state:
case neg
x y z : ℤ
h : PythagoreanTriple x y z
hc : Int.gcd x y = 1
hyo : y % 2 = 1
hzpos : 0 < z
h0 : ¬x = 0
v : ℚ := ↑x / ↑z
⊢ IsPrimitiveClassified h

Next tactic:
let w := (y : ℚ) / z
theorem isPrimitiveClassified_of_coprime_of_odd_of_pos (hc : Int.gcd x y = 1) (hyo : y % 2 = 1)
    (hzpos : 0 < z) : h.IsPrimitiveClassified := by
  by_cases h0 : x = 0
  · exact h.isPrimitiveClassified_of_coprime_of_zero_left hc h0
  let v := (x : ℚ) / z
  let w := (y : ℚ) / z
  

Proof state:
case neg
x y z : ℤ
h : PythagoreanTriple x y z
hc : Int.gcd x y = 1
hyo : y % 2 = 1
hzpos : 0 < z
h0 : ¬x = 0
v : ℚ := ↑x / ↑z
w : ℚ := ↑y / ↑z
⊢ IsPrimitiveClassified h

Next tactic:
have hq : v ^ 2 + w ^ 2 = 1 := by
    field_simp [sq]
    norm_cast
theorem isPrimitiveClassified_of_coprime_of_odd_of_pos (hc : Int.gcd x y = 1) (hyo : y % 2 = 1)
    (hzpos : 0 < z) : h.IsPrimitiveClassified := by
  by_cases h0 : x = 0
  · exact h.isPrimitiveClassified_of_coprime_of_zero_left hc h0
  let v := (x : ℚ) / z
  let w := (y : ℚ) / z
  have hq : v ^ 2 + w ^ 2 = 1 := by
    

Proof state:
x y z : ℤ
h : PythagoreanTriple x y z
hc : Int.gcd x y = 1
hyo : y % 2 = 1
hzpos : 0 < z
h0 : ¬x = 0
v : ℚ := ↑x / ↑z
w : ℚ := ↑y / ↑z
⊢ v ^ 2 + w ^ 2 = 1

Next tactic:
field_simp [sq]
theorem isPrimitiveClassified_of_coprime_of_odd_of_pos (hc : Int.gcd x y = 1) (hyo : y % 2 = 1)
    (hzpos : 0 < z) : h.IsPrimitiveClassified := by
  by_cases h0 : x = 0
  · exact h.isPrimitiveClassified_of_coprime_of_zero_left hc h0
  let v := (x : ℚ) / z
  let w := (y : ℚ) / z
  have hq : v ^ 2 + w ^ 2 = 1 := by
    field_simp [sq]
    

Proof state:
x y z : ℤ
h : PythagoreanTriple x y z
hc : Int.gcd x y = 1
hyo : y % 2 = 1
hzpos : 0 < z
h0 : ¬x = 0
v : ℚ := ↑x / ↑z
w : ℚ := ↑y / ↑z
⊢ ↑x * ↑x + ↑y * ↑y = ↑z * ↑z

Next tactic:
norm_cast
theorem isPrimitiveClassified_of_coprime_of_odd_of_pos (hc : Int.gcd x y = 1) (hyo : y % 2 = 1)
    (hzpos : 0 < z) : h.IsPrimitiveClassified := by
  by_cases h0 : x = 0
  · exact h.isPrimitiveClassified_of_coprime_of_zero_left hc h0
  let v := (x : ℚ) / z
  let w := (y : ℚ) / z
  have hq : v ^ 2 + w ^ 2 = 1 := by
    field_simp [sq]
    norm_cast
  

Proof state:
case neg
x y z : ℤ
h : PythagoreanTriple x y z
hc : Int.gcd x y = 1
hyo : y % 2 = 1
hzpos : 0 < z
h0 : ¬x = 0
v : ℚ := ↑x / ↑z
w : ℚ := ↑y / ↑z
hq : v ^ 2 + w ^ 2 = 1
⊢ IsPrimitiveClassified h

Next tactic:
have hvz : v ≠ 0 := by
    field_simp
    exact h0
theorem isPrimitiveClassified_of_coprime_of_odd_of_pos (hc : Int.gcd x y = 1) (hyo : y % 2 = 1)
    (hzpos : 0 < z) : h.IsPrimitiveClassified := by
  by_cases h0 : x = 0
  · exact h.isPrimitiveClassified_of_coprime_of_zero_left hc h0
  let v := (x : ℚ) / z
  let w := (y : ℚ) / z
  have hq : v ^ 2 + w ^ 2 = 1 := by
    field_simp [sq]
    norm_cast
  have hvz : v ≠ 0 := by
    

Proof state:
x y z : ℤ
h : PythagoreanTriple x y z
hc : Int.gcd x y = 1
hyo : y % 2 = 1
hzpos : 0 < z
h0 : ¬x = 0
v : ℚ := ↑x / ↑z
w : ℚ := ↑y / ↑z
hq : v ^ 2 + w ^ 2 = 1
⊢ v ≠ 0

Next tactic:
field_simp
theorem isPrimitiveClassified_of_coprime_of_odd_of_pos (hc : Int.gcd x y = 1) (hyo : y % 2 = 1)
    (hzpos : 0 < z) : h.IsPrimitiveClassified := by
  by_cases h0 : x = 0
  · exact h.isPrimitiveClassified_of_coprime_of_zero_left hc h0
  let v := (x : ℚ) / z
  let w := (y : ℚ) / z
  have hq : v ^ 2 + w ^ 2 = 1 := by
    field_simp [sq]
    norm_cast
  have hvz : v ≠ 0 := by
    field_simp
    

Proof state:
x y z : ℤ
h : PythagoreanTriple x y z
hc : Int.gcd x y = 1
hyo : y % 2 = 1
hzpos : 0 < z
h0 : ¬x = 0
v : ℚ := ↑x / ↑z
w : ℚ := ↑y / ↑z
hq : v ^ 2 + w ^ 2 = 1
⊢ ¬x = 0

Next tactic:
exact h0