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
    exact h0
  have hw1 : w ≠ -1 := by
    contrapose! hvz with hw1
    -- porting note: `contrapose` unfolds local names, refold them
    replace hw1 : w = -1 := hw1; show v = 0
    rw [hw1, neg_sq, one_pow, add_left_eq_self] at hq
    exact pow_eq_zero hq
  have hQ : ∀ x : ℚ, 1 + x ^ 2 ≠ 0 := by
    intro q
    apply ne_of_gt
    exact lt_add_of_pos_of_le zero_lt_one (sq_nonneg q)
  have hp : (⟨v, w⟩ : ℚ × ℚ) ∈ { p : ℚ × ℚ | p.1 ^ 2 + p.2 ^ 2 = 1 ∧ p.2 ≠ -1 } := ⟨hq, hw1⟩
  let q := (circleEquivGen hQ).symm ⟨⟨v, w⟩, hp⟩
  have ht4 : v = 2 * q / (1 + q ^ 2) ∧ w = (1 - q ^ 2) / (1 + q ^ 2) := by
    apply Prod.mk.inj
    have := ((circleEquivGen hQ).apply_symm_apply ⟨⟨v, w⟩, hp⟩).symm
    exact congr_arg Subtype.val this
  let m := (q.den : ℤ)
  let n := q.num
  have hm0 : m ≠ 0 := by
    -- Added to adapt to leanprover/lean4#2734.
    -- Without `unfold_let`, `norm_cast` can't see the coercion.
    -- One might try `zeta := true` in `Tactic.NormCast.derive`,
    -- but that seems to break many other things.
    unfold_let m
    norm_cast
    apply Rat.den_nz q
  have hq2 : q = n / m := (Rat.num_div_den q).symm
  have hm2n2 : 0 < m ^ 2 + n ^ 2 := by
    apply lt_add_of_pos_of_le _ (sq_nonneg n)
    exact lt_of_le_of_ne (sq_nonneg m) (Ne.symm (pow_ne_zero 2 hm0))
  have hm2n20 : (m : ℚ) ^ 2 + (n : ℚ) ^ 2 ≠ 0 := by
    norm_cast
    simpa only [Int.coe_nat_pow] using ne_of_gt hm2n2
  have hx1 {j k : ℚ} (h₁ : k ≠ 0) (h₂ : k ^ 2 + j ^ 2 ≠ 0) :
      (1 - (j / k) ^ 2) / (1 + (j / k) ^ 2) = (k ^ 2 - j ^ 2) / (k ^ 2 + j ^ 2) :=
    by field_simp
  have hw2 : w = ((m : ℚ) ^ 2 - (n : ℚ) ^ 2) / ((m : ℚ) ^ 2 + (n : ℚ) ^ 2) := by
    calc
      w = (1 - q ^ 2) / (1 + q ^ 2) := by apply ht4.2
      _ = (1 - (↑n / ↑m) ^ 2) / (1 + (↑n / ↑m) ^ 2) := by rw [hq2]
      _ = _ := by exact hx1 (Int.cast_ne_zero.mpr hm0) hm2n20
  have hx2 {j k : ℚ} (h₁ : k ≠ 0) (h₂ : k ^ 2 + j ^ 2 ≠ 0) :
      2 * (j / k) / (1 + (j / k) ^ 2) = 2 * k * j / (k ^ 2 + j ^ 2) :=
    have h₃ : k * (k ^ 2 + j ^ 2) ≠ 0 := mul_ne_zero h₁ h₂
    by field_simp; ring
  have hv2 : v = 2 * m * n / ((m : ℚ) ^ 2 + (n : ℚ) ^ 2) := by
    calc
      v = 2 * q / (1 + q ^ 2) := by apply ht4.1
      _ = 2 * (n / m) / (1 + (↑n / ↑m) ^ 2) := by rw [hq2]
      _ = _ := by exact hx2 (Int.cast_ne_zero.mpr hm0) hm2n20
  have hnmcp : Int.gcd n m = 1 := q.reduced
  have hmncp : Int.gcd m n = 1 := by
    rw [Int.gcd_comm]
    exact hnmcp
  cases' Int.emod_two_eq_zero_or_one m with hm2 hm2 <;>
    cases' Int.emod_two_eq_zero_or_one n with hn2 hn2
  · -- m even, n even
    exfalso
    have h1 : 2 ∣ (Int.gcd n m : ℤ) :=
      Int.dvd_gcd (Int.dvd_of_emod_eq_zero hn2) (Int.dvd_of_emod_eq_zero hm2)
    rw [hnmcp] at h1
    revert h1
    decide
  · -- m even, n odd
    apply h.isPrimitiveClassified_aux hc hzpos hm2n2 hv2 hw2 _ hmncp
    · apply Or.intro_left
      exact And.intro hm2 hn2
    · apply coprime_sq_sub_sq_add_of_even_odd hmncp hm2 hn2
  · -- m odd, n even
    apply h.isPrimitiveClassified_aux hc hzpos hm2n2 hv2 hw2 _ hmncp
    · apply Or.intro_right
      exact And.intro hm2 hn2
    

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
hvz : v ≠ 0
hw1 : w ≠ -1
hQ : ∀ (x : ℚ), 1 + x ^ 2 ≠ 0
hp : (v, w) ∈ {p | p.1 ^ 2 + p.2 ^ 2 = 1 ∧ p.2 ≠ -1}
q : ℚ := (circleEquivGen hQ).symm { val := (v, w), property := hp }
ht4 : v = 2 * q / (1 + q ^ 2) ∧ w = (1 - q ^ 2) / (1 + q ^ 2)
m : ℤ := ↑q.den
n : ℤ := q.num
hm0 : m ≠ 0
hq2 : q = ↑n / ↑m
hm2n2 : 0 < m ^ 2 + n ^ 2
hm2n20 : ↑m ^ 2 + ↑n ^ 2 ≠ 0
hx1 : ∀ {j k : ℚ}, k ≠ 0 → k ^ 2 + j ^ 2 ≠ 0 → (1 - (j / k) ^ 2) / (1 + (j / k) ^ 2) = (k ^ 2 - j ^ 2) / (k ^ 2 + j ^ 2)
hw2 : w = (↑m ^ 2 - ↑n ^ 2) / (↑m ^ 2 + ↑n ^ 2)
hx2 : ∀ {j k : ℚ}, k ≠ 0 → k ^ 2 + j ^ 2 ≠ 0 → 2 * (j / k) / (1 + (j / k) ^ 2) = 2 * k * j / (k ^ 2 + j ^ 2)
hv2 : v = 2 * ↑m * ↑n / (↑m ^ 2 + ↑n ^ 2)
hnmcp : Int.gcd n m = 1
hmncp : Int.gcd m n = 1
hm2 : m % 2 = 1
hn2 : n % 2 = 0
⊢ Int.gcd (m ^ 2 - n ^ 2) (m ^ 2 + n ^ 2) = 1

Next tactic:
apply coprime_sq_sub_sq_add_of_odd_even hmncp hm2 hn2