-- Mathlib/NumberTheory/PythagoreanTriples.lean
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

Proof state:
case intro.intro
m n : ℤ
h : m.gcd n = 1
hm : m % 2 = 1
hn : n % 2 = 1
m0 : ℤ
hm2 : m - 1 = m0 * 2
n0 : ℤ
hn2 : n - 1 = n0 * 2
⊢ 2 ∣ m ^ 2 + n ^ 2 ∧
    2 ∣ m ^ 2 - n ^ 2 ∧ (m ^ 2 - n ^ 2) / 2 % 2 = 0 ∧ ((m ^ 2 - n ^ 2) / 2).gcd ((m ^ 2 + n ^ 2) / 2) = 1

Tactic:
rw [sub_eq_iff_eq_add] at hm2 hn2

Result:
case intro.intro
m n : ℤ
h : m.gcd n = 1
hm : m % 2 = 1
hn : n % 2 = 1
m0 : ℤ
hm2 : m = m0 * 2 + 1
n0 : ℤ
hn2 : n = n0 * 2 + 1
⊢ 2 ∣ m ^ 2 + n ^ 2 ∧
    2 ∣ m ^ 2 - n ^ 2 ∧ (m ^ 2 - n ^ 2) / 2 % 2 = 0 ∧ ((m ^ 2 - n ^ 2) / 2).gcd ((m ^ 2 + n ^ 2) / 2) = 1

Proof state:
case intro.intro
m0 n0 : ℤ
hm : (m0 * 2 + 1) % 2 = 1
hn : (n0 * 2 + 1) % 2 = 1
h : (m0 * 2 + 1).gcd (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
⊢ (((m0 * 2 + 1) ^ 2 - (n0 * 2 + 1) ^ 2) / 2).gcd (((m0 * 2 + 1) ^ 2 + (n0 * 2 + 1) ^ 2) / 2) = 1

Tactic:
have h20 : (2 : ℤ) ≠ 0 := by decide

Result:
case intro.intro
m0 n0 : ℤ
hm : (m0 * 2 + 1) % 2 = 1
hn : (n0 * 2 + 1) % 2 = 1
h : (m0 * 2 + 1).gcd (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
⊢ (((m0 * 2 + 1) ^ 2 - (n0 * 2 + 1) ^ 2) / 2).gcd (((m0 * 2 + 1) ^ 2 + (n0 * 2 + 1) ^ 2) / 2) = 1

Proof state:
case intro.intro
m0 n0 : ℤ
hm : (m0 * 2 + 1) % 2 = 1
hn : (n0 * 2 + 1) % 2 = 1
h : (m0 * 2 + 1).gcd (n0 * 2 + 1) = 1
h1 : (m0 * 2 + 1) ^ 2 + (n0 * 2 + 1) ^ 2 = 2 * (2 * (m0 ^ 2 + n0 ^ 2 + m0 + n0) + 1)
⊢ 2 ∣ (m0 * 2 + 1) ^ 2 + (n0 * 2 + 1) ^ 2 ∧
    2 ∣ (m0 * 2 + 1) ^ 2 - (n0 * 2 + 1) ^ 2 ∧
      ((m0 * 2 + 1) ^ 2 - (n0 * 2 + 1) ^ 2) / 2 % 2 = 0 ∧
        (((m0 * 2 + 1) ^ 2 - (n0 * 2 + 1) ^ 2) / 2).gcd (((m0 * 2 + 1) ^ 2 + (n0 * 2 + 1) ^ 2) / 2) = 1

Tactic:
have h2 : (m0 * 2 + 1) ^ 2 - (n0 * 2 + 1) ^ 2 = 2 * (2 * (m0 ^ 2 - n0 ^ 2 + m0 - n0)) := by ring

Result:
case intro.intro
m0 n0 : ℤ
hm : (m0 * 2 + 1) % 2 = 1
hn : (n0 * 2 + 1) % 2 = 1
h : (m0 * 2 + 1).gcd (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))
⊢ 2 ∣ (m0 * 2 + 1) ^ 2 + (n0 * 2 + 1) ^ 2 ∧
    2 ∣ (m0 * 2 + 1) ^ 2 - (n0 * 2 + 1) ^ 2 ∧
      ((m0 * 2 + 1) ^ 2 - (n0 * 2 + 1) ^ 2) / 2 % 2 = 0 ∧
        (((m0 * 2 + 1) ^ 2 - (n0 * 2 + 1) ^ 2) / 2).gcd (((m0 * 2 + 1) ^ 2 + (n0 * 2 + 1) ^ 2) / 2) = 1

Proof state:
case intro.intro.intro.intro.intro
m0 n0 : ℤ
hm : (m0 * 2 + 1) % 2 = 1
hn : (n0 * 2 + 1) % 2 = 1
h : (m0 * 2 + 1).gcd (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 : ¬(2 * (m0 ^ 2 - n0 ^ 2 + m0 - n0)).gcd (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 ∣ (m0 * 2 + 1).gcd (n0 * 2 + 1)

Tactic:
apply Nat.dvd_gcd

Result:
case intro.intro.intro.intro.intro.a
m0 n0 : ℤ
hm : (m0 * 2 + 1) % 2 = 1
hn : (n0 * 2 + 1) % 2 = 1
h : (m0 * 2 + 1).gcd (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 : ¬(2 * (m0 ^ 2 - n0 ^ 2 + m0 - n0)).gcd (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 ∣ (m0 * 2 + 1).natAbs

case intro.intro.intro.intro.intro.a
m0 n0 : ℤ
hm : (m0 * 2 + 1) % 2 = 1
hn : (n0 * 2 + 1) % 2 = 1
h : (m0 * 2 + 1).gcd (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 : ¬(2 * (m0 ^ 2 - n0 ^ 2 + m0 - n0)).gcd (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).natAbs

Proof state:
m0 n0 : ℤ
hm : (m0 * 2 + 1) % 2 = 1
hn : (n0 * 2 + 1) % 2 = 1
h : (m0 * 2 + 1).gcd (n0 * 2 + 1) = 1
h1 : (m0 * 2 + 1) ^ 2 + (n0 * 2 + 1) ^ 2 = 2 * (2 * (m0 ^ 2 + n0 ^ 2 + m0 + n0) + 1)
⊢ (m0 * 2 + 1) ^ 2 - (n0 * 2 + 1) ^ 2 = 2 * (2 * (m0 ^ 2 - n0 ^ 2 + m0 - n0))

Tactic:
ring

Result:
no goals

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 : (m0 * 2 + 1).gcd (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 : ¬(2 * (m0 ^ 2 - n0 ^ 2 + m0 - n0)).gcd (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 ∣ (m0 * 2 + 1) ^ 2

Tactic:
convert dvd_add hp1 hp2

Result:
case h.e'_4
m0 n0 : ℤ
hm : (m0 * 2 + 1) % 2 = 1
hn : (n0 * 2 + 1) % 2 = 1
h : (m0 * 2 + 1).gcd (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 : ¬(2 * (m0 ^ 2 - n0 ^ 2 + m0 - n0)).gcd (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
⊢ (m0 * 2 + 1) ^ 2 = 2 * (m0 ^ 2 - n0 ^ 2 + m0 - n0) + (2 * (m0 ^ 2 + n0 ^ 2 + m0 + n0) + 1)

Proof state:
case intro.intro
n : ℤ
hn : n % 2 = 1
m0 n0 : ℤ
hn2 : n = n0 * 2 + 1
h : (m0 * 2 + 1).gcd n = 1
hm : (m0 * 2 + 1) % 2 = 1
⊢ 2 ∣ (m0 * 2 + 1) ^ 2 + n ^ 2 ∧
    2 ∣ (m0 * 2 + 1) ^ 2 - n ^ 2 ∧
      ((m0 * 2 + 1) ^ 2 - n ^ 2) / 2 % 2 = 0 ∧ (((m0 * 2 + 1) ^ 2 - n ^ 2) / 2).gcd (((m0 * 2 + 1) ^ 2 + n ^ 2) / 2) = 1

Tactic:
subst n

Result:
case intro.intro
m0 n0 : ℤ
hm : (m0 * 2 + 1) % 2 = 1
hn : (n0 * 2 + 1) % 2 = 1
h : (m0 * 2 + 1).gcd (n0 * 2 + 1) = 1
⊢ 2 ∣ (m0 * 2 + 1) ^ 2 + (n0 * 2 + 1) ^ 2 ∧
    2 ∣ (m0 * 2 + 1) ^ 2 - (n0 * 2 + 1) ^ 2 ∧
      ((m0 * 2 + 1) ^ 2 - (n0 * 2 + 1) ^ 2) / 2 % 2 = 0 ∧
        (((m0 * 2 + 1) ^ 2 - (n0 * 2 + 1) ^ 2) / 2).gcd (((m0 * 2 + 1) ^ 2 + (n0 * 2 + 1) ^ 2) / 2) = 1