-- Mathlib/NumberTheory/PythagoreanTriples.lean
theorem coprime_classification' {x y z : ℤ} (h : PythagoreanTriple x y z)
    (h_coprime : Int.gcd x y = 1) (h_parity : x % 2 = 1) (h_pos : 0 < z) :
    ∃ m n,
      x = m ^ 2 - n ^ 2 ∧
        y = 2 * m * n ∧
          z = m ^ 2 + n ^ 2 ∧
            Int.gcd m n = 1 ∧ (m % 2 = 0 ∧ n % 2 = 1 ∨ m % 2 = 1 ∧ n % 2 = 0) ∧ 0 ≤ m

Proof state:
case h.inl.right.inl.left
x y z : ℤ
h : PythagoreanTriple x y z
h_coprime : x.gcd y = 1
h_parity : x % 2 = 1
h_pos✝ : 0 < z
m n : ℤ
ht3 : m.gcd n = 1
ht4 : m % 2 = 0 ∧ n % 2 = 1 ∨ m % 2 = 1 ∧ n % 2 = 0
hm : m < 0
h_odd : x = m ^ 2 - n ^ 2 ∧ y = 2 * m * n
h_pos : z = m ^ 2 + n ^ 2
⊢ (-m).natAbs.gcd (-n).natAbs = 1

Tactic:
rw [Int.natAbs_neg, Int.natAbs_neg]

Result:
case h.inl.right.inl.left
x y z : ℤ
h : PythagoreanTriple x y z
h_coprime : x.gcd y = 1
h_parity : x % 2 = 1
h_pos✝ : 0 < z
m n : ℤ
ht3 : m.gcd n = 1
ht4 : m % 2 = 0 ∧ n % 2 = 1 ∨ m % 2 = 1 ∧ n % 2 = 0
hm : m < 0
h_odd : x = m ^ 2 - n ^ 2 ∧ y = 2 * m * n
h_pos : z = m ^ 2 + n ^ 2
⊢ m.natAbs.gcd n.natAbs = 1

Proof state:
x y z : ℤ
h : PythagoreanTriple x y z
h_coprime : x.gcd y = 1
h_parity : x % 2 = 1
h_pos : 0 < z
⊢ ∃ m n,
    x = m ^ 2 - n ^ 2 ∧
      y = 2 * m * n ∧ z = m ^ 2 + n ^ 2 ∧ m.gcd n = 1 ∧ (m % 2 = 0 ∧ n % 2 = 1 ∨ m % 2 = 1 ∧ n % 2 = 0) ∧ 0 ≤ m

Tactic:
obtain ⟨m, n, ht1, ht2, ht3, ht4⟩ :=
  PythagoreanTriple.coprime_classification.mp (And.intro h h_coprime)

Result:
case intro.intro.intro.intro.intro
x y z : ℤ
h : PythagoreanTriple x y z
h_coprime : x.gcd y = 1
h_parity : x % 2 = 1
h_pos : 0 < z
m n : ℤ
ht1 : x = m ^ 2 - n ^ 2 ∧ y = 2 * m * n ∨ x = 2 * m * n ∧ y = m ^ 2 - n ^ 2
ht2 : z = m ^ 2 + n ^ 2 ∨ z = -(m ^ 2 + n ^ 2)
ht3 : m.gcd n = 1
ht4 : m % 2 = 0 ∧ n % 2 = 1 ∨ m % 2 = 1 ∧ n % 2 = 0
⊢ ∃ m n,
    x = m ^ 2 - n ^ 2 ∧
      y = 2 * m * n ∧ z = m ^ 2 + n ^ 2 ∧ m.gcd n = 1 ∧ (m % 2 = 0 ∧ n % 2 = 1 ∨ m % 2 = 1 ∧ n % 2 = 0) ∧ 0 ≤ m

Proof state:
case h.inl.inr
x y z : ℤ
h : PythagoreanTriple x y z
h_coprime : x.gcd y = 1
h_parity : x % 2 = 1
h_pos : 0 < z
m n : ℤ
ht3 : m.gcd n = 1
ht4 : m % 2 = 0 ∧ n % 2 = 1 ∨ m % 2 = 1 ∧ n % 2 = 0
hm : 0 ≤ m
h_odd : x = m ^ 2 - n ^ 2 ∧ y = 2 * m * n
h_neg : z = -(m ^ 2 + n ^ 2)
⊢ z = m ^ 2 + n ^ 2 ∧ m.gcd n = 1 ∧ (m % 2 = 0 ∧ n % 2 = 1 ∨ m % 2 = 1 ∧ n % 2 = 0) ∧ 0 ≤ m

Tactic:
exfalso

Result:
case h.inl.inr
x y z : ℤ
h : PythagoreanTriple x y z
h_coprime : x.gcd y = 1
h_parity : x % 2 = 1
h_pos : 0 < z
m n : ℤ
ht3 : m.gcd n = 1
ht4 : m % 2 = 0 ∧ n % 2 = 1 ∨ m % 2 = 1 ∧ n % 2 = 0
hm : 0 ≤ m
h_odd : x = m ^ 2 - n ^ 2 ∧ y = 2 * m * n
h_neg : z = -(m ^ 2 + n ^ 2)
⊢ False

Proof state:
case h.inl.left
x y z : ℤ
h : PythagoreanTriple x y z
h_coprime : x.gcd y = 1
h_parity : x % 2 = 1
h_pos : 0 < z
m n : ℤ
ht2 : z = m ^ 2 + n ^ 2 ∨ z = -(m ^ 2 + n ^ 2)
ht3 : m.gcd n = 1
ht4 : m % 2 = 0 ∧ n % 2 = 1 ∨ m % 2 = 1 ∧ n % 2 = 0
hm : m < 0
h_odd : x = m ^ 2 - n ^ 2 ∧ y = 2 * m * n
⊢ y = 2 * -m * -n

Tactic:
rw [h_odd.2]

Result:
case h.inl.left
x y z : ℤ
h : PythagoreanTriple x y z
h_coprime : x.gcd y = 1
h_parity : x % 2 = 1
h_pos : 0 < z
m n : ℤ
ht2 : z = m ^ 2 + n ^ 2 ∨ z = -(m ^ 2 + n ^ 2)
ht3 : m.gcd n = 1
ht4 : m % 2 = 0 ∧ n % 2 = 1 ∨ m % 2 = 1 ∧ n % 2 = 0
hm : m < 0
h_odd : x = m ^ 2 - n ^ 2 ∧ y = 2 * m * n
⊢ 2 * m * n = 2 * -m * -n

Proof state:
case h.inr.intro
y z : ℤ
h_pos : 0 < z
m n : ℤ
ht2 : z = m ^ 2 + n ^ 2 ∨ z = -(m ^ 2 + n ^ 2)
ht3 : m.gcd n = 1
ht4 : m % 2 = 0 ∧ n % 2 = 1 ∨ m % 2 = 1 ∧ n % 2 = 0
hm : m < 0
h : PythagoreanTriple (2 * m * n) y z
h_coprime : (2 * m * n).gcd y = 1
h_parity : 0 = 1
⊢ False

Tactic:
exact zero_ne_one h_parity

Result:
no goals

Proof state:
case h.inl
x y z : ℤ
h : PythagoreanTriple x y z
h_coprime : x.gcd y = 1
h_parity : x % 2 = 1
h_pos : 0 < z
m n : ℤ
ht2 : z = m ^ 2 + n ^ 2 ∨ z = -(m ^ 2 + n ^ 2)
ht3 : m.gcd n = 1
ht4 : m % 2 = 0 ∧ n % 2 = 1 ∨ m % 2 = 1 ∧ n % 2 = 0
hm : m < 0
h_odd : x = m ^ 2 - n ^ 2 ∧ y = 2 * m * n
⊢ x = (-m) ^ 2 - (-n) ^ 2 ∧
    y = 2 * -m * -n ∧
      z = (-m) ^ 2 + (-n) ^ 2 ∧ (-m).gcd (-n) = 1 ∧ (-m % 2 = 0 ∧ -n % 2 = 1 ∨ -m % 2 = 1 ∧ -n % 2 = 0) ∧ 0 ≤ -m

Tactic:
rw [neg_sq m]

Result:
case h.inl
x y z : ℤ
h : PythagoreanTriple x y z
h_coprime : x.gcd y = 1
h_parity : x % 2 = 1
h_pos : 0 < z
m n : ℤ
ht2 : z = m ^ 2 + n ^ 2 ∨ z = -(m ^ 2 + n ^ 2)
ht3 : m.gcd n = 1
ht4 : m % 2 = 0 ∧ n % 2 = 1 ∨ m % 2 = 1 ∧ n % 2 = 0
hm : m < 0
h_odd : x = m ^ 2 - n ^ 2 ∧ y = 2 * m * n
⊢ x = m ^ 2 - (-n) ^ 2 ∧
    y = 2 * -m * -n ∧
      z = m ^ 2 + (-n) ^ 2 ∧ (-m).gcd (-n) = 1 ∧ (-m % 2 = 0 ∧ -n % 2 = 1 ∨ -m % 2 = 1 ∧ -n % 2 = 0) ∧ 0 ≤ -m

Proof state:
case h.inr.intro
y z : ℤ
h_pos : 0 < z
m n : ℤ
ht2 : z = m ^ 2 + n ^ 2 ∨ z = -(m ^ 2 + n ^ 2)
ht3 : m.gcd n = 1
ht4 : m % 2 = 0 ∧ n % 2 = 1 ∨ m % 2 = 1 ∧ n % 2 = 0
hm : m < 0
h : PythagoreanTriple (2 * m * n) y z
h_coprime : (2 * m * n).gcd y = 1
h_parity : 2 * m * n % 2 = 1
⊢ False

Tactic:
rw [mul_assoc, Int.mul_emod_right] at h_parity

Result:
case h.inr.intro
y z : ℤ
h_pos : 0 < z
m n : ℤ
ht2 : z = m ^ 2 + n ^ 2 ∨ z = -(m ^ 2 + n ^ 2)
ht3 : m.gcd n = 1
ht4 : m % 2 = 0 ∧ n % 2 = 1 ∨ m % 2 = 1 ∧ n % 2 = 0
hm : m < 0
h : PythagoreanTriple (2 * m * n) y z
h_coprime : (2 * m * n).gcd y = 1
h_parity : 0 = 1
⊢ False

Proof state:
case h.inl.right.inl.left
x y z : ℤ
h : PythagoreanTriple x y z
h_coprime : x.gcd y = 1
h_parity : x % 2 = 1
h_pos✝ : 0 < z
m n : ℤ
ht3 : m.gcd n = 1
ht4 : m % 2 = 0 ∧ n % 2 = 1 ∨ m % 2 = 1 ∧ n % 2 = 0
hm : m < 0
h_odd : x = m ^ 2 - n ^ 2 ∧ y = 2 * m * n
h_pos : z = m ^ 2 + n ^ 2
⊢ m.natAbs.gcd n.natAbs = 1

Tactic:
exact ht3

Result:
no goals

Proof state:
case h.inr
x y z : ℤ
h : PythagoreanTriple x y z
h_coprime : x.gcd y = 1
h_parity : x % 2 = 1
h_pos : 0 < z
m n : ℤ
ht2 : z = m ^ 2 + n ^ 2 ∨ z = -(m ^ 2 + n ^ 2)
ht3 : m.gcd n = 1
ht4 : m % 2 = 0 ∧ n % 2 = 1 ∨ m % 2 = 1 ∧ n % 2 = 0
hm : 0 ≤ m
h_even : x = 2 * m * n ∧ y = m ^ 2 - n ^ 2
⊢ False

Tactic:
rcases h_even with ⟨rfl, -⟩

Result:
case h.inr.intro
y z : ℤ
h_pos : 0 < z
m n : ℤ
ht2 : z = m ^ 2 + n ^ 2 ∨ z = -(m ^ 2 + n ^ 2)
ht3 : m.gcd n = 1
ht4 : m % 2 = 0 ∧ n % 2 = 1 ∨ m % 2 = 1 ∧ n % 2 = 0
hm : 0 ≤ m
h : PythagoreanTriple (2 * m * n) y z
h_coprime : (2 * m * n).gcd y = 1
h_parity : 2 * m * n % 2 = 1
⊢ False

Proof state:
case h.inl.inr
x y z : ℤ
h : PythagoreanTriple x y z
h_coprime : x.gcd y = 1
h_parity : x % 2 = 1
m n : ℤ
ht3 : m.gcd n = 1
ht4 : m % 2 = 0 ∧ n % 2 = 1 ∨ m % 2 = 1 ∧ n % 2 = 0
hm : 0 ≤ m
h_odd : x = m ^ 2 - n ^ 2 ∧ y = 2 * m * n
h_neg : z = -(m ^ 2 + n ^ 2)
⊢ 0 < z → False

Tactic:
rw [h_neg]

Result:
case h.inl.inr
x y z : ℤ
h : PythagoreanTriple x y z
h_coprime : x.gcd y = 1
h_parity : x % 2 = 1
m n : ℤ
ht3 : m.gcd n = 1
ht4 : m % 2 = 0 ∧ n % 2 = 1 ∨ m % 2 = 1 ∧ n % 2 = 0
hm : 0 ≤ m
h_odd : x = m ^ 2 - n ^ 2 ∧ y = 2 * m * n
h_neg : z = -(m ^ 2 + n ^ 2)
⊢ 0 < -(m ^ 2 + n ^ 2) → False

Proof state:
case h.inl.inr
x y z : ℤ
h : PythagoreanTriple x y z
h_coprime : x.gcd y = 1
h_parity : x % 2 = 1
h_pos : 0 < z
m n : ℤ
ht3 : m.gcd n = 1
ht4 : m % 2 = 0 ∧ n % 2 = 1 ∨ m % 2 = 1 ∧ n % 2 = 0
hm : 0 ≤ m
h_odd : x = m ^ 2 - n ^ 2 ∧ y = 2 * m * n
h_neg : z = -(m ^ 2 + n ^ 2)
⊢ False

Tactic:
revert h_pos

Result:
case h.inl.inr
x y z : ℤ
h : PythagoreanTriple x y z
h_coprime : x.gcd y = 1
h_parity : x % 2 = 1
m n : ℤ
ht3 : m.gcd n = 1
ht4 : m % 2 = 0 ∧ n % 2 = 1 ∨ m % 2 = 1 ∧ n % 2 = 0
hm : 0 ≤ m
h_odd : x = m ^ 2 - n ^ 2 ∧ y = 2 * m * n
h_neg : z = -(m ^ 2 + n ^ 2)
⊢ 0 < z → False

Proof state:
case h.inl.left
x y z : ℤ
h : PythagoreanTriple x y z
h_coprime : x.gcd y = 1
h_parity : x % 2 = 1
h_pos : 0 < z
m n : ℤ
ht2 : z = m ^ 2 + n ^ 2 ∨ z = -(m ^ 2 + n ^ 2)
ht3 : m.gcd n = 1
ht4 : m % 2 = 0 ∧ n % 2 = 1 ∨ m % 2 = 1 ∧ n % 2 = 0
hm : m < 0
h_odd : x = m ^ 2 - n ^ 2 ∧ y = 2 * m * n
⊢ 2 * m * n = 2 * -m * -n

Tactic:
ring

Result:
no goals