l3lab/ntp-mathlib huggingface dataset:
/-- by assuming `x` is odd and `z` is positive we get a slightly more precise classification of
the Pythagorean triple `x ^ 2 + y ^ 2 = z ^ 2`-/
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 := by
  obtain ⟨m, n, ht1, ht2, ht3, ht4⟩ :=
    PythagoreanTriple.coprime_classification.mp (And.intro h h_coprime)
  rcases le_or_lt 0 m with hm | hm
  · use m, n
    cases' ht1 with h_odd h_even
    · apply And.intro h_odd.1
      apply And.intro h_odd.2
      cases' ht2 with h_pos h_neg
      · apply And.intro h_pos (And.intro ht3 (And.intro ht4 hm))
      · exfalso
        revert h_pos
        rw [h_neg]
        exact imp_false.mpr (not_lt.mpr (neg_nonpos.mpr (add_nonneg (sq_nonneg m) (sq_nonneg n))))
    exfalso
    rcases h_even with ⟨rfl, -⟩
    rw [mul_assoc, Int.mul_emod_right] at h_parity
    exact zero_ne_one h_parity
  · use -m, -n
    cases' ht1 with h_odd h_even
    · rw [neg_sq m]
      rw [neg_sq n]
      apply And.intro h_odd.1
      constructor
      · rw [h_odd.2]
        ring
      cases' ht2 with h_pos h_neg
      · apply And.intro h_pos
        constructor
        · delta Int.gcd
          rw [Int.natAbs_neg, Int.natAbs_neg]
          exact ht3
        · rw [Int.neg_emod_two, Int.neg_emod_two]
          apply And.intro ht4
          linarith
      · exfalso
        revert h_pos
        rw [h_neg]
        exact imp_false.mpr (not_lt.mpr (neg_nonpos.mpr (add_nonneg (sq_nonneg m) (sq_nonneg n))))
    exfalso
    

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

Next tactic:
rcases h_even with ⟨rfl, -⟩
/-- by assuming `x` is odd and `z` is positive we get a slightly more precise classification of
the Pythagorean triple `x ^ 2 + y ^ 2 = z ^ 2`-/
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 := by
  obtain ⟨m, n, ht1, ht2, ht3, ht4⟩ :=
    PythagoreanTriple.coprime_classification.mp (And.intro h h_coprime)
  rcases le_or_lt 0 m with hm | hm
  · use m, n
    cases' ht1 with h_odd h_even
    · apply And.intro h_odd.1
      apply And.intro h_odd.2
      cases' ht2 with h_pos h_neg
      · apply And.intro h_pos (And.intro ht3 (And.intro ht4 hm))
      · exfalso
        revert h_pos
        rw [h_neg]
        exact imp_false.mpr (not_lt.mpr (neg_nonpos.mpr (add_nonneg (sq_nonneg m) (sq_nonneg n))))
    exfalso
    rcases h_even with ⟨rfl, -⟩
    rw [mul_assoc, Int.mul_emod_right] at h_parity
    exact zero_ne_one h_parity
  · use -m, -n
    cases' ht1 with h_odd h_even
    · rw [neg_sq m]
      rw [neg_sq n]
      apply And.intro h_odd.1
      constructor
      · rw [h_odd.2]
        ring
      cases' ht2 with h_pos h_neg
      · apply And.intro h_pos
        constructor
        · delta Int.gcd
          rw [Int.natAbs_neg, Int.natAbs_neg]
          exact ht3
        · rw [Int.neg_emod_two, Int.neg_emod_two]
          apply And.intro ht4
          linarith
      · exfalso
        revert h_pos
        rw [h_neg]
        exact imp_false.mpr (not_lt.mpr (neg_nonpos.mpr (add_nonneg (sq_nonneg m) (sq_nonneg n))))
    exfalso
    rcases h_even with ⟨rfl, -⟩
    

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

Next tactic:
rw [mul_assoc, Int.mul_emod_right] at h_parity
/-- by assuming `x` is odd and `z` is positive we get a slightly more precise classification of
the Pythagorean triple `x ^ 2 + y ^ 2 = z ^ 2`-/
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 := by
  obtain ⟨m, n, ht1, ht2, ht3, ht4⟩ :=
    PythagoreanTriple.coprime_classification.mp (And.intro h h_coprime)
  rcases le_or_lt 0 m with hm | hm
  · use m, n
    cases' ht1 with h_odd h_even
    · apply And.intro h_odd.1
      apply And.intro h_odd.2
      cases' ht2 with h_pos h_neg
      · apply And.intro h_pos (And.intro ht3 (And.intro ht4 hm))
      · exfalso
        revert h_pos
        rw [h_neg]
        exact imp_false.mpr (not_lt.mpr (neg_nonpos.mpr (add_nonneg (sq_nonneg m) (sq_nonneg n))))
    exfalso
    rcases h_even with ⟨rfl, -⟩
    rw [mul_assoc, Int.mul_emod_right] at h_parity
    exact zero_ne_one h_parity
  · use -m, -n
    cases' ht1 with h_odd h_even
    · rw [neg_sq m]
      rw [neg_sq n]
      apply And.intro h_odd.1
      constructor
      · rw [h_odd.2]
        ring
      cases' ht2 with h_pos h_neg
      · apply And.intro h_pos
        constructor
        · delta Int.gcd
          rw [Int.natAbs_neg, Int.natAbs_neg]
          exact ht3
        · rw [Int.neg_emod_two, Int.neg_emod_two]
          apply And.intro ht4
          linarith
      · exfalso
        revert h_pos
        rw [h_neg]
        exact imp_false.mpr (not_lt.mpr (neg_nonpos.mpr (add_nonneg (sq_nonneg m) (sq_nonneg n))))
    exfalso
    rcases h_even with ⟨rfl, -⟩
    rw [mul_assoc, Int.mul_emod_right] at h_parity
    

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

Next tactic:
exact zero_ne_one h_parity
/-- **Formula for Pythagorean Triples** -/
theorem classification :
    PythagoreanTriple x y z ↔
      ∃ k m n,
        (x = k * (m ^ 2 - n ^ 2) ∧ y = k * (2 * m * n) ∨
            x = k * (2 * m * n) ∧ y = k * (m ^ 2 - n ^ 2)) ∧
          (z = k * (m ^ 2 + n ^ 2) ∨ z = -k * (m ^ 2 + n ^ 2)) := by
  

Proof state:
x y z : ℤ
h : PythagoreanTriple x y z
⊢ PythagoreanTriple x y z ↔
    ∃ k m n,
      (x = k * (m ^ 2 - n ^ 2) ∧ y = k * (2 * m * n) ∨ x = k * (2 * m * n) ∧ y = k * (m ^ 2 - n ^ 2)) ∧
        (z = k * (m ^ 2 + n ^ 2) ∨ z = -k * (m ^ 2 + n ^ 2))

Next tactic:
clear h
/-- **Formula for Pythagorean Triples** -/
theorem classification :
    PythagoreanTriple x y z ↔
      ∃ k m n,
        (x = k * (m ^ 2 - n ^ 2) ∧ y = k * (2 * m * n) ∨
            x = k * (2 * m * n) ∧ y = k * (m ^ 2 - n ^ 2)) ∧
          (z = k * (m ^ 2 + n ^ 2) ∨ z = -k * (m ^ 2 + n ^ 2)) := by
  clear h
  

Proof state:
x y z : ℤ
⊢ PythagoreanTriple x y z ↔
    ∃ k m n,
      (x = k * (m ^ 2 - n ^ 2) ∧ y = k * (2 * m * n) ∨ x = k * (2 * m * n) ∧ y = k * (m ^ 2 - n ^ 2)) ∧
        (z = k * (m ^ 2 + n ^ 2) ∨ z = -k * (m ^ 2 + n ^ 2))

Next tactic:
constructor
/-- **Formula for Pythagorean Triples** -/
theorem classification :
    PythagoreanTriple x y z ↔
      ∃ k m n,
        (x = k * (m ^ 2 - n ^ 2) ∧ y = k * (2 * m * n) ∨
            x = k * (2 * m * n) ∧ y = k * (m ^ 2 - n ^ 2)) ∧
          (z = k * (m ^ 2 + n ^ 2) ∨ z = -k * (m ^ 2 + n ^ 2)) := by
  clear h
  constructor
  · 

Proof state:
case mp
x y z : ℤ
⊢ PythagoreanTriple x y z →
    ∃ k m n,
      (x = k * (m ^ 2 - n ^ 2) ∧ y = k * (2 * m * n) ∨ x = k * (2 * m * n) ∧ y = k * (m ^ 2 - n ^ 2)) ∧
        (z = k * (m ^ 2 + n ^ 2) ∨ z = -k * (m ^ 2 + n ^ 2))

Next tactic:
intro h
/-- **Formula for Pythagorean Triples** -/
theorem classification :
    PythagoreanTriple x y z ↔
      ∃ k m n,
        (x = k * (m ^ 2 - n ^ 2) ∧ y = k * (2 * m * n) ∨
            x = k * (2 * m * n) ∧ y = k * (m ^ 2 - n ^ 2)) ∧
          (z = k * (m ^ 2 + n ^ 2) ∨ z = -k * (m ^ 2 + n ^ 2)) := by
  clear h
  constructor
  · intro h
    

Proof state:
case mp
x y z : ℤ
h : PythagoreanTriple x y z
⊢ ∃ k m n,
    (x = k * (m ^ 2 - n ^ 2) ∧ y = k * (2 * m * n) ∨ x = k * (2 * m * n) ∧ y = k * (m ^ 2 - n ^ 2)) ∧
      (z = k * (m ^ 2 + n ^ 2) ∨ z = -k * (m ^ 2 + n ^ 2))

Next tactic:
obtain ⟨k, m, n, H⟩ := h.classified
/-- **Formula for Pythagorean Triples** -/
theorem classification :
    PythagoreanTriple x y z ↔
      ∃ k m n,
        (x = k * (m ^ 2 - n ^ 2) ∧ y = k * (2 * m * n) ∨
            x = k * (2 * m * n) ∧ y = k * (m ^ 2 - n ^ 2)) ∧
          (z = k * (m ^ 2 + n ^ 2) ∨ z = -k * (m ^ 2 + n ^ 2)) := by
  clear h
  constructor
  · intro h
    obtain ⟨k, m, n, H⟩ := h.classified
    

Proof state:
case mp.intro.intro.intro
x y z : ℤ
h : PythagoreanTriple x y z
k m n : ℤ
H : (x = k * (m ^ 2 - n ^ 2) ∧ y = k * (2 * m * n) ∨ x = k * (2 * m * n) ∧ y = k * (m ^ 2 - n ^ 2)) ∧ Int.gcd m n = 1
⊢ ∃ k m n,
    (x = k * (m ^ 2 - n ^ 2) ∧ y = k * (2 * m * n) ∨ x = k * (2 * m * n) ∧ y = k * (m ^ 2 - n ^ 2)) ∧
      (z = k * (m ^ 2 + n ^ 2) ∨ z = -k * (m ^ 2 + n ^ 2))

Next tactic:
use k, m, n
/-- **Formula for Pythagorean Triples** -/
theorem classification :
    PythagoreanTriple x y z ↔
      ∃ k m n,
        (x = k * (m ^ 2 - n ^ 2) ∧ y = k * (2 * m * n) ∨
            x = k * (2 * m * n) ∧ y = k * (m ^ 2 - n ^ 2)) ∧
          (z = k * (m ^ 2 + n ^ 2) ∨ z = -k * (m ^ 2 + n ^ 2)) := by
  clear h
  constructor
  · intro h
    obtain ⟨k, m, n, H⟩ := h.classified
    use k, m, n
    

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

Next tactic:
rcases H with (⟨rfl, rfl⟩ | ⟨rfl, rfl⟩)
/-- **Formula for Pythagorean Triples** -/
theorem classification :
    PythagoreanTriple x y z ↔
      ∃ k m n,
        (x = k * (m ^ 2 - n ^ 2) ∧ y = k * (2 * m * n) ∨
            x = k * (2 * m * n) ∧ y = k * (m ^ 2 - n ^ 2)) ∧
          (z = k * (m ^ 2 + n ^ 2) ∨ z = -k * (m ^ 2 + n ^ 2)) := by
  clear h
  constructor
  · intro h
    obtain ⟨k, m, n, H⟩ := h.classified
    use k, m, n
    rcases H with (⟨rfl, rfl⟩ | ⟨rfl, rfl⟩)
    · 

Proof state:
case h.intro.inl.intro
z k m n : ℤ
right✝ : Int.gcd m n = 1
h : PythagoreanTriple (k * (m ^ 2 - n ^ 2)) (k * (2 * m * n)) z
⊢ (k * (m ^ 2 - n ^ 2) = k * (m ^ 2 - n ^ 2) ∧ k * (2 * m * n) = k * (2 * m * n) ∨
      k * (m ^ 2 - n ^ 2) = k * (2 * m * n) ∧ k * (2 * m * n) = k * (m ^ 2 - n ^ 2)) ∧
    (z = k * (m ^ 2 + n ^ 2) ∨ z = -k * (m ^ 2 + n ^ 2))

Next tactic:
refine' ⟨Or.inl ⟨rfl, rfl⟩, _⟩
/-- **Formula for Pythagorean Triples** -/
theorem classification :
    PythagoreanTriple x y z ↔
      ∃ k m n,
        (x = k * (m ^ 2 - n ^ 2) ∧ y = k * (2 * m * n) ∨
            x = k * (2 * m * n) ∧ y = k * (m ^ 2 - n ^ 2)) ∧
          (z = k * (m ^ 2 + n ^ 2) ∨ z = -k * (m ^ 2 + n ^ 2)) := by
  clear h
  constructor
  · intro h
    obtain ⟨k, m, n, H⟩ := h.classified
    use k, m, n
    rcases H with (⟨rfl, rfl⟩ | ⟨rfl, rfl⟩)
    · refine' ⟨Or.inl ⟨rfl, rfl⟩, _⟩
      

Proof state:
case h.intro.inl.intro
z k m n : ℤ
right✝ : Int.gcd m n = 1
h : PythagoreanTriple (k * (m ^ 2 - n ^ 2)) (k * (2 * m * n)) z
⊢ z = k * (m ^ 2 + n ^ 2) ∨ z = -k * (m ^ 2 + n ^ 2)

Next tactic:
have : z ^ 2 = (k * (m ^ 2 + n ^ 2)) ^ 2 := by
        rw [sq, ← h.eq]
        ring
/-- **Formula for Pythagorean Triples** -/
theorem classification :
    PythagoreanTriple x y z ↔
      ∃ k m n,
        (x = k * (m ^ 2 - n ^ 2) ∧ y = k * (2 * m * n) ∨
            x = k * (2 * m * n) ∧ y = k * (m ^ 2 - n ^ 2)) ∧
          (z = k * (m ^ 2 + n ^ 2) ∨ z = -k * (m ^ 2 + n ^ 2)) := by
  clear h
  constructor
  · intro h
    obtain ⟨k, m, n, H⟩ := h.classified
    use k, m, n
    rcases H with (⟨rfl, rfl⟩ | ⟨rfl, rfl⟩)
    · refine' ⟨Or.inl ⟨rfl, rfl⟩, _⟩
      have : z ^ 2 = (k * (m ^ 2 + n ^ 2)) ^ 2 := by
        

Proof state:
z k m n : ℤ
right✝ : Int.gcd m n = 1
h : PythagoreanTriple (k * (m ^ 2 - n ^ 2)) (k * (2 * m * n)) z
⊢ z ^ 2 = (k * (m ^ 2 + n ^ 2)) ^ 2

Next tactic:
rw [sq, ← h.eq]
/-- **Formula for Pythagorean Triples** -/
theorem classification :
    PythagoreanTriple x y z ↔
      ∃ k m n,
        (x = k * (m ^ 2 - n ^ 2) ∧ y = k * (2 * m * n) ∨
            x = k * (2 * m * n) ∧ y = k * (m ^ 2 - n ^ 2)) ∧
          (z = k * (m ^ 2 + n ^ 2) ∨ z = -k * (m ^ 2 + n ^ 2)) := by
  clear h
  constructor
  · intro h
    obtain ⟨k, m, n, H⟩ := h.classified
    use k, m, n
    rcases H with (⟨rfl, rfl⟩ | ⟨rfl, rfl⟩)
    · refine' ⟨Or.inl ⟨rfl, rfl⟩, _⟩
      have : z ^ 2 = (k * (m ^ 2 + n ^ 2)) ^ 2 := by
        rw [sq, ← h.eq]
        

Proof state:
z k m n : ℤ
right✝ : Int.gcd m n = 1
h : PythagoreanTriple (k * (m ^ 2 - n ^ 2)) (k * (2 * m * n)) z
⊢ k * (m ^ 2 - n ^ 2) * (k * (m ^ 2 - n ^ 2)) + k * (2 * m * n) * (k * (2 * m * n)) = (k * (m ^ 2 + n ^ 2)) ^ 2

Next tactic:
ring
/-- **Formula for Pythagorean Triples** -/
theorem classification :
    PythagoreanTriple x y z ↔
      ∃ k m n,
        (x = k * (m ^ 2 - n ^ 2) ∧ y = k * (2 * m * n) ∨
            x = k * (2 * m * n) ∧ y = k * (m ^ 2 - n ^ 2)) ∧
          (z = k * (m ^ 2 + n ^ 2) ∨ z = -k * (m ^ 2 + n ^ 2)) := by
  clear h
  constructor
  · intro h
    obtain ⟨k, m, n, H⟩ := h.classified
    use k, m, n
    rcases H with (⟨rfl, rfl⟩ | ⟨rfl, rfl⟩)
    · refine' ⟨Or.inl ⟨rfl, rfl⟩, _⟩
      have : z ^ 2 = (k * (m ^ 2 + n ^ 2)) ^ 2 := by
        rw [sq, ← h.eq]
        ring
      

Proof state:
case h.intro.inl.intro
z k m n : ℤ
right✝ : Int.gcd m n = 1
h : PythagoreanTriple (k * (m ^ 2 - n ^ 2)) (k * (2 * m * n)) z
this : z ^ 2 = (k * (m ^ 2 + n ^ 2)) ^ 2
⊢ z = k * (m ^ 2 + n ^ 2) ∨ z = -k * (m ^ 2 + n ^ 2)

Next tactic:
simpa using eq_or_eq_neg_of_sq_eq_sq _ _ this
/-- **Formula for Pythagorean Triples** -/
theorem classification :
    PythagoreanTriple x y z ↔
      ∃ k m n,
        (x = k * (m ^ 2 - n ^ 2) ∧ y = k * (2 * m * n) ∨
            x = k * (2 * m * n) ∧ y = k * (m ^ 2 - n ^ 2)) ∧
          (z = k * (m ^ 2 + n ^ 2) ∨ z = -k * (m ^ 2 + n ^ 2)) := by
  clear h
  constructor
  · intro h
    obtain ⟨k, m, n, H⟩ := h.classified
    use k, m, n
    rcases H with (⟨rfl, rfl⟩ | ⟨rfl, rfl⟩)
    · refine' ⟨Or.inl ⟨rfl, rfl⟩, _⟩
      have : z ^ 2 = (k * (m ^ 2 + n ^ 2)) ^ 2 := by
        rw [sq, ← h.eq]
        ring
      simpa using eq_or_eq_neg_of_sq_eq_sq _ _ this
    · 

Proof state:
case h.intro.inr.intro
z k m n : ℤ
right✝ : Int.gcd m n = 1
h : PythagoreanTriple (k * (2 * m * n)) (k * (m ^ 2 - n ^ 2)) z
⊢ (k * (2 * m * n) = k * (m ^ 2 - n ^ 2) ∧ k * (m ^ 2 - n ^ 2) = k * (2 * m * n) ∨
      k * (2 * m * n) = k * (2 * m * n) ∧ k * (m ^ 2 - n ^ 2) = k * (m ^ 2 - n ^ 2)) ∧
    (z = k * (m ^ 2 + n ^ 2) ∨ z = -k * (m ^ 2 + n ^ 2))

Next tactic:
refine' ⟨Or.inr ⟨rfl, rfl⟩, _⟩
/-- **Formula for Pythagorean Triples** -/
theorem classification :
    PythagoreanTriple x y z ↔
      ∃ k m n,
        (x = k * (m ^ 2 - n ^ 2) ∧ y = k * (2 * m * n) ∨
            x = k * (2 * m * n) ∧ y = k * (m ^ 2 - n ^ 2)) ∧
          (z = k * (m ^ 2 + n ^ 2) ∨ z = -k * (m ^ 2 + n ^ 2)) := by
  clear h
  constructor
  · intro h
    obtain ⟨k, m, n, H⟩ := h.classified
    use k, m, n
    rcases H with (⟨rfl, rfl⟩ | ⟨rfl, rfl⟩)
    · refine' ⟨Or.inl ⟨rfl, rfl⟩, _⟩
      have : z ^ 2 = (k * (m ^ 2 + n ^ 2)) ^ 2 := by
        rw [sq, ← h.eq]
        ring
      simpa using eq_or_eq_neg_of_sq_eq_sq _ _ this
    · refine' ⟨Or.inr ⟨rfl, rfl⟩, _⟩
      

Proof state:
case h.intro.inr.intro
z k m n : ℤ
right✝ : Int.gcd m n = 1
h : PythagoreanTriple (k * (2 * m * n)) (k * (m ^ 2 - n ^ 2)) z
⊢ z = k * (m ^ 2 + n ^ 2) ∨ z = -k * (m ^ 2 + n ^ 2)

Next tactic:
have : z ^ 2 = (k * (m ^ 2 + n ^ 2)) ^ 2 := by
        rw [sq, ← h.eq]
        ring
/-- **Formula for Pythagorean Triples** -/
theorem classification :
    PythagoreanTriple x y z ↔
      ∃ k m n,
        (x = k * (m ^ 2 - n ^ 2) ∧ y = k * (2 * m * n) ∨
            x = k * (2 * m * n) ∧ y = k * (m ^ 2 - n ^ 2)) ∧
          (z = k * (m ^ 2 + n ^ 2) ∨ z = -k * (m ^ 2 + n ^ 2)) := by
  clear h
  constructor
  · intro h
    obtain ⟨k, m, n, H⟩ := h.classified
    use k, m, n
    rcases H with (⟨rfl, rfl⟩ | ⟨rfl, rfl⟩)
    · refine' ⟨Or.inl ⟨rfl, rfl⟩, _⟩
      have : z ^ 2 = (k * (m ^ 2 + n ^ 2)) ^ 2 := by
        rw [sq, ← h.eq]
        ring
      simpa using eq_or_eq_neg_of_sq_eq_sq _ _ this
    · refine' ⟨Or.inr ⟨rfl, rfl⟩, _⟩
      have : z ^ 2 = (k * (m ^ 2 + n ^ 2)) ^ 2 := by
        

Proof state:
z k m n : ℤ
right✝ : Int.gcd m n = 1
h : PythagoreanTriple (k * (2 * m * n)) (k * (m ^ 2 - n ^ 2)) z
⊢ z ^ 2 = (k * (m ^ 2 + n ^ 2)) ^ 2

Next tactic:
rw [sq, ← h.eq]
/-- **Formula for Pythagorean Triples** -/
theorem classification :
    PythagoreanTriple x y z ↔
      ∃ k m n,
        (x = k * (m ^ 2 - n ^ 2) ∧ y = k * (2 * m * n) ∨
            x = k * (2 * m * n) ∧ y = k * (m ^ 2 - n ^ 2)) ∧
          (z = k * (m ^ 2 + n ^ 2) ∨ z = -k * (m ^ 2 + n ^ 2)) := by
  clear h
  constructor
  · intro h
    obtain ⟨k, m, n, H⟩ := h.classified
    use k, m, n
    rcases H with (⟨rfl, rfl⟩ | ⟨rfl, rfl⟩)
    · refine' ⟨Or.inl ⟨rfl, rfl⟩, _⟩
      have : z ^ 2 = (k * (m ^ 2 + n ^ 2)) ^ 2 := by
        rw [sq, ← h.eq]
        ring
      simpa using eq_or_eq_neg_of_sq_eq_sq _ _ this
    · refine' ⟨Or.inr ⟨rfl, rfl⟩, _⟩
      have : z ^ 2 = (k * (m ^ 2 + n ^ 2)) ^ 2 := by
        rw [sq, ← h.eq]
        

Proof state:
z k m n : ℤ
right✝ : Int.gcd m n = 1
h : PythagoreanTriple (k * (2 * m * n)) (k * (m ^ 2 - n ^ 2)) z
⊢ k * (2 * m * n) * (k * (2 * m * n)) + k * (m ^ 2 - n ^ 2) * (k * (m ^ 2 - n ^ 2)) = (k * (m ^ 2 + n ^ 2)) ^ 2

Next tactic:
ring
/-- **Formula for Pythagorean Triples** -/
theorem classification :
    PythagoreanTriple x y z ↔
      ∃ k m n,
        (x = k * (m ^ 2 - n ^ 2) ∧ y = k * (2 * m * n) ∨
            x = k * (2 * m * n) ∧ y = k * (m ^ 2 - n ^ 2)) ∧
          (z = k * (m ^ 2 + n ^ 2) ∨ z = -k * (m ^ 2 + n ^ 2)) := by
  clear h
  constructor
  · intro h
    obtain ⟨k, m, n, H⟩ := h.classified
    use k, m, n
    rcases H with (⟨rfl, rfl⟩ | ⟨rfl, rfl⟩)
    · refine' ⟨Or.inl ⟨rfl, rfl⟩, _⟩
      have : z ^ 2 = (k * (m ^ 2 + n ^ 2)) ^ 2 := by
        rw [sq, ← h.eq]
        ring
      simpa using eq_or_eq_neg_of_sq_eq_sq _ _ this
    · refine' ⟨Or.inr ⟨rfl, rfl⟩, _⟩
      have : z ^ 2 = (k * (m ^ 2 + n ^ 2)) ^ 2 := by
        rw [sq, ← h.eq]
        ring
      

Proof state:
case h.intro.inr.intro
z k m n : ℤ
right✝ : Int.gcd m n = 1
h : PythagoreanTriple (k * (2 * m * n)) (k * (m ^ 2 - n ^ 2)) z
this : z ^ 2 = (k * (m ^ 2 + n ^ 2)) ^ 2
⊢ z = k * (m ^ 2 + n ^ 2) ∨ z = -k * (m ^ 2 + n ^ 2)

Next tactic:
simpa using eq_or_eq_neg_of_sq_eq_sq _ _ this
/-- **Formula for Pythagorean Triples** -/
theorem classification :
    PythagoreanTriple x y z ↔
      ∃ k m n,
        (x = k * (m ^ 2 - n ^ 2) ∧ y = k * (2 * m * n) ∨
            x = k * (2 * m * n) ∧ y = k * (m ^ 2 - n ^ 2)) ∧
          (z = k * (m ^ 2 + n ^ 2) ∨ z = -k * (m ^ 2 + n ^ 2)) := by
  clear h
  constructor
  · intro h
    obtain ⟨k, m, n, H⟩ := h.classified
    use k, m, n
    rcases H with (⟨rfl, rfl⟩ | ⟨rfl, rfl⟩)
    · refine' ⟨Or.inl ⟨rfl, rfl⟩, _⟩
      have : z ^ 2 = (k * (m ^ 2 + n ^ 2)) ^ 2 := by
        rw [sq, ← h.eq]
        ring
      simpa using eq_or_eq_neg_of_sq_eq_sq _ _ this
    · refine' ⟨Or.inr ⟨rfl, rfl⟩, _⟩
      have : z ^ 2 = (k * (m ^ 2 + n ^ 2)) ^ 2 := by
        rw [sq, ← h.eq]
        ring
      simpa using eq_or_eq_neg_of_sq_eq_sq _ _ this
  · 

Proof state:
case mpr
x y z : ℤ
⊢ (∃ k m n,
      (x = k * (m ^ 2 - n ^ 2) ∧ y = k * (2 * m * n) ∨ x = k * (2 * m * n) ∧ y = k * (m ^ 2 - n ^ 2)) ∧
        (z = k * (m ^ 2 + n ^ 2) ∨ z = -k * (m ^ 2 + n ^ 2))) →
    PythagoreanTriple x y z

Next tactic:
rintro ⟨k, m, n, ⟨rfl, rfl⟩ | ⟨rfl, rfl⟩, rfl | rfl⟩
/-- **Formula for Pythagorean Triples** -/
theorem classification :
    PythagoreanTriple x y z ↔
      ∃ k m n,
        (x = k * (m ^ 2 - n ^ 2) ∧ y = k * (2 * m * n) ∨
            x = k * (2 * m * n) ∧ y = k * (m ^ 2 - n ^ 2)) ∧
          (z = k * (m ^ 2 + n ^ 2) ∨ z = -k * (m ^ 2 + n ^ 2)) := by
  clear h
  constructor
  · intro h
    obtain ⟨k, m, n, H⟩ := h.classified
    use k, m, n
    rcases H with (⟨rfl, rfl⟩ | ⟨rfl, rfl⟩)
    · refine' ⟨Or.inl ⟨rfl, rfl⟩, _⟩
      have : z ^ 2 = (k * (m ^ 2 + n ^ 2)) ^ 2 := by
        rw [sq, ← h.eq]
        ring
      simpa using eq_or_eq_neg_of_sq_eq_sq _ _ this
    · refine' ⟨Or.inr ⟨rfl, rfl⟩, _⟩
      have : z ^ 2 = (k * (m ^ 2 + n ^ 2)) ^ 2 := by
        rw [sq, ← h.eq]
        ring
      simpa using eq_or_eq_neg_of_sq_eq_sq _ _ this
  · rintro ⟨k, m, n, ⟨rfl, rfl⟩ | ⟨rfl, rfl⟩, rfl | rfl⟩ <;> 

Proof state:
case mpr.intro.intro.intro.intro.inl.intro.inl
k m n : ℤ
⊢ PythagoreanTriple (k * (m ^ 2 - n ^ 2)) (k * (2 * m * n)) (k * (m ^ 2 + n ^ 2))

Next tactic:
delta PythagoreanTriple
/-- **Formula for Pythagorean Triples** -/
theorem classification :
    PythagoreanTriple x y z ↔
      ∃ k m n,
        (x = k * (m ^ 2 - n ^ 2) ∧ y = k * (2 * m * n) ∨
            x = k * (2 * m * n) ∧ y = k * (m ^ 2 - n ^ 2)) ∧
          (z = k * (m ^ 2 + n ^ 2) ∨ z = -k * (m ^ 2 + n ^ 2)) := by
  clear h
  constructor
  · intro h
    obtain ⟨k, m, n, H⟩ := h.classified
    use k, m, n
    rcases H with (⟨rfl, rfl⟩ | ⟨rfl, rfl⟩)
    · refine' ⟨Or.inl ⟨rfl, rfl⟩, _⟩
      have : z ^ 2 = (k * (m ^ 2 + n ^ 2)) ^ 2 := by
        rw [sq, ← h.eq]
        ring
      simpa using eq_or_eq_neg_of_sq_eq_sq _ _ this
    · refine' ⟨Or.inr ⟨rfl, rfl⟩, _⟩
      have : z ^ 2 = (k * (m ^ 2 + n ^ 2)) ^ 2 := by
        rw [sq, ← h.eq]
        ring
      simpa using eq_or_eq_neg_of_sq_eq_sq _ _ this
  · rintro ⟨k, m, n, ⟨rfl, rfl⟩ | ⟨rfl, rfl⟩, rfl | rfl⟩ <;> 

Proof state:
case mpr.intro.intro.intro.intro.inl.intro.inr
k m n : ℤ
⊢ PythagoreanTriple (k * (m ^ 2 - n ^ 2)) (k * (2 * m * n)) (-k * (m ^ 2 + n ^ 2))

Next tactic:
delta PythagoreanTriple
/-- **Formula for Pythagorean Triples** -/
theorem classification :
    PythagoreanTriple x y z ↔
      ∃ k m n,
        (x = k * (m ^ 2 - n ^ 2) ∧ y = k * (2 * m * n) ∨
            x = k * (2 * m * n) ∧ y = k * (m ^ 2 - n ^ 2)) ∧
          (z = k * (m ^ 2 + n ^ 2) ∨ z = -k * (m ^ 2 + n ^ 2)) := by
  clear h
  constructor
  · intro h
    obtain ⟨k, m, n, H⟩ := h.classified
    use k, m, n
    rcases H with (⟨rfl, rfl⟩ | ⟨rfl, rfl⟩)
    · refine' ⟨Or.inl ⟨rfl, rfl⟩, _⟩
      have : z ^ 2 = (k * (m ^ 2 + n ^ 2)) ^ 2 := by
        rw [sq, ← h.eq]
        ring
      simpa using eq_or_eq_neg_of_sq_eq_sq _ _ this
    · refine' ⟨Or.inr ⟨rfl, rfl⟩, _⟩
      have : z ^ 2 = (k * (m ^ 2 + n ^ 2)) ^ 2 := by
        rw [sq, ← h.eq]
        ring
      simpa using eq_or_eq_neg_of_sq_eq_sq _ _ this
  · rintro ⟨k, m, n, ⟨rfl, rfl⟩ | ⟨rfl, rfl⟩, rfl | rfl⟩ <;> 

Proof state:
case mpr.intro.intro.intro.intro.inr.intro.inl
k m n : ℤ
⊢ PythagoreanTriple (k * (2 * m * n)) (k * (m ^ 2 - n ^ 2)) (k * (m ^ 2 + n ^ 2))

Next tactic:
delta PythagoreanTriple