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
        

Proof state:
case h.inl.right.inl
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
ht3 : Int.gcd m 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
⊢ Int.gcd (-m) (-n) = 1 ∧ (-m % 2 = 0 ∧ -n % 2 = 1 ∨ -m % 2 = 1 ∧ -n % 2 = 0) ∧ 0 ≤ -m

Next tactic:
constructor
/-- 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
        · 

Proof state:
case h.inl.right.inl.left
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
ht3 : Int.gcd m 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
⊢ Int.gcd (-m) (-n) = 1

Next tactic:
delta Int.gcd
/-- 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
          

Proof state:
case h.inl.right.inl.left
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
ht3 : Int.gcd m 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
⊢ Nat.gcd (Int.natAbs (-m)) (Int.natAbs (-n)) = 1

Next tactic:
rw [Int.natAbs_neg, Int.natAbs_neg]
/-- 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]
          

Proof state:
case h.inl.right.inl.left
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
ht3 : Int.gcd m 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
⊢ Nat.gcd (Int.natAbs m) (Int.natAbs n) = 1

Next tactic:
exact ht3
/-- 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
        · 

Proof state:
case h.inl.right.inl.right
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
ht3 : Int.gcd m 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 % 2 = 0 ∧ -n % 2 = 1 ∨ -m % 2 = 1 ∧ -n % 2 = 0) ∧ 0 ≤ -m

Next tactic:
rw [Int.neg_emod_two, Int.neg_emod_two]
/-- 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]
          

Proof state:
case h.inl.right.inl.right
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
ht3 : Int.gcd m 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 % 2 = 0 ∧ n % 2 = 1 ∨ m % 2 = 1 ∧ n % 2 = 0) ∧ 0 ≤ -m

Next tactic:
apply And.intro ht4
/-- 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
          

Proof state:
case h.inl.right.inl.right
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
ht3 : Int.gcd m 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
⊢ 0 ≤ -m

Next tactic:
linarith
/-- 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
      · 

Proof state:
case h.inl.right.inr
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
ht3 : Int.gcd m 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_neg : z = -(m ^ 2 + n ^ 2)
⊢ z = m ^ 2 + n ^ 2 ∧ Int.gcd (-m) (-n) = 1 ∧ (-m % 2 = 0 ∧ -n % 2 = 1 ∨ -m % 2 = 1 ∧ -n % 2 = 0) ∧ 0 ≤ -m

Next tactic:
exfalso
/-- 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
        

Proof state:
case h.inl.right.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
ht3 : Int.gcd m 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_neg : z = -(m ^ 2 + n ^ 2)
⊢ False

Next tactic:
revert h_pos
/-- 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
        

Proof state:
case h.inl.right.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
m n : ℤ
ht3 : Int.gcd m 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_neg : z = -(m ^ 2 + n ^ 2)
⊢ 0 < z → False

Next tactic:
rw [h_neg]
/-- 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]
        

Proof state:
case h.inl.right.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
m n : ℤ
ht3 : Int.gcd m 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_neg : z = -(m ^ 2 + n ^ 2)
⊢ 0 < -(m ^ 2 + n ^ 2) → False

Next tactic:
exact imp_false.mpr (not_lt.mpr (neg_nonpos.mpr (add_nonneg (sq_nonneg m) (sq_nonneg n))))
/-- 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))))
    

Proof state:
case h.inr
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
⊢ 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

Next tactic:
exfalso