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))))
    

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 : 0 ≤ m
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
/-- 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
    

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 : 0 ≤ m
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, -⟩
    

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 : 0 ≤ m
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
    

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 : 0 ≤ m
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
/-- 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
  · 

Proof state:
case intro.intro.intro.intro.intro.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
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 : Int.gcd m n = 1
ht4 : m % 2 = 0 ∧ n % 2 = 1 ∨ m % 2 = 1 ∧ n % 2 = 0
hm : m < 0
⊢ ∃ 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

Next tactic:
use -m, -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
    

Proof state:
case 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
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 : Int.gcd m n = 1
ht4 : m % 2 = 0 ∧ n % 2 = 1 ∨ m % 2 = 1 ∧ n % 2 = 0
hm : m < 0
⊢ 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:
cases' ht1 with h_odd h_even
/-- 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
    · 

Proof state:
case h.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
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_odd : x = m ^ 2 - n ^ 2 ∧ y = 2 * 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

Next tactic:
rw [neg_sq m]
/-- 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]
      

Proof state:
case h.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
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_odd : x = m ^ 2 - n ^ 2 ∧ y = 2 * 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

Next tactic:
rw [neg_sq 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]
      

Proof state:
case h.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
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_odd : x = m ^ 2 - n ^ 2 ∧ y = 2 * 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

Next tactic:
apply And.intro h_odd.1
/-- 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
      

Proof state:
case h.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
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_odd : x = m ^ 2 - n ^ 2 ∧ y = 2 * m * n
⊢ 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:
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
      · 

Proof state:
case h.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
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_odd : x = m ^ 2 - n ^ 2 ∧ y = 2 * m * n
⊢ y = 2 * -m * -n

Next tactic:
rw [h_odd.2]
/-- 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]
        

Proof state:
case h.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
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_odd : x = m ^ 2 - n ^ 2 ∧ y = 2 * m * n
⊢ 2 * m * n = 2 * -m * -n

Next tactic:
ring
/-- 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
      

Proof state:
case h.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
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_odd : 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:
cases' ht2 with h_pos 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
      · 

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
⊢ 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:
apply And.intro h_pos