l3lab/ntp-mathlib huggingface dataset:
private 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 := by
  cases' exists_eq_mul_left_of_dvd (Int.dvd_sub_of_emod_eq hm) with m0 hm2
  cases' exists_eq_mul_left_of_dvd (Int.dvd_sub_of_emod_eq hn) with n0 hn2
  rw [sub_eq_iff_eq_add] at hm2 hn2
  subst m
  subst n
  have h1 : (m0 * 2 + 1) ^ 2 + (n0 * 2 + 1) ^ 2 = 2 * (2 * (m0 ^ 2 + n0 ^ 2 + m0 + n0) + 1) := by
    ring
  have h2 : (m0 * 2 + 1) ^ 2 - (n0 * 2 + 1) ^ 2 = 2 * (2 * (m0 ^ 2 - n0 ^ 2 + m0 - n0)) := by ring
  have h3 : ((m0 * 2 + 1) ^ 2 - (n0 * 2 + 1) ^ 2) / 2 % 2 = 0 := by
    rw [h2, Int.mul_ediv_cancel_left, Int.mul_emod_right]
    

Proof state:
case H
m0 n0 : ℤ
hm : (m0 * 2 + 1) % 2 = 1
hn : (n0 * 2 + 1) % 2 = 1
h : Int.gcd (m0 * 2 + 1) (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 ≠ 0

Next tactic:
exact by decide
private 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 := by
  cases' exists_eq_mul_left_of_dvd (Int.dvd_sub_of_emod_eq hm) with m0 hm2
  cases' exists_eq_mul_left_of_dvd (Int.dvd_sub_of_emod_eq hn) with n0 hn2
  rw [sub_eq_iff_eq_add] at hm2 hn2
  subst m
  subst n
  have h1 : (m0 * 2 + 1) ^ 2 + (n0 * 2 + 1) ^ 2 = 2 * (2 * (m0 ^ 2 + n0 ^ 2 + m0 + n0) + 1) := by
    ring
  have h2 : (m0 * 2 + 1) ^ 2 - (n0 * 2 + 1) ^ 2 = 2 * (2 * (m0 ^ 2 - n0 ^ 2 + m0 - n0)) := by ring
  have h3 : ((m0 * 2 + 1) ^ 2 - (n0 * 2 + 1) ^ 2) / 2 % 2 = 0 := by
    rw [h2, Int.mul_ediv_cancel_left, Int.mul_emod_right]
    exact by 

Proof state:
m0 n0 : ℤ
hm : (m0 * 2 + 1) % 2 = 1
hn : (n0 * 2 + 1) % 2 = 1
h : Int.gcd (m0 * 2 + 1) (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 ≠ 0

Next tactic:
decide
private 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 := by
  cases' exists_eq_mul_left_of_dvd (Int.dvd_sub_of_emod_eq hm) with m0 hm2
  cases' exists_eq_mul_left_of_dvd (Int.dvd_sub_of_emod_eq hn) with n0 hn2
  rw [sub_eq_iff_eq_add] at hm2 hn2
  subst m
  subst n
  have h1 : (m0 * 2 + 1) ^ 2 + (n0 * 2 + 1) ^ 2 = 2 * (2 * (m0 ^ 2 + n0 ^ 2 + m0 + n0) + 1) := by
    ring
  have h2 : (m0 * 2 + 1) ^ 2 - (n0 * 2 + 1) ^ 2 = 2 * (2 * (m0 ^ 2 - n0 ^ 2 + m0 - n0)) := by ring
  have h3 : ((m0 * 2 + 1) ^ 2 - (n0 * 2 + 1) ^ 2) / 2 % 2 = 0 := by
    rw [h2, Int.mul_ediv_cancel_left, Int.mul_emod_right]
    exact by decide
  

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

Next tactic:
refine' ⟨⟨_, h1⟩, ⟨_, h2⟩, h3, _⟩
private 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 := by
  cases' exists_eq_mul_left_of_dvd (Int.dvd_sub_of_emod_eq hm) with m0 hm2
  cases' exists_eq_mul_left_of_dvd (Int.dvd_sub_of_emod_eq hn) with n0 hn2
  rw [sub_eq_iff_eq_add] at hm2 hn2
  subst m
  subst n
  have h1 : (m0 * 2 + 1) ^ 2 + (n0 * 2 + 1) ^ 2 = 2 * (2 * (m0 ^ 2 + n0 ^ 2 + m0 + n0) + 1) := by
    ring
  have h2 : (m0 * 2 + 1) ^ 2 - (n0 * 2 + 1) ^ 2 = 2 * (2 * (m0 ^ 2 - n0 ^ 2 + m0 - n0)) := by ring
  have h3 : ((m0 * 2 + 1) ^ 2 - (n0 * 2 + 1) ^ 2) / 2 % 2 = 0 := by
    rw [h2, Int.mul_ediv_cancel_left, Int.mul_emod_right]
    exact by decide
  refine' ⟨⟨_, h1⟩, ⟨_, h2⟩, h3, _⟩
  

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

Next tactic:
have h20 : (2 : ℤ) ≠ 0 := by decide
private 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 := by
  cases' exists_eq_mul_left_of_dvd (Int.dvd_sub_of_emod_eq hm) with m0 hm2
  cases' exists_eq_mul_left_of_dvd (Int.dvd_sub_of_emod_eq hn) with n0 hn2
  rw [sub_eq_iff_eq_add] at hm2 hn2
  subst m
  subst n
  have h1 : (m0 * 2 + 1) ^ 2 + (n0 * 2 + 1) ^ 2 = 2 * (2 * (m0 ^ 2 + n0 ^ 2 + m0 + n0) + 1) := by
    ring
  have h2 : (m0 * 2 + 1) ^ 2 - (n0 * 2 + 1) ^ 2 = 2 * (2 * (m0 ^ 2 - n0 ^ 2 + m0 - n0)) := by ring
  have h3 : ((m0 * 2 + 1) ^ 2 - (n0 * 2 + 1) ^ 2) / 2 % 2 = 0 := by
    rw [h2, Int.mul_ediv_cancel_left, Int.mul_emod_right]
    exact by decide
  refine' ⟨⟨_, h1⟩, ⟨_, h2⟩, h3, _⟩
  have h20 : (2 : ℤ) ≠ 0 := by 

Proof state:
m0 n0 : ℤ
hm : (m0 * 2 + 1) % 2 = 1
hn : (n0 * 2 + 1) % 2 = 1
h : Int.gcd (m0 * 2 + 1) (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
⊢ 2 ≠ 0

Next tactic:
decide
private 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 := by
  cases' exists_eq_mul_left_of_dvd (Int.dvd_sub_of_emod_eq hm) with m0 hm2
  cases' exists_eq_mul_left_of_dvd (Int.dvd_sub_of_emod_eq hn) with n0 hn2
  rw [sub_eq_iff_eq_add] at hm2 hn2
  subst m
  subst n
  have h1 : (m0 * 2 + 1) ^ 2 + (n0 * 2 + 1) ^ 2 = 2 * (2 * (m0 ^ 2 + n0 ^ 2 + m0 + n0) + 1) := by
    ring
  have h2 : (m0 * 2 + 1) ^ 2 - (n0 * 2 + 1) ^ 2 = 2 * (2 * (m0 ^ 2 - n0 ^ 2 + m0 - n0)) := by ring
  have h3 : ((m0 * 2 + 1) ^ 2 - (n0 * 2 + 1) ^ 2) / 2 % 2 = 0 := by
    rw [h2, Int.mul_ediv_cancel_left, Int.mul_emod_right]
    exact by decide
  refine' ⟨⟨_, h1⟩, ⟨_, h2⟩, h3, _⟩
  have h20 : (2 : ℤ) ≠ 0 := by decide
  

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

Next tactic:
rw [h1, h2, Int.mul_ediv_cancel_left _ h20, Int.mul_ediv_cancel_left _ h20]
private 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 := by
  cases' exists_eq_mul_left_of_dvd (Int.dvd_sub_of_emod_eq hm) with m0 hm2
  cases' exists_eq_mul_left_of_dvd (Int.dvd_sub_of_emod_eq hn) with n0 hn2
  rw [sub_eq_iff_eq_add] at hm2 hn2
  subst m
  subst n
  have h1 : (m0 * 2 + 1) ^ 2 + (n0 * 2 + 1) ^ 2 = 2 * (2 * (m0 ^ 2 + n0 ^ 2 + m0 + n0) + 1) := by
    ring
  have h2 : (m0 * 2 + 1) ^ 2 - (n0 * 2 + 1) ^ 2 = 2 * (2 * (m0 ^ 2 - n0 ^ 2 + m0 - n0)) := by ring
  have h3 : ((m0 * 2 + 1) ^ 2 - (n0 * 2 + 1) ^ 2) / 2 % 2 = 0 := by
    rw [h2, Int.mul_ediv_cancel_left, Int.mul_emod_right]
    exact by decide
  refine' ⟨⟨_, h1⟩, ⟨_, h2⟩, h3, _⟩
  have h20 : (2 : ℤ) ≠ 0 := by decide
  rw [h1, h2, Int.mul_ediv_cancel_left _ h20, Int.mul_ediv_cancel_left _ h20]
  

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

Next tactic:
by_contra h4
private 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 := by
  cases' exists_eq_mul_left_of_dvd (Int.dvd_sub_of_emod_eq hm) with m0 hm2
  cases' exists_eq_mul_left_of_dvd (Int.dvd_sub_of_emod_eq hn) with n0 hn2
  rw [sub_eq_iff_eq_add] at hm2 hn2
  subst m
  subst n
  have h1 : (m0 * 2 + 1) ^ 2 + (n0 * 2 + 1) ^ 2 = 2 * (2 * (m0 ^ 2 + n0 ^ 2 + m0 + n0) + 1) := by
    ring
  have h2 : (m0 * 2 + 1) ^ 2 - (n0 * 2 + 1) ^ 2 = 2 * (2 * (m0 ^ 2 - n0 ^ 2 + m0 - n0)) := by ring
  have h3 : ((m0 * 2 + 1) ^ 2 - (n0 * 2 + 1) ^ 2) / 2 % 2 = 0 := by
    rw [h2, Int.mul_ediv_cancel_left, Int.mul_emod_right]
    exact by decide
  refine' ⟨⟨_, h1⟩, ⟨_, h2⟩, h3, _⟩
  have h20 : (2 : ℤ) ≠ 0 := by decide
  rw [h1, h2, Int.mul_ediv_cancel_left _ h20, Int.mul_ediv_cancel_left _ h20]
  by_contra h4
  

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

Next tactic:
obtain ⟨p, hp, hp1, hp2⟩ := Nat.Prime.not_coprime_iff_dvd.mp h4
private 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 := by
  cases' exists_eq_mul_left_of_dvd (Int.dvd_sub_of_emod_eq hm) with m0 hm2
  cases' exists_eq_mul_left_of_dvd (Int.dvd_sub_of_emod_eq hn) with n0 hn2
  rw [sub_eq_iff_eq_add] at hm2 hn2
  subst m
  subst n
  have h1 : (m0 * 2 + 1) ^ 2 + (n0 * 2 + 1) ^ 2 = 2 * (2 * (m0 ^ 2 + n0 ^ 2 + m0 + n0) + 1) := by
    ring
  have h2 : (m0 * 2 + 1) ^ 2 - (n0 * 2 + 1) ^ 2 = 2 * (2 * (m0 ^ 2 - n0 ^ 2 + m0 - n0)) := by ring
  have h3 : ((m0 * 2 + 1) ^ 2 - (n0 * 2 + 1) ^ 2) / 2 % 2 = 0 := by
    rw [h2, Int.mul_ediv_cancel_left, Int.mul_emod_right]
    exact by decide
  refine' ⟨⟨_, h1⟩, ⟨_, h2⟩, h3, _⟩
  have h20 : (2 : ℤ) ≠ 0 := by decide
  rw [h1, h2, Int.mul_ediv_cancel_left _ h20, Int.mul_ediv_cancel_left _ h20]
  by_contra h4
  obtain ⟨p, hp, hp1, hp2⟩ := Nat.Prime.not_coprime_iff_dvd.mp h4
  

Proof state:
case intro.intro.intro.intro.intro
m0 n0 : ℤ
hm : (m0 * 2 + 1) % 2 = 1
hn : (n0 * 2 + 1) % 2 = 1
h : Int.gcd (m0 * 2 + 1) (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 : ¬Int.gcd (2 * (m0 ^ 2 - n0 ^ 2 + m0 - n0)) (2 * (m0 ^ 2 + n0 ^ 2 + m0 + n0) + 1) = 1
p : ℕ
hp : Nat.Prime p
hp1 : p ∣ Int.natAbs (2 * (m0 ^ 2 - n0 ^ 2 + m0 - n0))
hp2 : p ∣ Int.natAbs (2 * (m0 ^ 2 + n0 ^ 2 + m0 + n0) + 1)
⊢ False

Next tactic:
apply hp.not_dvd_one
private 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 := by
  cases' exists_eq_mul_left_of_dvd (Int.dvd_sub_of_emod_eq hm) with m0 hm2
  cases' exists_eq_mul_left_of_dvd (Int.dvd_sub_of_emod_eq hn) with n0 hn2
  rw [sub_eq_iff_eq_add] at hm2 hn2
  subst m
  subst n
  have h1 : (m0 * 2 + 1) ^ 2 + (n0 * 2 + 1) ^ 2 = 2 * (2 * (m0 ^ 2 + n0 ^ 2 + m0 + n0) + 1) := by
    ring
  have h2 : (m0 * 2 + 1) ^ 2 - (n0 * 2 + 1) ^ 2 = 2 * (2 * (m0 ^ 2 - n0 ^ 2 + m0 - n0)) := by ring
  have h3 : ((m0 * 2 + 1) ^ 2 - (n0 * 2 + 1) ^ 2) / 2 % 2 = 0 := by
    rw [h2, Int.mul_ediv_cancel_left, Int.mul_emod_right]
    exact by decide
  refine' ⟨⟨_, h1⟩, ⟨_, h2⟩, h3, _⟩
  have h20 : (2 : ℤ) ≠ 0 := by decide
  rw [h1, h2, Int.mul_ediv_cancel_left _ h20, Int.mul_ediv_cancel_left _ h20]
  by_contra h4
  obtain ⟨p, hp, hp1, hp2⟩ := Nat.Prime.not_coprime_iff_dvd.mp h4
  apply hp.not_dvd_one
  

Proof state:
case intro.intro.intro.intro.intro
m0 n0 : ℤ
hm : (m0 * 2 + 1) % 2 = 1
hn : (n0 * 2 + 1) % 2 = 1
h : Int.gcd (m0 * 2 + 1) (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 : ¬Int.gcd (2 * (m0 ^ 2 - n0 ^ 2 + m0 - n0)) (2 * (m0 ^ 2 + n0 ^ 2 + m0 + n0) + 1) = 1
p : ℕ
hp : Nat.Prime p
hp1 : p ∣ Int.natAbs (2 * (m0 ^ 2 - n0 ^ 2 + m0 - n0))
hp2 : p ∣ Int.natAbs (2 * (m0 ^ 2 + n0 ^ 2 + m0 + n0) + 1)
⊢ p ∣ 1

Next tactic:
rw [← h]
private 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 := by
  cases' exists_eq_mul_left_of_dvd (Int.dvd_sub_of_emod_eq hm) with m0 hm2
  cases' exists_eq_mul_left_of_dvd (Int.dvd_sub_of_emod_eq hn) with n0 hn2
  rw [sub_eq_iff_eq_add] at hm2 hn2
  subst m
  subst n
  have h1 : (m0 * 2 + 1) ^ 2 + (n0 * 2 + 1) ^ 2 = 2 * (2 * (m0 ^ 2 + n0 ^ 2 + m0 + n0) + 1) := by
    ring
  have h2 : (m0 * 2 + 1) ^ 2 - (n0 * 2 + 1) ^ 2 = 2 * (2 * (m0 ^ 2 - n0 ^ 2 + m0 - n0)) := by ring
  have h3 : ((m0 * 2 + 1) ^ 2 - (n0 * 2 + 1) ^ 2) / 2 % 2 = 0 := by
    rw [h2, Int.mul_ediv_cancel_left, Int.mul_emod_right]
    exact by decide
  refine' ⟨⟨_, h1⟩, ⟨_, h2⟩, h3, _⟩
  have h20 : (2 : ℤ) ≠ 0 := by decide
  rw [h1, h2, Int.mul_ediv_cancel_left _ h20, Int.mul_ediv_cancel_left _ h20]
  by_contra h4
  obtain ⟨p, hp, hp1, hp2⟩ := Nat.Prime.not_coprime_iff_dvd.mp h4
  apply hp.not_dvd_one
  rw [← h]
  

Proof state:
case intro.intro.intro.intro.intro
m0 n0 : ℤ
hm : (m0 * 2 + 1) % 2 = 1
hn : (n0 * 2 + 1) % 2 = 1
h : Int.gcd (m0 * 2 + 1) (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 : ¬Int.gcd (2 * (m0 ^ 2 - n0 ^ 2 + m0 - n0)) (2 * (m0 ^ 2 + n0 ^ 2 + m0 + n0) + 1) = 1
p : ℕ
hp : Nat.Prime p
hp1 : p ∣ Int.natAbs (2 * (m0 ^ 2 - n0 ^ 2 + m0 - n0))
hp2 : p ∣ Int.natAbs (2 * (m0 ^ 2 + n0 ^ 2 + m0 + n0) + 1)
⊢ p ∣ Int.gcd (m0 * 2 + 1) (n0 * 2 + 1)

Next tactic:
rw [← Int.coe_nat_dvd_left] at hp1 hp2
private 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 := by
  cases' exists_eq_mul_left_of_dvd (Int.dvd_sub_of_emod_eq hm) with m0 hm2
  cases' exists_eq_mul_left_of_dvd (Int.dvd_sub_of_emod_eq hn) with n0 hn2
  rw [sub_eq_iff_eq_add] at hm2 hn2
  subst m
  subst n
  have h1 : (m0 * 2 + 1) ^ 2 + (n0 * 2 + 1) ^ 2 = 2 * (2 * (m0 ^ 2 + n0 ^ 2 + m0 + n0) + 1) := by
    ring
  have h2 : (m0 * 2 + 1) ^ 2 - (n0 * 2 + 1) ^ 2 = 2 * (2 * (m0 ^ 2 - n0 ^ 2 + m0 - n0)) := by ring
  have h3 : ((m0 * 2 + 1) ^ 2 - (n0 * 2 + 1) ^ 2) / 2 % 2 = 0 := by
    rw [h2, Int.mul_ediv_cancel_left, Int.mul_emod_right]
    exact by decide
  refine' ⟨⟨_, h1⟩, ⟨_, h2⟩, h3, _⟩
  have h20 : (2 : ℤ) ≠ 0 := by decide
  rw [h1, h2, Int.mul_ediv_cancel_left _ h20, Int.mul_ediv_cancel_left _ h20]
  by_contra h4
  obtain ⟨p, hp, hp1, hp2⟩ := Nat.Prime.not_coprime_iff_dvd.mp h4
  apply hp.not_dvd_one
  rw [← h]
  rw [← Int.coe_nat_dvd_left] at hp1 hp2
  

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

Next tactic:
apply Nat.dvd_gcd
private 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 := by
  cases' exists_eq_mul_left_of_dvd (Int.dvd_sub_of_emod_eq hm) with m0 hm2
  cases' exists_eq_mul_left_of_dvd (Int.dvd_sub_of_emod_eq hn) with n0 hn2
  rw [sub_eq_iff_eq_add] at hm2 hn2
  subst m
  subst n
  have h1 : (m0 * 2 + 1) ^ 2 + (n0 * 2 + 1) ^ 2 = 2 * (2 * (m0 ^ 2 + n0 ^ 2 + m0 + n0) + 1) := by
    ring
  have h2 : (m0 * 2 + 1) ^ 2 - (n0 * 2 + 1) ^ 2 = 2 * (2 * (m0 ^ 2 - n0 ^ 2 + m0 - n0)) := by ring
  have h3 : ((m0 * 2 + 1) ^ 2 - (n0 * 2 + 1) ^ 2) / 2 % 2 = 0 := by
    rw [h2, Int.mul_ediv_cancel_left, Int.mul_emod_right]
    exact by decide
  refine' ⟨⟨_, h1⟩, ⟨_, h2⟩, h3, _⟩
  have h20 : (2 : ℤ) ≠ 0 := by decide
  rw [h1, h2, Int.mul_ediv_cancel_left _ h20, Int.mul_ediv_cancel_left _ h20]
  by_contra h4
  obtain ⟨p, hp, hp1, hp2⟩ := Nat.Prime.not_coprime_iff_dvd.mp h4
  apply hp.not_dvd_one
  rw [← h]
  rw [← Int.coe_nat_dvd_left] at hp1 hp2
  apply Nat.dvd_gcd
  · 

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

Next tactic:
apply Int.Prime.dvd_natAbs_of_coe_dvd_sq hp
private 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 := by
  cases' exists_eq_mul_left_of_dvd (Int.dvd_sub_of_emod_eq hm) with m0 hm2
  cases' exists_eq_mul_left_of_dvd (Int.dvd_sub_of_emod_eq hn) with n0 hn2
  rw [sub_eq_iff_eq_add] at hm2 hn2
  subst m
  subst n
  have h1 : (m0 * 2 + 1) ^ 2 + (n0 * 2 + 1) ^ 2 = 2 * (2 * (m0 ^ 2 + n0 ^ 2 + m0 + n0) + 1) := by
    ring
  have h2 : (m0 * 2 + 1) ^ 2 - (n0 * 2 + 1) ^ 2 = 2 * (2 * (m0 ^ 2 - n0 ^ 2 + m0 - n0)) := by ring
  have h3 : ((m0 * 2 + 1) ^ 2 - (n0 * 2 + 1) ^ 2) / 2 % 2 = 0 := by
    rw [h2, Int.mul_ediv_cancel_left, Int.mul_emod_right]
    exact by decide
  refine' ⟨⟨_, h1⟩, ⟨_, h2⟩, h3, _⟩
  have h20 : (2 : ℤ) ≠ 0 := by decide
  rw [h1, h2, Int.mul_ediv_cancel_left _ h20, Int.mul_ediv_cancel_left _ h20]
  by_contra h4
  obtain ⟨p, hp, hp1, hp2⟩ := Nat.Prime.not_coprime_iff_dvd.mp h4
  apply hp.not_dvd_one
  rw [← h]
  rw [← Int.coe_nat_dvd_left] at hp1 hp2
  apply Nat.dvd_gcd
  · apply Int.Prime.dvd_natAbs_of_coe_dvd_sq hp
    

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

Next tactic:
convert dvd_add hp1 hp2
private 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 := by
  cases' exists_eq_mul_left_of_dvd (Int.dvd_sub_of_emod_eq hm) with m0 hm2
  cases' exists_eq_mul_left_of_dvd (Int.dvd_sub_of_emod_eq hn) with n0 hn2
  rw [sub_eq_iff_eq_add] at hm2 hn2
  subst m
  subst n
  have h1 : (m0 * 2 + 1) ^ 2 + (n0 * 2 + 1) ^ 2 = 2 * (2 * (m0 ^ 2 + n0 ^ 2 + m0 + n0) + 1) := by
    ring
  have h2 : (m0 * 2 + 1) ^ 2 - (n0 * 2 + 1) ^ 2 = 2 * (2 * (m0 ^ 2 - n0 ^ 2 + m0 - n0)) := by ring
  have h3 : ((m0 * 2 + 1) ^ 2 - (n0 * 2 + 1) ^ 2) / 2 % 2 = 0 := by
    rw [h2, Int.mul_ediv_cancel_left, Int.mul_emod_right]
    exact by decide
  refine' ⟨⟨_, h1⟩, ⟨_, h2⟩, h3, _⟩
  have h20 : (2 : ℤ) ≠ 0 := by decide
  rw [h1, h2, Int.mul_ediv_cancel_left _ h20, Int.mul_ediv_cancel_left _ h20]
  by_contra h4
  obtain ⟨p, hp, hp1, hp2⟩ := Nat.Prime.not_coprime_iff_dvd.mp h4
  apply hp.not_dvd_one
  rw [← h]
  rw [← Int.coe_nat_dvd_left] at hp1 hp2
  apply Nat.dvd_gcd
  · apply Int.Prime.dvd_natAbs_of_coe_dvd_sq hp
    convert dvd_add hp1 hp2
    

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

Next tactic:
ring