l3lab/ntp-mathlib huggingface dataset:
theorem isPrimitiveClassified_of_coprime_of_odd_of_pos (hc : Int.gcd x y = 1) (hyo : y % 2 = 1)
    (hzpos : 0 < z) : h.IsPrimitiveClassified := by
  by_cases h0 : x = 0
  · exact h.isPrimitiveClassified_of_coprime_of_zero_left hc h0
  let v := (x : ℚ) / z
  let w := (y : ℚ) / z
  have hq : v ^ 2 + w ^ 2 = 1 := by
    field_simp [sq]
    norm_cast
  have hvz : v ≠ 0 := by
    field_simp
    exact h0
  

Proof state:
case neg
x y z : ℤ
h : PythagoreanTriple x y z
hc : Int.gcd x y = 1
hyo : y % 2 = 1
hzpos : 0 < z
h0 : ¬x = 0
v : ℚ := ↑x / ↑z
w : ℚ := ↑y / ↑z
hq : v ^ 2 + w ^ 2 = 1
hvz : v ≠ 0
⊢ IsPrimitiveClassified h

Next tactic:
have hw1 : w ≠ -1 := by
    contrapose! hvz with hw1
    -- porting note: `contrapose` unfolds local names, refold them
    replace hw1 : w = -1 := hw1; show v = 0
    rw [hw1, neg_sq, one_pow, add_left_eq_self] at hq
    exact pow_eq_zero hq
theorem isPrimitiveClassified_of_coprime_of_odd_of_pos (hc : Int.gcd x y = 1) (hyo : y % 2 = 1)
    (hzpos : 0 < z) : h.IsPrimitiveClassified := by
  by_cases h0 : x = 0
  · exact h.isPrimitiveClassified_of_coprime_of_zero_left hc h0
  let v := (x : ℚ) / z
  let w := (y : ℚ) / z
  have hq : v ^ 2 + w ^ 2 = 1 := by
    field_simp [sq]
    norm_cast
  have hvz : v ≠ 0 := by
    field_simp
    exact h0
  have hw1 : w ≠ -1 := by
    

Proof state:
x y z : ℤ
h : PythagoreanTriple x y z
hc : Int.gcd x y = 1
hyo : y % 2 = 1
hzpos : 0 < z
h0 : ¬x = 0
v : ℚ := ↑x / ↑z
w : ℚ := ↑y / ↑z
hq : v ^ 2 + w ^ 2 = 1
hvz : v ≠ 0
⊢ w ≠ -1

Next tactic:
contrapose! hvz with hw1
theorem isPrimitiveClassified_of_coprime_of_odd_of_pos (hc : Int.gcd x y = 1) (hyo : y % 2 = 1)
    (hzpos : 0 < z) : h.IsPrimitiveClassified := by
  by_cases h0 : x = 0
  · exact h.isPrimitiveClassified_of_coprime_of_zero_left hc h0
  let v := (x : ℚ) / z
  let w := (y : ℚ) / z
  have hq : v ^ 2 + w ^ 2 = 1 := by
    field_simp [sq]
    norm_cast
  have hvz : v ≠ 0 := by
    field_simp
    exact h0
  have hw1 : w ≠ -1 := by
    contrapose! hvz with hw1
    -- porting note: `contrapose` unfolds local names, refold them
    

Proof state:
x y z : ℤ
h : PythagoreanTriple x y z
hc : Int.gcd x y = 1
hyo : y % 2 = 1
hzpos : 0 < z
h0 : ¬x = 0
v : ℚ := ↑x / ↑z
w : ℚ := ↑y / ↑z
hq : v ^ 2 + w ^ 2 = 1
hw1 : w = -1
⊢ v = 0

Next tactic:
replace hw1 : w = -1 := hw1
theorem isPrimitiveClassified_of_coprime_of_odd_of_pos (hc : Int.gcd x y = 1) (hyo : y % 2 = 1)
    (hzpos : 0 < z) : h.IsPrimitiveClassified := by
  by_cases h0 : x = 0
  · exact h.isPrimitiveClassified_of_coprime_of_zero_left hc h0
  let v := (x : ℚ) / z
  let w := (y : ℚ) / z
  have hq : v ^ 2 + w ^ 2 = 1 := by
    field_simp [sq]
    norm_cast
  have hvz : v ≠ 0 := by
    field_simp
    exact h0
  have hw1 : w ≠ -1 := by
    contrapose! hvz with hw1
    -- porting note: `contrapose` unfolds local names, refold them
    replace hw1 : w = -1 := hw1; 

Proof state:
x y z : ℤ
h : PythagoreanTriple x y z
hc : Int.gcd x y = 1
hyo : y % 2 = 1
hzpos : 0 < z
h0 : ¬x = 0
v : ℚ := ↑x / ↑z
w : ℚ := ↑y / ↑z
hq : v ^ 2 + w ^ 2 = 1
hw1 : w = -1
⊢ v = 0

Next tactic:
show v = 0
theorem isPrimitiveClassified_of_coprime_of_odd_of_pos (hc : Int.gcd x y = 1) (hyo : y % 2 = 1)
    (hzpos : 0 < z) : h.IsPrimitiveClassified := by
  by_cases h0 : x = 0
  · exact h.isPrimitiveClassified_of_coprime_of_zero_left hc h0
  let v := (x : ℚ) / z
  let w := (y : ℚ) / z
  have hq : v ^ 2 + w ^ 2 = 1 := by
    field_simp [sq]
    norm_cast
  have hvz : v ≠ 0 := by
    field_simp
    exact h0
  have hw1 : w ≠ -1 := by
    contrapose! hvz with hw1
    -- porting note: `contrapose` unfolds local names, refold them
    replace hw1 : w = -1 := hw1; show v = 0
    

Proof state:
x y z : ℤ
h : PythagoreanTriple x y z
hc : Int.gcd x y = 1
hyo : y % 2 = 1
hzpos : 0 < z
h0 : ¬x = 0
v : ℚ := ↑x / ↑z
w : ℚ := ↑y / ↑z
hq : v ^ 2 + w ^ 2 = 1
hw1 : w = -1
⊢ v = 0

Next tactic:
rw [hw1, neg_sq, one_pow, add_left_eq_self] at hq
theorem isPrimitiveClassified_of_coprime_of_odd_of_pos (hc : Int.gcd x y = 1) (hyo : y % 2 = 1)
    (hzpos : 0 < z) : h.IsPrimitiveClassified := by
  by_cases h0 : x = 0
  · exact h.isPrimitiveClassified_of_coprime_of_zero_left hc h0
  let v := (x : ℚ) / z
  let w := (y : ℚ) / z
  have hq : v ^ 2 + w ^ 2 = 1 := by
    field_simp [sq]
    norm_cast
  have hvz : v ≠ 0 := by
    field_simp
    exact h0
  have hw1 : w ≠ -1 := by
    contrapose! hvz with hw1
    -- porting note: `contrapose` unfolds local names, refold them
    replace hw1 : w = -1 := hw1; show v = 0
    rw [hw1, neg_sq, one_pow, add_left_eq_self] at hq
    

Proof state:
x y z : ℤ
h : PythagoreanTriple x y z
hc : Int.gcd x y = 1
hyo : y % 2 = 1
hzpos : 0 < z
h0 : ¬x = 0
v : ℚ := ↑x / ↑z
w : ℚ := ↑y / ↑z
hq : v ^ 2 = 0
hw1 : w = -1
⊢ v = 0

Next tactic:
exact pow_eq_zero hq
theorem isPrimitiveClassified_of_coprime_of_odd_of_pos (hc : Int.gcd x y = 1) (hyo : y % 2 = 1)
    (hzpos : 0 < z) : h.IsPrimitiveClassified := by
  by_cases h0 : x = 0
  · exact h.isPrimitiveClassified_of_coprime_of_zero_left hc h0
  let v := (x : ℚ) / z
  let w := (y : ℚ) / z
  have hq : v ^ 2 + w ^ 2 = 1 := by
    field_simp [sq]
    norm_cast
  have hvz : v ≠ 0 := by
    field_simp
    exact h0
  have hw1 : w ≠ -1 := by
    contrapose! hvz with hw1
    -- porting note: `contrapose` unfolds local names, refold them
    replace hw1 : w = -1 := hw1; show v = 0
    rw [hw1, neg_sq, one_pow, add_left_eq_self] at hq
    exact pow_eq_zero hq
  

Proof state:
case neg
x y z : ℤ
h : PythagoreanTriple x y z
hc : Int.gcd x y = 1
hyo : y % 2 = 1
hzpos : 0 < z
h0 : ¬x = 0
v : ℚ := ↑x / ↑z
w : ℚ := ↑y / ↑z
hq : v ^ 2 + w ^ 2 = 1
hvz : v ≠ 0
hw1 : w ≠ -1
⊢ IsPrimitiveClassified h

Next tactic:
have hQ : ∀ x : ℚ, 1 + x ^ 2 ≠ 0 := by
    intro q
    apply ne_of_gt
    exact lt_add_of_pos_of_le zero_lt_one (sq_nonneg q)
theorem isPrimitiveClassified_of_coprime_of_odd_of_pos (hc : Int.gcd x y = 1) (hyo : y % 2 = 1)
    (hzpos : 0 < z) : h.IsPrimitiveClassified := by
  by_cases h0 : x = 0
  · exact h.isPrimitiveClassified_of_coprime_of_zero_left hc h0
  let v := (x : ℚ) / z
  let w := (y : ℚ) / z
  have hq : v ^ 2 + w ^ 2 = 1 := by
    field_simp [sq]
    norm_cast
  have hvz : v ≠ 0 := by
    field_simp
    exact h0
  have hw1 : w ≠ -1 := by
    contrapose! hvz with hw1
    -- porting note: `contrapose` unfolds local names, refold them
    replace hw1 : w = -1 := hw1; show v = 0
    rw [hw1, neg_sq, one_pow, add_left_eq_self] at hq
    exact pow_eq_zero hq
  have hQ : ∀ x : ℚ, 1 + x ^ 2 ≠ 0 := by
    

Proof state:
x y z : ℤ
h : PythagoreanTriple x y z
hc : Int.gcd x y = 1
hyo : y % 2 = 1
hzpos : 0 < z
h0 : ¬x = 0
v : ℚ := ↑x / ↑z
w : ℚ := ↑y / ↑z
hq : v ^ 2 + w ^ 2 = 1
hvz : v ≠ 0
hw1 : w ≠ -1
⊢ ∀ (x : ℚ), 1 + x ^ 2 ≠ 0

Next tactic:
intro q
theorem isPrimitiveClassified_of_coprime_of_odd_of_pos (hc : Int.gcd x y = 1) (hyo : y % 2 = 1)
    (hzpos : 0 < z) : h.IsPrimitiveClassified := by
  by_cases h0 : x = 0
  · exact h.isPrimitiveClassified_of_coprime_of_zero_left hc h0
  let v := (x : ℚ) / z
  let w := (y : ℚ) / z
  have hq : v ^ 2 + w ^ 2 = 1 := by
    field_simp [sq]
    norm_cast
  have hvz : v ≠ 0 := by
    field_simp
    exact h0
  have hw1 : w ≠ -1 := by
    contrapose! hvz with hw1
    -- porting note: `contrapose` unfolds local names, refold them
    replace hw1 : w = -1 := hw1; show v = 0
    rw [hw1, neg_sq, one_pow, add_left_eq_self] at hq
    exact pow_eq_zero hq
  have hQ : ∀ x : ℚ, 1 + x ^ 2 ≠ 0 := by
    intro q
    

Proof state:
x y z : ℤ
h : PythagoreanTriple x y z
hc : Int.gcd x y = 1
hyo : y % 2 = 1
hzpos : 0 < z
h0 : ¬x = 0
v : ℚ := ↑x / ↑z
w : ℚ := ↑y / ↑z
hq : v ^ 2 + w ^ 2 = 1
hvz : v ≠ 0
hw1 : w ≠ -1
q : ℚ
⊢ 1 + q ^ 2 ≠ 0

Next tactic:
apply ne_of_gt
theorem isPrimitiveClassified_of_coprime_of_odd_of_pos (hc : Int.gcd x y = 1) (hyo : y % 2 = 1)
    (hzpos : 0 < z) : h.IsPrimitiveClassified := by
  by_cases h0 : x = 0
  · exact h.isPrimitiveClassified_of_coprime_of_zero_left hc h0
  let v := (x : ℚ) / z
  let w := (y : ℚ) / z
  have hq : v ^ 2 + w ^ 2 = 1 := by
    field_simp [sq]
    norm_cast
  have hvz : v ≠ 0 := by
    field_simp
    exact h0
  have hw1 : w ≠ -1 := by
    contrapose! hvz with hw1
    -- porting note: `contrapose` unfolds local names, refold them
    replace hw1 : w = -1 := hw1; show v = 0
    rw [hw1, neg_sq, one_pow, add_left_eq_self] at hq
    exact pow_eq_zero hq
  have hQ : ∀ x : ℚ, 1 + x ^ 2 ≠ 0 := by
    intro q
    apply ne_of_gt
    

Proof state:
case h
x y z : ℤ
h : PythagoreanTriple x y z
hc : Int.gcd x y = 1
hyo : y % 2 = 1
hzpos : 0 < z
h0 : ¬x = 0
v : ℚ := ↑x / ↑z
w : ℚ := ↑y / ↑z
hq : v ^ 2 + w ^ 2 = 1
hvz : v ≠ 0
hw1 : w ≠ -1
q : ℚ
⊢ 0 < 1 + q ^ 2

Next tactic:
exact lt_add_of_pos_of_le zero_lt_one (sq_nonneg q)
theorem isPrimitiveClassified_of_coprime_of_odd_of_pos (hc : Int.gcd x y = 1) (hyo : y % 2 = 1)
    (hzpos : 0 < z) : h.IsPrimitiveClassified := by
  by_cases h0 : x = 0
  · exact h.isPrimitiveClassified_of_coprime_of_zero_left hc h0
  let v := (x : ℚ) / z
  let w := (y : ℚ) / z
  have hq : v ^ 2 + w ^ 2 = 1 := by
    field_simp [sq]
    norm_cast
  have hvz : v ≠ 0 := by
    field_simp
    exact h0
  have hw1 : w ≠ -1 := by
    contrapose! hvz with hw1
    -- porting note: `contrapose` unfolds local names, refold them
    replace hw1 : w = -1 := hw1; show v = 0
    rw [hw1, neg_sq, one_pow, add_left_eq_self] at hq
    exact pow_eq_zero hq
  have hQ : ∀ x : ℚ, 1 + x ^ 2 ≠ 0 := by
    intro q
    apply ne_of_gt
    exact lt_add_of_pos_of_le zero_lt_one (sq_nonneg q)
  

Proof state:
case neg
x y z : ℤ
h : PythagoreanTriple x y z
hc : Int.gcd x y = 1
hyo : y % 2 = 1
hzpos : 0 < z
h0 : ¬x = 0
v : ℚ := ↑x / ↑z
w : ℚ := ↑y / ↑z
hq : v ^ 2 + w ^ 2 = 1
hvz : v ≠ 0
hw1 : w ≠ -1
hQ : ∀ (x : ℚ), 1 + x ^ 2 ≠ 0
⊢ IsPrimitiveClassified h

Next tactic:
have hp : (⟨v, w⟩ : ℚ × ℚ) ∈ { p : ℚ × ℚ | p.1 ^ 2 + p.2 ^ 2 = 1 ∧ p.2 ≠ -1 } := ⟨hq, hw1⟩
theorem isPrimitiveClassified_of_coprime_of_odd_of_pos (hc : Int.gcd x y = 1) (hyo : y % 2 = 1)
    (hzpos : 0 < z) : h.IsPrimitiveClassified := by
  by_cases h0 : x = 0
  · exact h.isPrimitiveClassified_of_coprime_of_zero_left hc h0
  let v := (x : ℚ) / z
  let w := (y : ℚ) / z
  have hq : v ^ 2 + w ^ 2 = 1 := by
    field_simp [sq]
    norm_cast
  have hvz : v ≠ 0 := by
    field_simp
    exact h0
  have hw1 : w ≠ -1 := by
    contrapose! hvz with hw1
    -- porting note: `contrapose` unfolds local names, refold them
    replace hw1 : w = -1 := hw1; show v = 0
    rw [hw1, neg_sq, one_pow, add_left_eq_self] at hq
    exact pow_eq_zero hq
  have hQ : ∀ x : ℚ, 1 + x ^ 2 ≠ 0 := by
    intro q
    apply ne_of_gt
    exact lt_add_of_pos_of_le zero_lt_one (sq_nonneg q)
  have hp : (⟨v, w⟩ : ℚ × ℚ) ∈ { p : ℚ × ℚ | p.1 ^ 2 + p.2 ^ 2 = 1 ∧ p.2 ≠ -1 } := ⟨hq, hw1⟩
  

Proof state:
case neg
x y z : ℤ
h : PythagoreanTriple x y z
hc : Int.gcd x y = 1
hyo : y % 2 = 1
hzpos : 0 < z
h0 : ¬x = 0
v : ℚ := ↑x / ↑z
w : ℚ := ↑y / ↑z
hq : v ^ 2 + w ^ 2 = 1
hvz : v ≠ 0
hw1 : w ≠ -1
hQ : ∀ (x : ℚ), 1 + x ^ 2 ≠ 0
hp : (v, w) ∈ {p | p.1 ^ 2 + p.2 ^ 2 = 1 ∧ p.2 ≠ -1}
⊢ IsPrimitiveClassified h

Next tactic:
let q := (circleEquivGen hQ).symm ⟨⟨v, w⟩, hp⟩
theorem isPrimitiveClassified_of_coprime_of_odd_of_pos (hc : Int.gcd x y = 1) (hyo : y % 2 = 1)
    (hzpos : 0 < z) : h.IsPrimitiveClassified := by
  by_cases h0 : x = 0
  · exact h.isPrimitiveClassified_of_coprime_of_zero_left hc h0
  let v := (x : ℚ) / z
  let w := (y : ℚ) / z
  have hq : v ^ 2 + w ^ 2 = 1 := by
    field_simp [sq]
    norm_cast
  have hvz : v ≠ 0 := by
    field_simp
    exact h0
  have hw1 : w ≠ -1 := by
    contrapose! hvz with hw1
    -- porting note: `contrapose` unfolds local names, refold them
    replace hw1 : w = -1 := hw1; show v = 0
    rw [hw1, neg_sq, one_pow, add_left_eq_self] at hq
    exact pow_eq_zero hq
  have hQ : ∀ x : ℚ, 1 + x ^ 2 ≠ 0 := by
    intro q
    apply ne_of_gt
    exact lt_add_of_pos_of_le zero_lt_one (sq_nonneg q)
  have hp : (⟨v, w⟩ : ℚ × ℚ) ∈ { p : ℚ × ℚ | p.1 ^ 2 + p.2 ^ 2 = 1 ∧ p.2 ≠ -1 } := ⟨hq, hw1⟩
  let q := (circleEquivGen hQ).symm ⟨⟨v, w⟩, hp⟩
  

Proof state:
case neg
x y z : ℤ
h : PythagoreanTriple x y z
hc : Int.gcd x y = 1
hyo : y % 2 = 1
hzpos : 0 < z
h0 : ¬x = 0
v : ℚ := ↑x / ↑z
w : ℚ := ↑y / ↑z
hq : v ^ 2 + w ^ 2 = 1
hvz : v ≠ 0
hw1 : w ≠ -1
hQ : ∀ (x : ℚ), 1 + x ^ 2 ≠ 0
hp : (v, w) ∈ {p | p.1 ^ 2 + p.2 ^ 2 = 1 ∧ p.2 ≠ -1}
q : ℚ := (circleEquivGen hQ).symm { val := (v, w), property := hp }
⊢ IsPrimitiveClassified h

Next tactic:
have ht4 : v = 2 * q / (1 + q ^ 2) ∧ w = (1 - q ^ 2) / (1 + q ^ 2) := by
    apply Prod.mk.inj
    have := ((circleEquivGen hQ).apply_symm_apply ⟨⟨v, w⟩, hp⟩).symm
    exact congr_arg Subtype.val this
theorem isPrimitiveClassified_of_coprime_of_odd_of_pos (hc : Int.gcd x y = 1) (hyo : y % 2 = 1)
    (hzpos : 0 < z) : h.IsPrimitiveClassified := by
  by_cases h0 : x = 0
  · exact h.isPrimitiveClassified_of_coprime_of_zero_left hc h0
  let v := (x : ℚ) / z
  let w := (y : ℚ) / z
  have hq : v ^ 2 + w ^ 2 = 1 := by
    field_simp [sq]
    norm_cast
  have hvz : v ≠ 0 := by
    field_simp
    exact h0
  have hw1 : w ≠ -1 := by
    contrapose! hvz with hw1
    -- porting note: `contrapose` unfolds local names, refold them
    replace hw1 : w = -1 := hw1; show v = 0
    rw [hw1, neg_sq, one_pow, add_left_eq_self] at hq
    exact pow_eq_zero hq
  have hQ : ∀ x : ℚ, 1 + x ^ 2 ≠ 0 := by
    intro q
    apply ne_of_gt
    exact lt_add_of_pos_of_le zero_lt_one (sq_nonneg q)
  have hp : (⟨v, w⟩ : ℚ × ℚ) ∈ { p : ℚ × ℚ | p.1 ^ 2 + p.2 ^ 2 = 1 ∧ p.2 ≠ -1 } := ⟨hq, hw1⟩
  let q := (circleEquivGen hQ).symm ⟨⟨v, w⟩, hp⟩
  have ht4 : v = 2 * q / (1 + q ^ 2) ∧ w = (1 - q ^ 2) / (1 + q ^ 2) := by
    

Proof state:
x y z : ℤ
h : PythagoreanTriple x y z
hc : Int.gcd x y = 1
hyo : y % 2 = 1
hzpos : 0 < z
h0 : ¬x = 0
v : ℚ := ↑x / ↑z
w : ℚ := ↑y / ↑z
hq : v ^ 2 + w ^ 2 = 1
hvz : v ≠ 0
hw1 : w ≠ -1
hQ : ∀ (x : ℚ), 1 + x ^ 2 ≠ 0
hp : (v, w) ∈ {p | p.1 ^ 2 + p.2 ^ 2 = 1 ∧ p.2 ≠ -1}
q : ℚ := (circleEquivGen hQ).symm { val := (v, w), property := hp }
⊢ v = 2 * q / (1 + q ^ 2) ∧ w = (1 - q ^ 2) / (1 + q ^ 2)

Next tactic:
apply Prod.mk.inj
theorem isPrimitiveClassified_of_coprime_of_odd_of_pos (hc : Int.gcd x y = 1) (hyo : y % 2 = 1)
    (hzpos : 0 < z) : h.IsPrimitiveClassified := by
  by_cases h0 : x = 0
  · exact h.isPrimitiveClassified_of_coprime_of_zero_left hc h0
  let v := (x : ℚ) / z
  let w := (y : ℚ) / z
  have hq : v ^ 2 + w ^ 2 = 1 := by
    field_simp [sq]
    norm_cast
  have hvz : v ≠ 0 := by
    field_simp
    exact h0
  have hw1 : w ≠ -1 := by
    contrapose! hvz with hw1
    -- porting note: `contrapose` unfolds local names, refold them
    replace hw1 : w = -1 := hw1; show v = 0
    rw [hw1, neg_sq, one_pow, add_left_eq_self] at hq
    exact pow_eq_zero hq
  have hQ : ∀ x : ℚ, 1 + x ^ 2 ≠ 0 := by
    intro q
    apply ne_of_gt
    exact lt_add_of_pos_of_le zero_lt_one (sq_nonneg q)
  have hp : (⟨v, w⟩ : ℚ × ℚ) ∈ { p : ℚ × ℚ | p.1 ^ 2 + p.2 ^ 2 = 1 ∧ p.2 ≠ -1 } := ⟨hq, hw1⟩
  let q := (circleEquivGen hQ).symm ⟨⟨v, w⟩, hp⟩
  have ht4 : v = 2 * q / (1 + q ^ 2) ∧ w = (1 - q ^ 2) / (1 + q ^ 2) := by
    apply Prod.mk.inj
    

Proof state:
case x
x y z : ℤ
h : PythagoreanTriple x y z
hc : Int.gcd x y = 1
hyo : y % 2 = 1
hzpos : 0 < z
h0 : ¬x = 0
v : ℚ := ↑x / ↑z
w : ℚ := ↑y / ↑z
hq : v ^ 2 + w ^ 2 = 1
hvz : v ≠ 0
hw1 : w ≠ -1
hQ : ∀ (x : ℚ), 1 + x ^ 2 ≠ 0
hp : (v, w) ∈ {p | p.1 ^ 2 + p.2 ^ 2 = 1 ∧ p.2 ≠ -1}
q : ℚ := (circleEquivGen hQ).symm { val := (v, w), property := hp }
⊢ (v, w) = (2 * q / (1 + q ^ 2), (1 - q ^ 2) / (1 + q ^ 2))

Next tactic:
have := ((circleEquivGen hQ).apply_symm_apply ⟨⟨v, w⟩, hp⟩).symm
theorem isPrimitiveClassified_of_coprime_of_odd_of_pos (hc : Int.gcd x y = 1) (hyo : y % 2 = 1)
    (hzpos : 0 < z) : h.IsPrimitiveClassified := by
  by_cases h0 : x = 0
  · exact h.isPrimitiveClassified_of_coprime_of_zero_left hc h0
  let v := (x : ℚ) / z
  let w := (y : ℚ) / z
  have hq : v ^ 2 + w ^ 2 = 1 := by
    field_simp [sq]
    norm_cast
  have hvz : v ≠ 0 := by
    field_simp
    exact h0
  have hw1 : w ≠ -1 := by
    contrapose! hvz with hw1
    -- porting note: `contrapose` unfolds local names, refold them
    replace hw1 : w = -1 := hw1; show v = 0
    rw [hw1, neg_sq, one_pow, add_left_eq_self] at hq
    exact pow_eq_zero hq
  have hQ : ∀ x : ℚ, 1 + x ^ 2 ≠ 0 := by
    intro q
    apply ne_of_gt
    exact lt_add_of_pos_of_le zero_lt_one (sq_nonneg q)
  have hp : (⟨v, w⟩ : ℚ × ℚ) ∈ { p : ℚ × ℚ | p.1 ^ 2 + p.2 ^ 2 = 1 ∧ p.2 ≠ -1 } := ⟨hq, hw1⟩
  let q := (circleEquivGen hQ).symm ⟨⟨v, w⟩, hp⟩
  have ht4 : v = 2 * q / (1 + q ^ 2) ∧ w = (1 - q ^ 2) / (1 + q ^ 2) := by
    apply Prod.mk.inj
    have := ((circleEquivGen hQ).apply_symm_apply ⟨⟨v, w⟩, hp⟩).symm
    

Proof state:
case x
x y z : ℤ
h : PythagoreanTriple x y z
hc : Int.gcd x y = 1
hyo : y % 2 = 1
hzpos : 0 < z
h0 : ¬x = 0
v : ℚ := ↑x / ↑z
w : ℚ := ↑y / ↑z
hq : v ^ 2 + w ^ 2 = 1
hvz : v ≠ 0
hw1 : w ≠ -1
hQ : ∀ (x : ℚ), 1 + x ^ 2 ≠ 0
hp : (v, w) ∈ {p | p.1 ^ 2 + p.2 ^ 2 = 1 ∧ p.2 ≠ -1}
q : ℚ := (circleEquivGen hQ).symm { val := (v, w), property := hp }
this :
  { val := (v, w), property := hp } = (circleEquivGen hQ) ((circleEquivGen hQ).symm { val := (v, w), property := hp })
⊢ (v, w) = (2 * q / (1 + q ^ 2), (1 - q ^ 2) / (1 + q ^ 2))

Next tactic:
exact congr_arg Subtype.val this
theorem isPrimitiveClassified_of_coprime_of_odd_of_pos (hc : Int.gcd x y = 1) (hyo : y % 2 = 1)
    (hzpos : 0 < z) : h.IsPrimitiveClassified := by
  by_cases h0 : x = 0
  · exact h.isPrimitiveClassified_of_coprime_of_zero_left hc h0
  let v := (x : ℚ) / z
  let w := (y : ℚ) / z
  have hq : v ^ 2 + w ^ 2 = 1 := by
    field_simp [sq]
    norm_cast
  have hvz : v ≠ 0 := by
    field_simp
    exact h0
  have hw1 : w ≠ -1 := by
    contrapose! hvz with hw1
    -- porting note: `contrapose` unfolds local names, refold them
    replace hw1 : w = -1 := hw1; show v = 0
    rw [hw1, neg_sq, one_pow, add_left_eq_self] at hq
    exact pow_eq_zero hq
  have hQ : ∀ x : ℚ, 1 + x ^ 2 ≠ 0 := by
    intro q
    apply ne_of_gt
    exact lt_add_of_pos_of_le zero_lt_one (sq_nonneg q)
  have hp : (⟨v, w⟩ : ℚ × ℚ) ∈ { p : ℚ × ℚ | p.1 ^ 2 + p.2 ^ 2 = 1 ∧ p.2 ≠ -1 } := ⟨hq, hw1⟩
  let q := (circleEquivGen hQ).symm ⟨⟨v, w⟩, hp⟩
  have ht4 : v = 2 * q / (1 + q ^ 2) ∧ w = (1 - q ^ 2) / (1 + q ^ 2) := by
    apply Prod.mk.inj
    have := ((circleEquivGen hQ).apply_symm_apply ⟨⟨v, w⟩, hp⟩).symm
    exact congr_arg Subtype.val this
  

Proof state:
case neg
x y z : ℤ
h : PythagoreanTriple x y z
hc : Int.gcd x y = 1
hyo : y % 2 = 1
hzpos : 0 < z
h0 : ¬x = 0
v : ℚ := ↑x / ↑z
w : ℚ := ↑y / ↑z
hq : v ^ 2 + w ^ 2 = 1
hvz : v ≠ 0
hw1 : w ≠ -1
hQ : ∀ (x : ℚ), 1 + x ^ 2 ≠ 0
hp : (v, w) ∈ {p | p.1 ^ 2 + p.2 ^ 2 = 1 ∧ p.2 ≠ -1}
q : ℚ := (circleEquivGen hQ).symm { val := (v, w), property := hp }
ht4 : v = 2 * q / (1 + q ^ 2) ∧ w = (1 - q ^ 2) / (1 + q ^ 2)
⊢ IsPrimitiveClassified h

Next tactic:
let m := (q.den : ℤ)
theorem isPrimitiveClassified_of_coprime_of_odd_of_pos (hc : Int.gcd x y = 1) (hyo : y % 2 = 1)
    (hzpos : 0 < z) : h.IsPrimitiveClassified := by
  by_cases h0 : x = 0
  · exact h.isPrimitiveClassified_of_coprime_of_zero_left hc h0
  let v := (x : ℚ) / z
  let w := (y : ℚ) / z
  have hq : v ^ 2 + w ^ 2 = 1 := by
    field_simp [sq]
    norm_cast
  have hvz : v ≠ 0 := by
    field_simp
    exact h0
  have hw1 : w ≠ -1 := by
    contrapose! hvz with hw1
    -- porting note: `contrapose` unfolds local names, refold them
    replace hw1 : w = -1 := hw1; show v = 0
    rw [hw1, neg_sq, one_pow, add_left_eq_self] at hq
    exact pow_eq_zero hq
  have hQ : ∀ x : ℚ, 1 + x ^ 2 ≠ 0 := by
    intro q
    apply ne_of_gt
    exact lt_add_of_pos_of_le zero_lt_one (sq_nonneg q)
  have hp : (⟨v, w⟩ : ℚ × ℚ) ∈ { p : ℚ × ℚ | p.1 ^ 2 + p.2 ^ 2 = 1 ∧ p.2 ≠ -1 } := ⟨hq, hw1⟩
  let q := (circleEquivGen hQ).symm ⟨⟨v, w⟩, hp⟩
  have ht4 : v = 2 * q / (1 + q ^ 2) ∧ w = (1 - q ^ 2) / (1 + q ^ 2) := by
    apply Prod.mk.inj
    have := ((circleEquivGen hQ).apply_symm_apply ⟨⟨v, w⟩, hp⟩).symm
    exact congr_arg Subtype.val this
  let m := (q.den : ℤ)
  

Proof state:
case neg
x y z : ℤ
h : PythagoreanTriple x y z
hc : Int.gcd x y = 1
hyo : y % 2 = 1
hzpos : 0 < z
h0 : ¬x = 0
v : ℚ := ↑x / ↑z
w : ℚ := ↑y / ↑z
hq : v ^ 2 + w ^ 2 = 1
hvz : v ≠ 0
hw1 : w ≠ -1
hQ : ∀ (x : ℚ), 1 + x ^ 2 ≠ 0
hp : (v, w) ∈ {p | p.1 ^ 2 + p.2 ^ 2 = 1 ∧ p.2 ≠ -1}
q : ℚ := (circleEquivGen hQ).symm { val := (v, w), property := hp }
ht4 : v = 2 * q / (1 + q ^ 2) ∧ w = (1 - q ^ 2) / (1 + q ^ 2)
m : ℤ := ↑q.den
⊢ IsPrimitiveClassified h

Next tactic:
let n := q.num
theorem isPrimitiveClassified_of_coprime_of_odd_of_pos (hc : Int.gcd x y = 1) (hyo : y % 2 = 1)
    (hzpos : 0 < z) : h.IsPrimitiveClassified := by
  by_cases h0 : x = 0
  · exact h.isPrimitiveClassified_of_coprime_of_zero_left hc h0
  let v := (x : ℚ) / z
  let w := (y : ℚ) / z
  have hq : v ^ 2 + w ^ 2 = 1 := by
    field_simp [sq]
    norm_cast
  have hvz : v ≠ 0 := by
    field_simp
    exact h0
  have hw1 : w ≠ -1 := by
    contrapose! hvz with hw1
    -- porting note: `contrapose` unfolds local names, refold them
    replace hw1 : w = -1 := hw1; show v = 0
    rw [hw1, neg_sq, one_pow, add_left_eq_self] at hq
    exact pow_eq_zero hq
  have hQ : ∀ x : ℚ, 1 + x ^ 2 ≠ 0 := by
    intro q
    apply ne_of_gt
    exact lt_add_of_pos_of_le zero_lt_one (sq_nonneg q)
  have hp : (⟨v, w⟩ : ℚ × ℚ) ∈ { p : ℚ × ℚ | p.1 ^ 2 + p.2 ^ 2 = 1 ∧ p.2 ≠ -1 } := ⟨hq, hw1⟩
  let q := (circleEquivGen hQ).symm ⟨⟨v, w⟩, hp⟩
  have ht4 : v = 2 * q / (1 + q ^ 2) ∧ w = (1 - q ^ 2) / (1 + q ^ 2) := by
    apply Prod.mk.inj
    have := ((circleEquivGen hQ).apply_symm_apply ⟨⟨v, w⟩, hp⟩).symm
    exact congr_arg Subtype.val this
  let m := (q.den : ℤ)
  let n := q.num
  

Proof state:
case neg
x y z : ℤ
h : PythagoreanTriple x y z
hc : Int.gcd x y = 1
hyo : y % 2 = 1
hzpos : 0 < z
h0 : ¬x = 0
v : ℚ := ↑x / ↑z
w : ℚ := ↑y / ↑z
hq : v ^ 2 + w ^ 2 = 1
hvz : v ≠ 0
hw1 : w ≠ -1
hQ : ∀ (x : ℚ), 1 + x ^ 2 ≠ 0
hp : (v, w) ∈ {p | p.1 ^ 2 + p.2 ^ 2 = 1 ∧ p.2 ≠ -1}
q : ℚ := (circleEquivGen hQ).symm { val := (v, w), property := hp }
ht4 : v = 2 * q / (1 + q ^ 2) ∧ w = (1 - q ^ 2) / (1 + q ^ 2)
m : ℤ := ↑q.den
n : ℤ := q.num
⊢ IsPrimitiveClassified h

Next tactic:
have hm0 : m ≠ 0 := by
    -- Added to adapt to leanprover/lean4#2734.
    -- Without `unfold_let`, `norm_cast` can't see the coercion.
    -- One might try `zeta := true` in `Tactic.NormCast.derive`,
    -- but that seems to break many other things.
    unfold_let m
    norm_cast
    apply Rat.den_nz q
theorem isPrimitiveClassified_of_coprime_of_odd_of_pos (hc : Int.gcd x y = 1) (hyo : y % 2 = 1)
    (hzpos : 0 < z) : h.IsPrimitiveClassified := by
  by_cases h0 : x = 0
  · exact h.isPrimitiveClassified_of_coprime_of_zero_left hc h0
  let v := (x : ℚ) / z
  let w := (y : ℚ) / z
  have hq : v ^ 2 + w ^ 2 = 1 := by
    field_simp [sq]
    norm_cast
  have hvz : v ≠ 0 := by
    field_simp
    exact h0
  have hw1 : w ≠ -1 := by
    contrapose! hvz with hw1
    -- porting note: `contrapose` unfolds local names, refold them
    replace hw1 : w = -1 := hw1; show v = 0
    rw [hw1, neg_sq, one_pow, add_left_eq_self] at hq
    exact pow_eq_zero hq
  have hQ : ∀ x : ℚ, 1 + x ^ 2 ≠ 0 := by
    intro q
    apply ne_of_gt
    exact lt_add_of_pos_of_le zero_lt_one (sq_nonneg q)
  have hp : (⟨v, w⟩ : ℚ × ℚ) ∈ { p : ℚ × ℚ | p.1 ^ 2 + p.2 ^ 2 = 1 ∧ p.2 ≠ -1 } := ⟨hq, hw1⟩
  let q := (circleEquivGen hQ).symm ⟨⟨v, w⟩, hp⟩
  have ht4 : v = 2 * q / (1 + q ^ 2) ∧ w = (1 - q ^ 2) / (1 + q ^ 2) := by
    apply Prod.mk.inj
    have := ((circleEquivGen hQ).apply_symm_apply ⟨⟨v, w⟩, hp⟩).symm
    exact congr_arg Subtype.val this
  let m := (q.den : ℤ)
  let n := q.num
  have hm0 : m ≠ 0 := by
    -- Added to adapt to leanprover/lean4#2734.
    -- Without `unfold_let`, `norm_cast` can't see the coercion.
    -- One might try `zeta := true` in `Tactic.NormCast.derive`,
    -- but that seems to break many other things.
    

Proof state:
x y z : ℤ
h : PythagoreanTriple x y z
hc : Int.gcd x y = 1
hyo : y % 2 = 1
hzpos : 0 < z
h0 : ¬x = 0
v : ℚ := ↑x / ↑z
w : ℚ := ↑y / ↑z
hq : v ^ 2 + w ^ 2 = 1
hvz : v ≠ 0
hw1 : w ≠ -1
hQ : ∀ (x : ℚ), 1 + x ^ 2 ≠ 0
hp : (v, w) ∈ {p | p.1 ^ 2 + p.2 ^ 2 = 1 ∧ p.2 ≠ -1}
q : ℚ := (circleEquivGen hQ).symm { val := (v, w), property := hp }
ht4 : v = 2 * q / (1 + q ^ 2) ∧ w = (1 - q ^ 2) / (1 + q ^ 2)
m : ℤ := ↑q.den
n : ℤ := q.num
⊢ m ≠ 0

Next tactic:
unfold_let m