l3lab/ntp-mathlib huggingface dataset:
/-- If `d` is positive and not a square, then a fundamental solution exists. -/
theorem exists_of_not_isSquare (h₀ : 0 < d) (hd : ¬IsSquare d) :
    ∃ a : Solution₁ d, IsFundamental a := by
  obtain ⟨a, ha₁, ha₂⟩ := exists_pos_of_not_isSquare h₀ hd
  -- convert to `x : ℕ` to be able to use `Nat.find`
  have P : ∃ x' : ℕ, 1 < x' ∧ ∃ y' : ℤ, 0 < y' ∧ (x' : ℤ) ^ 2 - d * y' ^ 2 = 1 := by
    have hax := a.prop
    lift a.x to ℕ using by 

Proof state:
d : ℤ
h₀ : 0 < d
hd : ¬IsSquare d
a : Solution₁ d
ha₁ : 1 < Solution₁.x a
ha₂ : 0 < Solution₁.y a
hax : Solution₁.x a ^ 2 - d * Solution₁.y a ^ 2 = 1
⊢ 0 ≤ Solution₁.x a

Next tactic:
positivity
/-- If `d` is positive and not a square, then a fundamental solution exists. -/
theorem exists_of_not_isSquare (h₀ : 0 < d) (hd : ¬IsSquare d) :
    ∃ a : Solution₁ d, IsFundamental a := by
  obtain ⟨a, ha₁, ha₂⟩ := exists_pos_of_not_isSquare h₀ hd
  -- convert to `x : ℕ` to be able to use `Nat.find`
  have P : ∃ x' : ℕ, 1 < x' ∧ ∃ y' : ℤ, 0 < y' ∧ (x' : ℤ) ^ 2 - d * y' ^ 2 = 1 := by
    have hax := a.prop
    lift a.x to ℕ using by positivity with ax
    

Proof state:
case intro
d : ℤ
h₀ : 0 < d
hd : ¬IsSquare d
a : Solution₁ d
ha₂ : 0 < Solution₁.y a
ax : ℕ
ha₁✝ ha₁ : 1 < ↑ax
hax✝ hax : ↑ax ^ 2 - d * Solution₁.y a ^ 2 = 1
⊢ ∃ x', 1 < x' ∧ ∃ y', 0 < y' ∧ ↑x' ^ 2 - d * y' ^ 2 = 1

Next tactic:
norm_cast at ha₁
/-- If `d` is positive and not a square, then a fundamental solution exists. -/
theorem exists_of_not_isSquare (h₀ : 0 < d) (hd : ¬IsSquare d) :
    ∃ a : Solution₁ d, IsFundamental a := by
  obtain ⟨a, ha₁, ha₂⟩ := exists_pos_of_not_isSquare h₀ hd
  -- convert to `x : ℕ` to be able to use `Nat.find`
  have P : ∃ x' : ℕ, 1 < x' ∧ ∃ y' : ℤ, 0 < y' ∧ (x' : ℤ) ^ 2 - d * y' ^ 2 = 1 := by
    have hax := a.prop
    lift a.x to ℕ using by positivity with ax
    norm_cast at ha₁
    

Proof state:
case intro
d : ℤ
h₀ : 0 < d
hd : ¬IsSquare d
a : Solution₁ d
ha₂ : 0 < Solution₁.y a
ax : ℕ
ha₁✝ : 1 < ↑ax
hax✝ hax : ↑ax ^ 2 - d * Solution₁.y a ^ 2 = 1
ha₁ : 1 < ax
⊢ ∃ x', 1 < x' ∧ ∃ y', 0 < y' ∧ ↑x' ^ 2 - d * y' ^ 2 = 1

Next tactic:
exact ⟨ax, ha₁, a.y, ha₂, hax⟩
/-- If `d` is positive and not a square, then a fundamental solution exists. -/
theorem exists_of_not_isSquare (h₀ : 0 < d) (hd : ¬IsSquare d) :
    ∃ a : Solution₁ d, IsFundamental a := by
  obtain ⟨a, ha₁, ha₂⟩ := exists_pos_of_not_isSquare h₀ hd
  -- convert to `x : ℕ` to be able to use `Nat.find`
  have P : ∃ x' : ℕ, 1 < x' ∧ ∃ y' : ℤ, 0 < y' ∧ (x' : ℤ) ^ 2 - d * y' ^ 2 = 1 := by
    have hax := a.prop
    lift a.x to ℕ using by positivity with ax
    norm_cast at ha₁
    exact ⟨ax, ha₁, a.y, ha₂, hax⟩
  

Proof state:
case intro.intro
d : ℤ
h₀ : 0 < d
hd : ¬IsSquare d
a : Solution₁ d
ha₁ : 1 < Solution₁.x a
ha₂ : 0 < Solution₁.y a
P : ∃ x', 1 < x' ∧ ∃ y', 0 < y' ∧ ↑x' ^ 2 - d * y' ^ 2 = 1
⊢ ∃ a, IsFundamental a

Next tactic:
classical
  -- to avoid having to show that the predicate is decidable
  let x₁ := Nat.find P
  obtain ⟨hx, y₁, hy₀, hy₁⟩ := Nat.find_spec P
  refine' ⟨mk x₁ y₁ hy₁, by rw [x_mk]; exact mod_cast hx, hy₀, fun {b} hb => _⟩
  rw [x_mk]
  have hb' := (Int.toNat_of_nonneg <| zero_le_one.trans hb.le).symm
  have hb'' := hb
  rw [hb'] at hb ⊢
  norm_cast at hb ⊢
  refine' Nat.find_min' P ⟨hb, |b.y|, abs_pos.mpr <| y_ne_zero_of_one_lt_x hb'', _⟩
  rw [← hb', sq_abs]
  exact b.prop
/-- If `d` is positive and not a square, then a fundamental solution exists. -/
theorem exists_of_not_isSquare (h₀ : 0 < d) (hd : ¬IsSquare d) :
    ∃ a : Solution₁ d, IsFundamental a := by
  obtain ⟨a, ha₁, ha₂⟩ := exists_pos_of_not_isSquare h₀ hd
  -- convert to `x : ℕ` to be able to use `Nat.find`
  have P : ∃ x' : ℕ, 1 < x' ∧ ∃ y' : ℤ, 0 < y' ∧ (x' : ℤ) ^ 2 - d * y' ^ 2 = 1 := by
    have hax := a.prop
    lift a.x to ℕ using by positivity with ax
    norm_cast at ha₁
    exact ⟨ax, ha₁, a.y, ha₂, hax⟩
  classical
  -- to avoid having to show that the predicate is decidable
  

Proof state:
case intro.intro
d : ℤ
h₀ : 0 < d
hd : ¬IsSquare d
a : Solution₁ d
ha₁ : 1 < Solution₁.x a
ha₂ : 0 < Solution₁.y a
P : ∃ x', 1 < x' ∧ ∃ y', 0 < y' ∧ ↑x' ^ 2 - d * y' ^ 2 = 1
⊢ ∃ a, IsFundamental a

Next tactic:
let x₁ := Nat.find P
/-- If `d` is positive and not a square, then a fundamental solution exists. -/
theorem exists_of_not_isSquare (h₀ : 0 < d) (hd : ¬IsSquare d) :
    ∃ a : Solution₁ d, IsFundamental a := by
  obtain ⟨a, ha₁, ha₂⟩ := exists_pos_of_not_isSquare h₀ hd
  -- convert to `x : ℕ` to be able to use `Nat.find`
  have P : ∃ x' : ℕ, 1 < x' ∧ ∃ y' : ℤ, 0 < y' ∧ (x' : ℤ) ^ 2 - d * y' ^ 2 = 1 := by
    have hax := a.prop
    lift a.x to ℕ using by positivity with ax
    norm_cast at ha₁
    exact ⟨ax, ha₁, a.y, ha₂, hax⟩
  classical
  -- to avoid having to show that the predicate is decidable
  let x₁ := Nat.find P
  

Proof state:
case intro.intro
d : ℤ
h₀ : 0 < d
hd : ¬IsSquare d
a : Solution₁ d
ha₁ : 1 < Solution₁.x a
ha₂ : 0 < Solution₁.y a
P : ∃ x', 1 < x' ∧ ∃ y', 0 < y' ∧ ↑x' ^ 2 - d * y' ^ 2 = 1
x₁ : ℕ := Nat.find P
⊢ ∃ a, IsFundamental a

Next tactic:
obtain ⟨hx, y₁, hy₀, hy₁⟩ := Nat.find_spec P
/-- If `d` is positive and not a square, then a fundamental solution exists. -/
theorem exists_of_not_isSquare (h₀ : 0 < d) (hd : ¬IsSquare d) :
    ∃ a : Solution₁ d, IsFundamental a := by
  obtain ⟨a, ha₁, ha₂⟩ := exists_pos_of_not_isSquare h₀ hd
  -- convert to `x : ℕ` to be able to use `Nat.find`
  have P : ∃ x' : ℕ, 1 < x' ∧ ∃ y' : ℤ, 0 < y' ∧ (x' : ℤ) ^ 2 - d * y' ^ 2 = 1 := by
    have hax := a.prop
    lift a.x to ℕ using by positivity with ax
    norm_cast at ha₁
    exact ⟨ax, ha₁, a.y, ha₂, hax⟩
  classical
  -- to avoid having to show that the predicate is decidable
  let x₁ := Nat.find P
  obtain ⟨hx, y₁, hy₀, hy₁⟩ := Nat.find_spec P
  

Proof state:
case intro.intro.intro.intro.intro
d : ℤ
h₀ : 0 < d
hd : ¬IsSquare d
a : Solution₁ d
ha₁ : 1 < Solution₁.x a
ha₂ : 0 < Solution₁.y a
P : ∃ x', 1 < x' ∧ ∃ y', 0 < y' ∧ ↑x' ^ 2 - d * y' ^ 2 = 1
x₁ : ℕ := Nat.find P
hx : 1 < Nat.find P
y₁ : ℤ
hy₀ : 0 < y₁
hy₁ : ↑(Nat.find P) ^ 2 - d * y₁ ^ 2 = 1
⊢ ∃ a, IsFundamental a

Next tactic:
refine' ⟨mk x₁ y₁ hy₁, by rw [x_mk]; exact mod_cast hx, hy₀, fun {b} hb => _⟩
/-- If `d` is positive and not a square, then a fundamental solution exists. -/
theorem exists_of_not_isSquare (h₀ : 0 < d) (hd : ¬IsSquare d) :
    ∃ a : Solution₁ d, IsFundamental a := by
  obtain ⟨a, ha₁, ha₂⟩ := exists_pos_of_not_isSquare h₀ hd
  -- convert to `x : ℕ` to be able to use `Nat.find`
  have P : ∃ x' : ℕ, 1 < x' ∧ ∃ y' : ℤ, 0 < y' ∧ (x' : ℤ) ^ 2 - d * y' ^ 2 = 1 := by
    have hax := a.prop
    lift a.x to ℕ using by positivity with ax
    norm_cast at ha₁
    exact ⟨ax, ha₁, a.y, ha₂, hax⟩
  classical
  -- to avoid having to show that the predicate is decidable
  let x₁ := Nat.find P
  obtain ⟨hx, y₁, hy₀, hy₁⟩ := Nat.find_spec P
  refine' ⟨mk x₁ y₁ hy₁, by 

Proof state:
d : ℤ
h₀ : 0 < d
hd : ¬IsSquare d
a : Solution₁ d
ha₁ : 1 < Solution₁.x a
ha₂ : 0 < Solution₁.y a
P : ∃ x', 1 < x' ∧ ∃ y', 0 < y' ∧ ↑x' ^ 2 - d * y' ^ 2 = 1
x₁ : ℕ := Nat.find P
hx : 1 < Nat.find P
y₁ : ℤ
hy₀ : 0 < y₁
hy₁ : ↑(Nat.find P) ^ 2 - d * y₁ ^ 2 = 1
⊢ 1 < Solution₁.x (Solution₁.mk (↑x₁) y₁ hy₁)

Next tactic:
rw [x_mk]
/-- If `d` is positive and not a square, then a fundamental solution exists. -/
theorem exists_of_not_isSquare (h₀ : 0 < d) (hd : ¬IsSquare d) :
    ∃ a : Solution₁ d, IsFundamental a := by
  obtain ⟨a, ha₁, ha₂⟩ := exists_pos_of_not_isSquare h₀ hd
  -- convert to `x : ℕ` to be able to use `Nat.find`
  have P : ∃ x' : ℕ, 1 < x' ∧ ∃ y' : ℤ, 0 < y' ∧ (x' : ℤ) ^ 2 - d * y' ^ 2 = 1 := by
    have hax := a.prop
    lift a.x to ℕ using by positivity with ax
    norm_cast at ha₁
    exact ⟨ax, ha₁, a.y, ha₂, hax⟩
  classical
  -- to avoid having to show that the predicate is decidable
  let x₁ := Nat.find P
  obtain ⟨hx, y₁, hy₀, hy₁⟩ := Nat.find_spec P
  refine' ⟨mk x₁ y₁ hy₁, by rw [x_mk]; 

Proof state:
d : ℤ
h₀ : 0 < d
hd : ¬IsSquare d
a : Solution₁ d
ha₁ : 1 < Solution₁.x a
ha₂ : 0 < Solution₁.y a
P : ∃ x', 1 < x' ∧ ∃ y', 0 < y' ∧ ↑x' ^ 2 - d * y' ^ 2 = 1
x₁ : ℕ := Nat.find P
hx : 1 < Nat.find P
y₁ : ℤ
hy₀ : 0 < y₁
hy₁ : ↑(Nat.find P) ^ 2 - d * y₁ ^ 2 = 1
⊢ 1 < ↑x₁

Next tactic:
exact mod_cast hx
/-- If `d` is positive and not a square, then a fundamental solution exists. -/
theorem exists_of_not_isSquare (h₀ : 0 < d) (hd : ¬IsSquare d) :
    ∃ a : Solution₁ d, IsFundamental a := by
  obtain ⟨a, ha₁, ha₂⟩ := exists_pos_of_not_isSquare h₀ hd
  -- convert to `x : ℕ` to be able to use `Nat.find`
  have P : ∃ x' : ℕ, 1 < x' ∧ ∃ y' : ℤ, 0 < y' ∧ (x' : ℤ) ^ 2 - d * y' ^ 2 = 1 := by
    have hax := a.prop
    lift a.x to ℕ using by positivity with ax
    norm_cast at ha₁
    exact ⟨ax, ha₁, a.y, ha₂, hax⟩
  classical
  -- to avoid having to show that the predicate is decidable
  let x₁ := Nat.find P
  obtain ⟨hx, y₁, hy₀, hy₁⟩ := Nat.find_spec P
  refine' ⟨mk x₁ y₁ hy₁, by rw [x_mk]; exact mod_cast hx, hy₀, fun {b} hb => _⟩
  

Proof state:
case intro.intro.intro.intro.intro
d : ℤ
h₀ : 0 < d
hd : ¬IsSquare d
a : Solution₁ d
ha₁ : 1 < Solution₁.x a
ha₂ : 0 < Solution₁.y a
P : ∃ x', 1 < x' ∧ ∃ y', 0 < y' ∧ ↑x' ^ 2 - d * y' ^ 2 = 1
x₁ : ℕ := Nat.find P
hx : 1 < Nat.find P
y₁ : ℤ
hy₀ : 0 < y₁
hy₁ : ↑(Nat.find P) ^ 2 - d * y₁ ^ 2 = 1
b : Solution₁ d
hb : 1 < Solution₁.x b
⊢ Solution₁.x (Solution₁.mk (↑x₁) y₁ hy₁) ≤ Solution₁.x b

Next tactic:
rw [x_mk]
/-- If `d` is positive and not a square, then a fundamental solution exists. -/
theorem exists_of_not_isSquare (h₀ : 0 < d) (hd : ¬IsSquare d) :
    ∃ a : Solution₁ d, IsFundamental a := by
  obtain ⟨a, ha₁, ha₂⟩ := exists_pos_of_not_isSquare h₀ hd
  -- convert to `x : ℕ` to be able to use `Nat.find`
  have P : ∃ x' : ℕ, 1 < x' ∧ ∃ y' : ℤ, 0 < y' ∧ (x' : ℤ) ^ 2 - d * y' ^ 2 = 1 := by
    have hax := a.prop
    lift a.x to ℕ using by positivity with ax
    norm_cast at ha₁
    exact ⟨ax, ha₁, a.y, ha₂, hax⟩
  classical
  -- to avoid having to show that the predicate is decidable
  let x₁ := Nat.find P
  obtain ⟨hx, y₁, hy₀, hy₁⟩ := Nat.find_spec P
  refine' ⟨mk x₁ y₁ hy₁, by rw [x_mk]; exact mod_cast hx, hy₀, fun {b} hb => _⟩
  rw [x_mk]
  

Proof state:
case intro.intro.intro.intro.intro
d : ℤ
h₀ : 0 < d
hd : ¬IsSquare d
a : Solution₁ d
ha₁ : 1 < Solution₁.x a
ha₂ : 0 < Solution₁.y a
P : ∃ x', 1 < x' ∧ ∃ y', 0 < y' ∧ ↑x' ^ 2 - d * y' ^ 2 = 1
x₁ : ℕ := Nat.find P
hx : 1 < Nat.find P
y₁ : ℤ
hy₀ : 0 < y₁
hy₁ : ↑(Nat.find P) ^ 2 - d * y₁ ^ 2 = 1
b : Solution₁ d
hb : 1 < Solution₁.x b
⊢ ↑x₁ ≤ Solution₁.x b

Next tactic:
have hb' := (Int.toNat_of_nonneg <| zero_le_one.trans hb.le).symm
/-- If `d` is positive and not a square, then a fundamental solution exists. -/
theorem exists_of_not_isSquare (h₀ : 0 < d) (hd : ¬IsSquare d) :
    ∃ a : Solution₁ d, IsFundamental a := by
  obtain ⟨a, ha₁, ha₂⟩ := exists_pos_of_not_isSquare h₀ hd
  -- convert to `x : ℕ` to be able to use `Nat.find`
  have P : ∃ x' : ℕ, 1 < x' ∧ ∃ y' : ℤ, 0 < y' ∧ (x' : ℤ) ^ 2 - d * y' ^ 2 = 1 := by
    have hax := a.prop
    lift a.x to ℕ using by positivity with ax
    norm_cast at ha₁
    exact ⟨ax, ha₁, a.y, ha₂, hax⟩
  classical
  -- to avoid having to show that the predicate is decidable
  let x₁ := Nat.find P
  obtain ⟨hx, y₁, hy₀, hy₁⟩ := Nat.find_spec P
  refine' ⟨mk x₁ y₁ hy₁, by rw [x_mk]; exact mod_cast hx, hy₀, fun {b} hb => _⟩
  rw [x_mk]
  have hb' := (Int.toNat_of_nonneg <| zero_le_one.trans hb.le).symm
  

Proof state:
case intro.intro.intro.intro.intro
d : ℤ
h₀ : 0 < d
hd : ¬IsSquare d
a : Solution₁ d
ha₁ : 1 < Solution₁.x a
ha₂ : 0 < Solution₁.y a
P : ∃ x', 1 < x' ∧ ∃ y', 0 < y' ∧ ↑x' ^ 2 - d * y' ^ 2 = 1
x₁ : ℕ := Nat.find P
hx : 1 < Nat.find P
y₁ : ℤ
hy₀ : 0 < y₁
hy₁ : ↑(Nat.find P) ^ 2 - d * y₁ ^ 2 = 1
b : Solution₁ d
hb : 1 < Solution₁.x b
hb' : Solution₁.x b = ↑(Int.toNat (Solution₁.x b))
⊢ ↑x₁ ≤ Solution₁.x b

Next tactic:
have hb'' := hb
/-- If `d` is positive and not a square, then a fundamental solution exists. -/
theorem exists_of_not_isSquare (h₀ : 0 < d) (hd : ¬IsSquare d) :
    ∃ a : Solution₁ d, IsFundamental a := by
  obtain ⟨a, ha₁, ha₂⟩ := exists_pos_of_not_isSquare h₀ hd
  -- convert to `x : ℕ` to be able to use `Nat.find`
  have P : ∃ x' : ℕ, 1 < x' ∧ ∃ y' : ℤ, 0 < y' ∧ (x' : ℤ) ^ 2 - d * y' ^ 2 = 1 := by
    have hax := a.prop
    lift a.x to ℕ using by positivity with ax
    norm_cast at ha₁
    exact ⟨ax, ha₁, a.y, ha₂, hax⟩
  classical
  -- to avoid having to show that the predicate is decidable
  let x₁ := Nat.find P
  obtain ⟨hx, y₁, hy₀, hy₁⟩ := Nat.find_spec P
  refine' ⟨mk x₁ y₁ hy₁, by rw [x_mk]; exact mod_cast hx, hy₀, fun {b} hb => _⟩
  rw [x_mk]
  have hb' := (Int.toNat_of_nonneg <| zero_le_one.trans hb.le).symm
  have hb'' := hb
  

Proof state:
case intro.intro.intro.intro.intro
d : ℤ
h₀ : 0 < d
hd : ¬IsSquare d
a : Solution₁ d
ha₁ : 1 < Solution₁.x a
ha₂ : 0 < Solution₁.y a
P : ∃ x', 1 < x' ∧ ∃ y', 0 < y' ∧ ↑x' ^ 2 - d * y' ^ 2 = 1
x₁ : ℕ := Nat.find P
hx : 1 < Nat.find P
y₁ : ℤ
hy₀ : 0 < y₁
hy₁ : ↑(Nat.find P) ^ 2 - d * y₁ ^ 2 = 1
b : Solution₁ d
hb : 1 < Solution₁.x b
hb' : Solution₁.x b = ↑(Int.toNat (Solution₁.x b))
hb'' : 1 < Solution₁.x b
⊢ ↑x₁ ≤ Solution₁.x b

Next tactic:
rw [hb'] at hb ⊢
/-- If `d` is positive and not a square, then a fundamental solution exists. -/
theorem exists_of_not_isSquare (h₀ : 0 < d) (hd : ¬IsSquare d) :
    ∃ a : Solution₁ d, IsFundamental a := by
  obtain ⟨a, ha₁, ha₂⟩ := exists_pos_of_not_isSquare h₀ hd
  -- convert to `x : ℕ` to be able to use `Nat.find`
  have P : ∃ x' : ℕ, 1 < x' ∧ ∃ y' : ℤ, 0 < y' ∧ (x' : ℤ) ^ 2 - d * y' ^ 2 = 1 := by
    have hax := a.prop
    lift a.x to ℕ using by positivity with ax
    norm_cast at ha₁
    exact ⟨ax, ha₁, a.y, ha₂, hax⟩
  classical
  -- to avoid having to show that the predicate is decidable
  let x₁ := Nat.find P
  obtain ⟨hx, y₁, hy₀, hy₁⟩ := Nat.find_spec P
  refine' ⟨mk x₁ y₁ hy₁, by rw [x_mk]; exact mod_cast hx, hy₀, fun {b} hb => _⟩
  rw [x_mk]
  have hb' := (Int.toNat_of_nonneg <| zero_le_one.trans hb.le).symm
  have hb'' := hb
  rw [hb'] at hb ⊢
  

Proof state:
case intro.intro.intro.intro.intro
d : ℤ
h₀ : 0 < d
hd : ¬IsSquare d
a : Solution₁ d
ha₁ : 1 < Solution₁.x a
ha₂ : 0 < Solution₁.y a
P : ∃ x', 1 < x' ∧ ∃ y', 0 < y' ∧ ↑x' ^ 2 - d * y' ^ 2 = 1
x₁ : ℕ := Nat.find P
hx : 1 < Nat.find P
y₁ : ℤ
hy₀ : 0 < y₁
hy₁ : ↑(Nat.find P) ^ 2 - d * y₁ ^ 2 = 1
b : Solution₁ d
hb : 1 < ↑(Int.toNat (Solution₁.x b))
hb' : Solution₁.x b = ↑(Int.toNat (Solution₁.x b))
hb'' : 1 < Solution₁.x b
⊢ ↑x₁ ≤ ↑(Int.toNat (Solution₁.x b))

Next tactic:
norm_cast at hb ⊢
/-- If `d` is positive and not a square, then a fundamental solution exists. -/
theorem exists_of_not_isSquare (h₀ : 0 < d) (hd : ¬IsSquare d) :
    ∃ a : Solution₁ d, IsFundamental a := by
  obtain ⟨a, ha₁, ha₂⟩ := exists_pos_of_not_isSquare h₀ hd
  -- convert to `x : ℕ` to be able to use `Nat.find`
  have P : ∃ x' : ℕ, 1 < x' ∧ ∃ y' : ℤ, 0 < y' ∧ (x' : ℤ) ^ 2 - d * y' ^ 2 = 1 := by
    have hax := a.prop
    lift a.x to ℕ using by positivity with ax
    norm_cast at ha₁
    exact ⟨ax, ha₁, a.y, ha₂, hax⟩
  classical
  -- to avoid having to show that the predicate is decidable
  let x₁ := Nat.find P
  obtain ⟨hx, y₁, hy₀, hy₁⟩ := Nat.find_spec P
  refine' ⟨mk x₁ y₁ hy₁, by rw [x_mk]; exact mod_cast hx, hy₀, fun {b} hb => _⟩
  rw [x_mk]
  have hb' := (Int.toNat_of_nonneg <| zero_le_one.trans hb.le).symm
  have hb'' := hb
  rw [hb'] at hb ⊢
  norm_cast at hb ⊢
  

Proof state:
case intro.intro.intro.intro.intro
d : ℤ
h₀ : 0 < d
hd : ¬IsSquare d
a : Solution₁ d
ha₁ : 1 < Solution₁.x a
ha₂ : 0 < Solution₁.y a
P : ∃ x', 1 < x' ∧ ∃ y', 0 < y' ∧ ↑x' ^ 2 - d * y' ^ 2 = 1
x₁ : ℕ := Nat.find P
hx : 1 < Nat.find P
y₁ : ℤ
hy₀ : 0 < y₁
hy₁ : ↑(Nat.find P) ^ 2 - d * y₁ ^ 2 = 1
b : Solution₁ d
hb' : Solution₁.x b = ↑(Int.toNat (Solution₁.x b))
hb'' : 1 < Solution₁.x b
hb : 1 < Int.toNat (Solution₁.x b)
⊢ x₁ ≤ Int.toNat (Solution₁.x b)

Next tactic:
refine' Nat.find_min' P ⟨hb, |b.y|, abs_pos.mpr <| y_ne_zero_of_one_lt_x hb'', _⟩
/-- If `d` is positive and not a square, then a fundamental solution exists. -/
theorem exists_of_not_isSquare (h₀ : 0 < d) (hd : ¬IsSquare d) :
    ∃ a : Solution₁ d, IsFundamental a := by
  obtain ⟨a, ha₁, ha₂⟩ := exists_pos_of_not_isSquare h₀ hd
  -- convert to `x : ℕ` to be able to use `Nat.find`
  have P : ∃ x' : ℕ, 1 < x' ∧ ∃ y' : ℤ, 0 < y' ∧ (x' : ℤ) ^ 2 - d * y' ^ 2 = 1 := by
    have hax := a.prop
    lift a.x to ℕ using by positivity with ax
    norm_cast at ha₁
    exact ⟨ax, ha₁, a.y, ha₂, hax⟩
  classical
  -- to avoid having to show that the predicate is decidable
  let x₁ := Nat.find P
  obtain ⟨hx, y₁, hy₀, hy₁⟩ := Nat.find_spec P
  refine' ⟨mk x₁ y₁ hy₁, by rw [x_mk]; exact mod_cast hx, hy₀, fun {b} hb => _⟩
  rw [x_mk]
  have hb' := (Int.toNat_of_nonneg <| zero_le_one.trans hb.le).symm
  have hb'' := hb
  rw [hb'] at hb ⊢
  norm_cast at hb ⊢
  refine' Nat.find_min' P ⟨hb, |b.y|, abs_pos.mpr <| y_ne_zero_of_one_lt_x hb'', _⟩
  

Proof state:
case intro.intro.intro.intro.intro
d : ℤ
h₀ : 0 < d
hd : ¬IsSquare d
a : Solution₁ d
ha₁ : 1 < Solution₁.x a
ha₂ : 0 < Solution₁.y a
P : ∃ x', 1 < x' ∧ ∃ y', 0 < y' ∧ ↑x' ^ 2 - d * y' ^ 2 = 1
x₁ : ℕ := Nat.find P
hx : 1 < Nat.find P
y₁ : ℤ
hy₀ : 0 < y₁
hy₁ : ↑(Nat.find P) ^ 2 - d * y₁ ^ 2 = 1
b : Solution₁ d
hb' : Solution₁.x b = ↑(Int.toNat (Solution₁.x b))
hb'' : 1 < Solution₁.x b
hb : 1 < Int.toNat (Solution₁.x b)
⊢ ↑(Int.toNat (Solution₁.x b)) ^ 2 - d * |Solution₁.y b| ^ 2 = 1

Next tactic:
rw [← hb', sq_abs]
/-- If `d` is positive and not a square, then a fundamental solution exists. -/
theorem exists_of_not_isSquare (h₀ : 0 < d) (hd : ¬IsSquare d) :
    ∃ a : Solution₁ d, IsFundamental a := by
  obtain ⟨a, ha₁, ha₂⟩ := exists_pos_of_not_isSquare h₀ hd
  -- convert to `x : ℕ` to be able to use `Nat.find`
  have P : ∃ x' : ℕ, 1 < x' ∧ ∃ y' : ℤ, 0 < y' ∧ (x' : ℤ) ^ 2 - d * y' ^ 2 = 1 := by
    have hax := a.prop
    lift a.x to ℕ using by positivity with ax
    norm_cast at ha₁
    exact ⟨ax, ha₁, a.y, ha₂, hax⟩
  classical
  -- to avoid having to show that the predicate is decidable
  let x₁ := Nat.find P
  obtain ⟨hx, y₁, hy₀, hy₁⟩ := Nat.find_spec P
  refine' ⟨mk x₁ y₁ hy₁, by rw [x_mk]; exact mod_cast hx, hy₀, fun {b} hb => _⟩
  rw [x_mk]
  have hb' := (Int.toNat_of_nonneg <| zero_le_one.trans hb.le).symm
  have hb'' := hb
  rw [hb'] at hb ⊢
  norm_cast at hb ⊢
  refine' Nat.find_min' P ⟨hb, |b.y|, abs_pos.mpr <| y_ne_zero_of_one_lt_x hb'', _⟩
  rw [← hb', sq_abs]
  

Proof state:
case intro.intro.intro.intro.intro
d : ℤ
h₀ : 0 < d
hd : ¬IsSquare d
a : Solution₁ d
ha₁ : 1 < Solution₁.x a
ha₂ : 0 < Solution₁.y a
P : ∃ x', 1 < x' ∧ ∃ y', 0 < y' ∧ ↑x' ^ 2 - d * y' ^ 2 = 1
x₁ : ℕ := Nat.find P
hx : 1 < Nat.find P
y₁ : ℤ
hy₀ : 0 < y₁
hy₁ : ↑(Nat.find P) ^ 2 - d * y₁ ^ 2 = 1
b : Solution₁ d
hb' : Solution₁.x b = ↑(Int.toNat (Solution₁.x b))
hb'' : 1 < Solution₁.x b
hb : 1 < Int.toNat (Solution₁.x b)
⊢ Solution₁.x b ^ 2 - d * Solution₁.y b ^ 2 = 1

Next tactic:
exact b.prop
/-- The map sending an integer `n` to the `y`-coordinate of `a^n` for a fundamental
solution `a` is stritcly increasing. -/
theorem y_strictMono {a : Solution₁ d} (h : IsFundamental a) :
    StrictMono fun n : ℤ => (a ^ n).y := by
  

Proof state:
d : ℤ
a : Solution₁ d
h : IsFundamental a
⊢ StrictMono fun n => Solution₁.y (a ^ n)

Next tactic:
have H : ∀ n : ℤ, 0 ≤ n → (a ^ n).y < (a ^ (n + 1)).y := by
    intro n hn
    rw [← sub_pos, zpow_add, zpow_one, y_mul, add_sub_assoc]
    rw [show (a ^ n).y * a.x - (a ^ n).y = (a ^ n).y * (a.x - 1) by ring]
    refine'
      add_pos_of_pos_of_nonneg (mul_pos (x_zpow_pos h.x_pos _) h.2.1)
        (mul_nonneg _ (by rw [sub_nonneg]; exact h.1.le))
    rcases hn.eq_or_lt with (rfl | hn)
    · simp only [zpow_zero, y_one, le_refl]
    · exact (y_zpow_pos h.x_pos h.2.1 hn).le
/-- The map sending an integer `n` to the `y`-coordinate of `a^n` for a fundamental
solution `a` is stritcly increasing. -/
theorem y_strictMono {a : Solution₁ d} (h : IsFundamental a) :
    StrictMono fun n : ℤ => (a ^ n).y := by
  have H : ∀ n : ℤ, 0 ≤ n → (a ^ n).y < (a ^ (n + 1)).y := by
    

Proof state:
d : ℤ
a : Solution₁ d
h : IsFundamental a
⊢ ∀ (n : ℤ), 0 ≤ n → Solution₁.y (a ^ n) < Solution₁.y (a ^ (n + 1))

Next tactic:
intro n hn
/-- The map sending an integer `n` to the `y`-coordinate of `a^n` for a fundamental
solution `a` is stritcly increasing. -/
theorem y_strictMono {a : Solution₁ d} (h : IsFundamental a) :
    StrictMono fun n : ℤ => (a ^ n).y := by
  have H : ∀ n : ℤ, 0 ≤ n → (a ^ n).y < (a ^ (n + 1)).y := by
    intro n hn
    

Proof state:
d : ℤ
a : Solution₁ d
h : IsFundamental a
n : ℤ
hn : 0 ≤ n
⊢ Solution₁.y (a ^ n) < Solution₁.y (a ^ (n + 1))

Next tactic:
rw [← sub_pos, zpow_add, zpow_one, y_mul, add_sub_assoc]
/-- The map sending an integer `n` to the `y`-coordinate of `a^n` for a fundamental
solution `a` is stritcly increasing. -/
theorem y_strictMono {a : Solution₁ d} (h : IsFundamental a) :
    StrictMono fun n : ℤ => (a ^ n).y := by
  have H : ∀ n : ℤ, 0 ≤ n → (a ^ n).y < (a ^ (n + 1)).y := by
    intro n hn
    rw [← sub_pos, zpow_add, zpow_one, y_mul, add_sub_assoc]
    

Proof state:
d : ℤ
a : Solution₁ d
h : IsFundamental a
n : ℤ
hn : 0 ≤ n
⊢ 0 < Solution₁.x (a ^ n) * Solution₁.y a + (Solution₁.y (a ^ n) * Solution₁.x a - Solution₁.y (a ^ n))

Next tactic:
rw [show (a ^ n).y * a.x - (a ^ n).y = (a ^ n).y * (a.x - 1) by ring]
/-- The map sending an integer `n` to the `y`-coordinate of `a^n` for a fundamental
solution `a` is stritcly increasing. -/
theorem y_strictMono {a : Solution₁ d} (h : IsFundamental a) :
    StrictMono fun n : ℤ => (a ^ n).y := by
  have H : ∀ n : ℤ, 0 ≤ n → (a ^ n).y < (a ^ (n + 1)).y := by
    intro n hn
    rw [← sub_pos, zpow_add, zpow_one, y_mul, add_sub_assoc]
    rw [show (a ^ n).y * a.x - (a ^ n).y = (a ^ n).y * (a.x - 1) by 

Proof state:
d : ℤ
a : Solution₁ d
h : IsFundamental a
n : ℤ
hn : 0 ≤ n
⊢ Solution₁.y (a ^ n) * Solution₁.x a - Solution₁.y (a ^ n) = Solution₁.y (a ^ n) * (Solution₁.x a - 1)

Next tactic:
ring
/-- The map sending an integer `n` to the `y`-coordinate of `a^n` for a fundamental
solution `a` is stritcly increasing. -/
theorem y_strictMono {a : Solution₁ d} (h : IsFundamental a) :
    StrictMono fun n : ℤ => (a ^ n).y := by
  have H : ∀ n : ℤ, 0 ≤ n → (a ^ n).y < (a ^ (n + 1)).y := by
    intro n hn
    rw [← sub_pos, zpow_add, zpow_one, y_mul, add_sub_assoc]
    rw [show (a ^ n).y * a.x - (a ^ n).y = (a ^ n).y * (a.x - 1) by ring]
    

Proof state:
d : ℤ
a : Solution₁ d
h : IsFundamental a
n : ℤ
hn : 0 ≤ n
⊢ 0 < Solution₁.x (a ^ n) * Solution₁.y a + Solution₁.y (a ^ n) * (Solution₁.x a - 1)

Next tactic:
refine'
      add_pos_of_pos_of_nonneg (mul_pos (x_zpow_pos h.x_pos _) h.2.1)
        (mul_nonneg _ (by rw [sub_nonneg]; exact h.1.le))
/-- The map sending an integer `n` to the `y`-coordinate of `a^n` for a fundamental
solution `a` is stritcly increasing. -/
theorem y_strictMono {a : Solution₁ d} (h : IsFundamental a) :
    StrictMono fun n : ℤ => (a ^ n).y := by
  have H : ∀ n : ℤ, 0 ≤ n → (a ^ n).y < (a ^ (n + 1)).y := by
    intro n hn
    rw [← sub_pos, zpow_add, zpow_one, y_mul, add_sub_assoc]
    rw [show (a ^ n).y * a.x - (a ^ n).y = (a ^ n).y * (a.x - 1) by ring]
    refine'
      add_pos_of_pos_of_nonneg (mul_pos (x_zpow_pos h.x_pos _) h.2.1)
        (mul_nonneg _ (by 

Proof state:
d : ℤ
a : Solution₁ d
h : IsFundamental a
n : ℤ
hn : 0 ≤ n
⊢ 0 ≤ Solution₁.x a - 1

Next tactic:
rw [sub_nonneg]
/-- The map sending an integer `n` to the `y`-coordinate of `a^n` for a fundamental
solution `a` is stritcly increasing. -/
theorem y_strictMono {a : Solution₁ d} (h : IsFundamental a) :
    StrictMono fun n : ℤ => (a ^ n).y := by
  have H : ∀ n : ℤ, 0 ≤ n → (a ^ n).y < (a ^ (n + 1)).y := by
    intro n hn
    rw [← sub_pos, zpow_add, zpow_one, y_mul, add_sub_assoc]
    rw [show (a ^ n).y * a.x - (a ^ n).y = (a ^ n).y * (a.x - 1) by ring]
    refine'
      add_pos_of_pos_of_nonneg (mul_pos (x_zpow_pos h.x_pos _) h.2.1)
        (mul_nonneg _ (by rw [sub_nonneg]; 

Proof state:
d : ℤ
a : Solution₁ d
h : IsFundamental a
n : ℤ
hn : 0 ≤ n
⊢ 1 ≤ Solution₁.x a

Next tactic:
exact h.1.le