l3lab/ntp-mathlib huggingface dataset:
/-- If we multiply a positive solution with the inverse of a fundamental solution,
the `x`-coordinate stays positive. -/
theorem mul_inv_x_pos {a₁ : Solution₁ d} (h : IsFundamental a₁) {a : Solution₁ d} (hax : 1 < a.x)
    (hay : 0 < a.y) : 0 < (a * a₁⁻¹).x := by
  simp only [x_mul, x_inv, y_inv, mul_neg, lt_add_neg_iff_add_lt, zero_add]
  refine' (mul_lt_mul_left <| zero_lt_one.trans hax).mp _
  

Proof state:
d : ℤ
a₁ : Solution₁ d
h : IsFundamental a₁
a : Solution₁ d
hax : 1 < Solution₁.x a
hay : 0 < Solution₁.y a
⊢ Solution₁.x a * (d * (Solution₁.y a * Solution₁.y a₁)) < Solution₁.x a * (Solution₁.x a * Solution₁.x a₁)

Next tactic:
rw [(by ring : a.x * (d * (a.y * a₁.y)) = d * a.y * (a.x * a₁.y))]
/-- If we multiply a positive solution with the inverse of a fundamental solution,
the `x`-coordinate stays positive. -/
theorem mul_inv_x_pos {a₁ : Solution₁ d} (h : IsFundamental a₁) {a : Solution₁ d} (hax : 1 < a.x)
    (hay : 0 < a.y) : 0 < (a * a₁⁻¹).x := by
  simp only [x_mul, x_inv, y_inv, mul_neg, lt_add_neg_iff_add_lt, zero_add]
  refine' (mul_lt_mul_left <| zero_lt_one.trans hax).mp _
  rw [(by 

Proof state:
d : ℤ
a₁ : Solution₁ d
h : IsFundamental a₁
a : Solution₁ d
hax : 1 < Solution₁.x a
hay : 0 < Solution₁.y a
⊢ Solution₁.x a * (d * (Solution₁.y a * Solution₁.y a₁)) = d * Solution₁.y a * (Solution₁.x a * Solution₁.y a₁)

Next tactic:
ring
/-- If we multiply a positive solution with the inverse of a fundamental solution,
the `x`-coordinate stays positive. -/
theorem mul_inv_x_pos {a₁ : Solution₁ d} (h : IsFundamental a₁) {a : Solution₁ d} (hax : 1 < a.x)
    (hay : 0 < a.y) : 0 < (a * a₁⁻¹).x := by
  simp only [x_mul, x_inv, y_inv, mul_neg, lt_add_neg_iff_add_lt, zero_add]
  refine' (mul_lt_mul_left <| zero_lt_one.trans hax).mp _
  rw [(by ring : a.x * (d * (a.y * a₁.y)) = d * a.y * (a.x * a₁.y))]
  

Proof state:
d : ℤ
a₁ : Solution₁ d
h : IsFundamental a₁
a : Solution₁ d
hax : 1 < Solution₁.x a
hay : 0 < Solution₁.y a
⊢ d * Solution₁.y a * (Solution₁.x a * Solution₁.y a₁) < Solution₁.x a * (Solution₁.x a * Solution₁.x a₁)

Next tactic:
refine' ((mul_le_mul_left <| mul_pos h.d_pos hay).mpr <| x_mul_y_le_y_mul_x h hax hay).trans_lt _
/-- If we multiply a positive solution with the inverse of a fundamental solution,
the `x`-coordinate stays positive. -/
theorem mul_inv_x_pos {a₁ : Solution₁ d} (h : IsFundamental a₁) {a : Solution₁ d} (hax : 1 < a.x)
    (hay : 0 < a.y) : 0 < (a * a₁⁻¹).x := by
  simp only [x_mul, x_inv, y_inv, mul_neg, lt_add_neg_iff_add_lt, zero_add]
  refine' (mul_lt_mul_left <| zero_lt_one.trans hax).mp _
  rw [(by ring : a.x * (d * (a.y * a₁.y)) = d * a.y * (a.x * a₁.y))]
  refine' ((mul_le_mul_left <| mul_pos h.d_pos hay).mpr <| x_mul_y_le_y_mul_x h hax hay).trans_lt _
  

Proof state:
d : ℤ
a₁ : Solution₁ d
h : IsFundamental a₁
a : Solution₁ d
hax : 1 < Solution₁.x a
hay : 0 < Solution₁.y a
⊢ d * Solution₁.y a * (Solution₁.y a * Solution₁.x a₁) < Solution₁.x a * (Solution₁.x a * Solution₁.x a₁)

Next tactic:
rw [← mul_assoc, mul_assoc d, ← sq, a.prop_y, ← sub_pos]
/-- If we multiply a positive solution with the inverse of a fundamental solution,
the `x`-coordinate stays positive. -/
theorem mul_inv_x_pos {a₁ : Solution₁ d} (h : IsFundamental a₁) {a : Solution₁ d} (hax : 1 < a.x)
    (hay : 0 < a.y) : 0 < (a * a₁⁻¹).x := by
  simp only [x_mul, x_inv, y_inv, mul_neg, lt_add_neg_iff_add_lt, zero_add]
  refine' (mul_lt_mul_left <| zero_lt_one.trans hax).mp _
  rw [(by ring : a.x * (d * (a.y * a₁.y)) = d * a.y * (a.x * a₁.y))]
  refine' ((mul_le_mul_left <| mul_pos h.d_pos hay).mpr <| x_mul_y_le_y_mul_x h hax hay).trans_lt _
  rw [← mul_assoc, mul_assoc d, ← sq, a.prop_y, ← sub_pos]
  

Proof state:
d : ℤ
a₁ : Solution₁ d
h : IsFundamental a₁
a : Solution₁ d
hax : 1 < Solution₁.x a
hay : 0 < Solution₁.y a
⊢ 0 < Solution₁.x a * (Solution₁.x a * Solution₁.x a₁) - (Solution₁.x a ^ 2 - 1) * Solution₁.x a₁

Next tactic:
ring_nf
/-- If we multiply a positive solution with the inverse of a fundamental solution,
the `x`-coordinate stays positive. -/
theorem mul_inv_x_pos {a₁ : Solution₁ d} (h : IsFundamental a₁) {a : Solution₁ d} (hax : 1 < a.x)
    (hay : 0 < a.y) : 0 < (a * a₁⁻¹).x := by
  simp only [x_mul, x_inv, y_inv, mul_neg, lt_add_neg_iff_add_lt, zero_add]
  refine' (mul_lt_mul_left <| zero_lt_one.trans hax).mp _
  rw [(by ring : a.x * (d * (a.y * a₁.y)) = d * a.y * (a.x * a₁.y))]
  refine' ((mul_le_mul_left <| mul_pos h.d_pos hay).mpr <| x_mul_y_le_y_mul_x h hax hay).trans_lt _
  rw [← mul_assoc, mul_assoc d, ← sq, a.prop_y, ← sub_pos]
  ring_nf
  

Proof state:
d : ℤ
a₁ : Solution₁ d
h : IsFundamental a₁
a : Solution₁ d
hax : 1 < Solution₁.x a
hay : 0 < Solution₁.y a
⊢ 0 < Solution₁.x a₁

Next tactic:
exact zero_lt_one.trans h.1
/-- If we multiply a positive solution with the inverse of a fundamental solution,
the `x`-coordinate decreases. -/
theorem mul_inv_x_lt_x {a₁ : Solution₁ d} (h : IsFundamental a₁) {a : Solution₁ d} (hax : 1 < a.x)
    (hay : 0 < a.y) : (a * a₁⁻¹).x < a.x := by
  

Proof state:
d : ℤ
a₁ : Solution₁ d
h : IsFundamental a₁
a : Solution₁ d
hax : 1 < Solution₁.x a
hay : 0 < Solution₁.y a
⊢ Solution₁.x (a * a₁⁻¹) < Solution₁.x a

Next tactic:
simp only [x_mul, x_inv, y_inv, mul_neg, add_neg_lt_iff_le_add']
/-- If we multiply a positive solution with the inverse of a fundamental solution,
the `x`-coordinate decreases. -/
theorem mul_inv_x_lt_x {a₁ : Solution₁ d} (h : IsFundamental a₁) {a : Solution₁ d} (hax : 1 < a.x)
    (hay : 0 < a.y) : (a * a₁⁻¹).x < a.x := by
  simp only [x_mul, x_inv, y_inv, mul_neg, add_neg_lt_iff_le_add']
  

Proof state:
d : ℤ
a₁ : Solution₁ d
h : IsFundamental a₁
a : Solution₁ d
hax : 1 < Solution₁.x a
hay : 0 < Solution₁.y a
⊢ Solution₁.x a * Solution₁.x a₁ < d * (Solution₁.y a * Solution₁.y a₁) + Solution₁.x a

Next tactic:
refine' (mul_lt_mul_left h.2.1).mp _
/-- If we multiply a positive solution with the inverse of a fundamental solution,
the `x`-coordinate decreases. -/
theorem mul_inv_x_lt_x {a₁ : Solution₁ d} (h : IsFundamental a₁) {a : Solution₁ d} (hax : 1 < a.x)
    (hay : 0 < a.y) : (a * a₁⁻¹).x < a.x := by
  simp only [x_mul, x_inv, y_inv, mul_neg, add_neg_lt_iff_le_add']
  refine' (mul_lt_mul_left h.2.1).mp _
  

Proof state:
d : ℤ
a₁ : Solution₁ d
h : IsFundamental a₁
a : Solution₁ d
hax : 1 < Solution₁.x a
hay : 0 < Solution₁.y a
⊢ Solution₁.y a₁ * (Solution₁.x a * Solution₁.x a₁) <
    Solution₁.y a₁ * (d * (Solution₁.y a * Solution₁.y a₁) + Solution₁.x a)

Next tactic:
rw [(by ring : a₁.y * (a.x * a₁.x) = a.x * a₁.y * a₁.x)]
/-- If we multiply a positive solution with the inverse of a fundamental solution,
the `x`-coordinate decreases. -/
theorem mul_inv_x_lt_x {a₁ : Solution₁ d} (h : IsFundamental a₁) {a : Solution₁ d} (hax : 1 < a.x)
    (hay : 0 < a.y) : (a * a₁⁻¹).x < a.x := by
  simp only [x_mul, x_inv, y_inv, mul_neg, add_neg_lt_iff_le_add']
  refine' (mul_lt_mul_left h.2.1).mp _
  rw [(by 

Proof state:
d : ℤ
a₁ : Solution₁ d
h : IsFundamental a₁
a : Solution₁ d
hax : 1 < Solution₁.x a
hay : 0 < Solution₁.y a
⊢ Solution₁.y a₁ * (Solution₁.x a * Solution₁.x a₁) = Solution₁.x a * Solution₁.y a₁ * Solution₁.x a₁

Next tactic:
ring
/-- If we multiply a positive solution with the inverse of a fundamental solution,
the `x`-coordinate decreases. -/
theorem mul_inv_x_lt_x {a₁ : Solution₁ d} (h : IsFundamental a₁) {a : Solution₁ d} (hax : 1 < a.x)
    (hay : 0 < a.y) : (a * a₁⁻¹).x < a.x := by
  simp only [x_mul, x_inv, y_inv, mul_neg, add_neg_lt_iff_le_add']
  refine' (mul_lt_mul_left h.2.1).mp _
  rw [(by ring : a₁.y * (a.x * a₁.x) = a.x * a₁.y * a₁.x)]
  

Proof state:
d : ℤ
a₁ : Solution₁ d
h : IsFundamental a₁
a : Solution₁ d
hax : 1 < Solution₁.x a
hay : 0 < Solution₁.y a
⊢ Solution₁.x a * Solution₁.y a₁ * Solution₁.x a₁ <
    Solution₁.y a₁ * (d * (Solution₁.y a * Solution₁.y a₁) + Solution₁.x a)

Next tactic:
refine'
    ((mul_le_mul_right <| zero_lt_one.trans h.1).mpr <| x_mul_y_le_y_mul_x h hax hay).trans_lt _
/-- If we multiply a positive solution with the inverse of a fundamental solution,
the `x`-coordinate decreases. -/
theorem mul_inv_x_lt_x {a₁ : Solution₁ d} (h : IsFundamental a₁) {a : Solution₁ d} (hax : 1 < a.x)
    (hay : 0 < a.y) : (a * a₁⁻¹).x < a.x := by
  simp only [x_mul, x_inv, y_inv, mul_neg, add_neg_lt_iff_le_add']
  refine' (mul_lt_mul_left h.2.1).mp _
  rw [(by ring : a₁.y * (a.x * a₁.x) = a.x * a₁.y * a₁.x)]
  refine'
    ((mul_le_mul_right <| zero_lt_one.trans h.1).mpr <| x_mul_y_le_y_mul_x h hax hay).trans_lt _
  

Proof state:
d : ℤ
a₁ : Solution₁ d
h : IsFundamental a₁
a : Solution₁ d
hax : 1 < Solution₁.x a
hay : 0 < Solution₁.y a
⊢ Solution₁.y a * Solution₁.x a₁ * Solution₁.x a₁ <
    Solution₁.y a₁ * (d * (Solution₁.y a * Solution₁.y a₁) + Solution₁.x a)

Next tactic:
rw [mul_assoc, ← sq, a₁.prop_x, ← sub_neg]
/-- If we multiply a positive solution with the inverse of a fundamental solution,
the `x`-coordinate decreases. -/
theorem mul_inv_x_lt_x {a₁ : Solution₁ d} (h : IsFundamental a₁) {a : Solution₁ d} (hax : 1 < a.x)
    (hay : 0 < a.y) : (a * a₁⁻¹).x < a.x := by
  simp only [x_mul, x_inv, y_inv, mul_neg, add_neg_lt_iff_le_add']
  refine' (mul_lt_mul_left h.2.1).mp _
  rw [(by ring : a₁.y * (a.x * a₁.x) = a.x * a₁.y * a₁.x)]
  refine'
    ((mul_le_mul_right <| zero_lt_one.trans h.1).mpr <| x_mul_y_le_y_mul_x h hax hay).trans_lt _
  rw [mul_assoc, ← sq, a₁.prop_x, ← sub_neg]
  -- Porting note: was `ring_nf`
  

Proof state:
d : ℤ
a₁ : Solution₁ d
h : IsFundamental a₁
a : Solution₁ d
hax : 1 < Solution₁.x a
hay : 0 < Solution₁.y a
⊢ Solution₁.y a * (1 + d * Solution₁.y a₁ ^ 2) -
      Solution₁.y a₁ * (d * (Solution₁.y a * Solution₁.y a₁) + Solution₁.x a) <
    0

Next tactic:
suffices a.y - a.x * a₁.y < 0 by convert this using 1; ring
/-- If we multiply a positive solution with the inverse of a fundamental solution,
the `x`-coordinate decreases. -/
theorem mul_inv_x_lt_x {a₁ : Solution₁ d} (h : IsFundamental a₁) {a : Solution₁ d} (hax : 1 < a.x)
    (hay : 0 < a.y) : (a * a₁⁻¹).x < a.x := by
  simp only [x_mul, x_inv, y_inv, mul_neg, add_neg_lt_iff_le_add']
  refine' (mul_lt_mul_left h.2.1).mp _
  rw [(by ring : a₁.y * (a.x * a₁.x) = a.x * a₁.y * a₁.x)]
  refine'
    ((mul_le_mul_right <| zero_lt_one.trans h.1).mpr <| x_mul_y_le_y_mul_x h hax hay).trans_lt _
  rw [mul_assoc, ← sq, a₁.prop_x, ← sub_neg]
  -- Porting note: was `ring_nf`
  suffices a.y - a.x * a₁.y < 0 by 

Proof state:
d : ℤ
a₁ : Solution₁ d
h : IsFundamental a₁
a : Solution₁ d
hax : 1 < Solution₁.x a
hay : 0 < Solution₁.y a
this : Solution₁.y a - Solution₁.x a * Solution₁.y a₁ < 0
⊢ Solution₁.y a * (1 + d * Solution₁.y a₁ ^ 2) -
      Solution₁.y a₁ * (d * (Solution₁.y a * Solution₁.y a₁) + Solution₁.x a) <
    0

Next tactic:
convert this using 1
/-- If we multiply a positive solution with the inverse of a fundamental solution,
the `x`-coordinate decreases. -/
theorem mul_inv_x_lt_x {a₁ : Solution₁ d} (h : IsFundamental a₁) {a : Solution₁ d} (hax : 1 < a.x)
    (hay : 0 < a.y) : (a * a₁⁻¹).x < a.x := by
  simp only [x_mul, x_inv, y_inv, mul_neg, add_neg_lt_iff_le_add']
  refine' (mul_lt_mul_left h.2.1).mp _
  rw [(by ring : a₁.y * (a.x * a₁.x) = a.x * a₁.y * a₁.x)]
  refine'
    ((mul_le_mul_right <| zero_lt_one.trans h.1).mpr <| x_mul_y_le_y_mul_x h hax hay).trans_lt _
  rw [mul_assoc, ← sq, a₁.prop_x, ← sub_neg]
  -- Porting note: was `ring_nf`
  suffices a.y - a.x * a₁.y < 0 by convert this using 1; 

Proof state:
case h.e'_3
d : ℤ
a₁ : Solution₁ d
h : IsFundamental a₁
a : Solution₁ d
hax : 1 < Solution₁.x a
hay : 0 < Solution₁.y a
this : Solution₁.y a - Solution₁.x a * Solution₁.y a₁ < 0
⊢ Solution₁.y a * (1 + d * Solution₁.y a₁ ^ 2) -
      Solution₁.y a₁ * (d * (Solution₁.y a * Solution₁.y a₁) + Solution₁.x a) =
    Solution₁.y a - Solution₁.x a * Solution₁.y a₁

Next tactic:
ring
/-- If we multiply a positive solution with the inverse of a fundamental solution,
the `x`-coordinate decreases. -/
theorem mul_inv_x_lt_x {a₁ : Solution₁ d} (h : IsFundamental a₁) {a : Solution₁ d} (hax : 1 < a.x)
    (hay : 0 < a.y) : (a * a₁⁻¹).x < a.x := by
  simp only [x_mul, x_inv, y_inv, mul_neg, add_neg_lt_iff_le_add']
  refine' (mul_lt_mul_left h.2.1).mp _
  rw [(by ring : a₁.y * (a.x * a₁.x) = a.x * a₁.y * a₁.x)]
  refine'
    ((mul_le_mul_right <| zero_lt_one.trans h.1).mpr <| x_mul_y_le_y_mul_x h hax hay).trans_lt _
  rw [mul_assoc, ← sq, a₁.prop_x, ← sub_neg]
  -- Porting note: was `ring_nf`
  suffices a.y - a.x * a₁.y < 0 by convert this using 1; ring
  

Proof state:
d : ℤ
a₁ : Solution₁ d
h : IsFundamental a₁
a : Solution₁ d
hax : 1 < Solution₁.x a
hay : 0 < Solution₁.y a
⊢ Solution₁.y a - Solution₁.x a * Solution₁.y a₁ < 0

Next tactic:
rw [sub_neg, ← abs_of_pos hay, ← abs_of_pos h.2.1, ← abs_of_pos <| zero_lt_one.trans hax, ←
    abs_mul, ← sq_lt_sq, mul_pow, a.prop_x]
/-- If we multiply a positive solution with the inverse of a fundamental solution,
the `x`-coordinate decreases. -/
theorem mul_inv_x_lt_x {a₁ : Solution₁ d} (h : IsFundamental a₁) {a : Solution₁ d} (hax : 1 < a.x)
    (hay : 0 < a.y) : (a * a₁⁻¹).x < a.x := by
  simp only [x_mul, x_inv, y_inv, mul_neg, add_neg_lt_iff_le_add']
  refine' (mul_lt_mul_left h.2.1).mp _
  rw [(by ring : a₁.y * (a.x * a₁.x) = a.x * a₁.y * a₁.x)]
  refine'
    ((mul_le_mul_right <| zero_lt_one.trans h.1).mpr <| x_mul_y_le_y_mul_x h hax hay).trans_lt _
  rw [mul_assoc, ← sq, a₁.prop_x, ← sub_neg]
  -- Porting note: was `ring_nf`
  suffices a.y - a.x * a₁.y < 0 by convert this using 1; ring
  rw [sub_neg, ← abs_of_pos hay, ← abs_of_pos h.2.1, ← abs_of_pos <| zero_lt_one.trans hax, ←
    abs_mul, ← sq_lt_sq, mul_pow, a.prop_x]
  

Proof state:
d : ℤ
a₁ : Solution₁ d
h : IsFundamental a₁
a : Solution₁ d
hax : 1 < Solution₁.x a
hay : 0 < Solution₁.y a
⊢ Solution₁.y a ^ 2 < (1 + d * Solution₁.y a ^ 2) * Solution₁.y a₁ ^ 2

Next tactic:
calc
    a.y ^ 2 = 1 * a.y ^ 2 := (one_mul _).symm
    _ ≤ d * a.y ^ 2 := ((mul_le_mul_right <| sq_pos_of_pos hay).mpr h.d_pos)
    _ < d * a.y ^ 2 + 1 := (lt_add_one _)
    _ = (1 + d * a.y ^ 2) * 1 := by rw [add_comm, mul_one]
    _ ≤ (1 + d * a.y ^ 2) * a₁.y ^ 2 :=
      (mul_le_mul_left (by have := h.d_pos; positivity)).mpr (sq_pos_of_pos h.2.1)
/-- If we multiply a positive solution with the inverse of a fundamental solution,
the `x`-coordinate decreases. -/
theorem mul_inv_x_lt_x {a₁ : Solution₁ d} (h : IsFundamental a₁) {a : Solution₁ d} (hax : 1 < a.x)
    (hay : 0 < a.y) : (a * a₁⁻¹).x < a.x := by
  simp only [x_mul, x_inv, y_inv, mul_neg, add_neg_lt_iff_le_add']
  refine' (mul_lt_mul_left h.2.1).mp _
  rw [(by ring : a₁.y * (a.x * a₁.x) = a.x * a₁.y * a₁.x)]
  refine'
    ((mul_le_mul_right <| zero_lt_one.trans h.1).mpr <| x_mul_y_le_y_mul_x h hax hay).trans_lt _
  rw [mul_assoc, ← sq, a₁.prop_x, ← sub_neg]
  -- Porting note: was `ring_nf`
  suffices a.y - a.x * a₁.y < 0 by convert this using 1; ring
  rw [sub_neg, ← abs_of_pos hay, ← abs_of_pos h.2.1, ← abs_of_pos <| zero_lt_one.trans hax, ←
    abs_mul, ← sq_lt_sq, mul_pow, a.prop_x]
  calc
    a.y ^ 2 = 1 * a.y ^ 2 := (one_mul _).symm
    _ ≤ d * a.y ^ 2 := ((mul_le_mul_right <| sq_pos_of_pos hay).mpr h.d_pos)
    _ < d * a.y ^ 2 + 1 := (lt_add_one _)
    _ = (1 + d * a.y ^ 2) * 1 := by 

Proof state:
d : ℤ
a₁ : Solution₁ d
h : IsFundamental a₁
a : Solution₁ d
hax : 1 < Solution₁.x a
hay : 0 < Solution₁.y a
⊢ d * Solution₁.y a ^ 2 + 1 = (1 + d * Solution₁.y a ^ 2) * 1

Next tactic:
rw [add_comm, mul_one]
/-- If we multiply a positive solution with the inverse of a fundamental solution,
the `x`-coordinate decreases. -/
theorem mul_inv_x_lt_x {a₁ : Solution₁ d} (h : IsFundamental a₁) {a : Solution₁ d} (hax : 1 < a.x)
    (hay : 0 < a.y) : (a * a₁⁻¹).x < a.x := by
  simp only [x_mul, x_inv, y_inv, mul_neg, add_neg_lt_iff_le_add']
  refine' (mul_lt_mul_left h.2.1).mp _
  rw [(by ring : a₁.y * (a.x * a₁.x) = a.x * a₁.y * a₁.x)]
  refine'
    ((mul_le_mul_right <| zero_lt_one.trans h.1).mpr <| x_mul_y_le_y_mul_x h hax hay).trans_lt _
  rw [mul_assoc, ← sq, a₁.prop_x, ← sub_neg]
  -- Porting note: was `ring_nf`
  suffices a.y - a.x * a₁.y < 0 by convert this using 1; ring
  rw [sub_neg, ← abs_of_pos hay, ← abs_of_pos h.2.1, ← abs_of_pos <| zero_lt_one.trans hax, ←
    abs_mul, ← sq_lt_sq, mul_pow, a.prop_x]
  calc
    a.y ^ 2 = 1 * a.y ^ 2 := (one_mul _).symm
    _ ≤ d * a.y ^ 2 := ((mul_le_mul_right <| sq_pos_of_pos hay).mpr h.d_pos)
    _ < d * a.y ^ 2 + 1 := (lt_add_one _)
    _ = (1 + d * a.y ^ 2) * 1 := by rw [add_comm, mul_one]
    _ ≤ (1 + d * a.y ^ 2) * a₁.y ^ 2 :=
      (mul_le_mul_left (by 

Proof state:
d : ℤ
a₁ : Solution₁ d
h : IsFundamental a₁
a : Solution₁ d
hax : 1 < Solution₁.x a
hay : 0 < Solution₁.y a
⊢ 0 < 1 + d * Solution₁.y a ^ 2

Next tactic:
have := h.d_pos
/-- If we multiply a positive solution with the inverse of a fundamental solution,
the `x`-coordinate decreases. -/
theorem mul_inv_x_lt_x {a₁ : Solution₁ d} (h : IsFundamental a₁) {a : Solution₁ d} (hax : 1 < a.x)
    (hay : 0 < a.y) : (a * a₁⁻¹).x < a.x := by
  simp only [x_mul, x_inv, y_inv, mul_neg, add_neg_lt_iff_le_add']
  refine' (mul_lt_mul_left h.2.1).mp _
  rw [(by ring : a₁.y * (a.x * a₁.x) = a.x * a₁.y * a₁.x)]
  refine'
    ((mul_le_mul_right <| zero_lt_one.trans h.1).mpr <| x_mul_y_le_y_mul_x h hax hay).trans_lt _
  rw [mul_assoc, ← sq, a₁.prop_x, ← sub_neg]
  -- Porting note: was `ring_nf`
  suffices a.y - a.x * a₁.y < 0 by convert this using 1; ring
  rw [sub_neg, ← abs_of_pos hay, ← abs_of_pos h.2.1, ← abs_of_pos <| zero_lt_one.trans hax, ←
    abs_mul, ← sq_lt_sq, mul_pow, a.prop_x]
  calc
    a.y ^ 2 = 1 * a.y ^ 2 := (one_mul _).symm
    _ ≤ d * a.y ^ 2 := ((mul_le_mul_right <| sq_pos_of_pos hay).mpr h.d_pos)
    _ < d * a.y ^ 2 + 1 := (lt_add_one _)
    _ = (1 + d * a.y ^ 2) * 1 := by rw [add_comm, mul_one]
    _ ≤ (1 + d * a.y ^ 2) * a₁.y ^ 2 :=
      (mul_le_mul_left (by have := h.d_pos; 

Proof state:
d : ℤ
a₁ : Solution₁ d
h : IsFundamental a₁
a : Solution₁ d
hax : 1 < Solution₁.x a
hay : 0 < Solution₁.y a
this : 0 < d
⊢ 0 < 1 + d * Solution₁.y a ^ 2

Next tactic:
positivity
/-- Any nonnegative solution is a power with nonnegative exponent of a fundamental solution. -/
theorem eq_pow_of_nonneg {a₁ : Solution₁ d} (h : IsFundamental a₁) {a : Solution₁ d} (hax : 0 < a.x)
    (hay : 0 ≤ a.y) : ∃ n : ℕ, a = a₁ ^ n := by
  

Proof state:
d : ℤ
a₁ : Solution₁ d
h : IsFundamental a₁
a : Solution₁ d
hax : 0 < Solution₁.x a
hay : 0 ≤ Solution₁.y a
⊢ ∃ n, a = a₁ ^ n

Next tactic:
lift a.x to ℕ using hax.le with ax hax'
/-- Any nonnegative solution is a power with nonnegative exponent of a fundamental solution. -/
theorem eq_pow_of_nonneg {a₁ : Solution₁ d} (h : IsFundamental a₁) {a : Solution₁ d} (hax : 0 < a.x)
    (hay : 0 ≤ a.y) : ∃ n : ℕ, a = a₁ ^ n := by
  lift a.x to ℕ using hax.le with ax hax'
  -- Porting note: added
  

Proof state:
case intro
d : ℤ
a₁ : Solution₁ d
h : IsFundamental a₁
a : Solution₁ d
hay : 0 ≤ Solution₁.y a
ax : ℕ
hax' : ↑ax = Solution₁.x a
hax✝ hax : 0 < ↑ax
⊢ ∃ n, a = a₁ ^ n

Next tactic:
clear hax
/-- Any nonnegative solution is a power with nonnegative exponent of a fundamental solution. -/
theorem eq_pow_of_nonneg {a₁ : Solution₁ d} (h : IsFundamental a₁) {a : Solution₁ d} (hax : 0 < a.x)
    (hay : 0 ≤ a.y) : ∃ n : ℕ, a = a₁ ^ n := by
  lift a.x to ℕ using hax.le with ax hax'
  -- Porting note: added
  clear hax
  

Proof state:
case intro
d : ℤ
a₁ : Solution₁ d
h : IsFundamental a₁
a : Solution₁ d
hay : 0 ≤ Solution₁.y a
ax : ℕ
hax' : ↑ax = Solution₁.x a
hax : 0 < ↑ax
⊢ ∃ n, a = a₁ ^ n

Next tactic:
induction' ax using Nat.strong_induction_on with x ih generalizing a
/-- Any nonnegative solution is a power with nonnegative exponent of a fundamental solution. -/
theorem eq_pow_of_nonneg {a₁ : Solution₁ d} (h : IsFundamental a₁) {a : Solution₁ d} (hax : 0 < a.x)
    (hay : 0 ≤ a.y) : ∃ n : ℕ, a = a₁ ^ n := by
  lift a.x to ℕ using hax.le with ax hax'
  -- Porting note: added
  clear hax
  induction' ax using Nat.strong_induction_on with x ih generalizing a
  

Proof state:
case intro.h
d : ℤ
a₁ : Solution₁ d
h : IsFundamental a₁
x : ℕ
ih : ∀ m < x, ∀ {a : Solution₁ d}, 0 ≤ Solution₁.y a → ↑m = Solution₁.x a → 0 < ↑m → ∃ n, a = a₁ ^ n
a : Solution₁ d
hay : 0 ≤ Solution₁.y a
hax' : ↑x = Solution₁.x a
hax : 0 < ↑x
⊢ ∃ n, a = a₁ ^ n

Next tactic:
rcases hay.eq_or_lt with hy | hy
/-- Any nonnegative solution is a power with nonnegative exponent of a fundamental solution. -/
theorem eq_pow_of_nonneg {a₁ : Solution₁ d} (h : IsFundamental a₁) {a : Solution₁ d} (hax : 0 < a.x)
    (hay : 0 ≤ a.y) : ∃ n : ℕ, a = a₁ ^ n := by
  lift a.x to ℕ using hax.le with ax hax'
  -- Porting note: added
  clear hax
  induction' ax using Nat.strong_induction_on with x ih generalizing a
  rcases hay.eq_or_lt with hy | hy
  · -- case 1: `a = 1`
    

Proof state:
case intro.h.inl
d : ℤ
a₁ : Solution₁ d
h : IsFundamental a₁
x : ℕ
ih : ∀ m < x, ∀ {a : Solution₁ d}, 0 ≤ Solution₁.y a → ↑m = Solution₁.x a → 0 < ↑m → ∃ n, a = a₁ ^ n
a : Solution₁ d
hay : 0 ≤ Solution₁.y a
hax' : ↑x = Solution₁.x a
hax : 0 < ↑x
hy : 0 = Solution₁.y a
⊢ ∃ n, a = a₁ ^ n

Next tactic:
refine' ⟨0, _⟩
/-- Any nonnegative solution is a power with nonnegative exponent of a fundamental solution. -/
theorem eq_pow_of_nonneg {a₁ : Solution₁ d} (h : IsFundamental a₁) {a : Solution₁ d} (hax : 0 < a.x)
    (hay : 0 ≤ a.y) : ∃ n : ℕ, a = a₁ ^ n := by
  lift a.x to ℕ using hax.le with ax hax'
  -- Porting note: added
  clear hax
  induction' ax using Nat.strong_induction_on with x ih generalizing a
  rcases hay.eq_or_lt with hy | hy
  · -- case 1: `a = 1`
    refine' ⟨0, _⟩
    

Proof state:
case intro.h.inl
d : ℤ
a₁ : Solution₁ d
h : IsFundamental a₁
x : ℕ
ih : ∀ m < x, ∀ {a : Solution₁ d}, 0 ≤ Solution₁.y a → ↑m = Solution₁.x a → 0 < ↑m → ∃ n, a = a₁ ^ n
a : Solution₁ d
hay : 0 ≤ Solution₁.y a
hax' : ↑x = Solution₁.x a
hax : 0 < ↑x
hy : 0 = Solution₁.y a
⊢ a = a₁ ^ 0

Next tactic:
simp only [pow_zero]
/-- Any nonnegative solution is a power with nonnegative exponent of a fundamental solution. -/
theorem eq_pow_of_nonneg {a₁ : Solution₁ d} (h : IsFundamental a₁) {a : Solution₁ d} (hax : 0 < a.x)
    (hay : 0 ≤ a.y) : ∃ n : ℕ, a = a₁ ^ n := by
  lift a.x to ℕ using hax.le with ax hax'
  -- Porting note: added
  clear hax
  induction' ax using Nat.strong_induction_on with x ih generalizing a
  rcases hay.eq_or_lt with hy | hy
  · -- case 1: `a = 1`
    refine' ⟨0, _⟩
    simp only [pow_zero]
    

Proof state:
case intro.h.inl
d : ℤ
a₁ : Solution₁ d
h : IsFundamental a₁
x : ℕ
ih : ∀ m < x, ∀ {a : Solution₁ d}, 0 ≤ Solution₁.y a → ↑m = Solution₁.x a → 0 < ↑m → ∃ n, a = a₁ ^ n
a : Solution₁ d
hay : 0 ≤ Solution₁.y a
hax' : ↑x = Solution₁.x a
hax : 0 < ↑x
hy : 0 = Solution₁.y a
⊢ a = 1

Next tactic:
ext
/-- Any nonnegative solution is a power with nonnegative exponent of a fundamental solution. -/
theorem eq_pow_of_nonneg {a₁ : Solution₁ d} (h : IsFundamental a₁) {a : Solution₁ d} (hax : 0 < a.x)
    (hay : 0 ≤ a.y) : ∃ n : ℕ, a = a₁ ^ n := by
  lift a.x to ℕ using hax.le with ax hax'
  -- Porting note: added
  clear hax
  induction' ax using Nat.strong_induction_on with x ih generalizing a
  rcases hay.eq_or_lt with hy | hy
  · -- case 1: `a = 1`
    refine' ⟨0, _⟩
    simp only [pow_zero]
    ext <;> 

Proof state:
case intro.h.inl.hx
d : ℤ
a₁ : Solution₁ d
h : IsFundamental a₁
x : ℕ
ih : ∀ m < x, ∀ {a : Solution₁ d}, 0 ≤ Solution₁.y a → ↑m = Solution₁.x a → 0 < ↑m → ∃ n, a = a₁ ^ n
a : Solution₁ d
hay : 0 ≤ Solution₁.y a
hax' : ↑x = Solution₁.x a
hax : 0 < ↑x
hy : 0 = Solution₁.y a
⊢ Solution₁.x a = Solution₁.x 1

Next tactic:
simp only [x_one, y_one]
/-- Any nonnegative solution is a power with nonnegative exponent of a fundamental solution. -/
theorem eq_pow_of_nonneg {a₁ : Solution₁ d} (h : IsFundamental a₁) {a : Solution₁ d} (hax : 0 < a.x)
    (hay : 0 ≤ a.y) : ∃ n : ℕ, a = a₁ ^ n := by
  lift a.x to ℕ using hax.le with ax hax'
  -- Porting note: added
  clear hax
  induction' ax using Nat.strong_induction_on with x ih generalizing a
  rcases hay.eq_or_lt with hy | hy
  · -- case 1: `a = 1`
    refine' ⟨0, _⟩
    simp only [pow_zero]
    ext <;> 

Proof state:
case intro.h.inl.hy
d : ℤ
a₁ : Solution₁ d
h : IsFundamental a₁
x : ℕ
ih : ∀ m < x, ∀ {a : Solution₁ d}, 0 ≤ Solution₁.y a → ↑m = Solution₁.x a → 0 < ↑m → ∃ n, a = a₁ ^ n
a : Solution₁ d
hay : 0 ≤ Solution₁.y a
hax' : ↑x = Solution₁.x a
hax : 0 < ↑x
hy : 0 = Solution₁.y a
⊢ Solution₁.y a = Solution₁.y 1

Next tactic:
simp only [x_one, y_one]