l3lab/ntp-mathlib huggingface dataset:
/-- 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 <;> simp only [x_one, y_one]
    · 

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 = 1

Next tactic:
have prop := a.prop
/-- 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 <;> simp only [x_one, y_one]
    · have prop := a.prop
      

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
prop : Solution₁.x a ^ 2 - d * Solution₁.y a ^ 2 = 1
⊢ Solution₁.x a = 1

Next tactic:
rw [← hy, sq (0 : ℤ), zero_mul, mul_zero, sub_zero,
        sq_eq_one_iff] at prop
/-- 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 <;> simp only [x_one, y_one]
    · have prop := a.prop
      rw [← hy, sq (0 : ℤ), zero_mul, mul_zero, sub_zero,
        sq_eq_one_iff] at prop
      

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
prop : Solution₁.x a = 1 ∨ Solution₁.x a = -1
⊢ Solution₁.x a = 1

Next tactic:
refine' prop.resolve_right fun hf => _
/-- 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 <;> simp only [x_one, y_one]
    · have prop := a.prop
      rw [← hy, sq (0 : ℤ), zero_mul, mul_zero, sub_zero,
        sq_eq_one_iff] at prop
      refine' prop.resolve_right fun hf => _
      

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
prop : Solution₁.x a = 1 ∨ Solution₁.x a = -1
hf : Solution₁.x a = -1
⊢ False

Next tactic:
have := (hax.trans_eq hax').le.trans_eq hf
/-- 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 <;> simp only [x_one, y_one]
    · have prop := a.prop
      rw [← hy, sq (0 : ℤ), zero_mul, mul_zero, sub_zero,
        sq_eq_one_iff] at prop
      refine' prop.resolve_right fun hf => _
      have := (hax.trans_eq hax').le.trans_eq hf
      

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
prop : Solution₁.x a = 1 ∨ Solution₁.x a = -1
hf : Solution₁.x a = -1
this : 0 ≤ -1
⊢ False

Next tactic:
norm_num at this
/-- 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 <;> simp only [x_one, y_one]
    · have prop := a.prop
      rw [← hy, sq (0 : ℤ), zero_mul, mul_zero, sub_zero,
        sq_eq_one_iff] at prop
      refine' prop.resolve_right fun hf => _
      have := (hax.trans_eq hax').le.trans_eq hf
      norm_num at this
    · 

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 = 0

Next tactic:
exact hy.symm
/-- 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 <;> simp only [x_one, y_one]
    · have prop := a.prop
      rw [← hy, sq (0 : ℤ), zero_mul, mul_zero, sub_zero,
        sq_eq_one_iff] at prop
      refine' prop.resolve_right fun hf => _
      have := (hax.trans_eq hax').le.trans_eq hf
      norm_num at this
    · exact hy.symm
  · -- case 2: `a ≥ a₁`
    

Proof state:
case intro.h.inr
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:
have hx₁ : 1 < a.x := by nlinarith [a.prop, h.d_pos]
/-- 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 <;> simp only [x_one, y_one]
    · have prop := a.prop
      rw [← hy, sq (0 : ℤ), zero_mul, mul_zero, sub_zero,
        sq_eq_one_iff] at prop
      refine' prop.resolve_right fun hf => _
      have := (hax.trans_eq hax').le.trans_eq hf
      norm_num at this
    · exact hy.symm
  · -- case 2: `a ≥ a₁`
    have hx₁ : 1 < a.x := by 

Proof state:
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
⊢ 1 < Solution₁.x a

Next tactic:
nlinarith [a.prop, h.d_pos]
/-- 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 <;> simp only [x_one, y_one]
    · have prop := a.prop
      rw [← hy, sq (0 : ℤ), zero_mul, mul_zero, sub_zero,
        sq_eq_one_iff] at prop
      refine' prop.resolve_right fun hf => _
      have := (hax.trans_eq hax').le.trans_eq hf
      norm_num at this
    · exact hy.symm
  · -- case 2: `a ≥ a₁`
    have hx₁ : 1 < a.x := by nlinarith [a.prop, h.d_pos]
    

Proof state:
case intro.h.inr
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
hx₁ : 1 < Solution₁.x a
⊢ ∃ n, a = a₁ ^ n

Next tactic:
have hxx₁ := h.mul_inv_x_pos hx₁ 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`
    refine' ⟨0, _⟩
    simp only [pow_zero]
    ext <;> simp only [x_one, y_one]
    · have prop := a.prop
      rw [← hy, sq (0 : ℤ), zero_mul, mul_zero, sub_zero,
        sq_eq_one_iff] at prop
      refine' prop.resolve_right fun hf => _
      have := (hax.trans_eq hax').le.trans_eq hf
      norm_num at this
    · exact hy.symm
  · -- case 2: `a ≥ a₁`
    have hx₁ : 1 < a.x := by nlinarith [a.prop, h.d_pos]
    have hxx₁ := h.mul_inv_x_pos hx₁ hy
    

Proof state:
case intro.h.inr
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
hx₁ : 1 < Solution₁.x a
hxx₁ : 0 < Solution₁.x (a * a₁⁻¹)
⊢ ∃ n, a = a₁ ^ n

Next tactic:
have hxx₂ := h.mul_inv_x_lt_x hx₁ 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`
    refine' ⟨0, _⟩
    simp only [pow_zero]
    ext <;> simp only [x_one, y_one]
    · have prop := a.prop
      rw [← hy, sq (0 : ℤ), zero_mul, mul_zero, sub_zero,
        sq_eq_one_iff] at prop
      refine' prop.resolve_right fun hf => _
      have := (hax.trans_eq hax').le.trans_eq hf
      norm_num at this
    · exact hy.symm
  · -- case 2: `a ≥ a₁`
    have hx₁ : 1 < a.x := by nlinarith [a.prop, h.d_pos]
    have hxx₁ := h.mul_inv_x_pos hx₁ hy
    have hxx₂ := h.mul_inv_x_lt_x hx₁ hy
    

Proof state:
case intro.h.inr
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
hx₁ : 1 < Solution₁.x a
hxx₁ : 0 < Solution₁.x (a * a₁⁻¹)
hxx₂ : Solution₁.x (a * a₁⁻¹) < Solution₁.x a
⊢ ∃ n, a = a₁ ^ n

Next tactic:
have hyy := h.mul_inv_y_nonneg hx₁ 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`
    refine' ⟨0, _⟩
    simp only [pow_zero]
    ext <;> simp only [x_one, y_one]
    · have prop := a.prop
      rw [← hy, sq (0 : ℤ), zero_mul, mul_zero, sub_zero,
        sq_eq_one_iff] at prop
      refine' prop.resolve_right fun hf => _
      have := (hax.trans_eq hax').le.trans_eq hf
      norm_num at this
    · exact hy.symm
  · -- case 2: `a ≥ a₁`
    have hx₁ : 1 < a.x := by nlinarith [a.prop, h.d_pos]
    have hxx₁ := h.mul_inv_x_pos hx₁ hy
    have hxx₂ := h.mul_inv_x_lt_x hx₁ hy
    have hyy := h.mul_inv_y_nonneg hx₁ hy
    

Proof state:
case intro.h.inr
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
hx₁ : 1 < Solution₁.x a
hxx₁ : 0 < Solution₁.x (a * a₁⁻¹)
hxx₂ : Solution₁.x (a * a₁⁻¹) < Solution₁.x a
hyy : 0 ≤ Solution₁.y (a * a₁⁻¹)
⊢ ∃ n, a = a₁ ^ n

Next tactic:
lift (a * a₁⁻¹).x to ℕ using hxx₁.le with x' hx'
/-- 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 <;> simp only [x_one, y_one]
    · have prop := a.prop
      rw [← hy, sq (0 : ℤ), zero_mul, mul_zero, sub_zero,
        sq_eq_one_iff] at prop
      refine' prop.resolve_right fun hf => _
      have := (hax.trans_eq hax').le.trans_eq hf
      norm_num at this
    · exact hy.symm
  · -- case 2: `a ≥ a₁`
    have hx₁ : 1 < a.x := by nlinarith [a.prop, h.d_pos]
    have hxx₁ := h.mul_inv_x_pos hx₁ hy
    have hxx₂ := h.mul_inv_x_lt_x hx₁ hy
    have hyy := h.mul_inv_y_nonneg hx₁ hy
    lift (a * a₁⁻¹).x to ℕ using hxx₁.le with x' hx'
    -- Porting note: `ih` has its arguments in a different order compared to lean 3.
    

Proof state:
case intro.h.inr.intro
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
hx₁ : 1 < Solution₁.x a
hyy : 0 ≤ Solution₁.y (a * a₁⁻¹)
x' : ℕ
hx' : ↑x' = Solution₁.x (a * a₁⁻¹)
hxx₁✝ hxx₁ : 0 < ↑x'
hxx₂✝ hxx₂ : ↑x' < Solution₁.x a
⊢ ∃ n, a = a₁ ^ n

Next tactic:
obtain ⟨n, hn⟩ := ih x' (mod_cast hxx₂.trans_eq hax'.symm) hyy hx' hxx₁
/-- 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 <;> simp only [x_one, y_one]
    · have prop := a.prop
      rw [← hy, sq (0 : ℤ), zero_mul, mul_zero, sub_zero,
        sq_eq_one_iff] at prop
      refine' prop.resolve_right fun hf => _
      have := (hax.trans_eq hax').le.trans_eq hf
      norm_num at this
    · exact hy.symm
  · -- case 2: `a ≥ a₁`
    have hx₁ : 1 < a.x := by nlinarith [a.prop, h.d_pos]
    have hxx₁ := h.mul_inv_x_pos hx₁ hy
    have hxx₂ := h.mul_inv_x_lt_x hx₁ hy
    have hyy := h.mul_inv_y_nonneg hx₁ hy
    lift (a * a₁⁻¹).x to ℕ using hxx₁.le with x' hx'
    -- Porting note: `ih` has its arguments in a different order compared to lean 3.
    obtain ⟨n, hn⟩ := ih x' (mod_cast hxx₂.trans_eq hax'.symm) hyy hx' hxx₁
    

Proof state:
case intro.h.inr.intro.intro
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
hx₁ : 1 < Solution₁.x a
hyy : 0 ≤ Solution₁.y (a * a₁⁻¹)
x' : ℕ
hx' : ↑x' = Solution₁.x (a * a₁⁻¹)
hxx₁✝ hxx₁ : 0 < ↑x'
hxx₂✝ hxx₂ : ↑x' < Solution₁.x a
n : ℕ
hn : a * a₁⁻¹ = a₁ ^ n
⊢ ∃ n, a = a₁ ^ n

Next tactic:
exact ⟨n + 1, by rw [pow_succ, ← hn, mul_comm a, ← mul_assoc, mul_inv_self, one_mul]⟩
/-- 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 <;> simp only [x_one, y_one]
    · have prop := a.prop
      rw [← hy, sq (0 : ℤ), zero_mul, mul_zero, sub_zero,
        sq_eq_one_iff] at prop
      refine' prop.resolve_right fun hf => _
      have := (hax.trans_eq hax').le.trans_eq hf
      norm_num at this
    · exact hy.symm
  · -- case 2: `a ≥ a₁`
    have hx₁ : 1 < a.x := by nlinarith [a.prop, h.d_pos]
    have hxx₁ := h.mul_inv_x_pos hx₁ hy
    have hxx₂ := h.mul_inv_x_lt_x hx₁ hy
    have hyy := h.mul_inv_y_nonneg hx₁ hy
    lift (a * a₁⁻¹).x to ℕ using hxx₁.le with x' hx'
    -- Porting note: `ih` has its arguments in a different order compared to lean 3.
    obtain ⟨n, hn⟩ := ih x' (mod_cast hxx₂.trans_eq hax'.symm) hyy hx' hxx₁
    exact ⟨n + 1, by 

Proof state:
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
hx₁ : 1 < Solution₁.x a
hyy : 0 ≤ Solution₁.y (a * a₁⁻¹)
x' : ℕ
hx' : ↑x' = Solution₁.x (a * a₁⁻¹)
hxx₁✝ hxx₁ : 0 < ↑x'
hxx₂✝ hxx₂ : ↑x' < Solution₁.x a
n : ℕ
hn : a * a₁⁻¹ = a₁ ^ n
⊢ a = a₁ ^ (n + 1)

Next tactic:
rw [pow_succ, ← hn, mul_comm a, ← mul_assoc, mul_inv_self, one_mul]
/-- Every solution is, up to a sign, a power of a given fundamental solution. -/
theorem eq_zpow_or_neg_zpow {a₁ : Solution₁ d} (h : IsFundamental a₁) (a : Solution₁ d) :
    ∃ n : ℤ, a = a₁ ^ n ∨ a = -a₁ ^ n := by
  

Proof state:
d : ℤ
a₁ : Solution₁ d
h : IsFundamental a₁
a : Solution₁ d
⊢ ∃ n, a = a₁ ^ n ∨ a = -a₁ ^ n

Next tactic:
obtain ⟨b, hbx, hby, hb⟩ := exists_pos_variant h.d_pos a
/-- Every solution is, up to a sign, a power of a given fundamental solution. -/
theorem eq_zpow_or_neg_zpow {a₁ : Solution₁ d} (h : IsFundamental a₁) (a : Solution₁ d) :
    ∃ n : ℤ, a = a₁ ^ n ∨ a = -a₁ ^ n := by
  obtain ⟨b, hbx, hby, hb⟩ := exists_pos_variant h.d_pos a
  

Proof state:
case intro.intro.intro
d : ℤ
a₁ : Solution₁ d
h : IsFundamental a₁
a b : Solution₁ d
hbx : 0 < Solution₁.x b
hby : 0 ≤ Solution₁.y b
hb : a ∈ {b, b⁻¹, -b, -b⁻¹}
⊢ ∃ n, a = a₁ ^ n ∨ a = -a₁ ^ n

Next tactic:
obtain ⟨n, hn⟩ := h.eq_pow_of_nonneg hbx hby
/-- Every solution is, up to a sign, a power of a given fundamental solution. -/
theorem eq_zpow_or_neg_zpow {a₁ : Solution₁ d} (h : IsFundamental a₁) (a : Solution₁ d) :
    ∃ n : ℤ, a = a₁ ^ n ∨ a = -a₁ ^ n := by
  obtain ⟨b, hbx, hby, hb⟩ := exists_pos_variant h.d_pos a
  obtain ⟨n, hn⟩ := h.eq_pow_of_nonneg hbx hby
  

Proof state:
case intro.intro.intro.intro
d : ℤ
a₁ : Solution₁ d
h : IsFundamental a₁
a b : Solution₁ d
hbx : 0 < Solution₁.x b
hby : 0 ≤ Solution₁.y b
hb : a ∈ {b, b⁻¹, -b, -b⁻¹}
n : ℕ
hn : b = a₁ ^ n
⊢ ∃ n, a = a₁ ^ n ∨ a = -a₁ ^ n

Next tactic:
rcases hb with (rfl | rfl | rfl | hb)
/-- Every solution is, up to a sign, a power of a given fundamental solution. -/
theorem eq_zpow_or_neg_zpow {a₁ : Solution₁ d} (h : IsFundamental a₁) (a : Solution₁ d) :
    ∃ n : ℤ, a = a₁ ^ n ∨ a = -a₁ ^ n := by
  obtain ⟨b, hbx, hby, hb⟩ := exists_pos_variant h.d_pos a
  obtain ⟨n, hn⟩ := h.eq_pow_of_nonneg hbx hby
  rcases hb with (rfl | rfl | rfl | hb)
  · 

Proof state:
case intro.intro.intro.intro.inl
d : ℤ
a₁ : Solution₁ d
h : IsFundamental a₁
a : Solution₁ d
n : ℕ
hbx : 0 < Solution₁.x a
hby : 0 ≤ Solution₁.y a
hn : a = a₁ ^ n
⊢ ∃ n, a = a₁ ^ n ∨ a = -a₁ ^ n

Next tactic:
exact ⟨n, Or.inl (mod_cast hn)⟩
/-- Every solution is, up to a sign, a power of a given fundamental solution. -/
theorem eq_zpow_or_neg_zpow {a₁ : Solution₁ d} (h : IsFundamental a₁) (a : Solution₁ d) :
    ∃ n : ℤ, a = a₁ ^ n ∨ a = -a₁ ^ n := by
  obtain ⟨b, hbx, hby, hb⟩ := exists_pos_variant h.d_pos a
  obtain ⟨n, hn⟩ := h.eq_pow_of_nonneg hbx hby
  rcases hb with (rfl | rfl | rfl | hb)
  · exact ⟨n, Or.inl (mod_cast hn)⟩
  · 

Proof state:
case intro.intro.intro.intro.inr.inl
d : ℤ
a₁ : Solution₁ d
h : IsFundamental a₁
b : Solution₁ d
hbx : 0 < Solution₁.x b
hby : 0 ≤ Solution₁.y b
n : ℕ
hn : b = a₁ ^ n
⊢ ∃ n, b⁻¹ = a₁ ^ n ∨ b⁻¹ = -a₁ ^ n

Next tactic:
exact ⟨-n, Or.inl (by simp [hn])⟩
/-- Every solution is, up to a sign, a power of a given fundamental solution. -/
theorem eq_zpow_or_neg_zpow {a₁ : Solution₁ d} (h : IsFundamental a₁) (a : Solution₁ d) :
    ∃ n : ℤ, a = a₁ ^ n ∨ a = -a₁ ^ n := by
  obtain ⟨b, hbx, hby, hb⟩ := exists_pos_variant h.d_pos a
  obtain ⟨n, hn⟩ := h.eq_pow_of_nonneg hbx hby
  rcases hb with (rfl | rfl | rfl | hb)
  · exact ⟨n, Or.inl (mod_cast hn)⟩
  · exact ⟨-n, Or.inl (by 

Proof state:
d : ℤ
a₁ : Solution₁ d
h : IsFundamental a₁
b : Solution₁ d
hbx : 0 < Solution₁.x b
hby : 0 ≤ Solution₁.y b
n : ℕ
hn : b = a₁ ^ n
⊢ b⁻¹ = a₁ ^ (-↑n)

Next tactic:
simp [hn]
/-- Every solution is, up to a sign, a power of a given fundamental solution. -/
theorem eq_zpow_or_neg_zpow {a₁ : Solution₁ d} (h : IsFundamental a₁) (a : Solution₁ d) :
    ∃ n : ℤ, a = a₁ ^ n ∨ a = -a₁ ^ n := by
  obtain ⟨b, hbx, hby, hb⟩ := exists_pos_variant h.d_pos a
  obtain ⟨n, hn⟩ := h.eq_pow_of_nonneg hbx hby
  rcases hb with (rfl | rfl | rfl | hb)
  · exact ⟨n, Or.inl (mod_cast hn)⟩
  · exact ⟨-n, Or.inl (by simp [hn])⟩
  · 

Proof state:
case intro.intro.intro.intro.inr.inr.inl
d : ℤ
a₁ : Solution₁ d
h : IsFundamental a₁
b : Solution₁ d
hbx : 0 < Solution₁.x b
hby : 0 ≤ Solution₁.y b
n : ℕ
hn : b = a₁ ^ n
⊢ ∃ n, -b = a₁ ^ n ∨ -b = -a₁ ^ n

Next tactic:
exact ⟨n, Or.inr (by simp [hn])⟩
/-- Every solution is, up to a sign, a power of a given fundamental solution. -/
theorem eq_zpow_or_neg_zpow {a₁ : Solution₁ d} (h : IsFundamental a₁) (a : Solution₁ d) :
    ∃ n : ℤ, a = a₁ ^ n ∨ a = -a₁ ^ n := by
  obtain ⟨b, hbx, hby, hb⟩ := exists_pos_variant h.d_pos a
  obtain ⟨n, hn⟩ := h.eq_pow_of_nonneg hbx hby
  rcases hb with (rfl | rfl | rfl | hb)
  · exact ⟨n, Or.inl (mod_cast hn)⟩
  · exact ⟨-n, Or.inl (by simp [hn])⟩
  · exact ⟨n, Or.inr (by 

Proof state:
d : ℤ
a₁ : Solution₁ d
h : IsFundamental a₁
b : Solution₁ d
hbx : 0 < Solution₁.x b
hby : 0 ≤ Solution₁.y b
n : ℕ
hn : b = a₁ ^ n
⊢ -b = -a₁ ^ ↑n

Next tactic:
simp [hn]