awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.NumberTheory.PellMatiyasevic (goal)
Goal:
a : ℕ
a1 : 1 < a
n k : ℕ
hx : xn a1 (n * k) ≡ xn a1 n ^ k [MOD HPow.hPow (β := ℕ) (yn a1 n) 2]
hy : yn a1 (n * k) ≡ k * xn a1 n ^ (k - 1) * yn a1 n [MOD HPow.hPow (β := ℕ) (yn a1 n) 3]
L :
  xn a1 (n * k) * xn a1 n + d a1 * yn a1 (n * k) * yn a1 n ≡ xn a1 n ^ k * xn a1 n + 0 [MOD
    HPow.hPow (β := ℕ) (yn a1 n) 2]
R :
  xn a1 (n * k) * yn a1 n + yn a1 (n * k) * xn a1 n ≡ xn a1 n ^ k * yn a1 n + k * xn a1 n ^ k * yn a1 n [MOD
    HPow.hPow (β := ℕ) (yn a1 n) 3]
⊢ xn a1 (n * (k + 1)) ≡ xn a1 n ^ (k + 1) [MOD HPow.hPow (β := ℕ) (yn a1 n) 2] ∧
    yn a1 (n * (k + 1)) ≡ (k + 1) * xn a1 n ^ (k + 1 - 1) * yn a1 n [MOD HPow.hPow (β := ℕ) (yn a1 n) 3]
have rw : xn a1 (n * k) * xn a1 n + d a1 * yn a1 (n * k) * yn a1 n ≡ xn a1 n ^ k * xn a1 n [MOD HPow.hPow (β := ℕ) (yn a1 n) 2] ∧
  xn a1 (n * k) * yn a1 n + yn a1 (n * k) * xn a1 n ≡ xn a1 n ^ k * yn a1 n + k * xn a1 n ^ k * yn a1 n [MOD
    HPow.hPow (β := ℕ) (yn a1 n) 3]
Tactic: rw [add_tsub_cancel_right, Nat.mul_succ, xn_add, yn_add, pow_succ (xn _ n), Nat.succ_mul,
      add_comm (k * xn _ n ^ k) (xn _ n ^ k), right_distrib]
-- Lean 4 proof step from Mathlib.NumberTheory.PellMatiyasevic (goal)
Goal:
a : ℕ
a1 : 1 < a
n t : ℕ
h : yn a1 n * yn a1 n ∣ yn a1 t
nt : n ∣ t
n0 : n = 0
⊢ yn a1 n ∣ t
have rwa : ∀ (nt : 0 ∣ t) (n0 : n = 0), yn a1 0 ∣ t
Tactic: rwa [n0] at nt ⊢
Cleared hypotheses: nt, n0
-- Lean 4 proof step from Mathlib.NumberTheory.PellMatiyasevic (goal)
Goal:
a : ℕ
a1 : 1 < a
n t : ℕ
h : yn a1 n * yn a1 n ∣ yn a1 t
nt : n ∣ t
n0l : 0 < n
⊢ yn a1 n ∣ t
have let : ∀ (k : ℕ) (ke : t = n * k), yn a1 n ∣ t
Tactic: let ⟨k, ke⟩ := nt
-- Lean 4 proof step from Mathlib.NumberTheory.PellMatiyasevic (hypothesis)
Goal:
a : ℕ
a1 : 1 < a
n t : ℕ
h : yn a1 n * yn a1 n ∣ yn a1 t
nt : n ∣ t
n0l : 0 < n
k : ℕ
ke : t = n * k
⊢ k * xn a1 n ^ (k - 1) * yn a1 n ≡ 0 [MOD yn a1 n * yn a1 n]
have xm : yn a1 (n * k) ≡ k * xn a1 n ^ (k - 1) * yn a1 n [MOD HPow.hPow (β := ℕ) (yn a1 n) 3]
Tactic: have xm := (xy_modEq_yn a1 n k).right
-- Lean 4 proof step from Mathlib.NumberTheory.PellMatiyasevic (hypothesis)
Goal:
a : ℕ
a1 : 1 < a
n t : ℕ
h : yn a1 n * yn a1 n ∣ yn a1 t
nt : n ∣ t
n0l : 0 < n
k : ℕ
ke : t = n * k
xm : yn a1 (n * k) ≡ k * xn a1 n ^ (k - 1) * yn a1 n [MOD HPow.hPow (β := ℕ) (yn a1 n) 3]
⊢ k * xn a1 n ^ (k - 1) * yn a1 n ≡ 0 [MOD yn a1 n * yn a1 n]
have xm₁ : yn a1 t ≡ k * xn a1 n ^ (k - 1) * yn a1 n [MOD HPow.hPow (β := ℕ) (yn a1 n) 3]
Tactic: rw [← ke] at xm
Cleared hypotheses: xm
-- Lean 4 proof step from Mathlib.NumberTheory.PellMatiyasevic (hypothesis)
Goal:
a : ℕ
a1 : 1 < a
n t : ℕ
h : yn a1 n * yn a1 n ∣ yn a1 t
nt : n ∣ t
n0l : 0 < n
k : ℕ
ke : t = n * k
⊢ yn a1 n ∣ t
have this : yn a1 n ∣ k * xn a1 n ^ (k - 1)
Tactic: have : yn a1 n ∣ k * xn a1 n ^ (k - 1) :=
      Nat.dvd_of_mul_dvd_mul_right (strictMono_y a1 n0l) <|
        modEq_zero_iff_dvd.1 <| by
          have xm := (xy_modEq_yn a1 n k).right; rw [← ke] at xm
          exact (xm.of_dvd <| by simp [_root_.pow_succ]).symm.trans h.modEq_zero_nat
-- Lean 4 proof step from Mathlib.NumberTheory.PellMatiyasevic (goal)
Goal:
a : ℕ
a1 : 1 < a
n t : ℕ
h : yn a1 n * yn a1 n ∣ yn a1 t
nt : n ∣ t
n0l : 0 < n
k : ℕ
ke : t = n * k
this : yn a1 n ∣ k * xn a1 n ^ (k - 1)
⊢ yn a1 n ∣ t
have rw : yn a1 n ∣ n * k
Tactic: rw [ke]
-- Lean 4 proof step from Mathlib.NumberTheory.PellMatiyasevic (goal)
Goal:
a : ℕ
a1 : 1 < a
n : ℕ
⊢ pellZd a1 (n + 2) + pellZd a1 n = (↑(2 * a) : ℤ√(↑(d a1) : ℤ)) * pellZd a1 (n + 1)
have re : (pellZd a1 (n + 2) + pellZd a1 n).re = ((↑(2 * a) : ℤ√(↑(d a1) : ℤ)) * pellZd a1 (n + 1)).re
Tactic: ext
-- Lean 4 proof step from Mathlib.NumberTheory.PellMatiyasevic (goal)
Goal:
a : ℕ
a1 : 1 < a
n : ℕ
re : (pellZd a1 (n + 2) + pellZd a1 n).re = ((↑(2 * a) : ℤ√(↑(d a1) : ℤ)) * pellZd a1 (n + 1)).re
⊢ pellZd a1 (n + 2) + pellZd a1 n = (↑(2 * a) : ℤ√(↑(d a1) : ℤ)) * pellZd a1 (n + 1)
have im : (pellZd a1 (n + 2) + pellZd a1 n).im = ((↑(2 * a) : ℤ√(↑(d a1) : ℤ)) * pellZd a1 (n + 1)).im
Tactic: ext
-- Lean 4 proof step from Mathlib.NumberTheory.PellMatiyasevic (goal)
Goal:
a : ℕ
a1 : 1 < a
n : ℕ
⊢ (pellZd a1 (n + 2) + pellZd a1 n).re = ((↑(2 * a) : ℤ√(↑(d a1) : ℤ)) * pellZd a1 (n + 1)).re
have re : (HMul.hMul (α := ℤ) (↑(xn a1 n) : ℤ) (↑a : ℤ) + (HMul.hMul (α := ℤ) (↑a : ℤ) (↑a : ℤ) - 1) * (↑(yn a1 n) : ℤ)) *
        (↑a : ℤ) +
      (HMul.hMul (α := ℤ) (↑a : ℤ) (↑a : ℤ) - 1) * ((↑(xn a1 n) : ℤ) + HMul.hMul (α := ℤ) (↑(yn a1 n) : ℤ) (↑a : ℤ)) +
    (↑(xn a1 n) : ℤ) =
  HMul.hMul (α := ℤ) 2 (↑a : ℤ) *
    (HMul.hMul (α := ℤ) (↑(xn a1 n) : ℤ) (↑a : ℤ) + (HMul.hMul (α := ℤ) (↑a : ℤ) (↑a : ℤ) - 1) * (↑(yn a1 n) : ℤ))
Tactic: simp [dz_val, az]
-- Lean 4 proof step from Mathlib.NumberTheory.PellMatiyasevic (goal)
Goal:
a : ℕ
a1 : 1 < a
n : ℕ
⊢ (pellZd a1 (n + 2) + pellZd a1 n).im = ((↑(2 * a) : ℤ√(↑(d a1) : ℤ)) * pellZd a1 (n + 1)).im
have im : HMul.hMul (α := ℤ) (↑(xn a1 n) : ℤ) (↑a : ℤ) + (HMul.hMul (α := ℤ) (↑a : ℤ) (↑a : ℤ) - 1) * (↑(yn a1 n) : ℤ) +
      ((↑(xn a1 n) : ℤ) + HMul.hMul (α := ℤ) (↑(yn a1 n) : ℤ) (↑a : ℤ)) * (↑a : ℤ) +
    (↑(yn a1 n) : ℤ) =
  HMul.hMul (α := ℤ) 2 (↑a : ℤ) * ((↑(xn a1 n) : ℤ) + HMul.hMul (α := ℤ) (↑(yn a1 n) : ℤ) (↑a : ℤ))
Tactic: simp [dz_val, az]
-- Lean 4 proof step from Mathlib.NumberTheory.PellMatiyasevic (hypothesis)
Goal:
a : ℕ
a1 : 1 < a
n : ℕ
⊢ xn a1 (n + 2) + xn a1 n = 2 * a * xn a1 (n + 1) ∧ yn a1 (n + 2) + yn a1 n = 2 * a * yn a1 (n + 1)
have this : pellZd a1 (n + 2) + pellZd a1 n = (↑(2 * a) : ℤ√(↑(d a1) : ℤ)) * pellZd a1 (n + 1)
Tactic: have := pellZd_succ_succ a1 n
-- Lean 4 proof step from Mathlib.NumberTheory.PellMatiyasevic (hypothesis)
Goal:
a : ℕ
a1 : 1 < a
n : ℕ
this :
  HAdd.hAdd (α := ℤ√(↑(d a1) : ℤ)) { re := (↑(xn a1 (n + 2)) : ℤ), im := (↑(yn a1 (n + 2)) : ℤ) : ℤ√(↑(d a1) : ℤ) }
      { re := (↑(xn a1 n) : ℤ), im := (↑(yn a1 n) : ℤ) : ℤ√(↑(d a1) : ℤ) } =
    HMul.hMul (α := ℤ√(↑(d a1) : ℤ)) (↑(2 * a) : ℤ√(↑(d a1) : ℤ))
      { re := (↑(xn a1 (n + 1)) : ℤ), im := (↑(yn a1 (n + 1)) : ℤ) }
⊢ xn a1 (n + 2) + xn a1 n = 2 * a * xn a1 (n + 1) ∧ yn a1 (n + 2) + yn a1 n = 2 * a * yn a1 (n + 1)
have this₁ : HAdd.hAdd (α := ℤ√(↑(d a1) : ℤ)) { re := (↑(xn a1 (n + 2)) : ℤ), im := (↑(yn a1 (n + 2)) : ℤ) : ℤ√(↑(d a1) : ℤ) }
    { re := (↑(xn a1 n) : ℤ), im := (↑(yn a1 n) : ℤ) : ℤ√(↑(d a1) : ℤ) } =
  { re := HMul.hMul (α := ℤ) (↑(2 * a) : ℤ) (↑(xn a1 (n + 1)) : ℤ),
    im := HMul.hMul (α := ℤ) (↑(2 * a) : ℤ) (↑(yn a1 (n + 1)) : ℤ) }
Tactic: rw [Zsqrtd.nsmul_val (2 * a : ℕ)] at this
Cleared hypotheses: this
-- Lean 4 proof step from Mathlib.NumberTheory.PellMatiyasevic (hypothesis)
Goal:
a : ℕ
a1 : 1 < a
n : ℕ
this :
  HAdd.hAdd (α := ℤ√(↑(d a1) : ℤ)) { re := (↑(xn a1 (n + 2)) : ℤ), im := (↑(yn a1 (n + 2)) : ℤ) : ℤ√(↑(d a1) : ℤ) }
      { re := (↑(xn a1 n) : ℤ), im := (↑(yn a1 n) : ℤ) : ℤ√(↑(d a1) : ℤ) } =
    { re := HMul.hMul (α := ℤ) (↑(2 * a) : ℤ) (↑(xn a1 (n + 1)) : ℤ),
      im := HMul.hMul (α := ℤ) (↑(2 * a) : ℤ) (↑(yn a1 (n + 1)) : ℤ) }
⊢ xn a1 (n + 2) + xn a1 n = 2 * a * xn a1 (n + 1) ∧ yn a1 (n + 2) + yn a1 n = 2 * a * yn a1 (n + 1)
have h₁ : Zsqrtd.re (d := (↑(d a1) : ℤ)) { re := (↑(xn a1 (n + 2)) : ℤ), im := (↑(yn a1 (n + 2)) : ℤ) : ℤ√(↑(d a1) : ℤ) } +
    Zsqrtd.re (d := (↑(d a1) : ℤ)) { re := (↑(xn a1 n) : ℤ), im := (↑(yn a1 n) : ℤ) : ℤ√(↑(d a1) : ℤ) } =
  HMul.hMul (α := ℤ) (↑(2 * a) : ℤ) (↑(xn a1 (n + 1)) : ℤ)
Tactic: injection this with h₁ h₂
Cleared hypotheses: this
-- Lean 4 proof step from Mathlib.NumberTheory.PellMatiyasevic (hypothesis)
Goal:
a : ℕ
a1 : 1 < a
n : ℕ
this :
  HAdd.hAdd (α := ℤ√(↑(d a1) : ℤ)) { re := (↑(xn a1 (n + 2)) : ℤ), im := (↑(yn a1 (n + 2)) : ℤ) : ℤ√(↑(d a1) : ℤ) }
      { re := (↑(xn a1 n) : ℤ), im := (↑(yn a1 n) : ℤ) : ℤ√(↑(d a1) : ℤ) } =
    { re := HMul.hMul (α := ℤ) (↑(2 * a) : ℤ) (↑(xn a1 (n + 1)) : ℤ),
      im := HMul.hMul (α := ℤ) (↑(2 * a) : ℤ) (↑(yn a1 (n + 1)) : ℤ) }
h₁ :
  Zsqrtd.re (d := (↑(d a1) : ℤ)) { re := (↑(xn a1 (n + 2)) : ℤ), im := (↑(yn a1 (n + 2)) : ℤ) : ℤ√(↑(d a1) : ℤ) } +
      Zsqrtd.re (d := (↑(d a1) : ℤ)) { re := (↑(xn a1 n) : ℤ), im := (↑(yn a1 n) : ℤ) : ℤ√(↑(d a1) : ℤ) } =
    HMul.hMul (α := ℤ) (↑(2 * a) : ℤ) (↑(xn a1 (n + 1)) : ℤ)
⊢ xn a1 (n + 2) + xn a1 n = 2 * a * xn a1 (n + 1) ∧ yn a1 (n + 2) + yn a1 n = 2 * a * yn a1 (n + 1)
have h₂ : Zsqrtd.im (d := (↑(d a1) : ℤ)) { re := (↑(xn a1 (n + 2)) : ℤ), im := (↑(yn a1 (n + 2)) : ℤ) : ℤ√(↑(d a1) : ℤ) } +
    Zsqrtd.im (d := (↑(d a1) : ℤ)) { re := (↑(xn a1 n) : ℤ), im := (↑(yn a1 n) : ℤ) : ℤ√(↑(d a1) : ℤ) } =
  HMul.hMul (α := ℤ) (↑(2 * a) : ℤ) (↑(yn a1 (n + 1)) : ℤ)
Tactic: injection this with h₁ h₂
Cleared hypotheses: this
-- Lean 4 proof step from Mathlib.NumberTheory.PellMatiyasevic (goal)
Goal:
a : ℕ
a1 : 1 < a
n : ℕ
⊢ HAdd.hAdd (α := ℤ) (↑(xn a1 (n + 2)) : ℤ) (↑(xn a1 n) : ℤ) = HMul.hMul (α := ℤ) (↑(2 * a) : ℤ) (↑(xn a1 (n + 1)) : ℤ)
have rw : (↑(xn a1 (n + 2) + xn a1 n) : ℤ) = HMul.hMul (α := ℤ) (↑(2 * a) : ℤ) (↑(xn a1 (n + 1)) : ℤ)
Tactic: rw [← Int.natCast_add, ← Int.natCast_mul, xn_succ_succ]
-- Lean 4 proof step from Mathlib.NumberTheory.PellMatiyasevic (goal)
Goal:
a : ℕ
a1 : 1 < a
n : ℕ
rw : (↑(xn a1 (n + 2) + xn a1 n) : ℤ) = HMul.hMul (α := ℤ) (↑(2 * a) : ℤ) (↑(xn a1 (n + 1)) : ℤ)
⊢ HAdd.hAdd (α := ℤ) (↑(xn a1 (n + 2)) : ℤ) (↑(xn a1 n) : ℤ) = HMul.hMul (α := ℤ) (↑(2 * a) : ℤ) (↑(xn a1 (n + 1)) : ℤ)
have rw₁ : Eq (α := ℤ) (↑(xn a1 (n + 2) + xn a1 n) : ℤ) (↑(2 * a * xn a1 (n + 1)) : ℤ)
Tactic: rw [← Int.natCast_add, ← Int.natCast_mul, xn_succ_succ]
-- Lean 4 proof step from Mathlib.NumberTheory.PellMatiyasevic (goal)
Goal:
a : ℕ
a1 : 1 < a
n : ℕ
rw : (↑(xn a1 (n + 2) + xn a1 n) : ℤ) = HMul.hMul (α := ℤ) (↑(2 * a) : ℤ) (↑(xn a1 (n + 1)) : ℤ)
rw₁ : Eq (α := ℤ) (↑(xn a1 (n + 2) + xn a1 n) : ℤ) (↑(2 * a * xn a1 (n + 1)) : ℤ)
⊢ HAdd.hAdd (α := ℤ) (↑(xn a1 (n + 2)) : ℤ) (↑(xn a1 n) : ℤ) = HMul.hMul (α := ℤ) (↑(2 * a) : ℤ) (↑(xn a1 (n + 1)) : ℤ)
have rw₂ : Eq (α := ℤ) (↑(2 * a * xn a1 (n + 1)) : ℤ) (↑(2 * a * xn a1 (n + 1)) : ℤ)
Tactic: rw [← Int.natCast_add, ← Int.natCast_mul, xn_succ_succ]
-- Lean 4 proof step from Mathlib.NumberTheory.PellMatiyasevic (goal)
Goal:
a : ℕ
a1 : 1 < a
n : ℕ
⊢ HAdd.hAdd (α := ℤ) (↑(yn a1 (n + 2)) : ℤ) (↑(yn a1 n) : ℤ) = HMul.hMul (α := ℤ) (↑(2 * a) : ℤ) (↑(yn a1 (n + 1)) : ℤ)
have rw : (↑(yn a1 (n + 2) + yn a1 n) : ℤ) = HMul.hMul (α := ℤ) (↑(2 * a) : ℤ) (↑(yn a1 (n + 1)) : ℤ)
Tactic: rw [← Int.natCast_add, ← Int.natCast_mul, yn_succ_succ]
-- Lean 4 proof step from Mathlib.NumberTheory.PellMatiyasevic (goal)
Goal:
a : ℕ
a1 : 1 < a
n : ℕ
rw : (↑(yn a1 (n + 2) + yn a1 n) : ℤ) = HMul.hMul (α := ℤ) (↑(2 * a) : ℤ) (↑(yn a1 (n + 1)) : ℤ)
⊢ HAdd.hAdd (α := ℤ) (↑(yn a1 (n + 2)) : ℤ) (↑(yn a1 n) : ℤ) = HMul.hMul (α := ℤ) (↑(2 * a) : ℤ) (↑(yn a1 (n + 1)) : ℤ)
have rw₁ : Eq (α := ℤ) (↑(yn a1 (n + 2) + yn a1 n) : ℤ) (↑(2 * a * yn a1 (n + 1)) : ℤ)
Tactic: rw [← Int.natCast_add, ← Int.natCast_mul, yn_succ_succ]
-- Lean 4 proof step from Mathlib.NumberTheory.PellMatiyasevic (goal)
Goal:
a : ℕ
a1 : 1 < a
n : ℕ
rw : (↑(yn a1 (n + 2) + yn a1 n) : ℤ) = HMul.hMul (α := ℤ) (↑(2 * a) : ℤ) (↑(yn a1 (n + 1)) : ℤ)
rw₁ : Eq (α := ℤ) (↑(yn a1 (n + 2) + yn a1 n) : ℤ) (↑(2 * a * yn a1 (n + 1)) : ℤ)
⊢ HAdd.hAdd (α := ℤ) (↑(yn a1 (n + 2)) : ℤ) (↑(yn a1 n) : ℤ) = HMul.hMul (α := ℤ) (↑(2 * a) : ℤ) (↑(yn a1 (n + 1)) : ℤ)
have rw₂ : Eq (α := ℤ) (↑(2 * a * yn a1 (n + 1)) : ℤ) (↑(2 * a * yn a1 (n + 1)) : ℤ)
Tactic: rw [← Int.natCast_add, ← Int.natCast_mul, yn_succ_succ]
-- Lean 4 proof step from Mathlib.NumberTheory.PellMatiyasevic (goal)
Goal:
a : ℕ
a1 : 1 < a
n : ℕ
⊢ yn a1 (n + 2) + yn a1 n ≡ n + 2 + n [MOD a - 1]
have rw : 2 * a * yn a1 (n + 1) ≡ 2 * (n + 1) [MOD a - 1]
Tactic: rw [yn_succ_succ, (by ring : n + 2 + n = 2 * (n + 1))]
-- Lean 4 proof step from Mathlib.NumberTheory.PellMatiyasevic (goal)
Goal:
a : ℕ
a1 : 1 < a
n : ℕ
⊢ yn a1 (n + 2) + yn a1 n ≡ n + 2 + n [MOD 2]
have rw : 2 * (a * yn a1 (n + 1)) ≡ 2 * (n + 1) [MOD 2]
Tactic: rw [yn_succ_succ, mul_assoc, (by ring : n + 2 + n = 2 * (n + 1))]
-- Lean 4 proof step from Mathlib.NumberTheory.PellMatiyasevic (goal)
Goal:
a : ℕ
a1 : 1 < a
y n : ℕ
⊢ (↑(y ^ (n + 2)) : ℤ) - HMul.hMul (α := ℤ) (↑(2 * a) : ℤ) (↑(y ^ (n + 1)) : ℤ) + (↑(y ^ n) : ℤ) =
    (HMul.hMul (α := ℤ) 2 (↑a : ℤ) * (↑y : ℤ) - HMul.hMul (α := ℤ) (↑y : ℤ) (↑y : ℤ) - 1) * -(↑(y ^ n) : ℤ)
have simp : (↑y : ℤ) * ((↑y : ℤ) * HPow.hPow (α := ℤ) (↑y : ℤ) n) - (↑a : ℤ) * ((↑y : ℤ) * HPow.hPow (α := ℤ) (↑y : ℤ) n * 2) +
    HPow.hPow (α := ℤ) (↑y : ℤ) n =
  -(HPow.hPow (α := ℤ) (↑y : ℤ) n *
      ((↑a : ℤ) * HMul.hMul (α := ℤ) (↑y : ℤ) 2 - HMul.hMul (α := ℤ) (↑y : ℤ) (↑y : ℤ) - 1))
Tactic: simp [_root_.pow_succ, mul_comm, mul_left_comm]
-- Lean 4 proof step from Mathlib.NumberTheory.PellMatiyasevic (hypothesis)
Goal:
a : ℕ
a1 : 1 < a
y n : ℕ
⊢ HMul.hMul (α := ℤ) 2 (↑a : ℤ) * (↑y : ℤ) - HMul.hMul (α := ℤ) (↑y : ℤ) (↑y : ℤ) - 1 ∣
    yz a1 (n + 2) * HSub.hSub (α := ℤ) (↑a : ℤ) (↑y : ℤ) + (↑(y ^ (n + 2)) : ℤ) - xz a1 (n + 2)
have this : HMul.hMul (α := ℤ) 2 (↑a : ℤ) * (↑y : ℤ) - HMul.hMul (α := ℤ) (↑y : ℤ) (↑y : ℤ) - 1 ∣
  (↑(y ^ (n + 2)) : ℤ) - HMul.hMul (α := ℤ) (↑(2 * a) : ℤ) (↑(y ^ (n + 1)) : ℤ) + (↑(y ^ n) : ℤ)
Tactic: have : (2 * a * y - y * y - 1 : ℤ) ∣ ↑(y ^ (n + 2)) - ↑(2 * a) * ↑(y ^ (n + 1)) + ↑(y ^ n) :=
      ⟨-↑(y ^ n), by
        simp [_root_.pow_succ, mul_comm, mul_left_comm]
        ring⟩
-- Lean 4 proof step from Mathlib.NumberTheory.PellMatiyasevic (goal)
Goal:
a : ℕ
a1 : 1 < a
y n : ℕ
this :
  HMul.hMul (α := ℤ) 2 (↑a : ℤ) * (↑y : ℤ) - HMul.hMul (α := ℤ) (↑y : ℤ) (↑y : ℤ) - 1 ∣
    (↑(y ^ (n + 2)) : ℤ) - HMul.hMul (α := ℤ) (↑(2 * a) : ℤ) (↑(y ^ (n + 1)) : ℤ) + (↑(y ^ n) : ℤ)
⊢ HMul.hMul (α := ℤ) 2 (↑a : ℤ) * (↑y : ℤ) - HMul.hMul (α := ℤ) (↑y : ℤ) (↑y : ℤ) - 1 ∣
    yz a1 (n + 2) * HSub.hSub (α := ℤ) (↑a : ℤ) (↑y : ℤ) + (↑(y ^ (n + 2)) : ℤ) - xz a1 (n + 2)
have rw : HMul.hMul (α := ℤ) 2 (↑a : ℤ) * (↑y : ℤ) - HMul.hMul (α := ℤ) (↑y : ℤ) (↑y : ℤ) - 1 ∣
  (↑(y ^ (n + 2)) : ℤ) - HMul.hMul (α := ℤ) (↑(2 * a) : ℤ) (↑(y ^ (n + 1)) : ℤ) + (↑(y ^ n) : ℤ) +
      (↑(2 * a) : ℤ) * (yz a1 (n + 1) * HSub.hSub (α := ℤ) (↑a : ℤ) (↑y : ℤ) + (↑(y ^ (n + 1)) : ℤ) - xz a1 (n + 1)) -
    (yz a1 n * HSub.hSub (α := ℤ) (↑a : ℤ) (↑y : ℤ) + (↑(y ^ n) : ℤ) - xz a1 n)
Tactic: rw [xz_succ_succ, yz_succ_succ, x_sub_y_dvd_pow_lem ↑(y ^ (n + 2)) ↑(y ^ (n + 1)) ↑(y ^ n)]
-- Lean 4 proof step from Mathlib.NumberTheory.PellMatiyasevic (hypothesis)
Goal:
a : ℕ
a1 : 1 < a
n j : ℕ
⊢ xn a1 n ∣ d a1 * yn a1 n * (yn a1 n * xn a1 j) + xn a1 j
have h1 : d a1 * yn a1 n * (yn a1 n * xn a1 j) + xn a1 j = (d a1 * yn a1 n * yn a1 n + 1) * xn a1 j
Tactic: have h1 : d a1 * yn a1 n * (yn a1 n * xn a1 j) + xn a1 j =
      (d a1 * yn a1 n * yn a1 n + 1) * xn a1 j := by
    simp [add_mul, mul_assoc]
-- Lean 4 proof step from Mathlib.NumberTheory.PellMatiyasevic (goal)
Goal:
a : ℕ
a1 : 1 < a
n j : ℕ
h1 : d a1 * yn a1 n * (yn a1 n * xn a1 j) + xn a1 j = (d a1 * yn a1 n * yn a1 n + 1) * xn a1 j
⊢ d a1 * yn a1 n * yn a1 n + 1 = xn a1 n * xn a1 n
have zify : ∀
  (h1 :
    HMul.hMul (α := ℤ) (↑(d a1) : ℤ) (↑(yn a1 n) : ℤ) * HMul.hMul (α := ℤ) (↑(yn a1 n) : ℤ) (↑(xn a1 j) : ℤ) +
        (↑(xn a1 j) : ℤ) =
      (HMul.hMul (α := ℤ) (↑(d a1) : ℤ) (↑(yn a1 n) : ℤ) * (↑(yn a1 n) : ℤ) + 1) * (↑(xn a1 j) : ℤ)),
  HMul.hMul (α := ℤ) (↑(d a1) : ℤ) (↑(yn a1 n) : ℤ) * (↑(yn a1 n) : ℤ) + 1 =
    HMul.hMul (α := ℤ) (↑(xn a1 n) : ℤ) (↑(xn a1 n) : ℤ)
Tactic: zify at *
Cleared hypotheses: h1
-- Lean 4 proof step from Mathlib.NumberTheory.PellMatiyasevic (hypothesis)
Goal:
a : ℕ
a1 : 1 < a
n j : ℕ
h1 : d a1 * yn a1 n * (yn a1 n * xn a1 j) + xn a1 j = (d a1 * yn a1 n * yn a1 n + 1) * xn a1 j
⊢ xn a1 n ∣ d a1 * yn a1 n * (yn a1 n * xn a1 j) + xn a1 j
have h2 : d a1 * yn a1 n * yn a1 n + 1 = xn a1 n * xn a1 n
Tactic: have h2 : d a1 * yn a1 n * yn a1 n + 1 = xn a1 n * xn a1 n := by
    zify at *
    apply add_eq_of_eq_sub' (Eq.symm (pell_eqz a1 n))
-- Lean 4 proof step from Mathlib.NumberTheory.PellMatiyasevic (hypothesis)
Goal:
a : ℕ
a1 : 1 < a
n j : ℕ
h1 : d a1 * yn a1 n * (yn a1 n * xn a1 j) + xn a1 j = (d a1 * yn a1 n * yn a1 n + 1) * xn a1 j
h2 : d a1 * yn a1 n * yn a1 n + 1 = xn a1 n * xn a1 n
⊢ xn a1 n ∣ d a1 * yn a1 n * (yn a1 n * xn a1 j) + xn a1 j
have h1₁ : d a1 * yn a1 n * (yn a1 n * xn a1 j) + xn a1 j = xn a1 n * xn a1 n * xn a1 j
Tactic: rw [h2] at h1
Cleared hypotheses: h1, h2
-- Lean 4 proof step from Mathlib.NumberTheory.PellMatiyasevic (goal)
Goal:
a : ℕ
a1 : 1 < a
n j : ℕ
h1 : d a1 * yn a1 n * (yn a1 n * xn a1 j) + xn a1 j = xn a1 n * xn a1 n * xn a1 j
h2 : d a1 * yn a1 n * yn a1 n + 1 = xn a1 n * xn a1 n
⊢ xn a1 n ∣ d a1 * yn a1 n * (yn a1 n * xn a1 j) + xn a1 j
have rw : xn a1 n ∣ xn a1 n * (xn a1 n * xn a1 j)
Tactic: rw [h1, mul_assoc]
-- Lean 4 proof step from Mathlib.NumberTheory.PellMatiyasevic (goal)
Goal:
a : ℕ
a1 : 1 < a
n j : ℕ
⊢ xn a1 (2 * n + j) + xn a1 j ≡ 0 [MOD xn a1 n]
have rw : xn a1 n * xn a1 (n + j) + (d a1 * yn a1 n * yn a1 (n + j) + xn a1 j) ≡ HAdd.hAdd (α := ℕ) 0 0 [MOD xn a1 n]
Tactic: rw [two_mul, add_assoc, xn_add, add_assoc, ← zero_add 0]
-- Lean 4 proof step from Mathlib.NumberTheory.PellMatiyasevic (goal)
Goal:
a : ℕ
a1 : 1 < a
n j : ℕ
⊢ xn a1 n * xn a1 (n + j) + (d a1 * yn a1 n * yn a1 (n + j) + xn a1 j) ≡ HAdd.hAdd (α := ℕ) 0 0 [MOD xn a1 n]
have refine : d a1 * yn a1 n * yn a1 (n + j) + xn a1 j ≡ 0 [MOD xn a1 n]
Tactic: refine (dvd_mul_right (xn a1 n) (xn a1 (n + j))).modEq_zero_nat.add ?_
-- Lean 4 proof step from Mathlib.NumberTheory.PellMatiyasevic (goal)
Goal:
a : ℕ
a1 : 1 < a
n j : ℕ
⊢ d a1 * yn a1 n * yn a1 (n + j) + xn a1 j ≡ 0 [MOD xn a1 n]
have rw : d a1 * yn a1 n * (xn a1 n * yn a1 j) + (d a1 * yn a1 n * (yn a1 n * xn a1 j) + xn a1 j) ≡ HAdd.hAdd (α := ℕ) 0 0 [MOD
  xn a1 n]
Tactic: rw [yn_add, left_distrib, add_assoc, ← zero_add 0]
-- Lean 4 proof step from Mathlib.NumberTheory.PellMatiyasevic (goal)
Goal:
a : ℕ
a1 : 1 < a
n j : ℕ
h : j ≤ n
⊢ xz a1 n ∣ (↑(d a1) : ℤ) * yz a1 n * yz a1 (n - j) + xz a1 j
have rw : xz a1 n ∣ (↑(d a1) : ℤ) * yz a1 n * (xz a1 j * yz a1 n) + xz a1 j - (↑(d a1) : ℤ) * yz a1 n * (xz a1 n * yz a1 j)
Tactic: rw [yz_sub _ h, mul_sub_left_distrib, sub_add_eq_add_sub]
-- Lean 4 proof step from Mathlib.NumberTheory.PellMatiyasevic (goal)
Goal:
a : ℕ
a1 : 1 < a
n j : ℕ
h : j ≤ n
⊢ (↑(xn a1 n) : ℤ) ∣
    HMul.hMul (α := ℤ) (↑(d a1) : ℤ) (↑(yn a1 n) : ℤ) * HMul.hMul (α := ℤ) (↑(xn a1 j) : ℤ) (↑(yn a1 n) : ℤ) +
      (↑(xn a1 j) : ℤ)
have rw : (↑(xn a1 n) : ℤ) ∣
  HMul.hMul (α := ℤ) (↑(d a1) : ℤ) (↑(yn a1 n) : ℤ) * HMul.hMul (α := ℤ) (↑(yn a1 n) : ℤ) (↑(xn a1 j) : ℤ) +
    (↑(xn a1 j) : ℤ)
Tactic: rw [mul_comm (xn _ _ : ℤ)]
-- Lean 4 proof step from Mathlib.NumberTheory.PellMatiyasevic (hypothesis)
Goal:
a : ℕ
a1 : 1 < a
n j : ℕ
h : j ≤ n
⊢ xn a1 (2 * n - j) + xn a1 j ≡ 0 [MOD xn a1 n]
have h1 : xz a1 n ∣ (↑(d a1) : ℤ) * yz a1 n * yz a1 (n - j) + xz a1 j
Tactic: have h1 : xz a1 n ∣ d a1 * yz a1 n * yz a1 (n - j) + xz a1 j := by
    rw [yz_sub _ h, mul_sub_left_distrib, sub_add_eq_add_sub]
    exact
      dvd_sub
        (by
          delta xz; delta yz
          rw [mul_comm (xn _ _ : ℤ)]
          exact mod_cast (xn_modEq_x2n_add_lem _ n j))
        ((dvd_mul_right _ _).mul_left _)
-- Lean 4 proof step from Mathlib.NumberTheory.PellMatiyasevic (goal)
Goal:
a : ℕ
a1 : 1 < a
n j : ℕ
h : j ≤ n
h1 : xz a1 n ∣ (↑(d a1) : ℤ) * yz a1 n * yz a1 (n - j) + xz a1 j
⊢ xn a1 (2 * n - j) + xn a1 j ≡ 0 [MOD xn a1 n]
have rw : xn a1 n * xn a1 (n - j) + (d a1 * yn a1 n * yn a1 (n - j) + xn a1 j) ≡ HAdd.hAdd (α := ℕ) 0 0 [MOD xn a1 n]
Tactic: rw [two_mul, add_tsub_assoc_of_le h, xn_add, add_assoc, ← zero_add 0]
-- Lean 4 proof step from Mathlib.NumberTheory.PellMatiyasevic (goal)
Goal:
a : ℕ
a1 : 1 < a
n j : ℕ
h : j ≤ 2 * n
jn : n ≤ j
⊢ 2 * n - j + j ≤ n + j
have rw : n + n ≤ n + j
Tactic: rw [tsub_add_cancel_of_le h, two_mul]
-- Lean 4 proof step from Mathlib.NumberTheory.PellMatiyasevic (hypothesis)
Goal:
a : ℕ
a1 : 1 < a
n j : ℕ
h : j ≤ 2 * n
jn : n ≤ j
⊢ xn a1 (2 * n - j) + xn a1 j ≡ 0 [MOD xn a1 n]
have this : 2 * n - j + j ≤ n + j
Tactic: have : 2 * n - j + j ≤ n + j := by
      rw [tsub_add_cancel_of_le h, two_mul]; exact Nat.add_le_add_left jn _
-- Lean 4 proof step from Mathlib.NumberTheory.PellMatiyasevic (goal)
Goal:
a : ℕ
a1 : 1 < a
n j : ℕ
h : j ≤ 2 * n
jn : n ≤ j
this : 2 * n - j + j ≤ n + j
t : xn a1 (2 * n - (2 * n - j)) + xn a1 (2 * n - j) ≡ 0 [MOD xn a1 n] := ⋯
⊢ xn a1 (2 * n - j) + xn a1 j ≡ 0 [MOD xn a1 n]
have rwa : ∀ (t : xn a1 j + xn a1 (2 * n - j) ≡ 0 [MOD xn a1 n]), xn a1 (2 * n - j) + xn a1 j ≡ 0 [MOD xn a1 n]
Tactic: rwa [tsub_tsub_cancel_of_le h, add_comm] at t
Cleared hypotheses: t
-- Lean 4 proof step from Mathlib.NumberTheory.PellMatiyasevic (goal)
Goal:
a : ℕ
a1 : 1 < a
n j : ℕ
h : j ≤ 2 * n
jn : n ≤ j
this : 2 * n - j + j ≤ n + j
t : xn a1 (2 * n - (2 * n - j)) + xn a1 (2 * n - j) ≡ 0 [MOD xn a1 n] := ⋯
rwa : ∀ (t : xn a1 j + xn a1 (2 * n - j) ≡ 0 [MOD xn a1 n]), xn a1 (2 * n - j) + xn a1 j ≡ 0 [MOD xn a1 n]
⊢ xn a1 (2 * n - j) + xn a1 j ≡ 0 [MOD xn a1 n]
have rwa₁ : ∀ (t : xn a1 (2 * n - j) + xn a1 j ≡ 0 [MOD xn a1 n]), xn a1 (2 * n - j) + xn a1 j ≡ 0 [MOD xn a1 n]
Tactic: rwa [tsub_tsub_cancel_of_le h, add_comm] at t
Cleared hypotheses: t
-- Lean 4 proof step from Mathlib.NumberTheory.PellMatiyasevic (goal)
Goal:
a : ℕ
a1 : 1 < a
n j : ℕ
⊢ 0 ≡ xn a1 j + xn a1 (2 * n + j) [MOD xn a1 n]
have rw : 0 ≡ xn a1 (2 * n + j) + xn a1 j [MOD xn a1 n]
Tactic: rw [add_comm]
-- Lean 4 proof step from Mathlib.NumberTheory.PellMatiyasevic (goal)
Goal:
a : ℕ
a1 : 1 < a
n j : ℕ
⊢ xn a1 (4 * n + j) + xn a1 (2 * n + j) ≡ xn a1 j + xn a1 (2 * n + j) [MOD xn a1 n]
have refine : xn a1 (4 * n + j) + xn a1 (2 * n + j) ≡ 0 [MOD xn a1 n]
Tactic: refine @ModEq.trans _ _ 0 _ ?_ (by rw [add_comm]; exact (xn_modEq_x2n_add _ _ _).symm)
-- Lean 4 proof step from Mathlib.NumberTheory.PellMatiyasevic (goal)
Goal:
a : ℕ
a1 : 1 < a
n j : ℕ
⊢ xn a1 (4 * n + j) + xn a1 (2 * n + j) ≡ 0 [MOD xn a1 n]
have rw : xn a1 (2 * n + (2 * n + j)) + xn a1 (2 * n + j) ≡ 0 [MOD xn a1 n]
Tactic: rw [show 4 * n = 2 * n + 2 * n from right_distrib 2 2 n, add_assoc]
-- Lean 4 proof step from Mathlib.NumberTheory.PellMatiyasevic (goal)
Goal:
a : ℕ
a1 : 1 < a
n j : ℕ
h : j ≤ 2 * n
⊢ 2 * n ≤ 2 * n
have rw : 1 * n + n ≤ 1 * n + n
Tactic: rw [Nat.succ_mul]
-- Lean 4 proof step from Mathlib.NumberTheory.PellMatiyasevic (goal)
Goal:
a : ℕ
a1 : 1 < a
n j : ℕ
h h' : j ≤ 2 * n
⊢ 0 ≡ xn a1 j + xn a1 (2 * n - j) [MOD xn a1 n]
have rw : 0 ≡ xn a1 (2 * n - j) + xn a1 j [MOD xn a1 n]
Tactic: rw [add_comm]
-- Lean 4 proof step from Mathlib.NumberTheory.PellMatiyasevic (goal)
Goal:
a : ℕ
a1 : 1 < a
n j : ℕ
h h' : j ≤ 2 * n
⊢ xn a1 (4 * n - j) + xn a1 (2 * n - j) ≡ xn a1 j + xn a1 (2 * n - j) [MOD xn a1 n]
have refine : xn a1 (4 * n - j) + xn a1 (2 * n - j) ≡ 0 [MOD xn a1 n]
Tactic: refine @ModEq.trans _ _ 0 _ ?_ (by rw [add_comm]; exact (xn_modEq_x2n_sub _ h).symm)
-- Lean 4 proof step from Mathlib.NumberTheory.PellMatiyasevic (goal)
Goal:
a : ℕ
a1 : 1 < a
n j : ℕ
h h' : j ≤ 2 * n
⊢ xn a1 (4 * n - j) + xn a1 (2 * n - j) ≡ 0 [MOD xn a1 n]
have rw : xn a1 (2 * n + (2 * n - j)) + xn a1 (2 * n - j) ≡ 0 [MOD xn a1 n]
Tactic: rw [show 4 * n = 2 * n + 2 * n from right_distrib 2 2 n, add_tsub_assoc_of_le h']
-- Lean 4 proof step from Mathlib.NumberTheory.PellMatiyasevic (goal)
Goal:
a : ℕ
a1 : 1 < a
i n j : ℕ
ij : i < j + 1
jn : j + 1 < n
⊢ xn a1 i % xn a1 n < xn a1 (j + 1) % xn a1 n
have suffices : xn a1 j % xn a1 n < xn a1 (j + 1) % xn a1 n
Tactic: suffices xn a1 j % xn a1 n < xn a1 (j + 1) % xn a1 n from
      (lt_or_eq_of_le (Nat.le_of_succ_le_succ ij)).elim
        (fun h => lt_trans (eq_of_xn_modEq_lem1 h (le_of_lt jn)) this) fun h => by
        rw [h]; exact this
-- Lean 4 proof step from Mathlib.NumberTheory.PellMatiyasevic (goal)
Goal:
a : ℕ
a1 : 1 < a
i n j : ℕ
ij : i < j + 1
jn : j + 1 < n
⊢ xn a1 j % xn a1 n < xn a1 (j + 1) % xn a1 n
have rw : xn a1 j < xn a1 (j + 1)
Tactic: rw [Nat.mod_eq_of_lt (strictMono_x _ (Nat.lt_of_succ_lt jn)),
        Nat.mod_eq_of_lt (strictMono_x _ jn)]
-- Lean 4 proof step from Mathlib.NumberTheory.PellMatiyasevic (hypothesis)
Goal:
a : ℕ
a1 : 1 < a
n : ℕ
h : 2 * xn a1 n = xn a1 (n + 1)
⊢ a = 2 ∧ n = 0
have h₁ : xn a1 n * 2 = xn a1 n * a + d a1 * yn a1 n
Tactic: rw [xn_succ, mul_comm] at h
Cleared hypotheses: h