awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.NumberTheory.PellMatiyasevic (hypothesis)
Goal:
a : ℕ
a1 : 1 < a
n : ℕ
h : xn a1 n * 2 = xn a1 n * a + d a1 * yn a1 n
⊢ a = 2 ∧ n = 0
have this : n = 0
Tactic: have : n = 0 :=
    n.eq_zero_or_pos.resolve_right fun np =>
      _root_.ne_of_lt
        (lt_of_le_of_lt (Nat.mul_le_mul_left _ a1)
          (Nat.lt_add_of_pos_right <| mul_pos (d_pos a1) (strictMono_y a1 np)))
        h
-- Lean 4 proof step from Mathlib.NumberTheory.PellMatiyasevic (goal)
Goal:
a : ℕ
a1 : 1 < a
n : ℕ
h : xn a1 n * 2 = xn a1 n * a + d a1 * yn a1 n
this : n = 0
⊢ a = 2 ∧ n = 0
have refl : ∀ (h : xn a1 0 * 2 = xn a1 0 * a + d a1 * yn a1 0), a = 2 ∧ Eq (α := ℕ) 0 0
Tactic: cases this
Cleared hypotheses: n, h, this
-- Lean 4 proof step from Mathlib.NumberTheory.PellMatiyasevic (hypothesis)
Goal:
a : ℕ
a1 : 1 < a
h : xn a1 0 * 2 = xn a1 0 * a + d a1 * yn a1 0
⊢ a = 2 ∧ Eq (α := ℕ) 0 0
have h₁ : 2 = a
Tactic: simp at h
Cleared hypotheses: h
-- Lean 4 proof step from Mathlib.NumberTheory.PellMatiyasevic (hypothesis)
Goal:
a : ℕ
a1 : 1 < a
i n : ℕ
npos : 0 < n
j : ℕ
ij : i < j + 1
j2n : j + 1 ≤ 2 * n
jnn : j + 1 ≠ n
ntriv : ¬(a = 2 ∧ n = 1 ∧ i = 0 ∧ j + 1 = 2)
k : ℕ
kn : k > n
k2n : k ≤ 2 * n
k2nl : 2 * n - k < n := ⋯
⊢ (↑(xn a1 k % xn a1 n) : ℤ) = HSub.hSub (α := ℤ) (↑(xn a1 n) : ℤ) (↑(xn a1 (2 * n - k)) : ℤ)
have xle : xn a1 (2 * n - k) ≤ xn a1 n
Tactic: have xle : xn a1 (2 * n - k) ≤ xn a1 n := le_of_lt <| strictMono_x a1 k2nl
-- Lean 4 proof step from Mathlib.NumberTheory.PellMatiyasevic (goal)
Goal:
a : ℕ
a1 : 1 < a
i n : ℕ
npos : 0 < n
j : ℕ
ij : i < j + 1
j2n : j + 1 ≤ 2 * n
jnn : j + 1 ≠ n
ntriv : ¬(a = 2 ∧ n = 1 ∧ i = 0 ∧ j + 1 = 2)
k : ℕ
kn : k > n
k2n : k ≤ 2 * n
k2nl : 2 * n - k < n := ⋯
xle : xn a1 (2 * n - k) ≤ xn a1 n
this : xn a1 k % xn a1 n = xn a1 n - xn a1 (2 * n - k)
⊢ (↑(xn a1 k % xn a1 n) : ℤ) = HSub.hSub (α := ℤ) (↑(xn a1 n) : ℤ) (↑(xn a1 (2 * n - k)) : ℤ)
have rw : (↑(xn a1 n - xn a1 (2 * n - k)) : ℤ) = HSub.hSub (α := ℤ) (↑(xn a1 n) : ℤ) (↑(xn a1 (2 * n - k)) : ℤ)
Tactic: rw [this, Int.ofNat_sub xle]
-- Lean 4 proof step from Mathlib.NumberTheory.PellMatiyasevic (goal)
Goal:
a : ℕ
a1 : 1 < a
i n : ℕ
npos : 0 < n
j : ℕ
ij : i < j + 1
j2n : j + 1 ≤ 2 * n
jnn : j + 1 ≠ n
ntriv : ¬(a = 2 ∧ n = 1 ∧ i = 0 ∧ j + 1 = 2)
k : ℕ
kn : k > n
k2n : k ≤ 2 * n
k2nl : 2 * n - k < n := ⋯
xle : xn a1 (2 * n - k) ≤ xn a1 n
this : xn a1 k % xn a1 n = xn a1 n - xn a1 (2 * n - k)
rw : (↑(xn a1 n - xn a1 (2 * n - k)) : ℤ) = HSub.hSub (α := ℤ) (↑(xn a1 n) : ℤ) (↑(xn a1 (2 * n - k)) : ℤ)
⊢ (↑(xn a1 k % xn a1 n) : ℤ) = HSub.hSub (α := ℤ) (↑(xn a1 n) : ℤ) (↑(xn a1 (2 * n - k)) : ℤ)
have rw₁ : HSub.hSub (α := ℤ) (↑(xn a1 n) : ℤ) (↑(xn a1 (2 * n - k)) : ℤ) =
  HSub.hSub (α := ℤ) (↑(xn a1 n) : ℤ) (↑(xn a1 (2 * n - k)) : ℤ)
Tactic: rw [this, Int.ofNat_sub xle]
-- Lean 4 proof step from Mathlib.NumberTheory.PellMatiyasevic (goal)
Goal:
a : ℕ
a1 : 1 < a
i n : ℕ
npos : 0 < n
j : ℕ
ij : i < j + 1
j2n : j + 1 ≤ 2 * n
jnn : j + 1 ≠ n
ntriv : ¬(a = 2 ∧ n = 1 ∧ i = 0 ∧ j + 1 = 2)
k : ℕ
kn : k > n
k2n : k ≤ 2 * n
k2nl : 2 * n - k < n := ⋯
xle : xn a1 (2 * n - k) ≤ xn a1 n
⊢ (↑(xn a1 k % xn a1 n) : ℤ) = HSub.hSub (α := ℤ) (↑(xn a1 n) : ℤ) (↑(xn a1 (2 * n - k)) : ℤ)
have suffices : xn a1 k % xn a1 n = xn a1 n - xn a1 (2 * n - k)
Tactic: suffices xn a1 k % xn a1 n = xn a1 n - xn a1 (2 * n - k) by rw [this, Int.ofNat_sub xle]
-- Lean 4 proof step from Mathlib.NumberTheory.PellMatiyasevic (goal)
Goal:
a : ℕ
a1 : 1 < a
i n : ℕ
npos : 0 < n
j : ℕ
ij : i < j + 1
j2n : j + 1 ≤ 2 * n
jnn : j + 1 ≠ n
ntriv : ¬(a = 2 ∧ n = 1 ∧ i = 0 ∧ j + 1 = 2)
k : ℕ
kn : k > n
k2n : k ≤ 2 * n
k2nl : 2 * n - k < n := ⋯
xle : xn a1 (2 * n - k) ≤ xn a1 n
⊢ xn a1 k % xn a1 n = xn a1 n - xn a1 (2 * n - k)
have rw : xn a1 k % xn a1 n = (xn a1 n - xn a1 (2 * n - k)) % xn a1 n
Tactic: rw [← Nat.mod_eq_of_lt (Nat.sub_lt (x_pos a1 n) (x_pos a1 (2 * n - k)))]
-- Lean 4 proof step from Mathlib.NumberTheory.PellMatiyasevic (goal)
Goal:
a : ℕ
a1 : 1 < a
i n : ℕ
npos : 0 < n
j : ℕ
ij : i < j + 1
j2n : j + 1 ≤ 2 * n
jnn : j + 1 ≠ n
ntriv : ¬(a = 2 ∧ n = 1 ∧ i = 0 ∧ j + 1 = 2)
k : ℕ
kn : k > n
k2n : k ≤ 2 * n
k2nl : 2 * n - k < n := ⋯
xle : xn a1 (2 * n - k) ≤ xn a1 n
⊢ xn a1 k % xn a1 n = (xn a1 n - xn a1 (2 * n - k)) % xn a1 n
have apply : xn a1 k + xn a1 (2 * n - k) ≡ xn a1 n - xn a1 (2 * n - k) + xn a1 (2 * n - k) [MOD xn a1 n]
Tactic: apply ModEq.add_right_cancel' (xn a1 (2 * n - k))
-- Lean 4 proof step from Mathlib.NumberTheory.PellMatiyasevic (goal)
Goal:
a : ℕ
a1 : 1 < a
i n : ℕ
npos : 0 < n
j : ℕ
ij : i < j + 1
j2n : j + 1 ≤ 2 * n
jnn : j + 1 ≠ n
ntriv : ¬(a = 2 ∧ n = 1 ∧ i = 0 ∧ j + 1 = 2)
k : ℕ
kn : k > n
k2n : k ≤ 2 * n
k2nl : 2 * n - k < n := ⋯
xle : xn a1 (2 * n - k) ≤ xn a1 n
⊢ xn a1 k + xn a1 (2 * n - k) ≡ xn a1 n - xn a1 (2 * n - k) + xn a1 (2 * n - k) [MOD xn a1 n]
have rw : xn a1 k + xn a1 (2 * n - k) ≡ xn a1 n [MOD xn a1 n]
Tactic: rw [tsub_add_cancel_of_le xle]
-- Lean 4 proof step from Mathlib.NumberTheory.PellMatiyasevic (hypothesis)
Goal:
a : ℕ
a1 : 1 < a
i n : ℕ
npos : 0 < n
j : ℕ
ij : i < j + 1
j2n : j + 1 ≤ 2 * n
jnn : j + 1 ≠ n
ntriv : ¬(a = 2 ∧ n = 1 ∧ i = 0 ∧ j + 1 = 2)
k : ℕ
kn : k > n
k2n : k ≤ 2 * n
k2nl : 2 * n - k < n := ⋯
xle : xn a1 (2 * n - k) ≤ xn a1 n
⊢ xn a1 k + xn a1 (2 * n - k) ≡ xn a1 n [MOD xn a1 n]
have t : xn a1 (2 * n - (2 * n - k)) + xn a1 (2 * n - k) ≡ 0 [MOD xn a1 n]
Tactic: have t := xn_modEq_x2n_sub_lem a1 k2nl.le
-- Lean 4 proof step from Mathlib.NumberTheory.PellMatiyasevic (hypothesis)
Goal:
a : ℕ
a1 : 1 < a
i n : ℕ
npos : 0 < n
j : ℕ
ij : i < j + 1
j2n : j + 1 ≤ 2 * n
jnn : j + 1 ≠ n
ntriv : ¬(a = 2 ∧ n = 1 ∧ i = 0 ∧ j + 1 = 2)
k : ℕ
kn : k > n
k2n : k ≤ 2 * n
k2nl : 2 * n - k < n := ⋯
xle : xn a1 (2 * n - k) ≤ xn a1 n
t : xn a1 (2 * n - (2 * n - k)) + xn a1 (2 * n - k) ≡ 0 [MOD xn a1 n]
⊢ xn a1 k + xn a1 (2 * n - k) ≡ xn a1 n [MOD xn a1 n]
have t₁ : xn a1 k + xn a1 (2 * n - k) ≡ 0 [MOD xn a1 n]
Tactic: rw [tsub_tsub_cancel_of_le k2n] at t
Cleared hypotheses: t
-- Lean 4 proof step from Mathlib.NumberTheory.PellMatiyasevic (goal)
Goal:
a : ℕ
a1 : 1 < a
i n : ℕ
npos : 0 < n
j : ℕ
ij : i < j + 1
j2n : j + 1 ≤ 2 * n
jnn : j + 1 ≠ n
ntriv : ¬(a = 2 ∧ n = 1 ∧ i = 0 ∧ j + 1 = 2)
lem2 : ∀ k > n, k ≤ 2 * n → (↑(xn a1 k % xn a1 n) : ℤ) = HSub.hSub (α := ℤ) (↑(xn a1 n) : ℤ) (↑(xn a1 (2 * n - k)) : ℤ)
o : j = n ∨ n < j
jn : j = n
⊢ xn a1 i % xn a1 n < xn a1 (j + 1) % xn a1 n
have refl : ∀ (ij : i < n + 1) (j2n : n + 1 ≤ 2 * n) (jnn : n + 1 ≠ n) (ntriv : ¬(a = 2 ∧ n = 1 ∧ i = 0 ∧ n + 1 = 2))
  (o : n = n ∨ n < n), xn a1 i % xn a1 n < xn a1 (n + 1) % xn a1 n
Tactic: cases jn
Cleared hypotheses: j, ij, j2n, jnn, ntriv, o, jn
-- Lean 4 proof step from Mathlib.NumberTheory.PellMatiyasevic (goal)
Goal:
a : ℕ
a1 : 1 < a
i n : ℕ
npos : 0 < n
lem2 : ∀ k > n, k ≤ 2 * n → (↑(xn a1 k % xn a1 n) : ℤ) = HSub.hSub (α := ℤ) (↑(xn a1 n) : ℤ) (↑(xn a1 (2 * n - k)) : ℤ)
ij : i < n + 1
j2n : n + 1 ≤ 2 * n
jnn : n + 1 ≠ n
ntriv : ¬(a = 2 ∧ n = 1 ∧ i = 0 ∧ n + 1 = 2)
o : n = n ∨ n < n
⊢ xn a1 i % xn a1 n < xn a1 (n + 1) % xn a1 n
have refl : LT.lt (α := ℤ) (↑(xn a1 i % xn a1 n) : ℤ) (↑(xn a1 (n + 1) % xn a1 n) : ℤ)
Tactic: apply Int.lt_of_ofNat_lt_ofNat
-- Lean 4 proof step from Mathlib.NumberTheory.PellMatiyasevic (goal)
Goal:
a : ℕ
a1 : 1 < a
i n : ℕ
npos : 0 < n
lem2 : ∀ k > n, k ≤ 2 * n → (↑(xn a1 k % xn a1 n) : ℤ) = HSub.hSub (α := ℤ) (↑(xn a1 n) : ℤ) (↑(xn a1 (2 * n - k)) : ℤ)
ij : i < n + 1
j2n : n + 1 ≤ 2 * n
jnn : n + 1 ≠ n
ntriv : ¬(a = 2 ∧ n = 1 ∧ i = 0 ∧ n + 1 = 2)
o : n = n ∨ n < n
⊢ 2 * n - (n + 1) = n - 1
have rw : n + n - (n + 1) = n - 1
Tactic: rw [two_mul, tsub_add_eq_tsub_tsub, add_tsub_cancel_right]
-- Lean 4 proof step from Mathlib.NumberTheory.PellMatiyasevic (goal)
Goal:
a : ℕ
a1 : 1 < a
i n : ℕ
npos : 0 < n
lem2 : ∀ k > n, k ≤ 2 * n → (↑(xn a1 k % xn a1 n) : ℤ) = HSub.hSub (α := ℤ) (↑(xn a1 n) : ℤ) (↑(xn a1 (2 * n - k)) : ℤ)
ij : i < n + 1
j2n : n + 1 ≤ 2 * n
jnn : n + 1 ≠ n
ntriv : ¬(a = 2 ∧ n = 1 ∧ i = 0 ∧ n + 1 = 2)
o : n = n ∨ n < n
rw : n + n - (n + 1) = n - 1
⊢ 2 * n - (n + 1) = n - 1
have rw₁ : n + n - n - 1 = n - 1
Tactic: rw [two_mul, tsub_add_eq_tsub_tsub, add_tsub_cancel_right]
-- Lean 4 proof step from Mathlib.NumberTheory.PellMatiyasevic (goal)
Goal:
a : ℕ
a1 : 1 < a
i n : ℕ
npos : 0 < n
lem2 : ∀ k > n, k ≤ 2 * n → (↑(xn a1 k % xn a1 n) : ℤ) = HSub.hSub (α := ℤ) (↑(xn a1 n) : ℤ) (↑(xn a1 (2 * n - k)) : ℤ)
ij : i < n + 1
j2n : n + 1 ≤ 2 * n
jnn : n + 1 ≠ n
ntriv : ¬(a = 2 ∧ n = 1 ∧ i = 0 ∧ n + 1 = 2)
o : n = n ∨ n < n
rw : n + n - (n + 1) = n - 1
rw₁ : n + n - n - 1 = n - 1
⊢ 2 * n - (n + 1) = n - 1
have rw₂ : n - 1 = n - 1
Tactic: rw [two_mul, tsub_add_eq_tsub_tsub, add_tsub_cancel_right]
-- Lean 4 proof step from Mathlib.NumberTheory.PellMatiyasevic (goal)
Goal:
a : ℕ
a1 : 1 < a
i n : ℕ
npos : 0 < n
lem2 : ∀ k > n, k ≤ 2 * n → (↑(xn a1 k % xn a1 n) : ℤ) = HSub.hSub (α := ℤ) (↑(xn a1 n) : ℤ) (↑(xn a1 (2 * n - k)) : ℤ)
ij : i < n + 1
j2n : n + 1 ≤ 2 * n
jnn : n + 1 ≠ n
ntriv : ¬(a = 2 ∧ n = 1 ∧ i = 0 ∧ n + 1 = 2)
o : n = n ∨ n < n
⊢ LT.lt (α := ℤ) (↑(xn a1 i % xn a1 n) : ℤ) (↑(xn a1 (n + 1) % xn a1 n) : ℤ)
have refl : (↑(xn a1 i % xn a1 n) : ℤ) < HSub.hSub (α := ℤ) (↑(xn a1 n) : ℤ) (↑(xn a1 (n - 1)) : ℤ)
Tactic: rw [lem2 (n + 1) (Nat.lt_succ_self _) j2n,
            show 2 * n - (n + 1) = n - 1 by
              rw [two_mul, tsub_add_eq_tsub_tsub, add_tsub_cancel_right]]
-- Lean 4 proof step from Mathlib.NumberTheory.PellMatiyasevic (goal)
Goal:
a : ℕ
a1 : 1 < a
i n : ℕ
npos : 0 < n
lem2 : ∀ k > n, k ≤ 2 * n → (↑(xn a1 k % xn a1 n) : ℤ) = HSub.hSub (α := ℤ) (↑(xn a1 n) : ℤ) (↑(xn a1 (2 * n - k)) : ℤ)
ij : i < n + 1
j2n : n + 1 ≤ 2 * n
jnn : n + 1 ≠ n
ntriv : ¬(a = 2 ∧ n = 1 ∧ i = 0 ∧ n + 1 = 2)
o : n = n ∨ n < n
⊢ (↑(xn a1 i % xn a1 n) : ℤ) < HSub.hSub (α := ℤ) (↑(xn a1 n) : ℤ) (↑(xn a1 (n - 1)) : ℤ)
have refl : xn a1 (n - 1) + xn a1 i % xn a1 n < xn a1 n
Tactic: refine lt_sub_left_of_add_lt (Int.ofNat_lt_ofNat_of_lt ?_)
-- Lean 4 proof step from Mathlib.NumberTheory.PellMatiyasevic (goal)
Goal:
a : ℕ
a1 : 1 < a
i n : ℕ
npos : 0 < n
lem2 : ∀ k > n, k ≤ 2 * n → (↑(xn a1 k % xn a1 n) : ℤ) = HSub.hSub (α := ℤ) (↑(xn a1 n) : ℤ) (↑(xn a1 (2 * n - k)) : ℤ)
ij : i < n + 1
j2n : n + 1 ≤ 2 * n
jnn : n + 1 ≠ n
ntriv : ¬(a = 2 ∧ n = 1 ∧ i = 0 ∧ n + 1 = 2)
o : n = n ∨ n < n
⊢ xn a1 (n - 1) + xn a1 i % xn a1 n < xn a1 n
have refl : ∀ (lin : i < n), xn a1 (n - 1) + xn a1 i % xn a1 n < xn a1 n
Tactic: rcases lt_or_eq_of_le <| Nat.le_of_succ_le_succ ij with lin | ein
-- Lean 4 proof step from Mathlib.NumberTheory.PellMatiyasevic (goal)
Goal:
a : ℕ
a1 : 1 < a
i n : ℕ
npos : 0 < n
lem2 : ∀ k > n, k ≤ 2 * n → (↑(xn a1 k % xn a1 n) : ℤ) = HSub.hSub (α := ℤ) (↑(xn a1 n) : ℤ) (↑(xn a1 (2 * n - k)) : ℤ)
ij : i < n + 1
j2n : n + 1 ≤ 2 * n
jnn : n + 1 ≠ n
ntriv : ¬(a = 2 ∧ n = 1 ∧ i = 0 ∧ n + 1 = 2)
o : n = n ∨ n < n
refl : ∀ (lin : i < n), xn a1 (n - 1) + xn a1 i % xn a1 n < xn a1 n
⊢ xn a1 (n - 1) + xn a1 i % xn a1 n < xn a1 n
have refl₁ : ∀ (ein : i = n), xn a1 (n - 1) + xn a1 i % xn a1 n < xn a1 n
Tactic: rcases lt_or_eq_of_le <| Nat.le_of_succ_le_succ ij with lin | ein
-- Lean 4 proof step from Mathlib.NumberTheory.PellMatiyasevic (goal)
Goal:
a : ℕ
a1 : 1 < a
i n : ℕ
npos : 0 < n
lem2 : ∀ k > n, k ≤ 2 * n → (↑(xn a1 k % xn a1 n) : ℤ) = HSub.hSub (α := ℤ) (↑(xn a1 n) : ℤ) (↑(xn a1 (2 * n - k)) : ℤ)
ij : i < n + 1
j2n : n + 1 ≤ 2 * n
jnn : n + 1 ≠ n
ntriv : ¬(a = 2 ∧ n = 1 ∧ i = 0 ∧ n + 1 = 2)
o : n = n ∨ n < n
lin : i < n
⊢ xn a1 (n - 1) + xn a1 i % xn a1 n < xn a1 n
have refl : xn a1 (n - 1) + xn a1 i < xn a1 n
Tactic: rw [Nat.mod_eq_of_lt (strictMono_x _ lin)]
-- Lean 4 proof step from Mathlib.NumberTheory.PellMatiyasevic (goal)
Goal:
a : ℕ
a1 : 1 < a
i n : ℕ
npos : 0 < n
lem2 : ∀ k > n, k ≤ 2 * n → (↑(xn a1 k % xn a1 n) : ℤ) = HSub.hSub (α := ℤ) (↑(xn a1 n) : ℤ) (↑(xn a1 (2 * n - k)) : ℤ)
ij : i < n + 1
j2n : n + 1 ≤ 2 * n
jnn : n + 1 ≠ n
ntriv : ¬(a = 2 ∧ n = 1 ∧ i = 0 ∧ n + 1 = 2)
o : n = n ∨ n < n
lin : i < n
⊢ xn a1 n = xn a1 (n - 1 + 1)
have rw : xn a1 n = xn a1 n
Tactic: rw [tsub_add_cancel_of_le (succ_le_of_lt npos)]
-- Lean 4 proof step from Mathlib.NumberTheory.PellMatiyasevic (goal)
Goal:
a : ℕ
a1 : 1 < a
i n : ℕ
npos : 0 < n
lem2 : ∀ k > n, k ≤ 2 * n → (↑(xn a1 k % xn a1 n) : ℤ) = HSub.hSub (α := ℤ) (↑(xn a1 n) : ℤ) (↑(xn a1 (2 * n - k)) : ℤ)
ij : i < n + 1
j2n : n + 1 ≤ 2 * n
jnn : n + 1 ≠ n
ntriv : ¬(a = 2 ∧ n = 1 ∧ i = 0 ∧ n + 1 = 2)
o : n = n ∨ n < n
lin : i < n
⊢ xn a1 (n - 1) + xn a1 (n - 1) ≤ xn a1 n
have rw : xn a1 (n - 1) * 2 ≤ xn a1 (n - 1) * a + d a1 * yn a1 (n - 1)
Tactic: rw [← two_mul, mul_comm,
                show xn a1 n = xn a1 (n - 1 + 1) by rw [tsub_add_cancel_of_le (succ_le_of_lt npos)],
                xn_succ]
-- Lean 4 proof step from Mathlib.NumberTheory.PellMatiyasevic (hypothesis)
Goal:
a : ℕ
a1 : 1 < a
i n : ℕ
npos : 0 < n
lem2 : ∀ k > n, k ≤ 2 * n → (↑(xn a1 k % xn a1 n) : ℤ) = HSub.hSub (α := ℤ) (↑(xn a1 n) : ℤ) (↑(xn a1 (2 * n - k)) : ℤ)
ij : i < n + 1
j2n : n + 1 ≤ 2 * n
jnn : n + 1 ≠ n
ntriv : ¬(a = 2 ∧ n = 1 ∧ i = 0 ∧ n + 1 = 2)
o : n = n ∨ n < n
lin : i < n
⊢ xn a1 (n - 1) + xn a1 i < xn a1 n
have ll : xn a1 (n - 1) + xn a1 (n - 1) ≤ xn a1 n
Tactic: have ll : xn a1 (n - 1) + xn a1 (n - 1) ≤ xn a1 n := by
              rw [← two_mul, mul_comm,
                show xn a1 n = xn a1 (n - 1 + 1) by rw [tsub_add_cancel_of_le (succ_le_of_lt npos)],
                xn_succ]
              exact le_trans (Nat.mul_le_mul_left _ a1) (Nat.le_add_right _ _)
-- Lean 4 proof step from Mathlib.NumberTheory.PellMatiyasevic (hypothesis)
Goal:
a : ℕ
a1 : 1 < a
i n : ℕ
npos : 0 < n
lem2 : ∀ k > n, k ≤ 2 * n → (↑(xn a1 k % xn a1 n) : ℤ) = HSub.hSub (α := ℤ) (↑(xn a1 n) : ℤ) (↑(xn a1 (2 * n - k)) : ℤ)
ij : i < n + 1
j2n : n + 1 ≤ 2 * n
jnn : n + 1 ≠ n
ntriv : ¬(a = 2 ∧ n = 1 ∧ i = 0 ∧ n + 1 = 2)
o : n = n ∨ n < n
lin : i < n
ll : xn a1 (n - 1) + xn a1 (n - 1) ≤ xn a1 n
⊢ xn a1 (n - 1) + xn a1 i < xn a1 n
have npm : (n - 1).succ = n
Tactic: have npm : (n - 1).succ = n := Nat.succ_pred_eq_of_pos npos
-- Lean 4 proof step from Mathlib.NumberTheory.PellMatiyasevic (goal)
Goal:
a : ℕ
a1 : 1 < a
i n : ℕ
npos : 0 < n
lem2 : ∀ k > n, k ≤ 2 * n → (↑(xn a1 k % xn a1 n) : ℤ) = HSub.hSub (α := ℤ) (↑(xn a1 n) : ℤ) (↑(xn a1 (2 * n - k)) : ℤ)
ij : i < n + 1
j2n : n + 1 ≤ 2 * n
jnn : n + 1 ≠ n
ntriv : ¬(a = 2 ∧ n = 1 ∧ i = 0 ∧ n + 1 = 2)
o : n = n ∨ n < n
lin : i < n
ll : xn a1 (n - 1) + xn a1 (n - 1) ≤ xn a1 n
npm : (n - 1).succ = n
⊢ i ≤ n - 1
have a₁ : i.succ ≤ (n - 1).succ
Tactic: apply Nat.le_of_succ_le_succ
-- Lean 4 proof step from Mathlib.NumberTheory.PellMatiyasevic (goal)
Goal:
a : ℕ
a1 : 1 < a
i n : ℕ
npos : 0 < n
lem2 : ∀ k > n, k ≤ 2 * n → (↑(xn a1 k % xn a1 n) : ℤ) = HSub.hSub (α := ℤ) (↑(xn a1 n) : ℤ) (↑(xn a1 (2 * n - k)) : ℤ)
ij : i < n + 1
j2n : n + 1 ≤ 2 * n
jnn : n + 1 ≠ n
ntriv : ¬(a = 2 ∧ n = 1 ∧ i = 0 ∧ n + 1 = 2)
o : n = n ∨ n < n
lin : i < n
ll : xn a1 (n - 1) + xn a1 (n - 1) ≤ xn a1 n
npm : (n - 1).succ = n
⊢ i.succ ≤ (n - 1).succ
have a₁ : i.succ ≤ n
Tactic: rw [npm]
-- Lean 4 proof step from Mathlib.NumberTheory.PellMatiyasevic (hypothesis)
Goal:
a : ℕ
a1 : 1 < a
i n : ℕ
npos : 0 < n
lem2 : ∀ k > n, k ≤ 2 * n → (↑(xn a1 k % xn a1 n) : ℤ) = HSub.hSub (α := ℤ) (↑(xn a1 n) : ℤ) (↑(xn a1 (2 * n - k)) : ℤ)
ij : i < n + 1
j2n : n + 1 ≤ 2 * n
jnn : n + 1 ≠ n
ntriv : ¬(a = 2 ∧ n = 1 ∧ i = 0 ∧ n + 1 = 2)
o : n = n ∨ n < n
lin : i < n
ll : xn a1 (n - 1) + xn a1 (n - 1) ≤ xn a1 n
npm : (n - 1).succ = n
⊢ xn a1 (n - 1) + xn a1 i < xn a1 n
have il : i ≤ n - 1
Tactic: have il : i ≤ n - 1 := by
              apply Nat.le_of_succ_le_succ
              rw [npm]
              exact lin
-- Lean 4 proof step from Mathlib.NumberTheory.PellMatiyasevic (goal)
Goal:
a : ℕ
a1 : 1 < a
i n : ℕ
npos : 0 < n
lem2 : ∀ k > n, k ≤ 2 * n → (↑(xn a1 k % xn a1 n) : ℤ) = HSub.hSub (α := ℤ) (↑(xn a1 n) : ℤ) (↑(xn a1 (2 * n - k)) : ℤ)
ij : i < n + 1
j2n : n + 1 ≤ 2 * n
jnn : n + 1 ≠ n
ntriv : ¬(a = 2 ∧ n = 1 ∧ i = 0 ∧ n + 1 = 2)
o : n = n ∨ n < n
lin : i < n
ll : xn a1 (n - 1) + xn a1 (n - 1) ≤ xn a1 n
npm : (n - 1).succ = n
il : i ≤ n - 1
⊢ xn a1 (n - 1) + xn a1 i < xn a1 n
have refl : ∀ (ill : i < n - 1), xn a1 (n - 1) + xn a1 i < xn a1 n
Tactic: rcases lt_or_eq_of_le il with ill | ile
-- Lean 4 proof step from Mathlib.NumberTheory.PellMatiyasevic (goal)
Goal:
a : ℕ
a1 : 1 < a
i n : ℕ
npos : 0 < n
lem2 : ∀ k > n, k ≤ 2 * n → (↑(xn a1 k % xn a1 n) : ℤ) = HSub.hSub (α := ℤ) (↑(xn a1 n) : ℤ) (↑(xn a1 (2 * n - k)) : ℤ)
ij : i < n + 1
j2n : n + 1 ≤ 2 * n
jnn : n + 1 ≠ n
ntriv : ¬(a = 2 ∧ n = 1 ∧ i = 0 ∧ n + 1 = 2)
o : n = n ∨ n < n
lin : i < n
ll : xn a1 (n - 1) + xn a1 (n - 1) ≤ xn a1 n
npm : (n - 1).succ = n
il : i ≤ n - 1
refl : ∀ (ill : i < n - 1), xn a1 (n - 1) + xn a1 i < xn a1 n
⊢ xn a1 (n - 1) + xn a1 i < xn a1 n
have refl₁ : ∀ (ile : i = n - 1), xn a1 (n - 1) + xn a1 i < xn a1 n
Tactic: rcases lt_or_eq_of_le il with ill | ile
-- Lean 4 proof step from Mathlib.NumberTheory.PellMatiyasevic (goal)
Goal:
a : ℕ
a1 : 1 < a
i n : ℕ
npos : 0 < n
lem2 : ∀ k > n, k ≤ 2 * n → (↑(xn a1 k % xn a1 n) : ℤ) = HSub.hSub (α := ℤ) (↑(xn a1 n) : ℤ) (↑(xn a1 (2 * n - k)) : ℤ)
ij : i < n + 1
j2n : n + 1 ≤ 2 * n
jnn : n + 1 ≠ n
ntriv : ¬(a = 2 ∧ n = 1 ∧ i = 0 ∧ n + 1 = 2)
o : n = n ∨ n < n
lin : i < n
ll : xn a1 (n - 1) + xn a1 (n - 1) ≤ xn a1 n
npm : (n - 1).succ = n
il : i ≤ n - 1
ile : i = n - 1
⊢ xn a1 (n - 1) + xn a1 i < xn a1 n
have refl : xn a1 (n - 1) + xn a1 (n - 1) < xn a1 n
Tactic: rw [ile]
-- Lean 4 proof step from Mathlib.NumberTheory.PellMatiyasevic (goal)
Goal:
a : ℕ
a1 : 1 < a
i n : ℕ
npos : 0 < n
lem2 : ∀ k > n, k ≤ 2 * n → (↑(xn a1 k % xn a1 n) : ℤ) = HSub.hSub (α := ℤ) (↑(xn a1 n) : ℤ) (↑(xn a1 (2 * n - k)) : ℤ)
ij : i < n + 1
j2n : n + 1 ≤ 2 * n
jnn : n + 1 ≠ n
ntriv : ¬(a = 2 ∧ n = 1 ∧ i = 0 ∧ n + 1 = 2)
o : n = n ∨ n < n
lin : i < n
ll : xn a1 (n - 1) + xn a1 (n - 1) ≤ xn a1 n
npm : (n - 1).succ = n
il : i ≤ n - 1
ile : i = n - 1
⊢ xn a1 (n - 1) + xn a1 (n - 1) < xn a1 n
have refl : xn a1 (n - 1) + xn a1 (n - 1) ≠ xn a1 n
Tactic: apply lt_of_le_of_ne ll
-- Lean 4 proof step from Mathlib.NumberTheory.PellMatiyasevic (goal)
Goal:
a : ℕ
a1 : 1 < a
i n : ℕ
npos : 0 < n
lem2 : ∀ k > n, k ≤ 2 * n → (↑(xn a1 k % xn a1 n) : ℤ) = HSub.hSub (α := ℤ) (↑(xn a1 n) : ℤ) (↑(xn a1 (2 * n - k)) : ℤ)
ij : i < n + 1
j2n : n + 1 ≤ 2 * n
jnn : n + 1 ≠ n
ntriv : ¬(a = 2 ∧ n = 1 ∧ i = 0 ∧ n + 1 = 2)
o : n = n ∨ n < n
lin : i < n
ll : xn a1 (n - 1) + xn a1 (n - 1) ≤ xn a1 n
npm : (n - 1).succ = n
il : i ≤ n - 1
ile : i = n - 1
⊢ xn a1 (n - 1) + xn a1 (n - 1) ≠ xn a1 n
have refl : 2 * xn a1 (n - 1) ≠ xn a1 n
Tactic: rw [← two_mul]
-- Lean 4 proof step from Mathlib.NumberTheory.PellMatiyasevic (hypothesis)
Goal:
a : ℕ
a1 : 1 < a
i n : ℕ
npos : 0 < n
lem2 : ∀ k > n, k ≤ 2 * n → (↑(xn a1 k % xn a1 n) : ℤ) = HSub.hSub (α := ℤ) (↑(xn a1 n) : ℤ) (↑(xn a1 (2 * n - k)) : ℤ)
ij : i < n + 1
j2n : n + 1 ≤ 2 * n
jnn : n + 1 ≠ n
ntriv : ¬(a = 2 ∧ n = 1 ∧ i = 0 ∧ n + 1 = 2)
o : n = n ∨ n < n
lin : i < n
ll : xn a1 (n - 1) + xn a1 (n - 1) ≤ xn a1 n
npm : (n - 1).succ = n
il : i ≤ n - 1
ile : i = n - 1
e : 2 * xn a1 (n - 1) = xn a1 n
⊢ a = 2 ∧ n = 1 ∧ i = 0 ∧ n + 1 = 2
have a2 : a = 2
Tactic: let ⟨a2, s1⟩ :=
                    @eq_of_xn_modEq_lem2 _ a1 (n - 1)
                      (by rwa [tsub_add_cancel_of_le (succ_le_of_lt npos)])
-- Lean 4 proof step from Mathlib.NumberTheory.PellMatiyasevic (hypothesis)
Goal:
a : ℕ
a1 : 1 < a
i n : ℕ
npos : 0 < n
lem2 : ∀ k > n, k ≤ 2 * n → (↑(xn a1 k % xn a1 n) : ℤ) = HSub.hSub (α := ℤ) (↑(xn a1 n) : ℤ) (↑(xn a1 (2 * n - k)) : ℤ)
ij : i < n + 1
j2n : n + 1 ≤ 2 * n
jnn : n + 1 ≠ n
ntriv : ¬(a = 2 ∧ n = 1 ∧ i = 0 ∧ n + 1 = 2)
o : n = n ∨ n < n
lin : i < n
ll : xn a1 (n - 1) + xn a1 (n - 1) ≤ xn a1 n
npm : (n - 1).succ = n
il : i ≤ n - 1
ile : i = n - 1
e : 2 * xn a1 (n - 1) = xn a1 n
a2 : a = 2
⊢ a = 2 ∧ n = 1 ∧ i = 0 ∧ n + 1 = 2
have s1 : n - 1 = 0
Tactic: let ⟨a2, s1⟩ :=
                    @eq_of_xn_modEq_lem2 _ a1 (n - 1)
                      (by rwa [tsub_add_cancel_of_le (succ_le_of_lt npos)])
-- Lean 4 proof step from Mathlib.NumberTheory.PellMatiyasevic (hypothesis)
Goal:
a : ℕ
a1 : 1 < a
i n : ℕ
npos : 0 < n
lem2 : ∀ k > n, k ≤ 2 * n → (↑(xn a1 k % xn a1 n) : ℤ) = HSub.hSub (α := ℤ) (↑(xn a1 n) : ℤ) (↑(xn a1 (2 * n - k)) : ℤ)
ij : i < n + 1
j2n : n + 1 ≤ 2 * n
jnn : n + 1 ≠ n
ntriv : ¬(a = 2 ∧ n = 1 ∧ i = 0 ∧ n + 1 = 2)
o : n = n ∨ n < n
lin : i < n
ll : xn a1 (n - 1) + xn a1 (n - 1) ≤ xn a1 n
npm : (n - 1).succ = n
il : i ≤ n - 1
ile : i = n - 1
e : 2 * xn a1 (n - 1) = xn a1 n
a2 : a = 2
s1 : n - 1 = 0
⊢ a = 2 ∧ n = 1 ∧ i = 0 ∧ n + 1 = 2
have n1 : n = 1
Tactic: have n1 : n = 1 := le_antisymm (tsub_eq_zero_iff_le.mp s1) npos
-- Lean 4 proof step from Mathlib.NumberTheory.PellMatiyasevic (goal)
Goal:
a : ℕ
a1 : 1 < a
i n : ℕ
npos : 0 < n
lem2 : ∀ k > n, k ≤ 2 * n → (↑(xn a1 k % xn a1 n) : ℤ) = HSub.hSub (α := ℤ) (↑(xn a1 n) : ℤ) (↑(xn a1 (2 * n - k)) : ℤ)
ij : i < n + 1
j2n : n + 1 ≤ 2 * n
jnn : n + 1 ≠ n
ntriv : ¬(a = 2 ∧ n = 1 ∧ i = 0 ∧ n + 1 = 2)
o : n = n ∨ n < n
lin : i < n
ll : xn a1 (n - 1) + xn a1 (n - 1) ≤ xn a1 n
npm : (n - 1).succ = n
il : i ≤ n - 1
ile : i = n - 1
e : 2 * xn a1 (n - 1) = xn a1 n
a2 : a = 2
s1 : n - 1 = 0
n1 : n = 1
⊢ a = 2 ∧ n = 1 ∧ i = 0 ∧ n + 1 = 2
have rw : Eq (α := ℕ) 2 2 ∧ Eq (α := ℕ) 1 1 ∧ HSub.hSub (α := ℕ) 1 1 = 0 ∧ HAdd.hAdd (α := ℕ) 1 1 = 2
Tactic: rw [ile, a2, n1]
-- Lean 4 proof step from Mathlib.NumberTheory.PellMatiyasevic (goal)
Goal:
a : ℕ
a1 : 1 < a
i n : ℕ
npos : 0 < n
lem2 : ∀ k > n, k ≤ 2 * n → (↑(xn a1 k % xn a1 n) : ℤ) = HSub.hSub (α := ℤ) (↑(xn a1 n) : ℤ) (↑(xn a1 (2 * n - k)) : ℤ)
ij : i < n + 1
j2n : n + 1 ≤ 2 * n
jnn : n + 1 ≠ n
ntriv : ¬(a = 2 ∧ n = 1 ∧ i = 0 ∧ n + 1 = 2)
o : n = n ∨ n < n
ein : i = n
⊢ xn a1 (n - 1) + xn a1 i % xn a1 n < xn a1 n
have refl : xn a1 (n - 1) < xn a1 n
Tactic: rw [ein, Nat.mod_self, add_zero]
-- Lean 4 proof step from Mathlib.NumberTheory.PellMatiyasevic (hypothesis)
Goal:
a : ℕ
a1 : 1 < a
i n : ℕ
npos : 0 < n
j : ℕ
ij : i < j + 1
j2n : j + 1 ≤ 2 * n
jnn : j + 1 ≠ n
ntriv : ¬(a = 2 ∧ n = 1 ∧ i = 0 ∧ j + 1 = 2)
lem2 : ∀ k > n, k ≤ 2 * n → (↑(xn a1 k % xn a1 n) : ℤ) = HSub.hSub (α := ℤ) (↑(xn a1 n) : ℤ) (↑(xn a1 (2 * n - k)) : ℤ)
o : j = n ∨ n < j
jn_1 : j > n
jn : j ≠ n
s : xn a1 j % xn a1 n < xn a1 (j + 1) % xn a1 n
h : i < j
h_1 : a = 2 ∧ n = 1 ∧ i = 0 ∧ j = 2
left : a = 2
n1 : n = 1
left_1 : i = 0
j2 : j = 2
⊢ False
have j2n₁ : HAdd.hAdd (α := ℕ) 2 1 ≤ HMul.hMul (α := ℕ) 2 1
Tactic: rw [n1, j2] at j2n
Cleared hypotheses: j2n, jnn, ntriv, lem2, o, jn_1, jn, s, h, h_1, left, n1, left_1, j2
-- Lean 4 proof step from Mathlib.NumberTheory.PellMatiyasevic (goal)
Goal:
a : ℕ
a1 : 1 < a
i n : ℕ
npos : 0 < n
j : ℕ
ij : i < j + 1
j2n : j + 1 ≤ 2 * n
jnn : j + 1 ≠ n
ntriv : ¬(a = 2 ∧ n = 1 ∧ i = 0 ∧ j + 1 = 2)
lem2 : ∀ k > n, k ≤ 2 * n → (↑(xn a1 k % xn a1 n) : ℤ) = HSub.hSub (α := ℤ) (↑(xn a1 n) : ℤ) (↑(xn a1 (2 * n - k)) : ℤ)
o : j = n ∨ n < j
jn : j > n
lem1 : j ≠ n → xn a1 j % xn a1 n < xn a1 (j + 1) % xn a1 n → xn a1 i % xn a1 n < xn a1 (j + 1) % xn a1 n
⊢ LT.lt (α := ℤ) (↑(xn a1 j % xn a1 n) : ℤ) (↑(xn a1 (j + 1) % xn a1 n) : ℤ)
have rw : HSub.hSub (α := ℤ) (↑(xn a1 n) : ℤ) (↑(xn a1 (2 * n - j)) : ℤ) <
  HSub.hSub (α := ℤ) (↑(xn a1 n) : ℤ) (↑(xn a1 (2 * n - (j + 1))) : ℤ)
Tactic: rw [lem2 j jn (le_of_lt j2n), lem2 (j + 1) (Nat.le_succ_of_le jn) j2n]
-- Lean 4 proof step from Mathlib.NumberTheory.PellMatiyasevic (goal)
Goal:
a : ℕ
a1 : 1 < a
i n : ℕ
npos : 0 < n
j : ℕ
ij : i < j + 1
j2n : j + 1 ≤ 2 * n
jnn : j + 1 ≠ n
ntriv : ¬(a = 2 ∧ n = 1 ∧ i = 0 ∧ j + 1 = 2)
lem2 : ∀ k > n, k ≤ 2 * n → (↑(xn a1 k % xn a1 n) : ℤ) = HSub.hSub (α := ℤ) (↑(xn a1 n) : ℤ) (↑(xn a1 (2 * n - k)) : ℤ)
o : j = n ∨ n < j
jn : j > n
lem1 : j ≠ n → xn a1 j % xn a1 n < xn a1 (j + 1) % xn a1 n → xn a1 i % xn a1 n < xn a1 (j + 1) % xn a1 n
⊢ HSub.hSub (α := ℤ) (↑(xn a1 n) : ℤ) (↑(xn a1 (2 * n - j)) : ℤ) <
    HSub.hSub (α := ℤ) (↑(xn a1 n) : ℤ) (↑(xn a1 (2 * n - (j + 1))) : ℤ)
have refine : 2 * n - (j + 1) < 2 * n - j
Tactic: refine sub_lt_sub_left (Int.ofNat_lt_ofNat_of_lt <| strictMono_x _ ?_) _