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Proof step 18 from proof sorted-next-1-swap (previous step)

n % 2 = 1, 2 * n0 < n, n + 1 < # x[n, x[n + 1]], 2 * n0 + 1 < n, n + 1 < # x, x[n + 1] < x[n], x[n, x[n + 1]][2 + 2 * n0] < x[1 + 2 * n0], ∀ n0. 2 + 2 * n0 < n → ¬ x[2 + 2 * n0] < x[1 + 2 * n0] ⊦ x[n, x[n + 1]] = [], n = 2 * n0 + 1, n = 2 * n0, x = []

User applied rule insert rewrite lemma

And obtained the following new goals

Rule argument

spec: list-del name: put-at seq: m < # x ⊦ m ≠ n → x[m, a][n] = x[n] subst: [n, a, m, x] → [2 + 2 * n0, x[n + 1], n, x] flags: <all paths>

Additional information

Used simplifier rules: ⊦ ¬ m < n → ¬ m + m0 < n; (s)%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic%c: ⊦ m + n = n + m; from specification nat-basic% ⊦ m = m0 + n0 ∧ n < n0 → m0 + n < m; (s)% ⊦ n < # x → # x[n, a] = # x; (s)%lf-02: ⊦ m < n ∧ m = n0 → n0 < n; (forward) from specification nat-basic%lf: ⊦ m < n ∧ n < n0 → m +1 < n0; (forward) from specification nat-basic%lf-01: ⊦ m < n ∧ n = n0 → m < n0; (forward) from specification nat-basic%trans: ⊦ n < n0 ∧ n0 < n1 → n < n1; (forward) from specification nat-basic%n ≠ 0 ⊦ (m + n0 * n) % n = m % n; (s)%mult-a: ⊦ m0 * m * n = (m0 * m) * n; from specification nat-mult%mult-c: ⊦ m * n = n * m; from specification nat-mult Used terms: 2 + 2 * n0, x[n + 1], n, x