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Proof step 19 from proof heap-order-pull-up-child (previous step)

kar[m1 / 2] = key-index(key4, m6), kar[m1 + 1] = key-index(key3, m5), kar[2 * m1 / 2 + 1] = key-index(key1, m3), kar[2 * m1 / 2 + 2] = key-index(key0, m4), 2 * m1 / 2 + 2 < # kar, key4 < key0, 2 * m1 / 2 + 1 < # kar, m1 / 2 < # kar, m1 + 1 < # kar, m1 + 1 < n, key3 < key0, 2 * m1 / 2 + 1 < n, m1 / 2 < n, 2 * m1 / 2 + 2 < n, key0 < key1, key4 < key1, key3 < key1, n ≤ # kar, ∀ m. m < n ∧ m ≠ 0 → ¬ kar[m].key < kar[(m - 1) / 2].key ⊦ key3 < key4

User applied rule insert rewrite lemma

And obtained the following new goals

Rule argument

spec: nat-div name: div-two-mult seq: ⊦ m % 2 = 1 → 2 * m / 2 = m - 1 subst: [m] → [m1] flags: <all paths>

Additional information

Used simplifier rules: ⊦ m % 2 = 1 ↔ m % 2 ≠ 0; (ss)%n ≠ 0 ⊦ m % n = 0 → m / n * n = m; (s)%mult-a: ⊦ m0 * m * n = (m0 * m) * n; from specification nat-mult%mult-c: ⊦ m * n = n * m; from specification nat-mult% ⊦ m % 2 = 0 → (m / 2 < n ↔ m < n * 2); (s)%fle: ⊦ m < n ∧ n ≤ n0 → m < n0; (forward) from specification nat%f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic% ⊦ key-index(key, m) = key-index(key0, m0) ↔ key = key0 ∧ m = m0; (ws)% ⊦ 2 * m = 0 ↔ m = 0; (s)% ⊦ m < n → n ≠ 0; (s)%lel: ⊦ key < key0 ∧ key = key1 → key1 < key0; (forward) from specification oelem[binheap-key]% ⊦ ¬ key < key; (s) Used terms: m1