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Proof step 15 from proof heap-order-overwrite-children-ok (previous step)

kar[m1 + 1] = key-index(key0, m2), kar[2 * m1 / 2 + 2] = key-index(key3, m5), kar[2 * m1 / 2 + 1] = key-index(key4, m6), kar[m1 / 2] = key-index(key2, m4), m1 / 2 < # kar, m1 / 2 < n, key0 < key1, m1 + 1 < n, m1 + 1 < # kar, n ≤ # kar, 2 * m1 / 2 + 2 < n → ¬ key3 < key1, 2 * m1 / 2 + 1 < n → ¬ key4 < key1, ∀ m. m < n ∧ m ≠ 0 → ¬ kar[m].key < kar[(m - 1) / 2].key ⊦ key1 < key2

User applied rule insert elim lemma

And obtained the following new goals

Proof step 16:
kar[m0] = key-index(key2, m4), kar[2 * m0 + 1] = key-index(key4, m6), kar[2 * m0 + 2] = key-index(key3, m5), kar[m1 + 1] = key-index(key0, m2), m1 = m0 * 2 + n0, m1 + 1 < # kar, m1 + 1 < n, key0 < key1, m0 < n, m0 < # kar, n0 < 2, n ≤ # kar, 2 * m0 + 1 < n → ¬ key4 < key1, 2 * m0 + 2 < n → ¬ key3 < key1, ∀ m. m < n ∧ m ≠ 0 → ¬ kar[m].key < kar[(m - 1) / 2].key ⊦ 2 = 0, key1 < key2

Rule argument

spec: nat-div name: div-elim seq: n ≠ 0 ⊦ m0 = m / n ∧ n0 = m % n ↔ m = m0 * n + n0 ∧ n0 < n subst: [m, n, m0, n0] → [m1, 2, m0, n0]

Additional information

Used simplifier rules: n ≠ 0 ⊦ m0 = m / n ∧ n0 = m % n ↔ m = m0 * n + n0 ∧ n0 < n; (elim) Used terms: m1, 2, m0, n0