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Proof step 13 from proof heap-order-pull-up-child (previous step)

kar[2 * m + 2] = key-index(key2, m4), kar[2 * m + 1] = key-index(key1, m3), kar[m, (2 * m + 2 < n ∧ key2 < key1 ⊃ key-index(key2, m4);key-index(key1, m3))][m1 + 1] = key-index(key, m0), kar[m, (2 * m + 2 < n ∧ key2 < key1 ⊃ key-index(key2, m4);key-index(key1, m3))][m1 / 2] = key-index(key0, m2), m < n, 2 * m + 1 < n, key < key0, m1 + 1 < n, m < # kar, 2 * m + 1 < # kar, m1 + 1 < # kar, n ≤ # kar, ∀ m. m < n ∧ m ≠ 0 → ¬ kar[m].key < kar[(m - 1) / 2].key ⊦ kar[m1 + 1].key < kar[m1 / 2].key

Heuristic elimination applied rule insert elim lemma

And obtained the following new goals

Proof step 14:
kar[m, (2 * m + 2 < n ∧ key2 < key1 ⊃ key-index(key2, m4);key-index(key1, m3))][m1 / 2] = key-index(key0, m2), kar[m, (2 * m + 2 < n ∧ key2 < key1 ⊃ key-index(key2, m4);key-index(key1, m3))][m1 + 1] = key-index(key, m0), kar[2 * m + 1] = key-index(key1, m3), kar[2 * m + 2] = key-index(key2, m4), kar[m1 + 1] = key-index(key3, m5), m1 + 1 < # kar, 2 * m + 1 < # kar, m < # kar, m1 + 1 < n, key < key0, 2 * m + 1 < n, m < n, n ≤ # kar, ∀ m. m < n ∧ m ≠ 0 → ¬ kar[m].key < kar[(m - 1) / 2].key ⊦ key3 < kar[m1 / 2].key

Rule argument

spec: key-index name: elim seq: ⊦ key = ki.key ∧ m = ki.idx ↔ ki = key-index(key, m) subst: [ki, key, m] → [kar[m1 + 1], key3, m5]

Additional information

Used simplifier rules: ⊦ key = ki.key ∧ m = ki.idx ↔ ki = key-index(key, m); (elim) Used terms: kar[m1 + 1], key3, m5