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Proof step 8 from proof heap-order-overwrite0 (previous step)

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

Heuristic elimination applied rule insert elim lemma

And obtained the following new goals

Proof step 9:
kar[m0 + 1] = key-index(key0, m2), kar[0, key-index(key, m1)][m0 / 2] = key-index(key1, m3), key0 < key1, m0 + 1 < n, m0 + 1 < # kar, n ≤ # kar, ∀ m. m < n ∧ m ≠ 0 → ¬ kar[m].key < kar[(m - 1) / 2].key ⊦ kar[0].key < 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[0, key-index(key, m1)][m0 / 2], key1, m3]

Additional information

Used simplifier rules: ⊦ key = ki.key ∧ m = ki.idx ↔ ki = key-index(key, m); (elim) Used terms: kar[0, key-index(key, m1)][m0 / 2], key1, m3