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Proof step 9 from proof heap-order-add-smaller (previous step)

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

Heuristic elimination applied rule insert elim lemma

And obtained the following new goals

Proof step 10:
kar[m1 + 1, key-index(key, m2)][m0 + 1] = key-index(key0, m3), key0 < kar[m0 / 2].key, m0 < # kar, m0 + 1 < # kar, m1 + 1 < # kar, m0 ≤ m1, ∀ m. ¬ m1 < m ∧ m ≠ 0 → ¬ kar[m].key < kar[(m - 1) / 2].key ⊦ key < 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, key-index(key, m2)][m0 + 1], key0, m3]

Additional information

Used simplifier rules: ⊦ key = ki.key ∧ m = ki.idx ↔ ki = key-index(key, m); (elim) Used terms: kar[m1 + 1, key-index(key, m2)][m0 + 1], key0, m3