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Proof step 8 from proof heap-order-pull-down-parent (previous step)

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

Heuristic elimination applied rule insert elim lemma

And obtained the following new goals

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

Additional information

Used simplifier rules: ⊦ key = ki.key ∧ m = ki.idx ↔ ki = key-index(key, m); (elim) Used terms: kar[m1 + 1, kar[m1 / 2]][m2 / 2], key0, m3