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Proof step 5 from proof heap-order-overwrite-children-ok (previous step)

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

Heuristic elimination applied rule insert elim lemma

And obtained the following new goals

Proof step 6:
ki = key-index(key, m0), m1 + 1 < # kar, m < # kar, m1 + 1 < n, kar[m, ki][m1 + 1].key < kar[m, ki][m1 / 2].key, m < n, n ≤ # kar, 2 * m + 1 < n → ¬ kar[2 * m + 1].key < key, 2 * m + 2 < n → ¬ kar[2 * m + 2].key < key, ∀ m. m < n ∧ m ≠ 0 → ¬ kar[m].key < kar[(m - 1) / 2].key ⊦ key < kar[m].key

Rule argument

spec: key-index name: elim seq: ⊦ key = ki.key ∧ m = ki.idx ↔ ki = key-index(key, m) subst: [ki, key, m] → [ki, key, m0]

Additional information

Used simplifier rules: ⊦ key = ki.key ∧ m = ki.idx ↔ ki = key-index(key, m); (elim) Used terms: ki, key, m0