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Proof step 61 from proof prepost-update (previous step)

kar[m3 + 1, key-index(key0, m2)][m4 + 1] = key-index(key2, m1), ar[m] = elemindex(a, m3 + 1), kar[m3 + 1] = key-index(key1, m), kar[m3 / 2] = key-index(key0, m2), kar[m3 + 1, key-index(key0, m2)][m4 / 2] = key-index(key3, m5), kar[m4 + 1] = key-index(key4, m6), 2exp-size(# kar), key2 < key3, m4 + 1 < m0, m3 + 1 < m0, key < key1, key < key0, m3 + 1 < # kar, m4 + 1 < # kar, no-dups-idx(kar, m0), valid-idx(kar, ar, m0), key-index(key1, m) ∈ keyset(kar, m0), m0 ≤ # kar, ∀ m. m < m0 ∧ m ≠ 0 → ¬ kar[m].key < kar[(m - 1) / 2].key ⊦ # keyset(kar, m0) = 0, keyset(kar, m0) = ∅, key4 < kar[m4 / 2].key

Heuristic elimination applied rule insert elim lemma

And obtained the following new goals

Proof step 62:
kar[m4 + 1] = key-index(key4, m6), kar[m3 + 1, key-index(key0, m2)][m4 / 2] = key-index(key3, m5), kar[m3 / 2] = key-index(key0, m2), kar[m3 + 1] = key-index(key1, m), ar[m] = elemindex(a, m3 + 1), kar[m3 + 1, key-index(key0, m2)][m4 + 1] = key-index(key2, m1), kar[m4 / 2] = key-index(key5, m7), 2exp-size(# kar), m4 + 1 < # kar, m3 + 1 < # kar, key < key0, key < key1, m3 + 1 < m0, m4 + 1 < m0, key2 < key3, no-dups-idx(kar, m0), valid-idx(kar, ar, m0), key-index(key1, m) ∈ keyset(kar, m0), m0 ≤ # kar, ∀ m. m < m0 ∧ m ≠ 0 → ¬ kar[m].key < kar[(m - 1) / 2].key ⊦ # keyset(kar, m0) = 0, keyset(kar, m0) = ∅, key4 < key5

Rule argument

spec: key-index name: elim seq: ⊦ key = ki.key ∧ m = ki.idx ↔ ki = key-index(key, m) subst: [ki, key, m] → [kar[m4 / 2], key5, m7]

Additional information

Used simplifier rules: ⊦ key = ki.key ∧ m = ki.idx ↔ ki = key-index(key, m); (elim) Used terms: kar[m4 / 2], key5, m7