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Proof step 54 from proof prepost-update (previous step)

ar[m] = elemindex(a, m3 + 1), kar[m3 + 1] = key-index(key1, m), kar[m3 / 2] = key-index(key0, m2), 2exp-size(# kar), kar[m3 + 1, key-index(key0, m2)][m4 + 1].key < kar[m3 + 1, key-index(key0, m2)][m4 / 2].key, m4 + 1 < m0, m3 + 1 < m0, key < key1, key < key0, m3 + 1 < # kar, m4 + 1 < # kar, heap-order(kar, m0), no-dups-idx(kar, m0), valid-idx(kar, ar, m0), key-index(key1, m) ∈ keyset(kar, m0), m0 ≤ # kar ⊦ # keyset(kar, m0) = 0, keyset(kar, m0) = ∅

Heuristic elimination applied rule insert elim lemma

And obtained the following new goals

Proof step 55:
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), 2exp-size(# kar), m4 + 1 < # kar, m3 + 1 < # kar, key < key0, key < key1, m3 + 1 < m0, m4 + 1 < m0, key2 < kar[m3 + 1, key-index(key0, m2)][m4 / 2].key, heap-order(kar, m0), no-dups-idx(kar, m0), valid-idx(kar, ar, m0), key-index(key1, m) ∈ keyset(kar, m0), m0 ≤ # kar ⊦ # keyset(kar, m0) = 0, keyset(kar, m0) = ∅

Rule argument

spec: key-index name: elim seq: ⊦ key = ki.key ∧ m = ki.idx ↔ ki = key-index(key, m) subst: [ki, key, m] → [kar[m3 + 1, key-index(key0, m2)][m4 + 1], key2, m1]

Additional information

Used simplifier rules: ⊦ key = ki.key ∧ m = ki.idx ↔ ki = key-index(key, m); (elim) Used terms: kar[m3 + 1, key-index(key0, m2)][m4 + 1], key2, m1