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Proof step 36 from proof prepost-update (previous step)

kar[m1] = key-index(key1, m), ar[m] = elemindex(a, m1), kar[m1, key-index(key, m)][(m1 - 1) / 2] = key-index(key0, m2), kar[m1, key-index(key, m)][m4 + 1] = key-index(key2, m3), 2exp-size(# kar), m4 + 1 < # kar, key < key1, m1 < # kar, m1 < m0, m4 + 1 < m0, key2 < kar[m1, key-index(key, m)][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, m1 = 0 ∨ ¬ key < key0 ⊦ # keyset(kar, m0) = 0, keyset(kar, m0) = ∅

Heuristic elimination applied rule insert elim lemma

And obtained the following new goals

Proof step 37:
kar[m1, key-index(key, m)][m4 + 1] = key-index(key2, m3), kar[m1, key-index(key, m)][(m1 - 1) / 2] = key-index(key0, m2), ar[m] = elemindex(a, m1), kar[m1] = key-index(key1, m), kar[m1, key-index(key, m)][m4 / 2] = key-index(key3, m5), 2exp-size(# kar), key2 < key3, m4 + 1 < m0, m1 < m0, m1 < # kar, key < key1, 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, m1 = 0 ∨ ¬ key < key0 ⊦ # 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[m1, key-index(key, m)][m4 / 2], key3, m5]

Additional information

Used simplifier rules: ⊦ key = ki.key ∧ m = ki.idx ↔ ki = key-index(key, m); (elim) Used terms: kar[m1, key-index(key, m)][m4 / 2], key3, m5