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Proof step 76 from proof prepost-update (previous step)

ar[m] = elemindex(a, m1), kar[m1] = key-index(key1, m), 2exp-size(# kar), m1 < # kar, m1 < m0, 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) = ∅, key < key1, ( (2 * m1 + 1 < m0 → ¬ kar[2 * m1 + 1].key < key) ∧ (2 * m1 + 2 < m0 → ¬ kar[2 * m1 + 2].key < key) → heap-order(kar[m1, key-index(key, m)], m0)) ∧ ( 2 * m1 + 1 < m0 ∧ kar[2 * m1 + 1].key < key ∨ 2 * m1 + 2 < m0 ∧ kar[2 * m1 + 2].key < key → heap-order(kar[m1, (2 * m1 + 2 < m0 ∧ kar[2 * m1 + 2].key < kar[2 * m1 + 1].key ⊃ kar[2 * m1 + 2];kar[2 * m1 + 1])], m0))

Heuristic elimination applied rule insert elim lemma

And obtained the following new goals

Proof step 77:
kar[m1] = key-index(key1, m), ar[m] = elemindex(a, m1), kar[2 * m1 + 2] = key-index(key0, m2), 2exp-size(# kar), m1 < m0, m1 < # 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) = ∅, key < key1, ((2 * m1 + 1 < m0 → ¬ kar[2 * m1 + 1].key < key) ∧ (2 * m1 + 2 < m0 → ¬ key0 < key) → heap-order(kar[m1, key-index(key, m)], m0)) ∧ ( 2 * m1 + 1 < m0 ∧ kar[2 * m1 + 1].key < key ∨ 2 * m1 + 2 < m0 ∧ key0 < key → heap-order(kar[m1, (2 * m1 + 2 < m0 ∧ key0 < kar[2 * m1 + 1].key ⊃ kar[2 * m1 + 2];kar[2 * m1 + 1])], 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[2 * m1 + 2], key0, m2]

Additional information

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