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Proof step 161 from proof prepost-bubble-down (previous step)

# kar0 = # kar, kar[2 * n1 + 1] = key-index(key, m0), kar[2 * n1 + 2] = key-index(key0, m1), kar[n1] = key-index(key1, m2), kar0[n] = key-index(key2, m3), keyset(kar0, m) = keyset(kar, m), 2 * n + 1 < m, n < # kar, 2 * n + 1 < # kar, heap-order(kar0[n, kar0[2 * n + 1]], m), id-idx(ar1, ar), no-dups-idx(kar0, m), no-dups-idx(kar, m), valid-idx(kar0, ar, m), valid-idx(kar, ar1, m), # ar ≤ kar0[2 * n + 1].idx, m ≤ # kar, ¬ kar0[2 * n + 1].key < key2 ∧ (2 * n + 2 < m → ¬ kar0[2 * n + 2].key < key2) → heap-order(kar0, m), 2 * n + 2 < m → 2 * n + 2 < # kar, (2 * n1 + 1 < m → ¬ key < key1) ∧ (2 * n1 + 2 < m → ¬ key0 < key1) → heap-order(kar, m), 2 * n1 + 1 < m ∧ key < key1 ∨ 2 * n1 + 2 < m ∧ key0 < key1 → heap-order(kar[n1, (2 * n1 + 2 < m ∧ key0 < key ⊃ key-index(key0, m1);key-index(key, m0))], m), kar0[2 * n + 1].key < key2 ∨ 2 * n + 2 < m ∧ kar0[2 * n + 2].key < key2 ⊦ 2 * n + 2 < m ∧ kar0[2 * n + 2].key < kar0[2 * n + 1].key

Heuristic elimination applied rule insert elim lemma

And obtained the following new goals

Proof step 162:
# kar0 = # kar, kar0[n] = key-index(key2, m3), kar[n1] = key-index(key1, m2), kar[2 * n1 + 2] = key-index(key0, m1), kar[2 * n1 + 1] = key-index(key, m0), kar0[2 * n + 2] = key-index(key3, m4), keyset(kar0, m) = keyset(kar, m), 2 * n + 1 < # kar, n < # kar, 2 * n + 1 < m, heap-order(kar0[n, kar0[2 * n + 1]], m), id-idx(ar1, ar), no-dups-idx(kar, m), no-dups-idx(kar0, m), valid-idx(kar, ar1, m), valid-idx(kar0, ar, m), m ≤ # kar, # ar ≤ kar0[2 * n + 1].idx, 2 * n1 + 1 < m ∧ key < key1 ∨ 2 * n1 + 2 < m ∧ key0 < key1 → heap-order(kar[n1, (2 * n1 + 2 < m ∧ key0 < key ⊃ key-index(key0, m1);key-index(key, m0))], m), (2 * n1 + 1 < m → ¬ key < key1) ∧ (2 * n1 + 2 < m → ¬ key0 < key1) → heap-order(kar, m), 2 * n + 2 < m → 2 * n + 2 < # kar, ¬ kar0[2 * n + 1].key < key2 ∧ (2 * n + 2 < m → ¬ key3 < key2) → heap-order(kar0, m), kar0[2 * n + 1].key < key2 ∨ 2 * n + 2 < m ∧ key3 < key2 ⊦ 2 * n + 2 < m ∧ key3 < kar0[2 * n + 1].key

Rule argument

spec: key-index name: elim seq: ⊦ key = ki.key ∧ m = ki.idx ↔ ki = key-index(key, m) subst: [ki, key, m] → [kar0[2 * n + 2], key3, m4]

Additional information

Used simplifier rules: ⊦ key = ki.key ∧ m = ki.idx ↔ ki = key-index(key, m); (elim) Used terms: kar0[2 * n + 2], key3, m4