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Proof step 66 from proof prepost-remove (previous step)

kar[m1, key-index(key0, m0)][(m1 - 1) / 2] = key-index(key1, m3), ar1[m] = elemindex(a, m1), kar[m1] = key-index(key, m), kar[m2] = key-index(key0, m0), 2exp-size(# kar), m1 + 1 < # kar, m1 < # kar, m2 < # kar, m0 < # ar1, m1 < m2, heap-order(kar, m2 + 1), no-dups-idx(kar, m2 + 1), valid-idx(kar, ar1, m2 + 1), m1 ≠ 0 → (m1 - 1) / 2 < # kar ∧ 1 ≤ m1, key-index(key, m) = key-index(key0, m0) ∨ key-index(key, m) ∈ keyset(kar, m2) ⊦ #(keyset(kar, m2) ++ key-index(key0, m0)) = 0, m2 = m1, m1 ≠ 0 ∧ key0 < key1, ( (2 * m1 + 1 < m2 → ¬ kar[2 * m1 + 1].key < key0) ∧ (2 * m1 + 2 < m2 → ¬ kar[2 * m1 + 2].key < key0) → heap-order(kar[m1, key-index(key0, m0)], m2)) ∧ ( 2 * m1 + 1 < m2 ∧ kar[2 * m1 + 1].key < key0 ∨ 2 * m1 + 2 < m2 ∧ kar[2 * m1 + 2].key < key0 → heap-order(kar[m1, (2 * m1 + 2 < m2 ∧ kar[2 * m1 + 2].key < kar[2 * m1 + 1].key ⊃ kar[2 * m1 + 2];kar[2 * m1 + 1])], m2))

Heuristic elimination applied rule insert elim lemma

And obtained the following new goals

Proof step 67:
kar[m2] = key-index(key0, m0), kar[m1] = key-index(key, m), ar1[m] = elemindex(a, m1), kar[m1, key-index(key0, m0)][(m1 - 1) / 2] = key-index(key1, m3), kar[2 * m1 + 2] = key-index(key2, m4), 2exp-size(# kar), m1 < m2, m0 < # ar1, m2 < # kar, m1 < # kar, m1 + 1 < # kar, heap-order(kar, m2 + 1), no-dups-idx(kar, m2 + 1), valid-idx(kar, ar1, m2 + 1), m1 ≠ 0 → (m1 - 1) / 2 < # kar ∧ 1 ≤ m1, key-index(key, m) = key-index(key0, m0) ∨ key-index(key, m) ∈ keyset(kar, m2) ⊦ #(keyset(kar, m2) ++ key-index(key0, m0)) = 0, m2 = m1, ((2 * m1 + 1 < m2 → ¬ kar[2 * m1 + 1].key < key0) ∧ (2 * m1 + 2 < m2 → ¬ key2 < key0) → heap-order(kar[m1, key-index(key0, m0)], m2)) ∧ ( 2 * m1 + 1 < m2 ∧ kar[2 * m1 + 1].key < key0 ∨ 2 * m1 + 2 < m2 ∧ key2 < key0 → heap-order(kar[m1, (2 * m1 + 2 < m2 ∧ key2 < kar[2 * m1 + 1].key ⊃ kar[2 * m1 + 2];kar[2 * m1 + 1])], m2)), m1 ≠ 0 ∧ key0 < key1

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], key2, m4]

Additional information

Used simplifier rules: ⊦ key = ki.key ∧ m = ki.idx ↔ ki = key-index(key, m); (elim) Used terms: kar[2 * m1 + 2], key2, m4