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Proof step 44 from proof prepost-update (previous step)

kar[m4 / 2] = key-index(key5, m7), 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), kar[m4 + 1] = key-index(key4, m6), 2exp-size(# kar), key2 < key3, m4 + 1 < m0, m1 < m0, m1 < # kar, key < key1, m4 + 1 < # kar, 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, m4 / 2 = m1, keyset(kar, m0) = ∅, key4 < key5

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 45:
kar[m4 + 1] = key-index(key4, m6), kar[m4 / 2] = key-index(key5, m7), 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 < key5, 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, m4 / 2 = m1, keyset(kar, m0) = ∅, key4 < key5

Rule argument

Additional information

Used simplifier rules: nen: ⊦ ¬ key < key0 ∧ key0 = key1 → ¬ key < key1; (forward) from specification oelem[binheap-key]% ⊦ key-index(key, m) = key-index(key0, m0) ↔ key = key0 ∧ m = m0; (ws)% ⊦ m ≠ n → kar[n, ki][m] = kar[m]; (s)