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Proof step 53 from proof prepost-update (previous step)

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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 54:
ar[m] = elemindex(a, m3 + 1), kar[m3 + 1] = key-index(key1, m), kar[m3 / 2] = key-index(key0, m2), 2exp-size(# kar), kar[m3 + 1, key-index(key0, m2)][m4 + 1].key < kar[m3 + 1, key-index(key0, m2)][m4 / 2].key, m4 + 1 < m0, m3 + 1 < m0, key < key1, key < key0, m3 + 1 < # kar, 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 ⊦ # keyset(kar, m0) = 0, keyset(kar, m0) = ∅

Rule argument

Additional information

Used simplifier rules: fle: ⊦ m < n ∧ n ≤ n0 → m < n0; (forward) from specification nat%c: ⊦ m + n = n + m; from specification nat-basic%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic% ⊦ m + 1 ≠ 0; (s)