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Proof step 40 from proof prepost-update (previous step)

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), 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, m4 + 1 < m0 ∧ m4 + 1 ≠ 0 → ¬ kar[m4 + 1].key < kar[(m4 + 1 - 1) / 2].key, m1 = 0 ∨ ¬ key < key0 ⊦ # keyset(kar, m0) = 0, keyset(kar, m0) = ∅

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 41:
kar[m1, key-index(key, m)][m4 / 2] = key-index(key3, m5), 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 < key3, 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, keyset(kar, m0) = ∅, kar[m4 + 1].key < kar[m4 / 2].key

Rule argument

Additional information

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