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Proof step 48 from proof prepost-update (previous step)

kar[m1] = key-index(key1, m), ar[m] = elemindex(a, m1), kar[m1, key-index(key, m)][m3 / 2] = key-index(key0, m2), m1 = m3 + 1, 2exp-size(# kar), key < key0, key < key1, m1 < # kar, m1 < m0, 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, heap-order(kar[m1, key-index(key0, m2)], m0)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 49:
ar[m] = elemindex(a, m3 + 1), kar[m3 + 1] = key-index(key1, m), kar[m3 / 2] = key-index(key0, m2), 2exp-size(# kar), m3 + 1 < m0, key < key1, key < key0, m3 + 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) = ∅, heap-order(kar[m3 + 1, key-index(key0, m2)], m0)

Rule argument

Additional information

Used simplifier rules: fle: ⊦ m < n ∧ n ≤ n0 → m < n0; (forward) from specification nat%key-index-in-keyset-array-extract: ⊦ ar[m] = elemindex(a, m0) ∧ valid-idx(kar, ar, n) ∧ key-index(key, m) ∈ keyset(kar, n) → kar[m0] = key-index(key, m); (forward) from specification bin-min-heap-inv% ⊦ m + 1 ≠ 0; (s)%m ≠ n ⊦ kar[m, ki][n] = kar[n]; (s)%c: ⊦ m + n = n + m; from specification nat-basic%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic% ⊦ m / 2 ≠ m + 1; (s)