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Proof step 45 from proof prepost-remove (previous step)

# kar0 = # kar, kar[m2 / 2] = key-index(key1, m3), kar[m2 + 1] = key-index(key, m), ar1[m] = elemindex(a, m2 + 1), kar[m1] = key-index(key0, m0), keyset(kar[m2 + 1, key-index(key0, m0)], m1) = keyset(kar0, m1), 2exp-size(# kar), 2 + m2 < # kar, m2 < # kar, m2 < m1, m0 < # ar1, m1 < # kar, m2 + 1 < # kar, key0 < key1, m2 / 2 < # kar, m2 + 1 < m1, heap-order(kar[m2 + 1, key-index(key1, m3)], m1), heap-order(kar, m1 + 1), heap-order(kar0, m1), id-idx(ar1, ar0), no-dups-idx(kar, m1 + 1), no-dups-idx(kar0, m1), valid-idx(kar, ar1, m1 + 1), valid-idx(kar0, ar0, m1), ar1[m] = elemindex(a, m0 + 1) ∧ valid-idx(kar, ar1, m1 + 1) ∧ key-index(key, m) ∈ keyset(kar, m1 + 1) → kar[m0 + 1] = key-index(key, m), key-index(key, m) = key-index(key0, m0) ∨ key-index(key, m) ∈ keyset(kar, m1) ⊦ #(keyset(kar, m1) ++ key-index(key0, m0)) = 0, # kar = 1, keyset(kar0, m1) = (keyset(kar, m1) ++ key-index(key0, m0)) -- key-index(key, m), m1 = m2 + 1

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 46:
# kar0 = # kar, kar[m2 / 2] = key-index(key1, m3), kar[m2 + 1] = key-index(key, m), ar1[m] = elemindex(a, m2 + 1), kar[m1] = key-index(key0, m0), keyset(kar[m2 + 1, key-index(key0, m0)], m1) = keyset(kar0, m1), 2exp-size(# kar), 2 + m2 < # kar, m2 < # kar, m2 < m1, m0 < # ar1, m1 < # kar, m2 + 1 < # kar, key0 < key1, m2 / 2 < # kar, m2 + 1 < m1, heap-order(kar[m2 + 1, key-index(key1, m3)], m1), heap-order(kar, m1 + 1), heap-order(kar0, m1), id-idx(ar1, ar0), no-dups-idx(kar, m1 + 1), no-dups-idx(kar0, m1), valid-idx(kar, ar1, m1 + 1), valid-idx(kar0, ar0, m1), elemindex(a, m2 + 1) = elemindex(a, m0 + 1) → kar[m0 + 1] = key-index(key, m), key-index(key, m) = key-index(key0, m0) ∨ key-index(key, m) ∈ keyset(kar, m1) ⊦ #(keyset(kar, m1) ++ key-index(key0, m0)) = 0, # kar = 1, keyset(kar0, m1) = (keyset(kar, m1) ++ key-index(key0, m0)) -- key-index(key, m), m1 = m2 + 1

Rule argument

Additional information

Used simplifier rules: ⊦ ki ∈ s ++ ki0 ↔ ki = ki0 ∨ ki ∈ s; (s)%c: ⊦ m + n = n + m; from specification nat-basic%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic% ⊦ n < # kar → keyset(kar, n + 1) = keyset(kar, n) ++ kar[n]; (s)