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Proof step 5 from proof max-sum-next-less-zero (previous step)

n1 + m0 = n + 1, sum(ar, n0, m) < sum(ar, n1, m0), ar[n] < 0, n0 + m < # ar, n < # ar, upper-bound(subarray(ar, 0, n), sum(ar, n0, m)), n0 + m ≤ n, n0 + m ≤ n + 1 ⊦

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 6:
⊦ upper-bound(ar, i) ↔ (∀ n, m. n + m ≤ # ar → sum(ar, n, m) ≤ i)
Proof step 7:
n1 + m0 = n + 1, n < # ar, n0 + m < # ar, ar[n] < 0, sum(ar, n0, m) < sum(ar, n1, m0), n0 + m ≤ n + 1, n0 + m ≤ n, ∀ n1, m0. n1 + m0 ≤ # subarray(ar, 0, n) → sum(subarray(ar, 0, n), n1, m0) ≤ sum(ar, n0, m) ⊦

Rule argument

spec: <toplevel> name: upper-bound seq: ⊦ upper-bound(ar, i) ↔ (∀ n, m. n + m ≤ # ar → sum(ar, n, m) ≤ i) subst: [i, ar] → [sum(ar, n0, m), subarray(ar, 0, n)] flags: <all paths>

Additional information

Used terms: sum(ar, n0, m), subarray(ar, 0, n)