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Proof step 17 from proof max-sum-new-current-max (previous step)

sum(ar, n0, m0) < 0, sum(ar, n0, m0) < ~1, sum(ar, m, n1) + ar[n1 + m] < 0, sum(ar, n0, m0) < sum(ar, m, n1) + ar[n1 + m], n0 + m0 < # ar, m < # ar, n1 + m < # ar, upper-bound(subarray(ar, 0, n1 + m), sum(ar, n0, m0)), n0 + m0 ≤ n1 + m, ∀ n2. n2 ≤ n1 + m → sum(ar, n2, n1 + m - n2) ≤ sum(ar, m, n1) ⊦

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 18:
⊦ upper-bound(ar, i) ↔ (∀ n, m. n + m ≤ # ar → sum(ar, n, m) ≤ i)
Proof step 19:
n1 + m < # ar, m < # ar, n0 + m0 < # ar, sum(ar, n0, m0) < sum(ar, m, n1) + ar[n1 + m], sum(ar, m, n1) + ar[n1 + m] < 0, sum(ar, n0, m0) < ~1, sum(ar, n0, m0) < 0, n0 + m0 ≤ n1 + m, ∀ n2. n2 ≤ n1 + m → sum(ar, n2, n1 + m - n2) ≤ sum(ar, m, n1), ∀ n, m1. n + m1 ≤ # subarray(ar, 0, n1 + m) → sum(subarray(ar, 0, n1 + m), n, m1) ≤ sum(ar, n0, m0) ⊦

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, m0), subarray(ar, 0, n1 + m)] flags: <all paths>

Additional information

Used terms: sum(ar, n0, m0), subarray(ar, 0, n1 + m)