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Proof step 11 from proof max-sum-new-current-max (previous step)

n + m1 = (n1 + m) + 1, sum(ar, n0, m0) < sum(ar, m, n1) + ar[n1 + m], n0 + m0 < # ar, m < # ar, sum(ar, n0, m0) + 1 < sum(ar, n, m1), n1 + m < n + m1, sum(ar, n0, m0) < sum(ar, n, m1), sum(ar, m, n1) + ar[n1 + m] < sum(ar, n, m1), n0 + m0 < n + m1, n1 + m < # ar, suffix-upper-bound(subarray(ar, 0, n1 + m), sum(ar, m, n1)), upper-bound(subarray(ar, 0, n1 + m), sum(ar, n0, m0)), n0 + m0 ≤ n1 + m, n + m1 ≤ # ar ⊦

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 12:
⊦ suffix-upper-bound(ar, i) ↔ (∀ n. n ≤ # ar → sum(ar, n, # ar - n) ≤ i)
Proof step 13:
n + m1 = (n1 + m) + 1, n1 + m < # ar, n0 + m0 < n + m1, sum(ar, m, n1) + ar[n1 + m] < sum(ar, n, m1), sum(ar, n0, m0) < sum(ar, n, m1), n1 + m < n + m1, sum(ar, n0, m0) + 1 < sum(ar, n, m1), m < # ar, n0 + m0 < # ar, sum(ar, n0, m0) < sum(ar, m, n1) + ar[n1 + m], upper-bound(subarray(ar, 0, n1 + m), sum(ar, n0, m0)), n + m1 ≤ # ar, n0 + m0 ≤ n1 + m, ∀ n. n ≤ # subarray(ar, 0, n1 + m) → sum(subarray(ar, 0, n1 + m), n, # subarray(ar, 0, n1 + m) - n) ≤ sum(ar, m, n1) ⊦

Rule argument

spec: <toplevel> name: suffix-upper-bound seq: ⊦ suffix-upper-bound(ar, i) ↔ (∀ n. n ≤ # ar → sum(ar, n, # ar - n) ≤ i) subst: [i, ar] → [sum(ar, m, n1), subarray(ar, 0, n1 + m)] flags: <all paths>

Additional information

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