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Proof step 15 from proof max-sum-new-current-max (previous step)

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 ≤ n1 + m → sum(ar, n, n1 + m - n) ≤ sum(ar, m, n1), ∀ n2. n2 ≤ n1 + m → sum(ar, n2, n1 + m - n2) ≤ sum(ar, m, n1) ⊦

User applied rule constructor cut

And obtained the following new goals

Proof step 16:
n + m1 = (n1 + m) + 1, m1 = 0, 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, upper-bound(subarray(ar, 0, n1 + m), sum(ar, n0, m0)), n0 + m0 ≤ n1 + m, n + m1 ≤ # ar, n ≤ n1 + m → sum(ar, n, n1 + m - n) ≤ sum(ar, m, n1), ∀ n2. n2 ≤ n1 + m → sum(ar, n2, n1 + m - n2) ≤ sum(ar, m, n1) ⊦
Proof step 21:
n + m1 = (n1 + m) + 1, m1 = m2 +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, upper-bound(subarray(ar, 0, n1 + m), sum(ar, n0, m0)), n0 + m0 ≤ n1 + m, n + m1 ≤ # ar, n ≤ n1 + m → sum(ar, n, n1 + m - n) ≤ sum(ar, m, n1), ∀ n2. n2 ≤ n1 + m → sum(ar, n2, n1 + m - n2) ≤ sum(ar, m, n1) ⊦

Rule argument

var: m1

Additional information

Used simplifier rules: nat freely generated by 0, +1;;