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

n1 + m1 = (n2 + m) + 1, sum(ar, m, n2) + ar[n2 + m] < sum(ar, n1, m1), n2 + m < # ar, n0 + m0 < # ar, m < # ar, sum(ar, n0, m0) < sum(ar, n1, m1), upper-bound(subarray(ar, 0, n2 + m), sum(ar, n0, m0)), n0 + m0 ≤ n1 + m1, sum(ar, m, n2) + ar[n2 + m] ≤ sum(ar, n0, m0), n0 + m0 ≤ n2 + m, n1 ≤ n2 + m → sum(ar, n1, n2 + m - n1) ≤ sum(ar, m, n2), ∀ n. n ≤ n2 + m → sum(ar, n, n2 + m - n) ≤ sum(ar, m, n2) ⊦

User applied rule constructor cut

And obtained the following new goals

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

Rule argument

var: m1

Additional information

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