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Proof step 9 from proof max-sum-next-less-zero (previous step)

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 ≤ n → sum(ar, n1, m0) ≤ sum(ar, n0, m), ∀ n2, m1. n2 + m1 ≤ n → sum(ar, n2, m1) ≤ sum(ar, n0, m) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 10:
n1 + m0 = n + 1, sum(ar, n0, m) < sum(ar, n1, m0), ar[n] < 0, n0 + m < # ar, n < # ar, n0 + m ≤ n, n0 + m ≤ n + 1, ∀ n2, m1. n2 + m1 ≤ n → sum(ar, n2, m1) ≤ sum(ar, n0, m) ⊦

Rule argument

Additional information

Used simplifier rules: ⊦ m = n + 1 → ¬ m ≤ n; (s)%c: ⊦ m + n = n + m; from specification nat-basic%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic