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Proof step 12 from proof max-sum-next-less-zero (previous step)

sum(ar, n0, m) < 0, n0 + m < # ar, ar[n] < 0, n < # ar, n0 + m ≤ n + 1, n0 + m ≤ n, ∀ n2, m1. n2 + m1 ≤ n → sum(ar, n2, m1) ≤ sum(ar, n0, m) ⊦

User applied rule all left

And closed the goal!

Rule argument

pos: left 7 terms: n0, 0 abort: F discard: F computed: F

Additional information

Used simplifier rules: sum-zero: ⊦ sum(ar, n, 0) = 0;% ⊦ n + 0 = n; (s)%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic%c: ⊦ m + n = n + m; from specification nat-basic% ⊦ i < j → (j ≤ i ↔ false); (as)% ⊦ m ≤ n ↔ ¬ n < m; (ss)%f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic%fle-01: ⊦ m ≤ n ∧ n < n0 → m < n0; (forward) from specification nat%le-one: ⊦ m ≤ n + 1 ∧ n < n0 → m ≤ n0; (forward) from specification nat% ⊦ m < n → n ≠ 0; (s)% ⊦ n + m ≤ m0 → ¬ n0 + m0 < m; (s) Used terms: n0, 0