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Proof step 20 from proof max-sum-keep-current-max (previous step)

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

User applied rule all left

And closed the goal!

Rule argument

pos: left 10 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%le-one: ⊦ m ≤ n + 1 ∧ n < n0 → m ≤ n0; (forward) from specification nat%fle-01: ⊦ m ≤ n ∧ n < n0 → m < n0; (forward) from specification nat% ⊦ m < n → n ≠ 0; (s)% ⊦ m + m0 ≤ m ↔ m0 = 0; (s)% ⊦ ¬ m + n < m; (s) Used terms: n0, 0