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Proof step 13 from proof upper-bound-reduce (previous step)

i0 < sum(ar0, n, m), n + m ≤ # ar0, (n + m ≤ # ar → sum(ar, n, m) ≤ i) → n + m ≤ # ar0 → sum(ar0, n, m) ≤ i0, ∀ n, m. n + m ≤ # ar → sum(ar, n, m) ≤ i, ∀ n, m. (n + m ≤ # ar → sum(ar, n, m) ≤ i) → n + m ≤ # ar0 → sum(ar0, n, m) ≤ i0 ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 14:
i0 < sum(ar0, n, m), i < sum(ar, n, m), n + m ≤ # ar0, n + m ≤ # ar, ∀ n, m. (n + m ≤ # ar → sum(ar, n, m) ≤ i) → n + m ≤ # ar0 → sum(ar0, n, m) ≤ i0, ∀ n, m. n + m ≤ # ar → sum(ar, n, m) ≤ i ⊦

Rule argument

Additional information

Used simplifier rules: ⊦ i ≤ j ↔ ¬ j < i; (ss)% ⊦ i < j → (j ≤ i ↔ false); (as)