WORST_CASE(?,O(n^1)) proof of input_nhlZi6fxRf.trs # AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^1). (0) CpxTRS (1) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] (2) CpxTRS (3) CpxTrsMatchBoundsProof [FINISHED, 0 ms] (4) BOUNDS(1, n^1) ---------------------------------------- (0) Obligation: The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^1). The TRS R consists of the following rules: odd(S(x)) -> even(x) even(S(x)) -> odd(x) odd(0) -> 0 even(0) -> S(0) S is empty. Rewrite Strategy: PARALLEL_INNERMOST ---------------------------------------- (1) RelTrsToTrsProof (UPPER BOUND(ID)) transformed relative TRS to TRS ---------------------------------------- (2) Obligation: The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^1). The TRS R consists of the following rules: odd(S(x)) -> even(x) even(S(x)) -> odd(x) odd(0) -> 0 even(0) -> S(0) S is empty. Rewrite Strategy: PARALLEL_INNERMOST ---------------------------------------- (3) CpxTrsMatchBoundsProof (FINISHED) A linear upper bound on the runtime complexity of the TRS R could be shown with a Match Bound [MATCHBOUNDS1,MATCHBOUNDS2] of 1. The certificate found is represented by the following graph. "[1, 2, 3] {(1,2,[odd_1|0, even_1|0, even_1|1, 0|1, odd_1|1]), (1,3,[S_1|1]), (2,2,[S_1|0, 0|0]), (3,2,[0|1])}" ---------------------------------------- (4) BOUNDS(1, n^1)