WORST_CASE(?,O(n^1)) proof of input_OGpbwTgaD7.trs # AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^1).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: f(g(x)) -> g(g(f(x))) f(g(x)) -> g(g(g(x))) 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: f(g(x)) -> g(g(f(x))) f(g(x)) -> g(g(g(x))) 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, 4, 5, 6] {(1,2,[f_1|0]), (1,3,[g_1|1]), (1,5,[g_1|1]), (2,2,[g_1|0]), (3,4,[g_1|1]), (4,2,[f_1|1]), (4,3,[g_1|1]), (4,5,[g_1|1]), (5,6,[g_1|1]), (6,2,[g_1|1])}" ---------------------------------------- (4) BOUNDS(1, n^1)