WORST_CASE(?,O(n^1)) proof of input_KeY9alYLGg.trs # AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^1).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(0) -> s(0) f(s(0)) -> s(0) f(s(s(x))) -> f(f(s(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(0) -> s(0) f(s(0)) -> s(0) f(s(s(x))) -> f(f(s(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 2. The certificate found is represented by the following graph. "[1, 2, 3, 4, 5, 6, 7] {(1,2,[f_1|0]), (1,3,[s_1|1]), (1,4,[f_1|1]), (1,7,[s_1|2]), (2,2,[0|0, s_1|0]), (3,2,[0|1]), (4,5,[f_1|1]), (4,6,[s_1|1]), (4,4,[f_1|1]), (4,7,[s_1|2]), (5,2,[s_1|1]), (6,2,[0|1]), (7,2,[0|2])}" ---------------------------------------- (4) BOUNDS(1, n^1)