WORST_CASE(?,O(n^1)) proof of input_hYZdn33mdh.trs # AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^1).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(s(x)) -> s(s(f(p(s(x))))) f(0) -> 0 p(s(x)) -> 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(s(x)) -> s(s(f(p(s(x))))) f(0) -> 0 p(s(x)) -> 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, 8, 9, 10] {(1,2,[f_1|0, p_1|0, 0|1, s_1|1]), (1,3,[s_1|1]), (2,2,[s_1|0, 0|0]), (3,4,[s_1|1]), (4,5,[f_1|1]), (4,7,[s_1|2]), (4,2,[0|2]), (5,6,[p_1|1]), (5,2,[s_1|1, 0|1]), (6,2,[s_1|1]), (7,8,[s_1|2]), (8,9,[f_1|2]), (8,7,[s_1|2]), (8,2,[0|2]), (9,10,[p_1|2]), (9,2,[s_1|1, 0|1]), (10,2,[s_1|2])}" ---------------------------------------- (4) BOUNDS(1, n^1)