WORST_CASE(?,O(n^1)) proof of input_2JKwJmADqS.trs # AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^1).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) CpxTrsMatchBoundsTAProof [FINISHED, 68 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: +(0, y) -> y +(s(x), 0) -> s(x) +(s(x), s(y)) -> s(+(s(x), +(y, 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: +(0, y) -> y +(s(x), 0) -> s(x) +(s(x), s(y)) -> s(+(s(x), +(y, 0))) S is empty. Rewrite Strategy: PARALLEL_INNERMOST ---------------------------------------- (3) CpxTrsMatchBoundsTAProof (FINISHED) A linear upper bound on the runtime complexity of the TRS R could be shown with a Match-Bound[TAB_LEFTLINEAR,TAB_NONLEFTLINEAR] (for contructor-based start-terms) of 3. The compatible tree automaton used to show the Match-Boundedness (for constructor-based start-terms) is represented by: final states : [1] transitions: 00() -> 0 s0(0) -> 0 +0(0, 0) -> 1 s1(0) -> 1 s1(0) -> 3 01() -> 5 +1(0, 5) -> 4 +1(3, 4) -> 2 s1(2) -> 1 s1(0) -> 4 s2(0) -> 2 s2(0) -> 7 02() -> 9 +2(0, 9) -> 8 +2(7, 8) -> 6 s2(6) -> 2 s1(0) -> 8 s3(0) -> 6 s2(6) -> 6 0 -> 1 5 -> 4 9 -> 8 ---------------------------------------- (4) BOUNDS(1, n^1)