WORST_CASE(?,O(n^1)) proof of input_KWe1ALO0Lf.trs # AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^1).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, 51 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: g(S(x), y) -> g(x, S(y)) f(y, S(x)) -> f(S(y), x) g(0, x2) -> x2 f(x1, 0) -> g(x1, 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: g(S(x), y) -> g(x, S(y)) f(y, S(x)) -> f(S(y), x) g(0, x2) -> x2 f(x1, 0) -> g(x1, 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 2. The compatible tree automaton used to show the Match-Boundedness (for constructor-based start-terms) is represented by: final states : [1, 2] transitions: S0(0) -> 0 00() -> 0 g0(0, 0) -> 1 f0(0, 0) -> 2 S1(0) -> 3 g1(0, 3) -> 1 S1(0) -> 4 f1(4, 0) -> 2 01() -> 5 g1(0, 5) -> 2 S1(3) -> 3 S1(5) -> 3 g1(0, 3) -> 2 S1(4) -> 4 g1(4, 5) -> 2 S2(5) -> 6 g2(0, 6) -> 2 g2(4, 6) -> 2 S1(6) -> 3 S2(6) -> 6 0 -> 1 3 -> 1 3 -> 2 5 -> 2 6 -> 2 ---------------------------------------- (4) BOUNDS(1, n^1)