WORST_CASE(?,O(n^1)) proof of input_zUO3vJAULT.trs # AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^1).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, 29 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: h(f(x, y)) -> f(f(a, h(h(y))), 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: h(f(x, y)) -> f(f(a, h(h(y))), x) 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] transitions: f0(0, 0) -> 0 a0() -> 0 h0(0) -> 1 a1() -> 3 h1(0) -> 5 h1(5) -> 4 f1(3, 4) -> 2 f1(2, 0) -> 1 f1(2, 0) -> 5 a2() -> 7 h2(0) -> 9 h2(9) -> 8 f2(7, 8) -> 6 f2(6, 2) -> 4 f1(2, 0) -> 9 f2(6, 2) -> 8 ---------------------------------------- (4) BOUNDS(1, n^1)