WORST_CASE(?,O(n^1)) proof of input_dA8mHFdJtk.trs # AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^1).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, 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: h(f(x), y) -> f(g(x, y)) g(x, y) -> h(x, y) 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(g(x, y)) g(x, y) -> h(x, y) 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: f0(0) -> 0 h0(0, 0) -> 1 g0(0, 0) -> 2 g1(0, 0) -> 3 f1(3) -> 1 h1(0, 0) -> 2 f1(3) -> 2 h2(0, 0) -> 3 f1(3) -> 3 ---------------------------------------- (4) BOUNDS(1, n^1)