WORST_CASE(?,O(n^1)) proof of input_ufv7G9K9aL.trs # AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^1).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: f(s(X)) -> f(X) g(cons(0, Y)) -> g(Y) g(cons(s(X), Y)) -> s(X) h(cons(X, Y)) -> h(g(cons(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: f(s(X)) -> f(X) g(cons(0, Y)) -> g(Y) g(cons(s(X), Y)) -> s(X) h(cons(X, Y)) -> h(g(cons(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 1. The compatible tree automaton used to show the Match-Boundedness (for constructor-based start-terms) is represented by: final states : [1, 2, 3] transitions: s0(0) -> 0 cons0(0, 0) -> 0 00() -> 0 f0(0) -> 1 g0(0) -> 2 h0(0) -> 3 f1(0) -> 1 g1(0) -> 2 s1(0) -> 2 cons1(0, 0) -> 5 g1(5) -> 4 h1(4) -> 3 g1(0) -> 4 s1(0) -> 4 ---------------------------------------- (4) BOUNDS(1, n^1)