WORST_CASE(?,O(n^1)) proof of input_wB9NhdZvRz.trs # AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^1).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(a, empty) -> g(a, empty) f(a, cons(x, k)) -> f(cons(x, a), k) g(empty, d) -> d g(cons(x, k), d) -> g(k, cons(x, d)) 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(a, empty) -> g(a, empty) f(a, cons(x, k)) -> f(cons(x, a), k) g(empty, d) -> d g(cons(x, k), d) -> g(k, cons(x, d)) 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: empty0() -> 0 cons0(0, 0) -> 0 f0(0, 0) -> 1 g0(0, 0) -> 2 empty1() -> 3 g1(0, 3) -> 1 cons1(0, 0) -> 4 f1(4, 0) -> 1 cons1(0, 0) -> 5 g1(0, 5) -> 2 g1(4, 3) -> 1 cons1(0, 4) -> 4 cons1(0, 3) -> 5 g1(0, 5) -> 1 cons1(0, 5) -> 5 cons2(0, 3) -> 6 g2(0, 6) -> 1 g2(4, 6) -> 1 cons1(0, 6) -> 5 cons2(0, 6) -> 6 0 -> 2 3 -> 1 5 -> 2 5 -> 1 6 -> 1 ---------------------------------------- (4) BOUNDS(1, n^1)