WORST_CASE(?,O(n^1)) proof of input_F2mqry8XS9.trs # AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^1).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, 35 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(nil) -> nil f(.(nil, y)) -> .(nil, f(y)) f(.(.(x, y), z)) -> f(.(x, .(y, z))) g(nil) -> nil g(.(x, nil)) -> .(g(x), nil) g(.(x, .(y, z))) -> g(.(.(x, y), z)) 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(nil) -> nil f(.(nil, y)) -> .(nil, f(y)) f(.(.(x, y), z)) -> f(.(x, .(y, z))) g(nil) -> nil g(.(x, nil)) -> .(g(x), nil) g(.(x, .(y, z))) -> g(.(.(x, y), z)) 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] transitions: nil0() -> 0 .0(0, 0) -> 0 f0(0) -> 1 g0(0) -> 2 nil1() -> 1 nil1() -> 3 f1(0) -> 4 .1(3, 4) -> 1 .1(0, 0) -> 6 .1(0, 6) -> 5 f1(5) -> 1 nil1() -> 2 g1(0) -> 7 nil1() -> 8 .1(7, 8) -> 2 .1(0, 0) -> 10 .1(10, 0) -> 9 g1(9) -> 2 nil1() -> 4 .1(3, 4) -> 4 f1(6) -> 4 f1(5) -> 4 .1(0, 6) -> 6 nil1() -> 7 .1(7, 8) -> 7 g1(10) -> 7 g1(9) -> 7 .1(10, 0) -> 10 ---------------------------------------- (4) BOUNDS(1, n^1)