WORST_CASE(?,O(n^1)) proof of input_hDl7xbwJil.trs # AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^1).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, 229 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(x, c(y, z)) -> h(c(s(y), x), z) h(c(s(x), c(s(0), y)), z) -> h(y, c(s(0), c(x, 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: h(x, c(y, z)) -> h(c(s(y), x), z) h(c(s(x), c(s(0), y)), z) -> h(y, c(s(0), c(x, 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 2. The compatible tree automaton used to show the Match-Boundedness (for constructor-based start-terms) is represented by: final states : [1] transitions: c0(0, 0) -> 0 s0(0) -> 0 00() -> 0 h0(0, 0) -> 1 s1(0) -> 3 c1(3, 0) -> 2 h1(2, 0) -> 1 01() -> 6 s1(6) -> 5 c1(0, 0) -> 7 c1(5, 7) -> 4 h1(0, 4) -> 1 c1(3, 2) -> 2 s2(5) -> 9 c2(9, 0) -> 8 h2(8, 7) -> 1 c1(0, 4) -> 7 h1(2, 4) -> 1 s2(0) -> 9 c2(9, 8) -> 8 h2(8, 0) -> 1 h2(8, 4) -> 1 c1(5, 7) -> 7 c2(9, 2) -> 8 c1(3, 8) -> 2 c1(0, 7) -> 7 c1(5, 0) -> 7 c1(5, 4) -> 7 h1(8, 4) -> 1 ---------------------------------------- (4) BOUNDS(1, n^1)