WORST_CASE(?,O(n^1)) proof of input_lNZcJfndbZ.trs # AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^1).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, 53 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: max(L(x)) -> x max(N(L(0), L(y))) -> y max(N(L(s(x)), L(s(y)))) -> s(max(N(L(x), L(y)))) max(N(L(x), N(y, z))) -> max(N(L(x), L(max(N(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: max(L(x)) -> x max(N(L(0), L(y))) -> y max(N(L(s(x)), L(s(y)))) -> s(max(N(L(x), L(y)))) max(N(L(x), N(y, z))) -> max(N(L(x), L(max(N(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] transitions: L0(0) -> 0 N0(0, 0) -> 0 00() -> 0 s0(0) -> 0 max0(0) -> 1 L1(0) -> 4 L1(0) -> 5 N1(4, 5) -> 3 max1(3) -> 2 s1(2) -> 1 N1(0, 0) -> 8 max1(8) -> 7 L1(7) -> 6 N1(4, 6) -> 3 max1(3) -> 1 s1(2) -> 7 max1(3) -> 7 L1(2) -> 5 0 -> 1 0 -> 7 0 -> 2 7 -> 1 7 -> 2 2 -> 1 2 -> 7 ---------------------------------------- (4) BOUNDS(1, n^1)