WORST_CASE(?,O(n^1)) proof of input_z2U9Lff0CR.trs # AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^1).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: compS_f#1(compS_f(x2), x1) -> compS_f#1(x2, S(x1)) compS_f#1(id, x3) -> S(x3) iter#3(0) -> id iter#3(S(x6)) -> compS_f(iter#3(x6)) main(0) -> 0 main(S(x9)) -> compS_f#1(iter#3(x9), 0) 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: compS_f#1(compS_f(x2), x1) -> compS_f#1(x2, S(x1)) compS_f#1(id, x3) -> S(x3) iter#3(0) -> id iter#3(S(x6)) -> compS_f(iter#3(x6)) main(0) -> 0 main(S(x9)) -> compS_f#1(iter#3(x9), 0) 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, 3] transitions: compS_f0(0) -> 0 S0(0) -> 0 id0() -> 0 00() -> 0 compS_f#10(0, 0) -> 1 iter#30(0) -> 2 main0(0) -> 3 S1(0) -> 4 compS_f#11(0, 4) -> 1 S1(0) -> 1 id1() -> 2 iter#31(0) -> 5 compS_f1(5) -> 2 01() -> 3 iter#31(0) -> 6 01() -> 7 compS_f#11(6, 7) -> 3 S1(4) -> 4 S1(4) -> 1 id1() -> 5 id1() -> 6 compS_f1(5) -> 5 compS_f1(5) -> 6 S2(7) -> 8 compS_f#12(5, 8) -> 3 S2(7) -> 3 S2(8) -> 8 S2(8) -> 3 ---------------------------------------- (4) BOUNDS(1, n^1)