WORST_CASE(?,O(n^1)) proof of input_FO0EE39H7t.trs # AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^1).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, 38 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: plus#2(0, x12) -> x12 plus#2(S(x4), x2) -> S(plus#2(x4, x2)) fold#3(Nil) -> 0 fold#3(Cons(x4, x2)) -> plus#2(x4, fold#3(x2)) main(x1) -> fold#3(x1) 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: plus#2(0, x12) -> x12 plus#2(S(x4), x2) -> S(plus#2(x4, x2)) fold#3(Nil) -> 0 fold#3(Cons(x4, x2)) -> plus#2(x4, fold#3(x2)) main(x1) -> fold#3(x1) 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, 3] transitions: 00() -> 0 S0(0) -> 0 Nil0() -> 0 Cons0(0, 0) -> 0 plus#20(0, 0) -> 1 fold#30(0) -> 2 main0(0) -> 3 plus#21(0, 0) -> 4 S1(4) -> 1 01() -> 2 fold#31(0) -> 5 plus#21(0, 5) -> 2 fold#31(0) -> 3 plus#21(0, 5) -> 4 S1(4) -> 2 S1(4) -> 4 01() -> 3 01() -> 5 plus#21(0, 5) -> 3 plus#21(0, 5) -> 5 S1(4) -> 3 S1(4) -> 5 0 -> 1 0 -> 4 5 -> 2 5 -> 3 5 -> 4 ---------------------------------------- (4) BOUNDS(1, n^1)