WORST_CASE(?,O(n^1)) proof of input_MahBZETzXO.trs # AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^1).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: foldl#3(x2, Nil) -> x2 foldl#3(x16, Cons(x24, x6)) -> foldl#3(Cons(x24, x16), x6) main(x1) -> foldl#3(Nil, 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: foldl#3(x2, Nil) -> x2 foldl#3(x16, Cons(x24, x6)) -> foldl#3(Cons(x24, x16), x6) main(x1) -> foldl#3(Nil, 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] transitions: Nil0() -> 0 Cons0(0, 0) -> 0 foldl#30(0, 0) -> 1 main0(0) -> 2 Cons1(0, 0) -> 3 foldl#31(3, 0) -> 1 Nil1() -> 4 foldl#31(4, 0) -> 2 Cons1(0, 3) -> 3 Cons1(0, 4) -> 3 foldl#31(3, 0) -> 2 0 -> 1 3 -> 1 3 -> 2 4 -> 2 ---------------------------------------- (4) BOUNDS(1, n^1)