WORST_CASE(?,O(n^1)) proof of input_DkX67lU7to.trs # AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^1).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, 50 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: walk#1(Nil) -> walk_xs walk#1(Cons(x4, x3)) -> comp_f_g(walk#1(x3), walk_xs_3(x4)) comp_f_g#1(comp_f_g(x7, x9), walk_xs_3(x8), x12) -> comp_f_g#1(x7, x9, Cons(x8, x12)) comp_f_g#1(walk_xs, walk_xs_3(x8), x12) -> Cons(x8, x12) main(Nil) -> Nil main(Cons(x4, x5)) -> comp_f_g#1(walk#1(x5), walk_xs_3(x4), Nil) 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: walk#1(Nil) -> walk_xs walk#1(Cons(x4, x3)) -> comp_f_g(walk#1(x3), walk_xs_3(x4)) comp_f_g#1(comp_f_g(x7, x9), walk_xs_3(x8), x12) -> comp_f_g#1(x7, x9, Cons(x8, x12)) comp_f_g#1(walk_xs, walk_xs_3(x8), x12) -> Cons(x8, x12) main(Nil) -> Nil main(Cons(x4, x5)) -> comp_f_g#1(walk#1(x5), walk_xs_3(x4), Nil) 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: Nil0() -> 0 walk_xs0() -> 0 Cons0(0, 0) -> 0 comp_f_g0(0, 0) -> 0 walk_xs_30(0) -> 0 walk#10(0) -> 1 comp_f_g#10(0, 0, 0) -> 2 main0(0) -> 3 walk_xs1() -> 1 walk#11(0) -> 4 walk_xs_31(0) -> 5 comp_f_g1(4, 5) -> 1 Cons1(0, 0) -> 6 comp_f_g#11(0, 0, 6) -> 2 Cons1(0, 0) -> 2 Nil1() -> 3 walk#11(0) -> 7 walk_xs_31(0) -> 8 Nil1() -> 9 comp_f_g#11(7, 8, 9) -> 3 walk_xs1() -> 4 walk_xs1() -> 7 comp_f_g1(4, 5) -> 4 comp_f_g1(4, 5) -> 7 Cons1(0, 6) -> 6 Cons1(0, 6) -> 2 Cons2(0, 9) -> 10 comp_f_g#12(4, 5, 10) -> 3 Cons2(0, 9) -> 3 Cons2(0, 10) -> 10 Cons2(0, 10) -> 3 ---------------------------------------- (4) BOUNDS(1, n^1)