WORST_CASE(?,O(n^1)) proof of input_Y5WZmne4pp.trs # AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^1).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: addlist(Cons(x, xs'), Cons(S(0), xs)) -> Cons(S(x), addlist(xs', xs)) addlist(Cons(S(0), xs'), Cons(x, xs)) -> Cons(S(x), addlist(xs', xs)) addlist(Nil, ys) -> Nil notEmpty(Cons(x, xs)) -> True notEmpty(Nil) -> False goal(xs, ys) -> addlist(xs, ys) 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: addlist(Cons(x, xs'), Cons(S(0), xs)) -> Cons(S(x), addlist(xs', xs)) addlist(Cons(S(0), xs'), Cons(x, xs)) -> Cons(S(x), addlist(xs', xs)) addlist(Nil, ys) -> Nil notEmpty(Cons(x, xs)) -> True notEmpty(Nil) -> False goal(xs, ys) -> addlist(xs, ys) 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: Cons0(0, 0) -> 0 S0(0) -> 0 00() -> 0 Nil0() -> 0 True0() -> 0 False0() -> 0 addlist0(0, 0) -> 1 notEmpty0(0) -> 2 goal0(0, 0) -> 3 S1(0) -> 4 addlist1(0, 0) -> 5 Cons1(4, 5) -> 1 Nil1() -> 1 True1() -> 2 False1() -> 2 addlist1(0, 0) -> 3 Cons1(4, 5) -> 3 Cons1(4, 5) -> 5 Nil1() -> 3 Nil1() -> 5 ---------------------------------------- (4) BOUNDS(1, n^1)