WORST_CASE(?,O(n^1)) proof of input_TMEtzimbns.trs # AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^1).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: odd(Cons(x, xs)) -> even(xs) odd(Nil) -> False even(Cons(x, xs)) -> odd(xs) notEmpty(Cons(x, xs)) -> True notEmpty(Nil) -> False even(Nil) -> True evenodd(x) -> even(x) 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: odd(Cons(x, xs)) -> even(xs) odd(Nil) -> False even(Cons(x, xs)) -> odd(xs) notEmpty(Cons(x, xs)) -> True notEmpty(Nil) -> False even(Nil) -> True evenodd(x) -> even(x) 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, 4] transitions: Cons0(0, 0) -> 0 Nil0() -> 0 False0() -> 0 True0() -> 0 odd0(0) -> 1 even0(0) -> 2 notEmpty0(0) -> 3 evenodd0(0) -> 4 even1(0) -> 1 False1() -> 1 odd1(0) -> 2 True1() -> 3 False1() -> 3 True1() -> 2 even1(0) -> 4 even1(0) -> 2 False1() -> 2 odd1(0) -> 1 odd1(0) -> 4 True1() -> 1 True1() -> 4 False1() -> 4 ---------------------------------------- (4) BOUNDS(1, n^1)