WORST_CASE(?,O(n^1)) proof of input_N7kFNhDwYJ.trs # AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^1).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, 44 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: dec(Cons(Nil, Nil)) -> Nil dec(Cons(Nil, Cons(x, xs))) -> dec(Cons(x, xs)) dec(Cons(Cons(x, xs), Nil)) -> dec(Nil) dec(Cons(Cons(x', xs'), Cons(x, xs))) -> dec(Cons(x, xs)) isNilNil(Cons(Nil, Nil)) -> True isNilNil(Cons(Nil, Cons(x, xs))) -> False isNilNil(Cons(Cons(x, xs), Nil)) -> False isNilNil(Cons(Cons(x', xs'), Cons(x, xs))) -> False nestdec(Nil) -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil))))))))))))))))) nestdec(Cons(x, xs)) -> nestdec(dec(Cons(x, xs))) number17(n) -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil))))))))))))))))) goal(x) -> nestdec(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: dec(Cons(Nil, Nil)) -> Nil dec(Cons(Nil, Cons(x, xs))) -> dec(Cons(x, xs)) dec(Cons(Cons(x, xs), Nil)) -> dec(Nil) dec(Cons(Cons(x', xs'), Cons(x, xs))) -> dec(Cons(x, xs)) isNilNil(Cons(Nil, Nil)) -> True isNilNil(Cons(Nil, Cons(x, xs))) -> False isNilNil(Cons(Cons(x, xs), Nil)) -> False isNilNil(Cons(Cons(x', xs'), Cons(x, xs))) -> False nestdec(Nil) -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil))))))))))))))))) nestdec(Cons(x, xs)) -> nestdec(dec(Cons(x, xs))) number17(n) -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil))))))))))))))))) goal(x) -> nestdec(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 2. The compatible tree automaton used to show the Match-Boundedness (for constructor-based start-terms) is represented by: final states : [1, 2, 3, 4, 5] transitions: Cons0(0, 0) -> 0 Nil0() -> 0 True0() -> 0 False0() -> 0 dec0(0) -> 1 isNilNil0(0) -> 2 nestdec0(0) -> 3 number170(0) -> 4 goal0(0) -> 5 Nil1() -> 1 Cons1(0, 0) -> 6 dec1(6) -> 1 Nil1() -> 7 dec1(7) -> 1 True1() -> 2 False1() -> 2 Nil1() -> 8 Nil1() -> 11 Cons1(8, 11) -> 10 Cons1(8, 10) -> 9 Cons1(8, 9) -> 9 Cons1(8, 9) -> 3 dec1(6) -> 12 nestdec1(12) -> 3 Cons1(8, 9) -> 4 nestdec1(0) -> 5 Nil1() -> 12 dec1(7) -> 12 Cons1(8, 9) -> 5 nestdec1(12) -> 5 Nil2() -> 13 Nil2() -> 16 Cons2(13, 16) -> 15 Cons2(13, 15) -> 14 Cons2(13, 14) -> 14 Cons2(13, 14) -> 3 Cons2(13, 14) -> 5 ---------------------------------------- (4) BOUNDS(1, n^1)