/export/starexec/sandbox/solver/bin/starexec_run_complexity /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files -------------------------------------------------------------------------------- WORST_CASE(Omega(n^1), O(n^1)) proof of /export/starexec/sandbox/benchmark/theBenchmark.xml # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). (0) CpxTRS (1) NestedDefinedSymbolProof [UPPER BOUND(ID), 0 ms] (2) CpxTRS (3) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] (4) CpxTRS (5) CpxTrsMatchBoundsTAProof [FINISHED, 401 ms] (6) BOUNDS(1, n^1) (7) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] (8) TRS for Loop Detection (9) DecreasingLoopProof [LOWER BOUND(ID), 0 ms] (10) BEST (11) proven lower bound (12) LowerBoundPropagationProof [FINISHED, 0 ms] (13) BOUNDS(n^1, INF) (14) TRS for Loop Detection ---------------------------------------- (0) Obligation: The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). The TRS R consists of the following rules: active(U11(tt, N, X, XS)) -> mark(U12(splitAt(N, XS), X)) active(U12(pair(YS, ZS), X)) -> mark(pair(cons(X, YS), ZS)) active(afterNth(N, XS)) -> mark(snd(splitAt(N, XS))) active(and(tt, X)) -> mark(X) active(fst(pair(X, Y))) -> mark(X) active(head(cons(N, XS))) -> mark(N) active(natsFrom(N)) -> mark(cons(N, natsFrom(s(N)))) active(sel(N, XS)) -> mark(head(afterNth(N, XS))) active(snd(pair(X, Y))) -> mark(Y) active(splitAt(0, XS)) -> mark(pair(nil, XS)) active(splitAt(s(N), cons(X, XS))) -> mark(U11(tt, N, X, XS)) active(tail(cons(N, XS))) -> mark(XS) active(take(N, XS)) -> mark(fst(splitAt(N, XS))) active(U11(X1, X2, X3, X4)) -> U11(active(X1), X2, X3, X4) active(U12(X1, X2)) -> U12(active(X1), X2) active(splitAt(X1, X2)) -> splitAt(active(X1), X2) active(splitAt(X1, X2)) -> splitAt(X1, active(X2)) active(pair(X1, X2)) -> pair(active(X1), X2) active(pair(X1, X2)) -> pair(X1, active(X2)) active(cons(X1, X2)) -> cons(active(X1), X2) active(afterNth(X1, X2)) -> afterNth(active(X1), X2) active(afterNth(X1, X2)) -> afterNth(X1, active(X2)) active(snd(X)) -> snd(active(X)) active(and(X1, X2)) -> and(active(X1), X2) active(fst(X)) -> fst(active(X)) active(head(X)) -> head(active(X)) active(natsFrom(X)) -> natsFrom(active(X)) active(s(X)) -> s(active(X)) active(sel(X1, X2)) -> sel(active(X1), X2) active(sel(X1, X2)) -> sel(X1, active(X2)) active(tail(X)) -> tail(active(X)) active(take(X1, X2)) -> take(active(X1), X2) active(take(X1, X2)) -> take(X1, active(X2)) U11(mark(X1), X2, X3, X4) -> mark(U11(X1, X2, X3, X4)) U12(mark(X1), X2) -> mark(U12(X1, X2)) splitAt(mark(X1), X2) -> mark(splitAt(X1, X2)) splitAt(X1, mark(X2)) -> mark(splitAt(X1, X2)) pair(mark(X1), X2) -> mark(pair(X1, X2)) pair(X1, mark(X2)) -> mark(pair(X1, X2)) cons(mark(X1), X2) -> mark(cons(X1, X2)) afterNth(mark(X1), X2) -> mark(afterNth(X1, X2)) afterNth(X1, mark(X2)) -> mark(afterNth(X1, X2)) snd(mark(X)) -> mark(snd(X)) and(mark(X1), X2) -> mark(and(X1, X2)) fst(mark(X)) -> mark(fst(X)) head(mark(X)) -> mark(head(X)) natsFrom(mark(X)) -> mark(natsFrom(X)) s(mark(X)) -> mark(s(X)) sel(mark(X1), X2) -> mark(sel(X1, X2)) sel(X1, mark(X2)) -> mark(sel(X1, X2)) tail(mark(X)) -> mark(tail(X)) take(mark(X1), X2) -> mark(take(X1, X2)) take(X1, mark(X2)) -> mark(take(X1, X2)) proper(U11(X1, X2, X3, X4)) -> U11(proper(X1), proper(X2), proper(X3), proper(X4)) proper(tt) -> ok(tt) proper(U12(X1, X2)) -> U12(proper(X1), proper(X2)) proper(splitAt(X1, X2)) -> splitAt(proper(X1), proper(X2)) proper(pair(X1, X2)) -> pair(proper(X1), proper(X2)) proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) proper(afterNth(X1, X2)) -> afterNth(proper(X1), proper(X2)) proper(snd(X)) -> snd(proper(X)) proper(and(X1, X2)) -> and(proper(X1), proper(X2)) proper(fst(X)) -> fst(proper(X)) proper(head(X)) -> head(proper(X)) proper(natsFrom(X)) -> natsFrom(proper(X)) proper(s(X)) -> s(proper(X)) proper(sel(X1, X2)) -> sel(proper(X1), proper(X2)) proper(0) -> ok(0) proper(nil) -> ok(nil) proper(tail(X)) -> tail(proper(X)) proper(take(X1, X2)) -> take(proper(X1), proper(X2)) U11(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U11(X1, X2, X3, X4)) U12(ok(X1), ok(X2)) -> ok(U12(X1, X2)) splitAt(ok(X1), ok(X2)) -> ok(splitAt(X1, X2)) pair(ok(X1), ok(X2)) -> ok(pair(X1, X2)) cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) afterNth(ok(X1), ok(X2)) -> ok(afterNth(X1, X2)) snd(ok(X)) -> ok(snd(X)) and(ok(X1), ok(X2)) -> ok(and(X1, X2)) fst(ok(X)) -> ok(fst(X)) head(ok(X)) -> ok(head(X)) natsFrom(ok(X)) -> ok(natsFrom(X)) s(ok(X)) -> ok(s(X)) sel(ok(X1), ok(X2)) -> ok(sel(X1, X2)) tail(ok(X)) -> ok(tail(X)) take(ok(X1), ok(X2)) -> ok(take(X1, X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) S is empty. Rewrite Strategy: FULL ---------------------------------------- (1) NestedDefinedSymbolProof (UPPER BOUND(ID)) The following defined symbols can occur below the 0th argument of top: proper, active The following defined symbols can occur below the 0th argument of proper: proper, active The following defined symbols can occur below the 0th argument of active: proper, active Hence, the left-hand sides of the following rules are not basic-reachable and can be removed: active(U11(tt, N, X, XS)) -> mark(U12(splitAt(N, XS), X)) active(U12(pair(YS, ZS), X)) -> mark(pair(cons(X, YS), ZS)) active(afterNth(N, XS)) -> mark(snd(splitAt(N, XS))) active(and(tt, X)) -> mark(X) active(fst(pair(X, Y))) -> mark(X) active(head(cons(N, XS))) -> mark(N) active(natsFrom(N)) -> mark(cons(N, natsFrom(s(N)))) active(sel(N, XS)) -> mark(head(afterNth(N, XS))) active(snd(pair(X, Y))) -> mark(Y) active(splitAt(0, XS)) -> mark(pair(nil, XS)) active(splitAt(s(N), cons(X, XS))) -> mark(U11(tt, N, X, XS)) active(tail(cons(N, XS))) -> mark(XS) active(take(N, XS)) -> mark(fst(splitAt(N, XS))) active(U11(X1, X2, X3, X4)) -> U11(active(X1), X2, X3, X4) active(U12(X1, X2)) -> U12(active(X1), X2) active(splitAt(X1, X2)) -> splitAt(active(X1), X2) active(splitAt(X1, X2)) -> splitAt(X1, active(X2)) active(pair(X1, X2)) -> pair(active(X1), X2) active(pair(X1, X2)) -> pair(X1, active(X2)) active(cons(X1, X2)) -> cons(active(X1), X2) active(afterNth(X1, X2)) -> afterNth(active(X1), X2) active(afterNth(X1, X2)) -> afterNth(X1, active(X2)) active(snd(X)) -> snd(active(X)) active(and(X1, X2)) -> and(active(X1), X2) active(fst(X)) -> fst(active(X)) active(head(X)) -> head(active(X)) active(natsFrom(X)) -> natsFrom(active(X)) active(s(X)) -> s(active(X)) active(sel(X1, X2)) -> sel(active(X1), X2) active(sel(X1, X2)) -> sel(X1, active(X2)) active(tail(X)) -> tail(active(X)) active(take(X1, X2)) -> take(active(X1), X2) active(take(X1, X2)) -> take(X1, active(X2)) proper(U11(X1, X2, X3, X4)) -> U11(proper(X1), proper(X2), proper(X3), proper(X4)) proper(U12(X1, X2)) -> U12(proper(X1), proper(X2)) proper(splitAt(X1, X2)) -> splitAt(proper(X1), proper(X2)) proper(pair(X1, X2)) -> pair(proper(X1), proper(X2)) proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) proper(afterNth(X1, X2)) -> afterNth(proper(X1), proper(X2)) proper(snd(X)) -> snd(proper(X)) proper(and(X1, X2)) -> and(proper(X1), proper(X2)) proper(fst(X)) -> fst(proper(X)) proper(head(X)) -> head(proper(X)) proper(natsFrom(X)) -> natsFrom(proper(X)) proper(s(X)) -> s(proper(X)) proper(sel(X1, X2)) -> sel(proper(X1), proper(X2)) proper(tail(X)) -> tail(proper(X)) proper(take(X1, X2)) -> take(proper(X1), proper(X2)) ---------------------------------------- (2) Obligation: The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^1). The TRS R consists of the following rules: U11(mark(X1), X2, X3, X4) -> mark(U11(X1, X2, X3, X4)) U12(mark(X1), X2) -> mark(U12(X1, X2)) splitAt(mark(X1), X2) -> mark(splitAt(X1, X2)) splitAt(X1, mark(X2)) -> mark(splitAt(X1, X2)) pair(mark(X1), X2) -> mark(pair(X1, X2)) pair(X1, mark(X2)) -> mark(pair(X1, X2)) cons(mark(X1), X2) -> mark(cons(X1, X2)) afterNth(mark(X1), X2) -> mark(afterNth(X1, X2)) afterNth(X1, mark(X2)) -> mark(afterNth(X1, X2)) snd(mark(X)) -> mark(snd(X)) and(mark(X1), X2) -> mark(and(X1, X2)) fst(mark(X)) -> mark(fst(X)) head(mark(X)) -> mark(head(X)) natsFrom(mark(X)) -> mark(natsFrom(X)) s(mark(X)) -> mark(s(X)) sel(mark(X1), X2) -> mark(sel(X1, X2)) sel(X1, mark(X2)) -> mark(sel(X1, X2)) tail(mark(X)) -> mark(tail(X)) take(mark(X1), X2) -> mark(take(X1, X2)) take(X1, mark(X2)) -> mark(take(X1, X2)) proper(tt) -> ok(tt) proper(0) -> ok(0) proper(nil) -> ok(nil) U11(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U11(X1, X2, X3, X4)) U12(ok(X1), ok(X2)) -> ok(U12(X1, X2)) splitAt(ok(X1), ok(X2)) -> ok(splitAt(X1, X2)) pair(ok(X1), ok(X2)) -> ok(pair(X1, X2)) cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) afterNth(ok(X1), ok(X2)) -> ok(afterNth(X1, X2)) snd(ok(X)) -> ok(snd(X)) and(ok(X1), ok(X2)) -> ok(and(X1, X2)) fst(ok(X)) -> ok(fst(X)) head(ok(X)) -> ok(head(X)) natsFrom(ok(X)) -> ok(natsFrom(X)) s(ok(X)) -> ok(s(X)) sel(ok(X1), ok(X2)) -> ok(sel(X1, X2)) tail(ok(X)) -> ok(tail(X)) take(ok(X1), ok(X2)) -> ok(take(X1, X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) S is empty. Rewrite Strategy: FULL ---------------------------------------- (3) RelTrsToTrsProof (UPPER BOUND(ID)) transformed relative TRS to TRS ---------------------------------------- (4) Obligation: The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^1). The TRS R consists of the following rules: U11(mark(X1), X2, X3, X4) -> mark(U11(X1, X2, X3, X4)) U12(mark(X1), X2) -> mark(U12(X1, X2)) splitAt(mark(X1), X2) -> mark(splitAt(X1, X2)) splitAt(X1, mark(X2)) -> mark(splitAt(X1, X2)) pair(mark(X1), X2) -> mark(pair(X1, X2)) pair(X1, mark(X2)) -> mark(pair(X1, X2)) cons(mark(X1), X2) -> mark(cons(X1, X2)) afterNth(mark(X1), X2) -> mark(afterNth(X1, X2)) afterNth(X1, mark(X2)) -> mark(afterNth(X1, X2)) snd(mark(X)) -> mark(snd(X)) and(mark(X1), X2) -> mark(and(X1, X2)) fst(mark(X)) -> mark(fst(X)) head(mark(X)) -> mark(head(X)) natsFrom(mark(X)) -> mark(natsFrom(X)) s(mark(X)) -> mark(s(X)) sel(mark(X1), X2) -> mark(sel(X1, X2)) sel(X1, mark(X2)) -> mark(sel(X1, X2)) tail(mark(X)) -> mark(tail(X)) take(mark(X1), X2) -> mark(take(X1, X2)) take(X1, mark(X2)) -> mark(take(X1, X2)) proper(tt) -> ok(tt) proper(0) -> ok(0) proper(nil) -> ok(nil) U11(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U11(X1, X2, X3, X4)) U12(ok(X1), ok(X2)) -> ok(U12(X1, X2)) splitAt(ok(X1), ok(X2)) -> ok(splitAt(X1, X2)) pair(ok(X1), ok(X2)) -> ok(pair(X1, X2)) cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) afterNth(ok(X1), ok(X2)) -> ok(afterNth(X1, X2)) snd(ok(X)) -> ok(snd(X)) and(ok(X1), ok(X2)) -> ok(and(X1, X2)) fst(ok(X)) -> ok(fst(X)) head(ok(X)) -> ok(head(X)) natsFrom(ok(X)) -> ok(natsFrom(X)) s(ok(X)) -> ok(s(X)) sel(ok(X1), ok(X2)) -> ok(sel(X1, X2)) tail(ok(X)) -> ok(tail(X)) take(ok(X1), ok(X2)) -> ok(take(X1, X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) S is empty. Rewrite Strategy: FULL ---------------------------------------- (5) 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, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17] transitions: mark0(0) -> 0 tt0() -> 0 ok0(0) -> 0 00() -> 0 nil0() -> 0 active0(0) -> 0 U110(0, 0, 0, 0) -> 1 U120(0, 0) -> 2 splitAt0(0, 0) -> 3 pair0(0, 0) -> 4 cons0(0, 0) -> 5 afterNth0(0, 0) -> 6 snd0(0) -> 7 and0(0, 0) -> 8 fst0(0) -> 9 head0(0) -> 10 natsFrom0(0) -> 11 s0(0) -> 12 sel0(0, 0) -> 13 tail0(0) -> 14 take0(0, 0) -> 15 proper0(0) -> 16 top0(0) -> 17 U111(0, 0, 0, 0) -> 18 mark1(18) -> 1 U121(0, 0) -> 19 mark1(19) -> 2 splitAt1(0, 0) -> 20 mark1(20) -> 3 pair1(0, 0) -> 21 mark1(21) -> 4 cons1(0, 0) -> 22 mark1(22) -> 5 afterNth1(0, 0) -> 23 mark1(23) -> 6 snd1(0) -> 24 mark1(24) -> 7 and1(0, 0) -> 25 mark1(25) -> 8 fst1(0) -> 26 mark1(26) -> 9 head1(0) -> 27 mark1(27) -> 10 natsFrom1(0) -> 28 mark1(28) -> 11 s1(0) -> 29 mark1(29) -> 12 sel1(0, 0) -> 30 mark1(30) -> 13 tail1(0) -> 31 mark1(31) -> 14 take1(0, 0) -> 32 mark1(32) -> 15 tt1() -> 33 ok1(33) -> 16 01() -> 34 ok1(34) -> 16 nil1() -> 35 ok1(35) -> 16 U111(0, 0, 0, 0) -> 36 ok1(36) -> 1 U121(0, 0) -> 37 ok1(37) -> 2 splitAt1(0, 0) -> 38 ok1(38) -> 3 pair1(0, 0) -> 39 ok1(39) -> 4 cons1(0, 0) -> 40 ok1(40) -> 5 afterNth1(0, 0) -> 41 ok1(41) -> 6 snd1(0) -> 42 ok1(42) -> 7 and1(0, 0) -> 43 ok1(43) -> 8 fst1(0) -> 44 ok1(44) -> 9 head1(0) -> 45 ok1(45) -> 10 natsFrom1(0) -> 46 ok1(46) -> 11 s1(0) -> 47 ok1(47) -> 12 sel1(0, 0) -> 48 ok1(48) -> 13 tail1(0) -> 49 ok1(49) -> 14 take1(0, 0) -> 50 ok1(50) -> 15 proper1(0) -> 51 top1(51) -> 17 active1(0) -> 52 top1(52) -> 17 mark1(18) -> 18 mark1(18) -> 36 mark1(19) -> 19 mark1(19) -> 37 mark1(20) -> 20 mark1(20) -> 38 mark1(21) -> 21 mark1(21) -> 39 mark1(22) -> 22 mark1(22) -> 40 mark1(23) -> 23 mark1(23) -> 41 mark1(24) -> 24 mark1(24) -> 42 mark1(25) -> 25 mark1(25) -> 43 mark1(26) -> 26 mark1(26) -> 44 mark1(27) -> 27 mark1(27) -> 45 mark1(28) -> 28 mark1(28) -> 46 mark1(29) -> 29 mark1(29) -> 47 mark1(30) -> 30 mark1(30) -> 48 mark1(31) -> 31 mark1(31) -> 49 mark1(32) -> 32 mark1(32) -> 50 ok1(33) -> 51 ok1(34) -> 51 ok1(35) -> 51 ok1(36) -> 18 ok1(36) -> 36 ok1(37) -> 19 ok1(37) -> 37 ok1(38) -> 20 ok1(38) -> 38 ok1(39) -> 21 ok1(39) -> 39 ok1(40) -> 22 ok1(40) -> 40 ok1(41) -> 23 ok1(41) -> 41 ok1(42) -> 24 ok1(42) -> 42 ok1(43) -> 25 ok1(43) -> 43 ok1(44) -> 26 ok1(44) -> 44 ok1(45) -> 27 ok1(45) -> 45 ok1(46) -> 28 ok1(46) -> 46 ok1(47) -> 29 ok1(47) -> 47 ok1(48) -> 30 ok1(48) -> 48 ok1(49) -> 31 ok1(49) -> 49 ok1(50) -> 32 ok1(50) -> 50 active2(33) -> 53 top2(53) -> 17 active2(34) -> 53 active2(35) -> 53 ---------------------------------------- (6) BOUNDS(1, n^1) ---------------------------------------- (7) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) Transformed a relative TRS into a decreasing-loop problem. ---------------------------------------- (8) Obligation: Analyzing the following TRS for decreasing loops: The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). The TRS R consists of the following rules: active(U11(tt, N, X, XS)) -> mark(U12(splitAt(N, XS), X)) active(U12(pair(YS, ZS), X)) -> mark(pair(cons(X, YS), ZS)) active(afterNth(N, XS)) -> mark(snd(splitAt(N, XS))) active(and(tt, X)) -> mark(X) active(fst(pair(X, Y))) -> mark(X) active(head(cons(N, XS))) -> mark(N) active(natsFrom(N)) -> mark(cons(N, natsFrom(s(N)))) active(sel(N, XS)) -> mark(head(afterNth(N, XS))) active(snd(pair(X, Y))) -> mark(Y) active(splitAt(0, XS)) -> mark(pair(nil, XS)) active(splitAt(s(N), cons(X, XS))) -> mark(U11(tt, N, X, XS)) active(tail(cons(N, XS))) -> mark(XS) active(take(N, XS)) -> mark(fst(splitAt(N, XS))) active(U11(X1, X2, X3, X4)) -> U11(active(X1), X2, X3, X4) active(U12(X1, X2)) -> U12(active(X1), X2) active(splitAt(X1, X2)) -> splitAt(active(X1), X2) active(splitAt(X1, X2)) -> splitAt(X1, active(X2)) active(pair(X1, X2)) -> pair(active(X1), X2) active(pair(X1, X2)) -> pair(X1, active(X2)) active(cons(X1, X2)) -> cons(active(X1), X2) active(afterNth(X1, X2)) -> afterNth(active(X1), X2) active(afterNth(X1, X2)) -> afterNth(X1, active(X2)) active(snd(X)) -> snd(active(X)) active(and(X1, X2)) -> and(active(X1), X2) active(fst(X)) -> fst(active(X)) active(head(X)) -> head(active(X)) active(natsFrom(X)) -> natsFrom(active(X)) active(s(X)) -> s(active(X)) active(sel(X1, X2)) -> sel(active(X1), X2) active(sel(X1, X2)) -> sel(X1, active(X2)) active(tail(X)) -> tail(active(X)) active(take(X1, X2)) -> take(active(X1), X2) active(take(X1, X2)) -> take(X1, active(X2)) U11(mark(X1), X2, X3, X4) -> mark(U11(X1, X2, X3, X4)) U12(mark(X1), X2) -> mark(U12(X1, X2)) splitAt(mark(X1), X2) -> mark(splitAt(X1, X2)) splitAt(X1, mark(X2)) -> mark(splitAt(X1, X2)) pair(mark(X1), X2) -> mark(pair(X1, X2)) pair(X1, mark(X2)) -> mark(pair(X1, X2)) cons(mark(X1), X2) -> mark(cons(X1, X2)) afterNth(mark(X1), X2) -> mark(afterNth(X1, X2)) afterNth(X1, mark(X2)) -> mark(afterNth(X1, X2)) snd(mark(X)) -> mark(snd(X)) and(mark(X1), X2) -> mark(and(X1, X2)) fst(mark(X)) -> mark(fst(X)) head(mark(X)) -> mark(head(X)) natsFrom(mark(X)) -> mark(natsFrom(X)) s(mark(X)) -> mark(s(X)) sel(mark(X1), X2) -> mark(sel(X1, X2)) sel(X1, mark(X2)) -> mark(sel(X1, X2)) tail(mark(X)) -> mark(tail(X)) take(mark(X1), X2) -> mark(take(X1, X2)) take(X1, mark(X2)) -> mark(take(X1, X2)) proper(U11(X1, X2, X3, X4)) -> U11(proper(X1), proper(X2), proper(X3), proper(X4)) proper(tt) -> ok(tt) proper(U12(X1, X2)) -> U12(proper(X1), proper(X2)) proper(splitAt(X1, X2)) -> splitAt(proper(X1), proper(X2)) proper(pair(X1, X2)) -> pair(proper(X1), proper(X2)) proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) proper(afterNth(X1, X2)) -> afterNth(proper(X1), proper(X2)) proper(snd(X)) -> snd(proper(X)) proper(and(X1, X2)) -> and(proper(X1), proper(X2)) proper(fst(X)) -> fst(proper(X)) proper(head(X)) -> head(proper(X)) proper(natsFrom(X)) -> natsFrom(proper(X)) proper(s(X)) -> s(proper(X)) proper(sel(X1, X2)) -> sel(proper(X1), proper(X2)) proper(0) -> ok(0) proper(nil) -> ok(nil) proper(tail(X)) -> tail(proper(X)) proper(take(X1, X2)) -> take(proper(X1), proper(X2)) U11(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U11(X1, X2, X3, X4)) U12(ok(X1), ok(X2)) -> ok(U12(X1, X2)) splitAt(ok(X1), ok(X2)) -> ok(splitAt(X1, X2)) pair(ok(X1), ok(X2)) -> ok(pair(X1, X2)) cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) afterNth(ok(X1), ok(X2)) -> ok(afterNth(X1, X2)) snd(ok(X)) -> ok(snd(X)) and(ok(X1), ok(X2)) -> ok(and(X1, X2)) fst(ok(X)) -> ok(fst(X)) head(ok(X)) -> ok(head(X)) natsFrom(ok(X)) -> ok(natsFrom(X)) s(ok(X)) -> ok(s(X)) sel(ok(X1), ok(X2)) -> ok(sel(X1, X2)) tail(ok(X)) -> ok(tail(X)) take(ok(X1), ok(X2)) -> ok(take(X1, X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) S is empty. Rewrite Strategy: FULL ---------------------------------------- (9) DecreasingLoopProof (LOWER BOUND(ID)) The following loop(s) give(s) rise to the lower bound Omega(n^1): The rewrite sequence take(ok(X1), ok(X2)) ->^+ ok(take(X1, X2)) gives rise to a decreasing loop by considering the right hand sides subterm at position [0]. The pumping substitution is [X1 / ok(X1), X2 / ok(X2)]. The result substitution is [ ]. ---------------------------------------- (10) Complex Obligation (BEST) ---------------------------------------- (11) Obligation: Proved the lower bound n^1 for the following obligation: The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). The TRS R consists of the following rules: active(U11(tt, N, X, XS)) -> mark(U12(splitAt(N, XS), X)) active(U12(pair(YS, ZS), X)) -> mark(pair(cons(X, YS), ZS)) active(afterNth(N, XS)) -> mark(snd(splitAt(N, XS))) active(and(tt, X)) -> mark(X) active(fst(pair(X, Y))) -> mark(X) active(head(cons(N, XS))) -> mark(N) active(natsFrom(N)) -> mark(cons(N, natsFrom(s(N)))) active(sel(N, XS)) -> mark(head(afterNth(N, XS))) active(snd(pair(X, Y))) -> mark(Y) active(splitAt(0, XS)) -> mark(pair(nil, XS)) active(splitAt(s(N), cons(X, XS))) -> mark(U11(tt, N, X, XS)) active(tail(cons(N, XS))) -> mark(XS) active(take(N, XS)) -> mark(fst(splitAt(N, XS))) active(U11(X1, X2, X3, X4)) -> U11(active(X1), X2, X3, X4) active(U12(X1, X2)) -> U12(active(X1), X2) active(splitAt(X1, X2)) -> splitAt(active(X1), X2) active(splitAt(X1, X2)) -> splitAt(X1, active(X2)) active(pair(X1, X2)) -> pair(active(X1), X2) active(pair(X1, X2)) -> pair(X1, active(X2)) active(cons(X1, X2)) -> cons(active(X1), X2) active(afterNth(X1, X2)) -> afterNth(active(X1), X2) active(afterNth(X1, X2)) -> afterNth(X1, active(X2)) active(snd(X)) -> snd(active(X)) active(and(X1, X2)) -> and(active(X1), X2) active(fst(X)) -> fst(active(X)) active(head(X)) -> head(active(X)) active(natsFrom(X)) -> natsFrom(active(X)) active(s(X)) -> s(active(X)) active(sel(X1, X2)) -> sel(active(X1), X2) active(sel(X1, X2)) -> sel(X1, active(X2)) active(tail(X)) -> tail(active(X)) active(take(X1, X2)) -> take(active(X1), X2) active(take(X1, X2)) -> take(X1, active(X2)) U11(mark(X1), X2, X3, X4) -> mark(U11(X1, X2, X3, X4)) U12(mark(X1), X2) -> mark(U12(X1, X2)) splitAt(mark(X1), X2) -> mark(splitAt(X1, X2)) splitAt(X1, mark(X2)) -> mark(splitAt(X1, X2)) pair(mark(X1), X2) -> mark(pair(X1, X2)) pair(X1, mark(X2)) -> mark(pair(X1, X2)) cons(mark(X1), X2) -> mark(cons(X1, X2)) afterNth(mark(X1), X2) -> mark(afterNth(X1, X2)) afterNth(X1, mark(X2)) -> mark(afterNth(X1, X2)) snd(mark(X)) -> mark(snd(X)) and(mark(X1), X2) -> mark(and(X1, X2)) fst(mark(X)) -> mark(fst(X)) head(mark(X)) -> mark(head(X)) natsFrom(mark(X)) -> mark(natsFrom(X)) s(mark(X)) -> mark(s(X)) sel(mark(X1), X2) -> mark(sel(X1, X2)) sel(X1, mark(X2)) -> mark(sel(X1, X2)) tail(mark(X)) -> mark(tail(X)) take(mark(X1), X2) -> mark(take(X1, X2)) take(X1, mark(X2)) -> mark(take(X1, X2)) proper(U11(X1, X2, X3, X4)) -> U11(proper(X1), proper(X2), proper(X3), proper(X4)) proper(tt) -> ok(tt) proper(U12(X1, X2)) -> U12(proper(X1), proper(X2)) proper(splitAt(X1, X2)) -> splitAt(proper(X1), proper(X2)) proper(pair(X1, X2)) -> pair(proper(X1), proper(X2)) proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) proper(afterNth(X1, X2)) -> afterNth(proper(X1), proper(X2)) proper(snd(X)) -> snd(proper(X)) proper(and(X1, X2)) -> and(proper(X1), proper(X2)) proper(fst(X)) -> fst(proper(X)) proper(head(X)) -> head(proper(X)) proper(natsFrom(X)) -> natsFrom(proper(X)) proper(s(X)) -> s(proper(X)) proper(sel(X1, X2)) -> sel(proper(X1), proper(X2)) proper(0) -> ok(0) proper(nil) -> ok(nil) proper(tail(X)) -> tail(proper(X)) proper(take(X1, X2)) -> take(proper(X1), proper(X2)) U11(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U11(X1, X2, X3, X4)) U12(ok(X1), ok(X2)) -> ok(U12(X1, X2)) splitAt(ok(X1), ok(X2)) -> ok(splitAt(X1, X2)) pair(ok(X1), ok(X2)) -> ok(pair(X1, X2)) cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) afterNth(ok(X1), ok(X2)) -> ok(afterNth(X1, X2)) snd(ok(X)) -> ok(snd(X)) and(ok(X1), ok(X2)) -> ok(and(X1, X2)) fst(ok(X)) -> ok(fst(X)) head(ok(X)) -> ok(head(X)) natsFrom(ok(X)) -> ok(natsFrom(X)) s(ok(X)) -> ok(s(X)) sel(ok(X1), ok(X2)) -> ok(sel(X1, X2)) tail(ok(X)) -> ok(tail(X)) take(ok(X1), ok(X2)) -> ok(take(X1, X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) S is empty. Rewrite Strategy: FULL ---------------------------------------- (12) LowerBoundPropagationProof (FINISHED) Propagated lower bound. ---------------------------------------- (13) BOUNDS(n^1, INF) ---------------------------------------- (14) Obligation: Analyzing the following TRS for decreasing loops: The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). The TRS R consists of the following rules: active(U11(tt, N, X, XS)) -> mark(U12(splitAt(N, XS), X)) active(U12(pair(YS, ZS), X)) -> mark(pair(cons(X, YS), ZS)) active(afterNth(N, XS)) -> mark(snd(splitAt(N, XS))) active(and(tt, X)) -> mark(X) active(fst(pair(X, Y))) -> mark(X) active(head(cons(N, XS))) -> mark(N) active(natsFrom(N)) -> mark(cons(N, natsFrom(s(N)))) active(sel(N, XS)) -> mark(head(afterNth(N, XS))) active(snd(pair(X, Y))) -> mark(Y) active(splitAt(0, XS)) -> mark(pair(nil, XS)) active(splitAt(s(N), cons(X, XS))) -> mark(U11(tt, N, X, XS)) active(tail(cons(N, XS))) -> mark(XS) active(take(N, XS)) -> mark(fst(splitAt(N, XS))) active(U11(X1, X2, X3, X4)) -> U11(active(X1), X2, X3, X4) active(U12(X1, X2)) -> U12(active(X1), X2) active(splitAt(X1, X2)) -> splitAt(active(X1), X2) active(splitAt(X1, X2)) -> splitAt(X1, active(X2)) active(pair(X1, X2)) -> pair(active(X1), X2) active(pair(X1, X2)) -> pair(X1, active(X2)) active(cons(X1, X2)) -> cons(active(X1), X2) active(afterNth(X1, X2)) -> afterNth(active(X1), X2) active(afterNth(X1, X2)) -> afterNth(X1, active(X2)) active(snd(X)) -> snd(active(X)) active(and(X1, X2)) -> and(active(X1), X2) active(fst(X)) -> fst(active(X)) active(head(X)) -> head(active(X)) active(natsFrom(X)) -> natsFrom(active(X)) active(s(X)) -> s(active(X)) active(sel(X1, X2)) -> sel(active(X1), X2) active(sel(X1, X2)) -> sel(X1, active(X2)) active(tail(X)) -> tail(active(X)) active(take(X1, X2)) -> take(active(X1), X2) active(take(X1, X2)) -> take(X1, active(X2)) U11(mark(X1), X2, X3, X4) -> mark(U11(X1, X2, X3, X4)) U12(mark(X1), X2) -> mark(U12(X1, X2)) splitAt(mark(X1), X2) -> mark(splitAt(X1, X2)) splitAt(X1, mark(X2)) -> mark(splitAt(X1, X2)) pair(mark(X1), X2) -> mark(pair(X1, X2)) pair(X1, mark(X2)) -> mark(pair(X1, X2)) cons(mark(X1), X2) -> mark(cons(X1, X2)) afterNth(mark(X1), X2) -> mark(afterNth(X1, X2)) afterNth(X1, mark(X2)) -> mark(afterNth(X1, X2)) snd(mark(X)) -> mark(snd(X)) and(mark(X1), X2) -> mark(and(X1, X2)) fst(mark(X)) -> mark(fst(X)) head(mark(X)) -> mark(head(X)) natsFrom(mark(X)) -> mark(natsFrom(X)) s(mark(X)) -> mark(s(X)) sel(mark(X1), X2) -> mark(sel(X1, X2)) sel(X1, mark(X2)) -> mark(sel(X1, X2)) tail(mark(X)) -> mark(tail(X)) take(mark(X1), X2) -> mark(take(X1, X2)) take(X1, mark(X2)) -> mark(take(X1, X2)) proper(U11(X1, X2, X3, X4)) -> U11(proper(X1), proper(X2), proper(X3), proper(X4)) proper(tt) -> ok(tt) proper(U12(X1, X2)) -> U12(proper(X1), proper(X2)) proper(splitAt(X1, X2)) -> splitAt(proper(X1), proper(X2)) proper(pair(X1, X2)) -> pair(proper(X1), proper(X2)) proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) proper(afterNth(X1, X2)) -> afterNth(proper(X1), proper(X2)) proper(snd(X)) -> snd(proper(X)) proper(and(X1, X2)) -> and(proper(X1), proper(X2)) proper(fst(X)) -> fst(proper(X)) proper(head(X)) -> head(proper(X)) proper(natsFrom(X)) -> natsFrom(proper(X)) proper(s(X)) -> s(proper(X)) proper(sel(X1, X2)) -> sel(proper(X1), proper(X2)) proper(0) -> ok(0) proper(nil) -> ok(nil) proper(tail(X)) -> tail(proper(X)) proper(take(X1, X2)) -> take(proper(X1), proper(X2)) U11(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U11(X1, X2, X3, X4)) U12(ok(X1), ok(X2)) -> ok(U12(X1, X2)) splitAt(ok(X1), ok(X2)) -> ok(splitAt(X1, X2)) pair(ok(X1), ok(X2)) -> ok(pair(X1, X2)) cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) afterNth(ok(X1), ok(X2)) -> ok(afterNth(X1, X2)) snd(ok(X)) -> ok(snd(X)) and(ok(X1), ok(X2)) -> ok(and(X1, X2)) fst(ok(X)) -> ok(fst(X)) head(ok(X)) -> ok(head(X)) natsFrom(ok(X)) -> ok(natsFrom(X)) s(ok(X)) -> ok(s(X)) sel(ok(X1), ok(X2)) -> ok(sel(X1, X2)) tail(ok(X)) -> ok(tail(X)) take(ok(X1), ok(X2)) -> ok(take(X1, X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) S is empty. Rewrite Strategy: FULL