36.78/10.90 WORST_CASE(Omega(n^1), O(n^1)) 36.78/10.91 proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml 36.78/10.91 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 36.78/10.91 36.78/10.91 36.78/10.91 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 36.78/10.91 36.78/10.91 (0) CpxTRS 36.78/10.91 (1) NestedDefinedSymbolProof [UPPER BOUND(ID), 0 ms] 36.78/10.91 (2) CpxTRS 36.78/10.91 (3) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] 36.78/10.91 (4) CpxTRS 36.78/10.91 (5) CpxTrsMatchBoundsTAProof [FINISHED, 430 ms] 36.78/10.91 (6) BOUNDS(1, n^1) 36.78/10.91 (7) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] 36.78/10.91 (8) TRS for Loop Detection 36.78/10.91 (9) DecreasingLoopProof [LOWER BOUND(ID), 0 ms] 36.78/10.91 (10) BEST 36.78/10.91 (11) proven lower bound 36.78/10.91 (12) LowerBoundPropagationProof [FINISHED, 0 ms] 36.78/10.91 (13) BOUNDS(n^1, INF) 36.78/10.91 (14) TRS for Loop Detection 36.78/10.91 36.78/10.91 36.78/10.91 ---------------------------------------- 36.78/10.91 36.78/10.91 (0) 36.78/10.91 Obligation: 36.78/10.91 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 36.78/10.91 36.78/10.91 36.78/10.91 The TRS R consists of the following rules: 36.78/10.91 36.78/10.91 active(U11(tt, N, X, XS)) -> mark(U12(splitAt(N, XS), X)) 36.78/10.92 active(U12(pair(YS, ZS), X)) -> mark(pair(cons(X, YS), ZS)) 36.78/10.92 active(afterNth(N, XS)) -> mark(snd(splitAt(N, XS))) 36.78/10.92 active(and(tt, X)) -> mark(X) 36.78/10.92 active(fst(pair(X, Y))) -> mark(X) 36.78/10.92 active(head(cons(N, XS))) -> mark(N) 36.78/10.92 active(natsFrom(N)) -> mark(cons(N, natsFrom(s(N)))) 36.78/10.92 active(sel(N, XS)) -> mark(head(afterNth(N, XS))) 36.78/10.92 active(snd(pair(X, Y))) -> mark(Y) 36.78/10.92 active(splitAt(0, XS)) -> mark(pair(nil, XS)) 36.78/10.92 active(splitAt(s(N), cons(X, XS))) -> mark(U11(tt, N, X, XS)) 36.78/10.92 active(tail(cons(N, XS))) -> mark(XS) 36.78/10.92 active(take(N, XS)) -> mark(fst(splitAt(N, XS))) 36.78/10.92 active(U11(X1, X2, X3, X4)) -> U11(active(X1), X2, X3, X4) 36.78/10.92 active(U12(X1, X2)) -> U12(active(X1), X2) 36.78/10.92 active(splitAt(X1, X2)) -> splitAt(active(X1), X2) 36.78/10.92 active(splitAt(X1, X2)) -> splitAt(X1, active(X2)) 36.78/10.92 active(pair(X1, X2)) -> pair(active(X1), X2) 36.78/10.92 active(pair(X1, X2)) -> pair(X1, active(X2)) 36.78/10.92 active(cons(X1, X2)) -> cons(active(X1), X2) 36.78/10.92 active(afterNth(X1, X2)) -> afterNth(active(X1), X2) 36.78/10.92 active(afterNth(X1, X2)) -> afterNth(X1, active(X2)) 36.78/10.92 active(snd(X)) -> snd(active(X)) 36.78/10.92 active(and(X1, X2)) -> and(active(X1), X2) 36.78/10.92 active(fst(X)) -> fst(active(X)) 36.78/10.92 active(head(X)) -> head(active(X)) 36.78/10.92 active(natsFrom(X)) -> natsFrom(active(X)) 36.78/10.92 active(s(X)) -> s(active(X)) 36.78/10.92 active(sel(X1, X2)) -> sel(active(X1), X2) 36.78/10.92 active(sel(X1, X2)) -> sel(X1, active(X2)) 36.78/10.92 active(tail(X)) -> tail(active(X)) 36.78/10.92 active(take(X1, X2)) -> take(active(X1), X2) 36.78/10.92 active(take(X1, X2)) -> take(X1, active(X2)) 36.78/10.92 U11(mark(X1), X2, X3, X4) -> mark(U11(X1, X2, X3, X4)) 36.78/10.92 U12(mark(X1), X2) -> mark(U12(X1, X2)) 36.78/10.92 splitAt(mark(X1), X2) -> mark(splitAt(X1, X2)) 36.78/10.92 splitAt(X1, mark(X2)) -> mark(splitAt(X1, X2)) 36.78/10.92 pair(mark(X1), X2) -> mark(pair(X1, X2)) 36.78/10.92 pair(X1, mark(X2)) -> mark(pair(X1, X2)) 36.78/10.92 cons(mark(X1), X2) -> mark(cons(X1, X2)) 36.78/10.92 afterNth(mark(X1), X2) -> mark(afterNth(X1, X2)) 36.78/10.92 afterNth(X1, mark(X2)) -> mark(afterNth(X1, X2)) 36.78/10.92 snd(mark(X)) -> mark(snd(X)) 36.78/10.92 and(mark(X1), X2) -> mark(and(X1, X2)) 36.78/10.92 fst(mark(X)) -> mark(fst(X)) 36.78/10.92 head(mark(X)) -> mark(head(X)) 36.78/10.92 natsFrom(mark(X)) -> mark(natsFrom(X)) 36.78/10.92 s(mark(X)) -> mark(s(X)) 36.78/10.92 sel(mark(X1), X2) -> mark(sel(X1, X2)) 36.78/10.92 sel(X1, mark(X2)) -> mark(sel(X1, X2)) 36.78/10.92 tail(mark(X)) -> mark(tail(X)) 36.78/10.92 take(mark(X1), X2) -> mark(take(X1, X2)) 36.78/10.92 take(X1, mark(X2)) -> mark(take(X1, X2)) 36.78/10.92 proper(U11(X1, X2, X3, X4)) -> U11(proper(X1), proper(X2), proper(X3), proper(X4)) 36.78/10.92 proper(tt) -> ok(tt) 36.78/10.92 proper(U12(X1, X2)) -> U12(proper(X1), proper(X2)) 36.78/10.92 proper(splitAt(X1, X2)) -> splitAt(proper(X1), proper(X2)) 36.78/10.92 proper(pair(X1, X2)) -> pair(proper(X1), proper(X2)) 36.78/10.92 proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) 36.78/10.92 proper(afterNth(X1, X2)) -> afterNth(proper(X1), proper(X2)) 36.78/10.92 proper(snd(X)) -> snd(proper(X)) 36.78/10.92 proper(and(X1, X2)) -> and(proper(X1), proper(X2)) 36.78/10.92 proper(fst(X)) -> fst(proper(X)) 36.78/10.92 proper(head(X)) -> head(proper(X)) 36.78/10.92 proper(natsFrom(X)) -> natsFrom(proper(X)) 36.78/10.92 proper(s(X)) -> s(proper(X)) 36.78/10.92 proper(sel(X1, X2)) -> sel(proper(X1), proper(X2)) 36.78/10.92 proper(0) -> ok(0) 36.78/10.92 proper(nil) -> ok(nil) 36.78/10.92 proper(tail(X)) -> tail(proper(X)) 36.78/10.92 proper(take(X1, X2)) -> take(proper(X1), proper(X2)) 36.78/10.92 U11(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U11(X1, X2, X3, X4)) 36.78/10.92 U12(ok(X1), ok(X2)) -> ok(U12(X1, X2)) 36.78/10.92 splitAt(ok(X1), ok(X2)) -> ok(splitAt(X1, X2)) 36.78/10.92 pair(ok(X1), ok(X2)) -> ok(pair(X1, X2)) 36.78/10.92 cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) 36.78/10.92 afterNth(ok(X1), ok(X2)) -> ok(afterNth(X1, X2)) 36.78/10.92 snd(ok(X)) -> ok(snd(X)) 36.78/10.92 and(ok(X1), ok(X2)) -> ok(and(X1, X2)) 36.78/10.92 fst(ok(X)) -> ok(fst(X)) 36.78/10.92 head(ok(X)) -> ok(head(X)) 36.78/10.92 natsFrom(ok(X)) -> ok(natsFrom(X)) 36.78/10.92 s(ok(X)) -> ok(s(X)) 36.78/10.92 sel(ok(X1), ok(X2)) -> ok(sel(X1, X2)) 36.78/10.92 tail(ok(X)) -> ok(tail(X)) 36.78/10.92 take(ok(X1), ok(X2)) -> ok(take(X1, X2)) 36.78/10.92 top(mark(X)) -> top(proper(X)) 36.78/10.92 top(ok(X)) -> top(active(X)) 36.78/10.92 36.78/10.92 S is empty. 36.78/10.92 Rewrite Strategy: FULL 36.78/10.92 ---------------------------------------- 36.78/10.92 36.78/10.92 (1) NestedDefinedSymbolProof (UPPER BOUND(ID)) 36.78/10.92 The following defined symbols can occur below the 0th argument of top: proper, active 36.78/10.92 The following defined symbols can occur below the 0th argument of proper: proper, active 36.78/10.92 The following defined symbols can occur below the 0th argument of active: proper, active 36.78/10.92 36.78/10.92 Hence, the left-hand sides of the following rules are not basic-reachable and can be removed: 36.78/10.92 active(U11(tt, N, X, XS)) -> mark(U12(splitAt(N, XS), X)) 36.78/10.92 active(U12(pair(YS, ZS), X)) -> mark(pair(cons(X, YS), ZS)) 36.78/10.92 active(afterNth(N, XS)) -> mark(snd(splitAt(N, XS))) 36.78/10.92 active(and(tt, X)) -> mark(X) 36.78/10.92 active(fst(pair(X, Y))) -> mark(X) 36.78/10.92 active(head(cons(N, XS))) -> mark(N) 36.78/10.92 active(natsFrom(N)) -> mark(cons(N, natsFrom(s(N)))) 36.78/10.92 active(sel(N, XS)) -> mark(head(afterNth(N, XS))) 36.78/10.92 active(snd(pair(X, Y))) -> mark(Y) 36.78/10.92 active(splitAt(0, XS)) -> mark(pair(nil, XS)) 36.78/10.92 active(splitAt(s(N), cons(X, XS))) -> mark(U11(tt, N, X, XS)) 36.78/10.92 active(tail(cons(N, XS))) -> mark(XS) 36.78/10.92 active(take(N, XS)) -> mark(fst(splitAt(N, XS))) 36.78/10.92 active(U11(X1, X2, X3, X4)) -> U11(active(X1), X2, X3, X4) 36.78/10.92 active(U12(X1, X2)) -> U12(active(X1), X2) 36.78/10.92 active(splitAt(X1, X2)) -> splitAt(active(X1), X2) 36.78/10.92 active(splitAt(X1, X2)) -> splitAt(X1, active(X2)) 36.78/10.92 active(pair(X1, X2)) -> pair(active(X1), X2) 36.78/10.92 active(pair(X1, X2)) -> pair(X1, active(X2)) 36.78/10.92 active(cons(X1, X2)) -> cons(active(X1), X2) 36.78/10.92 active(afterNth(X1, X2)) -> afterNth(active(X1), X2) 36.78/10.92 active(afterNth(X1, X2)) -> afterNth(X1, active(X2)) 36.78/10.92 active(snd(X)) -> snd(active(X)) 36.78/10.92 active(and(X1, X2)) -> and(active(X1), X2) 36.78/10.92 active(fst(X)) -> fst(active(X)) 36.78/10.92 active(head(X)) -> head(active(X)) 36.78/10.92 active(natsFrom(X)) -> natsFrom(active(X)) 36.78/10.92 active(s(X)) -> s(active(X)) 36.78/10.92 active(sel(X1, X2)) -> sel(active(X1), X2) 36.78/10.92 active(sel(X1, X2)) -> sel(X1, active(X2)) 36.78/10.92 active(tail(X)) -> tail(active(X)) 36.78/10.92 active(take(X1, X2)) -> take(active(X1), X2) 36.78/10.92 active(take(X1, X2)) -> take(X1, active(X2)) 36.78/10.92 proper(U11(X1, X2, X3, X4)) -> U11(proper(X1), proper(X2), proper(X3), proper(X4)) 36.78/10.92 proper(U12(X1, X2)) -> U12(proper(X1), proper(X2)) 36.78/10.92 proper(splitAt(X1, X2)) -> splitAt(proper(X1), proper(X2)) 36.78/10.92 proper(pair(X1, X2)) -> pair(proper(X1), proper(X2)) 36.78/10.92 proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) 36.78/10.92 proper(afterNth(X1, X2)) -> afterNth(proper(X1), proper(X2)) 36.78/10.92 proper(snd(X)) -> snd(proper(X)) 36.78/10.92 proper(and(X1, X2)) -> and(proper(X1), proper(X2)) 36.78/10.92 proper(fst(X)) -> fst(proper(X)) 36.78/10.92 proper(head(X)) -> head(proper(X)) 36.78/10.92 proper(natsFrom(X)) -> natsFrom(proper(X)) 36.78/10.92 proper(s(X)) -> s(proper(X)) 36.78/10.92 proper(sel(X1, X2)) -> sel(proper(X1), proper(X2)) 36.78/10.92 proper(tail(X)) -> tail(proper(X)) 36.78/10.92 proper(take(X1, X2)) -> take(proper(X1), proper(X2)) 36.78/10.92 36.78/10.92 ---------------------------------------- 36.78/10.92 36.78/10.92 (2) 36.78/10.92 Obligation: 36.78/10.92 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^1). 36.78/10.92 36.78/10.92 36.78/10.92 The TRS R consists of the following rules: 36.78/10.92 36.78/10.92 U11(mark(X1), X2, X3, X4) -> mark(U11(X1, X2, X3, X4)) 36.78/10.92 U12(mark(X1), X2) -> mark(U12(X1, X2)) 36.78/10.92 splitAt(mark(X1), X2) -> mark(splitAt(X1, X2)) 36.78/10.92 splitAt(X1, mark(X2)) -> mark(splitAt(X1, X2)) 36.78/10.92 pair(mark(X1), X2) -> mark(pair(X1, X2)) 36.78/10.92 pair(X1, mark(X2)) -> mark(pair(X1, X2)) 36.78/10.92 cons(mark(X1), X2) -> mark(cons(X1, X2)) 36.78/10.92 afterNth(mark(X1), X2) -> mark(afterNth(X1, X2)) 36.78/10.92 afterNth(X1, mark(X2)) -> mark(afterNth(X1, X2)) 36.78/10.92 snd(mark(X)) -> mark(snd(X)) 36.78/10.92 and(mark(X1), X2) -> mark(and(X1, X2)) 36.78/10.92 fst(mark(X)) -> mark(fst(X)) 36.78/10.92 head(mark(X)) -> mark(head(X)) 36.78/10.92 natsFrom(mark(X)) -> mark(natsFrom(X)) 36.78/10.92 s(mark(X)) -> mark(s(X)) 36.78/10.92 sel(mark(X1), X2) -> mark(sel(X1, X2)) 36.78/10.92 sel(X1, mark(X2)) -> mark(sel(X1, X2)) 36.78/10.92 tail(mark(X)) -> mark(tail(X)) 36.78/10.92 take(mark(X1), X2) -> mark(take(X1, X2)) 36.78/10.92 take(X1, mark(X2)) -> mark(take(X1, X2)) 36.78/10.92 proper(tt) -> ok(tt) 36.78/10.92 proper(0) -> ok(0) 36.78/10.92 proper(nil) -> ok(nil) 36.78/10.92 U11(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U11(X1, X2, X3, X4)) 36.78/10.92 U12(ok(X1), ok(X2)) -> ok(U12(X1, X2)) 36.78/10.92 splitAt(ok(X1), ok(X2)) -> ok(splitAt(X1, X2)) 36.78/10.92 pair(ok(X1), ok(X2)) -> ok(pair(X1, X2)) 36.78/10.92 cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) 36.78/10.92 afterNth(ok(X1), ok(X2)) -> ok(afterNth(X1, X2)) 36.78/10.92 snd(ok(X)) -> ok(snd(X)) 36.78/10.92 and(ok(X1), ok(X2)) -> ok(and(X1, X2)) 36.78/10.92 fst(ok(X)) -> ok(fst(X)) 36.78/10.92 head(ok(X)) -> ok(head(X)) 36.78/10.92 natsFrom(ok(X)) -> ok(natsFrom(X)) 36.78/10.92 s(ok(X)) -> ok(s(X)) 36.78/10.92 sel(ok(X1), ok(X2)) -> ok(sel(X1, X2)) 36.78/10.92 tail(ok(X)) -> ok(tail(X)) 36.78/10.92 take(ok(X1), ok(X2)) -> ok(take(X1, X2)) 36.78/10.92 top(mark(X)) -> top(proper(X)) 36.78/10.92 top(ok(X)) -> top(active(X)) 36.78/10.92 36.78/10.92 S is empty. 36.78/10.92 Rewrite Strategy: FULL 36.78/10.92 ---------------------------------------- 36.78/10.92 36.78/10.92 (3) RelTrsToTrsProof (UPPER BOUND(ID)) 36.78/10.92 transformed relative TRS to TRS 36.78/10.92 ---------------------------------------- 36.78/10.92 36.78/10.92 (4) 36.78/10.92 Obligation: 36.78/10.92 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^1). 36.78/10.92 36.78/10.92 36.78/10.92 The TRS R consists of the following rules: 36.78/10.92 36.78/10.92 U11(mark(X1), X2, X3, X4) -> mark(U11(X1, X2, X3, X4)) 36.78/10.92 U12(mark(X1), X2) -> mark(U12(X1, X2)) 36.78/10.92 splitAt(mark(X1), X2) -> mark(splitAt(X1, X2)) 36.78/10.92 splitAt(X1, mark(X2)) -> mark(splitAt(X1, X2)) 36.78/10.92 pair(mark(X1), X2) -> mark(pair(X1, X2)) 36.78/10.92 pair(X1, mark(X2)) -> mark(pair(X1, X2)) 36.78/10.92 cons(mark(X1), X2) -> mark(cons(X1, X2)) 36.78/10.92 afterNth(mark(X1), X2) -> mark(afterNth(X1, X2)) 36.78/10.92 afterNth(X1, mark(X2)) -> mark(afterNth(X1, X2)) 36.78/10.92 snd(mark(X)) -> mark(snd(X)) 36.78/10.92 and(mark(X1), X2) -> mark(and(X1, X2)) 36.78/10.92 fst(mark(X)) -> mark(fst(X)) 36.78/10.92 head(mark(X)) -> mark(head(X)) 36.78/10.92 natsFrom(mark(X)) -> mark(natsFrom(X)) 36.78/10.92 s(mark(X)) -> mark(s(X)) 36.78/10.92 sel(mark(X1), X2) -> mark(sel(X1, X2)) 36.78/10.92 sel(X1, mark(X2)) -> mark(sel(X1, X2)) 36.78/10.92 tail(mark(X)) -> mark(tail(X)) 36.78/10.92 take(mark(X1), X2) -> mark(take(X1, X2)) 36.78/10.92 take(X1, mark(X2)) -> mark(take(X1, X2)) 36.78/10.92 proper(tt) -> ok(tt) 36.78/10.92 proper(0) -> ok(0) 36.78/10.92 proper(nil) -> ok(nil) 36.78/10.92 U11(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U11(X1, X2, X3, X4)) 36.78/10.92 U12(ok(X1), ok(X2)) -> ok(U12(X1, X2)) 36.78/10.92 splitAt(ok(X1), ok(X2)) -> ok(splitAt(X1, X2)) 36.78/10.92 pair(ok(X1), ok(X2)) -> ok(pair(X1, X2)) 36.78/10.92 cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) 36.78/10.92 afterNth(ok(X1), ok(X2)) -> ok(afterNth(X1, X2)) 36.78/10.92 snd(ok(X)) -> ok(snd(X)) 36.78/10.92 and(ok(X1), ok(X2)) -> ok(and(X1, X2)) 36.78/10.92 fst(ok(X)) -> ok(fst(X)) 36.78/10.92 head(ok(X)) -> ok(head(X)) 36.78/10.92 natsFrom(ok(X)) -> ok(natsFrom(X)) 36.78/10.92 s(ok(X)) -> ok(s(X)) 36.78/10.92 sel(ok(X1), ok(X2)) -> ok(sel(X1, X2)) 36.78/10.92 tail(ok(X)) -> ok(tail(X)) 36.78/10.92 take(ok(X1), ok(X2)) -> ok(take(X1, X2)) 36.78/10.92 top(mark(X)) -> top(proper(X)) 36.78/10.92 top(ok(X)) -> top(active(X)) 36.78/10.92 36.78/10.92 S is empty. 36.78/10.92 Rewrite Strategy: FULL 36.78/10.92 ---------------------------------------- 36.78/10.92 36.78/10.92 (5) CpxTrsMatchBoundsTAProof (FINISHED) 36.78/10.92 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. 36.78/10.92 36.78/10.92 The compatible tree automaton used to show the Match-Boundedness (for constructor-based start-terms) is represented by: 36.78/10.92 final states : [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17] 36.78/10.92 transitions: 36.78/10.92 mark0(0) -> 0 36.78/10.92 tt0() -> 0 36.78/10.92 ok0(0) -> 0 36.78/10.92 00() -> 0 36.78/10.92 nil0() -> 0 36.78/10.92 active0(0) -> 0 36.78/10.92 U110(0, 0, 0, 0) -> 1 36.78/10.92 U120(0, 0) -> 2 36.78/10.92 splitAt0(0, 0) -> 3 36.78/10.92 pair0(0, 0) -> 4 36.78/10.92 cons0(0, 0) -> 5 36.78/10.92 afterNth0(0, 0) -> 6 36.78/10.92 snd0(0) -> 7 36.78/10.92 and0(0, 0) -> 8 36.78/10.92 fst0(0) -> 9 36.78/10.92 head0(0) -> 10 36.78/10.92 natsFrom0(0) -> 11 36.78/10.92 s0(0) -> 12 36.78/10.92 sel0(0, 0) -> 13 36.78/10.92 tail0(0) -> 14 36.78/10.92 take0(0, 0) -> 15 36.78/10.92 proper0(0) -> 16 36.78/10.92 top0(0) -> 17 36.78/10.92 U111(0, 0, 0, 0) -> 18 36.78/10.92 mark1(18) -> 1 36.78/10.92 U121(0, 0) -> 19 36.78/10.92 mark1(19) -> 2 36.78/10.92 splitAt1(0, 0) -> 20 36.78/10.92 mark1(20) -> 3 36.78/10.92 pair1(0, 0) -> 21 36.78/10.92 mark1(21) -> 4 36.78/10.92 cons1(0, 0) -> 22 36.78/10.92 mark1(22) -> 5 36.78/10.92 afterNth1(0, 0) -> 23 36.78/10.92 mark1(23) -> 6 36.78/10.92 snd1(0) -> 24 36.78/10.92 mark1(24) -> 7 36.78/10.92 and1(0, 0) -> 25 36.78/10.92 mark1(25) -> 8 36.78/10.92 fst1(0) -> 26 36.78/10.92 mark1(26) -> 9 36.78/10.92 head1(0) -> 27 36.78/10.92 mark1(27) -> 10 36.78/10.92 natsFrom1(0) -> 28 36.78/10.92 mark1(28) -> 11 36.78/10.92 s1(0) -> 29 36.78/10.92 mark1(29) -> 12 36.78/10.92 sel1(0, 0) -> 30 36.78/10.92 mark1(30) -> 13 36.78/10.92 tail1(0) -> 31 36.78/10.92 mark1(31) -> 14 36.78/10.92 take1(0, 0) -> 32 36.78/10.92 mark1(32) -> 15 36.78/10.92 tt1() -> 33 36.78/10.92 ok1(33) -> 16 36.78/10.92 01() -> 34 36.78/10.92 ok1(34) -> 16 36.78/10.92 nil1() -> 35 36.78/10.92 ok1(35) -> 16 36.78/10.92 U111(0, 0, 0, 0) -> 36 36.78/10.92 ok1(36) -> 1 36.78/10.92 U121(0, 0) -> 37 36.78/10.92 ok1(37) -> 2 36.78/10.92 splitAt1(0, 0) -> 38 36.78/10.92 ok1(38) -> 3 36.78/10.92 pair1(0, 0) -> 39 36.78/10.92 ok1(39) -> 4 36.78/10.92 cons1(0, 0) -> 40 36.78/10.92 ok1(40) -> 5 36.78/10.92 afterNth1(0, 0) -> 41 36.78/10.92 ok1(41) -> 6 36.78/10.92 snd1(0) -> 42 36.78/10.92 ok1(42) -> 7 36.78/10.92 and1(0, 0) -> 43 36.78/10.92 ok1(43) -> 8 36.78/10.92 fst1(0) -> 44 36.78/10.92 ok1(44) -> 9 36.78/10.92 head1(0) -> 45 36.78/10.92 ok1(45) -> 10 36.78/10.92 natsFrom1(0) -> 46 36.78/10.92 ok1(46) -> 11 36.78/10.92 s1(0) -> 47 36.78/10.92 ok1(47) -> 12 36.78/10.92 sel1(0, 0) -> 48 36.78/10.92 ok1(48) -> 13 36.78/10.92 tail1(0) -> 49 36.78/10.92 ok1(49) -> 14 36.78/10.92 take1(0, 0) -> 50 36.78/10.92 ok1(50) -> 15 36.78/10.92 proper1(0) -> 51 36.78/10.92 top1(51) -> 17 36.78/10.92 active1(0) -> 52 36.78/10.92 top1(52) -> 17 36.78/10.92 mark1(18) -> 18 36.78/10.92 mark1(18) -> 36 36.78/10.92 mark1(19) -> 19 36.78/10.92 mark1(19) -> 37 36.78/10.92 mark1(20) -> 20 36.78/10.92 mark1(20) -> 38 36.78/10.92 mark1(21) -> 21 36.78/10.92 mark1(21) -> 39 36.78/10.92 mark1(22) -> 22 36.78/10.92 mark1(22) -> 40 36.78/10.92 mark1(23) -> 23 36.78/10.92 mark1(23) -> 41 36.78/10.92 mark1(24) -> 24 36.78/10.92 mark1(24) -> 42 36.78/10.92 mark1(25) -> 25 36.78/10.92 mark1(25) -> 43 36.78/10.92 mark1(26) -> 26 36.78/10.92 mark1(26) -> 44 36.78/10.92 mark1(27) -> 27 36.78/10.92 mark1(27) -> 45 36.78/10.92 mark1(28) -> 28 36.78/10.92 mark1(28) -> 46 36.78/10.92 mark1(29) -> 29 36.78/10.92 mark1(29) -> 47 36.78/10.92 mark1(30) -> 30 36.78/10.92 mark1(30) -> 48 36.78/10.92 mark1(31) -> 31 36.78/10.92 mark1(31) -> 49 36.78/10.92 mark1(32) -> 32 36.78/10.92 mark1(32) -> 50 36.78/10.92 ok1(33) -> 51 36.78/10.92 ok1(34) -> 51 36.78/10.92 ok1(35) -> 51 36.78/10.92 ok1(36) -> 18 36.78/10.92 ok1(36) -> 36 36.78/10.92 ok1(37) -> 19 36.78/10.92 ok1(37) -> 37 36.78/10.92 ok1(38) -> 20 36.78/10.92 ok1(38) -> 38 36.78/10.92 ok1(39) -> 21 36.78/10.92 ok1(39) -> 39 36.78/10.92 ok1(40) -> 22 36.78/10.92 ok1(40) -> 40 36.78/10.92 ok1(41) -> 23 36.78/10.92 ok1(41) -> 41 36.78/10.92 ok1(42) -> 24 36.78/10.92 ok1(42) -> 42 36.78/10.92 ok1(43) -> 25 36.78/10.92 ok1(43) -> 43 36.78/10.92 ok1(44) -> 26 36.78/10.92 ok1(44) -> 44 36.78/10.92 ok1(45) -> 27 36.78/10.92 ok1(45) -> 45 36.78/10.92 ok1(46) -> 28 36.78/10.92 ok1(46) -> 46 36.78/10.92 ok1(47) -> 29 36.78/10.92 ok1(47) -> 47 36.78/10.92 ok1(48) -> 30 36.78/10.92 ok1(48) -> 48 36.78/10.92 ok1(49) -> 31 36.78/10.92 ok1(49) -> 49 36.78/10.92 ok1(50) -> 32 36.78/10.92 ok1(50) -> 50 36.78/10.92 active2(33) -> 53 36.78/10.92 top2(53) -> 17 36.78/10.92 active2(34) -> 53 36.78/10.92 active2(35) -> 53 36.78/10.92 36.78/10.92 ---------------------------------------- 36.78/10.92 36.78/10.92 (6) 36.78/10.92 BOUNDS(1, n^1) 36.78/10.92 36.78/10.92 ---------------------------------------- 36.78/10.92 36.78/10.92 (7) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) 36.78/10.92 Transformed a relative TRS into a decreasing-loop problem. 36.78/10.92 ---------------------------------------- 36.78/10.92 36.78/10.92 (8) 36.78/10.92 Obligation: 36.78/10.92 Analyzing the following TRS for decreasing loops: 36.78/10.92 36.78/10.92 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 36.78/10.92 36.78/10.92 36.78/10.92 The TRS R consists of the following rules: 36.78/10.92 36.78/10.92 active(U11(tt, N, X, XS)) -> mark(U12(splitAt(N, XS), X)) 36.78/10.92 active(U12(pair(YS, ZS), X)) -> mark(pair(cons(X, YS), ZS)) 36.78/10.92 active(afterNth(N, XS)) -> mark(snd(splitAt(N, XS))) 36.78/10.92 active(and(tt, X)) -> mark(X) 36.78/10.92 active(fst(pair(X, Y))) -> mark(X) 36.78/10.92 active(head(cons(N, XS))) -> mark(N) 36.78/10.92 active(natsFrom(N)) -> mark(cons(N, natsFrom(s(N)))) 36.78/10.92 active(sel(N, XS)) -> mark(head(afterNth(N, XS))) 36.78/10.92 active(snd(pair(X, Y))) -> mark(Y) 36.78/10.92 active(splitAt(0, XS)) -> mark(pair(nil, XS)) 36.78/10.92 active(splitAt(s(N), cons(X, XS))) -> mark(U11(tt, N, X, XS)) 36.78/10.92 active(tail(cons(N, XS))) -> mark(XS) 36.78/10.92 active(take(N, XS)) -> mark(fst(splitAt(N, XS))) 36.78/10.92 active(U11(X1, X2, X3, X4)) -> U11(active(X1), X2, X3, X4) 36.78/10.92 active(U12(X1, X2)) -> U12(active(X1), X2) 36.78/10.92 active(splitAt(X1, X2)) -> splitAt(active(X1), X2) 36.78/10.92 active(splitAt(X1, X2)) -> splitAt(X1, active(X2)) 36.78/10.92 active(pair(X1, X2)) -> pair(active(X1), X2) 36.78/10.92 active(pair(X1, X2)) -> pair(X1, active(X2)) 36.78/10.92 active(cons(X1, X2)) -> cons(active(X1), X2) 36.78/10.92 active(afterNth(X1, X2)) -> afterNth(active(X1), X2) 36.78/10.92 active(afterNth(X1, X2)) -> afterNth(X1, active(X2)) 36.78/10.92 active(snd(X)) -> snd(active(X)) 36.78/10.92 active(and(X1, X2)) -> and(active(X1), X2) 36.78/10.92 active(fst(X)) -> fst(active(X)) 36.78/10.92 active(head(X)) -> head(active(X)) 36.78/10.92 active(natsFrom(X)) -> natsFrom(active(X)) 36.78/10.92 active(s(X)) -> s(active(X)) 36.78/10.92 active(sel(X1, X2)) -> sel(active(X1), X2) 36.78/10.92 active(sel(X1, X2)) -> sel(X1, active(X2)) 36.78/10.92 active(tail(X)) -> tail(active(X)) 36.78/10.92 active(take(X1, X2)) -> take(active(X1), X2) 36.78/10.92 active(take(X1, X2)) -> take(X1, active(X2)) 36.78/10.92 U11(mark(X1), X2, X3, X4) -> mark(U11(X1, X2, X3, X4)) 36.78/10.92 U12(mark(X1), X2) -> mark(U12(X1, X2)) 36.78/10.92 splitAt(mark(X1), X2) -> mark(splitAt(X1, X2)) 36.78/10.92 splitAt(X1, mark(X2)) -> mark(splitAt(X1, X2)) 36.78/10.92 pair(mark(X1), X2) -> mark(pair(X1, X2)) 36.78/10.92 pair(X1, mark(X2)) -> mark(pair(X1, X2)) 36.78/10.92 cons(mark(X1), X2) -> mark(cons(X1, X2)) 36.78/10.92 afterNth(mark(X1), X2) -> mark(afterNth(X1, X2)) 36.78/10.92 afterNth(X1, mark(X2)) -> mark(afterNth(X1, X2)) 36.78/10.92 snd(mark(X)) -> mark(snd(X)) 36.78/10.92 and(mark(X1), X2) -> mark(and(X1, X2)) 36.78/10.92 fst(mark(X)) -> mark(fst(X)) 36.78/10.92 head(mark(X)) -> mark(head(X)) 36.78/10.92 natsFrom(mark(X)) -> mark(natsFrom(X)) 36.78/10.92 s(mark(X)) -> mark(s(X)) 36.78/10.92 sel(mark(X1), X2) -> mark(sel(X1, X2)) 36.78/10.92 sel(X1, mark(X2)) -> mark(sel(X1, X2)) 36.78/10.92 tail(mark(X)) -> mark(tail(X)) 36.78/10.92 take(mark(X1), X2) -> mark(take(X1, X2)) 36.78/10.92 take(X1, mark(X2)) -> mark(take(X1, X2)) 36.78/10.92 proper(U11(X1, X2, X3, X4)) -> U11(proper(X1), proper(X2), proper(X3), proper(X4)) 36.78/10.92 proper(tt) -> ok(tt) 36.78/10.92 proper(U12(X1, X2)) -> U12(proper(X1), proper(X2)) 36.78/10.92 proper(splitAt(X1, X2)) -> splitAt(proper(X1), proper(X2)) 36.78/10.92 proper(pair(X1, X2)) -> pair(proper(X1), proper(X2)) 36.78/10.92 proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) 36.78/10.92 proper(afterNth(X1, X2)) -> afterNth(proper(X1), proper(X2)) 36.78/10.92 proper(snd(X)) -> snd(proper(X)) 36.78/10.92 proper(and(X1, X2)) -> and(proper(X1), proper(X2)) 36.78/10.92 proper(fst(X)) -> fst(proper(X)) 36.78/10.92 proper(head(X)) -> head(proper(X)) 36.78/10.92 proper(natsFrom(X)) -> natsFrom(proper(X)) 36.78/10.92 proper(s(X)) -> s(proper(X)) 36.78/10.92 proper(sel(X1, X2)) -> sel(proper(X1), proper(X2)) 36.78/10.92 proper(0) -> ok(0) 36.78/10.92 proper(nil) -> ok(nil) 36.78/10.92 proper(tail(X)) -> tail(proper(X)) 36.78/10.92 proper(take(X1, X2)) -> take(proper(X1), proper(X2)) 36.78/10.92 U11(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U11(X1, X2, X3, X4)) 36.78/10.92 U12(ok(X1), ok(X2)) -> ok(U12(X1, X2)) 36.78/10.92 splitAt(ok(X1), ok(X2)) -> ok(splitAt(X1, X2)) 36.78/10.92 pair(ok(X1), ok(X2)) -> ok(pair(X1, X2)) 36.78/10.92 cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) 36.78/10.92 afterNth(ok(X1), ok(X2)) -> ok(afterNth(X1, X2)) 36.78/10.92 snd(ok(X)) -> ok(snd(X)) 36.78/10.92 and(ok(X1), ok(X2)) -> ok(and(X1, X2)) 36.78/10.92 fst(ok(X)) -> ok(fst(X)) 36.78/10.92 head(ok(X)) -> ok(head(X)) 36.78/10.92 natsFrom(ok(X)) -> ok(natsFrom(X)) 36.78/10.92 s(ok(X)) -> ok(s(X)) 36.78/10.92 sel(ok(X1), ok(X2)) -> ok(sel(X1, X2)) 36.78/10.92 tail(ok(X)) -> ok(tail(X)) 36.78/10.92 take(ok(X1), ok(X2)) -> ok(take(X1, X2)) 36.78/10.92 top(mark(X)) -> top(proper(X)) 36.78/10.92 top(ok(X)) -> top(active(X)) 36.78/10.92 36.78/10.92 S is empty. 36.78/10.92 Rewrite Strategy: FULL 36.78/10.92 ---------------------------------------- 36.78/10.92 36.78/10.92 (9) DecreasingLoopProof (LOWER BOUND(ID)) 36.78/10.92 The following loop(s) give(s) rise to the lower bound Omega(n^1): 36.78/10.92 36.78/10.92 The rewrite sequence 36.78/10.92 36.78/10.92 take(ok(X1), ok(X2)) ->^+ ok(take(X1, X2)) 36.78/10.92 36.78/10.92 gives rise to a decreasing loop by considering the right hand sides subterm at position [0]. 36.78/10.92 36.78/10.92 The pumping substitution is [X1 / ok(X1), X2 / ok(X2)]. 36.78/10.92 36.78/10.92 The result substitution is [ ]. 36.78/10.92 36.78/10.92 36.78/10.92 36.78/10.92 36.78/10.92 ---------------------------------------- 36.78/10.92 36.78/10.92 (10) 36.78/10.92 Complex Obligation (BEST) 36.78/10.92 36.78/10.92 ---------------------------------------- 36.78/10.92 36.78/10.92 (11) 36.78/10.92 Obligation: 36.78/10.92 Proved the lower bound n^1 for the following obligation: 36.78/10.92 36.78/10.92 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 36.78/10.92 36.78/10.92 36.78/10.92 The TRS R consists of the following rules: 36.78/10.92 36.78/10.92 active(U11(tt, N, X, XS)) -> mark(U12(splitAt(N, XS), X)) 36.78/10.92 active(U12(pair(YS, ZS), X)) -> mark(pair(cons(X, YS), ZS)) 36.78/10.92 active(afterNth(N, XS)) -> mark(snd(splitAt(N, XS))) 36.78/10.92 active(and(tt, X)) -> mark(X) 36.78/10.92 active(fst(pair(X, Y))) -> mark(X) 36.78/10.92 active(head(cons(N, XS))) -> mark(N) 36.78/10.92 active(natsFrom(N)) -> mark(cons(N, natsFrom(s(N)))) 36.78/10.92 active(sel(N, XS)) -> mark(head(afterNth(N, XS))) 36.78/10.92 active(snd(pair(X, Y))) -> mark(Y) 36.78/10.92 active(splitAt(0, XS)) -> mark(pair(nil, XS)) 36.78/10.92 active(splitAt(s(N), cons(X, XS))) -> mark(U11(tt, N, X, XS)) 36.78/10.92 active(tail(cons(N, XS))) -> mark(XS) 36.78/10.92 active(take(N, XS)) -> mark(fst(splitAt(N, XS))) 36.78/10.92 active(U11(X1, X2, X3, X4)) -> U11(active(X1), X2, X3, X4) 36.78/10.92 active(U12(X1, X2)) -> U12(active(X1), X2) 36.78/10.92 active(splitAt(X1, X2)) -> splitAt(active(X1), X2) 36.78/10.92 active(splitAt(X1, X2)) -> splitAt(X1, active(X2)) 36.78/10.92 active(pair(X1, X2)) -> pair(active(X1), X2) 36.78/10.92 active(pair(X1, X2)) -> pair(X1, active(X2)) 36.78/10.92 active(cons(X1, X2)) -> cons(active(X1), X2) 36.78/10.92 active(afterNth(X1, X2)) -> afterNth(active(X1), X2) 36.78/10.93 active(afterNth(X1, X2)) -> afterNth(X1, active(X2)) 36.78/10.93 active(snd(X)) -> snd(active(X)) 36.78/10.93 active(and(X1, X2)) -> and(active(X1), X2) 36.78/10.93 active(fst(X)) -> fst(active(X)) 36.78/10.93 active(head(X)) -> head(active(X)) 36.78/10.93 active(natsFrom(X)) -> natsFrom(active(X)) 36.78/10.93 active(s(X)) -> s(active(X)) 36.78/10.93 active(sel(X1, X2)) -> sel(active(X1), X2) 36.78/10.93 active(sel(X1, X2)) -> sel(X1, active(X2)) 36.78/10.93 active(tail(X)) -> tail(active(X)) 36.78/10.93 active(take(X1, X2)) -> take(active(X1), X2) 36.78/10.93 active(take(X1, X2)) -> take(X1, active(X2)) 36.78/10.93 U11(mark(X1), X2, X3, X4) -> mark(U11(X1, X2, X3, X4)) 36.78/10.93 U12(mark(X1), X2) -> mark(U12(X1, X2)) 36.78/10.93 splitAt(mark(X1), X2) -> mark(splitAt(X1, X2)) 36.78/10.93 splitAt(X1, mark(X2)) -> mark(splitAt(X1, X2)) 36.78/10.93 pair(mark(X1), X2) -> mark(pair(X1, X2)) 36.78/10.93 pair(X1, mark(X2)) -> mark(pair(X1, X2)) 36.78/10.93 cons(mark(X1), X2) -> mark(cons(X1, X2)) 36.78/10.93 afterNth(mark(X1), X2) -> mark(afterNth(X1, X2)) 36.78/10.93 afterNth(X1, mark(X2)) -> mark(afterNth(X1, X2)) 36.78/10.93 snd(mark(X)) -> mark(snd(X)) 36.78/10.93 and(mark(X1), X2) -> mark(and(X1, X2)) 36.78/10.93 fst(mark(X)) -> mark(fst(X)) 36.78/10.93 head(mark(X)) -> mark(head(X)) 36.78/10.93 natsFrom(mark(X)) -> mark(natsFrom(X)) 36.78/10.93 s(mark(X)) -> mark(s(X)) 36.78/10.93 sel(mark(X1), X2) -> mark(sel(X1, X2)) 36.78/10.93 sel(X1, mark(X2)) -> mark(sel(X1, X2)) 36.78/10.93 tail(mark(X)) -> mark(tail(X)) 36.78/10.93 take(mark(X1), X2) -> mark(take(X1, X2)) 36.78/10.93 take(X1, mark(X2)) -> mark(take(X1, X2)) 36.78/10.93 proper(U11(X1, X2, X3, X4)) -> U11(proper(X1), proper(X2), proper(X3), proper(X4)) 36.78/10.93 proper(tt) -> ok(tt) 36.78/10.93 proper(U12(X1, X2)) -> U12(proper(X1), proper(X2)) 36.78/10.93 proper(splitAt(X1, X2)) -> splitAt(proper(X1), proper(X2)) 36.78/10.93 proper(pair(X1, X2)) -> pair(proper(X1), proper(X2)) 36.78/10.93 proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) 36.78/10.93 proper(afterNth(X1, X2)) -> afterNth(proper(X1), proper(X2)) 36.78/10.93 proper(snd(X)) -> snd(proper(X)) 36.78/10.93 proper(and(X1, X2)) -> and(proper(X1), proper(X2)) 36.78/10.93 proper(fst(X)) -> fst(proper(X)) 36.78/10.93 proper(head(X)) -> head(proper(X)) 36.78/10.93 proper(natsFrom(X)) -> natsFrom(proper(X)) 36.78/10.93 proper(s(X)) -> s(proper(X)) 36.78/10.93 proper(sel(X1, X2)) -> sel(proper(X1), proper(X2)) 36.78/10.93 proper(0) -> ok(0) 36.78/10.93 proper(nil) -> ok(nil) 36.78/10.93 proper(tail(X)) -> tail(proper(X)) 36.78/10.93 proper(take(X1, X2)) -> take(proper(X1), proper(X2)) 36.78/10.93 U11(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U11(X1, X2, X3, X4)) 36.78/10.93 U12(ok(X1), ok(X2)) -> ok(U12(X1, X2)) 36.78/10.93 splitAt(ok(X1), ok(X2)) -> ok(splitAt(X1, X2)) 36.78/10.93 pair(ok(X1), ok(X2)) -> ok(pair(X1, X2)) 36.78/10.93 cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) 36.78/10.93 afterNth(ok(X1), ok(X2)) -> ok(afterNth(X1, X2)) 36.78/10.93 snd(ok(X)) -> ok(snd(X)) 36.78/10.93 and(ok(X1), ok(X2)) -> ok(and(X1, X2)) 36.78/10.93 fst(ok(X)) -> ok(fst(X)) 36.78/10.93 head(ok(X)) -> ok(head(X)) 36.78/10.93 natsFrom(ok(X)) -> ok(natsFrom(X)) 36.78/10.93 s(ok(X)) -> ok(s(X)) 36.78/10.93 sel(ok(X1), ok(X2)) -> ok(sel(X1, X2)) 36.78/10.93 tail(ok(X)) -> ok(tail(X)) 36.78/10.93 take(ok(X1), ok(X2)) -> ok(take(X1, X2)) 36.78/10.93 top(mark(X)) -> top(proper(X)) 36.78/10.93 top(ok(X)) -> top(active(X)) 36.78/10.93 36.78/10.93 S is empty. 36.78/10.93 Rewrite Strategy: FULL 36.78/10.93 ---------------------------------------- 36.78/10.93 36.78/10.93 (12) LowerBoundPropagationProof (FINISHED) 36.78/10.93 Propagated lower bound. 36.78/10.93 ---------------------------------------- 36.78/10.93 36.78/10.93 (13) 36.78/10.93 BOUNDS(n^1, INF) 36.78/10.93 36.78/10.93 ---------------------------------------- 36.78/10.93 36.78/10.93 (14) 36.78/10.93 Obligation: 36.78/10.93 Analyzing the following TRS for decreasing loops: 36.78/10.93 36.78/10.93 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 36.78/10.93 36.78/10.93 36.78/10.93 The TRS R consists of the following rules: 36.78/10.93 36.78/10.93 active(U11(tt, N, X, XS)) -> mark(U12(splitAt(N, XS), X)) 36.78/10.93 active(U12(pair(YS, ZS), X)) -> mark(pair(cons(X, YS), ZS)) 36.78/10.93 active(afterNth(N, XS)) -> mark(snd(splitAt(N, XS))) 36.78/10.93 active(and(tt, X)) -> mark(X) 36.78/10.93 active(fst(pair(X, Y))) -> mark(X) 36.78/10.93 active(head(cons(N, XS))) -> mark(N) 36.78/10.93 active(natsFrom(N)) -> mark(cons(N, natsFrom(s(N)))) 36.78/10.93 active(sel(N, XS)) -> mark(head(afterNth(N, XS))) 36.78/10.93 active(snd(pair(X, Y))) -> mark(Y) 36.78/10.93 active(splitAt(0, XS)) -> mark(pair(nil, XS)) 36.78/10.93 active(splitAt(s(N), cons(X, XS))) -> mark(U11(tt, N, X, XS)) 36.78/10.93 active(tail(cons(N, XS))) -> mark(XS) 36.78/10.93 active(take(N, XS)) -> mark(fst(splitAt(N, XS))) 36.78/10.93 active(U11(X1, X2, X3, X4)) -> U11(active(X1), X2, X3, X4) 36.78/10.93 active(U12(X1, X2)) -> U12(active(X1), X2) 36.78/10.93 active(splitAt(X1, X2)) -> splitAt(active(X1), X2) 36.78/10.93 active(splitAt(X1, X2)) -> splitAt(X1, active(X2)) 36.78/10.93 active(pair(X1, X2)) -> pair(active(X1), X2) 36.78/10.93 active(pair(X1, X2)) -> pair(X1, active(X2)) 36.78/10.93 active(cons(X1, X2)) -> cons(active(X1), X2) 36.78/10.93 active(afterNth(X1, X2)) -> afterNth(active(X1), X2) 36.78/10.93 active(afterNth(X1, X2)) -> afterNth(X1, active(X2)) 36.78/10.93 active(snd(X)) -> snd(active(X)) 36.78/10.93 active(and(X1, X2)) -> and(active(X1), X2) 36.78/10.93 active(fst(X)) -> fst(active(X)) 36.78/10.93 active(head(X)) -> head(active(X)) 36.78/10.93 active(natsFrom(X)) -> natsFrom(active(X)) 36.78/10.93 active(s(X)) -> s(active(X)) 36.78/10.93 active(sel(X1, X2)) -> sel(active(X1), X2) 36.78/10.93 active(sel(X1, X2)) -> sel(X1, active(X2)) 36.78/10.93 active(tail(X)) -> tail(active(X)) 36.78/10.93 active(take(X1, X2)) -> take(active(X1), X2) 36.78/10.93 active(take(X1, X2)) -> take(X1, active(X2)) 36.78/10.93 U11(mark(X1), X2, X3, X4) -> mark(U11(X1, X2, X3, X4)) 36.78/10.93 U12(mark(X1), X2) -> mark(U12(X1, X2)) 36.78/10.93 splitAt(mark(X1), X2) -> mark(splitAt(X1, X2)) 36.78/10.93 splitAt(X1, mark(X2)) -> mark(splitAt(X1, X2)) 36.78/10.93 pair(mark(X1), X2) -> mark(pair(X1, X2)) 36.78/10.93 pair(X1, mark(X2)) -> mark(pair(X1, X2)) 36.78/10.93 cons(mark(X1), X2) -> mark(cons(X1, X2)) 36.78/10.93 afterNth(mark(X1), X2) -> mark(afterNth(X1, X2)) 36.78/10.93 afterNth(X1, mark(X2)) -> mark(afterNth(X1, X2)) 36.78/10.93 snd(mark(X)) -> mark(snd(X)) 36.78/10.93 and(mark(X1), X2) -> mark(and(X1, X2)) 36.78/10.93 fst(mark(X)) -> mark(fst(X)) 36.78/10.93 head(mark(X)) -> mark(head(X)) 36.78/10.93 natsFrom(mark(X)) -> mark(natsFrom(X)) 36.78/10.93 s(mark(X)) -> mark(s(X)) 36.78/10.93 sel(mark(X1), X2) -> mark(sel(X1, X2)) 36.78/10.93 sel(X1, mark(X2)) -> mark(sel(X1, X2)) 36.78/10.93 tail(mark(X)) -> mark(tail(X)) 36.78/10.93 take(mark(X1), X2) -> mark(take(X1, X2)) 36.78/10.93 take(X1, mark(X2)) -> mark(take(X1, X2)) 36.78/10.93 proper(U11(X1, X2, X3, X4)) -> U11(proper(X1), proper(X2), proper(X3), proper(X4)) 36.78/10.93 proper(tt) -> ok(tt) 36.78/10.93 proper(U12(X1, X2)) -> U12(proper(X1), proper(X2)) 36.78/10.93 proper(splitAt(X1, X2)) -> splitAt(proper(X1), proper(X2)) 36.78/10.93 proper(pair(X1, X2)) -> pair(proper(X1), proper(X2)) 36.78/10.93 proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) 36.78/10.93 proper(afterNth(X1, X2)) -> afterNth(proper(X1), proper(X2)) 36.78/10.93 proper(snd(X)) -> snd(proper(X)) 36.78/10.93 proper(and(X1, X2)) -> and(proper(X1), proper(X2)) 36.78/10.93 proper(fst(X)) -> fst(proper(X)) 36.78/10.93 proper(head(X)) -> head(proper(X)) 36.78/10.93 proper(natsFrom(X)) -> natsFrom(proper(X)) 36.78/10.93 proper(s(X)) -> s(proper(X)) 36.78/10.93 proper(sel(X1, X2)) -> sel(proper(X1), proper(X2)) 36.78/10.93 proper(0) -> ok(0) 36.78/10.93 proper(nil) -> ok(nil) 36.78/10.93 proper(tail(X)) -> tail(proper(X)) 36.78/10.93 proper(take(X1, X2)) -> take(proper(X1), proper(X2)) 36.78/10.93 U11(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U11(X1, X2, X3, X4)) 36.78/10.93 U12(ok(X1), ok(X2)) -> ok(U12(X1, X2)) 36.78/10.93 splitAt(ok(X1), ok(X2)) -> ok(splitAt(X1, X2)) 36.78/10.93 pair(ok(X1), ok(X2)) -> ok(pair(X1, X2)) 36.78/10.93 cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) 36.78/10.93 afterNth(ok(X1), ok(X2)) -> ok(afterNth(X1, X2)) 36.78/10.93 snd(ok(X)) -> ok(snd(X)) 36.78/10.93 and(ok(X1), ok(X2)) -> ok(and(X1, X2)) 36.78/10.93 fst(ok(X)) -> ok(fst(X)) 36.78/10.93 head(ok(X)) -> ok(head(X)) 36.78/10.93 natsFrom(ok(X)) -> ok(natsFrom(X)) 36.78/10.93 s(ok(X)) -> ok(s(X)) 36.78/10.93 sel(ok(X1), ok(X2)) -> ok(sel(X1, X2)) 36.78/10.93 tail(ok(X)) -> ok(tail(X)) 36.78/10.93 take(ok(X1), ok(X2)) -> ok(take(X1, X2)) 36.78/10.93 top(mark(X)) -> top(proper(X)) 36.78/10.93 top(ok(X)) -> top(active(X)) 36.78/10.93 36.78/10.93 S is empty. 36.78/10.93 Rewrite Strategy: FULL 36.87/10.96 EOF