/export/starexec/sandbox2/solver/bin/starexec_run_complexity /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files


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WORST_CASE(Omega(n^1), O(n^1))
proof of /export/starexec/sandbox2/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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), 30 ms]
(2) CpxTRS
    (3) RcToIrcProof [BOTH BOUNDS(ID, ID), 118 ms]
    (4) CpxTRS
    (5) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
    (6) CdtProblem
    (7) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (8) CdtProblem
    (9) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (10) CdtProblem
    (11) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
    (12) CdtProblem
    (13) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 430 ms]
    (14) CdtProblem
    (15) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 102 ms]
    (16) CdtProblem
    (17) SIsEmptyProof [BOTH BOUNDS(ID, ID), 0 ms]
    (18) BOUNDS(1, 1)
(19) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
(20) TRS for Loop Detection
    (21) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
    (22) BEST
        (23) proven lower bound
            (24) LowerBoundPropagationProof [FINISHED, 0 ms]
            (25) BOUNDS(n^1, INF)
        (26) 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(U101(tt, V1, V2)) -> mark(U102(isNatural(V1), V2))
   active(U102(tt, V2)) -> mark(U103(isLNat(V2)))
   active(U103(tt)) -> mark(tt)
   active(U11(tt, N, XS)) -> mark(snd(splitAt(N, XS)))
   active(U111(tt, V1)) -> mark(U112(isLNat(V1)))
   active(U112(tt)) -> mark(tt)
   active(U121(tt, V1)) -> mark(U122(isNatural(V1)))
   active(U122(tt)) -> mark(tt)
   active(U131(tt, V1, V2)) -> mark(U132(isNatural(V1), V2))
   active(U132(tt, V2)) -> mark(U133(isLNat(V2)))
   active(U133(tt)) -> mark(tt)
   active(U141(tt, V1, V2)) -> mark(U142(isLNat(V1), V2))
   active(U142(tt, V2)) -> mark(U143(isLNat(V2)))
   active(U143(tt)) -> mark(tt)
   active(U151(tt, V1, V2)) -> mark(U152(isNatural(V1), V2))
   active(U152(tt, V2)) -> mark(U153(isLNat(V2)))
   active(U153(tt)) -> mark(tt)
   active(U161(tt, N)) -> mark(cons(N, natsFrom(s(N))))
   active(U171(tt, N, XS)) -> mark(head(afterNth(N, XS)))
   active(U181(tt, Y)) -> mark(Y)
   active(U191(tt, XS)) -> mark(pair(nil, XS))
   active(U201(tt, N, X, XS)) -> mark(U202(splitAt(N, XS), X))
   active(U202(pair(YS, ZS), X)) -> mark(pair(cons(X, YS), ZS))
   active(U21(tt, X)) -> mark(X)
   active(U211(tt, XS)) -> mark(XS)
   active(U221(tt, N, XS)) -> mark(fst(splitAt(N, XS)))
   active(U31(tt, N)) -> mark(N)
   active(U41(tt, V1, V2)) -> mark(U42(isNatural(V1), V2))
   active(U42(tt, V2)) -> mark(U43(isLNat(V2)))
   active(U43(tt)) -> mark(tt)
   active(U51(tt, V1, V2)) -> mark(U52(isNatural(V1), V2))
   active(U52(tt, V2)) -> mark(U53(isLNat(V2)))
   active(U53(tt)) -> mark(tt)
   active(U61(tt, V1)) -> mark(U62(isPLNat(V1)))
   active(U62(tt)) -> mark(tt)
   active(U71(tt, V1)) -> mark(U72(isNatural(V1)))
   active(U72(tt)) -> mark(tt)
   active(U81(tt, V1)) -> mark(U82(isPLNat(V1)))
   active(U82(tt)) -> mark(tt)
   active(U91(tt, V1)) -> mark(U92(isLNat(V1)))
   active(U92(tt)) -> mark(tt)
   active(afterNth(N, XS)) -> mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
   active(and(tt, X)) -> mark(X)
   active(fst(pair(X, Y))) -> mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
   active(head(cons(N, XS))) -> mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
   active(isLNat(nil)) -> mark(tt)
   active(isLNat(afterNth(V1, V2))) -> mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
   active(isLNat(cons(V1, V2))) -> mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
   active(isLNat(fst(V1))) -> mark(U61(isPLNatKind(V1), V1))
   active(isLNat(natsFrom(V1))) -> mark(U71(isNaturalKind(V1), V1))
   active(isLNat(snd(V1))) -> mark(U81(isPLNatKind(V1), V1))
   active(isLNat(tail(V1))) -> mark(U91(isLNatKind(V1), V1))
   active(isLNat(take(V1, V2))) -> mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
   active(isLNatKind(nil)) -> mark(tt)
   active(isLNatKind(afterNth(V1, V2))) -> mark(and(isNaturalKind(V1), isLNatKind(V2)))
   active(isLNatKind(cons(V1, V2))) -> mark(and(isNaturalKind(V1), isLNatKind(V2)))
   active(isLNatKind(fst(V1))) -> mark(isPLNatKind(V1))
   active(isLNatKind(natsFrom(V1))) -> mark(isNaturalKind(V1))
   active(isLNatKind(snd(V1))) -> mark(isPLNatKind(V1))
   active(isLNatKind(tail(V1))) -> mark(isLNatKind(V1))
   active(isLNatKind(take(V1, V2))) -> mark(and(isNaturalKind(V1), isLNatKind(V2)))
   active(isNatural(0)) -> mark(tt)
   active(isNatural(head(V1))) -> mark(U111(isLNatKind(V1), V1))
   active(isNatural(s(V1))) -> mark(U121(isNaturalKind(V1), V1))
   active(isNatural(sel(V1, V2))) -> mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
   active(isNaturalKind(0)) -> mark(tt)
   active(isNaturalKind(head(V1))) -> mark(isLNatKind(V1))
   active(isNaturalKind(s(V1))) -> mark(isNaturalKind(V1))
   active(isNaturalKind(sel(V1, V2))) -> mark(and(isNaturalKind(V1), isLNatKind(V2)))
   active(isPLNat(pair(V1, V2))) -> mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
   active(isPLNat(splitAt(V1, V2))) -> mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
   active(isPLNatKind(pair(V1, V2))) -> mark(and(isLNatKind(V1), isLNatKind(V2)))
   active(isPLNatKind(splitAt(V1, V2))) -> mark(and(isNaturalKind(V1), isLNatKind(V2)))
   active(natsFrom(N)) -> mark(U161(and(isNatural(N), isNaturalKind(N)), N))
   active(sel(N, XS)) -> mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
   active(snd(pair(X, Y))) -> mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
   active(splitAt(0, XS)) -> mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
   active(splitAt(s(N), cons(X, XS))) -> mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
   active(tail(cons(N, XS))) -> mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
   active(take(N, XS)) -> mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
   active(U101(X1, X2, X3)) -> U101(active(X1), X2, X3)
   active(U102(X1, X2)) -> U102(active(X1), X2)
   active(U103(X)) -> U103(active(X))
   active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3)
   active(snd(X)) -> snd(active(X))
   active(splitAt(X1, X2)) -> splitAt(active(X1), X2)
   active(splitAt(X1, X2)) -> splitAt(X1, active(X2))
   active(U111(X1, X2)) -> U111(active(X1), X2)
   active(U112(X)) -> U112(active(X))
   active(U121(X1, X2)) -> U121(active(X1), X2)
   active(U122(X)) -> U122(active(X))
   active(U131(X1, X2, X3)) -> U131(active(X1), X2, X3)
   active(U132(X1, X2)) -> U132(active(X1), X2)
   active(U133(X)) -> U133(active(X))
   active(U141(X1, X2, X3)) -> U141(active(X1), X2, X3)
   active(U142(X1, X2)) -> U142(active(X1), X2)
   active(U143(X)) -> U143(active(X))
   active(U151(X1, X2, X3)) -> U151(active(X1), X2, X3)
   active(U152(X1, X2)) -> U152(active(X1), X2)
   active(U153(X)) -> U153(active(X))
   active(U161(X1, X2)) -> U161(active(X1), X2)
   active(cons(X1, X2)) -> cons(active(X1), X2)
   active(natsFrom(X)) -> natsFrom(active(X))
   active(s(X)) -> s(active(X))
   active(U171(X1, X2, X3)) -> U171(active(X1), X2, X3)
   active(head(X)) -> head(active(X))
   active(afterNth(X1, X2)) -> afterNth(active(X1), X2)
   active(afterNth(X1, X2)) -> afterNth(X1, active(X2))
   active(U181(X1, X2)) -> U181(active(X1), X2)
   active(U191(X1, X2)) -> U191(active(X1), X2)
   active(pair(X1, X2)) -> pair(active(X1), X2)
   active(pair(X1, X2)) -> pair(X1, active(X2))
   active(U201(X1, X2, X3, X4)) -> U201(active(X1), X2, X3, X4)
   active(U202(X1, X2)) -> U202(active(X1), X2)
   active(U21(X1, X2)) -> U21(active(X1), X2)
   active(U211(X1, X2)) -> U211(active(X1), X2)
   active(U221(X1, X2, X3)) -> U221(active(X1), X2, X3)
   active(fst(X)) -> fst(active(X))
   active(U31(X1, X2)) -> U31(active(X1), X2)
   active(U41(X1, X2, X3)) -> U41(active(X1), X2, X3)
   active(U42(X1, X2)) -> U42(active(X1), X2)
   active(U43(X)) -> U43(active(X))
   active(U51(X1, X2, X3)) -> U51(active(X1), X2, X3)
   active(U52(X1, X2)) -> U52(active(X1), X2)
   active(U53(X)) -> U53(active(X))
   active(U61(X1, X2)) -> U61(active(X1), X2)
   active(U62(X)) -> U62(active(X))
   active(U71(X1, X2)) -> U71(active(X1), X2)
   active(U72(X)) -> U72(active(X))
   active(U81(X1, X2)) -> U81(active(X1), X2)
   active(U82(X)) -> U82(active(X))
   active(U91(X1, X2)) -> U91(active(X1), X2)
   active(U92(X)) -> U92(active(X))
   active(and(X1, X2)) -> and(active(X1), X2)
   active(tail(X)) -> tail(active(X))
   active(take(X1, X2)) -> take(active(X1), X2)
   active(take(X1, X2)) -> take(X1, active(X2))
   active(sel(X1, X2)) -> sel(active(X1), X2)
   active(sel(X1, X2)) -> sel(X1, active(X2))
   U101(mark(X1), X2, X3) -> mark(U101(X1, X2, X3))
   U102(mark(X1), X2) -> mark(U102(X1, X2))
   U103(mark(X)) -> mark(U103(X))
   U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3))
   snd(mark(X)) -> mark(snd(X))
   splitAt(mark(X1), X2) -> mark(splitAt(X1, X2))
   splitAt(X1, mark(X2)) -> mark(splitAt(X1, X2))
   U111(mark(X1), X2) -> mark(U111(X1, X2))
   U112(mark(X)) -> mark(U112(X))
   U121(mark(X1), X2) -> mark(U121(X1, X2))
   U122(mark(X)) -> mark(U122(X))
   U131(mark(X1), X2, X3) -> mark(U131(X1, X2, X3))
   U132(mark(X1), X2) -> mark(U132(X1, X2))
   U133(mark(X)) -> mark(U133(X))
   U141(mark(X1), X2, X3) -> mark(U141(X1, X2, X3))
   U142(mark(X1), X2) -> mark(U142(X1, X2))
   U143(mark(X)) -> mark(U143(X))
   U151(mark(X1), X2, X3) -> mark(U151(X1, X2, X3))
   U152(mark(X1), X2) -> mark(U152(X1, X2))
   U153(mark(X)) -> mark(U153(X))
   U161(mark(X1), X2) -> mark(U161(X1, X2))
   cons(mark(X1), X2) -> mark(cons(X1, X2))
   natsFrom(mark(X)) -> mark(natsFrom(X))
   s(mark(X)) -> mark(s(X))
   U171(mark(X1), X2, X3) -> mark(U171(X1, X2, X3))
   head(mark(X)) -> mark(head(X))
   afterNth(mark(X1), X2) -> mark(afterNth(X1, X2))
   afterNth(X1, mark(X2)) -> mark(afterNth(X1, X2))
   U181(mark(X1), X2) -> mark(U181(X1, X2))
   U191(mark(X1), X2) -> mark(U191(X1, X2))
   pair(mark(X1), X2) -> mark(pair(X1, X2))
   pair(X1, mark(X2)) -> mark(pair(X1, X2))
   U201(mark(X1), X2, X3, X4) -> mark(U201(X1, X2, X3, X4))
   U202(mark(X1), X2) -> mark(U202(X1, X2))
   U21(mark(X1), X2) -> mark(U21(X1, X2))
   U211(mark(X1), X2) -> mark(U211(X1, X2))
   U221(mark(X1), X2, X3) -> mark(U221(X1, X2, X3))
   fst(mark(X)) -> mark(fst(X))
   U31(mark(X1), X2) -> mark(U31(X1, X2))
   U41(mark(X1), X2, X3) -> mark(U41(X1, X2, X3))
   U42(mark(X1), X2) -> mark(U42(X1, X2))
   U43(mark(X)) -> mark(U43(X))
   U51(mark(X1), X2, X3) -> mark(U51(X1, X2, X3))
   U52(mark(X1), X2) -> mark(U52(X1, X2))
   U53(mark(X)) -> mark(U53(X))
   U61(mark(X1), X2) -> mark(U61(X1, X2))
   U62(mark(X)) -> mark(U62(X))
   U71(mark(X1), X2) -> mark(U71(X1, X2))
   U72(mark(X)) -> mark(U72(X))
   U81(mark(X1), X2) -> mark(U81(X1, X2))
   U82(mark(X)) -> mark(U82(X))
   U91(mark(X1), X2) -> mark(U91(X1, X2))
   U92(mark(X)) -> mark(U92(X))
   and(mark(X1), X2) -> mark(and(X1, X2))
   tail(mark(X)) -> mark(tail(X))
   take(mark(X1), X2) -> mark(take(X1, X2))
   take(X1, mark(X2)) -> mark(take(X1, X2))
   sel(mark(X1), X2) -> mark(sel(X1, X2))
   sel(X1, mark(X2)) -> mark(sel(X1, X2))
   proper(U101(X1, X2, X3)) -> U101(proper(X1), proper(X2), proper(X3))
   proper(tt) -> ok(tt)
   proper(U102(X1, X2)) -> U102(proper(X1), proper(X2))
   proper(isNatural(X)) -> isNatural(proper(X))
   proper(U103(X)) -> U103(proper(X))
   proper(isLNat(X)) -> isLNat(proper(X))
   proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3))
   proper(snd(X)) -> snd(proper(X))
   proper(splitAt(X1, X2)) -> splitAt(proper(X1), proper(X2))
   proper(U111(X1, X2)) -> U111(proper(X1), proper(X2))
   proper(U112(X)) -> U112(proper(X))
   proper(U121(X1, X2)) -> U121(proper(X1), proper(X2))
   proper(U122(X)) -> U122(proper(X))
   proper(U131(X1, X2, X3)) -> U131(proper(X1), proper(X2), proper(X3))
   proper(U132(X1, X2)) -> U132(proper(X1), proper(X2))
   proper(U133(X)) -> U133(proper(X))
   proper(U141(X1, X2, X3)) -> U141(proper(X1), proper(X2), proper(X3))
   proper(U142(X1, X2)) -> U142(proper(X1), proper(X2))
   proper(U143(X)) -> U143(proper(X))
   proper(U151(X1, X2, X3)) -> U151(proper(X1), proper(X2), proper(X3))
   proper(U152(X1, X2)) -> U152(proper(X1), proper(X2))
   proper(U153(X)) -> U153(proper(X))
   proper(U161(X1, X2)) -> U161(proper(X1), proper(X2))
   proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
   proper(natsFrom(X)) -> natsFrom(proper(X))
   proper(s(X)) -> s(proper(X))
   proper(U171(X1, X2, X3)) -> U171(proper(X1), proper(X2), proper(X3))
   proper(head(X)) -> head(proper(X))
   proper(afterNth(X1, X2)) -> afterNth(proper(X1), proper(X2))
   proper(U181(X1, X2)) -> U181(proper(X1), proper(X2))
   proper(U191(X1, X2)) -> U191(proper(X1), proper(X2))
   proper(pair(X1, X2)) -> pair(proper(X1), proper(X2))
   proper(nil) -> ok(nil)
   proper(U201(X1, X2, X3, X4)) -> U201(proper(X1), proper(X2), proper(X3), proper(X4))
   proper(U202(X1, X2)) -> U202(proper(X1), proper(X2))
   proper(U21(X1, X2)) -> U21(proper(X1), proper(X2))
   proper(U211(X1, X2)) -> U211(proper(X1), proper(X2))
   proper(U221(X1, X2, X3)) -> U221(proper(X1), proper(X2), proper(X3))
   proper(fst(X)) -> fst(proper(X))
   proper(U31(X1, X2)) -> U31(proper(X1), proper(X2))
   proper(U41(X1, X2, X3)) -> U41(proper(X1), proper(X2), proper(X3))
   proper(U42(X1, X2)) -> U42(proper(X1), proper(X2))
   proper(U43(X)) -> U43(proper(X))
   proper(U51(X1, X2, X3)) -> U51(proper(X1), proper(X2), proper(X3))
   proper(U52(X1, X2)) -> U52(proper(X1), proper(X2))
   proper(U53(X)) -> U53(proper(X))
   proper(U61(X1, X2)) -> U61(proper(X1), proper(X2))
   proper(U62(X)) -> U62(proper(X))
   proper(isPLNat(X)) -> isPLNat(proper(X))
   proper(U71(X1, X2)) -> U71(proper(X1), proper(X2))
   proper(U72(X)) -> U72(proper(X))
   proper(U81(X1, X2)) -> U81(proper(X1), proper(X2))
   proper(U82(X)) -> U82(proper(X))
   proper(U91(X1, X2)) -> U91(proper(X1), proper(X2))
   proper(U92(X)) -> U92(proper(X))
   proper(and(X1, X2)) -> and(proper(X1), proper(X2))
   proper(isNaturalKind(X)) -> isNaturalKind(proper(X))
   proper(isLNatKind(X)) -> isLNatKind(proper(X))
   proper(isPLNatKind(X)) -> isPLNatKind(proper(X))
   proper(tail(X)) -> tail(proper(X))
   proper(take(X1, X2)) -> take(proper(X1), proper(X2))
   proper(0) -> ok(0)
   proper(sel(X1, X2)) -> sel(proper(X1), proper(X2))
   U101(ok(X1), ok(X2), ok(X3)) -> ok(U101(X1, X2, X3))
   U102(ok(X1), ok(X2)) -> ok(U102(X1, X2))
   isNatural(ok(X)) -> ok(isNatural(X))
   U103(ok(X)) -> ok(U103(X))
   isLNat(ok(X)) -> ok(isLNat(X))
   U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
   snd(ok(X)) -> ok(snd(X))
   splitAt(ok(X1), ok(X2)) -> ok(splitAt(X1, X2))
   U111(ok(X1), ok(X2)) -> ok(U111(X1, X2))
   U112(ok(X)) -> ok(U112(X))
   U121(ok(X1), ok(X2)) -> ok(U121(X1, X2))
   U122(ok(X)) -> ok(U122(X))
   U131(ok(X1), ok(X2), ok(X3)) -> ok(U131(X1, X2, X3))
   U132(ok(X1), ok(X2)) -> ok(U132(X1, X2))
   U133(ok(X)) -> ok(U133(X))
   U141(ok(X1), ok(X2), ok(X3)) -> ok(U141(X1, X2, X3))
   U142(ok(X1), ok(X2)) -> ok(U142(X1, X2))
   U143(ok(X)) -> ok(U143(X))
   U151(ok(X1), ok(X2), ok(X3)) -> ok(U151(X1, X2, X3))
   U152(ok(X1), ok(X2)) -> ok(U152(X1, X2))
   U153(ok(X)) -> ok(U153(X))
   U161(ok(X1), ok(X2)) -> ok(U161(X1, X2))
   cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
   natsFrom(ok(X)) -> ok(natsFrom(X))
   s(ok(X)) -> ok(s(X))
   U171(ok(X1), ok(X2), ok(X3)) -> ok(U171(X1, X2, X3))
   head(ok(X)) -> ok(head(X))
   afterNth(ok(X1), ok(X2)) -> ok(afterNth(X1, X2))
   U181(ok(X1), ok(X2)) -> ok(U181(X1, X2))
   U191(ok(X1), ok(X2)) -> ok(U191(X1, X2))
   pair(ok(X1), ok(X2)) -> ok(pair(X1, X2))
   U201(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U201(X1, X2, X3, X4))
   U202(ok(X1), ok(X2)) -> ok(U202(X1, X2))
   U21(ok(X1), ok(X2)) -> ok(U21(X1, X2))
   U211(ok(X1), ok(X2)) -> ok(U211(X1, X2))
   U221(ok(X1), ok(X2), ok(X3)) -> ok(U221(X1, X2, X3))
   fst(ok(X)) -> ok(fst(X))
   U31(ok(X1), ok(X2)) -> ok(U31(X1, X2))
   U41(ok(X1), ok(X2), ok(X3)) -> ok(U41(X1, X2, X3))
   U42(ok(X1), ok(X2)) -> ok(U42(X1, X2))
   U43(ok(X)) -> ok(U43(X))
   U51(ok(X1), ok(X2), ok(X3)) -> ok(U51(X1, X2, X3))
   U52(ok(X1), ok(X2)) -> ok(U52(X1, X2))
   U53(ok(X)) -> ok(U53(X))
   U61(ok(X1), ok(X2)) -> ok(U61(X1, X2))
   U62(ok(X)) -> ok(U62(X))
   isPLNat(ok(X)) -> ok(isPLNat(X))
   U71(ok(X1), ok(X2)) -> ok(U71(X1, X2))
   U72(ok(X)) -> ok(U72(X))
   U81(ok(X1), ok(X2)) -> ok(U81(X1, X2))
   U82(ok(X)) -> ok(U82(X))
   U91(ok(X1), ok(X2)) -> ok(U91(X1, X2))
   U92(ok(X)) -> ok(U92(X))
   and(ok(X1), ok(X2)) -> ok(and(X1, X2))
   isNaturalKind(ok(X)) -> ok(isNaturalKind(X))
   isLNatKind(ok(X)) -> ok(isLNatKind(X))
   isPLNatKind(ok(X)) -> ok(isPLNatKind(X))
   tail(ok(X)) -> ok(tail(X))
   take(ok(X1), ok(X2)) -> ok(take(X1, X2))
   sel(ok(X1), ok(X2)) -> ok(sel(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(U101(tt, V1, V2)) -> mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) -> mark(U103(isLNat(V2)))
active(U103(tt)) -> mark(tt)
active(U11(tt, N, XS)) -> mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) -> mark(U112(isLNat(V1)))
active(U112(tt)) -> mark(tt)
active(U121(tt, V1)) -> mark(U122(isNatural(V1)))
active(U122(tt)) -> mark(tt)
active(U131(tt, V1, V2)) -> mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) -> mark(U133(isLNat(V2)))
active(U133(tt)) -> mark(tt)
active(U141(tt, V1, V2)) -> mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) -> mark(U143(isLNat(V2)))
active(U143(tt)) -> mark(tt)
active(U151(tt, V1, V2)) -> mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) -> mark(U153(isLNat(V2)))
active(U153(tt)) -> mark(tt)
active(U161(tt, N)) -> mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) -> mark(head(afterNth(N, XS)))
active(U181(tt, Y)) -> mark(Y)
active(U191(tt, XS)) -> mark(pair(nil, XS))
active(U201(tt, N, X, XS)) -> mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) -> mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) -> mark(X)
active(U211(tt, XS)) -> mark(XS)
active(U221(tt, N, XS)) -> mark(fst(splitAt(N, XS)))
active(U31(tt, N)) -> mark(N)
active(U41(tt, V1, V2)) -> mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) -> mark(U43(isLNat(V2)))
active(U43(tt)) -> mark(tt)
active(U51(tt, V1, V2)) -> mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) -> mark(U53(isLNat(V2)))
active(U53(tt)) -> mark(tt)
active(U61(tt, V1)) -> mark(U62(isPLNat(V1)))
active(U62(tt)) -> mark(tt)
active(U71(tt, V1)) -> mark(U72(isNatural(V1)))
active(U72(tt)) -> mark(tt)
active(U81(tt, V1)) -> mark(U82(isPLNat(V1)))
active(U82(tt)) -> mark(tt)
active(U91(tt, V1)) -> mark(U92(isLNat(V1)))
active(U92(tt)) -> mark(tt)
active(afterNth(N, XS)) -> mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) -> mark(X)
active(fst(pair(X, Y))) -> mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) -> mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) -> mark(tt)
active(isLNat(afterNth(V1, V2))) -> mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) -> mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) -> mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) -> mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) -> mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) -> mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) -> mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) -> mark(tt)
active(isLNatKind(afterNth(V1, V2))) -> mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) -> mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) -> mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) -> mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) -> mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) -> mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) -> mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) -> mark(tt)
active(isNatural(head(V1))) -> mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) -> mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) -> mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) -> mark(tt)
active(isNaturalKind(head(V1))) -> mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) -> mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) -> mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) -> mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) -> mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) -> mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) -> mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) -> mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) -> mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) -> mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) -> mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) -> mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) -> mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) -> mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(U101(X1, X2, X3)) -> U101(active(X1), X2, X3)
active(U102(X1, X2)) -> U102(active(X1), X2)
active(U103(X)) -> U103(active(X))
active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3)
active(snd(X)) -> snd(active(X))
active(splitAt(X1, X2)) -> splitAt(active(X1), X2)
active(splitAt(X1, X2)) -> splitAt(X1, active(X2))
active(U111(X1, X2)) -> U111(active(X1), X2)
active(U112(X)) -> U112(active(X))
active(U121(X1, X2)) -> U121(active(X1), X2)
active(U122(X)) -> U122(active(X))
active(U131(X1, X2, X3)) -> U131(active(X1), X2, X3)
active(U132(X1, X2)) -> U132(active(X1), X2)
active(U133(X)) -> U133(active(X))
active(U141(X1, X2, X3)) -> U141(active(X1), X2, X3)
active(U142(X1, X2)) -> U142(active(X1), X2)
active(U143(X)) -> U143(active(X))
active(U151(X1, X2, X3)) -> U151(active(X1), X2, X3)
active(U152(X1, X2)) -> U152(active(X1), X2)
active(U153(X)) -> U153(active(X))
active(U161(X1, X2)) -> U161(active(X1), X2)
active(cons(X1, X2)) -> cons(active(X1), X2)
active(natsFrom(X)) -> natsFrom(active(X))
active(s(X)) -> s(active(X))
active(U171(X1, X2, X3)) -> U171(active(X1), X2, X3)
active(head(X)) -> head(active(X))
active(afterNth(X1, X2)) -> afterNth(active(X1), X2)
active(afterNth(X1, X2)) -> afterNth(X1, active(X2))
active(U181(X1, X2)) -> U181(active(X1), X2)
active(U191(X1, X2)) -> U191(active(X1), X2)
active(pair(X1, X2)) -> pair(active(X1), X2)
active(pair(X1, X2)) -> pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) -> U201(active(X1), X2, X3, X4)
active(U202(X1, X2)) -> U202(active(X1), X2)
active(U21(X1, X2)) -> U21(active(X1), X2)
active(U211(X1, X2)) -> U211(active(X1), X2)
active(U221(X1, X2, X3)) -> U221(active(X1), X2, X3)
active(fst(X)) -> fst(active(X))
active(U31(X1, X2)) -> U31(active(X1), X2)
active(U41(X1, X2, X3)) -> U41(active(X1), X2, X3)
active(U42(X1, X2)) -> U42(active(X1), X2)
active(U43(X)) -> U43(active(X))
active(U51(X1, X2, X3)) -> U51(active(X1), X2, X3)
active(U52(X1, X2)) -> U52(active(X1), X2)
active(U53(X)) -> U53(active(X))
active(U61(X1, X2)) -> U61(active(X1), X2)
active(U62(X)) -> U62(active(X))
active(U71(X1, X2)) -> U71(active(X1), X2)
active(U72(X)) -> U72(active(X))
active(U81(X1, X2)) -> U81(active(X1), X2)
active(U82(X)) -> U82(active(X))
active(U91(X1, X2)) -> U91(active(X1), X2)
active(U92(X)) -> U92(active(X))
active(and(X1, X2)) -> and(active(X1), X2)
active(tail(X)) -> tail(active(X))
active(take(X1, X2)) -> take(active(X1), X2)
active(take(X1, X2)) -> take(X1, active(X2))
active(sel(X1, X2)) -> sel(active(X1), X2)
active(sel(X1, X2)) -> sel(X1, active(X2))
proper(U101(X1, X2, X3)) -> U101(proper(X1), proper(X2), proper(X3))
proper(U102(X1, X2)) -> U102(proper(X1), proper(X2))
proper(isNatural(X)) -> isNatural(proper(X))
proper(U103(X)) -> U103(proper(X))
proper(isLNat(X)) -> isLNat(proper(X))
proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) -> snd(proper(X))
proper(splitAt(X1, X2)) -> splitAt(proper(X1), proper(X2))
proper(U111(X1, X2)) -> U111(proper(X1), proper(X2))
proper(U112(X)) -> U112(proper(X))
proper(U121(X1, X2)) -> U121(proper(X1), proper(X2))
proper(U122(X)) -> U122(proper(X))
proper(U131(X1, X2, X3)) -> U131(proper(X1), proper(X2), proper(X3))
proper(U132(X1, X2)) -> U132(proper(X1), proper(X2))
proper(U133(X)) -> U133(proper(X))
proper(U141(X1, X2, X3)) -> U141(proper(X1), proper(X2), proper(X3))
proper(U142(X1, X2)) -> U142(proper(X1), proper(X2))
proper(U143(X)) -> U143(proper(X))
proper(U151(X1, X2, X3)) -> U151(proper(X1), proper(X2), proper(X3))
proper(U152(X1, X2)) -> U152(proper(X1), proper(X2))
proper(U153(X)) -> U153(proper(X))
proper(U161(X1, X2)) -> U161(proper(X1), proper(X2))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(natsFrom(X)) -> natsFrom(proper(X))
proper(s(X)) -> s(proper(X))
proper(U171(X1, X2, X3)) -> U171(proper(X1), proper(X2), proper(X3))
proper(head(X)) -> head(proper(X))
proper(afterNth(X1, X2)) -> afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) -> U181(proper(X1), proper(X2))
proper(U191(X1, X2)) -> U191(proper(X1), proper(X2))
proper(pair(X1, X2)) -> pair(proper(X1), proper(X2))
proper(U201(X1, X2, X3, X4)) -> U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2)) -> U202(proper(X1), proper(X2))
proper(U21(X1, X2)) -> U21(proper(X1), proper(X2))
proper(U211(X1, X2)) -> U211(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) -> U221(proper(X1), proper(X2), proper(X3))
proper(fst(X)) -> fst(proper(X))
proper(U31(X1, X2)) -> U31(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) -> U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) -> U42(proper(X1), proper(X2))
proper(U43(X)) -> U43(proper(X))
proper(U51(X1, X2, X3)) -> U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) -> U52(proper(X1), proper(X2))
proper(U53(X)) -> U53(proper(X))
proper(U61(X1, X2)) -> U61(proper(X1), proper(X2))
proper(U62(X)) -> U62(proper(X))
proper(isPLNat(X)) -> isPLNat(proper(X))
proper(U71(X1, X2)) -> U71(proper(X1), proper(X2))
proper(U72(X)) -> U72(proper(X))
proper(U81(X1, X2)) -> U81(proper(X1), proper(X2))
proper(U82(X)) -> U82(proper(X))
proper(U91(X1, X2)) -> U91(proper(X1), proper(X2))
proper(U92(X)) -> U92(proper(X))
proper(and(X1, X2)) -> and(proper(X1), proper(X2))
proper(isNaturalKind(X)) -> isNaturalKind(proper(X))
proper(isLNatKind(X)) -> isLNatKind(proper(X))
proper(isPLNatKind(X)) -> isPLNatKind(proper(X))
proper(tail(X)) -> tail(proper(X))
proper(take(X1, X2)) -> take(proper(X1), proper(X2))
proper(sel(X1, X2)) -> sel(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:

   U101(mark(X1), X2, X3) -> mark(U101(X1, X2, X3))
   U102(mark(X1), X2) -> mark(U102(X1, X2))
   U103(mark(X)) -> mark(U103(X))
   U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3))
   snd(mark(X)) -> mark(snd(X))
   splitAt(mark(X1), X2) -> mark(splitAt(X1, X2))
   splitAt(X1, mark(X2)) -> mark(splitAt(X1, X2))
   U111(mark(X1), X2) -> mark(U111(X1, X2))
   U112(mark(X)) -> mark(U112(X))
   U121(mark(X1), X2) -> mark(U121(X1, X2))
   U122(mark(X)) -> mark(U122(X))
   U131(mark(X1), X2, X3) -> mark(U131(X1, X2, X3))
   U132(mark(X1), X2) -> mark(U132(X1, X2))
   U133(mark(X)) -> mark(U133(X))
   U141(mark(X1), X2, X3) -> mark(U141(X1, X2, X3))
   U142(mark(X1), X2) -> mark(U142(X1, X2))
   U143(mark(X)) -> mark(U143(X))
   U151(mark(X1), X2, X3) -> mark(U151(X1, X2, X3))
   U152(mark(X1), X2) -> mark(U152(X1, X2))
   U153(mark(X)) -> mark(U153(X))
   U161(mark(X1), X2) -> mark(U161(X1, X2))
   cons(mark(X1), X2) -> mark(cons(X1, X2))
   natsFrom(mark(X)) -> mark(natsFrom(X))
   s(mark(X)) -> mark(s(X))
   U171(mark(X1), X2, X3) -> mark(U171(X1, X2, X3))
   head(mark(X)) -> mark(head(X))
   afterNth(mark(X1), X2) -> mark(afterNth(X1, X2))
   afterNth(X1, mark(X2)) -> mark(afterNth(X1, X2))
   U181(mark(X1), X2) -> mark(U181(X1, X2))
   U191(mark(X1), X2) -> mark(U191(X1, X2))
   pair(mark(X1), X2) -> mark(pair(X1, X2))
   pair(X1, mark(X2)) -> mark(pair(X1, X2))
   U201(mark(X1), X2, X3, X4) -> mark(U201(X1, X2, X3, X4))
   U202(mark(X1), X2) -> mark(U202(X1, X2))
   U21(mark(X1), X2) -> mark(U21(X1, X2))
   U211(mark(X1), X2) -> mark(U211(X1, X2))
   U221(mark(X1), X2, X3) -> mark(U221(X1, X2, X3))
   fst(mark(X)) -> mark(fst(X))
   U31(mark(X1), X2) -> mark(U31(X1, X2))
   U41(mark(X1), X2, X3) -> mark(U41(X1, X2, X3))
   U42(mark(X1), X2) -> mark(U42(X1, X2))
   U43(mark(X)) -> mark(U43(X))
   U51(mark(X1), X2, X3) -> mark(U51(X1, X2, X3))
   U52(mark(X1), X2) -> mark(U52(X1, X2))
   U53(mark(X)) -> mark(U53(X))
   U61(mark(X1), X2) -> mark(U61(X1, X2))
   U62(mark(X)) -> mark(U62(X))
   U71(mark(X1), X2) -> mark(U71(X1, X2))
   U72(mark(X)) -> mark(U72(X))
   U81(mark(X1), X2) -> mark(U81(X1, X2))
   U82(mark(X)) -> mark(U82(X))
   U91(mark(X1), X2) -> mark(U91(X1, X2))
   U92(mark(X)) -> mark(U92(X))
   and(mark(X1), X2) -> mark(and(X1, X2))
   tail(mark(X)) -> mark(tail(X))
   take(mark(X1), X2) -> mark(take(X1, X2))
   take(X1, mark(X2)) -> mark(take(X1, X2))
   sel(mark(X1), X2) -> mark(sel(X1, X2))
   sel(X1, mark(X2)) -> mark(sel(X1, X2))
   proper(tt) -> ok(tt)
   proper(nil) -> ok(nil)
   proper(0) -> ok(0)
   U101(ok(X1), ok(X2), ok(X3)) -> ok(U101(X1, X2, X3))
   U102(ok(X1), ok(X2)) -> ok(U102(X1, X2))
   isNatural(ok(X)) -> ok(isNatural(X))
   U103(ok(X)) -> ok(U103(X))
   isLNat(ok(X)) -> ok(isLNat(X))
   U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
   snd(ok(X)) -> ok(snd(X))
   splitAt(ok(X1), ok(X2)) -> ok(splitAt(X1, X2))
   U111(ok(X1), ok(X2)) -> ok(U111(X1, X2))
   U112(ok(X)) -> ok(U112(X))
   U121(ok(X1), ok(X2)) -> ok(U121(X1, X2))
   U122(ok(X)) -> ok(U122(X))
   U131(ok(X1), ok(X2), ok(X3)) -> ok(U131(X1, X2, X3))
   U132(ok(X1), ok(X2)) -> ok(U132(X1, X2))
   U133(ok(X)) -> ok(U133(X))
   U141(ok(X1), ok(X2), ok(X3)) -> ok(U141(X1, X2, X3))
   U142(ok(X1), ok(X2)) -> ok(U142(X1, X2))
   U143(ok(X)) -> ok(U143(X))
   U151(ok(X1), ok(X2), ok(X3)) -> ok(U151(X1, X2, X3))
   U152(ok(X1), ok(X2)) -> ok(U152(X1, X2))
   U153(ok(X)) -> ok(U153(X))
   U161(ok(X1), ok(X2)) -> ok(U161(X1, X2))
   cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
   natsFrom(ok(X)) -> ok(natsFrom(X))
   s(ok(X)) -> ok(s(X))
   U171(ok(X1), ok(X2), ok(X3)) -> ok(U171(X1, X2, X3))
   head(ok(X)) -> ok(head(X))
   afterNth(ok(X1), ok(X2)) -> ok(afterNth(X1, X2))
   U181(ok(X1), ok(X2)) -> ok(U181(X1, X2))
   U191(ok(X1), ok(X2)) -> ok(U191(X1, X2))
   pair(ok(X1), ok(X2)) -> ok(pair(X1, X2))
   U201(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U201(X1, X2, X3, X4))
   U202(ok(X1), ok(X2)) -> ok(U202(X1, X2))
   U21(ok(X1), ok(X2)) -> ok(U21(X1, X2))
   U211(ok(X1), ok(X2)) -> ok(U211(X1, X2))
   U221(ok(X1), ok(X2), ok(X3)) -> ok(U221(X1, X2, X3))
   fst(ok(X)) -> ok(fst(X))
   U31(ok(X1), ok(X2)) -> ok(U31(X1, X2))
   U41(ok(X1), ok(X2), ok(X3)) -> ok(U41(X1, X2, X3))
   U42(ok(X1), ok(X2)) -> ok(U42(X1, X2))
   U43(ok(X)) -> ok(U43(X))
   U51(ok(X1), ok(X2), ok(X3)) -> ok(U51(X1, X2, X3))
   U52(ok(X1), ok(X2)) -> ok(U52(X1, X2))
   U53(ok(X)) -> ok(U53(X))
   U61(ok(X1), ok(X2)) -> ok(U61(X1, X2))
   U62(ok(X)) -> ok(U62(X))
   isPLNat(ok(X)) -> ok(isPLNat(X))
   U71(ok(X1), ok(X2)) -> ok(U71(X1, X2))
   U72(ok(X)) -> ok(U72(X))
   U81(ok(X1), ok(X2)) -> ok(U81(X1, X2))
   U82(ok(X)) -> ok(U82(X))
   U91(ok(X1), ok(X2)) -> ok(U91(X1, X2))
   U92(ok(X)) -> ok(U92(X))
   and(ok(X1), ok(X2)) -> ok(and(X1, X2))
   isNaturalKind(ok(X)) -> ok(isNaturalKind(X))
   isLNatKind(ok(X)) -> ok(isLNatKind(X))
   isPLNatKind(ok(X)) -> ok(isPLNatKind(X))
   tail(ok(X)) -> ok(tail(X))
   take(ok(X1), ok(X2)) -> ok(take(X1, X2))
   sel(ok(X1), ok(X2)) -> ok(sel(X1, X2))
   top(mark(X)) -> top(proper(X))
   top(ok(X)) -> top(active(X))

S is empty.
Rewrite Strategy: FULL
----------------------------------------

(3) RcToIrcProof (BOTH BOUNDS(ID, ID))
Converted rc-obligation to irc-obligation.

As the TRS is a non-duplicating overlay system, we have rc = irc.
----------------------------------------

(4)
Obligation:
The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^1).


The TRS R consists of the following rules:

   U101(mark(X1), X2, X3) -> mark(U101(X1, X2, X3))
   U102(mark(X1), X2) -> mark(U102(X1, X2))
   U103(mark(X)) -> mark(U103(X))
   U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3))
   snd(mark(X)) -> mark(snd(X))
   splitAt(mark(X1), X2) -> mark(splitAt(X1, X2))
   splitAt(X1, mark(X2)) -> mark(splitAt(X1, X2))
   U111(mark(X1), X2) -> mark(U111(X1, X2))
   U112(mark(X)) -> mark(U112(X))
   U121(mark(X1), X2) -> mark(U121(X1, X2))
   U122(mark(X)) -> mark(U122(X))
   U131(mark(X1), X2, X3) -> mark(U131(X1, X2, X3))
   U132(mark(X1), X2) -> mark(U132(X1, X2))
   U133(mark(X)) -> mark(U133(X))
   U141(mark(X1), X2, X3) -> mark(U141(X1, X2, X3))
   U142(mark(X1), X2) -> mark(U142(X1, X2))
   U143(mark(X)) -> mark(U143(X))
   U151(mark(X1), X2, X3) -> mark(U151(X1, X2, X3))
   U152(mark(X1), X2) -> mark(U152(X1, X2))
   U153(mark(X)) -> mark(U153(X))
   U161(mark(X1), X2) -> mark(U161(X1, X2))
   cons(mark(X1), X2) -> mark(cons(X1, X2))
   natsFrom(mark(X)) -> mark(natsFrom(X))
   s(mark(X)) -> mark(s(X))
   U171(mark(X1), X2, X3) -> mark(U171(X1, X2, X3))
   head(mark(X)) -> mark(head(X))
   afterNth(mark(X1), X2) -> mark(afterNth(X1, X2))
   afterNth(X1, mark(X2)) -> mark(afterNth(X1, X2))
   U181(mark(X1), X2) -> mark(U181(X1, X2))
   U191(mark(X1), X2) -> mark(U191(X1, X2))
   pair(mark(X1), X2) -> mark(pair(X1, X2))
   pair(X1, mark(X2)) -> mark(pair(X1, X2))
   U201(mark(X1), X2, X3, X4) -> mark(U201(X1, X2, X3, X4))
   U202(mark(X1), X2) -> mark(U202(X1, X2))
   U21(mark(X1), X2) -> mark(U21(X1, X2))
   U211(mark(X1), X2) -> mark(U211(X1, X2))
   U221(mark(X1), X2, X3) -> mark(U221(X1, X2, X3))
   fst(mark(X)) -> mark(fst(X))
   U31(mark(X1), X2) -> mark(U31(X1, X2))
   U41(mark(X1), X2, X3) -> mark(U41(X1, X2, X3))
   U42(mark(X1), X2) -> mark(U42(X1, X2))
   U43(mark(X)) -> mark(U43(X))
   U51(mark(X1), X2, X3) -> mark(U51(X1, X2, X3))
   U52(mark(X1), X2) -> mark(U52(X1, X2))
   U53(mark(X)) -> mark(U53(X))
   U61(mark(X1), X2) -> mark(U61(X1, X2))
   U62(mark(X)) -> mark(U62(X))
   U71(mark(X1), X2) -> mark(U71(X1, X2))
   U72(mark(X)) -> mark(U72(X))
   U81(mark(X1), X2) -> mark(U81(X1, X2))
   U82(mark(X)) -> mark(U82(X))
   U91(mark(X1), X2) -> mark(U91(X1, X2))
   U92(mark(X)) -> mark(U92(X))
   and(mark(X1), X2) -> mark(and(X1, X2))
   tail(mark(X)) -> mark(tail(X))
   take(mark(X1), X2) -> mark(take(X1, X2))
   take(X1, mark(X2)) -> mark(take(X1, X2))
   sel(mark(X1), X2) -> mark(sel(X1, X2))
   sel(X1, mark(X2)) -> mark(sel(X1, X2))
   proper(tt) -> ok(tt)
   proper(nil) -> ok(nil)
   proper(0) -> ok(0)
   U101(ok(X1), ok(X2), ok(X3)) -> ok(U101(X1, X2, X3))
   U102(ok(X1), ok(X2)) -> ok(U102(X1, X2))
   isNatural(ok(X)) -> ok(isNatural(X))
   U103(ok(X)) -> ok(U103(X))
   isLNat(ok(X)) -> ok(isLNat(X))
   U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
   snd(ok(X)) -> ok(snd(X))
   splitAt(ok(X1), ok(X2)) -> ok(splitAt(X1, X2))
   U111(ok(X1), ok(X2)) -> ok(U111(X1, X2))
   U112(ok(X)) -> ok(U112(X))
   U121(ok(X1), ok(X2)) -> ok(U121(X1, X2))
   U122(ok(X)) -> ok(U122(X))
   U131(ok(X1), ok(X2), ok(X3)) -> ok(U131(X1, X2, X3))
   U132(ok(X1), ok(X2)) -> ok(U132(X1, X2))
   U133(ok(X)) -> ok(U133(X))
   U141(ok(X1), ok(X2), ok(X3)) -> ok(U141(X1, X2, X3))
   U142(ok(X1), ok(X2)) -> ok(U142(X1, X2))
   U143(ok(X)) -> ok(U143(X))
   U151(ok(X1), ok(X2), ok(X3)) -> ok(U151(X1, X2, X3))
   U152(ok(X1), ok(X2)) -> ok(U152(X1, X2))
   U153(ok(X)) -> ok(U153(X))
   U161(ok(X1), ok(X2)) -> ok(U161(X1, X2))
   cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
   natsFrom(ok(X)) -> ok(natsFrom(X))
   s(ok(X)) -> ok(s(X))
   U171(ok(X1), ok(X2), ok(X3)) -> ok(U171(X1, X2, X3))
   head(ok(X)) -> ok(head(X))
   afterNth(ok(X1), ok(X2)) -> ok(afterNth(X1, X2))
   U181(ok(X1), ok(X2)) -> ok(U181(X1, X2))
   U191(ok(X1), ok(X2)) -> ok(U191(X1, X2))
   pair(ok(X1), ok(X2)) -> ok(pair(X1, X2))
   U201(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U201(X1, X2, X3, X4))
   U202(ok(X1), ok(X2)) -> ok(U202(X1, X2))
   U21(ok(X1), ok(X2)) -> ok(U21(X1, X2))
   U211(ok(X1), ok(X2)) -> ok(U211(X1, X2))
   U221(ok(X1), ok(X2), ok(X3)) -> ok(U221(X1, X2, X3))
   fst(ok(X)) -> ok(fst(X))
   U31(ok(X1), ok(X2)) -> ok(U31(X1, X2))
   U41(ok(X1), ok(X2), ok(X3)) -> ok(U41(X1, X2, X3))
   U42(ok(X1), ok(X2)) -> ok(U42(X1, X2))
   U43(ok(X)) -> ok(U43(X))
   U51(ok(X1), ok(X2), ok(X3)) -> ok(U51(X1, X2, X3))
   U52(ok(X1), ok(X2)) -> ok(U52(X1, X2))
   U53(ok(X)) -> ok(U53(X))
   U61(ok(X1), ok(X2)) -> ok(U61(X1, X2))
   U62(ok(X)) -> ok(U62(X))
   isPLNat(ok(X)) -> ok(isPLNat(X))
   U71(ok(X1), ok(X2)) -> ok(U71(X1, X2))
   U72(ok(X)) -> ok(U72(X))
   U81(ok(X1), ok(X2)) -> ok(U81(X1, X2))
   U82(ok(X)) -> ok(U82(X))
   U91(ok(X1), ok(X2)) -> ok(U91(X1, X2))
   U92(ok(X)) -> ok(U92(X))
   and(ok(X1), ok(X2)) -> ok(and(X1, X2))
   isNaturalKind(ok(X)) -> ok(isNaturalKind(X))
   isLNatKind(ok(X)) -> ok(isLNatKind(X))
   isPLNatKind(ok(X)) -> ok(isPLNatKind(X))
   tail(ok(X)) -> ok(tail(X))
   take(ok(X1), ok(X2)) -> ok(take(X1, X2))
   sel(ok(X1), ok(X2)) -> ok(sel(X1, X2))
   top(mark(X)) -> top(proper(X))
   top(ok(X)) -> top(active(X))

S is empty.
Rewrite Strategy: INNERMOST
----------------------------------------

(5) CpxTrsToCdtProof (UPPER BOUND(ID))
Converted Cpx (relative) TRS to CDT
----------------------------------------

(6)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   U101(mark(z0), z1, z2) -> mark(U101(z0, z1, z2))
   U101(ok(z0), ok(z1), ok(z2)) -> ok(U101(z0, z1, z2))
   U102(mark(z0), z1) -> mark(U102(z0, z1))
   U102(ok(z0), ok(z1)) -> ok(U102(z0, z1))
   U103(mark(z0)) -> mark(U103(z0))
   U103(ok(z0)) -> ok(U103(z0))
   U11(mark(z0), z1, z2) -> mark(U11(z0, z1, z2))
   U11(ok(z0), ok(z1), ok(z2)) -> ok(U11(z0, z1, z2))
   snd(mark(z0)) -> mark(snd(z0))
   snd(ok(z0)) -> ok(snd(z0))
   splitAt(mark(z0), z1) -> mark(splitAt(z0, z1))
   splitAt(z0, mark(z1)) -> mark(splitAt(z0, z1))
   splitAt(ok(z0), ok(z1)) -> ok(splitAt(z0, z1))
   U111(mark(z0), z1) -> mark(U111(z0, z1))
   U111(ok(z0), ok(z1)) -> ok(U111(z0, z1))
   U112(mark(z0)) -> mark(U112(z0))
   U112(ok(z0)) -> ok(U112(z0))
   U121(mark(z0), z1) -> mark(U121(z0, z1))
   U121(ok(z0), ok(z1)) -> ok(U121(z0, z1))
   U122(mark(z0)) -> mark(U122(z0))
   U122(ok(z0)) -> ok(U122(z0))
   U131(mark(z0), z1, z2) -> mark(U131(z0, z1, z2))
   U131(ok(z0), ok(z1), ok(z2)) -> ok(U131(z0, z1, z2))
   U132(mark(z0), z1) -> mark(U132(z0, z1))
   U132(ok(z0), ok(z1)) -> ok(U132(z0, z1))
   U133(mark(z0)) -> mark(U133(z0))
   U133(ok(z0)) -> ok(U133(z0))
   U141(mark(z0), z1, z2) -> mark(U141(z0, z1, z2))
   U141(ok(z0), ok(z1), ok(z2)) -> ok(U141(z0, z1, z2))
   U142(mark(z0), z1) -> mark(U142(z0, z1))
   U142(ok(z0), ok(z1)) -> ok(U142(z0, z1))
   U143(mark(z0)) -> mark(U143(z0))
   U143(ok(z0)) -> ok(U143(z0))
   U151(mark(z0), z1, z2) -> mark(U151(z0, z1, z2))
   U151(ok(z0), ok(z1), ok(z2)) -> ok(U151(z0, z1, z2))
   U152(mark(z0), z1) -> mark(U152(z0, z1))
   U152(ok(z0), ok(z1)) -> ok(U152(z0, z1))
   U153(mark(z0)) -> mark(U153(z0))
   U153(ok(z0)) -> ok(U153(z0))
   U161(mark(z0), z1) -> mark(U161(z0, z1))
   U161(ok(z0), ok(z1)) -> ok(U161(z0, z1))
   cons(mark(z0), z1) -> mark(cons(z0, z1))
   cons(ok(z0), ok(z1)) -> ok(cons(z0, z1))
   natsFrom(mark(z0)) -> mark(natsFrom(z0))
   natsFrom(ok(z0)) -> ok(natsFrom(z0))
   s(mark(z0)) -> mark(s(z0))
   s(ok(z0)) -> ok(s(z0))
   U171(mark(z0), z1, z2) -> mark(U171(z0, z1, z2))
   U171(ok(z0), ok(z1), ok(z2)) -> ok(U171(z0, z1, z2))
   head(mark(z0)) -> mark(head(z0))
   head(ok(z0)) -> ok(head(z0))
   afterNth(mark(z0), z1) -> mark(afterNth(z0, z1))
   afterNth(z0, mark(z1)) -> mark(afterNth(z0, z1))
   afterNth(ok(z0), ok(z1)) -> ok(afterNth(z0, z1))
   U181(mark(z0), z1) -> mark(U181(z0, z1))
   U181(ok(z0), ok(z1)) -> ok(U181(z0, z1))
   U191(mark(z0), z1) -> mark(U191(z0, z1))
   U191(ok(z0), ok(z1)) -> ok(U191(z0, z1))
   pair(mark(z0), z1) -> mark(pair(z0, z1))
   pair(z0, mark(z1)) -> mark(pair(z0, z1))
   pair(ok(z0), ok(z1)) -> ok(pair(z0, z1))
   U201(mark(z0), z1, z2, z3) -> mark(U201(z0, z1, z2, z3))
   U201(ok(z0), ok(z1), ok(z2), ok(z3)) -> ok(U201(z0, z1, z2, z3))
   U202(mark(z0), z1) -> mark(U202(z0, z1))
   U202(ok(z0), ok(z1)) -> ok(U202(z0, z1))
   U21(mark(z0), z1) -> mark(U21(z0, z1))
   U21(ok(z0), ok(z1)) -> ok(U21(z0, z1))
   U211(mark(z0), z1) -> mark(U211(z0, z1))
   U211(ok(z0), ok(z1)) -> ok(U211(z0, z1))
   U221(mark(z0), z1, z2) -> mark(U221(z0, z1, z2))
   U221(ok(z0), ok(z1), ok(z2)) -> ok(U221(z0, z1, z2))
   fst(mark(z0)) -> mark(fst(z0))
   fst(ok(z0)) -> ok(fst(z0))
   U31(mark(z0), z1) -> mark(U31(z0, z1))
   U31(ok(z0), ok(z1)) -> ok(U31(z0, z1))
   U41(mark(z0), z1, z2) -> mark(U41(z0, z1, z2))
   U41(ok(z0), ok(z1), ok(z2)) -> ok(U41(z0, z1, z2))
   U42(mark(z0), z1) -> mark(U42(z0, z1))
   U42(ok(z0), ok(z1)) -> ok(U42(z0, z1))
   U43(mark(z0)) -> mark(U43(z0))
   U43(ok(z0)) -> ok(U43(z0))
   U51(mark(z0), z1, z2) -> mark(U51(z0, z1, z2))
   U51(ok(z0), ok(z1), ok(z2)) -> ok(U51(z0, z1, z2))
   U52(mark(z0), z1) -> mark(U52(z0, z1))
   U52(ok(z0), ok(z1)) -> ok(U52(z0, z1))
   U53(mark(z0)) -> mark(U53(z0))
   U53(ok(z0)) -> ok(U53(z0))
   U61(mark(z0), z1) -> mark(U61(z0, z1))
   U61(ok(z0), ok(z1)) -> ok(U61(z0, z1))
   U62(mark(z0)) -> mark(U62(z0))
   U62(ok(z0)) -> ok(U62(z0))
   U71(mark(z0), z1) -> mark(U71(z0, z1))
   U71(ok(z0), ok(z1)) -> ok(U71(z0, z1))
   U72(mark(z0)) -> mark(U72(z0))
   U72(ok(z0)) -> ok(U72(z0))
   U81(mark(z0), z1) -> mark(U81(z0, z1))
   U81(ok(z0), ok(z1)) -> ok(U81(z0, z1))
   U82(mark(z0)) -> mark(U82(z0))
   U82(ok(z0)) -> ok(U82(z0))
   U91(mark(z0), z1) -> mark(U91(z0, z1))
   U91(ok(z0), ok(z1)) -> ok(U91(z0, z1))
   U92(mark(z0)) -> mark(U92(z0))
   U92(ok(z0)) -> ok(U92(z0))
   and(mark(z0), z1) -> mark(and(z0, z1))
   and(ok(z0), ok(z1)) -> ok(and(z0, z1))
   tail(mark(z0)) -> mark(tail(z0))
   tail(ok(z0)) -> ok(tail(z0))
   take(mark(z0), z1) -> mark(take(z0, z1))
   take(z0, mark(z1)) -> mark(take(z0, z1))
   take(ok(z0), ok(z1)) -> ok(take(z0, z1))
   sel(mark(z0), z1) -> mark(sel(z0, z1))
   sel(z0, mark(z1)) -> mark(sel(z0, z1))
   sel(ok(z0), ok(z1)) -> ok(sel(z0, z1))
   proper(tt) -> ok(tt)
   proper(nil) -> ok(nil)
   proper(0) -> ok(0)
   isNatural(ok(z0)) -> ok(isNatural(z0))
   isLNat(ok(z0)) -> ok(isLNat(z0))
   isPLNat(ok(z0)) -> ok(isPLNat(z0))
   isNaturalKind(ok(z0)) -> ok(isNaturalKind(z0))
   isLNatKind(ok(z0)) -> ok(isLNatKind(z0))
   isPLNatKind(ok(z0)) -> ok(isPLNatKind(z0))
   top(mark(z0)) -> top(proper(z0))
   top(ok(z0)) -> top(active(z0))
Tuples:
   U101'(mark(z0), z1, z2) -> c(U101'(z0, z1, z2))
   U101'(ok(z0), ok(z1), ok(z2)) -> c1(U101'(z0, z1, z2))
   U102'(mark(z0), z1) -> c2(U102'(z0, z1))
   U102'(ok(z0), ok(z1)) -> c3(U102'(z0, z1))
   U103'(mark(z0)) -> c4(U103'(z0))
   U103'(ok(z0)) -> c5(U103'(z0))
   U11'(mark(z0), z1, z2) -> c6(U11'(z0, z1, z2))
   U11'(ok(z0), ok(z1), ok(z2)) -> c7(U11'(z0, z1, z2))
   SND(mark(z0)) -> c8(SND(z0))
   SND(ok(z0)) -> c9(SND(z0))
   SPLITAT(mark(z0), z1) -> c10(SPLITAT(z0, z1))
   SPLITAT(z0, mark(z1)) -> c11(SPLITAT(z0, z1))
   SPLITAT(ok(z0), ok(z1)) -> c12(SPLITAT(z0, z1))
   U111'(mark(z0), z1) -> c13(U111'(z0, z1))
   U111'(ok(z0), ok(z1)) -> c14(U111'(z0, z1))
   U112'(mark(z0)) -> c15(U112'(z0))
   U112'(ok(z0)) -> c16(U112'(z0))
   U121'(mark(z0), z1) -> c17(U121'(z0, z1))
   U121'(ok(z0), ok(z1)) -> c18(U121'(z0, z1))
   U122'(mark(z0)) -> c19(U122'(z0))
   U122'(ok(z0)) -> c20(U122'(z0))
   U131'(mark(z0), z1, z2) -> c21(U131'(z0, z1, z2))
   U131'(ok(z0), ok(z1), ok(z2)) -> c22(U131'(z0, z1, z2))
   U132'(mark(z0), z1) -> c23(U132'(z0, z1))
   U132'(ok(z0), ok(z1)) -> c24(U132'(z0, z1))
   U133'(mark(z0)) -> c25(U133'(z0))
   U133'(ok(z0)) -> c26(U133'(z0))
   U141'(mark(z0), z1, z2) -> c27(U141'(z0, z1, z2))
   U141'(ok(z0), ok(z1), ok(z2)) -> c28(U141'(z0, z1, z2))
   U142'(mark(z0), z1) -> c29(U142'(z0, z1))
   U142'(ok(z0), ok(z1)) -> c30(U142'(z0, z1))
   U143'(mark(z0)) -> c31(U143'(z0))
   U143'(ok(z0)) -> c32(U143'(z0))
   U151'(mark(z0), z1, z2) -> c33(U151'(z0, z1, z2))
   U151'(ok(z0), ok(z1), ok(z2)) -> c34(U151'(z0, z1, z2))
   U152'(mark(z0), z1) -> c35(U152'(z0, z1))
   U152'(ok(z0), ok(z1)) -> c36(U152'(z0, z1))
   U153'(mark(z0)) -> c37(U153'(z0))
   U153'(ok(z0)) -> c38(U153'(z0))
   U161'(mark(z0), z1) -> c39(U161'(z0, z1))
   U161'(ok(z0), ok(z1)) -> c40(U161'(z0, z1))
   CONS(mark(z0), z1) -> c41(CONS(z0, z1))
   CONS(ok(z0), ok(z1)) -> c42(CONS(z0, z1))
   NATSFROM(mark(z0)) -> c43(NATSFROM(z0))
   NATSFROM(ok(z0)) -> c44(NATSFROM(z0))
   S(mark(z0)) -> c45(S(z0))
   S(ok(z0)) -> c46(S(z0))
   U171'(mark(z0), z1, z2) -> c47(U171'(z0, z1, z2))
   U171'(ok(z0), ok(z1), ok(z2)) -> c48(U171'(z0, z1, z2))
   HEAD(mark(z0)) -> c49(HEAD(z0))
   HEAD(ok(z0)) -> c50(HEAD(z0))
   AFTERNTH(mark(z0), z1) -> c51(AFTERNTH(z0, z1))
   AFTERNTH(z0, mark(z1)) -> c52(AFTERNTH(z0, z1))
   AFTERNTH(ok(z0), ok(z1)) -> c53(AFTERNTH(z0, z1))
   U181'(mark(z0), z1) -> c54(U181'(z0, z1))
   U181'(ok(z0), ok(z1)) -> c55(U181'(z0, z1))
   U191'(mark(z0), z1) -> c56(U191'(z0, z1))
   U191'(ok(z0), ok(z1)) -> c57(U191'(z0, z1))
   PAIR(mark(z0), z1) -> c58(PAIR(z0, z1))
   PAIR(z0, mark(z1)) -> c59(PAIR(z0, z1))
   PAIR(ok(z0), ok(z1)) -> c60(PAIR(z0, z1))
   U201'(mark(z0), z1, z2, z3) -> c61(U201'(z0, z1, z2, z3))
   U201'(ok(z0), ok(z1), ok(z2), ok(z3)) -> c62(U201'(z0, z1, z2, z3))
   U202'(mark(z0), z1) -> c63(U202'(z0, z1))
   U202'(ok(z0), ok(z1)) -> c64(U202'(z0, z1))
   U21'(mark(z0), z1) -> c65(U21'(z0, z1))
   U21'(ok(z0), ok(z1)) -> c66(U21'(z0, z1))
   U211'(mark(z0), z1) -> c67(U211'(z0, z1))
   U211'(ok(z0), ok(z1)) -> c68(U211'(z0, z1))
   U221'(mark(z0), z1, z2) -> c69(U221'(z0, z1, z2))
   U221'(ok(z0), ok(z1), ok(z2)) -> c70(U221'(z0, z1, z2))
   FST(mark(z0)) -> c71(FST(z0))
   FST(ok(z0)) -> c72(FST(z0))
   U31'(mark(z0), z1) -> c73(U31'(z0, z1))
   U31'(ok(z0), ok(z1)) -> c74(U31'(z0, z1))
   U41'(mark(z0), z1, z2) -> c75(U41'(z0, z1, z2))
   U41'(ok(z0), ok(z1), ok(z2)) -> c76(U41'(z0, z1, z2))
   U42'(mark(z0), z1) -> c77(U42'(z0, z1))
   U42'(ok(z0), ok(z1)) -> c78(U42'(z0, z1))
   U43'(mark(z0)) -> c79(U43'(z0))
   U43'(ok(z0)) -> c80(U43'(z0))
   U51'(mark(z0), z1, z2) -> c81(U51'(z0, z1, z2))
   U51'(ok(z0), ok(z1), ok(z2)) -> c82(U51'(z0, z1, z2))
   U52'(mark(z0), z1) -> c83(U52'(z0, z1))
   U52'(ok(z0), ok(z1)) -> c84(U52'(z0, z1))
   U53'(mark(z0)) -> c85(U53'(z0))
   U53'(ok(z0)) -> c86(U53'(z0))
   U61'(mark(z0), z1) -> c87(U61'(z0, z1))
   U61'(ok(z0), ok(z1)) -> c88(U61'(z0, z1))
   U62'(mark(z0)) -> c89(U62'(z0))
   U62'(ok(z0)) -> c90(U62'(z0))
   U71'(mark(z0), z1) -> c91(U71'(z0, z1))
   U71'(ok(z0), ok(z1)) -> c92(U71'(z0, z1))
   U72'(mark(z0)) -> c93(U72'(z0))
   U72'(ok(z0)) -> c94(U72'(z0))
   U81'(mark(z0), z1) -> c95(U81'(z0, z1))
   U81'(ok(z0), ok(z1)) -> c96(U81'(z0, z1))
   U82'(mark(z0)) -> c97(U82'(z0))
   U82'(ok(z0)) -> c98(U82'(z0))
   U91'(mark(z0), z1) -> c99(U91'(z0, z1))
   U91'(ok(z0), ok(z1)) -> c100(U91'(z0, z1))
   U92'(mark(z0)) -> c101(U92'(z0))
   U92'(ok(z0)) -> c102(U92'(z0))
   AND(mark(z0), z1) -> c103(AND(z0, z1))
   AND(ok(z0), ok(z1)) -> c104(AND(z0, z1))
   TAIL(mark(z0)) -> c105(TAIL(z0))
   TAIL(ok(z0)) -> c106(TAIL(z0))
   TAKE(mark(z0), z1) -> c107(TAKE(z0, z1))
   TAKE(z0, mark(z1)) -> c108(TAKE(z0, z1))
   TAKE(ok(z0), ok(z1)) -> c109(TAKE(z0, z1))
   SEL(mark(z0), z1) -> c110(SEL(z0, z1))
   SEL(z0, mark(z1)) -> c111(SEL(z0, z1))
   SEL(ok(z0), ok(z1)) -> c112(SEL(z0, z1))
   PROPER(tt) -> c113
   PROPER(nil) -> c114
   PROPER(0) -> c115
   ISNATURAL(ok(z0)) -> c116(ISNATURAL(z0))
   ISLNAT(ok(z0)) -> c117(ISLNAT(z0))
   ISPLNAT(ok(z0)) -> c118(ISPLNAT(z0))
   ISNATURALKIND(ok(z0)) -> c119(ISNATURALKIND(z0))
   ISLNATKIND(ok(z0)) -> c120(ISLNATKIND(z0))
   ISPLNATKIND(ok(z0)) -> c121(ISPLNATKIND(z0))
   TOP(mark(z0)) -> c122(TOP(proper(z0)), PROPER(z0))
   TOP(ok(z0)) -> c123(TOP(active(z0)))
S tuples:
   U101'(mark(z0), z1, z2) -> c(U101'(z0, z1, z2))
   U101'(ok(z0), ok(z1), ok(z2)) -> c1(U101'(z0, z1, z2))
   U102'(mark(z0), z1) -> c2(U102'(z0, z1))
   U102'(ok(z0), ok(z1)) -> c3(U102'(z0, z1))
   U103'(mark(z0)) -> c4(U103'(z0))
   U103'(ok(z0)) -> c5(U103'(z0))
   U11'(mark(z0), z1, z2) -> c6(U11'(z0, z1, z2))
   U11'(ok(z0), ok(z1), ok(z2)) -> c7(U11'(z0, z1, z2))
   SND(mark(z0)) -> c8(SND(z0))
   SND(ok(z0)) -> c9(SND(z0))
   SPLITAT(mark(z0), z1) -> c10(SPLITAT(z0, z1))
   SPLITAT(z0, mark(z1)) -> c11(SPLITAT(z0, z1))
   SPLITAT(ok(z0), ok(z1)) -> c12(SPLITAT(z0, z1))
   U111'(mark(z0), z1) -> c13(U111'(z0, z1))
   U111'(ok(z0), ok(z1)) -> c14(U111'(z0, z1))
   U112'(mark(z0)) -> c15(U112'(z0))
   U112'(ok(z0)) -> c16(U112'(z0))
   U121'(mark(z0), z1) -> c17(U121'(z0, z1))
   U121'(ok(z0), ok(z1)) -> c18(U121'(z0, z1))
   U122'(mark(z0)) -> c19(U122'(z0))
   U122'(ok(z0)) -> c20(U122'(z0))
   U131'(mark(z0), z1, z2) -> c21(U131'(z0, z1, z2))
   U131'(ok(z0), ok(z1), ok(z2)) -> c22(U131'(z0, z1, z2))
   U132'(mark(z0), z1) -> c23(U132'(z0, z1))
   U132'(ok(z0), ok(z1)) -> c24(U132'(z0, z1))
   U133'(mark(z0)) -> c25(U133'(z0))
   U133'(ok(z0)) -> c26(U133'(z0))
   U141'(mark(z0), z1, z2) -> c27(U141'(z0, z1, z2))
   U141'(ok(z0), ok(z1), ok(z2)) -> c28(U141'(z0, z1, z2))
   U142'(mark(z0), z1) -> c29(U142'(z0, z1))
   U142'(ok(z0), ok(z1)) -> c30(U142'(z0, z1))
   U143'(mark(z0)) -> c31(U143'(z0))
   U143'(ok(z0)) -> c32(U143'(z0))
   U151'(mark(z0), z1, z2) -> c33(U151'(z0, z1, z2))
   U151'(ok(z0), ok(z1), ok(z2)) -> c34(U151'(z0, z1, z2))
   U152'(mark(z0), z1) -> c35(U152'(z0, z1))
   U152'(ok(z0), ok(z1)) -> c36(U152'(z0, z1))
   U153'(mark(z0)) -> c37(U153'(z0))
   U153'(ok(z0)) -> c38(U153'(z0))
   U161'(mark(z0), z1) -> c39(U161'(z0, z1))
   U161'(ok(z0), ok(z1)) -> c40(U161'(z0, z1))
   CONS(mark(z0), z1) -> c41(CONS(z0, z1))
   CONS(ok(z0), ok(z1)) -> c42(CONS(z0, z1))
   NATSFROM(mark(z0)) -> c43(NATSFROM(z0))
   NATSFROM(ok(z0)) -> c44(NATSFROM(z0))
   S(mark(z0)) -> c45(S(z0))
   S(ok(z0)) -> c46(S(z0))
   U171'(mark(z0), z1, z2) -> c47(U171'(z0, z1, z2))
   U171'(ok(z0), ok(z1), ok(z2)) -> c48(U171'(z0, z1, z2))
   HEAD(mark(z0)) -> c49(HEAD(z0))
   HEAD(ok(z0)) -> c50(HEAD(z0))
   AFTERNTH(mark(z0), z1) -> c51(AFTERNTH(z0, z1))
   AFTERNTH(z0, mark(z1)) -> c52(AFTERNTH(z0, z1))
   AFTERNTH(ok(z0), ok(z1)) -> c53(AFTERNTH(z0, z1))
   U181'(mark(z0), z1) -> c54(U181'(z0, z1))
   U181'(ok(z0), ok(z1)) -> c55(U181'(z0, z1))
   U191'(mark(z0), z1) -> c56(U191'(z0, z1))
   U191'(ok(z0), ok(z1)) -> c57(U191'(z0, z1))
   PAIR(mark(z0), z1) -> c58(PAIR(z0, z1))
   PAIR(z0, mark(z1)) -> c59(PAIR(z0, z1))
   PAIR(ok(z0), ok(z1)) -> c60(PAIR(z0, z1))
   U201'(mark(z0), z1, z2, z3) -> c61(U201'(z0, z1, z2, z3))
   U201'(ok(z0), ok(z1), ok(z2), ok(z3)) -> c62(U201'(z0, z1, z2, z3))
   U202'(mark(z0), z1) -> c63(U202'(z0, z1))
   U202'(ok(z0), ok(z1)) -> c64(U202'(z0, z1))
   U21'(mark(z0), z1) -> c65(U21'(z0, z1))
   U21'(ok(z0), ok(z1)) -> c66(U21'(z0, z1))
   U211'(mark(z0), z1) -> c67(U211'(z0, z1))
   U211'(ok(z0), ok(z1)) -> c68(U211'(z0, z1))
   U221'(mark(z0), z1, z2) -> c69(U221'(z0, z1, z2))
   U221'(ok(z0), ok(z1), ok(z2)) -> c70(U221'(z0, z1, z2))
   FST(mark(z0)) -> c71(FST(z0))
   FST(ok(z0)) -> c72(FST(z0))
   U31'(mark(z0), z1) -> c73(U31'(z0, z1))
   U31'(ok(z0), ok(z1)) -> c74(U31'(z0, z1))
   U41'(mark(z0), z1, z2) -> c75(U41'(z0, z1, z2))
   U41'(ok(z0), ok(z1), ok(z2)) -> c76(U41'(z0, z1, z2))
   U42'(mark(z0), z1) -> c77(U42'(z0, z1))
   U42'(ok(z0), ok(z1)) -> c78(U42'(z0, z1))
   U43'(mark(z0)) -> c79(U43'(z0))
   U43'(ok(z0)) -> c80(U43'(z0))
   U51'(mark(z0), z1, z2) -> c81(U51'(z0, z1, z2))
   U51'(ok(z0), ok(z1), ok(z2)) -> c82(U51'(z0, z1, z2))
   U52'(mark(z0), z1) -> c83(U52'(z0, z1))
   U52'(ok(z0), ok(z1)) -> c84(U52'(z0, z1))
   U53'(mark(z0)) -> c85(U53'(z0))
   U53'(ok(z0)) -> c86(U53'(z0))
   U61'(mark(z0), z1) -> c87(U61'(z0, z1))
   U61'(ok(z0), ok(z1)) -> c88(U61'(z0, z1))
   U62'(mark(z0)) -> c89(U62'(z0))
   U62'(ok(z0)) -> c90(U62'(z0))
   U71'(mark(z0), z1) -> c91(U71'(z0, z1))
   U71'(ok(z0), ok(z1)) -> c92(U71'(z0, z1))
   U72'(mark(z0)) -> c93(U72'(z0))
   U72'(ok(z0)) -> c94(U72'(z0))
   U81'(mark(z0), z1) -> c95(U81'(z0, z1))
   U81'(ok(z0), ok(z1)) -> c96(U81'(z0, z1))
   U82'(mark(z0)) -> c97(U82'(z0))
   U82'(ok(z0)) -> c98(U82'(z0))
   U91'(mark(z0), z1) -> c99(U91'(z0, z1))
   U91'(ok(z0), ok(z1)) -> c100(U91'(z0, z1))
   U92'(mark(z0)) -> c101(U92'(z0))
   U92'(ok(z0)) -> c102(U92'(z0))
   AND(mark(z0), z1) -> c103(AND(z0, z1))
   AND(ok(z0), ok(z1)) -> c104(AND(z0, z1))
   TAIL(mark(z0)) -> c105(TAIL(z0))
   TAIL(ok(z0)) -> c106(TAIL(z0))
   TAKE(mark(z0), z1) -> c107(TAKE(z0, z1))
   TAKE(z0, mark(z1)) -> c108(TAKE(z0, z1))
   TAKE(ok(z0), ok(z1)) -> c109(TAKE(z0, z1))
   SEL(mark(z0), z1) -> c110(SEL(z0, z1))
   SEL(z0, mark(z1)) -> c111(SEL(z0, z1))
   SEL(ok(z0), ok(z1)) -> c112(SEL(z0, z1))
   PROPER(tt) -> c113
   PROPER(nil) -> c114
   PROPER(0) -> c115
   ISNATURAL(ok(z0)) -> c116(ISNATURAL(z0))
   ISLNAT(ok(z0)) -> c117(ISLNAT(z0))
   ISPLNAT(ok(z0)) -> c118(ISPLNAT(z0))
   ISNATURALKIND(ok(z0)) -> c119(ISNATURALKIND(z0))
   ISLNATKIND(ok(z0)) -> c120(ISLNATKIND(z0))
   ISPLNATKIND(ok(z0)) -> c121(ISPLNATKIND(z0))
   TOP(mark(z0)) -> c122(TOP(proper(z0)), PROPER(z0))
   TOP(ok(z0)) -> c123(TOP(active(z0)))
K tuples:none
Defined Rule Symbols:   U101_3, U102_2, U103_1, U11_3, snd_1, splitAt_2, U111_2, U112_1, U121_2, U122_1, U131_3, U132_2, U133_1, U141_3, U142_2, U143_1, U151_3, U152_2, U153_1, U161_2, cons_2, natsFrom_1, s_1, U171_3, head_1, afterNth_2, U181_2, U191_2, pair_2, U201_4, U202_2, U21_2, U211_2, U221_3, fst_1, U31_2, U41_3, U42_2, U43_1, U51_3, U52_2, U53_1, U61_2, U62_1, U71_2, U72_1, U81_2, U82_1, U91_2, U92_1, and_2, tail_1, take_2, sel_2, proper_1, isNatural_1, isLNat_1, isPLNat_1, isNaturalKind_1, isLNatKind_1, isPLNatKind_1, top_1

Defined Pair Symbols:   U101'_3, U102'_2, U103'_1, U11'_3, SND_1, SPLITAT_2, U111'_2, U112'_1, U121'_2, U122'_1, U131'_3, U132'_2, U133'_1, U141'_3, U142'_2, U143'_1, U151'_3, U152'_2, U153'_1, U161'_2, CONS_2, NATSFROM_1, S_1, U171'_3, HEAD_1, AFTERNTH_2, U181'_2, U191'_2, PAIR_2, U201'_4, U202'_2, U21'_2, U211'_2, U221'_3, FST_1, U31'_2, U41'_3, U42'_2, U43'_1, U51'_3, U52'_2, U53'_1, U61'_2, U62'_1, U71'_2, U72'_1, U81'_2, U82'_1, U91'_2, U92'_1, AND_2, TAIL_1, TAKE_2, SEL_2, PROPER_1, ISNATURAL_1, ISLNAT_1, ISPLNAT_1, ISNATURALKIND_1, ISLNATKIND_1, ISPLNATKIND_1, TOP_1

Compound Symbols:   c_1, c1_1, c2_1, c3_1, c4_1, c5_1, c6_1, c7_1, c8_1, c9_1, c10_1, c11_1, c12_1, c13_1, c14_1, c15_1, c16_1, c17_1, c18_1, c19_1, c20_1, c21_1, c22_1, c23_1, c24_1, c25_1, c26_1, c27_1, c28_1, c29_1, c30_1, c31_1, c32_1, c33_1, c34_1, c35_1, c36_1, c37_1, c38_1, c39_1, c40_1, c41_1, c42_1, c43_1, c44_1, c45_1, c46_1, c47_1, c48_1, c49_1, c50_1, c51_1, c52_1, c53_1, c54_1, c55_1, c56_1, c57_1, c58_1, c59_1, c60_1, c61_1, c62_1, c63_1, c64_1, c65_1, c66_1, c67_1, c68_1, c69_1, c70_1, c71_1, c72_1, c73_1, c74_1, c75_1, c76_1, c77_1, c78_1, c79_1, c80_1, c81_1, c82_1, c83_1, c84_1, c85_1, c86_1, c87_1, c88_1, c89_1, c90_1, c91_1, c92_1, c93_1, c94_1, c95_1, c96_1, c97_1, c98_1, c99_1, c100_1, c101_1, c102_1, c103_1, c104_1, c105_1, c106_1, c107_1, c108_1, c109_1, c110_1, c111_1, c112_1, c113, c114, c115, c116_1, c117_1, c118_1, c119_1, c120_1, c121_1, c122_2, c123_1


----------------------------------------

(7) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 4 trailing nodes:
   PROPER(nil) -> c114
   TOP(ok(z0)) -> c123(TOP(active(z0)))
   PROPER(0) -> c115
   PROPER(tt) -> c113

----------------------------------------

(8)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   U101(mark(z0), z1, z2) -> mark(U101(z0, z1, z2))
   U101(ok(z0), ok(z1), ok(z2)) -> ok(U101(z0, z1, z2))
   U102(mark(z0), z1) -> mark(U102(z0, z1))
   U102(ok(z0), ok(z1)) -> ok(U102(z0, z1))
   U103(mark(z0)) -> mark(U103(z0))
   U103(ok(z0)) -> ok(U103(z0))
   U11(mark(z0), z1, z2) -> mark(U11(z0, z1, z2))
   U11(ok(z0), ok(z1), ok(z2)) -> ok(U11(z0, z1, z2))
   snd(mark(z0)) -> mark(snd(z0))
   snd(ok(z0)) -> ok(snd(z0))
   splitAt(mark(z0), z1) -> mark(splitAt(z0, z1))
   splitAt(z0, mark(z1)) -> mark(splitAt(z0, z1))
   splitAt(ok(z0), ok(z1)) -> ok(splitAt(z0, z1))
   U111(mark(z0), z1) -> mark(U111(z0, z1))
   U111(ok(z0), ok(z1)) -> ok(U111(z0, z1))
   U112(mark(z0)) -> mark(U112(z0))
   U112(ok(z0)) -> ok(U112(z0))
   U121(mark(z0), z1) -> mark(U121(z0, z1))
   U121(ok(z0), ok(z1)) -> ok(U121(z0, z1))
   U122(mark(z0)) -> mark(U122(z0))
   U122(ok(z0)) -> ok(U122(z0))
   U131(mark(z0), z1, z2) -> mark(U131(z0, z1, z2))
   U131(ok(z0), ok(z1), ok(z2)) -> ok(U131(z0, z1, z2))
   U132(mark(z0), z1) -> mark(U132(z0, z1))
   U132(ok(z0), ok(z1)) -> ok(U132(z0, z1))
   U133(mark(z0)) -> mark(U133(z0))
   U133(ok(z0)) -> ok(U133(z0))
   U141(mark(z0), z1, z2) -> mark(U141(z0, z1, z2))
   U141(ok(z0), ok(z1), ok(z2)) -> ok(U141(z0, z1, z2))
   U142(mark(z0), z1) -> mark(U142(z0, z1))
   U142(ok(z0), ok(z1)) -> ok(U142(z0, z1))
   U143(mark(z0)) -> mark(U143(z0))
   U143(ok(z0)) -> ok(U143(z0))
   U151(mark(z0), z1, z2) -> mark(U151(z0, z1, z2))
   U151(ok(z0), ok(z1), ok(z2)) -> ok(U151(z0, z1, z2))
   U152(mark(z0), z1) -> mark(U152(z0, z1))
   U152(ok(z0), ok(z1)) -> ok(U152(z0, z1))
   U153(mark(z0)) -> mark(U153(z0))
   U153(ok(z0)) -> ok(U153(z0))
   U161(mark(z0), z1) -> mark(U161(z0, z1))
   U161(ok(z0), ok(z1)) -> ok(U161(z0, z1))
   cons(mark(z0), z1) -> mark(cons(z0, z1))
   cons(ok(z0), ok(z1)) -> ok(cons(z0, z1))
   natsFrom(mark(z0)) -> mark(natsFrom(z0))
   natsFrom(ok(z0)) -> ok(natsFrom(z0))
   s(mark(z0)) -> mark(s(z0))
   s(ok(z0)) -> ok(s(z0))
   U171(mark(z0), z1, z2) -> mark(U171(z0, z1, z2))
   U171(ok(z0), ok(z1), ok(z2)) -> ok(U171(z0, z1, z2))
   head(mark(z0)) -> mark(head(z0))
   head(ok(z0)) -> ok(head(z0))
   afterNth(mark(z0), z1) -> mark(afterNth(z0, z1))
   afterNth(z0, mark(z1)) -> mark(afterNth(z0, z1))
   afterNth(ok(z0), ok(z1)) -> ok(afterNth(z0, z1))
   U181(mark(z0), z1) -> mark(U181(z0, z1))
   U181(ok(z0), ok(z1)) -> ok(U181(z0, z1))
   U191(mark(z0), z1) -> mark(U191(z0, z1))
   U191(ok(z0), ok(z1)) -> ok(U191(z0, z1))
   pair(mark(z0), z1) -> mark(pair(z0, z1))
   pair(z0, mark(z1)) -> mark(pair(z0, z1))
   pair(ok(z0), ok(z1)) -> ok(pair(z0, z1))
   U201(mark(z0), z1, z2, z3) -> mark(U201(z0, z1, z2, z3))
   U201(ok(z0), ok(z1), ok(z2), ok(z3)) -> ok(U201(z0, z1, z2, z3))
   U202(mark(z0), z1) -> mark(U202(z0, z1))
   U202(ok(z0), ok(z1)) -> ok(U202(z0, z1))
   U21(mark(z0), z1) -> mark(U21(z0, z1))
   U21(ok(z0), ok(z1)) -> ok(U21(z0, z1))
   U211(mark(z0), z1) -> mark(U211(z0, z1))
   U211(ok(z0), ok(z1)) -> ok(U211(z0, z1))
   U221(mark(z0), z1, z2) -> mark(U221(z0, z1, z2))
   U221(ok(z0), ok(z1), ok(z2)) -> ok(U221(z0, z1, z2))
   fst(mark(z0)) -> mark(fst(z0))
   fst(ok(z0)) -> ok(fst(z0))
   U31(mark(z0), z1) -> mark(U31(z0, z1))
   U31(ok(z0), ok(z1)) -> ok(U31(z0, z1))
   U41(mark(z0), z1, z2) -> mark(U41(z0, z1, z2))
   U41(ok(z0), ok(z1), ok(z2)) -> ok(U41(z0, z1, z2))
   U42(mark(z0), z1) -> mark(U42(z0, z1))
   U42(ok(z0), ok(z1)) -> ok(U42(z0, z1))
   U43(mark(z0)) -> mark(U43(z0))
   U43(ok(z0)) -> ok(U43(z0))
   U51(mark(z0), z1, z2) -> mark(U51(z0, z1, z2))
   U51(ok(z0), ok(z1), ok(z2)) -> ok(U51(z0, z1, z2))
   U52(mark(z0), z1) -> mark(U52(z0, z1))
   U52(ok(z0), ok(z1)) -> ok(U52(z0, z1))
   U53(mark(z0)) -> mark(U53(z0))
   U53(ok(z0)) -> ok(U53(z0))
   U61(mark(z0), z1) -> mark(U61(z0, z1))
   U61(ok(z0), ok(z1)) -> ok(U61(z0, z1))
   U62(mark(z0)) -> mark(U62(z0))
   U62(ok(z0)) -> ok(U62(z0))
   U71(mark(z0), z1) -> mark(U71(z0, z1))
   U71(ok(z0), ok(z1)) -> ok(U71(z0, z1))
   U72(mark(z0)) -> mark(U72(z0))
   U72(ok(z0)) -> ok(U72(z0))
   U81(mark(z0), z1) -> mark(U81(z0, z1))
   U81(ok(z0), ok(z1)) -> ok(U81(z0, z1))
   U82(mark(z0)) -> mark(U82(z0))
   U82(ok(z0)) -> ok(U82(z0))
   U91(mark(z0), z1) -> mark(U91(z0, z1))
   U91(ok(z0), ok(z1)) -> ok(U91(z0, z1))
   U92(mark(z0)) -> mark(U92(z0))
   U92(ok(z0)) -> ok(U92(z0))
   and(mark(z0), z1) -> mark(and(z0, z1))
   and(ok(z0), ok(z1)) -> ok(and(z0, z1))
   tail(mark(z0)) -> mark(tail(z0))
   tail(ok(z0)) -> ok(tail(z0))
   take(mark(z0), z1) -> mark(take(z0, z1))
   take(z0, mark(z1)) -> mark(take(z0, z1))
   take(ok(z0), ok(z1)) -> ok(take(z0, z1))
   sel(mark(z0), z1) -> mark(sel(z0, z1))
   sel(z0, mark(z1)) -> mark(sel(z0, z1))
   sel(ok(z0), ok(z1)) -> ok(sel(z0, z1))
   proper(tt) -> ok(tt)
   proper(nil) -> ok(nil)
   proper(0) -> ok(0)
   isNatural(ok(z0)) -> ok(isNatural(z0))
   isLNat(ok(z0)) -> ok(isLNat(z0))
   isPLNat(ok(z0)) -> ok(isPLNat(z0))
   isNaturalKind(ok(z0)) -> ok(isNaturalKind(z0))
   isLNatKind(ok(z0)) -> ok(isLNatKind(z0))
   isPLNatKind(ok(z0)) -> ok(isPLNatKind(z0))
   top(mark(z0)) -> top(proper(z0))
   top(ok(z0)) -> top(active(z0))
Tuples:
   U101'(mark(z0), z1, z2) -> c(U101'(z0, z1, z2))
   U101'(ok(z0), ok(z1), ok(z2)) -> c1(U101'(z0, z1, z2))
   U102'(mark(z0), z1) -> c2(U102'(z0, z1))
   U102'(ok(z0), ok(z1)) -> c3(U102'(z0, z1))
   U103'(mark(z0)) -> c4(U103'(z0))
   U103'(ok(z0)) -> c5(U103'(z0))
   U11'(mark(z0), z1, z2) -> c6(U11'(z0, z1, z2))
   U11'(ok(z0), ok(z1), ok(z2)) -> c7(U11'(z0, z1, z2))
   SND(mark(z0)) -> c8(SND(z0))
   SND(ok(z0)) -> c9(SND(z0))
   SPLITAT(mark(z0), z1) -> c10(SPLITAT(z0, z1))
   SPLITAT(z0, mark(z1)) -> c11(SPLITAT(z0, z1))
   SPLITAT(ok(z0), ok(z1)) -> c12(SPLITAT(z0, z1))
   U111'(mark(z0), z1) -> c13(U111'(z0, z1))
   U111'(ok(z0), ok(z1)) -> c14(U111'(z0, z1))
   U112'(mark(z0)) -> c15(U112'(z0))
   U112'(ok(z0)) -> c16(U112'(z0))
   U121'(mark(z0), z1) -> c17(U121'(z0, z1))
   U121'(ok(z0), ok(z1)) -> c18(U121'(z0, z1))
   U122'(mark(z0)) -> c19(U122'(z0))
   U122'(ok(z0)) -> c20(U122'(z0))
   U131'(mark(z0), z1, z2) -> c21(U131'(z0, z1, z2))
   U131'(ok(z0), ok(z1), ok(z2)) -> c22(U131'(z0, z1, z2))
   U132'(mark(z0), z1) -> c23(U132'(z0, z1))
   U132'(ok(z0), ok(z1)) -> c24(U132'(z0, z1))
   U133'(mark(z0)) -> c25(U133'(z0))
   U133'(ok(z0)) -> c26(U133'(z0))
   U141'(mark(z0), z1, z2) -> c27(U141'(z0, z1, z2))
   U141'(ok(z0), ok(z1), ok(z2)) -> c28(U141'(z0, z1, z2))
   U142'(mark(z0), z1) -> c29(U142'(z0, z1))
   U142'(ok(z0), ok(z1)) -> c30(U142'(z0, z1))
   U143'(mark(z0)) -> c31(U143'(z0))
   U143'(ok(z0)) -> c32(U143'(z0))
   U151'(mark(z0), z1, z2) -> c33(U151'(z0, z1, z2))
   U151'(ok(z0), ok(z1), ok(z2)) -> c34(U151'(z0, z1, z2))
   U152'(mark(z0), z1) -> c35(U152'(z0, z1))
   U152'(ok(z0), ok(z1)) -> c36(U152'(z0, z1))
   U153'(mark(z0)) -> c37(U153'(z0))
   U153'(ok(z0)) -> c38(U153'(z0))
   U161'(mark(z0), z1) -> c39(U161'(z0, z1))
   U161'(ok(z0), ok(z1)) -> c40(U161'(z0, z1))
   CONS(mark(z0), z1) -> c41(CONS(z0, z1))
   CONS(ok(z0), ok(z1)) -> c42(CONS(z0, z1))
   NATSFROM(mark(z0)) -> c43(NATSFROM(z0))
   NATSFROM(ok(z0)) -> c44(NATSFROM(z0))
   S(mark(z0)) -> c45(S(z0))
   S(ok(z0)) -> c46(S(z0))
   U171'(mark(z0), z1, z2) -> c47(U171'(z0, z1, z2))
   U171'(ok(z0), ok(z1), ok(z2)) -> c48(U171'(z0, z1, z2))
   HEAD(mark(z0)) -> c49(HEAD(z0))
   HEAD(ok(z0)) -> c50(HEAD(z0))
   AFTERNTH(mark(z0), z1) -> c51(AFTERNTH(z0, z1))
   AFTERNTH(z0, mark(z1)) -> c52(AFTERNTH(z0, z1))
   AFTERNTH(ok(z0), ok(z1)) -> c53(AFTERNTH(z0, z1))
   U181'(mark(z0), z1) -> c54(U181'(z0, z1))
   U181'(ok(z0), ok(z1)) -> c55(U181'(z0, z1))
   U191'(mark(z0), z1) -> c56(U191'(z0, z1))
   U191'(ok(z0), ok(z1)) -> c57(U191'(z0, z1))
   PAIR(mark(z0), z1) -> c58(PAIR(z0, z1))
   PAIR(z0, mark(z1)) -> c59(PAIR(z0, z1))
   PAIR(ok(z0), ok(z1)) -> c60(PAIR(z0, z1))
   U201'(mark(z0), z1, z2, z3) -> c61(U201'(z0, z1, z2, z3))
   U201'(ok(z0), ok(z1), ok(z2), ok(z3)) -> c62(U201'(z0, z1, z2, z3))
   U202'(mark(z0), z1) -> c63(U202'(z0, z1))
   U202'(ok(z0), ok(z1)) -> c64(U202'(z0, z1))
   U21'(mark(z0), z1) -> c65(U21'(z0, z1))
   U21'(ok(z0), ok(z1)) -> c66(U21'(z0, z1))
   U211'(mark(z0), z1) -> c67(U211'(z0, z1))
   U211'(ok(z0), ok(z1)) -> c68(U211'(z0, z1))
   U221'(mark(z0), z1, z2) -> c69(U221'(z0, z1, z2))
   U221'(ok(z0), ok(z1), ok(z2)) -> c70(U221'(z0, z1, z2))
   FST(mark(z0)) -> c71(FST(z0))
   FST(ok(z0)) -> c72(FST(z0))
   U31'(mark(z0), z1) -> c73(U31'(z0, z1))
   U31'(ok(z0), ok(z1)) -> c74(U31'(z0, z1))
   U41'(mark(z0), z1, z2) -> c75(U41'(z0, z1, z2))
   U41'(ok(z0), ok(z1), ok(z2)) -> c76(U41'(z0, z1, z2))
   U42'(mark(z0), z1) -> c77(U42'(z0, z1))
   U42'(ok(z0), ok(z1)) -> c78(U42'(z0, z1))
   U43'(mark(z0)) -> c79(U43'(z0))
   U43'(ok(z0)) -> c80(U43'(z0))
   U51'(mark(z0), z1, z2) -> c81(U51'(z0, z1, z2))
   U51'(ok(z0), ok(z1), ok(z2)) -> c82(U51'(z0, z1, z2))
   U52'(mark(z0), z1) -> c83(U52'(z0, z1))
   U52'(ok(z0), ok(z1)) -> c84(U52'(z0, z1))
   U53'(mark(z0)) -> c85(U53'(z0))
   U53'(ok(z0)) -> c86(U53'(z0))
   U61'(mark(z0), z1) -> c87(U61'(z0, z1))
   U61'(ok(z0), ok(z1)) -> c88(U61'(z0, z1))
   U62'(mark(z0)) -> c89(U62'(z0))
   U62'(ok(z0)) -> c90(U62'(z0))
   U71'(mark(z0), z1) -> c91(U71'(z0, z1))
   U71'(ok(z0), ok(z1)) -> c92(U71'(z0, z1))
   U72'(mark(z0)) -> c93(U72'(z0))
   U72'(ok(z0)) -> c94(U72'(z0))
   U81'(mark(z0), z1) -> c95(U81'(z0, z1))
   U81'(ok(z0), ok(z1)) -> c96(U81'(z0, z1))
   U82'(mark(z0)) -> c97(U82'(z0))
   U82'(ok(z0)) -> c98(U82'(z0))
   U91'(mark(z0), z1) -> c99(U91'(z0, z1))
   U91'(ok(z0), ok(z1)) -> c100(U91'(z0, z1))
   U92'(mark(z0)) -> c101(U92'(z0))
   U92'(ok(z0)) -> c102(U92'(z0))
   AND(mark(z0), z1) -> c103(AND(z0, z1))
   AND(ok(z0), ok(z1)) -> c104(AND(z0, z1))
   TAIL(mark(z0)) -> c105(TAIL(z0))
   TAIL(ok(z0)) -> c106(TAIL(z0))
   TAKE(mark(z0), z1) -> c107(TAKE(z0, z1))
   TAKE(z0, mark(z1)) -> c108(TAKE(z0, z1))
   TAKE(ok(z0), ok(z1)) -> c109(TAKE(z0, z1))
   SEL(mark(z0), z1) -> c110(SEL(z0, z1))
   SEL(z0, mark(z1)) -> c111(SEL(z0, z1))
   SEL(ok(z0), ok(z1)) -> c112(SEL(z0, z1))
   ISNATURAL(ok(z0)) -> c116(ISNATURAL(z0))
   ISLNAT(ok(z0)) -> c117(ISLNAT(z0))
   ISPLNAT(ok(z0)) -> c118(ISPLNAT(z0))
   ISNATURALKIND(ok(z0)) -> c119(ISNATURALKIND(z0))
   ISLNATKIND(ok(z0)) -> c120(ISLNATKIND(z0))
   ISPLNATKIND(ok(z0)) -> c121(ISPLNATKIND(z0))
   TOP(mark(z0)) -> c122(TOP(proper(z0)), PROPER(z0))
S tuples:
   U101'(mark(z0), z1, z2) -> c(U101'(z0, z1, z2))
   U101'(ok(z0), ok(z1), ok(z2)) -> c1(U101'(z0, z1, z2))
   U102'(mark(z0), z1) -> c2(U102'(z0, z1))
   U102'(ok(z0), ok(z1)) -> c3(U102'(z0, z1))
   U103'(mark(z0)) -> c4(U103'(z0))
   U103'(ok(z0)) -> c5(U103'(z0))
   U11'(mark(z0), z1, z2) -> c6(U11'(z0, z1, z2))
   U11'(ok(z0), ok(z1), ok(z2)) -> c7(U11'(z0, z1, z2))
   SND(mark(z0)) -> c8(SND(z0))
   SND(ok(z0)) -> c9(SND(z0))
   SPLITAT(mark(z0), z1) -> c10(SPLITAT(z0, z1))
   SPLITAT(z0, mark(z1)) -> c11(SPLITAT(z0, z1))
   SPLITAT(ok(z0), ok(z1)) -> c12(SPLITAT(z0, z1))
   U111'(mark(z0), z1) -> c13(U111'(z0, z1))
   U111'(ok(z0), ok(z1)) -> c14(U111'(z0, z1))
   U112'(mark(z0)) -> c15(U112'(z0))
   U112'(ok(z0)) -> c16(U112'(z0))
   U121'(mark(z0), z1) -> c17(U121'(z0, z1))
   U121'(ok(z0), ok(z1)) -> c18(U121'(z0, z1))
   U122'(mark(z0)) -> c19(U122'(z0))
   U122'(ok(z0)) -> c20(U122'(z0))
   U131'(mark(z0), z1, z2) -> c21(U131'(z0, z1, z2))
   U131'(ok(z0), ok(z1), ok(z2)) -> c22(U131'(z0, z1, z2))
   U132'(mark(z0), z1) -> c23(U132'(z0, z1))
   U132'(ok(z0), ok(z1)) -> c24(U132'(z0, z1))
   U133'(mark(z0)) -> c25(U133'(z0))
   U133'(ok(z0)) -> c26(U133'(z0))
   U141'(mark(z0), z1, z2) -> c27(U141'(z0, z1, z2))
   U141'(ok(z0), ok(z1), ok(z2)) -> c28(U141'(z0, z1, z2))
   U142'(mark(z0), z1) -> c29(U142'(z0, z1))
   U142'(ok(z0), ok(z1)) -> c30(U142'(z0, z1))
   U143'(mark(z0)) -> c31(U143'(z0))
   U143'(ok(z0)) -> c32(U143'(z0))
   U151'(mark(z0), z1, z2) -> c33(U151'(z0, z1, z2))
   U151'(ok(z0), ok(z1), ok(z2)) -> c34(U151'(z0, z1, z2))
   U152'(mark(z0), z1) -> c35(U152'(z0, z1))
   U152'(ok(z0), ok(z1)) -> c36(U152'(z0, z1))
   U153'(mark(z0)) -> c37(U153'(z0))
   U153'(ok(z0)) -> c38(U153'(z0))
   U161'(mark(z0), z1) -> c39(U161'(z0, z1))
   U161'(ok(z0), ok(z1)) -> c40(U161'(z0, z1))
   CONS(mark(z0), z1) -> c41(CONS(z0, z1))
   CONS(ok(z0), ok(z1)) -> c42(CONS(z0, z1))
   NATSFROM(mark(z0)) -> c43(NATSFROM(z0))
   NATSFROM(ok(z0)) -> c44(NATSFROM(z0))
   S(mark(z0)) -> c45(S(z0))
   S(ok(z0)) -> c46(S(z0))
   U171'(mark(z0), z1, z2) -> c47(U171'(z0, z1, z2))
   U171'(ok(z0), ok(z1), ok(z2)) -> c48(U171'(z0, z1, z2))
   HEAD(mark(z0)) -> c49(HEAD(z0))
   HEAD(ok(z0)) -> c50(HEAD(z0))
   AFTERNTH(mark(z0), z1) -> c51(AFTERNTH(z0, z1))
   AFTERNTH(z0, mark(z1)) -> c52(AFTERNTH(z0, z1))
   AFTERNTH(ok(z0), ok(z1)) -> c53(AFTERNTH(z0, z1))
   U181'(mark(z0), z1) -> c54(U181'(z0, z1))
   U181'(ok(z0), ok(z1)) -> c55(U181'(z0, z1))
   U191'(mark(z0), z1) -> c56(U191'(z0, z1))
   U191'(ok(z0), ok(z1)) -> c57(U191'(z0, z1))
   PAIR(mark(z0), z1) -> c58(PAIR(z0, z1))
   PAIR(z0, mark(z1)) -> c59(PAIR(z0, z1))
   PAIR(ok(z0), ok(z1)) -> c60(PAIR(z0, z1))
   U201'(mark(z0), z1, z2, z3) -> c61(U201'(z0, z1, z2, z3))
   U201'(ok(z0), ok(z1), ok(z2), ok(z3)) -> c62(U201'(z0, z1, z2, z3))
   U202'(mark(z0), z1) -> c63(U202'(z0, z1))
   U202'(ok(z0), ok(z1)) -> c64(U202'(z0, z1))
   U21'(mark(z0), z1) -> c65(U21'(z0, z1))
   U21'(ok(z0), ok(z1)) -> c66(U21'(z0, z1))
   U211'(mark(z0), z1) -> c67(U211'(z0, z1))
   U211'(ok(z0), ok(z1)) -> c68(U211'(z0, z1))
   U221'(mark(z0), z1, z2) -> c69(U221'(z0, z1, z2))
   U221'(ok(z0), ok(z1), ok(z2)) -> c70(U221'(z0, z1, z2))
   FST(mark(z0)) -> c71(FST(z0))
   FST(ok(z0)) -> c72(FST(z0))
   U31'(mark(z0), z1) -> c73(U31'(z0, z1))
   U31'(ok(z0), ok(z1)) -> c74(U31'(z0, z1))
   U41'(mark(z0), z1, z2) -> c75(U41'(z0, z1, z2))
   U41'(ok(z0), ok(z1), ok(z2)) -> c76(U41'(z0, z1, z2))
   U42'(mark(z0), z1) -> c77(U42'(z0, z1))
   U42'(ok(z0), ok(z1)) -> c78(U42'(z0, z1))
   U43'(mark(z0)) -> c79(U43'(z0))
   U43'(ok(z0)) -> c80(U43'(z0))
   U51'(mark(z0), z1, z2) -> c81(U51'(z0, z1, z2))
   U51'(ok(z0), ok(z1), ok(z2)) -> c82(U51'(z0, z1, z2))
   U52'(mark(z0), z1) -> c83(U52'(z0, z1))
   U52'(ok(z0), ok(z1)) -> c84(U52'(z0, z1))
   U53'(mark(z0)) -> c85(U53'(z0))
   U53'(ok(z0)) -> c86(U53'(z0))
   U61'(mark(z0), z1) -> c87(U61'(z0, z1))
   U61'(ok(z0), ok(z1)) -> c88(U61'(z0, z1))
   U62'(mark(z0)) -> c89(U62'(z0))
   U62'(ok(z0)) -> c90(U62'(z0))
   U71'(mark(z0), z1) -> c91(U71'(z0, z1))
   U71'(ok(z0), ok(z1)) -> c92(U71'(z0, z1))
   U72'(mark(z0)) -> c93(U72'(z0))
   U72'(ok(z0)) -> c94(U72'(z0))
   U81'(mark(z0), z1) -> c95(U81'(z0, z1))
   U81'(ok(z0), ok(z1)) -> c96(U81'(z0, z1))
   U82'(mark(z0)) -> c97(U82'(z0))
   U82'(ok(z0)) -> c98(U82'(z0))
   U91'(mark(z0), z1) -> c99(U91'(z0, z1))
   U91'(ok(z0), ok(z1)) -> c100(U91'(z0, z1))
   U92'(mark(z0)) -> c101(U92'(z0))
   U92'(ok(z0)) -> c102(U92'(z0))
   AND(mark(z0), z1) -> c103(AND(z0, z1))
   AND(ok(z0), ok(z1)) -> c104(AND(z0, z1))
   TAIL(mark(z0)) -> c105(TAIL(z0))
   TAIL(ok(z0)) -> c106(TAIL(z0))
   TAKE(mark(z0), z1) -> c107(TAKE(z0, z1))
   TAKE(z0, mark(z1)) -> c108(TAKE(z0, z1))
   TAKE(ok(z0), ok(z1)) -> c109(TAKE(z0, z1))
   SEL(mark(z0), z1) -> c110(SEL(z0, z1))
   SEL(z0, mark(z1)) -> c111(SEL(z0, z1))
   SEL(ok(z0), ok(z1)) -> c112(SEL(z0, z1))
   ISNATURAL(ok(z0)) -> c116(ISNATURAL(z0))
   ISLNAT(ok(z0)) -> c117(ISLNAT(z0))
   ISPLNAT(ok(z0)) -> c118(ISPLNAT(z0))
   ISNATURALKIND(ok(z0)) -> c119(ISNATURALKIND(z0))
   ISLNATKIND(ok(z0)) -> c120(ISLNATKIND(z0))
   ISPLNATKIND(ok(z0)) -> c121(ISPLNATKIND(z0))
   TOP(mark(z0)) -> c122(TOP(proper(z0)), PROPER(z0))
K tuples:none
Defined Rule Symbols:   U101_3, U102_2, U103_1, U11_3, snd_1, splitAt_2, U111_2, U112_1, U121_2, U122_1, U131_3, U132_2, U133_1, U141_3, U142_2, U143_1, U151_3, U152_2, U153_1, U161_2, cons_2, natsFrom_1, s_1, U171_3, head_1, afterNth_2, U181_2, U191_2, pair_2, U201_4, U202_2, U21_2, U211_2, U221_3, fst_1, U31_2, U41_3, U42_2, U43_1, U51_3, U52_2, U53_1, U61_2, U62_1, U71_2, U72_1, U81_2, U82_1, U91_2, U92_1, and_2, tail_1, take_2, sel_2, proper_1, isNatural_1, isLNat_1, isPLNat_1, isNaturalKind_1, isLNatKind_1, isPLNatKind_1, top_1

Defined Pair Symbols:   U101'_3, U102'_2, U103'_1, U11'_3, SND_1, SPLITAT_2, U111'_2, U112'_1, U121'_2, U122'_1, U131'_3, U132'_2, U133'_1, U141'_3, U142'_2, U143'_1, U151'_3, U152'_2, U153'_1, U161'_2, CONS_2, NATSFROM_1, S_1, U171'_3, HEAD_1, AFTERNTH_2, U181'_2, U191'_2, PAIR_2, U201'_4, U202'_2, U21'_2, U211'_2, U221'_3, FST_1, U31'_2, U41'_3, U42'_2, U43'_1, U51'_3, U52'_2, U53'_1, U61'_2, U62'_1, U71'_2, U72'_1, U81'_2, U82'_1, U91'_2, U92'_1, AND_2, TAIL_1, TAKE_2, SEL_2, ISNATURAL_1, ISLNAT_1, ISPLNAT_1, ISNATURALKIND_1, ISLNATKIND_1, ISPLNATKIND_1, TOP_1

Compound Symbols:   c_1, c1_1, c2_1, c3_1, c4_1, c5_1, c6_1, c7_1, c8_1, c9_1, c10_1, c11_1, c12_1, c13_1, c14_1, c15_1, c16_1, c17_1, c18_1, c19_1, c20_1, c21_1, c22_1, c23_1, c24_1, c25_1, c26_1, c27_1, c28_1, c29_1, c30_1, c31_1, c32_1, c33_1, c34_1, c35_1, c36_1, c37_1, c38_1, c39_1, c40_1, c41_1, c42_1, c43_1, c44_1, c45_1, c46_1, c47_1, c48_1, c49_1, c50_1, c51_1, c52_1, c53_1, c54_1, c55_1, c56_1, c57_1, c58_1, c59_1, c60_1, c61_1, c62_1, c63_1, c64_1, c65_1, c66_1, c67_1, c68_1, c69_1, c70_1, c71_1, c72_1, c73_1, c74_1, c75_1, c76_1, c77_1, c78_1, c79_1, c80_1, c81_1, c82_1, c83_1, c84_1, c85_1, c86_1, c87_1, c88_1, c89_1, c90_1, c91_1, c92_1, c93_1, c94_1, c95_1, c96_1, c97_1, c98_1, c99_1, c100_1, c101_1, c102_1, c103_1, c104_1, c105_1, c106_1, c107_1, c108_1, c109_1, c110_1, c111_1, c112_1, c116_1, c117_1, c118_1, c119_1, c120_1, c121_1, c122_2


----------------------------------------

(9) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing tuple parts
----------------------------------------

(10)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   U101(mark(z0), z1, z2) -> mark(U101(z0, z1, z2))
   U101(ok(z0), ok(z1), ok(z2)) -> ok(U101(z0, z1, z2))
   U102(mark(z0), z1) -> mark(U102(z0, z1))
   U102(ok(z0), ok(z1)) -> ok(U102(z0, z1))
   U103(mark(z0)) -> mark(U103(z0))
   U103(ok(z0)) -> ok(U103(z0))
   U11(mark(z0), z1, z2) -> mark(U11(z0, z1, z2))
   U11(ok(z0), ok(z1), ok(z2)) -> ok(U11(z0, z1, z2))
   snd(mark(z0)) -> mark(snd(z0))
   snd(ok(z0)) -> ok(snd(z0))
   splitAt(mark(z0), z1) -> mark(splitAt(z0, z1))
   splitAt(z0, mark(z1)) -> mark(splitAt(z0, z1))
   splitAt(ok(z0), ok(z1)) -> ok(splitAt(z0, z1))
   U111(mark(z0), z1) -> mark(U111(z0, z1))
   U111(ok(z0), ok(z1)) -> ok(U111(z0, z1))
   U112(mark(z0)) -> mark(U112(z0))
   U112(ok(z0)) -> ok(U112(z0))
   U121(mark(z0), z1) -> mark(U121(z0, z1))
   U121(ok(z0), ok(z1)) -> ok(U121(z0, z1))
   U122(mark(z0)) -> mark(U122(z0))
   U122(ok(z0)) -> ok(U122(z0))
   U131(mark(z0), z1, z2) -> mark(U131(z0, z1, z2))
   U131(ok(z0), ok(z1), ok(z2)) -> ok(U131(z0, z1, z2))
   U132(mark(z0), z1) -> mark(U132(z0, z1))
   U132(ok(z0), ok(z1)) -> ok(U132(z0, z1))
   U133(mark(z0)) -> mark(U133(z0))
   U133(ok(z0)) -> ok(U133(z0))
   U141(mark(z0), z1, z2) -> mark(U141(z0, z1, z2))
   U141(ok(z0), ok(z1), ok(z2)) -> ok(U141(z0, z1, z2))
   U142(mark(z0), z1) -> mark(U142(z0, z1))
   U142(ok(z0), ok(z1)) -> ok(U142(z0, z1))
   U143(mark(z0)) -> mark(U143(z0))
   U143(ok(z0)) -> ok(U143(z0))
   U151(mark(z0), z1, z2) -> mark(U151(z0, z1, z2))
   U151(ok(z0), ok(z1), ok(z2)) -> ok(U151(z0, z1, z2))
   U152(mark(z0), z1) -> mark(U152(z0, z1))
   U152(ok(z0), ok(z1)) -> ok(U152(z0, z1))
   U153(mark(z0)) -> mark(U153(z0))
   U153(ok(z0)) -> ok(U153(z0))
   U161(mark(z0), z1) -> mark(U161(z0, z1))
   U161(ok(z0), ok(z1)) -> ok(U161(z0, z1))
   cons(mark(z0), z1) -> mark(cons(z0, z1))
   cons(ok(z0), ok(z1)) -> ok(cons(z0, z1))
   natsFrom(mark(z0)) -> mark(natsFrom(z0))
   natsFrom(ok(z0)) -> ok(natsFrom(z0))
   s(mark(z0)) -> mark(s(z0))
   s(ok(z0)) -> ok(s(z0))
   U171(mark(z0), z1, z2) -> mark(U171(z0, z1, z2))
   U171(ok(z0), ok(z1), ok(z2)) -> ok(U171(z0, z1, z2))
   head(mark(z0)) -> mark(head(z0))
   head(ok(z0)) -> ok(head(z0))
   afterNth(mark(z0), z1) -> mark(afterNth(z0, z1))
   afterNth(z0, mark(z1)) -> mark(afterNth(z0, z1))
   afterNth(ok(z0), ok(z1)) -> ok(afterNth(z0, z1))
   U181(mark(z0), z1) -> mark(U181(z0, z1))
   U181(ok(z0), ok(z1)) -> ok(U181(z0, z1))
   U191(mark(z0), z1) -> mark(U191(z0, z1))
   U191(ok(z0), ok(z1)) -> ok(U191(z0, z1))
   pair(mark(z0), z1) -> mark(pair(z0, z1))
   pair(z0, mark(z1)) -> mark(pair(z0, z1))
   pair(ok(z0), ok(z1)) -> ok(pair(z0, z1))
   U201(mark(z0), z1, z2, z3) -> mark(U201(z0, z1, z2, z3))
   U201(ok(z0), ok(z1), ok(z2), ok(z3)) -> ok(U201(z0, z1, z2, z3))
   U202(mark(z0), z1) -> mark(U202(z0, z1))
   U202(ok(z0), ok(z1)) -> ok(U202(z0, z1))
   U21(mark(z0), z1) -> mark(U21(z0, z1))
   U21(ok(z0), ok(z1)) -> ok(U21(z0, z1))
   U211(mark(z0), z1) -> mark(U211(z0, z1))
   U211(ok(z0), ok(z1)) -> ok(U211(z0, z1))
   U221(mark(z0), z1, z2) -> mark(U221(z0, z1, z2))
   U221(ok(z0), ok(z1), ok(z2)) -> ok(U221(z0, z1, z2))
   fst(mark(z0)) -> mark(fst(z0))
   fst(ok(z0)) -> ok(fst(z0))
   U31(mark(z0), z1) -> mark(U31(z0, z1))
   U31(ok(z0), ok(z1)) -> ok(U31(z0, z1))
   U41(mark(z0), z1, z2) -> mark(U41(z0, z1, z2))
   U41(ok(z0), ok(z1), ok(z2)) -> ok(U41(z0, z1, z2))
   U42(mark(z0), z1) -> mark(U42(z0, z1))
   U42(ok(z0), ok(z1)) -> ok(U42(z0, z1))
   U43(mark(z0)) -> mark(U43(z0))
   U43(ok(z0)) -> ok(U43(z0))
   U51(mark(z0), z1, z2) -> mark(U51(z0, z1, z2))
   U51(ok(z0), ok(z1), ok(z2)) -> ok(U51(z0, z1, z2))
   U52(mark(z0), z1) -> mark(U52(z0, z1))
   U52(ok(z0), ok(z1)) -> ok(U52(z0, z1))
   U53(mark(z0)) -> mark(U53(z0))
   U53(ok(z0)) -> ok(U53(z0))
   U61(mark(z0), z1) -> mark(U61(z0, z1))
   U61(ok(z0), ok(z1)) -> ok(U61(z0, z1))
   U62(mark(z0)) -> mark(U62(z0))
   U62(ok(z0)) -> ok(U62(z0))
   U71(mark(z0), z1) -> mark(U71(z0, z1))
   U71(ok(z0), ok(z1)) -> ok(U71(z0, z1))
   U72(mark(z0)) -> mark(U72(z0))
   U72(ok(z0)) -> ok(U72(z0))
   U81(mark(z0), z1) -> mark(U81(z0, z1))
   U81(ok(z0), ok(z1)) -> ok(U81(z0, z1))
   U82(mark(z0)) -> mark(U82(z0))
   U82(ok(z0)) -> ok(U82(z0))
   U91(mark(z0), z1) -> mark(U91(z0, z1))
   U91(ok(z0), ok(z1)) -> ok(U91(z0, z1))
   U92(mark(z0)) -> mark(U92(z0))
   U92(ok(z0)) -> ok(U92(z0))
   and(mark(z0), z1) -> mark(and(z0, z1))
   and(ok(z0), ok(z1)) -> ok(and(z0, z1))
   tail(mark(z0)) -> mark(tail(z0))
   tail(ok(z0)) -> ok(tail(z0))
   take(mark(z0), z1) -> mark(take(z0, z1))
   take(z0, mark(z1)) -> mark(take(z0, z1))
   take(ok(z0), ok(z1)) -> ok(take(z0, z1))
   sel(mark(z0), z1) -> mark(sel(z0, z1))
   sel(z0, mark(z1)) -> mark(sel(z0, z1))
   sel(ok(z0), ok(z1)) -> ok(sel(z0, z1))
   proper(tt) -> ok(tt)
   proper(nil) -> ok(nil)
   proper(0) -> ok(0)
   isNatural(ok(z0)) -> ok(isNatural(z0))
   isLNat(ok(z0)) -> ok(isLNat(z0))
   isPLNat(ok(z0)) -> ok(isPLNat(z0))
   isNaturalKind(ok(z0)) -> ok(isNaturalKind(z0))
   isLNatKind(ok(z0)) -> ok(isLNatKind(z0))
   isPLNatKind(ok(z0)) -> ok(isPLNatKind(z0))
   top(mark(z0)) -> top(proper(z0))
   top(ok(z0)) -> top(active(z0))
Tuples:
   U101'(mark(z0), z1, z2) -> c(U101'(z0, z1, z2))
   U101'(ok(z0), ok(z1), ok(z2)) -> c1(U101'(z0, z1, z2))
   U102'(mark(z0), z1) -> c2(U102'(z0, z1))
   U102'(ok(z0), ok(z1)) -> c3(U102'(z0, z1))
   U103'(mark(z0)) -> c4(U103'(z0))
   U103'(ok(z0)) -> c5(U103'(z0))
   U11'(mark(z0), z1, z2) -> c6(U11'(z0, z1, z2))
   U11'(ok(z0), ok(z1), ok(z2)) -> c7(U11'(z0, z1, z2))
   SND(mark(z0)) -> c8(SND(z0))
   SND(ok(z0)) -> c9(SND(z0))
   SPLITAT(mark(z0), z1) -> c10(SPLITAT(z0, z1))
   SPLITAT(z0, mark(z1)) -> c11(SPLITAT(z0, z1))
   SPLITAT(ok(z0), ok(z1)) -> c12(SPLITAT(z0, z1))
   U111'(mark(z0), z1) -> c13(U111'(z0, z1))
   U111'(ok(z0), ok(z1)) -> c14(U111'(z0, z1))
   U112'(mark(z0)) -> c15(U112'(z0))
   U112'(ok(z0)) -> c16(U112'(z0))
   U121'(mark(z0), z1) -> c17(U121'(z0, z1))
   U121'(ok(z0), ok(z1)) -> c18(U121'(z0, z1))
   U122'(mark(z0)) -> c19(U122'(z0))
   U122'(ok(z0)) -> c20(U122'(z0))
   U131'(mark(z0), z1, z2) -> c21(U131'(z0, z1, z2))
   U131'(ok(z0), ok(z1), ok(z2)) -> c22(U131'(z0, z1, z2))
   U132'(mark(z0), z1) -> c23(U132'(z0, z1))
   U132'(ok(z0), ok(z1)) -> c24(U132'(z0, z1))
   U133'(mark(z0)) -> c25(U133'(z0))
   U133'(ok(z0)) -> c26(U133'(z0))
   U141'(mark(z0), z1, z2) -> c27(U141'(z0, z1, z2))
   U141'(ok(z0), ok(z1), ok(z2)) -> c28(U141'(z0, z1, z2))
   U142'(mark(z0), z1) -> c29(U142'(z0, z1))
   U142'(ok(z0), ok(z1)) -> c30(U142'(z0, z1))
   U143'(mark(z0)) -> c31(U143'(z0))
   U143'(ok(z0)) -> c32(U143'(z0))
   U151'(mark(z0), z1, z2) -> c33(U151'(z0, z1, z2))
   U151'(ok(z0), ok(z1), ok(z2)) -> c34(U151'(z0, z1, z2))
   U152'(mark(z0), z1) -> c35(U152'(z0, z1))
   U152'(ok(z0), ok(z1)) -> c36(U152'(z0, z1))
   U153'(mark(z0)) -> c37(U153'(z0))
   U153'(ok(z0)) -> c38(U153'(z0))
   U161'(mark(z0), z1) -> c39(U161'(z0, z1))
   U161'(ok(z0), ok(z1)) -> c40(U161'(z0, z1))
   CONS(mark(z0), z1) -> c41(CONS(z0, z1))
   CONS(ok(z0), ok(z1)) -> c42(CONS(z0, z1))
   NATSFROM(mark(z0)) -> c43(NATSFROM(z0))
   NATSFROM(ok(z0)) -> c44(NATSFROM(z0))
   S(mark(z0)) -> c45(S(z0))
   S(ok(z0)) -> c46(S(z0))
   U171'(mark(z0), z1, z2) -> c47(U171'(z0, z1, z2))
   U171'(ok(z0), ok(z1), ok(z2)) -> c48(U171'(z0, z1, z2))
   HEAD(mark(z0)) -> c49(HEAD(z0))
   HEAD(ok(z0)) -> c50(HEAD(z0))
   AFTERNTH(mark(z0), z1) -> c51(AFTERNTH(z0, z1))
   AFTERNTH(z0, mark(z1)) -> c52(AFTERNTH(z0, z1))
   AFTERNTH(ok(z0), ok(z1)) -> c53(AFTERNTH(z0, z1))
   U181'(mark(z0), z1) -> c54(U181'(z0, z1))
   U181'(ok(z0), ok(z1)) -> c55(U181'(z0, z1))
   U191'(mark(z0), z1) -> c56(U191'(z0, z1))
   U191'(ok(z0), ok(z1)) -> c57(U191'(z0, z1))
   PAIR(mark(z0), z1) -> c58(PAIR(z0, z1))
   PAIR(z0, mark(z1)) -> c59(PAIR(z0, z1))
   PAIR(ok(z0), ok(z1)) -> c60(PAIR(z0, z1))
   U201'(mark(z0), z1, z2, z3) -> c61(U201'(z0, z1, z2, z3))
   U201'(ok(z0), ok(z1), ok(z2), ok(z3)) -> c62(U201'(z0, z1, z2, z3))
   U202'(mark(z0), z1) -> c63(U202'(z0, z1))
   U202'(ok(z0), ok(z1)) -> c64(U202'(z0, z1))
   U21'(mark(z0), z1) -> c65(U21'(z0, z1))
   U21'(ok(z0), ok(z1)) -> c66(U21'(z0, z1))
   U211'(mark(z0), z1) -> c67(U211'(z0, z1))
   U211'(ok(z0), ok(z1)) -> c68(U211'(z0, z1))
   U221'(mark(z0), z1, z2) -> c69(U221'(z0, z1, z2))
   U221'(ok(z0), ok(z1), ok(z2)) -> c70(U221'(z0, z1, z2))
   FST(mark(z0)) -> c71(FST(z0))
   FST(ok(z0)) -> c72(FST(z0))
   U31'(mark(z0), z1) -> c73(U31'(z0, z1))
   U31'(ok(z0), ok(z1)) -> c74(U31'(z0, z1))
   U41'(mark(z0), z1, z2) -> c75(U41'(z0, z1, z2))
   U41'(ok(z0), ok(z1), ok(z2)) -> c76(U41'(z0, z1, z2))
   U42'(mark(z0), z1) -> c77(U42'(z0, z1))
   U42'(ok(z0), ok(z1)) -> c78(U42'(z0, z1))
   U43'(mark(z0)) -> c79(U43'(z0))
   U43'(ok(z0)) -> c80(U43'(z0))
   U51'(mark(z0), z1, z2) -> c81(U51'(z0, z1, z2))
   U51'(ok(z0), ok(z1), ok(z2)) -> c82(U51'(z0, z1, z2))
   U52'(mark(z0), z1) -> c83(U52'(z0, z1))
   U52'(ok(z0), ok(z1)) -> c84(U52'(z0, z1))
   U53'(mark(z0)) -> c85(U53'(z0))
   U53'(ok(z0)) -> c86(U53'(z0))
   U61'(mark(z0), z1) -> c87(U61'(z0, z1))
   U61'(ok(z0), ok(z1)) -> c88(U61'(z0, z1))
   U62'(mark(z0)) -> c89(U62'(z0))
   U62'(ok(z0)) -> c90(U62'(z0))
   U71'(mark(z0), z1) -> c91(U71'(z0, z1))
   U71'(ok(z0), ok(z1)) -> c92(U71'(z0, z1))
   U72'(mark(z0)) -> c93(U72'(z0))
   U72'(ok(z0)) -> c94(U72'(z0))
   U81'(mark(z0), z1) -> c95(U81'(z0, z1))
   U81'(ok(z0), ok(z1)) -> c96(U81'(z0, z1))
   U82'(mark(z0)) -> c97(U82'(z0))
   U82'(ok(z0)) -> c98(U82'(z0))
   U91'(mark(z0), z1) -> c99(U91'(z0, z1))
   U91'(ok(z0), ok(z1)) -> c100(U91'(z0, z1))
   U92'(mark(z0)) -> c101(U92'(z0))
   U92'(ok(z0)) -> c102(U92'(z0))
   AND(mark(z0), z1) -> c103(AND(z0, z1))
   AND(ok(z0), ok(z1)) -> c104(AND(z0, z1))
   TAIL(mark(z0)) -> c105(TAIL(z0))
   TAIL(ok(z0)) -> c106(TAIL(z0))
   TAKE(mark(z0), z1) -> c107(TAKE(z0, z1))
   TAKE(z0, mark(z1)) -> c108(TAKE(z0, z1))
   TAKE(ok(z0), ok(z1)) -> c109(TAKE(z0, z1))
   SEL(mark(z0), z1) -> c110(SEL(z0, z1))
   SEL(z0, mark(z1)) -> c111(SEL(z0, z1))
   SEL(ok(z0), ok(z1)) -> c112(SEL(z0, z1))
   ISNATURAL(ok(z0)) -> c116(ISNATURAL(z0))
   ISLNAT(ok(z0)) -> c117(ISLNAT(z0))
   ISPLNAT(ok(z0)) -> c118(ISPLNAT(z0))
   ISNATURALKIND(ok(z0)) -> c119(ISNATURALKIND(z0))
   ISLNATKIND(ok(z0)) -> c120(ISLNATKIND(z0))
   ISPLNATKIND(ok(z0)) -> c121(ISPLNATKIND(z0))
   TOP(mark(z0)) -> c122(TOP(proper(z0)))
S tuples:
   U101'(mark(z0), z1, z2) -> c(U101'(z0, z1, z2))
   U101'(ok(z0), ok(z1), ok(z2)) -> c1(U101'(z0, z1, z2))
   U102'(mark(z0), z1) -> c2(U102'(z0, z1))
   U102'(ok(z0), ok(z1)) -> c3(U102'(z0, z1))
   U103'(mark(z0)) -> c4(U103'(z0))
   U103'(ok(z0)) -> c5(U103'(z0))
   U11'(mark(z0), z1, z2) -> c6(U11'(z0, z1, z2))
   U11'(ok(z0), ok(z1), ok(z2)) -> c7(U11'(z0, z1, z2))
   SND(mark(z0)) -> c8(SND(z0))
   SND(ok(z0)) -> c9(SND(z0))
   SPLITAT(mark(z0), z1) -> c10(SPLITAT(z0, z1))
   SPLITAT(z0, mark(z1)) -> c11(SPLITAT(z0, z1))
   SPLITAT(ok(z0), ok(z1)) -> c12(SPLITAT(z0, z1))
   U111'(mark(z0), z1) -> c13(U111'(z0, z1))
   U111'(ok(z0), ok(z1)) -> c14(U111'(z0, z1))
   U112'(mark(z0)) -> c15(U112'(z0))
   U112'(ok(z0)) -> c16(U112'(z0))
   U121'(mark(z0), z1) -> c17(U121'(z0, z1))
   U121'(ok(z0), ok(z1)) -> c18(U121'(z0, z1))
   U122'(mark(z0)) -> c19(U122'(z0))
   U122'(ok(z0)) -> c20(U122'(z0))
   U131'(mark(z0), z1, z2) -> c21(U131'(z0, z1, z2))
   U131'(ok(z0), ok(z1), ok(z2)) -> c22(U131'(z0, z1, z2))
   U132'(mark(z0), z1) -> c23(U132'(z0, z1))
   U132'(ok(z0), ok(z1)) -> c24(U132'(z0, z1))
   U133'(mark(z0)) -> c25(U133'(z0))
   U133'(ok(z0)) -> c26(U133'(z0))
   U141'(mark(z0), z1, z2) -> c27(U141'(z0, z1, z2))
   U141'(ok(z0), ok(z1), ok(z2)) -> c28(U141'(z0, z1, z2))
   U142'(mark(z0), z1) -> c29(U142'(z0, z1))
   U142'(ok(z0), ok(z1)) -> c30(U142'(z0, z1))
   U143'(mark(z0)) -> c31(U143'(z0))
   U143'(ok(z0)) -> c32(U143'(z0))
   U151'(mark(z0), z1, z2) -> c33(U151'(z0, z1, z2))
   U151'(ok(z0), ok(z1), ok(z2)) -> c34(U151'(z0, z1, z2))
   U152'(mark(z0), z1) -> c35(U152'(z0, z1))
   U152'(ok(z0), ok(z1)) -> c36(U152'(z0, z1))
   U153'(mark(z0)) -> c37(U153'(z0))
   U153'(ok(z0)) -> c38(U153'(z0))
   U161'(mark(z0), z1) -> c39(U161'(z0, z1))
   U161'(ok(z0), ok(z1)) -> c40(U161'(z0, z1))
   CONS(mark(z0), z1) -> c41(CONS(z0, z1))
   CONS(ok(z0), ok(z1)) -> c42(CONS(z0, z1))
   NATSFROM(mark(z0)) -> c43(NATSFROM(z0))
   NATSFROM(ok(z0)) -> c44(NATSFROM(z0))
   S(mark(z0)) -> c45(S(z0))
   S(ok(z0)) -> c46(S(z0))
   U171'(mark(z0), z1, z2) -> c47(U171'(z0, z1, z2))
   U171'(ok(z0), ok(z1), ok(z2)) -> c48(U171'(z0, z1, z2))
   HEAD(mark(z0)) -> c49(HEAD(z0))
   HEAD(ok(z0)) -> c50(HEAD(z0))
   AFTERNTH(mark(z0), z1) -> c51(AFTERNTH(z0, z1))
   AFTERNTH(z0, mark(z1)) -> c52(AFTERNTH(z0, z1))
   AFTERNTH(ok(z0), ok(z1)) -> c53(AFTERNTH(z0, z1))
   U181'(mark(z0), z1) -> c54(U181'(z0, z1))
   U181'(ok(z0), ok(z1)) -> c55(U181'(z0, z1))
   U191'(mark(z0), z1) -> c56(U191'(z0, z1))
   U191'(ok(z0), ok(z1)) -> c57(U191'(z0, z1))
   PAIR(mark(z0), z1) -> c58(PAIR(z0, z1))
   PAIR(z0, mark(z1)) -> c59(PAIR(z0, z1))
   PAIR(ok(z0), ok(z1)) -> c60(PAIR(z0, z1))
   U201'(mark(z0), z1, z2, z3) -> c61(U201'(z0, z1, z2, z3))
   U201'(ok(z0), ok(z1), ok(z2), ok(z3)) -> c62(U201'(z0, z1, z2, z3))
   U202'(mark(z0), z1) -> c63(U202'(z0, z1))
   U202'(ok(z0), ok(z1)) -> c64(U202'(z0, z1))
   U21'(mark(z0), z1) -> c65(U21'(z0, z1))
   U21'(ok(z0), ok(z1)) -> c66(U21'(z0, z1))
   U211'(mark(z0), z1) -> c67(U211'(z0, z1))
   U211'(ok(z0), ok(z1)) -> c68(U211'(z0, z1))
   U221'(mark(z0), z1, z2) -> c69(U221'(z0, z1, z2))
   U221'(ok(z0), ok(z1), ok(z2)) -> c70(U221'(z0, z1, z2))
   FST(mark(z0)) -> c71(FST(z0))
   FST(ok(z0)) -> c72(FST(z0))
   U31'(mark(z0), z1) -> c73(U31'(z0, z1))
   U31'(ok(z0), ok(z1)) -> c74(U31'(z0, z1))
   U41'(mark(z0), z1, z2) -> c75(U41'(z0, z1, z2))
   U41'(ok(z0), ok(z1), ok(z2)) -> c76(U41'(z0, z1, z2))
   U42'(mark(z0), z1) -> c77(U42'(z0, z1))
   U42'(ok(z0), ok(z1)) -> c78(U42'(z0, z1))
   U43'(mark(z0)) -> c79(U43'(z0))
   U43'(ok(z0)) -> c80(U43'(z0))
   U51'(mark(z0), z1, z2) -> c81(U51'(z0, z1, z2))
   U51'(ok(z0), ok(z1), ok(z2)) -> c82(U51'(z0, z1, z2))
   U52'(mark(z0), z1) -> c83(U52'(z0, z1))
   U52'(ok(z0), ok(z1)) -> c84(U52'(z0, z1))
   U53'(mark(z0)) -> c85(U53'(z0))
   U53'(ok(z0)) -> c86(U53'(z0))
   U61'(mark(z0), z1) -> c87(U61'(z0, z1))
   U61'(ok(z0), ok(z1)) -> c88(U61'(z0, z1))
   U62'(mark(z0)) -> c89(U62'(z0))
   U62'(ok(z0)) -> c90(U62'(z0))
   U71'(mark(z0), z1) -> c91(U71'(z0, z1))
   U71'(ok(z0), ok(z1)) -> c92(U71'(z0, z1))
   U72'(mark(z0)) -> c93(U72'(z0))
   U72'(ok(z0)) -> c94(U72'(z0))
   U81'(mark(z0), z1) -> c95(U81'(z0, z1))
   U81'(ok(z0), ok(z1)) -> c96(U81'(z0, z1))
   U82'(mark(z0)) -> c97(U82'(z0))
   U82'(ok(z0)) -> c98(U82'(z0))
   U91'(mark(z0), z1) -> c99(U91'(z0, z1))
   U91'(ok(z0), ok(z1)) -> c100(U91'(z0, z1))
   U92'(mark(z0)) -> c101(U92'(z0))
   U92'(ok(z0)) -> c102(U92'(z0))
   AND(mark(z0), z1) -> c103(AND(z0, z1))
   AND(ok(z0), ok(z1)) -> c104(AND(z0, z1))
   TAIL(mark(z0)) -> c105(TAIL(z0))
   TAIL(ok(z0)) -> c106(TAIL(z0))
   TAKE(mark(z0), z1) -> c107(TAKE(z0, z1))
   TAKE(z0, mark(z1)) -> c108(TAKE(z0, z1))
   TAKE(ok(z0), ok(z1)) -> c109(TAKE(z0, z1))
   SEL(mark(z0), z1) -> c110(SEL(z0, z1))
   SEL(z0, mark(z1)) -> c111(SEL(z0, z1))
   SEL(ok(z0), ok(z1)) -> c112(SEL(z0, z1))
   ISNATURAL(ok(z0)) -> c116(ISNATURAL(z0))
   ISLNAT(ok(z0)) -> c117(ISLNAT(z0))
   ISPLNAT(ok(z0)) -> c118(ISPLNAT(z0))
   ISNATURALKIND(ok(z0)) -> c119(ISNATURALKIND(z0))
   ISLNATKIND(ok(z0)) -> c120(ISLNATKIND(z0))
   ISPLNATKIND(ok(z0)) -> c121(ISPLNATKIND(z0))
   TOP(mark(z0)) -> c122(TOP(proper(z0)))
K tuples:none
Defined Rule Symbols:   U101_3, U102_2, U103_1, U11_3, snd_1, splitAt_2, U111_2, U112_1, U121_2, U122_1, U131_3, U132_2, U133_1, U141_3, U142_2, U143_1, U151_3, U152_2, U153_1, U161_2, cons_2, natsFrom_1, s_1, U171_3, head_1, afterNth_2, U181_2, U191_2, pair_2, U201_4, U202_2, U21_2, U211_2, U221_3, fst_1, U31_2, U41_3, U42_2, U43_1, U51_3, U52_2, U53_1, U61_2, U62_1, U71_2, U72_1, U81_2, U82_1, U91_2, U92_1, and_2, tail_1, take_2, sel_2, proper_1, isNatural_1, isLNat_1, isPLNat_1, isNaturalKind_1, isLNatKind_1, isPLNatKind_1, top_1

Defined Pair Symbols:   U101'_3, U102'_2, U103'_1, U11'_3, SND_1, SPLITAT_2, U111'_2, U112'_1, U121'_2, U122'_1, U131'_3, U132'_2, U133'_1, U141'_3, U142'_2, U143'_1, U151'_3, U152'_2, U153'_1, U161'_2, CONS_2, NATSFROM_1, S_1, U171'_3, HEAD_1, AFTERNTH_2, U181'_2, U191'_2, PAIR_2, U201'_4, U202'_2, U21'_2, U211'_2, U221'_3, FST_1, U31'_2, U41'_3, U42'_2, U43'_1, U51'_3, U52'_2, U53'_1, U61'_2, U62'_1, U71'_2, U72'_1, U81'_2, U82'_1, U91'_2, U92'_1, AND_2, TAIL_1, TAKE_2, SEL_2, ISNATURAL_1, ISLNAT_1, ISPLNAT_1, ISNATURALKIND_1, ISLNATKIND_1, ISPLNATKIND_1, TOP_1

Compound Symbols:   c_1, c1_1, c2_1, c3_1, c4_1, c5_1, c6_1, c7_1, c8_1, c9_1, c10_1, c11_1, c12_1, c13_1, c14_1, c15_1, c16_1, c17_1, c18_1, c19_1, c20_1, c21_1, c22_1, c23_1, c24_1, c25_1, c26_1, c27_1, c28_1, c29_1, c30_1, c31_1, c32_1, c33_1, c34_1, c35_1, c36_1, c37_1, c38_1, c39_1, c40_1, c41_1, c42_1, c43_1, c44_1, c45_1, c46_1, c47_1, c48_1, c49_1, c50_1, c51_1, c52_1, c53_1, c54_1, c55_1, c56_1, c57_1, c58_1, c59_1, c60_1, c61_1, c62_1, c63_1, c64_1, c65_1, c66_1, c67_1, c68_1, c69_1, c70_1, c71_1, c72_1, c73_1, c74_1, c75_1, c76_1, c77_1, c78_1, c79_1, c80_1, c81_1, c82_1, c83_1, c84_1, c85_1, c86_1, c87_1, c88_1, c89_1, c90_1, c91_1, c92_1, c93_1, c94_1, c95_1, c96_1, c97_1, c98_1, c99_1, c100_1, c101_1, c102_1, c103_1, c104_1, c105_1, c106_1, c107_1, c108_1, c109_1, c110_1, c111_1, c112_1, c116_1, c117_1, c118_1, c119_1, c120_1, c121_1, c122_1


----------------------------------------

(11) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   U101(mark(z0), z1, z2) -> mark(U101(z0, z1, z2))
   U101(ok(z0), ok(z1), ok(z2)) -> ok(U101(z0, z1, z2))
   U102(mark(z0), z1) -> mark(U102(z0, z1))
   U102(ok(z0), ok(z1)) -> ok(U102(z0, z1))
   U103(mark(z0)) -> mark(U103(z0))
   U103(ok(z0)) -> ok(U103(z0))
   U11(mark(z0), z1, z2) -> mark(U11(z0, z1, z2))
   U11(ok(z0), ok(z1), ok(z2)) -> ok(U11(z0, z1, z2))
   snd(mark(z0)) -> mark(snd(z0))
   snd(ok(z0)) -> ok(snd(z0))
   splitAt(mark(z0), z1) -> mark(splitAt(z0, z1))
   splitAt(z0, mark(z1)) -> mark(splitAt(z0, z1))
   splitAt(ok(z0), ok(z1)) -> ok(splitAt(z0, z1))
   U111(mark(z0), z1) -> mark(U111(z0, z1))
   U111(ok(z0), ok(z1)) -> ok(U111(z0, z1))
   U112(mark(z0)) -> mark(U112(z0))
   U112(ok(z0)) -> ok(U112(z0))
   U121(mark(z0), z1) -> mark(U121(z0, z1))
   U121(ok(z0), ok(z1)) -> ok(U121(z0, z1))
   U122(mark(z0)) -> mark(U122(z0))
   U122(ok(z0)) -> ok(U122(z0))
   U131(mark(z0), z1, z2) -> mark(U131(z0, z1, z2))
   U131(ok(z0), ok(z1), ok(z2)) -> ok(U131(z0, z1, z2))
   U132(mark(z0), z1) -> mark(U132(z0, z1))
   U132(ok(z0), ok(z1)) -> ok(U132(z0, z1))
   U133(mark(z0)) -> mark(U133(z0))
   U133(ok(z0)) -> ok(U133(z0))
   U141(mark(z0), z1, z2) -> mark(U141(z0, z1, z2))
   U141(ok(z0), ok(z1), ok(z2)) -> ok(U141(z0, z1, z2))
   U142(mark(z0), z1) -> mark(U142(z0, z1))
   U142(ok(z0), ok(z1)) -> ok(U142(z0, z1))
   U143(mark(z0)) -> mark(U143(z0))
   U143(ok(z0)) -> ok(U143(z0))
   U151(mark(z0), z1, z2) -> mark(U151(z0, z1, z2))
   U151(ok(z0), ok(z1), ok(z2)) -> ok(U151(z0, z1, z2))
   U152(mark(z0), z1) -> mark(U152(z0, z1))
   U152(ok(z0), ok(z1)) -> ok(U152(z0, z1))
   U153(mark(z0)) -> mark(U153(z0))
   U153(ok(z0)) -> ok(U153(z0))
   U161(mark(z0), z1) -> mark(U161(z0, z1))
   U161(ok(z0), ok(z1)) -> ok(U161(z0, z1))
   cons(mark(z0), z1) -> mark(cons(z0, z1))
   cons(ok(z0), ok(z1)) -> ok(cons(z0, z1))
   natsFrom(mark(z0)) -> mark(natsFrom(z0))
   natsFrom(ok(z0)) -> ok(natsFrom(z0))
   s(mark(z0)) -> mark(s(z0))
   s(ok(z0)) -> ok(s(z0))
   U171(mark(z0), z1, z2) -> mark(U171(z0, z1, z2))
   U171(ok(z0), ok(z1), ok(z2)) -> ok(U171(z0, z1, z2))
   head(mark(z0)) -> mark(head(z0))
   head(ok(z0)) -> ok(head(z0))
   afterNth(mark(z0), z1) -> mark(afterNth(z0, z1))
   afterNth(z0, mark(z1)) -> mark(afterNth(z0, z1))
   afterNth(ok(z0), ok(z1)) -> ok(afterNth(z0, z1))
   U181(mark(z0), z1) -> mark(U181(z0, z1))
   U181(ok(z0), ok(z1)) -> ok(U181(z0, z1))
   U191(mark(z0), z1) -> mark(U191(z0, z1))
   U191(ok(z0), ok(z1)) -> ok(U191(z0, z1))
   pair(mark(z0), z1) -> mark(pair(z0, z1))
   pair(z0, mark(z1)) -> mark(pair(z0, z1))
   pair(ok(z0), ok(z1)) -> ok(pair(z0, z1))
   U201(mark(z0), z1, z2, z3) -> mark(U201(z0, z1, z2, z3))
   U201(ok(z0), ok(z1), ok(z2), ok(z3)) -> ok(U201(z0, z1, z2, z3))
   U202(mark(z0), z1) -> mark(U202(z0, z1))
   U202(ok(z0), ok(z1)) -> ok(U202(z0, z1))
   U21(mark(z0), z1) -> mark(U21(z0, z1))
   U21(ok(z0), ok(z1)) -> ok(U21(z0, z1))
   U211(mark(z0), z1) -> mark(U211(z0, z1))
   U211(ok(z0), ok(z1)) -> ok(U211(z0, z1))
   U221(mark(z0), z1, z2) -> mark(U221(z0, z1, z2))
   U221(ok(z0), ok(z1), ok(z2)) -> ok(U221(z0, z1, z2))
   fst(mark(z0)) -> mark(fst(z0))
   fst(ok(z0)) -> ok(fst(z0))
   U31(mark(z0), z1) -> mark(U31(z0, z1))
   U31(ok(z0), ok(z1)) -> ok(U31(z0, z1))
   U41(mark(z0), z1, z2) -> mark(U41(z0, z1, z2))
   U41(ok(z0), ok(z1), ok(z2)) -> ok(U41(z0, z1, z2))
   U42(mark(z0), z1) -> mark(U42(z0, z1))
   U42(ok(z0), ok(z1)) -> ok(U42(z0, z1))
   U43(mark(z0)) -> mark(U43(z0))
   U43(ok(z0)) -> ok(U43(z0))
   U51(mark(z0), z1, z2) -> mark(U51(z0, z1, z2))
   U51(ok(z0), ok(z1), ok(z2)) -> ok(U51(z0, z1, z2))
   U52(mark(z0), z1) -> mark(U52(z0, z1))
   U52(ok(z0), ok(z1)) -> ok(U52(z0, z1))
   U53(mark(z0)) -> mark(U53(z0))
   U53(ok(z0)) -> ok(U53(z0))
   U61(mark(z0), z1) -> mark(U61(z0, z1))
   U61(ok(z0), ok(z1)) -> ok(U61(z0, z1))
   U62(mark(z0)) -> mark(U62(z0))
   U62(ok(z0)) -> ok(U62(z0))
   U71(mark(z0), z1) -> mark(U71(z0, z1))
   U71(ok(z0), ok(z1)) -> ok(U71(z0, z1))
   U72(mark(z0)) -> mark(U72(z0))
   U72(ok(z0)) -> ok(U72(z0))
   U81(mark(z0), z1) -> mark(U81(z0, z1))
   U81(ok(z0), ok(z1)) -> ok(U81(z0, z1))
   U82(mark(z0)) -> mark(U82(z0))
   U82(ok(z0)) -> ok(U82(z0))
   U91(mark(z0), z1) -> mark(U91(z0, z1))
   U91(ok(z0), ok(z1)) -> ok(U91(z0, z1))
   U92(mark(z0)) -> mark(U92(z0))
   U92(ok(z0)) -> ok(U92(z0))
   and(mark(z0), z1) -> mark(and(z0, z1))
   and(ok(z0), ok(z1)) -> ok(and(z0, z1))
   tail(mark(z0)) -> mark(tail(z0))
   tail(ok(z0)) -> ok(tail(z0))
   take(mark(z0), z1) -> mark(take(z0, z1))
   take(z0, mark(z1)) -> mark(take(z0, z1))
   take(ok(z0), ok(z1)) -> ok(take(z0, z1))
   sel(mark(z0), z1) -> mark(sel(z0, z1))
   sel(z0, mark(z1)) -> mark(sel(z0, z1))
   sel(ok(z0), ok(z1)) -> ok(sel(z0, z1))
   isNatural(ok(z0)) -> ok(isNatural(z0))
   isLNat(ok(z0)) -> ok(isLNat(z0))
   isPLNat(ok(z0)) -> ok(isPLNat(z0))
   isNaturalKind(ok(z0)) -> ok(isNaturalKind(z0))
   isLNatKind(ok(z0)) -> ok(isLNatKind(z0))
   isPLNatKind(ok(z0)) -> ok(isPLNatKind(z0))
   top(mark(z0)) -> top(proper(z0))
   top(ok(z0)) -> top(active(z0))

----------------------------------------

(12)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   proper(tt) -> ok(tt)
   proper(nil) -> ok(nil)
   proper(0) -> ok(0)
Tuples:
   U101'(mark(z0), z1, z2) -> c(U101'(z0, z1, z2))
   U101'(ok(z0), ok(z1), ok(z2)) -> c1(U101'(z0, z1, z2))
   U102'(mark(z0), z1) -> c2(U102'(z0, z1))
   U102'(ok(z0), ok(z1)) -> c3(U102'(z0, z1))
   U103'(mark(z0)) -> c4(U103'(z0))
   U103'(ok(z0)) -> c5(U103'(z0))
   U11'(mark(z0), z1, z2) -> c6(U11'(z0, z1, z2))
   U11'(ok(z0), ok(z1), ok(z2)) -> c7(U11'(z0, z1, z2))
   SND(mark(z0)) -> c8(SND(z0))
   SND(ok(z0)) -> c9(SND(z0))
   SPLITAT(mark(z0), z1) -> c10(SPLITAT(z0, z1))
   SPLITAT(z0, mark(z1)) -> c11(SPLITAT(z0, z1))
   SPLITAT(ok(z0), ok(z1)) -> c12(SPLITAT(z0, z1))
   U111'(mark(z0), z1) -> c13(U111'(z0, z1))
   U111'(ok(z0), ok(z1)) -> c14(U111'(z0, z1))
   U112'(mark(z0)) -> c15(U112'(z0))
   U112'(ok(z0)) -> c16(U112'(z0))
   U121'(mark(z0), z1) -> c17(U121'(z0, z1))
   U121'(ok(z0), ok(z1)) -> c18(U121'(z0, z1))
   U122'(mark(z0)) -> c19(U122'(z0))
   U122'(ok(z0)) -> c20(U122'(z0))
   U131'(mark(z0), z1, z2) -> c21(U131'(z0, z1, z2))
   U131'(ok(z0), ok(z1), ok(z2)) -> c22(U131'(z0, z1, z2))
   U132'(mark(z0), z1) -> c23(U132'(z0, z1))
   U132'(ok(z0), ok(z1)) -> c24(U132'(z0, z1))
   U133'(mark(z0)) -> c25(U133'(z0))
   U133'(ok(z0)) -> c26(U133'(z0))
   U141'(mark(z0), z1, z2) -> c27(U141'(z0, z1, z2))
   U141'(ok(z0), ok(z1), ok(z2)) -> c28(U141'(z0, z1, z2))
   U142'(mark(z0), z1) -> c29(U142'(z0, z1))
   U142'(ok(z0), ok(z1)) -> c30(U142'(z0, z1))
   U143'(mark(z0)) -> c31(U143'(z0))
   U143'(ok(z0)) -> c32(U143'(z0))
   U151'(mark(z0), z1, z2) -> c33(U151'(z0, z1, z2))
   U151'(ok(z0), ok(z1), ok(z2)) -> c34(U151'(z0, z1, z2))
   U152'(mark(z0), z1) -> c35(U152'(z0, z1))
   U152'(ok(z0), ok(z1)) -> c36(U152'(z0, z1))
   U153'(mark(z0)) -> c37(U153'(z0))
   U153'(ok(z0)) -> c38(U153'(z0))
   U161'(mark(z0), z1) -> c39(U161'(z0, z1))
   U161'(ok(z0), ok(z1)) -> c40(U161'(z0, z1))
   CONS(mark(z0), z1) -> c41(CONS(z0, z1))
   CONS(ok(z0), ok(z1)) -> c42(CONS(z0, z1))
   NATSFROM(mark(z0)) -> c43(NATSFROM(z0))
   NATSFROM(ok(z0)) -> c44(NATSFROM(z0))
   S(mark(z0)) -> c45(S(z0))
   S(ok(z0)) -> c46(S(z0))
   U171'(mark(z0), z1, z2) -> c47(U171'(z0, z1, z2))
   U171'(ok(z0), ok(z1), ok(z2)) -> c48(U171'(z0, z1, z2))
   HEAD(mark(z0)) -> c49(HEAD(z0))
   HEAD(ok(z0)) -> c50(HEAD(z0))
   AFTERNTH(mark(z0), z1) -> c51(AFTERNTH(z0, z1))
   AFTERNTH(z0, mark(z1)) -> c52(AFTERNTH(z0, z1))
   AFTERNTH(ok(z0), ok(z1)) -> c53(AFTERNTH(z0, z1))
   U181'(mark(z0), z1) -> c54(U181'(z0, z1))
   U181'(ok(z0), ok(z1)) -> c55(U181'(z0, z1))
   U191'(mark(z0), z1) -> c56(U191'(z0, z1))
   U191'(ok(z0), ok(z1)) -> c57(U191'(z0, z1))
   PAIR(mark(z0), z1) -> c58(PAIR(z0, z1))
   PAIR(z0, mark(z1)) -> c59(PAIR(z0, z1))
   PAIR(ok(z0), ok(z1)) -> c60(PAIR(z0, z1))
   U201'(mark(z0), z1, z2, z3) -> c61(U201'(z0, z1, z2, z3))
   U201'(ok(z0), ok(z1), ok(z2), ok(z3)) -> c62(U201'(z0, z1, z2, z3))
   U202'(mark(z0), z1) -> c63(U202'(z0, z1))
   U202'(ok(z0), ok(z1)) -> c64(U202'(z0, z1))
   U21'(mark(z0), z1) -> c65(U21'(z0, z1))
   U21'(ok(z0), ok(z1)) -> c66(U21'(z0, z1))
   U211'(mark(z0), z1) -> c67(U211'(z0, z1))
   U211'(ok(z0), ok(z1)) -> c68(U211'(z0, z1))
   U221'(mark(z0), z1, z2) -> c69(U221'(z0, z1, z2))
   U221'(ok(z0), ok(z1), ok(z2)) -> c70(U221'(z0, z1, z2))
   FST(mark(z0)) -> c71(FST(z0))
   FST(ok(z0)) -> c72(FST(z0))
   U31'(mark(z0), z1) -> c73(U31'(z0, z1))
   U31'(ok(z0), ok(z1)) -> c74(U31'(z0, z1))
   U41'(mark(z0), z1, z2) -> c75(U41'(z0, z1, z2))
   U41'(ok(z0), ok(z1), ok(z2)) -> c76(U41'(z0, z1, z2))
   U42'(mark(z0), z1) -> c77(U42'(z0, z1))
   U42'(ok(z0), ok(z1)) -> c78(U42'(z0, z1))
   U43'(mark(z0)) -> c79(U43'(z0))
   U43'(ok(z0)) -> c80(U43'(z0))
   U51'(mark(z0), z1, z2) -> c81(U51'(z0, z1, z2))
   U51'(ok(z0), ok(z1), ok(z2)) -> c82(U51'(z0, z1, z2))
   U52'(mark(z0), z1) -> c83(U52'(z0, z1))
   U52'(ok(z0), ok(z1)) -> c84(U52'(z0, z1))
   U53'(mark(z0)) -> c85(U53'(z0))
   U53'(ok(z0)) -> c86(U53'(z0))
   U61'(mark(z0), z1) -> c87(U61'(z0, z1))
   U61'(ok(z0), ok(z1)) -> c88(U61'(z0, z1))
   U62'(mark(z0)) -> c89(U62'(z0))
   U62'(ok(z0)) -> c90(U62'(z0))
   U71'(mark(z0), z1) -> c91(U71'(z0, z1))
   U71'(ok(z0), ok(z1)) -> c92(U71'(z0, z1))
   U72'(mark(z0)) -> c93(U72'(z0))
   U72'(ok(z0)) -> c94(U72'(z0))
   U81'(mark(z0), z1) -> c95(U81'(z0, z1))
   U81'(ok(z0), ok(z1)) -> c96(U81'(z0, z1))
   U82'(mark(z0)) -> c97(U82'(z0))
   U82'(ok(z0)) -> c98(U82'(z0))
   U91'(mark(z0), z1) -> c99(U91'(z0, z1))
   U91'(ok(z0), ok(z1)) -> c100(U91'(z0, z1))
   U92'(mark(z0)) -> c101(U92'(z0))
   U92'(ok(z0)) -> c102(U92'(z0))
   AND(mark(z0), z1) -> c103(AND(z0, z1))
   AND(ok(z0), ok(z1)) -> c104(AND(z0, z1))
   TAIL(mark(z0)) -> c105(TAIL(z0))
   TAIL(ok(z0)) -> c106(TAIL(z0))
   TAKE(mark(z0), z1) -> c107(TAKE(z0, z1))
   TAKE(z0, mark(z1)) -> c108(TAKE(z0, z1))
   TAKE(ok(z0), ok(z1)) -> c109(TAKE(z0, z1))
   SEL(mark(z0), z1) -> c110(SEL(z0, z1))
   SEL(z0, mark(z1)) -> c111(SEL(z0, z1))
   SEL(ok(z0), ok(z1)) -> c112(SEL(z0, z1))
   ISNATURAL(ok(z0)) -> c116(ISNATURAL(z0))
   ISLNAT(ok(z0)) -> c117(ISLNAT(z0))
   ISPLNAT(ok(z0)) -> c118(ISPLNAT(z0))
   ISNATURALKIND(ok(z0)) -> c119(ISNATURALKIND(z0))
   ISLNATKIND(ok(z0)) -> c120(ISLNATKIND(z0))
   ISPLNATKIND(ok(z0)) -> c121(ISPLNATKIND(z0))
   TOP(mark(z0)) -> c122(TOP(proper(z0)))
S tuples:
   U101'(mark(z0), z1, z2) -> c(U101'(z0, z1, z2))
   U101'(ok(z0), ok(z1), ok(z2)) -> c1(U101'(z0, z1, z2))
   U102'(mark(z0), z1) -> c2(U102'(z0, z1))
   U102'(ok(z0), ok(z1)) -> c3(U102'(z0, z1))
   U103'(mark(z0)) -> c4(U103'(z0))
   U103'(ok(z0)) -> c5(U103'(z0))
   U11'(mark(z0), z1, z2) -> c6(U11'(z0, z1, z2))
   U11'(ok(z0), ok(z1), ok(z2)) -> c7(U11'(z0, z1, z2))
   SND(mark(z0)) -> c8(SND(z0))
   SND(ok(z0)) -> c9(SND(z0))
   SPLITAT(mark(z0), z1) -> c10(SPLITAT(z0, z1))
   SPLITAT(z0, mark(z1)) -> c11(SPLITAT(z0, z1))
   SPLITAT(ok(z0), ok(z1)) -> c12(SPLITAT(z0, z1))
   U111'(mark(z0), z1) -> c13(U111'(z0, z1))
   U111'(ok(z0), ok(z1)) -> c14(U111'(z0, z1))
   U112'(mark(z0)) -> c15(U112'(z0))
   U112'(ok(z0)) -> c16(U112'(z0))
   U121'(mark(z0), z1) -> c17(U121'(z0, z1))
   U121'(ok(z0), ok(z1)) -> c18(U121'(z0, z1))
   U122'(mark(z0)) -> c19(U122'(z0))
   U122'(ok(z0)) -> c20(U122'(z0))
   U131'(mark(z0), z1, z2) -> c21(U131'(z0, z1, z2))
   U131'(ok(z0), ok(z1), ok(z2)) -> c22(U131'(z0, z1, z2))
   U132'(mark(z0), z1) -> c23(U132'(z0, z1))
   U132'(ok(z0), ok(z1)) -> c24(U132'(z0, z1))
   U133'(mark(z0)) -> c25(U133'(z0))
   U133'(ok(z0)) -> c26(U133'(z0))
   U141'(mark(z0), z1, z2) -> c27(U141'(z0, z1, z2))
   U141'(ok(z0), ok(z1), ok(z2)) -> c28(U141'(z0, z1, z2))
   U142'(mark(z0), z1) -> c29(U142'(z0, z1))
   U142'(ok(z0), ok(z1)) -> c30(U142'(z0, z1))
   U143'(mark(z0)) -> c31(U143'(z0))
   U143'(ok(z0)) -> c32(U143'(z0))
   U151'(mark(z0), z1, z2) -> c33(U151'(z0, z1, z2))
   U151'(ok(z0), ok(z1), ok(z2)) -> c34(U151'(z0, z1, z2))
   U152'(mark(z0), z1) -> c35(U152'(z0, z1))
   U152'(ok(z0), ok(z1)) -> c36(U152'(z0, z1))
   U153'(mark(z0)) -> c37(U153'(z0))
   U153'(ok(z0)) -> c38(U153'(z0))
   U161'(mark(z0), z1) -> c39(U161'(z0, z1))
   U161'(ok(z0), ok(z1)) -> c40(U161'(z0, z1))
   CONS(mark(z0), z1) -> c41(CONS(z0, z1))
   CONS(ok(z0), ok(z1)) -> c42(CONS(z0, z1))
   NATSFROM(mark(z0)) -> c43(NATSFROM(z0))
   NATSFROM(ok(z0)) -> c44(NATSFROM(z0))
   S(mark(z0)) -> c45(S(z0))
   S(ok(z0)) -> c46(S(z0))
   U171'(mark(z0), z1, z2) -> c47(U171'(z0, z1, z2))
   U171'(ok(z0), ok(z1), ok(z2)) -> c48(U171'(z0, z1, z2))
   HEAD(mark(z0)) -> c49(HEAD(z0))
   HEAD(ok(z0)) -> c50(HEAD(z0))
   AFTERNTH(mark(z0), z1) -> c51(AFTERNTH(z0, z1))
   AFTERNTH(z0, mark(z1)) -> c52(AFTERNTH(z0, z1))
   AFTERNTH(ok(z0), ok(z1)) -> c53(AFTERNTH(z0, z1))
   U181'(mark(z0), z1) -> c54(U181'(z0, z1))
   U181'(ok(z0), ok(z1)) -> c55(U181'(z0, z1))
   U191'(mark(z0), z1) -> c56(U191'(z0, z1))
   U191'(ok(z0), ok(z1)) -> c57(U191'(z0, z1))
   PAIR(mark(z0), z1) -> c58(PAIR(z0, z1))
   PAIR(z0, mark(z1)) -> c59(PAIR(z0, z1))
   PAIR(ok(z0), ok(z1)) -> c60(PAIR(z0, z1))
   U201'(mark(z0), z1, z2, z3) -> c61(U201'(z0, z1, z2, z3))
   U201'(ok(z0), ok(z1), ok(z2), ok(z3)) -> c62(U201'(z0, z1, z2, z3))
   U202'(mark(z0), z1) -> c63(U202'(z0, z1))
   U202'(ok(z0), ok(z1)) -> c64(U202'(z0, z1))
   U21'(mark(z0), z1) -> c65(U21'(z0, z1))
   U21'(ok(z0), ok(z1)) -> c66(U21'(z0, z1))
   U211'(mark(z0), z1) -> c67(U211'(z0, z1))
   U211'(ok(z0), ok(z1)) -> c68(U211'(z0, z1))
   U221'(mark(z0), z1, z2) -> c69(U221'(z0, z1, z2))
   U221'(ok(z0), ok(z1), ok(z2)) -> c70(U221'(z0, z1, z2))
   FST(mark(z0)) -> c71(FST(z0))
   FST(ok(z0)) -> c72(FST(z0))
   U31'(mark(z0), z1) -> c73(U31'(z0, z1))
   U31'(ok(z0), ok(z1)) -> c74(U31'(z0, z1))
   U41'(mark(z0), z1, z2) -> c75(U41'(z0, z1, z2))
   U41'(ok(z0), ok(z1), ok(z2)) -> c76(U41'(z0, z1, z2))
   U42'(mark(z0), z1) -> c77(U42'(z0, z1))
   U42'(ok(z0), ok(z1)) -> c78(U42'(z0, z1))
   U43'(mark(z0)) -> c79(U43'(z0))
   U43'(ok(z0)) -> c80(U43'(z0))
   U51'(mark(z0), z1, z2) -> c81(U51'(z0, z1, z2))
   U51'(ok(z0), ok(z1), ok(z2)) -> c82(U51'(z0, z1, z2))
   U52'(mark(z0), z1) -> c83(U52'(z0, z1))
   U52'(ok(z0), ok(z1)) -> c84(U52'(z0, z1))
   U53'(mark(z0)) -> c85(U53'(z0))
   U53'(ok(z0)) -> c86(U53'(z0))
   U61'(mark(z0), z1) -> c87(U61'(z0, z1))
   U61'(ok(z0), ok(z1)) -> c88(U61'(z0, z1))
   U62'(mark(z0)) -> c89(U62'(z0))
   U62'(ok(z0)) -> c90(U62'(z0))
   U71'(mark(z0), z1) -> c91(U71'(z0, z1))
   U71'(ok(z0), ok(z1)) -> c92(U71'(z0, z1))
   U72'(mark(z0)) -> c93(U72'(z0))
   U72'(ok(z0)) -> c94(U72'(z0))
   U81'(mark(z0), z1) -> c95(U81'(z0, z1))
   U81'(ok(z0), ok(z1)) -> c96(U81'(z0, z1))
   U82'(mark(z0)) -> c97(U82'(z0))
   U82'(ok(z0)) -> c98(U82'(z0))
   U91'(mark(z0), z1) -> c99(U91'(z0, z1))
   U91'(ok(z0), ok(z1)) -> c100(U91'(z0, z1))
   U92'(mark(z0)) -> c101(U92'(z0))
   U92'(ok(z0)) -> c102(U92'(z0))
   AND(mark(z0), z1) -> c103(AND(z0, z1))
   AND(ok(z0), ok(z1)) -> c104(AND(z0, z1))
   TAIL(mark(z0)) -> c105(TAIL(z0))
   TAIL(ok(z0)) -> c106(TAIL(z0))
   TAKE(mark(z0), z1) -> c107(TAKE(z0, z1))
   TAKE(z0, mark(z1)) -> c108(TAKE(z0, z1))
   TAKE(ok(z0), ok(z1)) -> c109(TAKE(z0, z1))
   SEL(mark(z0), z1) -> c110(SEL(z0, z1))
   SEL(z0, mark(z1)) -> c111(SEL(z0, z1))
   SEL(ok(z0), ok(z1)) -> c112(SEL(z0, z1))
   ISNATURAL(ok(z0)) -> c116(ISNATURAL(z0))
   ISLNAT(ok(z0)) -> c117(ISLNAT(z0))
   ISPLNAT(ok(z0)) -> c118(ISPLNAT(z0))
   ISNATURALKIND(ok(z0)) -> c119(ISNATURALKIND(z0))
   ISLNATKIND(ok(z0)) -> c120(ISLNATKIND(z0))
   ISPLNATKIND(ok(z0)) -> c121(ISPLNATKIND(z0))
   TOP(mark(z0)) -> c122(TOP(proper(z0)))
K tuples:none
Defined Rule Symbols:   proper_1

Defined Pair Symbols:   U101'_3, U102'_2, U103'_1, U11'_3, SND_1, SPLITAT_2, U111'_2, U112'_1, U121'_2, U122'_1, U131'_3, U132'_2, U133'_1, U141'_3, U142'_2, U143'_1, U151'_3, U152'_2, U153'_1, U161'_2, CONS_2, NATSFROM_1, S_1, U171'_3, HEAD_1, AFTERNTH_2, U181'_2, U191'_2, PAIR_2, U201'_4, U202'_2, U21'_2, U211'_2, U221'_3, FST_1, U31'_2, U41'_3, U42'_2, U43'_1, U51'_3, U52'_2, U53'_1, U61'_2, U62'_1, U71'_2, U72'_1, U81'_2, U82'_1, U91'_2, U92'_1, AND_2, TAIL_1, TAKE_2, SEL_2, ISNATURAL_1, ISLNAT_1, ISPLNAT_1, ISNATURALKIND_1, ISLNATKIND_1, ISPLNATKIND_1, TOP_1

Compound Symbols:   c_1, c1_1, c2_1, c3_1, c4_1, c5_1, c6_1, c7_1, c8_1, c9_1, c10_1, c11_1, c12_1, c13_1, c14_1, c15_1, c16_1, c17_1, c18_1, c19_1, c20_1, c21_1, c22_1, c23_1, c24_1, c25_1, c26_1, c27_1, c28_1, c29_1, c30_1, c31_1, c32_1, c33_1, c34_1, c35_1, c36_1, c37_1, c38_1, c39_1, c40_1, c41_1, c42_1, c43_1, c44_1, c45_1, c46_1, c47_1, c48_1, c49_1, c50_1, c51_1, c52_1, c53_1, c54_1, c55_1, c56_1, c57_1, c58_1, c59_1, c60_1, c61_1, c62_1, c63_1, c64_1, c65_1, c66_1, c67_1, c68_1, c69_1, c70_1, c71_1, c72_1, c73_1, c74_1, c75_1, c76_1, c77_1, c78_1, c79_1, c80_1, c81_1, c82_1, c83_1, c84_1, c85_1, c86_1, c87_1, c88_1, c89_1, c90_1, c91_1, c92_1, c93_1, c94_1, c95_1, c96_1, c97_1, c98_1, c99_1, c100_1, c101_1, c102_1, c103_1, c104_1, c105_1, c106_1, c107_1, c108_1, c109_1, c110_1, c111_1, c112_1, c116_1, c117_1, c118_1, c119_1, c120_1, c121_1, c122_1


----------------------------------------

(13) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   U101'(mark(z0), z1, z2) -> c(U101'(z0, z1, z2))
   U101'(ok(z0), ok(z1), ok(z2)) -> c1(U101'(z0, z1, z2))
   U102'(mark(z0), z1) -> c2(U102'(z0, z1))
   U102'(ok(z0), ok(z1)) -> c3(U102'(z0, z1))
   U103'(mark(z0)) -> c4(U103'(z0))
   U103'(ok(z0)) -> c5(U103'(z0))
   U11'(mark(z0), z1, z2) -> c6(U11'(z0, z1, z2))
   U11'(ok(z0), ok(z1), ok(z2)) -> c7(U11'(z0, z1, z2))
   SND(mark(z0)) -> c8(SND(z0))
   SND(ok(z0)) -> c9(SND(z0))
   SPLITAT(mark(z0), z1) -> c10(SPLITAT(z0, z1))
   SPLITAT(z0, mark(z1)) -> c11(SPLITAT(z0, z1))
   SPLITAT(ok(z0), ok(z1)) -> c12(SPLITAT(z0, z1))
   U111'(mark(z0), z1) -> c13(U111'(z0, z1))
   U111'(ok(z0), ok(z1)) -> c14(U111'(z0, z1))
   U112'(mark(z0)) -> c15(U112'(z0))
   U112'(ok(z0)) -> c16(U112'(z0))
   U121'(mark(z0), z1) -> c17(U121'(z0, z1))
   U121'(ok(z0), ok(z1)) -> c18(U121'(z0, z1))
   U122'(mark(z0)) -> c19(U122'(z0))
   U122'(ok(z0)) -> c20(U122'(z0))
   U131'(mark(z0), z1, z2) -> c21(U131'(z0, z1, z2))
   U131'(ok(z0), ok(z1), ok(z2)) -> c22(U131'(z0, z1, z2))
   U132'(mark(z0), z1) -> c23(U132'(z0, z1))
   U132'(ok(z0), ok(z1)) -> c24(U132'(z0, z1))
   U133'(mark(z0)) -> c25(U133'(z0))
   U133'(ok(z0)) -> c26(U133'(z0))
   U141'(mark(z0), z1, z2) -> c27(U141'(z0, z1, z2))
   U141'(ok(z0), ok(z1), ok(z2)) -> c28(U141'(z0, z1, z2))
   U142'(mark(z0), z1) -> c29(U142'(z0, z1))
   U142'(ok(z0), ok(z1)) -> c30(U142'(z0, z1))
   U143'(mark(z0)) -> c31(U143'(z0))
   U143'(ok(z0)) -> c32(U143'(z0))
   U151'(mark(z0), z1, z2) -> c33(U151'(z0, z1, z2))
   U151'(ok(z0), ok(z1), ok(z2)) -> c34(U151'(z0, z1, z2))
   U152'(mark(z0), z1) -> c35(U152'(z0, z1))
   U152'(ok(z0), ok(z1)) -> c36(U152'(z0, z1))
   U153'(mark(z0)) -> c37(U153'(z0))
   U153'(ok(z0)) -> c38(U153'(z0))
   U161'(mark(z0), z1) -> c39(U161'(z0, z1))
   U161'(ok(z0), ok(z1)) -> c40(U161'(z0, z1))
   CONS(mark(z0), z1) -> c41(CONS(z0, z1))
   CONS(ok(z0), ok(z1)) -> c42(CONS(z0, z1))
   NATSFROM(mark(z0)) -> c43(NATSFROM(z0))
   NATSFROM(ok(z0)) -> c44(NATSFROM(z0))
   S(mark(z0)) -> c45(S(z0))
   S(ok(z0)) -> c46(S(z0))
   U171'(mark(z0), z1, z2) -> c47(U171'(z0, z1, z2))
   U171'(ok(z0), ok(z1), ok(z2)) -> c48(U171'(z0, z1, z2))
   HEAD(mark(z0)) -> c49(HEAD(z0))
   HEAD(ok(z0)) -> c50(HEAD(z0))
   AFTERNTH(mark(z0), z1) -> c51(AFTERNTH(z0, z1))
   AFTERNTH(z0, mark(z1)) -> c52(AFTERNTH(z0, z1))
   AFTERNTH(ok(z0), ok(z1)) -> c53(AFTERNTH(z0, z1))
   U181'(mark(z0), z1) -> c54(U181'(z0, z1))
   U181'(ok(z0), ok(z1)) -> c55(U181'(z0, z1))
   U191'(mark(z0), z1) -> c56(U191'(z0, z1))
   U191'(ok(z0), ok(z1)) -> c57(U191'(z0, z1))
   PAIR(mark(z0), z1) -> c58(PAIR(z0, z1))
   PAIR(z0, mark(z1)) -> c59(PAIR(z0, z1))
   PAIR(ok(z0), ok(z1)) -> c60(PAIR(z0, z1))
   U201'(mark(z0), z1, z2, z3) -> c61(U201'(z0, z1, z2, z3))
   U201'(ok(z0), ok(z1), ok(z2), ok(z3)) -> c62(U201'(z0, z1, z2, z3))
   U202'(mark(z0), z1) -> c63(U202'(z0, z1))
   U202'(ok(z0), ok(z1)) -> c64(U202'(z0, z1))
   U21'(mark(z0), z1) -> c65(U21'(z0, z1))
   U21'(ok(z0), ok(z1)) -> c66(U21'(z0, z1))
   U211'(mark(z0), z1) -> c67(U211'(z0, z1))
   U211'(ok(z0), ok(z1)) -> c68(U211'(z0, z1))
   U221'(mark(z0), z1, z2) -> c69(U221'(z0, z1, z2))
   U221'(ok(z0), ok(z1), ok(z2)) -> c70(U221'(z0, z1, z2))
   FST(mark(z0)) -> c71(FST(z0))
   FST(ok(z0)) -> c72(FST(z0))
   U31'(mark(z0), z1) -> c73(U31'(z0, z1))
   U31'(ok(z0), ok(z1)) -> c74(U31'(z0, z1))
   U41'(mark(z0), z1, z2) -> c75(U41'(z0, z1, z2))
   U41'(ok(z0), ok(z1), ok(z2)) -> c76(U41'(z0, z1, z2))
   U42'(mark(z0), z1) -> c77(U42'(z0, z1))
   U42'(ok(z0), ok(z1)) -> c78(U42'(z0, z1))
   U43'(mark(z0)) -> c79(U43'(z0))
   U43'(ok(z0)) -> c80(U43'(z0))
   U51'(mark(z0), z1, z2) -> c81(U51'(z0, z1, z2))
   U51'(ok(z0), ok(z1), ok(z2)) -> c82(U51'(z0, z1, z2))
   U52'(mark(z0), z1) -> c83(U52'(z0, z1))
   U52'(ok(z0), ok(z1)) -> c84(U52'(z0, z1))
   U53'(mark(z0)) -> c85(U53'(z0))
   U53'(ok(z0)) -> c86(U53'(z0))
   U61'(mark(z0), z1) -> c87(U61'(z0, z1))
   U61'(ok(z0), ok(z1)) -> c88(U61'(z0, z1))
   U62'(mark(z0)) -> c89(U62'(z0))
   U62'(ok(z0)) -> c90(U62'(z0))
   U71'(mark(z0), z1) -> c91(U71'(z0, z1))
   U71'(ok(z0), ok(z1)) -> c92(U71'(z0, z1))
   U72'(mark(z0)) -> c93(U72'(z0))
   U72'(ok(z0)) -> c94(U72'(z0))
   U81'(mark(z0), z1) -> c95(U81'(z0, z1))
   U81'(ok(z0), ok(z1)) -> c96(U81'(z0, z1))
   U82'(mark(z0)) -> c97(U82'(z0))
   U82'(ok(z0)) -> c98(U82'(z0))
   U91'(mark(z0), z1) -> c99(U91'(z0, z1))
   U91'(ok(z0), ok(z1)) -> c100(U91'(z0, z1))
   U92'(mark(z0)) -> c101(U92'(z0))
   U92'(ok(z0)) -> c102(U92'(z0))
   AND(mark(z0), z1) -> c103(AND(z0, z1))
   AND(ok(z0), ok(z1)) -> c104(AND(z0, z1))
   TAIL(mark(z0)) -> c105(TAIL(z0))
   TAIL(ok(z0)) -> c106(TAIL(z0))
   TAKE(mark(z0), z1) -> c107(TAKE(z0, z1))
   TAKE(z0, mark(z1)) -> c108(TAKE(z0, z1))
   TAKE(ok(z0), ok(z1)) -> c109(TAKE(z0, z1))
   SEL(mark(z0), z1) -> c110(SEL(z0, z1))
   SEL(z0, mark(z1)) -> c111(SEL(z0, z1))
   SEL(ok(z0), ok(z1)) -> c112(SEL(z0, z1))
   ISNATURAL(ok(z0)) -> c116(ISNATURAL(z0))
   ISLNAT(ok(z0)) -> c117(ISLNAT(z0))
   ISPLNAT(ok(z0)) -> c118(ISPLNAT(z0))
   ISNATURALKIND(ok(z0)) -> c119(ISNATURALKIND(z0))
   ISLNATKIND(ok(z0)) -> c120(ISLNATKIND(z0))
   ISPLNATKIND(ok(z0)) -> c121(ISPLNATKIND(z0))
We considered the (Usable) Rules:
   proper(tt) -> ok(tt)
   proper(nil) -> ok(nil)
   proper(0) -> ok(0)
And the Tuples:
   U101'(mark(z0), z1, z2) -> c(U101'(z0, z1, z2))
   U101'(ok(z0), ok(z1), ok(z2)) -> c1(U101'(z0, z1, z2))
   U102'(mark(z0), z1) -> c2(U102'(z0, z1))
   U102'(ok(z0), ok(z1)) -> c3(U102'(z0, z1))
   U103'(mark(z0)) -> c4(U103'(z0))
   U103'(ok(z0)) -> c5(U103'(z0))
   U11'(mark(z0), z1, z2) -> c6(U11'(z0, z1, z2))
   U11'(ok(z0), ok(z1), ok(z2)) -> c7(U11'(z0, z1, z2))
   SND(mark(z0)) -> c8(SND(z0))
   SND(ok(z0)) -> c9(SND(z0))
   SPLITAT(mark(z0), z1) -> c10(SPLITAT(z0, z1))
   SPLITAT(z0, mark(z1)) -> c11(SPLITAT(z0, z1))
   SPLITAT(ok(z0), ok(z1)) -> c12(SPLITAT(z0, z1))
   U111'(mark(z0), z1) -> c13(U111'(z0, z1))
   U111'(ok(z0), ok(z1)) -> c14(U111'(z0, z1))
   U112'(mark(z0)) -> c15(U112'(z0))
   U112'(ok(z0)) -> c16(U112'(z0))
   U121'(mark(z0), z1) -> c17(U121'(z0, z1))
   U121'(ok(z0), ok(z1)) -> c18(U121'(z0, z1))
   U122'(mark(z0)) -> c19(U122'(z0))
   U122'(ok(z0)) -> c20(U122'(z0))
   U131'(mark(z0), z1, z2) -> c21(U131'(z0, z1, z2))
   U131'(ok(z0), ok(z1), ok(z2)) -> c22(U131'(z0, z1, z2))
   U132'(mark(z0), z1) -> c23(U132'(z0, z1))
   U132'(ok(z0), ok(z1)) -> c24(U132'(z0, z1))
   U133'(mark(z0)) -> c25(U133'(z0))
   U133'(ok(z0)) -> c26(U133'(z0))
   U141'(mark(z0), z1, z2) -> c27(U141'(z0, z1, z2))
   U141'(ok(z0), ok(z1), ok(z2)) -> c28(U141'(z0, z1, z2))
   U142'(mark(z0), z1) -> c29(U142'(z0, z1))
   U142'(ok(z0), ok(z1)) -> c30(U142'(z0, z1))
   U143'(mark(z0)) -> c31(U143'(z0))
   U143'(ok(z0)) -> c32(U143'(z0))
   U151'(mark(z0), z1, z2) -> c33(U151'(z0, z1, z2))
   U151'(ok(z0), ok(z1), ok(z2)) -> c34(U151'(z0, z1, z2))
   U152'(mark(z0), z1) -> c35(U152'(z0, z1))
   U152'(ok(z0), ok(z1)) -> c36(U152'(z0, z1))
   U153'(mark(z0)) -> c37(U153'(z0))
   U153'(ok(z0)) -> c38(U153'(z0))
   U161'(mark(z0), z1) -> c39(U161'(z0, z1))
   U161'(ok(z0), ok(z1)) -> c40(U161'(z0, z1))
   CONS(mark(z0), z1) -> c41(CONS(z0, z1))
   CONS(ok(z0), ok(z1)) -> c42(CONS(z0, z1))
   NATSFROM(mark(z0)) -> c43(NATSFROM(z0))
   NATSFROM(ok(z0)) -> c44(NATSFROM(z0))
   S(mark(z0)) -> c45(S(z0))
   S(ok(z0)) -> c46(S(z0))
   U171'(mark(z0), z1, z2) -> c47(U171'(z0, z1, z2))
   U171'(ok(z0), ok(z1), ok(z2)) -> c48(U171'(z0, z1, z2))
   HEAD(mark(z0)) -> c49(HEAD(z0))
   HEAD(ok(z0)) -> c50(HEAD(z0))
   AFTERNTH(mark(z0), z1) -> c51(AFTERNTH(z0, z1))
   AFTERNTH(z0, mark(z1)) -> c52(AFTERNTH(z0, z1))
   AFTERNTH(ok(z0), ok(z1)) -> c53(AFTERNTH(z0, z1))
   U181'(mark(z0), z1) -> c54(U181'(z0, z1))
   U181'(ok(z0), ok(z1)) -> c55(U181'(z0, z1))
   U191'(mark(z0), z1) -> c56(U191'(z0, z1))
   U191'(ok(z0), ok(z1)) -> c57(U191'(z0, z1))
   PAIR(mark(z0), z1) -> c58(PAIR(z0, z1))
   PAIR(z0, mark(z1)) -> c59(PAIR(z0, z1))
   PAIR(ok(z0), ok(z1)) -> c60(PAIR(z0, z1))
   U201'(mark(z0), z1, z2, z3) -> c61(U201'(z0, z1, z2, z3))
   U201'(ok(z0), ok(z1), ok(z2), ok(z3)) -> c62(U201'(z0, z1, z2, z3))
   U202'(mark(z0), z1) -> c63(U202'(z0, z1))
   U202'(ok(z0), ok(z1)) -> c64(U202'(z0, z1))
   U21'(mark(z0), z1) -> c65(U21'(z0, z1))
   U21'(ok(z0), ok(z1)) -> c66(U21'(z0, z1))
   U211'(mark(z0), z1) -> c67(U211'(z0, z1))
   U211'(ok(z0), ok(z1)) -> c68(U211'(z0, z1))
   U221'(mark(z0), z1, z2) -> c69(U221'(z0, z1, z2))
   U221'(ok(z0), ok(z1), ok(z2)) -> c70(U221'(z0, z1, z2))
   FST(mark(z0)) -> c71(FST(z0))
   FST(ok(z0)) -> c72(FST(z0))
   U31'(mark(z0), z1) -> c73(U31'(z0, z1))
   U31'(ok(z0), ok(z1)) -> c74(U31'(z0, z1))
   U41'(mark(z0), z1, z2) -> c75(U41'(z0, z1, z2))
   U41'(ok(z0), ok(z1), ok(z2)) -> c76(U41'(z0, z1, z2))
   U42'(mark(z0), z1) -> c77(U42'(z0, z1))
   U42'(ok(z0), ok(z1)) -> c78(U42'(z0, z1))
   U43'(mark(z0)) -> c79(U43'(z0))
   U43'(ok(z0)) -> c80(U43'(z0))
   U51'(mark(z0), z1, z2) -> c81(U51'(z0, z1, z2))
   U51'(ok(z0), ok(z1), ok(z2)) -> c82(U51'(z0, z1, z2))
   U52'(mark(z0), z1) -> c83(U52'(z0, z1))
   U52'(ok(z0), ok(z1)) -> c84(U52'(z0, z1))
   U53'(mark(z0)) -> c85(U53'(z0))
   U53'(ok(z0)) -> c86(U53'(z0))
   U61'(mark(z0), z1) -> c87(U61'(z0, z1))
   U61'(ok(z0), ok(z1)) -> c88(U61'(z0, z1))
   U62'(mark(z0)) -> c89(U62'(z0))
   U62'(ok(z0)) -> c90(U62'(z0))
   U71'(mark(z0), z1) -> c91(U71'(z0, z1))
   U71'(ok(z0), ok(z1)) -> c92(U71'(z0, z1))
   U72'(mark(z0)) -> c93(U72'(z0))
   U72'(ok(z0)) -> c94(U72'(z0))
   U81'(mark(z0), z1) -> c95(U81'(z0, z1))
   U81'(ok(z0), ok(z1)) -> c96(U81'(z0, z1))
   U82'(mark(z0)) -> c97(U82'(z0))
   U82'(ok(z0)) -> c98(U82'(z0))
   U91'(mark(z0), z1) -> c99(U91'(z0, z1))
   U91'(ok(z0), ok(z1)) -> c100(U91'(z0, z1))
   U92'(mark(z0)) -> c101(U92'(z0))
   U92'(ok(z0)) -> c102(U92'(z0))
   AND(mark(z0), z1) -> c103(AND(z0, z1))
   AND(ok(z0), ok(z1)) -> c104(AND(z0, z1))
   TAIL(mark(z0)) -> c105(TAIL(z0))
   TAIL(ok(z0)) -> c106(TAIL(z0))
   TAKE(mark(z0), z1) -> c107(TAKE(z0, z1))
   TAKE(z0, mark(z1)) -> c108(TAKE(z0, z1))
   TAKE(ok(z0), ok(z1)) -> c109(TAKE(z0, z1))
   SEL(mark(z0), z1) -> c110(SEL(z0, z1))
   SEL(z0, mark(z1)) -> c111(SEL(z0, z1))
   SEL(ok(z0), ok(z1)) -> c112(SEL(z0, z1))
   ISNATURAL(ok(z0)) -> c116(ISNATURAL(z0))
   ISLNAT(ok(z0)) -> c117(ISLNAT(z0))
   ISPLNAT(ok(z0)) -> c118(ISPLNAT(z0))
   ISNATURALKIND(ok(z0)) -> c119(ISNATURALKIND(z0))
   ISLNATKIND(ok(z0)) -> c120(ISLNATKIND(z0))
   ISPLNATKIND(ok(z0)) -> c121(ISPLNATKIND(z0))
   TOP(mark(z0)) -> c122(TOP(proper(z0)))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = [1]
   POL(AFTERNTH(x_1, x_2)) = x_1 + x_2
   POL(AND(x_1, x_2)) = x_1 + x_2
   POL(CONS(x_1, x_2)) = x_1 + x_2
   POL(FST(x_1)) = x_1
   POL(HEAD(x_1)) = x_1
   POL(ISLNAT(x_1)) = x_1
   POL(ISLNATKIND(x_1)) = x_1
   POL(ISNATURAL(x_1)) = x_1
   POL(ISNATURALKIND(x_1)) = x_1
   POL(ISPLNAT(x_1)) = x_1
   POL(ISPLNATKIND(x_1)) = x_1
   POL(NATSFROM(x_1)) = x_1
   POL(PAIR(x_1, x_2)) = x_1 + x_2
   POL(S(x_1)) = x_1
   POL(SEL(x_1, x_2)) = x_1 + x_2
   POL(SND(x_1)) = x_1
   POL(SPLITAT(x_1, x_2)) = x_1 + x_2
   POL(TAIL(x_1)) = x_1
   POL(TAKE(x_1, x_2)) = x_1 + x_2
   POL(TOP(x_1)) = x_1
   POL(U101'(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(U102'(x_1, x_2)) = x_1 + x_2
   POL(U103'(x_1)) = x_1
   POL(U11'(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(U111'(x_1, x_2)) = x_1 + x_2
   POL(U112'(x_1)) = x_1
   POL(U121'(x_1, x_2)) = x_1 + x_2
   POL(U122'(x_1)) = x_1
   POL(U131'(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(U132'(x_1, x_2)) = x_1 + x_2
   POL(U133'(x_1)) = x_1
   POL(U141'(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(U142'(x_1, x_2)) = x_1 + x_2
   POL(U143'(x_1)) = x_1
   POL(U151'(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(U152'(x_1, x_2)) = x_1 + x_2
   POL(U153'(x_1)) = x_1
   POL(U161'(x_1, x_2)) = x_1 + x_2
   POL(U171'(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(U181'(x_1, x_2)) = x_1 + x_2
   POL(U191'(x_1, x_2)) = x_1 + x_2
   POL(U201'(x_1, x_2, x_3, x_4)) = x_1 + x_2 + x_3 + x_4
   POL(U202'(x_1, x_2)) = x_1 + x_2
   POL(U21'(x_1, x_2)) = x_1 + x_2
   POL(U211'(x_1, x_2)) = x_1 + x_2
   POL(U221'(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(U31'(x_1, x_2)) = x_1 + x_2
   POL(U41'(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(U42'(x_1, x_2)) = x_1 + x_2
   POL(U43'(x_1)) = x_1
   POL(U51'(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(U52'(x_1, x_2)) = x_1 + x_2
   POL(U53'(x_1)) = x_1
   POL(U61'(x_1, x_2)) = x_1 + x_2
   POL(U62'(x_1)) = x_1
   POL(U71'(x_1, x_2)) = x_1 + x_2
   POL(U72'(x_1)) = x_1
   POL(U81'(x_1, x_2)) = x_1 + x_2
   POL(U82'(x_1)) = x_1
   POL(U91'(x_1, x_2)) = x_1 + x_2
   POL(U92'(x_1)) = x_1
   POL(c(x_1)) = x_1
   POL(c1(x_1)) = x_1
   POL(c10(x_1)) = x_1
   POL(c100(x_1)) = x_1
   POL(c101(x_1)) = x_1
   POL(c102(x_1)) = x_1
   POL(c103(x_1)) = x_1
   POL(c104(x_1)) = x_1
   POL(c105(x_1)) = x_1
   POL(c106(x_1)) = x_1
   POL(c107(x_1)) = x_1
   POL(c108(x_1)) = x_1
   POL(c109(x_1)) = x_1
   POL(c11(x_1)) = x_1
   POL(c110(x_1)) = x_1
   POL(c111(x_1)) = x_1
   POL(c112(x_1)) = x_1
   POL(c116(x_1)) = x_1
   POL(c117(x_1)) = x_1
   POL(c118(x_1)) = x_1
   POL(c119(x_1)) = x_1
   POL(c12(x_1)) = x_1
   POL(c120(x_1)) = x_1
   POL(c121(x_1)) = x_1
   POL(c122(x_1)) = x_1
   POL(c13(x_1)) = x_1
   POL(c14(x_1)) = x_1
   POL(c15(x_1)) = x_1
   POL(c16(x_1)) = x_1
   POL(c17(x_1)) = x_1
   POL(c18(x_1)) = x_1
   POL(c19(x_1)) = x_1
   POL(c2(x_1)) = x_1
   POL(c20(x_1)) = x_1
   POL(c21(x_1)) = x_1
   POL(c22(x_1)) = x_1
   POL(c23(x_1)) = x_1
   POL(c24(x_1)) = x_1
   POL(c25(x_1)) = x_1
   POL(c26(x_1)) = x_1
   POL(c27(x_1)) = x_1
   POL(c28(x_1)) = x_1
   POL(c29(x_1)) = x_1
   POL(c3(x_1)) = x_1
   POL(c30(x_1)) = x_1
   POL(c31(x_1)) = x_1
   POL(c32(x_1)) = x_1
   POL(c33(x_1)) = x_1
   POL(c34(x_1)) = x_1
   POL(c35(x_1)) = x_1
   POL(c36(x_1)) = x_1
   POL(c37(x_1)) = x_1
   POL(c38(x_1)) = x_1
   POL(c39(x_1)) = x_1
   POL(c4(x_1)) = x_1
   POL(c40(x_1)) = x_1
   POL(c41(x_1)) = x_1
   POL(c42(x_1)) = x_1
   POL(c43(x_1)) = x_1
   POL(c44(x_1)) = x_1
   POL(c45(x_1)) = x_1
   POL(c46(x_1)) = x_1
   POL(c47(x_1)) = x_1
   POL(c48(x_1)) = x_1
   POL(c49(x_1)) = x_1
   POL(c5(x_1)) = x_1
   POL(c50(x_1)) = x_1
   POL(c51(x_1)) = x_1
   POL(c52(x_1)) = x_1
   POL(c53(x_1)) = x_1
   POL(c54(x_1)) = x_1
   POL(c55(x_1)) = x_1
   POL(c56(x_1)) = x_1
   POL(c57(x_1)) = x_1
   POL(c58(x_1)) = x_1
   POL(c59(x_1)) = x_1
   POL(c6(x_1)) = x_1
   POL(c60(x_1)) = x_1
   POL(c61(x_1)) = x_1
   POL(c62(x_1)) = x_1
   POL(c63(x_1)) = x_1
   POL(c64(x_1)) = x_1
   POL(c65(x_1)) = x_1
   POL(c66(x_1)) = x_1
   POL(c67(x_1)) = x_1
   POL(c68(x_1)) = x_1
   POL(c69(x_1)) = x_1
   POL(c7(x_1)) = x_1
   POL(c70(x_1)) = x_1
   POL(c71(x_1)) = x_1
   POL(c72(x_1)) = x_1
   POL(c73(x_1)) = x_1
   POL(c74(x_1)) = x_1
   POL(c75(x_1)) = x_1
   POL(c76(x_1)) = x_1
   POL(c77(x_1)) = x_1
   POL(c78(x_1)) = x_1
   POL(c79(x_1)) = x_1
   POL(c8(x_1)) = x_1
   POL(c80(x_1)) = x_1
   POL(c81(x_1)) = x_1
   POL(c82(x_1)) = x_1
   POL(c83(x_1)) = x_1
   POL(c84(x_1)) = x_1
   POL(c85(x_1)) = x_1
   POL(c86(x_1)) = x_1
   POL(c87(x_1)) = x_1
   POL(c88(x_1)) = x_1
   POL(c89(x_1)) = x_1
   POL(c9(x_1)) = x_1
   POL(c90(x_1)) = x_1
   POL(c91(x_1)) = x_1
   POL(c92(x_1)) = x_1
   POL(c93(x_1)) = x_1
   POL(c94(x_1)) = x_1
   POL(c95(x_1)) = x_1
   POL(c96(x_1)) = x_1
   POL(c97(x_1)) = x_1
   POL(c98(x_1)) = x_1
   POL(c99(x_1)) = x_1
   POL(mark(x_1)) = [1] + x_1
   POL(nil) = [1]
   POL(ok(x_1)) = [1] + x_1
   POL(proper(x_1)) = [1] + x_1
   POL(tt) = [1]

----------------------------------------

(14)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   proper(tt) -> ok(tt)
   proper(nil) -> ok(nil)
   proper(0) -> ok(0)
Tuples:
   U101'(mark(z0), z1, z2) -> c(U101'(z0, z1, z2))
   U101'(ok(z0), ok(z1), ok(z2)) -> c1(U101'(z0, z1, z2))
   U102'(mark(z0), z1) -> c2(U102'(z0, z1))
   U102'(ok(z0), ok(z1)) -> c3(U102'(z0, z1))
   U103'(mark(z0)) -> c4(U103'(z0))
   U103'(ok(z0)) -> c5(U103'(z0))
   U11'(mark(z0), z1, z2) -> c6(U11'(z0, z1, z2))
   U11'(ok(z0), ok(z1), ok(z2)) -> c7(U11'(z0, z1, z2))
   SND(mark(z0)) -> c8(SND(z0))
   SND(ok(z0)) -> c9(SND(z0))
   SPLITAT(mark(z0), z1) -> c10(SPLITAT(z0, z1))
   SPLITAT(z0, mark(z1)) -> c11(SPLITAT(z0, z1))
   SPLITAT(ok(z0), ok(z1)) -> c12(SPLITAT(z0, z1))
   U111'(mark(z0), z1) -> c13(U111'(z0, z1))
   U111'(ok(z0), ok(z1)) -> c14(U111'(z0, z1))
   U112'(mark(z0)) -> c15(U112'(z0))
   U112'(ok(z0)) -> c16(U112'(z0))
   U121'(mark(z0), z1) -> c17(U121'(z0, z1))
   U121'(ok(z0), ok(z1)) -> c18(U121'(z0, z1))
   U122'(mark(z0)) -> c19(U122'(z0))
   U122'(ok(z0)) -> c20(U122'(z0))
   U131'(mark(z0), z1, z2) -> c21(U131'(z0, z1, z2))
   U131'(ok(z0), ok(z1), ok(z2)) -> c22(U131'(z0, z1, z2))
   U132'(mark(z0), z1) -> c23(U132'(z0, z1))
   U132'(ok(z0), ok(z1)) -> c24(U132'(z0, z1))
   U133'(mark(z0)) -> c25(U133'(z0))
   U133'(ok(z0)) -> c26(U133'(z0))
   U141'(mark(z0), z1, z2) -> c27(U141'(z0, z1, z2))
   U141'(ok(z0), ok(z1), ok(z2)) -> c28(U141'(z0, z1, z2))
   U142'(mark(z0), z1) -> c29(U142'(z0, z1))
   U142'(ok(z0), ok(z1)) -> c30(U142'(z0, z1))
   U143'(mark(z0)) -> c31(U143'(z0))
   U143'(ok(z0)) -> c32(U143'(z0))
   U151'(mark(z0), z1, z2) -> c33(U151'(z0, z1, z2))
   U151'(ok(z0), ok(z1), ok(z2)) -> c34(U151'(z0, z1, z2))
   U152'(mark(z0), z1) -> c35(U152'(z0, z1))
   U152'(ok(z0), ok(z1)) -> c36(U152'(z0, z1))
   U153'(mark(z0)) -> c37(U153'(z0))
   U153'(ok(z0)) -> c38(U153'(z0))
   U161'(mark(z0), z1) -> c39(U161'(z0, z1))
   U161'(ok(z0), ok(z1)) -> c40(U161'(z0, z1))
   CONS(mark(z0), z1) -> c41(CONS(z0, z1))
   CONS(ok(z0), ok(z1)) -> c42(CONS(z0, z1))
   NATSFROM(mark(z0)) -> c43(NATSFROM(z0))
   NATSFROM(ok(z0)) -> c44(NATSFROM(z0))
   S(mark(z0)) -> c45(S(z0))
   S(ok(z0)) -> c46(S(z0))
   U171'(mark(z0), z1, z2) -> c47(U171'(z0, z1, z2))
   U171'(ok(z0), ok(z1), ok(z2)) -> c48(U171'(z0, z1, z2))
   HEAD(mark(z0)) -> c49(HEAD(z0))
   HEAD(ok(z0)) -> c50(HEAD(z0))
   AFTERNTH(mark(z0), z1) -> c51(AFTERNTH(z0, z1))
   AFTERNTH(z0, mark(z1)) -> c52(AFTERNTH(z0, z1))
   AFTERNTH(ok(z0), ok(z1)) -> c53(AFTERNTH(z0, z1))
   U181'(mark(z0), z1) -> c54(U181'(z0, z1))
   U181'(ok(z0), ok(z1)) -> c55(U181'(z0, z1))
   U191'(mark(z0), z1) -> c56(U191'(z0, z1))
   U191'(ok(z0), ok(z1)) -> c57(U191'(z0, z1))
   PAIR(mark(z0), z1) -> c58(PAIR(z0, z1))
   PAIR(z0, mark(z1)) -> c59(PAIR(z0, z1))
   PAIR(ok(z0), ok(z1)) -> c60(PAIR(z0, z1))
   U201'(mark(z0), z1, z2, z3) -> c61(U201'(z0, z1, z2, z3))
   U201'(ok(z0), ok(z1), ok(z2), ok(z3)) -> c62(U201'(z0, z1, z2, z3))
   U202'(mark(z0), z1) -> c63(U202'(z0, z1))
   U202'(ok(z0), ok(z1)) -> c64(U202'(z0, z1))
   U21'(mark(z0), z1) -> c65(U21'(z0, z1))
   U21'(ok(z0), ok(z1)) -> c66(U21'(z0, z1))
   U211'(mark(z0), z1) -> c67(U211'(z0, z1))
   U211'(ok(z0), ok(z1)) -> c68(U211'(z0, z1))
   U221'(mark(z0), z1, z2) -> c69(U221'(z0, z1, z2))
   U221'(ok(z0), ok(z1), ok(z2)) -> c70(U221'(z0, z1, z2))
   FST(mark(z0)) -> c71(FST(z0))
   FST(ok(z0)) -> c72(FST(z0))
   U31'(mark(z0), z1) -> c73(U31'(z0, z1))
   U31'(ok(z0), ok(z1)) -> c74(U31'(z0, z1))
   U41'(mark(z0), z1, z2) -> c75(U41'(z0, z1, z2))
   U41'(ok(z0), ok(z1), ok(z2)) -> c76(U41'(z0, z1, z2))
   U42'(mark(z0), z1) -> c77(U42'(z0, z1))
   U42'(ok(z0), ok(z1)) -> c78(U42'(z0, z1))
   U43'(mark(z0)) -> c79(U43'(z0))
   U43'(ok(z0)) -> c80(U43'(z0))
   U51'(mark(z0), z1, z2) -> c81(U51'(z0, z1, z2))
   U51'(ok(z0), ok(z1), ok(z2)) -> c82(U51'(z0, z1, z2))
   U52'(mark(z0), z1) -> c83(U52'(z0, z1))
   U52'(ok(z0), ok(z1)) -> c84(U52'(z0, z1))
   U53'(mark(z0)) -> c85(U53'(z0))
   U53'(ok(z0)) -> c86(U53'(z0))
   U61'(mark(z0), z1) -> c87(U61'(z0, z1))
   U61'(ok(z0), ok(z1)) -> c88(U61'(z0, z1))
   U62'(mark(z0)) -> c89(U62'(z0))
   U62'(ok(z0)) -> c90(U62'(z0))
   U71'(mark(z0), z1) -> c91(U71'(z0, z1))
   U71'(ok(z0), ok(z1)) -> c92(U71'(z0, z1))
   U72'(mark(z0)) -> c93(U72'(z0))
   U72'(ok(z0)) -> c94(U72'(z0))
   U81'(mark(z0), z1) -> c95(U81'(z0, z1))
   U81'(ok(z0), ok(z1)) -> c96(U81'(z0, z1))
   U82'(mark(z0)) -> c97(U82'(z0))
   U82'(ok(z0)) -> c98(U82'(z0))
   U91'(mark(z0), z1) -> c99(U91'(z0, z1))
   U91'(ok(z0), ok(z1)) -> c100(U91'(z0, z1))
   U92'(mark(z0)) -> c101(U92'(z0))
   U92'(ok(z0)) -> c102(U92'(z0))
   AND(mark(z0), z1) -> c103(AND(z0, z1))
   AND(ok(z0), ok(z1)) -> c104(AND(z0, z1))
   TAIL(mark(z0)) -> c105(TAIL(z0))
   TAIL(ok(z0)) -> c106(TAIL(z0))
   TAKE(mark(z0), z1) -> c107(TAKE(z0, z1))
   TAKE(z0, mark(z1)) -> c108(TAKE(z0, z1))
   TAKE(ok(z0), ok(z1)) -> c109(TAKE(z0, z1))
   SEL(mark(z0), z1) -> c110(SEL(z0, z1))
   SEL(z0, mark(z1)) -> c111(SEL(z0, z1))
   SEL(ok(z0), ok(z1)) -> c112(SEL(z0, z1))
   ISNATURAL(ok(z0)) -> c116(ISNATURAL(z0))
   ISLNAT(ok(z0)) -> c117(ISLNAT(z0))
   ISPLNAT(ok(z0)) -> c118(ISPLNAT(z0))
   ISNATURALKIND(ok(z0)) -> c119(ISNATURALKIND(z0))
   ISLNATKIND(ok(z0)) -> c120(ISLNATKIND(z0))
   ISPLNATKIND(ok(z0)) -> c121(ISPLNATKIND(z0))
   TOP(mark(z0)) -> c122(TOP(proper(z0)))
S tuples:
   TOP(mark(z0)) -> c122(TOP(proper(z0)))
K tuples:
   U101'(mark(z0), z1, z2) -> c(U101'(z0, z1, z2))
   U101'(ok(z0), ok(z1), ok(z2)) -> c1(U101'(z0, z1, z2))
   U102'(mark(z0), z1) -> c2(U102'(z0, z1))
   U102'(ok(z0), ok(z1)) -> c3(U102'(z0, z1))
   U103'(mark(z0)) -> c4(U103'(z0))
   U103'(ok(z0)) -> c5(U103'(z0))
   U11'(mark(z0), z1, z2) -> c6(U11'(z0, z1, z2))
   U11'(ok(z0), ok(z1), ok(z2)) -> c7(U11'(z0, z1, z2))
   SND(mark(z0)) -> c8(SND(z0))
   SND(ok(z0)) -> c9(SND(z0))
   SPLITAT(mark(z0), z1) -> c10(SPLITAT(z0, z1))
   SPLITAT(z0, mark(z1)) -> c11(SPLITAT(z0, z1))
   SPLITAT(ok(z0), ok(z1)) -> c12(SPLITAT(z0, z1))
   U111'(mark(z0), z1) -> c13(U111'(z0, z1))
   U111'(ok(z0), ok(z1)) -> c14(U111'(z0, z1))
   U112'(mark(z0)) -> c15(U112'(z0))
   U112'(ok(z0)) -> c16(U112'(z0))
   U121'(mark(z0), z1) -> c17(U121'(z0, z1))
   U121'(ok(z0), ok(z1)) -> c18(U121'(z0, z1))
   U122'(mark(z0)) -> c19(U122'(z0))
   U122'(ok(z0)) -> c20(U122'(z0))
   U131'(mark(z0), z1, z2) -> c21(U131'(z0, z1, z2))
   U131'(ok(z0), ok(z1), ok(z2)) -> c22(U131'(z0, z1, z2))
   U132'(mark(z0), z1) -> c23(U132'(z0, z1))
   U132'(ok(z0), ok(z1)) -> c24(U132'(z0, z1))
   U133'(mark(z0)) -> c25(U133'(z0))
   U133'(ok(z0)) -> c26(U133'(z0))
   U141'(mark(z0), z1, z2) -> c27(U141'(z0, z1, z2))
   U141'(ok(z0), ok(z1), ok(z2)) -> c28(U141'(z0, z1, z2))
   U142'(mark(z0), z1) -> c29(U142'(z0, z1))
   U142'(ok(z0), ok(z1)) -> c30(U142'(z0, z1))
   U143'(mark(z0)) -> c31(U143'(z0))
   U143'(ok(z0)) -> c32(U143'(z0))
   U151'(mark(z0), z1, z2) -> c33(U151'(z0, z1, z2))
   U151'(ok(z0), ok(z1), ok(z2)) -> c34(U151'(z0, z1, z2))
   U152'(mark(z0), z1) -> c35(U152'(z0, z1))
   U152'(ok(z0), ok(z1)) -> c36(U152'(z0, z1))
   U153'(mark(z0)) -> c37(U153'(z0))
   U153'(ok(z0)) -> c38(U153'(z0))
   U161'(mark(z0), z1) -> c39(U161'(z0, z1))
   U161'(ok(z0), ok(z1)) -> c40(U161'(z0, z1))
   CONS(mark(z0), z1) -> c41(CONS(z0, z1))
   CONS(ok(z0), ok(z1)) -> c42(CONS(z0, z1))
   NATSFROM(mark(z0)) -> c43(NATSFROM(z0))
   NATSFROM(ok(z0)) -> c44(NATSFROM(z0))
   S(mark(z0)) -> c45(S(z0))
   S(ok(z0)) -> c46(S(z0))
   U171'(mark(z0), z1, z2) -> c47(U171'(z0, z1, z2))
   U171'(ok(z0), ok(z1), ok(z2)) -> c48(U171'(z0, z1, z2))
   HEAD(mark(z0)) -> c49(HEAD(z0))
   HEAD(ok(z0)) -> c50(HEAD(z0))
   AFTERNTH(mark(z0), z1) -> c51(AFTERNTH(z0, z1))
   AFTERNTH(z0, mark(z1)) -> c52(AFTERNTH(z0, z1))
   AFTERNTH(ok(z0), ok(z1)) -> c53(AFTERNTH(z0, z1))
   U181'(mark(z0), z1) -> c54(U181'(z0, z1))
   U181'(ok(z0), ok(z1)) -> c55(U181'(z0, z1))
   U191'(mark(z0), z1) -> c56(U191'(z0, z1))
   U191'(ok(z0), ok(z1)) -> c57(U191'(z0, z1))
   PAIR(mark(z0), z1) -> c58(PAIR(z0, z1))
   PAIR(z0, mark(z1)) -> c59(PAIR(z0, z1))
   PAIR(ok(z0), ok(z1)) -> c60(PAIR(z0, z1))
   U201'(mark(z0), z1, z2, z3) -> c61(U201'(z0, z1, z2, z3))
   U201'(ok(z0), ok(z1), ok(z2), ok(z3)) -> c62(U201'(z0, z1, z2, z3))
   U202'(mark(z0), z1) -> c63(U202'(z0, z1))
   U202'(ok(z0), ok(z1)) -> c64(U202'(z0, z1))
   U21'(mark(z0), z1) -> c65(U21'(z0, z1))
   U21'(ok(z0), ok(z1)) -> c66(U21'(z0, z1))
   U211'(mark(z0), z1) -> c67(U211'(z0, z1))
   U211'(ok(z0), ok(z1)) -> c68(U211'(z0, z1))
   U221'(mark(z0), z1, z2) -> c69(U221'(z0, z1, z2))
   U221'(ok(z0), ok(z1), ok(z2)) -> c70(U221'(z0, z1, z2))
   FST(mark(z0)) -> c71(FST(z0))
   FST(ok(z0)) -> c72(FST(z0))
   U31'(mark(z0), z1) -> c73(U31'(z0, z1))
   U31'(ok(z0), ok(z1)) -> c74(U31'(z0, z1))
   U41'(mark(z0), z1, z2) -> c75(U41'(z0, z1, z2))
   U41'(ok(z0), ok(z1), ok(z2)) -> c76(U41'(z0, z1, z2))
   U42'(mark(z0), z1) -> c77(U42'(z0, z1))
   U42'(ok(z0), ok(z1)) -> c78(U42'(z0, z1))
   U43'(mark(z0)) -> c79(U43'(z0))
   U43'(ok(z0)) -> c80(U43'(z0))
   U51'(mark(z0), z1, z2) -> c81(U51'(z0, z1, z2))
   U51'(ok(z0), ok(z1), ok(z2)) -> c82(U51'(z0, z1, z2))
   U52'(mark(z0), z1) -> c83(U52'(z0, z1))
   U52'(ok(z0), ok(z1)) -> c84(U52'(z0, z1))
   U53'(mark(z0)) -> c85(U53'(z0))
   U53'(ok(z0)) -> c86(U53'(z0))
   U61'(mark(z0), z1) -> c87(U61'(z0, z1))
   U61'(ok(z0), ok(z1)) -> c88(U61'(z0, z1))
   U62'(mark(z0)) -> c89(U62'(z0))
   U62'(ok(z0)) -> c90(U62'(z0))
   U71'(mark(z0), z1) -> c91(U71'(z0, z1))
   U71'(ok(z0), ok(z1)) -> c92(U71'(z0, z1))
   U72'(mark(z0)) -> c93(U72'(z0))
   U72'(ok(z0)) -> c94(U72'(z0))
   U81'(mark(z0), z1) -> c95(U81'(z0, z1))
   U81'(ok(z0), ok(z1)) -> c96(U81'(z0, z1))
   U82'(mark(z0)) -> c97(U82'(z0))
   U82'(ok(z0)) -> c98(U82'(z0))
   U91'(mark(z0), z1) -> c99(U91'(z0, z1))
   U91'(ok(z0), ok(z1)) -> c100(U91'(z0, z1))
   U92'(mark(z0)) -> c101(U92'(z0))
   U92'(ok(z0)) -> c102(U92'(z0))
   AND(mark(z0), z1) -> c103(AND(z0, z1))
   AND(ok(z0), ok(z1)) -> c104(AND(z0, z1))
   TAIL(mark(z0)) -> c105(TAIL(z0))
   TAIL(ok(z0)) -> c106(TAIL(z0))
   TAKE(mark(z0), z1) -> c107(TAKE(z0, z1))
   TAKE(z0, mark(z1)) -> c108(TAKE(z0, z1))
   TAKE(ok(z0), ok(z1)) -> c109(TAKE(z0, z1))
   SEL(mark(z0), z1) -> c110(SEL(z0, z1))
   SEL(z0, mark(z1)) -> c111(SEL(z0, z1))
   SEL(ok(z0), ok(z1)) -> c112(SEL(z0, z1))
   ISNATURAL(ok(z0)) -> c116(ISNATURAL(z0))
   ISLNAT(ok(z0)) -> c117(ISLNAT(z0))
   ISPLNAT(ok(z0)) -> c118(ISPLNAT(z0))
   ISNATURALKIND(ok(z0)) -> c119(ISNATURALKIND(z0))
   ISLNATKIND(ok(z0)) -> c120(ISLNATKIND(z0))
   ISPLNATKIND(ok(z0)) -> c121(ISPLNATKIND(z0))
Defined Rule Symbols:   proper_1

Defined Pair Symbols:   U101'_3, U102'_2, U103'_1, U11'_3, SND_1, SPLITAT_2, U111'_2, U112'_1, U121'_2, U122'_1, U131'_3, U132'_2, U133'_1, U141'_3, U142'_2, U143'_1, U151'_3, U152'_2, U153'_1, U161'_2, CONS_2, NATSFROM_1, S_1, U171'_3, HEAD_1, AFTERNTH_2, U181'_2, U191'_2, PAIR_2, U201'_4, U202'_2, U21'_2, U211'_2, U221'_3, FST_1, U31'_2, U41'_3, U42'_2, U43'_1, U51'_3, U52'_2, U53'_1, U61'_2, U62'_1, U71'_2, U72'_1, U81'_2, U82'_1, U91'_2, U92'_1, AND_2, TAIL_1, TAKE_2, SEL_2, ISNATURAL_1, ISLNAT_1, ISPLNAT_1, ISNATURALKIND_1, ISLNATKIND_1, ISPLNATKIND_1, TOP_1

Compound Symbols:   c_1, c1_1, c2_1, c3_1, c4_1, c5_1, c6_1, c7_1, c8_1, c9_1, c10_1, c11_1, c12_1, c13_1, c14_1, c15_1, c16_1, c17_1, c18_1, c19_1, c20_1, c21_1, c22_1, c23_1, c24_1, c25_1, c26_1, c27_1, c28_1, c29_1, c30_1, c31_1, c32_1, c33_1, c34_1, c35_1, c36_1, c37_1, c38_1, c39_1, c40_1, c41_1, c42_1, c43_1, c44_1, c45_1, c46_1, c47_1, c48_1, c49_1, c50_1, c51_1, c52_1, c53_1, c54_1, c55_1, c56_1, c57_1, c58_1, c59_1, c60_1, c61_1, c62_1, c63_1, c64_1, c65_1, c66_1, c67_1, c68_1, c69_1, c70_1, c71_1, c72_1, c73_1, c74_1, c75_1, c76_1, c77_1, c78_1, c79_1, c80_1, c81_1, c82_1, c83_1, c84_1, c85_1, c86_1, c87_1, c88_1, c89_1, c90_1, c91_1, c92_1, c93_1, c94_1, c95_1, c96_1, c97_1, c98_1, c99_1, c100_1, c101_1, c102_1, c103_1, c104_1, c105_1, c106_1, c107_1, c108_1, c109_1, c110_1, c111_1, c112_1, c116_1, c117_1, c118_1, c119_1, c120_1, c121_1, c122_1


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(15) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   TOP(mark(z0)) -> c122(TOP(proper(z0)))
We considered the (Usable) Rules:
   proper(tt) -> ok(tt)
   proper(nil) -> ok(nil)
   proper(0) -> ok(0)
And the Tuples:
   U101'(mark(z0), z1, z2) -> c(U101'(z0, z1, z2))
   U101'(ok(z0), ok(z1), ok(z2)) -> c1(U101'(z0, z1, z2))
   U102'(mark(z0), z1) -> c2(U102'(z0, z1))
   U102'(ok(z0), ok(z1)) -> c3(U102'(z0, z1))
   U103'(mark(z0)) -> c4(U103'(z0))
   U103'(ok(z0)) -> c5(U103'(z0))
   U11'(mark(z0), z1, z2) -> c6(U11'(z0, z1, z2))
   U11'(ok(z0), ok(z1), ok(z2)) -> c7(U11'(z0, z1, z2))
   SND(mark(z0)) -> c8(SND(z0))
   SND(ok(z0)) -> c9(SND(z0))
   SPLITAT(mark(z0), z1) -> c10(SPLITAT(z0, z1))
   SPLITAT(z0, mark(z1)) -> c11(SPLITAT(z0, z1))
   SPLITAT(ok(z0), ok(z1)) -> c12(SPLITAT(z0, z1))
   U111'(mark(z0), z1) -> c13(U111'(z0, z1))
   U111'(ok(z0), ok(z1)) -> c14(U111'(z0, z1))
   U112'(mark(z0)) -> c15(U112'(z0))
   U112'(ok(z0)) -> c16(U112'(z0))
   U121'(mark(z0), z1) -> c17(U121'(z0, z1))
   U121'(ok(z0), ok(z1)) -> c18(U121'(z0, z1))
   U122'(mark(z0)) -> c19(U122'(z0))
   U122'(ok(z0)) -> c20(U122'(z0))
   U131'(mark(z0), z1, z2) -> c21(U131'(z0, z1, z2))
   U131'(ok(z0), ok(z1), ok(z2)) -> c22(U131'(z0, z1, z2))
   U132'(mark(z0), z1) -> c23(U132'(z0, z1))
   U132'(ok(z0), ok(z1)) -> c24(U132'(z0, z1))
   U133'(mark(z0)) -> c25(U133'(z0))
   U133'(ok(z0)) -> c26(U133'(z0))
   U141'(mark(z0), z1, z2) -> c27(U141'(z0, z1, z2))
   U141'(ok(z0), ok(z1), ok(z2)) -> c28(U141'(z0, z1, z2))
   U142'(mark(z0), z1) -> c29(U142'(z0, z1))
   U142'(ok(z0), ok(z1)) -> c30(U142'(z0, z1))
   U143'(mark(z0)) -> c31(U143'(z0))
   U143'(ok(z0)) -> c32(U143'(z0))
   U151'(mark(z0), z1, z2) -> c33(U151'(z0, z1, z2))
   U151'(ok(z0), ok(z1), ok(z2)) -> c34(U151'(z0, z1, z2))
   U152'(mark(z0), z1) -> c35(U152'(z0, z1))
   U152'(ok(z0), ok(z1)) -> c36(U152'(z0, z1))
   U153'(mark(z0)) -> c37(U153'(z0))
   U153'(ok(z0)) -> c38(U153'(z0))
   U161'(mark(z0), z1) -> c39(U161'(z0, z1))
   U161'(ok(z0), ok(z1)) -> c40(U161'(z0, z1))
   CONS(mark(z0), z1) -> c41(CONS(z0, z1))
   CONS(ok(z0), ok(z1)) -> c42(CONS(z0, z1))
   NATSFROM(mark(z0)) -> c43(NATSFROM(z0))
   NATSFROM(ok(z0)) -> c44(NATSFROM(z0))
   S(mark(z0)) -> c45(S(z0))
   S(ok(z0)) -> c46(S(z0))
   U171'(mark(z0), z1, z2) -> c47(U171'(z0, z1, z2))
   U171'(ok(z0), ok(z1), ok(z2)) -> c48(U171'(z0, z1, z2))
   HEAD(mark(z0)) -> c49(HEAD(z0))
   HEAD(ok(z0)) -> c50(HEAD(z0))
   AFTERNTH(mark(z0), z1) -> c51(AFTERNTH(z0, z1))
   AFTERNTH(z0, mark(z1)) -> c52(AFTERNTH(z0, z1))
   AFTERNTH(ok(z0), ok(z1)) -> c53(AFTERNTH(z0, z1))
   U181'(mark(z0), z1) -> c54(U181'(z0, z1))
   U181'(ok(z0), ok(z1)) -> c55(U181'(z0, z1))
   U191'(mark(z0), z1) -> c56(U191'(z0, z1))
   U191'(ok(z0), ok(z1)) -> c57(U191'(z0, z1))
   PAIR(mark(z0), z1) -> c58(PAIR(z0, z1))
   PAIR(z0, mark(z1)) -> c59(PAIR(z0, z1))
   PAIR(ok(z0), ok(z1)) -> c60(PAIR(z0, z1))
   U201'(mark(z0), z1, z2, z3) -> c61(U201'(z0, z1, z2, z3))
   U201'(ok(z0), ok(z1), ok(z2), ok(z3)) -> c62(U201'(z0, z1, z2, z3))
   U202'(mark(z0), z1) -> c63(U202'(z0, z1))
   U202'(ok(z0), ok(z1)) -> c64(U202'(z0, z1))
   U21'(mark(z0), z1) -> c65(U21'(z0, z1))
   U21'(ok(z0), ok(z1)) -> c66(U21'(z0, z1))
   U211'(mark(z0), z1) -> c67(U211'(z0, z1))
   U211'(ok(z0), ok(z1)) -> c68(U211'(z0, z1))
   U221'(mark(z0), z1, z2) -> c69(U221'(z0, z1, z2))
   U221'(ok(z0), ok(z1), ok(z2)) -> c70(U221'(z0, z1, z2))
   FST(mark(z0)) -> c71(FST(z0))
   FST(ok(z0)) -> c72(FST(z0))
   U31'(mark(z0), z1) -> c73(U31'(z0, z1))
   U31'(ok(z0), ok(z1)) -> c74(U31'(z0, z1))
   U41'(mark(z0), z1, z2) -> c75(U41'(z0, z1, z2))
   U41'(ok(z0), ok(z1), ok(z2)) -> c76(U41'(z0, z1, z2))
   U42'(mark(z0), z1) -> c77(U42'(z0, z1))
   U42'(ok(z0), ok(z1)) -> c78(U42'(z0, z1))
   U43'(mark(z0)) -> c79(U43'(z0))
   U43'(ok(z0)) -> c80(U43'(z0))
   U51'(mark(z0), z1, z2) -> c81(U51'(z0, z1, z2))
   U51'(ok(z0), ok(z1), ok(z2)) -> c82(U51'(z0, z1, z2))
   U52'(mark(z0), z1) -> c83(U52'(z0, z1))
   U52'(ok(z0), ok(z1)) -> c84(U52'(z0, z1))
   U53'(mark(z0)) -> c85(U53'(z0))
   U53'(ok(z0)) -> c86(U53'(z0))
   U61'(mark(z0), z1) -> c87(U61'(z0, z1))
   U61'(ok(z0), ok(z1)) -> c88(U61'(z0, z1))
   U62'(mark(z0)) -> c89(U62'(z0))
   U62'(ok(z0)) -> c90(U62'(z0))
   U71'(mark(z0), z1) -> c91(U71'(z0, z1))
   U71'(ok(z0), ok(z1)) -> c92(U71'(z0, z1))
   U72'(mark(z0)) -> c93(U72'(z0))
   U72'(ok(z0)) -> c94(U72'(z0))
   U81'(mark(z0), z1) -> c95(U81'(z0, z1))
   U81'(ok(z0), ok(z1)) -> c96(U81'(z0, z1))
   U82'(mark(z0)) -> c97(U82'(z0))
   U82'(ok(z0)) -> c98(U82'(z0))
   U91'(mark(z0), z1) -> c99(U91'(z0, z1))
   U91'(ok(z0), ok(z1)) -> c100(U91'(z0, z1))
   U92'(mark(z0)) -> c101(U92'(z0))
   U92'(ok(z0)) -> c102(U92'(z0))
   AND(mark(z0), z1) -> c103(AND(z0, z1))
   AND(ok(z0), ok(z1)) -> c104(AND(z0, z1))
   TAIL(mark(z0)) -> c105(TAIL(z0))
   TAIL(ok(z0)) -> c106(TAIL(z0))
   TAKE(mark(z0), z1) -> c107(TAKE(z0, z1))
   TAKE(z0, mark(z1)) -> c108(TAKE(z0, z1))
   TAKE(ok(z0), ok(z1)) -> c109(TAKE(z0, z1))
   SEL(mark(z0), z1) -> c110(SEL(z0, z1))
   SEL(z0, mark(z1)) -> c111(SEL(z0, z1))
   SEL(ok(z0), ok(z1)) -> c112(SEL(z0, z1))
   ISNATURAL(ok(z0)) -> c116(ISNATURAL(z0))
   ISLNAT(ok(z0)) -> c117(ISLNAT(z0))
   ISPLNAT(ok(z0)) -> c118(ISPLNAT(z0))
   ISNATURALKIND(ok(z0)) -> c119(ISNATURALKIND(z0))
   ISLNATKIND(ok(z0)) -> c120(ISLNATKIND(z0))
   ISPLNATKIND(ok(z0)) -> c121(ISPLNATKIND(z0))
   TOP(mark(z0)) -> c122(TOP(proper(z0)))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = 0
   POL(AFTERNTH(x_1, x_2)) = 0
   POL(AND(x_1, x_2)) = 0
   POL(CONS(x_1, x_2)) = 0
   POL(FST(x_1)) = 0
   POL(HEAD(x_1)) = 0
   POL(ISLNAT(x_1)) = 0
   POL(ISLNATKIND(x_1)) = 0
   POL(ISNATURAL(x_1)) = 0
   POL(ISNATURALKIND(x_1)) = 0
   POL(ISPLNAT(x_1)) = 0
   POL(ISPLNATKIND(x_1)) = 0
   POL(NATSFROM(x_1)) = 0
   POL(PAIR(x_1, x_2)) = 0
   POL(S(x_1)) = 0
   POL(SEL(x_1, x_2)) = 0
   POL(SND(x_1)) = 0
   POL(SPLITAT(x_1, x_2)) = 0
   POL(TAIL(x_1)) = 0
   POL(TAKE(x_1, x_2)) = 0
   POL(TOP(x_1)) = x_1
   POL(U101'(x_1, x_2, x_3)) = 0
   POL(U102'(x_1, x_2)) = 0
   POL(U103'(x_1)) = 0
   POL(U11'(x_1, x_2, x_3)) = 0
   POL(U111'(x_1, x_2)) = 0
   POL(U112'(x_1)) = 0
   POL(U121'(x_1, x_2)) = 0
   POL(U122'(x_1)) = 0
   POL(U131'(x_1, x_2, x_3)) = 0
   POL(U132'(x_1, x_2)) = 0
   POL(U133'(x_1)) = 0
   POL(U141'(x_1, x_2, x_3)) = 0
   POL(U142'(x_1, x_2)) = 0
   POL(U143'(x_1)) = 0
   POL(U151'(x_1, x_2, x_3)) = 0
   POL(U152'(x_1, x_2)) = 0
   POL(U153'(x_1)) = 0
   POL(U161'(x_1, x_2)) = 0
   POL(U171'(x_1, x_2, x_3)) = 0
   POL(U181'(x_1, x_2)) = 0
   POL(U191'(x_1, x_2)) = 0
   POL(U201'(x_1, x_2, x_3, x_4)) = 0
   POL(U202'(x_1, x_2)) = 0
   POL(U21'(x_1, x_2)) = 0
   POL(U211'(x_1, x_2)) = 0
   POL(U221'(x_1, x_2, x_3)) = 0
   POL(U31'(x_1, x_2)) = 0
   POL(U41'(x_1, x_2, x_3)) = 0
   POL(U42'(x_1, x_2)) = 0
   POL(U43'(x_1)) = 0
   POL(U51'(x_1, x_2, x_3)) = 0
   POL(U52'(x_1, x_2)) = 0
   POL(U53'(x_1)) = 0
   POL(U61'(x_1, x_2)) = 0
   POL(U62'(x_1)) = 0
   POL(U71'(x_1, x_2)) = 0
   POL(U72'(x_1)) = 0
   POL(U81'(x_1, x_2)) = 0
   POL(U82'(x_1)) = 0
   POL(U91'(x_1, x_2)) = 0
   POL(U92'(x_1)) = 0
   POL(c(x_1)) = x_1
   POL(c1(x_1)) = x_1
   POL(c10(x_1)) = x_1
   POL(c100(x_1)) = x_1
   POL(c101(x_1)) = x_1
   POL(c102(x_1)) = x_1
   POL(c103(x_1)) = x_1
   POL(c104(x_1)) = x_1
   POL(c105(x_1)) = x_1
   POL(c106(x_1)) = x_1
   POL(c107(x_1)) = x_1
   POL(c108(x_1)) = x_1
   POL(c109(x_1)) = x_1
   POL(c11(x_1)) = x_1
   POL(c110(x_1)) = x_1
   POL(c111(x_1)) = x_1
   POL(c112(x_1)) = x_1
   POL(c116(x_1)) = x_1
   POL(c117(x_1)) = x_1
   POL(c118(x_1)) = x_1
   POL(c119(x_1)) = x_1
   POL(c12(x_1)) = x_1
   POL(c120(x_1)) = x_1
   POL(c121(x_1)) = x_1
   POL(c122(x_1)) = x_1
   POL(c13(x_1)) = x_1
   POL(c14(x_1)) = x_1
   POL(c15(x_1)) = x_1
   POL(c16(x_1)) = x_1
   POL(c17(x_1)) = x_1
   POL(c18(x_1)) = x_1
   POL(c19(x_1)) = x_1
   POL(c2(x_1)) = x_1
   POL(c20(x_1)) = x_1
   POL(c21(x_1)) = x_1
   POL(c22(x_1)) = x_1
   POL(c23(x_1)) = x_1
   POL(c24(x_1)) = x_1
   POL(c25(x_1)) = x_1
   POL(c26(x_1)) = x_1
   POL(c27(x_1)) = x_1
   POL(c28(x_1)) = x_1
   POL(c29(x_1)) = x_1
   POL(c3(x_1)) = x_1
   POL(c30(x_1)) = x_1
   POL(c31(x_1)) = x_1
   POL(c32(x_1)) = x_1
   POL(c33(x_1)) = x_1
   POL(c34(x_1)) = x_1
   POL(c35(x_1)) = x_1
   POL(c36(x_1)) = x_1
   POL(c37(x_1)) = x_1
   POL(c38(x_1)) = x_1
   POL(c39(x_1)) = x_1
   POL(c4(x_1)) = x_1
   POL(c40(x_1)) = x_1
   POL(c41(x_1)) = x_1
   POL(c42(x_1)) = x_1
   POL(c43(x_1)) = x_1
   POL(c44(x_1)) = x_1
   POL(c45(x_1)) = x_1
   POL(c46(x_1)) = x_1
   POL(c47(x_1)) = x_1
   POL(c48(x_1)) = x_1
   POL(c49(x_1)) = x_1
   POL(c5(x_1)) = x_1
   POL(c50(x_1)) = x_1
   POL(c51(x_1)) = x_1
   POL(c52(x_1)) = x_1
   POL(c53(x_1)) = x_1
   POL(c54(x_1)) = x_1
   POL(c55(x_1)) = x_1
   POL(c56(x_1)) = x_1
   POL(c57(x_1)) = x_1
   POL(c58(x_1)) = x_1
   POL(c59(x_1)) = x_1
   POL(c6(x_1)) = x_1
   POL(c60(x_1)) = x_1
   POL(c61(x_1)) = x_1
   POL(c62(x_1)) = x_1
   POL(c63(x_1)) = x_1
   POL(c64(x_1)) = x_1
   POL(c65(x_1)) = x_1
   POL(c66(x_1)) = x_1
   POL(c67(x_1)) = x_1
   POL(c68(x_1)) = x_1
   POL(c69(x_1)) = x_1
   POL(c7(x_1)) = x_1
   POL(c70(x_1)) = x_1
   POL(c71(x_1)) = x_1
   POL(c72(x_1)) = x_1
   POL(c73(x_1)) = x_1
   POL(c74(x_1)) = x_1
   POL(c75(x_1)) = x_1
   POL(c76(x_1)) = x_1
   POL(c77(x_1)) = x_1
   POL(c78(x_1)) = x_1
   POL(c79(x_1)) = x_1
   POL(c8(x_1)) = x_1
   POL(c80(x_1)) = x_1
   POL(c81(x_1)) = x_1
   POL(c82(x_1)) = x_1
   POL(c83(x_1)) = x_1
   POL(c84(x_1)) = x_1
   POL(c85(x_1)) = x_1
   POL(c86(x_1)) = x_1
   POL(c87(x_1)) = x_1
   POL(c88(x_1)) = x_1
   POL(c89(x_1)) = x_1
   POL(c9(x_1)) = x_1
   POL(c90(x_1)) = x_1
   POL(c91(x_1)) = x_1
   POL(c92(x_1)) = x_1
   POL(c93(x_1)) = x_1
   POL(c94(x_1)) = x_1
   POL(c95(x_1)) = x_1
   POL(c96(x_1)) = x_1
   POL(c97(x_1)) = x_1
   POL(c98(x_1)) = x_1
   POL(c99(x_1)) = x_1
   POL(mark(x_1)) = [1] + x_1
   POL(nil) = 0
   POL(ok(x_1)) = 0
   POL(proper(x_1)) = x_1
   POL(tt) = [2]

----------------------------------------

(16)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   proper(tt) -> ok(tt)
   proper(nil) -> ok(nil)
   proper(0) -> ok(0)
Tuples:
   U101'(mark(z0), z1, z2) -> c(U101'(z0, z1, z2))
   U101'(ok(z0), ok(z1), ok(z2)) -> c1(U101'(z0, z1, z2))
   U102'(mark(z0), z1) -> c2(U102'(z0, z1))
   U102'(ok(z0), ok(z1)) -> c3(U102'(z0, z1))
   U103'(mark(z0)) -> c4(U103'(z0))
   U103'(ok(z0)) -> c5(U103'(z0))
   U11'(mark(z0), z1, z2) -> c6(U11'(z0, z1, z2))
   U11'(ok(z0), ok(z1), ok(z2)) -> c7(U11'(z0, z1, z2))
   SND(mark(z0)) -> c8(SND(z0))
   SND(ok(z0)) -> c9(SND(z0))
   SPLITAT(mark(z0), z1) -> c10(SPLITAT(z0, z1))
   SPLITAT(z0, mark(z1)) -> c11(SPLITAT(z0, z1))
   SPLITAT(ok(z0), ok(z1)) -> c12(SPLITAT(z0, z1))
   U111'(mark(z0), z1) -> c13(U111'(z0, z1))
   U111'(ok(z0), ok(z1)) -> c14(U111'(z0, z1))
   U112'(mark(z0)) -> c15(U112'(z0))
   U112'(ok(z0)) -> c16(U112'(z0))
   U121'(mark(z0), z1) -> c17(U121'(z0, z1))
   U121'(ok(z0), ok(z1)) -> c18(U121'(z0, z1))
   U122'(mark(z0)) -> c19(U122'(z0))
   U122'(ok(z0)) -> c20(U122'(z0))
   U131'(mark(z0), z1, z2) -> c21(U131'(z0, z1, z2))
   U131'(ok(z0), ok(z1), ok(z2)) -> c22(U131'(z0, z1, z2))
   U132'(mark(z0), z1) -> c23(U132'(z0, z1))
   U132'(ok(z0), ok(z1)) -> c24(U132'(z0, z1))
   U133'(mark(z0)) -> c25(U133'(z0))
   U133'(ok(z0)) -> c26(U133'(z0))
   U141'(mark(z0), z1, z2) -> c27(U141'(z0, z1, z2))
   U141'(ok(z0), ok(z1), ok(z2)) -> c28(U141'(z0, z1, z2))
   U142'(mark(z0), z1) -> c29(U142'(z0, z1))
   U142'(ok(z0), ok(z1)) -> c30(U142'(z0, z1))
   U143'(mark(z0)) -> c31(U143'(z0))
   U143'(ok(z0)) -> c32(U143'(z0))
   U151'(mark(z0), z1, z2) -> c33(U151'(z0, z1, z2))
   U151'(ok(z0), ok(z1), ok(z2)) -> c34(U151'(z0, z1, z2))
   U152'(mark(z0), z1) -> c35(U152'(z0, z1))
   U152'(ok(z0), ok(z1)) -> c36(U152'(z0, z1))
   U153'(mark(z0)) -> c37(U153'(z0))
   U153'(ok(z0)) -> c38(U153'(z0))
   U161'(mark(z0), z1) -> c39(U161'(z0, z1))
   U161'(ok(z0), ok(z1)) -> c40(U161'(z0, z1))
   CONS(mark(z0), z1) -> c41(CONS(z0, z1))
   CONS(ok(z0), ok(z1)) -> c42(CONS(z0, z1))
   NATSFROM(mark(z0)) -> c43(NATSFROM(z0))
   NATSFROM(ok(z0)) -> c44(NATSFROM(z0))
   S(mark(z0)) -> c45(S(z0))
   S(ok(z0)) -> c46(S(z0))
   U171'(mark(z0), z1, z2) -> c47(U171'(z0, z1, z2))
   U171'(ok(z0), ok(z1), ok(z2)) -> c48(U171'(z0, z1, z2))
   HEAD(mark(z0)) -> c49(HEAD(z0))
   HEAD(ok(z0)) -> c50(HEAD(z0))
   AFTERNTH(mark(z0), z1) -> c51(AFTERNTH(z0, z1))
   AFTERNTH(z0, mark(z1)) -> c52(AFTERNTH(z0, z1))
   AFTERNTH(ok(z0), ok(z1)) -> c53(AFTERNTH(z0, z1))
   U181'(mark(z0), z1) -> c54(U181'(z0, z1))
   U181'(ok(z0), ok(z1)) -> c55(U181'(z0, z1))
   U191'(mark(z0), z1) -> c56(U191'(z0, z1))
   U191'(ok(z0), ok(z1)) -> c57(U191'(z0, z1))
   PAIR(mark(z0), z1) -> c58(PAIR(z0, z1))
   PAIR(z0, mark(z1)) -> c59(PAIR(z0, z1))
   PAIR(ok(z0), ok(z1)) -> c60(PAIR(z0, z1))
   U201'(mark(z0), z1, z2, z3) -> c61(U201'(z0, z1, z2, z3))
   U201'(ok(z0), ok(z1), ok(z2), ok(z3)) -> c62(U201'(z0, z1, z2, z3))
   U202'(mark(z0), z1) -> c63(U202'(z0, z1))
   U202'(ok(z0), ok(z1)) -> c64(U202'(z0, z1))
   U21'(mark(z0), z1) -> c65(U21'(z0, z1))
   U21'(ok(z0), ok(z1)) -> c66(U21'(z0, z1))
   U211'(mark(z0), z1) -> c67(U211'(z0, z1))
   U211'(ok(z0), ok(z1)) -> c68(U211'(z0, z1))
   U221'(mark(z0), z1, z2) -> c69(U221'(z0, z1, z2))
   U221'(ok(z0), ok(z1), ok(z2)) -> c70(U221'(z0, z1, z2))
   FST(mark(z0)) -> c71(FST(z0))
   FST(ok(z0)) -> c72(FST(z0))
   U31'(mark(z0), z1) -> c73(U31'(z0, z1))
   U31'(ok(z0), ok(z1)) -> c74(U31'(z0, z1))
   U41'(mark(z0), z1, z2) -> c75(U41'(z0, z1, z2))
   U41'(ok(z0), ok(z1), ok(z2)) -> c76(U41'(z0, z1, z2))
   U42'(mark(z0), z1) -> c77(U42'(z0, z1))
   U42'(ok(z0), ok(z1)) -> c78(U42'(z0, z1))
   U43'(mark(z0)) -> c79(U43'(z0))
   U43'(ok(z0)) -> c80(U43'(z0))
   U51'(mark(z0), z1, z2) -> c81(U51'(z0, z1, z2))
   U51'(ok(z0), ok(z1), ok(z2)) -> c82(U51'(z0, z1, z2))
   U52'(mark(z0), z1) -> c83(U52'(z0, z1))
   U52'(ok(z0), ok(z1)) -> c84(U52'(z0, z1))
   U53'(mark(z0)) -> c85(U53'(z0))
   U53'(ok(z0)) -> c86(U53'(z0))
   U61'(mark(z0), z1) -> c87(U61'(z0, z1))
   U61'(ok(z0), ok(z1)) -> c88(U61'(z0, z1))
   U62'(mark(z0)) -> c89(U62'(z0))
   U62'(ok(z0)) -> c90(U62'(z0))
   U71'(mark(z0), z1) -> c91(U71'(z0, z1))
   U71'(ok(z0), ok(z1)) -> c92(U71'(z0, z1))
   U72'(mark(z0)) -> c93(U72'(z0))
   U72'(ok(z0)) -> c94(U72'(z0))
   U81'(mark(z0), z1) -> c95(U81'(z0, z1))
   U81'(ok(z0), ok(z1)) -> c96(U81'(z0, z1))
   U82'(mark(z0)) -> c97(U82'(z0))
   U82'(ok(z0)) -> c98(U82'(z0))
   U91'(mark(z0), z1) -> c99(U91'(z0, z1))
   U91'(ok(z0), ok(z1)) -> c100(U91'(z0, z1))
   U92'(mark(z0)) -> c101(U92'(z0))
   U92'(ok(z0)) -> c102(U92'(z0))
   AND(mark(z0), z1) -> c103(AND(z0, z1))
   AND(ok(z0), ok(z1)) -> c104(AND(z0, z1))
   TAIL(mark(z0)) -> c105(TAIL(z0))
   TAIL(ok(z0)) -> c106(TAIL(z0))
   TAKE(mark(z0), z1) -> c107(TAKE(z0, z1))
   TAKE(z0, mark(z1)) -> c108(TAKE(z0, z1))
   TAKE(ok(z0), ok(z1)) -> c109(TAKE(z0, z1))
   SEL(mark(z0), z1) -> c110(SEL(z0, z1))
   SEL(z0, mark(z1)) -> c111(SEL(z0, z1))
   SEL(ok(z0), ok(z1)) -> c112(SEL(z0, z1))
   ISNATURAL(ok(z0)) -> c116(ISNATURAL(z0))
   ISLNAT(ok(z0)) -> c117(ISLNAT(z0))
   ISPLNAT(ok(z0)) -> c118(ISPLNAT(z0))
   ISNATURALKIND(ok(z0)) -> c119(ISNATURALKIND(z0))
   ISLNATKIND(ok(z0)) -> c120(ISLNATKIND(z0))
   ISPLNATKIND(ok(z0)) -> c121(ISPLNATKIND(z0))
   TOP(mark(z0)) -> c122(TOP(proper(z0)))
S tuples:none
K tuples:
   U101'(mark(z0), z1, z2) -> c(U101'(z0, z1, z2))
   U101'(ok(z0), ok(z1), ok(z2)) -> c1(U101'(z0, z1, z2))
   U102'(mark(z0), z1) -> c2(U102'(z0, z1))
   U102'(ok(z0), ok(z1)) -> c3(U102'(z0, z1))
   U103'(mark(z0)) -> c4(U103'(z0))
   U103'(ok(z0)) -> c5(U103'(z0))
   U11'(mark(z0), z1, z2) -> c6(U11'(z0, z1, z2))
   U11'(ok(z0), ok(z1), ok(z2)) -> c7(U11'(z0, z1, z2))
   SND(mark(z0)) -> c8(SND(z0))
   SND(ok(z0)) -> c9(SND(z0))
   SPLITAT(mark(z0), z1) -> c10(SPLITAT(z0, z1))
   SPLITAT(z0, mark(z1)) -> c11(SPLITAT(z0, z1))
   SPLITAT(ok(z0), ok(z1)) -> c12(SPLITAT(z0, z1))
   U111'(mark(z0), z1) -> c13(U111'(z0, z1))
   U111'(ok(z0), ok(z1)) -> c14(U111'(z0, z1))
   U112'(mark(z0)) -> c15(U112'(z0))
   U112'(ok(z0)) -> c16(U112'(z0))
   U121'(mark(z0), z1) -> c17(U121'(z0, z1))
   U121'(ok(z0), ok(z1)) -> c18(U121'(z0, z1))
   U122'(mark(z0)) -> c19(U122'(z0))
   U122'(ok(z0)) -> c20(U122'(z0))
   U131'(mark(z0), z1, z2) -> c21(U131'(z0, z1, z2))
   U131'(ok(z0), ok(z1), ok(z2)) -> c22(U131'(z0, z1, z2))
   U132'(mark(z0), z1) -> c23(U132'(z0, z1))
   U132'(ok(z0), ok(z1)) -> c24(U132'(z0, z1))
   U133'(mark(z0)) -> c25(U133'(z0))
   U133'(ok(z0)) -> c26(U133'(z0))
   U141'(mark(z0), z1, z2) -> c27(U141'(z0, z1, z2))
   U141'(ok(z0), ok(z1), ok(z2)) -> c28(U141'(z0, z1, z2))
   U142'(mark(z0), z1) -> c29(U142'(z0, z1))
   U142'(ok(z0), ok(z1)) -> c30(U142'(z0, z1))
   U143'(mark(z0)) -> c31(U143'(z0))
   U143'(ok(z0)) -> c32(U143'(z0))
   U151'(mark(z0), z1, z2) -> c33(U151'(z0, z1, z2))
   U151'(ok(z0), ok(z1), ok(z2)) -> c34(U151'(z0, z1, z2))
   U152'(mark(z0), z1) -> c35(U152'(z0, z1))
   U152'(ok(z0), ok(z1)) -> c36(U152'(z0, z1))
   U153'(mark(z0)) -> c37(U153'(z0))
   U153'(ok(z0)) -> c38(U153'(z0))
   U161'(mark(z0), z1) -> c39(U161'(z0, z1))
   U161'(ok(z0), ok(z1)) -> c40(U161'(z0, z1))
   CONS(mark(z0), z1) -> c41(CONS(z0, z1))
   CONS(ok(z0), ok(z1)) -> c42(CONS(z0, z1))
   NATSFROM(mark(z0)) -> c43(NATSFROM(z0))
   NATSFROM(ok(z0)) -> c44(NATSFROM(z0))
   S(mark(z0)) -> c45(S(z0))
   S(ok(z0)) -> c46(S(z0))
   U171'(mark(z0), z1, z2) -> c47(U171'(z0, z1, z2))
   U171'(ok(z0), ok(z1), ok(z2)) -> c48(U171'(z0, z1, z2))
   HEAD(mark(z0)) -> c49(HEAD(z0))
   HEAD(ok(z0)) -> c50(HEAD(z0))
   AFTERNTH(mark(z0), z1) -> c51(AFTERNTH(z0, z1))
   AFTERNTH(z0, mark(z1)) -> c52(AFTERNTH(z0, z1))
   AFTERNTH(ok(z0), ok(z1)) -> c53(AFTERNTH(z0, z1))
   U181'(mark(z0), z1) -> c54(U181'(z0, z1))
   U181'(ok(z0), ok(z1)) -> c55(U181'(z0, z1))
   U191'(mark(z0), z1) -> c56(U191'(z0, z1))
   U191'(ok(z0), ok(z1)) -> c57(U191'(z0, z1))
   PAIR(mark(z0), z1) -> c58(PAIR(z0, z1))
   PAIR(z0, mark(z1)) -> c59(PAIR(z0, z1))
   PAIR(ok(z0), ok(z1)) -> c60(PAIR(z0, z1))
   U201'(mark(z0), z1, z2, z3) -> c61(U201'(z0, z1, z2, z3))
   U201'(ok(z0), ok(z1), ok(z2), ok(z3)) -> c62(U201'(z0, z1, z2, z3))
   U202'(mark(z0), z1) -> c63(U202'(z0, z1))
   U202'(ok(z0), ok(z1)) -> c64(U202'(z0, z1))
   U21'(mark(z0), z1) -> c65(U21'(z0, z1))
   U21'(ok(z0), ok(z1)) -> c66(U21'(z0, z1))
   U211'(mark(z0), z1) -> c67(U211'(z0, z1))
   U211'(ok(z0), ok(z1)) -> c68(U211'(z0, z1))
   U221'(mark(z0), z1, z2) -> c69(U221'(z0, z1, z2))
   U221'(ok(z0), ok(z1), ok(z2)) -> c70(U221'(z0, z1, z2))
   FST(mark(z0)) -> c71(FST(z0))
   FST(ok(z0)) -> c72(FST(z0))
   U31'(mark(z0), z1) -> c73(U31'(z0, z1))
   U31'(ok(z0), ok(z1)) -> c74(U31'(z0, z1))
   U41'(mark(z0), z1, z2) -> c75(U41'(z0, z1, z2))
   U41'(ok(z0), ok(z1), ok(z2)) -> c76(U41'(z0, z1, z2))
   U42'(mark(z0), z1) -> c77(U42'(z0, z1))
   U42'(ok(z0), ok(z1)) -> c78(U42'(z0, z1))
   U43'(mark(z0)) -> c79(U43'(z0))
   U43'(ok(z0)) -> c80(U43'(z0))
   U51'(mark(z0), z1, z2) -> c81(U51'(z0, z1, z2))
   U51'(ok(z0), ok(z1), ok(z2)) -> c82(U51'(z0, z1, z2))
   U52'(mark(z0), z1) -> c83(U52'(z0, z1))
   U52'(ok(z0), ok(z1)) -> c84(U52'(z0, z1))
   U53'(mark(z0)) -> c85(U53'(z0))
   U53'(ok(z0)) -> c86(U53'(z0))
   U61'(mark(z0), z1) -> c87(U61'(z0, z1))
   U61'(ok(z0), ok(z1)) -> c88(U61'(z0, z1))
   U62'(mark(z0)) -> c89(U62'(z0))
   U62'(ok(z0)) -> c90(U62'(z0))
   U71'(mark(z0), z1) -> c91(U71'(z0, z1))
   U71'(ok(z0), ok(z1)) -> c92(U71'(z0, z1))
   U72'(mark(z0)) -> c93(U72'(z0))
   U72'(ok(z0)) -> c94(U72'(z0))
   U81'(mark(z0), z1) -> c95(U81'(z0, z1))
   U81'(ok(z0), ok(z1)) -> c96(U81'(z0, z1))
   U82'(mark(z0)) -> c97(U82'(z0))
   U82'(ok(z0)) -> c98(U82'(z0))
   U91'(mark(z0), z1) -> c99(U91'(z0, z1))
   U91'(ok(z0), ok(z1)) -> c100(U91'(z0, z1))
   U92'(mark(z0)) -> c101(U92'(z0))
   U92'(ok(z0)) -> c102(U92'(z0))
   AND(mark(z0), z1) -> c103(AND(z0, z1))
   AND(ok(z0), ok(z1)) -> c104(AND(z0, z1))
   TAIL(mark(z0)) -> c105(TAIL(z0))
   TAIL(ok(z0)) -> c106(TAIL(z0))
   TAKE(mark(z0), z1) -> c107(TAKE(z0, z1))
   TAKE(z0, mark(z1)) -> c108(TAKE(z0, z1))
   TAKE(ok(z0), ok(z1)) -> c109(TAKE(z0, z1))
   SEL(mark(z0), z1) -> c110(SEL(z0, z1))
   SEL(z0, mark(z1)) -> c111(SEL(z0, z1))
   SEL(ok(z0), ok(z1)) -> c112(SEL(z0, z1))
   ISNATURAL(ok(z0)) -> c116(ISNATURAL(z0))
   ISLNAT(ok(z0)) -> c117(ISLNAT(z0))
   ISPLNAT(ok(z0)) -> c118(ISPLNAT(z0))
   ISNATURALKIND(ok(z0)) -> c119(ISNATURALKIND(z0))
   ISLNATKIND(ok(z0)) -> c120(ISLNATKIND(z0))
   ISPLNATKIND(ok(z0)) -> c121(ISPLNATKIND(z0))
   TOP(mark(z0)) -> c122(TOP(proper(z0)))
Defined Rule Symbols:   proper_1

Defined Pair Symbols:   U101'_3, U102'_2, U103'_1, U11'_3, SND_1, SPLITAT_2, U111'_2, U112'_1, U121'_2, U122'_1, U131'_3, U132'_2, U133'_1, U141'_3, U142'_2, U143'_1, U151'_3, U152'_2, U153'_1, U161'_2, CONS_2, NATSFROM_1, S_1, U171'_3, HEAD_1, AFTERNTH_2, U181'_2, U191'_2, PAIR_2, U201'_4, U202'_2, U21'_2, U211'_2, U221'_3, FST_1, U31'_2, U41'_3, U42'_2, U43'_1, U51'_3, U52'_2, U53'_1, U61'_2, U62'_1, U71'_2, U72'_1, U81'_2, U82'_1, U91'_2, U92'_1, AND_2, TAIL_1, TAKE_2, SEL_2, ISNATURAL_1, ISLNAT_1, ISPLNAT_1, ISNATURALKIND_1, ISLNATKIND_1, ISPLNATKIND_1, TOP_1

Compound Symbols:   c_1, c1_1, c2_1, c3_1, c4_1, c5_1, c6_1, c7_1, c8_1, c9_1, c10_1, c11_1, c12_1, c13_1, c14_1, c15_1, c16_1, c17_1, c18_1, c19_1, c20_1, c21_1, c22_1, c23_1, c24_1, c25_1, c26_1, c27_1, c28_1, c29_1, c30_1, c31_1, c32_1, c33_1, c34_1, c35_1, c36_1, c37_1, c38_1, c39_1, c40_1, c41_1, c42_1, c43_1, c44_1, c45_1, c46_1, c47_1, c48_1, c49_1, c50_1, c51_1, c52_1, c53_1, c54_1, c55_1, c56_1, c57_1, c58_1, c59_1, c60_1, c61_1, c62_1, c63_1, c64_1, c65_1, c66_1, c67_1, c68_1, c69_1, c70_1, c71_1, c72_1, c73_1, c74_1, c75_1, c76_1, c77_1, c78_1, c79_1, c80_1, c81_1, c82_1, c83_1, c84_1, c85_1, c86_1, c87_1, c88_1, c89_1, c90_1, c91_1, c92_1, c93_1, c94_1, c95_1, c96_1, c97_1, c98_1, c99_1, c100_1, c101_1, c102_1, c103_1, c104_1, c105_1, c106_1, c107_1, c108_1, c109_1, c110_1, c111_1, c112_1, c116_1, c117_1, c118_1, c119_1, c120_1, c121_1, c122_1


----------------------------------------

(17) SIsEmptyProof (BOTH BOUNDS(ID, ID))
The set S is empty
----------------------------------------

(18)
BOUNDS(1, 1)

----------------------------------------

(19) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
Transformed a relative TRS into a decreasing-loop problem.
----------------------------------------

(20)
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(U101(tt, V1, V2)) -> mark(U102(isNatural(V1), V2))
   active(U102(tt, V2)) -> mark(U103(isLNat(V2)))
   active(U103(tt)) -> mark(tt)
   active(U11(tt, N, XS)) -> mark(snd(splitAt(N, XS)))
   active(U111(tt, V1)) -> mark(U112(isLNat(V1)))
   active(U112(tt)) -> mark(tt)
   active(U121(tt, V1)) -> mark(U122(isNatural(V1)))
   active(U122(tt)) -> mark(tt)
   active(U131(tt, V1, V2)) -> mark(U132(isNatural(V1), V2))
   active(U132(tt, V2)) -> mark(U133(isLNat(V2)))
   active(U133(tt)) -> mark(tt)
   active(U141(tt, V1, V2)) -> mark(U142(isLNat(V1), V2))
   active(U142(tt, V2)) -> mark(U143(isLNat(V2)))
   active(U143(tt)) -> mark(tt)
   active(U151(tt, V1, V2)) -> mark(U152(isNatural(V1), V2))
   active(U152(tt, V2)) -> mark(U153(isLNat(V2)))
   active(U153(tt)) -> mark(tt)
   active(U161(tt, N)) -> mark(cons(N, natsFrom(s(N))))
   active(U171(tt, N, XS)) -> mark(head(afterNth(N, XS)))
   active(U181(tt, Y)) -> mark(Y)
   active(U191(tt, XS)) -> mark(pair(nil, XS))
   active(U201(tt, N, X, XS)) -> mark(U202(splitAt(N, XS), X))
   active(U202(pair(YS, ZS), X)) -> mark(pair(cons(X, YS), ZS))
   active(U21(tt, X)) -> mark(X)
   active(U211(tt, XS)) -> mark(XS)
   active(U221(tt, N, XS)) -> mark(fst(splitAt(N, XS)))
   active(U31(tt, N)) -> mark(N)
   active(U41(tt, V1, V2)) -> mark(U42(isNatural(V1), V2))
   active(U42(tt, V2)) -> mark(U43(isLNat(V2)))
   active(U43(tt)) -> mark(tt)
   active(U51(tt, V1, V2)) -> mark(U52(isNatural(V1), V2))
   active(U52(tt, V2)) -> mark(U53(isLNat(V2)))
   active(U53(tt)) -> mark(tt)
   active(U61(tt, V1)) -> mark(U62(isPLNat(V1)))
   active(U62(tt)) -> mark(tt)
   active(U71(tt, V1)) -> mark(U72(isNatural(V1)))
   active(U72(tt)) -> mark(tt)
   active(U81(tt, V1)) -> mark(U82(isPLNat(V1)))
   active(U82(tt)) -> mark(tt)
   active(U91(tt, V1)) -> mark(U92(isLNat(V1)))
   active(U92(tt)) -> mark(tt)
   active(afterNth(N, XS)) -> mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
   active(and(tt, X)) -> mark(X)
   active(fst(pair(X, Y))) -> mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
   active(head(cons(N, XS))) -> mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
   active(isLNat(nil)) -> mark(tt)
   active(isLNat(afterNth(V1, V2))) -> mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
   active(isLNat(cons(V1, V2))) -> mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
   active(isLNat(fst(V1))) -> mark(U61(isPLNatKind(V1), V1))
   active(isLNat(natsFrom(V1))) -> mark(U71(isNaturalKind(V1), V1))
   active(isLNat(snd(V1))) -> mark(U81(isPLNatKind(V1), V1))
   active(isLNat(tail(V1))) -> mark(U91(isLNatKind(V1), V1))
   active(isLNat(take(V1, V2))) -> mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
   active(isLNatKind(nil)) -> mark(tt)
   active(isLNatKind(afterNth(V1, V2))) -> mark(and(isNaturalKind(V1), isLNatKind(V2)))
   active(isLNatKind(cons(V1, V2))) -> mark(and(isNaturalKind(V1), isLNatKind(V2)))
   active(isLNatKind(fst(V1))) -> mark(isPLNatKind(V1))
   active(isLNatKind(natsFrom(V1))) -> mark(isNaturalKind(V1))
   active(isLNatKind(snd(V1))) -> mark(isPLNatKind(V1))
   active(isLNatKind(tail(V1))) -> mark(isLNatKind(V1))
   active(isLNatKind(take(V1, V2))) -> mark(and(isNaturalKind(V1), isLNatKind(V2)))
   active(isNatural(0)) -> mark(tt)
   active(isNatural(head(V1))) -> mark(U111(isLNatKind(V1), V1))
   active(isNatural(s(V1))) -> mark(U121(isNaturalKind(V1), V1))
   active(isNatural(sel(V1, V2))) -> mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
   active(isNaturalKind(0)) -> mark(tt)
   active(isNaturalKind(head(V1))) -> mark(isLNatKind(V1))
   active(isNaturalKind(s(V1))) -> mark(isNaturalKind(V1))
   active(isNaturalKind(sel(V1, V2))) -> mark(and(isNaturalKind(V1), isLNatKind(V2)))
   active(isPLNat(pair(V1, V2))) -> mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
   active(isPLNat(splitAt(V1, V2))) -> mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
   active(isPLNatKind(pair(V1, V2))) -> mark(and(isLNatKind(V1), isLNatKind(V2)))
   active(isPLNatKind(splitAt(V1, V2))) -> mark(and(isNaturalKind(V1), isLNatKind(V2)))
   active(natsFrom(N)) -> mark(U161(and(isNatural(N), isNaturalKind(N)), N))
   active(sel(N, XS)) -> mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
   active(snd(pair(X, Y))) -> mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
   active(splitAt(0, XS)) -> mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
   active(splitAt(s(N), cons(X, XS))) -> mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
   active(tail(cons(N, XS))) -> mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
   active(take(N, XS)) -> mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
   active(U101(X1, X2, X3)) -> U101(active(X1), X2, X3)
   active(U102(X1, X2)) -> U102(active(X1), X2)
   active(U103(X)) -> U103(active(X))
   active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3)
   active(snd(X)) -> snd(active(X))
   active(splitAt(X1, X2)) -> splitAt(active(X1), X2)
   active(splitAt(X1, X2)) -> splitAt(X1, active(X2))
   active(U111(X1, X2)) -> U111(active(X1), X2)
   active(U112(X)) -> U112(active(X))
   active(U121(X1, X2)) -> U121(active(X1), X2)
   active(U122(X)) -> U122(active(X))
   active(U131(X1, X2, X3)) -> U131(active(X1), X2, X3)
   active(U132(X1, X2)) -> U132(active(X1), X2)
   active(U133(X)) -> U133(active(X))
   active(U141(X1, X2, X3)) -> U141(active(X1), X2, X3)
   active(U142(X1, X2)) -> U142(active(X1), X2)
   active(U143(X)) -> U143(active(X))
   active(U151(X1, X2, X3)) -> U151(active(X1), X2, X3)
   active(U152(X1, X2)) -> U152(active(X1), X2)
   active(U153(X)) -> U153(active(X))
   active(U161(X1, X2)) -> U161(active(X1), X2)
   active(cons(X1, X2)) -> cons(active(X1), X2)
   active(natsFrom(X)) -> natsFrom(active(X))
   active(s(X)) -> s(active(X))
   active(U171(X1, X2, X3)) -> U171(active(X1), X2, X3)
   active(head(X)) -> head(active(X))
   active(afterNth(X1, X2)) -> afterNth(active(X1), X2)
   active(afterNth(X1, X2)) -> afterNth(X1, active(X2))
   active(U181(X1, X2)) -> U181(active(X1), X2)
   active(U191(X1, X2)) -> U191(active(X1), X2)
   active(pair(X1, X2)) -> pair(active(X1), X2)
   active(pair(X1, X2)) -> pair(X1, active(X2))
   active(U201(X1, X2, X3, X4)) -> U201(active(X1), X2, X3, X4)
   active(U202(X1, X2)) -> U202(active(X1), X2)
   active(U21(X1, X2)) -> U21(active(X1), X2)
   active(U211(X1, X2)) -> U211(active(X1), X2)
   active(U221(X1, X2, X3)) -> U221(active(X1), X2, X3)
   active(fst(X)) -> fst(active(X))
   active(U31(X1, X2)) -> U31(active(X1), X2)
   active(U41(X1, X2, X3)) -> U41(active(X1), X2, X3)
   active(U42(X1, X2)) -> U42(active(X1), X2)
   active(U43(X)) -> U43(active(X))
   active(U51(X1, X2, X3)) -> U51(active(X1), X2, X3)
   active(U52(X1, X2)) -> U52(active(X1), X2)
   active(U53(X)) -> U53(active(X))
   active(U61(X1, X2)) -> U61(active(X1), X2)
   active(U62(X)) -> U62(active(X))
   active(U71(X1, X2)) -> U71(active(X1), X2)
   active(U72(X)) -> U72(active(X))
   active(U81(X1, X2)) -> U81(active(X1), X2)
   active(U82(X)) -> U82(active(X))
   active(U91(X1, X2)) -> U91(active(X1), X2)
   active(U92(X)) -> U92(active(X))
   active(and(X1, X2)) -> and(active(X1), X2)
   active(tail(X)) -> tail(active(X))
   active(take(X1, X2)) -> take(active(X1), X2)
   active(take(X1, X2)) -> take(X1, active(X2))
   active(sel(X1, X2)) -> sel(active(X1), X2)
   active(sel(X1, X2)) -> sel(X1, active(X2))
   U101(mark(X1), X2, X3) -> mark(U101(X1, X2, X3))
   U102(mark(X1), X2) -> mark(U102(X1, X2))
   U103(mark(X)) -> mark(U103(X))
   U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3))
   snd(mark(X)) -> mark(snd(X))
   splitAt(mark(X1), X2) -> mark(splitAt(X1, X2))
   splitAt(X1, mark(X2)) -> mark(splitAt(X1, X2))
   U111(mark(X1), X2) -> mark(U111(X1, X2))
   U112(mark(X)) -> mark(U112(X))
   U121(mark(X1), X2) -> mark(U121(X1, X2))
   U122(mark(X)) -> mark(U122(X))
   U131(mark(X1), X2, X3) -> mark(U131(X1, X2, X3))
   U132(mark(X1), X2) -> mark(U132(X1, X2))
   U133(mark(X)) -> mark(U133(X))
   U141(mark(X1), X2, X3) -> mark(U141(X1, X2, X3))
   U142(mark(X1), X2) -> mark(U142(X1, X2))
   U143(mark(X)) -> mark(U143(X))
   U151(mark(X1), X2, X3) -> mark(U151(X1, X2, X3))
   U152(mark(X1), X2) -> mark(U152(X1, X2))
   U153(mark(X)) -> mark(U153(X))
   U161(mark(X1), X2) -> mark(U161(X1, X2))
   cons(mark(X1), X2) -> mark(cons(X1, X2))
   natsFrom(mark(X)) -> mark(natsFrom(X))
   s(mark(X)) -> mark(s(X))
   U171(mark(X1), X2, X3) -> mark(U171(X1, X2, X3))
   head(mark(X)) -> mark(head(X))
   afterNth(mark(X1), X2) -> mark(afterNth(X1, X2))
   afterNth(X1, mark(X2)) -> mark(afterNth(X1, X2))
   U181(mark(X1), X2) -> mark(U181(X1, X2))
   U191(mark(X1), X2) -> mark(U191(X1, X2))
   pair(mark(X1), X2) -> mark(pair(X1, X2))
   pair(X1, mark(X2)) -> mark(pair(X1, X2))
   U201(mark(X1), X2, X3, X4) -> mark(U201(X1, X2, X3, X4))
   U202(mark(X1), X2) -> mark(U202(X1, X2))
   U21(mark(X1), X2) -> mark(U21(X1, X2))
   U211(mark(X1), X2) -> mark(U211(X1, X2))
   U221(mark(X1), X2, X3) -> mark(U221(X1, X2, X3))
   fst(mark(X)) -> mark(fst(X))
   U31(mark(X1), X2) -> mark(U31(X1, X2))
   U41(mark(X1), X2, X3) -> mark(U41(X1, X2, X3))
   U42(mark(X1), X2) -> mark(U42(X1, X2))
   U43(mark(X)) -> mark(U43(X))
   U51(mark(X1), X2, X3) -> mark(U51(X1, X2, X3))
   U52(mark(X1), X2) -> mark(U52(X1, X2))
   U53(mark(X)) -> mark(U53(X))
   U61(mark(X1), X2) -> mark(U61(X1, X2))
   U62(mark(X)) -> mark(U62(X))
   U71(mark(X1), X2) -> mark(U71(X1, X2))
   U72(mark(X)) -> mark(U72(X))
   U81(mark(X1), X2) -> mark(U81(X1, X2))
   U82(mark(X)) -> mark(U82(X))
   U91(mark(X1), X2) -> mark(U91(X1, X2))
   U92(mark(X)) -> mark(U92(X))
   and(mark(X1), X2) -> mark(and(X1, X2))
   tail(mark(X)) -> mark(tail(X))
   take(mark(X1), X2) -> mark(take(X1, X2))
   take(X1, mark(X2)) -> mark(take(X1, X2))
   sel(mark(X1), X2) -> mark(sel(X1, X2))
   sel(X1, mark(X2)) -> mark(sel(X1, X2))
   proper(U101(X1, X2, X3)) -> U101(proper(X1), proper(X2), proper(X3))
   proper(tt) -> ok(tt)
   proper(U102(X1, X2)) -> U102(proper(X1), proper(X2))
   proper(isNatural(X)) -> isNatural(proper(X))
   proper(U103(X)) -> U103(proper(X))
   proper(isLNat(X)) -> isLNat(proper(X))
   proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3))
   proper(snd(X)) -> snd(proper(X))
   proper(splitAt(X1, X2)) -> splitAt(proper(X1), proper(X2))
   proper(U111(X1, X2)) -> U111(proper(X1), proper(X2))
   proper(U112(X)) -> U112(proper(X))
   proper(U121(X1, X2)) -> U121(proper(X1), proper(X2))
   proper(U122(X)) -> U122(proper(X))
   proper(U131(X1, X2, X3)) -> U131(proper(X1), proper(X2), proper(X3))
   proper(U132(X1, X2)) -> U132(proper(X1), proper(X2))
   proper(U133(X)) -> U133(proper(X))
   proper(U141(X1, X2, X3)) -> U141(proper(X1), proper(X2), proper(X3))
   proper(U142(X1, X2)) -> U142(proper(X1), proper(X2))
   proper(U143(X)) -> U143(proper(X))
   proper(U151(X1, X2, X3)) -> U151(proper(X1), proper(X2), proper(X3))
   proper(U152(X1, X2)) -> U152(proper(X1), proper(X2))
   proper(U153(X)) -> U153(proper(X))
   proper(U161(X1, X2)) -> U161(proper(X1), proper(X2))
   proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
   proper(natsFrom(X)) -> natsFrom(proper(X))
   proper(s(X)) -> s(proper(X))
   proper(U171(X1, X2, X3)) -> U171(proper(X1), proper(X2), proper(X3))
   proper(head(X)) -> head(proper(X))
   proper(afterNth(X1, X2)) -> afterNth(proper(X1), proper(X2))
   proper(U181(X1, X2)) -> U181(proper(X1), proper(X2))
   proper(U191(X1, X2)) -> U191(proper(X1), proper(X2))
   proper(pair(X1, X2)) -> pair(proper(X1), proper(X2))
   proper(nil) -> ok(nil)
   proper(U201(X1, X2, X3, X4)) -> U201(proper(X1), proper(X2), proper(X3), proper(X4))
   proper(U202(X1, X2)) -> U202(proper(X1), proper(X2))
   proper(U21(X1, X2)) -> U21(proper(X1), proper(X2))
   proper(U211(X1, X2)) -> U211(proper(X1), proper(X2))
   proper(U221(X1, X2, X3)) -> U221(proper(X1), proper(X2), proper(X3))
   proper(fst(X)) -> fst(proper(X))
   proper(U31(X1, X2)) -> U31(proper(X1), proper(X2))
   proper(U41(X1, X2, X3)) -> U41(proper(X1), proper(X2), proper(X3))
   proper(U42(X1, X2)) -> U42(proper(X1), proper(X2))
   proper(U43(X)) -> U43(proper(X))
   proper(U51(X1, X2, X3)) -> U51(proper(X1), proper(X2), proper(X3))
   proper(U52(X1, X2)) -> U52(proper(X1), proper(X2))
   proper(U53(X)) -> U53(proper(X))
   proper(U61(X1, X2)) -> U61(proper(X1), proper(X2))
   proper(U62(X)) -> U62(proper(X))
   proper(isPLNat(X)) -> isPLNat(proper(X))
   proper(U71(X1, X2)) -> U71(proper(X1), proper(X2))
   proper(U72(X)) -> U72(proper(X))
   proper(U81(X1, X2)) -> U81(proper(X1), proper(X2))
   proper(U82(X)) -> U82(proper(X))
   proper(U91(X1, X2)) -> U91(proper(X1), proper(X2))
   proper(U92(X)) -> U92(proper(X))
   proper(and(X1, X2)) -> and(proper(X1), proper(X2))
   proper(isNaturalKind(X)) -> isNaturalKind(proper(X))
   proper(isLNatKind(X)) -> isLNatKind(proper(X))
   proper(isPLNatKind(X)) -> isPLNatKind(proper(X))
   proper(tail(X)) -> tail(proper(X))
   proper(take(X1, X2)) -> take(proper(X1), proper(X2))
   proper(0) -> ok(0)
   proper(sel(X1, X2)) -> sel(proper(X1), proper(X2))
   U101(ok(X1), ok(X2), ok(X3)) -> ok(U101(X1, X2, X3))
   U102(ok(X1), ok(X2)) -> ok(U102(X1, X2))
   isNatural(ok(X)) -> ok(isNatural(X))
   U103(ok(X)) -> ok(U103(X))
   isLNat(ok(X)) -> ok(isLNat(X))
   U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
   snd(ok(X)) -> ok(snd(X))
   splitAt(ok(X1), ok(X2)) -> ok(splitAt(X1, X2))
   U111(ok(X1), ok(X2)) -> ok(U111(X1, X2))
   U112(ok(X)) -> ok(U112(X))
   U121(ok(X1), ok(X2)) -> ok(U121(X1, X2))
   U122(ok(X)) -> ok(U122(X))
   U131(ok(X1), ok(X2), ok(X3)) -> ok(U131(X1, X2, X3))
   U132(ok(X1), ok(X2)) -> ok(U132(X1, X2))
   U133(ok(X)) -> ok(U133(X))
   U141(ok(X1), ok(X2), ok(X3)) -> ok(U141(X1, X2, X3))
   U142(ok(X1), ok(X2)) -> ok(U142(X1, X2))
   U143(ok(X)) -> ok(U143(X))
   U151(ok(X1), ok(X2), ok(X3)) -> ok(U151(X1, X2, X3))
   U152(ok(X1), ok(X2)) -> ok(U152(X1, X2))
   U153(ok(X)) -> ok(U153(X))
   U161(ok(X1), ok(X2)) -> ok(U161(X1, X2))
   cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
   natsFrom(ok(X)) -> ok(natsFrom(X))
   s(ok(X)) -> ok(s(X))
   U171(ok(X1), ok(X2), ok(X3)) -> ok(U171(X1, X2, X3))
   head(ok(X)) -> ok(head(X))
   afterNth(ok(X1), ok(X2)) -> ok(afterNth(X1, X2))
   U181(ok(X1), ok(X2)) -> ok(U181(X1, X2))
   U191(ok(X1), ok(X2)) -> ok(U191(X1, X2))
   pair(ok(X1), ok(X2)) -> ok(pair(X1, X2))
   U201(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U201(X1, X2, X3, X4))
   U202(ok(X1), ok(X2)) -> ok(U202(X1, X2))
   U21(ok(X1), ok(X2)) -> ok(U21(X1, X2))
   U211(ok(X1), ok(X2)) -> ok(U211(X1, X2))
   U221(ok(X1), ok(X2), ok(X3)) -> ok(U221(X1, X2, X3))
   fst(ok(X)) -> ok(fst(X))
   U31(ok(X1), ok(X2)) -> ok(U31(X1, X2))
   U41(ok(X1), ok(X2), ok(X3)) -> ok(U41(X1, X2, X3))
   U42(ok(X1), ok(X2)) -> ok(U42(X1, X2))
   U43(ok(X)) -> ok(U43(X))
   U51(ok(X1), ok(X2), ok(X3)) -> ok(U51(X1, X2, X3))
   U52(ok(X1), ok(X2)) -> ok(U52(X1, X2))
   U53(ok(X)) -> ok(U53(X))
   U61(ok(X1), ok(X2)) -> ok(U61(X1, X2))
   U62(ok(X)) -> ok(U62(X))
   isPLNat(ok(X)) -> ok(isPLNat(X))
   U71(ok(X1), ok(X2)) -> ok(U71(X1, X2))
   U72(ok(X)) -> ok(U72(X))
   U81(ok(X1), ok(X2)) -> ok(U81(X1, X2))
   U82(ok(X)) -> ok(U82(X))
   U91(ok(X1), ok(X2)) -> ok(U91(X1, X2))
   U92(ok(X)) -> ok(U92(X))
   and(ok(X1), ok(X2)) -> ok(and(X1, X2))
   isNaturalKind(ok(X)) -> ok(isNaturalKind(X))
   isLNatKind(ok(X)) -> ok(isLNatKind(X))
   isPLNatKind(ok(X)) -> ok(isPLNatKind(X))
   tail(ok(X)) -> ok(tail(X))
   take(ok(X1), ok(X2)) -> ok(take(X1, X2))
   sel(ok(X1), ok(X2)) -> ok(sel(X1, X2))
   top(mark(X)) -> top(proper(X))
   top(ok(X)) -> top(active(X))

S is empty.
Rewrite Strategy: FULL
----------------------------------------

(21) DecreasingLoopProof (LOWER BOUND(ID))
The following loop(s) give(s) rise to the lower bound Omega(n^1):

The rewrite sequence

U133(mark(X)) ->^+ mark(U133(X))

gives rise to a decreasing loop by considering the right hand sides subterm at position [0].

The pumping substitution is [X / mark(X)].

The result substitution is [ ].




----------------------------------------

(22)
Complex Obligation (BEST)

----------------------------------------

(23)
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(U101(tt, V1, V2)) -> mark(U102(isNatural(V1), V2))
   active(U102(tt, V2)) -> mark(U103(isLNat(V2)))
   active(U103(tt)) -> mark(tt)
   active(U11(tt, N, XS)) -> mark(snd(splitAt(N, XS)))
   active(U111(tt, V1)) -> mark(U112(isLNat(V1)))
   active(U112(tt)) -> mark(tt)
   active(U121(tt, V1)) -> mark(U122(isNatural(V1)))
   active(U122(tt)) -> mark(tt)
   active(U131(tt, V1, V2)) -> mark(U132(isNatural(V1), V2))
   active(U132(tt, V2)) -> mark(U133(isLNat(V2)))
   active(U133(tt)) -> mark(tt)
   active(U141(tt, V1, V2)) -> mark(U142(isLNat(V1), V2))
   active(U142(tt, V2)) -> mark(U143(isLNat(V2)))
   active(U143(tt)) -> mark(tt)
   active(U151(tt, V1, V2)) -> mark(U152(isNatural(V1), V2))
   active(U152(tt, V2)) -> mark(U153(isLNat(V2)))
   active(U153(tt)) -> mark(tt)
   active(U161(tt, N)) -> mark(cons(N, natsFrom(s(N))))
   active(U171(tt, N, XS)) -> mark(head(afterNth(N, XS)))
   active(U181(tt, Y)) -> mark(Y)
   active(U191(tt, XS)) -> mark(pair(nil, XS))
   active(U201(tt, N, X, XS)) -> mark(U202(splitAt(N, XS), X))
   active(U202(pair(YS, ZS), X)) -> mark(pair(cons(X, YS), ZS))
   active(U21(tt, X)) -> mark(X)
   active(U211(tt, XS)) -> mark(XS)
   active(U221(tt, N, XS)) -> mark(fst(splitAt(N, XS)))
   active(U31(tt, N)) -> mark(N)
   active(U41(tt, V1, V2)) -> mark(U42(isNatural(V1), V2))
   active(U42(tt, V2)) -> mark(U43(isLNat(V2)))
   active(U43(tt)) -> mark(tt)
   active(U51(tt, V1, V2)) -> mark(U52(isNatural(V1), V2))
   active(U52(tt, V2)) -> mark(U53(isLNat(V2)))
   active(U53(tt)) -> mark(tt)
   active(U61(tt, V1)) -> mark(U62(isPLNat(V1)))
   active(U62(tt)) -> mark(tt)
   active(U71(tt, V1)) -> mark(U72(isNatural(V1)))
   active(U72(tt)) -> mark(tt)
   active(U81(tt, V1)) -> mark(U82(isPLNat(V1)))
   active(U82(tt)) -> mark(tt)
   active(U91(tt, V1)) -> mark(U92(isLNat(V1)))
   active(U92(tt)) -> mark(tt)
   active(afterNth(N, XS)) -> mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
   active(and(tt, X)) -> mark(X)
   active(fst(pair(X, Y))) -> mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
   active(head(cons(N, XS))) -> mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
   active(isLNat(nil)) -> mark(tt)
   active(isLNat(afterNth(V1, V2))) -> mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
   active(isLNat(cons(V1, V2))) -> mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
   active(isLNat(fst(V1))) -> mark(U61(isPLNatKind(V1), V1))
   active(isLNat(natsFrom(V1))) -> mark(U71(isNaturalKind(V1), V1))
   active(isLNat(snd(V1))) -> mark(U81(isPLNatKind(V1), V1))
   active(isLNat(tail(V1))) -> mark(U91(isLNatKind(V1), V1))
   active(isLNat(take(V1, V2))) -> mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
   active(isLNatKind(nil)) -> mark(tt)
   active(isLNatKind(afterNth(V1, V2))) -> mark(and(isNaturalKind(V1), isLNatKind(V2)))
   active(isLNatKind(cons(V1, V2))) -> mark(and(isNaturalKind(V1), isLNatKind(V2)))
   active(isLNatKind(fst(V1))) -> mark(isPLNatKind(V1))
   active(isLNatKind(natsFrom(V1))) -> mark(isNaturalKind(V1))
   active(isLNatKind(snd(V1))) -> mark(isPLNatKind(V1))
   active(isLNatKind(tail(V1))) -> mark(isLNatKind(V1))
   active(isLNatKind(take(V1, V2))) -> mark(and(isNaturalKind(V1), isLNatKind(V2)))
   active(isNatural(0)) -> mark(tt)
   active(isNatural(head(V1))) -> mark(U111(isLNatKind(V1), V1))
   active(isNatural(s(V1))) -> mark(U121(isNaturalKind(V1), V1))
   active(isNatural(sel(V1, V2))) -> mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
   active(isNaturalKind(0)) -> mark(tt)
   active(isNaturalKind(head(V1))) -> mark(isLNatKind(V1))
   active(isNaturalKind(s(V1))) -> mark(isNaturalKind(V1))
   active(isNaturalKind(sel(V1, V2))) -> mark(and(isNaturalKind(V1), isLNatKind(V2)))
   active(isPLNat(pair(V1, V2))) -> mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
   active(isPLNat(splitAt(V1, V2))) -> mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
   active(isPLNatKind(pair(V1, V2))) -> mark(and(isLNatKind(V1), isLNatKind(V2)))
   active(isPLNatKind(splitAt(V1, V2))) -> mark(and(isNaturalKind(V1), isLNatKind(V2)))
   active(natsFrom(N)) -> mark(U161(and(isNatural(N), isNaturalKind(N)), N))
   active(sel(N, XS)) -> mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
   active(snd(pair(X, Y))) -> mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
   active(splitAt(0, XS)) -> mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
   active(splitAt(s(N), cons(X, XS))) -> mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
   active(tail(cons(N, XS))) -> mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
   active(take(N, XS)) -> mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
   active(U101(X1, X2, X3)) -> U101(active(X1), X2, X3)
   active(U102(X1, X2)) -> U102(active(X1), X2)
   active(U103(X)) -> U103(active(X))
   active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3)
   active(snd(X)) -> snd(active(X))
   active(splitAt(X1, X2)) -> splitAt(active(X1), X2)
   active(splitAt(X1, X2)) -> splitAt(X1, active(X2))
   active(U111(X1, X2)) -> U111(active(X1), X2)
   active(U112(X)) -> U112(active(X))
   active(U121(X1, X2)) -> U121(active(X1), X2)
   active(U122(X)) -> U122(active(X))
   active(U131(X1, X2, X3)) -> U131(active(X1), X2, X3)
   active(U132(X1, X2)) -> U132(active(X1), X2)
   active(U133(X)) -> U133(active(X))
   active(U141(X1, X2, X3)) -> U141(active(X1), X2, X3)
   active(U142(X1, X2)) -> U142(active(X1), X2)
   active(U143(X)) -> U143(active(X))
   active(U151(X1, X2, X3)) -> U151(active(X1), X2, X3)
   active(U152(X1, X2)) -> U152(active(X1), X2)
   active(U153(X)) -> U153(active(X))
   active(U161(X1, X2)) -> U161(active(X1), X2)
   active(cons(X1, X2)) -> cons(active(X1), X2)
   active(natsFrom(X)) -> natsFrom(active(X))
   active(s(X)) -> s(active(X))
   active(U171(X1, X2, X3)) -> U171(active(X1), X2, X3)
   active(head(X)) -> head(active(X))
   active(afterNth(X1, X2)) -> afterNth(active(X1), X2)
   active(afterNth(X1, X2)) -> afterNth(X1, active(X2))
   active(U181(X1, X2)) -> U181(active(X1), X2)
   active(U191(X1, X2)) -> U191(active(X1), X2)
   active(pair(X1, X2)) -> pair(active(X1), X2)
   active(pair(X1, X2)) -> pair(X1, active(X2))
   active(U201(X1, X2, X3, X4)) -> U201(active(X1), X2, X3, X4)
   active(U202(X1, X2)) -> U202(active(X1), X2)
   active(U21(X1, X2)) -> U21(active(X1), X2)
   active(U211(X1, X2)) -> U211(active(X1), X2)
   active(U221(X1, X2, X3)) -> U221(active(X1), X2, X3)
   active(fst(X)) -> fst(active(X))
   active(U31(X1, X2)) -> U31(active(X1), X2)
   active(U41(X1, X2, X3)) -> U41(active(X1), X2, X3)
   active(U42(X1, X2)) -> U42(active(X1), X2)
   active(U43(X)) -> U43(active(X))
   active(U51(X1, X2, X3)) -> U51(active(X1), X2, X3)
   active(U52(X1, X2)) -> U52(active(X1), X2)
   active(U53(X)) -> U53(active(X))
   active(U61(X1, X2)) -> U61(active(X1), X2)
   active(U62(X)) -> U62(active(X))
   active(U71(X1, X2)) -> U71(active(X1), X2)
   active(U72(X)) -> U72(active(X))
   active(U81(X1, X2)) -> U81(active(X1), X2)
   active(U82(X)) -> U82(active(X))
   active(U91(X1, X2)) -> U91(active(X1), X2)
   active(U92(X)) -> U92(active(X))
   active(and(X1, X2)) -> and(active(X1), X2)
   active(tail(X)) -> tail(active(X))
   active(take(X1, X2)) -> take(active(X1), X2)
   active(take(X1, X2)) -> take(X1, active(X2))
   active(sel(X1, X2)) -> sel(active(X1), X2)
   active(sel(X1, X2)) -> sel(X1, active(X2))
   U101(mark(X1), X2, X3) -> mark(U101(X1, X2, X3))
   U102(mark(X1), X2) -> mark(U102(X1, X2))
   U103(mark(X)) -> mark(U103(X))
   U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3))
   snd(mark(X)) -> mark(snd(X))
   splitAt(mark(X1), X2) -> mark(splitAt(X1, X2))
   splitAt(X1, mark(X2)) -> mark(splitAt(X1, X2))
   U111(mark(X1), X2) -> mark(U111(X1, X2))
   U112(mark(X)) -> mark(U112(X))
   U121(mark(X1), X2) -> mark(U121(X1, X2))
   U122(mark(X)) -> mark(U122(X))
   U131(mark(X1), X2, X3) -> mark(U131(X1, X2, X3))
   U132(mark(X1), X2) -> mark(U132(X1, X2))
   U133(mark(X)) -> mark(U133(X))
   U141(mark(X1), X2, X3) -> mark(U141(X1, X2, X3))
   U142(mark(X1), X2) -> mark(U142(X1, X2))
   U143(mark(X)) -> mark(U143(X))
   U151(mark(X1), X2, X3) -> mark(U151(X1, X2, X3))
   U152(mark(X1), X2) -> mark(U152(X1, X2))
   U153(mark(X)) -> mark(U153(X))
   U161(mark(X1), X2) -> mark(U161(X1, X2))
   cons(mark(X1), X2) -> mark(cons(X1, X2))
   natsFrom(mark(X)) -> mark(natsFrom(X))
   s(mark(X)) -> mark(s(X))
   U171(mark(X1), X2, X3) -> mark(U171(X1, X2, X3))
   head(mark(X)) -> mark(head(X))
   afterNth(mark(X1), X2) -> mark(afterNth(X1, X2))
   afterNth(X1, mark(X2)) -> mark(afterNth(X1, X2))
   U181(mark(X1), X2) -> mark(U181(X1, X2))
   U191(mark(X1), X2) -> mark(U191(X1, X2))
   pair(mark(X1), X2) -> mark(pair(X1, X2))
   pair(X1, mark(X2)) -> mark(pair(X1, X2))
   U201(mark(X1), X2, X3, X4) -> mark(U201(X1, X2, X3, X4))
   U202(mark(X1), X2) -> mark(U202(X1, X2))
   U21(mark(X1), X2) -> mark(U21(X1, X2))
   U211(mark(X1), X2) -> mark(U211(X1, X2))
   U221(mark(X1), X2, X3) -> mark(U221(X1, X2, X3))
   fst(mark(X)) -> mark(fst(X))
   U31(mark(X1), X2) -> mark(U31(X1, X2))
   U41(mark(X1), X2, X3) -> mark(U41(X1, X2, X3))
   U42(mark(X1), X2) -> mark(U42(X1, X2))
   U43(mark(X)) -> mark(U43(X))
   U51(mark(X1), X2, X3) -> mark(U51(X1, X2, X3))
   U52(mark(X1), X2) -> mark(U52(X1, X2))
   U53(mark(X)) -> mark(U53(X))
   U61(mark(X1), X2) -> mark(U61(X1, X2))
   U62(mark(X)) -> mark(U62(X))
   U71(mark(X1), X2) -> mark(U71(X1, X2))
   U72(mark(X)) -> mark(U72(X))
   U81(mark(X1), X2) -> mark(U81(X1, X2))
   U82(mark(X)) -> mark(U82(X))
   U91(mark(X1), X2) -> mark(U91(X1, X2))
   U92(mark(X)) -> mark(U92(X))
   and(mark(X1), X2) -> mark(and(X1, X2))
   tail(mark(X)) -> mark(tail(X))
   take(mark(X1), X2) -> mark(take(X1, X2))
   take(X1, mark(X2)) -> mark(take(X1, X2))
   sel(mark(X1), X2) -> mark(sel(X1, X2))
   sel(X1, mark(X2)) -> mark(sel(X1, X2))
   proper(U101(X1, X2, X3)) -> U101(proper(X1), proper(X2), proper(X3))
   proper(tt) -> ok(tt)
   proper(U102(X1, X2)) -> U102(proper(X1), proper(X2))
   proper(isNatural(X)) -> isNatural(proper(X))
   proper(U103(X)) -> U103(proper(X))
   proper(isLNat(X)) -> isLNat(proper(X))
   proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3))
   proper(snd(X)) -> snd(proper(X))
   proper(splitAt(X1, X2)) -> splitAt(proper(X1), proper(X2))
   proper(U111(X1, X2)) -> U111(proper(X1), proper(X2))
   proper(U112(X)) -> U112(proper(X))
   proper(U121(X1, X2)) -> U121(proper(X1), proper(X2))
   proper(U122(X)) -> U122(proper(X))
   proper(U131(X1, X2, X3)) -> U131(proper(X1), proper(X2), proper(X3))
   proper(U132(X1, X2)) -> U132(proper(X1), proper(X2))
   proper(U133(X)) -> U133(proper(X))
   proper(U141(X1, X2, X3)) -> U141(proper(X1), proper(X2), proper(X3))
   proper(U142(X1, X2)) -> U142(proper(X1), proper(X2))
   proper(U143(X)) -> U143(proper(X))
   proper(U151(X1, X2, X3)) -> U151(proper(X1), proper(X2), proper(X3))
   proper(U152(X1, X2)) -> U152(proper(X1), proper(X2))
   proper(U153(X)) -> U153(proper(X))
   proper(U161(X1, X2)) -> U161(proper(X1), proper(X2))
   proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
   proper(natsFrom(X)) -> natsFrom(proper(X))
   proper(s(X)) -> s(proper(X))
   proper(U171(X1, X2, X3)) -> U171(proper(X1), proper(X2), proper(X3))
   proper(head(X)) -> head(proper(X))
   proper(afterNth(X1, X2)) -> afterNth(proper(X1), proper(X2))
   proper(U181(X1, X2)) -> U181(proper(X1), proper(X2))
   proper(U191(X1, X2)) -> U191(proper(X1), proper(X2))
   proper(pair(X1, X2)) -> pair(proper(X1), proper(X2))
   proper(nil) -> ok(nil)
   proper(U201(X1, X2, X3, X4)) -> U201(proper(X1), proper(X2), proper(X3), proper(X4))
   proper(U202(X1, X2)) -> U202(proper(X1), proper(X2))
   proper(U21(X1, X2)) -> U21(proper(X1), proper(X2))
   proper(U211(X1, X2)) -> U211(proper(X1), proper(X2))
   proper(U221(X1, X2, X3)) -> U221(proper(X1), proper(X2), proper(X3))
   proper(fst(X)) -> fst(proper(X))
   proper(U31(X1, X2)) -> U31(proper(X1), proper(X2))
   proper(U41(X1, X2, X3)) -> U41(proper(X1), proper(X2), proper(X3))
   proper(U42(X1, X2)) -> U42(proper(X1), proper(X2))
   proper(U43(X)) -> U43(proper(X))
   proper(U51(X1, X2, X3)) -> U51(proper(X1), proper(X2), proper(X3))
   proper(U52(X1, X2)) -> U52(proper(X1), proper(X2))
   proper(U53(X)) -> U53(proper(X))
   proper(U61(X1, X2)) -> U61(proper(X1), proper(X2))
   proper(U62(X)) -> U62(proper(X))
   proper(isPLNat(X)) -> isPLNat(proper(X))
   proper(U71(X1, X2)) -> U71(proper(X1), proper(X2))
   proper(U72(X)) -> U72(proper(X))
   proper(U81(X1, X2)) -> U81(proper(X1), proper(X2))
   proper(U82(X)) -> U82(proper(X))
   proper(U91(X1, X2)) -> U91(proper(X1), proper(X2))
   proper(U92(X)) -> U92(proper(X))
   proper(and(X1, X2)) -> and(proper(X1), proper(X2))
   proper(isNaturalKind(X)) -> isNaturalKind(proper(X))
   proper(isLNatKind(X)) -> isLNatKind(proper(X))
   proper(isPLNatKind(X)) -> isPLNatKind(proper(X))
   proper(tail(X)) -> tail(proper(X))
   proper(take(X1, X2)) -> take(proper(X1), proper(X2))
   proper(0) -> ok(0)
   proper(sel(X1, X2)) -> sel(proper(X1), proper(X2))
   U101(ok(X1), ok(X2), ok(X3)) -> ok(U101(X1, X2, X3))
   U102(ok(X1), ok(X2)) -> ok(U102(X1, X2))
   isNatural(ok(X)) -> ok(isNatural(X))
   U103(ok(X)) -> ok(U103(X))
   isLNat(ok(X)) -> ok(isLNat(X))
   U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
   snd(ok(X)) -> ok(snd(X))
   splitAt(ok(X1), ok(X2)) -> ok(splitAt(X1, X2))
   U111(ok(X1), ok(X2)) -> ok(U111(X1, X2))
   U112(ok(X)) -> ok(U112(X))
   U121(ok(X1), ok(X2)) -> ok(U121(X1, X2))
   U122(ok(X)) -> ok(U122(X))
   U131(ok(X1), ok(X2), ok(X3)) -> ok(U131(X1, X2, X3))
   U132(ok(X1), ok(X2)) -> ok(U132(X1, X2))
   U133(ok(X)) -> ok(U133(X))
   U141(ok(X1), ok(X2), ok(X3)) -> ok(U141(X1, X2, X3))
   U142(ok(X1), ok(X2)) -> ok(U142(X1, X2))
   U143(ok(X)) -> ok(U143(X))
   U151(ok(X1), ok(X2), ok(X3)) -> ok(U151(X1, X2, X3))
   U152(ok(X1), ok(X2)) -> ok(U152(X1, X2))
   U153(ok(X)) -> ok(U153(X))
   U161(ok(X1), ok(X2)) -> ok(U161(X1, X2))
   cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
   natsFrom(ok(X)) -> ok(natsFrom(X))
   s(ok(X)) -> ok(s(X))
   U171(ok(X1), ok(X2), ok(X3)) -> ok(U171(X1, X2, X3))
   head(ok(X)) -> ok(head(X))
   afterNth(ok(X1), ok(X2)) -> ok(afterNth(X1, X2))
   U181(ok(X1), ok(X2)) -> ok(U181(X1, X2))
   U191(ok(X1), ok(X2)) -> ok(U191(X1, X2))
   pair(ok(X1), ok(X2)) -> ok(pair(X1, X2))
   U201(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U201(X1, X2, X3, X4))
   U202(ok(X1), ok(X2)) -> ok(U202(X1, X2))
   U21(ok(X1), ok(X2)) -> ok(U21(X1, X2))
   U211(ok(X1), ok(X2)) -> ok(U211(X1, X2))
   U221(ok(X1), ok(X2), ok(X3)) -> ok(U221(X1, X2, X3))
   fst(ok(X)) -> ok(fst(X))
   U31(ok(X1), ok(X2)) -> ok(U31(X1, X2))
   U41(ok(X1), ok(X2), ok(X3)) -> ok(U41(X1, X2, X3))
   U42(ok(X1), ok(X2)) -> ok(U42(X1, X2))
   U43(ok(X)) -> ok(U43(X))
   U51(ok(X1), ok(X2), ok(X3)) -> ok(U51(X1, X2, X3))
   U52(ok(X1), ok(X2)) -> ok(U52(X1, X2))
   U53(ok(X)) -> ok(U53(X))
   U61(ok(X1), ok(X2)) -> ok(U61(X1, X2))
   U62(ok(X)) -> ok(U62(X))
   isPLNat(ok(X)) -> ok(isPLNat(X))
   U71(ok(X1), ok(X2)) -> ok(U71(X1, X2))
   U72(ok(X)) -> ok(U72(X))
   U81(ok(X1), ok(X2)) -> ok(U81(X1, X2))
   U82(ok(X)) -> ok(U82(X))
   U91(ok(X1), ok(X2)) -> ok(U91(X1, X2))
   U92(ok(X)) -> ok(U92(X))
   and(ok(X1), ok(X2)) -> ok(and(X1, X2))
   isNaturalKind(ok(X)) -> ok(isNaturalKind(X))
   isLNatKind(ok(X)) -> ok(isLNatKind(X))
   isPLNatKind(ok(X)) -> ok(isPLNatKind(X))
   tail(ok(X)) -> ok(tail(X))
   take(ok(X1), ok(X2)) -> ok(take(X1, X2))
   sel(ok(X1), ok(X2)) -> ok(sel(X1, X2))
   top(mark(X)) -> top(proper(X))
   top(ok(X)) -> top(active(X))

S is empty.
Rewrite Strategy: FULL
----------------------------------------

(24) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
----------------------------------------

(25)
BOUNDS(n^1, INF)

----------------------------------------

(26)
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(U101(tt, V1, V2)) -> mark(U102(isNatural(V1), V2))
   active(U102(tt, V2)) -> mark(U103(isLNat(V2)))
   active(U103(tt)) -> mark(tt)
   active(U11(tt, N, XS)) -> mark(snd(splitAt(N, XS)))
   active(U111(tt, V1)) -> mark(U112(isLNat(V1)))
   active(U112(tt)) -> mark(tt)
   active(U121(tt, V1)) -> mark(U122(isNatural(V1)))
   active(U122(tt)) -> mark(tt)
   active(U131(tt, V1, V2)) -> mark(U132(isNatural(V1), V2))
   active(U132(tt, V2)) -> mark(U133(isLNat(V2)))
   active(U133(tt)) -> mark(tt)
   active(U141(tt, V1, V2)) -> mark(U142(isLNat(V1), V2))
   active(U142(tt, V2)) -> mark(U143(isLNat(V2)))
   active(U143(tt)) -> mark(tt)
   active(U151(tt, V1, V2)) -> mark(U152(isNatural(V1), V2))
   active(U152(tt, V2)) -> mark(U153(isLNat(V2)))
   active(U153(tt)) -> mark(tt)
   active(U161(tt, N)) -> mark(cons(N, natsFrom(s(N))))
   active(U171(tt, N, XS)) -> mark(head(afterNth(N, XS)))
   active(U181(tt, Y)) -> mark(Y)
   active(U191(tt, XS)) -> mark(pair(nil, XS))
   active(U201(tt, N, X, XS)) -> mark(U202(splitAt(N, XS), X))
   active(U202(pair(YS, ZS), X)) -> mark(pair(cons(X, YS), ZS))
   active(U21(tt, X)) -> mark(X)
   active(U211(tt, XS)) -> mark(XS)
   active(U221(tt, N, XS)) -> mark(fst(splitAt(N, XS)))
   active(U31(tt, N)) -> mark(N)
   active(U41(tt, V1, V2)) -> mark(U42(isNatural(V1), V2))
   active(U42(tt, V2)) -> mark(U43(isLNat(V2)))
   active(U43(tt)) -> mark(tt)
   active(U51(tt, V1, V2)) -> mark(U52(isNatural(V1), V2))
   active(U52(tt, V2)) -> mark(U53(isLNat(V2)))
   active(U53(tt)) -> mark(tt)
   active(U61(tt, V1)) -> mark(U62(isPLNat(V1)))
   active(U62(tt)) -> mark(tt)
   active(U71(tt, V1)) -> mark(U72(isNatural(V1)))
   active(U72(tt)) -> mark(tt)
   active(U81(tt, V1)) -> mark(U82(isPLNat(V1)))
   active(U82(tt)) -> mark(tt)
   active(U91(tt, V1)) -> mark(U92(isLNat(V1)))
   active(U92(tt)) -> mark(tt)
   active(afterNth(N, XS)) -> mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
   active(and(tt, X)) -> mark(X)
   active(fst(pair(X, Y))) -> mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
   active(head(cons(N, XS))) -> mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
   active(isLNat(nil)) -> mark(tt)
   active(isLNat(afterNth(V1, V2))) -> mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
   active(isLNat(cons(V1, V2))) -> mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
   active(isLNat(fst(V1))) -> mark(U61(isPLNatKind(V1), V1))
   active(isLNat(natsFrom(V1))) -> mark(U71(isNaturalKind(V1), V1))
   active(isLNat(snd(V1))) -> mark(U81(isPLNatKind(V1), V1))
   active(isLNat(tail(V1))) -> mark(U91(isLNatKind(V1), V1))
   active(isLNat(take(V1, V2))) -> mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
   active(isLNatKind(nil)) -> mark(tt)
   active(isLNatKind(afterNth(V1, V2))) -> mark(and(isNaturalKind(V1), isLNatKind(V2)))
   active(isLNatKind(cons(V1, V2))) -> mark(and(isNaturalKind(V1), isLNatKind(V2)))
   active(isLNatKind(fst(V1))) -> mark(isPLNatKind(V1))
   active(isLNatKind(natsFrom(V1))) -> mark(isNaturalKind(V1))
   active(isLNatKind(snd(V1))) -> mark(isPLNatKind(V1))
   active(isLNatKind(tail(V1))) -> mark(isLNatKind(V1))
   active(isLNatKind(take(V1, V2))) -> mark(and(isNaturalKind(V1), isLNatKind(V2)))
   active(isNatural(0)) -> mark(tt)
   active(isNatural(head(V1))) -> mark(U111(isLNatKind(V1), V1))
   active(isNatural(s(V1))) -> mark(U121(isNaturalKind(V1), V1))
   active(isNatural(sel(V1, V2))) -> mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
   active(isNaturalKind(0)) -> mark(tt)
   active(isNaturalKind(head(V1))) -> mark(isLNatKind(V1))
   active(isNaturalKind(s(V1))) -> mark(isNaturalKind(V1))
   active(isNaturalKind(sel(V1, V2))) -> mark(and(isNaturalKind(V1), isLNatKind(V2)))
   active(isPLNat(pair(V1, V2))) -> mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
   active(isPLNat(splitAt(V1, V2))) -> mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
   active(isPLNatKind(pair(V1, V2))) -> mark(and(isLNatKind(V1), isLNatKind(V2)))
   active(isPLNatKind(splitAt(V1, V2))) -> mark(and(isNaturalKind(V1), isLNatKind(V2)))
   active(natsFrom(N)) -> mark(U161(and(isNatural(N), isNaturalKind(N)), N))
   active(sel(N, XS)) -> mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
   active(snd(pair(X, Y))) -> mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
   active(splitAt(0, XS)) -> mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
   active(splitAt(s(N), cons(X, XS))) -> mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
   active(tail(cons(N, XS))) -> mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
   active(take(N, XS)) -> mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
   active(U101(X1, X2, X3)) -> U101(active(X1), X2, X3)
   active(U102(X1, X2)) -> U102(active(X1), X2)
   active(U103(X)) -> U103(active(X))
   active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3)
   active(snd(X)) -> snd(active(X))
   active(splitAt(X1, X2)) -> splitAt(active(X1), X2)
   active(splitAt(X1, X2)) -> splitAt(X1, active(X2))
   active(U111(X1, X2)) -> U111(active(X1), X2)
   active(U112(X)) -> U112(active(X))
   active(U121(X1, X2)) -> U121(active(X1), X2)
   active(U122(X)) -> U122(active(X))
   active(U131(X1, X2, X3)) -> U131(active(X1), X2, X3)
   active(U132(X1, X2)) -> U132(active(X1), X2)
   active(U133(X)) -> U133(active(X))
   active(U141(X1, X2, X3)) -> U141(active(X1), X2, X3)
   active(U142(X1, X2)) -> U142(active(X1), X2)
   active(U143(X)) -> U143(active(X))
   active(U151(X1, X2, X3)) -> U151(active(X1), X2, X3)
   active(U152(X1, X2)) -> U152(active(X1), X2)
   active(U153(X)) -> U153(active(X))
   active(U161(X1, X2)) -> U161(active(X1), X2)
   active(cons(X1, X2)) -> cons(active(X1), X2)
   active(natsFrom(X)) -> natsFrom(active(X))
   active(s(X)) -> s(active(X))
   active(U171(X1, X2, X3)) -> U171(active(X1), X2, X3)
   active(head(X)) -> head(active(X))
   active(afterNth(X1, X2)) -> afterNth(active(X1), X2)
   active(afterNth(X1, X2)) -> afterNth(X1, active(X2))
   active(U181(X1, X2)) -> U181(active(X1), X2)
   active(U191(X1, X2)) -> U191(active(X1), X2)
   active(pair(X1, X2)) -> pair(active(X1), X2)
   active(pair(X1, X2)) -> pair(X1, active(X2))
   active(U201(X1, X2, X3, X4)) -> U201(active(X1), X2, X3, X4)
   active(U202(X1, X2)) -> U202(active(X1), X2)
   active(U21(X1, X2)) -> U21(active(X1), X2)
   active(U211(X1, X2)) -> U211(active(X1), X2)
   active(U221(X1, X2, X3)) -> U221(active(X1), X2, X3)
   active(fst(X)) -> fst(active(X))
   active(U31(X1, X2)) -> U31(active(X1), X2)
   active(U41(X1, X2, X3)) -> U41(active(X1), X2, X3)
   active(U42(X1, X2)) -> U42(active(X1), X2)
   active(U43(X)) -> U43(active(X))
   active(U51(X1, X2, X3)) -> U51(active(X1), X2, X3)
   active(U52(X1, X2)) -> U52(active(X1), X2)
   active(U53(X)) -> U53(active(X))
   active(U61(X1, X2)) -> U61(active(X1), X2)
   active(U62(X)) -> U62(active(X))
   active(U71(X1, X2)) -> U71(active(X1), X2)
   active(U72(X)) -> U72(active(X))
   active(U81(X1, X2)) -> U81(active(X1), X2)
   active(U82(X)) -> U82(active(X))
   active(U91(X1, X2)) -> U91(active(X1), X2)
   active(U92(X)) -> U92(active(X))
   active(and(X1, X2)) -> and(active(X1), X2)
   active(tail(X)) -> tail(active(X))
   active(take(X1, X2)) -> take(active(X1), X2)
   active(take(X1, X2)) -> take(X1, active(X2))
   active(sel(X1, X2)) -> sel(active(X1), X2)
   active(sel(X1, X2)) -> sel(X1, active(X2))
   U101(mark(X1), X2, X3) -> mark(U101(X1, X2, X3))
   U102(mark(X1), X2) -> mark(U102(X1, X2))
   U103(mark(X)) -> mark(U103(X))
   U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3))
   snd(mark(X)) -> mark(snd(X))
   splitAt(mark(X1), X2) -> mark(splitAt(X1, X2))
   splitAt(X1, mark(X2)) -> mark(splitAt(X1, X2))
   U111(mark(X1), X2) -> mark(U111(X1, X2))
   U112(mark(X)) -> mark(U112(X))
   U121(mark(X1), X2) -> mark(U121(X1, X2))
   U122(mark(X)) -> mark(U122(X))
   U131(mark(X1), X2, X3) -> mark(U131(X1, X2, X3))
   U132(mark(X1), X2) -> mark(U132(X1, X2))
   U133(mark(X)) -> mark(U133(X))
   U141(mark(X1), X2, X3) -> mark(U141(X1, X2, X3))
   U142(mark(X1), X2) -> mark(U142(X1, X2))
   U143(mark(X)) -> mark(U143(X))
   U151(mark(X1), X2, X3) -> mark(U151(X1, X2, X3))
   U152(mark(X1), X2) -> mark(U152(X1, X2))
   U153(mark(X)) -> mark(U153(X))
   U161(mark(X1), X2) -> mark(U161(X1, X2))
   cons(mark(X1), X2) -> mark(cons(X1, X2))
   natsFrom(mark(X)) -> mark(natsFrom(X))
   s(mark(X)) -> mark(s(X))
   U171(mark(X1), X2, X3) -> mark(U171(X1, X2, X3))
   head(mark(X)) -> mark(head(X))
   afterNth(mark(X1), X2) -> mark(afterNth(X1, X2))
   afterNth(X1, mark(X2)) -> mark(afterNth(X1, X2))
   U181(mark(X1), X2) -> mark(U181(X1, X2))
   U191(mark(X1), X2) -> mark(U191(X1, X2))
   pair(mark(X1), X2) -> mark(pair(X1, X2))
   pair(X1, mark(X2)) -> mark(pair(X1, X2))
   U201(mark(X1), X2, X3, X4) -> mark(U201(X1, X2, X3, X4))
   U202(mark(X1), X2) -> mark(U202(X1, X2))
   U21(mark(X1), X2) -> mark(U21(X1, X2))
   U211(mark(X1), X2) -> mark(U211(X1, X2))
   U221(mark(X1), X2, X3) -> mark(U221(X1, X2, X3))
   fst(mark(X)) -> mark(fst(X))
   U31(mark(X1), X2) -> mark(U31(X1, X2))
   U41(mark(X1), X2, X3) -> mark(U41(X1, X2, X3))
   U42(mark(X1), X2) -> mark(U42(X1, X2))
   U43(mark(X)) -> mark(U43(X))
   U51(mark(X1), X2, X3) -> mark(U51(X1, X2, X3))
   U52(mark(X1), X2) -> mark(U52(X1, X2))
   U53(mark(X)) -> mark(U53(X))
   U61(mark(X1), X2) -> mark(U61(X1, X2))
   U62(mark(X)) -> mark(U62(X))
   U71(mark(X1), X2) -> mark(U71(X1, X2))
   U72(mark(X)) -> mark(U72(X))
   U81(mark(X1), X2) -> mark(U81(X1, X2))
   U82(mark(X)) -> mark(U82(X))
   U91(mark(X1), X2) -> mark(U91(X1, X2))
   U92(mark(X)) -> mark(U92(X))
   and(mark(X1), X2) -> mark(and(X1, X2))
   tail(mark(X)) -> mark(tail(X))
   take(mark(X1), X2) -> mark(take(X1, X2))
   take(X1, mark(X2)) -> mark(take(X1, X2))
   sel(mark(X1), X2) -> mark(sel(X1, X2))
   sel(X1, mark(X2)) -> mark(sel(X1, X2))
   proper(U101(X1, X2, X3)) -> U101(proper(X1), proper(X2), proper(X3))
   proper(tt) -> ok(tt)
   proper(U102(X1, X2)) -> U102(proper(X1), proper(X2))
   proper(isNatural(X)) -> isNatural(proper(X))
   proper(U103(X)) -> U103(proper(X))
   proper(isLNat(X)) -> isLNat(proper(X))
   proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3))
   proper(snd(X)) -> snd(proper(X))
   proper(splitAt(X1, X2)) -> splitAt(proper(X1), proper(X2))
   proper(U111(X1, X2)) -> U111(proper(X1), proper(X2))
   proper(U112(X)) -> U112(proper(X))
   proper(U121(X1, X2)) -> U121(proper(X1), proper(X2))
   proper(U122(X)) -> U122(proper(X))
   proper(U131(X1, X2, X3)) -> U131(proper(X1), proper(X2), proper(X3))
   proper(U132(X1, X2)) -> U132(proper(X1), proper(X2))
   proper(U133(X)) -> U133(proper(X))
   proper(U141(X1, X2, X3)) -> U141(proper(X1), proper(X2), proper(X3))
   proper(U142(X1, X2)) -> U142(proper(X1), proper(X2))
   proper(U143(X)) -> U143(proper(X))
   proper(U151(X1, X2, X3)) -> U151(proper(X1), proper(X2), proper(X3))
   proper(U152(X1, X2)) -> U152(proper(X1), proper(X2))
   proper(U153(X)) -> U153(proper(X))
   proper(U161(X1, X2)) -> U161(proper(X1), proper(X2))
   proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
   proper(natsFrom(X)) -> natsFrom(proper(X))
   proper(s(X)) -> s(proper(X))
   proper(U171(X1, X2, X3)) -> U171(proper(X1), proper(X2), proper(X3))
   proper(head(X)) -> head(proper(X))
   proper(afterNth(X1, X2)) -> afterNth(proper(X1), proper(X2))
   proper(U181(X1, X2)) -> U181(proper(X1), proper(X2))
   proper(U191(X1, X2)) -> U191(proper(X1), proper(X2))
   proper(pair(X1, X2)) -> pair(proper(X1), proper(X2))
   proper(nil) -> ok(nil)
   proper(U201(X1, X2, X3, X4)) -> U201(proper(X1), proper(X2), proper(X3), proper(X4))
   proper(U202(X1, X2)) -> U202(proper(X1), proper(X2))
   proper(U21(X1, X2)) -> U21(proper(X1), proper(X2))
   proper(U211(X1, X2)) -> U211(proper(X1), proper(X2))
   proper(U221(X1, X2, X3)) -> U221(proper(X1), proper(X2), proper(X3))
   proper(fst(X)) -> fst(proper(X))
   proper(U31(X1, X2)) -> U31(proper(X1), proper(X2))
   proper(U41(X1, X2, X3)) -> U41(proper(X1), proper(X2), proper(X3))
   proper(U42(X1, X2)) -> U42(proper(X1), proper(X2))
   proper(U43(X)) -> U43(proper(X))
   proper(U51(X1, X2, X3)) -> U51(proper(X1), proper(X2), proper(X3))
   proper(U52(X1, X2)) -> U52(proper(X1), proper(X2))
   proper(U53(X)) -> U53(proper(X))
   proper(U61(X1, X2)) -> U61(proper(X1), proper(X2))
   proper(U62(X)) -> U62(proper(X))
   proper(isPLNat(X)) -> isPLNat(proper(X))
   proper(U71(X1, X2)) -> U71(proper(X1), proper(X2))
   proper(U72(X)) -> U72(proper(X))
   proper(U81(X1, X2)) -> U81(proper(X1), proper(X2))
   proper(U82(X)) -> U82(proper(X))
   proper(U91(X1, X2)) -> U91(proper(X1), proper(X2))
   proper(U92(X)) -> U92(proper(X))
   proper(and(X1, X2)) -> and(proper(X1), proper(X2))
   proper(isNaturalKind(X)) -> isNaturalKind(proper(X))
   proper(isLNatKind(X)) -> isLNatKind(proper(X))
   proper(isPLNatKind(X)) -> isPLNatKind(proper(X))
   proper(tail(X)) -> tail(proper(X))
   proper(take(X1, X2)) -> take(proper(X1), proper(X2))
   proper(0) -> ok(0)
   proper(sel(X1, X2)) -> sel(proper(X1), proper(X2))
   U101(ok(X1), ok(X2), ok(X3)) -> ok(U101(X1, X2, X3))
   U102(ok(X1), ok(X2)) -> ok(U102(X1, X2))
   isNatural(ok(X)) -> ok(isNatural(X))
   U103(ok(X)) -> ok(U103(X))
   isLNat(ok(X)) -> ok(isLNat(X))
   U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
   snd(ok(X)) -> ok(snd(X))
   splitAt(ok(X1), ok(X2)) -> ok(splitAt(X1, X2))
   U111(ok(X1), ok(X2)) -> ok(U111(X1, X2))
   U112(ok(X)) -> ok(U112(X))
   U121(ok(X1), ok(X2)) -> ok(U121(X1, X2))
   U122(ok(X)) -> ok(U122(X))
   U131(ok(X1), ok(X2), ok(X3)) -> ok(U131(X1, X2, X3))
   U132(ok(X1), ok(X2)) -> ok(U132(X1, X2))
   U133(ok(X)) -> ok(U133(X))
   U141(ok(X1), ok(X2), ok(X3)) -> ok(U141(X1, X2, X3))
   U142(ok(X1), ok(X2)) -> ok(U142(X1, X2))
   U143(ok(X)) -> ok(U143(X))
   U151(ok(X1), ok(X2), ok(X3)) -> ok(U151(X1, X2, X3))
   U152(ok(X1), ok(X2)) -> ok(U152(X1, X2))
   U153(ok(X)) -> ok(U153(X))
   U161(ok(X1), ok(X2)) -> ok(U161(X1, X2))
   cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
   natsFrom(ok(X)) -> ok(natsFrom(X))
   s(ok(X)) -> ok(s(X))
   U171(ok(X1), ok(X2), ok(X3)) -> ok(U171(X1, X2, X3))
   head(ok(X)) -> ok(head(X))
   afterNth(ok(X1), ok(X2)) -> ok(afterNth(X1, X2))
   U181(ok(X1), ok(X2)) -> ok(U181(X1, X2))
   U191(ok(X1), ok(X2)) -> ok(U191(X1, X2))
   pair(ok(X1), ok(X2)) -> ok(pair(X1, X2))
   U201(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U201(X1, X2, X3, X4))
   U202(ok(X1), ok(X2)) -> ok(U202(X1, X2))
   U21(ok(X1), ok(X2)) -> ok(U21(X1, X2))
   U211(ok(X1), ok(X2)) -> ok(U211(X1, X2))
   U221(ok(X1), ok(X2), ok(X3)) -> ok(U221(X1, X2, X3))
   fst(ok(X)) -> ok(fst(X))
   U31(ok(X1), ok(X2)) -> ok(U31(X1, X2))
   U41(ok(X1), ok(X2), ok(X3)) -> ok(U41(X1, X2, X3))
   U42(ok(X1), ok(X2)) -> ok(U42(X1, X2))
   U43(ok(X)) -> ok(U43(X))
   U51(ok(X1), ok(X2), ok(X3)) -> ok(U51(X1, X2, X3))
   U52(ok(X1), ok(X2)) -> ok(U52(X1, X2))
   U53(ok(X)) -> ok(U53(X))
   U61(ok(X1), ok(X2)) -> ok(U61(X1, X2))
   U62(ok(X)) -> ok(U62(X))
   isPLNat(ok(X)) -> ok(isPLNat(X))
   U71(ok(X1), ok(X2)) -> ok(U71(X1, X2))
   U72(ok(X)) -> ok(U72(X))
   U81(ok(X1), ok(X2)) -> ok(U81(X1, X2))
   U82(ok(X)) -> ok(U82(X))
   U91(ok(X1), ok(X2)) -> ok(U91(X1, X2))
   U92(ok(X)) -> ok(U92(X))
   and(ok(X1), ok(X2)) -> ok(and(X1, X2))
   isNaturalKind(ok(X)) -> ok(isNaturalKind(X))
   isLNatKind(ok(X)) -> ok(isLNatKind(X))
   isPLNatKind(ok(X)) -> ok(isPLNatKind(X))
   tail(ok(X)) -> ok(tail(X))
   take(ok(X1), ok(X2)) -> ok(take(X1, X2))
   sel(ok(X1), ok(X2)) -> ok(sel(X1, X2))
   top(mark(X)) -> top(proper(X))
   top(ok(X)) -> top(active(X))

S is empty.
Rewrite Strategy: FULL