/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).

(0) CpxTRS
(1) NestedDefinedSymbolProof [UPPER BOUND(ID), 0 ms]
(2) CpxTRS
    (3) RcToIrcProof [BOTH BOUNDS(ID, ID), 43 ms]
    (4) CpxTRS
    (5) CpxTrsToCdtProof [UPPER BOUND(ID), 14 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)), 347 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) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(20) CpxTRS
    (21) TypeInferenceProof [BOTH BOUNDS(ID, ID), 9 ms]
    (22) typed CpxTrs
    (23) OrderProof [LOWER BOUND(ID), 0 ms]
    (24) typed CpxTrs
    (25) RewriteLemmaProof [LOWER BOUND(ID), 459 ms]
    (26) BEST
        (27) proven lower bound
            (28) LowerBoundPropagationProof [FINISHED, 0 ms]
            (29) BOUNDS(n^1, INF)
        (30) typed CpxTrs
            (31) RewriteLemmaProof [LOWER BOUND(ID), 214 ms]
            (32) typed CpxTrs
            (33) RewriteLemmaProof [LOWER BOUND(ID), 156 ms]
            (34) typed CpxTrs
            (35) RewriteLemmaProof [LOWER BOUND(ID), 134 ms]
            (36) typed CpxTrs
            (37) RewriteLemmaProof [LOWER BOUND(ID), 95 ms]
            (38) typed CpxTrs
            (39) RewriteLemmaProof [LOWER BOUND(ID), 112 ms]
            (40) typed CpxTrs
            (41) RewriteLemmaProof [LOWER BOUND(ID), 131 ms]
            (42) typed CpxTrs
            (43) RewriteLemmaProof [LOWER BOUND(ID), 178 ms]
            (44) typed CpxTrs
            (45) RewriteLemmaProof [LOWER BOUND(ID), 107 ms]
            (46) typed CpxTrs
            (47) RewriteLemmaProof [LOWER BOUND(ID), 127 ms]
            (48) typed CpxTrs
            (49) RewriteLemmaProof [LOWER BOUND(ID), 249 ms]
            (50) typed CpxTrs
            (51) RewriteLemmaProof [LOWER BOUND(ID), 144 ms]
            (52) typed CpxTrs
            (53) RewriteLemmaProof [LOWER BOUND(ID), 125 ms]
            (54) typed CpxTrs
            (55) RewriteLemmaProof [LOWER BOUND(ID), 157 ms]
            (56) typed CpxTrs
            (57) RewriteLemmaProof [LOWER BOUND(ID), 207 ms]
            (58) typed CpxTrs
            (59) RewriteLemmaProof [LOWER BOUND(ID), 348 ms]
            (60) typed CpxTrs
            (61) RewriteLemmaProof [LOWER BOUND(ID), 344 ms]
            (62) typed CpxTrs
            (63) RewriteLemmaProof [LOWER BOUND(ID), 214 ms]
            (64) typed CpxTrs
            (65) RewriteLemmaProof [LOWER BOUND(ID), 223 ms]
            (66) typed CpxTrs
            (67) RewriteLemmaProof [LOWER BOUND(ID), 164 ms]
            (68) typed CpxTrs
            (69) RewriteLemmaProof [LOWER BOUND(ID), 233 ms]
            (70) typed CpxTrs
            (71) RewriteLemmaProof [LOWER BOUND(ID), 118 ms]
            (72) typed CpxTrs
            (73) RewriteLemmaProof [LOWER BOUND(ID), 161 ms]
            (74) typed CpxTrs
            (75) RewriteLemmaProof [LOWER BOUND(ID), 124 ms]
            (76) typed CpxTrs
            (77) RewriteLemmaProof [LOWER BOUND(ID), 53 ms]
            (78) typed CpxTrs
            (79) RewriteLemmaProof [LOWER BOUND(ID), 301 ms]
            (80) typed CpxTrs
            (81) RewriteLemmaProof [LOWER BOUND(ID), 252 ms]
            (82) typed CpxTrs
            (83) RewriteLemmaProof [LOWER BOUND(ID), 323 ms]
            (84) typed CpxTrs
            (85) RewriteLemmaProof [LOWER BOUND(ID), 186 ms]
            (86) typed CpxTrs
            (87) RewriteLemmaProof [LOWER BOUND(ID), 246 ms]
            (88) typed CpxTrs
            (89) RewriteLemmaProof [LOWER BOUND(ID), 131 ms]
            (90) typed CpxTrs
            (91) RewriteLemmaProof [LOWER BOUND(ID), 160 ms]
            (92) typed CpxTrs
            (93) RewriteLemmaProof [LOWER BOUND(ID), 138 ms]
            (94) typed CpxTrs
            (95) RewriteLemmaProof [LOWER BOUND(ID), 99 ms]
            (96) typed CpxTrs
            (97) RewriteLemmaProof [LOWER BOUND(ID), 205 ms]
            (98) typed CpxTrs
            (99) RewriteLemmaProof [LOWER BOUND(ID), 101 ms]
            (100) typed CpxTrs


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

(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, V2)) -> mark(U102(isLNat(V2)))
   active(U102(tt)) -> mark(tt)
   active(U11(tt, N, XS)) -> mark(U12(isLNat(XS), N, XS))
   active(U111(tt)) -> mark(tt)
   active(U12(tt, N, XS)) -> mark(snd(splitAt(N, XS)))
   active(U121(tt)) -> mark(tt)
   active(U131(tt, V2)) -> mark(U132(isLNat(V2)))
   active(U132(tt)) -> mark(tt)
   active(U141(tt, V2)) -> mark(U142(isLNat(V2)))
   active(U142(tt)) -> mark(tt)
   active(U151(tt, V2)) -> mark(U152(isLNat(V2)))
   active(U152(tt)) -> mark(tt)
   active(U161(tt, N)) -> mark(cons(N, natsFrom(s(N))))
   active(U171(tt, N, XS)) -> mark(U172(isLNat(XS), N, XS))
   active(U172(tt, N, XS)) -> mark(head(afterNth(N, XS)))
   active(U181(tt, Y)) -> mark(U182(isLNat(Y), Y))
   active(U182(tt, Y)) -> mark(Y)
   active(U191(tt, XS)) -> mark(pair(nil, XS))
   active(U201(tt, N, X, XS)) -> mark(U202(isNatural(X), N, X, XS))
   active(U202(tt, N, X, XS)) -> mark(U203(isLNat(XS), N, X, XS))
   active(U203(tt, N, X, XS)) -> mark(U204(splitAt(N, XS), X))
   active(U204(pair(YS, ZS), X)) -> mark(pair(cons(X, YS), ZS))
   active(U21(tt, X, Y)) -> mark(U22(isLNat(Y), X))
   active(U211(tt, XS)) -> mark(U212(isLNat(XS), XS))
   active(U212(tt, XS)) -> mark(XS)
   active(U22(tt, X)) -> mark(X)
   active(U221(tt, N, XS)) -> mark(U222(isLNat(XS), N, XS))
   active(U222(tt, N, XS)) -> mark(fst(splitAt(N, XS)))
   active(U31(tt, N, XS)) -> mark(U32(isLNat(XS), N))
   active(U32(tt, N)) -> mark(N)
   active(U41(tt, V2)) -> mark(U42(isLNat(V2)))
   active(U42(tt)) -> mark(tt)
   active(U51(tt, V2)) -> mark(U52(isLNat(V2)))
   active(U52(tt)) -> mark(tt)
   active(U61(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt)) -> mark(tt)
   active(U91(tt)) -> mark(tt)
   active(afterNth(N, XS)) -> mark(U11(isNatural(N), N, XS))
   active(fst(pair(X, Y))) -> mark(U21(isLNat(X), X, Y))
   active(head(cons(N, XS))) -> mark(U31(isNatural(N), N, XS))
   active(isLNat(nil)) -> mark(tt)
   active(isLNat(afterNth(V1, V2))) -> mark(U41(isNatural(V1), V2))
   active(isLNat(cons(V1, V2))) -> mark(U51(isNatural(V1), V2))
   active(isLNat(fst(V1))) -> mark(U61(isPLNat(V1)))
   active(isLNat(natsFrom(V1))) -> mark(U71(isNatural(V1)))
   active(isLNat(snd(V1))) -> mark(U81(isPLNat(V1)))
   active(isLNat(tail(V1))) -> mark(U91(isLNat(V1)))
   active(isLNat(take(V1, V2))) -> mark(U101(isNatural(V1), V2))
   active(isNatural(0)) -> mark(tt)
   active(isNatural(head(V1))) -> mark(U111(isLNat(V1)))
   active(isNatural(s(V1))) -> mark(U121(isNatural(V1)))
   active(isNatural(sel(V1, V2))) -> mark(U131(isNatural(V1), V2))
   active(isPLNat(pair(V1, V2))) -> mark(U141(isLNat(V1), V2))
   active(isPLNat(splitAt(V1, V2))) -> mark(U151(isNatural(V1), V2))
   active(natsFrom(N)) -> mark(U161(isNatural(N), N))
   active(sel(N, XS)) -> mark(U171(isNatural(N), N, XS))
   active(snd(pair(X, Y))) -> mark(U181(isLNat(X), Y))
   active(splitAt(0, XS)) -> mark(U191(isLNat(XS), XS))
   active(splitAt(s(N), cons(X, XS))) -> mark(U201(isNatural(N), N, X, XS))
   active(tail(cons(N, XS))) -> mark(U211(isNatural(N), XS))
   active(take(N, XS)) -> mark(U221(isNatural(N), N, XS))
   active(U101(X1, X2)) -> U101(active(X1), X2)
   active(U102(X)) -> U102(active(X))
   active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3)
   active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3)
   active(U111(X)) -> U111(active(X))
   active(snd(X)) -> snd(active(X))
   active(splitAt(X1, X2)) -> splitAt(active(X1), X2)
   active(splitAt(X1, X2)) -> splitAt(X1, active(X2))
   active(U121(X)) -> U121(active(X))
   active(U131(X1, X2)) -> U131(active(X1), X2)
   active(U132(X)) -> U132(active(X))
   active(U141(X1, X2)) -> U141(active(X1), X2)
   active(U142(X)) -> U142(active(X))
   active(U151(X1, X2)) -> U151(active(X1), X2)
   active(U152(X)) -> U152(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(U172(X1, X2, X3)) -> U172(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(U182(X1, X2)) -> U182(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, X3, X4)) -> U202(active(X1), X2, X3, X4)
   active(U203(X1, X2, X3, X4)) -> U203(active(X1), X2, X3, X4)
   active(U204(X1, X2)) -> U204(active(X1), X2)
   active(U21(X1, X2, X3)) -> U21(active(X1), X2, X3)
   active(U22(X1, X2)) -> U22(active(X1), X2)
   active(U211(X1, X2)) -> U211(active(X1), X2)
   active(U212(X1, X2)) -> U212(active(X1), X2)
   active(U221(X1, X2, X3)) -> U221(active(X1), X2, X3)
   active(U222(X1, X2, X3)) -> U222(active(X1), X2, X3)
   active(fst(X)) -> fst(active(X))
   active(U31(X1, X2, X3)) -> U31(active(X1), X2, X3)
   active(U32(X1, X2)) -> U32(active(X1), X2)
   active(U41(X1, X2)) -> U41(active(X1), X2)
   active(U42(X)) -> U42(active(X))
   active(U51(X1, X2)) -> U51(active(X1), X2)
   active(U52(X)) -> U52(active(X))
   active(U61(X)) -> U61(active(X))
   active(U71(X)) -> U71(active(X))
   active(U81(X)) -> U81(active(X))
   active(U91(X)) -> U91(active(X))
   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) -> mark(U101(X1, X2))
   U102(mark(X)) -> mark(U102(X))
   U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3))
   U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3))
   U111(mark(X)) -> mark(U111(X))
   snd(mark(X)) -> mark(snd(X))
   splitAt(mark(X1), X2) -> mark(splitAt(X1, X2))
   splitAt(X1, mark(X2)) -> mark(splitAt(X1, X2))
   U121(mark(X)) -> mark(U121(X))
   U131(mark(X1), X2) -> mark(U131(X1, X2))
   U132(mark(X)) -> mark(U132(X))
   U141(mark(X1), X2) -> mark(U141(X1, X2))
   U142(mark(X)) -> mark(U142(X))
   U151(mark(X1), X2) -> mark(U151(X1, X2))
   U152(mark(X)) -> mark(U152(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))
   U172(mark(X1), X2, X3) -> mark(U172(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))
   U182(mark(X1), X2) -> mark(U182(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, X3, X4) -> mark(U202(X1, X2, X3, X4))
   U203(mark(X1), X2, X3, X4) -> mark(U203(X1, X2, X3, X4))
   U204(mark(X1), X2) -> mark(U204(X1, X2))
   U21(mark(X1), X2, X3) -> mark(U21(X1, X2, X3))
   U22(mark(X1), X2) -> mark(U22(X1, X2))
   U211(mark(X1), X2) -> mark(U211(X1, X2))
   U212(mark(X1), X2) -> mark(U212(X1, X2))
   U221(mark(X1), X2, X3) -> mark(U221(X1, X2, X3))
   U222(mark(X1), X2, X3) -> mark(U222(X1, X2, X3))
   fst(mark(X)) -> mark(fst(X))
   U31(mark(X1), X2, X3) -> mark(U31(X1, X2, X3))
   U32(mark(X1), X2) -> mark(U32(X1, X2))
   U41(mark(X1), X2) -> mark(U41(X1, X2))
   U42(mark(X)) -> mark(U42(X))
   U51(mark(X1), X2) -> mark(U51(X1, X2))
   U52(mark(X)) -> mark(U52(X))
   U61(mark(X)) -> mark(U61(X))
   U71(mark(X)) -> mark(U71(X))
   U81(mark(X)) -> mark(U81(X))
   U91(mark(X)) -> mark(U91(X))
   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)) -> U101(proper(X1), proper(X2))
   proper(tt) -> ok(tt)
   proper(U102(X)) -> U102(proper(X))
   proper(isLNat(X)) -> isLNat(proper(X))
   proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3))
   proper(U12(X1, X2, X3)) -> U12(proper(X1), proper(X2), proper(X3))
   proper(U111(X)) -> U111(proper(X))
   proper(snd(X)) -> snd(proper(X))
   proper(splitAt(X1, X2)) -> splitAt(proper(X1), proper(X2))
   proper(U121(X)) -> U121(proper(X))
   proper(U131(X1, X2)) -> U131(proper(X1), proper(X2))
   proper(U132(X)) -> U132(proper(X))
   proper(U141(X1, X2)) -> U141(proper(X1), proper(X2))
   proper(U142(X)) -> U142(proper(X))
   proper(U151(X1, X2)) -> U151(proper(X1), proper(X2))
   proper(U152(X)) -> U152(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(U172(X1, X2, X3)) -> U172(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(U182(X1, X2)) -> U182(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, X3, X4)) -> U202(proper(X1), proper(X2), proper(X3), proper(X4))
   proper(isNatural(X)) -> isNatural(proper(X))
   proper(U203(X1, X2, X3, X4)) -> U203(proper(X1), proper(X2), proper(X3), proper(X4))
   proper(U204(X1, X2)) -> U204(proper(X1), proper(X2))
   proper(U21(X1, X2, X3)) -> U21(proper(X1), proper(X2), proper(X3))
   proper(U22(X1, X2)) -> U22(proper(X1), proper(X2))
   proper(U211(X1, X2)) -> U211(proper(X1), proper(X2))
   proper(U212(X1, X2)) -> U212(proper(X1), proper(X2))
   proper(U221(X1, X2, X3)) -> U221(proper(X1), proper(X2), proper(X3))
   proper(U222(X1, X2, X3)) -> U222(proper(X1), proper(X2), proper(X3))
   proper(fst(X)) -> fst(proper(X))
   proper(U31(X1, X2, X3)) -> U31(proper(X1), proper(X2), proper(X3))
   proper(U32(X1, X2)) -> U32(proper(X1), proper(X2))
   proper(U41(X1, X2)) -> U41(proper(X1), proper(X2))
   proper(U42(X)) -> U42(proper(X))
   proper(U51(X1, X2)) -> U51(proper(X1), proper(X2))
   proper(U52(X)) -> U52(proper(X))
   proper(U61(X)) -> U61(proper(X))
   proper(U71(X)) -> U71(proper(X))
   proper(U81(X)) -> U81(proper(X))
   proper(U91(X)) -> U91(proper(X))
   proper(isPLNat(X)) -> isPLNat(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(U101(X1, X2))
   U102(ok(X)) -> ok(U102(X))
   isLNat(ok(X)) -> ok(isLNat(X))
   U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
   U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))
   U111(ok(X)) -> ok(U111(X))
   snd(ok(X)) -> ok(snd(X))
   splitAt(ok(X1), ok(X2)) -> ok(splitAt(X1, X2))
   U121(ok(X)) -> ok(U121(X))
   U131(ok(X1), ok(X2)) -> ok(U131(X1, X2))
   U132(ok(X)) -> ok(U132(X))
   U141(ok(X1), ok(X2)) -> ok(U141(X1, X2))
   U142(ok(X)) -> ok(U142(X))
   U151(ok(X1), ok(X2)) -> ok(U151(X1, X2))
   U152(ok(X)) -> ok(U152(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))
   U172(ok(X1), ok(X2), ok(X3)) -> ok(U172(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))
   U182(ok(X1), ok(X2)) -> ok(U182(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(X3), ok(X4)) -> ok(U202(X1, X2, X3, X4))
   isNatural(ok(X)) -> ok(isNatural(X))
   U203(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U203(X1, X2, X3, X4))
   U204(ok(X1), ok(X2)) -> ok(U204(X1, X2))
   U21(ok(X1), ok(X2), ok(X3)) -> ok(U21(X1, X2, X3))
   U22(ok(X1), ok(X2)) -> ok(U22(X1, X2))
   U211(ok(X1), ok(X2)) -> ok(U211(X1, X2))
   U212(ok(X1), ok(X2)) -> ok(U212(X1, X2))
   U221(ok(X1), ok(X2), ok(X3)) -> ok(U221(X1, X2, X3))
   U222(ok(X1), ok(X2), ok(X3)) -> ok(U222(X1, X2, X3))
   fst(ok(X)) -> ok(fst(X))
   U31(ok(X1), ok(X2), ok(X3)) -> ok(U31(X1, X2, X3))
   U32(ok(X1), ok(X2)) -> ok(U32(X1, X2))
   U41(ok(X1), ok(X2)) -> ok(U41(X1, X2))
   U42(ok(X)) -> ok(U42(X))
   U51(ok(X1), ok(X2)) -> ok(U51(X1, X2))
   U52(ok(X)) -> ok(U52(X))
   U61(ok(X)) -> ok(U61(X))
   U71(ok(X)) -> ok(U71(X))
   U81(ok(X)) -> ok(U81(X))
   U91(ok(X)) -> ok(U91(X))
   isPLNat(ok(X)) -> ok(isPLNat(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, V2)) -> mark(U102(isLNat(V2)))
active(U102(tt)) -> mark(tt)
active(U11(tt, N, XS)) -> mark(U12(isLNat(XS), N, XS))
active(U111(tt)) -> mark(tt)
active(U12(tt, N, XS)) -> mark(snd(splitAt(N, XS)))
active(U121(tt)) -> mark(tt)
active(U131(tt, V2)) -> mark(U132(isLNat(V2)))
active(U132(tt)) -> mark(tt)
active(U141(tt, V2)) -> mark(U142(isLNat(V2)))
active(U142(tt)) -> mark(tt)
active(U151(tt, V2)) -> mark(U152(isLNat(V2)))
active(U152(tt)) -> mark(tt)
active(U161(tt, N)) -> mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) -> mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) -> mark(head(afterNth(N, XS)))
active(U181(tt, Y)) -> mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) -> mark(Y)
active(U191(tt, XS)) -> mark(pair(nil, XS))
active(U201(tt, N, X, XS)) -> mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) -> mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) -> mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) -> mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) -> mark(U22(isLNat(Y), X))
active(U211(tt, XS)) -> mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) -> mark(XS)
active(U22(tt, X)) -> mark(X)
active(U221(tt, N, XS)) -> mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) -> mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) -> mark(U32(isLNat(XS), N))
active(U32(tt, N)) -> mark(N)
active(U41(tt, V2)) -> mark(U42(isLNat(V2)))
active(U42(tt)) -> mark(tt)
active(U51(tt, V2)) -> mark(U52(isLNat(V2)))
active(U52(tt)) -> mark(tt)
active(U61(tt)) -> mark(tt)
active(U71(tt)) -> mark(tt)
active(U81(tt)) -> mark(tt)
active(U91(tt)) -> mark(tt)
active(afterNth(N, XS)) -> mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) -> mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) -> mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) -> mark(tt)
active(isLNat(afterNth(V1, V2))) -> mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) -> mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) -> mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) -> mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) -> mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) -> mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) -> mark(U101(isNatural(V1), V2))
active(isNatural(0)) -> mark(tt)
active(isNatural(head(V1))) -> mark(U111(isLNat(V1)))
active(isNatural(s(V1))) -> mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) -> mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) -> mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) -> mark(U151(isNatural(V1), V2))
active(natsFrom(N)) -> mark(U161(isNatural(N), N))
active(sel(N, XS)) -> mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) -> mark(U181(isLNat(X), Y))
active(splitAt(0, XS)) -> mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) -> mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) -> mark(U211(isNatural(N), XS))
active(take(N, XS)) -> mark(U221(isNatural(N), N, XS))
active(U101(X1, X2)) -> U101(active(X1), X2)
active(U102(X)) -> U102(active(X))
active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3)
active(U111(X)) -> U111(active(X))
active(snd(X)) -> snd(active(X))
active(splitAt(X1, X2)) -> splitAt(active(X1), X2)
active(splitAt(X1, X2)) -> splitAt(X1, active(X2))
active(U121(X)) -> U121(active(X))
active(U131(X1, X2)) -> U131(active(X1), X2)
active(U132(X)) -> U132(active(X))
active(U141(X1, X2)) -> U141(active(X1), X2)
active(U142(X)) -> U142(active(X))
active(U151(X1, X2)) -> U151(active(X1), X2)
active(U152(X)) -> U152(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(U172(X1, X2, X3)) -> U172(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(U182(X1, X2)) -> U182(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, X3, X4)) -> U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) -> U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) -> U204(active(X1), X2)
active(U21(X1, X2, X3)) -> U21(active(X1), X2, X3)
active(U22(X1, X2)) -> U22(active(X1), X2)
active(U211(X1, X2)) -> U211(active(X1), X2)
active(U212(X1, X2)) -> U212(active(X1), X2)
active(U221(X1, X2, X3)) -> U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) -> U222(active(X1), X2, X3)
active(fst(X)) -> fst(active(X))
active(U31(X1, X2, X3)) -> U31(active(X1), X2, X3)
active(U32(X1, X2)) -> U32(active(X1), X2)
active(U41(X1, X2)) -> U41(active(X1), X2)
active(U42(X)) -> U42(active(X))
active(U51(X1, X2)) -> U51(active(X1), X2)
active(U52(X)) -> U52(active(X))
active(U61(X)) -> U61(active(X))
active(U71(X)) -> U71(active(X))
active(U81(X)) -> U81(active(X))
active(U91(X)) -> U91(active(X))
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)) -> U101(proper(X1), proper(X2))
proper(U102(X)) -> U102(proper(X))
proper(isLNat(X)) -> isLNat(proper(X))
proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) -> U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) -> U111(proper(X))
proper(snd(X)) -> snd(proper(X))
proper(splitAt(X1, X2)) -> splitAt(proper(X1), proper(X2))
proper(U121(X)) -> U121(proper(X))
proper(U131(X1, X2)) -> U131(proper(X1), proper(X2))
proper(U132(X)) -> U132(proper(X))
proper(U141(X1, X2)) -> U141(proper(X1), proper(X2))
proper(U142(X)) -> U142(proper(X))
proper(U151(X1, X2)) -> U151(proper(X1), proper(X2))
proper(U152(X)) -> U152(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(U172(X1, X2, X3)) -> U172(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(U182(X1, X2)) -> U182(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, X3, X4)) -> U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) -> isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) -> U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) -> U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) -> U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) -> U22(proper(X1), proper(X2))
proper(U211(X1, X2)) -> U211(proper(X1), proper(X2))
proper(U212(X1, X2)) -> U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) -> U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) -> U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) -> fst(proper(X))
proper(U31(X1, X2, X3)) -> U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) -> U32(proper(X1), proper(X2))
proper(U41(X1, X2)) -> U41(proper(X1), proper(X2))
proper(U42(X)) -> U42(proper(X))
proper(U51(X1, X2)) -> U51(proper(X1), proper(X2))
proper(U52(X)) -> U52(proper(X))
proper(U61(X)) -> U61(proper(X))
proper(U71(X)) -> U71(proper(X))
proper(U81(X)) -> U81(proper(X))
proper(U91(X)) -> U91(proper(X))
proper(isPLNat(X)) -> isPLNat(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) -> mark(U101(X1, X2))
   U102(mark(X)) -> mark(U102(X))
   U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3))
   U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3))
   U111(mark(X)) -> mark(U111(X))
   snd(mark(X)) -> mark(snd(X))
   splitAt(mark(X1), X2) -> mark(splitAt(X1, X2))
   splitAt(X1, mark(X2)) -> mark(splitAt(X1, X2))
   U121(mark(X)) -> mark(U121(X))
   U131(mark(X1), X2) -> mark(U131(X1, X2))
   U132(mark(X)) -> mark(U132(X))
   U141(mark(X1), X2) -> mark(U141(X1, X2))
   U142(mark(X)) -> mark(U142(X))
   U151(mark(X1), X2) -> mark(U151(X1, X2))
   U152(mark(X)) -> mark(U152(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))
   U172(mark(X1), X2, X3) -> mark(U172(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))
   U182(mark(X1), X2) -> mark(U182(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, X3, X4) -> mark(U202(X1, X2, X3, X4))
   U203(mark(X1), X2, X3, X4) -> mark(U203(X1, X2, X3, X4))
   U204(mark(X1), X2) -> mark(U204(X1, X2))
   U21(mark(X1), X2, X3) -> mark(U21(X1, X2, X3))
   U22(mark(X1), X2) -> mark(U22(X1, X2))
   U211(mark(X1), X2) -> mark(U211(X1, X2))
   U212(mark(X1), X2) -> mark(U212(X1, X2))
   U221(mark(X1), X2, X3) -> mark(U221(X1, X2, X3))
   U222(mark(X1), X2, X3) -> mark(U222(X1, X2, X3))
   fst(mark(X)) -> mark(fst(X))
   U31(mark(X1), X2, X3) -> mark(U31(X1, X2, X3))
   U32(mark(X1), X2) -> mark(U32(X1, X2))
   U41(mark(X1), X2) -> mark(U41(X1, X2))
   U42(mark(X)) -> mark(U42(X))
   U51(mark(X1), X2) -> mark(U51(X1, X2))
   U52(mark(X)) -> mark(U52(X))
   U61(mark(X)) -> mark(U61(X))
   U71(mark(X)) -> mark(U71(X))
   U81(mark(X)) -> mark(U81(X))
   U91(mark(X)) -> mark(U91(X))
   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(U101(X1, X2))
   U102(ok(X)) -> ok(U102(X))
   isLNat(ok(X)) -> ok(isLNat(X))
   U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
   U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))
   U111(ok(X)) -> ok(U111(X))
   snd(ok(X)) -> ok(snd(X))
   splitAt(ok(X1), ok(X2)) -> ok(splitAt(X1, X2))
   U121(ok(X)) -> ok(U121(X))
   U131(ok(X1), ok(X2)) -> ok(U131(X1, X2))
   U132(ok(X)) -> ok(U132(X))
   U141(ok(X1), ok(X2)) -> ok(U141(X1, X2))
   U142(ok(X)) -> ok(U142(X))
   U151(ok(X1), ok(X2)) -> ok(U151(X1, X2))
   U152(ok(X)) -> ok(U152(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))
   U172(ok(X1), ok(X2), ok(X3)) -> ok(U172(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))
   U182(ok(X1), ok(X2)) -> ok(U182(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(X3), ok(X4)) -> ok(U202(X1, X2, X3, X4))
   isNatural(ok(X)) -> ok(isNatural(X))
   U203(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U203(X1, X2, X3, X4))
   U204(ok(X1), ok(X2)) -> ok(U204(X1, X2))
   U21(ok(X1), ok(X2), ok(X3)) -> ok(U21(X1, X2, X3))
   U22(ok(X1), ok(X2)) -> ok(U22(X1, X2))
   U211(ok(X1), ok(X2)) -> ok(U211(X1, X2))
   U212(ok(X1), ok(X2)) -> ok(U212(X1, X2))
   U221(ok(X1), ok(X2), ok(X3)) -> ok(U221(X1, X2, X3))
   U222(ok(X1), ok(X2), ok(X3)) -> ok(U222(X1, X2, X3))
   fst(ok(X)) -> ok(fst(X))
   U31(ok(X1), ok(X2), ok(X3)) -> ok(U31(X1, X2, X3))
   U32(ok(X1), ok(X2)) -> ok(U32(X1, X2))
   U41(ok(X1), ok(X2)) -> ok(U41(X1, X2))
   U42(ok(X)) -> ok(U42(X))
   U51(ok(X1), ok(X2)) -> ok(U51(X1, X2))
   U52(ok(X)) -> ok(U52(X))
   U61(ok(X)) -> ok(U61(X))
   U71(ok(X)) -> ok(U71(X))
   U81(ok(X)) -> ok(U81(X))
   U91(ok(X)) -> ok(U91(X))
   isPLNat(ok(X)) -> ok(isPLNat(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) -> mark(U101(X1, X2))
   U102(mark(X)) -> mark(U102(X))
   U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3))
   U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3))
   U111(mark(X)) -> mark(U111(X))
   snd(mark(X)) -> mark(snd(X))
   splitAt(mark(X1), X2) -> mark(splitAt(X1, X2))
   splitAt(X1, mark(X2)) -> mark(splitAt(X1, X2))
   U121(mark(X)) -> mark(U121(X))
   U131(mark(X1), X2) -> mark(U131(X1, X2))
   U132(mark(X)) -> mark(U132(X))
   U141(mark(X1), X2) -> mark(U141(X1, X2))
   U142(mark(X)) -> mark(U142(X))
   U151(mark(X1), X2) -> mark(U151(X1, X2))
   U152(mark(X)) -> mark(U152(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))
   U172(mark(X1), X2, X3) -> mark(U172(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))
   U182(mark(X1), X2) -> mark(U182(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, X3, X4) -> mark(U202(X1, X2, X3, X4))
   U203(mark(X1), X2, X3, X4) -> mark(U203(X1, X2, X3, X4))
   U204(mark(X1), X2) -> mark(U204(X1, X2))
   U21(mark(X1), X2, X3) -> mark(U21(X1, X2, X3))
   U22(mark(X1), X2) -> mark(U22(X1, X2))
   U211(mark(X1), X2) -> mark(U211(X1, X2))
   U212(mark(X1), X2) -> mark(U212(X1, X2))
   U221(mark(X1), X2, X3) -> mark(U221(X1, X2, X3))
   U222(mark(X1), X2, X3) -> mark(U222(X1, X2, X3))
   fst(mark(X)) -> mark(fst(X))
   U31(mark(X1), X2, X3) -> mark(U31(X1, X2, X3))
   U32(mark(X1), X2) -> mark(U32(X1, X2))
   U41(mark(X1), X2) -> mark(U41(X1, X2))
   U42(mark(X)) -> mark(U42(X))
   U51(mark(X1), X2) -> mark(U51(X1, X2))
   U52(mark(X)) -> mark(U52(X))
   U61(mark(X)) -> mark(U61(X))
   U71(mark(X)) -> mark(U71(X))
   U81(mark(X)) -> mark(U81(X))
   U91(mark(X)) -> mark(U91(X))
   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(U101(X1, X2))
   U102(ok(X)) -> ok(U102(X))
   isLNat(ok(X)) -> ok(isLNat(X))
   U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
   U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))
   U111(ok(X)) -> ok(U111(X))
   snd(ok(X)) -> ok(snd(X))
   splitAt(ok(X1), ok(X2)) -> ok(splitAt(X1, X2))
   U121(ok(X)) -> ok(U121(X))
   U131(ok(X1), ok(X2)) -> ok(U131(X1, X2))
   U132(ok(X)) -> ok(U132(X))
   U141(ok(X1), ok(X2)) -> ok(U141(X1, X2))
   U142(ok(X)) -> ok(U142(X))
   U151(ok(X1), ok(X2)) -> ok(U151(X1, X2))
   U152(ok(X)) -> ok(U152(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))
   U172(ok(X1), ok(X2), ok(X3)) -> ok(U172(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))
   U182(ok(X1), ok(X2)) -> ok(U182(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(X3), ok(X4)) -> ok(U202(X1, X2, X3, X4))
   isNatural(ok(X)) -> ok(isNatural(X))
   U203(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U203(X1, X2, X3, X4))
   U204(ok(X1), ok(X2)) -> ok(U204(X1, X2))
   U21(ok(X1), ok(X2), ok(X3)) -> ok(U21(X1, X2, X3))
   U22(ok(X1), ok(X2)) -> ok(U22(X1, X2))
   U211(ok(X1), ok(X2)) -> ok(U211(X1, X2))
   U212(ok(X1), ok(X2)) -> ok(U212(X1, X2))
   U221(ok(X1), ok(X2), ok(X3)) -> ok(U221(X1, X2, X3))
   U222(ok(X1), ok(X2), ok(X3)) -> ok(U222(X1, X2, X3))
   fst(ok(X)) -> ok(fst(X))
   U31(ok(X1), ok(X2), ok(X3)) -> ok(U31(X1, X2, X3))
   U32(ok(X1), ok(X2)) -> ok(U32(X1, X2))
   U41(ok(X1), ok(X2)) -> ok(U41(X1, X2))
   U42(ok(X)) -> ok(U42(X))
   U51(ok(X1), ok(X2)) -> ok(U51(X1, X2))
   U52(ok(X)) -> ok(U52(X))
   U61(ok(X)) -> ok(U61(X))
   U71(ok(X)) -> ok(U71(X))
   U81(ok(X)) -> ok(U81(X))
   U91(ok(X)) -> ok(U91(X))
   isPLNat(ok(X)) -> ok(isPLNat(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) -> mark(U101(z0, z1))
   U101(ok(z0), ok(z1)) -> ok(U101(z0, z1))
   U102(mark(z0)) -> mark(U102(z0))
   U102(ok(z0)) -> ok(U102(z0))
   U11(mark(z0), z1, z2) -> mark(U11(z0, z1, z2))
   U11(ok(z0), ok(z1), ok(z2)) -> ok(U11(z0, z1, z2))
   U12(mark(z0), z1, z2) -> mark(U12(z0, z1, z2))
   U12(ok(z0), ok(z1), ok(z2)) -> ok(U12(z0, z1, z2))
   U111(mark(z0)) -> mark(U111(z0))
   U111(ok(z0)) -> ok(U111(z0))
   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))
   U121(mark(z0)) -> mark(U121(z0))
   U121(ok(z0)) -> ok(U121(z0))
   U131(mark(z0), z1) -> mark(U131(z0, z1))
   U131(ok(z0), ok(z1)) -> ok(U131(z0, z1))
   U132(mark(z0)) -> mark(U132(z0))
   U132(ok(z0)) -> ok(U132(z0))
   U141(mark(z0), z1) -> mark(U141(z0, z1))
   U141(ok(z0), ok(z1)) -> ok(U141(z0, z1))
   U142(mark(z0)) -> mark(U142(z0))
   U142(ok(z0)) -> ok(U142(z0))
   U151(mark(z0), z1) -> mark(U151(z0, z1))
   U151(ok(z0), ok(z1)) -> ok(U151(z0, z1))
   U152(mark(z0)) -> mark(U152(z0))
   U152(ok(z0)) -> ok(U152(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))
   U172(mark(z0), z1, z2) -> mark(U172(z0, z1, z2))
   U172(ok(z0), ok(z1), ok(z2)) -> ok(U172(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))
   U182(mark(z0), z1) -> mark(U182(z0, z1))
   U182(ok(z0), ok(z1)) -> ok(U182(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, z2, z3) -> mark(U202(z0, z1, z2, z3))
   U202(ok(z0), ok(z1), ok(z2), ok(z3)) -> ok(U202(z0, z1, z2, z3))
   U203(mark(z0), z1, z2, z3) -> mark(U203(z0, z1, z2, z3))
   U203(ok(z0), ok(z1), ok(z2), ok(z3)) -> ok(U203(z0, z1, z2, z3))
   U204(mark(z0), z1) -> mark(U204(z0, z1))
   U204(ok(z0), ok(z1)) -> ok(U204(z0, z1))
   U21(mark(z0), z1, z2) -> mark(U21(z0, z1, z2))
   U21(ok(z0), ok(z1), ok(z2)) -> ok(U21(z0, z1, z2))
   U22(mark(z0), z1) -> mark(U22(z0, z1))
   U22(ok(z0), ok(z1)) -> ok(U22(z0, z1))
   U211(mark(z0), z1) -> mark(U211(z0, z1))
   U211(ok(z0), ok(z1)) -> ok(U211(z0, z1))
   U212(mark(z0), z1) -> mark(U212(z0, z1))
   U212(ok(z0), ok(z1)) -> ok(U212(z0, z1))
   U221(mark(z0), z1, z2) -> mark(U221(z0, z1, z2))
   U221(ok(z0), ok(z1), ok(z2)) -> ok(U221(z0, z1, z2))
   U222(mark(z0), z1, z2) -> mark(U222(z0, z1, z2))
   U222(ok(z0), ok(z1), ok(z2)) -> ok(U222(z0, z1, z2))
   fst(mark(z0)) -> mark(fst(z0))
   fst(ok(z0)) -> ok(fst(z0))
   U31(mark(z0), z1, z2) -> mark(U31(z0, z1, z2))
   U31(ok(z0), ok(z1), ok(z2)) -> ok(U31(z0, z1, z2))
   U32(mark(z0), z1) -> mark(U32(z0, z1))
   U32(ok(z0), ok(z1)) -> ok(U32(z0, z1))
   U41(mark(z0), z1) -> mark(U41(z0, z1))
   U41(ok(z0), ok(z1)) -> ok(U41(z0, z1))
   U42(mark(z0)) -> mark(U42(z0))
   U42(ok(z0)) -> ok(U42(z0))
   U51(mark(z0), z1) -> mark(U51(z0, z1))
   U51(ok(z0), ok(z1)) -> ok(U51(z0, z1))
   U52(mark(z0)) -> mark(U52(z0))
   U52(ok(z0)) -> ok(U52(z0))
   U61(mark(z0)) -> mark(U61(z0))
   U61(ok(z0)) -> ok(U61(z0))
   U71(mark(z0)) -> mark(U71(z0))
   U71(ok(z0)) -> ok(U71(z0))
   U81(mark(z0)) -> mark(U81(z0))
   U81(ok(z0)) -> ok(U81(z0))
   U91(mark(z0)) -> mark(U91(z0))
   U91(ok(z0)) -> ok(U91(z0))
   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)
   isLNat(ok(z0)) -> ok(isLNat(z0))
   isNatural(ok(z0)) -> ok(isNatural(z0))
   isPLNat(ok(z0)) -> ok(isPLNat(z0))
   top(mark(z0)) -> top(proper(z0))
   top(ok(z0)) -> top(active(z0))
Tuples:
   U101'(mark(z0), z1) -> c(U101'(z0, z1))
   U101'(ok(z0), ok(z1)) -> c1(U101'(z0, z1))
   U102'(mark(z0)) -> c2(U102'(z0))
   U102'(ok(z0)) -> c3(U102'(z0))
   U11'(mark(z0), z1, z2) -> c4(U11'(z0, z1, z2))
   U11'(ok(z0), ok(z1), ok(z2)) -> c5(U11'(z0, z1, z2))
   U12'(mark(z0), z1, z2) -> c6(U12'(z0, z1, z2))
   U12'(ok(z0), ok(z1), ok(z2)) -> c7(U12'(z0, z1, z2))
   U111'(mark(z0)) -> c8(U111'(z0))
   U111'(ok(z0)) -> c9(U111'(z0))
   SND(mark(z0)) -> c10(SND(z0))
   SND(ok(z0)) -> c11(SND(z0))
   SPLITAT(mark(z0), z1) -> c12(SPLITAT(z0, z1))
   SPLITAT(z0, mark(z1)) -> c13(SPLITAT(z0, z1))
   SPLITAT(ok(z0), ok(z1)) -> c14(SPLITAT(z0, z1))
   U121'(mark(z0)) -> c15(U121'(z0))
   U121'(ok(z0)) -> c16(U121'(z0))
   U131'(mark(z0), z1) -> c17(U131'(z0, z1))
   U131'(ok(z0), ok(z1)) -> c18(U131'(z0, z1))
   U132'(mark(z0)) -> c19(U132'(z0))
   U132'(ok(z0)) -> c20(U132'(z0))
   U141'(mark(z0), z1) -> c21(U141'(z0, z1))
   U141'(ok(z0), ok(z1)) -> c22(U141'(z0, z1))
   U142'(mark(z0)) -> c23(U142'(z0))
   U142'(ok(z0)) -> c24(U142'(z0))
   U151'(mark(z0), z1) -> c25(U151'(z0, z1))
   U151'(ok(z0), ok(z1)) -> c26(U151'(z0, z1))
   U152'(mark(z0)) -> c27(U152'(z0))
   U152'(ok(z0)) -> c28(U152'(z0))
   U161'(mark(z0), z1) -> c29(U161'(z0, z1))
   U161'(ok(z0), ok(z1)) -> c30(U161'(z0, z1))
   CONS(mark(z0), z1) -> c31(CONS(z0, z1))
   CONS(ok(z0), ok(z1)) -> c32(CONS(z0, z1))
   NATSFROM(mark(z0)) -> c33(NATSFROM(z0))
   NATSFROM(ok(z0)) -> c34(NATSFROM(z0))
   S(mark(z0)) -> c35(S(z0))
   S(ok(z0)) -> c36(S(z0))
   U171'(mark(z0), z1, z2) -> c37(U171'(z0, z1, z2))
   U171'(ok(z0), ok(z1), ok(z2)) -> c38(U171'(z0, z1, z2))
   U172'(mark(z0), z1, z2) -> c39(U172'(z0, z1, z2))
   U172'(ok(z0), ok(z1), ok(z2)) -> c40(U172'(z0, z1, z2))
   HEAD(mark(z0)) -> c41(HEAD(z0))
   HEAD(ok(z0)) -> c42(HEAD(z0))
   AFTERNTH(mark(z0), z1) -> c43(AFTERNTH(z0, z1))
   AFTERNTH(z0, mark(z1)) -> c44(AFTERNTH(z0, z1))
   AFTERNTH(ok(z0), ok(z1)) -> c45(AFTERNTH(z0, z1))
   U181'(mark(z0), z1) -> c46(U181'(z0, z1))
   U181'(ok(z0), ok(z1)) -> c47(U181'(z0, z1))
   U182'(mark(z0), z1) -> c48(U182'(z0, z1))
   U182'(ok(z0), ok(z1)) -> c49(U182'(z0, z1))
   U191'(mark(z0), z1) -> c50(U191'(z0, z1))
   U191'(ok(z0), ok(z1)) -> c51(U191'(z0, z1))
   PAIR(mark(z0), z1) -> c52(PAIR(z0, z1))
   PAIR(z0, mark(z1)) -> c53(PAIR(z0, z1))
   PAIR(ok(z0), ok(z1)) -> c54(PAIR(z0, z1))
   U201'(mark(z0), z1, z2, z3) -> c55(U201'(z0, z1, z2, z3))
   U201'(ok(z0), ok(z1), ok(z2), ok(z3)) -> c56(U201'(z0, z1, z2, z3))
   U202'(mark(z0), z1, z2, z3) -> c57(U202'(z0, z1, z2, z3))
   U202'(ok(z0), ok(z1), ok(z2), ok(z3)) -> c58(U202'(z0, z1, z2, z3))
   U203'(mark(z0), z1, z2, z3) -> c59(U203'(z0, z1, z2, z3))
   U203'(ok(z0), ok(z1), ok(z2), ok(z3)) -> c60(U203'(z0, z1, z2, z3))
   U204'(mark(z0), z1) -> c61(U204'(z0, z1))
   U204'(ok(z0), ok(z1)) -> c62(U204'(z0, z1))
   U21'(mark(z0), z1, z2) -> c63(U21'(z0, z1, z2))
   U21'(ok(z0), ok(z1), ok(z2)) -> c64(U21'(z0, z1, z2))
   U22'(mark(z0), z1) -> c65(U22'(z0, z1))
   U22'(ok(z0), ok(z1)) -> c66(U22'(z0, z1))
   U211'(mark(z0), z1) -> c67(U211'(z0, z1))
   U211'(ok(z0), ok(z1)) -> c68(U211'(z0, z1))
   U212'(mark(z0), z1) -> c69(U212'(z0, z1))
   U212'(ok(z0), ok(z1)) -> c70(U212'(z0, z1))
   U221'(mark(z0), z1, z2) -> c71(U221'(z0, z1, z2))
   U221'(ok(z0), ok(z1), ok(z2)) -> c72(U221'(z0, z1, z2))
   U222'(mark(z0), z1, z2) -> c73(U222'(z0, z1, z2))
   U222'(ok(z0), ok(z1), ok(z2)) -> c74(U222'(z0, z1, z2))
   FST(mark(z0)) -> c75(FST(z0))
   FST(ok(z0)) -> c76(FST(z0))
   U31'(mark(z0), z1, z2) -> c77(U31'(z0, z1, z2))
   U31'(ok(z0), ok(z1), ok(z2)) -> c78(U31'(z0, z1, z2))
   U32'(mark(z0), z1) -> c79(U32'(z0, z1))
   U32'(ok(z0), ok(z1)) -> c80(U32'(z0, z1))
   U41'(mark(z0), z1) -> c81(U41'(z0, z1))
   U41'(ok(z0), ok(z1)) -> c82(U41'(z0, z1))
   U42'(mark(z0)) -> c83(U42'(z0))
   U42'(ok(z0)) -> c84(U42'(z0))
   U51'(mark(z0), z1) -> c85(U51'(z0, z1))
   U51'(ok(z0), ok(z1)) -> c86(U51'(z0, z1))
   U52'(mark(z0)) -> c87(U52'(z0))
   U52'(ok(z0)) -> c88(U52'(z0))
   U61'(mark(z0)) -> c89(U61'(z0))
   U61'(ok(z0)) -> c90(U61'(z0))
   U71'(mark(z0)) -> c91(U71'(z0))
   U71'(ok(z0)) -> c92(U71'(z0))
   U81'(mark(z0)) -> c93(U81'(z0))
   U81'(ok(z0)) -> c94(U81'(z0))
   U91'(mark(z0)) -> c95(U91'(z0))
   U91'(ok(z0)) -> c96(U91'(z0))
   TAIL(mark(z0)) -> c97(TAIL(z0))
   TAIL(ok(z0)) -> c98(TAIL(z0))
   TAKE(mark(z0), z1) -> c99(TAKE(z0, z1))
   TAKE(z0, mark(z1)) -> c100(TAKE(z0, z1))
   TAKE(ok(z0), ok(z1)) -> c101(TAKE(z0, z1))
   SEL(mark(z0), z1) -> c102(SEL(z0, z1))
   SEL(z0, mark(z1)) -> c103(SEL(z0, z1))
   SEL(ok(z0), ok(z1)) -> c104(SEL(z0, z1))
   PROPER(tt) -> c105
   PROPER(nil) -> c106
   PROPER(0) -> c107
   ISLNAT(ok(z0)) -> c108(ISLNAT(z0))
   ISNATURAL(ok(z0)) -> c109(ISNATURAL(z0))
   ISPLNAT(ok(z0)) -> c110(ISPLNAT(z0))
   TOP(mark(z0)) -> c111(TOP(proper(z0)), PROPER(z0))
   TOP(ok(z0)) -> c112(TOP(active(z0)))
S tuples:
   U101'(mark(z0), z1) -> c(U101'(z0, z1))
   U101'(ok(z0), ok(z1)) -> c1(U101'(z0, z1))
   U102'(mark(z0)) -> c2(U102'(z0))
   U102'(ok(z0)) -> c3(U102'(z0))
   U11'(mark(z0), z1, z2) -> c4(U11'(z0, z1, z2))
   U11'(ok(z0), ok(z1), ok(z2)) -> c5(U11'(z0, z1, z2))
   U12'(mark(z0), z1, z2) -> c6(U12'(z0, z1, z2))
   U12'(ok(z0), ok(z1), ok(z2)) -> c7(U12'(z0, z1, z2))
   U111'(mark(z0)) -> c8(U111'(z0))
   U111'(ok(z0)) -> c9(U111'(z0))
   SND(mark(z0)) -> c10(SND(z0))
   SND(ok(z0)) -> c11(SND(z0))
   SPLITAT(mark(z0), z1) -> c12(SPLITAT(z0, z1))
   SPLITAT(z0, mark(z1)) -> c13(SPLITAT(z0, z1))
   SPLITAT(ok(z0), ok(z1)) -> c14(SPLITAT(z0, z1))
   U121'(mark(z0)) -> c15(U121'(z0))
   U121'(ok(z0)) -> c16(U121'(z0))
   U131'(mark(z0), z1) -> c17(U131'(z0, z1))
   U131'(ok(z0), ok(z1)) -> c18(U131'(z0, z1))
   U132'(mark(z0)) -> c19(U132'(z0))
   U132'(ok(z0)) -> c20(U132'(z0))
   U141'(mark(z0), z1) -> c21(U141'(z0, z1))
   U141'(ok(z0), ok(z1)) -> c22(U141'(z0, z1))
   U142'(mark(z0)) -> c23(U142'(z0))
   U142'(ok(z0)) -> c24(U142'(z0))
   U151'(mark(z0), z1) -> c25(U151'(z0, z1))
   U151'(ok(z0), ok(z1)) -> c26(U151'(z0, z1))
   U152'(mark(z0)) -> c27(U152'(z0))
   U152'(ok(z0)) -> c28(U152'(z0))
   U161'(mark(z0), z1) -> c29(U161'(z0, z1))
   U161'(ok(z0), ok(z1)) -> c30(U161'(z0, z1))
   CONS(mark(z0), z1) -> c31(CONS(z0, z1))
   CONS(ok(z0), ok(z1)) -> c32(CONS(z0, z1))
   NATSFROM(mark(z0)) -> c33(NATSFROM(z0))
   NATSFROM(ok(z0)) -> c34(NATSFROM(z0))
   S(mark(z0)) -> c35(S(z0))
   S(ok(z0)) -> c36(S(z0))
   U171'(mark(z0), z1, z2) -> c37(U171'(z0, z1, z2))
   U171'(ok(z0), ok(z1), ok(z2)) -> c38(U171'(z0, z1, z2))
   U172'(mark(z0), z1, z2) -> c39(U172'(z0, z1, z2))
   U172'(ok(z0), ok(z1), ok(z2)) -> c40(U172'(z0, z1, z2))
   HEAD(mark(z0)) -> c41(HEAD(z0))
   HEAD(ok(z0)) -> c42(HEAD(z0))
   AFTERNTH(mark(z0), z1) -> c43(AFTERNTH(z0, z1))
   AFTERNTH(z0, mark(z1)) -> c44(AFTERNTH(z0, z1))
   AFTERNTH(ok(z0), ok(z1)) -> c45(AFTERNTH(z0, z1))
   U181'(mark(z0), z1) -> c46(U181'(z0, z1))
   U181'(ok(z0), ok(z1)) -> c47(U181'(z0, z1))
   U182'(mark(z0), z1) -> c48(U182'(z0, z1))
   U182'(ok(z0), ok(z1)) -> c49(U182'(z0, z1))
   U191'(mark(z0), z1) -> c50(U191'(z0, z1))
   U191'(ok(z0), ok(z1)) -> c51(U191'(z0, z1))
   PAIR(mark(z0), z1) -> c52(PAIR(z0, z1))
   PAIR(z0, mark(z1)) -> c53(PAIR(z0, z1))
   PAIR(ok(z0), ok(z1)) -> c54(PAIR(z0, z1))
   U201'(mark(z0), z1, z2, z3) -> c55(U201'(z0, z1, z2, z3))
   U201'(ok(z0), ok(z1), ok(z2), ok(z3)) -> c56(U201'(z0, z1, z2, z3))
   U202'(mark(z0), z1, z2, z3) -> c57(U202'(z0, z1, z2, z3))
   U202'(ok(z0), ok(z1), ok(z2), ok(z3)) -> c58(U202'(z0, z1, z2, z3))
   U203'(mark(z0), z1, z2, z3) -> c59(U203'(z0, z1, z2, z3))
   U203'(ok(z0), ok(z1), ok(z2), ok(z3)) -> c60(U203'(z0, z1, z2, z3))
   U204'(mark(z0), z1) -> c61(U204'(z0, z1))
   U204'(ok(z0), ok(z1)) -> c62(U204'(z0, z1))
   U21'(mark(z0), z1, z2) -> c63(U21'(z0, z1, z2))
   U21'(ok(z0), ok(z1), ok(z2)) -> c64(U21'(z0, z1, z2))
   U22'(mark(z0), z1) -> c65(U22'(z0, z1))
   U22'(ok(z0), ok(z1)) -> c66(U22'(z0, z1))
   U211'(mark(z0), z1) -> c67(U211'(z0, z1))
   U211'(ok(z0), ok(z1)) -> c68(U211'(z0, z1))
   U212'(mark(z0), z1) -> c69(U212'(z0, z1))
   U212'(ok(z0), ok(z1)) -> c70(U212'(z0, z1))
   U221'(mark(z0), z1, z2) -> c71(U221'(z0, z1, z2))
   U221'(ok(z0), ok(z1), ok(z2)) -> c72(U221'(z0, z1, z2))
   U222'(mark(z0), z1, z2) -> c73(U222'(z0, z1, z2))
   U222'(ok(z0), ok(z1), ok(z2)) -> c74(U222'(z0, z1, z2))
   FST(mark(z0)) -> c75(FST(z0))
   FST(ok(z0)) -> c76(FST(z0))
   U31'(mark(z0), z1, z2) -> c77(U31'(z0, z1, z2))
   U31'(ok(z0), ok(z1), ok(z2)) -> c78(U31'(z0, z1, z2))
   U32'(mark(z0), z1) -> c79(U32'(z0, z1))
   U32'(ok(z0), ok(z1)) -> c80(U32'(z0, z1))
   U41'(mark(z0), z1) -> c81(U41'(z0, z1))
   U41'(ok(z0), ok(z1)) -> c82(U41'(z0, z1))
   U42'(mark(z0)) -> c83(U42'(z0))
   U42'(ok(z0)) -> c84(U42'(z0))
   U51'(mark(z0), z1) -> c85(U51'(z0, z1))
   U51'(ok(z0), ok(z1)) -> c86(U51'(z0, z1))
   U52'(mark(z0)) -> c87(U52'(z0))
   U52'(ok(z0)) -> c88(U52'(z0))
   U61'(mark(z0)) -> c89(U61'(z0))
   U61'(ok(z0)) -> c90(U61'(z0))
   U71'(mark(z0)) -> c91(U71'(z0))
   U71'(ok(z0)) -> c92(U71'(z0))
   U81'(mark(z0)) -> c93(U81'(z0))
   U81'(ok(z0)) -> c94(U81'(z0))
   U91'(mark(z0)) -> c95(U91'(z0))
   U91'(ok(z0)) -> c96(U91'(z0))
   TAIL(mark(z0)) -> c97(TAIL(z0))
   TAIL(ok(z0)) -> c98(TAIL(z0))
   TAKE(mark(z0), z1) -> c99(TAKE(z0, z1))
   TAKE(z0, mark(z1)) -> c100(TAKE(z0, z1))
   TAKE(ok(z0), ok(z1)) -> c101(TAKE(z0, z1))
   SEL(mark(z0), z1) -> c102(SEL(z0, z1))
   SEL(z0, mark(z1)) -> c103(SEL(z0, z1))
   SEL(ok(z0), ok(z1)) -> c104(SEL(z0, z1))
   PROPER(tt) -> c105
   PROPER(nil) -> c106
   PROPER(0) -> c107
   ISLNAT(ok(z0)) -> c108(ISLNAT(z0))
   ISNATURAL(ok(z0)) -> c109(ISNATURAL(z0))
   ISPLNAT(ok(z0)) -> c110(ISPLNAT(z0))
   TOP(mark(z0)) -> c111(TOP(proper(z0)), PROPER(z0))
   TOP(ok(z0)) -> c112(TOP(active(z0)))
K tuples:none
Defined Rule Symbols:   U101_2, U102_1, U11_3, U12_3, U111_1, snd_1, splitAt_2, U121_1, U131_2, U132_1, U141_2, U142_1, U151_2, U152_1, U161_2, cons_2, natsFrom_1, s_1, U171_3, U172_3, head_1, afterNth_2, U181_2, U182_2, U191_2, pair_2, U201_4, U202_4, U203_4, U204_2, U21_3, U22_2, U211_2, U212_2, U221_3, U222_3, fst_1, U31_3, U32_2, U41_2, U42_1, U51_2, U52_1, U61_1, U71_1, U81_1, U91_1, tail_1, take_2, sel_2, proper_1, isLNat_1, isNatural_1, isPLNat_1, top_1

Defined Pair Symbols:   U101'_2, U102'_1, U11'_3, U12'_3, U111'_1, SND_1, SPLITAT_2, U121'_1, U131'_2, U132'_1, U141'_2, U142'_1, U151'_2, U152'_1, U161'_2, CONS_2, NATSFROM_1, S_1, U171'_3, U172'_3, HEAD_1, AFTERNTH_2, U181'_2, U182'_2, U191'_2, PAIR_2, U201'_4, U202'_4, U203'_4, U204'_2, U21'_3, U22'_2, U211'_2, U212'_2, U221'_3, U222'_3, FST_1, U31'_3, U32'_2, U41'_2, U42'_1, U51'_2, U52'_1, U61'_1, U71'_1, U81'_1, U91'_1, TAIL_1, TAKE_2, SEL_2, PROPER_1, ISLNAT_1, ISNATURAL_1, ISPLNAT_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, c106, c107, c108_1, c109_1, c110_1, c111_2, c112_1


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

(7) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 4 trailing nodes:
   PROPER(nil) -> c106
   PROPER(tt) -> c105
   TOP(ok(z0)) -> c112(TOP(active(z0)))
   PROPER(0) -> c107

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

(8)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   U101(mark(z0), z1) -> mark(U101(z0, z1))
   U101(ok(z0), ok(z1)) -> ok(U101(z0, z1))
   U102(mark(z0)) -> mark(U102(z0))
   U102(ok(z0)) -> ok(U102(z0))
   U11(mark(z0), z1, z2) -> mark(U11(z0, z1, z2))
   U11(ok(z0), ok(z1), ok(z2)) -> ok(U11(z0, z1, z2))
   U12(mark(z0), z1, z2) -> mark(U12(z0, z1, z2))
   U12(ok(z0), ok(z1), ok(z2)) -> ok(U12(z0, z1, z2))
   U111(mark(z0)) -> mark(U111(z0))
   U111(ok(z0)) -> ok(U111(z0))
   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))
   U121(mark(z0)) -> mark(U121(z0))
   U121(ok(z0)) -> ok(U121(z0))
   U131(mark(z0), z1) -> mark(U131(z0, z1))
   U131(ok(z0), ok(z1)) -> ok(U131(z0, z1))
   U132(mark(z0)) -> mark(U132(z0))
   U132(ok(z0)) -> ok(U132(z0))
   U141(mark(z0), z1) -> mark(U141(z0, z1))
   U141(ok(z0), ok(z1)) -> ok(U141(z0, z1))
   U142(mark(z0)) -> mark(U142(z0))
   U142(ok(z0)) -> ok(U142(z0))
   U151(mark(z0), z1) -> mark(U151(z0, z1))
   U151(ok(z0), ok(z1)) -> ok(U151(z0, z1))
   U152(mark(z0)) -> mark(U152(z0))
   U152(ok(z0)) -> ok(U152(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))
   U172(mark(z0), z1, z2) -> mark(U172(z0, z1, z2))
   U172(ok(z0), ok(z1), ok(z2)) -> ok(U172(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))
   U182(mark(z0), z1) -> mark(U182(z0, z1))
   U182(ok(z0), ok(z1)) -> ok(U182(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, z2, z3) -> mark(U202(z0, z1, z2, z3))
   U202(ok(z0), ok(z1), ok(z2), ok(z3)) -> ok(U202(z0, z1, z2, z3))
   U203(mark(z0), z1, z2, z3) -> mark(U203(z0, z1, z2, z3))
   U203(ok(z0), ok(z1), ok(z2), ok(z3)) -> ok(U203(z0, z1, z2, z3))
   U204(mark(z0), z1) -> mark(U204(z0, z1))
   U204(ok(z0), ok(z1)) -> ok(U204(z0, z1))
   U21(mark(z0), z1, z2) -> mark(U21(z0, z1, z2))
   U21(ok(z0), ok(z1), ok(z2)) -> ok(U21(z0, z1, z2))
   U22(mark(z0), z1) -> mark(U22(z0, z1))
   U22(ok(z0), ok(z1)) -> ok(U22(z0, z1))
   U211(mark(z0), z1) -> mark(U211(z0, z1))
   U211(ok(z0), ok(z1)) -> ok(U211(z0, z1))
   U212(mark(z0), z1) -> mark(U212(z0, z1))
   U212(ok(z0), ok(z1)) -> ok(U212(z0, z1))
   U221(mark(z0), z1, z2) -> mark(U221(z0, z1, z2))
   U221(ok(z0), ok(z1), ok(z2)) -> ok(U221(z0, z1, z2))
   U222(mark(z0), z1, z2) -> mark(U222(z0, z1, z2))
   U222(ok(z0), ok(z1), ok(z2)) -> ok(U222(z0, z1, z2))
   fst(mark(z0)) -> mark(fst(z0))
   fst(ok(z0)) -> ok(fst(z0))
   U31(mark(z0), z1, z2) -> mark(U31(z0, z1, z2))
   U31(ok(z0), ok(z1), ok(z2)) -> ok(U31(z0, z1, z2))
   U32(mark(z0), z1) -> mark(U32(z0, z1))
   U32(ok(z0), ok(z1)) -> ok(U32(z0, z1))
   U41(mark(z0), z1) -> mark(U41(z0, z1))
   U41(ok(z0), ok(z1)) -> ok(U41(z0, z1))
   U42(mark(z0)) -> mark(U42(z0))
   U42(ok(z0)) -> ok(U42(z0))
   U51(mark(z0), z1) -> mark(U51(z0, z1))
   U51(ok(z0), ok(z1)) -> ok(U51(z0, z1))
   U52(mark(z0)) -> mark(U52(z0))
   U52(ok(z0)) -> ok(U52(z0))
   U61(mark(z0)) -> mark(U61(z0))
   U61(ok(z0)) -> ok(U61(z0))
   U71(mark(z0)) -> mark(U71(z0))
   U71(ok(z0)) -> ok(U71(z0))
   U81(mark(z0)) -> mark(U81(z0))
   U81(ok(z0)) -> ok(U81(z0))
   U91(mark(z0)) -> mark(U91(z0))
   U91(ok(z0)) -> ok(U91(z0))
   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)
   isLNat(ok(z0)) -> ok(isLNat(z0))
   isNatural(ok(z0)) -> ok(isNatural(z0))
   isPLNat(ok(z0)) -> ok(isPLNat(z0))
   top(mark(z0)) -> top(proper(z0))
   top(ok(z0)) -> top(active(z0))
Tuples:
   U101'(mark(z0), z1) -> c(U101'(z0, z1))
   U101'(ok(z0), ok(z1)) -> c1(U101'(z0, z1))
   U102'(mark(z0)) -> c2(U102'(z0))
   U102'(ok(z0)) -> c3(U102'(z0))
   U11'(mark(z0), z1, z2) -> c4(U11'(z0, z1, z2))
   U11'(ok(z0), ok(z1), ok(z2)) -> c5(U11'(z0, z1, z2))
   U12'(mark(z0), z1, z2) -> c6(U12'(z0, z1, z2))
   U12'(ok(z0), ok(z1), ok(z2)) -> c7(U12'(z0, z1, z2))
   U111'(mark(z0)) -> c8(U111'(z0))
   U111'(ok(z0)) -> c9(U111'(z0))
   SND(mark(z0)) -> c10(SND(z0))
   SND(ok(z0)) -> c11(SND(z0))
   SPLITAT(mark(z0), z1) -> c12(SPLITAT(z0, z1))
   SPLITAT(z0, mark(z1)) -> c13(SPLITAT(z0, z1))
   SPLITAT(ok(z0), ok(z1)) -> c14(SPLITAT(z0, z1))
   U121'(mark(z0)) -> c15(U121'(z0))
   U121'(ok(z0)) -> c16(U121'(z0))
   U131'(mark(z0), z1) -> c17(U131'(z0, z1))
   U131'(ok(z0), ok(z1)) -> c18(U131'(z0, z1))
   U132'(mark(z0)) -> c19(U132'(z0))
   U132'(ok(z0)) -> c20(U132'(z0))
   U141'(mark(z0), z1) -> c21(U141'(z0, z1))
   U141'(ok(z0), ok(z1)) -> c22(U141'(z0, z1))
   U142'(mark(z0)) -> c23(U142'(z0))
   U142'(ok(z0)) -> c24(U142'(z0))
   U151'(mark(z0), z1) -> c25(U151'(z0, z1))
   U151'(ok(z0), ok(z1)) -> c26(U151'(z0, z1))
   U152'(mark(z0)) -> c27(U152'(z0))
   U152'(ok(z0)) -> c28(U152'(z0))
   U161'(mark(z0), z1) -> c29(U161'(z0, z1))
   U161'(ok(z0), ok(z1)) -> c30(U161'(z0, z1))
   CONS(mark(z0), z1) -> c31(CONS(z0, z1))
   CONS(ok(z0), ok(z1)) -> c32(CONS(z0, z1))
   NATSFROM(mark(z0)) -> c33(NATSFROM(z0))
   NATSFROM(ok(z0)) -> c34(NATSFROM(z0))
   S(mark(z0)) -> c35(S(z0))
   S(ok(z0)) -> c36(S(z0))
   U171'(mark(z0), z1, z2) -> c37(U171'(z0, z1, z2))
   U171'(ok(z0), ok(z1), ok(z2)) -> c38(U171'(z0, z1, z2))
   U172'(mark(z0), z1, z2) -> c39(U172'(z0, z1, z2))
   U172'(ok(z0), ok(z1), ok(z2)) -> c40(U172'(z0, z1, z2))
   HEAD(mark(z0)) -> c41(HEAD(z0))
   HEAD(ok(z0)) -> c42(HEAD(z0))
   AFTERNTH(mark(z0), z1) -> c43(AFTERNTH(z0, z1))
   AFTERNTH(z0, mark(z1)) -> c44(AFTERNTH(z0, z1))
   AFTERNTH(ok(z0), ok(z1)) -> c45(AFTERNTH(z0, z1))
   U181'(mark(z0), z1) -> c46(U181'(z0, z1))
   U181'(ok(z0), ok(z1)) -> c47(U181'(z0, z1))
   U182'(mark(z0), z1) -> c48(U182'(z0, z1))
   U182'(ok(z0), ok(z1)) -> c49(U182'(z0, z1))
   U191'(mark(z0), z1) -> c50(U191'(z0, z1))
   U191'(ok(z0), ok(z1)) -> c51(U191'(z0, z1))
   PAIR(mark(z0), z1) -> c52(PAIR(z0, z1))
   PAIR(z0, mark(z1)) -> c53(PAIR(z0, z1))
   PAIR(ok(z0), ok(z1)) -> c54(PAIR(z0, z1))
   U201'(mark(z0), z1, z2, z3) -> c55(U201'(z0, z1, z2, z3))
   U201'(ok(z0), ok(z1), ok(z2), ok(z3)) -> c56(U201'(z0, z1, z2, z3))
   U202'(mark(z0), z1, z2, z3) -> c57(U202'(z0, z1, z2, z3))
   U202'(ok(z0), ok(z1), ok(z2), ok(z3)) -> c58(U202'(z0, z1, z2, z3))
   U203'(mark(z0), z1, z2, z3) -> c59(U203'(z0, z1, z2, z3))
   U203'(ok(z0), ok(z1), ok(z2), ok(z3)) -> c60(U203'(z0, z1, z2, z3))
   U204'(mark(z0), z1) -> c61(U204'(z0, z1))
   U204'(ok(z0), ok(z1)) -> c62(U204'(z0, z1))
   U21'(mark(z0), z1, z2) -> c63(U21'(z0, z1, z2))
   U21'(ok(z0), ok(z1), ok(z2)) -> c64(U21'(z0, z1, z2))
   U22'(mark(z0), z1) -> c65(U22'(z0, z1))
   U22'(ok(z0), ok(z1)) -> c66(U22'(z0, z1))
   U211'(mark(z0), z1) -> c67(U211'(z0, z1))
   U211'(ok(z0), ok(z1)) -> c68(U211'(z0, z1))
   U212'(mark(z0), z1) -> c69(U212'(z0, z1))
   U212'(ok(z0), ok(z1)) -> c70(U212'(z0, z1))
   U221'(mark(z0), z1, z2) -> c71(U221'(z0, z1, z2))
   U221'(ok(z0), ok(z1), ok(z2)) -> c72(U221'(z0, z1, z2))
   U222'(mark(z0), z1, z2) -> c73(U222'(z0, z1, z2))
   U222'(ok(z0), ok(z1), ok(z2)) -> c74(U222'(z0, z1, z2))
   FST(mark(z0)) -> c75(FST(z0))
   FST(ok(z0)) -> c76(FST(z0))
   U31'(mark(z0), z1, z2) -> c77(U31'(z0, z1, z2))
   U31'(ok(z0), ok(z1), ok(z2)) -> c78(U31'(z0, z1, z2))
   U32'(mark(z0), z1) -> c79(U32'(z0, z1))
   U32'(ok(z0), ok(z1)) -> c80(U32'(z0, z1))
   U41'(mark(z0), z1) -> c81(U41'(z0, z1))
   U41'(ok(z0), ok(z1)) -> c82(U41'(z0, z1))
   U42'(mark(z0)) -> c83(U42'(z0))
   U42'(ok(z0)) -> c84(U42'(z0))
   U51'(mark(z0), z1) -> c85(U51'(z0, z1))
   U51'(ok(z0), ok(z1)) -> c86(U51'(z0, z1))
   U52'(mark(z0)) -> c87(U52'(z0))
   U52'(ok(z0)) -> c88(U52'(z0))
   U61'(mark(z0)) -> c89(U61'(z0))
   U61'(ok(z0)) -> c90(U61'(z0))
   U71'(mark(z0)) -> c91(U71'(z0))
   U71'(ok(z0)) -> c92(U71'(z0))
   U81'(mark(z0)) -> c93(U81'(z0))
   U81'(ok(z0)) -> c94(U81'(z0))
   U91'(mark(z0)) -> c95(U91'(z0))
   U91'(ok(z0)) -> c96(U91'(z0))
   TAIL(mark(z0)) -> c97(TAIL(z0))
   TAIL(ok(z0)) -> c98(TAIL(z0))
   TAKE(mark(z0), z1) -> c99(TAKE(z0, z1))
   TAKE(z0, mark(z1)) -> c100(TAKE(z0, z1))
   TAKE(ok(z0), ok(z1)) -> c101(TAKE(z0, z1))
   SEL(mark(z0), z1) -> c102(SEL(z0, z1))
   SEL(z0, mark(z1)) -> c103(SEL(z0, z1))
   SEL(ok(z0), ok(z1)) -> c104(SEL(z0, z1))
   ISLNAT(ok(z0)) -> c108(ISLNAT(z0))
   ISNATURAL(ok(z0)) -> c109(ISNATURAL(z0))
   ISPLNAT(ok(z0)) -> c110(ISPLNAT(z0))
   TOP(mark(z0)) -> c111(TOP(proper(z0)), PROPER(z0))
S tuples:
   U101'(mark(z0), z1) -> c(U101'(z0, z1))
   U101'(ok(z0), ok(z1)) -> c1(U101'(z0, z1))
   U102'(mark(z0)) -> c2(U102'(z0))
   U102'(ok(z0)) -> c3(U102'(z0))
   U11'(mark(z0), z1, z2) -> c4(U11'(z0, z1, z2))
   U11'(ok(z0), ok(z1), ok(z2)) -> c5(U11'(z0, z1, z2))
   U12'(mark(z0), z1, z2) -> c6(U12'(z0, z1, z2))
   U12'(ok(z0), ok(z1), ok(z2)) -> c7(U12'(z0, z1, z2))
   U111'(mark(z0)) -> c8(U111'(z0))
   U111'(ok(z0)) -> c9(U111'(z0))
   SND(mark(z0)) -> c10(SND(z0))
   SND(ok(z0)) -> c11(SND(z0))
   SPLITAT(mark(z0), z1) -> c12(SPLITAT(z0, z1))
   SPLITAT(z0, mark(z1)) -> c13(SPLITAT(z0, z1))
   SPLITAT(ok(z0), ok(z1)) -> c14(SPLITAT(z0, z1))
   U121'(mark(z0)) -> c15(U121'(z0))
   U121'(ok(z0)) -> c16(U121'(z0))
   U131'(mark(z0), z1) -> c17(U131'(z0, z1))
   U131'(ok(z0), ok(z1)) -> c18(U131'(z0, z1))
   U132'(mark(z0)) -> c19(U132'(z0))
   U132'(ok(z0)) -> c20(U132'(z0))
   U141'(mark(z0), z1) -> c21(U141'(z0, z1))
   U141'(ok(z0), ok(z1)) -> c22(U141'(z0, z1))
   U142'(mark(z0)) -> c23(U142'(z0))
   U142'(ok(z0)) -> c24(U142'(z0))
   U151'(mark(z0), z1) -> c25(U151'(z0, z1))
   U151'(ok(z0), ok(z1)) -> c26(U151'(z0, z1))
   U152'(mark(z0)) -> c27(U152'(z0))
   U152'(ok(z0)) -> c28(U152'(z0))
   U161'(mark(z0), z1) -> c29(U161'(z0, z1))
   U161'(ok(z0), ok(z1)) -> c30(U161'(z0, z1))
   CONS(mark(z0), z1) -> c31(CONS(z0, z1))
   CONS(ok(z0), ok(z1)) -> c32(CONS(z0, z1))
   NATSFROM(mark(z0)) -> c33(NATSFROM(z0))
   NATSFROM(ok(z0)) -> c34(NATSFROM(z0))
   S(mark(z0)) -> c35(S(z0))
   S(ok(z0)) -> c36(S(z0))
   U171'(mark(z0), z1, z2) -> c37(U171'(z0, z1, z2))
   U171'(ok(z0), ok(z1), ok(z2)) -> c38(U171'(z0, z1, z2))
   U172'(mark(z0), z1, z2) -> c39(U172'(z0, z1, z2))
   U172'(ok(z0), ok(z1), ok(z2)) -> c40(U172'(z0, z1, z2))
   HEAD(mark(z0)) -> c41(HEAD(z0))
   HEAD(ok(z0)) -> c42(HEAD(z0))
   AFTERNTH(mark(z0), z1) -> c43(AFTERNTH(z0, z1))
   AFTERNTH(z0, mark(z1)) -> c44(AFTERNTH(z0, z1))
   AFTERNTH(ok(z0), ok(z1)) -> c45(AFTERNTH(z0, z1))
   U181'(mark(z0), z1) -> c46(U181'(z0, z1))
   U181'(ok(z0), ok(z1)) -> c47(U181'(z0, z1))
   U182'(mark(z0), z1) -> c48(U182'(z0, z1))
   U182'(ok(z0), ok(z1)) -> c49(U182'(z0, z1))
   U191'(mark(z0), z1) -> c50(U191'(z0, z1))
   U191'(ok(z0), ok(z1)) -> c51(U191'(z0, z1))
   PAIR(mark(z0), z1) -> c52(PAIR(z0, z1))
   PAIR(z0, mark(z1)) -> c53(PAIR(z0, z1))
   PAIR(ok(z0), ok(z1)) -> c54(PAIR(z0, z1))
   U201'(mark(z0), z1, z2, z3) -> c55(U201'(z0, z1, z2, z3))
   U201'(ok(z0), ok(z1), ok(z2), ok(z3)) -> c56(U201'(z0, z1, z2, z3))
   U202'(mark(z0), z1, z2, z3) -> c57(U202'(z0, z1, z2, z3))
   U202'(ok(z0), ok(z1), ok(z2), ok(z3)) -> c58(U202'(z0, z1, z2, z3))
   U203'(mark(z0), z1, z2, z3) -> c59(U203'(z0, z1, z2, z3))
   U203'(ok(z0), ok(z1), ok(z2), ok(z3)) -> c60(U203'(z0, z1, z2, z3))
   U204'(mark(z0), z1) -> c61(U204'(z0, z1))
   U204'(ok(z0), ok(z1)) -> c62(U204'(z0, z1))
   U21'(mark(z0), z1, z2) -> c63(U21'(z0, z1, z2))
   U21'(ok(z0), ok(z1), ok(z2)) -> c64(U21'(z0, z1, z2))
   U22'(mark(z0), z1) -> c65(U22'(z0, z1))
   U22'(ok(z0), ok(z1)) -> c66(U22'(z0, z1))
   U211'(mark(z0), z1) -> c67(U211'(z0, z1))
   U211'(ok(z0), ok(z1)) -> c68(U211'(z0, z1))
   U212'(mark(z0), z1) -> c69(U212'(z0, z1))
   U212'(ok(z0), ok(z1)) -> c70(U212'(z0, z1))
   U221'(mark(z0), z1, z2) -> c71(U221'(z0, z1, z2))
   U221'(ok(z0), ok(z1), ok(z2)) -> c72(U221'(z0, z1, z2))
   U222'(mark(z0), z1, z2) -> c73(U222'(z0, z1, z2))
   U222'(ok(z0), ok(z1), ok(z2)) -> c74(U222'(z0, z1, z2))
   FST(mark(z0)) -> c75(FST(z0))
   FST(ok(z0)) -> c76(FST(z0))
   U31'(mark(z0), z1, z2) -> c77(U31'(z0, z1, z2))
   U31'(ok(z0), ok(z1), ok(z2)) -> c78(U31'(z0, z1, z2))
   U32'(mark(z0), z1) -> c79(U32'(z0, z1))
   U32'(ok(z0), ok(z1)) -> c80(U32'(z0, z1))
   U41'(mark(z0), z1) -> c81(U41'(z0, z1))
   U41'(ok(z0), ok(z1)) -> c82(U41'(z0, z1))
   U42'(mark(z0)) -> c83(U42'(z0))
   U42'(ok(z0)) -> c84(U42'(z0))
   U51'(mark(z0), z1) -> c85(U51'(z0, z1))
   U51'(ok(z0), ok(z1)) -> c86(U51'(z0, z1))
   U52'(mark(z0)) -> c87(U52'(z0))
   U52'(ok(z0)) -> c88(U52'(z0))
   U61'(mark(z0)) -> c89(U61'(z0))
   U61'(ok(z0)) -> c90(U61'(z0))
   U71'(mark(z0)) -> c91(U71'(z0))
   U71'(ok(z0)) -> c92(U71'(z0))
   U81'(mark(z0)) -> c93(U81'(z0))
   U81'(ok(z0)) -> c94(U81'(z0))
   U91'(mark(z0)) -> c95(U91'(z0))
   U91'(ok(z0)) -> c96(U91'(z0))
   TAIL(mark(z0)) -> c97(TAIL(z0))
   TAIL(ok(z0)) -> c98(TAIL(z0))
   TAKE(mark(z0), z1) -> c99(TAKE(z0, z1))
   TAKE(z0, mark(z1)) -> c100(TAKE(z0, z1))
   TAKE(ok(z0), ok(z1)) -> c101(TAKE(z0, z1))
   SEL(mark(z0), z1) -> c102(SEL(z0, z1))
   SEL(z0, mark(z1)) -> c103(SEL(z0, z1))
   SEL(ok(z0), ok(z1)) -> c104(SEL(z0, z1))
   ISLNAT(ok(z0)) -> c108(ISLNAT(z0))
   ISNATURAL(ok(z0)) -> c109(ISNATURAL(z0))
   ISPLNAT(ok(z0)) -> c110(ISPLNAT(z0))
   TOP(mark(z0)) -> c111(TOP(proper(z0)), PROPER(z0))
K tuples:none
Defined Rule Symbols:   U101_2, U102_1, U11_3, U12_3, U111_1, snd_1, splitAt_2, U121_1, U131_2, U132_1, U141_2, U142_1, U151_2, U152_1, U161_2, cons_2, natsFrom_1, s_1, U171_3, U172_3, head_1, afterNth_2, U181_2, U182_2, U191_2, pair_2, U201_4, U202_4, U203_4, U204_2, U21_3, U22_2, U211_2, U212_2, U221_3, U222_3, fst_1, U31_3, U32_2, U41_2, U42_1, U51_2, U52_1, U61_1, U71_1, U81_1, U91_1, tail_1, take_2, sel_2, proper_1, isLNat_1, isNatural_1, isPLNat_1, top_1

Defined Pair Symbols:   U101'_2, U102'_1, U11'_3, U12'_3, U111'_1, SND_1, SPLITAT_2, U121'_1, U131'_2, U132'_1, U141'_2, U142'_1, U151'_2, U152'_1, U161'_2, CONS_2, NATSFROM_1, S_1, U171'_3, U172'_3, HEAD_1, AFTERNTH_2, U181'_2, U182'_2, U191'_2, PAIR_2, U201'_4, U202'_4, U203'_4, U204'_2, U21'_3, U22'_2, U211'_2, U212'_2, U221'_3, U222'_3, FST_1, U31'_3, U32'_2, U41'_2, U42'_1, U51'_2, U52'_1, U61'_1, U71'_1, U81'_1, U91'_1, TAIL_1, TAKE_2, SEL_2, ISLNAT_1, ISNATURAL_1, ISPLNAT_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, c108_1, c109_1, c110_1, c111_2


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

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

(10)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   U101(mark(z0), z1) -> mark(U101(z0, z1))
   U101(ok(z0), ok(z1)) -> ok(U101(z0, z1))
   U102(mark(z0)) -> mark(U102(z0))
   U102(ok(z0)) -> ok(U102(z0))
   U11(mark(z0), z1, z2) -> mark(U11(z0, z1, z2))
   U11(ok(z0), ok(z1), ok(z2)) -> ok(U11(z0, z1, z2))
   U12(mark(z0), z1, z2) -> mark(U12(z0, z1, z2))
   U12(ok(z0), ok(z1), ok(z2)) -> ok(U12(z0, z1, z2))
   U111(mark(z0)) -> mark(U111(z0))
   U111(ok(z0)) -> ok(U111(z0))
   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))
   U121(mark(z0)) -> mark(U121(z0))
   U121(ok(z0)) -> ok(U121(z0))
   U131(mark(z0), z1) -> mark(U131(z0, z1))
   U131(ok(z0), ok(z1)) -> ok(U131(z0, z1))
   U132(mark(z0)) -> mark(U132(z0))
   U132(ok(z0)) -> ok(U132(z0))
   U141(mark(z0), z1) -> mark(U141(z0, z1))
   U141(ok(z0), ok(z1)) -> ok(U141(z0, z1))
   U142(mark(z0)) -> mark(U142(z0))
   U142(ok(z0)) -> ok(U142(z0))
   U151(mark(z0), z1) -> mark(U151(z0, z1))
   U151(ok(z0), ok(z1)) -> ok(U151(z0, z1))
   U152(mark(z0)) -> mark(U152(z0))
   U152(ok(z0)) -> ok(U152(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))
   U172(mark(z0), z1, z2) -> mark(U172(z0, z1, z2))
   U172(ok(z0), ok(z1), ok(z2)) -> ok(U172(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))
   U182(mark(z0), z1) -> mark(U182(z0, z1))
   U182(ok(z0), ok(z1)) -> ok(U182(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, z2, z3) -> mark(U202(z0, z1, z2, z3))
   U202(ok(z0), ok(z1), ok(z2), ok(z3)) -> ok(U202(z0, z1, z2, z3))
   U203(mark(z0), z1, z2, z3) -> mark(U203(z0, z1, z2, z3))
   U203(ok(z0), ok(z1), ok(z2), ok(z3)) -> ok(U203(z0, z1, z2, z3))
   U204(mark(z0), z1) -> mark(U204(z0, z1))
   U204(ok(z0), ok(z1)) -> ok(U204(z0, z1))
   U21(mark(z0), z1, z2) -> mark(U21(z0, z1, z2))
   U21(ok(z0), ok(z1), ok(z2)) -> ok(U21(z0, z1, z2))
   U22(mark(z0), z1) -> mark(U22(z0, z1))
   U22(ok(z0), ok(z1)) -> ok(U22(z0, z1))
   U211(mark(z0), z1) -> mark(U211(z0, z1))
   U211(ok(z0), ok(z1)) -> ok(U211(z0, z1))
   U212(mark(z0), z1) -> mark(U212(z0, z1))
   U212(ok(z0), ok(z1)) -> ok(U212(z0, z1))
   U221(mark(z0), z1, z2) -> mark(U221(z0, z1, z2))
   U221(ok(z0), ok(z1), ok(z2)) -> ok(U221(z0, z1, z2))
   U222(mark(z0), z1, z2) -> mark(U222(z0, z1, z2))
   U222(ok(z0), ok(z1), ok(z2)) -> ok(U222(z0, z1, z2))
   fst(mark(z0)) -> mark(fst(z0))
   fst(ok(z0)) -> ok(fst(z0))
   U31(mark(z0), z1, z2) -> mark(U31(z0, z1, z2))
   U31(ok(z0), ok(z1), ok(z2)) -> ok(U31(z0, z1, z2))
   U32(mark(z0), z1) -> mark(U32(z0, z1))
   U32(ok(z0), ok(z1)) -> ok(U32(z0, z1))
   U41(mark(z0), z1) -> mark(U41(z0, z1))
   U41(ok(z0), ok(z1)) -> ok(U41(z0, z1))
   U42(mark(z0)) -> mark(U42(z0))
   U42(ok(z0)) -> ok(U42(z0))
   U51(mark(z0), z1) -> mark(U51(z0, z1))
   U51(ok(z0), ok(z1)) -> ok(U51(z0, z1))
   U52(mark(z0)) -> mark(U52(z0))
   U52(ok(z0)) -> ok(U52(z0))
   U61(mark(z0)) -> mark(U61(z0))
   U61(ok(z0)) -> ok(U61(z0))
   U71(mark(z0)) -> mark(U71(z0))
   U71(ok(z0)) -> ok(U71(z0))
   U81(mark(z0)) -> mark(U81(z0))
   U81(ok(z0)) -> ok(U81(z0))
   U91(mark(z0)) -> mark(U91(z0))
   U91(ok(z0)) -> ok(U91(z0))
   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)
   isLNat(ok(z0)) -> ok(isLNat(z0))
   isNatural(ok(z0)) -> ok(isNatural(z0))
   isPLNat(ok(z0)) -> ok(isPLNat(z0))
   top(mark(z0)) -> top(proper(z0))
   top(ok(z0)) -> top(active(z0))
Tuples:
   U101'(mark(z0), z1) -> c(U101'(z0, z1))
   U101'(ok(z0), ok(z1)) -> c1(U101'(z0, z1))
   U102'(mark(z0)) -> c2(U102'(z0))
   U102'(ok(z0)) -> c3(U102'(z0))
   U11'(mark(z0), z1, z2) -> c4(U11'(z0, z1, z2))
   U11'(ok(z0), ok(z1), ok(z2)) -> c5(U11'(z0, z1, z2))
   U12'(mark(z0), z1, z2) -> c6(U12'(z0, z1, z2))
   U12'(ok(z0), ok(z1), ok(z2)) -> c7(U12'(z0, z1, z2))
   U111'(mark(z0)) -> c8(U111'(z0))
   U111'(ok(z0)) -> c9(U111'(z0))
   SND(mark(z0)) -> c10(SND(z0))
   SND(ok(z0)) -> c11(SND(z0))
   SPLITAT(mark(z0), z1) -> c12(SPLITAT(z0, z1))
   SPLITAT(z0, mark(z1)) -> c13(SPLITAT(z0, z1))
   SPLITAT(ok(z0), ok(z1)) -> c14(SPLITAT(z0, z1))
   U121'(mark(z0)) -> c15(U121'(z0))
   U121'(ok(z0)) -> c16(U121'(z0))
   U131'(mark(z0), z1) -> c17(U131'(z0, z1))
   U131'(ok(z0), ok(z1)) -> c18(U131'(z0, z1))
   U132'(mark(z0)) -> c19(U132'(z0))
   U132'(ok(z0)) -> c20(U132'(z0))
   U141'(mark(z0), z1) -> c21(U141'(z0, z1))
   U141'(ok(z0), ok(z1)) -> c22(U141'(z0, z1))
   U142'(mark(z0)) -> c23(U142'(z0))
   U142'(ok(z0)) -> c24(U142'(z0))
   U151'(mark(z0), z1) -> c25(U151'(z0, z1))
   U151'(ok(z0), ok(z1)) -> c26(U151'(z0, z1))
   U152'(mark(z0)) -> c27(U152'(z0))
   U152'(ok(z0)) -> c28(U152'(z0))
   U161'(mark(z0), z1) -> c29(U161'(z0, z1))
   U161'(ok(z0), ok(z1)) -> c30(U161'(z0, z1))
   CONS(mark(z0), z1) -> c31(CONS(z0, z1))
   CONS(ok(z0), ok(z1)) -> c32(CONS(z0, z1))
   NATSFROM(mark(z0)) -> c33(NATSFROM(z0))
   NATSFROM(ok(z0)) -> c34(NATSFROM(z0))
   S(mark(z0)) -> c35(S(z0))
   S(ok(z0)) -> c36(S(z0))
   U171'(mark(z0), z1, z2) -> c37(U171'(z0, z1, z2))
   U171'(ok(z0), ok(z1), ok(z2)) -> c38(U171'(z0, z1, z2))
   U172'(mark(z0), z1, z2) -> c39(U172'(z0, z1, z2))
   U172'(ok(z0), ok(z1), ok(z2)) -> c40(U172'(z0, z1, z2))
   HEAD(mark(z0)) -> c41(HEAD(z0))
   HEAD(ok(z0)) -> c42(HEAD(z0))
   AFTERNTH(mark(z0), z1) -> c43(AFTERNTH(z0, z1))
   AFTERNTH(z0, mark(z1)) -> c44(AFTERNTH(z0, z1))
   AFTERNTH(ok(z0), ok(z1)) -> c45(AFTERNTH(z0, z1))
   U181'(mark(z0), z1) -> c46(U181'(z0, z1))
   U181'(ok(z0), ok(z1)) -> c47(U181'(z0, z1))
   U182'(mark(z0), z1) -> c48(U182'(z0, z1))
   U182'(ok(z0), ok(z1)) -> c49(U182'(z0, z1))
   U191'(mark(z0), z1) -> c50(U191'(z0, z1))
   U191'(ok(z0), ok(z1)) -> c51(U191'(z0, z1))
   PAIR(mark(z0), z1) -> c52(PAIR(z0, z1))
   PAIR(z0, mark(z1)) -> c53(PAIR(z0, z1))
   PAIR(ok(z0), ok(z1)) -> c54(PAIR(z0, z1))
   U201'(mark(z0), z1, z2, z3) -> c55(U201'(z0, z1, z2, z3))
   U201'(ok(z0), ok(z1), ok(z2), ok(z3)) -> c56(U201'(z0, z1, z2, z3))
   U202'(mark(z0), z1, z2, z3) -> c57(U202'(z0, z1, z2, z3))
   U202'(ok(z0), ok(z1), ok(z2), ok(z3)) -> c58(U202'(z0, z1, z2, z3))
   U203'(mark(z0), z1, z2, z3) -> c59(U203'(z0, z1, z2, z3))
   U203'(ok(z0), ok(z1), ok(z2), ok(z3)) -> c60(U203'(z0, z1, z2, z3))
   U204'(mark(z0), z1) -> c61(U204'(z0, z1))
   U204'(ok(z0), ok(z1)) -> c62(U204'(z0, z1))
   U21'(mark(z0), z1, z2) -> c63(U21'(z0, z1, z2))
   U21'(ok(z0), ok(z1), ok(z2)) -> c64(U21'(z0, z1, z2))
   U22'(mark(z0), z1) -> c65(U22'(z0, z1))
   U22'(ok(z0), ok(z1)) -> c66(U22'(z0, z1))
   U211'(mark(z0), z1) -> c67(U211'(z0, z1))
   U211'(ok(z0), ok(z1)) -> c68(U211'(z0, z1))
   U212'(mark(z0), z1) -> c69(U212'(z0, z1))
   U212'(ok(z0), ok(z1)) -> c70(U212'(z0, z1))
   U221'(mark(z0), z1, z2) -> c71(U221'(z0, z1, z2))
   U221'(ok(z0), ok(z1), ok(z2)) -> c72(U221'(z0, z1, z2))
   U222'(mark(z0), z1, z2) -> c73(U222'(z0, z1, z2))
   U222'(ok(z0), ok(z1), ok(z2)) -> c74(U222'(z0, z1, z2))
   FST(mark(z0)) -> c75(FST(z0))
   FST(ok(z0)) -> c76(FST(z0))
   U31'(mark(z0), z1, z2) -> c77(U31'(z0, z1, z2))
   U31'(ok(z0), ok(z1), ok(z2)) -> c78(U31'(z0, z1, z2))
   U32'(mark(z0), z1) -> c79(U32'(z0, z1))
   U32'(ok(z0), ok(z1)) -> c80(U32'(z0, z1))
   U41'(mark(z0), z1) -> c81(U41'(z0, z1))
   U41'(ok(z0), ok(z1)) -> c82(U41'(z0, z1))
   U42'(mark(z0)) -> c83(U42'(z0))
   U42'(ok(z0)) -> c84(U42'(z0))
   U51'(mark(z0), z1) -> c85(U51'(z0, z1))
   U51'(ok(z0), ok(z1)) -> c86(U51'(z0, z1))
   U52'(mark(z0)) -> c87(U52'(z0))
   U52'(ok(z0)) -> c88(U52'(z0))
   U61'(mark(z0)) -> c89(U61'(z0))
   U61'(ok(z0)) -> c90(U61'(z0))
   U71'(mark(z0)) -> c91(U71'(z0))
   U71'(ok(z0)) -> c92(U71'(z0))
   U81'(mark(z0)) -> c93(U81'(z0))
   U81'(ok(z0)) -> c94(U81'(z0))
   U91'(mark(z0)) -> c95(U91'(z0))
   U91'(ok(z0)) -> c96(U91'(z0))
   TAIL(mark(z0)) -> c97(TAIL(z0))
   TAIL(ok(z0)) -> c98(TAIL(z0))
   TAKE(mark(z0), z1) -> c99(TAKE(z0, z1))
   TAKE(z0, mark(z1)) -> c100(TAKE(z0, z1))
   TAKE(ok(z0), ok(z1)) -> c101(TAKE(z0, z1))
   SEL(mark(z0), z1) -> c102(SEL(z0, z1))
   SEL(z0, mark(z1)) -> c103(SEL(z0, z1))
   SEL(ok(z0), ok(z1)) -> c104(SEL(z0, z1))
   ISLNAT(ok(z0)) -> c108(ISLNAT(z0))
   ISNATURAL(ok(z0)) -> c109(ISNATURAL(z0))
   ISPLNAT(ok(z0)) -> c110(ISPLNAT(z0))
   TOP(mark(z0)) -> c111(TOP(proper(z0)))
S tuples:
   U101'(mark(z0), z1) -> c(U101'(z0, z1))
   U101'(ok(z0), ok(z1)) -> c1(U101'(z0, z1))
   U102'(mark(z0)) -> c2(U102'(z0))
   U102'(ok(z0)) -> c3(U102'(z0))
   U11'(mark(z0), z1, z2) -> c4(U11'(z0, z1, z2))
   U11'(ok(z0), ok(z1), ok(z2)) -> c5(U11'(z0, z1, z2))
   U12'(mark(z0), z1, z2) -> c6(U12'(z0, z1, z2))
   U12'(ok(z0), ok(z1), ok(z2)) -> c7(U12'(z0, z1, z2))
   U111'(mark(z0)) -> c8(U111'(z0))
   U111'(ok(z0)) -> c9(U111'(z0))
   SND(mark(z0)) -> c10(SND(z0))
   SND(ok(z0)) -> c11(SND(z0))
   SPLITAT(mark(z0), z1) -> c12(SPLITAT(z0, z1))
   SPLITAT(z0, mark(z1)) -> c13(SPLITAT(z0, z1))
   SPLITAT(ok(z0), ok(z1)) -> c14(SPLITAT(z0, z1))
   U121'(mark(z0)) -> c15(U121'(z0))
   U121'(ok(z0)) -> c16(U121'(z0))
   U131'(mark(z0), z1) -> c17(U131'(z0, z1))
   U131'(ok(z0), ok(z1)) -> c18(U131'(z0, z1))
   U132'(mark(z0)) -> c19(U132'(z0))
   U132'(ok(z0)) -> c20(U132'(z0))
   U141'(mark(z0), z1) -> c21(U141'(z0, z1))
   U141'(ok(z0), ok(z1)) -> c22(U141'(z0, z1))
   U142'(mark(z0)) -> c23(U142'(z0))
   U142'(ok(z0)) -> c24(U142'(z0))
   U151'(mark(z0), z1) -> c25(U151'(z0, z1))
   U151'(ok(z0), ok(z1)) -> c26(U151'(z0, z1))
   U152'(mark(z0)) -> c27(U152'(z0))
   U152'(ok(z0)) -> c28(U152'(z0))
   U161'(mark(z0), z1) -> c29(U161'(z0, z1))
   U161'(ok(z0), ok(z1)) -> c30(U161'(z0, z1))
   CONS(mark(z0), z1) -> c31(CONS(z0, z1))
   CONS(ok(z0), ok(z1)) -> c32(CONS(z0, z1))
   NATSFROM(mark(z0)) -> c33(NATSFROM(z0))
   NATSFROM(ok(z0)) -> c34(NATSFROM(z0))
   S(mark(z0)) -> c35(S(z0))
   S(ok(z0)) -> c36(S(z0))
   U171'(mark(z0), z1, z2) -> c37(U171'(z0, z1, z2))
   U171'(ok(z0), ok(z1), ok(z2)) -> c38(U171'(z0, z1, z2))
   U172'(mark(z0), z1, z2) -> c39(U172'(z0, z1, z2))
   U172'(ok(z0), ok(z1), ok(z2)) -> c40(U172'(z0, z1, z2))
   HEAD(mark(z0)) -> c41(HEAD(z0))
   HEAD(ok(z0)) -> c42(HEAD(z0))
   AFTERNTH(mark(z0), z1) -> c43(AFTERNTH(z0, z1))
   AFTERNTH(z0, mark(z1)) -> c44(AFTERNTH(z0, z1))
   AFTERNTH(ok(z0), ok(z1)) -> c45(AFTERNTH(z0, z1))
   U181'(mark(z0), z1) -> c46(U181'(z0, z1))
   U181'(ok(z0), ok(z1)) -> c47(U181'(z0, z1))
   U182'(mark(z0), z1) -> c48(U182'(z0, z1))
   U182'(ok(z0), ok(z1)) -> c49(U182'(z0, z1))
   U191'(mark(z0), z1) -> c50(U191'(z0, z1))
   U191'(ok(z0), ok(z1)) -> c51(U191'(z0, z1))
   PAIR(mark(z0), z1) -> c52(PAIR(z0, z1))
   PAIR(z0, mark(z1)) -> c53(PAIR(z0, z1))
   PAIR(ok(z0), ok(z1)) -> c54(PAIR(z0, z1))
   U201'(mark(z0), z1, z2, z3) -> c55(U201'(z0, z1, z2, z3))
   U201'(ok(z0), ok(z1), ok(z2), ok(z3)) -> c56(U201'(z0, z1, z2, z3))
   U202'(mark(z0), z1, z2, z3) -> c57(U202'(z0, z1, z2, z3))
   U202'(ok(z0), ok(z1), ok(z2), ok(z3)) -> c58(U202'(z0, z1, z2, z3))
   U203'(mark(z0), z1, z2, z3) -> c59(U203'(z0, z1, z2, z3))
   U203'(ok(z0), ok(z1), ok(z2), ok(z3)) -> c60(U203'(z0, z1, z2, z3))
   U204'(mark(z0), z1) -> c61(U204'(z0, z1))
   U204'(ok(z0), ok(z1)) -> c62(U204'(z0, z1))
   U21'(mark(z0), z1, z2) -> c63(U21'(z0, z1, z2))
   U21'(ok(z0), ok(z1), ok(z2)) -> c64(U21'(z0, z1, z2))
   U22'(mark(z0), z1) -> c65(U22'(z0, z1))
   U22'(ok(z0), ok(z1)) -> c66(U22'(z0, z1))
   U211'(mark(z0), z1) -> c67(U211'(z0, z1))
   U211'(ok(z0), ok(z1)) -> c68(U211'(z0, z1))
   U212'(mark(z0), z1) -> c69(U212'(z0, z1))
   U212'(ok(z0), ok(z1)) -> c70(U212'(z0, z1))
   U221'(mark(z0), z1, z2) -> c71(U221'(z0, z1, z2))
   U221'(ok(z0), ok(z1), ok(z2)) -> c72(U221'(z0, z1, z2))
   U222'(mark(z0), z1, z2) -> c73(U222'(z0, z1, z2))
   U222'(ok(z0), ok(z1), ok(z2)) -> c74(U222'(z0, z1, z2))
   FST(mark(z0)) -> c75(FST(z0))
   FST(ok(z0)) -> c76(FST(z0))
   U31'(mark(z0), z1, z2) -> c77(U31'(z0, z1, z2))
   U31'(ok(z0), ok(z1), ok(z2)) -> c78(U31'(z0, z1, z2))
   U32'(mark(z0), z1) -> c79(U32'(z0, z1))
   U32'(ok(z0), ok(z1)) -> c80(U32'(z0, z1))
   U41'(mark(z0), z1) -> c81(U41'(z0, z1))
   U41'(ok(z0), ok(z1)) -> c82(U41'(z0, z1))
   U42'(mark(z0)) -> c83(U42'(z0))
   U42'(ok(z0)) -> c84(U42'(z0))
   U51'(mark(z0), z1) -> c85(U51'(z0, z1))
   U51'(ok(z0), ok(z1)) -> c86(U51'(z0, z1))
   U52'(mark(z0)) -> c87(U52'(z0))
   U52'(ok(z0)) -> c88(U52'(z0))
   U61'(mark(z0)) -> c89(U61'(z0))
   U61'(ok(z0)) -> c90(U61'(z0))
   U71'(mark(z0)) -> c91(U71'(z0))
   U71'(ok(z0)) -> c92(U71'(z0))
   U81'(mark(z0)) -> c93(U81'(z0))
   U81'(ok(z0)) -> c94(U81'(z0))
   U91'(mark(z0)) -> c95(U91'(z0))
   U91'(ok(z0)) -> c96(U91'(z0))
   TAIL(mark(z0)) -> c97(TAIL(z0))
   TAIL(ok(z0)) -> c98(TAIL(z0))
   TAKE(mark(z0), z1) -> c99(TAKE(z0, z1))
   TAKE(z0, mark(z1)) -> c100(TAKE(z0, z1))
   TAKE(ok(z0), ok(z1)) -> c101(TAKE(z0, z1))
   SEL(mark(z0), z1) -> c102(SEL(z0, z1))
   SEL(z0, mark(z1)) -> c103(SEL(z0, z1))
   SEL(ok(z0), ok(z1)) -> c104(SEL(z0, z1))
   ISLNAT(ok(z0)) -> c108(ISLNAT(z0))
   ISNATURAL(ok(z0)) -> c109(ISNATURAL(z0))
   ISPLNAT(ok(z0)) -> c110(ISPLNAT(z0))
   TOP(mark(z0)) -> c111(TOP(proper(z0)))
K tuples:none
Defined Rule Symbols:   U101_2, U102_1, U11_3, U12_3, U111_1, snd_1, splitAt_2, U121_1, U131_2, U132_1, U141_2, U142_1, U151_2, U152_1, U161_2, cons_2, natsFrom_1, s_1, U171_3, U172_3, head_1, afterNth_2, U181_2, U182_2, U191_2, pair_2, U201_4, U202_4, U203_4, U204_2, U21_3, U22_2, U211_2, U212_2, U221_3, U222_3, fst_1, U31_3, U32_2, U41_2, U42_1, U51_2, U52_1, U61_1, U71_1, U81_1, U91_1, tail_1, take_2, sel_2, proper_1, isLNat_1, isNatural_1, isPLNat_1, top_1

Defined Pair Symbols:   U101'_2, U102'_1, U11'_3, U12'_3, U111'_1, SND_1, SPLITAT_2, U121'_1, U131'_2, U132'_1, U141'_2, U142'_1, U151'_2, U152'_1, U161'_2, CONS_2, NATSFROM_1, S_1, U171'_3, U172'_3, HEAD_1, AFTERNTH_2, U181'_2, U182'_2, U191'_2, PAIR_2, U201'_4, U202'_4, U203'_4, U204'_2, U21'_3, U22'_2, U211'_2, U212'_2, U221'_3, U222'_3, FST_1, U31'_3, U32'_2, U41'_2, U42'_1, U51'_2, U52'_1, U61'_1, U71'_1, U81'_1, U91'_1, TAIL_1, TAKE_2, SEL_2, ISLNAT_1, ISNATURAL_1, ISPLNAT_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, c108_1, c109_1, c110_1, c111_1


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

(11) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   U101(mark(z0), z1) -> mark(U101(z0, z1))
   U101(ok(z0), ok(z1)) -> ok(U101(z0, z1))
   U102(mark(z0)) -> mark(U102(z0))
   U102(ok(z0)) -> ok(U102(z0))
   U11(mark(z0), z1, z2) -> mark(U11(z0, z1, z2))
   U11(ok(z0), ok(z1), ok(z2)) -> ok(U11(z0, z1, z2))
   U12(mark(z0), z1, z2) -> mark(U12(z0, z1, z2))
   U12(ok(z0), ok(z1), ok(z2)) -> ok(U12(z0, z1, z2))
   U111(mark(z0)) -> mark(U111(z0))
   U111(ok(z0)) -> ok(U111(z0))
   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))
   U121(mark(z0)) -> mark(U121(z0))
   U121(ok(z0)) -> ok(U121(z0))
   U131(mark(z0), z1) -> mark(U131(z0, z1))
   U131(ok(z0), ok(z1)) -> ok(U131(z0, z1))
   U132(mark(z0)) -> mark(U132(z0))
   U132(ok(z0)) -> ok(U132(z0))
   U141(mark(z0), z1) -> mark(U141(z0, z1))
   U141(ok(z0), ok(z1)) -> ok(U141(z0, z1))
   U142(mark(z0)) -> mark(U142(z0))
   U142(ok(z0)) -> ok(U142(z0))
   U151(mark(z0), z1) -> mark(U151(z0, z1))
   U151(ok(z0), ok(z1)) -> ok(U151(z0, z1))
   U152(mark(z0)) -> mark(U152(z0))
   U152(ok(z0)) -> ok(U152(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))
   U172(mark(z0), z1, z2) -> mark(U172(z0, z1, z2))
   U172(ok(z0), ok(z1), ok(z2)) -> ok(U172(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))
   U182(mark(z0), z1) -> mark(U182(z0, z1))
   U182(ok(z0), ok(z1)) -> ok(U182(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, z2, z3) -> mark(U202(z0, z1, z2, z3))
   U202(ok(z0), ok(z1), ok(z2), ok(z3)) -> ok(U202(z0, z1, z2, z3))
   U203(mark(z0), z1, z2, z3) -> mark(U203(z0, z1, z2, z3))
   U203(ok(z0), ok(z1), ok(z2), ok(z3)) -> ok(U203(z0, z1, z2, z3))
   U204(mark(z0), z1) -> mark(U204(z0, z1))
   U204(ok(z0), ok(z1)) -> ok(U204(z0, z1))
   U21(mark(z0), z1, z2) -> mark(U21(z0, z1, z2))
   U21(ok(z0), ok(z1), ok(z2)) -> ok(U21(z0, z1, z2))
   U22(mark(z0), z1) -> mark(U22(z0, z1))
   U22(ok(z0), ok(z1)) -> ok(U22(z0, z1))
   U211(mark(z0), z1) -> mark(U211(z0, z1))
   U211(ok(z0), ok(z1)) -> ok(U211(z0, z1))
   U212(mark(z0), z1) -> mark(U212(z0, z1))
   U212(ok(z0), ok(z1)) -> ok(U212(z0, z1))
   U221(mark(z0), z1, z2) -> mark(U221(z0, z1, z2))
   U221(ok(z0), ok(z1), ok(z2)) -> ok(U221(z0, z1, z2))
   U222(mark(z0), z1, z2) -> mark(U222(z0, z1, z2))
   U222(ok(z0), ok(z1), ok(z2)) -> ok(U222(z0, z1, z2))
   fst(mark(z0)) -> mark(fst(z0))
   fst(ok(z0)) -> ok(fst(z0))
   U31(mark(z0), z1, z2) -> mark(U31(z0, z1, z2))
   U31(ok(z0), ok(z1), ok(z2)) -> ok(U31(z0, z1, z2))
   U32(mark(z0), z1) -> mark(U32(z0, z1))
   U32(ok(z0), ok(z1)) -> ok(U32(z0, z1))
   U41(mark(z0), z1) -> mark(U41(z0, z1))
   U41(ok(z0), ok(z1)) -> ok(U41(z0, z1))
   U42(mark(z0)) -> mark(U42(z0))
   U42(ok(z0)) -> ok(U42(z0))
   U51(mark(z0), z1) -> mark(U51(z0, z1))
   U51(ok(z0), ok(z1)) -> ok(U51(z0, z1))
   U52(mark(z0)) -> mark(U52(z0))
   U52(ok(z0)) -> ok(U52(z0))
   U61(mark(z0)) -> mark(U61(z0))
   U61(ok(z0)) -> ok(U61(z0))
   U71(mark(z0)) -> mark(U71(z0))
   U71(ok(z0)) -> ok(U71(z0))
   U81(mark(z0)) -> mark(U81(z0))
   U81(ok(z0)) -> ok(U81(z0))
   U91(mark(z0)) -> mark(U91(z0))
   U91(ok(z0)) -> ok(U91(z0))
   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))
   isLNat(ok(z0)) -> ok(isLNat(z0))
   isNatural(ok(z0)) -> ok(isNatural(z0))
   isPLNat(ok(z0)) -> ok(isPLNat(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) -> c(U101'(z0, z1))
   U101'(ok(z0), ok(z1)) -> c1(U101'(z0, z1))
   U102'(mark(z0)) -> c2(U102'(z0))
   U102'(ok(z0)) -> c3(U102'(z0))
   U11'(mark(z0), z1, z2) -> c4(U11'(z0, z1, z2))
   U11'(ok(z0), ok(z1), ok(z2)) -> c5(U11'(z0, z1, z2))
   U12'(mark(z0), z1, z2) -> c6(U12'(z0, z1, z2))
   U12'(ok(z0), ok(z1), ok(z2)) -> c7(U12'(z0, z1, z2))
   U111'(mark(z0)) -> c8(U111'(z0))
   U111'(ok(z0)) -> c9(U111'(z0))
   SND(mark(z0)) -> c10(SND(z0))
   SND(ok(z0)) -> c11(SND(z0))
   SPLITAT(mark(z0), z1) -> c12(SPLITAT(z0, z1))
   SPLITAT(z0, mark(z1)) -> c13(SPLITAT(z0, z1))
   SPLITAT(ok(z0), ok(z1)) -> c14(SPLITAT(z0, z1))
   U121'(mark(z0)) -> c15(U121'(z0))
   U121'(ok(z0)) -> c16(U121'(z0))
   U131'(mark(z0), z1) -> c17(U131'(z0, z1))
   U131'(ok(z0), ok(z1)) -> c18(U131'(z0, z1))
   U132'(mark(z0)) -> c19(U132'(z0))
   U132'(ok(z0)) -> c20(U132'(z0))
   U141'(mark(z0), z1) -> c21(U141'(z0, z1))
   U141'(ok(z0), ok(z1)) -> c22(U141'(z0, z1))
   U142'(mark(z0)) -> c23(U142'(z0))
   U142'(ok(z0)) -> c24(U142'(z0))
   U151'(mark(z0), z1) -> c25(U151'(z0, z1))
   U151'(ok(z0), ok(z1)) -> c26(U151'(z0, z1))
   U152'(mark(z0)) -> c27(U152'(z0))
   U152'(ok(z0)) -> c28(U152'(z0))
   U161'(mark(z0), z1) -> c29(U161'(z0, z1))
   U161'(ok(z0), ok(z1)) -> c30(U161'(z0, z1))
   CONS(mark(z0), z1) -> c31(CONS(z0, z1))
   CONS(ok(z0), ok(z1)) -> c32(CONS(z0, z1))
   NATSFROM(mark(z0)) -> c33(NATSFROM(z0))
   NATSFROM(ok(z0)) -> c34(NATSFROM(z0))
   S(mark(z0)) -> c35(S(z0))
   S(ok(z0)) -> c36(S(z0))
   U171'(mark(z0), z1, z2) -> c37(U171'(z0, z1, z2))
   U171'(ok(z0), ok(z1), ok(z2)) -> c38(U171'(z0, z1, z2))
   U172'(mark(z0), z1, z2) -> c39(U172'(z0, z1, z2))
   U172'(ok(z0), ok(z1), ok(z2)) -> c40(U172'(z0, z1, z2))
   HEAD(mark(z0)) -> c41(HEAD(z0))
   HEAD(ok(z0)) -> c42(HEAD(z0))
   AFTERNTH(mark(z0), z1) -> c43(AFTERNTH(z0, z1))
   AFTERNTH(z0, mark(z1)) -> c44(AFTERNTH(z0, z1))
   AFTERNTH(ok(z0), ok(z1)) -> c45(AFTERNTH(z0, z1))
   U181'(mark(z0), z1) -> c46(U181'(z0, z1))
   U181'(ok(z0), ok(z1)) -> c47(U181'(z0, z1))
   U182'(mark(z0), z1) -> c48(U182'(z0, z1))
   U182'(ok(z0), ok(z1)) -> c49(U182'(z0, z1))
   U191'(mark(z0), z1) -> c50(U191'(z0, z1))
   U191'(ok(z0), ok(z1)) -> c51(U191'(z0, z1))
   PAIR(mark(z0), z1) -> c52(PAIR(z0, z1))
   PAIR(z0, mark(z1)) -> c53(PAIR(z0, z1))
   PAIR(ok(z0), ok(z1)) -> c54(PAIR(z0, z1))
   U201'(mark(z0), z1, z2, z3) -> c55(U201'(z0, z1, z2, z3))
   U201'(ok(z0), ok(z1), ok(z2), ok(z3)) -> c56(U201'(z0, z1, z2, z3))
   U202'(mark(z0), z1, z2, z3) -> c57(U202'(z0, z1, z2, z3))
   U202'(ok(z0), ok(z1), ok(z2), ok(z3)) -> c58(U202'(z0, z1, z2, z3))
   U203'(mark(z0), z1, z2, z3) -> c59(U203'(z0, z1, z2, z3))
   U203'(ok(z0), ok(z1), ok(z2), ok(z3)) -> c60(U203'(z0, z1, z2, z3))
   U204'(mark(z0), z1) -> c61(U204'(z0, z1))
   U204'(ok(z0), ok(z1)) -> c62(U204'(z0, z1))
   U21'(mark(z0), z1, z2) -> c63(U21'(z0, z1, z2))
   U21'(ok(z0), ok(z1), ok(z2)) -> c64(U21'(z0, z1, z2))
   U22'(mark(z0), z1) -> c65(U22'(z0, z1))
   U22'(ok(z0), ok(z1)) -> c66(U22'(z0, z1))
   U211'(mark(z0), z1) -> c67(U211'(z0, z1))
   U211'(ok(z0), ok(z1)) -> c68(U211'(z0, z1))
   U212'(mark(z0), z1) -> c69(U212'(z0, z1))
   U212'(ok(z0), ok(z1)) -> c70(U212'(z0, z1))
   U221'(mark(z0), z1, z2) -> c71(U221'(z0, z1, z2))
   U221'(ok(z0), ok(z1), ok(z2)) -> c72(U221'(z0, z1, z2))
   U222'(mark(z0), z1, z2) -> c73(U222'(z0, z1, z2))
   U222'(ok(z0), ok(z1), ok(z2)) -> c74(U222'(z0, z1, z2))
   FST(mark(z0)) -> c75(FST(z0))
   FST(ok(z0)) -> c76(FST(z0))
   U31'(mark(z0), z1, z2) -> c77(U31'(z0, z1, z2))
   U31'(ok(z0), ok(z1), ok(z2)) -> c78(U31'(z0, z1, z2))
   U32'(mark(z0), z1) -> c79(U32'(z0, z1))
   U32'(ok(z0), ok(z1)) -> c80(U32'(z0, z1))
   U41'(mark(z0), z1) -> c81(U41'(z0, z1))
   U41'(ok(z0), ok(z1)) -> c82(U41'(z0, z1))
   U42'(mark(z0)) -> c83(U42'(z0))
   U42'(ok(z0)) -> c84(U42'(z0))
   U51'(mark(z0), z1) -> c85(U51'(z0, z1))
   U51'(ok(z0), ok(z1)) -> c86(U51'(z0, z1))
   U52'(mark(z0)) -> c87(U52'(z0))
   U52'(ok(z0)) -> c88(U52'(z0))
   U61'(mark(z0)) -> c89(U61'(z0))
   U61'(ok(z0)) -> c90(U61'(z0))
   U71'(mark(z0)) -> c91(U71'(z0))
   U71'(ok(z0)) -> c92(U71'(z0))
   U81'(mark(z0)) -> c93(U81'(z0))
   U81'(ok(z0)) -> c94(U81'(z0))
   U91'(mark(z0)) -> c95(U91'(z0))
   U91'(ok(z0)) -> c96(U91'(z0))
   TAIL(mark(z0)) -> c97(TAIL(z0))
   TAIL(ok(z0)) -> c98(TAIL(z0))
   TAKE(mark(z0), z1) -> c99(TAKE(z0, z1))
   TAKE(z0, mark(z1)) -> c100(TAKE(z0, z1))
   TAKE(ok(z0), ok(z1)) -> c101(TAKE(z0, z1))
   SEL(mark(z0), z1) -> c102(SEL(z0, z1))
   SEL(z0, mark(z1)) -> c103(SEL(z0, z1))
   SEL(ok(z0), ok(z1)) -> c104(SEL(z0, z1))
   ISLNAT(ok(z0)) -> c108(ISLNAT(z0))
   ISNATURAL(ok(z0)) -> c109(ISNATURAL(z0))
   ISPLNAT(ok(z0)) -> c110(ISPLNAT(z0))
   TOP(mark(z0)) -> c111(TOP(proper(z0)))
S tuples:
   U101'(mark(z0), z1) -> c(U101'(z0, z1))
   U101'(ok(z0), ok(z1)) -> c1(U101'(z0, z1))
   U102'(mark(z0)) -> c2(U102'(z0))
   U102'(ok(z0)) -> c3(U102'(z0))
   U11'(mark(z0), z1, z2) -> c4(U11'(z0, z1, z2))
   U11'(ok(z0), ok(z1), ok(z2)) -> c5(U11'(z0, z1, z2))
   U12'(mark(z0), z1, z2) -> c6(U12'(z0, z1, z2))
   U12'(ok(z0), ok(z1), ok(z2)) -> c7(U12'(z0, z1, z2))
   U111'(mark(z0)) -> c8(U111'(z0))
   U111'(ok(z0)) -> c9(U111'(z0))
   SND(mark(z0)) -> c10(SND(z0))
   SND(ok(z0)) -> c11(SND(z0))
   SPLITAT(mark(z0), z1) -> c12(SPLITAT(z0, z1))
   SPLITAT(z0, mark(z1)) -> c13(SPLITAT(z0, z1))
   SPLITAT(ok(z0), ok(z1)) -> c14(SPLITAT(z0, z1))
   U121'(mark(z0)) -> c15(U121'(z0))
   U121'(ok(z0)) -> c16(U121'(z0))
   U131'(mark(z0), z1) -> c17(U131'(z0, z1))
   U131'(ok(z0), ok(z1)) -> c18(U131'(z0, z1))
   U132'(mark(z0)) -> c19(U132'(z0))
   U132'(ok(z0)) -> c20(U132'(z0))
   U141'(mark(z0), z1) -> c21(U141'(z0, z1))
   U141'(ok(z0), ok(z1)) -> c22(U141'(z0, z1))
   U142'(mark(z0)) -> c23(U142'(z0))
   U142'(ok(z0)) -> c24(U142'(z0))
   U151'(mark(z0), z1) -> c25(U151'(z0, z1))
   U151'(ok(z0), ok(z1)) -> c26(U151'(z0, z1))
   U152'(mark(z0)) -> c27(U152'(z0))
   U152'(ok(z0)) -> c28(U152'(z0))
   U161'(mark(z0), z1) -> c29(U161'(z0, z1))
   U161'(ok(z0), ok(z1)) -> c30(U161'(z0, z1))
   CONS(mark(z0), z1) -> c31(CONS(z0, z1))
   CONS(ok(z0), ok(z1)) -> c32(CONS(z0, z1))
   NATSFROM(mark(z0)) -> c33(NATSFROM(z0))
   NATSFROM(ok(z0)) -> c34(NATSFROM(z0))
   S(mark(z0)) -> c35(S(z0))
   S(ok(z0)) -> c36(S(z0))
   U171'(mark(z0), z1, z2) -> c37(U171'(z0, z1, z2))
   U171'(ok(z0), ok(z1), ok(z2)) -> c38(U171'(z0, z1, z2))
   U172'(mark(z0), z1, z2) -> c39(U172'(z0, z1, z2))
   U172'(ok(z0), ok(z1), ok(z2)) -> c40(U172'(z0, z1, z2))
   HEAD(mark(z0)) -> c41(HEAD(z0))
   HEAD(ok(z0)) -> c42(HEAD(z0))
   AFTERNTH(mark(z0), z1) -> c43(AFTERNTH(z0, z1))
   AFTERNTH(z0, mark(z1)) -> c44(AFTERNTH(z0, z1))
   AFTERNTH(ok(z0), ok(z1)) -> c45(AFTERNTH(z0, z1))
   U181'(mark(z0), z1) -> c46(U181'(z0, z1))
   U181'(ok(z0), ok(z1)) -> c47(U181'(z0, z1))
   U182'(mark(z0), z1) -> c48(U182'(z0, z1))
   U182'(ok(z0), ok(z1)) -> c49(U182'(z0, z1))
   U191'(mark(z0), z1) -> c50(U191'(z0, z1))
   U191'(ok(z0), ok(z1)) -> c51(U191'(z0, z1))
   PAIR(mark(z0), z1) -> c52(PAIR(z0, z1))
   PAIR(z0, mark(z1)) -> c53(PAIR(z0, z1))
   PAIR(ok(z0), ok(z1)) -> c54(PAIR(z0, z1))
   U201'(mark(z0), z1, z2, z3) -> c55(U201'(z0, z1, z2, z3))
   U201'(ok(z0), ok(z1), ok(z2), ok(z3)) -> c56(U201'(z0, z1, z2, z3))
   U202'(mark(z0), z1, z2, z3) -> c57(U202'(z0, z1, z2, z3))
   U202'(ok(z0), ok(z1), ok(z2), ok(z3)) -> c58(U202'(z0, z1, z2, z3))
   U203'(mark(z0), z1, z2, z3) -> c59(U203'(z0, z1, z2, z3))
   U203'(ok(z0), ok(z1), ok(z2), ok(z3)) -> c60(U203'(z0, z1, z2, z3))
   U204'(mark(z0), z1) -> c61(U204'(z0, z1))
   U204'(ok(z0), ok(z1)) -> c62(U204'(z0, z1))
   U21'(mark(z0), z1, z2) -> c63(U21'(z0, z1, z2))
   U21'(ok(z0), ok(z1), ok(z2)) -> c64(U21'(z0, z1, z2))
   U22'(mark(z0), z1) -> c65(U22'(z0, z1))
   U22'(ok(z0), ok(z1)) -> c66(U22'(z0, z1))
   U211'(mark(z0), z1) -> c67(U211'(z0, z1))
   U211'(ok(z0), ok(z1)) -> c68(U211'(z0, z1))
   U212'(mark(z0), z1) -> c69(U212'(z0, z1))
   U212'(ok(z0), ok(z1)) -> c70(U212'(z0, z1))
   U221'(mark(z0), z1, z2) -> c71(U221'(z0, z1, z2))
   U221'(ok(z0), ok(z1), ok(z2)) -> c72(U221'(z0, z1, z2))
   U222'(mark(z0), z1, z2) -> c73(U222'(z0, z1, z2))
   U222'(ok(z0), ok(z1), ok(z2)) -> c74(U222'(z0, z1, z2))
   FST(mark(z0)) -> c75(FST(z0))
   FST(ok(z0)) -> c76(FST(z0))
   U31'(mark(z0), z1, z2) -> c77(U31'(z0, z1, z2))
   U31'(ok(z0), ok(z1), ok(z2)) -> c78(U31'(z0, z1, z2))
   U32'(mark(z0), z1) -> c79(U32'(z0, z1))
   U32'(ok(z0), ok(z1)) -> c80(U32'(z0, z1))
   U41'(mark(z0), z1) -> c81(U41'(z0, z1))
   U41'(ok(z0), ok(z1)) -> c82(U41'(z0, z1))
   U42'(mark(z0)) -> c83(U42'(z0))
   U42'(ok(z0)) -> c84(U42'(z0))
   U51'(mark(z0), z1) -> c85(U51'(z0, z1))
   U51'(ok(z0), ok(z1)) -> c86(U51'(z0, z1))
   U52'(mark(z0)) -> c87(U52'(z0))
   U52'(ok(z0)) -> c88(U52'(z0))
   U61'(mark(z0)) -> c89(U61'(z0))
   U61'(ok(z0)) -> c90(U61'(z0))
   U71'(mark(z0)) -> c91(U71'(z0))
   U71'(ok(z0)) -> c92(U71'(z0))
   U81'(mark(z0)) -> c93(U81'(z0))
   U81'(ok(z0)) -> c94(U81'(z0))
   U91'(mark(z0)) -> c95(U91'(z0))
   U91'(ok(z0)) -> c96(U91'(z0))
   TAIL(mark(z0)) -> c97(TAIL(z0))
   TAIL(ok(z0)) -> c98(TAIL(z0))
   TAKE(mark(z0), z1) -> c99(TAKE(z0, z1))
   TAKE(z0, mark(z1)) -> c100(TAKE(z0, z1))
   TAKE(ok(z0), ok(z1)) -> c101(TAKE(z0, z1))
   SEL(mark(z0), z1) -> c102(SEL(z0, z1))
   SEL(z0, mark(z1)) -> c103(SEL(z0, z1))
   SEL(ok(z0), ok(z1)) -> c104(SEL(z0, z1))
   ISLNAT(ok(z0)) -> c108(ISLNAT(z0))
   ISNATURAL(ok(z0)) -> c109(ISNATURAL(z0))
   ISPLNAT(ok(z0)) -> c110(ISPLNAT(z0))
   TOP(mark(z0)) -> c111(TOP(proper(z0)))
K tuples:none
Defined Rule Symbols:   proper_1

Defined Pair Symbols:   U101'_2, U102'_1, U11'_3, U12'_3, U111'_1, SND_1, SPLITAT_2, U121'_1, U131'_2, U132'_1, U141'_2, U142'_1, U151'_2, U152'_1, U161'_2, CONS_2, NATSFROM_1, S_1, U171'_3, U172'_3, HEAD_1, AFTERNTH_2, U181'_2, U182'_2, U191'_2, PAIR_2, U201'_4, U202'_4, U203'_4, U204'_2, U21'_3, U22'_2, U211'_2, U212'_2, U221'_3, U222'_3, FST_1, U31'_3, U32'_2, U41'_2, U42'_1, U51'_2, U52'_1, U61'_1, U71'_1, U81'_1, U91'_1, TAIL_1, TAKE_2, SEL_2, ISLNAT_1, ISNATURAL_1, ISPLNAT_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, c108_1, c109_1, c110_1, c111_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) -> c(U101'(z0, z1))
   U101'(ok(z0), ok(z1)) -> c1(U101'(z0, z1))
   U102'(mark(z0)) -> c2(U102'(z0))
   U102'(ok(z0)) -> c3(U102'(z0))
   U11'(mark(z0), z1, z2) -> c4(U11'(z0, z1, z2))
   U11'(ok(z0), ok(z1), ok(z2)) -> c5(U11'(z0, z1, z2))
   U12'(mark(z0), z1, z2) -> c6(U12'(z0, z1, z2))
   U12'(ok(z0), ok(z1), ok(z2)) -> c7(U12'(z0, z1, z2))
   U111'(mark(z0)) -> c8(U111'(z0))
   U111'(ok(z0)) -> c9(U111'(z0))
   SND(mark(z0)) -> c10(SND(z0))
   SND(ok(z0)) -> c11(SND(z0))
   SPLITAT(mark(z0), z1) -> c12(SPLITAT(z0, z1))
   SPLITAT(z0, mark(z1)) -> c13(SPLITAT(z0, z1))
   SPLITAT(ok(z0), ok(z1)) -> c14(SPLITAT(z0, z1))
   U121'(mark(z0)) -> c15(U121'(z0))
   U121'(ok(z0)) -> c16(U121'(z0))
   U131'(mark(z0), z1) -> c17(U131'(z0, z1))
   U131'(ok(z0), ok(z1)) -> c18(U131'(z0, z1))
   U132'(mark(z0)) -> c19(U132'(z0))
   U132'(ok(z0)) -> c20(U132'(z0))
   U141'(mark(z0), z1) -> c21(U141'(z0, z1))
   U141'(ok(z0), ok(z1)) -> c22(U141'(z0, z1))
   U142'(mark(z0)) -> c23(U142'(z0))
   U142'(ok(z0)) -> c24(U142'(z0))
   U151'(mark(z0), z1) -> c25(U151'(z0, z1))
   U151'(ok(z0), ok(z1)) -> c26(U151'(z0, z1))
   U152'(mark(z0)) -> c27(U152'(z0))
   U152'(ok(z0)) -> c28(U152'(z0))
   U161'(mark(z0), z1) -> c29(U161'(z0, z1))
   U161'(ok(z0), ok(z1)) -> c30(U161'(z0, z1))
   CONS(mark(z0), z1) -> c31(CONS(z0, z1))
   CONS(ok(z0), ok(z1)) -> c32(CONS(z0, z1))
   NATSFROM(mark(z0)) -> c33(NATSFROM(z0))
   NATSFROM(ok(z0)) -> c34(NATSFROM(z0))
   S(mark(z0)) -> c35(S(z0))
   S(ok(z0)) -> c36(S(z0))
   U171'(mark(z0), z1, z2) -> c37(U171'(z0, z1, z2))
   U171'(ok(z0), ok(z1), ok(z2)) -> c38(U171'(z0, z1, z2))
   U172'(mark(z0), z1, z2) -> c39(U172'(z0, z1, z2))
   U172'(ok(z0), ok(z1), ok(z2)) -> c40(U172'(z0, z1, z2))
   HEAD(mark(z0)) -> c41(HEAD(z0))
   HEAD(ok(z0)) -> c42(HEAD(z0))
   AFTERNTH(mark(z0), z1) -> c43(AFTERNTH(z0, z1))
   AFTERNTH(z0, mark(z1)) -> c44(AFTERNTH(z0, z1))
   AFTERNTH(ok(z0), ok(z1)) -> c45(AFTERNTH(z0, z1))
   U181'(mark(z0), z1) -> c46(U181'(z0, z1))
   U181'(ok(z0), ok(z1)) -> c47(U181'(z0, z1))
   U182'(mark(z0), z1) -> c48(U182'(z0, z1))
   U182'(ok(z0), ok(z1)) -> c49(U182'(z0, z1))
   U191'(mark(z0), z1) -> c50(U191'(z0, z1))
   U191'(ok(z0), ok(z1)) -> c51(U191'(z0, z1))
   PAIR(mark(z0), z1) -> c52(PAIR(z0, z1))
   PAIR(z0, mark(z1)) -> c53(PAIR(z0, z1))
   PAIR(ok(z0), ok(z1)) -> c54(PAIR(z0, z1))
   U201'(mark(z0), z1, z2, z3) -> c55(U201'(z0, z1, z2, z3))
   U201'(ok(z0), ok(z1), ok(z2), ok(z3)) -> c56(U201'(z0, z1, z2, z3))
   U202'(mark(z0), z1, z2, z3) -> c57(U202'(z0, z1, z2, z3))
   U202'(ok(z0), ok(z1), ok(z2), ok(z3)) -> c58(U202'(z0, z1, z2, z3))
   U203'(mark(z0), z1, z2, z3) -> c59(U203'(z0, z1, z2, z3))
   U203'(ok(z0), ok(z1), ok(z2), ok(z3)) -> c60(U203'(z0, z1, z2, z3))
   U204'(mark(z0), z1) -> c61(U204'(z0, z1))
   U204'(ok(z0), ok(z1)) -> c62(U204'(z0, z1))
   U21'(mark(z0), z1, z2) -> c63(U21'(z0, z1, z2))
   U21'(ok(z0), ok(z1), ok(z2)) -> c64(U21'(z0, z1, z2))
   U22'(mark(z0), z1) -> c65(U22'(z0, z1))
   U22'(ok(z0), ok(z1)) -> c66(U22'(z0, z1))
   U211'(mark(z0), z1) -> c67(U211'(z0, z1))
   U211'(ok(z0), ok(z1)) -> c68(U211'(z0, z1))
   U212'(mark(z0), z1) -> c69(U212'(z0, z1))
   U212'(ok(z0), ok(z1)) -> c70(U212'(z0, z1))
   U221'(mark(z0), z1, z2) -> c71(U221'(z0, z1, z2))
   U221'(ok(z0), ok(z1), ok(z2)) -> c72(U221'(z0, z1, z2))
   U222'(mark(z0), z1, z2) -> c73(U222'(z0, z1, z2))
   U222'(ok(z0), ok(z1), ok(z2)) -> c74(U222'(z0, z1, z2))
   FST(mark(z0)) -> c75(FST(z0))
   FST(ok(z0)) -> c76(FST(z0))
   U31'(mark(z0), z1, z2) -> c77(U31'(z0, z1, z2))
   U31'(ok(z0), ok(z1), ok(z2)) -> c78(U31'(z0, z1, z2))
   U32'(mark(z0), z1) -> c79(U32'(z0, z1))
   U32'(ok(z0), ok(z1)) -> c80(U32'(z0, z1))
   U41'(mark(z0), z1) -> c81(U41'(z0, z1))
   U41'(ok(z0), ok(z1)) -> c82(U41'(z0, z1))
   U42'(mark(z0)) -> c83(U42'(z0))
   U42'(ok(z0)) -> c84(U42'(z0))
   U51'(mark(z0), z1) -> c85(U51'(z0, z1))
   U51'(ok(z0), ok(z1)) -> c86(U51'(z0, z1))
   U52'(mark(z0)) -> c87(U52'(z0))
   U52'(ok(z0)) -> c88(U52'(z0))
   U61'(mark(z0)) -> c89(U61'(z0))
   U61'(ok(z0)) -> c90(U61'(z0))
   U71'(mark(z0)) -> c91(U71'(z0))
   U71'(ok(z0)) -> c92(U71'(z0))
   U81'(mark(z0)) -> c93(U81'(z0))
   U81'(ok(z0)) -> c94(U81'(z0))
   U91'(mark(z0)) -> c95(U91'(z0))
   U91'(ok(z0)) -> c96(U91'(z0))
   TAIL(mark(z0)) -> c97(TAIL(z0))
   TAIL(ok(z0)) -> c98(TAIL(z0))
   TAKE(mark(z0), z1) -> c99(TAKE(z0, z1))
   TAKE(z0, mark(z1)) -> c100(TAKE(z0, z1))
   TAKE(ok(z0), ok(z1)) -> c101(TAKE(z0, z1))
   SEL(mark(z0), z1) -> c102(SEL(z0, z1))
   SEL(z0, mark(z1)) -> c103(SEL(z0, z1))
   SEL(ok(z0), ok(z1)) -> c104(SEL(z0, z1))
   ISLNAT(ok(z0)) -> c108(ISLNAT(z0))
   ISNATURAL(ok(z0)) -> c109(ISNATURAL(z0))
   ISPLNAT(ok(z0)) -> c110(ISPLNAT(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) -> c(U101'(z0, z1))
   U101'(ok(z0), ok(z1)) -> c1(U101'(z0, z1))
   U102'(mark(z0)) -> c2(U102'(z0))
   U102'(ok(z0)) -> c3(U102'(z0))
   U11'(mark(z0), z1, z2) -> c4(U11'(z0, z1, z2))
   U11'(ok(z0), ok(z1), ok(z2)) -> c5(U11'(z0, z1, z2))
   U12'(mark(z0), z1, z2) -> c6(U12'(z0, z1, z2))
   U12'(ok(z0), ok(z1), ok(z2)) -> c7(U12'(z0, z1, z2))
   U111'(mark(z0)) -> c8(U111'(z0))
   U111'(ok(z0)) -> c9(U111'(z0))
   SND(mark(z0)) -> c10(SND(z0))
   SND(ok(z0)) -> c11(SND(z0))
   SPLITAT(mark(z0), z1) -> c12(SPLITAT(z0, z1))
   SPLITAT(z0, mark(z1)) -> c13(SPLITAT(z0, z1))
   SPLITAT(ok(z0), ok(z1)) -> c14(SPLITAT(z0, z1))
   U121'(mark(z0)) -> c15(U121'(z0))
   U121'(ok(z0)) -> c16(U121'(z0))
   U131'(mark(z0), z1) -> c17(U131'(z0, z1))
   U131'(ok(z0), ok(z1)) -> c18(U131'(z0, z1))
   U132'(mark(z0)) -> c19(U132'(z0))
   U132'(ok(z0)) -> c20(U132'(z0))
   U141'(mark(z0), z1) -> c21(U141'(z0, z1))
   U141'(ok(z0), ok(z1)) -> c22(U141'(z0, z1))
   U142'(mark(z0)) -> c23(U142'(z0))
   U142'(ok(z0)) -> c24(U142'(z0))
   U151'(mark(z0), z1) -> c25(U151'(z0, z1))
   U151'(ok(z0), ok(z1)) -> c26(U151'(z0, z1))
   U152'(mark(z0)) -> c27(U152'(z0))
   U152'(ok(z0)) -> c28(U152'(z0))
   U161'(mark(z0), z1) -> c29(U161'(z0, z1))
   U161'(ok(z0), ok(z1)) -> c30(U161'(z0, z1))
   CONS(mark(z0), z1) -> c31(CONS(z0, z1))
   CONS(ok(z0), ok(z1)) -> c32(CONS(z0, z1))
   NATSFROM(mark(z0)) -> c33(NATSFROM(z0))
   NATSFROM(ok(z0)) -> c34(NATSFROM(z0))
   S(mark(z0)) -> c35(S(z0))
   S(ok(z0)) -> c36(S(z0))
   U171'(mark(z0), z1, z2) -> c37(U171'(z0, z1, z2))
   U171'(ok(z0), ok(z1), ok(z2)) -> c38(U171'(z0, z1, z2))
   U172'(mark(z0), z1, z2) -> c39(U172'(z0, z1, z2))
   U172'(ok(z0), ok(z1), ok(z2)) -> c40(U172'(z0, z1, z2))
   HEAD(mark(z0)) -> c41(HEAD(z0))
   HEAD(ok(z0)) -> c42(HEAD(z0))
   AFTERNTH(mark(z0), z1) -> c43(AFTERNTH(z0, z1))
   AFTERNTH(z0, mark(z1)) -> c44(AFTERNTH(z0, z1))
   AFTERNTH(ok(z0), ok(z1)) -> c45(AFTERNTH(z0, z1))
   U181'(mark(z0), z1) -> c46(U181'(z0, z1))
   U181'(ok(z0), ok(z1)) -> c47(U181'(z0, z1))
   U182'(mark(z0), z1) -> c48(U182'(z0, z1))
   U182'(ok(z0), ok(z1)) -> c49(U182'(z0, z1))
   U191'(mark(z0), z1) -> c50(U191'(z0, z1))
   U191'(ok(z0), ok(z1)) -> c51(U191'(z0, z1))
   PAIR(mark(z0), z1) -> c52(PAIR(z0, z1))
   PAIR(z0, mark(z1)) -> c53(PAIR(z0, z1))
   PAIR(ok(z0), ok(z1)) -> c54(PAIR(z0, z1))
   U201'(mark(z0), z1, z2, z3) -> c55(U201'(z0, z1, z2, z3))
   U201'(ok(z0), ok(z1), ok(z2), ok(z3)) -> c56(U201'(z0, z1, z2, z3))
   U202'(mark(z0), z1, z2, z3) -> c57(U202'(z0, z1, z2, z3))
   U202'(ok(z0), ok(z1), ok(z2), ok(z3)) -> c58(U202'(z0, z1, z2, z3))
   U203'(mark(z0), z1, z2, z3) -> c59(U203'(z0, z1, z2, z3))
   U203'(ok(z0), ok(z1), ok(z2), ok(z3)) -> c60(U203'(z0, z1, z2, z3))
   U204'(mark(z0), z1) -> c61(U204'(z0, z1))
   U204'(ok(z0), ok(z1)) -> c62(U204'(z0, z1))
   U21'(mark(z0), z1, z2) -> c63(U21'(z0, z1, z2))
   U21'(ok(z0), ok(z1), ok(z2)) -> c64(U21'(z0, z1, z2))
   U22'(mark(z0), z1) -> c65(U22'(z0, z1))
   U22'(ok(z0), ok(z1)) -> c66(U22'(z0, z1))
   U211'(mark(z0), z1) -> c67(U211'(z0, z1))
   U211'(ok(z0), ok(z1)) -> c68(U211'(z0, z1))
   U212'(mark(z0), z1) -> c69(U212'(z0, z1))
   U212'(ok(z0), ok(z1)) -> c70(U212'(z0, z1))
   U221'(mark(z0), z1, z2) -> c71(U221'(z0, z1, z2))
   U221'(ok(z0), ok(z1), ok(z2)) -> c72(U221'(z0, z1, z2))
   U222'(mark(z0), z1, z2) -> c73(U222'(z0, z1, z2))
   U222'(ok(z0), ok(z1), ok(z2)) -> c74(U222'(z0, z1, z2))
   FST(mark(z0)) -> c75(FST(z0))
   FST(ok(z0)) -> c76(FST(z0))
   U31'(mark(z0), z1, z2) -> c77(U31'(z0, z1, z2))
   U31'(ok(z0), ok(z1), ok(z2)) -> c78(U31'(z0, z1, z2))
   U32'(mark(z0), z1) -> c79(U32'(z0, z1))
   U32'(ok(z0), ok(z1)) -> c80(U32'(z0, z1))
   U41'(mark(z0), z1) -> c81(U41'(z0, z1))
   U41'(ok(z0), ok(z1)) -> c82(U41'(z0, z1))
   U42'(mark(z0)) -> c83(U42'(z0))
   U42'(ok(z0)) -> c84(U42'(z0))
   U51'(mark(z0), z1) -> c85(U51'(z0, z1))
   U51'(ok(z0), ok(z1)) -> c86(U51'(z0, z1))
   U52'(mark(z0)) -> c87(U52'(z0))
   U52'(ok(z0)) -> c88(U52'(z0))
   U61'(mark(z0)) -> c89(U61'(z0))
   U61'(ok(z0)) -> c90(U61'(z0))
   U71'(mark(z0)) -> c91(U71'(z0))
   U71'(ok(z0)) -> c92(U71'(z0))
   U81'(mark(z0)) -> c93(U81'(z0))
   U81'(ok(z0)) -> c94(U81'(z0))
   U91'(mark(z0)) -> c95(U91'(z0))
   U91'(ok(z0)) -> c96(U91'(z0))
   TAIL(mark(z0)) -> c97(TAIL(z0))
   TAIL(ok(z0)) -> c98(TAIL(z0))
   TAKE(mark(z0), z1) -> c99(TAKE(z0, z1))
   TAKE(z0, mark(z1)) -> c100(TAKE(z0, z1))
   TAKE(ok(z0), ok(z1)) -> c101(TAKE(z0, z1))
   SEL(mark(z0), z1) -> c102(SEL(z0, z1))
   SEL(z0, mark(z1)) -> c103(SEL(z0, z1))
   SEL(ok(z0), ok(z1)) -> c104(SEL(z0, z1))
   ISLNAT(ok(z0)) -> c108(ISLNAT(z0))
   ISNATURAL(ok(z0)) -> c109(ISNATURAL(z0))
   ISPLNAT(ok(z0)) -> c110(ISPLNAT(z0))
   TOP(mark(z0)) -> c111(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(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(ISNATURAL(x_1)) = x_1
   POL(ISPLNAT(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_1 + x_2
   POL(U102'(x_1)) = x_1
   POL(U11'(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(U111'(x_1)) = x_1
   POL(U12'(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(U121'(x_1)) = x_1
   POL(U131'(x_1, x_2)) = x_1 + x_2
   POL(U132'(x_1)) = x_1
   POL(U141'(x_1, x_2)) = x_1 + x_2
   POL(U142'(x_1)) = x_1
   POL(U151'(x_1, x_2)) = x_1 + x_2
   POL(U152'(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(U172'(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(U181'(x_1, x_2)) = x_1 + x_2
   POL(U182'(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_3, x_4)) = x_1 + x_2 + x_3 + x_4
   POL(U203'(x_1, x_2, x_3, x_4)) = x_1 + x_2 + x_3 + x_4
   POL(U204'(x_1, x_2)) = x_1 + x_2
   POL(U21'(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(U211'(x_1, x_2)) = x_1 + x_2
   POL(U212'(x_1, x_2)) = x_1 + x_2
   POL(U22'(x_1, x_2)) = x_1 + x_2
   POL(U221'(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(U222'(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(U31'(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(U32'(x_1, x_2)) = x_1 + x_2
   POL(U41'(x_1, x_2)) = x_1 + x_2
   POL(U42'(x_1)) = x_1
   POL(U51'(x_1, x_2)) = x_1 + x_2
   POL(U52'(x_1)) = x_1
   POL(U61'(x_1)) = x_1
   POL(U71'(x_1)) = x_1
   POL(U81'(x_1)) = x_1
   POL(U91'(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(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(c12(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) -> c(U101'(z0, z1))
   U101'(ok(z0), ok(z1)) -> c1(U101'(z0, z1))
   U102'(mark(z0)) -> c2(U102'(z0))
   U102'(ok(z0)) -> c3(U102'(z0))
   U11'(mark(z0), z1, z2) -> c4(U11'(z0, z1, z2))
   U11'(ok(z0), ok(z1), ok(z2)) -> c5(U11'(z0, z1, z2))
   U12'(mark(z0), z1, z2) -> c6(U12'(z0, z1, z2))
   U12'(ok(z0), ok(z1), ok(z2)) -> c7(U12'(z0, z1, z2))
   U111'(mark(z0)) -> c8(U111'(z0))
   U111'(ok(z0)) -> c9(U111'(z0))
   SND(mark(z0)) -> c10(SND(z0))
   SND(ok(z0)) -> c11(SND(z0))
   SPLITAT(mark(z0), z1) -> c12(SPLITAT(z0, z1))
   SPLITAT(z0, mark(z1)) -> c13(SPLITAT(z0, z1))
   SPLITAT(ok(z0), ok(z1)) -> c14(SPLITAT(z0, z1))
   U121'(mark(z0)) -> c15(U121'(z0))
   U121'(ok(z0)) -> c16(U121'(z0))
   U131'(mark(z0), z1) -> c17(U131'(z0, z1))
   U131'(ok(z0), ok(z1)) -> c18(U131'(z0, z1))
   U132'(mark(z0)) -> c19(U132'(z0))
   U132'(ok(z0)) -> c20(U132'(z0))
   U141'(mark(z0), z1) -> c21(U141'(z0, z1))
   U141'(ok(z0), ok(z1)) -> c22(U141'(z0, z1))
   U142'(mark(z0)) -> c23(U142'(z0))
   U142'(ok(z0)) -> c24(U142'(z0))
   U151'(mark(z0), z1) -> c25(U151'(z0, z1))
   U151'(ok(z0), ok(z1)) -> c26(U151'(z0, z1))
   U152'(mark(z0)) -> c27(U152'(z0))
   U152'(ok(z0)) -> c28(U152'(z0))
   U161'(mark(z0), z1) -> c29(U161'(z0, z1))
   U161'(ok(z0), ok(z1)) -> c30(U161'(z0, z1))
   CONS(mark(z0), z1) -> c31(CONS(z0, z1))
   CONS(ok(z0), ok(z1)) -> c32(CONS(z0, z1))
   NATSFROM(mark(z0)) -> c33(NATSFROM(z0))
   NATSFROM(ok(z0)) -> c34(NATSFROM(z0))
   S(mark(z0)) -> c35(S(z0))
   S(ok(z0)) -> c36(S(z0))
   U171'(mark(z0), z1, z2) -> c37(U171'(z0, z1, z2))
   U171'(ok(z0), ok(z1), ok(z2)) -> c38(U171'(z0, z1, z2))
   U172'(mark(z0), z1, z2) -> c39(U172'(z0, z1, z2))
   U172'(ok(z0), ok(z1), ok(z2)) -> c40(U172'(z0, z1, z2))
   HEAD(mark(z0)) -> c41(HEAD(z0))
   HEAD(ok(z0)) -> c42(HEAD(z0))
   AFTERNTH(mark(z0), z1) -> c43(AFTERNTH(z0, z1))
   AFTERNTH(z0, mark(z1)) -> c44(AFTERNTH(z0, z1))
   AFTERNTH(ok(z0), ok(z1)) -> c45(AFTERNTH(z0, z1))
   U181'(mark(z0), z1) -> c46(U181'(z0, z1))
   U181'(ok(z0), ok(z1)) -> c47(U181'(z0, z1))
   U182'(mark(z0), z1) -> c48(U182'(z0, z1))
   U182'(ok(z0), ok(z1)) -> c49(U182'(z0, z1))
   U191'(mark(z0), z1) -> c50(U191'(z0, z1))
   U191'(ok(z0), ok(z1)) -> c51(U191'(z0, z1))
   PAIR(mark(z0), z1) -> c52(PAIR(z0, z1))
   PAIR(z0, mark(z1)) -> c53(PAIR(z0, z1))
   PAIR(ok(z0), ok(z1)) -> c54(PAIR(z0, z1))
   U201'(mark(z0), z1, z2, z3) -> c55(U201'(z0, z1, z2, z3))
   U201'(ok(z0), ok(z1), ok(z2), ok(z3)) -> c56(U201'(z0, z1, z2, z3))
   U202'(mark(z0), z1, z2, z3) -> c57(U202'(z0, z1, z2, z3))
   U202'(ok(z0), ok(z1), ok(z2), ok(z3)) -> c58(U202'(z0, z1, z2, z3))
   U203'(mark(z0), z1, z2, z3) -> c59(U203'(z0, z1, z2, z3))
   U203'(ok(z0), ok(z1), ok(z2), ok(z3)) -> c60(U203'(z0, z1, z2, z3))
   U204'(mark(z0), z1) -> c61(U204'(z0, z1))
   U204'(ok(z0), ok(z1)) -> c62(U204'(z0, z1))
   U21'(mark(z0), z1, z2) -> c63(U21'(z0, z1, z2))
   U21'(ok(z0), ok(z1), ok(z2)) -> c64(U21'(z0, z1, z2))
   U22'(mark(z0), z1) -> c65(U22'(z0, z1))
   U22'(ok(z0), ok(z1)) -> c66(U22'(z0, z1))
   U211'(mark(z0), z1) -> c67(U211'(z0, z1))
   U211'(ok(z0), ok(z1)) -> c68(U211'(z0, z1))
   U212'(mark(z0), z1) -> c69(U212'(z0, z1))
   U212'(ok(z0), ok(z1)) -> c70(U212'(z0, z1))
   U221'(mark(z0), z1, z2) -> c71(U221'(z0, z1, z2))
   U221'(ok(z0), ok(z1), ok(z2)) -> c72(U221'(z0, z1, z2))
   U222'(mark(z0), z1, z2) -> c73(U222'(z0, z1, z2))
   U222'(ok(z0), ok(z1), ok(z2)) -> c74(U222'(z0, z1, z2))
   FST(mark(z0)) -> c75(FST(z0))
   FST(ok(z0)) -> c76(FST(z0))
   U31'(mark(z0), z1, z2) -> c77(U31'(z0, z1, z2))
   U31'(ok(z0), ok(z1), ok(z2)) -> c78(U31'(z0, z1, z2))
   U32'(mark(z0), z1) -> c79(U32'(z0, z1))
   U32'(ok(z0), ok(z1)) -> c80(U32'(z0, z1))
   U41'(mark(z0), z1) -> c81(U41'(z0, z1))
   U41'(ok(z0), ok(z1)) -> c82(U41'(z0, z1))
   U42'(mark(z0)) -> c83(U42'(z0))
   U42'(ok(z0)) -> c84(U42'(z0))
   U51'(mark(z0), z1) -> c85(U51'(z0, z1))
   U51'(ok(z0), ok(z1)) -> c86(U51'(z0, z1))
   U52'(mark(z0)) -> c87(U52'(z0))
   U52'(ok(z0)) -> c88(U52'(z0))
   U61'(mark(z0)) -> c89(U61'(z0))
   U61'(ok(z0)) -> c90(U61'(z0))
   U71'(mark(z0)) -> c91(U71'(z0))
   U71'(ok(z0)) -> c92(U71'(z0))
   U81'(mark(z0)) -> c93(U81'(z0))
   U81'(ok(z0)) -> c94(U81'(z0))
   U91'(mark(z0)) -> c95(U91'(z0))
   U91'(ok(z0)) -> c96(U91'(z0))
   TAIL(mark(z0)) -> c97(TAIL(z0))
   TAIL(ok(z0)) -> c98(TAIL(z0))
   TAKE(mark(z0), z1) -> c99(TAKE(z0, z1))
   TAKE(z0, mark(z1)) -> c100(TAKE(z0, z1))
   TAKE(ok(z0), ok(z1)) -> c101(TAKE(z0, z1))
   SEL(mark(z0), z1) -> c102(SEL(z0, z1))
   SEL(z0, mark(z1)) -> c103(SEL(z0, z1))
   SEL(ok(z0), ok(z1)) -> c104(SEL(z0, z1))
   ISLNAT(ok(z0)) -> c108(ISLNAT(z0))
   ISNATURAL(ok(z0)) -> c109(ISNATURAL(z0))
   ISPLNAT(ok(z0)) -> c110(ISPLNAT(z0))
   TOP(mark(z0)) -> c111(TOP(proper(z0)))
S tuples:
   TOP(mark(z0)) -> c111(TOP(proper(z0)))
K tuples:
   U101'(mark(z0), z1) -> c(U101'(z0, z1))
   U101'(ok(z0), ok(z1)) -> c1(U101'(z0, z1))
   U102'(mark(z0)) -> c2(U102'(z0))
   U102'(ok(z0)) -> c3(U102'(z0))
   U11'(mark(z0), z1, z2) -> c4(U11'(z0, z1, z2))
   U11'(ok(z0), ok(z1), ok(z2)) -> c5(U11'(z0, z1, z2))
   U12'(mark(z0), z1, z2) -> c6(U12'(z0, z1, z2))
   U12'(ok(z0), ok(z1), ok(z2)) -> c7(U12'(z0, z1, z2))
   U111'(mark(z0)) -> c8(U111'(z0))
   U111'(ok(z0)) -> c9(U111'(z0))
   SND(mark(z0)) -> c10(SND(z0))
   SND(ok(z0)) -> c11(SND(z0))
   SPLITAT(mark(z0), z1) -> c12(SPLITAT(z0, z1))
   SPLITAT(z0, mark(z1)) -> c13(SPLITAT(z0, z1))
   SPLITAT(ok(z0), ok(z1)) -> c14(SPLITAT(z0, z1))
   U121'(mark(z0)) -> c15(U121'(z0))
   U121'(ok(z0)) -> c16(U121'(z0))
   U131'(mark(z0), z1) -> c17(U131'(z0, z1))
   U131'(ok(z0), ok(z1)) -> c18(U131'(z0, z1))
   U132'(mark(z0)) -> c19(U132'(z0))
   U132'(ok(z0)) -> c20(U132'(z0))
   U141'(mark(z0), z1) -> c21(U141'(z0, z1))
   U141'(ok(z0), ok(z1)) -> c22(U141'(z0, z1))
   U142'(mark(z0)) -> c23(U142'(z0))
   U142'(ok(z0)) -> c24(U142'(z0))
   U151'(mark(z0), z1) -> c25(U151'(z0, z1))
   U151'(ok(z0), ok(z1)) -> c26(U151'(z0, z1))
   U152'(mark(z0)) -> c27(U152'(z0))
   U152'(ok(z0)) -> c28(U152'(z0))
   U161'(mark(z0), z1) -> c29(U161'(z0, z1))
   U161'(ok(z0), ok(z1)) -> c30(U161'(z0, z1))
   CONS(mark(z0), z1) -> c31(CONS(z0, z1))
   CONS(ok(z0), ok(z1)) -> c32(CONS(z0, z1))
   NATSFROM(mark(z0)) -> c33(NATSFROM(z0))
   NATSFROM(ok(z0)) -> c34(NATSFROM(z0))
   S(mark(z0)) -> c35(S(z0))
   S(ok(z0)) -> c36(S(z0))
   U171'(mark(z0), z1, z2) -> c37(U171'(z0, z1, z2))
   U171'(ok(z0), ok(z1), ok(z2)) -> c38(U171'(z0, z1, z2))
   U172'(mark(z0), z1, z2) -> c39(U172'(z0, z1, z2))
   U172'(ok(z0), ok(z1), ok(z2)) -> c40(U172'(z0, z1, z2))
   HEAD(mark(z0)) -> c41(HEAD(z0))
   HEAD(ok(z0)) -> c42(HEAD(z0))
   AFTERNTH(mark(z0), z1) -> c43(AFTERNTH(z0, z1))
   AFTERNTH(z0, mark(z1)) -> c44(AFTERNTH(z0, z1))
   AFTERNTH(ok(z0), ok(z1)) -> c45(AFTERNTH(z0, z1))
   U181'(mark(z0), z1) -> c46(U181'(z0, z1))
   U181'(ok(z0), ok(z1)) -> c47(U181'(z0, z1))
   U182'(mark(z0), z1) -> c48(U182'(z0, z1))
   U182'(ok(z0), ok(z1)) -> c49(U182'(z0, z1))
   U191'(mark(z0), z1) -> c50(U191'(z0, z1))
   U191'(ok(z0), ok(z1)) -> c51(U191'(z0, z1))
   PAIR(mark(z0), z1) -> c52(PAIR(z0, z1))
   PAIR(z0, mark(z1)) -> c53(PAIR(z0, z1))
   PAIR(ok(z0), ok(z1)) -> c54(PAIR(z0, z1))
   U201'(mark(z0), z1, z2, z3) -> c55(U201'(z0, z1, z2, z3))
   U201'(ok(z0), ok(z1), ok(z2), ok(z3)) -> c56(U201'(z0, z1, z2, z3))
   U202'(mark(z0), z1, z2, z3) -> c57(U202'(z0, z1, z2, z3))
   U202'(ok(z0), ok(z1), ok(z2), ok(z3)) -> c58(U202'(z0, z1, z2, z3))
   U203'(mark(z0), z1, z2, z3) -> c59(U203'(z0, z1, z2, z3))
   U203'(ok(z0), ok(z1), ok(z2), ok(z3)) -> c60(U203'(z0, z1, z2, z3))
   U204'(mark(z0), z1) -> c61(U204'(z0, z1))
   U204'(ok(z0), ok(z1)) -> c62(U204'(z0, z1))
   U21'(mark(z0), z1, z2) -> c63(U21'(z0, z1, z2))
   U21'(ok(z0), ok(z1), ok(z2)) -> c64(U21'(z0, z1, z2))
   U22'(mark(z0), z1) -> c65(U22'(z0, z1))
   U22'(ok(z0), ok(z1)) -> c66(U22'(z0, z1))
   U211'(mark(z0), z1) -> c67(U211'(z0, z1))
   U211'(ok(z0), ok(z1)) -> c68(U211'(z0, z1))
   U212'(mark(z0), z1) -> c69(U212'(z0, z1))
   U212'(ok(z0), ok(z1)) -> c70(U212'(z0, z1))
   U221'(mark(z0), z1, z2) -> c71(U221'(z0, z1, z2))
   U221'(ok(z0), ok(z1), ok(z2)) -> c72(U221'(z0, z1, z2))
   U222'(mark(z0), z1, z2) -> c73(U222'(z0, z1, z2))
   U222'(ok(z0), ok(z1), ok(z2)) -> c74(U222'(z0, z1, z2))
   FST(mark(z0)) -> c75(FST(z0))
   FST(ok(z0)) -> c76(FST(z0))
   U31'(mark(z0), z1, z2) -> c77(U31'(z0, z1, z2))
   U31'(ok(z0), ok(z1), ok(z2)) -> c78(U31'(z0, z1, z2))
   U32'(mark(z0), z1) -> c79(U32'(z0, z1))
   U32'(ok(z0), ok(z1)) -> c80(U32'(z0, z1))
   U41'(mark(z0), z1) -> c81(U41'(z0, z1))
   U41'(ok(z0), ok(z1)) -> c82(U41'(z0, z1))
   U42'(mark(z0)) -> c83(U42'(z0))
   U42'(ok(z0)) -> c84(U42'(z0))
   U51'(mark(z0), z1) -> c85(U51'(z0, z1))
   U51'(ok(z0), ok(z1)) -> c86(U51'(z0, z1))
   U52'(mark(z0)) -> c87(U52'(z0))
   U52'(ok(z0)) -> c88(U52'(z0))
   U61'(mark(z0)) -> c89(U61'(z0))
   U61'(ok(z0)) -> c90(U61'(z0))
   U71'(mark(z0)) -> c91(U71'(z0))
   U71'(ok(z0)) -> c92(U71'(z0))
   U81'(mark(z0)) -> c93(U81'(z0))
   U81'(ok(z0)) -> c94(U81'(z0))
   U91'(mark(z0)) -> c95(U91'(z0))
   U91'(ok(z0)) -> c96(U91'(z0))
   TAIL(mark(z0)) -> c97(TAIL(z0))
   TAIL(ok(z0)) -> c98(TAIL(z0))
   TAKE(mark(z0), z1) -> c99(TAKE(z0, z1))
   TAKE(z0, mark(z1)) -> c100(TAKE(z0, z1))
   TAKE(ok(z0), ok(z1)) -> c101(TAKE(z0, z1))
   SEL(mark(z0), z1) -> c102(SEL(z0, z1))
   SEL(z0, mark(z1)) -> c103(SEL(z0, z1))
   SEL(ok(z0), ok(z1)) -> c104(SEL(z0, z1))
   ISLNAT(ok(z0)) -> c108(ISLNAT(z0))
   ISNATURAL(ok(z0)) -> c109(ISNATURAL(z0))
   ISPLNAT(ok(z0)) -> c110(ISPLNAT(z0))
Defined Rule Symbols:   proper_1

Defined Pair Symbols:   U101'_2, U102'_1, U11'_3, U12'_3, U111'_1, SND_1, SPLITAT_2, U121'_1, U131'_2, U132'_1, U141'_2, U142'_1, U151'_2, U152'_1, U161'_2, CONS_2, NATSFROM_1, S_1, U171'_3, U172'_3, HEAD_1, AFTERNTH_2, U181'_2, U182'_2, U191'_2, PAIR_2, U201'_4, U202'_4, U203'_4, U204'_2, U21'_3, U22'_2, U211'_2, U212'_2, U221'_3, U222'_3, FST_1, U31'_3, U32'_2, U41'_2, U42'_1, U51'_2, U52'_1, U61'_1, U71'_1, U81'_1, U91'_1, TAIL_1, TAKE_2, SEL_2, ISLNAT_1, ISNATURAL_1, ISPLNAT_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, c108_1, c109_1, c110_1, c111_1


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

(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)) -> c111(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) -> c(U101'(z0, z1))
   U101'(ok(z0), ok(z1)) -> c1(U101'(z0, z1))
   U102'(mark(z0)) -> c2(U102'(z0))
   U102'(ok(z0)) -> c3(U102'(z0))
   U11'(mark(z0), z1, z2) -> c4(U11'(z0, z1, z2))
   U11'(ok(z0), ok(z1), ok(z2)) -> c5(U11'(z0, z1, z2))
   U12'(mark(z0), z1, z2) -> c6(U12'(z0, z1, z2))
   U12'(ok(z0), ok(z1), ok(z2)) -> c7(U12'(z0, z1, z2))
   U111'(mark(z0)) -> c8(U111'(z0))
   U111'(ok(z0)) -> c9(U111'(z0))
   SND(mark(z0)) -> c10(SND(z0))
   SND(ok(z0)) -> c11(SND(z0))
   SPLITAT(mark(z0), z1) -> c12(SPLITAT(z0, z1))
   SPLITAT(z0, mark(z1)) -> c13(SPLITAT(z0, z1))
   SPLITAT(ok(z0), ok(z1)) -> c14(SPLITAT(z0, z1))
   U121'(mark(z0)) -> c15(U121'(z0))
   U121'(ok(z0)) -> c16(U121'(z0))
   U131'(mark(z0), z1) -> c17(U131'(z0, z1))
   U131'(ok(z0), ok(z1)) -> c18(U131'(z0, z1))
   U132'(mark(z0)) -> c19(U132'(z0))
   U132'(ok(z0)) -> c20(U132'(z0))
   U141'(mark(z0), z1) -> c21(U141'(z0, z1))
   U141'(ok(z0), ok(z1)) -> c22(U141'(z0, z1))
   U142'(mark(z0)) -> c23(U142'(z0))
   U142'(ok(z0)) -> c24(U142'(z0))
   U151'(mark(z0), z1) -> c25(U151'(z0, z1))
   U151'(ok(z0), ok(z1)) -> c26(U151'(z0, z1))
   U152'(mark(z0)) -> c27(U152'(z0))
   U152'(ok(z0)) -> c28(U152'(z0))
   U161'(mark(z0), z1) -> c29(U161'(z0, z1))
   U161'(ok(z0), ok(z1)) -> c30(U161'(z0, z1))
   CONS(mark(z0), z1) -> c31(CONS(z0, z1))
   CONS(ok(z0), ok(z1)) -> c32(CONS(z0, z1))
   NATSFROM(mark(z0)) -> c33(NATSFROM(z0))
   NATSFROM(ok(z0)) -> c34(NATSFROM(z0))
   S(mark(z0)) -> c35(S(z0))
   S(ok(z0)) -> c36(S(z0))
   U171'(mark(z0), z1, z2) -> c37(U171'(z0, z1, z2))
   U171'(ok(z0), ok(z1), ok(z2)) -> c38(U171'(z0, z1, z2))
   U172'(mark(z0), z1, z2) -> c39(U172'(z0, z1, z2))
   U172'(ok(z0), ok(z1), ok(z2)) -> c40(U172'(z0, z1, z2))
   HEAD(mark(z0)) -> c41(HEAD(z0))
   HEAD(ok(z0)) -> c42(HEAD(z0))
   AFTERNTH(mark(z0), z1) -> c43(AFTERNTH(z0, z1))
   AFTERNTH(z0, mark(z1)) -> c44(AFTERNTH(z0, z1))
   AFTERNTH(ok(z0), ok(z1)) -> c45(AFTERNTH(z0, z1))
   U181'(mark(z0), z1) -> c46(U181'(z0, z1))
   U181'(ok(z0), ok(z1)) -> c47(U181'(z0, z1))
   U182'(mark(z0), z1) -> c48(U182'(z0, z1))
   U182'(ok(z0), ok(z1)) -> c49(U182'(z0, z1))
   U191'(mark(z0), z1) -> c50(U191'(z0, z1))
   U191'(ok(z0), ok(z1)) -> c51(U191'(z0, z1))
   PAIR(mark(z0), z1) -> c52(PAIR(z0, z1))
   PAIR(z0, mark(z1)) -> c53(PAIR(z0, z1))
   PAIR(ok(z0), ok(z1)) -> c54(PAIR(z0, z1))
   U201'(mark(z0), z1, z2, z3) -> c55(U201'(z0, z1, z2, z3))
   U201'(ok(z0), ok(z1), ok(z2), ok(z3)) -> c56(U201'(z0, z1, z2, z3))
   U202'(mark(z0), z1, z2, z3) -> c57(U202'(z0, z1, z2, z3))
   U202'(ok(z0), ok(z1), ok(z2), ok(z3)) -> c58(U202'(z0, z1, z2, z3))
   U203'(mark(z0), z1, z2, z3) -> c59(U203'(z0, z1, z2, z3))
   U203'(ok(z0), ok(z1), ok(z2), ok(z3)) -> c60(U203'(z0, z1, z2, z3))
   U204'(mark(z0), z1) -> c61(U204'(z0, z1))
   U204'(ok(z0), ok(z1)) -> c62(U204'(z0, z1))
   U21'(mark(z0), z1, z2) -> c63(U21'(z0, z1, z2))
   U21'(ok(z0), ok(z1), ok(z2)) -> c64(U21'(z0, z1, z2))
   U22'(mark(z0), z1) -> c65(U22'(z0, z1))
   U22'(ok(z0), ok(z1)) -> c66(U22'(z0, z1))
   U211'(mark(z0), z1) -> c67(U211'(z0, z1))
   U211'(ok(z0), ok(z1)) -> c68(U211'(z0, z1))
   U212'(mark(z0), z1) -> c69(U212'(z0, z1))
   U212'(ok(z0), ok(z1)) -> c70(U212'(z0, z1))
   U221'(mark(z0), z1, z2) -> c71(U221'(z0, z1, z2))
   U221'(ok(z0), ok(z1), ok(z2)) -> c72(U221'(z0, z1, z2))
   U222'(mark(z0), z1, z2) -> c73(U222'(z0, z1, z2))
   U222'(ok(z0), ok(z1), ok(z2)) -> c74(U222'(z0, z1, z2))
   FST(mark(z0)) -> c75(FST(z0))
   FST(ok(z0)) -> c76(FST(z0))
   U31'(mark(z0), z1, z2) -> c77(U31'(z0, z1, z2))
   U31'(ok(z0), ok(z1), ok(z2)) -> c78(U31'(z0, z1, z2))
   U32'(mark(z0), z1) -> c79(U32'(z0, z1))
   U32'(ok(z0), ok(z1)) -> c80(U32'(z0, z1))
   U41'(mark(z0), z1) -> c81(U41'(z0, z1))
   U41'(ok(z0), ok(z1)) -> c82(U41'(z0, z1))
   U42'(mark(z0)) -> c83(U42'(z0))
   U42'(ok(z0)) -> c84(U42'(z0))
   U51'(mark(z0), z1) -> c85(U51'(z0, z1))
   U51'(ok(z0), ok(z1)) -> c86(U51'(z0, z1))
   U52'(mark(z0)) -> c87(U52'(z0))
   U52'(ok(z0)) -> c88(U52'(z0))
   U61'(mark(z0)) -> c89(U61'(z0))
   U61'(ok(z0)) -> c90(U61'(z0))
   U71'(mark(z0)) -> c91(U71'(z0))
   U71'(ok(z0)) -> c92(U71'(z0))
   U81'(mark(z0)) -> c93(U81'(z0))
   U81'(ok(z0)) -> c94(U81'(z0))
   U91'(mark(z0)) -> c95(U91'(z0))
   U91'(ok(z0)) -> c96(U91'(z0))
   TAIL(mark(z0)) -> c97(TAIL(z0))
   TAIL(ok(z0)) -> c98(TAIL(z0))
   TAKE(mark(z0), z1) -> c99(TAKE(z0, z1))
   TAKE(z0, mark(z1)) -> c100(TAKE(z0, z1))
   TAKE(ok(z0), ok(z1)) -> c101(TAKE(z0, z1))
   SEL(mark(z0), z1) -> c102(SEL(z0, z1))
   SEL(z0, mark(z1)) -> c103(SEL(z0, z1))
   SEL(ok(z0), ok(z1)) -> c104(SEL(z0, z1))
   ISLNAT(ok(z0)) -> c108(ISLNAT(z0))
   ISNATURAL(ok(z0)) -> c109(ISNATURAL(z0))
   ISPLNAT(ok(z0)) -> c110(ISPLNAT(z0))
   TOP(mark(z0)) -> c111(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(CONS(x_1, x_2)) = 0
   POL(FST(x_1)) = 0
   POL(HEAD(x_1)) = 0
   POL(ISLNAT(x_1)) = 0
   POL(ISNATURAL(x_1)) = 0
   POL(ISPLNAT(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)) = 0
   POL(U102'(x_1)) = 0
   POL(U11'(x_1, x_2, x_3)) = 0
   POL(U111'(x_1)) = 0
   POL(U12'(x_1, x_2, x_3)) = 0
   POL(U121'(x_1)) = 0
   POL(U131'(x_1, x_2)) = 0
   POL(U132'(x_1)) = 0
   POL(U141'(x_1, x_2)) = 0
   POL(U142'(x_1)) = 0
   POL(U151'(x_1, x_2)) = 0
   POL(U152'(x_1)) = 0
   POL(U161'(x_1, x_2)) = 0
   POL(U171'(x_1, x_2, x_3)) = 0
   POL(U172'(x_1, x_2, x_3)) = 0
   POL(U181'(x_1, x_2)) = 0
   POL(U182'(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, x_3, x_4)) = 0
   POL(U203'(x_1, x_2, x_3, x_4)) = 0
   POL(U204'(x_1, x_2)) = 0
   POL(U21'(x_1, x_2, x_3)) = 0
   POL(U211'(x_1, x_2)) = 0
   POL(U212'(x_1, x_2)) = 0
   POL(U22'(x_1, x_2)) = 0
   POL(U221'(x_1, x_2, x_3)) = 0
   POL(U222'(x_1, x_2, x_3)) = 0
   POL(U31'(x_1, x_2, x_3)) = 0
   POL(U32'(x_1, x_2)) = 0
   POL(U41'(x_1, x_2)) = 0
   POL(U42'(x_1)) = 0
   POL(U51'(x_1, x_2)) = 0
   POL(U52'(x_1)) = 0
   POL(U61'(x_1)) = 0
   POL(U71'(x_1)) = 0
   POL(U81'(x_1)) = 0
   POL(U91'(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(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(c12(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) -> c(U101'(z0, z1))
   U101'(ok(z0), ok(z1)) -> c1(U101'(z0, z1))
   U102'(mark(z0)) -> c2(U102'(z0))
   U102'(ok(z0)) -> c3(U102'(z0))
   U11'(mark(z0), z1, z2) -> c4(U11'(z0, z1, z2))
   U11'(ok(z0), ok(z1), ok(z2)) -> c5(U11'(z0, z1, z2))
   U12'(mark(z0), z1, z2) -> c6(U12'(z0, z1, z2))
   U12'(ok(z0), ok(z1), ok(z2)) -> c7(U12'(z0, z1, z2))
   U111'(mark(z0)) -> c8(U111'(z0))
   U111'(ok(z0)) -> c9(U111'(z0))
   SND(mark(z0)) -> c10(SND(z0))
   SND(ok(z0)) -> c11(SND(z0))
   SPLITAT(mark(z0), z1) -> c12(SPLITAT(z0, z1))
   SPLITAT(z0, mark(z1)) -> c13(SPLITAT(z0, z1))
   SPLITAT(ok(z0), ok(z1)) -> c14(SPLITAT(z0, z1))
   U121'(mark(z0)) -> c15(U121'(z0))
   U121'(ok(z0)) -> c16(U121'(z0))
   U131'(mark(z0), z1) -> c17(U131'(z0, z1))
   U131'(ok(z0), ok(z1)) -> c18(U131'(z0, z1))
   U132'(mark(z0)) -> c19(U132'(z0))
   U132'(ok(z0)) -> c20(U132'(z0))
   U141'(mark(z0), z1) -> c21(U141'(z0, z1))
   U141'(ok(z0), ok(z1)) -> c22(U141'(z0, z1))
   U142'(mark(z0)) -> c23(U142'(z0))
   U142'(ok(z0)) -> c24(U142'(z0))
   U151'(mark(z0), z1) -> c25(U151'(z0, z1))
   U151'(ok(z0), ok(z1)) -> c26(U151'(z0, z1))
   U152'(mark(z0)) -> c27(U152'(z0))
   U152'(ok(z0)) -> c28(U152'(z0))
   U161'(mark(z0), z1) -> c29(U161'(z0, z1))
   U161'(ok(z0), ok(z1)) -> c30(U161'(z0, z1))
   CONS(mark(z0), z1) -> c31(CONS(z0, z1))
   CONS(ok(z0), ok(z1)) -> c32(CONS(z0, z1))
   NATSFROM(mark(z0)) -> c33(NATSFROM(z0))
   NATSFROM(ok(z0)) -> c34(NATSFROM(z0))
   S(mark(z0)) -> c35(S(z0))
   S(ok(z0)) -> c36(S(z0))
   U171'(mark(z0), z1, z2) -> c37(U171'(z0, z1, z2))
   U171'(ok(z0), ok(z1), ok(z2)) -> c38(U171'(z0, z1, z2))
   U172'(mark(z0), z1, z2) -> c39(U172'(z0, z1, z2))
   U172'(ok(z0), ok(z1), ok(z2)) -> c40(U172'(z0, z1, z2))
   HEAD(mark(z0)) -> c41(HEAD(z0))
   HEAD(ok(z0)) -> c42(HEAD(z0))
   AFTERNTH(mark(z0), z1) -> c43(AFTERNTH(z0, z1))
   AFTERNTH(z0, mark(z1)) -> c44(AFTERNTH(z0, z1))
   AFTERNTH(ok(z0), ok(z1)) -> c45(AFTERNTH(z0, z1))
   U181'(mark(z0), z1) -> c46(U181'(z0, z1))
   U181'(ok(z0), ok(z1)) -> c47(U181'(z0, z1))
   U182'(mark(z0), z1) -> c48(U182'(z0, z1))
   U182'(ok(z0), ok(z1)) -> c49(U182'(z0, z1))
   U191'(mark(z0), z1) -> c50(U191'(z0, z1))
   U191'(ok(z0), ok(z1)) -> c51(U191'(z0, z1))
   PAIR(mark(z0), z1) -> c52(PAIR(z0, z1))
   PAIR(z0, mark(z1)) -> c53(PAIR(z0, z1))
   PAIR(ok(z0), ok(z1)) -> c54(PAIR(z0, z1))
   U201'(mark(z0), z1, z2, z3) -> c55(U201'(z0, z1, z2, z3))
   U201'(ok(z0), ok(z1), ok(z2), ok(z3)) -> c56(U201'(z0, z1, z2, z3))
   U202'(mark(z0), z1, z2, z3) -> c57(U202'(z0, z1, z2, z3))
   U202'(ok(z0), ok(z1), ok(z2), ok(z3)) -> c58(U202'(z0, z1, z2, z3))
   U203'(mark(z0), z1, z2, z3) -> c59(U203'(z0, z1, z2, z3))
   U203'(ok(z0), ok(z1), ok(z2), ok(z3)) -> c60(U203'(z0, z1, z2, z3))
   U204'(mark(z0), z1) -> c61(U204'(z0, z1))
   U204'(ok(z0), ok(z1)) -> c62(U204'(z0, z1))
   U21'(mark(z0), z1, z2) -> c63(U21'(z0, z1, z2))
   U21'(ok(z0), ok(z1), ok(z2)) -> c64(U21'(z0, z1, z2))
   U22'(mark(z0), z1) -> c65(U22'(z0, z1))
   U22'(ok(z0), ok(z1)) -> c66(U22'(z0, z1))
   U211'(mark(z0), z1) -> c67(U211'(z0, z1))
   U211'(ok(z0), ok(z1)) -> c68(U211'(z0, z1))
   U212'(mark(z0), z1) -> c69(U212'(z0, z1))
   U212'(ok(z0), ok(z1)) -> c70(U212'(z0, z1))
   U221'(mark(z0), z1, z2) -> c71(U221'(z0, z1, z2))
   U221'(ok(z0), ok(z1), ok(z2)) -> c72(U221'(z0, z1, z2))
   U222'(mark(z0), z1, z2) -> c73(U222'(z0, z1, z2))
   U222'(ok(z0), ok(z1), ok(z2)) -> c74(U222'(z0, z1, z2))
   FST(mark(z0)) -> c75(FST(z0))
   FST(ok(z0)) -> c76(FST(z0))
   U31'(mark(z0), z1, z2) -> c77(U31'(z0, z1, z2))
   U31'(ok(z0), ok(z1), ok(z2)) -> c78(U31'(z0, z1, z2))
   U32'(mark(z0), z1) -> c79(U32'(z0, z1))
   U32'(ok(z0), ok(z1)) -> c80(U32'(z0, z1))
   U41'(mark(z0), z1) -> c81(U41'(z0, z1))
   U41'(ok(z0), ok(z1)) -> c82(U41'(z0, z1))
   U42'(mark(z0)) -> c83(U42'(z0))
   U42'(ok(z0)) -> c84(U42'(z0))
   U51'(mark(z0), z1) -> c85(U51'(z0, z1))
   U51'(ok(z0), ok(z1)) -> c86(U51'(z0, z1))
   U52'(mark(z0)) -> c87(U52'(z0))
   U52'(ok(z0)) -> c88(U52'(z0))
   U61'(mark(z0)) -> c89(U61'(z0))
   U61'(ok(z0)) -> c90(U61'(z0))
   U71'(mark(z0)) -> c91(U71'(z0))
   U71'(ok(z0)) -> c92(U71'(z0))
   U81'(mark(z0)) -> c93(U81'(z0))
   U81'(ok(z0)) -> c94(U81'(z0))
   U91'(mark(z0)) -> c95(U91'(z0))
   U91'(ok(z0)) -> c96(U91'(z0))
   TAIL(mark(z0)) -> c97(TAIL(z0))
   TAIL(ok(z0)) -> c98(TAIL(z0))
   TAKE(mark(z0), z1) -> c99(TAKE(z0, z1))
   TAKE(z0, mark(z1)) -> c100(TAKE(z0, z1))
   TAKE(ok(z0), ok(z1)) -> c101(TAKE(z0, z1))
   SEL(mark(z0), z1) -> c102(SEL(z0, z1))
   SEL(z0, mark(z1)) -> c103(SEL(z0, z1))
   SEL(ok(z0), ok(z1)) -> c104(SEL(z0, z1))
   ISLNAT(ok(z0)) -> c108(ISLNAT(z0))
   ISNATURAL(ok(z0)) -> c109(ISNATURAL(z0))
   ISPLNAT(ok(z0)) -> c110(ISPLNAT(z0))
   TOP(mark(z0)) -> c111(TOP(proper(z0)))
S tuples:none
K tuples:
   U101'(mark(z0), z1) -> c(U101'(z0, z1))
   U101'(ok(z0), ok(z1)) -> c1(U101'(z0, z1))
   U102'(mark(z0)) -> c2(U102'(z0))
   U102'(ok(z0)) -> c3(U102'(z0))
   U11'(mark(z0), z1, z2) -> c4(U11'(z0, z1, z2))
   U11'(ok(z0), ok(z1), ok(z2)) -> c5(U11'(z0, z1, z2))
   U12'(mark(z0), z1, z2) -> c6(U12'(z0, z1, z2))
   U12'(ok(z0), ok(z1), ok(z2)) -> c7(U12'(z0, z1, z2))
   U111'(mark(z0)) -> c8(U111'(z0))
   U111'(ok(z0)) -> c9(U111'(z0))
   SND(mark(z0)) -> c10(SND(z0))
   SND(ok(z0)) -> c11(SND(z0))
   SPLITAT(mark(z0), z1) -> c12(SPLITAT(z0, z1))
   SPLITAT(z0, mark(z1)) -> c13(SPLITAT(z0, z1))
   SPLITAT(ok(z0), ok(z1)) -> c14(SPLITAT(z0, z1))
   U121'(mark(z0)) -> c15(U121'(z0))
   U121'(ok(z0)) -> c16(U121'(z0))
   U131'(mark(z0), z1) -> c17(U131'(z0, z1))
   U131'(ok(z0), ok(z1)) -> c18(U131'(z0, z1))
   U132'(mark(z0)) -> c19(U132'(z0))
   U132'(ok(z0)) -> c20(U132'(z0))
   U141'(mark(z0), z1) -> c21(U141'(z0, z1))
   U141'(ok(z0), ok(z1)) -> c22(U141'(z0, z1))
   U142'(mark(z0)) -> c23(U142'(z0))
   U142'(ok(z0)) -> c24(U142'(z0))
   U151'(mark(z0), z1) -> c25(U151'(z0, z1))
   U151'(ok(z0), ok(z1)) -> c26(U151'(z0, z1))
   U152'(mark(z0)) -> c27(U152'(z0))
   U152'(ok(z0)) -> c28(U152'(z0))
   U161'(mark(z0), z1) -> c29(U161'(z0, z1))
   U161'(ok(z0), ok(z1)) -> c30(U161'(z0, z1))
   CONS(mark(z0), z1) -> c31(CONS(z0, z1))
   CONS(ok(z0), ok(z1)) -> c32(CONS(z0, z1))
   NATSFROM(mark(z0)) -> c33(NATSFROM(z0))
   NATSFROM(ok(z0)) -> c34(NATSFROM(z0))
   S(mark(z0)) -> c35(S(z0))
   S(ok(z0)) -> c36(S(z0))
   U171'(mark(z0), z1, z2) -> c37(U171'(z0, z1, z2))
   U171'(ok(z0), ok(z1), ok(z2)) -> c38(U171'(z0, z1, z2))
   U172'(mark(z0), z1, z2) -> c39(U172'(z0, z1, z2))
   U172'(ok(z0), ok(z1), ok(z2)) -> c40(U172'(z0, z1, z2))
   HEAD(mark(z0)) -> c41(HEAD(z0))
   HEAD(ok(z0)) -> c42(HEAD(z0))
   AFTERNTH(mark(z0), z1) -> c43(AFTERNTH(z0, z1))
   AFTERNTH(z0, mark(z1)) -> c44(AFTERNTH(z0, z1))
   AFTERNTH(ok(z0), ok(z1)) -> c45(AFTERNTH(z0, z1))
   U181'(mark(z0), z1) -> c46(U181'(z0, z1))
   U181'(ok(z0), ok(z1)) -> c47(U181'(z0, z1))
   U182'(mark(z0), z1) -> c48(U182'(z0, z1))
   U182'(ok(z0), ok(z1)) -> c49(U182'(z0, z1))
   U191'(mark(z0), z1) -> c50(U191'(z0, z1))
   U191'(ok(z0), ok(z1)) -> c51(U191'(z0, z1))
   PAIR(mark(z0), z1) -> c52(PAIR(z0, z1))
   PAIR(z0, mark(z1)) -> c53(PAIR(z0, z1))
   PAIR(ok(z0), ok(z1)) -> c54(PAIR(z0, z1))
   U201'(mark(z0), z1, z2, z3) -> c55(U201'(z0, z1, z2, z3))
   U201'(ok(z0), ok(z1), ok(z2), ok(z3)) -> c56(U201'(z0, z1, z2, z3))
   U202'(mark(z0), z1, z2, z3) -> c57(U202'(z0, z1, z2, z3))
   U202'(ok(z0), ok(z1), ok(z2), ok(z3)) -> c58(U202'(z0, z1, z2, z3))
   U203'(mark(z0), z1, z2, z3) -> c59(U203'(z0, z1, z2, z3))
   U203'(ok(z0), ok(z1), ok(z2), ok(z3)) -> c60(U203'(z0, z1, z2, z3))
   U204'(mark(z0), z1) -> c61(U204'(z0, z1))
   U204'(ok(z0), ok(z1)) -> c62(U204'(z0, z1))
   U21'(mark(z0), z1, z2) -> c63(U21'(z0, z1, z2))
   U21'(ok(z0), ok(z1), ok(z2)) -> c64(U21'(z0, z1, z2))
   U22'(mark(z0), z1) -> c65(U22'(z0, z1))
   U22'(ok(z0), ok(z1)) -> c66(U22'(z0, z1))
   U211'(mark(z0), z1) -> c67(U211'(z0, z1))
   U211'(ok(z0), ok(z1)) -> c68(U211'(z0, z1))
   U212'(mark(z0), z1) -> c69(U212'(z0, z1))
   U212'(ok(z0), ok(z1)) -> c70(U212'(z0, z1))
   U221'(mark(z0), z1, z2) -> c71(U221'(z0, z1, z2))
   U221'(ok(z0), ok(z1), ok(z2)) -> c72(U221'(z0, z1, z2))
   U222'(mark(z0), z1, z2) -> c73(U222'(z0, z1, z2))
   U222'(ok(z0), ok(z1), ok(z2)) -> c74(U222'(z0, z1, z2))
   FST(mark(z0)) -> c75(FST(z0))
   FST(ok(z0)) -> c76(FST(z0))
   U31'(mark(z0), z1, z2) -> c77(U31'(z0, z1, z2))
   U31'(ok(z0), ok(z1), ok(z2)) -> c78(U31'(z0, z1, z2))
   U32'(mark(z0), z1) -> c79(U32'(z0, z1))
   U32'(ok(z0), ok(z1)) -> c80(U32'(z0, z1))
   U41'(mark(z0), z1) -> c81(U41'(z0, z1))
   U41'(ok(z0), ok(z1)) -> c82(U41'(z0, z1))
   U42'(mark(z0)) -> c83(U42'(z0))
   U42'(ok(z0)) -> c84(U42'(z0))
   U51'(mark(z0), z1) -> c85(U51'(z0, z1))
   U51'(ok(z0), ok(z1)) -> c86(U51'(z0, z1))
   U52'(mark(z0)) -> c87(U52'(z0))
   U52'(ok(z0)) -> c88(U52'(z0))
   U61'(mark(z0)) -> c89(U61'(z0))
   U61'(ok(z0)) -> c90(U61'(z0))
   U71'(mark(z0)) -> c91(U71'(z0))
   U71'(ok(z0)) -> c92(U71'(z0))
   U81'(mark(z0)) -> c93(U81'(z0))
   U81'(ok(z0)) -> c94(U81'(z0))
   U91'(mark(z0)) -> c95(U91'(z0))
   U91'(ok(z0)) -> c96(U91'(z0))
   TAIL(mark(z0)) -> c97(TAIL(z0))
   TAIL(ok(z0)) -> c98(TAIL(z0))
   TAKE(mark(z0), z1) -> c99(TAKE(z0, z1))
   TAKE(z0, mark(z1)) -> c100(TAKE(z0, z1))
   TAKE(ok(z0), ok(z1)) -> c101(TAKE(z0, z1))
   SEL(mark(z0), z1) -> c102(SEL(z0, z1))
   SEL(z0, mark(z1)) -> c103(SEL(z0, z1))
   SEL(ok(z0), ok(z1)) -> c104(SEL(z0, z1))
   ISLNAT(ok(z0)) -> c108(ISLNAT(z0))
   ISNATURAL(ok(z0)) -> c109(ISNATURAL(z0))
   ISPLNAT(ok(z0)) -> c110(ISPLNAT(z0))
   TOP(mark(z0)) -> c111(TOP(proper(z0)))
Defined Rule Symbols:   proper_1

Defined Pair Symbols:   U101'_2, U102'_1, U11'_3, U12'_3, U111'_1, SND_1, SPLITAT_2, U121'_1, U131'_2, U132'_1, U141'_2, U142'_1, U151'_2, U152'_1, U161'_2, CONS_2, NATSFROM_1, S_1, U171'_3, U172'_3, HEAD_1, AFTERNTH_2, U181'_2, U182'_2, U191'_2, PAIR_2, U201'_4, U202'_4, U203'_4, U204'_2, U21'_3, U22'_2, U211'_2, U212'_2, U221'_3, U222'_3, FST_1, U31'_3, U32'_2, U41'_2, U42'_1, U51'_2, U52'_1, U61'_1, U71'_1, U81'_1, U91'_1, TAIL_1, TAKE_2, SEL_2, ISLNAT_1, ISNATURAL_1, ISPLNAT_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, c108_1, c109_1, c110_1, c111_1


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

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

(18)
BOUNDS(1, 1)

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

(19) RenamingProof (BOTH BOUNDS(ID, ID))
Renamed function symbols to avoid clashes with predefined symbol.
----------------------------------------

(20)
Obligation:
The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   active(U101(tt, V2)) -> mark(U102(isLNat(V2)))
   active(U102(tt)) -> mark(tt)
   active(U11(tt, N, XS)) -> mark(U12(isLNat(XS), N, XS))
   active(U111(tt)) -> mark(tt)
   active(U12(tt, N, XS)) -> mark(snd(splitAt(N, XS)))
   active(U121(tt)) -> mark(tt)
   active(U131(tt, V2)) -> mark(U132(isLNat(V2)))
   active(U132(tt)) -> mark(tt)
   active(U141(tt, V2)) -> mark(U142(isLNat(V2)))
   active(U142(tt)) -> mark(tt)
   active(U151(tt, V2)) -> mark(U152(isLNat(V2)))
   active(U152(tt)) -> mark(tt)
   active(U161(tt, N)) -> mark(cons(N, natsFrom(s(N))))
   active(U171(tt, N, XS)) -> mark(U172(isLNat(XS), N, XS))
   active(U172(tt, N, XS)) -> mark(head(afterNth(N, XS)))
   active(U181(tt, Y)) -> mark(U182(isLNat(Y), Y))
   active(U182(tt, Y)) -> mark(Y)
   active(U191(tt, XS)) -> mark(pair(nil, XS))
   active(U201(tt, N, X, XS)) -> mark(U202(isNatural(X), N, X, XS))
   active(U202(tt, N, X, XS)) -> mark(U203(isLNat(XS), N, X, XS))
   active(U203(tt, N, X, XS)) -> mark(U204(splitAt(N, XS), X))
   active(U204(pair(YS, ZS), X)) -> mark(pair(cons(X, YS), ZS))
   active(U21(tt, X, Y)) -> mark(U22(isLNat(Y), X))
   active(U211(tt, XS)) -> mark(U212(isLNat(XS), XS))
   active(U212(tt, XS)) -> mark(XS)
   active(U22(tt, X)) -> mark(X)
   active(U221(tt, N, XS)) -> mark(U222(isLNat(XS), N, XS))
   active(U222(tt, N, XS)) -> mark(fst(splitAt(N, XS)))
   active(U31(tt, N, XS)) -> mark(U32(isLNat(XS), N))
   active(U32(tt, N)) -> mark(N)
   active(U41(tt, V2)) -> mark(U42(isLNat(V2)))
   active(U42(tt)) -> mark(tt)
   active(U51(tt, V2)) -> mark(U52(isLNat(V2)))
   active(U52(tt)) -> mark(tt)
   active(U61(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt)) -> mark(tt)
   active(U91(tt)) -> mark(tt)
   active(afterNth(N, XS)) -> mark(U11(isNatural(N), N, XS))
   active(fst(pair(X, Y))) -> mark(U21(isLNat(X), X, Y))
   active(head(cons(N, XS))) -> mark(U31(isNatural(N), N, XS))
   active(isLNat(nil)) -> mark(tt)
   active(isLNat(afterNth(V1, V2))) -> mark(U41(isNatural(V1), V2))
   active(isLNat(cons(V1, V2))) -> mark(U51(isNatural(V1), V2))
   active(isLNat(fst(V1))) -> mark(U61(isPLNat(V1)))
   active(isLNat(natsFrom(V1))) -> mark(U71(isNatural(V1)))
   active(isLNat(snd(V1))) -> mark(U81(isPLNat(V1)))
   active(isLNat(tail(V1))) -> mark(U91(isLNat(V1)))
   active(isLNat(take(V1, V2))) -> mark(U101(isNatural(V1), V2))
   active(isNatural(0')) -> mark(tt)
   active(isNatural(head(V1))) -> mark(U111(isLNat(V1)))
   active(isNatural(s(V1))) -> mark(U121(isNatural(V1)))
   active(isNatural(sel(V1, V2))) -> mark(U131(isNatural(V1), V2))
   active(isPLNat(pair(V1, V2))) -> mark(U141(isLNat(V1), V2))
   active(isPLNat(splitAt(V1, V2))) -> mark(U151(isNatural(V1), V2))
   active(natsFrom(N)) -> mark(U161(isNatural(N), N))
   active(sel(N, XS)) -> mark(U171(isNatural(N), N, XS))
   active(snd(pair(X, Y))) -> mark(U181(isLNat(X), Y))
   active(splitAt(0', XS)) -> mark(U191(isLNat(XS), XS))
   active(splitAt(s(N), cons(X, XS))) -> mark(U201(isNatural(N), N, X, XS))
   active(tail(cons(N, XS))) -> mark(U211(isNatural(N), XS))
   active(take(N, XS)) -> mark(U221(isNatural(N), N, XS))
   active(U101(X1, X2)) -> U101(active(X1), X2)
   active(U102(X)) -> U102(active(X))
   active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3)
   active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3)
   active(U111(X)) -> U111(active(X))
   active(snd(X)) -> snd(active(X))
   active(splitAt(X1, X2)) -> splitAt(active(X1), X2)
   active(splitAt(X1, X2)) -> splitAt(X1, active(X2))
   active(U121(X)) -> U121(active(X))
   active(U131(X1, X2)) -> U131(active(X1), X2)
   active(U132(X)) -> U132(active(X))
   active(U141(X1, X2)) -> U141(active(X1), X2)
   active(U142(X)) -> U142(active(X))
   active(U151(X1, X2)) -> U151(active(X1), X2)
   active(U152(X)) -> U152(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(U172(X1, X2, X3)) -> U172(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(U182(X1, X2)) -> U182(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, X3, X4)) -> U202(active(X1), X2, X3, X4)
   active(U203(X1, X2, X3, X4)) -> U203(active(X1), X2, X3, X4)
   active(U204(X1, X2)) -> U204(active(X1), X2)
   active(U21(X1, X2, X3)) -> U21(active(X1), X2, X3)
   active(U22(X1, X2)) -> U22(active(X1), X2)
   active(U211(X1, X2)) -> U211(active(X1), X2)
   active(U212(X1, X2)) -> U212(active(X1), X2)
   active(U221(X1, X2, X3)) -> U221(active(X1), X2, X3)
   active(U222(X1, X2, X3)) -> U222(active(X1), X2, X3)
   active(fst(X)) -> fst(active(X))
   active(U31(X1, X2, X3)) -> U31(active(X1), X2, X3)
   active(U32(X1, X2)) -> U32(active(X1), X2)
   active(U41(X1, X2)) -> U41(active(X1), X2)
   active(U42(X)) -> U42(active(X))
   active(U51(X1, X2)) -> U51(active(X1), X2)
   active(U52(X)) -> U52(active(X))
   active(U61(X)) -> U61(active(X))
   active(U71(X)) -> U71(active(X))
   active(U81(X)) -> U81(active(X))
   active(U91(X)) -> U91(active(X))
   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) -> mark(U101(X1, X2))
   U102(mark(X)) -> mark(U102(X))
   U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3))
   U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3))
   U111(mark(X)) -> mark(U111(X))
   snd(mark(X)) -> mark(snd(X))
   splitAt(mark(X1), X2) -> mark(splitAt(X1, X2))
   splitAt(X1, mark(X2)) -> mark(splitAt(X1, X2))
   U121(mark(X)) -> mark(U121(X))
   U131(mark(X1), X2) -> mark(U131(X1, X2))
   U132(mark(X)) -> mark(U132(X))
   U141(mark(X1), X2) -> mark(U141(X1, X2))
   U142(mark(X)) -> mark(U142(X))
   U151(mark(X1), X2) -> mark(U151(X1, X2))
   U152(mark(X)) -> mark(U152(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))
   U172(mark(X1), X2, X3) -> mark(U172(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))
   U182(mark(X1), X2) -> mark(U182(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, X3, X4) -> mark(U202(X1, X2, X3, X4))
   U203(mark(X1), X2, X3, X4) -> mark(U203(X1, X2, X3, X4))
   U204(mark(X1), X2) -> mark(U204(X1, X2))
   U21(mark(X1), X2, X3) -> mark(U21(X1, X2, X3))
   U22(mark(X1), X2) -> mark(U22(X1, X2))
   U211(mark(X1), X2) -> mark(U211(X1, X2))
   U212(mark(X1), X2) -> mark(U212(X1, X2))
   U221(mark(X1), X2, X3) -> mark(U221(X1, X2, X3))
   U222(mark(X1), X2, X3) -> mark(U222(X1, X2, X3))
   fst(mark(X)) -> mark(fst(X))
   U31(mark(X1), X2, X3) -> mark(U31(X1, X2, X3))
   U32(mark(X1), X2) -> mark(U32(X1, X2))
   U41(mark(X1), X2) -> mark(U41(X1, X2))
   U42(mark(X)) -> mark(U42(X))
   U51(mark(X1), X2) -> mark(U51(X1, X2))
   U52(mark(X)) -> mark(U52(X))
   U61(mark(X)) -> mark(U61(X))
   U71(mark(X)) -> mark(U71(X))
   U81(mark(X)) -> mark(U81(X))
   U91(mark(X)) -> mark(U91(X))
   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)) -> U101(proper(X1), proper(X2))
   proper(tt) -> ok(tt)
   proper(U102(X)) -> U102(proper(X))
   proper(isLNat(X)) -> isLNat(proper(X))
   proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3))
   proper(U12(X1, X2, X3)) -> U12(proper(X1), proper(X2), proper(X3))
   proper(U111(X)) -> U111(proper(X))
   proper(snd(X)) -> snd(proper(X))
   proper(splitAt(X1, X2)) -> splitAt(proper(X1), proper(X2))
   proper(U121(X)) -> U121(proper(X))
   proper(U131(X1, X2)) -> U131(proper(X1), proper(X2))
   proper(U132(X)) -> U132(proper(X))
   proper(U141(X1, X2)) -> U141(proper(X1), proper(X2))
   proper(U142(X)) -> U142(proper(X))
   proper(U151(X1, X2)) -> U151(proper(X1), proper(X2))
   proper(U152(X)) -> U152(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(U172(X1, X2, X3)) -> U172(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(U182(X1, X2)) -> U182(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, X3, X4)) -> U202(proper(X1), proper(X2), proper(X3), proper(X4))
   proper(isNatural(X)) -> isNatural(proper(X))
   proper(U203(X1, X2, X3, X4)) -> U203(proper(X1), proper(X2), proper(X3), proper(X4))
   proper(U204(X1, X2)) -> U204(proper(X1), proper(X2))
   proper(U21(X1, X2, X3)) -> U21(proper(X1), proper(X2), proper(X3))
   proper(U22(X1, X2)) -> U22(proper(X1), proper(X2))
   proper(U211(X1, X2)) -> U211(proper(X1), proper(X2))
   proper(U212(X1, X2)) -> U212(proper(X1), proper(X2))
   proper(U221(X1, X2, X3)) -> U221(proper(X1), proper(X2), proper(X3))
   proper(U222(X1, X2, X3)) -> U222(proper(X1), proper(X2), proper(X3))
   proper(fst(X)) -> fst(proper(X))
   proper(U31(X1, X2, X3)) -> U31(proper(X1), proper(X2), proper(X3))
   proper(U32(X1, X2)) -> U32(proper(X1), proper(X2))
   proper(U41(X1, X2)) -> U41(proper(X1), proper(X2))
   proper(U42(X)) -> U42(proper(X))
   proper(U51(X1, X2)) -> U51(proper(X1), proper(X2))
   proper(U52(X)) -> U52(proper(X))
   proper(U61(X)) -> U61(proper(X))
   proper(U71(X)) -> U71(proper(X))
   proper(U81(X)) -> U81(proper(X))
   proper(U91(X)) -> U91(proper(X))
   proper(isPLNat(X)) -> isPLNat(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(U101(X1, X2))
   U102(ok(X)) -> ok(U102(X))
   isLNat(ok(X)) -> ok(isLNat(X))
   U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
   U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))
   U111(ok(X)) -> ok(U111(X))
   snd(ok(X)) -> ok(snd(X))
   splitAt(ok(X1), ok(X2)) -> ok(splitAt(X1, X2))
   U121(ok(X)) -> ok(U121(X))
   U131(ok(X1), ok(X2)) -> ok(U131(X1, X2))
   U132(ok(X)) -> ok(U132(X))
   U141(ok(X1), ok(X2)) -> ok(U141(X1, X2))
   U142(ok(X)) -> ok(U142(X))
   U151(ok(X1), ok(X2)) -> ok(U151(X1, X2))
   U152(ok(X)) -> ok(U152(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))
   U172(ok(X1), ok(X2), ok(X3)) -> ok(U172(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))
   U182(ok(X1), ok(X2)) -> ok(U182(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(X3), ok(X4)) -> ok(U202(X1, X2, X3, X4))
   isNatural(ok(X)) -> ok(isNatural(X))
   U203(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U203(X1, X2, X3, X4))
   U204(ok(X1), ok(X2)) -> ok(U204(X1, X2))
   U21(ok(X1), ok(X2), ok(X3)) -> ok(U21(X1, X2, X3))
   U22(ok(X1), ok(X2)) -> ok(U22(X1, X2))
   U211(ok(X1), ok(X2)) -> ok(U211(X1, X2))
   U212(ok(X1), ok(X2)) -> ok(U212(X1, X2))
   U221(ok(X1), ok(X2), ok(X3)) -> ok(U221(X1, X2, X3))
   U222(ok(X1), ok(X2), ok(X3)) -> ok(U222(X1, X2, X3))
   fst(ok(X)) -> ok(fst(X))
   U31(ok(X1), ok(X2), ok(X3)) -> ok(U31(X1, X2, X3))
   U32(ok(X1), ok(X2)) -> ok(U32(X1, X2))
   U41(ok(X1), ok(X2)) -> ok(U41(X1, X2))
   U42(ok(X)) -> ok(U42(X))
   U51(ok(X1), ok(X2)) -> ok(U51(X1, X2))
   U52(ok(X)) -> ok(U52(X))
   U61(ok(X)) -> ok(U61(X))
   U71(ok(X)) -> ok(U71(X))
   U81(ok(X)) -> ok(U81(X))
   U91(ok(X)) -> ok(U91(X))
   isPLNat(ok(X)) -> ok(isPLNat(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) TypeInferenceProof (BOTH BOUNDS(ID, ID))
Infered types.
----------------------------------------

(22)
Obligation:
TRS:
Rules:
active(U101(tt, V2)) -> mark(U102(isLNat(V2)))
active(U102(tt)) -> mark(tt)
active(U11(tt, N, XS)) -> mark(U12(isLNat(XS), N, XS))
active(U111(tt)) -> mark(tt)
active(U12(tt, N, XS)) -> mark(snd(splitAt(N, XS)))
active(U121(tt)) -> mark(tt)
active(U131(tt, V2)) -> mark(U132(isLNat(V2)))
active(U132(tt)) -> mark(tt)
active(U141(tt, V2)) -> mark(U142(isLNat(V2)))
active(U142(tt)) -> mark(tt)
active(U151(tt, V2)) -> mark(U152(isLNat(V2)))
active(U152(tt)) -> mark(tt)
active(U161(tt, N)) -> mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) -> mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) -> mark(head(afterNth(N, XS)))
active(U181(tt, Y)) -> mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) -> mark(Y)
active(U191(tt, XS)) -> mark(pair(nil, XS))
active(U201(tt, N, X, XS)) -> mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) -> mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) -> mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) -> mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) -> mark(U22(isLNat(Y), X))
active(U211(tt, XS)) -> mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) -> mark(XS)
active(U22(tt, X)) -> mark(X)
active(U221(tt, N, XS)) -> mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) -> mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) -> mark(U32(isLNat(XS), N))
active(U32(tt, N)) -> mark(N)
active(U41(tt, V2)) -> mark(U42(isLNat(V2)))
active(U42(tt)) -> mark(tt)
active(U51(tt, V2)) -> mark(U52(isLNat(V2)))
active(U52(tt)) -> mark(tt)
active(U61(tt)) -> mark(tt)
active(U71(tt)) -> mark(tt)
active(U81(tt)) -> mark(tt)
active(U91(tt)) -> mark(tt)
active(afterNth(N, XS)) -> mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) -> mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) -> mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) -> mark(tt)
active(isLNat(afterNth(V1, V2))) -> mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) -> mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) -> mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) -> mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) -> mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) -> mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) -> mark(U101(isNatural(V1), V2))
active(isNatural(0')) -> mark(tt)
active(isNatural(head(V1))) -> mark(U111(isLNat(V1)))
active(isNatural(s(V1))) -> mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) -> mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) -> mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) -> mark(U151(isNatural(V1), V2))
active(natsFrom(N)) -> mark(U161(isNatural(N), N))
active(sel(N, XS)) -> mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) -> mark(U181(isLNat(X), Y))
active(splitAt(0', XS)) -> mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) -> mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) -> mark(U211(isNatural(N), XS))
active(take(N, XS)) -> mark(U221(isNatural(N), N, XS))
active(U101(X1, X2)) -> U101(active(X1), X2)
active(U102(X)) -> U102(active(X))
active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3)
active(U111(X)) -> U111(active(X))
active(snd(X)) -> snd(active(X))
active(splitAt(X1, X2)) -> splitAt(active(X1), X2)
active(splitAt(X1, X2)) -> splitAt(X1, active(X2))
active(U121(X)) -> U121(active(X))
active(U131(X1, X2)) -> U131(active(X1), X2)
active(U132(X)) -> U132(active(X))
active(U141(X1, X2)) -> U141(active(X1), X2)
active(U142(X)) -> U142(active(X))
active(U151(X1, X2)) -> U151(active(X1), X2)
active(U152(X)) -> U152(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(U172(X1, X2, X3)) -> U172(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(U182(X1, X2)) -> U182(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, X3, X4)) -> U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) -> U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) -> U204(active(X1), X2)
active(U21(X1, X2, X3)) -> U21(active(X1), X2, X3)
active(U22(X1, X2)) -> U22(active(X1), X2)
active(U211(X1, X2)) -> U211(active(X1), X2)
active(U212(X1, X2)) -> U212(active(X1), X2)
active(U221(X1, X2, X3)) -> U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) -> U222(active(X1), X2, X3)
active(fst(X)) -> fst(active(X))
active(U31(X1, X2, X3)) -> U31(active(X1), X2, X3)
active(U32(X1, X2)) -> U32(active(X1), X2)
active(U41(X1, X2)) -> U41(active(X1), X2)
active(U42(X)) -> U42(active(X))
active(U51(X1, X2)) -> U51(active(X1), X2)
active(U52(X)) -> U52(active(X))
active(U61(X)) -> U61(active(X))
active(U71(X)) -> U71(active(X))
active(U81(X)) -> U81(active(X))
active(U91(X)) -> U91(active(X))
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) -> mark(U101(X1, X2))
U102(mark(X)) -> mark(U102(X))
U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3))
U111(mark(X)) -> mark(U111(X))
snd(mark(X)) -> mark(snd(X))
splitAt(mark(X1), X2) -> mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) -> mark(splitAt(X1, X2))
U121(mark(X)) -> mark(U121(X))
U131(mark(X1), X2) -> mark(U131(X1, X2))
U132(mark(X)) -> mark(U132(X))
U141(mark(X1), X2) -> mark(U141(X1, X2))
U142(mark(X)) -> mark(U142(X))
U151(mark(X1), X2) -> mark(U151(X1, X2))
U152(mark(X)) -> mark(U152(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))
U172(mark(X1), X2, X3) -> mark(U172(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))
U182(mark(X1), X2) -> mark(U182(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, X3, X4) -> mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) -> mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) -> mark(U204(X1, X2))
U21(mark(X1), X2, X3) -> mark(U21(X1, X2, X3))
U22(mark(X1), X2) -> mark(U22(X1, X2))
U211(mark(X1), X2) -> mark(U211(X1, X2))
U212(mark(X1), X2) -> mark(U212(X1, X2))
U221(mark(X1), X2, X3) -> mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) -> mark(U222(X1, X2, X3))
fst(mark(X)) -> mark(fst(X))
U31(mark(X1), X2, X3) -> mark(U31(X1, X2, X3))
U32(mark(X1), X2) -> mark(U32(X1, X2))
U41(mark(X1), X2) -> mark(U41(X1, X2))
U42(mark(X)) -> mark(U42(X))
U51(mark(X1), X2) -> mark(U51(X1, X2))
U52(mark(X)) -> mark(U52(X))
U61(mark(X)) -> mark(U61(X))
U71(mark(X)) -> mark(U71(X))
U81(mark(X)) -> mark(U81(X))
U91(mark(X)) -> mark(U91(X))
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)) -> U101(proper(X1), proper(X2))
proper(tt) -> ok(tt)
proper(U102(X)) -> U102(proper(X))
proper(isLNat(X)) -> isLNat(proper(X))
proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) -> U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) -> U111(proper(X))
proper(snd(X)) -> snd(proper(X))
proper(splitAt(X1, X2)) -> splitAt(proper(X1), proper(X2))
proper(U121(X)) -> U121(proper(X))
proper(U131(X1, X2)) -> U131(proper(X1), proper(X2))
proper(U132(X)) -> U132(proper(X))
proper(U141(X1, X2)) -> U141(proper(X1), proper(X2))
proper(U142(X)) -> U142(proper(X))
proper(U151(X1, X2)) -> U151(proper(X1), proper(X2))
proper(U152(X)) -> U152(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(U172(X1, X2, X3)) -> U172(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(U182(X1, X2)) -> U182(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, X3, X4)) -> U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) -> isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) -> U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) -> U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) -> U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) -> U22(proper(X1), proper(X2))
proper(U211(X1, X2)) -> U211(proper(X1), proper(X2))
proper(U212(X1, X2)) -> U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) -> U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) -> U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) -> fst(proper(X))
proper(U31(X1, X2, X3)) -> U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) -> U32(proper(X1), proper(X2))
proper(U41(X1, X2)) -> U41(proper(X1), proper(X2))
proper(U42(X)) -> U42(proper(X))
proper(U51(X1, X2)) -> U51(proper(X1), proper(X2))
proper(U52(X)) -> U52(proper(X))
proper(U61(X)) -> U61(proper(X))
proper(U71(X)) -> U71(proper(X))
proper(U81(X)) -> U81(proper(X))
proper(U91(X)) -> U91(proper(X))
proper(isPLNat(X)) -> isPLNat(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(U101(X1, X2))
U102(ok(X)) -> ok(U102(X))
isLNat(ok(X)) -> ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))
U111(ok(X)) -> ok(U111(X))
snd(ok(X)) -> ok(snd(X))
splitAt(ok(X1), ok(X2)) -> ok(splitAt(X1, X2))
U121(ok(X)) -> ok(U121(X))
U131(ok(X1), ok(X2)) -> ok(U131(X1, X2))
U132(ok(X)) -> ok(U132(X))
U141(ok(X1), ok(X2)) -> ok(U141(X1, X2))
U142(ok(X)) -> ok(U142(X))
U151(ok(X1), ok(X2)) -> ok(U151(X1, X2))
U152(ok(X)) -> ok(U152(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))
U172(ok(X1), ok(X2), ok(X3)) -> ok(U172(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))
U182(ok(X1), ok(X2)) -> ok(U182(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(X3), ok(X4)) -> ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) -> ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) -> ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) -> ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) -> ok(U22(X1, X2))
U211(ok(X1), ok(X2)) -> ok(U211(X1, X2))
U212(ok(X1), ok(X2)) -> ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) -> ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) -> ok(U222(X1, X2, X3))
fst(ok(X)) -> ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) -> ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) -> ok(U32(X1, X2))
U41(ok(X1), ok(X2)) -> ok(U41(X1, X2))
U42(ok(X)) -> ok(U42(X))
U51(ok(X1), ok(X2)) -> ok(U51(X1, X2))
U52(ok(X)) -> ok(U52(X))
U61(ok(X)) -> ok(U61(X))
U71(ok(X)) -> ok(U71(X))
U81(ok(X)) -> ok(U81(X))
U91(ok(X)) -> ok(U91(X))
isPLNat(ok(X)) -> ok(isPLNat(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))

Types:
active :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U12 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U172 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U182 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U203 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U204 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U22 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U212 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U222 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U32 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok -> top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat -> tt:mark:nil:0':ok

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

(23) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
active, U102, isLNat, U12, snd, splitAt, U132, U142, U152, cons, natsFrom, s, U172, head, afterNth, U182, pair, U202, isNatural, U203, U204, U22, U212, U222, fst, U32, U42, U52, U11, U21, U31, U41, U51, U61, isPLNat, U71, U81, U91, U101, U111, U121, U131, U141, U151, U161, U171, U181, U191, U201, U211, U221, tail, take, sel, proper, top

They will be analysed ascendingly in the following order:
U102 < active
isLNat < active
U12 < active
snd < active
splitAt < active
U132 < active
U142 < active
U152 < active
cons < active
natsFrom < active
s < active
U172 < active
head < active
afterNth < active
U182 < active
pair < active
U202 < active
isNatural < active
U203 < active
U204 < active
U22 < active
U212 < active
U222 < active
fst < active
U32 < active
U42 < active
U52 < active
U11 < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
isPLNat < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
U141 < active
U151 < active
U161 < active
U171 < active
U181 < active
U191 < active
U201 < active
U211 < active
U221 < active
tail < active
take < active
sel < active
active < top
U102 < proper
isLNat < proper
U12 < proper
snd < proper
splitAt < proper
U132 < proper
U142 < proper
U152 < proper
cons < proper
natsFrom < proper
s < proper
U172 < proper
head < proper
afterNth < proper
U182 < proper
pair < proper
U202 < proper
isNatural < proper
U203 < proper
U204 < proper
U22 < proper
U212 < proper
U222 < proper
fst < proper
U32 < proper
U42 < proper
U52 < proper
U11 < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
isPLNat < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
U141 < proper
U151 < proper
U161 < proper
U171 < proper
U181 < proper
U191 < proper
U201 < proper
U211 < proper
U221 < proper
tail < proper
take < proper
sel < proper
proper < top

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

(24)
Obligation:
TRS:
Rules:
active(U101(tt, V2)) -> mark(U102(isLNat(V2)))
active(U102(tt)) -> mark(tt)
active(U11(tt, N, XS)) -> mark(U12(isLNat(XS), N, XS))
active(U111(tt)) -> mark(tt)
active(U12(tt, N, XS)) -> mark(snd(splitAt(N, XS)))
active(U121(tt)) -> mark(tt)
active(U131(tt, V2)) -> mark(U132(isLNat(V2)))
active(U132(tt)) -> mark(tt)
active(U141(tt, V2)) -> mark(U142(isLNat(V2)))
active(U142(tt)) -> mark(tt)
active(U151(tt, V2)) -> mark(U152(isLNat(V2)))
active(U152(tt)) -> mark(tt)
active(U161(tt, N)) -> mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) -> mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) -> mark(head(afterNth(N, XS)))
active(U181(tt, Y)) -> mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) -> mark(Y)
active(U191(tt, XS)) -> mark(pair(nil, XS))
active(U201(tt, N, X, XS)) -> mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) -> mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) -> mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) -> mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) -> mark(U22(isLNat(Y), X))
active(U211(tt, XS)) -> mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) -> mark(XS)
active(U22(tt, X)) -> mark(X)
active(U221(tt, N, XS)) -> mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) -> mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) -> mark(U32(isLNat(XS), N))
active(U32(tt, N)) -> mark(N)
active(U41(tt, V2)) -> mark(U42(isLNat(V2)))
active(U42(tt)) -> mark(tt)
active(U51(tt, V2)) -> mark(U52(isLNat(V2)))
active(U52(tt)) -> mark(tt)
active(U61(tt)) -> mark(tt)
active(U71(tt)) -> mark(tt)
active(U81(tt)) -> mark(tt)
active(U91(tt)) -> mark(tt)
active(afterNth(N, XS)) -> mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) -> mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) -> mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) -> mark(tt)
active(isLNat(afterNth(V1, V2))) -> mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) -> mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) -> mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) -> mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) -> mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) -> mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) -> mark(U101(isNatural(V1), V2))
active(isNatural(0')) -> mark(tt)
active(isNatural(head(V1))) -> mark(U111(isLNat(V1)))
active(isNatural(s(V1))) -> mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) -> mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) -> mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) -> mark(U151(isNatural(V1), V2))
active(natsFrom(N)) -> mark(U161(isNatural(N), N))
active(sel(N, XS)) -> mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) -> mark(U181(isLNat(X), Y))
active(splitAt(0', XS)) -> mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) -> mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) -> mark(U211(isNatural(N), XS))
active(take(N, XS)) -> mark(U221(isNatural(N), N, XS))
active(U101(X1, X2)) -> U101(active(X1), X2)
active(U102(X)) -> U102(active(X))
active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3)
active(U111(X)) -> U111(active(X))
active(snd(X)) -> snd(active(X))
active(splitAt(X1, X2)) -> splitAt(active(X1), X2)
active(splitAt(X1, X2)) -> splitAt(X1, active(X2))
active(U121(X)) -> U121(active(X))
active(U131(X1, X2)) -> U131(active(X1), X2)
active(U132(X)) -> U132(active(X))
active(U141(X1, X2)) -> U141(active(X1), X2)
active(U142(X)) -> U142(active(X))
active(U151(X1, X2)) -> U151(active(X1), X2)
active(U152(X)) -> U152(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(U172(X1, X2, X3)) -> U172(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(U182(X1, X2)) -> U182(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, X3, X4)) -> U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) -> U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) -> U204(active(X1), X2)
active(U21(X1, X2, X3)) -> U21(active(X1), X2, X3)
active(U22(X1, X2)) -> U22(active(X1), X2)
active(U211(X1, X2)) -> U211(active(X1), X2)
active(U212(X1, X2)) -> U212(active(X1), X2)
active(U221(X1, X2, X3)) -> U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) -> U222(active(X1), X2, X3)
active(fst(X)) -> fst(active(X))
active(U31(X1, X2, X3)) -> U31(active(X1), X2, X3)
active(U32(X1, X2)) -> U32(active(X1), X2)
active(U41(X1, X2)) -> U41(active(X1), X2)
active(U42(X)) -> U42(active(X))
active(U51(X1, X2)) -> U51(active(X1), X2)
active(U52(X)) -> U52(active(X))
active(U61(X)) -> U61(active(X))
active(U71(X)) -> U71(active(X))
active(U81(X)) -> U81(active(X))
active(U91(X)) -> U91(active(X))
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) -> mark(U101(X1, X2))
U102(mark(X)) -> mark(U102(X))
U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3))
U111(mark(X)) -> mark(U111(X))
snd(mark(X)) -> mark(snd(X))
splitAt(mark(X1), X2) -> mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) -> mark(splitAt(X1, X2))
U121(mark(X)) -> mark(U121(X))
U131(mark(X1), X2) -> mark(U131(X1, X2))
U132(mark(X)) -> mark(U132(X))
U141(mark(X1), X2) -> mark(U141(X1, X2))
U142(mark(X)) -> mark(U142(X))
U151(mark(X1), X2) -> mark(U151(X1, X2))
U152(mark(X)) -> mark(U152(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))
U172(mark(X1), X2, X3) -> mark(U172(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))
U182(mark(X1), X2) -> mark(U182(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, X3, X4) -> mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) -> mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) -> mark(U204(X1, X2))
U21(mark(X1), X2, X3) -> mark(U21(X1, X2, X3))
U22(mark(X1), X2) -> mark(U22(X1, X2))
U211(mark(X1), X2) -> mark(U211(X1, X2))
U212(mark(X1), X2) -> mark(U212(X1, X2))
U221(mark(X1), X2, X3) -> mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) -> mark(U222(X1, X2, X3))
fst(mark(X)) -> mark(fst(X))
U31(mark(X1), X2, X3) -> mark(U31(X1, X2, X3))
U32(mark(X1), X2) -> mark(U32(X1, X2))
U41(mark(X1), X2) -> mark(U41(X1, X2))
U42(mark(X)) -> mark(U42(X))
U51(mark(X1), X2) -> mark(U51(X1, X2))
U52(mark(X)) -> mark(U52(X))
U61(mark(X)) -> mark(U61(X))
U71(mark(X)) -> mark(U71(X))
U81(mark(X)) -> mark(U81(X))
U91(mark(X)) -> mark(U91(X))
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)) -> U101(proper(X1), proper(X2))
proper(tt) -> ok(tt)
proper(U102(X)) -> U102(proper(X))
proper(isLNat(X)) -> isLNat(proper(X))
proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) -> U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) -> U111(proper(X))
proper(snd(X)) -> snd(proper(X))
proper(splitAt(X1, X2)) -> splitAt(proper(X1), proper(X2))
proper(U121(X)) -> U121(proper(X))
proper(U131(X1, X2)) -> U131(proper(X1), proper(X2))
proper(U132(X)) -> U132(proper(X))
proper(U141(X1, X2)) -> U141(proper(X1), proper(X2))
proper(U142(X)) -> U142(proper(X))
proper(U151(X1, X2)) -> U151(proper(X1), proper(X2))
proper(U152(X)) -> U152(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(U172(X1, X2, X3)) -> U172(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(U182(X1, X2)) -> U182(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, X3, X4)) -> U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) -> isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) -> U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) -> U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) -> U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) -> U22(proper(X1), proper(X2))
proper(U211(X1, X2)) -> U211(proper(X1), proper(X2))
proper(U212(X1, X2)) -> U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) -> U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) -> U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) -> fst(proper(X))
proper(U31(X1, X2, X3)) -> U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) -> U32(proper(X1), proper(X2))
proper(U41(X1, X2)) -> U41(proper(X1), proper(X2))
proper(U42(X)) -> U42(proper(X))
proper(U51(X1, X2)) -> U51(proper(X1), proper(X2))
proper(U52(X)) -> U52(proper(X))
proper(U61(X)) -> U61(proper(X))
proper(U71(X)) -> U71(proper(X))
proper(U81(X)) -> U81(proper(X))
proper(U91(X)) -> U91(proper(X))
proper(isPLNat(X)) -> isPLNat(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(U101(X1, X2))
U102(ok(X)) -> ok(U102(X))
isLNat(ok(X)) -> ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))
U111(ok(X)) -> ok(U111(X))
snd(ok(X)) -> ok(snd(X))
splitAt(ok(X1), ok(X2)) -> ok(splitAt(X1, X2))
U121(ok(X)) -> ok(U121(X))
U131(ok(X1), ok(X2)) -> ok(U131(X1, X2))
U132(ok(X)) -> ok(U132(X))
U141(ok(X1), ok(X2)) -> ok(U141(X1, X2))
U142(ok(X)) -> ok(U142(X))
U151(ok(X1), ok(X2)) -> ok(U151(X1, X2))
U152(ok(X)) -> ok(U152(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))
U172(ok(X1), ok(X2), ok(X3)) -> ok(U172(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))
U182(ok(X1), ok(X2)) -> ok(U182(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(X3), ok(X4)) -> ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) -> ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) -> ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) -> ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) -> ok(U22(X1, X2))
U211(ok(X1), ok(X2)) -> ok(U211(X1, X2))
U212(ok(X1), ok(X2)) -> ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) -> ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) -> ok(U222(X1, X2, X3))
fst(ok(X)) -> ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) -> ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) -> ok(U32(X1, X2))
U41(ok(X1), ok(X2)) -> ok(U41(X1, X2))
U42(ok(X)) -> ok(U42(X))
U51(ok(X1), ok(X2)) -> ok(U51(X1, X2))
U52(ok(X)) -> ok(U52(X))
U61(ok(X)) -> ok(U61(X))
U71(ok(X)) -> ok(U71(X))
U81(ok(X)) -> ok(U81(X))
U91(ok(X)) -> ok(U91(X))
isPLNat(ok(X)) -> ok(isPLNat(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))

Types:
active :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U12 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U172 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U182 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U203 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U204 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U22 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U212 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U222 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U32 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok -> top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat -> tt:mark:nil:0':ok


Generator Equations:
gen_tt:mark:nil:0':ok3_0(0) <=> tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) <=> mark(gen_tt:mark:nil:0':ok3_0(x))


The following defined symbols remain to be analysed:
U102, active, isLNat, U12, snd, splitAt, U132, U142, U152, cons, natsFrom, s, U172, head, afterNth, U182, pair, U202, isNatural, U203, U204, U22, U212, U222, fst, U32, U42, U52, U11, U21, U31, U41, U51, U61, isPLNat, U71, U81, U91, U101, U111, U121, U131, U141, U151, U161, U171, U181, U191, U201, U211, U221, tail, take, sel, proper, top

They will be analysed ascendingly in the following order:
U102 < active
isLNat < active
U12 < active
snd < active
splitAt < active
U132 < active
U142 < active
U152 < active
cons < active
natsFrom < active
s < active
U172 < active
head < active
afterNth < active
U182 < active
pair < active
U202 < active
isNatural < active
U203 < active
U204 < active
U22 < active
U212 < active
U222 < active
fst < active
U32 < active
U42 < active
U52 < active
U11 < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
isPLNat < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
U141 < active
U151 < active
U161 < active
U171 < active
U181 < active
U191 < active
U201 < active
U211 < active
U221 < active
tail < active
take < active
sel < active
active < top
U102 < proper
isLNat < proper
U12 < proper
snd < proper
splitAt < proper
U132 < proper
U142 < proper
U152 < proper
cons < proper
natsFrom < proper
s < proper
U172 < proper
head < proper
afterNth < proper
U182 < proper
pair < proper
U202 < proper
isNatural < proper
U203 < proper
U204 < proper
U22 < proper
U212 < proper
U222 < proper
fst < proper
U32 < proper
U42 < proper
U52 < proper
U11 < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
isPLNat < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
U141 < proper
U151 < proper
U161 < proper
U171 < proper
U181 < proper
U191 < proper
U201 < proper
U211 < proper
U221 < proper
tail < proper
take < proper
sel < proper
proper < top

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

(25) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
U102(gen_tt:mark:nil:0':ok3_0(+(1, n5_0))) -> *4_0, rt in Omega(n5_0)

Induction Base:
U102(gen_tt:mark:nil:0':ok3_0(+(1, 0)))

Induction Step:
U102(gen_tt:mark:nil:0':ok3_0(+(1, +(n5_0, 1)))) ->_R^Omega(1)
mark(U102(gen_tt:mark:nil:0':ok3_0(+(1, n5_0)))) ->_IH
mark(*4_0)

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
----------------------------------------

(26)
Complex Obligation (BEST)

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

(27)
Obligation:
Proved the lower bound n^1 for the following obligation:

TRS:
Rules:
active(U101(tt, V2)) -> mark(U102(isLNat(V2)))
active(U102(tt)) -> mark(tt)
active(U11(tt, N, XS)) -> mark(U12(isLNat(XS), N, XS))
active(U111(tt)) -> mark(tt)
active(U12(tt, N, XS)) -> mark(snd(splitAt(N, XS)))
active(U121(tt)) -> mark(tt)
active(U131(tt, V2)) -> mark(U132(isLNat(V2)))
active(U132(tt)) -> mark(tt)
active(U141(tt, V2)) -> mark(U142(isLNat(V2)))
active(U142(tt)) -> mark(tt)
active(U151(tt, V2)) -> mark(U152(isLNat(V2)))
active(U152(tt)) -> mark(tt)
active(U161(tt, N)) -> mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) -> mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) -> mark(head(afterNth(N, XS)))
active(U181(tt, Y)) -> mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) -> mark(Y)
active(U191(tt, XS)) -> mark(pair(nil, XS))
active(U201(tt, N, X, XS)) -> mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) -> mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) -> mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) -> mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) -> mark(U22(isLNat(Y), X))
active(U211(tt, XS)) -> mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) -> mark(XS)
active(U22(tt, X)) -> mark(X)
active(U221(tt, N, XS)) -> mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) -> mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) -> mark(U32(isLNat(XS), N))
active(U32(tt, N)) -> mark(N)
active(U41(tt, V2)) -> mark(U42(isLNat(V2)))
active(U42(tt)) -> mark(tt)
active(U51(tt, V2)) -> mark(U52(isLNat(V2)))
active(U52(tt)) -> mark(tt)
active(U61(tt)) -> mark(tt)
active(U71(tt)) -> mark(tt)
active(U81(tt)) -> mark(tt)
active(U91(tt)) -> mark(tt)
active(afterNth(N, XS)) -> mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) -> mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) -> mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) -> mark(tt)
active(isLNat(afterNth(V1, V2))) -> mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) -> mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) -> mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) -> mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) -> mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) -> mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) -> mark(U101(isNatural(V1), V2))
active(isNatural(0')) -> mark(tt)
active(isNatural(head(V1))) -> mark(U111(isLNat(V1)))
active(isNatural(s(V1))) -> mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) -> mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) -> mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) -> mark(U151(isNatural(V1), V2))
active(natsFrom(N)) -> mark(U161(isNatural(N), N))
active(sel(N, XS)) -> mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) -> mark(U181(isLNat(X), Y))
active(splitAt(0', XS)) -> mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) -> mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) -> mark(U211(isNatural(N), XS))
active(take(N, XS)) -> mark(U221(isNatural(N), N, XS))
active(U101(X1, X2)) -> U101(active(X1), X2)
active(U102(X)) -> U102(active(X))
active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3)
active(U111(X)) -> U111(active(X))
active(snd(X)) -> snd(active(X))
active(splitAt(X1, X2)) -> splitAt(active(X1), X2)
active(splitAt(X1, X2)) -> splitAt(X1, active(X2))
active(U121(X)) -> U121(active(X))
active(U131(X1, X2)) -> U131(active(X1), X2)
active(U132(X)) -> U132(active(X))
active(U141(X1, X2)) -> U141(active(X1), X2)
active(U142(X)) -> U142(active(X))
active(U151(X1, X2)) -> U151(active(X1), X2)
active(U152(X)) -> U152(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(U172(X1, X2, X3)) -> U172(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(U182(X1, X2)) -> U182(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, X3, X4)) -> U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) -> U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) -> U204(active(X1), X2)
active(U21(X1, X2, X3)) -> U21(active(X1), X2, X3)
active(U22(X1, X2)) -> U22(active(X1), X2)
active(U211(X1, X2)) -> U211(active(X1), X2)
active(U212(X1, X2)) -> U212(active(X1), X2)
active(U221(X1, X2, X3)) -> U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) -> U222(active(X1), X2, X3)
active(fst(X)) -> fst(active(X))
active(U31(X1, X2, X3)) -> U31(active(X1), X2, X3)
active(U32(X1, X2)) -> U32(active(X1), X2)
active(U41(X1, X2)) -> U41(active(X1), X2)
active(U42(X)) -> U42(active(X))
active(U51(X1, X2)) -> U51(active(X1), X2)
active(U52(X)) -> U52(active(X))
active(U61(X)) -> U61(active(X))
active(U71(X)) -> U71(active(X))
active(U81(X)) -> U81(active(X))
active(U91(X)) -> U91(active(X))
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) -> mark(U101(X1, X2))
U102(mark(X)) -> mark(U102(X))
U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3))
U111(mark(X)) -> mark(U111(X))
snd(mark(X)) -> mark(snd(X))
splitAt(mark(X1), X2) -> mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) -> mark(splitAt(X1, X2))
U121(mark(X)) -> mark(U121(X))
U131(mark(X1), X2) -> mark(U131(X1, X2))
U132(mark(X)) -> mark(U132(X))
U141(mark(X1), X2) -> mark(U141(X1, X2))
U142(mark(X)) -> mark(U142(X))
U151(mark(X1), X2) -> mark(U151(X1, X2))
U152(mark(X)) -> mark(U152(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))
U172(mark(X1), X2, X3) -> mark(U172(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))
U182(mark(X1), X2) -> mark(U182(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, X3, X4) -> mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) -> mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) -> mark(U204(X1, X2))
U21(mark(X1), X2, X3) -> mark(U21(X1, X2, X3))
U22(mark(X1), X2) -> mark(U22(X1, X2))
U211(mark(X1), X2) -> mark(U211(X1, X2))
U212(mark(X1), X2) -> mark(U212(X1, X2))
U221(mark(X1), X2, X3) -> mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) -> mark(U222(X1, X2, X3))
fst(mark(X)) -> mark(fst(X))
U31(mark(X1), X2, X3) -> mark(U31(X1, X2, X3))
U32(mark(X1), X2) -> mark(U32(X1, X2))
U41(mark(X1), X2) -> mark(U41(X1, X2))
U42(mark(X)) -> mark(U42(X))
U51(mark(X1), X2) -> mark(U51(X1, X2))
U52(mark(X)) -> mark(U52(X))
U61(mark(X)) -> mark(U61(X))
U71(mark(X)) -> mark(U71(X))
U81(mark(X)) -> mark(U81(X))
U91(mark(X)) -> mark(U91(X))
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)) -> U101(proper(X1), proper(X2))
proper(tt) -> ok(tt)
proper(U102(X)) -> U102(proper(X))
proper(isLNat(X)) -> isLNat(proper(X))
proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) -> U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) -> U111(proper(X))
proper(snd(X)) -> snd(proper(X))
proper(splitAt(X1, X2)) -> splitAt(proper(X1), proper(X2))
proper(U121(X)) -> U121(proper(X))
proper(U131(X1, X2)) -> U131(proper(X1), proper(X2))
proper(U132(X)) -> U132(proper(X))
proper(U141(X1, X2)) -> U141(proper(X1), proper(X2))
proper(U142(X)) -> U142(proper(X))
proper(U151(X1, X2)) -> U151(proper(X1), proper(X2))
proper(U152(X)) -> U152(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(U172(X1, X2, X3)) -> U172(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(U182(X1, X2)) -> U182(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, X3, X4)) -> U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) -> isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) -> U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) -> U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) -> U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) -> U22(proper(X1), proper(X2))
proper(U211(X1, X2)) -> U211(proper(X1), proper(X2))
proper(U212(X1, X2)) -> U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) -> U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) -> U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) -> fst(proper(X))
proper(U31(X1, X2, X3)) -> U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) -> U32(proper(X1), proper(X2))
proper(U41(X1, X2)) -> U41(proper(X1), proper(X2))
proper(U42(X)) -> U42(proper(X))
proper(U51(X1, X2)) -> U51(proper(X1), proper(X2))
proper(U52(X)) -> U52(proper(X))
proper(U61(X)) -> U61(proper(X))
proper(U71(X)) -> U71(proper(X))
proper(U81(X)) -> U81(proper(X))
proper(U91(X)) -> U91(proper(X))
proper(isPLNat(X)) -> isPLNat(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(U101(X1, X2))
U102(ok(X)) -> ok(U102(X))
isLNat(ok(X)) -> ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))
U111(ok(X)) -> ok(U111(X))
snd(ok(X)) -> ok(snd(X))
splitAt(ok(X1), ok(X2)) -> ok(splitAt(X1, X2))
U121(ok(X)) -> ok(U121(X))
U131(ok(X1), ok(X2)) -> ok(U131(X1, X2))
U132(ok(X)) -> ok(U132(X))
U141(ok(X1), ok(X2)) -> ok(U141(X1, X2))
U142(ok(X)) -> ok(U142(X))
U151(ok(X1), ok(X2)) -> ok(U151(X1, X2))
U152(ok(X)) -> ok(U152(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))
U172(ok(X1), ok(X2), ok(X3)) -> ok(U172(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))
U182(ok(X1), ok(X2)) -> ok(U182(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(X3), ok(X4)) -> ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) -> ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) -> ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) -> ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) -> ok(U22(X1, X2))
U211(ok(X1), ok(X2)) -> ok(U211(X1, X2))
U212(ok(X1), ok(X2)) -> ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) -> ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) -> ok(U222(X1, X2, X3))
fst(ok(X)) -> ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) -> ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) -> ok(U32(X1, X2))
U41(ok(X1), ok(X2)) -> ok(U41(X1, X2))
U42(ok(X)) -> ok(U42(X))
U51(ok(X1), ok(X2)) -> ok(U51(X1, X2))
U52(ok(X)) -> ok(U52(X))
U61(ok(X)) -> ok(U61(X))
U71(ok(X)) -> ok(U71(X))
U81(ok(X)) -> ok(U81(X))
U91(ok(X)) -> ok(U91(X))
isPLNat(ok(X)) -> ok(isPLNat(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))

Types:
active :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U12 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U172 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U182 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U203 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U204 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U22 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U212 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U222 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U32 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok -> top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat -> tt:mark:nil:0':ok


Generator Equations:
gen_tt:mark:nil:0':ok3_0(0) <=> tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) <=> mark(gen_tt:mark:nil:0':ok3_0(x))


The following defined symbols remain to be analysed:
U102, active, isLNat, U12, snd, splitAt, U132, U142, U152, cons, natsFrom, s, U172, head, afterNth, U182, pair, U202, isNatural, U203, U204, U22, U212, U222, fst, U32, U42, U52, U11, U21, U31, U41, U51, U61, isPLNat, U71, U81, U91, U101, U111, U121, U131, U141, U151, U161, U171, U181, U191, U201, U211, U221, tail, take, sel, proper, top

They will be analysed ascendingly in the following order:
U102 < active
isLNat < active
U12 < active
snd < active
splitAt < active
U132 < active
U142 < active
U152 < active
cons < active
natsFrom < active
s < active
U172 < active
head < active
afterNth < active
U182 < active
pair < active
U202 < active
isNatural < active
U203 < active
U204 < active
U22 < active
U212 < active
U222 < active
fst < active
U32 < active
U42 < active
U52 < active
U11 < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
isPLNat < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
U141 < active
U151 < active
U161 < active
U171 < active
U181 < active
U191 < active
U201 < active
U211 < active
U221 < active
tail < active
take < active
sel < active
active < top
U102 < proper
isLNat < proper
U12 < proper
snd < proper
splitAt < proper
U132 < proper
U142 < proper
U152 < proper
cons < proper
natsFrom < proper
s < proper
U172 < proper
head < proper
afterNth < proper
U182 < proper
pair < proper
U202 < proper
isNatural < proper
U203 < proper
U204 < proper
U22 < proper
U212 < proper
U222 < proper
fst < proper
U32 < proper
U42 < proper
U52 < proper
U11 < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
isPLNat < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
U141 < proper
U151 < proper
U161 < proper
U171 < proper
U181 < proper
U191 < proper
U201 < proper
U211 < proper
U221 < proper
tail < proper
take < proper
sel < proper
proper < top

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

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

(29)
BOUNDS(n^1, INF)

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

(30)
Obligation:
TRS:
Rules:
active(U101(tt, V2)) -> mark(U102(isLNat(V2)))
active(U102(tt)) -> mark(tt)
active(U11(tt, N, XS)) -> mark(U12(isLNat(XS), N, XS))
active(U111(tt)) -> mark(tt)
active(U12(tt, N, XS)) -> mark(snd(splitAt(N, XS)))
active(U121(tt)) -> mark(tt)
active(U131(tt, V2)) -> mark(U132(isLNat(V2)))
active(U132(tt)) -> mark(tt)
active(U141(tt, V2)) -> mark(U142(isLNat(V2)))
active(U142(tt)) -> mark(tt)
active(U151(tt, V2)) -> mark(U152(isLNat(V2)))
active(U152(tt)) -> mark(tt)
active(U161(tt, N)) -> mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) -> mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) -> mark(head(afterNth(N, XS)))
active(U181(tt, Y)) -> mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) -> mark(Y)
active(U191(tt, XS)) -> mark(pair(nil, XS))
active(U201(tt, N, X, XS)) -> mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) -> mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) -> mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) -> mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) -> mark(U22(isLNat(Y), X))
active(U211(tt, XS)) -> mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) -> mark(XS)
active(U22(tt, X)) -> mark(X)
active(U221(tt, N, XS)) -> mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) -> mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) -> mark(U32(isLNat(XS), N))
active(U32(tt, N)) -> mark(N)
active(U41(tt, V2)) -> mark(U42(isLNat(V2)))
active(U42(tt)) -> mark(tt)
active(U51(tt, V2)) -> mark(U52(isLNat(V2)))
active(U52(tt)) -> mark(tt)
active(U61(tt)) -> mark(tt)
active(U71(tt)) -> mark(tt)
active(U81(tt)) -> mark(tt)
active(U91(tt)) -> mark(tt)
active(afterNth(N, XS)) -> mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) -> mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) -> mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) -> mark(tt)
active(isLNat(afterNth(V1, V2))) -> mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) -> mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) -> mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) -> mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) -> mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) -> mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) -> mark(U101(isNatural(V1), V2))
active(isNatural(0')) -> mark(tt)
active(isNatural(head(V1))) -> mark(U111(isLNat(V1)))
active(isNatural(s(V1))) -> mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) -> mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) -> mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) -> mark(U151(isNatural(V1), V2))
active(natsFrom(N)) -> mark(U161(isNatural(N), N))
active(sel(N, XS)) -> mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) -> mark(U181(isLNat(X), Y))
active(splitAt(0', XS)) -> mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) -> mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) -> mark(U211(isNatural(N), XS))
active(take(N, XS)) -> mark(U221(isNatural(N), N, XS))
active(U101(X1, X2)) -> U101(active(X1), X2)
active(U102(X)) -> U102(active(X))
active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3)
active(U111(X)) -> U111(active(X))
active(snd(X)) -> snd(active(X))
active(splitAt(X1, X2)) -> splitAt(active(X1), X2)
active(splitAt(X1, X2)) -> splitAt(X1, active(X2))
active(U121(X)) -> U121(active(X))
active(U131(X1, X2)) -> U131(active(X1), X2)
active(U132(X)) -> U132(active(X))
active(U141(X1, X2)) -> U141(active(X1), X2)
active(U142(X)) -> U142(active(X))
active(U151(X1, X2)) -> U151(active(X1), X2)
active(U152(X)) -> U152(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(U172(X1, X2, X3)) -> U172(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(U182(X1, X2)) -> U182(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, X3, X4)) -> U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) -> U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) -> U204(active(X1), X2)
active(U21(X1, X2, X3)) -> U21(active(X1), X2, X3)
active(U22(X1, X2)) -> U22(active(X1), X2)
active(U211(X1, X2)) -> U211(active(X1), X2)
active(U212(X1, X2)) -> U212(active(X1), X2)
active(U221(X1, X2, X3)) -> U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) -> U222(active(X1), X2, X3)
active(fst(X)) -> fst(active(X))
active(U31(X1, X2, X3)) -> U31(active(X1), X2, X3)
active(U32(X1, X2)) -> U32(active(X1), X2)
active(U41(X1, X2)) -> U41(active(X1), X2)
active(U42(X)) -> U42(active(X))
active(U51(X1, X2)) -> U51(active(X1), X2)
active(U52(X)) -> U52(active(X))
active(U61(X)) -> U61(active(X))
active(U71(X)) -> U71(active(X))
active(U81(X)) -> U81(active(X))
active(U91(X)) -> U91(active(X))
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) -> mark(U101(X1, X2))
U102(mark(X)) -> mark(U102(X))
U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3))
U111(mark(X)) -> mark(U111(X))
snd(mark(X)) -> mark(snd(X))
splitAt(mark(X1), X2) -> mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) -> mark(splitAt(X1, X2))
U121(mark(X)) -> mark(U121(X))
U131(mark(X1), X2) -> mark(U131(X1, X2))
U132(mark(X)) -> mark(U132(X))
U141(mark(X1), X2) -> mark(U141(X1, X2))
U142(mark(X)) -> mark(U142(X))
U151(mark(X1), X2) -> mark(U151(X1, X2))
U152(mark(X)) -> mark(U152(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))
U172(mark(X1), X2, X3) -> mark(U172(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))
U182(mark(X1), X2) -> mark(U182(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, X3, X4) -> mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) -> mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) -> mark(U204(X1, X2))
U21(mark(X1), X2, X3) -> mark(U21(X1, X2, X3))
U22(mark(X1), X2) -> mark(U22(X1, X2))
U211(mark(X1), X2) -> mark(U211(X1, X2))
U212(mark(X1), X2) -> mark(U212(X1, X2))
U221(mark(X1), X2, X3) -> mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) -> mark(U222(X1, X2, X3))
fst(mark(X)) -> mark(fst(X))
U31(mark(X1), X2, X3) -> mark(U31(X1, X2, X3))
U32(mark(X1), X2) -> mark(U32(X1, X2))
U41(mark(X1), X2) -> mark(U41(X1, X2))
U42(mark(X)) -> mark(U42(X))
U51(mark(X1), X2) -> mark(U51(X1, X2))
U52(mark(X)) -> mark(U52(X))
U61(mark(X)) -> mark(U61(X))
U71(mark(X)) -> mark(U71(X))
U81(mark(X)) -> mark(U81(X))
U91(mark(X)) -> mark(U91(X))
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)) -> U101(proper(X1), proper(X2))
proper(tt) -> ok(tt)
proper(U102(X)) -> U102(proper(X))
proper(isLNat(X)) -> isLNat(proper(X))
proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) -> U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) -> U111(proper(X))
proper(snd(X)) -> snd(proper(X))
proper(splitAt(X1, X2)) -> splitAt(proper(X1), proper(X2))
proper(U121(X)) -> U121(proper(X))
proper(U131(X1, X2)) -> U131(proper(X1), proper(X2))
proper(U132(X)) -> U132(proper(X))
proper(U141(X1, X2)) -> U141(proper(X1), proper(X2))
proper(U142(X)) -> U142(proper(X))
proper(U151(X1, X2)) -> U151(proper(X1), proper(X2))
proper(U152(X)) -> U152(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(U172(X1, X2, X3)) -> U172(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(U182(X1, X2)) -> U182(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, X3, X4)) -> U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) -> isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) -> U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) -> U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) -> U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) -> U22(proper(X1), proper(X2))
proper(U211(X1, X2)) -> U211(proper(X1), proper(X2))
proper(U212(X1, X2)) -> U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) -> U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) -> U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) -> fst(proper(X))
proper(U31(X1, X2, X3)) -> U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) -> U32(proper(X1), proper(X2))
proper(U41(X1, X2)) -> U41(proper(X1), proper(X2))
proper(U42(X)) -> U42(proper(X))
proper(U51(X1, X2)) -> U51(proper(X1), proper(X2))
proper(U52(X)) -> U52(proper(X))
proper(U61(X)) -> U61(proper(X))
proper(U71(X)) -> U71(proper(X))
proper(U81(X)) -> U81(proper(X))
proper(U91(X)) -> U91(proper(X))
proper(isPLNat(X)) -> isPLNat(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(U101(X1, X2))
U102(ok(X)) -> ok(U102(X))
isLNat(ok(X)) -> ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))
U111(ok(X)) -> ok(U111(X))
snd(ok(X)) -> ok(snd(X))
splitAt(ok(X1), ok(X2)) -> ok(splitAt(X1, X2))
U121(ok(X)) -> ok(U121(X))
U131(ok(X1), ok(X2)) -> ok(U131(X1, X2))
U132(ok(X)) -> ok(U132(X))
U141(ok(X1), ok(X2)) -> ok(U141(X1, X2))
U142(ok(X)) -> ok(U142(X))
U151(ok(X1), ok(X2)) -> ok(U151(X1, X2))
U152(ok(X)) -> ok(U152(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))
U172(ok(X1), ok(X2), ok(X3)) -> ok(U172(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))
U182(ok(X1), ok(X2)) -> ok(U182(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(X3), ok(X4)) -> ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) -> ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) -> ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) -> ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) -> ok(U22(X1, X2))
U211(ok(X1), ok(X2)) -> ok(U211(X1, X2))
U212(ok(X1), ok(X2)) -> ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) -> ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) -> ok(U222(X1, X2, X3))
fst(ok(X)) -> ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) -> ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) -> ok(U32(X1, X2))
U41(ok(X1), ok(X2)) -> ok(U41(X1, X2))
U42(ok(X)) -> ok(U42(X))
U51(ok(X1), ok(X2)) -> ok(U51(X1, X2))
U52(ok(X)) -> ok(U52(X))
U61(ok(X)) -> ok(U61(X))
U71(ok(X)) -> ok(U71(X))
U81(ok(X)) -> ok(U81(X))
U91(ok(X)) -> ok(U91(X))
isPLNat(ok(X)) -> ok(isPLNat(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))

Types:
active :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U12 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U172 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U182 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U203 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U204 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U22 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U212 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U222 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U32 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok -> top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat -> tt:mark:nil:0':ok


Lemmas:
U102(gen_tt:mark:nil:0':ok3_0(+(1, n5_0))) -> *4_0, rt in Omega(n5_0)


Generator Equations:
gen_tt:mark:nil:0':ok3_0(0) <=> tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) <=> mark(gen_tt:mark:nil:0':ok3_0(x))


The following defined symbols remain to be analysed:
isLNat, active, U12, snd, splitAt, U132, U142, U152, cons, natsFrom, s, U172, head, afterNth, U182, pair, U202, isNatural, U203, U204, U22, U212, U222, fst, U32, U42, U52, U11, U21, U31, U41, U51, U61, isPLNat, U71, U81, U91, U101, U111, U121, U131, U141, U151, U161, U171, U181, U191, U201, U211, U221, tail, take, sel, proper, top

They will be analysed ascendingly in the following order:
isLNat < active
U12 < active
snd < active
splitAt < active
U132 < active
U142 < active
U152 < active
cons < active
natsFrom < active
s < active
U172 < active
head < active
afterNth < active
U182 < active
pair < active
U202 < active
isNatural < active
U203 < active
U204 < active
U22 < active
U212 < active
U222 < active
fst < active
U32 < active
U42 < active
U52 < active
U11 < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
isPLNat < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
U141 < active
U151 < active
U161 < active
U171 < active
U181 < active
U191 < active
U201 < active
U211 < active
U221 < active
tail < active
take < active
sel < active
active < top
isLNat < proper
U12 < proper
snd < proper
splitAt < proper
U132 < proper
U142 < proper
U152 < proper
cons < proper
natsFrom < proper
s < proper
U172 < proper
head < proper
afterNth < proper
U182 < proper
pair < proper
U202 < proper
isNatural < proper
U203 < proper
U204 < proper
U22 < proper
U212 < proper
U222 < proper
fst < proper
U32 < proper
U42 < proper
U52 < proper
U11 < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
isPLNat < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
U141 < proper
U151 < proper
U161 < proper
U171 < proper
U181 < proper
U191 < proper
U201 < proper
U211 < proper
U221 < proper
tail < proper
take < proper
sel < proper
proper < top

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

(31) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
U12(gen_tt:mark:nil:0':ok3_0(+(1, n1128_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) -> *4_0, rt in Omega(n1128_0)

Induction Base:
U12(gen_tt:mark:nil:0':ok3_0(+(1, 0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c))

Induction Step:
U12(gen_tt:mark:nil:0':ok3_0(+(1, +(n1128_0, 1))), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) ->_R^Omega(1)
mark(U12(gen_tt:mark:nil:0':ok3_0(+(1, n1128_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c))) ->_IH
mark(*4_0)

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
----------------------------------------

(32)
Obligation:
TRS:
Rules:
active(U101(tt, V2)) -> mark(U102(isLNat(V2)))
active(U102(tt)) -> mark(tt)
active(U11(tt, N, XS)) -> mark(U12(isLNat(XS), N, XS))
active(U111(tt)) -> mark(tt)
active(U12(tt, N, XS)) -> mark(snd(splitAt(N, XS)))
active(U121(tt)) -> mark(tt)
active(U131(tt, V2)) -> mark(U132(isLNat(V2)))
active(U132(tt)) -> mark(tt)
active(U141(tt, V2)) -> mark(U142(isLNat(V2)))
active(U142(tt)) -> mark(tt)
active(U151(tt, V2)) -> mark(U152(isLNat(V2)))
active(U152(tt)) -> mark(tt)
active(U161(tt, N)) -> mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) -> mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) -> mark(head(afterNth(N, XS)))
active(U181(tt, Y)) -> mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) -> mark(Y)
active(U191(tt, XS)) -> mark(pair(nil, XS))
active(U201(tt, N, X, XS)) -> mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) -> mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) -> mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) -> mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) -> mark(U22(isLNat(Y), X))
active(U211(tt, XS)) -> mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) -> mark(XS)
active(U22(tt, X)) -> mark(X)
active(U221(tt, N, XS)) -> mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) -> mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) -> mark(U32(isLNat(XS), N))
active(U32(tt, N)) -> mark(N)
active(U41(tt, V2)) -> mark(U42(isLNat(V2)))
active(U42(tt)) -> mark(tt)
active(U51(tt, V2)) -> mark(U52(isLNat(V2)))
active(U52(tt)) -> mark(tt)
active(U61(tt)) -> mark(tt)
active(U71(tt)) -> mark(tt)
active(U81(tt)) -> mark(tt)
active(U91(tt)) -> mark(tt)
active(afterNth(N, XS)) -> mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) -> mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) -> mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) -> mark(tt)
active(isLNat(afterNth(V1, V2))) -> mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) -> mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) -> mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) -> mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) -> mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) -> mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) -> mark(U101(isNatural(V1), V2))
active(isNatural(0')) -> mark(tt)
active(isNatural(head(V1))) -> mark(U111(isLNat(V1)))
active(isNatural(s(V1))) -> mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) -> mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) -> mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) -> mark(U151(isNatural(V1), V2))
active(natsFrom(N)) -> mark(U161(isNatural(N), N))
active(sel(N, XS)) -> mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) -> mark(U181(isLNat(X), Y))
active(splitAt(0', XS)) -> mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) -> mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) -> mark(U211(isNatural(N), XS))
active(take(N, XS)) -> mark(U221(isNatural(N), N, XS))
active(U101(X1, X2)) -> U101(active(X1), X2)
active(U102(X)) -> U102(active(X))
active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3)
active(U111(X)) -> U111(active(X))
active(snd(X)) -> snd(active(X))
active(splitAt(X1, X2)) -> splitAt(active(X1), X2)
active(splitAt(X1, X2)) -> splitAt(X1, active(X2))
active(U121(X)) -> U121(active(X))
active(U131(X1, X2)) -> U131(active(X1), X2)
active(U132(X)) -> U132(active(X))
active(U141(X1, X2)) -> U141(active(X1), X2)
active(U142(X)) -> U142(active(X))
active(U151(X1, X2)) -> U151(active(X1), X2)
active(U152(X)) -> U152(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(U172(X1, X2, X3)) -> U172(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(U182(X1, X2)) -> U182(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, X3, X4)) -> U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) -> U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) -> U204(active(X1), X2)
active(U21(X1, X2, X3)) -> U21(active(X1), X2, X3)
active(U22(X1, X2)) -> U22(active(X1), X2)
active(U211(X1, X2)) -> U211(active(X1), X2)
active(U212(X1, X2)) -> U212(active(X1), X2)
active(U221(X1, X2, X3)) -> U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) -> U222(active(X1), X2, X3)
active(fst(X)) -> fst(active(X))
active(U31(X1, X2, X3)) -> U31(active(X1), X2, X3)
active(U32(X1, X2)) -> U32(active(X1), X2)
active(U41(X1, X2)) -> U41(active(X1), X2)
active(U42(X)) -> U42(active(X))
active(U51(X1, X2)) -> U51(active(X1), X2)
active(U52(X)) -> U52(active(X))
active(U61(X)) -> U61(active(X))
active(U71(X)) -> U71(active(X))
active(U81(X)) -> U81(active(X))
active(U91(X)) -> U91(active(X))
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) -> mark(U101(X1, X2))
U102(mark(X)) -> mark(U102(X))
U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3))
U111(mark(X)) -> mark(U111(X))
snd(mark(X)) -> mark(snd(X))
splitAt(mark(X1), X2) -> mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) -> mark(splitAt(X1, X2))
U121(mark(X)) -> mark(U121(X))
U131(mark(X1), X2) -> mark(U131(X1, X2))
U132(mark(X)) -> mark(U132(X))
U141(mark(X1), X2) -> mark(U141(X1, X2))
U142(mark(X)) -> mark(U142(X))
U151(mark(X1), X2) -> mark(U151(X1, X2))
U152(mark(X)) -> mark(U152(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))
U172(mark(X1), X2, X3) -> mark(U172(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))
U182(mark(X1), X2) -> mark(U182(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, X3, X4) -> mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) -> mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) -> mark(U204(X1, X2))
U21(mark(X1), X2, X3) -> mark(U21(X1, X2, X3))
U22(mark(X1), X2) -> mark(U22(X1, X2))
U211(mark(X1), X2) -> mark(U211(X1, X2))
U212(mark(X1), X2) -> mark(U212(X1, X2))
U221(mark(X1), X2, X3) -> mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) -> mark(U222(X1, X2, X3))
fst(mark(X)) -> mark(fst(X))
U31(mark(X1), X2, X3) -> mark(U31(X1, X2, X3))
U32(mark(X1), X2) -> mark(U32(X1, X2))
U41(mark(X1), X2) -> mark(U41(X1, X2))
U42(mark(X)) -> mark(U42(X))
U51(mark(X1), X2) -> mark(U51(X1, X2))
U52(mark(X)) -> mark(U52(X))
U61(mark(X)) -> mark(U61(X))
U71(mark(X)) -> mark(U71(X))
U81(mark(X)) -> mark(U81(X))
U91(mark(X)) -> mark(U91(X))
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)) -> U101(proper(X1), proper(X2))
proper(tt) -> ok(tt)
proper(U102(X)) -> U102(proper(X))
proper(isLNat(X)) -> isLNat(proper(X))
proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) -> U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) -> U111(proper(X))
proper(snd(X)) -> snd(proper(X))
proper(splitAt(X1, X2)) -> splitAt(proper(X1), proper(X2))
proper(U121(X)) -> U121(proper(X))
proper(U131(X1, X2)) -> U131(proper(X1), proper(X2))
proper(U132(X)) -> U132(proper(X))
proper(U141(X1, X2)) -> U141(proper(X1), proper(X2))
proper(U142(X)) -> U142(proper(X))
proper(U151(X1, X2)) -> U151(proper(X1), proper(X2))
proper(U152(X)) -> U152(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(U172(X1, X2, X3)) -> U172(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(U182(X1, X2)) -> U182(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, X3, X4)) -> U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) -> isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) -> U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) -> U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) -> U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) -> U22(proper(X1), proper(X2))
proper(U211(X1, X2)) -> U211(proper(X1), proper(X2))
proper(U212(X1, X2)) -> U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) -> U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) -> U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) -> fst(proper(X))
proper(U31(X1, X2, X3)) -> U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) -> U32(proper(X1), proper(X2))
proper(U41(X1, X2)) -> U41(proper(X1), proper(X2))
proper(U42(X)) -> U42(proper(X))
proper(U51(X1, X2)) -> U51(proper(X1), proper(X2))
proper(U52(X)) -> U52(proper(X))
proper(U61(X)) -> U61(proper(X))
proper(U71(X)) -> U71(proper(X))
proper(U81(X)) -> U81(proper(X))
proper(U91(X)) -> U91(proper(X))
proper(isPLNat(X)) -> isPLNat(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(U101(X1, X2))
U102(ok(X)) -> ok(U102(X))
isLNat(ok(X)) -> ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))
U111(ok(X)) -> ok(U111(X))
snd(ok(X)) -> ok(snd(X))
splitAt(ok(X1), ok(X2)) -> ok(splitAt(X1, X2))
U121(ok(X)) -> ok(U121(X))
U131(ok(X1), ok(X2)) -> ok(U131(X1, X2))
U132(ok(X)) -> ok(U132(X))
U141(ok(X1), ok(X2)) -> ok(U141(X1, X2))
U142(ok(X)) -> ok(U142(X))
U151(ok(X1), ok(X2)) -> ok(U151(X1, X2))
U152(ok(X)) -> ok(U152(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))
U172(ok(X1), ok(X2), ok(X3)) -> ok(U172(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))
U182(ok(X1), ok(X2)) -> ok(U182(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(X3), ok(X4)) -> ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) -> ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) -> ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) -> ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) -> ok(U22(X1, X2))
U211(ok(X1), ok(X2)) -> ok(U211(X1, X2))
U212(ok(X1), ok(X2)) -> ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) -> ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) -> ok(U222(X1, X2, X3))
fst(ok(X)) -> ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) -> ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) -> ok(U32(X1, X2))
U41(ok(X1), ok(X2)) -> ok(U41(X1, X2))
U42(ok(X)) -> ok(U42(X))
U51(ok(X1), ok(X2)) -> ok(U51(X1, X2))
U52(ok(X)) -> ok(U52(X))
U61(ok(X)) -> ok(U61(X))
U71(ok(X)) -> ok(U71(X))
U81(ok(X)) -> ok(U81(X))
U91(ok(X)) -> ok(U91(X))
isPLNat(ok(X)) -> ok(isPLNat(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))

Types:
active :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U12 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U172 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U182 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U203 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U204 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U22 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U212 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U222 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U32 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok -> top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat -> tt:mark:nil:0':ok


Lemmas:
U102(gen_tt:mark:nil:0':ok3_0(+(1, n5_0))) -> *4_0, rt in Omega(n5_0)
U12(gen_tt:mark:nil:0':ok3_0(+(1, n1128_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) -> *4_0, rt in Omega(n1128_0)


Generator Equations:
gen_tt:mark:nil:0':ok3_0(0) <=> tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) <=> mark(gen_tt:mark:nil:0':ok3_0(x))


The following defined symbols remain to be analysed:
snd, active, splitAt, U132, U142, U152, cons, natsFrom, s, U172, head, afterNth, U182, pair, U202, isNatural, U203, U204, U22, U212, U222, fst, U32, U42, U52, U11, U21, U31, U41, U51, U61, isPLNat, U71, U81, U91, U101, U111, U121, U131, U141, U151, U161, U171, U181, U191, U201, U211, U221, tail, take, sel, proper, top

They will be analysed ascendingly in the following order:
snd < active
splitAt < active
U132 < active
U142 < active
U152 < active
cons < active
natsFrom < active
s < active
U172 < active
head < active
afterNth < active
U182 < active
pair < active
U202 < active
isNatural < active
U203 < active
U204 < active
U22 < active
U212 < active
U222 < active
fst < active
U32 < active
U42 < active
U52 < active
U11 < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
isPLNat < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
U141 < active
U151 < active
U161 < active
U171 < active
U181 < active
U191 < active
U201 < active
U211 < active
U221 < active
tail < active
take < active
sel < active
active < top
snd < proper
splitAt < proper
U132 < proper
U142 < proper
U152 < proper
cons < proper
natsFrom < proper
s < proper
U172 < proper
head < proper
afterNth < proper
U182 < proper
pair < proper
U202 < proper
isNatural < proper
U203 < proper
U204 < proper
U22 < proper
U212 < proper
U222 < proper
fst < proper
U32 < proper
U42 < proper
U52 < proper
U11 < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
isPLNat < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
U141 < proper
U151 < proper
U161 < proper
U171 < proper
U181 < proper
U191 < proper
U201 < proper
U211 < proper
U221 < proper
tail < proper
take < proper
sel < proper
proper < top

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

(33) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
snd(gen_tt:mark:nil:0':ok3_0(+(1, n9121_0))) -> *4_0, rt in Omega(n9121_0)

Induction Base:
snd(gen_tt:mark:nil:0':ok3_0(+(1, 0)))

Induction Step:
snd(gen_tt:mark:nil:0':ok3_0(+(1, +(n9121_0, 1)))) ->_R^Omega(1)
mark(snd(gen_tt:mark:nil:0':ok3_0(+(1, n9121_0)))) ->_IH
mark(*4_0)

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
----------------------------------------

(34)
Obligation:
TRS:
Rules:
active(U101(tt, V2)) -> mark(U102(isLNat(V2)))
active(U102(tt)) -> mark(tt)
active(U11(tt, N, XS)) -> mark(U12(isLNat(XS), N, XS))
active(U111(tt)) -> mark(tt)
active(U12(tt, N, XS)) -> mark(snd(splitAt(N, XS)))
active(U121(tt)) -> mark(tt)
active(U131(tt, V2)) -> mark(U132(isLNat(V2)))
active(U132(tt)) -> mark(tt)
active(U141(tt, V2)) -> mark(U142(isLNat(V2)))
active(U142(tt)) -> mark(tt)
active(U151(tt, V2)) -> mark(U152(isLNat(V2)))
active(U152(tt)) -> mark(tt)
active(U161(tt, N)) -> mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) -> mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) -> mark(head(afterNth(N, XS)))
active(U181(tt, Y)) -> mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) -> mark(Y)
active(U191(tt, XS)) -> mark(pair(nil, XS))
active(U201(tt, N, X, XS)) -> mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) -> mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) -> mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) -> mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) -> mark(U22(isLNat(Y), X))
active(U211(tt, XS)) -> mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) -> mark(XS)
active(U22(tt, X)) -> mark(X)
active(U221(tt, N, XS)) -> mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) -> mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) -> mark(U32(isLNat(XS), N))
active(U32(tt, N)) -> mark(N)
active(U41(tt, V2)) -> mark(U42(isLNat(V2)))
active(U42(tt)) -> mark(tt)
active(U51(tt, V2)) -> mark(U52(isLNat(V2)))
active(U52(tt)) -> mark(tt)
active(U61(tt)) -> mark(tt)
active(U71(tt)) -> mark(tt)
active(U81(tt)) -> mark(tt)
active(U91(tt)) -> mark(tt)
active(afterNth(N, XS)) -> mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) -> mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) -> mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) -> mark(tt)
active(isLNat(afterNth(V1, V2))) -> mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) -> mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) -> mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) -> mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) -> mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) -> mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) -> mark(U101(isNatural(V1), V2))
active(isNatural(0')) -> mark(tt)
active(isNatural(head(V1))) -> mark(U111(isLNat(V1)))
active(isNatural(s(V1))) -> mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) -> mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) -> mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) -> mark(U151(isNatural(V1), V2))
active(natsFrom(N)) -> mark(U161(isNatural(N), N))
active(sel(N, XS)) -> mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) -> mark(U181(isLNat(X), Y))
active(splitAt(0', XS)) -> mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) -> mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) -> mark(U211(isNatural(N), XS))
active(take(N, XS)) -> mark(U221(isNatural(N), N, XS))
active(U101(X1, X2)) -> U101(active(X1), X2)
active(U102(X)) -> U102(active(X))
active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3)
active(U111(X)) -> U111(active(X))
active(snd(X)) -> snd(active(X))
active(splitAt(X1, X2)) -> splitAt(active(X1), X2)
active(splitAt(X1, X2)) -> splitAt(X1, active(X2))
active(U121(X)) -> U121(active(X))
active(U131(X1, X2)) -> U131(active(X1), X2)
active(U132(X)) -> U132(active(X))
active(U141(X1, X2)) -> U141(active(X1), X2)
active(U142(X)) -> U142(active(X))
active(U151(X1, X2)) -> U151(active(X1), X2)
active(U152(X)) -> U152(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(U172(X1, X2, X3)) -> U172(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(U182(X1, X2)) -> U182(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, X3, X4)) -> U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) -> U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) -> U204(active(X1), X2)
active(U21(X1, X2, X3)) -> U21(active(X1), X2, X3)
active(U22(X1, X2)) -> U22(active(X1), X2)
active(U211(X1, X2)) -> U211(active(X1), X2)
active(U212(X1, X2)) -> U212(active(X1), X2)
active(U221(X1, X2, X3)) -> U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) -> U222(active(X1), X2, X3)
active(fst(X)) -> fst(active(X))
active(U31(X1, X2, X3)) -> U31(active(X1), X2, X3)
active(U32(X1, X2)) -> U32(active(X1), X2)
active(U41(X1, X2)) -> U41(active(X1), X2)
active(U42(X)) -> U42(active(X))
active(U51(X1, X2)) -> U51(active(X1), X2)
active(U52(X)) -> U52(active(X))
active(U61(X)) -> U61(active(X))
active(U71(X)) -> U71(active(X))
active(U81(X)) -> U81(active(X))
active(U91(X)) -> U91(active(X))
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) -> mark(U101(X1, X2))
U102(mark(X)) -> mark(U102(X))
U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3))
U111(mark(X)) -> mark(U111(X))
snd(mark(X)) -> mark(snd(X))
splitAt(mark(X1), X2) -> mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) -> mark(splitAt(X1, X2))
U121(mark(X)) -> mark(U121(X))
U131(mark(X1), X2) -> mark(U131(X1, X2))
U132(mark(X)) -> mark(U132(X))
U141(mark(X1), X2) -> mark(U141(X1, X2))
U142(mark(X)) -> mark(U142(X))
U151(mark(X1), X2) -> mark(U151(X1, X2))
U152(mark(X)) -> mark(U152(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))
U172(mark(X1), X2, X3) -> mark(U172(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))
U182(mark(X1), X2) -> mark(U182(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, X3, X4) -> mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) -> mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) -> mark(U204(X1, X2))
U21(mark(X1), X2, X3) -> mark(U21(X1, X2, X3))
U22(mark(X1), X2) -> mark(U22(X1, X2))
U211(mark(X1), X2) -> mark(U211(X1, X2))
U212(mark(X1), X2) -> mark(U212(X1, X2))
U221(mark(X1), X2, X3) -> mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) -> mark(U222(X1, X2, X3))
fst(mark(X)) -> mark(fst(X))
U31(mark(X1), X2, X3) -> mark(U31(X1, X2, X3))
U32(mark(X1), X2) -> mark(U32(X1, X2))
U41(mark(X1), X2) -> mark(U41(X1, X2))
U42(mark(X)) -> mark(U42(X))
U51(mark(X1), X2) -> mark(U51(X1, X2))
U52(mark(X)) -> mark(U52(X))
U61(mark(X)) -> mark(U61(X))
U71(mark(X)) -> mark(U71(X))
U81(mark(X)) -> mark(U81(X))
U91(mark(X)) -> mark(U91(X))
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)) -> U101(proper(X1), proper(X2))
proper(tt) -> ok(tt)
proper(U102(X)) -> U102(proper(X))
proper(isLNat(X)) -> isLNat(proper(X))
proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) -> U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) -> U111(proper(X))
proper(snd(X)) -> snd(proper(X))
proper(splitAt(X1, X2)) -> splitAt(proper(X1), proper(X2))
proper(U121(X)) -> U121(proper(X))
proper(U131(X1, X2)) -> U131(proper(X1), proper(X2))
proper(U132(X)) -> U132(proper(X))
proper(U141(X1, X2)) -> U141(proper(X1), proper(X2))
proper(U142(X)) -> U142(proper(X))
proper(U151(X1, X2)) -> U151(proper(X1), proper(X2))
proper(U152(X)) -> U152(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(U172(X1, X2, X3)) -> U172(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(U182(X1, X2)) -> U182(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, X3, X4)) -> U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) -> isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) -> U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) -> U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) -> U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) -> U22(proper(X1), proper(X2))
proper(U211(X1, X2)) -> U211(proper(X1), proper(X2))
proper(U212(X1, X2)) -> U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) -> U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) -> U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) -> fst(proper(X))
proper(U31(X1, X2, X3)) -> U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) -> U32(proper(X1), proper(X2))
proper(U41(X1, X2)) -> U41(proper(X1), proper(X2))
proper(U42(X)) -> U42(proper(X))
proper(U51(X1, X2)) -> U51(proper(X1), proper(X2))
proper(U52(X)) -> U52(proper(X))
proper(U61(X)) -> U61(proper(X))
proper(U71(X)) -> U71(proper(X))
proper(U81(X)) -> U81(proper(X))
proper(U91(X)) -> U91(proper(X))
proper(isPLNat(X)) -> isPLNat(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(U101(X1, X2))
U102(ok(X)) -> ok(U102(X))
isLNat(ok(X)) -> ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))
U111(ok(X)) -> ok(U111(X))
snd(ok(X)) -> ok(snd(X))
splitAt(ok(X1), ok(X2)) -> ok(splitAt(X1, X2))
U121(ok(X)) -> ok(U121(X))
U131(ok(X1), ok(X2)) -> ok(U131(X1, X2))
U132(ok(X)) -> ok(U132(X))
U141(ok(X1), ok(X2)) -> ok(U141(X1, X2))
U142(ok(X)) -> ok(U142(X))
U151(ok(X1), ok(X2)) -> ok(U151(X1, X2))
U152(ok(X)) -> ok(U152(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))
U172(ok(X1), ok(X2), ok(X3)) -> ok(U172(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))
U182(ok(X1), ok(X2)) -> ok(U182(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(X3), ok(X4)) -> ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) -> ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) -> ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) -> ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) -> ok(U22(X1, X2))
U211(ok(X1), ok(X2)) -> ok(U211(X1, X2))
U212(ok(X1), ok(X2)) -> ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) -> ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) -> ok(U222(X1, X2, X3))
fst(ok(X)) -> ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) -> ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) -> ok(U32(X1, X2))
U41(ok(X1), ok(X2)) -> ok(U41(X1, X2))
U42(ok(X)) -> ok(U42(X))
U51(ok(X1), ok(X2)) -> ok(U51(X1, X2))
U52(ok(X)) -> ok(U52(X))
U61(ok(X)) -> ok(U61(X))
U71(ok(X)) -> ok(U71(X))
U81(ok(X)) -> ok(U81(X))
U91(ok(X)) -> ok(U91(X))
isPLNat(ok(X)) -> ok(isPLNat(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))

Types:
active :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U12 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U172 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U182 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U203 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U204 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U22 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U212 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U222 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U32 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok -> top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat -> tt:mark:nil:0':ok


Lemmas:
U102(gen_tt:mark:nil:0':ok3_0(+(1, n5_0))) -> *4_0, rt in Omega(n5_0)
U12(gen_tt:mark:nil:0':ok3_0(+(1, n1128_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) -> *4_0, rt in Omega(n1128_0)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n9121_0))) -> *4_0, rt in Omega(n9121_0)


Generator Equations:
gen_tt:mark:nil:0':ok3_0(0) <=> tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) <=> mark(gen_tt:mark:nil:0':ok3_0(x))


The following defined symbols remain to be analysed:
splitAt, active, U132, U142, U152, cons, natsFrom, s, U172, head, afterNth, U182, pair, U202, isNatural, U203, U204, U22, U212, U222, fst, U32, U42, U52, U11, U21, U31, U41, U51, U61, isPLNat, U71, U81, U91, U101, U111, U121, U131, U141, U151, U161, U171, U181, U191, U201, U211, U221, tail, take, sel, proper, top

They will be analysed ascendingly in the following order:
splitAt < active
U132 < active
U142 < active
U152 < active
cons < active
natsFrom < active
s < active
U172 < active
head < active
afterNth < active
U182 < active
pair < active
U202 < active
isNatural < active
U203 < active
U204 < active
U22 < active
U212 < active
U222 < active
fst < active
U32 < active
U42 < active
U52 < active
U11 < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
isPLNat < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
U141 < active
U151 < active
U161 < active
U171 < active
U181 < active
U191 < active
U201 < active
U211 < active
U221 < active
tail < active
take < active
sel < active
active < top
splitAt < proper
U132 < proper
U142 < proper
U152 < proper
cons < proper
natsFrom < proper
s < proper
U172 < proper
head < proper
afterNth < proper
U182 < proper
pair < proper
U202 < proper
isNatural < proper
U203 < proper
U204 < proper
U22 < proper
U212 < proper
U222 < proper
fst < proper
U32 < proper
U42 < proper
U52 < proper
U11 < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
isPLNat < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
U141 < proper
U151 < proper
U161 < proper
U171 < proper
U181 < proper
U191 < proper
U201 < proper
U211 < proper
U221 < proper
tail < proper
take < proper
sel < proper
proper < top

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

(35) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n10537_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n10537_0)

Induction Base:
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, 0)), gen_tt:mark:nil:0':ok3_0(b))

Induction Step:
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, +(n10537_0, 1))), gen_tt:mark:nil:0':ok3_0(b)) ->_R^Omega(1)
mark(splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n10537_0)), gen_tt:mark:nil:0':ok3_0(b))) ->_IH
mark(*4_0)

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
----------------------------------------

(36)
Obligation:
TRS:
Rules:
active(U101(tt, V2)) -> mark(U102(isLNat(V2)))
active(U102(tt)) -> mark(tt)
active(U11(tt, N, XS)) -> mark(U12(isLNat(XS), N, XS))
active(U111(tt)) -> mark(tt)
active(U12(tt, N, XS)) -> mark(snd(splitAt(N, XS)))
active(U121(tt)) -> mark(tt)
active(U131(tt, V2)) -> mark(U132(isLNat(V2)))
active(U132(tt)) -> mark(tt)
active(U141(tt, V2)) -> mark(U142(isLNat(V2)))
active(U142(tt)) -> mark(tt)
active(U151(tt, V2)) -> mark(U152(isLNat(V2)))
active(U152(tt)) -> mark(tt)
active(U161(tt, N)) -> mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) -> mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) -> mark(head(afterNth(N, XS)))
active(U181(tt, Y)) -> mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) -> mark(Y)
active(U191(tt, XS)) -> mark(pair(nil, XS))
active(U201(tt, N, X, XS)) -> mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) -> mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) -> mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) -> mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) -> mark(U22(isLNat(Y), X))
active(U211(tt, XS)) -> mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) -> mark(XS)
active(U22(tt, X)) -> mark(X)
active(U221(tt, N, XS)) -> mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) -> mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) -> mark(U32(isLNat(XS), N))
active(U32(tt, N)) -> mark(N)
active(U41(tt, V2)) -> mark(U42(isLNat(V2)))
active(U42(tt)) -> mark(tt)
active(U51(tt, V2)) -> mark(U52(isLNat(V2)))
active(U52(tt)) -> mark(tt)
active(U61(tt)) -> mark(tt)
active(U71(tt)) -> mark(tt)
active(U81(tt)) -> mark(tt)
active(U91(tt)) -> mark(tt)
active(afterNth(N, XS)) -> mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) -> mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) -> mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) -> mark(tt)
active(isLNat(afterNth(V1, V2))) -> mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) -> mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) -> mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) -> mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) -> mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) -> mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) -> mark(U101(isNatural(V1), V2))
active(isNatural(0')) -> mark(tt)
active(isNatural(head(V1))) -> mark(U111(isLNat(V1)))
active(isNatural(s(V1))) -> mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) -> mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) -> mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) -> mark(U151(isNatural(V1), V2))
active(natsFrom(N)) -> mark(U161(isNatural(N), N))
active(sel(N, XS)) -> mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) -> mark(U181(isLNat(X), Y))
active(splitAt(0', XS)) -> mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) -> mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) -> mark(U211(isNatural(N), XS))
active(take(N, XS)) -> mark(U221(isNatural(N), N, XS))
active(U101(X1, X2)) -> U101(active(X1), X2)
active(U102(X)) -> U102(active(X))
active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3)
active(U111(X)) -> U111(active(X))
active(snd(X)) -> snd(active(X))
active(splitAt(X1, X2)) -> splitAt(active(X1), X2)
active(splitAt(X1, X2)) -> splitAt(X1, active(X2))
active(U121(X)) -> U121(active(X))
active(U131(X1, X2)) -> U131(active(X1), X2)
active(U132(X)) -> U132(active(X))
active(U141(X1, X2)) -> U141(active(X1), X2)
active(U142(X)) -> U142(active(X))
active(U151(X1, X2)) -> U151(active(X1), X2)
active(U152(X)) -> U152(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(U172(X1, X2, X3)) -> U172(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(U182(X1, X2)) -> U182(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, X3, X4)) -> U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) -> U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) -> U204(active(X1), X2)
active(U21(X1, X2, X3)) -> U21(active(X1), X2, X3)
active(U22(X1, X2)) -> U22(active(X1), X2)
active(U211(X1, X2)) -> U211(active(X1), X2)
active(U212(X1, X2)) -> U212(active(X1), X2)
active(U221(X1, X2, X3)) -> U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) -> U222(active(X1), X2, X3)
active(fst(X)) -> fst(active(X))
active(U31(X1, X2, X3)) -> U31(active(X1), X2, X3)
active(U32(X1, X2)) -> U32(active(X1), X2)
active(U41(X1, X2)) -> U41(active(X1), X2)
active(U42(X)) -> U42(active(X))
active(U51(X1, X2)) -> U51(active(X1), X2)
active(U52(X)) -> U52(active(X))
active(U61(X)) -> U61(active(X))
active(U71(X)) -> U71(active(X))
active(U81(X)) -> U81(active(X))
active(U91(X)) -> U91(active(X))
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) -> mark(U101(X1, X2))
U102(mark(X)) -> mark(U102(X))
U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3))
U111(mark(X)) -> mark(U111(X))
snd(mark(X)) -> mark(snd(X))
splitAt(mark(X1), X2) -> mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) -> mark(splitAt(X1, X2))
U121(mark(X)) -> mark(U121(X))
U131(mark(X1), X2) -> mark(U131(X1, X2))
U132(mark(X)) -> mark(U132(X))
U141(mark(X1), X2) -> mark(U141(X1, X2))
U142(mark(X)) -> mark(U142(X))
U151(mark(X1), X2) -> mark(U151(X1, X2))
U152(mark(X)) -> mark(U152(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))
U172(mark(X1), X2, X3) -> mark(U172(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))
U182(mark(X1), X2) -> mark(U182(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, X3, X4) -> mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) -> mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) -> mark(U204(X1, X2))
U21(mark(X1), X2, X3) -> mark(U21(X1, X2, X3))
U22(mark(X1), X2) -> mark(U22(X1, X2))
U211(mark(X1), X2) -> mark(U211(X1, X2))
U212(mark(X1), X2) -> mark(U212(X1, X2))
U221(mark(X1), X2, X3) -> mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) -> mark(U222(X1, X2, X3))
fst(mark(X)) -> mark(fst(X))
U31(mark(X1), X2, X3) -> mark(U31(X1, X2, X3))
U32(mark(X1), X2) -> mark(U32(X1, X2))
U41(mark(X1), X2) -> mark(U41(X1, X2))
U42(mark(X)) -> mark(U42(X))
U51(mark(X1), X2) -> mark(U51(X1, X2))
U52(mark(X)) -> mark(U52(X))
U61(mark(X)) -> mark(U61(X))
U71(mark(X)) -> mark(U71(X))
U81(mark(X)) -> mark(U81(X))
U91(mark(X)) -> mark(U91(X))
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)) -> U101(proper(X1), proper(X2))
proper(tt) -> ok(tt)
proper(U102(X)) -> U102(proper(X))
proper(isLNat(X)) -> isLNat(proper(X))
proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) -> U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) -> U111(proper(X))
proper(snd(X)) -> snd(proper(X))
proper(splitAt(X1, X2)) -> splitAt(proper(X1), proper(X2))
proper(U121(X)) -> U121(proper(X))
proper(U131(X1, X2)) -> U131(proper(X1), proper(X2))
proper(U132(X)) -> U132(proper(X))
proper(U141(X1, X2)) -> U141(proper(X1), proper(X2))
proper(U142(X)) -> U142(proper(X))
proper(U151(X1, X2)) -> U151(proper(X1), proper(X2))
proper(U152(X)) -> U152(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(U172(X1, X2, X3)) -> U172(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(U182(X1, X2)) -> U182(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, X3, X4)) -> U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) -> isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) -> U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) -> U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) -> U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) -> U22(proper(X1), proper(X2))
proper(U211(X1, X2)) -> U211(proper(X1), proper(X2))
proper(U212(X1, X2)) -> U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) -> U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) -> U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) -> fst(proper(X))
proper(U31(X1, X2, X3)) -> U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) -> U32(proper(X1), proper(X2))
proper(U41(X1, X2)) -> U41(proper(X1), proper(X2))
proper(U42(X)) -> U42(proper(X))
proper(U51(X1, X2)) -> U51(proper(X1), proper(X2))
proper(U52(X)) -> U52(proper(X))
proper(U61(X)) -> U61(proper(X))
proper(U71(X)) -> U71(proper(X))
proper(U81(X)) -> U81(proper(X))
proper(U91(X)) -> U91(proper(X))
proper(isPLNat(X)) -> isPLNat(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(U101(X1, X2))
U102(ok(X)) -> ok(U102(X))
isLNat(ok(X)) -> ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))
U111(ok(X)) -> ok(U111(X))
snd(ok(X)) -> ok(snd(X))
splitAt(ok(X1), ok(X2)) -> ok(splitAt(X1, X2))
U121(ok(X)) -> ok(U121(X))
U131(ok(X1), ok(X2)) -> ok(U131(X1, X2))
U132(ok(X)) -> ok(U132(X))
U141(ok(X1), ok(X2)) -> ok(U141(X1, X2))
U142(ok(X)) -> ok(U142(X))
U151(ok(X1), ok(X2)) -> ok(U151(X1, X2))
U152(ok(X)) -> ok(U152(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))
U172(ok(X1), ok(X2), ok(X3)) -> ok(U172(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))
U182(ok(X1), ok(X2)) -> ok(U182(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(X3), ok(X4)) -> ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) -> ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) -> ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) -> ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) -> ok(U22(X1, X2))
U211(ok(X1), ok(X2)) -> ok(U211(X1, X2))
U212(ok(X1), ok(X2)) -> ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) -> ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) -> ok(U222(X1, X2, X3))
fst(ok(X)) -> ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) -> ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) -> ok(U32(X1, X2))
U41(ok(X1), ok(X2)) -> ok(U41(X1, X2))
U42(ok(X)) -> ok(U42(X))
U51(ok(X1), ok(X2)) -> ok(U51(X1, X2))
U52(ok(X)) -> ok(U52(X))
U61(ok(X)) -> ok(U61(X))
U71(ok(X)) -> ok(U71(X))
U81(ok(X)) -> ok(U81(X))
U91(ok(X)) -> ok(U91(X))
isPLNat(ok(X)) -> ok(isPLNat(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))

Types:
active :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U12 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U172 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U182 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U203 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U204 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U22 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U212 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U222 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U32 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok -> top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat -> tt:mark:nil:0':ok


Lemmas:
U102(gen_tt:mark:nil:0':ok3_0(+(1, n5_0))) -> *4_0, rt in Omega(n5_0)
U12(gen_tt:mark:nil:0':ok3_0(+(1, n1128_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) -> *4_0, rt in Omega(n1128_0)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n9121_0))) -> *4_0, rt in Omega(n9121_0)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n10537_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n10537_0)


Generator Equations:
gen_tt:mark:nil:0':ok3_0(0) <=> tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) <=> mark(gen_tt:mark:nil:0':ok3_0(x))


The following defined symbols remain to be analysed:
U132, active, U142, U152, cons, natsFrom, s, U172, head, afterNth, U182, pair, U202, isNatural, U203, U204, U22, U212, U222, fst, U32, U42, U52, U11, U21, U31, U41, U51, U61, isPLNat, U71, U81, U91, U101, U111, U121, U131, U141, U151, U161, U171, U181, U191, U201, U211, U221, tail, take, sel, proper, top

They will be analysed ascendingly in the following order:
U132 < active
U142 < active
U152 < active
cons < active
natsFrom < active
s < active
U172 < active
head < active
afterNth < active
U182 < active
pair < active
U202 < active
isNatural < active
U203 < active
U204 < active
U22 < active
U212 < active
U222 < active
fst < active
U32 < active
U42 < active
U52 < active
U11 < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
isPLNat < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
U141 < active
U151 < active
U161 < active
U171 < active
U181 < active
U191 < active
U201 < active
U211 < active
U221 < active
tail < active
take < active
sel < active
active < top
U132 < proper
U142 < proper
U152 < proper
cons < proper
natsFrom < proper
s < proper
U172 < proper
head < proper
afterNth < proper
U182 < proper
pair < proper
U202 < proper
isNatural < proper
U203 < proper
U204 < proper
U22 < proper
U212 < proper
U222 < proper
fst < proper
U32 < proper
U42 < proper
U52 < proper
U11 < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
isPLNat < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
U141 < proper
U151 < proper
U161 < proper
U171 < proper
U181 < proper
U191 < proper
U201 < proper
U211 < proper
U221 < proper
tail < proper
take < proper
sel < proper
proper < top

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

(37) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
U132(gen_tt:mark:nil:0':ok3_0(+(1, n15953_0))) -> *4_0, rt in Omega(n15953_0)

Induction Base:
U132(gen_tt:mark:nil:0':ok3_0(+(1, 0)))

Induction Step:
U132(gen_tt:mark:nil:0':ok3_0(+(1, +(n15953_0, 1)))) ->_R^Omega(1)
mark(U132(gen_tt:mark:nil:0':ok3_0(+(1, n15953_0)))) ->_IH
mark(*4_0)

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
----------------------------------------

(38)
Obligation:
TRS:
Rules:
active(U101(tt, V2)) -> mark(U102(isLNat(V2)))
active(U102(tt)) -> mark(tt)
active(U11(tt, N, XS)) -> mark(U12(isLNat(XS), N, XS))
active(U111(tt)) -> mark(tt)
active(U12(tt, N, XS)) -> mark(snd(splitAt(N, XS)))
active(U121(tt)) -> mark(tt)
active(U131(tt, V2)) -> mark(U132(isLNat(V2)))
active(U132(tt)) -> mark(tt)
active(U141(tt, V2)) -> mark(U142(isLNat(V2)))
active(U142(tt)) -> mark(tt)
active(U151(tt, V2)) -> mark(U152(isLNat(V2)))
active(U152(tt)) -> mark(tt)
active(U161(tt, N)) -> mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) -> mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) -> mark(head(afterNth(N, XS)))
active(U181(tt, Y)) -> mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) -> mark(Y)
active(U191(tt, XS)) -> mark(pair(nil, XS))
active(U201(tt, N, X, XS)) -> mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) -> mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) -> mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) -> mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) -> mark(U22(isLNat(Y), X))
active(U211(tt, XS)) -> mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) -> mark(XS)
active(U22(tt, X)) -> mark(X)
active(U221(tt, N, XS)) -> mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) -> mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) -> mark(U32(isLNat(XS), N))
active(U32(tt, N)) -> mark(N)
active(U41(tt, V2)) -> mark(U42(isLNat(V2)))
active(U42(tt)) -> mark(tt)
active(U51(tt, V2)) -> mark(U52(isLNat(V2)))
active(U52(tt)) -> mark(tt)
active(U61(tt)) -> mark(tt)
active(U71(tt)) -> mark(tt)
active(U81(tt)) -> mark(tt)
active(U91(tt)) -> mark(tt)
active(afterNth(N, XS)) -> mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) -> mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) -> mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) -> mark(tt)
active(isLNat(afterNth(V1, V2))) -> mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) -> mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) -> mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) -> mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) -> mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) -> mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) -> mark(U101(isNatural(V1), V2))
active(isNatural(0')) -> mark(tt)
active(isNatural(head(V1))) -> mark(U111(isLNat(V1)))
active(isNatural(s(V1))) -> mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) -> mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) -> mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) -> mark(U151(isNatural(V1), V2))
active(natsFrom(N)) -> mark(U161(isNatural(N), N))
active(sel(N, XS)) -> mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) -> mark(U181(isLNat(X), Y))
active(splitAt(0', XS)) -> mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) -> mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) -> mark(U211(isNatural(N), XS))
active(take(N, XS)) -> mark(U221(isNatural(N), N, XS))
active(U101(X1, X2)) -> U101(active(X1), X2)
active(U102(X)) -> U102(active(X))
active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3)
active(U111(X)) -> U111(active(X))
active(snd(X)) -> snd(active(X))
active(splitAt(X1, X2)) -> splitAt(active(X1), X2)
active(splitAt(X1, X2)) -> splitAt(X1, active(X2))
active(U121(X)) -> U121(active(X))
active(U131(X1, X2)) -> U131(active(X1), X2)
active(U132(X)) -> U132(active(X))
active(U141(X1, X2)) -> U141(active(X1), X2)
active(U142(X)) -> U142(active(X))
active(U151(X1, X2)) -> U151(active(X1), X2)
active(U152(X)) -> U152(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(U172(X1, X2, X3)) -> U172(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(U182(X1, X2)) -> U182(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, X3, X4)) -> U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) -> U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) -> U204(active(X1), X2)
active(U21(X1, X2, X3)) -> U21(active(X1), X2, X3)
active(U22(X1, X2)) -> U22(active(X1), X2)
active(U211(X1, X2)) -> U211(active(X1), X2)
active(U212(X1, X2)) -> U212(active(X1), X2)
active(U221(X1, X2, X3)) -> U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) -> U222(active(X1), X2, X3)
active(fst(X)) -> fst(active(X))
active(U31(X1, X2, X3)) -> U31(active(X1), X2, X3)
active(U32(X1, X2)) -> U32(active(X1), X2)
active(U41(X1, X2)) -> U41(active(X1), X2)
active(U42(X)) -> U42(active(X))
active(U51(X1, X2)) -> U51(active(X1), X2)
active(U52(X)) -> U52(active(X))
active(U61(X)) -> U61(active(X))
active(U71(X)) -> U71(active(X))
active(U81(X)) -> U81(active(X))
active(U91(X)) -> U91(active(X))
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) -> mark(U101(X1, X2))
U102(mark(X)) -> mark(U102(X))
U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3))
U111(mark(X)) -> mark(U111(X))
snd(mark(X)) -> mark(snd(X))
splitAt(mark(X1), X2) -> mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) -> mark(splitAt(X1, X2))
U121(mark(X)) -> mark(U121(X))
U131(mark(X1), X2) -> mark(U131(X1, X2))
U132(mark(X)) -> mark(U132(X))
U141(mark(X1), X2) -> mark(U141(X1, X2))
U142(mark(X)) -> mark(U142(X))
U151(mark(X1), X2) -> mark(U151(X1, X2))
U152(mark(X)) -> mark(U152(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))
U172(mark(X1), X2, X3) -> mark(U172(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))
U182(mark(X1), X2) -> mark(U182(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, X3, X4) -> mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) -> mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) -> mark(U204(X1, X2))
U21(mark(X1), X2, X3) -> mark(U21(X1, X2, X3))
U22(mark(X1), X2) -> mark(U22(X1, X2))
U211(mark(X1), X2) -> mark(U211(X1, X2))
U212(mark(X1), X2) -> mark(U212(X1, X2))
U221(mark(X1), X2, X3) -> mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) -> mark(U222(X1, X2, X3))
fst(mark(X)) -> mark(fst(X))
U31(mark(X1), X2, X3) -> mark(U31(X1, X2, X3))
U32(mark(X1), X2) -> mark(U32(X1, X2))
U41(mark(X1), X2) -> mark(U41(X1, X2))
U42(mark(X)) -> mark(U42(X))
U51(mark(X1), X2) -> mark(U51(X1, X2))
U52(mark(X)) -> mark(U52(X))
U61(mark(X)) -> mark(U61(X))
U71(mark(X)) -> mark(U71(X))
U81(mark(X)) -> mark(U81(X))
U91(mark(X)) -> mark(U91(X))
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)) -> U101(proper(X1), proper(X2))
proper(tt) -> ok(tt)
proper(U102(X)) -> U102(proper(X))
proper(isLNat(X)) -> isLNat(proper(X))
proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) -> U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) -> U111(proper(X))
proper(snd(X)) -> snd(proper(X))
proper(splitAt(X1, X2)) -> splitAt(proper(X1), proper(X2))
proper(U121(X)) -> U121(proper(X))
proper(U131(X1, X2)) -> U131(proper(X1), proper(X2))
proper(U132(X)) -> U132(proper(X))
proper(U141(X1, X2)) -> U141(proper(X1), proper(X2))
proper(U142(X)) -> U142(proper(X))
proper(U151(X1, X2)) -> U151(proper(X1), proper(X2))
proper(U152(X)) -> U152(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(U172(X1, X2, X3)) -> U172(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(U182(X1, X2)) -> U182(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, X3, X4)) -> U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) -> isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) -> U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) -> U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) -> U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) -> U22(proper(X1), proper(X2))
proper(U211(X1, X2)) -> U211(proper(X1), proper(X2))
proper(U212(X1, X2)) -> U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) -> U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) -> U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) -> fst(proper(X))
proper(U31(X1, X2, X3)) -> U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) -> U32(proper(X1), proper(X2))
proper(U41(X1, X2)) -> U41(proper(X1), proper(X2))
proper(U42(X)) -> U42(proper(X))
proper(U51(X1, X2)) -> U51(proper(X1), proper(X2))
proper(U52(X)) -> U52(proper(X))
proper(U61(X)) -> U61(proper(X))
proper(U71(X)) -> U71(proper(X))
proper(U81(X)) -> U81(proper(X))
proper(U91(X)) -> U91(proper(X))
proper(isPLNat(X)) -> isPLNat(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(U101(X1, X2))
U102(ok(X)) -> ok(U102(X))
isLNat(ok(X)) -> ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))
U111(ok(X)) -> ok(U111(X))
snd(ok(X)) -> ok(snd(X))
splitAt(ok(X1), ok(X2)) -> ok(splitAt(X1, X2))
U121(ok(X)) -> ok(U121(X))
U131(ok(X1), ok(X2)) -> ok(U131(X1, X2))
U132(ok(X)) -> ok(U132(X))
U141(ok(X1), ok(X2)) -> ok(U141(X1, X2))
U142(ok(X)) -> ok(U142(X))
U151(ok(X1), ok(X2)) -> ok(U151(X1, X2))
U152(ok(X)) -> ok(U152(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))
U172(ok(X1), ok(X2), ok(X3)) -> ok(U172(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))
U182(ok(X1), ok(X2)) -> ok(U182(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(X3), ok(X4)) -> ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) -> ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) -> ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) -> ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) -> ok(U22(X1, X2))
U211(ok(X1), ok(X2)) -> ok(U211(X1, X2))
U212(ok(X1), ok(X2)) -> ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) -> ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) -> ok(U222(X1, X2, X3))
fst(ok(X)) -> ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) -> ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) -> ok(U32(X1, X2))
U41(ok(X1), ok(X2)) -> ok(U41(X1, X2))
U42(ok(X)) -> ok(U42(X))
U51(ok(X1), ok(X2)) -> ok(U51(X1, X2))
U52(ok(X)) -> ok(U52(X))
U61(ok(X)) -> ok(U61(X))
U71(ok(X)) -> ok(U71(X))
U81(ok(X)) -> ok(U81(X))
U91(ok(X)) -> ok(U91(X))
isPLNat(ok(X)) -> ok(isPLNat(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))

Types:
active :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U12 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U172 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U182 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U203 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U204 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U22 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U212 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U222 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U32 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok -> top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat -> tt:mark:nil:0':ok


Lemmas:
U102(gen_tt:mark:nil:0':ok3_0(+(1, n5_0))) -> *4_0, rt in Omega(n5_0)
U12(gen_tt:mark:nil:0':ok3_0(+(1, n1128_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) -> *4_0, rt in Omega(n1128_0)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n9121_0))) -> *4_0, rt in Omega(n9121_0)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n10537_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n10537_0)
U132(gen_tt:mark:nil:0':ok3_0(+(1, n15953_0))) -> *4_0, rt in Omega(n15953_0)


Generator Equations:
gen_tt:mark:nil:0':ok3_0(0) <=> tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) <=> mark(gen_tt:mark:nil:0':ok3_0(x))


The following defined symbols remain to be analysed:
U142, active, U152, cons, natsFrom, s, U172, head, afterNth, U182, pair, U202, isNatural, U203, U204, U22, U212, U222, fst, U32, U42, U52, U11, U21, U31, U41, U51, U61, isPLNat, U71, U81, U91, U101, U111, U121, U131, U141, U151, U161, U171, U181, U191, U201, U211, U221, tail, take, sel, proper, top

They will be analysed ascendingly in the following order:
U142 < active
U152 < active
cons < active
natsFrom < active
s < active
U172 < active
head < active
afterNth < active
U182 < active
pair < active
U202 < active
isNatural < active
U203 < active
U204 < active
U22 < active
U212 < active
U222 < active
fst < active
U32 < active
U42 < active
U52 < active
U11 < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
isPLNat < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
U141 < active
U151 < active
U161 < active
U171 < active
U181 < active
U191 < active
U201 < active
U211 < active
U221 < active
tail < active
take < active
sel < active
active < top
U142 < proper
U152 < proper
cons < proper
natsFrom < proper
s < proper
U172 < proper
head < proper
afterNth < proper
U182 < proper
pair < proper
U202 < proper
isNatural < proper
U203 < proper
U204 < proper
U22 < proper
U212 < proper
U222 < proper
fst < proper
U32 < proper
U42 < proper
U52 < proper
U11 < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
isPLNat < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
U141 < proper
U151 < proper
U161 < proper
U171 < proper
U181 < proper
U191 < proper
U201 < proper
U211 < proper
U221 < proper
tail < proper
take < proper
sel < proper
proper < top

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

(39) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
U142(gen_tt:mark:nil:0':ok3_0(+(1, n17620_0))) -> *4_0, rt in Omega(n17620_0)

Induction Base:
U142(gen_tt:mark:nil:0':ok3_0(+(1, 0)))

Induction Step:
U142(gen_tt:mark:nil:0':ok3_0(+(1, +(n17620_0, 1)))) ->_R^Omega(1)
mark(U142(gen_tt:mark:nil:0':ok3_0(+(1, n17620_0)))) ->_IH
mark(*4_0)

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
----------------------------------------

(40)
Obligation:
TRS:
Rules:
active(U101(tt, V2)) -> mark(U102(isLNat(V2)))
active(U102(tt)) -> mark(tt)
active(U11(tt, N, XS)) -> mark(U12(isLNat(XS), N, XS))
active(U111(tt)) -> mark(tt)
active(U12(tt, N, XS)) -> mark(snd(splitAt(N, XS)))
active(U121(tt)) -> mark(tt)
active(U131(tt, V2)) -> mark(U132(isLNat(V2)))
active(U132(tt)) -> mark(tt)
active(U141(tt, V2)) -> mark(U142(isLNat(V2)))
active(U142(tt)) -> mark(tt)
active(U151(tt, V2)) -> mark(U152(isLNat(V2)))
active(U152(tt)) -> mark(tt)
active(U161(tt, N)) -> mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) -> mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) -> mark(head(afterNth(N, XS)))
active(U181(tt, Y)) -> mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) -> mark(Y)
active(U191(tt, XS)) -> mark(pair(nil, XS))
active(U201(tt, N, X, XS)) -> mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) -> mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) -> mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) -> mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) -> mark(U22(isLNat(Y), X))
active(U211(tt, XS)) -> mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) -> mark(XS)
active(U22(tt, X)) -> mark(X)
active(U221(tt, N, XS)) -> mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) -> mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) -> mark(U32(isLNat(XS), N))
active(U32(tt, N)) -> mark(N)
active(U41(tt, V2)) -> mark(U42(isLNat(V2)))
active(U42(tt)) -> mark(tt)
active(U51(tt, V2)) -> mark(U52(isLNat(V2)))
active(U52(tt)) -> mark(tt)
active(U61(tt)) -> mark(tt)
active(U71(tt)) -> mark(tt)
active(U81(tt)) -> mark(tt)
active(U91(tt)) -> mark(tt)
active(afterNth(N, XS)) -> mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) -> mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) -> mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) -> mark(tt)
active(isLNat(afterNth(V1, V2))) -> mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) -> mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) -> mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) -> mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) -> mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) -> mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) -> mark(U101(isNatural(V1), V2))
active(isNatural(0')) -> mark(tt)
active(isNatural(head(V1))) -> mark(U111(isLNat(V1)))
active(isNatural(s(V1))) -> mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) -> mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) -> mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) -> mark(U151(isNatural(V1), V2))
active(natsFrom(N)) -> mark(U161(isNatural(N), N))
active(sel(N, XS)) -> mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) -> mark(U181(isLNat(X), Y))
active(splitAt(0', XS)) -> mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) -> mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) -> mark(U211(isNatural(N), XS))
active(take(N, XS)) -> mark(U221(isNatural(N), N, XS))
active(U101(X1, X2)) -> U101(active(X1), X2)
active(U102(X)) -> U102(active(X))
active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3)
active(U111(X)) -> U111(active(X))
active(snd(X)) -> snd(active(X))
active(splitAt(X1, X2)) -> splitAt(active(X1), X2)
active(splitAt(X1, X2)) -> splitAt(X1, active(X2))
active(U121(X)) -> U121(active(X))
active(U131(X1, X2)) -> U131(active(X1), X2)
active(U132(X)) -> U132(active(X))
active(U141(X1, X2)) -> U141(active(X1), X2)
active(U142(X)) -> U142(active(X))
active(U151(X1, X2)) -> U151(active(X1), X2)
active(U152(X)) -> U152(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(U172(X1, X2, X3)) -> U172(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(U182(X1, X2)) -> U182(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, X3, X4)) -> U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) -> U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) -> U204(active(X1), X2)
active(U21(X1, X2, X3)) -> U21(active(X1), X2, X3)
active(U22(X1, X2)) -> U22(active(X1), X2)
active(U211(X1, X2)) -> U211(active(X1), X2)
active(U212(X1, X2)) -> U212(active(X1), X2)
active(U221(X1, X2, X3)) -> U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) -> U222(active(X1), X2, X3)
active(fst(X)) -> fst(active(X))
active(U31(X1, X2, X3)) -> U31(active(X1), X2, X3)
active(U32(X1, X2)) -> U32(active(X1), X2)
active(U41(X1, X2)) -> U41(active(X1), X2)
active(U42(X)) -> U42(active(X))
active(U51(X1, X2)) -> U51(active(X1), X2)
active(U52(X)) -> U52(active(X))
active(U61(X)) -> U61(active(X))
active(U71(X)) -> U71(active(X))
active(U81(X)) -> U81(active(X))
active(U91(X)) -> U91(active(X))
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) -> mark(U101(X1, X2))
U102(mark(X)) -> mark(U102(X))
U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3))
U111(mark(X)) -> mark(U111(X))
snd(mark(X)) -> mark(snd(X))
splitAt(mark(X1), X2) -> mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) -> mark(splitAt(X1, X2))
U121(mark(X)) -> mark(U121(X))
U131(mark(X1), X2) -> mark(U131(X1, X2))
U132(mark(X)) -> mark(U132(X))
U141(mark(X1), X2) -> mark(U141(X1, X2))
U142(mark(X)) -> mark(U142(X))
U151(mark(X1), X2) -> mark(U151(X1, X2))
U152(mark(X)) -> mark(U152(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))
U172(mark(X1), X2, X3) -> mark(U172(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))
U182(mark(X1), X2) -> mark(U182(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, X3, X4) -> mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) -> mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) -> mark(U204(X1, X2))
U21(mark(X1), X2, X3) -> mark(U21(X1, X2, X3))
U22(mark(X1), X2) -> mark(U22(X1, X2))
U211(mark(X1), X2) -> mark(U211(X1, X2))
U212(mark(X1), X2) -> mark(U212(X1, X2))
U221(mark(X1), X2, X3) -> mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) -> mark(U222(X1, X2, X3))
fst(mark(X)) -> mark(fst(X))
U31(mark(X1), X2, X3) -> mark(U31(X1, X2, X3))
U32(mark(X1), X2) -> mark(U32(X1, X2))
U41(mark(X1), X2) -> mark(U41(X1, X2))
U42(mark(X)) -> mark(U42(X))
U51(mark(X1), X2) -> mark(U51(X1, X2))
U52(mark(X)) -> mark(U52(X))
U61(mark(X)) -> mark(U61(X))
U71(mark(X)) -> mark(U71(X))
U81(mark(X)) -> mark(U81(X))
U91(mark(X)) -> mark(U91(X))
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)) -> U101(proper(X1), proper(X2))
proper(tt) -> ok(tt)
proper(U102(X)) -> U102(proper(X))
proper(isLNat(X)) -> isLNat(proper(X))
proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) -> U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) -> U111(proper(X))
proper(snd(X)) -> snd(proper(X))
proper(splitAt(X1, X2)) -> splitAt(proper(X1), proper(X2))
proper(U121(X)) -> U121(proper(X))
proper(U131(X1, X2)) -> U131(proper(X1), proper(X2))
proper(U132(X)) -> U132(proper(X))
proper(U141(X1, X2)) -> U141(proper(X1), proper(X2))
proper(U142(X)) -> U142(proper(X))
proper(U151(X1, X2)) -> U151(proper(X1), proper(X2))
proper(U152(X)) -> U152(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(U172(X1, X2, X3)) -> U172(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(U182(X1, X2)) -> U182(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, X3, X4)) -> U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) -> isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) -> U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) -> U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) -> U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) -> U22(proper(X1), proper(X2))
proper(U211(X1, X2)) -> U211(proper(X1), proper(X2))
proper(U212(X1, X2)) -> U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) -> U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) -> U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) -> fst(proper(X))
proper(U31(X1, X2, X3)) -> U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) -> U32(proper(X1), proper(X2))
proper(U41(X1, X2)) -> U41(proper(X1), proper(X2))
proper(U42(X)) -> U42(proper(X))
proper(U51(X1, X2)) -> U51(proper(X1), proper(X2))
proper(U52(X)) -> U52(proper(X))
proper(U61(X)) -> U61(proper(X))
proper(U71(X)) -> U71(proper(X))
proper(U81(X)) -> U81(proper(X))
proper(U91(X)) -> U91(proper(X))
proper(isPLNat(X)) -> isPLNat(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(U101(X1, X2))
U102(ok(X)) -> ok(U102(X))
isLNat(ok(X)) -> ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))
U111(ok(X)) -> ok(U111(X))
snd(ok(X)) -> ok(snd(X))
splitAt(ok(X1), ok(X2)) -> ok(splitAt(X1, X2))
U121(ok(X)) -> ok(U121(X))
U131(ok(X1), ok(X2)) -> ok(U131(X1, X2))
U132(ok(X)) -> ok(U132(X))
U141(ok(X1), ok(X2)) -> ok(U141(X1, X2))
U142(ok(X)) -> ok(U142(X))
U151(ok(X1), ok(X2)) -> ok(U151(X1, X2))
U152(ok(X)) -> ok(U152(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))
U172(ok(X1), ok(X2), ok(X3)) -> ok(U172(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))
U182(ok(X1), ok(X2)) -> ok(U182(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(X3), ok(X4)) -> ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) -> ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) -> ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) -> ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) -> ok(U22(X1, X2))
U211(ok(X1), ok(X2)) -> ok(U211(X1, X2))
U212(ok(X1), ok(X2)) -> ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) -> ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) -> ok(U222(X1, X2, X3))
fst(ok(X)) -> ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) -> ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) -> ok(U32(X1, X2))
U41(ok(X1), ok(X2)) -> ok(U41(X1, X2))
U42(ok(X)) -> ok(U42(X))
U51(ok(X1), ok(X2)) -> ok(U51(X1, X2))
U52(ok(X)) -> ok(U52(X))
U61(ok(X)) -> ok(U61(X))
U71(ok(X)) -> ok(U71(X))
U81(ok(X)) -> ok(U81(X))
U91(ok(X)) -> ok(U91(X))
isPLNat(ok(X)) -> ok(isPLNat(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))

Types:
active :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U12 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U172 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U182 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U203 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U204 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U22 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U212 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U222 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U32 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok -> top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat -> tt:mark:nil:0':ok


Lemmas:
U102(gen_tt:mark:nil:0':ok3_0(+(1, n5_0))) -> *4_0, rt in Omega(n5_0)
U12(gen_tt:mark:nil:0':ok3_0(+(1, n1128_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) -> *4_0, rt in Omega(n1128_0)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n9121_0))) -> *4_0, rt in Omega(n9121_0)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n10537_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n10537_0)
U132(gen_tt:mark:nil:0':ok3_0(+(1, n15953_0))) -> *4_0, rt in Omega(n15953_0)
U142(gen_tt:mark:nil:0':ok3_0(+(1, n17620_0))) -> *4_0, rt in Omega(n17620_0)


Generator Equations:
gen_tt:mark:nil:0':ok3_0(0) <=> tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) <=> mark(gen_tt:mark:nil:0':ok3_0(x))


The following defined symbols remain to be analysed:
U152, active, cons, natsFrom, s, U172, head, afterNth, U182, pair, U202, isNatural, U203, U204, U22, U212, U222, fst, U32, U42, U52, U11, U21, U31, U41, U51, U61, isPLNat, U71, U81, U91, U101, U111, U121, U131, U141, U151, U161, U171, U181, U191, U201, U211, U221, tail, take, sel, proper, top

They will be analysed ascendingly in the following order:
U152 < active
cons < active
natsFrom < active
s < active
U172 < active
head < active
afterNth < active
U182 < active
pair < active
U202 < active
isNatural < active
U203 < active
U204 < active
U22 < active
U212 < active
U222 < active
fst < active
U32 < active
U42 < active
U52 < active
U11 < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
isPLNat < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
U141 < active
U151 < active
U161 < active
U171 < active
U181 < active
U191 < active
U201 < active
U211 < active
U221 < active
tail < active
take < active
sel < active
active < top
U152 < proper
cons < proper
natsFrom < proper
s < proper
U172 < proper
head < proper
afterNth < proper
U182 < proper
pair < proper
U202 < proper
isNatural < proper
U203 < proper
U204 < proper
U22 < proper
U212 < proper
U222 < proper
fst < proper
U32 < proper
U42 < proper
U52 < proper
U11 < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
isPLNat < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
U141 < proper
U151 < proper
U161 < proper
U171 < proper
U181 < proper
U191 < proper
U201 < proper
U211 < proper
U221 < proper
tail < proper
take < proper
sel < proper
proper < top

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

(41) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
U152(gen_tt:mark:nil:0':ok3_0(+(1, n19388_0))) -> *4_0, rt in Omega(n19388_0)

Induction Base:
U152(gen_tt:mark:nil:0':ok3_0(+(1, 0)))

Induction Step:
U152(gen_tt:mark:nil:0':ok3_0(+(1, +(n19388_0, 1)))) ->_R^Omega(1)
mark(U152(gen_tt:mark:nil:0':ok3_0(+(1, n19388_0)))) ->_IH
mark(*4_0)

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
----------------------------------------

(42)
Obligation:
TRS:
Rules:
active(U101(tt, V2)) -> mark(U102(isLNat(V2)))
active(U102(tt)) -> mark(tt)
active(U11(tt, N, XS)) -> mark(U12(isLNat(XS), N, XS))
active(U111(tt)) -> mark(tt)
active(U12(tt, N, XS)) -> mark(snd(splitAt(N, XS)))
active(U121(tt)) -> mark(tt)
active(U131(tt, V2)) -> mark(U132(isLNat(V2)))
active(U132(tt)) -> mark(tt)
active(U141(tt, V2)) -> mark(U142(isLNat(V2)))
active(U142(tt)) -> mark(tt)
active(U151(tt, V2)) -> mark(U152(isLNat(V2)))
active(U152(tt)) -> mark(tt)
active(U161(tt, N)) -> mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) -> mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) -> mark(head(afterNth(N, XS)))
active(U181(tt, Y)) -> mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) -> mark(Y)
active(U191(tt, XS)) -> mark(pair(nil, XS))
active(U201(tt, N, X, XS)) -> mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) -> mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) -> mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) -> mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) -> mark(U22(isLNat(Y), X))
active(U211(tt, XS)) -> mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) -> mark(XS)
active(U22(tt, X)) -> mark(X)
active(U221(tt, N, XS)) -> mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) -> mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) -> mark(U32(isLNat(XS), N))
active(U32(tt, N)) -> mark(N)
active(U41(tt, V2)) -> mark(U42(isLNat(V2)))
active(U42(tt)) -> mark(tt)
active(U51(tt, V2)) -> mark(U52(isLNat(V2)))
active(U52(tt)) -> mark(tt)
active(U61(tt)) -> mark(tt)
active(U71(tt)) -> mark(tt)
active(U81(tt)) -> mark(tt)
active(U91(tt)) -> mark(tt)
active(afterNth(N, XS)) -> mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) -> mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) -> mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) -> mark(tt)
active(isLNat(afterNth(V1, V2))) -> mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) -> mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) -> mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) -> mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) -> mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) -> mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) -> mark(U101(isNatural(V1), V2))
active(isNatural(0')) -> mark(tt)
active(isNatural(head(V1))) -> mark(U111(isLNat(V1)))
active(isNatural(s(V1))) -> mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) -> mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) -> mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) -> mark(U151(isNatural(V1), V2))
active(natsFrom(N)) -> mark(U161(isNatural(N), N))
active(sel(N, XS)) -> mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) -> mark(U181(isLNat(X), Y))
active(splitAt(0', XS)) -> mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) -> mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) -> mark(U211(isNatural(N), XS))
active(take(N, XS)) -> mark(U221(isNatural(N), N, XS))
active(U101(X1, X2)) -> U101(active(X1), X2)
active(U102(X)) -> U102(active(X))
active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3)
active(U111(X)) -> U111(active(X))
active(snd(X)) -> snd(active(X))
active(splitAt(X1, X2)) -> splitAt(active(X1), X2)
active(splitAt(X1, X2)) -> splitAt(X1, active(X2))
active(U121(X)) -> U121(active(X))
active(U131(X1, X2)) -> U131(active(X1), X2)
active(U132(X)) -> U132(active(X))
active(U141(X1, X2)) -> U141(active(X1), X2)
active(U142(X)) -> U142(active(X))
active(U151(X1, X2)) -> U151(active(X1), X2)
active(U152(X)) -> U152(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(U172(X1, X2, X3)) -> U172(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(U182(X1, X2)) -> U182(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, X3, X4)) -> U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) -> U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) -> U204(active(X1), X2)
active(U21(X1, X2, X3)) -> U21(active(X1), X2, X3)
active(U22(X1, X2)) -> U22(active(X1), X2)
active(U211(X1, X2)) -> U211(active(X1), X2)
active(U212(X1, X2)) -> U212(active(X1), X2)
active(U221(X1, X2, X3)) -> U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) -> U222(active(X1), X2, X3)
active(fst(X)) -> fst(active(X))
active(U31(X1, X2, X3)) -> U31(active(X1), X2, X3)
active(U32(X1, X2)) -> U32(active(X1), X2)
active(U41(X1, X2)) -> U41(active(X1), X2)
active(U42(X)) -> U42(active(X))
active(U51(X1, X2)) -> U51(active(X1), X2)
active(U52(X)) -> U52(active(X))
active(U61(X)) -> U61(active(X))
active(U71(X)) -> U71(active(X))
active(U81(X)) -> U81(active(X))
active(U91(X)) -> U91(active(X))
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) -> mark(U101(X1, X2))
U102(mark(X)) -> mark(U102(X))
U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3))
U111(mark(X)) -> mark(U111(X))
snd(mark(X)) -> mark(snd(X))
splitAt(mark(X1), X2) -> mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) -> mark(splitAt(X1, X2))
U121(mark(X)) -> mark(U121(X))
U131(mark(X1), X2) -> mark(U131(X1, X2))
U132(mark(X)) -> mark(U132(X))
U141(mark(X1), X2) -> mark(U141(X1, X2))
U142(mark(X)) -> mark(U142(X))
U151(mark(X1), X2) -> mark(U151(X1, X2))
U152(mark(X)) -> mark(U152(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))
U172(mark(X1), X2, X3) -> mark(U172(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))
U182(mark(X1), X2) -> mark(U182(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, X3, X4) -> mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) -> mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) -> mark(U204(X1, X2))
U21(mark(X1), X2, X3) -> mark(U21(X1, X2, X3))
U22(mark(X1), X2) -> mark(U22(X1, X2))
U211(mark(X1), X2) -> mark(U211(X1, X2))
U212(mark(X1), X2) -> mark(U212(X1, X2))
U221(mark(X1), X2, X3) -> mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) -> mark(U222(X1, X2, X3))
fst(mark(X)) -> mark(fst(X))
U31(mark(X1), X2, X3) -> mark(U31(X1, X2, X3))
U32(mark(X1), X2) -> mark(U32(X1, X2))
U41(mark(X1), X2) -> mark(U41(X1, X2))
U42(mark(X)) -> mark(U42(X))
U51(mark(X1), X2) -> mark(U51(X1, X2))
U52(mark(X)) -> mark(U52(X))
U61(mark(X)) -> mark(U61(X))
U71(mark(X)) -> mark(U71(X))
U81(mark(X)) -> mark(U81(X))
U91(mark(X)) -> mark(U91(X))
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)) -> U101(proper(X1), proper(X2))
proper(tt) -> ok(tt)
proper(U102(X)) -> U102(proper(X))
proper(isLNat(X)) -> isLNat(proper(X))
proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) -> U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) -> U111(proper(X))
proper(snd(X)) -> snd(proper(X))
proper(splitAt(X1, X2)) -> splitAt(proper(X1), proper(X2))
proper(U121(X)) -> U121(proper(X))
proper(U131(X1, X2)) -> U131(proper(X1), proper(X2))
proper(U132(X)) -> U132(proper(X))
proper(U141(X1, X2)) -> U141(proper(X1), proper(X2))
proper(U142(X)) -> U142(proper(X))
proper(U151(X1, X2)) -> U151(proper(X1), proper(X2))
proper(U152(X)) -> U152(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(U172(X1, X2, X3)) -> U172(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(U182(X1, X2)) -> U182(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, X3, X4)) -> U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) -> isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) -> U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) -> U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) -> U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) -> U22(proper(X1), proper(X2))
proper(U211(X1, X2)) -> U211(proper(X1), proper(X2))
proper(U212(X1, X2)) -> U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) -> U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) -> U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) -> fst(proper(X))
proper(U31(X1, X2, X3)) -> U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) -> U32(proper(X1), proper(X2))
proper(U41(X1, X2)) -> U41(proper(X1), proper(X2))
proper(U42(X)) -> U42(proper(X))
proper(U51(X1, X2)) -> U51(proper(X1), proper(X2))
proper(U52(X)) -> U52(proper(X))
proper(U61(X)) -> U61(proper(X))
proper(U71(X)) -> U71(proper(X))
proper(U81(X)) -> U81(proper(X))
proper(U91(X)) -> U91(proper(X))
proper(isPLNat(X)) -> isPLNat(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(U101(X1, X2))
U102(ok(X)) -> ok(U102(X))
isLNat(ok(X)) -> ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))
U111(ok(X)) -> ok(U111(X))
snd(ok(X)) -> ok(snd(X))
splitAt(ok(X1), ok(X2)) -> ok(splitAt(X1, X2))
U121(ok(X)) -> ok(U121(X))
U131(ok(X1), ok(X2)) -> ok(U131(X1, X2))
U132(ok(X)) -> ok(U132(X))
U141(ok(X1), ok(X2)) -> ok(U141(X1, X2))
U142(ok(X)) -> ok(U142(X))
U151(ok(X1), ok(X2)) -> ok(U151(X1, X2))
U152(ok(X)) -> ok(U152(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))
U172(ok(X1), ok(X2), ok(X3)) -> ok(U172(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))
U182(ok(X1), ok(X2)) -> ok(U182(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(X3), ok(X4)) -> ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) -> ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) -> ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) -> ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) -> ok(U22(X1, X2))
U211(ok(X1), ok(X2)) -> ok(U211(X1, X2))
U212(ok(X1), ok(X2)) -> ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) -> ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) -> ok(U222(X1, X2, X3))
fst(ok(X)) -> ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) -> ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) -> ok(U32(X1, X2))
U41(ok(X1), ok(X2)) -> ok(U41(X1, X2))
U42(ok(X)) -> ok(U42(X))
U51(ok(X1), ok(X2)) -> ok(U51(X1, X2))
U52(ok(X)) -> ok(U52(X))
U61(ok(X)) -> ok(U61(X))
U71(ok(X)) -> ok(U71(X))
U81(ok(X)) -> ok(U81(X))
U91(ok(X)) -> ok(U91(X))
isPLNat(ok(X)) -> ok(isPLNat(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))

Types:
active :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U12 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U172 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U182 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U203 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U204 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U22 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U212 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U222 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U32 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok -> top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat -> tt:mark:nil:0':ok


Lemmas:
U102(gen_tt:mark:nil:0':ok3_0(+(1, n5_0))) -> *4_0, rt in Omega(n5_0)
U12(gen_tt:mark:nil:0':ok3_0(+(1, n1128_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) -> *4_0, rt in Omega(n1128_0)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n9121_0))) -> *4_0, rt in Omega(n9121_0)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n10537_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n10537_0)
U132(gen_tt:mark:nil:0':ok3_0(+(1, n15953_0))) -> *4_0, rt in Omega(n15953_0)
U142(gen_tt:mark:nil:0':ok3_0(+(1, n17620_0))) -> *4_0, rt in Omega(n17620_0)
U152(gen_tt:mark:nil:0':ok3_0(+(1, n19388_0))) -> *4_0, rt in Omega(n19388_0)


Generator Equations:
gen_tt:mark:nil:0':ok3_0(0) <=> tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) <=> mark(gen_tt:mark:nil:0':ok3_0(x))


The following defined symbols remain to be analysed:
cons, active, natsFrom, s, U172, head, afterNth, U182, pair, U202, isNatural, U203, U204, U22, U212, U222, fst, U32, U42, U52, U11, U21, U31, U41, U51, U61, isPLNat, U71, U81, U91, U101, U111, U121, U131, U141, U151, U161, U171, U181, U191, U201, U211, U221, tail, take, sel, proper, top

They will be analysed ascendingly in the following order:
cons < active
natsFrom < active
s < active
U172 < active
head < active
afterNth < active
U182 < active
pair < active
U202 < active
isNatural < active
U203 < active
U204 < active
U22 < active
U212 < active
U222 < active
fst < active
U32 < active
U42 < active
U52 < active
U11 < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
isPLNat < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
U141 < active
U151 < active
U161 < active
U171 < active
U181 < active
U191 < active
U201 < active
U211 < active
U221 < active
tail < active
take < active
sel < active
active < top
cons < proper
natsFrom < proper
s < proper
U172 < proper
head < proper
afterNth < proper
U182 < proper
pair < proper
U202 < proper
isNatural < proper
U203 < proper
U204 < proper
U22 < proper
U212 < proper
U222 < proper
fst < proper
U32 < proper
U42 < proper
U52 < proper
U11 < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
isPLNat < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
U141 < proper
U151 < proper
U161 < proper
U171 < proper
U181 < proper
U191 < proper
U201 < proper
U211 < proper
U221 < proper
tail < proper
take < proper
sel < proper
proper < top

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

(43) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
cons(gen_tt:mark:nil:0':ok3_0(+(1, n21257_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n21257_0)

Induction Base:
cons(gen_tt:mark:nil:0':ok3_0(+(1, 0)), gen_tt:mark:nil:0':ok3_0(b))

Induction Step:
cons(gen_tt:mark:nil:0':ok3_0(+(1, +(n21257_0, 1))), gen_tt:mark:nil:0':ok3_0(b)) ->_R^Omega(1)
mark(cons(gen_tt:mark:nil:0':ok3_0(+(1, n21257_0)), gen_tt:mark:nil:0':ok3_0(b))) ->_IH
mark(*4_0)

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
----------------------------------------

(44)
Obligation:
TRS:
Rules:
active(U101(tt, V2)) -> mark(U102(isLNat(V2)))
active(U102(tt)) -> mark(tt)
active(U11(tt, N, XS)) -> mark(U12(isLNat(XS), N, XS))
active(U111(tt)) -> mark(tt)
active(U12(tt, N, XS)) -> mark(snd(splitAt(N, XS)))
active(U121(tt)) -> mark(tt)
active(U131(tt, V2)) -> mark(U132(isLNat(V2)))
active(U132(tt)) -> mark(tt)
active(U141(tt, V2)) -> mark(U142(isLNat(V2)))
active(U142(tt)) -> mark(tt)
active(U151(tt, V2)) -> mark(U152(isLNat(V2)))
active(U152(tt)) -> mark(tt)
active(U161(tt, N)) -> mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) -> mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) -> mark(head(afterNth(N, XS)))
active(U181(tt, Y)) -> mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) -> mark(Y)
active(U191(tt, XS)) -> mark(pair(nil, XS))
active(U201(tt, N, X, XS)) -> mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) -> mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) -> mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) -> mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) -> mark(U22(isLNat(Y), X))
active(U211(tt, XS)) -> mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) -> mark(XS)
active(U22(tt, X)) -> mark(X)
active(U221(tt, N, XS)) -> mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) -> mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) -> mark(U32(isLNat(XS), N))
active(U32(tt, N)) -> mark(N)
active(U41(tt, V2)) -> mark(U42(isLNat(V2)))
active(U42(tt)) -> mark(tt)
active(U51(tt, V2)) -> mark(U52(isLNat(V2)))
active(U52(tt)) -> mark(tt)
active(U61(tt)) -> mark(tt)
active(U71(tt)) -> mark(tt)
active(U81(tt)) -> mark(tt)
active(U91(tt)) -> mark(tt)
active(afterNth(N, XS)) -> mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) -> mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) -> mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) -> mark(tt)
active(isLNat(afterNth(V1, V2))) -> mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) -> mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) -> mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) -> mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) -> mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) -> mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) -> mark(U101(isNatural(V1), V2))
active(isNatural(0')) -> mark(tt)
active(isNatural(head(V1))) -> mark(U111(isLNat(V1)))
active(isNatural(s(V1))) -> mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) -> mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) -> mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) -> mark(U151(isNatural(V1), V2))
active(natsFrom(N)) -> mark(U161(isNatural(N), N))
active(sel(N, XS)) -> mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) -> mark(U181(isLNat(X), Y))
active(splitAt(0', XS)) -> mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) -> mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) -> mark(U211(isNatural(N), XS))
active(take(N, XS)) -> mark(U221(isNatural(N), N, XS))
active(U101(X1, X2)) -> U101(active(X1), X2)
active(U102(X)) -> U102(active(X))
active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3)
active(U111(X)) -> U111(active(X))
active(snd(X)) -> snd(active(X))
active(splitAt(X1, X2)) -> splitAt(active(X1), X2)
active(splitAt(X1, X2)) -> splitAt(X1, active(X2))
active(U121(X)) -> U121(active(X))
active(U131(X1, X2)) -> U131(active(X1), X2)
active(U132(X)) -> U132(active(X))
active(U141(X1, X2)) -> U141(active(X1), X2)
active(U142(X)) -> U142(active(X))
active(U151(X1, X2)) -> U151(active(X1), X2)
active(U152(X)) -> U152(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(U172(X1, X2, X3)) -> U172(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(U182(X1, X2)) -> U182(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, X3, X4)) -> U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) -> U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) -> U204(active(X1), X2)
active(U21(X1, X2, X3)) -> U21(active(X1), X2, X3)
active(U22(X1, X2)) -> U22(active(X1), X2)
active(U211(X1, X2)) -> U211(active(X1), X2)
active(U212(X1, X2)) -> U212(active(X1), X2)
active(U221(X1, X2, X3)) -> U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) -> U222(active(X1), X2, X3)
active(fst(X)) -> fst(active(X))
active(U31(X1, X2, X3)) -> U31(active(X1), X2, X3)
active(U32(X1, X2)) -> U32(active(X1), X2)
active(U41(X1, X2)) -> U41(active(X1), X2)
active(U42(X)) -> U42(active(X))
active(U51(X1, X2)) -> U51(active(X1), X2)
active(U52(X)) -> U52(active(X))
active(U61(X)) -> U61(active(X))
active(U71(X)) -> U71(active(X))
active(U81(X)) -> U81(active(X))
active(U91(X)) -> U91(active(X))
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) -> mark(U101(X1, X2))
U102(mark(X)) -> mark(U102(X))
U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3))
U111(mark(X)) -> mark(U111(X))
snd(mark(X)) -> mark(snd(X))
splitAt(mark(X1), X2) -> mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) -> mark(splitAt(X1, X2))
U121(mark(X)) -> mark(U121(X))
U131(mark(X1), X2) -> mark(U131(X1, X2))
U132(mark(X)) -> mark(U132(X))
U141(mark(X1), X2) -> mark(U141(X1, X2))
U142(mark(X)) -> mark(U142(X))
U151(mark(X1), X2) -> mark(U151(X1, X2))
U152(mark(X)) -> mark(U152(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))
U172(mark(X1), X2, X3) -> mark(U172(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))
U182(mark(X1), X2) -> mark(U182(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, X3, X4) -> mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) -> mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) -> mark(U204(X1, X2))
U21(mark(X1), X2, X3) -> mark(U21(X1, X2, X3))
U22(mark(X1), X2) -> mark(U22(X1, X2))
U211(mark(X1), X2) -> mark(U211(X1, X2))
U212(mark(X1), X2) -> mark(U212(X1, X2))
U221(mark(X1), X2, X3) -> mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) -> mark(U222(X1, X2, X3))
fst(mark(X)) -> mark(fst(X))
U31(mark(X1), X2, X3) -> mark(U31(X1, X2, X3))
U32(mark(X1), X2) -> mark(U32(X1, X2))
U41(mark(X1), X2) -> mark(U41(X1, X2))
U42(mark(X)) -> mark(U42(X))
U51(mark(X1), X2) -> mark(U51(X1, X2))
U52(mark(X)) -> mark(U52(X))
U61(mark(X)) -> mark(U61(X))
U71(mark(X)) -> mark(U71(X))
U81(mark(X)) -> mark(U81(X))
U91(mark(X)) -> mark(U91(X))
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)) -> U101(proper(X1), proper(X2))
proper(tt) -> ok(tt)
proper(U102(X)) -> U102(proper(X))
proper(isLNat(X)) -> isLNat(proper(X))
proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) -> U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) -> U111(proper(X))
proper(snd(X)) -> snd(proper(X))
proper(splitAt(X1, X2)) -> splitAt(proper(X1), proper(X2))
proper(U121(X)) -> U121(proper(X))
proper(U131(X1, X2)) -> U131(proper(X1), proper(X2))
proper(U132(X)) -> U132(proper(X))
proper(U141(X1, X2)) -> U141(proper(X1), proper(X2))
proper(U142(X)) -> U142(proper(X))
proper(U151(X1, X2)) -> U151(proper(X1), proper(X2))
proper(U152(X)) -> U152(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(U172(X1, X2, X3)) -> U172(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(U182(X1, X2)) -> U182(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, X3, X4)) -> U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) -> isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) -> U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) -> U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) -> U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) -> U22(proper(X1), proper(X2))
proper(U211(X1, X2)) -> U211(proper(X1), proper(X2))
proper(U212(X1, X2)) -> U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) -> U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) -> U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) -> fst(proper(X))
proper(U31(X1, X2, X3)) -> U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) -> U32(proper(X1), proper(X2))
proper(U41(X1, X2)) -> U41(proper(X1), proper(X2))
proper(U42(X)) -> U42(proper(X))
proper(U51(X1, X2)) -> U51(proper(X1), proper(X2))
proper(U52(X)) -> U52(proper(X))
proper(U61(X)) -> U61(proper(X))
proper(U71(X)) -> U71(proper(X))
proper(U81(X)) -> U81(proper(X))
proper(U91(X)) -> U91(proper(X))
proper(isPLNat(X)) -> isPLNat(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(U101(X1, X2))
U102(ok(X)) -> ok(U102(X))
isLNat(ok(X)) -> ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))
U111(ok(X)) -> ok(U111(X))
snd(ok(X)) -> ok(snd(X))
splitAt(ok(X1), ok(X2)) -> ok(splitAt(X1, X2))
U121(ok(X)) -> ok(U121(X))
U131(ok(X1), ok(X2)) -> ok(U131(X1, X2))
U132(ok(X)) -> ok(U132(X))
U141(ok(X1), ok(X2)) -> ok(U141(X1, X2))
U142(ok(X)) -> ok(U142(X))
U151(ok(X1), ok(X2)) -> ok(U151(X1, X2))
U152(ok(X)) -> ok(U152(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))
U172(ok(X1), ok(X2), ok(X3)) -> ok(U172(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))
U182(ok(X1), ok(X2)) -> ok(U182(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(X3), ok(X4)) -> ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) -> ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) -> ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) -> ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) -> ok(U22(X1, X2))
U211(ok(X1), ok(X2)) -> ok(U211(X1, X2))
U212(ok(X1), ok(X2)) -> ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) -> ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) -> ok(U222(X1, X2, X3))
fst(ok(X)) -> ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) -> ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) -> ok(U32(X1, X2))
U41(ok(X1), ok(X2)) -> ok(U41(X1, X2))
U42(ok(X)) -> ok(U42(X))
U51(ok(X1), ok(X2)) -> ok(U51(X1, X2))
U52(ok(X)) -> ok(U52(X))
U61(ok(X)) -> ok(U61(X))
U71(ok(X)) -> ok(U71(X))
U81(ok(X)) -> ok(U81(X))
U91(ok(X)) -> ok(U91(X))
isPLNat(ok(X)) -> ok(isPLNat(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))

Types:
active :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U12 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U172 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U182 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U203 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U204 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U22 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U212 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U222 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U32 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok -> top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat -> tt:mark:nil:0':ok


Lemmas:
U102(gen_tt:mark:nil:0':ok3_0(+(1, n5_0))) -> *4_0, rt in Omega(n5_0)
U12(gen_tt:mark:nil:0':ok3_0(+(1, n1128_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) -> *4_0, rt in Omega(n1128_0)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n9121_0))) -> *4_0, rt in Omega(n9121_0)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n10537_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n10537_0)
U132(gen_tt:mark:nil:0':ok3_0(+(1, n15953_0))) -> *4_0, rt in Omega(n15953_0)
U142(gen_tt:mark:nil:0':ok3_0(+(1, n17620_0))) -> *4_0, rt in Omega(n17620_0)
U152(gen_tt:mark:nil:0':ok3_0(+(1, n19388_0))) -> *4_0, rt in Omega(n19388_0)
cons(gen_tt:mark:nil:0':ok3_0(+(1, n21257_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n21257_0)


Generator Equations:
gen_tt:mark:nil:0':ok3_0(0) <=> tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) <=> mark(gen_tt:mark:nil:0':ok3_0(x))


The following defined symbols remain to be analysed:
natsFrom, active, s, U172, head, afterNth, U182, pair, U202, isNatural, U203, U204, U22, U212, U222, fst, U32, U42, U52, U11, U21, U31, U41, U51, U61, isPLNat, U71, U81, U91, U101, U111, U121, U131, U141, U151, U161, U171, U181, U191, U201, U211, U221, tail, take, sel, proper, top

They will be analysed ascendingly in the following order:
natsFrom < active
s < active
U172 < active
head < active
afterNth < active
U182 < active
pair < active
U202 < active
isNatural < active
U203 < active
U204 < active
U22 < active
U212 < active
U222 < active
fst < active
U32 < active
U42 < active
U52 < active
U11 < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
isPLNat < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
U141 < active
U151 < active
U161 < active
U171 < active
U181 < active
U191 < active
U201 < active
U211 < active
U221 < active
tail < active
take < active
sel < active
active < top
natsFrom < proper
s < proper
U172 < proper
head < proper
afterNth < proper
U182 < proper
pair < proper
U202 < proper
isNatural < proper
U203 < proper
U204 < proper
U22 < proper
U212 < proper
U222 < proper
fst < proper
U32 < proper
U42 < proper
U52 < proper
U11 < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
isPLNat < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
U141 < proper
U151 < proper
U161 < proper
U171 < proper
U181 < proper
U191 < proper
U201 < proper
U211 < proper
U221 < proper
tail < proper
take < proper
sel < proper
proper < top

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

(45) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
natsFrom(gen_tt:mark:nil:0':ok3_0(+(1, n27404_0))) -> *4_0, rt in Omega(n27404_0)

Induction Base:
natsFrom(gen_tt:mark:nil:0':ok3_0(+(1, 0)))

Induction Step:
natsFrom(gen_tt:mark:nil:0':ok3_0(+(1, +(n27404_0, 1)))) ->_R^Omega(1)
mark(natsFrom(gen_tt:mark:nil:0':ok3_0(+(1, n27404_0)))) ->_IH
mark(*4_0)

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
----------------------------------------

(46)
Obligation:
TRS:
Rules:
active(U101(tt, V2)) -> mark(U102(isLNat(V2)))
active(U102(tt)) -> mark(tt)
active(U11(tt, N, XS)) -> mark(U12(isLNat(XS), N, XS))
active(U111(tt)) -> mark(tt)
active(U12(tt, N, XS)) -> mark(snd(splitAt(N, XS)))
active(U121(tt)) -> mark(tt)
active(U131(tt, V2)) -> mark(U132(isLNat(V2)))
active(U132(tt)) -> mark(tt)
active(U141(tt, V2)) -> mark(U142(isLNat(V2)))
active(U142(tt)) -> mark(tt)
active(U151(tt, V2)) -> mark(U152(isLNat(V2)))
active(U152(tt)) -> mark(tt)
active(U161(tt, N)) -> mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) -> mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) -> mark(head(afterNth(N, XS)))
active(U181(tt, Y)) -> mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) -> mark(Y)
active(U191(tt, XS)) -> mark(pair(nil, XS))
active(U201(tt, N, X, XS)) -> mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) -> mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) -> mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) -> mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) -> mark(U22(isLNat(Y), X))
active(U211(tt, XS)) -> mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) -> mark(XS)
active(U22(tt, X)) -> mark(X)
active(U221(tt, N, XS)) -> mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) -> mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) -> mark(U32(isLNat(XS), N))
active(U32(tt, N)) -> mark(N)
active(U41(tt, V2)) -> mark(U42(isLNat(V2)))
active(U42(tt)) -> mark(tt)
active(U51(tt, V2)) -> mark(U52(isLNat(V2)))
active(U52(tt)) -> mark(tt)
active(U61(tt)) -> mark(tt)
active(U71(tt)) -> mark(tt)
active(U81(tt)) -> mark(tt)
active(U91(tt)) -> mark(tt)
active(afterNth(N, XS)) -> mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) -> mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) -> mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) -> mark(tt)
active(isLNat(afterNth(V1, V2))) -> mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) -> mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) -> mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) -> mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) -> mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) -> mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) -> mark(U101(isNatural(V1), V2))
active(isNatural(0')) -> mark(tt)
active(isNatural(head(V1))) -> mark(U111(isLNat(V1)))
active(isNatural(s(V1))) -> mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) -> mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) -> mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) -> mark(U151(isNatural(V1), V2))
active(natsFrom(N)) -> mark(U161(isNatural(N), N))
active(sel(N, XS)) -> mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) -> mark(U181(isLNat(X), Y))
active(splitAt(0', XS)) -> mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) -> mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) -> mark(U211(isNatural(N), XS))
active(take(N, XS)) -> mark(U221(isNatural(N), N, XS))
active(U101(X1, X2)) -> U101(active(X1), X2)
active(U102(X)) -> U102(active(X))
active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3)
active(U111(X)) -> U111(active(X))
active(snd(X)) -> snd(active(X))
active(splitAt(X1, X2)) -> splitAt(active(X1), X2)
active(splitAt(X1, X2)) -> splitAt(X1, active(X2))
active(U121(X)) -> U121(active(X))
active(U131(X1, X2)) -> U131(active(X1), X2)
active(U132(X)) -> U132(active(X))
active(U141(X1, X2)) -> U141(active(X1), X2)
active(U142(X)) -> U142(active(X))
active(U151(X1, X2)) -> U151(active(X1), X2)
active(U152(X)) -> U152(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(U172(X1, X2, X3)) -> U172(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(U182(X1, X2)) -> U182(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, X3, X4)) -> U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) -> U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) -> U204(active(X1), X2)
active(U21(X1, X2, X3)) -> U21(active(X1), X2, X3)
active(U22(X1, X2)) -> U22(active(X1), X2)
active(U211(X1, X2)) -> U211(active(X1), X2)
active(U212(X1, X2)) -> U212(active(X1), X2)
active(U221(X1, X2, X3)) -> U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) -> U222(active(X1), X2, X3)
active(fst(X)) -> fst(active(X))
active(U31(X1, X2, X3)) -> U31(active(X1), X2, X3)
active(U32(X1, X2)) -> U32(active(X1), X2)
active(U41(X1, X2)) -> U41(active(X1), X2)
active(U42(X)) -> U42(active(X))
active(U51(X1, X2)) -> U51(active(X1), X2)
active(U52(X)) -> U52(active(X))
active(U61(X)) -> U61(active(X))
active(U71(X)) -> U71(active(X))
active(U81(X)) -> U81(active(X))
active(U91(X)) -> U91(active(X))
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) -> mark(U101(X1, X2))
U102(mark(X)) -> mark(U102(X))
U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3))
U111(mark(X)) -> mark(U111(X))
snd(mark(X)) -> mark(snd(X))
splitAt(mark(X1), X2) -> mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) -> mark(splitAt(X1, X2))
U121(mark(X)) -> mark(U121(X))
U131(mark(X1), X2) -> mark(U131(X1, X2))
U132(mark(X)) -> mark(U132(X))
U141(mark(X1), X2) -> mark(U141(X1, X2))
U142(mark(X)) -> mark(U142(X))
U151(mark(X1), X2) -> mark(U151(X1, X2))
U152(mark(X)) -> mark(U152(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))
U172(mark(X1), X2, X3) -> mark(U172(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))
U182(mark(X1), X2) -> mark(U182(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, X3, X4) -> mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) -> mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) -> mark(U204(X1, X2))
U21(mark(X1), X2, X3) -> mark(U21(X1, X2, X3))
U22(mark(X1), X2) -> mark(U22(X1, X2))
U211(mark(X1), X2) -> mark(U211(X1, X2))
U212(mark(X1), X2) -> mark(U212(X1, X2))
U221(mark(X1), X2, X3) -> mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) -> mark(U222(X1, X2, X3))
fst(mark(X)) -> mark(fst(X))
U31(mark(X1), X2, X3) -> mark(U31(X1, X2, X3))
U32(mark(X1), X2) -> mark(U32(X1, X2))
U41(mark(X1), X2) -> mark(U41(X1, X2))
U42(mark(X)) -> mark(U42(X))
U51(mark(X1), X2) -> mark(U51(X1, X2))
U52(mark(X)) -> mark(U52(X))
U61(mark(X)) -> mark(U61(X))
U71(mark(X)) -> mark(U71(X))
U81(mark(X)) -> mark(U81(X))
U91(mark(X)) -> mark(U91(X))
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)) -> U101(proper(X1), proper(X2))
proper(tt) -> ok(tt)
proper(U102(X)) -> U102(proper(X))
proper(isLNat(X)) -> isLNat(proper(X))
proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) -> U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) -> U111(proper(X))
proper(snd(X)) -> snd(proper(X))
proper(splitAt(X1, X2)) -> splitAt(proper(X1), proper(X2))
proper(U121(X)) -> U121(proper(X))
proper(U131(X1, X2)) -> U131(proper(X1), proper(X2))
proper(U132(X)) -> U132(proper(X))
proper(U141(X1, X2)) -> U141(proper(X1), proper(X2))
proper(U142(X)) -> U142(proper(X))
proper(U151(X1, X2)) -> U151(proper(X1), proper(X2))
proper(U152(X)) -> U152(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(U172(X1, X2, X3)) -> U172(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(U182(X1, X2)) -> U182(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, X3, X4)) -> U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) -> isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) -> U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) -> U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) -> U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) -> U22(proper(X1), proper(X2))
proper(U211(X1, X2)) -> U211(proper(X1), proper(X2))
proper(U212(X1, X2)) -> U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) -> U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) -> U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) -> fst(proper(X))
proper(U31(X1, X2, X3)) -> U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) -> U32(proper(X1), proper(X2))
proper(U41(X1, X2)) -> U41(proper(X1), proper(X2))
proper(U42(X)) -> U42(proper(X))
proper(U51(X1, X2)) -> U51(proper(X1), proper(X2))
proper(U52(X)) -> U52(proper(X))
proper(U61(X)) -> U61(proper(X))
proper(U71(X)) -> U71(proper(X))
proper(U81(X)) -> U81(proper(X))
proper(U91(X)) -> U91(proper(X))
proper(isPLNat(X)) -> isPLNat(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(U101(X1, X2))
U102(ok(X)) -> ok(U102(X))
isLNat(ok(X)) -> ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))
U111(ok(X)) -> ok(U111(X))
snd(ok(X)) -> ok(snd(X))
splitAt(ok(X1), ok(X2)) -> ok(splitAt(X1, X2))
U121(ok(X)) -> ok(U121(X))
U131(ok(X1), ok(X2)) -> ok(U131(X1, X2))
U132(ok(X)) -> ok(U132(X))
U141(ok(X1), ok(X2)) -> ok(U141(X1, X2))
U142(ok(X)) -> ok(U142(X))
U151(ok(X1), ok(X2)) -> ok(U151(X1, X2))
U152(ok(X)) -> ok(U152(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))
U172(ok(X1), ok(X2), ok(X3)) -> ok(U172(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))
U182(ok(X1), ok(X2)) -> ok(U182(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(X3), ok(X4)) -> ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) -> ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) -> ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) -> ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) -> ok(U22(X1, X2))
U211(ok(X1), ok(X2)) -> ok(U211(X1, X2))
U212(ok(X1), ok(X2)) -> ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) -> ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) -> ok(U222(X1, X2, X3))
fst(ok(X)) -> ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) -> ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) -> ok(U32(X1, X2))
U41(ok(X1), ok(X2)) -> ok(U41(X1, X2))
U42(ok(X)) -> ok(U42(X))
U51(ok(X1), ok(X2)) -> ok(U51(X1, X2))
U52(ok(X)) -> ok(U52(X))
U61(ok(X)) -> ok(U61(X))
U71(ok(X)) -> ok(U71(X))
U81(ok(X)) -> ok(U81(X))
U91(ok(X)) -> ok(U91(X))
isPLNat(ok(X)) -> ok(isPLNat(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))

Types:
active :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U12 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U172 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U182 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U203 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U204 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U22 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U212 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U222 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U32 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok -> top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat -> tt:mark:nil:0':ok


Lemmas:
U102(gen_tt:mark:nil:0':ok3_0(+(1, n5_0))) -> *4_0, rt in Omega(n5_0)
U12(gen_tt:mark:nil:0':ok3_0(+(1, n1128_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) -> *4_0, rt in Omega(n1128_0)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n9121_0))) -> *4_0, rt in Omega(n9121_0)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n10537_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n10537_0)
U132(gen_tt:mark:nil:0':ok3_0(+(1, n15953_0))) -> *4_0, rt in Omega(n15953_0)
U142(gen_tt:mark:nil:0':ok3_0(+(1, n17620_0))) -> *4_0, rt in Omega(n17620_0)
U152(gen_tt:mark:nil:0':ok3_0(+(1, n19388_0))) -> *4_0, rt in Omega(n19388_0)
cons(gen_tt:mark:nil:0':ok3_0(+(1, n21257_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n21257_0)
natsFrom(gen_tt:mark:nil:0':ok3_0(+(1, n27404_0))) -> *4_0, rt in Omega(n27404_0)


Generator Equations:
gen_tt:mark:nil:0':ok3_0(0) <=> tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) <=> mark(gen_tt:mark:nil:0':ok3_0(x))


The following defined symbols remain to be analysed:
s, active, U172, head, afterNth, U182, pair, U202, isNatural, U203, U204, U22, U212, U222, fst, U32, U42, U52, U11, U21, U31, U41, U51, U61, isPLNat, U71, U81, U91, U101, U111, U121, U131, U141, U151, U161, U171, U181, U191, U201, U211, U221, tail, take, sel, proper, top

They will be analysed ascendingly in the following order:
s < active
U172 < active
head < active
afterNth < active
U182 < active
pair < active
U202 < active
isNatural < active
U203 < active
U204 < active
U22 < active
U212 < active
U222 < active
fst < active
U32 < active
U42 < active
U52 < active
U11 < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
isPLNat < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
U141 < active
U151 < active
U161 < active
U171 < active
U181 < active
U191 < active
U201 < active
U211 < active
U221 < active
tail < active
take < active
sel < active
active < top
s < proper
U172 < proper
head < proper
afterNth < proper
U182 < proper
pair < proper
U202 < proper
isNatural < proper
U203 < proper
U204 < proper
U22 < proper
U212 < proper
U222 < proper
fst < proper
U32 < proper
U42 < proper
U52 < proper
U11 < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
isPLNat < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
U141 < proper
U151 < proper
U161 < proper
U171 < proper
U181 < proper
U191 < proper
U201 < proper
U211 < proper
U221 < proper
tail < proper
take < proper
sel < proper
proper < top

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

(47) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
s(gen_tt:mark:nil:0':ok3_0(+(1, n29524_0))) -> *4_0, rt in Omega(n29524_0)

Induction Base:
s(gen_tt:mark:nil:0':ok3_0(+(1, 0)))

Induction Step:
s(gen_tt:mark:nil:0':ok3_0(+(1, +(n29524_0, 1)))) ->_R^Omega(1)
mark(s(gen_tt:mark:nil:0':ok3_0(+(1, n29524_0)))) ->_IH
mark(*4_0)

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
----------------------------------------

(48)
Obligation:
TRS:
Rules:
active(U101(tt, V2)) -> mark(U102(isLNat(V2)))
active(U102(tt)) -> mark(tt)
active(U11(tt, N, XS)) -> mark(U12(isLNat(XS), N, XS))
active(U111(tt)) -> mark(tt)
active(U12(tt, N, XS)) -> mark(snd(splitAt(N, XS)))
active(U121(tt)) -> mark(tt)
active(U131(tt, V2)) -> mark(U132(isLNat(V2)))
active(U132(tt)) -> mark(tt)
active(U141(tt, V2)) -> mark(U142(isLNat(V2)))
active(U142(tt)) -> mark(tt)
active(U151(tt, V2)) -> mark(U152(isLNat(V2)))
active(U152(tt)) -> mark(tt)
active(U161(tt, N)) -> mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) -> mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) -> mark(head(afterNth(N, XS)))
active(U181(tt, Y)) -> mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) -> mark(Y)
active(U191(tt, XS)) -> mark(pair(nil, XS))
active(U201(tt, N, X, XS)) -> mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) -> mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) -> mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) -> mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) -> mark(U22(isLNat(Y), X))
active(U211(tt, XS)) -> mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) -> mark(XS)
active(U22(tt, X)) -> mark(X)
active(U221(tt, N, XS)) -> mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) -> mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) -> mark(U32(isLNat(XS), N))
active(U32(tt, N)) -> mark(N)
active(U41(tt, V2)) -> mark(U42(isLNat(V2)))
active(U42(tt)) -> mark(tt)
active(U51(tt, V2)) -> mark(U52(isLNat(V2)))
active(U52(tt)) -> mark(tt)
active(U61(tt)) -> mark(tt)
active(U71(tt)) -> mark(tt)
active(U81(tt)) -> mark(tt)
active(U91(tt)) -> mark(tt)
active(afterNth(N, XS)) -> mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) -> mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) -> mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) -> mark(tt)
active(isLNat(afterNth(V1, V2))) -> mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) -> mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) -> mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) -> mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) -> mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) -> mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) -> mark(U101(isNatural(V1), V2))
active(isNatural(0')) -> mark(tt)
active(isNatural(head(V1))) -> mark(U111(isLNat(V1)))
active(isNatural(s(V1))) -> mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) -> mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) -> mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) -> mark(U151(isNatural(V1), V2))
active(natsFrom(N)) -> mark(U161(isNatural(N), N))
active(sel(N, XS)) -> mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) -> mark(U181(isLNat(X), Y))
active(splitAt(0', XS)) -> mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) -> mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) -> mark(U211(isNatural(N), XS))
active(take(N, XS)) -> mark(U221(isNatural(N), N, XS))
active(U101(X1, X2)) -> U101(active(X1), X2)
active(U102(X)) -> U102(active(X))
active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3)
active(U111(X)) -> U111(active(X))
active(snd(X)) -> snd(active(X))
active(splitAt(X1, X2)) -> splitAt(active(X1), X2)
active(splitAt(X1, X2)) -> splitAt(X1, active(X2))
active(U121(X)) -> U121(active(X))
active(U131(X1, X2)) -> U131(active(X1), X2)
active(U132(X)) -> U132(active(X))
active(U141(X1, X2)) -> U141(active(X1), X2)
active(U142(X)) -> U142(active(X))
active(U151(X1, X2)) -> U151(active(X1), X2)
active(U152(X)) -> U152(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(U172(X1, X2, X3)) -> U172(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(U182(X1, X2)) -> U182(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, X3, X4)) -> U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) -> U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) -> U204(active(X1), X2)
active(U21(X1, X2, X3)) -> U21(active(X1), X2, X3)
active(U22(X1, X2)) -> U22(active(X1), X2)
active(U211(X1, X2)) -> U211(active(X1), X2)
active(U212(X1, X2)) -> U212(active(X1), X2)
active(U221(X1, X2, X3)) -> U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) -> U222(active(X1), X2, X3)
active(fst(X)) -> fst(active(X))
active(U31(X1, X2, X3)) -> U31(active(X1), X2, X3)
active(U32(X1, X2)) -> U32(active(X1), X2)
active(U41(X1, X2)) -> U41(active(X1), X2)
active(U42(X)) -> U42(active(X))
active(U51(X1, X2)) -> U51(active(X1), X2)
active(U52(X)) -> U52(active(X))
active(U61(X)) -> U61(active(X))
active(U71(X)) -> U71(active(X))
active(U81(X)) -> U81(active(X))
active(U91(X)) -> U91(active(X))
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) -> mark(U101(X1, X2))
U102(mark(X)) -> mark(U102(X))
U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3))
U111(mark(X)) -> mark(U111(X))
snd(mark(X)) -> mark(snd(X))
splitAt(mark(X1), X2) -> mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) -> mark(splitAt(X1, X2))
U121(mark(X)) -> mark(U121(X))
U131(mark(X1), X2) -> mark(U131(X1, X2))
U132(mark(X)) -> mark(U132(X))
U141(mark(X1), X2) -> mark(U141(X1, X2))
U142(mark(X)) -> mark(U142(X))
U151(mark(X1), X2) -> mark(U151(X1, X2))
U152(mark(X)) -> mark(U152(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))
U172(mark(X1), X2, X3) -> mark(U172(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))
U182(mark(X1), X2) -> mark(U182(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, X3, X4) -> mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) -> mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) -> mark(U204(X1, X2))
U21(mark(X1), X2, X3) -> mark(U21(X1, X2, X3))
U22(mark(X1), X2) -> mark(U22(X1, X2))
U211(mark(X1), X2) -> mark(U211(X1, X2))
U212(mark(X1), X2) -> mark(U212(X1, X2))
U221(mark(X1), X2, X3) -> mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) -> mark(U222(X1, X2, X3))
fst(mark(X)) -> mark(fst(X))
U31(mark(X1), X2, X3) -> mark(U31(X1, X2, X3))
U32(mark(X1), X2) -> mark(U32(X1, X2))
U41(mark(X1), X2) -> mark(U41(X1, X2))
U42(mark(X)) -> mark(U42(X))
U51(mark(X1), X2) -> mark(U51(X1, X2))
U52(mark(X)) -> mark(U52(X))
U61(mark(X)) -> mark(U61(X))
U71(mark(X)) -> mark(U71(X))
U81(mark(X)) -> mark(U81(X))
U91(mark(X)) -> mark(U91(X))
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)) -> U101(proper(X1), proper(X2))
proper(tt) -> ok(tt)
proper(U102(X)) -> U102(proper(X))
proper(isLNat(X)) -> isLNat(proper(X))
proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) -> U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) -> U111(proper(X))
proper(snd(X)) -> snd(proper(X))
proper(splitAt(X1, X2)) -> splitAt(proper(X1), proper(X2))
proper(U121(X)) -> U121(proper(X))
proper(U131(X1, X2)) -> U131(proper(X1), proper(X2))
proper(U132(X)) -> U132(proper(X))
proper(U141(X1, X2)) -> U141(proper(X1), proper(X2))
proper(U142(X)) -> U142(proper(X))
proper(U151(X1, X2)) -> U151(proper(X1), proper(X2))
proper(U152(X)) -> U152(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(U172(X1, X2, X3)) -> U172(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(U182(X1, X2)) -> U182(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, X3, X4)) -> U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) -> isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) -> U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) -> U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) -> U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) -> U22(proper(X1), proper(X2))
proper(U211(X1, X2)) -> U211(proper(X1), proper(X2))
proper(U212(X1, X2)) -> U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) -> U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) -> U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) -> fst(proper(X))
proper(U31(X1, X2, X3)) -> U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) -> U32(proper(X1), proper(X2))
proper(U41(X1, X2)) -> U41(proper(X1), proper(X2))
proper(U42(X)) -> U42(proper(X))
proper(U51(X1, X2)) -> U51(proper(X1), proper(X2))
proper(U52(X)) -> U52(proper(X))
proper(U61(X)) -> U61(proper(X))
proper(U71(X)) -> U71(proper(X))
proper(U81(X)) -> U81(proper(X))
proper(U91(X)) -> U91(proper(X))
proper(isPLNat(X)) -> isPLNat(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(U101(X1, X2))
U102(ok(X)) -> ok(U102(X))
isLNat(ok(X)) -> ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))
U111(ok(X)) -> ok(U111(X))
snd(ok(X)) -> ok(snd(X))
splitAt(ok(X1), ok(X2)) -> ok(splitAt(X1, X2))
U121(ok(X)) -> ok(U121(X))
U131(ok(X1), ok(X2)) -> ok(U131(X1, X2))
U132(ok(X)) -> ok(U132(X))
U141(ok(X1), ok(X2)) -> ok(U141(X1, X2))
U142(ok(X)) -> ok(U142(X))
U151(ok(X1), ok(X2)) -> ok(U151(X1, X2))
U152(ok(X)) -> ok(U152(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))
U172(ok(X1), ok(X2), ok(X3)) -> ok(U172(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))
U182(ok(X1), ok(X2)) -> ok(U182(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(X3), ok(X4)) -> ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) -> ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) -> ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) -> ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) -> ok(U22(X1, X2))
U211(ok(X1), ok(X2)) -> ok(U211(X1, X2))
U212(ok(X1), ok(X2)) -> ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) -> ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) -> ok(U222(X1, X2, X3))
fst(ok(X)) -> ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) -> ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) -> ok(U32(X1, X2))
U41(ok(X1), ok(X2)) -> ok(U41(X1, X2))
U42(ok(X)) -> ok(U42(X))
U51(ok(X1), ok(X2)) -> ok(U51(X1, X2))
U52(ok(X)) -> ok(U52(X))
U61(ok(X)) -> ok(U61(X))
U71(ok(X)) -> ok(U71(X))
U81(ok(X)) -> ok(U81(X))
U91(ok(X)) -> ok(U91(X))
isPLNat(ok(X)) -> ok(isPLNat(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))

Types:
active :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U12 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U172 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U182 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U203 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U204 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U22 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U212 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U222 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U32 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok -> top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat -> tt:mark:nil:0':ok


Lemmas:
U102(gen_tt:mark:nil:0':ok3_0(+(1, n5_0))) -> *4_0, rt in Omega(n5_0)
U12(gen_tt:mark:nil:0':ok3_0(+(1, n1128_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) -> *4_0, rt in Omega(n1128_0)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n9121_0))) -> *4_0, rt in Omega(n9121_0)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n10537_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n10537_0)
U132(gen_tt:mark:nil:0':ok3_0(+(1, n15953_0))) -> *4_0, rt in Omega(n15953_0)
U142(gen_tt:mark:nil:0':ok3_0(+(1, n17620_0))) -> *4_0, rt in Omega(n17620_0)
U152(gen_tt:mark:nil:0':ok3_0(+(1, n19388_0))) -> *4_0, rt in Omega(n19388_0)
cons(gen_tt:mark:nil:0':ok3_0(+(1, n21257_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n21257_0)
natsFrom(gen_tt:mark:nil:0':ok3_0(+(1, n27404_0))) -> *4_0, rt in Omega(n27404_0)
s(gen_tt:mark:nil:0':ok3_0(+(1, n29524_0))) -> *4_0, rt in Omega(n29524_0)


Generator Equations:
gen_tt:mark:nil:0':ok3_0(0) <=> tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) <=> mark(gen_tt:mark:nil:0':ok3_0(x))


The following defined symbols remain to be analysed:
U172, active, head, afterNth, U182, pair, U202, isNatural, U203, U204, U22, U212, U222, fst, U32, U42, U52, U11, U21, U31, U41, U51, U61, isPLNat, U71, U81, U91, U101, U111, U121, U131, U141, U151, U161, U171, U181, U191, U201, U211, U221, tail, take, sel, proper, top

They will be analysed ascendingly in the following order:
U172 < active
head < active
afterNth < active
U182 < active
pair < active
U202 < active
isNatural < active
U203 < active
U204 < active
U22 < active
U212 < active
U222 < active
fst < active
U32 < active
U42 < active
U52 < active
U11 < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
isPLNat < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
U141 < active
U151 < active
U161 < active
U171 < active
U181 < active
U191 < active
U201 < active
U211 < active
U221 < active
tail < active
take < active
sel < active
active < top
U172 < proper
head < proper
afterNth < proper
U182 < proper
pair < proper
U202 < proper
isNatural < proper
U203 < proper
U204 < proper
U22 < proper
U212 < proper
U222 < proper
fst < proper
U32 < proper
U42 < proper
U52 < proper
U11 < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
isPLNat < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
U141 < proper
U151 < proper
U161 < proper
U171 < proper
U181 < proper
U191 < proper
U201 < proper
U211 < proper
U221 < proper
tail < proper
take < proper
sel < proper
proper < top

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

(49) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
U172(gen_tt:mark:nil:0':ok3_0(+(1, n31745_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) -> *4_0, rt in Omega(n31745_0)

Induction Base:
U172(gen_tt:mark:nil:0':ok3_0(+(1, 0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c))

Induction Step:
U172(gen_tt:mark:nil:0':ok3_0(+(1, +(n31745_0, 1))), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) ->_R^Omega(1)
mark(U172(gen_tt:mark:nil:0':ok3_0(+(1, n31745_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c))) ->_IH
mark(*4_0)

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
----------------------------------------

(50)
Obligation:
TRS:
Rules:
active(U101(tt, V2)) -> mark(U102(isLNat(V2)))
active(U102(tt)) -> mark(tt)
active(U11(tt, N, XS)) -> mark(U12(isLNat(XS), N, XS))
active(U111(tt)) -> mark(tt)
active(U12(tt, N, XS)) -> mark(snd(splitAt(N, XS)))
active(U121(tt)) -> mark(tt)
active(U131(tt, V2)) -> mark(U132(isLNat(V2)))
active(U132(tt)) -> mark(tt)
active(U141(tt, V2)) -> mark(U142(isLNat(V2)))
active(U142(tt)) -> mark(tt)
active(U151(tt, V2)) -> mark(U152(isLNat(V2)))
active(U152(tt)) -> mark(tt)
active(U161(tt, N)) -> mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) -> mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) -> mark(head(afterNth(N, XS)))
active(U181(tt, Y)) -> mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) -> mark(Y)
active(U191(tt, XS)) -> mark(pair(nil, XS))
active(U201(tt, N, X, XS)) -> mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) -> mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) -> mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) -> mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) -> mark(U22(isLNat(Y), X))
active(U211(tt, XS)) -> mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) -> mark(XS)
active(U22(tt, X)) -> mark(X)
active(U221(tt, N, XS)) -> mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) -> mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) -> mark(U32(isLNat(XS), N))
active(U32(tt, N)) -> mark(N)
active(U41(tt, V2)) -> mark(U42(isLNat(V2)))
active(U42(tt)) -> mark(tt)
active(U51(tt, V2)) -> mark(U52(isLNat(V2)))
active(U52(tt)) -> mark(tt)
active(U61(tt)) -> mark(tt)
active(U71(tt)) -> mark(tt)
active(U81(tt)) -> mark(tt)
active(U91(tt)) -> mark(tt)
active(afterNth(N, XS)) -> mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) -> mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) -> mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) -> mark(tt)
active(isLNat(afterNth(V1, V2))) -> mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) -> mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) -> mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) -> mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) -> mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) -> mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) -> mark(U101(isNatural(V1), V2))
active(isNatural(0')) -> mark(tt)
active(isNatural(head(V1))) -> mark(U111(isLNat(V1)))
active(isNatural(s(V1))) -> mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) -> mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) -> mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) -> mark(U151(isNatural(V1), V2))
active(natsFrom(N)) -> mark(U161(isNatural(N), N))
active(sel(N, XS)) -> mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) -> mark(U181(isLNat(X), Y))
active(splitAt(0', XS)) -> mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) -> mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) -> mark(U211(isNatural(N), XS))
active(take(N, XS)) -> mark(U221(isNatural(N), N, XS))
active(U101(X1, X2)) -> U101(active(X1), X2)
active(U102(X)) -> U102(active(X))
active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3)
active(U111(X)) -> U111(active(X))
active(snd(X)) -> snd(active(X))
active(splitAt(X1, X2)) -> splitAt(active(X1), X2)
active(splitAt(X1, X2)) -> splitAt(X1, active(X2))
active(U121(X)) -> U121(active(X))
active(U131(X1, X2)) -> U131(active(X1), X2)
active(U132(X)) -> U132(active(X))
active(U141(X1, X2)) -> U141(active(X1), X2)
active(U142(X)) -> U142(active(X))
active(U151(X1, X2)) -> U151(active(X1), X2)
active(U152(X)) -> U152(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(U172(X1, X2, X3)) -> U172(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(U182(X1, X2)) -> U182(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, X3, X4)) -> U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) -> U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) -> U204(active(X1), X2)
active(U21(X1, X2, X3)) -> U21(active(X1), X2, X3)
active(U22(X1, X2)) -> U22(active(X1), X2)
active(U211(X1, X2)) -> U211(active(X1), X2)
active(U212(X1, X2)) -> U212(active(X1), X2)
active(U221(X1, X2, X3)) -> U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) -> U222(active(X1), X2, X3)
active(fst(X)) -> fst(active(X))
active(U31(X1, X2, X3)) -> U31(active(X1), X2, X3)
active(U32(X1, X2)) -> U32(active(X1), X2)
active(U41(X1, X2)) -> U41(active(X1), X2)
active(U42(X)) -> U42(active(X))
active(U51(X1, X2)) -> U51(active(X1), X2)
active(U52(X)) -> U52(active(X))
active(U61(X)) -> U61(active(X))
active(U71(X)) -> U71(active(X))
active(U81(X)) -> U81(active(X))
active(U91(X)) -> U91(active(X))
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) -> mark(U101(X1, X2))
U102(mark(X)) -> mark(U102(X))
U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3))
U111(mark(X)) -> mark(U111(X))
snd(mark(X)) -> mark(snd(X))
splitAt(mark(X1), X2) -> mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) -> mark(splitAt(X1, X2))
U121(mark(X)) -> mark(U121(X))
U131(mark(X1), X2) -> mark(U131(X1, X2))
U132(mark(X)) -> mark(U132(X))
U141(mark(X1), X2) -> mark(U141(X1, X2))
U142(mark(X)) -> mark(U142(X))
U151(mark(X1), X2) -> mark(U151(X1, X2))
U152(mark(X)) -> mark(U152(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))
U172(mark(X1), X2, X3) -> mark(U172(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))
U182(mark(X1), X2) -> mark(U182(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, X3, X4) -> mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) -> mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) -> mark(U204(X1, X2))
U21(mark(X1), X2, X3) -> mark(U21(X1, X2, X3))
U22(mark(X1), X2) -> mark(U22(X1, X2))
U211(mark(X1), X2) -> mark(U211(X1, X2))
U212(mark(X1), X2) -> mark(U212(X1, X2))
U221(mark(X1), X2, X3) -> mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) -> mark(U222(X1, X2, X3))
fst(mark(X)) -> mark(fst(X))
U31(mark(X1), X2, X3) -> mark(U31(X1, X2, X3))
U32(mark(X1), X2) -> mark(U32(X1, X2))
U41(mark(X1), X2) -> mark(U41(X1, X2))
U42(mark(X)) -> mark(U42(X))
U51(mark(X1), X2) -> mark(U51(X1, X2))
U52(mark(X)) -> mark(U52(X))
U61(mark(X)) -> mark(U61(X))
U71(mark(X)) -> mark(U71(X))
U81(mark(X)) -> mark(U81(X))
U91(mark(X)) -> mark(U91(X))
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)) -> U101(proper(X1), proper(X2))
proper(tt) -> ok(tt)
proper(U102(X)) -> U102(proper(X))
proper(isLNat(X)) -> isLNat(proper(X))
proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) -> U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) -> U111(proper(X))
proper(snd(X)) -> snd(proper(X))
proper(splitAt(X1, X2)) -> splitAt(proper(X1), proper(X2))
proper(U121(X)) -> U121(proper(X))
proper(U131(X1, X2)) -> U131(proper(X1), proper(X2))
proper(U132(X)) -> U132(proper(X))
proper(U141(X1, X2)) -> U141(proper(X1), proper(X2))
proper(U142(X)) -> U142(proper(X))
proper(U151(X1, X2)) -> U151(proper(X1), proper(X2))
proper(U152(X)) -> U152(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(U172(X1, X2, X3)) -> U172(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(U182(X1, X2)) -> U182(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, X3, X4)) -> U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) -> isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) -> U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) -> U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) -> U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) -> U22(proper(X1), proper(X2))
proper(U211(X1, X2)) -> U211(proper(X1), proper(X2))
proper(U212(X1, X2)) -> U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) -> U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) -> U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) -> fst(proper(X))
proper(U31(X1, X2, X3)) -> U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) -> U32(proper(X1), proper(X2))
proper(U41(X1, X2)) -> U41(proper(X1), proper(X2))
proper(U42(X)) -> U42(proper(X))
proper(U51(X1, X2)) -> U51(proper(X1), proper(X2))
proper(U52(X)) -> U52(proper(X))
proper(U61(X)) -> U61(proper(X))
proper(U71(X)) -> U71(proper(X))
proper(U81(X)) -> U81(proper(X))
proper(U91(X)) -> U91(proper(X))
proper(isPLNat(X)) -> isPLNat(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(U101(X1, X2))
U102(ok(X)) -> ok(U102(X))
isLNat(ok(X)) -> ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))
U111(ok(X)) -> ok(U111(X))
snd(ok(X)) -> ok(snd(X))
splitAt(ok(X1), ok(X2)) -> ok(splitAt(X1, X2))
U121(ok(X)) -> ok(U121(X))
U131(ok(X1), ok(X2)) -> ok(U131(X1, X2))
U132(ok(X)) -> ok(U132(X))
U141(ok(X1), ok(X2)) -> ok(U141(X1, X2))
U142(ok(X)) -> ok(U142(X))
U151(ok(X1), ok(X2)) -> ok(U151(X1, X2))
U152(ok(X)) -> ok(U152(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))
U172(ok(X1), ok(X2), ok(X3)) -> ok(U172(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))
U182(ok(X1), ok(X2)) -> ok(U182(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(X3), ok(X4)) -> ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) -> ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) -> ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) -> ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) -> ok(U22(X1, X2))
U211(ok(X1), ok(X2)) -> ok(U211(X1, X2))
U212(ok(X1), ok(X2)) -> ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) -> ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) -> ok(U222(X1, X2, X3))
fst(ok(X)) -> ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) -> ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) -> ok(U32(X1, X2))
U41(ok(X1), ok(X2)) -> ok(U41(X1, X2))
U42(ok(X)) -> ok(U42(X))
U51(ok(X1), ok(X2)) -> ok(U51(X1, X2))
U52(ok(X)) -> ok(U52(X))
U61(ok(X)) -> ok(U61(X))
U71(ok(X)) -> ok(U71(X))
U81(ok(X)) -> ok(U81(X))
U91(ok(X)) -> ok(U91(X))
isPLNat(ok(X)) -> ok(isPLNat(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))

Types:
active :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U12 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U172 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U182 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U203 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U204 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U22 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U212 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U222 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U32 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok -> top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat -> tt:mark:nil:0':ok


Lemmas:
U102(gen_tt:mark:nil:0':ok3_0(+(1, n5_0))) -> *4_0, rt in Omega(n5_0)
U12(gen_tt:mark:nil:0':ok3_0(+(1, n1128_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) -> *4_0, rt in Omega(n1128_0)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n9121_0))) -> *4_0, rt in Omega(n9121_0)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n10537_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n10537_0)
U132(gen_tt:mark:nil:0':ok3_0(+(1, n15953_0))) -> *4_0, rt in Omega(n15953_0)
U142(gen_tt:mark:nil:0':ok3_0(+(1, n17620_0))) -> *4_0, rt in Omega(n17620_0)
U152(gen_tt:mark:nil:0':ok3_0(+(1, n19388_0))) -> *4_0, rt in Omega(n19388_0)
cons(gen_tt:mark:nil:0':ok3_0(+(1, n21257_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n21257_0)
natsFrom(gen_tt:mark:nil:0':ok3_0(+(1, n27404_0))) -> *4_0, rt in Omega(n27404_0)
s(gen_tt:mark:nil:0':ok3_0(+(1, n29524_0))) -> *4_0, rt in Omega(n29524_0)
U172(gen_tt:mark:nil:0':ok3_0(+(1, n31745_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) -> *4_0, rt in Omega(n31745_0)


Generator Equations:
gen_tt:mark:nil:0':ok3_0(0) <=> tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) <=> mark(gen_tt:mark:nil:0':ok3_0(x))


The following defined symbols remain to be analysed:
head, active, afterNth, U182, pair, U202, isNatural, U203, U204, U22, U212, U222, fst, U32, U42, U52, U11, U21, U31, U41, U51, U61, isPLNat, U71, U81, U91, U101, U111, U121, U131, U141, U151, U161, U171, U181, U191, U201, U211, U221, tail, take, sel, proper, top

They will be analysed ascendingly in the following order:
head < active
afterNth < active
U182 < active
pair < active
U202 < active
isNatural < active
U203 < active
U204 < active
U22 < active
U212 < active
U222 < active
fst < active
U32 < active
U42 < active
U52 < active
U11 < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
isPLNat < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
U141 < active
U151 < active
U161 < active
U171 < active
U181 < active
U191 < active
U201 < active
U211 < active
U221 < active
tail < active
take < active
sel < active
active < top
head < proper
afterNth < proper
U182 < proper
pair < proper
U202 < proper
isNatural < proper
U203 < proper
U204 < proper
U22 < proper
U212 < proper
U222 < proper
fst < proper
U32 < proper
U42 < proper
U52 < proper
U11 < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
isPLNat < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
U141 < proper
U151 < proper
U161 < proper
U171 < proper
U181 < proper
U191 < proper
U201 < proper
U211 < proper
U221 < proper
tail < proper
take < proper
sel < proper
proper < top

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

(51) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
head(gen_tt:mark:nil:0':ok3_0(+(1, n43161_0))) -> *4_0, rt in Omega(n43161_0)

Induction Base:
head(gen_tt:mark:nil:0':ok3_0(+(1, 0)))

Induction Step:
head(gen_tt:mark:nil:0':ok3_0(+(1, +(n43161_0, 1)))) ->_R^Omega(1)
mark(head(gen_tt:mark:nil:0':ok3_0(+(1, n43161_0)))) ->_IH
mark(*4_0)

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
----------------------------------------

(52)
Obligation:
TRS:
Rules:
active(U101(tt, V2)) -> mark(U102(isLNat(V2)))
active(U102(tt)) -> mark(tt)
active(U11(tt, N, XS)) -> mark(U12(isLNat(XS), N, XS))
active(U111(tt)) -> mark(tt)
active(U12(tt, N, XS)) -> mark(snd(splitAt(N, XS)))
active(U121(tt)) -> mark(tt)
active(U131(tt, V2)) -> mark(U132(isLNat(V2)))
active(U132(tt)) -> mark(tt)
active(U141(tt, V2)) -> mark(U142(isLNat(V2)))
active(U142(tt)) -> mark(tt)
active(U151(tt, V2)) -> mark(U152(isLNat(V2)))
active(U152(tt)) -> mark(tt)
active(U161(tt, N)) -> mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) -> mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) -> mark(head(afterNth(N, XS)))
active(U181(tt, Y)) -> mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) -> mark(Y)
active(U191(tt, XS)) -> mark(pair(nil, XS))
active(U201(tt, N, X, XS)) -> mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) -> mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) -> mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) -> mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) -> mark(U22(isLNat(Y), X))
active(U211(tt, XS)) -> mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) -> mark(XS)
active(U22(tt, X)) -> mark(X)
active(U221(tt, N, XS)) -> mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) -> mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) -> mark(U32(isLNat(XS), N))
active(U32(tt, N)) -> mark(N)
active(U41(tt, V2)) -> mark(U42(isLNat(V2)))
active(U42(tt)) -> mark(tt)
active(U51(tt, V2)) -> mark(U52(isLNat(V2)))
active(U52(tt)) -> mark(tt)
active(U61(tt)) -> mark(tt)
active(U71(tt)) -> mark(tt)
active(U81(tt)) -> mark(tt)
active(U91(tt)) -> mark(tt)
active(afterNth(N, XS)) -> mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) -> mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) -> mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) -> mark(tt)
active(isLNat(afterNth(V1, V2))) -> mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) -> mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) -> mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) -> mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) -> mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) -> mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) -> mark(U101(isNatural(V1), V2))
active(isNatural(0')) -> mark(tt)
active(isNatural(head(V1))) -> mark(U111(isLNat(V1)))
active(isNatural(s(V1))) -> mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) -> mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) -> mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) -> mark(U151(isNatural(V1), V2))
active(natsFrom(N)) -> mark(U161(isNatural(N), N))
active(sel(N, XS)) -> mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) -> mark(U181(isLNat(X), Y))
active(splitAt(0', XS)) -> mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) -> mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) -> mark(U211(isNatural(N), XS))
active(take(N, XS)) -> mark(U221(isNatural(N), N, XS))
active(U101(X1, X2)) -> U101(active(X1), X2)
active(U102(X)) -> U102(active(X))
active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3)
active(U111(X)) -> U111(active(X))
active(snd(X)) -> snd(active(X))
active(splitAt(X1, X2)) -> splitAt(active(X1), X2)
active(splitAt(X1, X2)) -> splitAt(X1, active(X2))
active(U121(X)) -> U121(active(X))
active(U131(X1, X2)) -> U131(active(X1), X2)
active(U132(X)) -> U132(active(X))
active(U141(X1, X2)) -> U141(active(X1), X2)
active(U142(X)) -> U142(active(X))
active(U151(X1, X2)) -> U151(active(X1), X2)
active(U152(X)) -> U152(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(U172(X1, X2, X3)) -> U172(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(U182(X1, X2)) -> U182(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, X3, X4)) -> U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) -> U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) -> U204(active(X1), X2)
active(U21(X1, X2, X3)) -> U21(active(X1), X2, X3)
active(U22(X1, X2)) -> U22(active(X1), X2)
active(U211(X1, X2)) -> U211(active(X1), X2)
active(U212(X1, X2)) -> U212(active(X1), X2)
active(U221(X1, X2, X3)) -> U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) -> U222(active(X1), X2, X3)
active(fst(X)) -> fst(active(X))
active(U31(X1, X2, X3)) -> U31(active(X1), X2, X3)
active(U32(X1, X2)) -> U32(active(X1), X2)
active(U41(X1, X2)) -> U41(active(X1), X2)
active(U42(X)) -> U42(active(X))
active(U51(X1, X2)) -> U51(active(X1), X2)
active(U52(X)) -> U52(active(X))
active(U61(X)) -> U61(active(X))
active(U71(X)) -> U71(active(X))
active(U81(X)) -> U81(active(X))
active(U91(X)) -> U91(active(X))
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) -> mark(U101(X1, X2))
U102(mark(X)) -> mark(U102(X))
U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3))
U111(mark(X)) -> mark(U111(X))
snd(mark(X)) -> mark(snd(X))
splitAt(mark(X1), X2) -> mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) -> mark(splitAt(X1, X2))
U121(mark(X)) -> mark(U121(X))
U131(mark(X1), X2) -> mark(U131(X1, X2))
U132(mark(X)) -> mark(U132(X))
U141(mark(X1), X2) -> mark(U141(X1, X2))
U142(mark(X)) -> mark(U142(X))
U151(mark(X1), X2) -> mark(U151(X1, X2))
U152(mark(X)) -> mark(U152(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))
U172(mark(X1), X2, X3) -> mark(U172(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))
U182(mark(X1), X2) -> mark(U182(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, X3, X4) -> mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) -> mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) -> mark(U204(X1, X2))
U21(mark(X1), X2, X3) -> mark(U21(X1, X2, X3))
U22(mark(X1), X2) -> mark(U22(X1, X2))
U211(mark(X1), X2) -> mark(U211(X1, X2))
U212(mark(X1), X2) -> mark(U212(X1, X2))
U221(mark(X1), X2, X3) -> mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) -> mark(U222(X1, X2, X3))
fst(mark(X)) -> mark(fst(X))
U31(mark(X1), X2, X3) -> mark(U31(X1, X2, X3))
U32(mark(X1), X2) -> mark(U32(X1, X2))
U41(mark(X1), X2) -> mark(U41(X1, X2))
U42(mark(X)) -> mark(U42(X))
U51(mark(X1), X2) -> mark(U51(X1, X2))
U52(mark(X)) -> mark(U52(X))
U61(mark(X)) -> mark(U61(X))
U71(mark(X)) -> mark(U71(X))
U81(mark(X)) -> mark(U81(X))
U91(mark(X)) -> mark(U91(X))
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)) -> U101(proper(X1), proper(X2))
proper(tt) -> ok(tt)
proper(U102(X)) -> U102(proper(X))
proper(isLNat(X)) -> isLNat(proper(X))
proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) -> U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) -> U111(proper(X))
proper(snd(X)) -> snd(proper(X))
proper(splitAt(X1, X2)) -> splitAt(proper(X1), proper(X2))
proper(U121(X)) -> U121(proper(X))
proper(U131(X1, X2)) -> U131(proper(X1), proper(X2))
proper(U132(X)) -> U132(proper(X))
proper(U141(X1, X2)) -> U141(proper(X1), proper(X2))
proper(U142(X)) -> U142(proper(X))
proper(U151(X1, X2)) -> U151(proper(X1), proper(X2))
proper(U152(X)) -> U152(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(U172(X1, X2, X3)) -> U172(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(U182(X1, X2)) -> U182(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, X3, X4)) -> U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) -> isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) -> U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) -> U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) -> U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) -> U22(proper(X1), proper(X2))
proper(U211(X1, X2)) -> U211(proper(X1), proper(X2))
proper(U212(X1, X2)) -> U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) -> U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) -> U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) -> fst(proper(X))
proper(U31(X1, X2, X3)) -> U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) -> U32(proper(X1), proper(X2))
proper(U41(X1, X2)) -> U41(proper(X1), proper(X2))
proper(U42(X)) -> U42(proper(X))
proper(U51(X1, X2)) -> U51(proper(X1), proper(X2))
proper(U52(X)) -> U52(proper(X))
proper(U61(X)) -> U61(proper(X))
proper(U71(X)) -> U71(proper(X))
proper(U81(X)) -> U81(proper(X))
proper(U91(X)) -> U91(proper(X))
proper(isPLNat(X)) -> isPLNat(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(U101(X1, X2))
U102(ok(X)) -> ok(U102(X))
isLNat(ok(X)) -> ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))
U111(ok(X)) -> ok(U111(X))
snd(ok(X)) -> ok(snd(X))
splitAt(ok(X1), ok(X2)) -> ok(splitAt(X1, X2))
U121(ok(X)) -> ok(U121(X))
U131(ok(X1), ok(X2)) -> ok(U131(X1, X2))
U132(ok(X)) -> ok(U132(X))
U141(ok(X1), ok(X2)) -> ok(U141(X1, X2))
U142(ok(X)) -> ok(U142(X))
U151(ok(X1), ok(X2)) -> ok(U151(X1, X2))
U152(ok(X)) -> ok(U152(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))
U172(ok(X1), ok(X2), ok(X3)) -> ok(U172(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))
U182(ok(X1), ok(X2)) -> ok(U182(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(X3), ok(X4)) -> ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) -> ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) -> ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) -> ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) -> ok(U22(X1, X2))
U211(ok(X1), ok(X2)) -> ok(U211(X1, X2))
U212(ok(X1), ok(X2)) -> ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) -> ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) -> ok(U222(X1, X2, X3))
fst(ok(X)) -> ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) -> ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) -> ok(U32(X1, X2))
U41(ok(X1), ok(X2)) -> ok(U41(X1, X2))
U42(ok(X)) -> ok(U42(X))
U51(ok(X1), ok(X2)) -> ok(U51(X1, X2))
U52(ok(X)) -> ok(U52(X))
U61(ok(X)) -> ok(U61(X))
U71(ok(X)) -> ok(U71(X))
U81(ok(X)) -> ok(U81(X))
U91(ok(X)) -> ok(U91(X))
isPLNat(ok(X)) -> ok(isPLNat(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))

Types:
active :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U12 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U172 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U182 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U203 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U204 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U22 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U212 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U222 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U32 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok -> top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat -> tt:mark:nil:0':ok


Lemmas:
U102(gen_tt:mark:nil:0':ok3_0(+(1, n5_0))) -> *4_0, rt in Omega(n5_0)
U12(gen_tt:mark:nil:0':ok3_0(+(1, n1128_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) -> *4_0, rt in Omega(n1128_0)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n9121_0))) -> *4_0, rt in Omega(n9121_0)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n10537_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n10537_0)
U132(gen_tt:mark:nil:0':ok3_0(+(1, n15953_0))) -> *4_0, rt in Omega(n15953_0)
U142(gen_tt:mark:nil:0':ok3_0(+(1, n17620_0))) -> *4_0, rt in Omega(n17620_0)
U152(gen_tt:mark:nil:0':ok3_0(+(1, n19388_0))) -> *4_0, rt in Omega(n19388_0)
cons(gen_tt:mark:nil:0':ok3_0(+(1, n21257_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n21257_0)
natsFrom(gen_tt:mark:nil:0':ok3_0(+(1, n27404_0))) -> *4_0, rt in Omega(n27404_0)
s(gen_tt:mark:nil:0':ok3_0(+(1, n29524_0))) -> *4_0, rt in Omega(n29524_0)
U172(gen_tt:mark:nil:0':ok3_0(+(1, n31745_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) -> *4_0, rt in Omega(n31745_0)
head(gen_tt:mark:nil:0':ok3_0(+(1, n43161_0))) -> *4_0, rt in Omega(n43161_0)


Generator Equations:
gen_tt:mark:nil:0':ok3_0(0) <=> tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) <=> mark(gen_tt:mark:nil:0':ok3_0(x))


The following defined symbols remain to be analysed:
afterNth, active, U182, pair, U202, isNatural, U203, U204, U22, U212, U222, fst, U32, U42, U52, U11, U21, U31, U41, U51, U61, isPLNat, U71, U81, U91, U101, U111, U121, U131, U141, U151, U161, U171, U181, U191, U201, U211, U221, tail, take, sel, proper, top

They will be analysed ascendingly in the following order:
afterNth < active
U182 < active
pair < active
U202 < active
isNatural < active
U203 < active
U204 < active
U22 < active
U212 < active
U222 < active
fst < active
U32 < active
U42 < active
U52 < active
U11 < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
isPLNat < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
U141 < active
U151 < active
U161 < active
U171 < active
U181 < active
U191 < active
U201 < active
U211 < active
U221 < active
tail < active
take < active
sel < active
active < top
afterNth < proper
U182 < proper
pair < proper
U202 < proper
isNatural < proper
U203 < proper
U204 < proper
U22 < proper
U212 < proper
U222 < proper
fst < proper
U32 < proper
U42 < proper
U52 < proper
U11 < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
isPLNat < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
U141 < proper
U151 < proper
U161 < proper
U171 < proper
U181 < proper
U191 < proper
U201 < proper
U211 < proper
U221 < proper
tail < proper
take < proper
sel < proper
proper < top

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

(53) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
afterNth(gen_tt:mark:nil:0':ok3_0(+(1, n45682_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n45682_0)

Induction Base:
afterNth(gen_tt:mark:nil:0':ok3_0(+(1, 0)), gen_tt:mark:nil:0':ok3_0(b))

Induction Step:
afterNth(gen_tt:mark:nil:0':ok3_0(+(1, +(n45682_0, 1))), gen_tt:mark:nil:0':ok3_0(b)) ->_R^Omega(1)
mark(afterNth(gen_tt:mark:nil:0':ok3_0(+(1, n45682_0)), gen_tt:mark:nil:0':ok3_0(b))) ->_IH
mark(*4_0)

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
----------------------------------------

(54)
Obligation:
TRS:
Rules:
active(U101(tt, V2)) -> mark(U102(isLNat(V2)))
active(U102(tt)) -> mark(tt)
active(U11(tt, N, XS)) -> mark(U12(isLNat(XS), N, XS))
active(U111(tt)) -> mark(tt)
active(U12(tt, N, XS)) -> mark(snd(splitAt(N, XS)))
active(U121(tt)) -> mark(tt)
active(U131(tt, V2)) -> mark(U132(isLNat(V2)))
active(U132(tt)) -> mark(tt)
active(U141(tt, V2)) -> mark(U142(isLNat(V2)))
active(U142(tt)) -> mark(tt)
active(U151(tt, V2)) -> mark(U152(isLNat(V2)))
active(U152(tt)) -> mark(tt)
active(U161(tt, N)) -> mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) -> mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) -> mark(head(afterNth(N, XS)))
active(U181(tt, Y)) -> mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) -> mark(Y)
active(U191(tt, XS)) -> mark(pair(nil, XS))
active(U201(tt, N, X, XS)) -> mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) -> mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) -> mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) -> mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) -> mark(U22(isLNat(Y), X))
active(U211(tt, XS)) -> mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) -> mark(XS)
active(U22(tt, X)) -> mark(X)
active(U221(tt, N, XS)) -> mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) -> mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) -> mark(U32(isLNat(XS), N))
active(U32(tt, N)) -> mark(N)
active(U41(tt, V2)) -> mark(U42(isLNat(V2)))
active(U42(tt)) -> mark(tt)
active(U51(tt, V2)) -> mark(U52(isLNat(V2)))
active(U52(tt)) -> mark(tt)
active(U61(tt)) -> mark(tt)
active(U71(tt)) -> mark(tt)
active(U81(tt)) -> mark(tt)
active(U91(tt)) -> mark(tt)
active(afterNth(N, XS)) -> mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) -> mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) -> mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) -> mark(tt)
active(isLNat(afterNth(V1, V2))) -> mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) -> mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) -> mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) -> mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) -> mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) -> mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) -> mark(U101(isNatural(V1), V2))
active(isNatural(0')) -> mark(tt)
active(isNatural(head(V1))) -> mark(U111(isLNat(V1)))
active(isNatural(s(V1))) -> mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) -> mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) -> mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) -> mark(U151(isNatural(V1), V2))
active(natsFrom(N)) -> mark(U161(isNatural(N), N))
active(sel(N, XS)) -> mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) -> mark(U181(isLNat(X), Y))
active(splitAt(0', XS)) -> mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) -> mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) -> mark(U211(isNatural(N), XS))
active(take(N, XS)) -> mark(U221(isNatural(N), N, XS))
active(U101(X1, X2)) -> U101(active(X1), X2)
active(U102(X)) -> U102(active(X))
active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3)
active(U111(X)) -> U111(active(X))
active(snd(X)) -> snd(active(X))
active(splitAt(X1, X2)) -> splitAt(active(X1), X2)
active(splitAt(X1, X2)) -> splitAt(X1, active(X2))
active(U121(X)) -> U121(active(X))
active(U131(X1, X2)) -> U131(active(X1), X2)
active(U132(X)) -> U132(active(X))
active(U141(X1, X2)) -> U141(active(X1), X2)
active(U142(X)) -> U142(active(X))
active(U151(X1, X2)) -> U151(active(X1), X2)
active(U152(X)) -> U152(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(U172(X1, X2, X3)) -> U172(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(U182(X1, X2)) -> U182(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, X3, X4)) -> U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) -> U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) -> U204(active(X1), X2)
active(U21(X1, X2, X3)) -> U21(active(X1), X2, X3)
active(U22(X1, X2)) -> U22(active(X1), X2)
active(U211(X1, X2)) -> U211(active(X1), X2)
active(U212(X1, X2)) -> U212(active(X1), X2)
active(U221(X1, X2, X3)) -> U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) -> U222(active(X1), X2, X3)
active(fst(X)) -> fst(active(X))
active(U31(X1, X2, X3)) -> U31(active(X1), X2, X3)
active(U32(X1, X2)) -> U32(active(X1), X2)
active(U41(X1, X2)) -> U41(active(X1), X2)
active(U42(X)) -> U42(active(X))
active(U51(X1, X2)) -> U51(active(X1), X2)
active(U52(X)) -> U52(active(X))
active(U61(X)) -> U61(active(X))
active(U71(X)) -> U71(active(X))
active(U81(X)) -> U81(active(X))
active(U91(X)) -> U91(active(X))
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) -> mark(U101(X1, X2))
U102(mark(X)) -> mark(U102(X))
U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3))
U111(mark(X)) -> mark(U111(X))
snd(mark(X)) -> mark(snd(X))
splitAt(mark(X1), X2) -> mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) -> mark(splitAt(X1, X2))
U121(mark(X)) -> mark(U121(X))
U131(mark(X1), X2) -> mark(U131(X1, X2))
U132(mark(X)) -> mark(U132(X))
U141(mark(X1), X2) -> mark(U141(X1, X2))
U142(mark(X)) -> mark(U142(X))
U151(mark(X1), X2) -> mark(U151(X1, X2))
U152(mark(X)) -> mark(U152(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))
U172(mark(X1), X2, X3) -> mark(U172(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))
U182(mark(X1), X2) -> mark(U182(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, X3, X4) -> mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) -> mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) -> mark(U204(X1, X2))
U21(mark(X1), X2, X3) -> mark(U21(X1, X2, X3))
U22(mark(X1), X2) -> mark(U22(X1, X2))
U211(mark(X1), X2) -> mark(U211(X1, X2))
U212(mark(X1), X2) -> mark(U212(X1, X2))
U221(mark(X1), X2, X3) -> mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) -> mark(U222(X1, X2, X3))
fst(mark(X)) -> mark(fst(X))
U31(mark(X1), X2, X3) -> mark(U31(X1, X2, X3))
U32(mark(X1), X2) -> mark(U32(X1, X2))
U41(mark(X1), X2) -> mark(U41(X1, X2))
U42(mark(X)) -> mark(U42(X))
U51(mark(X1), X2) -> mark(U51(X1, X2))
U52(mark(X)) -> mark(U52(X))
U61(mark(X)) -> mark(U61(X))
U71(mark(X)) -> mark(U71(X))
U81(mark(X)) -> mark(U81(X))
U91(mark(X)) -> mark(U91(X))
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)) -> U101(proper(X1), proper(X2))
proper(tt) -> ok(tt)
proper(U102(X)) -> U102(proper(X))
proper(isLNat(X)) -> isLNat(proper(X))
proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) -> U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) -> U111(proper(X))
proper(snd(X)) -> snd(proper(X))
proper(splitAt(X1, X2)) -> splitAt(proper(X1), proper(X2))
proper(U121(X)) -> U121(proper(X))
proper(U131(X1, X2)) -> U131(proper(X1), proper(X2))
proper(U132(X)) -> U132(proper(X))
proper(U141(X1, X2)) -> U141(proper(X1), proper(X2))
proper(U142(X)) -> U142(proper(X))
proper(U151(X1, X2)) -> U151(proper(X1), proper(X2))
proper(U152(X)) -> U152(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(U172(X1, X2, X3)) -> U172(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(U182(X1, X2)) -> U182(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, X3, X4)) -> U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) -> isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) -> U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) -> U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) -> U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) -> U22(proper(X1), proper(X2))
proper(U211(X1, X2)) -> U211(proper(X1), proper(X2))
proper(U212(X1, X2)) -> U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) -> U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) -> U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) -> fst(proper(X))
proper(U31(X1, X2, X3)) -> U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) -> U32(proper(X1), proper(X2))
proper(U41(X1, X2)) -> U41(proper(X1), proper(X2))
proper(U42(X)) -> U42(proper(X))
proper(U51(X1, X2)) -> U51(proper(X1), proper(X2))
proper(U52(X)) -> U52(proper(X))
proper(U61(X)) -> U61(proper(X))
proper(U71(X)) -> U71(proper(X))
proper(U81(X)) -> U81(proper(X))
proper(U91(X)) -> U91(proper(X))
proper(isPLNat(X)) -> isPLNat(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(U101(X1, X2))
U102(ok(X)) -> ok(U102(X))
isLNat(ok(X)) -> ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))
U111(ok(X)) -> ok(U111(X))
snd(ok(X)) -> ok(snd(X))
splitAt(ok(X1), ok(X2)) -> ok(splitAt(X1, X2))
U121(ok(X)) -> ok(U121(X))
U131(ok(X1), ok(X2)) -> ok(U131(X1, X2))
U132(ok(X)) -> ok(U132(X))
U141(ok(X1), ok(X2)) -> ok(U141(X1, X2))
U142(ok(X)) -> ok(U142(X))
U151(ok(X1), ok(X2)) -> ok(U151(X1, X2))
U152(ok(X)) -> ok(U152(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))
U172(ok(X1), ok(X2), ok(X3)) -> ok(U172(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))
U182(ok(X1), ok(X2)) -> ok(U182(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(X3), ok(X4)) -> ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) -> ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) -> ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) -> ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) -> ok(U22(X1, X2))
U211(ok(X1), ok(X2)) -> ok(U211(X1, X2))
U212(ok(X1), ok(X2)) -> ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) -> ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) -> ok(U222(X1, X2, X3))
fst(ok(X)) -> ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) -> ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) -> ok(U32(X1, X2))
U41(ok(X1), ok(X2)) -> ok(U41(X1, X2))
U42(ok(X)) -> ok(U42(X))
U51(ok(X1), ok(X2)) -> ok(U51(X1, X2))
U52(ok(X)) -> ok(U52(X))
U61(ok(X)) -> ok(U61(X))
U71(ok(X)) -> ok(U71(X))
U81(ok(X)) -> ok(U81(X))
U91(ok(X)) -> ok(U91(X))
isPLNat(ok(X)) -> ok(isPLNat(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))

Types:
active :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U12 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U172 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U182 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U203 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U204 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U22 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U212 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U222 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U32 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok -> top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat -> tt:mark:nil:0':ok


Lemmas:
U102(gen_tt:mark:nil:0':ok3_0(+(1, n5_0))) -> *4_0, rt in Omega(n5_0)
U12(gen_tt:mark:nil:0':ok3_0(+(1, n1128_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) -> *4_0, rt in Omega(n1128_0)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n9121_0))) -> *4_0, rt in Omega(n9121_0)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n10537_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n10537_0)
U132(gen_tt:mark:nil:0':ok3_0(+(1, n15953_0))) -> *4_0, rt in Omega(n15953_0)
U142(gen_tt:mark:nil:0':ok3_0(+(1, n17620_0))) -> *4_0, rt in Omega(n17620_0)
U152(gen_tt:mark:nil:0':ok3_0(+(1, n19388_0))) -> *4_0, rt in Omega(n19388_0)
cons(gen_tt:mark:nil:0':ok3_0(+(1, n21257_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n21257_0)
natsFrom(gen_tt:mark:nil:0':ok3_0(+(1, n27404_0))) -> *4_0, rt in Omega(n27404_0)
s(gen_tt:mark:nil:0':ok3_0(+(1, n29524_0))) -> *4_0, rt in Omega(n29524_0)
U172(gen_tt:mark:nil:0':ok3_0(+(1, n31745_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) -> *4_0, rt in Omega(n31745_0)
head(gen_tt:mark:nil:0':ok3_0(+(1, n43161_0))) -> *4_0, rt in Omega(n43161_0)
afterNth(gen_tt:mark:nil:0':ok3_0(+(1, n45682_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n45682_0)


Generator Equations:
gen_tt:mark:nil:0':ok3_0(0) <=> tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) <=> mark(gen_tt:mark:nil:0':ok3_0(x))


The following defined symbols remain to be analysed:
U182, active, pair, U202, isNatural, U203, U204, U22, U212, U222, fst, U32, U42, U52, U11, U21, U31, U41, U51, U61, isPLNat, U71, U81, U91, U101, U111, U121, U131, U141, U151, U161, U171, U181, U191, U201, U211, U221, tail, take, sel, proper, top

They will be analysed ascendingly in the following order:
U182 < active
pair < active
U202 < active
isNatural < active
U203 < active
U204 < active
U22 < active
U212 < active
U222 < active
fst < active
U32 < active
U42 < active
U52 < active
U11 < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
isPLNat < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
U141 < active
U151 < active
U161 < active
U171 < active
U181 < active
U191 < active
U201 < active
U211 < active
U221 < active
tail < active
take < active
sel < active
active < top
U182 < proper
pair < proper
U202 < proper
isNatural < proper
U203 < proper
U204 < proper
U22 < proper
U212 < proper
U222 < proper
fst < proper
U32 < proper
U42 < proper
U52 < proper
U11 < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
isPLNat < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
U141 < proper
U151 < proper
U161 < proper
U171 < proper
U181 < proper
U191 < proper
U201 < proper
U211 < proper
U221 < proper
tail < proper
take < proper
sel < proper
proper < top

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

(55) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
U182(gen_tt:mark:nil:0':ok3_0(+(1, n53362_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n53362_0)

Induction Base:
U182(gen_tt:mark:nil:0':ok3_0(+(1, 0)), gen_tt:mark:nil:0':ok3_0(b))

Induction Step:
U182(gen_tt:mark:nil:0':ok3_0(+(1, +(n53362_0, 1))), gen_tt:mark:nil:0':ok3_0(b)) ->_R^Omega(1)
mark(U182(gen_tt:mark:nil:0':ok3_0(+(1, n53362_0)), gen_tt:mark:nil:0':ok3_0(b))) ->_IH
mark(*4_0)

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
----------------------------------------

(56)
Obligation:
TRS:
Rules:
active(U101(tt, V2)) -> mark(U102(isLNat(V2)))
active(U102(tt)) -> mark(tt)
active(U11(tt, N, XS)) -> mark(U12(isLNat(XS), N, XS))
active(U111(tt)) -> mark(tt)
active(U12(tt, N, XS)) -> mark(snd(splitAt(N, XS)))
active(U121(tt)) -> mark(tt)
active(U131(tt, V2)) -> mark(U132(isLNat(V2)))
active(U132(tt)) -> mark(tt)
active(U141(tt, V2)) -> mark(U142(isLNat(V2)))
active(U142(tt)) -> mark(tt)
active(U151(tt, V2)) -> mark(U152(isLNat(V2)))
active(U152(tt)) -> mark(tt)
active(U161(tt, N)) -> mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) -> mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) -> mark(head(afterNth(N, XS)))
active(U181(tt, Y)) -> mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) -> mark(Y)
active(U191(tt, XS)) -> mark(pair(nil, XS))
active(U201(tt, N, X, XS)) -> mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) -> mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) -> mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) -> mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) -> mark(U22(isLNat(Y), X))
active(U211(tt, XS)) -> mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) -> mark(XS)
active(U22(tt, X)) -> mark(X)
active(U221(tt, N, XS)) -> mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) -> mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) -> mark(U32(isLNat(XS), N))
active(U32(tt, N)) -> mark(N)
active(U41(tt, V2)) -> mark(U42(isLNat(V2)))
active(U42(tt)) -> mark(tt)
active(U51(tt, V2)) -> mark(U52(isLNat(V2)))
active(U52(tt)) -> mark(tt)
active(U61(tt)) -> mark(tt)
active(U71(tt)) -> mark(tt)
active(U81(tt)) -> mark(tt)
active(U91(tt)) -> mark(tt)
active(afterNth(N, XS)) -> mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) -> mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) -> mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) -> mark(tt)
active(isLNat(afterNth(V1, V2))) -> mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) -> mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) -> mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) -> mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) -> mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) -> mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) -> mark(U101(isNatural(V1), V2))
active(isNatural(0')) -> mark(tt)
active(isNatural(head(V1))) -> mark(U111(isLNat(V1)))
active(isNatural(s(V1))) -> mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) -> mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) -> mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) -> mark(U151(isNatural(V1), V2))
active(natsFrom(N)) -> mark(U161(isNatural(N), N))
active(sel(N, XS)) -> mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) -> mark(U181(isLNat(X), Y))
active(splitAt(0', XS)) -> mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) -> mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) -> mark(U211(isNatural(N), XS))
active(take(N, XS)) -> mark(U221(isNatural(N), N, XS))
active(U101(X1, X2)) -> U101(active(X1), X2)
active(U102(X)) -> U102(active(X))
active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3)
active(U111(X)) -> U111(active(X))
active(snd(X)) -> snd(active(X))
active(splitAt(X1, X2)) -> splitAt(active(X1), X2)
active(splitAt(X1, X2)) -> splitAt(X1, active(X2))
active(U121(X)) -> U121(active(X))
active(U131(X1, X2)) -> U131(active(X1), X2)
active(U132(X)) -> U132(active(X))
active(U141(X1, X2)) -> U141(active(X1), X2)
active(U142(X)) -> U142(active(X))
active(U151(X1, X2)) -> U151(active(X1), X2)
active(U152(X)) -> U152(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(U172(X1, X2, X3)) -> U172(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(U182(X1, X2)) -> U182(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, X3, X4)) -> U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) -> U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) -> U204(active(X1), X2)
active(U21(X1, X2, X3)) -> U21(active(X1), X2, X3)
active(U22(X1, X2)) -> U22(active(X1), X2)
active(U211(X1, X2)) -> U211(active(X1), X2)
active(U212(X1, X2)) -> U212(active(X1), X2)
active(U221(X1, X2, X3)) -> U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) -> U222(active(X1), X2, X3)
active(fst(X)) -> fst(active(X))
active(U31(X1, X2, X3)) -> U31(active(X1), X2, X3)
active(U32(X1, X2)) -> U32(active(X1), X2)
active(U41(X1, X2)) -> U41(active(X1), X2)
active(U42(X)) -> U42(active(X))
active(U51(X1, X2)) -> U51(active(X1), X2)
active(U52(X)) -> U52(active(X))
active(U61(X)) -> U61(active(X))
active(U71(X)) -> U71(active(X))
active(U81(X)) -> U81(active(X))
active(U91(X)) -> U91(active(X))
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) -> mark(U101(X1, X2))
U102(mark(X)) -> mark(U102(X))
U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3))
U111(mark(X)) -> mark(U111(X))
snd(mark(X)) -> mark(snd(X))
splitAt(mark(X1), X2) -> mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) -> mark(splitAt(X1, X2))
U121(mark(X)) -> mark(U121(X))
U131(mark(X1), X2) -> mark(U131(X1, X2))
U132(mark(X)) -> mark(U132(X))
U141(mark(X1), X2) -> mark(U141(X1, X2))
U142(mark(X)) -> mark(U142(X))
U151(mark(X1), X2) -> mark(U151(X1, X2))
U152(mark(X)) -> mark(U152(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))
U172(mark(X1), X2, X3) -> mark(U172(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))
U182(mark(X1), X2) -> mark(U182(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, X3, X4) -> mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) -> mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) -> mark(U204(X1, X2))
U21(mark(X1), X2, X3) -> mark(U21(X1, X2, X3))
U22(mark(X1), X2) -> mark(U22(X1, X2))
U211(mark(X1), X2) -> mark(U211(X1, X2))
U212(mark(X1), X2) -> mark(U212(X1, X2))
U221(mark(X1), X2, X3) -> mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) -> mark(U222(X1, X2, X3))
fst(mark(X)) -> mark(fst(X))
U31(mark(X1), X2, X3) -> mark(U31(X1, X2, X3))
U32(mark(X1), X2) -> mark(U32(X1, X2))
U41(mark(X1), X2) -> mark(U41(X1, X2))
U42(mark(X)) -> mark(U42(X))
U51(mark(X1), X2) -> mark(U51(X1, X2))
U52(mark(X)) -> mark(U52(X))
U61(mark(X)) -> mark(U61(X))
U71(mark(X)) -> mark(U71(X))
U81(mark(X)) -> mark(U81(X))
U91(mark(X)) -> mark(U91(X))
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)) -> U101(proper(X1), proper(X2))
proper(tt) -> ok(tt)
proper(U102(X)) -> U102(proper(X))
proper(isLNat(X)) -> isLNat(proper(X))
proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) -> U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) -> U111(proper(X))
proper(snd(X)) -> snd(proper(X))
proper(splitAt(X1, X2)) -> splitAt(proper(X1), proper(X2))
proper(U121(X)) -> U121(proper(X))
proper(U131(X1, X2)) -> U131(proper(X1), proper(X2))
proper(U132(X)) -> U132(proper(X))
proper(U141(X1, X2)) -> U141(proper(X1), proper(X2))
proper(U142(X)) -> U142(proper(X))
proper(U151(X1, X2)) -> U151(proper(X1), proper(X2))
proper(U152(X)) -> U152(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(U172(X1, X2, X3)) -> U172(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(U182(X1, X2)) -> U182(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, X3, X4)) -> U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) -> isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) -> U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) -> U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) -> U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) -> U22(proper(X1), proper(X2))
proper(U211(X1, X2)) -> U211(proper(X1), proper(X2))
proper(U212(X1, X2)) -> U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) -> U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) -> U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) -> fst(proper(X))
proper(U31(X1, X2, X3)) -> U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) -> U32(proper(X1), proper(X2))
proper(U41(X1, X2)) -> U41(proper(X1), proper(X2))
proper(U42(X)) -> U42(proper(X))
proper(U51(X1, X2)) -> U51(proper(X1), proper(X2))
proper(U52(X)) -> U52(proper(X))
proper(U61(X)) -> U61(proper(X))
proper(U71(X)) -> U71(proper(X))
proper(U81(X)) -> U81(proper(X))
proper(U91(X)) -> U91(proper(X))
proper(isPLNat(X)) -> isPLNat(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(U101(X1, X2))
U102(ok(X)) -> ok(U102(X))
isLNat(ok(X)) -> ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))
U111(ok(X)) -> ok(U111(X))
snd(ok(X)) -> ok(snd(X))
splitAt(ok(X1), ok(X2)) -> ok(splitAt(X1, X2))
U121(ok(X)) -> ok(U121(X))
U131(ok(X1), ok(X2)) -> ok(U131(X1, X2))
U132(ok(X)) -> ok(U132(X))
U141(ok(X1), ok(X2)) -> ok(U141(X1, X2))
U142(ok(X)) -> ok(U142(X))
U151(ok(X1), ok(X2)) -> ok(U151(X1, X2))
U152(ok(X)) -> ok(U152(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))
U172(ok(X1), ok(X2), ok(X3)) -> ok(U172(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))
U182(ok(X1), ok(X2)) -> ok(U182(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(X3), ok(X4)) -> ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) -> ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) -> ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) -> ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) -> ok(U22(X1, X2))
U211(ok(X1), ok(X2)) -> ok(U211(X1, X2))
U212(ok(X1), ok(X2)) -> ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) -> ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) -> ok(U222(X1, X2, X3))
fst(ok(X)) -> ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) -> ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) -> ok(U32(X1, X2))
U41(ok(X1), ok(X2)) -> ok(U41(X1, X2))
U42(ok(X)) -> ok(U42(X))
U51(ok(X1), ok(X2)) -> ok(U51(X1, X2))
U52(ok(X)) -> ok(U52(X))
U61(ok(X)) -> ok(U61(X))
U71(ok(X)) -> ok(U71(X))
U81(ok(X)) -> ok(U81(X))
U91(ok(X)) -> ok(U91(X))
isPLNat(ok(X)) -> ok(isPLNat(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))

Types:
active :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U12 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U172 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U182 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U203 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U204 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U22 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U212 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U222 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U32 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok -> top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat -> tt:mark:nil:0':ok


Lemmas:
U102(gen_tt:mark:nil:0':ok3_0(+(1, n5_0))) -> *4_0, rt in Omega(n5_0)
U12(gen_tt:mark:nil:0':ok3_0(+(1, n1128_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) -> *4_0, rt in Omega(n1128_0)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n9121_0))) -> *4_0, rt in Omega(n9121_0)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n10537_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n10537_0)
U132(gen_tt:mark:nil:0':ok3_0(+(1, n15953_0))) -> *4_0, rt in Omega(n15953_0)
U142(gen_tt:mark:nil:0':ok3_0(+(1, n17620_0))) -> *4_0, rt in Omega(n17620_0)
U152(gen_tt:mark:nil:0':ok3_0(+(1, n19388_0))) -> *4_0, rt in Omega(n19388_0)
cons(gen_tt:mark:nil:0':ok3_0(+(1, n21257_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n21257_0)
natsFrom(gen_tt:mark:nil:0':ok3_0(+(1, n27404_0))) -> *4_0, rt in Omega(n27404_0)
s(gen_tt:mark:nil:0':ok3_0(+(1, n29524_0))) -> *4_0, rt in Omega(n29524_0)
U172(gen_tt:mark:nil:0':ok3_0(+(1, n31745_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) -> *4_0, rt in Omega(n31745_0)
head(gen_tt:mark:nil:0':ok3_0(+(1, n43161_0))) -> *4_0, rt in Omega(n43161_0)
afterNth(gen_tt:mark:nil:0':ok3_0(+(1, n45682_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n45682_0)
U182(gen_tt:mark:nil:0':ok3_0(+(1, n53362_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n53362_0)


Generator Equations:
gen_tt:mark:nil:0':ok3_0(0) <=> tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) <=> mark(gen_tt:mark:nil:0':ok3_0(x))


The following defined symbols remain to be analysed:
pair, active, U202, isNatural, U203, U204, U22, U212, U222, fst, U32, U42, U52, U11, U21, U31, U41, U51, U61, isPLNat, U71, U81, U91, U101, U111, U121, U131, U141, U151, U161, U171, U181, U191, U201, U211, U221, tail, take, sel, proper, top

They will be analysed ascendingly in the following order:
pair < active
U202 < active
isNatural < active
U203 < active
U204 < active
U22 < active
U212 < active
U222 < active
fst < active
U32 < active
U42 < active
U52 < active
U11 < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
isPLNat < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
U141 < active
U151 < active
U161 < active
U171 < active
U181 < active
U191 < active
U201 < active
U211 < active
U221 < active
tail < active
take < active
sel < active
active < top
pair < proper
U202 < proper
isNatural < proper
U203 < proper
U204 < proper
U22 < proper
U212 < proper
U222 < proper
fst < proper
U32 < proper
U42 < proper
U52 < proper
U11 < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
isPLNat < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
U141 < proper
U151 < proper
U161 < proper
U171 < proper
U181 < proper
U191 < proper
U201 < proper
U211 < proper
U221 < proper
tail < proper
take < proper
sel < proper
proper < top

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

(57) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
pair(gen_tt:mark:nil:0':ok3_0(+(1, n61149_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n61149_0)

Induction Base:
pair(gen_tt:mark:nil:0':ok3_0(+(1, 0)), gen_tt:mark:nil:0':ok3_0(b))

Induction Step:
pair(gen_tt:mark:nil:0':ok3_0(+(1, +(n61149_0, 1))), gen_tt:mark:nil:0':ok3_0(b)) ->_R^Omega(1)
mark(pair(gen_tt:mark:nil:0':ok3_0(+(1, n61149_0)), gen_tt:mark:nil:0':ok3_0(b))) ->_IH
mark(*4_0)

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
----------------------------------------

(58)
Obligation:
TRS:
Rules:
active(U101(tt, V2)) -> mark(U102(isLNat(V2)))
active(U102(tt)) -> mark(tt)
active(U11(tt, N, XS)) -> mark(U12(isLNat(XS), N, XS))
active(U111(tt)) -> mark(tt)
active(U12(tt, N, XS)) -> mark(snd(splitAt(N, XS)))
active(U121(tt)) -> mark(tt)
active(U131(tt, V2)) -> mark(U132(isLNat(V2)))
active(U132(tt)) -> mark(tt)
active(U141(tt, V2)) -> mark(U142(isLNat(V2)))
active(U142(tt)) -> mark(tt)
active(U151(tt, V2)) -> mark(U152(isLNat(V2)))
active(U152(tt)) -> mark(tt)
active(U161(tt, N)) -> mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) -> mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) -> mark(head(afterNth(N, XS)))
active(U181(tt, Y)) -> mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) -> mark(Y)
active(U191(tt, XS)) -> mark(pair(nil, XS))
active(U201(tt, N, X, XS)) -> mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) -> mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) -> mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) -> mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) -> mark(U22(isLNat(Y), X))
active(U211(tt, XS)) -> mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) -> mark(XS)
active(U22(tt, X)) -> mark(X)
active(U221(tt, N, XS)) -> mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) -> mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) -> mark(U32(isLNat(XS), N))
active(U32(tt, N)) -> mark(N)
active(U41(tt, V2)) -> mark(U42(isLNat(V2)))
active(U42(tt)) -> mark(tt)
active(U51(tt, V2)) -> mark(U52(isLNat(V2)))
active(U52(tt)) -> mark(tt)
active(U61(tt)) -> mark(tt)
active(U71(tt)) -> mark(tt)
active(U81(tt)) -> mark(tt)
active(U91(tt)) -> mark(tt)
active(afterNth(N, XS)) -> mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) -> mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) -> mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) -> mark(tt)
active(isLNat(afterNth(V1, V2))) -> mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) -> mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) -> mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) -> mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) -> mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) -> mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) -> mark(U101(isNatural(V1), V2))
active(isNatural(0')) -> mark(tt)
active(isNatural(head(V1))) -> mark(U111(isLNat(V1)))
active(isNatural(s(V1))) -> mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) -> mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) -> mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) -> mark(U151(isNatural(V1), V2))
active(natsFrom(N)) -> mark(U161(isNatural(N), N))
active(sel(N, XS)) -> mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) -> mark(U181(isLNat(X), Y))
active(splitAt(0', XS)) -> mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) -> mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) -> mark(U211(isNatural(N), XS))
active(take(N, XS)) -> mark(U221(isNatural(N), N, XS))
active(U101(X1, X2)) -> U101(active(X1), X2)
active(U102(X)) -> U102(active(X))
active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3)
active(U111(X)) -> U111(active(X))
active(snd(X)) -> snd(active(X))
active(splitAt(X1, X2)) -> splitAt(active(X1), X2)
active(splitAt(X1, X2)) -> splitAt(X1, active(X2))
active(U121(X)) -> U121(active(X))
active(U131(X1, X2)) -> U131(active(X1), X2)
active(U132(X)) -> U132(active(X))
active(U141(X1, X2)) -> U141(active(X1), X2)
active(U142(X)) -> U142(active(X))
active(U151(X1, X2)) -> U151(active(X1), X2)
active(U152(X)) -> U152(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(U172(X1, X2, X3)) -> U172(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(U182(X1, X2)) -> U182(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, X3, X4)) -> U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) -> U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) -> U204(active(X1), X2)
active(U21(X1, X2, X3)) -> U21(active(X1), X2, X3)
active(U22(X1, X2)) -> U22(active(X1), X2)
active(U211(X1, X2)) -> U211(active(X1), X2)
active(U212(X1, X2)) -> U212(active(X1), X2)
active(U221(X1, X2, X3)) -> U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) -> U222(active(X1), X2, X3)
active(fst(X)) -> fst(active(X))
active(U31(X1, X2, X3)) -> U31(active(X1), X2, X3)
active(U32(X1, X2)) -> U32(active(X1), X2)
active(U41(X1, X2)) -> U41(active(X1), X2)
active(U42(X)) -> U42(active(X))
active(U51(X1, X2)) -> U51(active(X1), X2)
active(U52(X)) -> U52(active(X))
active(U61(X)) -> U61(active(X))
active(U71(X)) -> U71(active(X))
active(U81(X)) -> U81(active(X))
active(U91(X)) -> U91(active(X))
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) -> mark(U101(X1, X2))
U102(mark(X)) -> mark(U102(X))
U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3))
U111(mark(X)) -> mark(U111(X))
snd(mark(X)) -> mark(snd(X))
splitAt(mark(X1), X2) -> mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) -> mark(splitAt(X1, X2))
U121(mark(X)) -> mark(U121(X))
U131(mark(X1), X2) -> mark(U131(X1, X2))
U132(mark(X)) -> mark(U132(X))
U141(mark(X1), X2) -> mark(U141(X1, X2))
U142(mark(X)) -> mark(U142(X))
U151(mark(X1), X2) -> mark(U151(X1, X2))
U152(mark(X)) -> mark(U152(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))
U172(mark(X1), X2, X3) -> mark(U172(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))
U182(mark(X1), X2) -> mark(U182(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, X3, X4) -> mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) -> mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) -> mark(U204(X1, X2))
U21(mark(X1), X2, X3) -> mark(U21(X1, X2, X3))
U22(mark(X1), X2) -> mark(U22(X1, X2))
U211(mark(X1), X2) -> mark(U211(X1, X2))
U212(mark(X1), X2) -> mark(U212(X1, X2))
U221(mark(X1), X2, X3) -> mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) -> mark(U222(X1, X2, X3))
fst(mark(X)) -> mark(fst(X))
U31(mark(X1), X2, X3) -> mark(U31(X1, X2, X3))
U32(mark(X1), X2) -> mark(U32(X1, X2))
U41(mark(X1), X2) -> mark(U41(X1, X2))
U42(mark(X)) -> mark(U42(X))
U51(mark(X1), X2) -> mark(U51(X1, X2))
U52(mark(X)) -> mark(U52(X))
U61(mark(X)) -> mark(U61(X))
U71(mark(X)) -> mark(U71(X))
U81(mark(X)) -> mark(U81(X))
U91(mark(X)) -> mark(U91(X))
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)) -> U101(proper(X1), proper(X2))
proper(tt) -> ok(tt)
proper(U102(X)) -> U102(proper(X))
proper(isLNat(X)) -> isLNat(proper(X))
proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) -> U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) -> U111(proper(X))
proper(snd(X)) -> snd(proper(X))
proper(splitAt(X1, X2)) -> splitAt(proper(X1), proper(X2))
proper(U121(X)) -> U121(proper(X))
proper(U131(X1, X2)) -> U131(proper(X1), proper(X2))
proper(U132(X)) -> U132(proper(X))
proper(U141(X1, X2)) -> U141(proper(X1), proper(X2))
proper(U142(X)) -> U142(proper(X))
proper(U151(X1, X2)) -> U151(proper(X1), proper(X2))
proper(U152(X)) -> U152(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(U172(X1, X2, X3)) -> U172(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(U182(X1, X2)) -> U182(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, X3, X4)) -> U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) -> isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) -> U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) -> U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) -> U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) -> U22(proper(X1), proper(X2))
proper(U211(X1, X2)) -> U211(proper(X1), proper(X2))
proper(U212(X1, X2)) -> U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) -> U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) -> U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) -> fst(proper(X))
proper(U31(X1, X2, X3)) -> U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) -> U32(proper(X1), proper(X2))
proper(U41(X1, X2)) -> U41(proper(X1), proper(X2))
proper(U42(X)) -> U42(proper(X))
proper(U51(X1, X2)) -> U51(proper(X1), proper(X2))
proper(U52(X)) -> U52(proper(X))
proper(U61(X)) -> U61(proper(X))
proper(U71(X)) -> U71(proper(X))
proper(U81(X)) -> U81(proper(X))
proper(U91(X)) -> U91(proper(X))
proper(isPLNat(X)) -> isPLNat(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(U101(X1, X2))
U102(ok(X)) -> ok(U102(X))
isLNat(ok(X)) -> ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))
U111(ok(X)) -> ok(U111(X))
snd(ok(X)) -> ok(snd(X))
splitAt(ok(X1), ok(X2)) -> ok(splitAt(X1, X2))
U121(ok(X)) -> ok(U121(X))
U131(ok(X1), ok(X2)) -> ok(U131(X1, X2))
U132(ok(X)) -> ok(U132(X))
U141(ok(X1), ok(X2)) -> ok(U141(X1, X2))
U142(ok(X)) -> ok(U142(X))
U151(ok(X1), ok(X2)) -> ok(U151(X1, X2))
U152(ok(X)) -> ok(U152(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))
U172(ok(X1), ok(X2), ok(X3)) -> ok(U172(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))
U182(ok(X1), ok(X2)) -> ok(U182(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(X3), ok(X4)) -> ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) -> ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) -> ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) -> ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) -> ok(U22(X1, X2))
U211(ok(X1), ok(X2)) -> ok(U211(X1, X2))
U212(ok(X1), ok(X2)) -> ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) -> ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) -> ok(U222(X1, X2, X3))
fst(ok(X)) -> ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) -> ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) -> ok(U32(X1, X2))
U41(ok(X1), ok(X2)) -> ok(U41(X1, X2))
U42(ok(X)) -> ok(U42(X))
U51(ok(X1), ok(X2)) -> ok(U51(X1, X2))
U52(ok(X)) -> ok(U52(X))
U61(ok(X)) -> ok(U61(X))
U71(ok(X)) -> ok(U71(X))
U81(ok(X)) -> ok(U81(X))
U91(ok(X)) -> ok(U91(X))
isPLNat(ok(X)) -> ok(isPLNat(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))

Types:
active :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U12 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U172 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U182 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U203 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U204 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U22 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U212 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U222 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U32 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok -> top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat -> tt:mark:nil:0':ok


Lemmas:
U102(gen_tt:mark:nil:0':ok3_0(+(1, n5_0))) -> *4_0, rt in Omega(n5_0)
U12(gen_tt:mark:nil:0':ok3_0(+(1, n1128_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) -> *4_0, rt in Omega(n1128_0)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n9121_0))) -> *4_0, rt in Omega(n9121_0)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n10537_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n10537_0)
U132(gen_tt:mark:nil:0':ok3_0(+(1, n15953_0))) -> *4_0, rt in Omega(n15953_0)
U142(gen_tt:mark:nil:0':ok3_0(+(1, n17620_0))) -> *4_0, rt in Omega(n17620_0)
U152(gen_tt:mark:nil:0':ok3_0(+(1, n19388_0))) -> *4_0, rt in Omega(n19388_0)
cons(gen_tt:mark:nil:0':ok3_0(+(1, n21257_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n21257_0)
natsFrom(gen_tt:mark:nil:0':ok3_0(+(1, n27404_0))) -> *4_0, rt in Omega(n27404_0)
s(gen_tt:mark:nil:0':ok3_0(+(1, n29524_0))) -> *4_0, rt in Omega(n29524_0)
U172(gen_tt:mark:nil:0':ok3_0(+(1, n31745_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) -> *4_0, rt in Omega(n31745_0)
head(gen_tt:mark:nil:0':ok3_0(+(1, n43161_0))) -> *4_0, rt in Omega(n43161_0)
afterNth(gen_tt:mark:nil:0':ok3_0(+(1, n45682_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n45682_0)
U182(gen_tt:mark:nil:0':ok3_0(+(1, n53362_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n53362_0)
pair(gen_tt:mark:nil:0':ok3_0(+(1, n61149_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n61149_0)


Generator Equations:
gen_tt:mark:nil:0':ok3_0(0) <=> tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) <=> mark(gen_tt:mark:nil:0':ok3_0(x))


The following defined symbols remain to be analysed:
U202, active, isNatural, U203, U204, U22, U212, U222, fst, U32, U42, U52, U11, U21, U31, U41, U51, U61, isPLNat, U71, U81, U91, U101, U111, U121, U131, U141, U151, U161, U171, U181, U191, U201, U211, U221, tail, take, sel, proper, top

They will be analysed ascendingly in the following order:
U202 < active
isNatural < active
U203 < active
U204 < active
U22 < active
U212 < active
U222 < active
fst < active
U32 < active
U42 < active
U52 < active
U11 < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
isPLNat < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
U141 < active
U151 < active
U161 < active
U171 < active
U181 < active
U191 < active
U201 < active
U211 < active
U221 < active
tail < active
take < active
sel < active
active < top
U202 < proper
isNatural < proper
U203 < proper
U204 < proper
U22 < proper
U212 < proper
U222 < proper
fst < proper
U32 < proper
U42 < proper
U52 < proper
U11 < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
isPLNat < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
U141 < proper
U151 < proper
U161 < proper
U171 < proper
U181 < proper
U191 < proper
U201 < proper
U211 < proper
U221 < proper
tail < proper
take < proper
sel < proper
proper < top

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

(59) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
U202(gen_tt:mark:nil:0':ok3_0(+(1, n69441_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c), gen_tt:mark:nil:0':ok3_0(d)) -> *4_0, rt in Omega(n69441_0)

Induction Base:
U202(gen_tt:mark:nil:0':ok3_0(+(1, 0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c), gen_tt:mark:nil:0':ok3_0(d))

Induction Step:
U202(gen_tt:mark:nil:0':ok3_0(+(1, +(n69441_0, 1))), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c), gen_tt:mark:nil:0':ok3_0(d)) ->_R^Omega(1)
mark(U202(gen_tt:mark:nil:0':ok3_0(+(1, n69441_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c), gen_tt:mark:nil:0':ok3_0(d))) ->_IH
mark(*4_0)

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
----------------------------------------

(60)
Obligation:
TRS:
Rules:
active(U101(tt, V2)) -> mark(U102(isLNat(V2)))
active(U102(tt)) -> mark(tt)
active(U11(tt, N, XS)) -> mark(U12(isLNat(XS), N, XS))
active(U111(tt)) -> mark(tt)
active(U12(tt, N, XS)) -> mark(snd(splitAt(N, XS)))
active(U121(tt)) -> mark(tt)
active(U131(tt, V2)) -> mark(U132(isLNat(V2)))
active(U132(tt)) -> mark(tt)
active(U141(tt, V2)) -> mark(U142(isLNat(V2)))
active(U142(tt)) -> mark(tt)
active(U151(tt, V2)) -> mark(U152(isLNat(V2)))
active(U152(tt)) -> mark(tt)
active(U161(tt, N)) -> mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) -> mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) -> mark(head(afterNth(N, XS)))
active(U181(tt, Y)) -> mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) -> mark(Y)
active(U191(tt, XS)) -> mark(pair(nil, XS))
active(U201(tt, N, X, XS)) -> mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) -> mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) -> mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) -> mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) -> mark(U22(isLNat(Y), X))
active(U211(tt, XS)) -> mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) -> mark(XS)
active(U22(tt, X)) -> mark(X)
active(U221(tt, N, XS)) -> mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) -> mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) -> mark(U32(isLNat(XS), N))
active(U32(tt, N)) -> mark(N)
active(U41(tt, V2)) -> mark(U42(isLNat(V2)))
active(U42(tt)) -> mark(tt)
active(U51(tt, V2)) -> mark(U52(isLNat(V2)))
active(U52(tt)) -> mark(tt)
active(U61(tt)) -> mark(tt)
active(U71(tt)) -> mark(tt)
active(U81(tt)) -> mark(tt)
active(U91(tt)) -> mark(tt)
active(afterNth(N, XS)) -> mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) -> mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) -> mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) -> mark(tt)
active(isLNat(afterNth(V1, V2))) -> mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) -> mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) -> mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) -> mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) -> mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) -> mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) -> mark(U101(isNatural(V1), V2))
active(isNatural(0')) -> mark(tt)
active(isNatural(head(V1))) -> mark(U111(isLNat(V1)))
active(isNatural(s(V1))) -> mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) -> mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) -> mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) -> mark(U151(isNatural(V1), V2))
active(natsFrom(N)) -> mark(U161(isNatural(N), N))
active(sel(N, XS)) -> mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) -> mark(U181(isLNat(X), Y))
active(splitAt(0', XS)) -> mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) -> mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) -> mark(U211(isNatural(N), XS))
active(take(N, XS)) -> mark(U221(isNatural(N), N, XS))
active(U101(X1, X2)) -> U101(active(X1), X2)
active(U102(X)) -> U102(active(X))
active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3)
active(U111(X)) -> U111(active(X))
active(snd(X)) -> snd(active(X))
active(splitAt(X1, X2)) -> splitAt(active(X1), X2)
active(splitAt(X1, X2)) -> splitAt(X1, active(X2))
active(U121(X)) -> U121(active(X))
active(U131(X1, X2)) -> U131(active(X1), X2)
active(U132(X)) -> U132(active(X))
active(U141(X1, X2)) -> U141(active(X1), X2)
active(U142(X)) -> U142(active(X))
active(U151(X1, X2)) -> U151(active(X1), X2)
active(U152(X)) -> U152(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(U172(X1, X2, X3)) -> U172(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(U182(X1, X2)) -> U182(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, X3, X4)) -> U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) -> U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) -> U204(active(X1), X2)
active(U21(X1, X2, X3)) -> U21(active(X1), X2, X3)
active(U22(X1, X2)) -> U22(active(X1), X2)
active(U211(X1, X2)) -> U211(active(X1), X2)
active(U212(X1, X2)) -> U212(active(X1), X2)
active(U221(X1, X2, X3)) -> U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) -> U222(active(X1), X2, X3)
active(fst(X)) -> fst(active(X))
active(U31(X1, X2, X3)) -> U31(active(X1), X2, X3)
active(U32(X1, X2)) -> U32(active(X1), X2)
active(U41(X1, X2)) -> U41(active(X1), X2)
active(U42(X)) -> U42(active(X))
active(U51(X1, X2)) -> U51(active(X1), X2)
active(U52(X)) -> U52(active(X))
active(U61(X)) -> U61(active(X))
active(U71(X)) -> U71(active(X))
active(U81(X)) -> U81(active(X))
active(U91(X)) -> U91(active(X))
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) -> mark(U101(X1, X2))
U102(mark(X)) -> mark(U102(X))
U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3))
U111(mark(X)) -> mark(U111(X))
snd(mark(X)) -> mark(snd(X))
splitAt(mark(X1), X2) -> mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) -> mark(splitAt(X1, X2))
U121(mark(X)) -> mark(U121(X))
U131(mark(X1), X2) -> mark(U131(X1, X2))
U132(mark(X)) -> mark(U132(X))
U141(mark(X1), X2) -> mark(U141(X1, X2))
U142(mark(X)) -> mark(U142(X))
U151(mark(X1), X2) -> mark(U151(X1, X2))
U152(mark(X)) -> mark(U152(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))
U172(mark(X1), X2, X3) -> mark(U172(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))
U182(mark(X1), X2) -> mark(U182(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, X3, X4) -> mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) -> mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) -> mark(U204(X1, X2))
U21(mark(X1), X2, X3) -> mark(U21(X1, X2, X3))
U22(mark(X1), X2) -> mark(U22(X1, X2))
U211(mark(X1), X2) -> mark(U211(X1, X2))
U212(mark(X1), X2) -> mark(U212(X1, X2))
U221(mark(X1), X2, X3) -> mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) -> mark(U222(X1, X2, X3))
fst(mark(X)) -> mark(fst(X))
U31(mark(X1), X2, X3) -> mark(U31(X1, X2, X3))
U32(mark(X1), X2) -> mark(U32(X1, X2))
U41(mark(X1), X2) -> mark(U41(X1, X2))
U42(mark(X)) -> mark(U42(X))
U51(mark(X1), X2) -> mark(U51(X1, X2))
U52(mark(X)) -> mark(U52(X))
U61(mark(X)) -> mark(U61(X))
U71(mark(X)) -> mark(U71(X))
U81(mark(X)) -> mark(U81(X))
U91(mark(X)) -> mark(U91(X))
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)) -> U101(proper(X1), proper(X2))
proper(tt) -> ok(tt)
proper(U102(X)) -> U102(proper(X))
proper(isLNat(X)) -> isLNat(proper(X))
proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) -> U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) -> U111(proper(X))
proper(snd(X)) -> snd(proper(X))
proper(splitAt(X1, X2)) -> splitAt(proper(X1), proper(X2))
proper(U121(X)) -> U121(proper(X))
proper(U131(X1, X2)) -> U131(proper(X1), proper(X2))
proper(U132(X)) -> U132(proper(X))
proper(U141(X1, X2)) -> U141(proper(X1), proper(X2))
proper(U142(X)) -> U142(proper(X))
proper(U151(X1, X2)) -> U151(proper(X1), proper(X2))
proper(U152(X)) -> U152(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(U172(X1, X2, X3)) -> U172(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(U182(X1, X2)) -> U182(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, X3, X4)) -> U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) -> isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) -> U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) -> U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) -> U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) -> U22(proper(X1), proper(X2))
proper(U211(X1, X2)) -> U211(proper(X1), proper(X2))
proper(U212(X1, X2)) -> U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) -> U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) -> U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) -> fst(proper(X))
proper(U31(X1, X2, X3)) -> U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) -> U32(proper(X1), proper(X2))
proper(U41(X1, X2)) -> U41(proper(X1), proper(X2))
proper(U42(X)) -> U42(proper(X))
proper(U51(X1, X2)) -> U51(proper(X1), proper(X2))
proper(U52(X)) -> U52(proper(X))
proper(U61(X)) -> U61(proper(X))
proper(U71(X)) -> U71(proper(X))
proper(U81(X)) -> U81(proper(X))
proper(U91(X)) -> U91(proper(X))
proper(isPLNat(X)) -> isPLNat(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(U101(X1, X2))
U102(ok(X)) -> ok(U102(X))
isLNat(ok(X)) -> ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))
U111(ok(X)) -> ok(U111(X))
snd(ok(X)) -> ok(snd(X))
splitAt(ok(X1), ok(X2)) -> ok(splitAt(X1, X2))
U121(ok(X)) -> ok(U121(X))
U131(ok(X1), ok(X2)) -> ok(U131(X1, X2))
U132(ok(X)) -> ok(U132(X))
U141(ok(X1), ok(X2)) -> ok(U141(X1, X2))
U142(ok(X)) -> ok(U142(X))
U151(ok(X1), ok(X2)) -> ok(U151(X1, X2))
U152(ok(X)) -> ok(U152(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))
U172(ok(X1), ok(X2), ok(X3)) -> ok(U172(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))
U182(ok(X1), ok(X2)) -> ok(U182(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(X3), ok(X4)) -> ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) -> ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) -> ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) -> ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) -> ok(U22(X1, X2))
U211(ok(X1), ok(X2)) -> ok(U211(X1, X2))
U212(ok(X1), ok(X2)) -> ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) -> ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) -> ok(U222(X1, X2, X3))
fst(ok(X)) -> ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) -> ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) -> ok(U32(X1, X2))
U41(ok(X1), ok(X2)) -> ok(U41(X1, X2))
U42(ok(X)) -> ok(U42(X))
U51(ok(X1), ok(X2)) -> ok(U51(X1, X2))
U52(ok(X)) -> ok(U52(X))
U61(ok(X)) -> ok(U61(X))
U71(ok(X)) -> ok(U71(X))
U81(ok(X)) -> ok(U81(X))
U91(ok(X)) -> ok(U91(X))
isPLNat(ok(X)) -> ok(isPLNat(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))

Types:
active :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U12 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U172 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U182 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U203 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U204 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U22 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U212 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U222 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U32 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok -> top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat -> tt:mark:nil:0':ok


Lemmas:
U102(gen_tt:mark:nil:0':ok3_0(+(1, n5_0))) -> *4_0, rt in Omega(n5_0)
U12(gen_tt:mark:nil:0':ok3_0(+(1, n1128_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) -> *4_0, rt in Omega(n1128_0)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n9121_0))) -> *4_0, rt in Omega(n9121_0)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n10537_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n10537_0)
U132(gen_tt:mark:nil:0':ok3_0(+(1, n15953_0))) -> *4_0, rt in Omega(n15953_0)
U142(gen_tt:mark:nil:0':ok3_0(+(1, n17620_0))) -> *4_0, rt in Omega(n17620_0)
U152(gen_tt:mark:nil:0':ok3_0(+(1, n19388_0))) -> *4_0, rt in Omega(n19388_0)
cons(gen_tt:mark:nil:0':ok3_0(+(1, n21257_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n21257_0)
natsFrom(gen_tt:mark:nil:0':ok3_0(+(1, n27404_0))) -> *4_0, rt in Omega(n27404_0)
s(gen_tt:mark:nil:0':ok3_0(+(1, n29524_0))) -> *4_0, rt in Omega(n29524_0)
U172(gen_tt:mark:nil:0':ok3_0(+(1, n31745_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) -> *4_0, rt in Omega(n31745_0)
head(gen_tt:mark:nil:0':ok3_0(+(1, n43161_0))) -> *4_0, rt in Omega(n43161_0)
afterNth(gen_tt:mark:nil:0':ok3_0(+(1, n45682_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n45682_0)
U182(gen_tt:mark:nil:0':ok3_0(+(1, n53362_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n53362_0)
pair(gen_tt:mark:nil:0':ok3_0(+(1, n61149_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n61149_0)
U202(gen_tt:mark:nil:0':ok3_0(+(1, n69441_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c), gen_tt:mark:nil:0':ok3_0(d)) -> *4_0, rt in Omega(n69441_0)


Generator Equations:
gen_tt:mark:nil:0':ok3_0(0) <=> tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) <=> mark(gen_tt:mark:nil:0':ok3_0(x))


The following defined symbols remain to be analysed:
isNatural, active, U203, U204, U22, U212, U222, fst, U32, U42, U52, U11, U21, U31, U41, U51, U61, isPLNat, U71, U81, U91, U101, U111, U121, U131, U141, U151, U161, U171, U181, U191, U201, U211, U221, tail, take, sel, proper, top

They will be analysed ascendingly in the following order:
isNatural < active
U203 < active
U204 < active
U22 < active
U212 < active
U222 < active
fst < active
U32 < active
U42 < active
U52 < active
U11 < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
isPLNat < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
U141 < active
U151 < active
U161 < active
U171 < active
U181 < active
U191 < active
U201 < active
U211 < active
U221 < active
tail < active
take < active
sel < active
active < top
isNatural < proper
U203 < proper
U204 < proper
U22 < proper
U212 < proper
U222 < proper
fst < proper
U32 < proper
U42 < proper
U52 < proper
U11 < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
isPLNat < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
U141 < proper
U151 < proper
U161 < proper
U171 < proper
U181 < proper
U191 < proper
U201 < proper
U211 < proper
U221 < proper
tail < proper
take < proper
sel < proper
proper < top

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

(61) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
U203(gen_tt:mark:nil:0':ok3_0(+(1, n88570_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c), gen_tt:mark:nil:0':ok3_0(d)) -> *4_0, rt in Omega(n88570_0)

Induction Base:
U203(gen_tt:mark:nil:0':ok3_0(+(1, 0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c), gen_tt:mark:nil:0':ok3_0(d))

Induction Step:
U203(gen_tt:mark:nil:0':ok3_0(+(1, +(n88570_0, 1))), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c), gen_tt:mark:nil:0':ok3_0(d)) ->_R^Omega(1)
mark(U203(gen_tt:mark:nil:0':ok3_0(+(1, n88570_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c), gen_tt:mark:nil:0':ok3_0(d))) ->_IH
mark(*4_0)

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
----------------------------------------

(62)
Obligation:
TRS:
Rules:
active(U101(tt, V2)) -> mark(U102(isLNat(V2)))
active(U102(tt)) -> mark(tt)
active(U11(tt, N, XS)) -> mark(U12(isLNat(XS), N, XS))
active(U111(tt)) -> mark(tt)
active(U12(tt, N, XS)) -> mark(snd(splitAt(N, XS)))
active(U121(tt)) -> mark(tt)
active(U131(tt, V2)) -> mark(U132(isLNat(V2)))
active(U132(tt)) -> mark(tt)
active(U141(tt, V2)) -> mark(U142(isLNat(V2)))
active(U142(tt)) -> mark(tt)
active(U151(tt, V2)) -> mark(U152(isLNat(V2)))
active(U152(tt)) -> mark(tt)
active(U161(tt, N)) -> mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) -> mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) -> mark(head(afterNth(N, XS)))
active(U181(tt, Y)) -> mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) -> mark(Y)
active(U191(tt, XS)) -> mark(pair(nil, XS))
active(U201(tt, N, X, XS)) -> mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) -> mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) -> mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) -> mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) -> mark(U22(isLNat(Y), X))
active(U211(tt, XS)) -> mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) -> mark(XS)
active(U22(tt, X)) -> mark(X)
active(U221(tt, N, XS)) -> mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) -> mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) -> mark(U32(isLNat(XS), N))
active(U32(tt, N)) -> mark(N)
active(U41(tt, V2)) -> mark(U42(isLNat(V2)))
active(U42(tt)) -> mark(tt)
active(U51(tt, V2)) -> mark(U52(isLNat(V2)))
active(U52(tt)) -> mark(tt)
active(U61(tt)) -> mark(tt)
active(U71(tt)) -> mark(tt)
active(U81(tt)) -> mark(tt)
active(U91(tt)) -> mark(tt)
active(afterNth(N, XS)) -> mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) -> mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) -> mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) -> mark(tt)
active(isLNat(afterNth(V1, V2))) -> mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) -> mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) -> mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) -> mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) -> mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) -> mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) -> mark(U101(isNatural(V1), V2))
active(isNatural(0')) -> mark(tt)
active(isNatural(head(V1))) -> mark(U111(isLNat(V1)))
active(isNatural(s(V1))) -> mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) -> mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) -> mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) -> mark(U151(isNatural(V1), V2))
active(natsFrom(N)) -> mark(U161(isNatural(N), N))
active(sel(N, XS)) -> mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) -> mark(U181(isLNat(X), Y))
active(splitAt(0', XS)) -> mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) -> mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) -> mark(U211(isNatural(N), XS))
active(take(N, XS)) -> mark(U221(isNatural(N), N, XS))
active(U101(X1, X2)) -> U101(active(X1), X2)
active(U102(X)) -> U102(active(X))
active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3)
active(U111(X)) -> U111(active(X))
active(snd(X)) -> snd(active(X))
active(splitAt(X1, X2)) -> splitAt(active(X1), X2)
active(splitAt(X1, X2)) -> splitAt(X1, active(X2))
active(U121(X)) -> U121(active(X))
active(U131(X1, X2)) -> U131(active(X1), X2)
active(U132(X)) -> U132(active(X))
active(U141(X1, X2)) -> U141(active(X1), X2)
active(U142(X)) -> U142(active(X))
active(U151(X1, X2)) -> U151(active(X1), X2)
active(U152(X)) -> U152(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(U172(X1, X2, X3)) -> U172(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(U182(X1, X2)) -> U182(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, X3, X4)) -> U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) -> U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) -> U204(active(X1), X2)
active(U21(X1, X2, X3)) -> U21(active(X1), X2, X3)
active(U22(X1, X2)) -> U22(active(X1), X2)
active(U211(X1, X2)) -> U211(active(X1), X2)
active(U212(X1, X2)) -> U212(active(X1), X2)
active(U221(X1, X2, X3)) -> U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) -> U222(active(X1), X2, X3)
active(fst(X)) -> fst(active(X))
active(U31(X1, X2, X3)) -> U31(active(X1), X2, X3)
active(U32(X1, X2)) -> U32(active(X1), X2)
active(U41(X1, X2)) -> U41(active(X1), X2)
active(U42(X)) -> U42(active(X))
active(U51(X1, X2)) -> U51(active(X1), X2)
active(U52(X)) -> U52(active(X))
active(U61(X)) -> U61(active(X))
active(U71(X)) -> U71(active(X))
active(U81(X)) -> U81(active(X))
active(U91(X)) -> U91(active(X))
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) -> mark(U101(X1, X2))
U102(mark(X)) -> mark(U102(X))
U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3))
U111(mark(X)) -> mark(U111(X))
snd(mark(X)) -> mark(snd(X))
splitAt(mark(X1), X2) -> mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) -> mark(splitAt(X1, X2))
U121(mark(X)) -> mark(U121(X))
U131(mark(X1), X2) -> mark(U131(X1, X2))
U132(mark(X)) -> mark(U132(X))
U141(mark(X1), X2) -> mark(U141(X1, X2))
U142(mark(X)) -> mark(U142(X))
U151(mark(X1), X2) -> mark(U151(X1, X2))
U152(mark(X)) -> mark(U152(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))
U172(mark(X1), X2, X3) -> mark(U172(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))
U182(mark(X1), X2) -> mark(U182(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, X3, X4) -> mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) -> mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) -> mark(U204(X1, X2))
U21(mark(X1), X2, X3) -> mark(U21(X1, X2, X3))
U22(mark(X1), X2) -> mark(U22(X1, X2))
U211(mark(X1), X2) -> mark(U211(X1, X2))
U212(mark(X1), X2) -> mark(U212(X1, X2))
U221(mark(X1), X2, X3) -> mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) -> mark(U222(X1, X2, X3))
fst(mark(X)) -> mark(fst(X))
U31(mark(X1), X2, X3) -> mark(U31(X1, X2, X3))
U32(mark(X1), X2) -> mark(U32(X1, X2))
U41(mark(X1), X2) -> mark(U41(X1, X2))
U42(mark(X)) -> mark(U42(X))
U51(mark(X1), X2) -> mark(U51(X1, X2))
U52(mark(X)) -> mark(U52(X))
U61(mark(X)) -> mark(U61(X))
U71(mark(X)) -> mark(U71(X))
U81(mark(X)) -> mark(U81(X))
U91(mark(X)) -> mark(U91(X))
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)) -> U101(proper(X1), proper(X2))
proper(tt) -> ok(tt)
proper(U102(X)) -> U102(proper(X))
proper(isLNat(X)) -> isLNat(proper(X))
proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) -> U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) -> U111(proper(X))
proper(snd(X)) -> snd(proper(X))
proper(splitAt(X1, X2)) -> splitAt(proper(X1), proper(X2))
proper(U121(X)) -> U121(proper(X))
proper(U131(X1, X2)) -> U131(proper(X1), proper(X2))
proper(U132(X)) -> U132(proper(X))
proper(U141(X1, X2)) -> U141(proper(X1), proper(X2))
proper(U142(X)) -> U142(proper(X))
proper(U151(X1, X2)) -> U151(proper(X1), proper(X2))
proper(U152(X)) -> U152(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(U172(X1, X2, X3)) -> U172(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(U182(X1, X2)) -> U182(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, X3, X4)) -> U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) -> isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) -> U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) -> U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) -> U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) -> U22(proper(X1), proper(X2))
proper(U211(X1, X2)) -> U211(proper(X1), proper(X2))
proper(U212(X1, X2)) -> U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) -> U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) -> U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) -> fst(proper(X))
proper(U31(X1, X2, X3)) -> U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) -> U32(proper(X1), proper(X2))
proper(U41(X1, X2)) -> U41(proper(X1), proper(X2))
proper(U42(X)) -> U42(proper(X))
proper(U51(X1, X2)) -> U51(proper(X1), proper(X2))
proper(U52(X)) -> U52(proper(X))
proper(U61(X)) -> U61(proper(X))
proper(U71(X)) -> U71(proper(X))
proper(U81(X)) -> U81(proper(X))
proper(U91(X)) -> U91(proper(X))
proper(isPLNat(X)) -> isPLNat(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(U101(X1, X2))
U102(ok(X)) -> ok(U102(X))
isLNat(ok(X)) -> ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))
U111(ok(X)) -> ok(U111(X))
snd(ok(X)) -> ok(snd(X))
splitAt(ok(X1), ok(X2)) -> ok(splitAt(X1, X2))
U121(ok(X)) -> ok(U121(X))
U131(ok(X1), ok(X2)) -> ok(U131(X1, X2))
U132(ok(X)) -> ok(U132(X))
U141(ok(X1), ok(X2)) -> ok(U141(X1, X2))
U142(ok(X)) -> ok(U142(X))
U151(ok(X1), ok(X2)) -> ok(U151(X1, X2))
U152(ok(X)) -> ok(U152(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))
U172(ok(X1), ok(X2), ok(X3)) -> ok(U172(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))
U182(ok(X1), ok(X2)) -> ok(U182(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(X3), ok(X4)) -> ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) -> ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) -> ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) -> ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) -> ok(U22(X1, X2))
U211(ok(X1), ok(X2)) -> ok(U211(X1, X2))
U212(ok(X1), ok(X2)) -> ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) -> ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) -> ok(U222(X1, X2, X3))
fst(ok(X)) -> ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) -> ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) -> ok(U32(X1, X2))
U41(ok(X1), ok(X2)) -> ok(U41(X1, X2))
U42(ok(X)) -> ok(U42(X))
U51(ok(X1), ok(X2)) -> ok(U51(X1, X2))
U52(ok(X)) -> ok(U52(X))
U61(ok(X)) -> ok(U61(X))
U71(ok(X)) -> ok(U71(X))
U81(ok(X)) -> ok(U81(X))
U91(ok(X)) -> ok(U91(X))
isPLNat(ok(X)) -> ok(isPLNat(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))

Types:
active :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U12 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U172 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U182 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U203 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U204 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U22 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U212 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U222 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U32 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok -> top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat -> tt:mark:nil:0':ok


Lemmas:
U102(gen_tt:mark:nil:0':ok3_0(+(1, n5_0))) -> *4_0, rt in Omega(n5_0)
U12(gen_tt:mark:nil:0':ok3_0(+(1, n1128_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) -> *4_0, rt in Omega(n1128_0)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n9121_0))) -> *4_0, rt in Omega(n9121_0)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n10537_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n10537_0)
U132(gen_tt:mark:nil:0':ok3_0(+(1, n15953_0))) -> *4_0, rt in Omega(n15953_0)
U142(gen_tt:mark:nil:0':ok3_0(+(1, n17620_0))) -> *4_0, rt in Omega(n17620_0)
U152(gen_tt:mark:nil:0':ok3_0(+(1, n19388_0))) -> *4_0, rt in Omega(n19388_0)
cons(gen_tt:mark:nil:0':ok3_0(+(1, n21257_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n21257_0)
natsFrom(gen_tt:mark:nil:0':ok3_0(+(1, n27404_0))) -> *4_0, rt in Omega(n27404_0)
s(gen_tt:mark:nil:0':ok3_0(+(1, n29524_0))) -> *4_0, rt in Omega(n29524_0)
U172(gen_tt:mark:nil:0':ok3_0(+(1, n31745_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) -> *4_0, rt in Omega(n31745_0)
head(gen_tt:mark:nil:0':ok3_0(+(1, n43161_0))) -> *4_0, rt in Omega(n43161_0)
afterNth(gen_tt:mark:nil:0':ok3_0(+(1, n45682_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n45682_0)
U182(gen_tt:mark:nil:0':ok3_0(+(1, n53362_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n53362_0)
pair(gen_tt:mark:nil:0':ok3_0(+(1, n61149_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n61149_0)
U202(gen_tt:mark:nil:0':ok3_0(+(1, n69441_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c), gen_tt:mark:nil:0':ok3_0(d)) -> *4_0, rt in Omega(n69441_0)
U203(gen_tt:mark:nil:0':ok3_0(+(1, n88570_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c), gen_tt:mark:nil:0':ok3_0(d)) -> *4_0, rt in Omega(n88570_0)


Generator Equations:
gen_tt:mark:nil:0':ok3_0(0) <=> tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) <=> mark(gen_tt:mark:nil:0':ok3_0(x))


The following defined symbols remain to be analysed:
U204, active, U22, U212, U222, fst, U32, U42, U52, U11, U21, U31, U41, U51, U61, isPLNat, U71, U81, U91, U101, U111, U121, U131, U141, U151, U161, U171, U181, U191, U201, U211, U221, tail, take, sel, proper, top

They will be analysed ascendingly in the following order:
U204 < active
U22 < active
U212 < active
U222 < active
fst < active
U32 < active
U42 < active
U52 < active
U11 < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
isPLNat < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
U141 < active
U151 < active
U161 < active
U171 < active
U181 < active
U191 < active
U201 < active
U211 < active
U221 < active
tail < active
take < active
sel < active
active < top
U204 < proper
U22 < proper
U212 < proper
U222 < proper
fst < proper
U32 < proper
U42 < proper
U52 < proper
U11 < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
isPLNat < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
U141 < proper
U151 < proper
U161 < proper
U171 < proper
U181 < proper
U191 < proper
U201 < proper
U211 < proper
U221 < proper
tail < proper
take < proper
sel < proper
proper < top

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

(63) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
U204(gen_tt:mark:nil:0':ok3_0(+(1, n108633_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n108633_0)

Induction Base:
U204(gen_tt:mark:nil:0':ok3_0(+(1, 0)), gen_tt:mark:nil:0':ok3_0(b))

Induction Step:
U204(gen_tt:mark:nil:0':ok3_0(+(1, +(n108633_0, 1))), gen_tt:mark:nil:0':ok3_0(b)) ->_R^Omega(1)
mark(U204(gen_tt:mark:nil:0':ok3_0(+(1, n108633_0)), gen_tt:mark:nil:0':ok3_0(b))) ->_IH
mark(*4_0)

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
----------------------------------------

(64)
Obligation:
TRS:
Rules:
active(U101(tt, V2)) -> mark(U102(isLNat(V2)))
active(U102(tt)) -> mark(tt)
active(U11(tt, N, XS)) -> mark(U12(isLNat(XS), N, XS))
active(U111(tt)) -> mark(tt)
active(U12(tt, N, XS)) -> mark(snd(splitAt(N, XS)))
active(U121(tt)) -> mark(tt)
active(U131(tt, V2)) -> mark(U132(isLNat(V2)))
active(U132(tt)) -> mark(tt)
active(U141(tt, V2)) -> mark(U142(isLNat(V2)))
active(U142(tt)) -> mark(tt)
active(U151(tt, V2)) -> mark(U152(isLNat(V2)))
active(U152(tt)) -> mark(tt)
active(U161(tt, N)) -> mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) -> mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) -> mark(head(afterNth(N, XS)))
active(U181(tt, Y)) -> mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) -> mark(Y)
active(U191(tt, XS)) -> mark(pair(nil, XS))
active(U201(tt, N, X, XS)) -> mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) -> mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) -> mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) -> mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) -> mark(U22(isLNat(Y), X))
active(U211(tt, XS)) -> mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) -> mark(XS)
active(U22(tt, X)) -> mark(X)
active(U221(tt, N, XS)) -> mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) -> mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) -> mark(U32(isLNat(XS), N))
active(U32(tt, N)) -> mark(N)
active(U41(tt, V2)) -> mark(U42(isLNat(V2)))
active(U42(tt)) -> mark(tt)
active(U51(tt, V2)) -> mark(U52(isLNat(V2)))
active(U52(tt)) -> mark(tt)
active(U61(tt)) -> mark(tt)
active(U71(tt)) -> mark(tt)
active(U81(tt)) -> mark(tt)
active(U91(tt)) -> mark(tt)
active(afterNth(N, XS)) -> mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) -> mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) -> mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) -> mark(tt)
active(isLNat(afterNth(V1, V2))) -> mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) -> mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) -> mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) -> mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) -> mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) -> mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) -> mark(U101(isNatural(V1), V2))
active(isNatural(0')) -> mark(tt)
active(isNatural(head(V1))) -> mark(U111(isLNat(V1)))
active(isNatural(s(V1))) -> mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) -> mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) -> mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) -> mark(U151(isNatural(V1), V2))
active(natsFrom(N)) -> mark(U161(isNatural(N), N))
active(sel(N, XS)) -> mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) -> mark(U181(isLNat(X), Y))
active(splitAt(0', XS)) -> mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) -> mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) -> mark(U211(isNatural(N), XS))
active(take(N, XS)) -> mark(U221(isNatural(N), N, XS))
active(U101(X1, X2)) -> U101(active(X1), X2)
active(U102(X)) -> U102(active(X))
active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3)
active(U111(X)) -> U111(active(X))
active(snd(X)) -> snd(active(X))
active(splitAt(X1, X2)) -> splitAt(active(X1), X2)
active(splitAt(X1, X2)) -> splitAt(X1, active(X2))
active(U121(X)) -> U121(active(X))
active(U131(X1, X2)) -> U131(active(X1), X2)
active(U132(X)) -> U132(active(X))
active(U141(X1, X2)) -> U141(active(X1), X2)
active(U142(X)) -> U142(active(X))
active(U151(X1, X2)) -> U151(active(X1), X2)
active(U152(X)) -> U152(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(U172(X1, X2, X3)) -> U172(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(U182(X1, X2)) -> U182(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, X3, X4)) -> U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) -> U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) -> U204(active(X1), X2)
active(U21(X1, X2, X3)) -> U21(active(X1), X2, X3)
active(U22(X1, X2)) -> U22(active(X1), X2)
active(U211(X1, X2)) -> U211(active(X1), X2)
active(U212(X1, X2)) -> U212(active(X1), X2)
active(U221(X1, X2, X3)) -> U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) -> U222(active(X1), X2, X3)
active(fst(X)) -> fst(active(X))
active(U31(X1, X2, X3)) -> U31(active(X1), X2, X3)
active(U32(X1, X2)) -> U32(active(X1), X2)
active(U41(X1, X2)) -> U41(active(X1), X2)
active(U42(X)) -> U42(active(X))
active(U51(X1, X2)) -> U51(active(X1), X2)
active(U52(X)) -> U52(active(X))
active(U61(X)) -> U61(active(X))
active(U71(X)) -> U71(active(X))
active(U81(X)) -> U81(active(X))
active(U91(X)) -> U91(active(X))
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) -> mark(U101(X1, X2))
U102(mark(X)) -> mark(U102(X))
U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3))
U111(mark(X)) -> mark(U111(X))
snd(mark(X)) -> mark(snd(X))
splitAt(mark(X1), X2) -> mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) -> mark(splitAt(X1, X2))
U121(mark(X)) -> mark(U121(X))
U131(mark(X1), X2) -> mark(U131(X1, X2))
U132(mark(X)) -> mark(U132(X))
U141(mark(X1), X2) -> mark(U141(X1, X2))
U142(mark(X)) -> mark(U142(X))
U151(mark(X1), X2) -> mark(U151(X1, X2))
U152(mark(X)) -> mark(U152(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))
U172(mark(X1), X2, X3) -> mark(U172(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))
U182(mark(X1), X2) -> mark(U182(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, X3, X4) -> mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) -> mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) -> mark(U204(X1, X2))
U21(mark(X1), X2, X3) -> mark(U21(X1, X2, X3))
U22(mark(X1), X2) -> mark(U22(X1, X2))
U211(mark(X1), X2) -> mark(U211(X1, X2))
U212(mark(X1), X2) -> mark(U212(X1, X2))
U221(mark(X1), X2, X3) -> mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) -> mark(U222(X1, X2, X3))
fst(mark(X)) -> mark(fst(X))
U31(mark(X1), X2, X3) -> mark(U31(X1, X2, X3))
U32(mark(X1), X2) -> mark(U32(X1, X2))
U41(mark(X1), X2) -> mark(U41(X1, X2))
U42(mark(X)) -> mark(U42(X))
U51(mark(X1), X2) -> mark(U51(X1, X2))
U52(mark(X)) -> mark(U52(X))
U61(mark(X)) -> mark(U61(X))
U71(mark(X)) -> mark(U71(X))
U81(mark(X)) -> mark(U81(X))
U91(mark(X)) -> mark(U91(X))
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)) -> U101(proper(X1), proper(X2))
proper(tt) -> ok(tt)
proper(U102(X)) -> U102(proper(X))
proper(isLNat(X)) -> isLNat(proper(X))
proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) -> U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) -> U111(proper(X))
proper(snd(X)) -> snd(proper(X))
proper(splitAt(X1, X2)) -> splitAt(proper(X1), proper(X2))
proper(U121(X)) -> U121(proper(X))
proper(U131(X1, X2)) -> U131(proper(X1), proper(X2))
proper(U132(X)) -> U132(proper(X))
proper(U141(X1, X2)) -> U141(proper(X1), proper(X2))
proper(U142(X)) -> U142(proper(X))
proper(U151(X1, X2)) -> U151(proper(X1), proper(X2))
proper(U152(X)) -> U152(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(U172(X1, X2, X3)) -> U172(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(U182(X1, X2)) -> U182(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, X3, X4)) -> U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) -> isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) -> U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) -> U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) -> U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) -> U22(proper(X1), proper(X2))
proper(U211(X1, X2)) -> U211(proper(X1), proper(X2))
proper(U212(X1, X2)) -> U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) -> U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) -> U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) -> fst(proper(X))
proper(U31(X1, X2, X3)) -> U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) -> U32(proper(X1), proper(X2))
proper(U41(X1, X2)) -> U41(proper(X1), proper(X2))
proper(U42(X)) -> U42(proper(X))
proper(U51(X1, X2)) -> U51(proper(X1), proper(X2))
proper(U52(X)) -> U52(proper(X))
proper(U61(X)) -> U61(proper(X))
proper(U71(X)) -> U71(proper(X))
proper(U81(X)) -> U81(proper(X))
proper(U91(X)) -> U91(proper(X))
proper(isPLNat(X)) -> isPLNat(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(U101(X1, X2))
U102(ok(X)) -> ok(U102(X))
isLNat(ok(X)) -> ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))
U111(ok(X)) -> ok(U111(X))
snd(ok(X)) -> ok(snd(X))
splitAt(ok(X1), ok(X2)) -> ok(splitAt(X1, X2))
U121(ok(X)) -> ok(U121(X))
U131(ok(X1), ok(X2)) -> ok(U131(X1, X2))
U132(ok(X)) -> ok(U132(X))
U141(ok(X1), ok(X2)) -> ok(U141(X1, X2))
U142(ok(X)) -> ok(U142(X))
U151(ok(X1), ok(X2)) -> ok(U151(X1, X2))
U152(ok(X)) -> ok(U152(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))
U172(ok(X1), ok(X2), ok(X3)) -> ok(U172(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))
U182(ok(X1), ok(X2)) -> ok(U182(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(X3), ok(X4)) -> ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) -> ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) -> ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) -> ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) -> ok(U22(X1, X2))
U211(ok(X1), ok(X2)) -> ok(U211(X1, X2))
U212(ok(X1), ok(X2)) -> ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) -> ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) -> ok(U222(X1, X2, X3))
fst(ok(X)) -> ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) -> ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) -> ok(U32(X1, X2))
U41(ok(X1), ok(X2)) -> ok(U41(X1, X2))
U42(ok(X)) -> ok(U42(X))
U51(ok(X1), ok(X2)) -> ok(U51(X1, X2))
U52(ok(X)) -> ok(U52(X))
U61(ok(X)) -> ok(U61(X))
U71(ok(X)) -> ok(U71(X))
U81(ok(X)) -> ok(U81(X))
U91(ok(X)) -> ok(U91(X))
isPLNat(ok(X)) -> ok(isPLNat(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))

Types:
active :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U12 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U172 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U182 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U203 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U204 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U22 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U212 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U222 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U32 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok -> top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat -> tt:mark:nil:0':ok


Lemmas:
U102(gen_tt:mark:nil:0':ok3_0(+(1, n5_0))) -> *4_0, rt in Omega(n5_0)
U12(gen_tt:mark:nil:0':ok3_0(+(1, n1128_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) -> *4_0, rt in Omega(n1128_0)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n9121_0))) -> *4_0, rt in Omega(n9121_0)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n10537_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n10537_0)
U132(gen_tt:mark:nil:0':ok3_0(+(1, n15953_0))) -> *4_0, rt in Omega(n15953_0)
U142(gen_tt:mark:nil:0':ok3_0(+(1, n17620_0))) -> *4_0, rt in Omega(n17620_0)
U152(gen_tt:mark:nil:0':ok3_0(+(1, n19388_0))) -> *4_0, rt in Omega(n19388_0)
cons(gen_tt:mark:nil:0':ok3_0(+(1, n21257_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n21257_0)
natsFrom(gen_tt:mark:nil:0':ok3_0(+(1, n27404_0))) -> *4_0, rt in Omega(n27404_0)
s(gen_tt:mark:nil:0':ok3_0(+(1, n29524_0))) -> *4_0, rt in Omega(n29524_0)
U172(gen_tt:mark:nil:0':ok3_0(+(1, n31745_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) -> *4_0, rt in Omega(n31745_0)
head(gen_tt:mark:nil:0':ok3_0(+(1, n43161_0))) -> *4_0, rt in Omega(n43161_0)
afterNth(gen_tt:mark:nil:0':ok3_0(+(1, n45682_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n45682_0)
U182(gen_tt:mark:nil:0':ok3_0(+(1, n53362_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n53362_0)
pair(gen_tt:mark:nil:0':ok3_0(+(1, n61149_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n61149_0)
U202(gen_tt:mark:nil:0':ok3_0(+(1, n69441_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c), gen_tt:mark:nil:0':ok3_0(d)) -> *4_0, rt in Omega(n69441_0)
U203(gen_tt:mark:nil:0':ok3_0(+(1, n88570_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c), gen_tt:mark:nil:0':ok3_0(d)) -> *4_0, rt in Omega(n88570_0)
U204(gen_tt:mark:nil:0':ok3_0(+(1, n108633_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n108633_0)


Generator Equations:
gen_tt:mark:nil:0':ok3_0(0) <=> tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) <=> mark(gen_tt:mark:nil:0':ok3_0(x))


The following defined symbols remain to be analysed:
U22, active, U212, U222, fst, U32, U42, U52, U11, U21, U31, U41, U51, U61, isPLNat, U71, U81, U91, U101, U111, U121, U131, U141, U151, U161, U171, U181, U191, U201, U211, U221, tail, take, sel, proper, top

They will be analysed ascendingly in the following order:
U22 < active
U212 < active
U222 < active
fst < active
U32 < active
U42 < active
U52 < active
U11 < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
isPLNat < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
U141 < active
U151 < active
U161 < active
U171 < active
U181 < active
U191 < active
U201 < active
U211 < active
U221 < active
tail < active
take < active
sel < active
active < top
U22 < proper
U212 < proper
U222 < proper
fst < proper
U32 < proper
U42 < proper
U52 < proper
U11 < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
isPLNat < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
U141 < proper
U151 < proper
U161 < proper
U171 < proper
U181 < proper
U191 < proper
U201 < proper
U211 < proper
U221 < proper
tail < proper
take < proper
sel < proper
proper < top

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

(65) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
U22(gen_tt:mark:nil:0':ok3_0(+(1, n118036_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n118036_0)

Induction Base:
U22(gen_tt:mark:nil:0':ok3_0(+(1, 0)), gen_tt:mark:nil:0':ok3_0(b))

Induction Step:
U22(gen_tt:mark:nil:0':ok3_0(+(1, +(n118036_0, 1))), gen_tt:mark:nil:0':ok3_0(b)) ->_R^Omega(1)
mark(U22(gen_tt:mark:nil:0':ok3_0(+(1, n118036_0)), gen_tt:mark:nil:0':ok3_0(b))) ->_IH
mark(*4_0)

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
----------------------------------------

(66)
Obligation:
TRS:
Rules:
active(U101(tt, V2)) -> mark(U102(isLNat(V2)))
active(U102(tt)) -> mark(tt)
active(U11(tt, N, XS)) -> mark(U12(isLNat(XS), N, XS))
active(U111(tt)) -> mark(tt)
active(U12(tt, N, XS)) -> mark(snd(splitAt(N, XS)))
active(U121(tt)) -> mark(tt)
active(U131(tt, V2)) -> mark(U132(isLNat(V2)))
active(U132(tt)) -> mark(tt)
active(U141(tt, V2)) -> mark(U142(isLNat(V2)))
active(U142(tt)) -> mark(tt)
active(U151(tt, V2)) -> mark(U152(isLNat(V2)))
active(U152(tt)) -> mark(tt)
active(U161(tt, N)) -> mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) -> mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) -> mark(head(afterNth(N, XS)))
active(U181(tt, Y)) -> mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) -> mark(Y)
active(U191(tt, XS)) -> mark(pair(nil, XS))
active(U201(tt, N, X, XS)) -> mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) -> mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) -> mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) -> mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) -> mark(U22(isLNat(Y), X))
active(U211(tt, XS)) -> mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) -> mark(XS)
active(U22(tt, X)) -> mark(X)
active(U221(tt, N, XS)) -> mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) -> mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) -> mark(U32(isLNat(XS), N))
active(U32(tt, N)) -> mark(N)
active(U41(tt, V2)) -> mark(U42(isLNat(V2)))
active(U42(tt)) -> mark(tt)
active(U51(tt, V2)) -> mark(U52(isLNat(V2)))
active(U52(tt)) -> mark(tt)
active(U61(tt)) -> mark(tt)
active(U71(tt)) -> mark(tt)
active(U81(tt)) -> mark(tt)
active(U91(tt)) -> mark(tt)
active(afterNth(N, XS)) -> mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) -> mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) -> mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) -> mark(tt)
active(isLNat(afterNth(V1, V2))) -> mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) -> mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) -> mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) -> mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) -> mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) -> mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) -> mark(U101(isNatural(V1), V2))
active(isNatural(0')) -> mark(tt)
active(isNatural(head(V1))) -> mark(U111(isLNat(V1)))
active(isNatural(s(V1))) -> mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) -> mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) -> mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) -> mark(U151(isNatural(V1), V2))
active(natsFrom(N)) -> mark(U161(isNatural(N), N))
active(sel(N, XS)) -> mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) -> mark(U181(isLNat(X), Y))
active(splitAt(0', XS)) -> mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) -> mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) -> mark(U211(isNatural(N), XS))
active(take(N, XS)) -> mark(U221(isNatural(N), N, XS))
active(U101(X1, X2)) -> U101(active(X1), X2)
active(U102(X)) -> U102(active(X))
active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3)
active(U111(X)) -> U111(active(X))
active(snd(X)) -> snd(active(X))
active(splitAt(X1, X2)) -> splitAt(active(X1), X2)
active(splitAt(X1, X2)) -> splitAt(X1, active(X2))
active(U121(X)) -> U121(active(X))
active(U131(X1, X2)) -> U131(active(X1), X2)
active(U132(X)) -> U132(active(X))
active(U141(X1, X2)) -> U141(active(X1), X2)
active(U142(X)) -> U142(active(X))
active(U151(X1, X2)) -> U151(active(X1), X2)
active(U152(X)) -> U152(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(U172(X1, X2, X3)) -> U172(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(U182(X1, X2)) -> U182(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, X3, X4)) -> U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) -> U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) -> U204(active(X1), X2)
active(U21(X1, X2, X3)) -> U21(active(X1), X2, X3)
active(U22(X1, X2)) -> U22(active(X1), X2)
active(U211(X1, X2)) -> U211(active(X1), X2)
active(U212(X1, X2)) -> U212(active(X1), X2)
active(U221(X1, X2, X3)) -> U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) -> U222(active(X1), X2, X3)
active(fst(X)) -> fst(active(X))
active(U31(X1, X2, X3)) -> U31(active(X1), X2, X3)
active(U32(X1, X2)) -> U32(active(X1), X2)
active(U41(X1, X2)) -> U41(active(X1), X2)
active(U42(X)) -> U42(active(X))
active(U51(X1, X2)) -> U51(active(X1), X2)
active(U52(X)) -> U52(active(X))
active(U61(X)) -> U61(active(X))
active(U71(X)) -> U71(active(X))
active(U81(X)) -> U81(active(X))
active(U91(X)) -> U91(active(X))
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) -> mark(U101(X1, X2))
U102(mark(X)) -> mark(U102(X))
U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3))
U111(mark(X)) -> mark(U111(X))
snd(mark(X)) -> mark(snd(X))
splitAt(mark(X1), X2) -> mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) -> mark(splitAt(X1, X2))
U121(mark(X)) -> mark(U121(X))
U131(mark(X1), X2) -> mark(U131(X1, X2))
U132(mark(X)) -> mark(U132(X))
U141(mark(X1), X2) -> mark(U141(X1, X2))
U142(mark(X)) -> mark(U142(X))
U151(mark(X1), X2) -> mark(U151(X1, X2))
U152(mark(X)) -> mark(U152(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))
U172(mark(X1), X2, X3) -> mark(U172(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))
U182(mark(X1), X2) -> mark(U182(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, X3, X4) -> mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) -> mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) -> mark(U204(X1, X2))
U21(mark(X1), X2, X3) -> mark(U21(X1, X2, X3))
U22(mark(X1), X2) -> mark(U22(X1, X2))
U211(mark(X1), X2) -> mark(U211(X1, X2))
U212(mark(X1), X2) -> mark(U212(X1, X2))
U221(mark(X1), X2, X3) -> mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) -> mark(U222(X1, X2, X3))
fst(mark(X)) -> mark(fst(X))
U31(mark(X1), X2, X3) -> mark(U31(X1, X2, X3))
U32(mark(X1), X2) -> mark(U32(X1, X2))
U41(mark(X1), X2) -> mark(U41(X1, X2))
U42(mark(X)) -> mark(U42(X))
U51(mark(X1), X2) -> mark(U51(X1, X2))
U52(mark(X)) -> mark(U52(X))
U61(mark(X)) -> mark(U61(X))
U71(mark(X)) -> mark(U71(X))
U81(mark(X)) -> mark(U81(X))
U91(mark(X)) -> mark(U91(X))
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)) -> U101(proper(X1), proper(X2))
proper(tt) -> ok(tt)
proper(U102(X)) -> U102(proper(X))
proper(isLNat(X)) -> isLNat(proper(X))
proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) -> U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) -> U111(proper(X))
proper(snd(X)) -> snd(proper(X))
proper(splitAt(X1, X2)) -> splitAt(proper(X1), proper(X2))
proper(U121(X)) -> U121(proper(X))
proper(U131(X1, X2)) -> U131(proper(X1), proper(X2))
proper(U132(X)) -> U132(proper(X))
proper(U141(X1, X2)) -> U141(proper(X1), proper(X2))
proper(U142(X)) -> U142(proper(X))
proper(U151(X1, X2)) -> U151(proper(X1), proper(X2))
proper(U152(X)) -> U152(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(U172(X1, X2, X3)) -> U172(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(U182(X1, X2)) -> U182(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, X3, X4)) -> U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) -> isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) -> U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) -> U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) -> U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) -> U22(proper(X1), proper(X2))
proper(U211(X1, X2)) -> U211(proper(X1), proper(X2))
proper(U212(X1, X2)) -> U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) -> U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) -> U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) -> fst(proper(X))
proper(U31(X1, X2, X3)) -> U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) -> U32(proper(X1), proper(X2))
proper(U41(X1, X2)) -> U41(proper(X1), proper(X2))
proper(U42(X)) -> U42(proper(X))
proper(U51(X1, X2)) -> U51(proper(X1), proper(X2))
proper(U52(X)) -> U52(proper(X))
proper(U61(X)) -> U61(proper(X))
proper(U71(X)) -> U71(proper(X))
proper(U81(X)) -> U81(proper(X))
proper(U91(X)) -> U91(proper(X))
proper(isPLNat(X)) -> isPLNat(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(U101(X1, X2))
U102(ok(X)) -> ok(U102(X))
isLNat(ok(X)) -> ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))
U111(ok(X)) -> ok(U111(X))
snd(ok(X)) -> ok(snd(X))
splitAt(ok(X1), ok(X2)) -> ok(splitAt(X1, X2))
U121(ok(X)) -> ok(U121(X))
U131(ok(X1), ok(X2)) -> ok(U131(X1, X2))
U132(ok(X)) -> ok(U132(X))
U141(ok(X1), ok(X2)) -> ok(U141(X1, X2))
U142(ok(X)) -> ok(U142(X))
U151(ok(X1), ok(X2)) -> ok(U151(X1, X2))
U152(ok(X)) -> ok(U152(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))
U172(ok(X1), ok(X2), ok(X3)) -> ok(U172(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))
U182(ok(X1), ok(X2)) -> ok(U182(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(X3), ok(X4)) -> ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) -> ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) -> ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) -> ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) -> ok(U22(X1, X2))
U211(ok(X1), ok(X2)) -> ok(U211(X1, X2))
U212(ok(X1), ok(X2)) -> ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) -> ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) -> ok(U222(X1, X2, X3))
fst(ok(X)) -> ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) -> ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) -> ok(U32(X1, X2))
U41(ok(X1), ok(X2)) -> ok(U41(X1, X2))
U42(ok(X)) -> ok(U42(X))
U51(ok(X1), ok(X2)) -> ok(U51(X1, X2))
U52(ok(X)) -> ok(U52(X))
U61(ok(X)) -> ok(U61(X))
U71(ok(X)) -> ok(U71(X))
U81(ok(X)) -> ok(U81(X))
U91(ok(X)) -> ok(U91(X))
isPLNat(ok(X)) -> ok(isPLNat(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))

Types:
active :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U12 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U172 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U182 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U203 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U204 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U22 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U212 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U222 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U32 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok -> top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat -> tt:mark:nil:0':ok


Lemmas:
U102(gen_tt:mark:nil:0':ok3_0(+(1, n5_0))) -> *4_0, rt in Omega(n5_0)
U12(gen_tt:mark:nil:0':ok3_0(+(1, n1128_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) -> *4_0, rt in Omega(n1128_0)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n9121_0))) -> *4_0, rt in Omega(n9121_0)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n10537_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n10537_0)
U132(gen_tt:mark:nil:0':ok3_0(+(1, n15953_0))) -> *4_0, rt in Omega(n15953_0)
U142(gen_tt:mark:nil:0':ok3_0(+(1, n17620_0))) -> *4_0, rt in Omega(n17620_0)
U152(gen_tt:mark:nil:0':ok3_0(+(1, n19388_0))) -> *4_0, rt in Omega(n19388_0)
cons(gen_tt:mark:nil:0':ok3_0(+(1, n21257_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n21257_0)
natsFrom(gen_tt:mark:nil:0':ok3_0(+(1, n27404_0))) -> *4_0, rt in Omega(n27404_0)
s(gen_tt:mark:nil:0':ok3_0(+(1, n29524_0))) -> *4_0, rt in Omega(n29524_0)
U172(gen_tt:mark:nil:0':ok3_0(+(1, n31745_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) -> *4_0, rt in Omega(n31745_0)
head(gen_tt:mark:nil:0':ok3_0(+(1, n43161_0))) -> *4_0, rt in Omega(n43161_0)
afterNth(gen_tt:mark:nil:0':ok3_0(+(1, n45682_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n45682_0)
U182(gen_tt:mark:nil:0':ok3_0(+(1, n53362_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n53362_0)
pair(gen_tt:mark:nil:0':ok3_0(+(1, n61149_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n61149_0)
U202(gen_tt:mark:nil:0':ok3_0(+(1, n69441_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c), gen_tt:mark:nil:0':ok3_0(d)) -> *4_0, rt in Omega(n69441_0)
U203(gen_tt:mark:nil:0':ok3_0(+(1, n88570_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c), gen_tt:mark:nil:0':ok3_0(d)) -> *4_0, rt in Omega(n88570_0)
U204(gen_tt:mark:nil:0':ok3_0(+(1, n108633_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n108633_0)
U22(gen_tt:mark:nil:0':ok3_0(+(1, n118036_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n118036_0)


Generator Equations:
gen_tt:mark:nil:0':ok3_0(0) <=> tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) <=> mark(gen_tt:mark:nil:0':ok3_0(x))


The following defined symbols remain to be analysed:
U212, active, U222, fst, U32, U42, U52, U11, U21, U31, U41, U51, U61, isPLNat, U71, U81, U91, U101, U111, U121, U131, U141, U151, U161, U171, U181, U191, U201, U211, U221, tail, take, sel, proper, top

They will be analysed ascendingly in the following order:
U212 < active
U222 < active
fst < active
U32 < active
U42 < active
U52 < active
U11 < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
isPLNat < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
U141 < active
U151 < active
U161 < active
U171 < active
U181 < active
U191 < active
U201 < active
U211 < active
U221 < active
tail < active
take < active
sel < active
active < top
U212 < proper
U222 < proper
fst < proper
U32 < proper
U42 < proper
U52 < proper
U11 < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
isPLNat < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
U141 < proper
U151 < proper
U161 < proper
U171 < proper
U181 < proper
U191 < proper
U201 < proper
U211 < proper
U221 < proper
tail < proper
take < proper
sel < proper
proper < top

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

(67) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
U212(gen_tt:mark:nil:0':ok3_0(+(1, n127745_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n127745_0)

Induction Base:
U212(gen_tt:mark:nil:0':ok3_0(+(1, 0)), gen_tt:mark:nil:0':ok3_0(b))

Induction Step:
U212(gen_tt:mark:nil:0':ok3_0(+(1, +(n127745_0, 1))), gen_tt:mark:nil:0':ok3_0(b)) ->_R^Omega(1)
mark(U212(gen_tt:mark:nil:0':ok3_0(+(1, n127745_0)), gen_tt:mark:nil:0':ok3_0(b))) ->_IH
mark(*4_0)

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
----------------------------------------

(68)
Obligation:
TRS:
Rules:
active(U101(tt, V2)) -> mark(U102(isLNat(V2)))
active(U102(tt)) -> mark(tt)
active(U11(tt, N, XS)) -> mark(U12(isLNat(XS), N, XS))
active(U111(tt)) -> mark(tt)
active(U12(tt, N, XS)) -> mark(snd(splitAt(N, XS)))
active(U121(tt)) -> mark(tt)
active(U131(tt, V2)) -> mark(U132(isLNat(V2)))
active(U132(tt)) -> mark(tt)
active(U141(tt, V2)) -> mark(U142(isLNat(V2)))
active(U142(tt)) -> mark(tt)
active(U151(tt, V2)) -> mark(U152(isLNat(V2)))
active(U152(tt)) -> mark(tt)
active(U161(tt, N)) -> mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) -> mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) -> mark(head(afterNth(N, XS)))
active(U181(tt, Y)) -> mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) -> mark(Y)
active(U191(tt, XS)) -> mark(pair(nil, XS))
active(U201(tt, N, X, XS)) -> mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) -> mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) -> mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) -> mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) -> mark(U22(isLNat(Y), X))
active(U211(tt, XS)) -> mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) -> mark(XS)
active(U22(tt, X)) -> mark(X)
active(U221(tt, N, XS)) -> mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) -> mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) -> mark(U32(isLNat(XS), N))
active(U32(tt, N)) -> mark(N)
active(U41(tt, V2)) -> mark(U42(isLNat(V2)))
active(U42(tt)) -> mark(tt)
active(U51(tt, V2)) -> mark(U52(isLNat(V2)))
active(U52(tt)) -> mark(tt)
active(U61(tt)) -> mark(tt)
active(U71(tt)) -> mark(tt)
active(U81(tt)) -> mark(tt)
active(U91(tt)) -> mark(tt)
active(afterNth(N, XS)) -> mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) -> mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) -> mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) -> mark(tt)
active(isLNat(afterNth(V1, V2))) -> mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) -> mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) -> mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) -> mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) -> mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) -> mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) -> mark(U101(isNatural(V1), V2))
active(isNatural(0')) -> mark(tt)
active(isNatural(head(V1))) -> mark(U111(isLNat(V1)))
active(isNatural(s(V1))) -> mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) -> mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) -> mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) -> mark(U151(isNatural(V1), V2))
active(natsFrom(N)) -> mark(U161(isNatural(N), N))
active(sel(N, XS)) -> mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) -> mark(U181(isLNat(X), Y))
active(splitAt(0', XS)) -> mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) -> mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) -> mark(U211(isNatural(N), XS))
active(take(N, XS)) -> mark(U221(isNatural(N), N, XS))
active(U101(X1, X2)) -> U101(active(X1), X2)
active(U102(X)) -> U102(active(X))
active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3)
active(U111(X)) -> U111(active(X))
active(snd(X)) -> snd(active(X))
active(splitAt(X1, X2)) -> splitAt(active(X1), X2)
active(splitAt(X1, X2)) -> splitAt(X1, active(X2))
active(U121(X)) -> U121(active(X))
active(U131(X1, X2)) -> U131(active(X1), X2)
active(U132(X)) -> U132(active(X))
active(U141(X1, X2)) -> U141(active(X1), X2)
active(U142(X)) -> U142(active(X))
active(U151(X1, X2)) -> U151(active(X1), X2)
active(U152(X)) -> U152(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(U172(X1, X2, X3)) -> U172(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(U182(X1, X2)) -> U182(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, X3, X4)) -> U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) -> U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) -> U204(active(X1), X2)
active(U21(X1, X2, X3)) -> U21(active(X1), X2, X3)
active(U22(X1, X2)) -> U22(active(X1), X2)
active(U211(X1, X2)) -> U211(active(X1), X2)
active(U212(X1, X2)) -> U212(active(X1), X2)
active(U221(X1, X2, X3)) -> U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) -> U222(active(X1), X2, X3)
active(fst(X)) -> fst(active(X))
active(U31(X1, X2, X3)) -> U31(active(X1), X2, X3)
active(U32(X1, X2)) -> U32(active(X1), X2)
active(U41(X1, X2)) -> U41(active(X1), X2)
active(U42(X)) -> U42(active(X))
active(U51(X1, X2)) -> U51(active(X1), X2)
active(U52(X)) -> U52(active(X))
active(U61(X)) -> U61(active(X))
active(U71(X)) -> U71(active(X))
active(U81(X)) -> U81(active(X))
active(U91(X)) -> U91(active(X))
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) -> mark(U101(X1, X2))
U102(mark(X)) -> mark(U102(X))
U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3))
U111(mark(X)) -> mark(U111(X))
snd(mark(X)) -> mark(snd(X))
splitAt(mark(X1), X2) -> mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) -> mark(splitAt(X1, X2))
U121(mark(X)) -> mark(U121(X))
U131(mark(X1), X2) -> mark(U131(X1, X2))
U132(mark(X)) -> mark(U132(X))
U141(mark(X1), X2) -> mark(U141(X1, X2))
U142(mark(X)) -> mark(U142(X))
U151(mark(X1), X2) -> mark(U151(X1, X2))
U152(mark(X)) -> mark(U152(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))
U172(mark(X1), X2, X3) -> mark(U172(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))
U182(mark(X1), X2) -> mark(U182(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, X3, X4) -> mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) -> mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) -> mark(U204(X1, X2))
U21(mark(X1), X2, X3) -> mark(U21(X1, X2, X3))
U22(mark(X1), X2) -> mark(U22(X1, X2))
U211(mark(X1), X2) -> mark(U211(X1, X2))
U212(mark(X1), X2) -> mark(U212(X1, X2))
U221(mark(X1), X2, X3) -> mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) -> mark(U222(X1, X2, X3))
fst(mark(X)) -> mark(fst(X))
U31(mark(X1), X2, X3) -> mark(U31(X1, X2, X3))
U32(mark(X1), X2) -> mark(U32(X1, X2))
U41(mark(X1), X2) -> mark(U41(X1, X2))
U42(mark(X)) -> mark(U42(X))
U51(mark(X1), X2) -> mark(U51(X1, X2))
U52(mark(X)) -> mark(U52(X))
U61(mark(X)) -> mark(U61(X))
U71(mark(X)) -> mark(U71(X))
U81(mark(X)) -> mark(U81(X))
U91(mark(X)) -> mark(U91(X))
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)) -> U101(proper(X1), proper(X2))
proper(tt) -> ok(tt)
proper(U102(X)) -> U102(proper(X))
proper(isLNat(X)) -> isLNat(proper(X))
proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) -> U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) -> U111(proper(X))
proper(snd(X)) -> snd(proper(X))
proper(splitAt(X1, X2)) -> splitAt(proper(X1), proper(X2))
proper(U121(X)) -> U121(proper(X))
proper(U131(X1, X2)) -> U131(proper(X1), proper(X2))
proper(U132(X)) -> U132(proper(X))
proper(U141(X1, X2)) -> U141(proper(X1), proper(X2))
proper(U142(X)) -> U142(proper(X))
proper(U151(X1, X2)) -> U151(proper(X1), proper(X2))
proper(U152(X)) -> U152(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(U172(X1, X2, X3)) -> U172(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(U182(X1, X2)) -> U182(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, X3, X4)) -> U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) -> isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) -> U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) -> U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) -> U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) -> U22(proper(X1), proper(X2))
proper(U211(X1, X2)) -> U211(proper(X1), proper(X2))
proper(U212(X1, X2)) -> U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) -> U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) -> U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) -> fst(proper(X))
proper(U31(X1, X2, X3)) -> U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) -> U32(proper(X1), proper(X2))
proper(U41(X1, X2)) -> U41(proper(X1), proper(X2))
proper(U42(X)) -> U42(proper(X))
proper(U51(X1, X2)) -> U51(proper(X1), proper(X2))
proper(U52(X)) -> U52(proper(X))
proper(U61(X)) -> U61(proper(X))
proper(U71(X)) -> U71(proper(X))
proper(U81(X)) -> U81(proper(X))
proper(U91(X)) -> U91(proper(X))
proper(isPLNat(X)) -> isPLNat(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(U101(X1, X2))
U102(ok(X)) -> ok(U102(X))
isLNat(ok(X)) -> ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))
U111(ok(X)) -> ok(U111(X))
snd(ok(X)) -> ok(snd(X))
splitAt(ok(X1), ok(X2)) -> ok(splitAt(X1, X2))
U121(ok(X)) -> ok(U121(X))
U131(ok(X1), ok(X2)) -> ok(U131(X1, X2))
U132(ok(X)) -> ok(U132(X))
U141(ok(X1), ok(X2)) -> ok(U141(X1, X2))
U142(ok(X)) -> ok(U142(X))
U151(ok(X1), ok(X2)) -> ok(U151(X1, X2))
U152(ok(X)) -> ok(U152(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))
U172(ok(X1), ok(X2), ok(X3)) -> ok(U172(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))
U182(ok(X1), ok(X2)) -> ok(U182(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(X3), ok(X4)) -> ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) -> ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) -> ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) -> ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) -> ok(U22(X1, X2))
U211(ok(X1), ok(X2)) -> ok(U211(X1, X2))
U212(ok(X1), ok(X2)) -> ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) -> ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) -> ok(U222(X1, X2, X3))
fst(ok(X)) -> ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) -> ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) -> ok(U32(X1, X2))
U41(ok(X1), ok(X2)) -> ok(U41(X1, X2))
U42(ok(X)) -> ok(U42(X))
U51(ok(X1), ok(X2)) -> ok(U51(X1, X2))
U52(ok(X)) -> ok(U52(X))
U61(ok(X)) -> ok(U61(X))
U71(ok(X)) -> ok(U71(X))
U81(ok(X)) -> ok(U81(X))
U91(ok(X)) -> ok(U91(X))
isPLNat(ok(X)) -> ok(isPLNat(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))

Types:
active :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U12 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U172 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U182 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U203 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U204 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U22 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U212 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U222 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U32 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok -> top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat -> tt:mark:nil:0':ok


Lemmas:
U102(gen_tt:mark:nil:0':ok3_0(+(1, n5_0))) -> *4_0, rt in Omega(n5_0)
U12(gen_tt:mark:nil:0':ok3_0(+(1, n1128_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) -> *4_0, rt in Omega(n1128_0)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n9121_0))) -> *4_0, rt in Omega(n9121_0)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n10537_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n10537_0)
U132(gen_tt:mark:nil:0':ok3_0(+(1, n15953_0))) -> *4_0, rt in Omega(n15953_0)
U142(gen_tt:mark:nil:0':ok3_0(+(1, n17620_0))) -> *4_0, rt in Omega(n17620_0)
U152(gen_tt:mark:nil:0':ok3_0(+(1, n19388_0))) -> *4_0, rt in Omega(n19388_0)
cons(gen_tt:mark:nil:0':ok3_0(+(1, n21257_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n21257_0)
natsFrom(gen_tt:mark:nil:0':ok3_0(+(1, n27404_0))) -> *4_0, rt in Omega(n27404_0)
s(gen_tt:mark:nil:0':ok3_0(+(1, n29524_0))) -> *4_0, rt in Omega(n29524_0)
U172(gen_tt:mark:nil:0':ok3_0(+(1, n31745_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) -> *4_0, rt in Omega(n31745_0)
head(gen_tt:mark:nil:0':ok3_0(+(1, n43161_0))) -> *4_0, rt in Omega(n43161_0)
afterNth(gen_tt:mark:nil:0':ok3_0(+(1, n45682_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n45682_0)
U182(gen_tt:mark:nil:0':ok3_0(+(1, n53362_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n53362_0)
pair(gen_tt:mark:nil:0':ok3_0(+(1, n61149_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n61149_0)
U202(gen_tt:mark:nil:0':ok3_0(+(1, n69441_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c), gen_tt:mark:nil:0':ok3_0(d)) -> *4_0, rt in Omega(n69441_0)
U203(gen_tt:mark:nil:0':ok3_0(+(1, n88570_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c), gen_tt:mark:nil:0':ok3_0(d)) -> *4_0, rt in Omega(n88570_0)
U204(gen_tt:mark:nil:0':ok3_0(+(1, n108633_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n108633_0)
U22(gen_tt:mark:nil:0':ok3_0(+(1, n118036_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n118036_0)
U212(gen_tt:mark:nil:0':ok3_0(+(1, n127745_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n127745_0)


Generator Equations:
gen_tt:mark:nil:0':ok3_0(0) <=> tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) <=> mark(gen_tt:mark:nil:0':ok3_0(x))


The following defined symbols remain to be analysed:
U222, active, fst, U32, U42, U52, U11, U21, U31, U41, U51, U61, isPLNat, U71, U81, U91, U101, U111, U121, U131, U141, U151, U161, U171, U181, U191, U201, U211, U221, tail, take, sel, proper, top

They will be analysed ascendingly in the following order:
U222 < active
fst < active
U32 < active
U42 < active
U52 < active
U11 < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
isPLNat < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
U141 < active
U151 < active
U161 < active
U171 < active
U181 < active
U191 < active
U201 < active
U211 < active
U221 < active
tail < active
take < active
sel < active
active < top
U222 < proper
fst < proper
U32 < proper
U42 < proper
U52 < proper
U11 < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
isPLNat < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
U141 < proper
U151 < proper
U161 < proper
U171 < proper
U181 < proper
U191 < proper
U201 < proper
U211 < proper
U221 < proper
tail < proper
take < proper
sel < proper
proper < top

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

(69) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
U222(gen_tt:mark:nil:0':ok3_0(+(1, n137760_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) -> *4_0, rt in Omega(n137760_0)

Induction Base:
U222(gen_tt:mark:nil:0':ok3_0(+(1, 0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c))

Induction Step:
U222(gen_tt:mark:nil:0':ok3_0(+(1, +(n137760_0, 1))), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) ->_R^Omega(1)
mark(U222(gen_tt:mark:nil:0':ok3_0(+(1, n137760_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c))) ->_IH
mark(*4_0)

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
----------------------------------------

(70)
Obligation:
TRS:
Rules:
active(U101(tt, V2)) -> mark(U102(isLNat(V2)))
active(U102(tt)) -> mark(tt)
active(U11(tt, N, XS)) -> mark(U12(isLNat(XS), N, XS))
active(U111(tt)) -> mark(tt)
active(U12(tt, N, XS)) -> mark(snd(splitAt(N, XS)))
active(U121(tt)) -> mark(tt)
active(U131(tt, V2)) -> mark(U132(isLNat(V2)))
active(U132(tt)) -> mark(tt)
active(U141(tt, V2)) -> mark(U142(isLNat(V2)))
active(U142(tt)) -> mark(tt)
active(U151(tt, V2)) -> mark(U152(isLNat(V2)))
active(U152(tt)) -> mark(tt)
active(U161(tt, N)) -> mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) -> mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) -> mark(head(afterNth(N, XS)))
active(U181(tt, Y)) -> mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) -> mark(Y)
active(U191(tt, XS)) -> mark(pair(nil, XS))
active(U201(tt, N, X, XS)) -> mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) -> mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) -> mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) -> mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) -> mark(U22(isLNat(Y), X))
active(U211(tt, XS)) -> mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) -> mark(XS)
active(U22(tt, X)) -> mark(X)
active(U221(tt, N, XS)) -> mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) -> mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) -> mark(U32(isLNat(XS), N))
active(U32(tt, N)) -> mark(N)
active(U41(tt, V2)) -> mark(U42(isLNat(V2)))
active(U42(tt)) -> mark(tt)
active(U51(tt, V2)) -> mark(U52(isLNat(V2)))
active(U52(tt)) -> mark(tt)
active(U61(tt)) -> mark(tt)
active(U71(tt)) -> mark(tt)
active(U81(tt)) -> mark(tt)
active(U91(tt)) -> mark(tt)
active(afterNth(N, XS)) -> mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) -> mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) -> mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) -> mark(tt)
active(isLNat(afterNth(V1, V2))) -> mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) -> mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) -> mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) -> mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) -> mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) -> mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) -> mark(U101(isNatural(V1), V2))
active(isNatural(0')) -> mark(tt)
active(isNatural(head(V1))) -> mark(U111(isLNat(V1)))
active(isNatural(s(V1))) -> mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) -> mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) -> mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) -> mark(U151(isNatural(V1), V2))
active(natsFrom(N)) -> mark(U161(isNatural(N), N))
active(sel(N, XS)) -> mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) -> mark(U181(isLNat(X), Y))
active(splitAt(0', XS)) -> mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) -> mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) -> mark(U211(isNatural(N), XS))
active(take(N, XS)) -> mark(U221(isNatural(N), N, XS))
active(U101(X1, X2)) -> U101(active(X1), X2)
active(U102(X)) -> U102(active(X))
active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3)
active(U111(X)) -> U111(active(X))
active(snd(X)) -> snd(active(X))
active(splitAt(X1, X2)) -> splitAt(active(X1), X2)
active(splitAt(X1, X2)) -> splitAt(X1, active(X2))
active(U121(X)) -> U121(active(X))
active(U131(X1, X2)) -> U131(active(X1), X2)
active(U132(X)) -> U132(active(X))
active(U141(X1, X2)) -> U141(active(X1), X2)
active(U142(X)) -> U142(active(X))
active(U151(X1, X2)) -> U151(active(X1), X2)
active(U152(X)) -> U152(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(U172(X1, X2, X3)) -> U172(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(U182(X1, X2)) -> U182(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, X3, X4)) -> U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) -> U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) -> U204(active(X1), X2)
active(U21(X1, X2, X3)) -> U21(active(X1), X2, X3)
active(U22(X1, X2)) -> U22(active(X1), X2)
active(U211(X1, X2)) -> U211(active(X1), X2)
active(U212(X1, X2)) -> U212(active(X1), X2)
active(U221(X1, X2, X3)) -> U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) -> U222(active(X1), X2, X3)
active(fst(X)) -> fst(active(X))
active(U31(X1, X2, X3)) -> U31(active(X1), X2, X3)
active(U32(X1, X2)) -> U32(active(X1), X2)
active(U41(X1, X2)) -> U41(active(X1), X2)
active(U42(X)) -> U42(active(X))
active(U51(X1, X2)) -> U51(active(X1), X2)
active(U52(X)) -> U52(active(X))
active(U61(X)) -> U61(active(X))
active(U71(X)) -> U71(active(X))
active(U81(X)) -> U81(active(X))
active(U91(X)) -> U91(active(X))
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) -> mark(U101(X1, X2))
U102(mark(X)) -> mark(U102(X))
U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3))
U111(mark(X)) -> mark(U111(X))
snd(mark(X)) -> mark(snd(X))
splitAt(mark(X1), X2) -> mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) -> mark(splitAt(X1, X2))
U121(mark(X)) -> mark(U121(X))
U131(mark(X1), X2) -> mark(U131(X1, X2))
U132(mark(X)) -> mark(U132(X))
U141(mark(X1), X2) -> mark(U141(X1, X2))
U142(mark(X)) -> mark(U142(X))
U151(mark(X1), X2) -> mark(U151(X1, X2))
U152(mark(X)) -> mark(U152(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))
U172(mark(X1), X2, X3) -> mark(U172(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))
U182(mark(X1), X2) -> mark(U182(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, X3, X4) -> mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) -> mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) -> mark(U204(X1, X2))
U21(mark(X1), X2, X3) -> mark(U21(X1, X2, X3))
U22(mark(X1), X2) -> mark(U22(X1, X2))
U211(mark(X1), X2) -> mark(U211(X1, X2))
U212(mark(X1), X2) -> mark(U212(X1, X2))
U221(mark(X1), X2, X3) -> mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) -> mark(U222(X1, X2, X3))
fst(mark(X)) -> mark(fst(X))
U31(mark(X1), X2, X3) -> mark(U31(X1, X2, X3))
U32(mark(X1), X2) -> mark(U32(X1, X2))
U41(mark(X1), X2) -> mark(U41(X1, X2))
U42(mark(X)) -> mark(U42(X))
U51(mark(X1), X2) -> mark(U51(X1, X2))
U52(mark(X)) -> mark(U52(X))
U61(mark(X)) -> mark(U61(X))
U71(mark(X)) -> mark(U71(X))
U81(mark(X)) -> mark(U81(X))
U91(mark(X)) -> mark(U91(X))
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)) -> U101(proper(X1), proper(X2))
proper(tt) -> ok(tt)
proper(U102(X)) -> U102(proper(X))
proper(isLNat(X)) -> isLNat(proper(X))
proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) -> U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) -> U111(proper(X))
proper(snd(X)) -> snd(proper(X))
proper(splitAt(X1, X2)) -> splitAt(proper(X1), proper(X2))
proper(U121(X)) -> U121(proper(X))
proper(U131(X1, X2)) -> U131(proper(X1), proper(X2))
proper(U132(X)) -> U132(proper(X))
proper(U141(X1, X2)) -> U141(proper(X1), proper(X2))
proper(U142(X)) -> U142(proper(X))
proper(U151(X1, X2)) -> U151(proper(X1), proper(X2))
proper(U152(X)) -> U152(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(U172(X1, X2, X3)) -> U172(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(U182(X1, X2)) -> U182(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, X3, X4)) -> U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) -> isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) -> U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) -> U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) -> U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) -> U22(proper(X1), proper(X2))
proper(U211(X1, X2)) -> U211(proper(X1), proper(X2))
proper(U212(X1, X2)) -> U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) -> U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) -> U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) -> fst(proper(X))
proper(U31(X1, X2, X3)) -> U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) -> U32(proper(X1), proper(X2))
proper(U41(X1, X2)) -> U41(proper(X1), proper(X2))
proper(U42(X)) -> U42(proper(X))
proper(U51(X1, X2)) -> U51(proper(X1), proper(X2))
proper(U52(X)) -> U52(proper(X))
proper(U61(X)) -> U61(proper(X))
proper(U71(X)) -> U71(proper(X))
proper(U81(X)) -> U81(proper(X))
proper(U91(X)) -> U91(proper(X))
proper(isPLNat(X)) -> isPLNat(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(U101(X1, X2))
U102(ok(X)) -> ok(U102(X))
isLNat(ok(X)) -> ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))
U111(ok(X)) -> ok(U111(X))
snd(ok(X)) -> ok(snd(X))
splitAt(ok(X1), ok(X2)) -> ok(splitAt(X1, X2))
U121(ok(X)) -> ok(U121(X))
U131(ok(X1), ok(X2)) -> ok(U131(X1, X2))
U132(ok(X)) -> ok(U132(X))
U141(ok(X1), ok(X2)) -> ok(U141(X1, X2))
U142(ok(X)) -> ok(U142(X))
U151(ok(X1), ok(X2)) -> ok(U151(X1, X2))
U152(ok(X)) -> ok(U152(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))
U172(ok(X1), ok(X2), ok(X3)) -> ok(U172(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))
U182(ok(X1), ok(X2)) -> ok(U182(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(X3), ok(X4)) -> ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) -> ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) -> ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) -> ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) -> ok(U22(X1, X2))
U211(ok(X1), ok(X2)) -> ok(U211(X1, X2))
U212(ok(X1), ok(X2)) -> ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) -> ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) -> ok(U222(X1, X2, X3))
fst(ok(X)) -> ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) -> ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) -> ok(U32(X1, X2))
U41(ok(X1), ok(X2)) -> ok(U41(X1, X2))
U42(ok(X)) -> ok(U42(X))
U51(ok(X1), ok(X2)) -> ok(U51(X1, X2))
U52(ok(X)) -> ok(U52(X))
U61(ok(X)) -> ok(U61(X))
U71(ok(X)) -> ok(U71(X))
U81(ok(X)) -> ok(U81(X))
U91(ok(X)) -> ok(U91(X))
isPLNat(ok(X)) -> ok(isPLNat(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))

Types:
active :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U12 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U172 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U182 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U203 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U204 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U22 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U212 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U222 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U32 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok -> top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat -> tt:mark:nil:0':ok


Lemmas:
U102(gen_tt:mark:nil:0':ok3_0(+(1, n5_0))) -> *4_0, rt in Omega(n5_0)
U12(gen_tt:mark:nil:0':ok3_0(+(1, n1128_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) -> *4_0, rt in Omega(n1128_0)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n9121_0))) -> *4_0, rt in Omega(n9121_0)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n10537_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n10537_0)
U132(gen_tt:mark:nil:0':ok3_0(+(1, n15953_0))) -> *4_0, rt in Omega(n15953_0)
U142(gen_tt:mark:nil:0':ok3_0(+(1, n17620_0))) -> *4_0, rt in Omega(n17620_0)
U152(gen_tt:mark:nil:0':ok3_0(+(1, n19388_0))) -> *4_0, rt in Omega(n19388_0)
cons(gen_tt:mark:nil:0':ok3_0(+(1, n21257_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n21257_0)
natsFrom(gen_tt:mark:nil:0':ok3_0(+(1, n27404_0))) -> *4_0, rt in Omega(n27404_0)
s(gen_tt:mark:nil:0':ok3_0(+(1, n29524_0))) -> *4_0, rt in Omega(n29524_0)
U172(gen_tt:mark:nil:0':ok3_0(+(1, n31745_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) -> *4_0, rt in Omega(n31745_0)
head(gen_tt:mark:nil:0':ok3_0(+(1, n43161_0))) -> *4_0, rt in Omega(n43161_0)
afterNth(gen_tt:mark:nil:0':ok3_0(+(1, n45682_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n45682_0)
U182(gen_tt:mark:nil:0':ok3_0(+(1, n53362_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n53362_0)
pair(gen_tt:mark:nil:0':ok3_0(+(1, n61149_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n61149_0)
U202(gen_tt:mark:nil:0':ok3_0(+(1, n69441_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c), gen_tt:mark:nil:0':ok3_0(d)) -> *4_0, rt in Omega(n69441_0)
U203(gen_tt:mark:nil:0':ok3_0(+(1, n88570_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c), gen_tt:mark:nil:0':ok3_0(d)) -> *4_0, rt in Omega(n88570_0)
U204(gen_tt:mark:nil:0':ok3_0(+(1, n108633_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n108633_0)
U22(gen_tt:mark:nil:0':ok3_0(+(1, n118036_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n118036_0)
U212(gen_tt:mark:nil:0':ok3_0(+(1, n127745_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n127745_0)
U222(gen_tt:mark:nil:0':ok3_0(+(1, n137760_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) -> *4_0, rt in Omega(n137760_0)


Generator Equations:
gen_tt:mark:nil:0':ok3_0(0) <=> tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) <=> mark(gen_tt:mark:nil:0':ok3_0(x))


The following defined symbols remain to be analysed:
fst, active, U32, U42, U52, U11, U21, U31, U41, U51, U61, isPLNat, U71, U81, U91, U101, U111, U121, U131, U141, U151, U161, U171, U181, U191, U201, U211, U221, tail, take, sel, proper, top

They will be analysed ascendingly in the following order:
fst < active
U32 < active
U42 < active
U52 < active
U11 < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
isPLNat < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
U141 < active
U151 < active
U161 < active
U171 < active
U181 < active
U191 < active
U201 < active
U211 < active
U221 < active
tail < active
take < active
sel < active
active < top
fst < proper
U32 < proper
U42 < proper
U52 < proper
U11 < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
isPLNat < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
U141 < proper
U151 < proper
U161 < proper
U171 < proper
U181 < proper
U191 < proper
U201 < proper
U211 < proper
U221 < proper
tail < proper
take < proper
sel < proper
proper < top

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

(71) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
fst(gen_tt:mark:nil:0':ok3_0(+(1, n154384_0))) -> *4_0, rt in Omega(n154384_0)

Induction Base:
fst(gen_tt:mark:nil:0':ok3_0(+(1, 0)))

Induction Step:
fst(gen_tt:mark:nil:0':ok3_0(+(1, +(n154384_0, 1)))) ->_R^Omega(1)
mark(fst(gen_tt:mark:nil:0':ok3_0(+(1, n154384_0)))) ->_IH
mark(*4_0)

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
----------------------------------------

(72)
Obligation:
TRS:
Rules:
active(U101(tt, V2)) -> mark(U102(isLNat(V2)))
active(U102(tt)) -> mark(tt)
active(U11(tt, N, XS)) -> mark(U12(isLNat(XS), N, XS))
active(U111(tt)) -> mark(tt)
active(U12(tt, N, XS)) -> mark(snd(splitAt(N, XS)))
active(U121(tt)) -> mark(tt)
active(U131(tt, V2)) -> mark(U132(isLNat(V2)))
active(U132(tt)) -> mark(tt)
active(U141(tt, V2)) -> mark(U142(isLNat(V2)))
active(U142(tt)) -> mark(tt)
active(U151(tt, V2)) -> mark(U152(isLNat(V2)))
active(U152(tt)) -> mark(tt)
active(U161(tt, N)) -> mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) -> mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) -> mark(head(afterNth(N, XS)))
active(U181(tt, Y)) -> mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) -> mark(Y)
active(U191(tt, XS)) -> mark(pair(nil, XS))
active(U201(tt, N, X, XS)) -> mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) -> mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) -> mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) -> mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) -> mark(U22(isLNat(Y), X))
active(U211(tt, XS)) -> mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) -> mark(XS)
active(U22(tt, X)) -> mark(X)
active(U221(tt, N, XS)) -> mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) -> mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) -> mark(U32(isLNat(XS), N))
active(U32(tt, N)) -> mark(N)
active(U41(tt, V2)) -> mark(U42(isLNat(V2)))
active(U42(tt)) -> mark(tt)
active(U51(tt, V2)) -> mark(U52(isLNat(V2)))
active(U52(tt)) -> mark(tt)
active(U61(tt)) -> mark(tt)
active(U71(tt)) -> mark(tt)
active(U81(tt)) -> mark(tt)
active(U91(tt)) -> mark(tt)
active(afterNth(N, XS)) -> mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) -> mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) -> mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) -> mark(tt)
active(isLNat(afterNth(V1, V2))) -> mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) -> mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) -> mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) -> mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) -> mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) -> mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) -> mark(U101(isNatural(V1), V2))
active(isNatural(0')) -> mark(tt)
active(isNatural(head(V1))) -> mark(U111(isLNat(V1)))
active(isNatural(s(V1))) -> mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) -> mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) -> mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) -> mark(U151(isNatural(V1), V2))
active(natsFrom(N)) -> mark(U161(isNatural(N), N))
active(sel(N, XS)) -> mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) -> mark(U181(isLNat(X), Y))
active(splitAt(0', XS)) -> mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) -> mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) -> mark(U211(isNatural(N), XS))
active(take(N, XS)) -> mark(U221(isNatural(N), N, XS))
active(U101(X1, X2)) -> U101(active(X1), X2)
active(U102(X)) -> U102(active(X))
active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3)
active(U111(X)) -> U111(active(X))
active(snd(X)) -> snd(active(X))
active(splitAt(X1, X2)) -> splitAt(active(X1), X2)
active(splitAt(X1, X2)) -> splitAt(X1, active(X2))
active(U121(X)) -> U121(active(X))
active(U131(X1, X2)) -> U131(active(X1), X2)
active(U132(X)) -> U132(active(X))
active(U141(X1, X2)) -> U141(active(X1), X2)
active(U142(X)) -> U142(active(X))
active(U151(X1, X2)) -> U151(active(X1), X2)
active(U152(X)) -> U152(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(U172(X1, X2, X3)) -> U172(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(U182(X1, X2)) -> U182(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, X3, X4)) -> U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) -> U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) -> U204(active(X1), X2)
active(U21(X1, X2, X3)) -> U21(active(X1), X2, X3)
active(U22(X1, X2)) -> U22(active(X1), X2)
active(U211(X1, X2)) -> U211(active(X1), X2)
active(U212(X1, X2)) -> U212(active(X1), X2)
active(U221(X1, X2, X3)) -> U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) -> U222(active(X1), X2, X3)
active(fst(X)) -> fst(active(X))
active(U31(X1, X2, X3)) -> U31(active(X1), X2, X3)
active(U32(X1, X2)) -> U32(active(X1), X2)
active(U41(X1, X2)) -> U41(active(X1), X2)
active(U42(X)) -> U42(active(X))
active(U51(X1, X2)) -> U51(active(X1), X2)
active(U52(X)) -> U52(active(X))
active(U61(X)) -> U61(active(X))
active(U71(X)) -> U71(active(X))
active(U81(X)) -> U81(active(X))
active(U91(X)) -> U91(active(X))
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) -> mark(U101(X1, X2))
U102(mark(X)) -> mark(U102(X))
U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3))
U111(mark(X)) -> mark(U111(X))
snd(mark(X)) -> mark(snd(X))
splitAt(mark(X1), X2) -> mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) -> mark(splitAt(X1, X2))
U121(mark(X)) -> mark(U121(X))
U131(mark(X1), X2) -> mark(U131(X1, X2))
U132(mark(X)) -> mark(U132(X))
U141(mark(X1), X2) -> mark(U141(X1, X2))
U142(mark(X)) -> mark(U142(X))
U151(mark(X1), X2) -> mark(U151(X1, X2))
U152(mark(X)) -> mark(U152(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))
U172(mark(X1), X2, X3) -> mark(U172(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))
U182(mark(X1), X2) -> mark(U182(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, X3, X4) -> mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) -> mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) -> mark(U204(X1, X2))
U21(mark(X1), X2, X3) -> mark(U21(X1, X2, X3))
U22(mark(X1), X2) -> mark(U22(X1, X2))
U211(mark(X1), X2) -> mark(U211(X1, X2))
U212(mark(X1), X2) -> mark(U212(X1, X2))
U221(mark(X1), X2, X3) -> mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) -> mark(U222(X1, X2, X3))
fst(mark(X)) -> mark(fst(X))
U31(mark(X1), X2, X3) -> mark(U31(X1, X2, X3))
U32(mark(X1), X2) -> mark(U32(X1, X2))
U41(mark(X1), X2) -> mark(U41(X1, X2))
U42(mark(X)) -> mark(U42(X))
U51(mark(X1), X2) -> mark(U51(X1, X2))
U52(mark(X)) -> mark(U52(X))
U61(mark(X)) -> mark(U61(X))
U71(mark(X)) -> mark(U71(X))
U81(mark(X)) -> mark(U81(X))
U91(mark(X)) -> mark(U91(X))
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)) -> U101(proper(X1), proper(X2))
proper(tt) -> ok(tt)
proper(U102(X)) -> U102(proper(X))
proper(isLNat(X)) -> isLNat(proper(X))
proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) -> U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) -> U111(proper(X))
proper(snd(X)) -> snd(proper(X))
proper(splitAt(X1, X2)) -> splitAt(proper(X1), proper(X2))
proper(U121(X)) -> U121(proper(X))
proper(U131(X1, X2)) -> U131(proper(X1), proper(X2))
proper(U132(X)) -> U132(proper(X))
proper(U141(X1, X2)) -> U141(proper(X1), proper(X2))
proper(U142(X)) -> U142(proper(X))
proper(U151(X1, X2)) -> U151(proper(X1), proper(X2))
proper(U152(X)) -> U152(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(U172(X1, X2, X3)) -> U172(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(U182(X1, X2)) -> U182(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, X3, X4)) -> U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) -> isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) -> U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) -> U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) -> U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) -> U22(proper(X1), proper(X2))
proper(U211(X1, X2)) -> U211(proper(X1), proper(X2))
proper(U212(X1, X2)) -> U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) -> U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) -> U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) -> fst(proper(X))
proper(U31(X1, X2, X3)) -> U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) -> U32(proper(X1), proper(X2))
proper(U41(X1, X2)) -> U41(proper(X1), proper(X2))
proper(U42(X)) -> U42(proper(X))
proper(U51(X1, X2)) -> U51(proper(X1), proper(X2))
proper(U52(X)) -> U52(proper(X))
proper(U61(X)) -> U61(proper(X))
proper(U71(X)) -> U71(proper(X))
proper(U81(X)) -> U81(proper(X))
proper(U91(X)) -> U91(proper(X))
proper(isPLNat(X)) -> isPLNat(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(U101(X1, X2))
U102(ok(X)) -> ok(U102(X))
isLNat(ok(X)) -> ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))
U111(ok(X)) -> ok(U111(X))
snd(ok(X)) -> ok(snd(X))
splitAt(ok(X1), ok(X2)) -> ok(splitAt(X1, X2))
U121(ok(X)) -> ok(U121(X))
U131(ok(X1), ok(X2)) -> ok(U131(X1, X2))
U132(ok(X)) -> ok(U132(X))
U141(ok(X1), ok(X2)) -> ok(U141(X1, X2))
U142(ok(X)) -> ok(U142(X))
U151(ok(X1), ok(X2)) -> ok(U151(X1, X2))
U152(ok(X)) -> ok(U152(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))
U172(ok(X1), ok(X2), ok(X3)) -> ok(U172(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))
U182(ok(X1), ok(X2)) -> ok(U182(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(X3), ok(X4)) -> ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) -> ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) -> ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) -> ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) -> ok(U22(X1, X2))
U211(ok(X1), ok(X2)) -> ok(U211(X1, X2))
U212(ok(X1), ok(X2)) -> ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) -> ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) -> ok(U222(X1, X2, X3))
fst(ok(X)) -> ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) -> ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) -> ok(U32(X1, X2))
U41(ok(X1), ok(X2)) -> ok(U41(X1, X2))
U42(ok(X)) -> ok(U42(X))
U51(ok(X1), ok(X2)) -> ok(U51(X1, X2))
U52(ok(X)) -> ok(U52(X))
U61(ok(X)) -> ok(U61(X))
U71(ok(X)) -> ok(U71(X))
U81(ok(X)) -> ok(U81(X))
U91(ok(X)) -> ok(U91(X))
isPLNat(ok(X)) -> ok(isPLNat(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))

Types:
active :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U12 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U172 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U182 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U203 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U204 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U22 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U212 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U222 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U32 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok -> top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat -> tt:mark:nil:0':ok


Lemmas:
U102(gen_tt:mark:nil:0':ok3_0(+(1, n5_0))) -> *4_0, rt in Omega(n5_0)
U12(gen_tt:mark:nil:0':ok3_0(+(1, n1128_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) -> *4_0, rt in Omega(n1128_0)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n9121_0))) -> *4_0, rt in Omega(n9121_0)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n10537_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n10537_0)
U132(gen_tt:mark:nil:0':ok3_0(+(1, n15953_0))) -> *4_0, rt in Omega(n15953_0)
U142(gen_tt:mark:nil:0':ok3_0(+(1, n17620_0))) -> *4_0, rt in Omega(n17620_0)
U152(gen_tt:mark:nil:0':ok3_0(+(1, n19388_0))) -> *4_0, rt in Omega(n19388_0)
cons(gen_tt:mark:nil:0':ok3_0(+(1, n21257_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n21257_0)
natsFrom(gen_tt:mark:nil:0':ok3_0(+(1, n27404_0))) -> *4_0, rt in Omega(n27404_0)
s(gen_tt:mark:nil:0':ok3_0(+(1, n29524_0))) -> *4_0, rt in Omega(n29524_0)
U172(gen_tt:mark:nil:0':ok3_0(+(1, n31745_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) -> *4_0, rt in Omega(n31745_0)
head(gen_tt:mark:nil:0':ok3_0(+(1, n43161_0))) -> *4_0, rt in Omega(n43161_0)
afterNth(gen_tt:mark:nil:0':ok3_0(+(1, n45682_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n45682_0)
U182(gen_tt:mark:nil:0':ok3_0(+(1, n53362_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n53362_0)
pair(gen_tt:mark:nil:0':ok3_0(+(1, n61149_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n61149_0)
U202(gen_tt:mark:nil:0':ok3_0(+(1, n69441_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c), gen_tt:mark:nil:0':ok3_0(d)) -> *4_0, rt in Omega(n69441_0)
U203(gen_tt:mark:nil:0':ok3_0(+(1, n88570_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c), gen_tt:mark:nil:0':ok3_0(d)) -> *4_0, rt in Omega(n88570_0)
U204(gen_tt:mark:nil:0':ok3_0(+(1, n108633_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n108633_0)
U22(gen_tt:mark:nil:0':ok3_0(+(1, n118036_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n118036_0)
U212(gen_tt:mark:nil:0':ok3_0(+(1, n127745_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n127745_0)
U222(gen_tt:mark:nil:0':ok3_0(+(1, n137760_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) -> *4_0, rt in Omega(n137760_0)
fst(gen_tt:mark:nil:0':ok3_0(+(1, n154384_0))) -> *4_0, rt in Omega(n154384_0)


Generator Equations:
gen_tt:mark:nil:0':ok3_0(0) <=> tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) <=> mark(gen_tt:mark:nil:0':ok3_0(x))


The following defined symbols remain to be analysed:
U32, active, U42, U52, U11, U21, U31, U41, U51, U61, isPLNat, U71, U81, U91, U101, U111, U121, U131, U141, U151, U161, U171, U181, U191, U201, U211, U221, tail, take, sel, proper, top

They will be analysed ascendingly in the following order:
U32 < active
U42 < active
U52 < active
U11 < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
isPLNat < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
U141 < active
U151 < active
U161 < active
U171 < active
U181 < active
U191 < active
U201 < active
U211 < active
U221 < active
tail < active
take < active
sel < active
active < top
U32 < proper
U42 < proper
U52 < proper
U11 < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
isPLNat < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
U141 < proper
U151 < proper
U161 < proper
U171 < proper
U181 < proper
U191 < proper
U201 < proper
U211 < proper
U221 < proper
tail < proper
take < proper
sel < proper
proper < top

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

(73) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
U32(gen_tt:mark:nil:0':ok3_0(+(1, n158601_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n158601_0)

Induction Base:
U32(gen_tt:mark:nil:0':ok3_0(+(1, 0)), gen_tt:mark:nil:0':ok3_0(b))

Induction Step:
U32(gen_tt:mark:nil:0':ok3_0(+(1, +(n158601_0, 1))), gen_tt:mark:nil:0':ok3_0(b)) ->_R^Omega(1)
mark(U32(gen_tt:mark:nil:0':ok3_0(+(1, n158601_0)), gen_tt:mark:nil:0':ok3_0(b))) ->_IH
mark(*4_0)

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
----------------------------------------

(74)
Obligation:
TRS:
Rules:
active(U101(tt, V2)) -> mark(U102(isLNat(V2)))
active(U102(tt)) -> mark(tt)
active(U11(tt, N, XS)) -> mark(U12(isLNat(XS), N, XS))
active(U111(tt)) -> mark(tt)
active(U12(tt, N, XS)) -> mark(snd(splitAt(N, XS)))
active(U121(tt)) -> mark(tt)
active(U131(tt, V2)) -> mark(U132(isLNat(V2)))
active(U132(tt)) -> mark(tt)
active(U141(tt, V2)) -> mark(U142(isLNat(V2)))
active(U142(tt)) -> mark(tt)
active(U151(tt, V2)) -> mark(U152(isLNat(V2)))
active(U152(tt)) -> mark(tt)
active(U161(tt, N)) -> mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) -> mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) -> mark(head(afterNth(N, XS)))
active(U181(tt, Y)) -> mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) -> mark(Y)
active(U191(tt, XS)) -> mark(pair(nil, XS))
active(U201(tt, N, X, XS)) -> mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) -> mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) -> mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) -> mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) -> mark(U22(isLNat(Y), X))
active(U211(tt, XS)) -> mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) -> mark(XS)
active(U22(tt, X)) -> mark(X)
active(U221(tt, N, XS)) -> mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) -> mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) -> mark(U32(isLNat(XS), N))
active(U32(tt, N)) -> mark(N)
active(U41(tt, V2)) -> mark(U42(isLNat(V2)))
active(U42(tt)) -> mark(tt)
active(U51(tt, V2)) -> mark(U52(isLNat(V2)))
active(U52(tt)) -> mark(tt)
active(U61(tt)) -> mark(tt)
active(U71(tt)) -> mark(tt)
active(U81(tt)) -> mark(tt)
active(U91(tt)) -> mark(tt)
active(afterNth(N, XS)) -> mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) -> mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) -> mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) -> mark(tt)
active(isLNat(afterNth(V1, V2))) -> mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) -> mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) -> mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) -> mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) -> mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) -> mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) -> mark(U101(isNatural(V1), V2))
active(isNatural(0')) -> mark(tt)
active(isNatural(head(V1))) -> mark(U111(isLNat(V1)))
active(isNatural(s(V1))) -> mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) -> mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) -> mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) -> mark(U151(isNatural(V1), V2))
active(natsFrom(N)) -> mark(U161(isNatural(N), N))
active(sel(N, XS)) -> mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) -> mark(U181(isLNat(X), Y))
active(splitAt(0', XS)) -> mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) -> mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) -> mark(U211(isNatural(N), XS))
active(take(N, XS)) -> mark(U221(isNatural(N), N, XS))
active(U101(X1, X2)) -> U101(active(X1), X2)
active(U102(X)) -> U102(active(X))
active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3)
active(U111(X)) -> U111(active(X))
active(snd(X)) -> snd(active(X))
active(splitAt(X1, X2)) -> splitAt(active(X1), X2)
active(splitAt(X1, X2)) -> splitAt(X1, active(X2))
active(U121(X)) -> U121(active(X))
active(U131(X1, X2)) -> U131(active(X1), X2)
active(U132(X)) -> U132(active(X))
active(U141(X1, X2)) -> U141(active(X1), X2)
active(U142(X)) -> U142(active(X))
active(U151(X1, X2)) -> U151(active(X1), X2)
active(U152(X)) -> U152(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(U172(X1, X2, X3)) -> U172(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(U182(X1, X2)) -> U182(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, X3, X4)) -> U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) -> U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) -> U204(active(X1), X2)
active(U21(X1, X2, X3)) -> U21(active(X1), X2, X3)
active(U22(X1, X2)) -> U22(active(X1), X2)
active(U211(X1, X2)) -> U211(active(X1), X2)
active(U212(X1, X2)) -> U212(active(X1), X2)
active(U221(X1, X2, X3)) -> U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) -> U222(active(X1), X2, X3)
active(fst(X)) -> fst(active(X))
active(U31(X1, X2, X3)) -> U31(active(X1), X2, X3)
active(U32(X1, X2)) -> U32(active(X1), X2)
active(U41(X1, X2)) -> U41(active(X1), X2)
active(U42(X)) -> U42(active(X))
active(U51(X1, X2)) -> U51(active(X1), X2)
active(U52(X)) -> U52(active(X))
active(U61(X)) -> U61(active(X))
active(U71(X)) -> U71(active(X))
active(U81(X)) -> U81(active(X))
active(U91(X)) -> U91(active(X))
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) -> mark(U101(X1, X2))
U102(mark(X)) -> mark(U102(X))
U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3))
U111(mark(X)) -> mark(U111(X))
snd(mark(X)) -> mark(snd(X))
splitAt(mark(X1), X2) -> mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) -> mark(splitAt(X1, X2))
U121(mark(X)) -> mark(U121(X))
U131(mark(X1), X2) -> mark(U131(X1, X2))
U132(mark(X)) -> mark(U132(X))
U141(mark(X1), X2) -> mark(U141(X1, X2))
U142(mark(X)) -> mark(U142(X))
U151(mark(X1), X2) -> mark(U151(X1, X2))
U152(mark(X)) -> mark(U152(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))
U172(mark(X1), X2, X3) -> mark(U172(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))
U182(mark(X1), X2) -> mark(U182(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, X3, X4) -> mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) -> mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) -> mark(U204(X1, X2))
U21(mark(X1), X2, X3) -> mark(U21(X1, X2, X3))
U22(mark(X1), X2) -> mark(U22(X1, X2))
U211(mark(X1), X2) -> mark(U211(X1, X2))
U212(mark(X1), X2) -> mark(U212(X1, X2))
U221(mark(X1), X2, X3) -> mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) -> mark(U222(X1, X2, X3))
fst(mark(X)) -> mark(fst(X))
U31(mark(X1), X2, X3) -> mark(U31(X1, X2, X3))
U32(mark(X1), X2) -> mark(U32(X1, X2))
U41(mark(X1), X2) -> mark(U41(X1, X2))
U42(mark(X)) -> mark(U42(X))
U51(mark(X1), X2) -> mark(U51(X1, X2))
U52(mark(X)) -> mark(U52(X))
U61(mark(X)) -> mark(U61(X))
U71(mark(X)) -> mark(U71(X))
U81(mark(X)) -> mark(U81(X))
U91(mark(X)) -> mark(U91(X))
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)) -> U101(proper(X1), proper(X2))
proper(tt) -> ok(tt)
proper(U102(X)) -> U102(proper(X))
proper(isLNat(X)) -> isLNat(proper(X))
proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) -> U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) -> U111(proper(X))
proper(snd(X)) -> snd(proper(X))
proper(splitAt(X1, X2)) -> splitAt(proper(X1), proper(X2))
proper(U121(X)) -> U121(proper(X))
proper(U131(X1, X2)) -> U131(proper(X1), proper(X2))
proper(U132(X)) -> U132(proper(X))
proper(U141(X1, X2)) -> U141(proper(X1), proper(X2))
proper(U142(X)) -> U142(proper(X))
proper(U151(X1, X2)) -> U151(proper(X1), proper(X2))
proper(U152(X)) -> U152(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(U172(X1, X2, X3)) -> U172(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(U182(X1, X2)) -> U182(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, X3, X4)) -> U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) -> isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) -> U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) -> U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) -> U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) -> U22(proper(X1), proper(X2))
proper(U211(X1, X2)) -> U211(proper(X1), proper(X2))
proper(U212(X1, X2)) -> U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) -> U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) -> U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) -> fst(proper(X))
proper(U31(X1, X2, X3)) -> U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) -> U32(proper(X1), proper(X2))
proper(U41(X1, X2)) -> U41(proper(X1), proper(X2))
proper(U42(X)) -> U42(proper(X))
proper(U51(X1, X2)) -> U51(proper(X1), proper(X2))
proper(U52(X)) -> U52(proper(X))
proper(U61(X)) -> U61(proper(X))
proper(U71(X)) -> U71(proper(X))
proper(U81(X)) -> U81(proper(X))
proper(U91(X)) -> U91(proper(X))
proper(isPLNat(X)) -> isPLNat(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(U101(X1, X2))
U102(ok(X)) -> ok(U102(X))
isLNat(ok(X)) -> ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))
U111(ok(X)) -> ok(U111(X))
snd(ok(X)) -> ok(snd(X))
splitAt(ok(X1), ok(X2)) -> ok(splitAt(X1, X2))
U121(ok(X)) -> ok(U121(X))
U131(ok(X1), ok(X2)) -> ok(U131(X1, X2))
U132(ok(X)) -> ok(U132(X))
U141(ok(X1), ok(X2)) -> ok(U141(X1, X2))
U142(ok(X)) -> ok(U142(X))
U151(ok(X1), ok(X2)) -> ok(U151(X1, X2))
U152(ok(X)) -> ok(U152(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))
U172(ok(X1), ok(X2), ok(X3)) -> ok(U172(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))
U182(ok(X1), ok(X2)) -> ok(U182(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(X3), ok(X4)) -> ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) -> ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) -> ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) -> ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) -> ok(U22(X1, X2))
U211(ok(X1), ok(X2)) -> ok(U211(X1, X2))
U212(ok(X1), ok(X2)) -> ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) -> ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) -> ok(U222(X1, X2, X3))
fst(ok(X)) -> ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) -> ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) -> ok(U32(X1, X2))
U41(ok(X1), ok(X2)) -> ok(U41(X1, X2))
U42(ok(X)) -> ok(U42(X))
U51(ok(X1), ok(X2)) -> ok(U51(X1, X2))
U52(ok(X)) -> ok(U52(X))
U61(ok(X)) -> ok(U61(X))
U71(ok(X)) -> ok(U71(X))
U81(ok(X)) -> ok(U81(X))
U91(ok(X)) -> ok(U91(X))
isPLNat(ok(X)) -> ok(isPLNat(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))

Types:
active :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U12 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U172 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U182 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U203 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U204 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U22 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U212 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U222 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U32 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok -> top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat -> tt:mark:nil:0':ok


Lemmas:
U102(gen_tt:mark:nil:0':ok3_0(+(1, n5_0))) -> *4_0, rt in Omega(n5_0)
U12(gen_tt:mark:nil:0':ok3_0(+(1, n1128_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) -> *4_0, rt in Omega(n1128_0)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n9121_0))) -> *4_0, rt in Omega(n9121_0)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n10537_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n10537_0)
U132(gen_tt:mark:nil:0':ok3_0(+(1, n15953_0))) -> *4_0, rt in Omega(n15953_0)
U142(gen_tt:mark:nil:0':ok3_0(+(1, n17620_0))) -> *4_0, rt in Omega(n17620_0)
U152(gen_tt:mark:nil:0':ok3_0(+(1, n19388_0))) -> *4_0, rt in Omega(n19388_0)
cons(gen_tt:mark:nil:0':ok3_0(+(1, n21257_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n21257_0)
natsFrom(gen_tt:mark:nil:0':ok3_0(+(1, n27404_0))) -> *4_0, rt in Omega(n27404_0)
s(gen_tt:mark:nil:0':ok3_0(+(1, n29524_0))) -> *4_0, rt in Omega(n29524_0)
U172(gen_tt:mark:nil:0':ok3_0(+(1, n31745_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) -> *4_0, rt in Omega(n31745_0)
head(gen_tt:mark:nil:0':ok3_0(+(1, n43161_0))) -> *4_0, rt in Omega(n43161_0)
afterNth(gen_tt:mark:nil:0':ok3_0(+(1, n45682_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n45682_0)
U182(gen_tt:mark:nil:0':ok3_0(+(1, n53362_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n53362_0)
pair(gen_tt:mark:nil:0':ok3_0(+(1, n61149_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n61149_0)
U202(gen_tt:mark:nil:0':ok3_0(+(1, n69441_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c), gen_tt:mark:nil:0':ok3_0(d)) -> *4_0, rt in Omega(n69441_0)
U203(gen_tt:mark:nil:0':ok3_0(+(1, n88570_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c), gen_tt:mark:nil:0':ok3_0(d)) -> *4_0, rt in Omega(n88570_0)
U204(gen_tt:mark:nil:0':ok3_0(+(1, n108633_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n108633_0)
U22(gen_tt:mark:nil:0':ok3_0(+(1, n118036_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n118036_0)
U212(gen_tt:mark:nil:0':ok3_0(+(1, n127745_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n127745_0)
U222(gen_tt:mark:nil:0':ok3_0(+(1, n137760_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) -> *4_0, rt in Omega(n137760_0)
fst(gen_tt:mark:nil:0':ok3_0(+(1, n154384_0))) -> *4_0, rt in Omega(n154384_0)
U32(gen_tt:mark:nil:0':ok3_0(+(1, n158601_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n158601_0)


Generator Equations:
gen_tt:mark:nil:0':ok3_0(0) <=> tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) <=> mark(gen_tt:mark:nil:0':ok3_0(x))


The following defined symbols remain to be analysed:
U42, active, U52, U11, U21, U31, U41, U51, U61, isPLNat, U71, U81, U91, U101, U111, U121, U131, U141, U151, U161, U171, U181, U191, U201, U211, U221, tail, take, sel, proper, top

They will be analysed ascendingly in the following order:
U42 < active
U52 < active
U11 < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
isPLNat < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
U141 < active
U151 < active
U161 < active
U171 < active
U181 < active
U191 < active
U201 < active
U211 < active
U221 < active
tail < active
take < active
sel < active
active < top
U42 < proper
U52 < proper
U11 < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
isPLNat < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
U141 < proper
U151 < proper
U161 < proper
U171 < proper
U181 < proper
U191 < proper
U201 < proper
U211 < proper
U221 < proper
tail < proper
take < proper
sel < proper
proper < top

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

(75) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
U42(gen_tt:mark:nil:0':ok3_0(+(1, n169534_0))) -> *4_0, rt in Omega(n169534_0)

Induction Base:
U42(gen_tt:mark:nil:0':ok3_0(+(1, 0)))

Induction Step:
U42(gen_tt:mark:nil:0':ok3_0(+(1, +(n169534_0, 1)))) ->_R^Omega(1)
mark(U42(gen_tt:mark:nil:0':ok3_0(+(1, n169534_0)))) ->_IH
mark(*4_0)

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
----------------------------------------

(76)
Obligation:
TRS:
Rules:
active(U101(tt, V2)) -> mark(U102(isLNat(V2)))
active(U102(tt)) -> mark(tt)
active(U11(tt, N, XS)) -> mark(U12(isLNat(XS), N, XS))
active(U111(tt)) -> mark(tt)
active(U12(tt, N, XS)) -> mark(snd(splitAt(N, XS)))
active(U121(tt)) -> mark(tt)
active(U131(tt, V2)) -> mark(U132(isLNat(V2)))
active(U132(tt)) -> mark(tt)
active(U141(tt, V2)) -> mark(U142(isLNat(V2)))
active(U142(tt)) -> mark(tt)
active(U151(tt, V2)) -> mark(U152(isLNat(V2)))
active(U152(tt)) -> mark(tt)
active(U161(tt, N)) -> mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) -> mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) -> mark(head(afterNth(N, XS)))
active(U181(tt, Y)) -> mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) -> mark(Y)
active(U191(tt, XS)) -> mark(pair(nil, XS))
active(U201(tt, N, X, XS)) -> mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) -> mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) -> mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) -> mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) -> mark(U22(isLNat(Y), X))
active(U211(tt, XS)) -> mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) -> mark(XS)
active(U22(tt, X)) -> mark(X)
active(U221(tt, N, XS)) -> mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) -> mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) -> mark(U32(isLNat(XS), N))
active(U32(tt, N)) -> mark(N)
active(U41(tt, V2)) -> mark(U42(isLNat(V2)))
active(U42(tt)) -> mark(tt)
active(U51(tt, V2)) -> mark(U52(isLNat(V2)))
active(U52(tt)) -> mark(tt)
active(U61(tt)) -> mark(tt)
active(U71(tt)) -> mark(tt)
active(U81(tt)) -> mark(tt)
active(U91(tt)) -> mark(tt)
active(afterNth(N, XS)) -> mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) -> mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) -> mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) -> mark(tt)
active(isLNat(afterNth(V1, V2))) -> mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) -> mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) -> mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) -> mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) -> mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) -> mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) -> mark(U101(isNatural(V1), V2))
active(isNatural(0')) -> mark(tt)
active(isNatural(head(V1))) -> mark(U111(isLNat(V1)))
active(isNatural(s(V1))) -> mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) -> mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) -> mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) -> mark(U151(isNatural(V1), V2))
active(natsFrom(N)) -> mark(U161(isNatural(N), N))
active(sel(N, XS)) -> mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) -> mark(U181(isLNat(X), Y))
active(splitAt(0', XS)) -> mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) -> mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) -> mark(U211(isNatural(N), XS))
active(take(N, XS)) -> mark(U221(isNatural(N), N, XS))
active(U101(X1, X2)) -> U101(active(X1), X2)
active(U102(X)) -> U102(active(X))
active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3)
active(U111(X)) -> U111(active(X))
active(snd(X)) -> snd(active(X))
active(splitAt(X1, X2)) -> splitAt(active(X1), X2)
active(splitAt(X1, X2)) -> splitAt(X1, active(X2))
active(U121(X)) -> U121(active(X))
active(U131(X1, X2)) -> U131(active(X1), X2)
active(U132(X)) -> U132(active(X))
active(U141(X1, X2)) -> U141(active(X1), X2)
active(U142(X)) -> U142(active(X))
active(U151(X1, X2)) -> U151(active(X1), X2)
active(U152(X)) -> U152(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(U172(X1, X2, X3)) -> U172(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(U182(X1, X2)) -> U182(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, X3, X4)) -> U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) -> U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) -> U204(active(X1), X2)
active(U21(X1, X2, X3)) -> U21(active(X1), X2, X3)
active(U22(X1, X2)) -> U22(active(X1), X2)
active(U211(X1, X2)) -> U211(active(X1), X2)
active(U212(X1, X2)) -> U212(active(X1), X2)
active(U221(X1, X2, X3)) -> U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) -> U222(active(X1), X2, X3)
active(fst(X)) -> fst(active(X))
active(U31(X1, X2, X3)) -> U31(active(X1), X2, X3)
active(U32(X1, X2)) -> U32(active(X1), X2)
active(U41(X1, X2)) -> U41(active(X1), X2)
active(U42(X)) -> U42(active(X))
active(U51(X1, X2)) -> U51(active(X1), X2)
active(U52(X)) -> U52(active(X))
active(U61(X)) -> U61(active(X))
active(U71(X)) -> U71(active(X))
active(U81(X)) -> U81(active(X))
active(U91(X)) -> U91(active(X))
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) -> mark(U101(X1, X2))
U102(mark(X)) -> mark(U102(X))
U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3))
U111(mark(X)) -> mark(U111(X))
snd(mark(X)) -> mark(snd(X))
splitAt(mark(X1), X2) -> mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) -> mark(splitAt(X1, X2))
U121(mark(X)) -> mark(U121(X))
U131(mark(X1), X2) -> mark(U131(X1, X2))
U132(mark(X)) -> mark(U132(X))
U141(mark(X1), X2) -> mark(U141(X1, X2))
U142(mark(X)) -> mark(U142(X))
U151(mark(X1), X2) -> mark(U151(X1, X2))
U152(mark(X)) -> mark(U152(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))
U172(mark(X1), X2, X3) -> mark(U172(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))
U182(mark(X1), X2) -> mark(U182(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, X3, X4) -> mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) -> mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) -> mark(U204(X1, X2))
U21(mark(X1), X2, X3) -> mark(U21(X1, X2, X3))
U22(mark(X1), X2) -> mark(U22(X1, X2))
U211(mark(X1), X2) -> mark(U211(X1, X2))
U212(mark(X1), X2) -> mark(U212(X1, X2))
U221(mark(X1), X2, X3) -> mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) -> mark(U222(X1, X2, X3))
fst(mark(X)) -> mark(fst(X))
U31(mark(X1), X2, X3) -> mark(U31(X1, X2, X3))
U32(mark(X1), X2) -> mark(U32(X1, X2))
U41(mark(X1), X2) -> mark(U41(X1, X2))
U42(mark(X)) -> mark(U42(X))
U51(mark(X1), X2) -> mark(U51(X1, X2))
U52(mark(X)) -> mark(U52(X))
U61(mark(X)) -> mark(U61(X))
U71(mark(X)) -> mark(U71(X))
U81(mark(X)) -> mark(U81(X))
U91(mark(X)) -> mark(U91(X))
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)) -> U101(proper(X1), proper(X2))
proper(tt) -> ok(tt)
proper(U102(X)) -> U102(proper(X))
proper(isLNat(X)) -> isLNat(proper(X))
proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) -> U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) -> U111(proper(X))
proper(snd(X)) -> snd(proper(X))
proper(splitAt(X1, X2)) -> splitAt(proper(X1), proper(X2))
proper(U121(X)) -> U121(proper(X))
proper(U131(X1, X2)) -> U131(proper(X1), proper(X2))
proper(U132(X)) -> U132(proper(X))
proper(U141(X1, X2)) -> U141(proper(X1), proper(X2))
proper(U142(X)) -> U142(proper(X))
proper(U151(X1, X2)) -> U151(proper(X1), proper(X2))
proper(U152(X)) -> U152(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(U172(X1, X2, X3)) -> U172(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(U182(X1, X2)) -> U182(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, X3, X4)) -> U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) -> isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) -> U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) -> U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) -> U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) -> U22(proper(X1), proper(X2))
proper(U211(X1, X2)) -> U211(proper(X1), proper(X2))
proper(U212(X1, X2)) -> U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) -> U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) -> U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) -> fst(proper(X))
proper(U31(X1, X2, X3)) -> U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) -> U32(proper(X1), proper(X2))
proper(U41(X1, X2)) -> U41(proper(X1), proper(X2))
proper(U42(X)) -> U42(proper(X))
proper(U51(X1, X2)) -> U51(proper(X1), proper(X2))
proper(U52(X)) -> U52(proper(X))
proper(U61(X)) -> U61(proper(X))
proper(U71(X)) -> U71(proper(X))
proper(U81(X)) -> U81(proper(X))
proper(U91(X)) -> U91(proper(X))
proper(isPLNat(X)) -> isPLNat(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(U101(X1, X2))
U102(ok(X)) -> ok(U102(X))
isLNat(ok(X)) -> ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))
U111(ok(X)) -> ok(U111(X))
snd(ok(X)) -> ok(snd(X))
splitAt(ok(X1), ok(X2)) -> ok(splitAt(X1, X2))
U121(ok(X)) -> ok(U121(X))
U131(ok(X1), ok(X2)) -> ok(U131(X1, X2))
U132(ok(X)) -> ok(U132(X))
U141(ok(X1), ok(X2)) -> ok(U141(X1, X2))
U142(ok(X)) -> ok(U142(X))
U151(ok(X1), ok(X2)) -> ok(U151(X1, X2))
U152(ok(X)) -> ok(U152(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))
U172(ok(X1), ok(X2), ok(X3)) -> ok(U172(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))
U182(ok(X1), ok(X2)) -> ok(U182(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(X3), ok(X4)) -> ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) -> ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) -> ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) -> ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) -> ok(U22(X1, X2))
U211(ok(X1), ok(X2)) -> ok(U211(X1, X2))
U212(ok(X1), ok(X2)) -> ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) -> ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) -> ok(U222(X1, X2, X3))
fst(ok(X)) -> ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) -> ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) -> ok(U32(X1, X2))
U41(ok(X1), ok(X2)) -> ok(U41(X1, X2))
U42(ok(X)) -> ok(U42(X))
U51(ok(X1), ok(X2)) -> ok(U51(X1, X2))
U52(ok(X)) -> ok(U52(X))
U61(ok(X)) -> ok(U61(X))
U71(ok(X)) -> ok(U71(X))
U81(ok(X)) -> ok(U81(X))
U91(ok(X)) -> ok(U91(X))
isPLNat(ok(X)) -> ok(isPLNat(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))

Types:
active :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U12 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U172 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U182 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U203 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U204 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U22 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U212 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U222 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U32 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok -> top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat -> tt:mark:nil:0':ok


Lemmas:
U102(gen_tt:mark:nil:0':ok3_0(+(1, n5_0))) -> *4_0, rt in Omega(n5_0)
U12(gen_tt:mark:nil:0':ok3_0(+(1, n1128_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) -> *4_0, rt in Omega(n1128_0)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n9121_0))) -> *4_0, rt in Omega(n9121_0)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n10537_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n10537_0)
U132(gen_tt:mark:nil:0':ok3_0(+(1, n15953_0))) -> *4_0, rt in Omega(n15953_0)
U142(gen_tt:mark:nil:0':ok3_0(+(1, n17620_0))) -> *4_0, rt in Omega(n17620_0)
U152(gen_tt:mark:nil:0':ok3_0(+(1, n19388_0))) -> *4_0, rt in Omega(n19388_0)
cons(gen_tt:mark:nil:0':ok3_0(+(1, n21257_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n21257_0)
natsFrom(gen_tt:mark:nil:0':ok3_0(+(1, n27404_0))) -> *4_0, rt in Omega(n27404_0)
s(gen_tt:mark:nil:0':ok3_0(+(1, n29524_0))) -> *4_0, rt in Omega(n29524_0)
U172(gen_tt:mark:nil:0':ok3_0(+(1, n31745_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) -> *4_0, rt in Omega(n31745_0)
head(gen_tt:mark:nil:0':ok3_0(+(1, n43161_0))) -> *4_0, rt in Omega(n43161_0)
afterNth(gen_tt:mark:nil:0':ok3_0(+(1, n45682_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n45682_0)
U182(gen_tt:mark:nil:0':ok3_0(+(1, n53362_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n53362_0)
pair(gen_tt:mark:nil:0':ok3_0(+(1, n61149_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n61149_0)
U202(gen_tt:mark:nil:0':ok3_0(+(1, n69441_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c), gen_tt:mark:nil:0':ok3_0(d)) -> *4_0, rt in Omega(n69441_0)
U203(gen_tt:mark:nil:0':ok3_0(+(1, n88570_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c), gen_tt:mark:nil:0':ok3_0(d)) -> *4_0, rt in Omega(n88570_0)
U204(gen_tt:mark:nil:0':ok3_0(+(1, n108633_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n108633_0)
U22(gen_tt:mark:nil:0':ok3_0(+(1, n118036_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n118036_0)
U212(gen_tt:mark:nil:0':ok3_0(+(1, n127745_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n127745_0)
U222(gen_tt:mark:nil:0':ok3_0(+(1, n137760_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) -> *4_0, rt in Omega(n137760_0)
fst(gen_tt:mark:nil:0':ok3_0(+(1, n154384_0))) -> *4_0, rt in Omega(n154384_0)
U32(gen_tt:mark:nil:0':ok3_0(+(1, n158601_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n158601_0)
U42(gen_tt:mark:nil:0':ok3_0(+(1, n169534_0))) -> *4_0, rt in Omega(n169534_0)


Generator Equations:
gen_tt:mark:nil:0':ok3_0(0) <=> tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) <=> mark(gen_tt:mark:nil:0':ok3_0(x))


The following defined symbols remain to be analysed:
U52, active, U11, U21, U31, U41, U51, U61, isPLNat, U71, U81, U91, U101, U111, U121, U131, U141, U151, U161, U171, U181, U191, U201, U211, U221, tail, take, sel, proper, top

They will be analysed ascendingly in the following order:
U52 < active
U11 < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
isPLNat < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
U141 < active
U151 < active
U161 < active
U171 < active
U181 < active
U191 < active
U201 < active
U211 < active
U221 < active
tail < active
take < active
sel < active
active < top
U52 < proper
U11 < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
isPLNat < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
U141 < proper
U151 < proper
U161 < proper
U171 < proper
U181 < proper
U191 < proper
U201 < proper
U211 < proper
U221 < proper
tail < proper
take < proper
sel < proper
proper < top

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

(77) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
U52(gen_tt:mark:nil:0':ok3_0(+(1, n174002_0))) -> *4_0, rt in Omega(n174002_0)

Induction Base:
U52(gen_tt:mark:nil:0':ok3_0(+(1, 0)))

Induction Step:
U52(gen_tt:mark:nil:0':ok3_0(+(1, +(n174002_0, 1)))) ->_R^Omega(1)
mark(U52(gen_tt:mark:nil:0':ok3_0(+(1, n174002_0)))) ->_IH
mark(*4_0)

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
----------------------------------------

(78)
Obligation:
TRS:
Rules:
active(U101(tt, V2)) -> mark(U102(isLNat(V2)))
active(U102(tt)) -> mark(tt)
active(U11(tt, N, XS)) -> mark(U12(isLNat(XS), N, XS))
active(U111(tt)) -> mark(tt)
active(U12(tt, N, XS)) -> mark(snd(splitAt(N, XS)))
active(U121(tt)) -> mark(tt)
active(U131(tt, V2)) -> mark(U132(isLNat(V2)))
active(U132(tt)) -> mark(tt)
active(U141(tt, V2)) -> mark(U142(isLNat(V2)))
active(U142(tt)) -> mark(tt)
active(U151(tt, V2)) -> mark(U152(isLNat(V2)))
active(U152(tt)) -> mark(tt)
active(U161(tt, N)) -> mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) -> mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) -> mark(head(afterNth(N, XS)))
active(U181(tt, Y)) -> mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) -> mark(Y)
active(U191(tt, XS)) -> mark(pair(nil, XS))
active(U201(tt, N, X, XS)) -> mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) -> mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) -> mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) -> mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) -> mark(U22(isLNat(Y), X))
active(U211(tt, XS)) -> mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) -> mark(XS)
active(U22(tt, X)) -> mark(X)
active(U221(tt, N, XS)) -> mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) -> mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) -> mark(U32(isLNat(XS), N))
active(U32(tt, N)) -> mark(N)
active(U41(tt, V2)) -> mark(U42(isLNat(V2)))
active(U42(tt)) -> mark(tt)
active(U51(tt, V2)) -> mark(U52(isLNat(V2)))
active(U52(tt)) -> mark(tt)
active(U61(tt)) -> mark(tt)
active(U71(tt)) -> mark(tt)
active(U81(tt)) -> mark(tt)
active(U91(tt)) -> mark(tt)
active(afterNth(N, XS)) -> mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) -> mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) -> mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) -> mark(tt)
active(isLNat(afterNth(V1, V2))) -> mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) -> mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) -> mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) -> mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) -> mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) -> mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) -> mark(U101(isNatural(V1), V2))
active(isNatural(0')) -> mark(tt)
active(isNatural(head(V1))) -> mark(U111(isLNat(V1)))
active(isNatural(s(V1))) -> mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) -> mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) -> mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) -> mark(U151(isNatural(V1), V2))
active(natsFrom(N)) -> mark(U161(isNatural(N), N))
active(sel(N, XS)) -> mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) -> mark(U181(isLNat(X), Y))
active(splitAt(0', XS)) -> mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) -> mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) -> mark(U211(isNatural(N), XS))
active(take(N, XS)) -> mark(U221(isNatural(N), N, XS))
active(U101(X1, X2)) -> U101(active(X1), X2)
active(U102(X)) -> U102(active(X))
active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3)
active(U111(X)) -> U111(active(X))
active(snd(X)) -> snd(active(X))
active(splitAt(X1, X2)) -> splitAt(active(X1), X2)
active(splitAt(X1, X2)) -> splitAt(X1, active(X2))
active(U121(X)) -> U121(active(X))
active(U131(X1, X2)) -> U131(active(X1), X2)
active(U132(X)) -> U132(active(X))
active(U141(X1, X2)) -> U141(active(X1), X2)
active(U142(X)) -> U142(active(X))
active(U151(X1, X2)) -> U151(active(X1), X2)
active(U152(X)) -> U152(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(U172(X1, X2, X3)) -> U172(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(U182(X1, X2)) -> U182(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, X3, X4)) -> U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) -> U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) -> U204(active(X1), X2)
active(U21(X1, X2, X3)) -> U21(active(X1), X2, X3)
active(U22(X1, X2)) -> U22(active(X1), X2)
active(U211(X1, X2)) -> U211(active(X1), X2)
active(U212(X1, X2)) -> U212(active(X1), X2)
active(U221(X1, X2, X3)) -> U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) -> U222(active(X1), X2, X3)
active(fst(X)) -> fst(active(X))
active(U31(X1, X2, X3)) -> U31(active(X1), X2, X3)
active(U32(X1, X2)) -> U32(active(X1), X2)
active(U41(X1, X2)) -> U41(active(X1), X2)
active(U42(X)) -> U42(active(X))
active(U51(X1, X2)) -> U51(active(X1), X2)
active(U52(X)) -> U52(active(X))
active(U61(X)) -> U61(active(X))
active(U71(X)) -> U71(active(X))
active(U81(X)) -> U81(active(X))
active(U91(X)) -> U91(active(X))
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) -> mark(U101(X1, X2))
U102(mark(X)) -> mark(U102(X))
U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3))
U111(mark(X)) -> mark(U111(X))
snd(mark(X)) -> mark(snd(X))
splitAt(mark(X1), X2) -> mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) -> mark(splitAt(X1, X2))
U121(mark(X)) -> mark(U121(X))
U131(mark(X1), X2) -> mark(U131(X1, X2))
U132(mark(X)) -> mark(U132(X))
U141(mark(X1), X2) -> mark(U141(X1, X2))
U142(mark(X)) -> mark(U142(X))
U151(mark(X1), X2) -> mark(U151(X1, X2))
U152(mark(X)) -> mark(U152(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))
U172(mark(X1), X2, X3) -> mark(U172(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))
U182(mark(X1), X2) -> mark(U182(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, X3, X4) -> mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) -> mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) -> mark(U204(X1, X2))
U21(mark(X1), X2, X3) -> mark(U21(X1, X2, X3))
U22(mark(X1), X2) -> mark(U22(X1, X2))
U211(mark(X1), X2) -> mark(U211(X1, X2))
U212(mark(X1), X2) -> mark(U212(X1, X2))
U221(mark(X1), X2, X3) -> mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) -> mark(U222(X1, X2, X3))
fst(mark(X)) -> mark(fst(X))
U31(mark(X1), X2, X3) -> mark(U31(X1, X2, X3))
U32(mark(X1), X2) -> mark(U32(X1, X2))
U41(mark(X1), X2) -> mark(U41(X1, X2))
U42(mark(X)) -> mark(U42(X))
U51(mark(X1), X2) -> mark(U51(X1, X2))
U52(mark(X)) -> mark(U52(X))
U61(mark(X)) -> mark(U61(X))
U71(mark(X)) -> mark(U71(X))
U81(mark(X)) -> mark(U81(X))
U91(mark(X)) -> mark(U91(X))
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)) -> U101(proper(X1), proper(X2))
proper(tt) -> ok(tt)
proper(U102(X)) -> U102(proper(X))
proper(isLNat(X)) -> isLNat(proper(X))
proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) -> U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) -> U111(proper(X))
proper(snd(X)) -> snd(proper(X))
proper(splitAt(X1, X2)) -> splitAt(proper(X1), proper(X2))
proper(U121(X)) -> U121(proper(X))
proper(U131(X1, X2)) -> U131(proper(X1), proper(X2))
proper(U132(X)) -> U132(proper(X))
proper(U141(X1, X2)) -> U141(proper(X1), proper(X2))
proper(U142(X)) -> U142(proper(X))
proper(U151(X1, X2)) -> U151(proper(X1), proper(X2))
proper(U152(X)) -> U152(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(U172(X1, X2, X3)) -> U172(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(U182(X1, X2)) -> U182(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, X3, X4)) -> U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) -> isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) -> U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) -> U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) -> U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) -> U22(proper(X1), proper(X2))
proper(U211(X1, X2)) -> U211(proper(X1), proper(X2))
proper(U212(X1, X2)) -> U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) -> U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) -> U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) -> fst(proper(X))
proper(U31(X1, X2, X3)) -> U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) -> U32(proper(X1), proper(X2))
proper(U41(X1, X2)) -> U41(proper(X1), proper(X2))
proper(U42(X)) -> U42(proper(X))
proper(U51(X1, X2)) -> U51(proper(X1), proper(X2))
proper(U52(X)) -> U52(proper(X))
proper(U61(X)) -> U61(proper(X))
proper(U71(X)) -> U71(proper(X))
proper(U81(X)) -> U81(proper(X))
proper(U91(X)) -> U91(proper(X))
proper(isPLNat(X)) -> isPLNat(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(U101(X1, X2))
U102(ok(X)) -> ok(U102(X))
isLNat(ok(X)) -> ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))
U111(ok(X)) -> ok(U111(X))
snd(ok(X)) -> ok(snd(X))
splitAt(ok(X1), ok(X2)) -> ok(splitAt(X1, X2))
U121(ok(X)) -> ok(U121(X))
U131(ok(X1), ok(X2)) -> ok(U131(X1, X2))
U132(ok(X)) -> ok(U132(X))
U141(ok(X1), ok(X2)) -> ok(U141(X1, X2))
U142(ok(X)) -> ok(U142(X))
U151(ok(X1), ok(X2)) -> ok(U151(X1, X2))
U152(ok(X)) -> ok(U152(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))
U172(ok(X1), ok(X2), ok(X3)) -> ok(U172(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))
U182(ok(X1), ok(X2)) -> ok(U182(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(X3), ok(X4)) -> ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) -> ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) -> ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) -> ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) -> ok(U22(X1, X2))
U211(ok(X1), ok(X2)) -> ok(U211(X1, X2))
U212(ok(X1), ok(X2)) -> ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) -> ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) -> ok(U222(X1, X2, X3))
fst(ok(X)) -> ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) -> ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) -> ok(U32(X1, X2))
U41(ok(X1), ok(X2)) -> ok(U41(X1, X2))
U42(ok(X)) -> ok(U42(X))
U51(ok(X1), ok(X2)) -> ok(U51(X1, X2))
U52(ok(X)) -> ok(U52(X))
U61(ok(X)) -> ok(U61(X))
U71(ok(X)) -> ok(U71(X))
U81(ok(X)) -> ok(U81(X))
U91(ok(X)) -> ok(U91(X))
isPLNat(ok(X)) -> ok(isPLNat(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))

Types:
active :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U12 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U172 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U182 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U203 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U204 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U22 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U212 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U222 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U32 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok -> top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat -> tt:mark:nil:0':ok


Lemmas:
U102(gen_tt:mark:nil:0':ok3_0(+(1, n5_0))) -> *4_0, rt in Omega(n5_0)
U12(gen_tt:mark:nil:0':ok3_0(+(1, n1128_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) -> *4_0, rt in Omega(n1128_0)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n9121_0))) -> *4_0, rt in Omega(n9121_0)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n10537_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n10537_0)
U132(gen_tt:mark:nil:0':ok3_0(+(1, n15953_0))) -> *4_0, rt in Omega(n15953_0)
U142(gen_tt:mark:nil:0':ok3_0(+(1, n17620_0))) -> *4_0, rt in Omega(n17620_0)
U152(gen_tt:mark:nil:0':ok3_0(+(1, n19388_0))) -> *4_0, rt in Omega(n19388_0)
cons(gen_tt:mark:nil:0':ok3_0(+(1, n21257_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n21257_0)
natsFrom(gen_tt:mark:nil:0':ok3_0(+(1, n27404_0))) -> *4_0, rt in Omega(n27404_0)
s(gen_tt:mark:nil:0':ok3_0(+(1, n29524_0))) -> *4_0, rt in Omega(n29524_0)
U172(gen_tt:mark:nil:0':ok3_0(+(1, n31745_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) -> *4_0, rt in Omega(n31745_0)
head(gen_tt:mark:nil:0':ok3_0(+(1, n43161_0))) -> *4_0, rt in Omega(n43161_0)
afterNth(gen_tt:mark:nil:0':ok3_0(+(1, n45682_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n45682_0)
U182(gen_tt:mark:nil:0':ok3_0(+(1, n53362_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n53362_0)
pair(gen_tt:mark:nil:0':ok3_0(+(1, n61149_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n61149_0)
U202(gen_tt:mark:nil:0':ok3_0(+(1, n69441_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c), gen_tt:mark:nil:0':ok3_0(d)) -> *4_0, rt in Omega(n69441_0)
U203(gen_tt:mark:nil:0':ok3_0(+(1, n88570_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c), gen_tt:mark:nil:0':ok3_0(d)) -> *4_0, rt in Omega(n88570_0)
U204(gen_tt:mark:nil:0':ok3_0(+(1, n108633_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n108633_0)
U22(gen_tt:mark:nil:0':ok3_0(+(1, n118036_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n118036_0)
U212(gen_tt:mark:nil:0':ok3_0(+(1, n127745_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n127745_0)
U222(gen_tt:mark:nil:0':ok3_0(+(1, n137760_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) -> *4_0, rt in Omega(n137760_0)
fst(gen_tt:mark:nil:0':ok3_0(+(1, n154384_0))) -> *4_0, rt in Omega(n154384_0)
U32(gen_tt:mark:nil:0':ok3_0(+(1, n158601_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n158601_0)
U42(gen_tt:mark:nil:0':ok3_0(+(1, n169534_0))) -> *4_0, rt in Omega(n169534_0)
U52(gen_tt:mark:nil:0':ok3_0(+(1, n174002_0))) -> *4_0, rt in Omega(n174002_0)


Generator Equations:
gen_tt:mark:nil:0':ok3_0(0) <=> tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) <=> mark(gen_tt:mark:nil:0':ok3_0(x))


The following defined symbols remain to be analysed:
U11, active, U21, U31, U41, U51, U61, isPLNat, U71, U81, U91, U101, U111, U121, U131, U141, U151, U161, U171, U181, U191, U201, U211, U221, tail, take, sel, proper, top

They will be analysed ascendingly in the following order:
U11 < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
isPLNat < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
U141 < active
U151 < active
U161 < active
U171 < active
U181 < active
U191 < active
U201 < active
U211 < active
U221 < active
tail < active
take < active
sel < active
active < top
U11 < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
isPLNat < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
U141 < proper
U151 < proper
U161 < proper
U171 < proper
U181 < proper
U191 < proper
U201 < proper
U211 < proper
U221 < proper
tail < proper
take < proper
sel < proper
proper < top

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

(79) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
U11(gen_tt:mark:nil:0':ok3_0(+(1, n178571_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) -> *4_0, rt in Omega(n178571_0)

Induction Base:
U11(gen_tt:mark:nil:0':ok3_0(+(1, 0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c))

Induction Step:
U11(gen_tt:mark:nil:0':ok3_0(+(1, +(n178571_0, 1))), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) ->_R^Omega(1)
mark(U11(gen_tt:mark:nil:0':ok3_0(+(1, n178571_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c))) ->_IH
mark(*4_0)

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
----------------------------------------

(80)
Obligation:
TRS:
Rules:
active(U101(tt, V2)) -> mark(U102(isLNat(V2)))
active(U102(tt)) -> mark(tt)
active(U11(tt, N, XS)) -> mark(U12(isLNat(XS), N, XS))
active(U111(tt)) -> mark(tt)
active(U12(tt, N, XS)) -> mark(snd(splitAt(N, XS)))
active(U121(tt)) -> mark(tt)
active(U131(tt, V2)) -> mark(U132(isLNat(V2)))
active(U132(tt)) -> mark(tt)
active(U141(tt, V2)) -> mark(U142(isLNat(V2)))
active(U142(tt)) -> mark(tt)
active(U151(tt, V2)) -> mark(U152(isLNat(V2)))
active(U152(tt)) -> mark(tt)
active(U161(tt, N)) -> mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) -> mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) -> mark(head(afterNth(N, XS)))
active(U181(tt, Y)) -> mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) -> mark(Y)
active(U191(tt, XS)) -> mark(pair(nil, XS))
active(U201(tt, N, X, XS)) -> mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) -> mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) -> mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) -> mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) -> mark(U22(isLNat(Y), X))
active(U211(tt, XS)) -> mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) -> mark(XS)
active(U22(tt, X)) -> mark(X)
active(U221(tt, N, XS)) -> mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) -> mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) -> mark(U32(isLNat(XS), N))
active(U32(tt, N)) -> mark(N)
active(U41(tt, V2)) -> mark(U42(isLNat(V2)))
active(U42(tt)) -> mark(tt)
active(U51(tt, V2)) -> mark(U52(isLNat(V2)))
active(U52(tt)) -> mark(tt)
active(U61(tt)) -> mark(tt)
active(U71(tt)) -> mark(tt)
active(U81(tt)) -> mark(tt)
active(U91(tt)) -> mark(tt)
active(afterNth(N, XS)) -> mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) -> mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) -> mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) -> mark(tt)
active(isLNat(afterNth(V1, V2))) -> mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) -> mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) -> mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) -> mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) -> mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) -> mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) -> mark(U101(isNatural(V1), V2))
active(isNatural(0')) -> mark(tt)
active(isNatural(head(V1))) -> mark(U111(isLNat(V1)))
active(isNatural(s(V1))) -> mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) -> mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) -> mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) -> mark(U151(isNatural(V1), V2))
active(natsFrom(N)) -> mark(U161(isNatural(N), N))
active(sel(N, XS)) -> mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) -> mark(U181(isLNat(X), Y))
active(splitAt(0', XS)) -> mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) -> mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) -> mark(U211(isNatural(N), XS))
active(take(N, XS)) -> mark(U221(isNatural(N), N, XS))
active(U101(X1, X2)) -> U101(active(X1), X2)
active(U102(X)) -> U102(active(X))
active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3)
active(U111(X)) -> U111(active(X))
active(snd(X)) -> snd(active(X))
active(splitAt(X1, X2)) -> splitAt(active(X1), X2)
active(splitAt(X1, X2)) -> splitAt(X1, active(X2))
active(U121(X)) -> U121(active(X))
active(U131(X1, X2)) -> U131(active(X1), X2)
active(U132(X)) -> U132(active(X))
active(U141(X1, X2)) -> U141(active(X1), X2)
active(U142(X)) -> U142(active(X))
active(U151(X1, X2)) -> U151(active(X1), X2)
active(U152(X)) -> U152(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(U172(X1, X2, X3)) -> U172(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(U182(X1, X2)) -> U182(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, X3, X4)) -> U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) -> U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) -> U204(active(X1), X2)
active(U21(X1, X2, X3)) -> U21(active(X1), X2, X3)
active(U22(X1, X2)) -> U22(active(X1), X2)
active(U211(X1, X2)) -> U211(active(X1), X2)
active(U212(X1, X2)) -> U212(active(X1), X2)
active(U221(X1, X2, X3)) -> U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) -> U222(active(X1), X2, X3)
active(fst(X)) -> fst(active(X))
active(U31(X1, X2, X3)) -> U31(active(X1), X2, X3)
active(U32(X1, X2)) -> U32(active(X1), X2)
active(U41(X1, X2)) -> U41(active(X1), X2)
active(U42(X)) -> U42(active(X))
active(U51(X1, X2)) -> U51(active(X1), X2)
active(U52(X)) -> U52(active(X))
active(U61(X)) -> U61(active(X))
active(U71(X)) -> U71(active(X))
active(U81(X)) -> U81(active(X))
active(U91(X)) -> U91(active(X))
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) -> mark(U101(X1, X2))
U102(mark(X)) -> mark(U102(X))
U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3))
U111(mark(X)) -> mark(U111(X))
snd(mark(X)) -> mark(snd(X))
splitAt(mark(X1), X2) -> mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) -> mark(splitAt(X1, X2))
U121(mark(X)) -> mark(U121(X))
U131(mark(X1), X2) -> mark(U131(X1, X2))
U132(mark(X)) -> mark(U132(X))
U141(mark(X1), X2) -> mark(U141(X1, X2))
U142(mark(X)) -> mark(U142(X))
U151(mark(X1), X2) -> mark(U151(X1, X2))
U152(mark(X)) -> mark(U152(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))
U172(mark(X1), X2, X3) -> mark(U172(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))
U182(mark(X1), X2) -> mark(U182(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, X3, X4) -> mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) -> mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) -> mark(U204(X1, X2))
U21(mark(X1), X2, X3) -> mark(U21(X1, X2, X3))
U22(mark(X1), X2) -> mark(U22(X1, X2))
U211(mark(X1), X2) -> mark(U211(X1, X2))
U212(mark(X1), X2) -> mark(U212(X1, X2))
U221(mark(X1), X2, X3) -> mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) -> mark(U222(X1, X2, X3))
fst(mark(X)) -> mark(fst(X))
U31(mark(X1), X2, X3) -> mark(U31(X1, X2, X3))
U32(mark(X1), X2) -> mark(U32(X1, X2))
U41(mark(X1), X2) -> mark(U41(X1, X2))
U42(mark(X)) -> mark(U42(X))
U51(mark(X1), X2) -> mark(U51(X1, X2))
U52(mark(X)) -> mark(U52(X))
U61(mark(X)) -> mark(U61(X))
U71(mark(X)) -> mark(U71(X))
U81(mark(X)) -> mark(U81(X))
U91(mark(X)) -> mark(U91(X))
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)) -> U101(proper(X1), proper(X2))
proper(tt) -> ok(tt)
proper(U102(X)) -> U102(proper(X))
proper(isLNat(X)) -> isLNat(proper(X))
proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) -> U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) -> U111(proper(X))
proper(snd(X)) -> snd(proper(X))
proper(splitAt(X1, X2)) -> splitAt(proper(X1), proper(X2))
proper(U121(X)) -> U121(proper(X))
proper(U131(X1, X2)) -> U131(proper(X1), proper(X2))
proper(U132(X)) -> U132(proper(X))
proper(U141(X1, X2)) -> U141(proper(X1), proper(X2))
proper(U142(X)) -> U142(proper(X))
proper(U151(X1, X2)) -> U151(proper(X1), proper(X2))
proper(U152(X)) -> U152(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(U172(X1, X2, X3)) -> U172(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(U182(X1, X2)) -> U182(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, X3, X4)) -> U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) -> isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) -> U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) -> U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) -> U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) -> U22(proper(X1), proper(X2))
proper(U211(X1, X2)) -> U211(proper(X1), proper(X2))
proper(U212(X1, X2)) -> U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) -> U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) -> U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) -> fst(proper(X))
proper(U31(X1, X2, X3)) -> U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) -> U32(proper(X1), proper(X2))
proper(U41(X1, X2)) -> U41(proper(X1), proper(X2))
proper(U42(X)) -> U42(proper(X))
proper(U51(X1, X2)) -> U51(proper(X1), proper(X2))
proper(U52(X)) -> U52(proper(X))
proper(U61(X)) -> U61(proper(X))
proper(U71(X)) -> U71(proper(X))
proper(U81(X)) -> U81(proper(X))
proper(U91(X)) -> U91(proper(X))
proper(isPLNat(X)) -> isPLNat(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(U101(X1, X2))
U102(ok(X)) -> ok(U102(X))
isLNat(ok(X)) -> ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))
U111(ok(X)) -> ok(U111(X))
snd(ok(X)) -> ok(snd(X))
splitAt(ok(X1), ok(X2)) -> ok(splitAt(X1, X2))
U121(ok(X)) -> ok(U121(X))
U131(ok(X1), ok(X2)) -> ok(U131(X1, X2))
U132(ok(X)) -> ok(U132(X))
U141(ok(X1), ok(X2)) -> ok(U141(X1, X2))
U142(ok(X)) -> ok(U142(X))
U151(ok(X1), ok(X2)) -> ok(U151(X1, X2))
U152(ok(X)) -> ok(U152(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))
U172(ok(X1), ok(X2), ok(X3)) -> ok(U172(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))
U182(ok(X1), ok(X2)) -> ok(U182(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(X3), ok(X4)) -> ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) -> ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) -> ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) -> ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) -> ok(U22(X1, X2))
U211(ok(X1), ok(X2)) -> ok(U211(X1, X2))
U212(ok(X1), ok(X2)) -> ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) -> ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) -> ok(U222(X1, X2, X3))
fst(ok(X)) -> ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) -> ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) -> ok(U32(X1, X2))
U41(ok(X1), ok(X2)) -> ok(U41(X1, X2))
U42(ok(X)) -> ok(U42(X))
U51(ok(X1), ok(X2)) -> ok(U51(X1, X2))
U52(ok(X)) -> ok(U52(X))
U61(ok(X)) -> ok(U61(X))
U71(ok(X)) -> ok(U71(X))
U81(ok(X)) -> ok(U81(X))
U91(ok(X)) -> ok(U91(X))
isPLNat(ok(X)) -> ok(isPLNat(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))

Types:
active :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U12 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U172 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U182 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U203 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U204 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U22 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U212 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U222 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U32 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok -> top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat -> tt:mark:nil:0':ok


Lemmas:
U102(gen_tt:mark:nil:0':ok3_0(+(1, n5_0))) -> *4_0, rt in Omega(n5_0)
U12(gen_tt:mark:nil:0':ok3_0(+(1, n1128_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) -> *4_0, rt in Omega(n1128_0)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n9121_0))) -> *4_0, rt in Omega(n9121_0)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n10537_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n10537_0)
U132(gen_tt:mark:nil:0':ok3_0(+(1, n15953_0))) -> *4_0, rt in Omega(n15953_0)
U142(gen_tt:mark:nil:0':ok3_0(+(1, n17620_0))) -> *4_0, rt in Omega(n17620_0)
U152(gen_tt:mark:nil:0':ok3_0(+(1, n19388_0))) -> *4_0, rt in Omega(n19388_0)
cons(gen_tt:mark:nil:0':ok3_0(+(1, n21257_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n21257_0)
natsFrom(gen_tt:mark:nil:0':ok3_0(+(1, n27404_0))) -> *4_0, rt in Omega(n27404_0)
s(gen_tt:mark:nil:0':ok3_0(+(1, n29524_0))) -> *4_0, rt in Omega(n29524_0)
U172(gen_tt:mark:nil:0':ok3_0(+(1, n31745_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) -> *4_0, rt in Omega(n31745_0)
head(gen_tt:mark:nil:0':ok3_0(+(1, n43161_0))) -> *4_0, rt in Omega(n43161_0)
afterNth(gen_tt:mark:nil:0':ok3_0(+(1, n45682_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n45682_0)
U182(gen_tt:mark:nil:0':ok3_0(+(1, n53362_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n53362_0)
pair(gen_tt:mark:nil:0':ok3_0(+(1, n61149_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n61149_0)
U202(gen_tt:mark:nil:0':ok3_0(+(1, n69441_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c), gen_tt:mark:nil:0':ok3_0(d)) -> *4_0, rt in Omega(n69441_0)
U203(gen_tt:mark:nil:0':ok3_0(+(1, n88570_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c), gen_tt:mark:nil:0':ok3_0(d)) -> *4_0, rt in Omega(n88570_0)
U204(gen_tt:mark:nil:0':ok3_0(+(1, n108633_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n108633_0)
U22(gen_tt:mark:nil:0':ok3_0(+(1, n118036_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n118036_0)
U212(gen_tt:mark:nil:0':ok3_0(+(1, n127745_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n127745_0)
U222(gen_tt:mark:nil:0':ok3_0(+(1, n137760_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) -> *4_0, rt in Omega(n137760_0)
fst(gen_tt:mark:nil:0':ok3_0(+(1, n154384_0))) -> *4_0, rt in Omega(n154384_0)
U32(gen_tt:mark:nil:0':ok3_0(+(1, n158601_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n158601_0)
U42(gen_tt:mark:nil:0':ok3_0(+(1, n169534_0))) -> *4_0, rt in Omega(n169534_0)
U52(gen_tt:mark:nil:0':ok3_0(+(1, n174002_0))) -> *4_0, rt in Omega(n174002_0)
U11(gen_tt:mark:nil:0':ok3_0(+(1, n178571_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) -> *4_0, rt in Omega(n178571_0)


Generator Equations:
gen_tt:mark:nil:0':ok3_0(0) <=> tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) <=> mark(gen_tt:mark:nil:0':ok3_0(x))


The following defined symbols remain to be analysed:
U21, active, U31, U41, U51, U61, isPLNat, U71, U81, U91, U101, U111, U121, U131, U141, U151, U161, U171, U181, U191, U201, U211, U221, tail, take, sel, proper, top

They will be analysed ascendingly in the following order:
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
isPLNat < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
U141 < active
U151 < active
U161 < active
U171 < active
U181 < active
U191 < active
U201 < active
U211 < active
U221 < active
tail < active
take < active
sel < active
active < top
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
isPLNat < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
U141 < proper
U151 < proper
U161 < proper
U171 < proper
U181 < proper
U191 < proper
U201 < proper
U211 < proper
U221 < proper
tail < proper
take < proper
sel < proper
proper < top

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(81) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
U21(gen_tt:mark:nil:0':ok3_0(+(1, n197211_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) -> *4_0, rt in Omega(n197211_0)

Induction Base:
U21(gen_tt:mark:nil:0':ok3_0(+(1, 0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c))

Induction Step:
U21(gen_tt:mark:nil:0':ok3_0(+(1, +(n197211_0, 1))), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) ->_R^Omega(1)
mark(U21(gen_tt:mark:nil:0':ok3_0(+(1, n197211_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c))) ->_IH
mark(*4_0)

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
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(82)
Obligation:
TRS:
Rules:
active(U101(tt, V2)) -> mark(U102(isLNat(V2)))
active(U102(tt)) -> mark(tt)
active(U11(tt, N, XS)) -> mark(U12(isLNat(XS), N, XS))
active(U111(tt)) -> mark(tt)
active(U12(tt, N, XS)) -> mark(snd(splitAt(N, XS)))
active(U121(tt)) -> mark(tt)
active(U131(tt, V2)) -> mark(U132(isLNat(V2)))
active(U132(tt)) -> mark(tt)
active(U141(tt, V2)) -> mark(U142(isLNat(V2)))
active(U142(tt)) -> mark(tt)
active(U151(tt, V2)) -> mark(U152(isLNat(V2)))
active(U152(tt)) -> mark(tt)
active(U161(tt, N)) -> mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) -> mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) -> mark(head(afterNth(N, XS)))
active(U181(tt, Y)) -> mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) -> mark(Y)
active(U191(tt, XS)) -> mark(pair(nil, XS))
active(U201(tt, N, X, XS)) -> mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) -> mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) -> mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) -> mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) -> mark(U22(isLNat(Y), X))
active(U211(tt, XS)) -> mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) -> mark(XS)
active(U22(tt, X)) -> mark(X)
active(U221(tt, N, XS)) -> mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) -> mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) -> mark(U32(isLNat(XS), N))
active(U32(tt, N)) -> mark(N)
active(U41(tt, V2)) -> mark(U42(isLNat(V2)))
active(U42(tt)) -> mark(tt)
active(U51(tt, V2)) -> mark(U52(isLNat(V2)))
active(U52(tt)) -> mark(tt)
active(U61(tt)) -> mark(tt)
active(U71(tt)) -> mark(tt)
active(U81(tt)) -> mark(tt)
active(U91(tt)) -> mark(tt)
active(afterNth(N, XS)) -> mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) -> mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) -> mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) -> mark(tt)
active(isLNat(afterNth(V1, V2))) -> mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) -> mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) -> mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) -> mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) -> mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) -> mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) -> mark(U101(isNatural(V1), V2))
active(isNatural(0')) -> mark(tt)
active(isNatural(head(V1))) -> mark(U111(isLNat(V1)))
active(isNatural(s(V1))) -> mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) -> mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) -> mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) -> mark(U151(isNatural(V1), V2))
active(natsFrom(N)) -> mark(U161(isNatural(N), N))
active(sel(N, XS)) -> mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) -> mark(U181(isLNat(X), Y))
active(splitAt(0', XS)) -> mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) -> mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) -> mark(U211(isNatural(N), XS))
active(take(N, XS)) -> mark(U221(isNatural(N), N, XS))
active(U101(X1, X2)) -> U101(active(X1), X2)
active(U102(X)) -> U102(active(X))
active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3)
active(U111(X)) -> U111(active(X))
active(snd(X)) -> snd(active(X))
active(splitAt(X1, X2)) -> splitAt(active(X1), X2)
active(splitAt(X1, X2)) -> splitAt(X1, active(X2))
active(U121(X)) -> U121(active(X))
active(U131(X1, X2)) -> U131(active(X1), X2)
active(U132(X)) -> U132(active(X))
active(U141(X1, X2)) -> U141(active(X1), X2)
active(U142(X)) -> U142(active(X))
active(U151(X1, X2)) -> U151(active(X1), X2)
active(U152(X)) -> U152(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(U172(X1, X2, X3)) -> U172(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(U182(X1, X2)) -> U182(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, X3, X4)) -> U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) -> U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) -> U204(active(X1), X2)
active(U21(X1, X2, X3)) -> U21(active(X1), X2, X3)
active(U22(X1, X2)) -> U22(active(X1), X2)
active(U211(X1, X2)) -> U211(active(X1), X2)
active(U212(X1, X2)) -> U212(active(X1), X2)
active(U221(X1, X2, X3)) -> U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) -> U222(active(X1), X2, X3)
active(fst(X)) -> fst(active(X))
active(U31(X1, X2, X3)) -> U31(active(X1), X2, X3)
active(U32(X1, X2)) -> U32(active(X1), X2)
active(U41(X1, X2)) -> U41(active(X1), X2)
active(U42(X)) -> U42(active(X))
active(U51(X1, X2)) -> U51(active(X1), X2)
active(U52(X)) -> U52(active(X))
active(U61(X)) -> U61(active(X))
active(U71(X)) -> U71(active(X))
active(U81(X)) -> U81(active(X))
active(U91(X)) -> U91(active(X))
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) -> mark(U101(X1, X2))
U102(mark(X)) -> mark(U102(X))
U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3))
U111(mark(X)) -> mark(U111(X))
snd(mark(X)) -> mark(snd(X))
splitAt(mark(X1), X2) -> mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) -> mark(splitAt(X1, X2))
U121(mark(X)) -> mark(U121(X))
U131(mark(X1), X2) -> mark(U131(X1, X2))
U132(mark(X)) -> mark(U132(X))
U141(mark(X1), X2) -> mark(U141(X1, X2))
U142(mark(X)) -> mark(U142(X))
U151(mark(X1), X2) -> mark(U151(X1, X2))
U152(mark(X)) -> mark(U152(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))
U172(mark(X1), X2, X3) -> mark(U172(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))
U182(mark(X1), X2) -> mark(U182(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, X3, X4) -> mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) -> mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) -> mark(U204(X1, X2))
U21(mark(X1), X2, X3) -> mark(U21(X1, X2, X3))
U22(mark(X1), X2) -> mark(U22(X1, X2))
U211(mark(X1), X2) -> mark(U211(X1, X2))
U212(mark(X1), X2) -> mark(U212(X1, X2))
U221(mark(X1), X2, X3) -> mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) -> mark(U222(X1, X2, X3))
fst(mark(X)) -> mark(fst(X))
U31(mark(X1), X2, X3) -> mark(U31(X1, X2, X3))
U32(mark(X1), X2) -> mark(U32(X1, X2))
U41(mark(X1), X2) -> mark(U41(X1, X2))
U42(mark(X)) -> mark(U42(X))
U51(mark(X1), X2) -> mark(U51(X1, X2))
U52(mark(X)) -> mark(U52(X))
U61(mark(X)) -> mark(U61(X))
U71(mark(X)) -> mark(U71(X))
U81(mark(X)) -> mark(U81(X))
U91(mark(X)) -> mark(U91(X))
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)) -> U101(proper(X1), proper(X2))
proper(tt) -> ok(tt)
proper(U102(X)) -> U102(proper(X))
proper(isLNat(X)) -> isLNat(proper(X))
proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) -> U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) -> U111(proper(X))
proper(snd(X)) -> snd(proper(X))
proper(splitAt(X1, X2)) -> splitAt(proper(X1), proper(X2))
proper(U121(X)) -> U121(proper(X))
proper(U131(X1, X2)) -> U131(proper(X1), proper(X2))
proper(U132(X)) -> U132(proper(X))
proper(U141(X1, X2)) -> U141(proper(X1), proper(X2))
proper(U142(X)) -> U142(proper(X))
proper(U151(X1, X2)) -> U151(proper(X1), proper(X2))
proper(U152(X)) -> U152(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(U172(X1, X2, X3)) -> U172(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(U182(X1, X2)) -> U182(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, X3, X4)) -> U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) -> isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) -> U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) -> U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) -> U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) -> U22(proper(X1), proper(X2))
proper(U211(X1, X2)) -> U211(proper(X1), proper(X2))
proper(U212(X1, X2)) -> U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) -> U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) -> U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) -> fst(proper(X))
proper(U31(X1, X2, X3)) -> U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) -> U32(proper(X1), proper(X2))
proper(U41(X1, X2)) -> U41(proper(X1), proper(X2))
proper(U42(X)) -> U42(proper(X))
proper(U51(X1, X2)) -> U51(proper(X1), proper(X2))
proper(U52(X)) -> U52(proper(X))
proper(U61(X)) -> U61(proper(X))
proper(U71(X)) -> U71(proper(X))
proper(U81(X)) -> U81(proper(X))
proper(U91(X)) -> U91(proper(X))
proper(isPLNat(X)) -> isPLNat(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(U101(X1, X2))
U102(ok(X)) -> ok(U102(X))
isLNat(ok(X)) -> ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))
U111(ok(X)) -> ok(U111(X))
snd(ok(X)) -> ok(snd(X))
splitAt(ok(X1), ok(X2)) -> ok(splitAt(X1, X2))
U121(ok(X)) -> ok(U121(X))
U131(ok(X1), ok(X2)) -> ok(U131(X1, X2))
U132(ok(X)) -> ok(U132(X))
U141(ok(X1), ok(X2)) -> ok(U141(X1, X2))
U142(ok(X)) -> ok(U142(X))
U151(ok(X1), ok(X2)) -> ok(U151(X1, X2))
U152(ok(X)) -> ok(U152(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))
U172(ok(X1), ok(X2), ok(X3)) -> ok(U172(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))
U182(ok(X1), ok(X2)) -> ok(U182(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(X3), ok(X4)) -> ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) -> ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) -> ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) -> ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) -> ok(U22(X1, X2))
U211(ok(X1), ok(X2)) -> ok(U211(X1, X2))
U212(ok(X1), ok(X2)) -> ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) -> ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) -> ok(U222(X1, X2, X3))
fst(ok(X)) -> ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) -> ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) -> ok(U32(X1, X2))
U41(ok(X1), ok(X2)) -> ok(U41(X1, X2))
U42(ok(X)) -> ok(U42(X))
U51(ok(X1), ok(X2)) -> ok(U51(X1, X2))
U52(ok(X)) -> ok(U52(X))
U61(ok(X)) -> ok(U61(X))
U71(ok(X)) -> ok(U71(X))
U81(ok(X)) -> ok(U81(X))
U91(ok(X)) -> ok(U91(X))
isPLNat(ok(X)) -> ok(isPLNat(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))

Types:
active :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U12 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U172 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U182 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U203 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U204 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U22 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U212 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U222 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U32 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok -> top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat -> tt:mark:nil:0':ok


Lemmas:
U102(gen_tt:mark:nil:0':ok3_0(+(1, n5_0))) -> *4_0, rt in Omega(n5_0)
U12(gen_tt:mark:nil:0':ok3_0(+(1, n1128_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) -> *4_0, rt in Omega(n1128_0)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n9121_0))) -> *4_0, rt in Omega(n9121_0)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n10537_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n10537_0)
U132(gen_tt:mark:nil:0':ok3_0(+(1, n15953_0))) -> *4_0, rt in Omega(n15953_0)
U142(gen_tt:mark:nil:0':ok3_0(+(1, n17620_0))) -> *4_0, rt in Omega(n17620_0)
U152(gen_tt:mark:nil:0':ok3_0(+(1, n19388_0))) -> *4_0, rt in Omega(n19388_0)
cons(gen_tt:mark:nil:0':ok3_0(+(1, n21257_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n21257_0)
natsFrom(gen_tt:mark:nil:0':ok3_0(+(1, n27404_0))) -> *4_0, rt in Omega(n27404_0)
s(gen_tt:mark:nil:0':ok3_0(+(1, n29524_0))) -> *4_0, rt in Omega(n29524_0)
U172(gen_tt:mark:nil:0':ok3_0(+(1, n31745_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) -> *4_0, rt in Omega(n31745_0)
head(gen_tt:mark:nil:0':ok3_0(+(1, n43161_0))) -> *4_0, rt in Omega(n43161_0)
afterNth(gen_tt:mark:nil:0':ok3_0(+(1, n45682_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n45682_0)
U182(gen_tt:mark:nil:0':ok3_0(+(1, n53362_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n53362_0)
pair(gen_tt:mark:nil:0':ok3_0(+(1, n61149_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n61149_0)
U202(gen_tt:mark:nil:0':ok3_0(+(1, n69441_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c), gen_tt:mark:nil:0':ok3_0(d)) -> *4_0, rt in Omega(n69441_0)
U203(gen_tt:mark:nil:0':ok3_0(+(1, n88570_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c), gen_tt:mark:nil:0':ok3_0(d)) -> *4_0, rt in Omega(n88570_0)
U204(gen_tt:mark:nil:0':ok3_0(+(1, n108633_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n108633_0)
U22(gen_tt:mark:nil:0':ok3_0(+(1, n118036_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n118036_0)
U212(gen_tt:mark:nil:0':ok3_0(+(1, n127745_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n127745_0)
U222(gen_tt:mark:nil:0':ok3_0(+(1, n137760_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) -> *4_0, rt in Omega(n137760_0)
fst(gen_tt:mark:nil:0':ok3_0(+(1, n154384_0))) -> *4_0, rt in Omega(n154384_0)
U32(gen_tt:mark:nil:0':ok3_0(+(1, n158601_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n158601_0)
U42(gen_tt:mark:nil:0':ok3_0(+(1, n169534_0))) -> *4_0, rt in Omega(n169534_0)
U52(gen_tt:mark:nil:0':ok3_0(+(1, n174002_0))) -> *4_0, rt in Omega(n174002_0)
U11(gen_tt:mark:nil:0':ok3_0(+(1, n178571_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) -> *4_0, rt in Omega(n178571_0)
U21(gen_tt:mark:nil:0':ok3_0(+(1, n197211_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) -> *4_0, rt in Omega(n197211_0)


Generator Equations:
gen_tt:mark:nil:0':ok3_0(0) <=> tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) <=> mark(gen_tt:mark:nil:0':ok3_0(x))


The following defined symbols remain to be analysed:
U31, active, U41, U51, U61, isPLNat, U71, U81, U91, U101, U111, U121, U131, U141, U151, U161, U171, U181, U191, U201, U211, U221, tail, take, sel, proper, top

They will be analysed ascendingly in the following order:
U31 < active
U41 < active
U51 < active
U61 < active
isPLNat < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
U141 < active
U151 < active
U161 < active
U171 < active
U181 < active
U191 < active
U201 < active
U211 < active
U221 < active
tail < active
take < active
sel < active
active < top
U31 < proper
U41 < proper
U51 < proper
U61 < proper
isPLNat < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
U141 < proper
U151 < proper
U161 < proper
U171 < proper
U181 < proper
U191 < proper
U201 < proper
U211 < proper
U221 < proper
tail < proper
take < proper
sel < proper
proper < top

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

(83) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
U31(gen_tt:mark:nil:0':ok3_0(+(1, n216460_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) -> *4_0, rt in Omega(n216460_0)

Induction Base:
U31(gen_tt:mark:nil:0':ok3_0(+(1, 0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c))

Induction Step:
U31(gen_tt:mark:nil:0':ok3_0(+(1, +(n216460_0, 1))), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) ->_R^Omega(1)
mark(U31(gen_tt:mark:nil:0':ok3_0(+(1, n216460_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c))) ->_IH
mark(*4_0)

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
----------------------------------------

(84)
Obligation:
TRS:
Rules:
active(U101(tt, V2)) -> mark(U102(isLNat(V2)))
active(U102(tt)) -> mark(tt)
active(U11(tt, N, XS)) -> mark(U12(isLNat(XS), N, XS))
active(U111(tt)) -> mark(tt)
active(U12(tt, N, XS)) -> mark(snd(splitAt(N, XS)))
active(U121(tt)) -> mark(tt)
active(U131(tt, V2)) -> mark(U132(isLNat(V2)))
active(U132(tt)) -> mark(tt)
active(U141(tt, V2)) -> mark(U142(isLNat(V2)))
active(U142(tt)) -> mark(tt)
active(U151(tt, V2)) -> mark(U152(isLNat(V2)))
active(U152(tt)) -> mark(tt)
active(U161(tt, N)) -> mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) -> mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) -> mark(head(afterNth(N, XS)))
active(U181(tt, Y)) -> mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) -> mark(Y)
active(U191(tt, XS)) -> mark(pair(nil, XS))
active(U201(tt, N, X, XS)) -> mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) -> mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) -> mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) -> mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) -> mark(U22(isLNat(Y), X))
active(U211(tt, XS)) -> mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) -> mark(XS)
active(U22(tt, X)) -> mark(X)
active(U221(tt, N, XS)) -> mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) -> mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) -> mark(U32(isLNat(XS), N))
active(U32(tt, N)) -> mark(N)
active(U41(tt, V2)) -> mark(U42(isLNat(V2)))
active(U42(tt)) -> mark(tt)
active(U51(tt, V2)) -> mark(U52(isLNat(V2)))
active(U52(tt)) -> mark(tt)
active(U61(tt)) -> mark(tt)
active(U71(tt)) -> mark(tt)
active(U81(tt)) -> mark(tt)
active(U91(tt)) -> mark(tt)
active(afterNth(N, XS)) -> mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) -> mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) -> mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) -> mark(tt)
active(isLNat(afterNth(V1, V2))) -> mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) -> mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) -> mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) -> mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) -> mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) -> mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) -> mark(U101(isNatural(V1), V2))
active(isNatural(0')) -> mark(tt)
active(isNatural(head(V1))) -> mark(U111(isLNat(V1)))
active(isNatural(s(V1))) -> mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) -> mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) -> mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) -> mark(U151(isNatural(V1), V2))
active(natsFrom(N)) -> mark(U161(isNatural(N), N))
active(sel(N, XS)) -> mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) -> mark(U181(isLNat(X), Y))
active(splitAt(0', XS)) -> mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) -> mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) -> mark(U211(isNatural(N), XS))
active(take(N, XS)) -> mark(U221(isNatural(N), N, XS))
active(U101(X1, X2)) -> U101(active(X1), X2)
active(U102(X)) -> U102(active(X))
active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3)
active(U111(X)) -> U111(active(X))
active(snd(X)) -> snd(active(X))
active(splitAt(X1, X2)) -> splitAt(active(X1), X2)
active(splitAt(X1, X2)) -> splitAt(X1, active(X2))
active(U121(X)) -> U121(active(X))
active(U131(X1, X2)) -> U131(active(X1), X2)
active(U132(X)) -> U132(active(X))
active(U141(X1, X2)) -> U141(active(X1), X2)
active(U142(X)) -> U142(active(X))
active(U151(X1, X2)) -> U151(active(X1), X2)
active(U152(X)) -> U152(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(U172(X1, X2, X3)) -> U172(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(U182(X1, X2)) -> U182(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, X3, X4)) -> U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) -> U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) -> U204(active(X1), X2)
active(U21(X1, X2, X3)) -> U21(active(X1), X2, X3)
active(U22(X1, X2)) -> U22(active(X1), X2)
active(U211(X1, X2)) -> U211(active(X1), X2)
active(U212(X1, X2)) -> U212(active(X1), X2)
active(U221(X1, X2, X3)) -> U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) -> U222(active(X1), X2, X3)
active(fst(X)) -> fst(active(X))
active(U31(X1, X2, X3)) -> U31(active(X1), X2, X3)
active(U32(X1, X2)) -> U32(active(X1), X2)
active(U41(X1, X2)) -> U41(active(X1), X2)
active(U42(X)) -> U42(active(X))
active(U51(X1, X2)) -> U51(active(X1), X2)
active(U52(X)) -> U52(active(X))
active(U61(X)) -> U61(active(X))
active(U71(X)) -> U71(active(X))
active(U81(X)) -> U81(active(X))
active(U91(X)) -> U91(active(X))
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) -> mark(U101(X1, X2))
U102(mark(X)) -> mark(U102(X))
U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3))
U111(mark(X)) -> mark(U111(X))
snd(mark(X)) -> mark(snd(X))
splitAt(mark(X1), X2) -> mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) -> mark(splitAt(X1, X2))
U121(mark(X)) -> mark(U121(X))
U131(mark(X1), X2) -> mark(U131(X1, X2))
U132(mark(X)) -> mark(U132(X))
U141(mark(X1), X2) -> mark(U141(X1, X2))
U142(mark(X)) -> mark(U142(X))
U151(mark(X1), X2) -> mark(U151(X1, X2))
U152(mark(X)) -> mark(U152(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))
U172(mark(X1), X2, X3) -> mark(U172(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))
U182(mark(X1), X2) -> mark(U182(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, X3, X4) -> mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) -> mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) -> mark(U204(X1, X2))
U21(mark(X1), X2, X3) -> mark(U21(X1, X2, X3))
U22(mark(X1), X2) -> mark(U22(X1, X2))
U211(mark(X1), X2) -> mark(U211(X1, X2))
U212(mark(X1), X2) -> mark(U212(X1, X2))
U221(mark(X1), X2, X3) -> mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) -> mark(U222(X1, X2, X3))
fst(mark(X)) -> mark(fst(X))
U31(mark(X1), X2, X3) -> mark(U31(X1, X2, X3))
U32(mark(X1), X2) -> mark(U32(X1, X2))
U41(mark(X1), X2) -> mark(U41(X1, X2))
U42(mark(X)) -> mark(U42(X))
U51(mark(X1), X2) -> mark(U51(X1, X2))
U52(mark(X)) -> mark(U52(X))
U61(mark(X)) -> mark(U61(X))
U71(mark(X)) -> mark(U71(X))
U81(mark(X)) -> mark(U81(X))
U91(mark(X)) -> mark(U91(X))
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)) -> U101(proper(X1), proper(X2))
proper(tt) -> ok(tt)
proper(U102(X)) -> U102(proper(X))
proper(isLNat(X)) -> isLNat(proper(X))
proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) -> U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) -> U111(proper(X))
proper(snd(X)) -> snd(proper(X))
proper(splitAt(X1, X2)) -> splitAt(proper(X1), proper(X2))
proper(U121(X)) -> U121(proper(X))
proper(U131(X1, X2)) -> U131(proper(X1), proper(X2))
proper(U132(X)) -> U132(proper(X))
proper(U141(X1, X2)) -> U141(proper(X1), proper(X2))
proper(U142(X)) -> U142(proper(X))
proper(U151(X1, X2)) -> U151(proper(X1), proper(X2))
proper(U152(X)) -> U152(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(U172(X1, X2, X3)) -> U172(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(U182(X1, X2)) -> U182(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, X3, X4)) -> U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) -> isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) -> U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) -> U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) -> U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) -> U22(proper(X1), proper(X2))
proper(U211(X1, X2)) -> U211(proper(X1), proper(X2))
proper(U212(X1, X2)) -> U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) -> U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) -> U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) -> fst(proper(X))
proper(U31(X1, X2, X3)) -> U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) -> U32(proper(X1), proper(X2))
proper(U41(X1, X2)) -> U41(proper(X1), proper(X2))
proper(U42(X)) -> U42(proper(X))
proper(U51(X1, X2)) -> U51(proper(X1), proper(X2))
proper(U52(X)) -> U52(proper(X))
proper(U61(X)) -> U61(proper(X))
proper(U71(X)) -> U71(proper(X))
proper(U81(X)) -> U81(proper(X))
proper(U91(X)) -> U91(proper(X))
proper(isPLNat(X)) -> isPLNat(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(U101(X1, X2))
U102(ok(X)) -> ok(U102(X))
isLNat(ok(X)) -> ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))
U111(ok(X)) -> ok(U111(X))
snd(ok(X)) -> ok(snd(X))
splitAt(ok(X1), ok(X2)) -> ok(splitAt(X1, X2))
U121(ok(X)) -> ok(U121(X))
U131(ok(X1), ok(X2)) -> ok(U131(X1, X2))
U132(ok(X)) -> ok(U132(X))
U141(ok(X1), ok(X2)) -> ok(U141(X1, X2))
U142(ok(X)) -> ok(U142(X))
U151(ok(X1), ok(X2)) -> ok(U151(X1, X2))
U152(ok(X)) -> ok(U152(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))
U172(ok(X1), ok(X2), ok(X3)) -> ok(U172(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))
U182(ok(X1), ok(X2)) -> ok(U182(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(X3), ok(X4)) -> ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) -> ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) -> ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) -> ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) -> ok(U22(X1, X2))
U211(ok(X1), ok(X2)) -> ok(U211(X1, X2))
U212(ok(X1), ok(X2)) -> ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) -> ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) -> ok(U222(X1, X2, X3))
fst(ok(X)) -> ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) -> ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) -> ok(U32(X1, X2))
U41(ok(X1), ok(X2)) -> ok(U41(X1, X2))
U42(ok(X)) -> ok(U42(X))
U51(ok(X1), ok(X2)) -> ok(U51(X1, X2))
U52(ok(X)) -> ok(U52(X))
U61(ok(X)) -> ok(U61(X))
U71(ok(X)) -> ok(U71(X))
U81(ok(X)) -> ok(U81(X))
U91(ok(X)) -> ok(U91(X))
isPLNat(ok(X)) -> ok(isPLNat(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))

Types:
active :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U12 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U172 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U182 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U203 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U204 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U22 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U212 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U222 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U32 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok -> top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat -> tt:mark:nil:0':ok


Lemmas:
U102(gen_tt:mark:nil:0':ok3_0(+(1, n5_0))) -> *4_0, rt in Omega(n5_0)
U12(gen_tt:mark:nil:0':ok3_0(+(1, n1128_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) -> *4_0, rt in Omega(n1128_0)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n9121_0))) -> *4_0, rt in Omega(n9121_0)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n10537_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n10537_0)
U132(gen_tt:mark:nil:0':ok3_0(+(1, n15953_0))) -> *4_0, rt in Omega(n15953_0)
U142(gen_tt:mark:nil:0':ok3_0(+(1, n17620_0))) -> *4_0, rt in Omega(n17620_0)
U152(gen_tt:mark:nil:0':ok3_0(+(1, n19388_0))) -> *4_0, rt in Omega(n19388_0)
cons(gen_tt:mark:nil:0':ok3_0(+(1, n21257_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n21257_0)
natsFrom(gen_tt:mark:nil:0':ok3_0(+(1, n27404_0))) -> *4_0, rt in Omega(n27404_0)
s(gen_tt:mark:nil:0':ok3_0(+(1, n29524_0))) -> *4_0, rt in Omega(n29524_0)
U172(gen_tt:mark:nil:0':ok3_0(+(1, n31745_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) -> *4_0, rt in Omega(n31745_0)
head(gen_tt:mark:nil:0':ok3_0(+(1, n43161_0))) -> *4_0, rt in Omega(n43161_0)
afterNth(gen_tt:mark:nil:0':ok3_0(+(1, n45682_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n45682_0)
U182(gen_tt:mark:nil:0':ok3_0(+(1, n53362_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n53362_0)
pair(gen_tt:mark:nil:0':ok3_0(+(1, n61149_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n61149_0)
U202(gen_tt:mark:nil:0':ok3_0(+(1, n69441_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c), gen_tt:mark:nil:0':ok3_0(d)) -> *4_0, rt in Omega(n69441_0)
U203(gen_tt:mark:nil:0':ok3_0(+(1, n88570_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c), gen_tt:mark:nil:0':ok3_0(d)) -> *4_0, rt in Omega(n88570_0)
U204(gen_tt:mark:nil:0':ok3_0(+(1, n108633_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n108633_0)
U22(gen_tt:mark:nil:0':ok3_0(+(1, n118036_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n118036_0)
U212(gen_tt:mark:nil:0':ok3_0(+(1, n127745_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n127745_0)
U222(gen_tt:mark:nil:0':ok3_0(+(1, n137760_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) -> *4_0, rt in Omega(n137760_0)
fst(gen_tt:mark:nil:0':ok3_0(+(1, n154384_0))) -> *4_0, rt in Omega(n154384_0)
U32(gen_tt:mark:nil:0':ok3_0(+(1, n158601_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n158601_0)
U42(gen_tt:mark:nil:0':ok3_0(+(1, n169534_0))) -> *4_0, rt in Omega(n169534_0)
U52(gen_tt:mark:nil:0':ok3_0(+(1, n174002_0))) -> *4_0, rt in Omega(n174002_0)
U11(gen_tt:mark:nil:0':ok3_0(+(1, n178571_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) -> *4_0, rt in Omega(n178571_0)
U21(gen_tt:mark:nil:0':ok3_0(+(1, n197211_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) -> *4_0, rt in Omega(n197211_0)
U31(gen_tt:mark:nil:0':ok3_0(+(1, n216460_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) -> *4_0, rt in Omega(n216460_0)


Generator Equations:
gen_tt:mark:nil:0':ok3_0(0) <=> tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) <=> mark(gen_tt:mark:nil:0':ok3_0(x))


The following defined symbols remain to be analysed:
U41, active, U51, U61, isPLNat, U71, U81, U91, U101, U111, U121, U131, U141, U151, U161, U171, U181, U191, U201, U211, U221, tail, take, sel, proper, top

They will be analysed ascendingly in the following order:
U41 < active
U51 < active
U61 < active
isPLNat < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
U141 < active
U151 < active
U161 < active
U171 < active
U181 < active
U191 < active
U201 < active
U211 < active
U221 < active
tail < active
take < active
sel < active
active < top
U41 < proper
U51 < proper
U61 < proper
isPLNat < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
U141 < proper
U151 < proper
U161 < proper
U171 < proper
U181 < proper
U191 < proper
U201 < proper
U211 < proper
U221 < proper
tail < proper
take < proper
sel < proper
proper < top

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

(85) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
U41(gen_tt:mark:nil:0':ok3_0(+(1, n236318_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n236318_0)

Induction Base:
U41(gen_tt:mark:nil:0':ok3_0(+(1, 0)), gen_tt:mark:nil:0':ok3_0(b))

Induction Step:
U41(gen_tt:mark:nil:0':ok3_0(+(1, +(n236318_0, 1))), gen_tt:mark:nil:0':ok3_0(b)) ->_R^Omega(1)
mark(U41(gen_tt:mark:nil:0':ok3_0(+(1, n236318_0)), gen_tt:mark:nil:0':ok3_0(b))) ->_IH
mark(*4_0)

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
----------------------------------------

(86)
Obligation:
TRS:
Rules:
active(U101(tt, V2)) -> mark(U102(isLNat(V2)))
active(U102(tt)) -> mark(tt)
active(U11(tt, N, XS)) -> mark(U12(isLNat(XS), N, XS))
active(U111(tt)) -> mark(tt)
active(U12(tt, N, XS)) -> mark(snd(splitAt(N, XS)))
active(U121(tt)) -> mark(tt)
active(U131(tt, V2)) -> mark(U132(isLNat(V2)))
active(U132(tt)) -> mark(tt)
active(U141(tt, V2)) -> mark(U142(isLNat(V2)))
active(U142(tt)) -> mark(tt)
active(U151(tt, V2)) -> mark(U152(isLNat(V2)))
active(U152(tt)) -> mark(tt)
active(U161(tt, N)) -> mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) -> mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) -> mark(head(afterNth(N, XS)))
active(U181(tt, Y)) -> mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) -> mark(Y)
active(U191(tt, XS)) -> mark(pair(nil, XS))
active(U201(tt, N, X, XS)) -> mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) -> mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) -> mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) -> mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) -> mark(U22(isLNat(Y), X))
active(U211(tt, XS)) -> mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) -> mark(XS)
active(U22(tt, X)) -> mark(X)
active(U221(tt, N, XS)) -> mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) -> mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) -> mark(U32(isLNat(XS), N))
active(U32(tt, N)) -> mark(N)
active(U41(tt, V2)) -> mark(U42(isLNat(V2)))
active(U42(tt)) -> mark(tt)
active(U51(tt, V2)) -> mark(U52(isLNat(V2)))
active(U52(tt)) -> mark(tt)
active(U61(tt)) -> mark(tt)
active(U71(tt)) -> mark(tt)
active(U81(tt)) -> mark(tt)
active(U91(tt)) -> mark(tt)
active(afterNth(N, XS)) -> mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) -> mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) -> mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) -> mark(tt)
active(isLNat(afterNth(V1, V2))) -> mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) -> mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) -> mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) -> mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) -> mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) -> mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) -> mark(U101(isNatural(V1), V2))
active(isNatural(0')) -> mark(tt)
active(isNatural(head(V1))) -> mark(U111(isLNat(V1)))
active(isNatural(s(V1))) -> mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) -> mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) -> mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) -> mark(U151(isNatural(V1), V2))
active(natsFrom(N)) -> mark(U161(isNatural(N), N))
active(sel(N, XS)) -> mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) -> mark(U181(isLNat(X), Y))
active(splitAt(0', XS)) -> mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) -> mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) -> mark(U211(isNatural(N), XS))
active(take(N, XS)) -> mark(U221(isNatural(N), N, XS))
active(U101(X1, X2)) -> U101(active(X1), X2)
active(U102(X)) -> U102(active(X))
active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3)
active(U111(X)) -> U111(active(X))
active(snd(X)) -> snd(active(X))
active(splitAt(X1, X2)) -> splitAt(active(X1), X2)
active(splitAt(X1, X2)) -> splitAt(X1, active(X2))
active(U121(X)) -> U121(active(X))
active(U131(X1, X2)) -> U131(active(X1), X2)
active(U132(X)) -> U132(active(X))
active(U141(X1, X2)) -> U141(active(X1), X2)
active(U142(X)) -> U142(active(X))
active(U151(X1, X2)) -> U151(active(X1), X2)
active(U152(X)) -> U152(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(U172(X1, X2, X3)) -> U172(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(U182(X1, X2)) -> U182(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, X3, X4)) -> U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) -> U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) -> U204(active(X1), X2)
active(U21(X1, X2, X3)) -> U21(active(X1), X2, X3)
active(U22(X1, X2)) -> U22(active(X1), X2)
active(U211(X1, X2)) -> U211(active(X1), X2)
active(U212(X1, X2)) -> U212(active(X1), X2)
active(U221(X1, X2, X3)) -> U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) -> U222(active(X1), X2, X3)
active(fst(X)) -> fst(active(X))
active(U31(X1, X2, X3)) -> U31(active(X1), X2, X3)
active(U32(X1, X2)) -> U32(active(X1), X2)
active(U41(X1, X2)) -> U41(active(X1), X2)
active(U42(X)) -> U42(active(X))
active(U51(X1, X2)) -> U51(active(X1), X2)
active(U52(X)) -> U52(active(X))
active(U61(X)) -> U61(active(X))
active(U71(X)) -> U71(active(X))
active(U81(X)) -> U81(active(X))
active(U91(X)) -> U91(active(X))
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) -> mark(U101(X1, X2))
U102(mark(X)) -> mark(U102(X))
U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3))
U111(mark(X)) -> mark(U111(X))
snd(mark(X)) -> mark(snd(X))
splitAt(mark(X1), X2) -> mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) -> mark(splitAt(X1, X2))
U121(mark(X)) -> mark(U121(X))
U131(mark(X1), X2) -> mark(U131(X1, X2))
U132(mark(X)) -> mark(U132(X))
U141(mark(X1), X2) -> mark(U141(X1, X2))
U142(mark(X)) -> mark(U142(X))
U151(mark(X1), X2) -> mark(U151(X1, X2))
U152(mark(X)) -> mark(U152(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))
U172(mark(X1), X2, X3) -> mark(U172(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))
U182(mark(X1), X2) -> mark(U182(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, X3, X4) -> mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) -> mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) -> mark(U204(X1, X2))
U21(mark(X1), X2, X3) -> mark(U21(X1, X2, X3))
U22(mark(X1), X2) -> mark(U22(X1, X2))
U211(mark(X1), X2) -> mark(U211(X1, X2))
U212(mark(X1), X2) -> mark(U212(X1, X2))
U221(mark(X1), X2, X3) -> mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) -> mark(U222(X1, X2, X3))
fst(mark(X)) -> mark(fst(X))
U31(mark(X1), X2, X3) -> mark(U31(X1, X2, X3))
U32(mark(X1), X2) -> mark(U32(X1, X2))
U41(mark(X1), X2) -> mark(U41(X1, X2))
U42(mark(X)) -> mark(U42(X))
U51(mark(X1), X2) -> mark(U51(X1, X2))
U52(mark(X)) -> mark(U52(X))
U61(mark(X)) -> mark(U61(X))
U71(mark(X)) -> mark(U71(X))
U81(mark(X)) -> mark(U81(X))
U91(mark(X)) -> mark(U91(X))
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)) -> U101(proper(X1), proper(X2))
proper(tt) -> ok(tt)
proper(U102(X)) -> U102(proper(X))
proper(isLNat(X)) -> isLNat(proper(X))
proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) -> U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) -> U111(proper(X))
proper(snd(X)) -> snd(proper(X))
proper(splitAt(X1, X2)) -> splitAt(proper(X1), proper(X2))
proper(U121(X)) -> U121(proper(X))
proper(U131(X1, X2)) -> U131(proper(X1), proper(X2))
proper(U132(X)) -> U132(proper(X))
proper(U141(X1, X2)) -> U141(proper(X1), proper(X2))
proper(U142(X)) -> U142(proper(X))
proper(U151(X1, X2)) -> U151(proper(X1), proper(X2))
proper(U152(X)) -> U152(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(U172(X1, X2, X3)) -> U172(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(U182(X1, X2)) -> U182(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, X3, X4)) -> U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) -> isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) -> U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) -> U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) -> U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) -> U22(proper(X1), proper(X2))
proper(U211(X1, X2)) -> U211(proper(X1), proper(X2))
proper(U212(X1, X2)) -> U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) -> U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) -> U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) -> fst(proper(X))
proper(U31(X1, X2, X3)) -> U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) -> U32(proper(X1), proper(X2))
proper(U41(X1, X2)) -> U41(proper(X1), proper(X2))
proper(U42(X)) -> U42(proper(X))
proper(U51(X1, X2)) -> U51(proper(X1), proper(X2))
proper(U52(X)) -> U52(proper(X))
proper(U61(X)) -> U61(proper(X))
proper(U71(X)) -> U71(proper(X))
proper(U81(X)) -> U81(proper(X))
proper(U91(X)) -> U91(proper(X))
proper(isPLNat(X)) -> isPLNat(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(U101(X1, X2))
U102(ok(X)) -> ok(U102(X))
isLNat(ok(X)) -> ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))
U111(ok(X)) -> ok(U111(X))
snd(ok(X)) -> ok(snd(X))
splitAt(ok(X1), ok(X2)) -> ok(splitAt(X1, X2))
U121(ok(X)) -> ok(U121(X))
U131(ok(X1), ok(X2)) -> ok(U131(X1, X2))
U132(ok(X)) -> ok(U132(X))
U141(ok(X1), ok(X2)) -> ok(U141(X1, X2))
U142(ok(X)) -> ok(U142(X))
U151(ok(X1), ok(X2)) -> ok(U151(X1, X2))
U152(ok(X)) -> ok(U152(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))
U172(ok(X1), ok(X2), ok(X3)) -> ok(U172(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))
U182(ok(X1), ok(X2)) -> ok(U182(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(X3), ok(X4)) -> ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) -> ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) -> ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) -> ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) -> ok(U22(X1, X2))
U211(ok(X1), ok(X2)) -> ok(U211(X1, X2))
U212(ok(X1), ok(X2)) -> ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) -> ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) -> ok(U222(X1, X2, X3))
fst(ok(X)) -> ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) -> ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) -> ok(U32(X1, X2))
U41(ok(X1), ok(X2)) -> ok(U41(X1, X2))
U42(ok(X)) -> ok(U42(X))
U51(ok(X1), ok(X2)) -> ok(U51(X1, X2))
U52(ok(X)) -> ok(U52(X))
U61(ok(X)) -> ok(U61(X))
U71(ok(X)) -> ok(U71(X))
U81(ok(X)) -> ok(U81(X))
U91(ok(X)) -> ok(U91(X))
isPLNat(ok(X)) -> ok(isPLNat(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))

Types:
active :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U12 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U172 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U182 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U203 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U204 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U22 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U212 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U222 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U32 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok -> top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat -> tt:mark:nil:0':ok


Lemmas:
U102(gen_tt:mark:nil:0':ok3_0(+(1, n5_0))) -> *4_0, rt in Omega(n5_0)
U12(gen_tt:mark:nil:0':ok3_0(+(1, n1128_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) -> *4_0, rt in Omega(n1128_0)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n9121_0))) -> *4_0, rt in Omega(n9121_0)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n10537_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n10537_0)
U132(gen_tt:mark:nil:0':ok3_0(+(1, n15953_0))) -> *4_0, rt in Omega(n15953_0)
U142(gen_tt:mark:nil:0':ok3_0(+(1, n17620_0))) -> *4_0, rt in Omega(n17620_0)
U152(gen_tt:mark:nil:0':ok3_0(+(1, n19388_0))) -> *4_0, rt in Omega(n19388_0)
cons(gen_tt:mark:nil:0':ok3_0(+(1, n21257_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n21257_0)
natsFrom(gen_tt:mark:nil:0':ok3_0(+(1, n27404_0))) -> *4_0, rt in Omega(n27404_0)
s(gen_tt:mark:nil:0':ok3_0(+(1, n29524_0))) -> *4_0, rt in Omega(n29524_0)
U172(gen_tt:mark:nil:0':ok3_0(+(1, n31745_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) -> *4_0, rt in Omega(n31745_0)
head(gen_tt:mark:nil:0':ok3_0(+(1, n43161_0))) -> *4_0, rt in Omega(n43161_0)
afterNth(gen_tt:mark:nil:0':ok3_0(+(1, n45682_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n45682_0)
U182(gen_tt:mark:nil:0':ok3_0(+(1, n53362_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n53362_0)
pair(gen_tt:mark:nil:0':ok3_0(+(1, n61149_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n61149_0)
U202(gen_tt:mark:nil:0':ok3_0(+(1, n69441_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c), gen_tt:mark:nil:0':ok3_0(d)) -> *4_0, rt in Omega(n69441_0)
U203(gen_tt:mark:nil:0':ok3_0(+(1, n88570_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c), gen_tt:mark:nil:0':ok3_0(d)) -> *4_0, rt in Omega(n88570_0)
U204(gen_tt:mark:nil:0':ok3_0(+(1, n108633_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n108633_0)
U22(gen_tt:mark:nil:0':ok3_0(+(1, n118036_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n118036_0)
U212(gen_tt:mark:nil:0':ok3_0(+(1, n127745_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n127745_0)
U222(gen_tt:mark:nil:0':ok3_0(+(1, n137760_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) -> *4_0, rt in Omega(n137760_0)
fst(gen_tt:mark:nil:0':ok3_0(+(1, n154384_0))) -> *4_0, rt in Omega(n154384_0)
U32(gen_tt:mark:nil:0':ok3_0(+(1, n158601_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n158601_0)
U42(gen_tt:mark:nil:0':ok3_0(+(1, n169534_0))) -> *4_0, rt in Omega(n169534_0)
U52(gen_tt:mark:nil:0':ok3_0(+(1, n174002_0))) -> *4_0, rt in Omega(n174002_0)
U11(gen_tt:mark:nil:0':ok3_0(+(1, n178571_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) -> *4_0, rt in Omega(n178571_0)
U21(gen_tt:mark:nil:0':ok3_0(+(1, n197211_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) -> *4_0, rt in Omega(n197211_0)
U31(gen_tt:mark:nil:0':ok3_0(+(1, n216460_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) -> *4_0, rt in Omega(n216460_0)
U41(gen_tt:mark:nil:0':ok3_0(+(1, n236318_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n236318_0)


Generator Equations:
gen_tt:mark:nil:0':ok3_0(0) <=> tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) <=> mark(gen_tt:mark:nil:0':ok3_0(x))


The following defined symbols remain to be analysed:
U51, active, U61, isPLNat, U71, U81, U91, U101, U111, U121, U131, U141, U151, U161, U171, U181, U191, U201, U211, U221, tail, take, sel, proper, top

They will be analysed ascendingly in the following order:
U51 < active
U61 < active
isPLNat < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
U141 < active
U151 < active
U161 < active
U171 < active
U181 < active
U191 < active
U201 < active
U211 < active
U221 < active
tail < active
take < active
sel < active
active < top
U51 < proper
U61 < proper
isPLNat < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
U141 < proper
U151 < proper
U161 < proper
U171 < proper
U181 < proper
U191 < proper
U201 < proper
U211 < proper
U221 < proper
tail < proper
take < proper
sel < proper
proper < top

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

(87) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
U51(gen_tt:mark:nil:0':ok3_0(+(1, n249185_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n249185_0)

Induction Base:
U51(gen_tt:mark:nil:0':ok3_0(+(1, 0)), gen_tt:mark:nil:0':ok3_0(b))

Induction Step:
U51(gen_tt:mark:nil:0':ok3_0(+(1, +(n249185_0, 1))), gen_tt:mark:nil:0':ok3_0(b)) ->_R^Omega(1)
mark(U51(gen_tt:mark:nil:0':ok3_0(+(1, n249185_0)), gen_tt:mark:nil:0':ok3_0(b))) ->_IH
mark(*4_0)

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
----------------------------------------

(88)
Obligation:
TRS:
Rules:
active(U101(tt, V2)) -> mark(U102(isLNat(V2)))
active(U102(tt)) -> mark(tt)
active(U11(tt, N, XS)) -> mark(U12(isLNat(XS), N, XS))
active(U111(tt)) -> mark(tt)
active(U12(tt, N, XS)) -> mark(snd(splitAt(N, XS)))
active(U121(tt)) -> mark(tt)
active(U131(tt, V2)) -> mark(U132(isLNat(V2)))
active(U132(tt)) -> mark(tt)
active(U141(tt, V2)) -> mark(U142(isLNat(V2)))
active(U142(tt)) -> mark(tt)
active(U151(tt, V2)) -> mark(U152(isLNat(V2)))
active(U152(tt)) -> mark(tt)
active(U161(tt, N)) -> mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) -> mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) -> mark(head(afterNth(N, XS)))
active(U181(tt, Y)) -> mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) -> mark(Y)
active(U191(tt, XS)) -> mark(pair(nil, XS))
active(U201(tt, N, X, XS)) -> mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) -> mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) -> mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) -> mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) -> mark(U22(isLNat(Y), X))
active(U211(tt, XS)) -> mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) -> mark(XS)
active(U22(tt, X)) -> mark(X)
active(U221(tt, N, XS)) -> mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) -> mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) -> mark(U32(isLNat(XS), N))
active(U32(tt, N)) -> mark(N)
active(U41(tt, V2)) -> mark(U42(isLNat(V2)))
active(U42(tt)) -> mark(tt)
active(U51(tt, V2)) -> mark(U52(isLNat(V2)))
active(U52(tt)) -> mark(tt)
active(U61(tt)) -> mark(tt)
active(U71(tt)) -> mark(tt)
active(U81(tt)) -> mark(tt)
active(U91(tt)) -> mark(tt)
active(afterNth(N, XS)) -> mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) -> mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) -> mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) -> mark(tt)
active(isLNat(afterNth(V1, V2))) -> mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) -> mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) -> mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) -> mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) -> mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) -> mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) -> mark(U101(isNatural(V1), V2))
active(isNatural(0')) -> mark(tt)
active(isNatural(head(V1))) -> mark(U111(isLNat(V1)))
active(isNatural(s(V1))) -> mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) -> mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) -> mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) -> mark(U151(isNatural(V1), V2))
active(natsFrom(N)) -> mark(U161(isNatural(N), N))
active(sel(N, XS)) -> mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) -> mark(U181(isLNat(X), Y))
active(splitAt(0', XS)) -> mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) -> mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) -> mark(U211(isNatural(N), XS))
active(take(N, XS)) -> mark(U221(isNatural(N), N, XS))
active(U101(X1, X2)) -> U101(active(X1), X2)
active(U102(X)) -> U102(active(X))
active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3)
active(U111(X)) -> U111(active(X))
active(snd(X)) -> snd(active(X))
active(splitAt(X1, X2)) -> splitAt(active(X1), X2)
active(splitAt(X1, X2)) -> splitAt(X1, active(X2))
active(U121(X)) -> U121(active(X))
active(U131(X1, X2)) -> U131(active(X1), X2)
active(U132(X)) -> U132(active(X))
active(U141(X1, X2)) -> U141(active(X1), X2)
active(U142(X)) -> U142(active(X))
active(U151(X1, X2)) -> U151(active(X1), X2)
active(U152(X)) -> U152(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(U172(X1, X2, X3)) -> U172(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(U182(X1, X2)) -> U182(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, X3, X4)) -> U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) -> U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) -> U204(active(X1), X2)
active(U21(X1, X2, X3)) -> U21(active(X1), X2, X3)
active(U22(X1, X2)) -> U22(active(X1), X2)
active(U211(X1, X2)) -> U211(active(X1), X2)
active(U212(X1, X2)) -> U212(active(X1), X2)
active(U221(X1, X2, X3)) -> U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) -> U222(active(X1), X2, X3)
active(fst(X)) -> fst(active(X))
active(U31(X1, X2, X3)) -> U31(active(X1), X2, X3)
active(U32(X1, X2)) -> U32(active(X1), X2)
active(U41(X1, X2)) -> U41(active(X1), X2)
active(U42(X)) -> U42(active(X))
active(U51(X1, X2)) -> U51(active(X1), X2)
active(U52(X)) -> U52(active(X))
active(U61(X)) -> U61(active(X))
active(U71(X)) -> U71(active(X))
active(U81(X)) -> U81(active(X))
active(U91(X)) -> U91(active(X))
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) -> mark(U101(X1, X2))
U102(mark(X)) -> mark(U102(X))
U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3))
U111(mark(X)) -> mark(U111(X))
snd(mark(X)) -> mark(snd(X))
splitAt(mark(X1), X2) -> mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) -> mark(splitAt(X1, X2))
U121(mark(X)) -> mark(U121(X))
U131(mark(X1), X2) -> mark(U131(X1, X2))
U132(mark(X)) -> mark(U132(X))
U141(mark(X1), X2) -> mark(U141(X1, X2))
U142(mark(X)) -> mark(U142(X))
U151(mark(X1), X2) -> mark(U151(X1, X2))
U152(mark(X)) -> mark(U152(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))
U172(mark(X1), X2, X3) -> mark(U172(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))
U182(mark(X1), X2) -> mark(U182(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, X3, X4) -> mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) -> mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) -> mark(U204(X1, X2))
U21(mark(X1), X2, X3) -> mark(U21(X1, X2, X3))
U22(mark(X1), X2) -> mark(U22(X1, X2))
U211(mark(X1), X2) -> mark(U211(X1, X2))
U212(mark(X1), X2) -> mark(U212(X1, X2))
U221(mark(X1), X2, X3) -> mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) -> mark(U222(X1, X2, X3))
fst(mark(X)) -> mark(fst(X))
U31(mark(X1), X2, X3) -> mark(U31(X1, X2, X3))
U32(mark(X1), X2) -> mark(U32(X1, X2))
U41(mark(X1), X2) -> mark(U41(X1, X2))
U42(mark(X)) -> mark(U42(X))
U51(mark(X1), X2) -> mark(U51(X1, X2))
U52(mark(X)) -> mark(U52(X))
U61(mark(X)) -> mark(U61(X))
U71(mark(X)) -> mark(U71(X))
U81(mark(X)) -> mark(U81(X))
U91(mark(X)) -> mark(U91(X))
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)) -> U101(proper(X1), proper(X2))
proper(tt) -> ok(tt)
proper(U102(X)) -> U102(proper(X))
proper(isLNat(X)) -> isLNat(proper(X))
proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) -> U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) -> U111(proper(X))
proper(snd(X)) -> snd(proper(X))
proper(splitAt(X1, X2)) -> splitAt(proper(X1), proper(X2))
proper(U121(X)) -> U121(proper(X))
proper(U131(X1, X2)) -> U131(proper(X1), proper(X2))
proper(U132(X)) -> U132(proper(X))
proper(U141(X1, X2)) -> U141(proper(X1), proper(X2))
proper(U142(X)) -> U142(proper(X))
proper(U151(X1, X2)) -> U151(proper(X1), proper(X2))
proper(U152(X)) -> U152(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(U172(X1, X2, X3)) -> U172(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(U182(X1, X2)) -> U182(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, X3, X4)) -> U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) -> isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) -> U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) -> U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) -> U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) -> U22(proper(X1), proper(X2))
proper(U211(X1, X2)) -> U211(proper(X1), proper(X2))
proper(U212(X1, X2)) -> U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) -> U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) -> U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) -> fst(proper(X))
proper(U31(X1, X2, X3)) -> U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) -> U32(proper(X1), proper(X2))
proper(U41(X1, X2)) -> U41(proper(X1), proper(X2))
proper(U42(X)) -> U42(proper(X))
proper(U51(X1, X2)) -> U51(proper(X1), proper(X2))
proper(U52(X)) -> U52(proper(X))
proper(U61(X)) -> U61(proper(X))
proper(U71(X)) -> U71(proper(X))
proper(U81(X)) -> U81(proper(X))
proper(U91(X)) -> U91(proper(X))
proper(isPLNat(X)) -> isPLNat(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(U101(X1, X2))
U102(ok(X)) -> ok(U102(X))
isLNat(ok(X)) -> ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))
U111(ok(X)) -> ok(U111(X))
snd(ok(X)) -> ok(snd(X))
splitAt(ok(X1), ok(X2)) -> ok(splitAt(X1, X2))
U121(ok(X)) -> ok(U121(X))
U131(ok(X1), ok(X2)) -> ok(U131(X1, X2))
U132(ok(X)) -> ok(U132(X))
U141(ok(X1), ok(X2)) -> ok(U141(X1, X2))
U142(ok(X)) -> ok(U142(X))
U151(ok(X1), ok(X2)) -> ok(U151(X1, X2))
U152(ok(X)) -> ok(U152(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))
U172(ok(X1), ok(X2), ok(X3)) -> ok(U172(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))
U182(ok(X1), ok(X2)) -> ok(U182(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(X3), ok(X4)) -> ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) -> ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) -> ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) -> ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) -> ok(U22(X1, X2))
U211(ok(X1), ok(X2)) -> ok(U211(X1, X2))
U212(ok(X1), ok(X2)) -> ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) -> ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) -> ok(U222(X1, X2, X3))
fst(ok(X)) -> ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) -> ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) -> ok(U32(X1, X2))
U41(ok(X1), ok(X2)) -> ok(U41(X1, X2))
U42(ok(X)) -> ok(U42(X))
U51(ok(X1), ok(X2)) -> ok(U51(X1, X2))
U52(ok(X)) -> ok(U52(X))
U61(ok(X)) -> ok(U61(X))
U71(ok(X)) -> ok(U71(X))
U81(ok(X)) -> ok(U81(X))
U91(ok(X)) -> ok(U91(X))
isPLNat(ok(X)) -> ok(isPLNat(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))

Types:
active :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U12 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U172 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U182 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U203 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U204 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U22 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U212 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U222 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U32 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok -> top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat -> tt:mark:nil:0':ok


Lemmas:
U102(gen_tt:mark:nil:0':ok3_0(+(1, n5_0))) -> *4_0, rt in Omega(n5_0)
U12(gen_tt:mark:nil:0':ok3_0(+(1, n1128_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) -> *4_0, rt in Omega(n1128_0)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n9121_0))) -> *4_0, rt in Omega(n9121_0)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n10537_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n10537_0)
U132(gen_tt:mark:nil:0':ok3_0(+(1, n15953_0))) -> *4_0, rt in Omega(n15953_0)
U142(gen_tt:mark:nil:0':ok3_0(+(1, n17620_0))) -> *4_0, rt in Omega(n17620_0)
U152(gen_tt:mark:nil:0':ok3_0(+(1, n19388_0))) -> *4_0, rt in Omega(n19388_0)
cons(gen_tt:mark:nil:0':ok3_0(+(1, n21257_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n21257_0)
natsFrom(gen_tt:mark:nil:0':ok3_0(+(1, n27404_0))) -> *4_0, rt in Omega(n27404_0)
s(gen_tt:mark:nil:0':ok3_0(+(1, n29524_0))) -> *4_0, rt in Omega(n29524_0)
U172(gen_tt:mark:nil:0':ok3_0(+(1, n31745_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) -> *4_0, rt in Omega(n31745_0)
head(gen_tt:mark:nil:0':ok3_0(+(1, n43161_0))) -> *4_0, rt in Omega(n43161_0)
afterNth(gen_tt:mark:nil:0':ok3_0(+(1, n45682_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n45682_0)
U182(gen_tt:mark:nil:0':ok3_0(+(1, n53362_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n53362_0)
pair(gen_tt:mark:nil:0':ok3_0(+(1, n61149_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n61149_0)
U202(gen_tt:mark:nil:0':ok3_0(+(1, n69441_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c), gen_tt:mark:nil:0':ok3_0(d)) -> *4_0, rt in Omega(n69441_0)
U203(gen_tt:mark:nil:0':ok3_0(+(1, n88570_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c), gen_tt:mark:nil:0':ok3_0(d)) -> *4_0, rt in Omega(n88570_0)
U204(gen_tt:mark:nil:0':ok3_0(+(1, n108633_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n108633_0)
U22(gen_tt:mark:nil:0':ok3_0(+(1, n118036_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n118036_0)
U212(gen_tt:mark:nil:0':ok3_0(+(1, n127745_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n127745_0)
U222(gen_tt:mark:nil:0':ok3_0(+(1, n137760_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) -> *4_0, rt in Omega(n137760_0)
fst(gen_tt:mark:nil:0':ok3_0(+(1, n154384_0))) -> *4_0, rt in Omega(n154384_0)
U32(gen_tt:mark:nil:0':ok3_0(+(1, n158601_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n158601_0)
U42(gen_tt:mark:nil:0':ok3_0(+(1, n169534_0))) -> *4_0, rt in Omega(n169534_0)
U52(gen_tt:mark:nil:0':ok3_0(+(1, n174002_0))) -> *4_0, rt in Omega(n174002_0)
U11(gen_tt:mark:nil:0':ok3_0(+(1, n178571_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) -> *4_0, rt in Omega(n178571_0)
U21(gen_tt:mark:nil:0':ok3_0(+(1, n197211_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) -> *4_0, rt in Omega(n197211_0)
U31(gen_tt:mark:nil:0':ok3_0(+(1, n216460_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) -> *4_0, rt in Omega(n216460_0)
U41(gen_tt:mark:nil:0':ok3_0(+(1, n236318_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n236318_0)
U51(gen_tt:mark:nil:0':ok3_0(+(1, n249185_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n249185_0)


Generator Equations:
gen_tt:mark:nil:0':ok3_0(0) <=> tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) <=> mark(gen_tt:mark:nil:0':ok3_0(x))


The following defined symbols remain to be analysed:
U61, active, isPLNat, U71, U81, U91, U101, U111, U121, U131, U141, U151, U161, U171, U181, U191, U201, U211, U221, tail, take, sel, proper, top

They will be analysed ascendingly in the following order:
U61 < active
isPLNat < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
U141 < active
U151 < active
U161 < active
U171 < active
U181 < active
U191 < active
U201 < active
U211 < active
U221 < active
tail < active
take < active
sel < active
active < top
U61 < proper
isPLNat < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
U141 < proper
U151 < proper
U161 < proper
U171 < proper
U181 < proper
U191 < proper
U201 < proper
U211 < proper
U221 < proper
tail < proper
take < proper
sel < proper
proper < top

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

(89) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
U61(gen_tt:mark:nil:0':ok3_0(+(1, n262358_0))) -> *4_0, rt in Omega(n262358_0)

Induction Base:
U61(gen_tt:mark:nil:0':ok3_0(+(1, 0)))

Induction Step:
U61(gen_tt:mark:nil:0':ok3_0(+(1, +(n262358_0, 1)))) ->_R^Omega(1)
mark(U61(gen_tt:mark:nil:0':ok3_0(+(1, n262358_0)))) ->_IH
mark(*4_0)

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
----------------------------------------

(90)
Obligation:
TRS:
Rules:
active(U101(tt, V2)) -> mark(U102(isLNat(V2)))
active(U102(tt)) -> mark(tt)
active(U11(tt, N, XS)) -> mark(U12(isLNat(XS), N, XS))
active(U111(tt)) -> mark(tt)
active(U12(tt, N, XS)) -> mark(snd(splitAt(N, XS)))
active(U121(tt)) -> mark(tt)
active(U131(tt, V2)) -> mark(U132(isLNat(V2)))
active(U132(tt)) -> mark(tt)
active(U141(tt, V2)) -> mark(U142(isLNat(V2)))
active(U142(tt)) -> mark(tt)
active(U151(tt, V2)) -> mark(U152(isLNat(V2)))
active(U152(tt)) -> mark(tt)
active(U161(tt, N)) -> mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) -> mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) -> mark(head(afterNth(N, XS)))
active(U181(tt, Y)) -> mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) -> mark(Y)
active(U191(tt, XS)) -> mark(pair(nil, XS))
active(U201(tt, N, X, XS)) -> mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) -> mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) -> mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) -> mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) -> mark(U22(isLNat(Y), X))
active(U211(tt, XS)) -> mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) -> mark(XS)
active(U22(tt, X)) -> mark(X)
active(U221(tt, N, XS)) -> mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) -> mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) -> mark(U32(isLNat(XS), N))
active(U32(tt, N)) -> mark(N)
active(U41(tt, V2)) -> mark(U42(isLNat(V2)))
active(U42(tt)) -> mark(tt)
active(U51(tt, V2)) -> mark(U52(isLNat(V2)))
active(U52(tt)) -> mark(tt)
active(U61(tt)) -> mark(tt)
active(U71(tt)) -> mark(tt)
active(U81(tt)) -> mark(tt)
active(U91(tt)) -> mark(tt)
active(afterNth(N, XS)) -> mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) -> mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) -> mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) -> mark(tt)
active(isLNat(afterNth(V1, V2))) -> mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) -> mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) -> mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) -> mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) -> mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) -> mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) -> mark(U101(isNatural(V1), V2))
active(isNatural(0')) -> mark(tt)
active(isNatural(head(V1))) -> mark(U111(isLNat(V1)))
active(isNatural(s(V1))) -> mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) -> mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) -> mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) -> mark(U151(isNatural(V1), V2))
active(natsFrom(N)) -> mark(U161(isNatural(N), N))
active(sel(N, XS)) -> mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) -> mark(U181(isLNat(X), Y))
active(splitAt(0', XS)) -> mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) -> mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) -> mark(U211(isNatural(N), XS))
active(take(N, XS)) -> mark(U221(isNatural(N), N, XS))
active(U101(X1, X2)) -> U101(active(X1), X2)
active(U102(X)) -> U102(active(X))
active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3)
active(U111(X)) -> U111(active(X))
active(snd(X)) -> snd(active(X))
active(splitAt(X1, X2)) -> splitAt(active(X1), X2)
active(splitAt(X1, X2)) -> splitAt(X1, active(X2))
active(U121(X)) -> U121(active(X))
active(U131(X1, X2)) -> U131(active(X1), X2)
active(U132(X)) -> U132(active(X))
active(U141(X1, X2)) -> U141(active(X1), X2)
active(U142(X)) -> U142(active(X))
active(U151(X1, X2)) -> U151(active(X1), X2)
active(U152(X)) -> U152(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(U172(X1, X2, X3)) -> U172(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(U182(X1, X2)) -> U182(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, X3, X4)) -> U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) -> U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) -> U204(active(X1), X2)
active(U21(X1, X2, X3)) -> U21(active(X1), X2, X3)
active(U22(X1, X2)) -> U22(active(X1), X2)
active(U211(X1, X2)) -> U211(active(X1), X2)
active(U212(X1, X2)) -> U212(active(X1), X2)
active(U221(X1, X2, X3)) -> U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) -> U222(active(X1), X2, X3)
active(fst(X)) -> fst(active(X))
active(U31(X1, X2, X3)) -> U31(active(X1), X2, X3)
active(U32(X1, X2)) -> U32(active(X1), X2)
active(U41(X1, X2)) -> U41(active(X1), X2)
active(U42(X)) -> U42(active(X))
active(U51(X1, X2)) -> U51(active(X1), X2)
active(U52(X)) -> U52(active(X))
active(U61(X)) -> U61(active(X))
active(U71(X)) -> U71(active(X))
active(U81(X)) -> U81(active(X))
active(U91(X)) -> U91(active(X))
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) -> mark(U101(X1, X2))
U102(mark(X)) -> mark(U102(X))
U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3))
U111(mark(X)) -> mark(U111(X))
snd(mark(X)) -> mark(snd(X))
splitAt(mark(X1), X2) -> mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) -> mark(splitAt(X1, X2))
U121(mark(X)) -> mark(U121(X))
U131(mark(X1), X2) -> mark(U131(X1, X2))
U132(mark(X)) -> mark(U132(X))
U141(mark(X1), X2) -> mark(U141(X1, X2))
U142(mark(X)) -> mark(U142(X))
U151(mark(X1), X2) -> mark(U151(X1, X2))
U152(mark(X)) -> mark(U152(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))
U172(mark(X1), X2, X3) -> mark(U172(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))
U182(mark(X1), X2) -> mark(U182(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, X3, X4) -> mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) -> mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) -> mark(U204(X1, X2))
U21(mark(X1), X2, X3) -> mark(U21(X1, X2, X3))
U22(mark(X1), X2) -> mark(U22(X1, X2))
U211(mark(X1), X2) -> mark(U211(X1, X2))
U212(mark(X1), X2) -> mark(U212(X1, X2))
U221(mark(X1), X2, X3) -> mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) -> mark(U222(X1, X2, X3))
fst(mark(X)) -> mark(fst(X))
U31(mark(X1), X2, X3) -> mark(U31(X1, X2, X3))
U32(mark(X1), X2) -> mark(U32(X1, X2))
U41(mark(X1), X2) -> mark(U41(X1, X2))
U42(mark(X)) -> mark(U42(X))
U51(mark(X1), X2) -> mark(U51(X1, X2))
U52(mark(X)) -> mark(U52(X))
U61(mark(X)) -> mark(U61(X))
U71(mark(X)) -> mark(U71(X))
U81(mark(X)) -> mark(U81(X))
U91(mark(X)) -> mark(U91(X))
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)) -> U101(proper(X1), proper(X2))
proper(tt) -> ok(tt)
proper(U102(X)) -> U102(proper(X))
proper(isLNat(X)) -> isLNat(proper(X))
proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) -> U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) -> U111(proper(X))
proper(snd(X)) -> snd(proper(X))
proper(splitAt(X1, X2)) -> splitAt(proper(X1), proper(X2))
proper(U121(X)) -> U121(proper(X))
proper(U131(X1, X2)) -> U131(proper(X1), proper(X2))
proper(U132(X)) -> U132(proper(X))
proper(U141(X1, X2)) -> U141(proper(X1), proper(X2))
proper(U142(X)) -> U142(proper(X))
proper(U151(X1, X2)) -> U151(proper(X1), proper(X2))
proper(U152(X)) -> U152(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(U172(X1, X2, X3)) -> U172(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(U182(X1, X2)) -> U182(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, X3, X4)) -> U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) -> isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) -> U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) -> U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) -> U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) -> U22(proper(X1), proper(X2))
proper(U211(X1, X2)) -> U211(proper(X1), proper(X2))
proper(U212(X1, X2)) -> U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) -> U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) -> U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) -> fst(proper(X))
proper(U31(X1, X2, X3)) -> U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) -> U32(proper(X1), proper(X2))
proper(U41(X1, X2)) -> U41(proper(X1), proper(X2))
proper(U42(X)) -> U42(proper(X))
proper(U51(X1, X2)) -> U51(proper(X1), proper(X2))
proper(U52(X)) -> U52(proper(X))
proper(U61(X)) -> U61(proper(X))
proper(U71(X)) -> U71(proper(X))
proper(U81(X)) -> U81(proper(X))
proper(U91(X)) -> U91(proper(X))
proper(isPLNat(X)) -> isPLNat(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(U101(X1, X2))
U102(ok(X)) -> ok(U102(X))
isLNat(ok(X)) -> ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))
U111(ok(X)) -> ok(U111(X))
snd(ok(X)) -> ok(snd(X))
splitAt(ok(X1), ok(X2)) -> ok(splitAt(X1, X2))
U121(ok(X)) -> ok(U121(X))
U131(ok(X1), ok(X2)) -> ok(U131(X1, X2))
U132(ok(X)) -> ok(U132(X))
U141(ok(X1), ok(X2)) -> ok(U141(X1, X2))
U142(ok(X)) -> ok(U142(X))
U151(ok(X1), ok(X2)) -> ok(U151(X1, X2))
U152(ok(X)) -> ok(U152(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))
U172(ok(X1), ok(X2), ok(X3)) -> ok(U172(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))
U182(ok(X1), ok(X2)) -> ok(U182(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(X3), ok(X4)) -> ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) -> ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) -> ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) -> ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) -> ok(U22(X1, X2))
U211(ok(X1), ok(X2)) -> ok(U211(X1, X2))
U212(ok(X1), ok(X2)) -> ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) -> ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) -> ok(U222(X1, X2, X3))
fst(ok(X)) -> ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) -> ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) -> ok(U32(X1, X2))
U41(ok(X1), ok(X2)) -> ok(U41(X1, X2))
U42(ok(X)) -> ok(U42(X))
U51(ok(X1), ok(X2)) -> ok(U51(X1, X2))
U52(ok(X)) -> ok(U52(X))
U61(ok(X)) -> ok(U61(X))
U71(ok(X)) -> ok(U71(X))
U81(ok(X)) -> ok(U81(X))
U91(ok(X)) -> ok(U91(X))
isPLNat(ok(X)) -> ok(isPLNat(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))

Types:
active :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U12 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U172 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U182 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U203 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U204 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U22 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U212 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U222 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U32 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok -> top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat -> tt:mark:nil:0':ok


Lemmas:
U102(gen_tt:mark:nil:0':ok3_0(+(1, n5_0))) -> *4_0, rt in Omega(n5_0)
U12(gen_tt:mark:nil:0':ok3_0(+(1, n1128_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) -> *4_0, rt in Omega(n1128_0)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n9121_0))) -> *4_0, rt in Omega(n9121_0)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n10537_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n10537_0)
U132(gen_tt:mark:nil:0':ok3_0(+(1, n15953_0))) -> *4_0, rt in Omega(n15953_0)
U142(gen_tt:mark:nil:0':ok3_0(+(1, n17620_0))) -> *4_0, rt in Omega(n17620_0)
U152(gen_tt:mark:nil:0':ok3_0(+(1, n19388_0))) -> *4_0, rt in Omega(n19388_0)
cons(gen_tt:mark:nil:0':ok3_0(+(1, n21257_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n21257_0)
natsFrom(gen_tt:mark:nil:0':ok3_0(+(1, n27404_0))) -> *4_0, rt in Omega(n27404_0)
s(gen_tt:mark:nil:0':ok3_0(+(1, n29524_0))) -> *4_0, rt in Omega(n29524_0)
U172(gen_tt:mark:nil:0':ok3_0(+(1, n31745_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) -> *4_0, rt in Omega(n31745_0)
head(gen_tt:mark:nil:0':ok3_0(+(1, n43161_0))) -> *4_0, rt in Omega(n43161_0)
afterNth(gen_tt:mark:nil:0':ok3_0(+(1, n45682_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n45682_0)
U182(gen_tt:mark:nil:0':ok3_0(+(1, n53362_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n53362_0)
pair(gen_tt:mark:nil:0':ok3_0(+(1, n61149_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n61149_0)
U202(gen_tt:mark:nil:0':ok3_0(+(1, n69441_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c), gen_tt:mark:nil:0':ok3_0(d)) -> *4_0, rt in Omega(n69441_0)
U203(gen_tt:mark:nil:0':ok3_0(+(1, n88570_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c), gen_tt:mark:nil:0':ok3_0(d)) -> *4_0, rt in Omega(n88570_0)
U204(gen_tt:mark:nil:0':ok3_0(+(1, n108633_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n108633_0)
U22(gen_tt:mark:nil:0':ok3_0(+(1, n118036_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n118036_0)
U212(gen_tt:mark:nil:0':ok3_0(+(1, n127745_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n127745_0)
U222(gen_tt:mark:nil:0':ok3_0(+(1, n137760_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) -> *4_0, rt in Omega(n137760_0)
fst(gen_tt:mark:nil:0':ok3_0(+(1, n154384_0))) -> *4_0, rt in Omega(n154384_0)
U32(gen_tt:mark:nil:0':ok3_0(+(1, n158601_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n158601_0)
U42(gen_tt:mark:nil:0':ok3_0(+(1, n169534_0))) -> *4_0, rt in Omega(n169534_0)
U52(gen_tt:mark:nil:0':ok3_0(+(1, n174002_0))) -> *4_0, rt in Omega(n174002_0)
U11(gen_tt:mark:nil:0':ok3_0(+(1, n178571_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) -> *4_0, rt in Omega(n178571_0)
U21(gen_tt:mark:nil:0':ok3_0(+(1, n197211_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) -> *4_0, rt in Omega(n197211_0)
U31(gen_tt:mark:nil:0':ok3_0(+(1, n216460_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) -> *4_0, rt in Omega(n216460_0)
U41(gen_tt:mark:nil:0':ok3_0(+(1, n236318_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n236318_0)
U51(gen_tt:mark:nil:0':ok3_0(+(1, n249185_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n249185_0)
U61(gen_tt:mark:nil:0':ok3_0(+(1, n262358_0))) -> *4_0, rt in Omega(n262358_0)


Generator Equations:
gen_tt:mark:nil:0':ok3_0(0) <=> tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) <=> mark(gen_tt:mark:nil:0':ok3_0(x))


The following defined symbols remain to be analysed:
isPLNat, active, U71, U81, U91, U101, U111, U121, U131, U141, U151, U161, U171, U181, U191, U201, U211, U221, tail, take, sel, proper, top

They will be analysed ascendingly in the following order:
isPLNat < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
U141 < active
U151 < active
U161 < active
U171 < active
U181 < active
U191 < active
U201 < active
U211 < active
U221 < active
tail < active
take < active
sel < active
active < top
isPLNat < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
U141 < proper
U151 < proper
U161 < proper
U171 < proper
U181 < proper
U191 < proper
U201 < proper
U211 < proper
U221 < proper
tail < proper
take < proper
sel < proper
proper < top

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

(91) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
U71(gen_tt:mark:nil:0':ok3_0(+(1, n268080_0))) -> *4_0, rt in Omega(n268080_0)

Induction Base:
U71(gen_tt:mark:nil:0':ok3_0(+(1, 0)))

Induction Step:
U71(gen_tt:mark:nil:0':ok3_0(+(1, +(n268080_0, 1)))) ->_R^Omega(1)
mark(U71(gen_tt:mark:nil:0':ok3_0(+(1, n268080_0)))) ->_IH
mark(*4_0)

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
----------------------------------------

(92)
Obligation:
TRS:
Rules:
active(U101(tt, V2)) -> mark(U102(isLNat(V2)))
active(U102(tt)) -> mark(tt)
active(U11(tt, N, XS)) -> mark(U12(isLNat(XS), N, XS))
active(U111(tt)) -> mark(tt)
active(U12(tt, N, XS)) -> mark(snd(splitAt(N, XS)))
active(U121(tt)) -> mark(tt)
active(U131(tt, V2)) -> mark(U132(isLNat(V2)))
active(U132(tt)) -> mark(tt)
active(U141(tt, V2)) -> mark(U142(isLNat(V2)))
active(U142(tt)) -> mark(tt)
active(U151(tt, V2)) -> mark(U152(isLNat(V2)))
active(U152(tt)) -> mark(tt)
active(U161(tt, N)) -> mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) -> mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) -> mark(head(afterNth(N, XS)))
active(U181(tt, Y)) -> mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) -> mark(Y)
active(U191(tt, XS)) -> mark(pair(nil, XS))
active(U201(tt, N, X, XS)) -> mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) -> mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) -> mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) -> mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) -> mark(U22(isLNat(Y), X))
active(U211(tt, XS)) -> mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) -> mark(XS)
active(U22(tt, X)) -> mark(X)
active(U221(tt, N, XS)) -> mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) -> mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) -> mark(U32(isLNat(XS), N))
active(U32(tt, N)) -> mark(N)
active(U41(tt, V2)) -> mark(U42(isLNat(V2)))
active(U42(tt)) -> mark(tt)
active(U51(tt, V2)) -> mark(U52(isLNat(V2)))
active(U52(tt)) -> mark(tt)
active(U61(tt)) -> mark(tt)
active(U71(tt)) -> mark(tt)
active(U81(tt)) -> mark(tt)
active(U91(tt)) -> mark(tt)
active(afterNth(N, XS)) -> mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) -> mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) -> mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) -> mark(tt)
active(isLNat(afterNth(V1, V2))) -> mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) -> mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) -> mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) -> mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) -> mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) -> mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) -> mark(U101(isNatural(V1), V2))
active(isNatural(0')) -> mark(tt)
active(isNatural(head(V1))) -> mark(U111(isLNat(V1)))
active(isNatural(s(V1))) -> mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) -> mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) -> mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) -> mark(U151(isNatural(V1), V2))
active(natsFrom(N)) -> mark(U161(isNatural(N), N))
active(sel(N, XS)) -> mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) -> mark(U181(isLNat(X), Y))
active(splitAt(0', XS)) -> mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) -> mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) -> mark(U211(isNatural(N), XS))
active(take(N, XS)) -> mark(U221(isNatural(N), N, XS))
active(U101(X1, X2)) -> U101(active(X1), X2)
active(U102(X)) -> U102(active(X))
active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3)
active(U111(X)) -> U111(active(X))
active(snd(X)) -> snd(active(X))
active(splitAt(X1, X2)) -> splitAt(active(X1), X2)
active(splitAt(X1, X2)) -> splitAt(X1, active(X2))
active(U121(X)) -> U121(active(X))
active(U131(X1, X2)) -> U131(active(X1), X2)
active(U132(X)) -> U132(active(X))
active(U141(X1, X2)) -> U141(active(X1), X2)
active(U142(X)) -> U142(active(X))
active(U151(X1, X2)) -> U151(active(X1), X2)
active(U152(X)) -> U152(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(U172(X1, X2, X3)) -> U172(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(U182(X1, X2)) -> U182(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, X3, X4)) -> U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) -> U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) -> U204(active(X1), X2)
active(U21(X1, X2, X3)) -> U21(active(X1), X2, X3)
active(U22(X1, X2)) -> U22(active(X1), X2)
active(U211(X1, X2)) -> U211(active(X1), X2)
active(U212(X1, X2)) -> U212(active(X1), X2)
active(U221(X1, X2, X3)) -> U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) -> U222(active(X1), X2, X3)
active(fst(X)) -> fst(active(X))
active(U31(X1, X2, X3)) -> U31(active(X1), X2, X3)
active(U32(X1, X2)) -> U32(active(X1), X2)
active(U41(X1, X2)) -> U41(active(X1), X2)
active(U42(X)) -> U42(active(X))
active(U51(X1, X2)) -> U51(active(X1), X2)
active(U52(X)) -> U52(active(X))
active(U61(X)) -> U61(active(X))
active(U71(X)) -> U71(active(X))
active(U81(X)) -> U81(active(X))
active(U91(X)) -> U91(active(X))
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) -> mark(U101(X1, X2))
U102(mark(X)) -> mark(U102(X))
U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3))
U111(mark(X)) -> mark(U111(X))
snd(mark(X)) -> mark(snd(X))
splitAt(mark(X1), X2) -> mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) -> mark(splitAt(X1, X2))
U121(mark(X)) -> mark(U121(X))
U131(mark(X1), X2) -> mark(U131(X1, X2))
U132(mark(X)) -> mark(U132(X))
U141(mark(X1), X2) -> mark(U141(X1, X2))
U142(mark(X)) -> mark(U142(X))
U151(mark(X1), X2) -> mark(U151(X1, X2))
U152(mark(X)) -> mark(U152(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))
U172(mark(X1), X2, X3) -> mark(U172(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))
U182(mark(X1), X2) -> mark(U182(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, X3, X4) -> mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) -> mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) -> mark(U204(X1, X2))
U21(mark(X1), X2, X3) -> mark(U21(X1, X2, X3))
U22(mark(X1), X2) -> mark(U22(X1, X2))
U211(mark(X1), X2) -> mark(U211(X1, X2))
U212(mark(X1), X2) -> mark(U212(X1, X2))
U221(mark(X1), X2, X3) -> mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) -> mark(U222(X1, X2, X3))
fst(mark(X)) -> mark(fst(X))
U31(mark(X1), X2, X3) -> mark(U31(X1, X2, X3))
U32(mark(X1), X2) -> mark(U32(X1, X2))
U41(mark(X1), X2) -> mark(U41(X1, X2))
U42(mark(X)) -> mark(U42(X))
U51(mark(X1), X2) -> mark(U51(X1, X2))
U52(mark(X)) -> mark(U52(X))
U61(mark(X)) -> mark(U61(X))
U71(mark(X)) -> mark(U71(X))
U81(mark(X)) -> mark(U81(X))
U91(mark(X)) -> mark(U91(X))
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)) -> U101(proper(X1), proper(X2))
proper(tt) -> ok(tt)
proper(U102(X)) -> U102(proper(X))
proper(isLNat(X)) -> isLNat(proper(X))
proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) -> U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) -> U111(proper(X))
proper(snd(X)) -> snd(proper(X))
proper(splitAt(X1, X2)) -> splitAt(proper(X1), proper(X2))
proper(U121(X)) -> U121(proper(X))
proper(U131(X1, X2)) -> U131(proper(X1), proper(X2))
proper(U132(X)) -> U132(proper(X))
proper(U141(X1, X2)) -> U141(proper(X1), proper(X2))
proper(U142(X)) -> U142(proper(X))
proper(U151(X1, X2)) -> U151(proper(X1), proper(X2))
proper(U152(X)) -> U152(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(U172(X1, X2, X3)) -> U172(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(U182(X1, X2)) -> U182(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, X3, X4)) -> U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) -> isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) -> U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) -> U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) -> U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) -> U22(proper(X1), proper(X2))
proper(U211(X1, X2)) -> U211(proper(X1), proper(X2))
proper(U212(X1, X2)) -> U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) -> U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) -> U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) -> fst(proper(X))
proper(U31(X1, X2, X3)) -> U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) -> U32(proper(X1), proper(X2))
proper(U41(X1, X2)) -> U41(proper(X1), proper(X2))
proper(U42(X)) -> U42(proper(X))
proper(U51(X1, X2)) -> U51(proper(X1), proper(X2))
proper(U52(X)) -> U52(proper(X))
proper(U61(X)) -> U61(proper(X))
proper(U71(X)) -> U71(proper(X))
proper(U81(X)) -> U81(proper(X))
proper(U91(X)) -> U91(proper(X))
proper(isPLNat(X)) -> isPLNat(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(U101(X1, X2))
U102(ok(X)) -> ok(U102(X))
isLNat(ok(X)) -> ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))
U111(ok(X)) -> ok(U111(X))
snd(ok(X)) -> ok(snd(X))
splitAt(ok(X1), ok(X2)) -> ok(splitAt(X1, X2))
U121(ok(X)) -> ok(U121(X))
U131(ok(X1), ok(X2)) -> ok(U131(X1, X2))
U132(ok(X)) -> ok(U132(X))
U141(ok(X1), ok(X2)) -> ok(U141(X1, X2))
U142(ok(X)) -> ok(U142(X))
U151(ok(X1), ok(X2)) -> ok(U151(X1, X2))
U152(ok(X)) -> ok(U152(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))
U172(ok(X1), ok(X2), ok(X3)) -> ok(U172(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))
U182(ok(X1), ok(X2)) -> ok(U182(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(X3), ok(X4)) -> ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) -> ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) -> ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) -> ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) -> ok(U22(X1, X2))
U211(ok(X1), ok(X2)) -> ok(U211(X1, X2))
U212(ok(X1), ok(X2)) -> ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) -> ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) -> ok(U222(X1, X2, X3))
fst(ok(X)) -> ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) -> ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) -> ok(U32(X1, X2))
U41(ok(X1), ok(X2)) -> ok(U41(X1, X2))
U42(ok(X)) -> ok(U42(X))
U51(ok(X1), ok(X2)) -> ok(U51(X1, X2))
U52(ok(X)) -> ok(U52(X))
U61(ok(X)) -> ok(U61(X))
U71(ok(X)) -> ok(U71(X))
U81(ok(X)) -> ok(U81(X))
U91(ok(X)) -> ok(U91(X))
isPLNat(ok(X)) -> ok(isPLNat(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))

Types:
active :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U12 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U172 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U182 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U203 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U204 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U22 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U212 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U222 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U32 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok -> top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat -> tt:mark:nil:0':ok


Lemmas:
U102(gen_tt:mark:nil:0':ok3_0(+(1, n5_0))) -> *4_0, rt in Omega(n5_0)
U12(gen_tt:mark:nil:0':ok3_0(+(1, n1128_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) -> *4_0, rt in Omega(n1128_0)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n9121_0))) -> *4_0, rt in Omega(n9121_0)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n10537_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n10537_0)
U132(gen_tt:mark:nil:0':ok3_0(+(1, n15953_0))) -> *4_0, rt in Omega(n15953_0)
U142(gen_tt:mark:nil:0':ok3_0(+(1, n17620_0))) -> *4_0, rt in Omega(n17620_0)
U152(gen_tt:mark:nil:0':ok3_0(+(1, n19388_0))) -> *4_0, rt in Omega(n19388_0)
cons(gen_tt:mark:nil:0':ok3_0(+(1, n21257_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n21257_0)
natsFrom(gen_tt:mark:nil:0':ok3_0(+(1, n27404_0))) -> *4_0, rt in Omega(n27404_0)
s(gen_tt:mark:nil:0':ok3_0(+(1, n29524_0))) -> *4_0, rt in Omega(n29524_0)
U172(gen_tt:mark:nil:0':ok3_0(+(1, n31745_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) -> *4_0, rt in Omega(n31745_0)
head(gen_tt:mark:nil:0':ok3_0(+(1, n43161_0))) -> *4_0, rt in Omega(n43161_0)
afterNth(gen_tt:mark:nil:0':ok3_0(+(1, n45682_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n45682_0)
U182(gen_tt:mark:nil:0':ok3_0(+(1, n53362_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n53362_0)
pair(gen_tt:mark:nil:0':ok3_0(+(1, n61149_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n61149_0)
U202(gen_tt:mark:nil:0':ok3_0(+(1, n69441_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c), gen_tt:mark:nil:0':ok3_0(d)) -> *4_0, rt in Omega(n69441_0)
U203(gen_tt:mark:nil:0':ok3_0(+(1, n88570_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c), gen_tt:mark:nil:0':ok3_0(d)) -> *4_0, rt in Omega(n88570_0)
U204(gen_tt:mark:nil:0':ok3_0(+(1, n108633_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n108633_0)
U22(gen_tt:mark:nil:0':ok3_0(+(1, n118036_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n118036_0)
U212(gen_tt:mark:nil:0':ok3_0(+(1, n127745_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n127745_0)
U222(gen_tt:mark:nil:0':ok3_0(+(1, n137760_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) -> *4_0, rt in Omega(n137760_0)
fst(gen_tt:mark:nil:0':ok3_0(+(1, n154384_0))) -> *4_0, rt in Omega(n154384_0)
U32(gen_tt:mark:nil:0':ok3_0(+(1, n158601_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n158601_0)
U42(gen_tt:mark:nil:0':ok3_0(+(1, n169534_0))) -> *4_0, rt in Omega(n169534_0)
U52(gen_tt:mark:nil:0':ok3_0(+(1, n174002_0))) -> *4_0, rt in Omega(n174002_0)
U11(gen_tt:mark:nil:0':ok3_0(+(1, n178571_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) -> *4_0, rt in Omega(n178571_0)
U21(gen_tt:mark:nil:0':ok3_0(+(1, n197211_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) -> *4_0, rt in Omega(n197211_0)
U31(gen_tt:mark:nil:0':ok3_0(+(1, n216460_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) -> *4_0, rt in Omega(n216460_0)
U41(gen_tt:mark:nil:0':ok3_0(+(1, n236318_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n236318_0)
U51(gen_tt:mark:nil:0':ok3_0(+(1, n249185_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n249185_0)
U61(gen_tt:mark:nil:0':ok3_0(+(1, n262358_0))) -> *4_0, rt in Omega(n262358_0)
U71(gen_tt:mark:nil:0':ok3_0(+(1, n268080_0))) -> *4_0, rt in Omega(n268080_0)


Generator Equations:
gen_tt:mark:nil:0':ok3_0(0) <=> tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) <=> mark(gen_tt:mark:nil:0':ok3_0(x))


The following defined symbols remain to be analysed:
U81, active, U91, U101, U111, U121, U131, U141, U151, U161, U171, U181, U191, U201, U211, U221, tail, take, sel, proper, top

They will be analysed ascendingly in the following order:
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
U141 < active
U151 < active
U161 < active
U171 < active
U181 < active
U191 < active
U201 < active
U211 < active
U221 < active
tail < active
take < active
sel < active
active < top
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
U141 < proper
U151 < proper
U161 < proper
U171 < proper
U181 < proper
U191 < proper
U201 < proper
U211 < proper
U221 < proper
tail < proper
take < proper
sel < proper
proper < top

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

(93) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
U81(gen_tt:mark:nil:0':ok3_0(+(1, n273748_0))) -> *4_0, rt in Omega(n273748_0)

Induction Base:
U81(gen_tt:mark:nil:0':ok3_0(+(1, 0)))

Induction Step:
U81(gen_tt:mark:nil:0':ok3_0(+(1, +(n273748_0, 1)))) ->_R^Omega(1)
mark(U81(gen_tt:mark:nil:0':ok3_0(+(1, n273748_0)))) ->_IH
mark(*4_0)

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
----------------------------------------

(94)
Obligation:
TRS:
Rules:
active(U101(tt, V2)) -> mark(U102(isLNat(V2)))
active(U102(tt)) -> mark(tt)
active(U11(tt, N, XS)) -> mark(U12(isLNat(XS), N, XS))
active(U111(tt)) -> mark(tt)
active(U12(tt, N, XS)) -> mark(snd(splitAt(N, XS)))
active(U121(tt)) -> mark(tt)
active(U131(tt, V2)) -> mark(U132(isLNat(V2)))
active(U132(tt)) -> mark(tt)
active(U141(tt, V2)) -> mark(U142(isLNat(V2)))
active(U142(tt)) -> mark(tt)
active(U151(tt, V2)) -> mark(U152(isLNat(V2)))
active(U152(tt)) -> mark(tt)
active(U161(tt, N)) -> mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) -> mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) -> mark(head(afterNth(N, XS)))
active(U181(tt, Y)) -> mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) -> mark(Y)
active(U191(tt, XS)) -> mark(pair(nil, XS))
active(U201(tt, N, X, XS)) -> mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) -> mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) -> mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) -> mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) -> mark(U22(isLNat(Y), X))
active(U211(tt, XS)) -> mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) -> mark(XS)
active(U22(tt, X)) -> mark(X)
active(U221(tt, N, XS)) -> mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) -> mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) -> mark(U32(isLNat(XS), N))
active(U32(tt, N)) -> mark(N)
active(U41(tt, V2)) -> mark(U42(isLNat(V2)))
active(U42(tt)) -> mark(tt)
active(U51(tt, V2)) -> mark(U52(isLNat(V2)))
active(U52(tt)) -> mark(tt)
active(U61(tt)) -> mark(tt)
active(U71(tt)) -> mark(tt)
active(U81(tt)) -> mark(tt)
active(U91(tt)) -> mark(tt)
active(afterNth(N, XS)) -> mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) -> mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) -> mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) -> mark(tt)
active(isLNat(afterNth(V1, V2))) -> mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) -> mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) -> mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) -> mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) -> mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) -> mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) -> mark(U101(isNatural(V1), V2))
active(isNatural(0')) -> mark(tt)
active(isNatural(head(V1))) -> mark(U111(isLNat(V1)))
active(isNatural(s(V1))) -> mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) -> mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) -> mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) -> mark(U151(isNatural(V1), V2))
active(natsFrom(N)) -> mark(U161(isNatural(N), N))
active(sel(N, XS)) -> mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) -> mark(U181(isLNat(X), Y))
active(splitAt(0', XS)) -> mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) -> mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) -> mark(U211(isNatural(N), XS))
active(take(N, XS)) -> mark(U221(isNatural(N), N, XS))
active(U101(X1, X2)) -> U101(active(X1), X2)
active(U102(X)) -> U102(active(X))
active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3)
active(U111(X)) -> U111(active(X))
active(snd(X)) -> snd(active(X))
active(splitAt(X1, X2)) -> splitAt(active(X1), X2)
active(splitAt(X1, X2)) -> splitAt(X1, active(X2))
active(U121(X)) -> U121(active(X))
active(U131(X1, X2)) -> U131(active(X1), X2)
active(U132(X)) -> U132(active(X))
active(U141(X1, X2)) -> U141(active(X1), X2)
active(U142(X)) -> U142(active(X))
active(U151(X1, X2)) -> U151(active(X1), X2)
active(U152(X)) -> U152(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(U172(X1, X2, X3)) -> U172(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(U182(X1, X2)) -> U182(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, X3, X4)) -> U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) -> U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) -> U204(active(X1), X2)
active(U21(X1, X2, X3)) -> U21(active(X1), X2, X3)
active(U22(X1, X2)) -> U22(active(X1), X2)
active(U211(X1, X2)) -> U211(active(X1), X2)
active(U212(X1, X2)) -> U212(active(X1), X2)
active(U221(X1, X2, X3)) -> U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) -> U222(active(X1), X2, X3)
active(fst(X)) -> fst(active(X))
active(U31(X1, X2, X3)) -> U31(active(X1), X2, X3)
active(U32(X1, X2)) -> U32(active(X1), X2)
active(U41(X1, X2)) -> U41(active(X1), X2)
active(U42(X)) -> U42(active(X))
active(U51(X1, X2)) -> U51(active(X1), X2)
active(U52(X)) -> U52(active(X))
active(U61(X)) -> U61(active(X))
active(U71(X)) -> U71(active(X))
active(U81(X)) -> U81(active(X))
active(U91(X)) -> U91(active(X))
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) -> mark(U101(X1, X2))
U102(mark(X)) -> mark(U102(X))
U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3))
U111(mark(X)) -> mark(U111(X))
snd(mark(X)) -> mark(snd(X))
splitAt(mark(X1), X2) -> mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) -> mark(splitAt(X1, X2))
U121(mark(X)) -> mark(U121(X))
U131(mark(X1), X2) -> mark(U131(X1, X2))
U132(mark(X)) -> mark(U132(X))
U141(mark(X1), X2) -> mark(U141(X1, X2))
U142(mark(X)) -> mark(U142(X))
U151(mark(X1), X2) -> mark(U151(X1, X2))
U152(mark(X)) -> mark(U152(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))
U172(mark(X1), X2, X3) -> mark(U172(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))
U182(mark(X1), X2) -> mark(U182(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, X3, X4) -> mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) -> mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) -> mark(U204(X1, X2))
U21(mark(X1), X2, X3) -> mark(U21(X1, X2, X3))
U22(mark(X1), X2) -> mark(U22(X1, X2))
U211(mark(X1), X2) -> mark(U211(X1, X2))
U212(mark(X1), X2) -> mark(U212(X1, X2))
U221(mark(X1), X2, X3) -> mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) -> mark(U222(X1, X2, X3))
fst(mark(X)) -> mark(fst(X))
U31(mark(X1), X2, X3) -> mark(U31(X1, X2, X3))
U32(mark(X1), X2) -> mark(U32(X1, X2))
U41(mark(X1), X2) -> mark(U41(X1, X2))
U42(mark(X)) -> mark(U42(X))
U51(mark(X1), X2) -> mark(U51(X1, X2))
U52(mark(X)) -> mark(U52(X))
U61(mark(X)) -> mark(U61(X))
U71(mark(X)) -> mark(U71(X))
U81(mark(X)) -> mark(U81(X))
U91(mark(X)) -> mark(U91(X))
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)) -> U101(proper(X1), proper(X2))
proper(tt) -> ok(tt)
proper(U102(X)) -> U102(proper(X))
proper(isLNat(X)) -> isLNat(proper(X))
proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) -> U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) -> U111(proper(X))
proper(snd(X)) -> snd(proper(X))
proper(splitAt(X1, X2)) -> splitAt(proper(X1), proper(X2))
proper(U121(X)) -> U121(proper(X))
proper(U131(X1, X2)) -> U131(proper(X1), proper(X2))
proper(U132(X)) -> U132(proper(X))
proper(U141(X1, X2)) -> U141(proper(X1), proper(X2))
proper(U142(X)) -> U142(proper(X))
proper(U151(X1, X2)) -> U151(proper(X1), proper(X2))
proper(U152(X)) -> U152(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(U172(X1, X2, X3)) -> U172(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(U182(X1, X2)) -> U182(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, X3, X4)) -> U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) -> isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) -> U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) -> U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) -> U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) -> U22(proper(X1), proper(X2))
proper(U211(X1, X2)) -> U211(proper(X1), proper(X2))
proper(U212(X1, X2)) -> U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) -> U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) -> U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) -> fst(proper(X))
proper(U31(X1, X2, X3)) -> U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) -> U32(proper(X1), proper(X2))
proper(U41(X1, X2)) -> U41(proper(X1), proper(X2))
proper(U42(X)) -> U42(proper(X))
proper(U51(X1, X2)) -> U51(proper(X1), proper(X2))
proper(U52(X)) -> U52(proper(X))
proper(U61(X)) -> U61(proper(X))
proper(U71(X)) -> U71(proper(X))
proper(U81(X)) -> U81(proper(X))
proper(U91(X)) -> U91(proper(X))
proper(isPLNat(X)) -> isPLNat(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(U101(X1, X2))
U102(ok(X)) -> ok(U102(X))
isLNat(ok(X)) -> ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))
U111(ok(X)) -> ok(U111(X))
snd(ok(X)) -> ok(snd(X))
splitAt(ok(X1), ok(X2)) -> ok(splitAt(X1, X2))
U121(ok(X)) -> ok(U121(X))
U131(ok(X1), ok(X2)) -> ok(U131(X1, X2))
U132(ok(X)) -> ok(U132(X))
U141(ok(X1), ok(X2)) -> ok(U141(X1, X2))
U142(ok(X)) -> ok(U142(X))
U151(ok(X1), ok(X2)) -> ok(U151(X1, X2))
U152(ok(X)) -> ok(U152(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))
U172(ok(X1), ok(X2), ok(X3)) -> ok(U172(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))
U182(ok(X1), ok(X2)) -> ok(U182(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(X3), ok(X4)) -> ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) -> ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) -> ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) -> ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) -> ok(U22(X1, X2))
U211(ok(X1), ok(X2)) -> ok(U211(X1, X2))
U212(ok(X1), ok(X2)) -> ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) -> ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) -> ok(U222(X1, X2, X3))
fst(ok(X)) -> ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) -> ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) -> ok(U32(X1, X2))
U41(ok(X1), ok(X2)) -> ok(U41(X1, X2))
U42(ok(X)) -> ok(U42(X))
U51(ok(X1), ok(X2)) -> ok(U51(X1, X2))
U52(ok(X)) -> ok(U52(X))
U61(ok(X)) -> ok(U61(X))
U71(ok(X)) -> ok(U71(X))
U81(ok(X)) -> ok(U81(X))
U91(ok(X)) -> ok(U91(X))
isPLNat(ok(X)) -> ok(isPLNat(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))

Types:
active :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U12 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U172 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U182 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U203 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U204 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U22 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U212 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U222 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U32 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok -> top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat -> tt:mark:nil:0':ok


Lemmas:
U102(gen_tt:mark:nil:0':ok3_0(+(1, n5_0))) -> *4_0, rt in Omega(n5_0)
U12(gen_tt:mark:nil:0':ok3_0(+(1, n1128_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) -> *4_0, rt in Omega(n1128_0)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n9121_0))) -> *4_0, rt in Omega(n9121_0)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n10537_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n10537_0)
U132(gen_tt:mark:nil:0':ok3_0(+(1, n15953_0))) -> *4_0, rt in Omega(n15953_0)
U142(gen_tt:mark:nil:0':ok3_0(+(1, n17620_0))) -> *4_0, rt in Omega(n17620_0)
U152(gen_tt:mark:nil:0':ok3_0(+(1, n19388_0))) -> *4_0, rt in Omega(n19388_0)
cons(gen_tt:mark:nil:0':ok3_0(+(1, n21257_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n21257_0)
natsFrom(gen_tt:mark:nil:0':ok3_0(+(1, n27404_0))) -> *4_0, rt in Omega(n27404_0)
s(gen_tt:mark:nil:0':ok3_0(+(1, n29524_0))) -> *4_0, rt in Omega(n29524_0)
U172(gen_tt:mark:nil:0':ok3_0(+(1, n31745_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) -> *4_0, rt in Omega(n31745_0)
head(gen_tt:mark:nil:0':ok3_0(+(1, n43161_0))) -> *4_0, rt in Omega(n43161_0)
afterNth(gen_tt:mark:nil:0':ok3_0(+(1, n45682_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n45682_0)
U182(gen_tt:mark:nil:0':ok3_0(+(1, n53362_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n53362_0)
pair(gen_tt:mark:nil:0':ok3_0(+(1, n61149_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n61149_0)
U202(gen_tt:mark:nil:0':ok3_0(+(1, n69441_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c), gen_tt:mark:nil:0':ok3_0(d)) -> *4_0, rt in Omega(n69441_0)
U203(gen_tt:mark:nil:0':ok3_0(+(1, n88570_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c), gen_tt:mark:nil:0':ok3_0(d)) -> *4_0, rt in Omega(n88570_0)
U204(gen_tt:mark:nil:0':ok3_0(+(1, n108633_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n108633_0)
U22(gen_tt:mark:nil:0':ok3_0(+(1, n118036_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n118036_0)
U212(gen_tt:mark:nil:0':ok3_0(+(1, n127745_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n127745_0)
U222(gen_tt:mark:nil:0':ok3_0(+(1, n137760_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) -> *4_0, rt in Omega(n137760_0)
fst(gen_tt:mark:nil:0':ok3_0(+(1, n154384_0))) -> *4_0, rt in Omega(n154384_0)
U32(gen_tt:mark:nil:0':ok3_0(+(1, n158601_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n158601_0)
U42(gen_tt:mark:nil:0':ok3_0(+(1, n169534_0))) -> *4_0, rt in Omega(n169534_0)
U52(gen_tt:mark:nil:0':ok3_0(+(1, n174002_0))) -> *4_0, rt in Omega(n174002_0)
U11(gen_tt:mark:nil:0':ok3_0(+(1, n178571_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) -> *4_0, rt in Omega(n178571_0)
U21(gen_tt:mark:nil:0':ok3_0(+(1, n197211_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) -> *4_0, rt in Omega(n197211_0)
U31(gen_tt:mark:nil:0':ok3_0(+(1, n216460_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) -> *4_0, rt in Omega(n216460_0)
U41(gen_tt:mark:nil:0':ok3_0(+(1, n236318_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n236318_0)
U51(gen_tt:mark:nil:0':ok3_0(+(1, n249185_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n249185_0)
U61(gen_tt:mark:nil:0':ok3_0(+(1, n262358_0))) -> *4_0, rt in Omega(n262358_0)
U71(gen_tt:mark:nil:0':ok3_0(+(1, n268080_0))) -> *4_0, rt in Omega(n268080_0)
U81(gen_tt:mark:nil:0':ok3_0(+(1, n273748_0))) -> *4_0, rt in Omega(n273748_0)


Generator Equations:
gen_tt:mark:nil:0':ok3_0(0) <=> tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) <=> mark(gen_tt:mark:nil:0':ok3_0(x))


The following defined symbols remain to be analysed:
U91, active, U101, U111, U121, U131, U141, U151, U161, U171, U181, U191, U201, U211, U221, tail, take, sel, proper, top

They will be analysed ascendingly in the following order:
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
U141 < active
U151 < active
U161 < active
U171 < active
U181 < active
U191 < active
U201 < active
U211 < active
U221 < active
tail < active
take < active
sel < active
active < top
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
U141 < proper
U151 < proper
U161 < proper
U171 < proper
U181 < proper
U191 < proper
U201 < proper
U211 < proper
U221 < proper
tail < proper
take < proper
sel < proper
proper < top

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

(95) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
U91(gen_tt:mark:nil:0':ok3_0(+(1, n279517_0))) -> *4_0, rt in Omega(n279517_0)

Induction Base:
U91(gen_tt:mark:nil:0':ok3_0(+(1, 0)))

Induction Step:
U91(gen_tt:mark:nil:0':ok3_0(+(1, +(n279517_0, 1)))) ->_R^Omega(1)
mark(U91(gen_tt:mark:nil:0':ok3_0(+(1, n279517_0)))) ->_IH
mark(*4_0)

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
----------------------------------------

(96)
Obligation:
TRS:
Rules:
active(U101(tt, V2)) -> mark(U102(isLNat(V2)))
active(U102(tt)) -> mark(tt)
active(U11(tt, N, XS)) -> mark(U12(isLNat(XS), N, XS))
active(U111(tt)) -> mark(tt)
active(U12(tt, N, XS)) -> mark(snd(splitAt(N, XS)))
active(U121(tt)) -> mark(tt)
active(U131(tt, V2)) -> mark(U132(isLNat(V2)))
active(U132(tt)) -> mark(tt)
active(U141(tt, V2)) -> mark(U142(isLNat(V2)))
active(U142(tt)) -> mark(tt)
active(U151(tt, V2)) -> mark(U152(isLNat(V2)))
active(U152(tt)) -> mark(tt)
active(U161(tt, N)) -> mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) -> mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) -> mark(head(afterNth(N, XS)))
active(U181(tt, Y)) -> mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) -> mark(Y)
active(U191(tt, XS)) -> mark(pair(nil, XS))
active(U201(tt, N, X, XS)) -> mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) -> mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) -> mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) -> mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) -> mark(U22(isLNat(Y), X))
active(U211(tt, XS)) -> mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) -> mark(XS)
active(U22(tt, X)) -> mark(X)
active(U221(tt, N, XS)) -> mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) -> mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) -> mark(U32(isLNat(XS), N))
active(U32(tt, N)) -> mark(N)
active(U41(tt, V2)) -> mark(U42(isLNat(V2)))
active(U42(tt)) -> mark(tt)
active(U51(tt, V2)) -> mark(U52(isLNat(V2)))
active(U52(tt)) -> mark(tt)
active(U61(tt)) -> mark(tt)
active(U71(tt)) -> mark(tt)
active(U81(tt)) -> mark(tt)
active(U91(tt)) -> mark(tt)
active(afterNth(N, XS)) -> mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) -> mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) -> mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) -> mark(tt)
active(isLNat(afterNth(V1, V2))) -> mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) -> mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) -> mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) -> mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) -> mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) -> mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) -> mark(U101(isNatural(V1), V2))
active(isNatural(0')) -> mark(tt)
active(isNatural(head(V1))) -> mark(U111(isLNat(V1)))
active(isNatural(s(V1))) -> mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) -> mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) -> mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) -> mark(U151(isNatural(V1), V2))
active(natsFrom(N)) -> mark(U161(isNatural(N), N))
active(sel(N, XS)) -> mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) -> mark(U181(isLNat(X), Y))
active(splitAt(0', XS)) -> mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) -> mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) -> mark(U211(isNatural(N), XS))
active(take(N, XS)) -> mark(U221(isNatural(N), N, XS))
active(U101(X1, X2)) -> U101(active(X1), X2)
active(U102(X)) -> U102(active(X))
active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3)
active(U111(X)) -> U111(active(X))
active(snd(X)) -> snd(active(X))
active(splitAt(X1, X2)) -> splitAt(active(X1), X2)
active(splitAt(X1, X2)) -> splitAt(X1, active(X2))
active(U121(X)) -> U121(active(X))
active(U131(X1, X2)) -> U131(active(X1), X2)
active(U132(X)) -> U132(active(X))
active(U141(X1, X2)) -> U141(active(X1), X2)
active(U142(X)) -> U142(active(X))
active(U151(X1, X2)) -> U151(active(X1), X2)
active(U152(X)) -> U152(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(U172(X1, X2, X3)) -> U172(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(U182(X1, X2)) -> U182(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, X3, X4)) -> U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) -> U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) -> U204(active(X1), X2)
active(U21(X1, X2, X3)) -> U21(active(X1), X2, X3)
active(U22(X1, X2)) -> U22(active(X1), X2)
active(U211(X1, X2)) -> U211(active(X1), X2)
active(U212(X1, X2)) -> U212(active(X1), X2)
active(U221(X1, X2, X3)) -> U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) -> U222(active(X1), X2, X3)
active(fst(X)) -> fst(active(X))
active(U31(X1, X2, X3)) -> U31(active(X1), X2, X3)
active(U32(X1, X2)) -> U32(active(X1), X2)
active(U41(X1, X2)) -> U41(active(X1), X2)
active(U42(X)) -> U42(active(X))
active(U51(X1, X2)) -> U51(active(X1), X2)
active(U52(X)) -> U52(active(X))
active(U61(X)) -> U61(active(X))
active(U71(X)) -> U71(active(X))
active(U81(X)) -> U81(active(X))
active(U91(X)) -> U91(active(X))
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) -> mark(U101(X1, X2))
U102(mark(X)) -> mark(U102(X))
U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3))
U111(mark(X)) -> mark(U111(X))
snd(mark(X)) -> mark(snd(X))
splitAt(mark(X1), X2) -> mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) -> mark(splitAt(X1, X2))
U121(mark(X)) -> mark(U121(X))
U131(mark(X1), X2) -> mark(U131(X1, X2))
U132(mark(X)) -> mark(U132(X))
U141(mark(X1), X2) -> mark(U141(X1, X2))
U142(mark(X)) -> mark(U142(X))
U151(mark(X1), X2) -> mark(U151(X1, X2))
U152(mark(X)) -> mark(U152(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))
U172(mark(X1), X2, X3) -> mark(U172(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))
U182(mark(X1), X2) -> mark(U182(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, X3, X4) -> mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) -> mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) -> mark(U204(X1, X2))
U21(mark(X1), X2, X3) -> mark(U21(X1, X2, X3))
U22(mark(X1), X2) -> mark(U22(X1, X2))
U211(mark(X1), X2) -> mark(U211(X1, X2))
U212(mark(X1), X2) -> mark(U212(X1, X2))
U221(mark(X1), X2, X3) -> mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) -> mark(U222(X1, X2, X3))
fst(mark(X)) -> mark(fst(X))
U31(mark(X1), X2, X3) -> mark(U31(X1, X2, X3))
U32(mark(X1), X2) -> mark(U32(X1, X2))
U41(mark(X1), X2) -> mark(U41(X1, X2))
U42(mark(X)) -> mark(U42(X))
U51(mark(X1), X2) -> mark(U51(X1, X2))
U52(mark(X)) -> mark(U52(X))
U61(mark(X)) -> mark(U61(X))
U71(mark(X)) -> mark(U71(X))
U81(mark(X)) -> mark(U81(X))
U91(mark(X)) -> mark(U91(X))
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)) -> U101(proper(X1), proper(X2))
proper(tt) -> ok(tt)
proper(U102(X)) -> U102(proper(X))
proper(isLNat(X)) -> isLNat(proper(X))
proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) -> U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) -> U111(proper(X))
proper(snd(X)) -> snd(proper(X))
proper(splitAt(X1, X2)) -> splitAt(proper(X1), proper(X2))
proper(U121(X)) -> U121(proper(X))
proper(U131(X1, X2)) -> U131(proper(X1), proper(X2))
proper(U132(X)) -> U132(proper(X))
proper(U141(X1, X2)) -> U141(proper(X1), proper(X2))
proper(U142(X)) -> U142(proper(X))
proper(U151(X1, X2)) -> U151(proper(X1), proper(X2))
proper(U152(X)) -> U152(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(U172(X1, X2, X3)) -> U172(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(U182(X1, X2)) -> U182(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, X3, X4)) -> U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) -> isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) -> U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) -> U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) -> U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) -> U22(proper(X1), proper(X2))
proper(U211(X1, X2)) -> U211(proper(X1), proper(X2))
proper(U212(X1, X2)) -> U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) -> U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) -> U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) -> fst(proper(X))
proper(U31(X1, X2, X3)) -> U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) -> U32(proper(X1), proper(X2))
proper(U41(X1, X2)) -> U41(proper(X1), proper(X2))
proper(U42(X)) -> U42(proper(X))
proper(U51(X1, X2)) -> U51(proper(X1), proper(X2))
proper(U52(X)) -> U52(proper(X))
proper(U61(X)) -> U61(proper(X))
proper(U71(X)) -> U71(proper(X))
proper(U81(X)) -> U81(proper(X))
proper(U91(X)) -> U91(proper(X))
proper(isPLNat(X)) -> isPLNat(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(U101(X1, X2))
U102(ok(X)) -> ok(U102(X))
isLNat(ok(X)) -> ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))
U111(ok(X)) -> ok(U111(X))
snd(ok(X)) -> ok(snd(X))
splitAt(ok(X1), ok(X2)) -> ok(splitAt(X1, X2))
U121(ok(X)) -> ok(U121(X))
U131(ok(X1), ok(X2)) -> ok(U131(X1, X2))
U132(ok(X)) -> ok(U132(X))
U141(ok(X1), ok(X2)) -> ok(U141(X1, X2))
U142(ok(X)) -> ok(U142(X))
U151(ok(X1), ok(X2)) -> ok(U151(X1, X2))
U152(ok(X)) -> ok(U152(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))
U172(ok(X1), ok(X2), ok(X3)) -> ok(U172(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))
U182(ok(X1), ok(X2)) -> ok(U182(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(X3), ok(X4)) -> ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) -> ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) -> ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) -> ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) -> ok(U22(X1, X2))
U211(ok(X1), ok(X2)) -> ok(U211(X1, X2))
U212(ok(X1), ok(X2)) -> ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) -> ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) -> ok(U222(X1, X2, X3))
fst(ok(X)) -> ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) -> ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) -> ok(U32(X1, X2))
U41(ok(X1), ok(X2)) -> ok(U41(X1, X2))
U42(ok(X)) -> ok(U42(X))
U51(ok(X1), ok(X2)) -> ok(U51(X1, X2))
U52(ok(X)) -> ok(U52(X))
U61(ok(X)) -> ok(U61(X))
U71(ok(X)) -> ok(U71(X))
U81(ok(X)) -> ok(U81(X))
U91(ok(X)) -> ok(U91(X))
isPLNat(ok(X)) -> ok(isPLNat(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))

Types:
active :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U12 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U172 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U182 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U203 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U204 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U22 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U212 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U222 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U32 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok -> top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat -> tt:mark:nil:0':ok


Lemmas:
U102(gen_tt:mark:nil:0':ok3_0(+(1, n5_0))) -> *4_0, rt in Omega(n5_0)
U12(gen_tt:mark:nil:0':ok3_0(+(1, n1128_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) -> *4_0, rt in Omega(n1128_0)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n9121_0))) -> *4_0, rt in Omega(n9121_0)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n10537_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n10537_0)
U132(gen_tt:mark:nil:0':ok3_0(+(1, n15953_0))) -> *4_0, rt in Omega(n15953_0)
U142(gen_tt:mark:nil:0':ok3_0(+(1, n17620_0))) -> *4_0, rt in Omega(n17620_0)
U152(gen_tt:mark:nil:0':ok3_0(+(1, n19388_0))) -> *4_0, rt in Omega(n19388_0)
cons(gen_tt:mark:nil:0':ok3_0(+(1, n21257_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n21257_0)
natsFrom(gen_tt:mark:nil:0':ok3_0(+(1, n27404_0))) -> *4_0, rt in Omega(n27404_0)
s(gen_tt:mark:nil:0':ok3_0(+(1, n29524_0))) -> *4_0, rt in Omega(n29524_0)
U172(gen_tt:mark:nil:0':ok3_0(+(1, n31745_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) -> *4_0, rt in Omega(n31745_0)
head(gen_tt:mark:nil:0':ok3_0(+(1, n43161_0))) -> *4_0, rt in Omega(n43161_0)
afterNth(gen_tt:mark:nil:0':ok3_0(+(1, n45682_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n45682_0)
U182(gen_tt:mark:nil:0':ok3_0(+(1, n53362_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n53362_0)
pair(gen_tt:mark:nil:0':ok3_0(+(1, n61149_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n61149_0)
U202(gen_tt:mark:nil:0':ok3_0(+(1, n69441_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c), gen_tt:mark:nil:0':ok3_0(d)) -> *4_0, rt in Omega(n69441_0)
U203(gen_tt:mark:nil:0':ok3_0(+(1, n88570_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c), gen_tt:mark:nil:0':ok3_0(d)) -> *4_0, rt in Omega(n88570_0)
U204(gen_tt:mark:nil:0':ok3_0(+(1, n108633_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n108633_0)
U22(gen_tt:mark:nil:0':ok3_0(+(1, n118036_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n118036_0)
U212(gen_tt:mark:nil:0':ok3_0(+(1, n127745_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n127745_0)
U222(gen_tt:mark:nil:0':ok3_0(+(1, n137760_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) -> *4_0, rt in Omega(n137760_0)
fst(gen_tt:mark:nil:0':ok3_0(+(1, n154384_0))) -> *4_0, rt in Omega(n154384_0)
U32(gen_tt:mark:nil:0':ok3_0(+(1, n158601_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n158601_0)
U42(gen_tt:mark:nil:0':ok3_0(+(1, n169534_0))) -> *4_0, rt in Omega(n169534_0)
U52(gen_tt:mark:nil:0':ok3_0(+(1, n174002_0))) -> *4_0, rt in Omega(n174002_0)
U11(gen_tt:mark:nil:0':ok3_0(+(1, n178571_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) -> *4_0, rt in Omega(n178571_0)
U21(gen_tt:mark:nil:0':ok3_0(+(1, n197211_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) -> *4_0, rt in Omega(n197211_0)
U31(gen_tt:mark:nil:0':ok3_0(+(1, n216460_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) -> *4_0, rt in Omega(n216460_0)
U41(gen_tt:mark:nil:0':ok3_0(+(1, n236318_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n236318_0)
U51(gen_tt:mark:nil:0':ok3_0(+(1, n249185_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n249185_0)
U61(gen_tt:mark:nil:0':ok3_0(+(1, n262358_0))) -> *4_0, rt in Omega(n262358_0)
U71(gen_tt:mark:nil:0':ok3_0(+(1, n268080_0))) -> *4_0, rt in Omega(n268080_0)
U81(gen_tt:mark:nil:0':ok3_0(+(1, n273748_0))) -> *4_0, rt in Omega(n273748_0)
U91(gen_tt:mark:nil:0':ok3_0(+(1, n279517_0))) -> *4_0, rt in Omega(n279517_0)


Generator Equations:
gen_tt:mark:nil:0':ok3_0(0) <=> tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) <=> mark(gen_tt:mark:nil:0':ok3_0(x))


The following defined symbols remain to be analysed:
U101, active, U111, U121, U131, U141, U151, U161, U171, U181, U191, U201, U211, U221, tail, take, sel, proper, top

They will be analysed ascendingly in the following order:
U101 < active
U111 < active
U121 < active
U131 < active
U141 < active
U151 < active
U161 < active
U171 < active
U181 < active
U191 < active
U201 < active
U211 < active
U221 < active
tail < active
take < active
sel < active
active < top
U101 < proper
U111 < proper
U121 < proper
U131 < proper
U141 < proper
U151 < proper
U161 < proper
U171 < proper
U181 < proper
U191 < proper
U201 < proper
U211 < proper
U221 < proper
tail < proper
take < proper
sel < proper
proper < top

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

(97) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
U101(gen_tt:mark:nil:0':ok3_0(+(1, n285387_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n285387_0)

Induction Base:
U101(gen_tt:mark:nil:0':ok3_0(+(1, 0)), gen_tt:mark:nil:0':ok3_0(b))

Induction Step:
U101(gen_tt:mark:nil:0':ok3_0(+(1, +(n285387_0, 1))), gen_tt:mark:nil:0':ok3_0(b)) ->_R^Omega(1)
mark(U101(gen_tt:mark:nil:0':ok3_0(+(1, n285387_0)), gen_tt:mark:nil:0':ok3_0(b))) ->_IH
mark(*4_0)

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
----------------------------------------

(98)
Obligation:
TRS:
Rules:
active(U101(tt, V2)) -> mark(U102(isLNat(V2)))
active(U102(tt)) -> mark(tt)
active(U11(tt, N, XS)) -> mark(U12(isLNat(XS), N, XS))
active(U111(tt)) -> mark(tt)
active(U12(tt, N, XS)) -> mark(snd(splitAt(N, XS)))
active(U121(tt)) -> mark(tt)
active(U131(tt, V2)) -> mark(U132(isLNat(V2)))
active(U132(tt)) -> mark(tt)
active(U141(tt, V2)) -> mark(U142(isLNat(V2)))
active(U142(tt)) -> mark(tt)
active(U151(tt, V2)) -> mark(U152(isLNat(V2)))
active(U152(tt)) -> mark(tt)
active(U161(tt, N)) -> mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) -> mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) -> mark(head(afterNth(N, XS)))
active(U181(tt, Y)) -> mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) -> mark(Y)
active(U191(tt, XS)) -> mark(pair(nil, XS))
active(U201(tt, N, X, XS)) -> mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) -> mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) -> mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) -> mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) -> mark(U22(isLNat(Y), X))
active(U211(tt, XS)) -> mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) -> mark(XS)
active(U22(tt, X)) -> mark(X)
active(U221(tt, N, XS)) -> mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) -> mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) -> mark(U32(isLNat(XS), N))
active(U32(tt, N)) -> mark(N)
active(U41(tt, V2)) -> mark(U42(isLNat(V2)))
active(U42(tt)) -> mark(tt)
active(U51(tt, V2)) -> mark(U52(isLNat(V2)))
active(U52(tt)) -> mark(tt)
active(U61(tt)) -> mark(tt)
active(U71(tt)) -> mark(tt)
active(U81(tt)) -> mark(tt)
active(U91(tt)) -> mark(tt)
active(afterNth(N, XS)) -> mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) -> mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) -> mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) -> mark(tt)
active(isLNat(afterNth(V1, V2))) -> mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) -> mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) -> mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) -> mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) -> mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) -> mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) -> mark(U101(isNatural(V1), V2))
active(isNatural(0')) -> mark(tt)
active(isNatural(head(V1))) -> mark(U111(isLNat(V1)))
active(isNatural(s(V1))) -> mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) -> mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) -> mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) -> mark(U151(isNatural(V1), V2))
active(natsFrom(N)) -> mark(U161(isNatural(N), N))
active(sel(N, XS)) -> mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) -> mark(U181(isLNat(X), Y))
active(splitAt(0', XS)) -> mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) -> mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) -> mark(U211(isNatural(N), XS))
active(take(N, XS)) -> mark(U221(isNatural(N), N, XS))
active(U101(X1, X2)) -> U101(active(X1), X2)
active(U102(X)) -> U102(active(X))
active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3)
active(U111(X)) -> U111(active(X))
active(snd(X)) -> snd(active(X))
active(splitAt(X1, X2)) -> splitAt(active(X1), X2)
active(splitAt(X1, X2)) -> splitAt(X1, active(X2))
active(U121(X)) -> U121(active(X))
active(U131(X1, X2)) -> U131(active(X1), X2)
active(U132(X)) -> U132(active(X))
active(U141(X1, X2)) -> U141(active(X1), X2)
active(U142(X)) -> U142(active(X))
active(U151(X1, X2)) -> U151(active(X1), X2)
active(U152(X)) -> U152(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(U172(X1, X2, X3)) -> U172(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(U182(X1, X2)) -> U182(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, X3, X4)) -> U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) -> U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) -> U204(active(X1), X2)
active(U21(X1, X2, X3)) -> U21(active(X1), X2, X3)
active(U22(X1, X2)) -> U22(active(X1), X2)
active(U211(X1, X2)) -> U211(active(X1), X2)
active(U212(X1, X2)) -> U212(active(X1), X2)
active(U221(X1, X2, X3)) -> U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) -> U222(active(X1), X2, X3)
active(fst(X)) -> fst(active(X))
active(U31(X1, X2, X3)) -> U31(active(X1), X2, X3)
active(U32(X1, X2)) -> U32(active(X1), X2)
active(U41(X1, X2)) -> U41(active(X1), X2)
active(U42(X)) -> U42(active(X))
active(U51(X1, X2)) -> U51(active(X1), X2)
active(U52(X)) -> U52(active(X))
active(U61(X)) -> U61(active(X))
active(U71(X)) -> U71(active(X))
active(U81(X)) -> U81(active(X))
active(U91(X)) -> U91(active(X))
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) -> mark(U101(X1, X2))
U102(mark(X)) -> mark(U102(X))
U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3))
U111(mark(X)) -> mark(U111(X))
snd(mark(X)) -> mark(snd(X))
splitAt(mark(X1), X2) -> mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) -> mark(splitAt(X1, X2))
U121(mark(X)) -> mark(U121(X))
U131(mark(X1), X2) -> mark(U131(X1, X2))
U132(mark(X)) -> mark(U132(X))
U141(mark(X1), X2) -> mark(U141(X1, X2))
U142(mark(X)) -> mark(U142(X))
U151(mark(X1), X2) -> mark(U151(X1, X2))
U152(mark(X)) -> mark(U152(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))
U172(mark(X1), X2, X3) -> mark(U172(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))
U182(mark(X1), X2) -> mark(U182(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, X3, X4) -> mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) -> mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) -> mark(U204(X1, X2))
U21(mark(X1), X2, X3) -> mark(U21(X1, X2, X3))
U22(mark(X1), X2) -> mark(U22(X1, X2))
U211(mark(X1), X2) -> mark(U211(X1, X2))
U212(mark(X1), X2) -> mark(U212(X1, X2))
U221(mark(X1), X2, X3) -> mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) -> mark(U222(X1, X2, X3))
fst(mark(X)) -> mark(fst(X))
U31(mark(X1), X2, X3) -> mark(U31(X1, X2, X3))
U32(mark(X1), X2) -> mark(U32(X1, X2))
U41(mark(X1), X2) -> mark(U41(X1, X2))
U42(mark(X)) -> mark(U42(X))
U51(mark(X1), X2) -> mark(U51(X1, X2))
U52(mark(X)) -> mark(U52(X))
U61(mark(X)) -> mark(U61(X))
U71(mark(X)) -> mark(U71(X))
U81(mark(X)) -> mark(U81(X))
U91(mark(X)) -> mark(U91(X))
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)) -> U101(proper(X1), proper(X2))
proper(tt) -> ok(tt)
proper(U102(X)) -> U102(proper(X))
proper(isLNat(X)) -> isLNat(proper(X))
proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) -> U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) -> U111(proper(X))
proper(snd(X)) -> snd(proper(X))
proper(splitAt(X1, X2)) -> splitAt(proper(X1), proper(X2))
proper(U121(X)) -> U121(proper(X))
proper(U131(X1, X2)) -> U131(proper(X1), proper(X2))
proper(U132(X)) -> U132(proper(X))
proper(U141(X1, X2)) -> U141(proper(X1), proper(X2))
proper(U142(X)) -> U142(proper(X))
proper(U151(X1, X2)) -> U151(proper(X1), proper(X2))
proper(U152(X)) -> U152(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(U172(X1, X2, X3)) -> U172(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(U182(X1, X2)) -> U182(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, X3, X4)) -> U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) -> isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) -> U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) -> U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) -> U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) -> U22(proper(X1), proper(X2))
proper(U211(X1, X2)) -> U211(proper(X1), proper(X2))
proper(U212(X1, X2)) -> U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) -> U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) -> U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) -> fst(proper(X))
proper(U31(X1, X2, X3)) -> U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) -> U32(proper(X1), proper(X2))
proper(U41(X1, X2)) -> U41(proper(X1), proper(X2))
proper(U42(X)) -> U42(proper(X))
proper(U51(X1, X2)) -> U51(proper(X1), proper(X2))
proper(U52(X)) -> U52(proper(X))
proper(U61(X)) -> U61(proper(X))
proper(U71(X)) -> U71(proper(X))
proper(U81(X)) -> U81(proper(X))
proper(U91(X)) -> U91(proper(X))
proper(isPLNat(X)) -> isPLNat(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(U101(X1, X2))
U102(ok(X)) -> ok(U102(X))
isLNat(ok(X)) -> ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))
U111(ok(X)) -> ok(U111(X))
snd(ok(X)) -> ok(snd(X))
splitAt(ok(X1), ok(X2)) -> ok(splitAt(X1, X2))
U121(ok(X)) -> ok(U121(X))
U131(ok(X1), ok(X2)) -> ok(U131(X1, X2))
U132(ok(X)) -> ok(U132(X))
U141(ok(X1), ok(X2)) -> ok(U141(X1, X2))
U142(ok(X)) -> ok(U142(X))
U151(ok(X1), ok(X2)) -> ok(U151(X1, X2))
U152(ok(X)) -> ok(U152(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))
U172(ok(X1), ok(X2), ok(X3)) -> ok(U172(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))
U182(ok(X1), ok(X2)) -> ok(U182(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(X3), ok(X4)) -> ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) -> ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) -> ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) -> ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) -> ok(U22(X1, X2))
U211(ok(X1), ok(X2)) -> ok(U211(X1, X2))
U212(ok(X1), ok(X2)) -> ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) -> ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) -> ok(U222(X1, X2, X3))
fst(ok(X)) -> ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) -> ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) -> ok(U32(X1, X2))
U41(ok(X1), ok(X2)) -> ok(U41(X1, X2))
U42(ok(X)) -> ok(U42(X))
U51(ok(X1), ok(X2)) -> ok(U51(X1, X2))
U52(ok(X)) -> ok(U52(X))
U61(ok(X)) -> ok(U61(X))
U71(ok(X)) -> ok(U71(X))
U81(ok(X)) -> ok(U81(X))
U91(ok(X)) -> ok(U91(X))
isPLNat(ok(X)) -> ok(isPLNat(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))

Types:
active :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U12 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U172 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U182 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U203 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U204 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U22 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U212 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U222 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U32 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok -> top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat -> tt:mark:nil:0':ok


Lemmas:
U102(gen_tt:mark:nil:0':ok3_0(+(1, n5_0))) -> *4_0, rt in Omega(n5_0)
U12(gen_tt:mark:nil:0':ok3_0(+(1, n1128_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) -> *4_0, rt in Omega(n1128_0)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n9121_0))) -> *4_0, rt in Omega(n9121_0)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n10537_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n10537_0)
U132(gen_tt:mark:nil:0':ok3_0(+(1, n15953_0))) -> *4_0, rt in Omega(n15953_0)
U142(gen_tt:mark:nil:0':ok3_0(+(1, n17620_0))) -> *4_0, rt in Omega(n17620_0)
U152(gen_tt:mark:nil:0':ok3_0(+(1, n19388_0))) -> *4_0, rt in Omega(n19388_0)
cons(gen_tt:mark:nil:0':ok3_0(+(1, n21257_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n21257_0)
natsFrom(gen_tt:mark:nil:0':ok3_0(+(1, n27404_0))) -> *4_0, rt in Omega(n27404_0)
s(gen_tt:mark:nil:0':ok3_0(+(1, n29524_0))) -> *4_0, rt in Omega(n29524_0)
U172(gen_tt:mark:nil:0':ok3_0(+(1, n31745_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) -> *4_0, rt in Omega(n31745_0)
head(gen_tt:mark:nil:0':ok3_0(+(1, n43161_0))) -> *4_0, rt in Omega(n43161_0)
afterNth(gen_tt:mark:nil:0':ok3_0(+(1, n45682_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n45682_0)
U182(gen_tt:mark:nil:0':ok3_0(+(1, n53362_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n53362_0)
pair(gen_tt:mark:nil:0':ok3_0(+(1, n61149_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n61149_0)
U202(gen_tt:mark:nil:0':ok3_0(+(1, n69441_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c), gen_tt:mark:nil:0':ok3_0(d)) -> *4_0, rt in Omega(n69441_0)
U203(gen_tt:mark:nil:0':ok3_0(+(1, n88570_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c), gen_tt:mark:nil:0':ok3_0(d)) -> *4_0, rt in Omega(n88570_0)
U204(gen_tt:mark:nil:0':ok3_0(+(1, n108633_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n108633_0)
U22(gen_tt:mark:nil:0':ok3_0(+(1, n118036_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n118036_0)
U212(gen_tt:mark:nil:0':ok3_0(+(1, n127745_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n127745_0)
U222(gen_tt:mark:nil:0':ok3_0(+(1, n137760_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) -> *4_0, rt in Omega(n137760_0)
fst(gen_tt:mark:nil:0':ok3_0(+(1, n154384_0))) -> *4_0, rt in Omega(n154384_0)
U32(gen_tt:mark:nil:0':ok3_0(+(1, n158601_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n158601_0)
U42(gen_tt:mark:nil:0':ok3_0(+(1, n169534_0))) -> *4_0, rt in Omega(n169534_0)
U52(gen_tt:mark:nil:0':ok3_0(+(1, n174002_0))) -> *4_0, rt in Omega(n174002_0)
U11(gen_tt:mark:nil:0':ok3_0(+(1, n178571_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) -> *4_0, rt in Omega(n178571_0)
U21(gen_tt:mark:nil:0':ok3_0(+(1, n197211_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) -> *4_0, rt in Omega(n197211_0)
U31(gen_tt:mark:nil:0':ok3_0(+(1, n216460_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) -> *4_0, rt in Omega(n216460_0)
U41(gen_tt:mark:nil:0':ok3_0(+(1, n236318_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n236318_0)
U51(gen_tt:mark:nil:0':ok3_0(+(1, n249185_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n249185_0)
U61(gen_tt:mark:nil:0':ok3_0(+(1, n262358_0))) -> *4_0, rt in Omega(n262358_0)
U71(gen_tt:mark:nil:0':ok3_0(+(1, n268080_0))) -> *4_0, rt in Omega(n268080_0)
U81(gen_tt:mark:nil:0':ok3_0(+(1, n273748_0))) -> *4_0, rt in Omega(n273748_0)
U91(gen_tt:mark:nil:0':ok3_0(+(1, n279517_0))) -> *4_0, rt in Omega(n279517_0)
U101(gen_tt:mark:nil:0':ok3_0(+(1, n285387_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n285387_0)


Generator Equations:
gen_tt:mark:nil:0':ok3_0(0) <=> tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) <=> mark(gen_tt:mark:nil:0':ok3_0(x))


The following defined symbols remain to be analysed:
U111, active, U121, U131, U141, U151, U161, U171, U181, U191, U201, U211, U221, tail, take, sel, proper, top

They will be analysed ascendingly in the following order:
U111 < active
U121 < active
U131 < active
U141 < active
U151 < active
U161 < active
U171 < active
U181 < active
U191 < active
U201 < active
U211 < active
U221 < active
tail < active
take < active
sel < active
active < top
U111 < proper
U121 < proper
U131 < proper
U141 < proper
U151 < proper
U161 < proper
U171 < proper
U181 < proper
U191 < proper
U201 < proper
U211 < proper
U221 < proper
tail < proper
take < proper
sel < proper
proper < top

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

(99) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
U111(gen_tt:mark:nil:0':ok3_0(+(1, n299698_0))) -> *4_0, rt in Omega(n299698_0)

Induction Base:
U111(gen_tt:mark:nil:0':ok3_0(+(1, 0)))

Induction Step:
U111(gen_tt:mark:nil:0':ok3_0(+(1, +(n299698_0, 1)))) ->_R^Omega(1)
mark(U111(gen_tt:mark:nil:0':ok3_0(+(1, n299698_0)))) ->_IH
mark(*4_0)

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
----------------------------------------

(100)
Obligation:
TRS:
Rules:
active(U101(tt, V2)) -> mark(U102(isLNat(V2)))
active(U102(tt)) -> mark(tt)
active(U11(tt, N, XS)) -> mark(U12(isLNat(XS), N, XS))
active(U111(tt)) -> mark(tt)
active(U12(tt, N, XS)) -> mark(snd(splitAt(N, XS)))
active(U121(tt)) -> mark(tt)
active(U131(tt, V2)) -> mark(U132(isLNat(V2)))
active(U132(tt)) -> mark(tt)
active(U141(tt, V2)) -> mark(U142(isLNat(V2)))
active(U142(tt)) -> mark(tt)
active(U151(tt, V2)) -> mark(U152(isLNat(V2)))
active(U152(tt)) -> mark(tt)
active(U161(tt, N)) -> mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) -> mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) -> mark(head(afterNth(N, XS)))
active(U181(tt, Y)) -> mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) -> mark(Y)
active(U191(tt, XS)) -> mark(pair(nil, XS))
active(U201(tt, N, X, XS)) -> mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) -> mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) -> mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) -> mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) -> mark(U22(isLNat(Y), X))
active(U211(tt, XS)) -> mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) -> mark(XS)
active(U22(tt, X)) -> mark(X)
active(U221(tt, N, XS)) -> mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) -> mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) -> mark(U32(isLNat(XS), N))
active(U32(tt, N)) -> mark(N)
active(U41(tt, V2)) -> mark(U42(isLNat(V2)))
active(U42(tt)) -> mark(tt)
active(U51(tt, V2)) -> mark(U52(isLNat(V2)))
active(U52(tt)) -> mark(tt)
active(U61(tt)) -> mark(tt)
active(U71(tt)) -> mark(tt)
active(U81(tt)) -> mark(tt)
active(U91(tt)) -> mark(tt)
active(afterNth(N, XS)) -> mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) -> mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) -> mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) -> mark(tt)
active(isLNat(afterNth(V1, V2))) -> mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) -> mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) -> mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) -> mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) -> mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) -> mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) -> mark(U101(isNatural(V1), V2))
active(isNatural(0')) -> mark(tt)
active(isNatural(head(V1))) -> mark(U111(isLNat(V1)))
active(isNatural(s(V1))) -> mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) -> mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) -> mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) -> mark(U151(isNatural(V1), V2))
active(natsFrom(N)) -> mark(U161(isNatural(N), N))
active(sel(N, XS)) -> mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) -> mark(U181(isLNat(X), Y))
active(splitAt(0', XS)) -> mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) -> mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) -> mark(U211(isNatural(N), XS))
active(take(N, XS)) -> mark(U221(isNatural(N), N, XS))
active(U101(X1, X2)) -> U101(active(X1), X2)
active(U102(X)) -> U102(active(X))
active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3)
active(U111(X)) -> U111(active(X))
active(snd(X)) -> snd(active(X))
active(splitAt(X1, X2)) -> splitAt(active(X1), X2)
active(splitAt(X1, X2)) -> splitAt(X1, active(X2))
active(U121(X)) -> U121(active(X))
active(U131(X1, X2)) -> U131(active(X1), X2)
active(U132(X)) -> U132(active(X))
active(U141(X1, X2)) -> U141(active(X1), X2)
active(U142(X)) -> U142(active(X))
active(U151(X1, X2)) -> U151(active(X1), X2)
active(U152(X)) -> U152(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(U172(X1, X2, X3)) -> U172(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(U182(X1, X2)) -> U182(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, X3, X4)) -> U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) -> U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) -> U204(active(X1), X2)
active(U21(X1, X2, X3)) -> U21(active(X1), X2, X3)
active(U22(X1, X2)) -> U22(active(X1), X2)
active(U211(X1, X2)) -> U211(active(X1), X2)
active(U212(X1, X2)) -> U212(active(X1), X2)
active(U221(X1, X2, X3)) -> U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) -> U222(active(X1), X2, X3)
active(fst(X)) -> fst(active(X))
active(U31(X1, X2, X3)) -> U31(active(X1), X2, X3)
active(U32(X1, X2)) -> U32(active(X1), X2)
active(U41(X1, X2)) -> U41(active(X1), X2)
active(U42(X)) -> U42(active(X))
active(U51(X1, X2)) -> U51(active(X1), X2)
active(U52(X)) -> U52(active(X))
active(U61(X)) -> U61(active(X))
active(U71(X)) -> U71(active(X))
active(U81(X)) -> U81(active(X))
active(U91(X)) -> U91(active(X))
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) -> mark(U101(X1, X2))
U102(mark(X)) -> mark(U102(X))
U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3))
U111(mark(X)) -> mark(U111(X))
snd(mark(X)) -> mark(snd(X))
splitAt(mark(X1), X2) -> mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) -> mark(splitAt(X1, X2))
U121(mark(X)) -> mark(U121(X))
U131(mark(X1), X2) -> mark(U131(X1, X2))
U132(mark(X)) -> mark(U132(X))
U141(mark(X1), X2) -> mark(U141(X1, X2))
U142(mark(X)) -> mark(U142(X))
U151(mark(X1), X2) -> mark(U151(X1, X2))
U152(mark(X)) -> mark(U152(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))
U172(mark(X1), X2, X3) -> mark(U172(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))
U182(mark(X1), X2) -> mark(U182(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, X3, X4) -> mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) -> mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) -> mark(U204(X1, X2))
U21(mark(X1), X2, X3) -> mark(U21(X1, X2, X3))
U22(mark(X1), X2) -> mark(U22(X1, X2))
U211(mark(X1), X2) -> mark(U211(X1, X2))
U212(mark(X1), X2) -> mark(U212(X1, X2))
U221(mark(X1), X2, X3) -> mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) -> mark(U222(X1, X2, X3))
fst(mark(X)) -> mark(fst(X))
U31(mark(X1), X2, X3) -> mark(U31(X1, X2, X3))
U32(mark(X1), X2) -> mark(U32(X1, X2))
U41(mark(X1), X2) -> mark(U41(X1, X2))
U42(mark(X)) -> mark(U42(X))
U51(mark(X1), X2) -> mark(U51(X1, X2))
U52(mark(X)) -> mark(U52(X))
U61(mark(X)) -> mark(U61(X))
U71(mark(X)) -> mark(U71(X))
U81(mark(X)) -> mark(U81(X))
U91(mark(X)) -> mark(U91(X))
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)) -> U101(proper(X1), proper(X2))
proper(tt) -> ok(tt)
proper(U102(X)) -> U102(proper(X))
proper(isLNat(X)) -> isLNat(proper(X))
proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) -> U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) -> U111(proper(X))
proper(snd(X)) -> snd(proper(X))
proper(splitAt(X1, X2)) -> splitAt(proper(X1), proper(X2))
proper(U121(X)) -> U121(proper(X))
proper(U131(X1, X2)) -> U131(proper(X1), proper(X2))
proper(U132(X)) -> U132(proper(X))
proper(U141(X1, X2)) -> U141(proper(X1), proper(X2))
proper(U142(X)) -> U142(proper(X))
proper(U151(X1, X2)) -> U151(proper(X1), proper(X2))
proper(U152(X)) -> U152(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(U172(X1, X2, X3)) -> U172(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(U182(X1, X2)) -> U182(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, X3, X4)) -> U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) -> isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) -> U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) -> U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) -> U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) -> U22(proper(X1), proper(X2))
proper(U211(X1, X2)) -> U211(proper(X1), proper(X2))
proper(U212(X1, X2)) -> U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) -> U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) -> U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) -> fst(proper(X))
proper(U31(X1, X2, X3)) -> U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) -> U32(proper(X1), proper(X2))
proper(U41(X1, X2)) -> U41(proper(X1), proper(X2))
proper(U42(X)) -> U42(proper(X))
proper(U51(X1, X2)) -> U51(proper(X1), proper(X2))
proper(U52(X)) -> U52(proper(X))
proper(U61(X)) -> U61(proper(X))
proper(U71(X)) -> U71(proper(X))
proper(U81(X)) -> U81(proper(X))
proper(U91(X)) -> U91(proper(X))
proper(isPLNat(X)) -> isPLNat(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(U101(X1, X2))
U102(ok(X)) -> ok(U102(X))
isLNat(ok(X)) -> ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))
U111(ok(X)) -> ok(U111(X))
snd(ok(X)) -> ok(snd(X))
splitAt(ok(X1), ok(X2)) -> ok(splitAt(X1, X2))
U121(ok(X)) -> ok(U121(X))
U131(ok(X1), ok(X2)) -> ok(U131(X1, X2))
U132(ok(X)) -> ok(U132(X))
U141(ok(X1), ok(X2)) -> ok(U141(X1, X2))
U142(ok(X)) -> ok(U142(X))
U151(ok(X1), ok(X2)) -> ok(U151(X1, X2))
U152(ok(X)) -> ok(U152(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))
U172(ok(X1), ok(X2), ok(X3)) -> ok(U172(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))
U182(ok(X1), ok(X2)) -> ok(U182(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(X3), ok(X4)) -> ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) -> ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) -> ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) -> ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) -> ok(U22(X1, X2))
U211(ok(X1), ok(X2)) -> ok(U211(X1, X2))
U212(ok(X1), ok(X2)) -> ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) -> ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) -> ok(U222(X1, X2, X3))
fst(ok(X)) -> ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) -> ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) -> ok(U32(X1, X2))
U41(ok(X1), ok(X2)) -> ok(U41(X1, X2))
U42(ok(X)) -> ok(U42(X))
U51(ok(X1), ok(X2)) -> ok(U51(X1, X2))
U52(ok(X)) -> ok(U52(X))
U61(ok(X)) -> ok(U61(X))
U71(ok(X)) -> ok(U71(X))
U81(ok(X)) -> ok(U81(X))
U91(ok(X)) -> ok(U91(X))
isPLNat(ok(X)) -> ok(isPLNat(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))

Types:
active :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U12 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U172 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U182 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U203 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U204 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U22 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U212 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U222 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U32 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok -> top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat -> tt:mark:nil:0':ok


Lemmas:
U102(gen_tt:mark:nil:0':ok3_0(+(1, n5_0))) -> *4_0, rt in Omega(n5_0)
U12(gen_tt:mark:nil:0':ok3_0(+(1, n1128_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) -> *4_0, rt in Omega(n1128_0)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n9121_0))) -> *4_0, rt in Omega(n9121_0)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n10537_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n10537_0)
U132(gen_tt:mark:nil:0':ok3_0(+(1, n15953_0))) -> *4_0, rt in Omega(n15953_0)
U142(gen_tt:mark:nil:0':ok3_0(+(1, n17620_0))) -> *4_0, rt in Omega(n17620_0)
U152(gen_tt:mark:nil:0':ok3_0(+(1, n19388_0))) -> *4_0, rt in Omega(n19388_0)
cons(gen_tt:mark:nil:0':ok3_0(+(1, n21257_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n21257_0)
natsFrom(gen_tt:mark:nil:0':ok3_0(+(1, n27404_0))) -> *4_0, rt in Omega(n27404_0)
s(gen_tt:mark:nil:0':ok3_0(+(1, n29524_0))) -> *4_0, rt in Omega(n29524_0)
U172(gen_tt:mark:nil:0':ok3_0(+(1, n31745_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) -> *4_0, rt in Omega(n31745_0)
head(gen_tt:mark:nil:0':ok3_0(+(1, n43161_0))) -> *4_0, rt in Omega(n43161_0)
afterNth(gen_tt:mark:nil:0':ok3_0(+(1, n45682_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n45682_0)
U182(gen_tt:mark:nil:0':ok3_0(+(1, n53362_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n53362_0)
pair(gen_tt:mark:nil:0':ok3_0(+(1, n61149_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n61149_0)
U202(gen_tt:mark:nil:0':ok3_0(+(1, n69441_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c), gen_tt:mark:nil:0':ok3_0(d)) -> *4_0, rt in Omega(n69441_0)
U203(gen_tt:mark:nil:0':ok3_0(+(1, n88570_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c), gen_tt:mark:nil:0':ok3_0(d)) -> *4_0, rt in Omega(n88570_0)
U204(gen_tt:mark:nil:0':ok3_0(+(1, n108633_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n108633_0)
U22(gen_tt:mark:nil:0':ok3_0(+(1, n118036_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n118036_0)
U212(gen_tt:mark:nil:0':ok3_0(+(1, n127745_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n127745_0)
U222(gen_tt:mark:nil:0':ok3_0(+(1, n137760_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) -> *4_0, rt in Omega(n137760_0)
fst(gen_tt:mark:nil:0':ok3_0(+(1, n154384_0))) -> *4_0, rt in Omega(n154384_0)
U32(gen_tt:mark:nil:0':ok3_0(+(1, n158601_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n158601_0)
U42(gen_tt:mark:nil:0':ok3_0(+(1, n169534_0))) -> *4_0, rt in Omega(n169534_0)
U52(gen_tt:mark:nil:0':ok3_0(+(1, n174002_0))) -> *4_0, rt in Omega(n174002_0)
U11(gen_tt:mark:nil:0':ok3_0(+(1, n178571_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) -> *4_0, rt in Omega(n178571_0)
U21(gen_tt:mark:nil:0':ok3_0(+(1, n197211_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) -> *4_0, rt in Omega(n197211_0)
U31(gen_tt:mark:nil:0':ok3_0(+(1, n216460_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) -> *4_0, rt in Omega(n216460_0)
U41(gen_tt:mark:nil:0':ok3_0(+(1, n236318_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n236318_0)
U51(gen_tt:mark:nil:0':ok3_0(+(1, n249185_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n249185_0)
U61(gen_tt:mark:nil:0':ok3_0(+(1, n262358_0))) -> *4_0, rt in Omega(n262358_0)
U71(gen_tt:mark:nil:0':ok3_0(+(1, n268080_0))) -> *4_0, rt in Omega(n268080_0)
U81(gen_tt:mark:nil:0':ok3_0(+(1, n273748_0))) -> *4_0, rt in Omega(n273748_0)
U91(gen_tt:mark:nil:0':ok3_0(+(1, n279517_0))) -> *4_0, rt in Omega(n279517_0)
U101(gen_tt:mark:nil:0':ok3_0(+(1, n285387_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n285387_0)
U111(gen_tt:mark:nil:0':ok3_0(+(1, n299698_0))) -> *4_0, rt in Omega(n299698_0)


Generator Equations:
gen_tt:mark:nil:0':ok3_0(0) <=> tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) <=> mark(gen_tt:mark:nil:0':ok3_0(x))


The following defined symbols remain to be analysed:
U121, active, U131, U141, U151, U161, U171, U181, U191, U201, U211, U221, tail, take, sel, proper, top

They will be analysed ascendingly in the following order:
U121 < active
U131 < active
U141 < active
U151 < active
U161 < active
U171 < active
U181 < active
U191 < active
U201 < active
U211 < active
U221 < active
tail < active
take < active
sel < active
active < top
U121 < proper
U131 < proper
U141 < proper
U151 < proper
U161 < proper
U171 < proper
U181 < proper
U191 < proper
U201 < proper
U211 < proper
U221 < proper
tail < proper
take < proper
sel < proper
proper < top