(0) 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:

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

Rewrite Strategy: FULL

(1) NestedDefinedSymbolProof (BOTH BOUNDS(ID, ID) transformation)

The following defined symbols can occur below the 0th argument of top: proper, active
The following defined symbols can occur below the 0th argument of proper: proper, active
The following defined symbols can occur below the 0th argument of active: proper, active

Hence, the left-hand sides of the following rules are not basic-reachable and can be removed:
active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2)) → U102(active(X1), X2)
active(U103(X)) → U103(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U111(X1, X2)) → U111(active(X1), X2)
active(U112(X)) → U112(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3)) → U131(active(X1), X2, X3)
active(U132(X1, X2)) → U132(active(X1), X2)
active(U133(X)) → U133(active(X))
active(U141(X1, X2, X3)) → U141(active(X1), X2, X3)
active(U142(X1, X2)) → U142(active(X1), X2)
active(U143(X)) → U143(active(X))
active(U151(X1, X2, X3)) → U151(active(X1), X2, X3)
active(U152(X1, X2)) → U152(active(X1), X2)
active(U153(X)) → U153(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2)) → U202(active(X1), X2)
active(U21(X1, X2)) → U21(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X1, X2)) → U81(active(X1), X2)
active(U82(X)) → U82(active(X))
active(U91(X1, X2)) → U91(active(X1), X2)
active(U92(X)) → U92(active(X))
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(U102(X1, X2)) → U102(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(U103(X)) → U103(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U111(X1, X2)) → U111(proper(X1), proper(X2))
proper(U112(X)) → U112(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(U131(X1, X2, X3)) → U131(proper(X1), proper(X2), proper(X3))
proper(U132(X1, X2)) → U132(proper(X1), proper(X2))
proper(U133(X)) → U133(proper(X))
proper(U141(X1, X2, X3)) → U141(proper(X1), proper(X2), proper(X3))
proper(U142(X1, X2)) → U142(proper(X1), proper(X2))
proper(U143(X)) → U143(proper(X))
proper(U151(X1, X2, X3)) → U151(proper(X1), proper(X2), proper(X3))
proper(U152(X1, X2)) → U152(proper(X1), proper(X2))
proper(U153(X)) → U153(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2)) → U202(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(U81(X1, X2)) → U81(proper(X1), proper(X2))
proper(U82(X)) → U82(proper(X))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(U92(X)) → U92(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNaturalKind(X)) → isNaturalKind(proper(X))
proper(isLNatKind(X)) → isLNatKind(proper(X))
proper(isPLNatKind(X)) → isPLNatKind(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))

(2) Obligation:

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


The TRS R consists of the following rules:

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

Rewrite Strategy: FULL

(3) RcToIrcProof (BOTH BOUNDS(ID, ID) transformation)

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:

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

Rewrite Strategy: INNERMOST

(5) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID) transformation)

Converted Cpx (relative) TRS to CDT

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

top(ok(z0)) → top(active(z0))
top(mark(z0)) → top(proper(z0))
U151(ok(z0), ok(z1), ok(z2)) → ok(U151(z0, z1, z2))
U151(mark(z0), z1, z2) → mark(U151(z0, z1, z2))
U43(ok(z0)) → ok(U43(z0))
U43(mark(z0)) → mark(U43(z0))
U202(mark(z0), z1) → mark(U202(z0, z1))
U202(ok(z0), ok(z1)) → ok(U202(z0, z1))
U143(ok(z0)) → ok(U143(z0))
U143(mark(z0)) → mark(U143(z0))
U52(ok(z0), ok(z1)) → ok(U52(z0, z1))
U52(mark(z0), z1) → mark(U52(z0, z1))
U51(mark(z0), z1, z2) → mark(U51(z0, z1, z2))
U51(ok(z0), ok(z1), ok(z2)) → ok(U51(z0, z1, z2))
U152(ok(z0), ok(z1)) → ok(U152(z0, z1))
U152(mark(z0), z1) → mark(U152(z0, z1))
pair(mark(z0), z1) → mark(pair(z0, z1))
pair(ok(z0), ok(z1)) → ok(pair(z0, z1))
pair(z0, mark(z1)) → mark(pair(z0, z1))
sel(ok(z0), ok(z1)) → ok(sel(z0, z1))
sel(mark(z0), z1) → mark(sel(z0, z1))
sel(z0, mark(z1)) → mark(sel(z0, z1))
U181(ok(z0), ok(z1)) → ok(U181(z0, z1))
U181(mark(z0), z1) → mark(U181(z0, z1))
proper(tt) → ok(tt)
proper(nil) → ok(nil)
proper(0) → ok(0)
U111(ok(z0), ok(z1)) → ok(U111(z0, z1))
U111(mark(z0), z1) → mark(U111(z0, z1))
isLNatKind(ok(z0)) → ok(isLNatKind(z0))
U11(mark(z0), z1, z2) → mark(U11(z0, z1, z2))
U11(ok(z0), ok(z1), ok(z2)) → ok(U11(z0, z1, z2))
U53(mark(z0)) → mark(U53(z0))
U53(ok(z0)) → ok(U53(z0))
U72(mark(z0)) → mark(U72(z0))
U72(ok(z0)) → ok(U72(z0))
U201(ok(z0), ok(z1), ok(z2), ok(z3)) → ok(U201(z0, z1, z2, z3))
U201(mark(z0), z1, z2, z3) → mark(U201(z0, z1, z2, z3))
splitAt(z0, mark(z1)) → mark(splitAt(z0, z1))
splitAt(mark(z0), z1) → mark(splitAt(z0, z1))
splitAt(ok(z0), ok(z1)) → ok(splitAt(z0, z1))
U112(mark(z0)) → mark(U112(z0))
U112(ok(z0)) → ok(U112(z0))
head(mark(z0)) → mark(head(z0))
head(ok(z0)) → ok(head(z0))
U82(mark(z0)) → mark(U82(z0))
U82(ok(z0)) → ok(U82(z0))
fst(mark(z0)) → mark(fst(z0))
fst(ok(z0)) → ok(fst(z0))
afterNth(ok(z0), ok(z1)) → ok(afterNth(z0, z1))
afterNth(z0, mark(z1)) → mark(afterNth(z0, z1))
afterNth(mark(z0), z1) → mark(afterNth(z0, z1))
tail(mark(z0)) → mark(tail(z0))
tail(ok(z0)) → ok(tail(z0))
U71(ok(z0), ok(z1)) → ok(U71(z0, z1))
U71(mark(z0), z1) → mark(U71(z0, z1))
isNaturalKind(ok(z0)) → ok(isNaturalKind(z0))
U42(ok(z0), ok(z1)) → ok(U42(z0, z1))
U42(mark(z0), z1) → mark(U42(z0, z1))
U142(mark(z0), z1) → mark(U142(z0, z1))
U142(ok(z0), ok(z1)) → ok(U142(z0, z1))
U62(ok(z0)) → ok(U62(z0))
U62(mark(z0)) → mark(U62(z0))
U122(mark(z0)) → mark(U122(z0))
U122(ok(z0)) → ok(U122(z0))
U121(mark(z0), z1) → mark(U121(z0, z1))
U121(ok(z0), ok(z1)) → ok(U121(z0, z1))
U21(mark(z0), z1) → mark(U21(z0, z1))
U21(ok(z0), ok(z1)) → ok(U21(z0, z1))
U131(ok(z0), ok(z1), ok(z2)) → ok(U131(z0, z1, z2))
U131(mark(z0), z1, z2) → mark(U131(z0, z1, z2))
U133(ok(z0)) → ok(U133(z0))
U133(mark(z0)) → mark(U133(z0))
take(z0, mark(z1)) → mark(take(z0, z1))
take(mark(z0), z1) → mark(take(z0, z1))
take(ok(z0), ok(z1)) → ok(take(z0, z1))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
natsFrom(mark(z0)) → mark(natsFrom(z0))
natsFrom(ok(z0)) → ok(natsFrom(z0))
U171(mark(z0), z1, z2) → mark(U171(z0, z1, z2))
U171(ok(z0), ok(z1), ok(z2)) → ok(U171(z0, z1, z2))
isPLNatKind(ok(z0)) → ok(isPLNatKind(z0))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
U41(mark(z0), z1, z2) → mark(U41(z0, z1, z2))
U41(ok(z0), ok(z1), ok(z2)) → ok(U41(z0, z1, z2))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
U91(ok(z0), ok(z1)) → ok(U91(z0, z1))
U91(mark(z0), z1) → mark(U91(z0, z1))
isNatural(ok(z0)) → ok(isNatural(z0))
U221(mark(z0), z1, z2) → mark(U221(z0, z1, z2))
U221(ok(z0), ok(z1), ok(z2)) → ok(U221(z0, z1, z2))
U101(ok(z0), ok(z1), ok(z2)) → ok(U101(z0, z1, z2))
U101(mark(z0), z1, z2) → mark(U101(z0, z1, z2))
U61(ok(z0), ok(z1)) → ok(U61(z0, z1))
U61(mark(z0), z1) → mark(U61(z0, z1))
snd(ok(z0)) → ok(snd(z0))
snd(mark(z0)) → mark(snd(z0))
U102(mark(z0), z1) → mark(U102(z0, z1))
U102(ok(z0), ok(z1)) → ok(U102(z0, z1))
U161(mark(z0), z1) → mark(U161(z0, z1))
U161(ok(z0), ok(z1)) → ok(U161(z0, z1))
U92(mark(z0)) → mark(U92(z0))
U92(ok(z0)) → ok(U92(z0))
isPLNat(ok(z0)) → ok(isPLNat(z0))
U141(mark(z0), z1, z2) → mark(U141(z0, z1, z2))
U141(ok(z0), ok(z1), ok(z2)) → ok(U141(z0, z1, z2))
U31(ok(z0), ok(z1)) → ok(U31(z0, z1))
U31(mark(z0), z1) → mark(U31(z0, z1))
U191(ok(z0), ok(z1)) → ok(U191(z0, z1))
U191(mark(z0), z1) → mark(U191(z0, z1))
U103(ok(z0)) → ok(U103(z0))
U103(mark(z0)) → mark(U103(z0))
U211(ok(z0), ok(z1)) → ok(U211(z0, z1))
U211(mark(z0), z1) → mark(U211(z0, z1))
U153(ok(z0)) → ok(U153(z0))
U153(mark(z0)) → mark(U153(z0))
U132(ok(z0), ok(z1)) → ok(U132(z0, z1))
U132(mark(z0), z1) → mark(U132(z0, z1))
U81(ok(z0), ok(z1)) → ok(U81(z0, z1))
U81(mark(z0), z1) → mark(U81(z0, z1))
isLNat(ok(z0)) → ok(isLNat(z0))
Tuples:

TOP(ok(z0)) → c(TOP(active(z0)))
TOP(mark(z0)) → c1(TOP(proper(z0)), PROPER(z0))
U151'(ok(z0), ok(z1), ok(z2)) → c2(U151'(z0, z1, z2))
U151'(mark(z0), z1, z2) → c3(U151'(z0, z1, z2))
U43'(ok(z0)) → c4(U43'(z0))
U43'(mark(z0)) → c5(U43'(z0))
U202'(mark(z0), z1) → c6(U202'(z0, z1))
U202'(ok(z0), ok(z1)) → c7(U202'(z0, z1))
U143'(ok(z0)) → c8(U143'(z0))
U143'(mark(z0)) → c9(U143'(z0))
U52'(ok(z0), ok(z1)) → c10(U52'(z0, z1))
U52'(mark(z0), z1) → c11(U52'(z0, z1))
U51'(mark(z0), z1, z2) → c12(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c13(U51'(z0, z1, z2))
U152'(ok(z0), ok(z1)) → c14(U152'(z0, z1))
U152'(mark(z0), z1) → c15(U152'(z0, z1))
PAIR(mark(z0), z1) → c16(PAIR(z0, z1))
PAIR(ok(z0), ok(z1)) → c17(PAIR(z0, z1))
PAIR(z0, mark(z1)) → c18(PAIR(z0, z1))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
SEL(mark(z0), z1) → c20(SEL(z0, z1))
SEL(z0, mark(z1)) → c21(SEL(z0, z1))
U181'(ok(z0), ok(z1)) → c22(U181'(z0, z1))
U181'(mark(z0), z1) → c23(U181'(z0, z1))
PROPER(tt) → c24
PROPER(nil) → c25
PROPER(0) → c26
U111'(ok(z0), ok(z1)) → c27(U111'(z0, z1))
U111'(mark(z0), z1) → c28(U111'(z0, z1))
ISLNATKIND(ok(z0)) → c29(ISLNATKIND(z0))
U11'(mark(z0), z1, z2) → c30(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c31(U11'(z0, z1, z2))
U53'(mark(z0)) → c32(U53'(z0))
U53'(ok(z0)) → c33(U53'(z0))
U72'(mark(z0)) → c34(U72'(z0))
U72'(ok(z0)) → c35(U72'(z0))
U201'(ok(z0), ok(z1), ok(z2), ok(z3)) → c36(U201'(z0, z1, z2, z3))
U201'(mark(z0), z1, z2, z3) → c37(U201'(z0, z1, z2, z3))
SPLITAT(z0, mark(z1)) → c38(SPLITAT(z0, z1))
SPLITAT(mark(z0), z1) → c39(SPLITAT(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c40(SPLITAT(z0, z1))
U112'(mark(z0)) → c41(U112'(z0))
U112'(ok(z0)) → c42(U112'(z0))
HEAD(mark(z0)) → c43(HEAD(z0))
HEAD(ok(z0)) → c44(HEAD(z0))
U82'(mark(z0)) → c45(U82'(z0))
U82'(ok(z0)) → c46(U82'(z0))
FST(mark(z0)) → c47(FST(z0))
FST(ok(z0)) → c48(FST(z0))
AFTERNTH(ok(z0), ok(z1)) → c49(AFTERNTH(z0, z1))
AFTERNTH(z0, mark(z1)) → c50(AFTERNTH(z0, z1))
AFTERNTH(mark(z0), z1) → c51(AFTERNTH(z0, z1))
TAIL(mark(z0)) → c52(TAIL(z0))
TAIL(ok(z0)) → c53(TAIL(z0))
U71'(ok(z0), ok(z1)) → c54(U71'(z0, z1))
U71'(mark(z0), z1) → c55(U71'(z0, z1))
ISNATURALKIND(ok(z0)) → c56(ISNATURALKIND(z0))
U42'(ok(z0), ok(z1)) → c57(U42'(z0, z1))
U42'(mark(z0), z1) → c58(U42'(z0, z1))
U142'(mark(z0), z1) → c59(U142'(z0, z1))
U142'(ok(z0), ok(z1)) → c60(U142'(z0, z1))
U62'(ok(z0)) → c61(U62'(z0))
U62'(mark(z0)) → c62(U62'(z0))
U122'(mark(z0)) → c63(U122'(z0))
U122'(ok(z0)) → c64(U122'(z0))
U121'(mark(z0), z1) → c65(U121'(z0, z1))
U121'(ok(z0), ok(z1)) → c66(U121'(z0, z1))
U21'(mark(z0), z1) → c67(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c68(U21'(z0, z1))
U131'(ok(z0), ok(z1), ok(z2)) → c69(U131'(z0, z1, z2))
U131'(mark(z0), z1, z2) → c70(U131'(z0, z1, z2))
U133'(ok(z0)) → c71(U133'(z0))
U133'(mark(z0)) → c72(U133'(z0))
TAKE(z0, mark(z1)) → c73(TAKE(z0, z1))
TAKE(mark(z0), z1) → c74(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c75(TAKE(z0, z1))
AND(mark(z0), z1) → c76(AND(z0, z1))
AND(ok(z0), ok(z1)) → c77(AND(z0, z1))
NATSFROM(mark(z0)) → c78(NATSFROM(z0))
NATSFROM(ok(z0)) → c79(NATSFROM(z0))
U171'(mark(z0), z1, z2) → c80(U171'(z0, z1, z2))
U171'(ok(z0), ok(z1), ok(z2)) → c81(U171'(z0, z1, z2))
ISPLNATKIND(ok(z0)) → c82(ISPLNATKIND(z0))
S(mark(z0)) → c83(S(z0))
S(ok(z0)) → c84(S(z0))
U41'(mark(z0), z1, z2) → c85(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c86(U41'(z0, z1, z2))
CONS(mark(z0), z1) → c87(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c88(CONS(z0, z1))
U91'(ok(z0), ok(z1)) → c89(U91'(z0, z1))
U91'(mark(z0), z1) → c90(U91'(z0, z1))
ISNATURAL(ok(z0)) → c91(ISNATURAL(z0))
U221'(mark(z0), z1, z2) → c92(U221'(z0, z1, z2))
U221'(ok(z0), ok(z1), ok(z2)) → c93(U221'(z0, z1, z2))
U101'(ok(z0), ok(z1), ok(z2)) → c94(U101'(z0, z1, z2))
U101'(mark(z0), z1, z2) → c95(U101'(z0, z1, z2))
U61'(ok(z0), ok(z1)) → c96(U61'(z0, z1))
U61'(mark(z0), z1) → c97(U61'(z0, z1))
SND(ok(z0)) → c98(SND(z0))
SND(mark(z0)) → c99(SND(z0))
U102'(mark(z0), z1) → c100(U102'(z0, z1))
U102'(ok(z0), ok(z1)) → c101(U102'(z0, z1))
U161'(mark(z0), z1) → c102(U161'(z0, z1))
U161'(ok(z0), ok(z1)) → c103(U161'(z0, z1))
U92'(mark(z0)) → c104(U92'(z0))
U92'(ok(z0)) → c105(U92'(z0))
ISPLNAT(ok(z0)) → c106(ISPLNAT(z0))
U141'(mark(z0), z1, z2) → c107(U141'(z0, z1, z2))
U141'(ok(z0), ok(z1), ok(z2)) → c108(U141'(z0, z1, z2))
U31'(ok(z0), ok(z1)) → c109(U31'(z0, z1))
U31'(mark(z0), z1) → c110(U31'(z0, z1))
U191'(ok(z0), ok(z1)) → c111(U191'(z0, z1))
U191'(mark(z0), z1) → c112(U191'(z0, z1))
U103'(ok(z0)) → c113(U103'(z0))
U103'(mark(z0)) → c114(U103'(z0))
U211'(ok(z0), ok(z1)) → c115(U211'(z0, z1))
U211'(mark(z0), z1) → c116(U211'(z0, z1))
U153'(ok(z0)) → c117(U153'(z0))
U153'(mark(z0)) → c118(U153'(z0))
U132'(ok(z0), ok(z1)) → c119(U132'(z0, z1))
U132'(mark(z0), z1) → c120(U132'(z0, z1))
U81'(ok(z0), ok(z1)) → c121(U81'(z0, z1))
U81'(mark(z0), z1) → c122(U81'(z0, z1))
ISLNAT(ok(z0)) → c123(ISLNAT(z0))
S tuples:

TOP(ok(z0)) → c(TOP(active(z0)))
TOP(mark(z0)) → c1(TOP(proper(z0)), PROPER(z0))
U151'(ok(z0), ok(z1), ok(z2)) → c2(U151'(z0, z1, z2))
U151'(mark(z0), z1, z2) → c3(U151'(z0, z1, z2))
U43'(ok(z0)) → c4(U43'(z0))
U43'(mark(z0)) → c5(U43'(z0))
U202'(mark(z0), z1) → c6(U202'(z0, z1))
U202'(ok(z0), ok(z1)) → c7(U202'(z0, z1))
U143'(ok(z0)) → c8(U143'(z0))
U143'(mark(z0)) → c9(U143'(z0))
U52'(ok(z0), ok(z1)) → c10(U52'(z0, z1))
U52'(mark(z0), z1) → c11(U52'(z0, z1))
U51'(mark(z0), z1, z2) → c12(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c13(U51'(z0, z1, z2))
U152'(ok(z0), ok(z1)) → c14(U152'(z0, z1))
U152'(mark(z0), z1) → c15(U152'(z0, z1))
PAIR(mark(z0), z1) → c16(PAIR(z0, z1))
PAIR(ok(z0), ok(z1)) → c17(PAIR(z0, z1))
PAIR(z0, mark(z1)) → c18(PAIR(z0, z1))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
SEL(mark(z0), z1) → c20(SEL(z0, z1))
SEL(z0, mark(z1)) → c21(SEL(z0, z1))
U181'(ok(z0), ok(z1)) → c22(U181'(z0, z1))
U181'(mark(z0), z1) → c23(U181'(z0, z1))
PROPER(tt) → c24
PROPER(nil) → c25
PROPER(0) → c26
U111'(ok(z0), ok(z1)) → c27(U111'(z0, z1))
U111'(mark(z0), z1) → c28(U111'(z0, z1))
ISLNATKIND(ok(z0)) → c29(ISLNATKIND(z0))
U11'(mark(z0), z1, z2) → c30(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c31(U11'(z0, z1, z2))
U53'(mark(z0)) → c32(U53'(z0))
U53'(ok(z0)) → c33(U53'(z0))
U72'(mark(z0)) → c34(U72'(z0))
U72'(ok(z0)) → c35(U72'(z0))
U201'(ok(z0), ok(z1), ok(z2), ok(z3)) → c36(U201'(z0, z1, z2, z3))
U201'(mark(z0), z1, z2, z3) → c37(U201'(z0, z1, z2, z3))
SPLITAT(z0, mark(z1)) → c38(SPLITAT(z0, z1))
SPLITAT(mark(z0), z1) → c39(SPLITAT(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c40(SPLITAT(z0, z1))
U112'(mark(z0)) → c41(U112'(z0))
U112'(ok(z0)) → c42(U112'(z0))
HEAD(mark(z0)) → c43(HEAD(z0))
HEAD(ok(z0)) → c44(HEAD(z0))
U82'(mark(z0)) → c45(U82'(z0))
U82'(ok(z0)) → c46(U82'(z0))
FST(mark(z0)) → c47(FST(z0))
FST(ok(z0)) → c48(FST(z0))
AFTERNTH(ok(z0), ok(z1)) → c49(AFTERNTH(z0, z1))
AFTERNTH(z0, mark(z1)) → c50(AFTERNTH(z0, z1))
AFTERNTH(mark(z0), z1) → c51(AFTERNTH(z0, z1))
TAIL(mark(z0)) → c52(TAIL(z0))
TAIL(ok(z0)) → c53(TAIL(z0))
U71'(ok(z0), ok(z1)) → c54(U71'(z0, z1))
U71'(mark(z0), z1) → c55(U71'(z0, z1))
ISNATURALKIND(ok(z0)) → c56(ISNATURALKIND(z0))
U42'(ok(z0), ok(z1)) → c57(U42'(z0, z1))
U42'(mark(z0), z1) → c58(U42'(z0, z1))
U142'(mark(z0), z1) → c59(U142'(z0, z1))
U142'(ok(z0), ok(z1)) → c60(U142'(z0, z1))
U62'(ok(z0)) → c61(U62'(z0))
U62'(mark(z0)) → c62(U62'(z0))
U122'(mark(z0)) → c63(U122'(z0))
U122'(ok(z0)) → c64(U122'(z0))
U121'(mark(z0), z1) → c65(U121'(z0, z1))
U121'(ok(z0), ok(z1)) → c66(U121'(z0, z1))
U21'(mark(z0), z1) → c67(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c68(U21'(z0, z1))
U131'(ok(z0), ok(z1), ok(z2)) → c69(U131'(z0, z1, z2))
U131'(mark(z0), z1, z2) → c70(U131'(z0, z1, z2))
U133'(ok(z0)) → c71(U133'(z0))
U133'(mark(z0)) → c72(U133'(z0))
TAKE(z0, mark(z1)) → c73(TAKE(z0, z1))
TAKE(mark(z0), z1) → c74(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c75(TAKE(z0, z1))
AND(mark(z0), z1) → c76(AND(z0, z1))
AND(ok(z0), ok(z1)) → c77(AND(z0, z1))
NATSFROM(mark(z0)) → c78(NATSFROM(z0))
NATSFROM(ok(z0)) → c79(NATSFROM(z0))
U171'(mark(z0), z1, z2) → c80(U171'(z0, z1, z2))
U171'(ok(z0), ok(z1), ok(z2)) → c81(U171'(z0, z1, z2))
ISPLNATKIND(ok(z0)) → c82(ISPLNATKIND(z0))
S(mark(z0)) → c83(S(z0))
S(ok(z0)) → c84(S(z0))
U41'(mark(z0), z1, z2) → c85(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c86(U41'(z0, z1, z2))
CONS(mark(z0), z1) → c87(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c88(CONS(z0, z1))
U91'(ok(z0), ok(z1)) → c89(U91'(z0, z1))
U91'(mark(z0), z1) → c90(U91'(z0, z1))
ISNATURAL(ok(z0)) → c91(ISNATURAL(z0))
U221'(mark(z0), z1, z2) → c92(U221'(z0, z1, z2))
U221'(ok(z0), ok(z1), ok(z2)) → c93(U221'(z0, z1, z2))
U101'(ok(z0), ok(z1), ok(z2)) → c94(U101'(z0, z1, z2))
U101'(mark(z0), z1, z2) → c95(U101'(z0, z1, z2))
U61'(ok(z0), ok(z1)) → c96(U61'(z0, z1))
U61'(mark(z0), z1) → c97(U61'(z0, z1))
SND(ok(z0)) → c98(SND(z0))
SND(mark(z0)) → c99(SND(z0))
U102'(mark(z0), z1) → c100(U102'(z0, z1))
U102'(ok(z0), ok(z1)) → c101(U102'(z0, z1))
U161'(mark(z0), z1) → c102(U161'(z0, z1))
U161'(ok(z0), ok(z1)) → c103(U161'(z0, z1))
U92'(mark(z0)) → c104(U92'(z0))
U92'(ok(z0)) → c105(U92'(z0))
ISPLNAT(ok(z0)) → c106(ISPLNAT(z0))
U141'(mark(z0), z1, z2) → c107(U141'(z0, z1, z2))
U141'(ok(z0), ok(z1), ok(z2)) → c108(U141'(z0, z1, z2))
U31'(ok(z0), ok(z1)) → c109(U31'(z0, z1))
U31'(mark(z0), z1) → c110(U31'(z0, z1))
U191'(ok(z0), ok(z1)) → c111(U191'(z0, z1))
U191'(mark(z0), z1) → c112(U191'(z0, z1))
U103'(ok(z0)) → c113(U103'(z0))
U103'(mark(z0)) → c114(U103'(z0))
U211'(ok(z0), ok(z1)) → c115(U211'(z0, z1))
U211'(mark(z0), z1) → c116(U211'(z0, z1))
U153'(ok(z0)) → c117(U153'(z0))
U153'(mark(z0)) → c118(U153'(z0))
U132'(ok(z0), ok(z1)) → c119(U132'(z0, z1))
U132'(mark(z0), z1) → c120(U132'(z0, z1))
U81'(ok(z0), ok(z1)) → c121(U81'(z0, z1))
U81'(mark(z0), z1) → c122(U81'(z0, z1))
ISLNAT(ok(z0)) → c123(ISLNAT(z0))
K tuples:none
Defined Rule Symbols:

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

Defined Pair Symbols:

TOP, U151', U43', U202', U143', U52', U51', U152', PAIR, SEL, U181', PROPER, U111', ISLNATKIND, U11', U53', U72', U201', SPLITAT, U112', HEAD, U82', FST, AFTERNTH, TAIL, U71', ISNATURALKIND, U42', U142', U62', U122', U121', U21', U131', U133', TAKE, AND, NATSFROM, U171', ISPLNATKIND, S, U41', CONS, U91', ISNATURAL, U221', U101', U61', SND, U102', U161', U92', ISPLNAT, U141', U31', U191', U103', U211', U153', U132', U81', ISLNAT

Compound Symbols:

c, c1, c2, c3, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c48, c49, c50, c51, c52, c53, c54, c55, c56, c57, c58, c59, c60, c61, c62, c63, c64, c65, c66, c67, c68, c69, c70, c71, c72, c73, c74, c75, c76, c77, c78, c79, c80, c81, c82, c83, c84, c85, c86, c87, c88, c89, c90, c91, c92, c93, c94, c95, c96, c97, c98, c99, c100, c101, c102, c103, c104, c105, c106, c107, c108, c109, c110, c111, c112, c113, c114, c115, c116, c117, c118, c119, c120, c121, c122, c123

(7) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)

Removed 4 trailing nodes:

PROPER(0) → c26
TOP(ok(z0)) → c(TOP(active(z0)))
PROPER(nil) → c25
PROPER(tt) → c24

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

top(ok(z0)) → top(active(z0))
top(mark(z0)) → top(proper(z0))
U151(ok(z0), ok(z1), ok(z2)) → ok(U151(z0, z1, z2))
U151(mark(z0), z1, z2) → mark(U151(z0, z1, z2))
U43(ok(z0)) → ok(U43(z0))
U43(mark(z0)) → mark(U43(z0))
U202(mark(z0), z1) → mark(U202(z0, z1))
U202(ok(z0), ok(z1)) → ok(U202(z0, z1))
U143(ok(z0)) → ok(U143(z0))
U143(mark(z0)) → mark(U143(z0))
U52(ok(z0), ok(z1)) → ok(U52(z0, z1))
U52(mark(z0), z1) → mark(U52(z0, z1))
U51(mark(z0), z1, z2) → mark(U51(z0, z1, z2))
U51(ok(z0), ok(z1), ok(z2)) → ok(U51(z0, z1, z2))
U152(ok(z0), ok(z1)) → ok(U152(z0, z1))
U152(mark(z0), z1) → mark(U152(z0, z1))
pair(mark(z0), z1) → mark(pair(z0, z1))
pair(ok(z0), ok(z1)) → ok(pair(z0, z1))
pair(z0, mark(z1)) → mark(pair(z0, z1))
sel(ok(z0), ok(z1)) → ok(sel(z0, z1))
sel(mark(z0), z1) → mark(sel(z0, z1))
sel(z0, mark(z1)) → mark(sel(z0, z1))
U181(ok(z0), ok(z1)) → ok(U181(z0, z1))
U181(mark(z0), z1) → mark(U181(z0, z1))
proper(tt) → ok(tt)
proper(nil) → ok(nil)
proper(0) → ok(0)
U111(ok(z0), ok(z1)) → ok(U111(z0, z1))
U111(mark(z0), z1) → mark(U111(z0, z1))
isLNatKind(ok(z0)) → ok(isLNatKind(z0))
U11(mark(z0), z1, z2) → mark(U11(z0, z1, z2))
U11(ok(z0), ok(z1), ok(z2)) → ok(U11(z0, z1, z2))
U53(mark(z0)) → mark(U53(z0))
U53(ok(z0)) → ok(U53(z0))
U72(mark(z0)) → mark(U72(z0))
U72(ok(z0)) → ok(U72(z0))
U201(ok(z0), ok(z1), ok(z2), ok(z3)) → ok(U201(z0, z1, z2, z3))
U201(mark(z0), z1, z2, z3) → mark(U201(z0, z1, z2, z3))
splitAt(z0, mark(z1)) → mark(splitAt(z0, z1))
splitAt(mark(z0), z1) → mark(splitAt(z0, z1))
splitAt(ok(z0), ok(z1)) → ok(splitAt(z0, z1))
U112(mark(z0)) → mark(U112(z0))
U112(ok(z0)) → ok(U112(z0))
head(mark(z0)) → mark(head(z0))
head(ok(z0)) → ok(head(z0))
U82(mark(z0)) → mark(U82(z0))
U82(ok(z0)) → ok(U82(z0))
fst(mark(z0)) → mark(fst(z0))
fst(ok(z0)) → ok(fst(z0))
afterNth(ok(z0), ok(z1)) → ok(afterNth(z0, z1))
afterNth(z0, mark(z1)) → mark(afterNth(z0, z1))
afterNth(mark(z0), z1) → mark(afterNth(z0, z1))
tail(mark(z0)) → mark(tail(z0))
tail(ok(z0)) → ok(tail(z0))
U71(ok(z0), ok(z1)) → ok(U71(z0, z1))
U71(mark(z0), z1) → mark(U71(z0, z1))
isNaturalKind(ok(z0)) → ok(isNaturalKind(z0))
U42(ok(z0), ok(z1)) → ok(U42(z0, z1))
U42(mark(z0), z1) → mark(U42(z0, z1))
U142(mark(z0), z1) → mark(U142(z0, z1))
U142(ok(z0), ok(z1)) → ok(U142(z0, z1))
U62(ok(z0)) → ok(U62(z0))
U62(mark(z0)) → mark(U62(z0))
U122(mark(z0)) → mark(U122(z0))
U122(ok(z0)) → ok(U122(z0))
U121(mark(z0), z1) → mark(U121(z0, z1))
U121(ok(z0), ok(z1)) → ok(U121(z0, z1))
U21(mark(z0), z1) → mark(U21(z0, z1))
U21(ok(z0), ok(z1)) → ok(U21(z0, z1))
U131(ok(z0), ok(z1), ok(z2)) → ok(U131(z0, z1, z2))
U131(mark(z0), z1, z2) → mark(U131(z0, z1, z2))
U133(ok(z0)) → ok(U133(z0))
U133(mark(z0)) → mark(U133(z0))
take(z0, mark(z1)) → mark(take(z0, z1))
take(mark(z0), z1) → mark(take(z0, z1))
take(ok(z0), ok(z1)) → ok(take(z0, z1))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
natsFrom(mark(z0)) → mark(natsFrom(z0))
natsFrom(ok(z0)) → ok(natsFrom(z0))
U171(mark(z0), z1, z2) → mark(U171(z0, z1, z2))
U171(ok(z0), ok(z1), ok(z2)) → ok(U171(z0, z1, z2))
isPLNatKind(ok(z0)) → ok(isPLNatKind(z0))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
U41(mark(z0), z1, z2) → mark(U41(z0, z1, z2))
U41(ok(z0), ok(z1), ok(z2)) → ok(U41(z0, z1, z2))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
U91(ok(z0), ok(z1)) → ok(U91(z0, z1))
U91(mark(z0), z1) → mark(U91(z0, z1))
isNatural(ok(z0)) → ok(isNatural(z0))
U221(mark(z0), z1, z2) → mark(U221(z0, z1, z2))
U221(ok(z0), ok(z1), ok(z2)) → ok(U221(z0, z1, z2))
U101(ok(z0), ok(z1), ok(z2)) → ok(U101(z0, z1, z2))
U101(mark(z0), z1, z2) → mark(U101(z0, z1, z2))
U61(ok(z0), ok(z1)) → ok(U61(z0, z1))
U61(mark(z0), z1) → mark(U61(z0, z1))
snd(ok(z0)) → ok(snd(z0))
snd(mark(z0)) → mark(snd(z0))
U102(mark(z0), z1) → mark(U102(z0, z1))
U102(ok(z0), ok(z1)) → ok(U102(z0, z1))
U161(mark(z0), z1) → mark(U161(z0, z1))
U161(ok(z0), ok(z1)) → ok(U161(z0, z1))
U92(mark(z0)) → mark(U92(z0))
U92(ok(z0)) → ok(U92(z0))
isPLNat(ok(z0)) → ok(isPLNat(z0))
U141(mark(z0), z1, z2) → mark(U141(z0, z1, z2))
U141(ok(z0), ok(z1), ok(z2)) → ok(U141(z0, z1, z2))
U31(ok(z0), ok(z1)) → ok(U31(z0, z1))
U31(mark(z0), z1) → mark(U31(z0, z1))
U191(ok(z0), ok(z1)) → ok(U191(z0, z1))
U191(mark(z0), z1) → mark(U191(z0, z1))
U103(ok(z0)) → ok(U103(z0))
U103(mark(z0)) → mark(U103(z0))
U211(ok(z0), ok(z1)) → ok(U211(z0, z1))
U211(mark(z0), z1) → mark(U211(z0, z1))
U153(ok(z0)) → ok(U153(z0))
U153(mark(z0)) → mark(U153(z0))
U132(ok(z0), ok(z1)) → ok(U132(z0, z1))
U132(mark(z0), z1) → mark(U132(z0, z1))
U81(ok(z0), ok(z1)) → ok(U81(z0, z1))
U81(mark(z0), z1) → mark(U81(z0, z1))
isLNat(ok(z0)) → ok(isLNat(z0))
Tuples:

TOP(mark(z0)) → c1(TOP(proper(z0)), PROPER(z0))
U151'(ok(z0), ok(z1), ok(z2)) → c2(U151'(z0, z1, z2))
U151'(mark(z0), z1, z2) → c3(U151'(z0, z1, z2))
U43'(ok(z0)) → c4(U43'(z0))
U43'(mark(z0)) → c5(U43'(z0))
U202'(mark(z0), z1) → c6(U202'(z0, z1))
U202'(ok(z0), ok(z1)) → c7(U202'(z0, z1))
U143'(ok(z0)) → c8(U143'(z0))
U143'(mark(z0)) → c9(U143'(z0))
U52'(ok(z0), ok(z1)) → c10(U52'(z0, z1))
U52'(mark(z0), z1) → c11(U52'(z0, z1))
U51'(mark(z0), z1, z2) → c12(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c13(U51'(z0, z1, z2))
U152'(ok(z0), ok(z1)) → c14(U152'(z0, z1))
U152'(mark(z0), z1) → c15(U152'(z0, z1))
PAIR(mark(z0), z1) → c16(PAIR(z0, z1))
PAIR(ok(z0), ok(z1)) → c17(PAIR(z0, z1))
PAIR(z0, mark(z1)) → c18(PAIR(z0, z1))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
SEL(mark(z0), z1) → c20(SEL(z0, z1))
SEL(z0, mark(z1)) → c21(SEL(z0, z1))
U181'(ok(z0), ok(z1)) → c22(U181'(z0, z1))
U181'(mark(z0), z1) → c23(U181'(z0, z1))
U111'(ok(z0), ok(z1)) → c27(U111'(z0, z1))
U111'(mark(z0), z1) → c28(U111'(z0, z1))
ISLNATKIND(ok(z0)) → c29(ISLNATKIND(z0))
U11'(mark(z0), z1, z2) → c30(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c31(U11'(z0, z1, z2))
U53'(mark(z0)) → c32(U53'(z0))
U53'(ok(z0)) → c33(U53'(z0))
U72'(mark(z0)) → c34(U72'(z0))
U72'(ok(z0)) → c35(U72'(z0))
U201'(ok(z0), ok(z1), ok(z2), ok(z3)) → c36(U201'(z0, z1, z2, z3))
U201'(mark(z0), z1, z2, z3) → c37(U201'(z0, z1, z2, z3))
SPLITAT(z0, mark(z1)) → c38(SPLITAT(z0, z1))
SPLITAT(mark(z0), z1) → c39(SPLITAT(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c40(SPLITAT(z0, z1))
U112'(mark(z0)) → c41(U112'(z0))
U112'(ok(z0)) → c42(U112'(z0))
HEAD(mark(z0)) → c43(HEAD(z0))
HEAD(ok(z0)) → c44(HEAD(z0))
U82'(mark(z0)) → c45(U82'(z0))
U82'(ok(z0)) → c46(U82'(z0))
FST(mark(z0)) → c47(FST(z0))
FST(ok(z0)) → c48(FST(z0))
AFTERNTH(ok(z0), ok(z1)) → c49(AFTERNTH(z0, z1))
AFTERNTH(z0, mark(z1)) → c50(AFTERNTH(z0, z1))
AFTERNTH(mark(z0), z1) → c51(AFTERNTH(z0, z1))
TAIL(mark(z0)) → c52(TAIL(z0))
TAIL(ok(z0)) → c53(TAIL(z0))
U71'(ok(z0), ok(z1)) → c54(U71'(z0, z1))
U71'(mark(z0), z1) → c55(U71'(z0, z1))
ISNATURALKIND(ok(z0)) → c56(ISNATURALKIND(z0))
U42'(ok(z0), ok(z1)) → c57(U42'(z0, z1))
U42'(mark(z0), z1) → c58(U42'(z0, z1))
U142'(mark(z0), z1) → c59(U142'(z0, z1))
U142'(ok(z0), ok(z1)) → c60(U142'(z0, z1))
U62'(ok(z0)) → c61(U62'(z0))
U62'(mark(z0)) → c62(U62'(z0))
U122'(mark(z0)) → c63(U122'(z0))
U122'(ok(z0)) → c64(U122'(z0))
U121'(mark(z0), z1) → c65(U121'(z0, z1))
U121'(ok(z0), ok(z1)) → c66(U121'(z0, z1))
U21'(mark(z0), z1) → c67(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c68(U21'(z0, z1))
U131'(ok(z0), ok(z1), ok(z2)) → c69(U131'(z0, z1, z2))
U131'(mark(z0), z1, z2) → c70(U131'(z0, z1, z2))
U133'(ok(z0)) → c71(U133'(z0))
U133'(mark(z0)) → c72(U133'(z0))
TAKE(z0, mark(z1)) → c73(TAKE(z0, z1))
TAKE(mark(z0), z1) → c74(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c75(TAKE(z0, z1))
AND(mark(z0), z1) → c76(AND(z0, z1))
AND(ok(z0), ok(z1)) → c77(AND(z0, z1))
NATSFROM(mark(z0)) → c78(NATSFROM(z0))
NATSFROM(ok(z0)) → c79(NATSFROM(z0))
U171'(mark(z0), z1, z2) → c80(U171'(z0, z1, z2))
U171'(ok(z0), ok(z1), ok(z2)) → c81(U171'(z0, z1, z2))
ISPLNATKIND(ok(z0)) → c82(ISPLNATKIND(z0))
S(mark(z0)) → c83(S(z0))
S(ok(z0)) → c84(S(z0))
U41'(mark(z0), z1, z2) → c85(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c86(U41'(z0, z1, z2))
CONS(mark(z0), z1) → c87(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c88(CONS(z0, z1))
U91'(ok(z0), ok(z1)) → c89(U91'(z0, z1))
U91'(mark(z0), z1) → c90(U91'(z0, z1))
ISNATURAL(ok(z0)) → c91(ISNATURAL(z0))
U221'(mark(z0), z1, z2) → c92(U221'(z0, z1, z2))
U221'(ok(z0), ok(z1), ok(z2)) → c93(U221'(z0, z1, z2))
U101'(ok(z0), ok(z1), ok(z2)) → c94(U101'(z0, z1, z2))
U101'(mark(z0), z1, z2) → c95(U101'(z0, z1, z2))
U61'(ok(z0), ok(z1)) → c96(U61'(z0, z1))
U61'(mark(z0), z1) → c97(U61'(z0, z1))
SND(ok(z0)) → c98(SND(z0))
SND(mark(z0)) → c99(SND(z0))
U102'(mark(z0), z1) → c100(U102'(z0, z1))
U102'(ok(z0), ok(z1)) → c101(U102'(z0, z1))
U161'(mark(z0), z1) → c102(U161'(z0, z1))
U161'(ok(z0), ok(z1)) → c103(U161'(z0, z1))
U92'(mark(z0)) → c104(U92'(z0))
U92'(ok(z0)) → c105(U92'(z0))
ISPLNAT(ok(z0)) → c106(ISPLNAT(z0))
U141'(mark(z0), z1, z2) → c107(U141'(z0, z1, z2))
U141'(ok(z0), ok(z1), ok(z2)) → c108(U141'(z0, z1, z2))
U31'(ok(z0), ok(z1)) → c109(U31'(z0, z1))
U31'(mark(z0), z1) → c110(U31'(z0, z1))
U191'(ok(z0), ok(z1)) → c111(U191'(z0, z1))
U191'(mark(z0), z1) → c112(U191'(z0, z1))
U103'(ok(z0)) → c113(U103'(z0))
U103'(mark(z0)) → c114(U103'(z0))
U211'(ok(z0), ok(z1)) → c115(U211'(z0, z1))
U211'(mark(z0), z1) → c116(U211'(z0, z1))
U153'(ok(z0)) → c117(U153'(z0))
U153'(mark(z0)) → c118(U153'(z0))
U132'(ok(z0), ok(z1)) → c119(U132'(z0, z1))
U132'(mark(z0), z1) → c120(U132'(z0, z1))
U81'(ok(z0), ok(z1)) → c121(U81'(z0, z1))
U81'(mark(z0), z1) → c122(U81'(z0, z1))
ISLNAT(ok(z0)) → c123(ISLNAT(z0))
S tuples:

TOP(mark(z0)) → c1(TOP(proper(z0)), PROPER(z0))
U151'(ok(z0), ok(z1), ok(z2)) → c2(U151'(z0, z1, z2))
U151'(mark(z0), z1, z2) → c3(U151'(z0, z1, z2))
U43'(ok(z0)) → c4(U43'(z0))
U43'(mark(z0)) → c5(U43'(z0))
U202'(mark(z0), z1) → c6(U202'(z0, z1))
U202'(ok(z0), ok(z1)) → c7(U202'(z0, z1))
U143'(ok(z0)) → c8(U143'(z0))
U143'(mark(z0)) → c9(U143'(z0))
U52'(ok(z0), ok(z1)) → c10(U52'(z0, z1))
U52'(mark(z0), z1) → c11(U52'(z0, z1))
U51'(mark(z0), z1, z2) → c12(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c13(U51'(z0, z1, z2))
U152'(ok(z0), ok(z1)) → c14(U152'(z0, z1))
U152'(mark(z0), z1) → c15(U152'(z0, z1))
PAIR(mark(z0), z1) → c16(PAIR(z0, z1))
PAIR(ok(z0), ok(z1)) → c17(PAIR(z0, z1))
PAIR(z0, mark(z1)) → c18(PAIR(z0, z1))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
SEL(mark(z0), z1) → c20(SEL(z0, z1))
SEL(z0, mark(z1)) → c21(SEL(z0, z1))
U181'(ok(z0), ok(z1)) → c22(U181'(z0, z1))
U181'(mark(z0), z1) → c23(U181'(z0, z1))
U111'(ok(z0), ok(z1)) → c27(U111'(z0, z1))
U111'(mark(z0), z1) → c28(U111'(z0, z1))
ISLNATKIND(ok(z0)) → c29(ISLNATKIND(z0))
U11'(mark(z0), z1, z2) → c30(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c31(U11'(z0, z1, z2))
U53'(mark(z0)) → c32(U53'(z0))
U53'(ok(z0)) → c33(U53'(z0))
U72'(mark(z0)) → c34(U72'(z0))
U72'(ok(z0)) → c35(U72'(z0))
U201'(ok(z0), ok(z1), ok(z2), ok(z3)) → c36(U201'(z0, z1, z2, z3))
U201'(mark(z0), z1, z2, z3) → c37(U201'(z0, z1, z2, z3))
SPLITAT(z0, mark(z1)) → c38(SPLITAT(z0, z1))
SPLITAT(mark(z0), z1) → c39(SPLITAT(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c40(SPLITAT(z0, z1))
U112'(mark(z0)) → c41(U112'(z0))
U112'(ok(z0)) → c42(U112'(z0))
HEAD(mark(z0)) → c43(HEAD(z0))
HEAD(ok(z0)) → c44(HEAD(z0))
U82'(mark(z0)) → c45(U82'(z0))
U82'(ok(z0)) → c46(U82'(z0))
FST(mark(z0)) → c47(FST(z0))
FST(ok(z0)) → c48(FST(z0))
AFTERNTH(ok(z0), ok(z1)) → c49(AFTERNTH(z0, z1))
AFTERNTH(z0, mark(z1)) → c50(AFTERNTH(z0, z1))
AFTERNTH(mark(z0), z1) → c51(AFTERNTH(z0, z1))
TAIL(mark(z0)) → c52(TAIL(z0))
TAIL(ok(z0)) → c53(TAIL(z0))
U71'(ok(z0), ok(z1)) → c54(U71'(z0, z1))
U71'(mark(z0), z1) → c55(U71'(z0, z1))
ISNATURALKIND(ok(z0)) → c56(ISNATURALKIND(z0))
U42'(ok(z0), ok(z1)) → c57(U42'(z0, z1))
U42'(mark(z0), z1) → c58(U42'(z0, z1))
U142'(mark(z0), z1) → c59(U142'(z0, z1))
U142'(ok(z0), ok(z1)) → c60(U142'(z0, z1))
U62'(ok(z0)) → c61(U62'(z0))
U62'(mark(z0)) → c62(U62'(z0))
U122'(mark(z0)) → c63(U122'(z0))
U122'(ok(z0)) → c64(U122'(z0))
U121'(mark(z0), z1) → c65(U121'(z0, z1))
U121'(ok(z0), ok(z1)) → c66(U121'(z0, z1))
U21'(mark(z0), z1) → c67(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c68(U21'(z0, z1))
U131'(ok(z0), ok(z1), ok(z2)) → c69(U131'(z0, z1, z2))
U131'(mark(z0), z1, z2) → c70(U131'(z0, z1, z2))
U133'(ok(z0)) → c71(U133'(z0))
U133'(mark(z0)) → c72(U133'(z0))
TAKE(z0, mark(z1)) → c73(TAKE(z0, z1))
TAKE(mark(z0), z1) → c74(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c75(TAKE(z0, z1))
AND(mark(z0), z1) → c76(AND(z0, z1))
AND(ok(z0), ok(z1)) → c77(AND(z0, z1))
NATSFROM(mark(z0)) → c78(NATSFROM(z0))
NATSFROM(ok(z0)) → c79(NATSFROM(z0))
U171'(mark(z0), z1, z2) → c80(U171'(z0, z1, z2))
U171'(ok(z0), ok(z1), ok(z2)) → c81(U171'(z0, z1, z2))
ISPLNATKIND(ok(z0)) → c82(ISPLNATKIND(z0))
S(mark(z0)) → c83(S(z0))
S(ok(z0)) → c84(S(z0))
U41'(mark(z0), z1, z2) → c85(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c86(U41'(z0, z1, z2))
CONS(mark(z0), z1) → c87(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c88(CONS(z0, z1))
U91'(ok(z0), ok(z1)) → c89(U91'(z0, z1))
U91'(mark(z0), z1) → c90(U91'(z0, z1))
ISNATURAL(ok(z0)) → c91(ISNATURAL(z0))
U221'(mark(z0), z1, z2) → c92(U221'(z0, z1, z2))
U221'(ok(z0), ok(z1), ok(z2)) → c93(U221'(z0, z1, z2))
U101'(ok(z0), ok(z1), ok(z2)) → c94(U101'(z0, z1, z2))
U101'(mark(z0), z1, z2) → c95(U101'(z0, z1, z2))
U61'(ok(z0), ok(z1)) → c96(U61'(z0, z1))
U61'(mark(z0), z1) → c97(U61'(z0, z1))
SND(ok(z0)) → c98(SND(z0))
SND(mark(z0)) → c99(SND(z0))
U102'(mark(z0), z1) → c100(U102'(z0, z1))
U102'(ok(z0), ok(z1)) → c101(U102'(z0, z1))
U161'(mark(z0), z1) → c102(U161'(z0, z1))
U161'(ok(z0), ok(z1)) → c103(U161'(z0, z1))
U92'(mark(z0)) → c104(U92'(z0))
U92'(ok(z0)) → c105(U92'(z0))
ISPLNAT(ok(z0)) → c106(ISPLNAT(z0))
U141'(mark(z0), z1, z2) → c107(U141'(z0, z1, z2))
U141'(ok(z0), ok(z1), ok(z2)) → c108(U141'(z0, z1, z2))
U31'(ok(z0), ok(z1)) → c109(U31'(z0, z1))
U31'(mark(z0), z1) → c110(U31'(z0, z1))
U191'(ok(z0), ok(z1)) → c111(U191'(z0, z1))
U191'(mark(z0), z1) → c112(U191'(z0, z1))
U103'(ok(z0)) → c113(U103'(z0))
U103'(mark(z0)) → c114(U103'(z0))
U211'(ok(z0), ok(z1)) → c115(U211'(z0, z1))
U211'(mark(z0), z1) → c116(U211'(z0, z1))
U153'(ok(z0)) → c117(U153'(z0))
U153'(mark(z0)) → c118(U153'(z0))
U132'(ok(z0), ok(z1)) → c119(U132'(z0, z1))
U132'(mark(z0), z1) → c120(U132'(z0, z1))
U81'(ok(z0), ok(z1)) → c121(U81'(z0, z1))
U81'(mark(z0), z1) → c122(U81'(z0, z1))
ISLNAT(ok(z0)) → c123(ISLNAT(z0))
K tuples:none
Defined Rule Symbols:

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

Defined Pair Symbols:

TOP, U151', U43', U202', U143', U52', U51', U152', PAIR, SEL, U181', U111', ISLNATKIND, U11', U53', U72', U201', SPLITAT, U112', HEAD, U82', FST, AFTERNTH, TAIL, U71', ISNATURALKIND, U42', U142', U62', U122', U121', U21', U131', U133', TAKE, AND, NATSFROM, U171', ISPLNATKIND, S, U41', CONS, U91', ISNATURAL, U221', U101', U61', SND, U102', U161', U92', ISPLNAT, U141', U31', U191', U103', U211', U153', U132', U81', ISLNAT

Compound Symbols:

c1, c2, c3, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c48, c49, c50, c51, c52, c53, c54, c55, c56, c57, c58, c59, c60, c61, c62, c63, c64, c65, c66, c67, c68, c69, c70, c71, c72, c73, c74, c75, c76, c77, c78, c79, c80, c81, c82, c83, c84, c85, c86, c87, c88, c89, c90, c91, c92, c93, c94, c95, c96, c97, c98, c99, c100, c101, c102, c103, c104, c105, c106, c107, c108, c109, c110, c111, c112, c113, c114, c115, c116, c117, c118, c119, c120, c121, c122, c123

(9) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)

Removed 1 trailing tuple parts

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

top(ok(z0)) → top(active(z0))
top(mark(z0)) → top(proper(z0))
U151(ok(z0), ok(z1), ok(z2)) → ok(U151(z0, z1, z2))
U151(mark(z0), z1, z2) → mark(U151(z0, z1, z2))
U43(ok(z0)) → ok(U43(z0))
U43(mark(z0)) → mark(U43(z0))
U202(mark(z0), z1) → mark(U202(z0, z1))
U202(ok(z0), ok(z1)) → ok(U202(z0, z1))
U143(ok(z0)) → ok(U143(z0))
U143(mark(z0)) → mark(U143(z0))
U52(ok(z0), ok(z1)) → ok(U52(z0, z1))
U52(mark(z0), z1) → mark(U52(z0, z1))
U51(mark(z0), z1, z2) → mark(U51(z0, z1, z2))
U51(ok(z0), ok(z1), ok(z2)) → ok(U51(z0, z1, z2))
U152(ok(z0), ok(z1)) → ok(U152(z0, z1))
U152(mark(z0), z1) → mark(U152(z0, z1))
pair(mark(z0), z1) → mark(pair(z0, z1))
pair(ok(z0), ok(z1)) → ok(pair(z0, z1))
pair(z0, mark(z1)) → mark(pair(z0, z1))
sel(ok(z0), ok(z1)) → ok(sel(z0, z1))
sel(mark(z0), z1) → mark(sel(z0, z1))
sel(z0, mark(z1)) → mark(sel(z0, z1))
U181(ok(z0), ok(z1)) → ok(U181(z0, z1))
U181(mark(z0), z1) → mark(U181(z0, z1))
proper(tt) → ok(tt)
proper(nil) → ok(nil)
proper(0) → ok(0)
U111(ok(z0), ok(z1)) → ok(U111(z0, z1))
U111(mark(z0), z1) → mark(U111(z0, z1))
isLNatKind(ok(z0)) → ok(isLNatKind(z0))
U11(mark(z0), z1, z2) → mark(U11(z0, z1, z2))
U11(ok(z0), ok(z1), ok(z2)) → ok(U11(z0, z1, z2))
U53(mark(z0)) → mark(U53(z0))
U53(ok(z0)) → ok(U53(z0))
U72(mark(z0)) → mark(U72(z0))
U72(ok(z0)) → ok(U72(z0))
U201(ok(z0), ok(z1), ok(z2), ok(z3)) → ok(U201(z0, z1, z2, z3))
U201(mark(z0), z1, z2, z3) → mark(U201(z0, z1, z2, z3))
splitAt(z0, mark(z1)) → mark(splitAt(z0, z1))
splitAt(mark(z0), z1) → mark(splitAt(z0, z1))
splitAt(ok(z0), ok(z1)) → ok(splitAt(z0, z1))
U112(mark(z0)) → mark(U112(z0))
U112(ok(z0)) → ok(U112(z0))
head(mark(z0)) → mark(head(z0))
head(ok(z0)) → ok(head(z0))
U82(mark(z0)) → mark(U82(z0))
U82(ok(z0)) → ok(U82(z0))
fst(mark(z0)) → mark(fst(z0))
fst(ok(z0)) → ok(fst(z0))
afterNth(ok(z0), ok(z1)) → ok(afterNth(z0, z1))
afterNth(z0, mark(z1)) → mark(afterNth(z0, z1))
afterNth(mark(z0), z1) → mark(afterNth(z0, z1))
tail(mark(z0)) → mark(tail(z0))
tail(ok(z0)) → ok(tail(z0))
U71(ok(z0), ok(z1)) → ok(U71(z0, z1))
U71(mark(z0), z1) → mark(U71(z0, z1))
isNaturalKind(ok(z0)) → ok(isNaturalKind(z0))
U42(ok(z0), ok(z1)) → ok(U42(z0, z1))
U42(mark(z0), z1) → mark(U42(z0, z1))
U142(mark(z0), z1) → mark(U142(z0, z1))
U142(ok(z0), ok(z1)) → ok(U142(z0, z1))
U62(ok(z0)) → ok(U62(z0))
U62(mark(z0)) → mark(U62(z0))
U122(mark(z0)) → mark(U122(z0))
U122(ok(z0)) → ok(U122(z0))
U121(mark(z0), z1) → mark(U121(z0, z1))
U121(ok(z0), ok(z1)) → ok(U121(z0, z1))
U21(mark(z0), z1) → mark(U21(z0, z1))
U21(ok(z0), ok(z1)) → ok(U21(z0, z1))
U131(ok(z0), ok(z1), ok(z2)) → ok(U131(z0, z1, z2))
U131(mark(z0), z1, z2) → mark(U131(z0, z1, z2))
U133(ok(z0)) → ok(U133(z0))
U133(mark(z0)) → mark(U133(z0))
take(z0, mark(z1)) → mark(take(z0, z1))
take(mark(z0), z1) → mark(take(z0, z1))
take(ok(z0), ok(z1)) → ok(take(z0, z1))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
natsFrom(mark(z0)) → mark(natsFrom(z0))
natsFrom(ok(z0)) → ok(natsFrom(z0))
U171(mark(z0), z1, z2) → mark(U171(z0, z1, z2))
U171(ok(z0), ok(z1), ok(z2)) → ok(U171(z0, z1, z2))
isPLNatKind(ok(z0)) → ok(isPLNatKind(z0))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
U41(mark(z0), z1, z2) → mark(U41(z0, z1, z2))
U41(ok(z0), ok(z1), ok(z2)) → ok(U41(z0, z1, z2))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
U91(ok(z0), ok(z1)) → ok(U91(z0, z1))
U91(mark(z0), z1) → mark(U91(z0, z1))
isNatural(ok(z0)) → ok(isNatural(z0))
U221(mark(z0), z1, z2) → mark(U221(z0, z1, z2))
U221(ok(z0), ok(z1), ok(z2)) → ok(U221(z0, z1, z2))
U101(ok(z0), ok(z1), ok(z2)) → ok(U101(z0, z1, z2))
U101(mark(z0), z1, z2) → mark(U101(z0, z1, z2))
U61(ok(z0), ok(z1)) → ok(U61(z0, z1))
U61(mark(z0), z1) → mark(U61(z0, z1))
snd(ok(z0)) → ok(snd(z0))
snd(mark(z0)) → mark(snd(z0))
U102(mark(z0), z1) → mark(U102(z0, z1))
U102(ok(z0), ok(z1)) → ok(U102(z0, z1))
U161(mark(z0), z1) → mark(U161(z0, z1))
U161(ok(z0), ok(z1)) → ok(U161(z0, z1))
U92(mark(z0)) → mark(U92(z0))
U92(ok(z0)) → ok(U92(z0))
isPLNat(ok(z0)) → ok(isPLNat(z0))
U141(mark(z0), z1, z2) → mark(U141(z0, z1, z2))
U141(ok(z0), ok(z1), ok(z2)) → ok(U141(z0, z1, z2))
U31(ok(z0), ok(z1)) → ok(U31(z0, z1))
U31(mark(z0), z1) → mark(U31(z0, z1))
U191(ok(z0), ok(z1)) → ok(U191(z0, z1))
U191(mark(z0), z1) → mark(U191(z0, z1))
U103(ok(z0)) → ok(U103(z0))
U103(mark(z0)) → mark(U103(z0))
U211(ok(z0), ok(z1)) → ok(U211(z0, z1))
U211(mark(z0), z1) → mark(U211(z0, z1))
U153(ok(z0)) → ok(U153(z0))
U153(mark(z0)) → mark(U153(z0))
U132(ok(z0), ok(z1)) → ok(U132(z0, z1))
U132(mark(z0), z1) → mark(U132(z0, z1))
U81(ok(z0), ok(z1)) → ok(U81(z0, z1))
U81(mark(z0), z1) → mark(U81(z0, z1))
isLNat(ok(z0)) → ok(isLNat(z0))
Tuples:

U151'(ok(z0), ok(z1), ok(z2)) → c2(U151'(z0, z1, z2))
U151'(mark(z0), z1, z2) → c3(U151'(z0, z1, z2))
U43'(ok(z0)) → c4(U43'(z0))
U43'(mark(z0)) → c5(U43'(z0))
U202'(mark(z0), z1) → c6(U202'(z0, z1))
U202'(ok(z0), ok(z1)) → c7(U202'(z0, z1))
U143'(ok(z0)) → c8(U143'(z0))
U143'(mark(z0)) → c9(U143'(z0))
U52'(ok(z0), ok(z1)) → c10(U52'(z0, z1))
U52'(mark(z0), z1) → c11(U52'(z0, z1))
U51'(mark(z0), z1, z2) → c12(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c13(U51'(z0, z1, z2))
U152'(ok(z0), ok(z1)) → c14(U152'(z0, z1))
U152'(mark(z0), z1) → c15(U152'(z0, z1))
PAIR(mark(z0), z1) → c16(PAIR(z0, z1))
PAIR(ok(z0), ok(z1)) → c17(PAIR(z0, z1))
PAIR(z0, mark(z1)) → c18(PAIR(z0, z1))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
SEL(mark(z0), z1) → c20(SEL(z0, z1))
SEL(z0, mark(z1)) → c21(SEL(z0, z1))
U181'(ok(z0), ok(z1)) → c22(U181'(z0, z1))
U181'(mark(z0), z1) → c23(U181'(z0, z1))
U111'(ok(z0), ok(z1)) → c27(U111'(z0, z1))
U111'(mark(z0), z1) → c28(U111'(z0, z1))
ISLNATKIND(ok(z0)) → c29(ISLNATKIND(z0))
U11'(mark(z0), z1, z2) → c30(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c31(U11'(z0, z1, z2))
U53'(mark(z0)) → c32(U53'(z0))
U53'(ok(z0)) → c33(U53'(z0))
U72'(mark(z0)) → c34(U72'(z0))
U72'(ok(z0)) → c35(U72'(z0))
U201'(ok(z0), ok(z1), ok(z2), ok(z3)) → c36(U201'(z0, z1, z2, z3))
U201'(mark(z0), z1, z2, z3) → c37(U201'(z0, z1, z2, z3))
SPLITAT(z0, mark(z1)) → c38(SPLITAT(z0, z1))
SPLITAT(mark(z0), z1) → c39(SPLITAT(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c40(SPLITAT(z0, z1))
U112'(mark(z0)) → c41(U112'(z0))
U112'(ok(z0)) → c42(U112'(z0))
HEAD(mark(z0)) → c43(HEAD(z0))
HEAD(ok(z0)) → c44(HEAD(z0))
U82'(mark(z0)) → c45(U82'(z0))
U82'(ok(z0)) → c46(U82'(z0))
FST(mark(z0)) → c47(FST(z0))
FST(ok(z0)) → c48(FST(z0))
AFTERNTH(ok(z0), ok(z1)) → c49(AFTERNTH(z0, z1))
AFTERNTH(z0, mark(z1)) → c50(AFTERNTH(z0, z1))
AFTERNTH(mark(z0), z1) → c51(AFTERNTH(z0, z1))
TAIL(mark(z0)) → c52(TAIL(z0))
TAIL(ok(z0)) → c53(TAIL(z0))
U71'(ok(z0), ok(z1)) → c54(U71'(z0, z1))
U71'(mark(z0), z1) → c55(U71'(z0, z1))
ISNATURALKIND(ok(z0)) → c56(ISNATURALKIND(z0))
U42'(ok(z0), ok(z1)) → c57(U42'(z0, z1))
U42'(mark(z0), z1) → c58(U42'(z0, z1))
U142'(mark(z0), z1) → c59(U142'(z0, z1))
U142'(ok(z0), ok(z1)) → c60(U142'(z0, z1))
U62'(ok(z0)) → c61(U62'(z0))
U62'(mark(z0)) → c62(U62'(z0))
U122'(mark(z0)) → c63(U122'(z0))
U122'(ok(z0)) → c64(U122'(z0))
U121'(mark(z0), z1) → c65(U121'(z0, z1))
U121'(ok(z0), ok(z1)) → c66(U121'(z0, z1))
U21'(mark(z0), z1) → c67(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c68(U21'(z0, z1))
U131'(ok(z0), ok(z1), ok(z2)) → c69(U131'(z0, z1, z2))
U131'(mark(z0), z1, z2) → c70(U131'(z0, z1, z2))
U133'(ok(z0)) → c71(U133'(z0))
U133'(mark(z0)) → c72(U133'(z0))
TAKE(z0, mark(z1)) → c73(TAKE(z0, z1))
TAKE(mark(z0), z1) → c74(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c75(TAKE(z0, z1))
AND(mark(z0), z1) → c76(AND(z0, z1))
AND(ok(z0), ok(z1)) → c77(AND(z0, z1))
NATSFROM(mark(z0)) → c78(NATSFROM(z0))
NATSFROM(ok(z0)) → c79(NATSFROM(z0))
U171'(mark(z0), z1, z2) → c80(U171'(z0, z1, z2))
U171'(ok(z0), ok(z1), ok(z2)) → c81(U171'(z0, z1, z2))
ISPLNATKIND(ok(z0)) → c82(ISPLNATKIND(z0))
S(mark(z0)) → c83(S(z0))
S(ok(z0)) → c84(S(z0))
U41'(mark(z0), z1, z2) → c85(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c86(U41'(z0, z1, z2))
CONS(mark(z0), z1) → c87(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c88(CONS(z0, z1))
U91'(ok(z0), ok(z1)) → c89(U91'(z0, z1))
U91'(mark(z0), z1) → c90(U91'(z0, z1))
ISNATURAL(ok(z0)) → c91(ISNATURAL(z0))
U221'(mark(z0), z1, z2) → c92(U221'(z0, z1, z2))
U221'(ok(z0), ok(z1), ok(z2)) → c93(U221'(z0, z1, z2))
U101'(ok(z0), ok(z1), ok(z2)) → c94(U101'(z0, z1, z2))
U101'(mark(z0), z1, z2) → c95(U101'(z0, z1, z2))
U61'(ok(z0), ok(z1)) → c96(U61'(z0, z1))
U61'(mark(z0), z1) → c97(U61'(z0, z1))
SND(ok(z0)) → c98(SND(z0))
SND(mark(z0)) → c99(SND(z0))
U102'(mark(z0), z1) → c100(U102'(z0, z1))
U102'(ok(z0), ok(z1)) → c101(U102'(z0, z1))
U161'(mark(z0), z1) → c102(U161'(z0, z1))
U161'(ok(z0), ok(z1)) → c103(U161'(z0, z1))
U92'(mark(z0)) → c104(U92'(z0))
U92'(ok(z0)) → c105(U92'(z0))
ISPLNAT(ok(z0)) → c106(ISPLNAT(z0))
U141'(mark(z0), z1, z2) → c107(U141'(z0, z1, z2))
U141'(ok(z0), ok(z1), ok(z2)) → c108(U141'(z0, z1, z2))
U31'(ok(z0), ok(z1)) → c109(U31'(z0, z1))
U31'(mark(z0), z1) → c110(U31'(z0, z1))
U191'(ok(z0), ok(z1)) → c111(U191'(z0, z1))
U191'(mark(z0), z1) → c112(U191'(z0, z1))
U103'(ok(z0)) → c113(U103'(z0))
U103'(mark(z0)) → c114(U103'(z0))
U211'(ok(z0), ok(z1)) → c115(U211'(z0, z1))
U211'(mark(z0), z1) → c116(U211'(z0, z1))
U153'(ok(z0)) → c117(U153'(z0))
U153'(mark(z0)) → c118(U153'(z0))
U132'(ok(z0), ok(z1)) → c119(U132'(z0, z1))
U132'(mark(z0), z1) → c120(U132'(z0, z1))
U81'(ok(z0), ok(z1)) → c121(U81'(z0, z1))
U81'(mark(z0), z1) → c122(U81'(z0, z1))
ISLNAT(ok(z0)) → c123(ISLNAT(z0))
TOP(mark(z0)) → c1(TOP(proper(z0)))
S tuples:

U151'(ok(z0), ok(z1), ok(z2)) → c2(U151'(z0, z1, z2))
U151'(mark(z0), z1, z2) → c3(U151'(z0, z1, z2))
U43'(ok(z0)) → c4(U43'(z0))
U43'(mark(z0)) → c5(U43'(z0))
U202'(mark(z0), z1) → c6(U202'(z0, z1))
U202'(ok(z0), ok(z1)) → c7(U202'(z0, z1))
U143'(ok(z0)) → c8(U143'(z0))
U143'(mark(z0)) → c9(U143'(z0))
U52'(ok(z0), ok(z1)) → c10(U52'(z0, z1))
U52'(mark(z0), z1) → c11(U52'(z0, z1))
U51'(mark(z0), z1, z2) → c12(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c13(U51'(z0, z1, z2))
U152'(ok(z0), ok(z1)) → c14(U152'(z0, z1))
U152'(mark(z0), z1) → c15(U152'(z0, z1))
PAIR(mark(z0), z1) → c16(PAIR(z0, z1))
PAIR(ok(z0), ok(z1)) → c17(PAIR(z0, z1))
PAIR(z0, mark(z1)) → c18(PAIR(z0, z1))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
SEL(mark(z0), z1) → c20(SEL(z0, z1))
SEL(z0, mark(z1)) → c21(SEL(z0, z1))
U181'(ok(z0), ok(z1)) → c22(U181'(z0, z1))
U181'(mark(z0), z1) → c23(U181'(z0, z1))
U111'(ok(z0), ok(z1)) → c27(U111'(z0, z1))
U111'(mark(z0), z1) → c28(U111'(z0, z1))
ISLNATKIND(ok(z0)) → c29(ISLNATKIND(z0))
U11'(mark(z0), z1, z2) → c30(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c31(U11'(z0, z1, z2))
U53'(mark(z0)) → c32(U53'(z0))
U53'(ok(z0)) → c33(U53'(z0))
U72'(mark(z0)) → c34(U72'(z0))
U72'(ok(z0)) → c35(U72'(z0))
U201'(ok(z0), ok(z1), ok(z2), ok(z3)) → c36(U201'(z0, z1, z2, z3))
U201'(mark(z0), z1, z2, z3) → c37(U201'(z0, z1, z2, z3))
SPLITAT(z0, mark(z1)) → c38(SPLITAT(z0, z1))
SPLITAT(mark(z0), z1) → c39(SPLITAT(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c40(SPLITAT(z0, z1))
U112'(mark(z0)) → c41(U112'(z0))
U112'(ok(z0)) → c42(U112'(z0))
HEAD(mark(z0)) → c43(HEAD(z0))
HEAD(ok(z0)) → c44(HEAD(z0))
U82'(mark(z0)) → c45(U82'(z0))
U82'(ok(z0)) → c46(U82'(z0))
FST(mark(z0)) → c47(FST(z0))
FST(ok(z0)) → c48(FST(z0))
AFTERNTH(ok(z0), ok(z1)) → c49(AFTERNTH(z0, z1))
AFTERNTH(z0, mark(z1)) → c50(AFTERNTH(z0, z1))
AFTERNTH(mark(z0), z1) → c51(AFTERNTH(z0, z1))
TAIL(mark(z0)) → c52(TAIL(z0))
TAIL(ok(z0)) → c53(TAIL(z0))
U71'(ok(z0), ok(z1)) → c54(U71'(z0, z1))
U71'(mark(z0), z1) → c55(U71'(z0, z1))
ISNATURALKIND(ok(z0)) → c56(ISNATURALKIND(z0))
U42'(ok(z0), ok(z1)) → c57(U42'(z0, z1))
U42'(mark(z0), z1) → c58(U42'(z0, z1))
U142'(mark(z0), z1) → c59(U142'(z0, z1))
U142'(ok(z0), ok(z1)) → c60(U142'(z0, z1))
U62'(ok(z0)) → c61(U62'(z0))
U62'(mark(z0)) → c62(U62'(z0))
U122'(mark(z0)) → c63(U122'(z0))
U122'(ok(z0)) → c64(U122'(z0))
U121'(mark(z0), z1) → c65(U121'(z0, z1))
U121'(ok(z0), ok(z1)) → c66(U121'(z0, z1))
U21'(mark(z0), z1) → c67(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c68(U21'(z0, z1))
U131'(ok(z0), ok(z1), ok(z2)) → c69(U131'(z0, z1, z2))
U131'(mark(z0), z1, z2) → c70(U131'(z0, z1, z2))
U133'(ok(z0)) → c71(U133'(z0))
U133'(mark(z0)) → c72(U133'(z0))
TAKE(z0, mark(z1)) → c73(TAKE(z0, z1))
TAKE(mark(z0), z1) → c74(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c75(TAKE(z0, z1))
AND(mark(z0), z1) → c76(AND(z0, z1))
AND(ok(z0), ok(z1)) → c77(AND(z0, z1))
NATSFROM(mark(z0)) → c78(NATSFROM(z0))
NATSFROM(ok(z0)) → c79(NATSFROM(z0))
U171'(mark(z0), z1, z2) → c80(U171'(z0, z1, z2))
U171'(ok(z0), ok(z1), ok(z2)) → c81(U171'(z0, z1, z2))
ISPLNATKIND(ok(z0)) → c82(ISPLNATKIND(z0))
S(mark(z0)) → c83(S(z0))
S(ok(z0)) → c84(S(z0))
U41'(mark(z0), z1, z2) → c85(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c86(U41'(z0, z1, z2))
CONS(mark(z0), z1) → c87(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c88(CONS(z0, z1))
U91'(ok(z0), ok(z1)) → c89(U91'(z0, z1))
U91'(mark(z0), z1) → c90(U91'(z0, z1))
ISNATURAL(ok(z0)) → c91(ISNATURAL(z0))
U221'(mark(z0), z1, z2) → c92(U221'(z0, z1, z2))
U221'(ok(z0), ok(z1), ok(z2)) → c93(U221'(z0, z1, z2))
U101'(ok(z0), ok(z1), ok(z2)) → c94(U101'(z0, z1, z2))
U101'(mark(z0), z1, z2) → c95(U101'(z0, z1, z2))
U61'(ok(z0), ok(z1)) → c96(U61'(z0, z1))
U61'(mark(z0), z1) → c97(U61'(z0, z1))
SND(ok(z0)) → c98(SND(z0))
SND(mark(z0)) → c99(SND(z0))
U102'(mark(z0), z1) → c100(U102'(z0, z1))
U102'(ok(z0), ok(z1)) → c101(U102'(z0, z1))
U161'(mark(z0), z1) → c102(U161'(z0, z1))
U161'(ok(z0), ok(z1)) → c103(U161'(z0, z1))
U92'(mark(z0)) → c104(U92'(z0))
U92'(ok(z0)) → c105(U92'(z0))
ISPLNAT(ok(z0)) → c106(ISPLNAT(z0))
U141'(mark(z0), z1, z2) → c107(U141'(z0, z1, z2))
U141'(ok(z0), ok(z1), ok(z2)) → c108(U141'(z0, z1, z2))
U31'(ok(z0), ok(z1)) → c109(U31'(z0, z1))
U31'(mark(z0), z1) → c110(U31'(z0, z1))
U191'(ok(z0), ok(z1)) → c111(U191'(z0, z1))
U191'(mark(z0), z1) → c112(U191'(z0, z1))
U103'(ok(z0)) → c113(U103'(z0))
U103'(mark(z0)) → c114(U103'(z0))
U211'(ok(z0), ok(z1)) → c115(U211'(z0, z1))
U211'(mark(z0), z1) → c116(U211'(z0, z1))
U153'(ok(z0)) → c117(U153'(z0))
U153'(mark(z0)) → c118(U153'(z0))
U132'(ok(z0), ok(z1)) → c119(U132'(z0, z1))
U132'(mark(z0), z1) → c120(U132'(z0, z1))
U81'(ok(z0), ok(z1)) → c121(U81'(z0, z1))
U81'(mark(z0), z1) → c122(U81'(z0, z1))
ISLNAT(ok(z0)) → c123(ISLNAT(z0))
TOP(mark(z0)) → c1(TOP(proper(z0)))
K tuples:none
Defined Rule Symbols:

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

Defined Pair Symbols:

U151', U43', U202', U143', U52', U51', U152', PAIR, SEL, U181', U111', ISLNATKIND, U11', U53', U72', U201', SPLITAT, U112', HEAD, U82', FST, AFTERNTH, TAIL, U71', ISNATURALKIND, U42', U142', U62', U122', U121', U21', U131', U133', TAKE, AND, NATSFROM, U171', ISPLNATKIND, S, U41', CONS, U91', ISNATURAL, U221', U101', U61', SND, U102', U161', U92', ISPLNAT, U141', U31', U191', U103', U211', U153', U132', U81', ISLNAT, TOP

Compound Symbols:

c2, c3, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c48, c49, c50, c51, c52, c53, c54, c55, c56, c57, c58, c59, c60, c61, c62, c63, c64, c65, c66, c67, c68, c69, c70, c71, c72, c73, c74, c75, c76, c77, c78, c79, c80, c81, c82, c83, c84, c85, c86, c87, c88, c89, c90, c91, c92, c93, c94, c95, c96, c97, c98, c99, c100, c101, c102, c103, c104, c105, c106, c107, c108, c109, c110, c111, c112, c113, c114, c115, c116, c117, c118, c119, c120, c121, c122, c123, c1

(11) CdtUsableRulesProof (EQUIVALENT transformation)

The following rules are not usable and were removed:

top(ok(z0)) → top(active(z0))
top(mark(z0)) → top(proper(z0))
U151(ok(z0), ok(z1), ok(z2)) → ok(U151(z0, z1, z2))
U151(mark(z0), z1, z2) → mark(U151(z0, z1, z2))
U43(ok(z0)) → ok(U43(z0))
U43(mark(z0)) → mark(U43(z0))
U202(mark(z0), z1) → mark(U202(z0, z1))
U202(ok(z0), ok(z1)) → ok(U202(z0, z1))
U143(ok(z0)) → ok(U143(z0))
U143(mark(z0)) → mark(U143(z0))
U52(ok(z0), ok(z1)) → ok(U52(z0, z1))
U52(mark(z0), z1) → mark(U52(z0, z1))
U51(mark(z0), z1, z2) → mark(U51(z0, z1, z2))
U51(ok(z0), ok(z1), ok(z2)) → ok(U51(z0, z1, z2))
U152(ok(z0), ok(z1)) → ok(U152(z0, z1))
U152(mark(z0), z1) → mark(U152(z0, z1))
pair(mark(z0), z1) → mark(pair(z0, z1))
pair(ok(z0), ok(z1)) → ok(pair(z0, z1))
pair(z0, mark(z1)) → mark(pair(z0, z1))
sel(ok(z0), ok(z1)) → ok(sel(z0, z1))
sel(mark(z0), z1) → mark(sel(z0, z1))
sel(z0, mark(z1)) → mark(sel(z0, z1))
U181(ok(z0), ok(z1)) → ok(U181(z0, z1))
U181(mark(z0), z1) → mark(U181(z0, z1))
U111(ok(z0), ok(z1)) → ok(U111(z0, z1))
U111(mark(z0), z1) → mark(U111(z0, z1))
isLNatKind(ok(z0)) → ok(isLNatKind(z0))
U11(mark(z0), z1, z2) → mark(U11(z0, z1, z2))
U11(ok(z0), ok(z1), ok(z2)) → ok(U11(z0, z1, z2))
U53(mark(z0)) → mark(U53(z0))
U53(ok(z0)) → ok(U53(z0))
U72(mark(z0)) → mark(U72(z0))
U72(ok(z0)) → ok(U72(z0))
U201(ok(z0), ok(z1), ok(z2), ok(z3)) → ok(U201(z0, z1, z2, z3))
U201(mark(z0), z1, z2, z3) → mark(U201(z0, z1, z2, z3))
splitAt(z0, mark(z1)) → mark(splitAt(z0, z1))
splitAt(mark(z0), z1) → mark(splitAt(z0, z1))
splitAt(ok(z0), ok(z1)) → ok(splitAt(z0, z1))
U112(mark(z0)) → mark(U112(z0))
U112(ok(z0)) → ok(U112(z0))
head(mark(z0)) → mark(head(z0))
head(ok(z0)) → ok(head(z0))
U82(mark(z0)) → mark(U82(z0))
U82(ok(z0)) → ok(U82(z0))
fst(mark(z0)) → mark(fst(z0))
fst(ok(z0)) → ok(fst(z0))
afterNth(ok(z0), ok(z1)) → ok(afterNth(z0, z1))
afterNth(z0, mark(z1)) → mark(afterNth(z0, z1))
afterNth(mark(z0), z1) → mark(afterNth(z0, z1))
tail(mark(z0)) → mark(tail(z0))
tail(ok(z0)) → ok(tail(z0))
U71(ok(z0), ok(z1)) → ok(U71(z0, z1))
U71(mark(z0), z1) → mark(U71(z0, z1))
isNaturalKind(ok(z0)) → ok(isNaturalKind(z0))
U42(ok(z0), ok(z1)) → ok(U42(z0, z1))
U42(mark(z0), z1) → mark(U42(z0, z1))
U142(mark(z0), z1) → mark(U142(z0, z1))
U142(ok(z0), ok(z1)) → ok(U142(z0, z1))
U62(ok(z0)) → ok(U62(z0))
U62(mark(z0)) → mark(U62(z0))
U122(mark(z0)) → mark(U122(z0))
U122(ok(z0)) → ok(U122(z0))
U121(mark(z0), z1) → mark(U121(z0, z1))
U121(ok(z0), ok(z1)) → ok(U121(z0, z1))
U21(mark(z0), z1) → mark(U21(z0, z1))
U21(ok(z0), ok(z1)) → ok(U21(z0, z1))
U131(ok(z0), ok(z1), ok(z2)) → ok(U131(z0, z1, z2))
U131(mark(z0), z1, z2) → mark(U131(z0, z1, z2))
U133(ok(z0)) → ok(U133(z0))
U133(mark(z0)) → mark(U133(z0))
take(z0, mark(z1)) → mark(take(z0, z1))
take(mark(z0), z1) → mark(take(z0, z1))
take(ok(z0), ok(z1)) → ok(take(z0, z1))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
natsFrom(mark(z0)) → mark(natsFrom(z0))
natsFrom(ok(z0)) → ok(natsFrom(z0))
U171(mark(z0), z1, z2) → mark(U171(z0, z1, z2))
U171(ok(z0), ok(z1), ok(z2)) → ok(U171(z0, z1, z2))
isPLNatKind(ok(z0)) → ok(isPLNatKind(z0))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
U41(mark(z0), z1, z2) → mark(U41(z0, z1, z2))
U41(ok(z0), ok(z1), ok(z2)) → ok(U41(z0, z1, z2))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
U91(ok(z0), ok(z1)) → ok(U91(z0, z1))
U91(mark(z0), z1) → mark(U91(z0, z1))
isNatural(ok(z0)) → ok(isNatural(z0))
U221(mark(z0), z1, z2) → mark(U221(z0, z1, z2))
U221(ok(z0), ok(z1), ok(z2)) → ok(U221(z0, z1, z2))
U101(ok(z0), ok(z1), ok(z2)) → ok(U101(z0, z1, z2))
U101(mark(z0), z1, z2) → mark(U101(z0, z1, z2))
U61(ok(z0), ok(z1)) → ok(U61(z0, z1))
U61(mark(z0), z1) → mark(U61(z0, z1))
snd(ok(z0)) → ok(snd(z0))
snd(mark(z0)) → mark(snd(z0))
U102(mark(z0), z1) → mark(U102(z0, z1))
U102(ok(z0), ok(z1)) → ok(U102(z0, z1))
U161(mark(z0), z1) → mark(U161(z0, z1))
U161(ok(z0), ok(z1)) → ok(U161(z0, z1))
U92(mark(z0)) → mark(U92(z0))
U92(ok(z0)) → ok(U92(z0))
isPLNat(ok(z0)) → ok(isPLNat(z0))
U141(mark(z0), z1, z2) → mark(U141(z0, z1, z2))
U141(ok(z0), ok(z1), ok(z2)) → ok(U141(z0, z1, z2))
U31(ok(z0), ok(z1)) → ok(U31(z0, z1))
U31(mark(z0), z1) → mark(U31(z0, z1))
U191(ok(z0), ok(z1)) → ok(U191(z0, z1))
U191(mark(z0), z1) → mark(U191(z0, z1))
U103(ok(z0)) → ok(U103(z0))
U103(mark(z0)) → mark(U103(z0))
U211(ok(z0), ok(z1)) → ok(U211(z0, z1))
U211(mark(z0), z1) → mark(U211(z0, z1))
U153(ok(z0)) → ok(U153(z0))
U153(mark(z0)) → mark(U153(z0))
U132(ok(z0), ok(z1)) → ok(U132(z0, z1))
U132(mark(z0), z1) → mark(U132(z0, z1))
U81(ok(z0), ok(z1)) → ok(U81(z0, z1))
U81(mark(z0), z1) → mark(U81(z0, z1))
isLNat(ok(z0)) → ok(isLNat(z0))

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:

proper(tt) → ok(tt)
proper(nil) → ok(nil)
proper(0) → ok(0)
Tuples:

U151'(ok(z0), ok(z1), ok(z2)) → c2(U151'(z0, z1, z2))
U151'(mark(z0), z1, z2) → c3(U151'(z0, z1, z2))
U43'(ok(z0)) → c4(U43'(z0))
U43'(mark(z0)) → c5(U43'(z0))
U202'(mark(z0), z1) → c6(U202'(z0, z1))
U202'(ok(z0), ok(z1)) → c7(U202'(z0, z1))
U143'(ok(z0)) → c8(U143'(z0))
U143'(mark(z0)) → c9(U143'(z0))
U52'(ok(z0), ok(z1)) → c10(U52'(z0, z1))
U52'(mark(z0), z1) → c11(U52'(z0, z1))
U51'(mark(z0), z1, z2) → c12(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c13(U51'(z0, z1, z2))
U152'(ok(z0), ok(z1)) → c14(U152'(z0, z1))
U152'(mark(z0), z1) → c15(U152'(z0, z1))
PAIR(mark(z0), z1) → c16(PAIR(z0, z1))
PAIR(ok(z0), ok(z1)) → c17(PAIR(z0, z1))
PAIR(z0, mark(z1)) → c18(PAIR(z0, z1))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
SEL(mark(z0), z1) → c20(SEL(z0, z1))
SEL(z0, mark(z1)) → c21(SEL(z0, z1))
U181'(ok(z0), ok(z1)) → c22(U181'(z0, z1))
U181'(mark(z0), z1) → c23(U181'(z0, z1))
U111'(ok(z0), ok(z1)) → c27(U111'(z0, z1))
U111'(mark(z0), z1) → c28(U111'(z0, z1))
ISLNATKIND(ok(z0)) → c29(ISLNATKIND(z0))
U11'(mark(z0), z1, z2) → c30(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c31(U11'(z0, z1, z2))
U53'(mark(z0)) → c32(U53'(z0))
U53'(ok(z0)) → c33(U53'(z0))
U72'(mark(z0)) → c34(U72'(z0))
U72'(ok(z0)) → c35(U72'(z0))
U201'(ok(z0), ok(z1), ok(z2), ok(z3)) → c36(U201'(z0, z1, z2, z3))
U201'(mark(z0), z1, z2, z3) → c37(U201'(z0, z1, z2, z3))
SPLITAT(z0, mark(z1)) → c38(SPLITAT(z0, z1))
SPLITAT(mark(z0), z1) → c39(SPLITAT(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c40(SPLITAT(z0, z1))
U112'(mark(z0)) → c41(U112'(z0))
U112'(ok(z0)) → c42(U112'(z0))
HEAD(mark(z0)) → c43(HEAD(z0))
HEAD(ok(z0)) → c44(HEAD(z0))
U82'(mark(z0)) → c45(U82'(z0))
U82'(ok(z0)) → c46(U82'(z0))
FST(mark(z0)) → c47(FST(z0))
FST(ok(z0)) → c48(FST(z0))
AFTERNTH(ok(z0), ok(z1)) → c49(AFTERNTH(z0, z1))
AFTERNTH(z0, mark(z1)) → c50(AFTERNTH(z0, z1))
AFTERNTH(mark(z0), z1) → c51(AFTERNTH(z0, z1))
TAIL(mark(z0)) → c52(TAIL(z0))
TAIL(ok(z0)) → c53(TAIL(z0))
U71'(ok(z0), ok(z1)) → c54(U71'(z0, z1))
U71'(mark(z0), z1) → c55(U71'(z0, z1))
ISNATURALKIND(ok(z0)) → c56(ISNATURALKIND(z0))
U42'(ok(z0), ok(z1)) → c57(U42'(z0, z1))
U42'(mark(z0), z1) → c58(U42'(z0, z1))
U142'(mark(z0), z1) → c59(U142'(z0, z1))
U142'(ok(z0), ok(z1)) → c60(U142'(z0, z1))
U62'(ok(z0)) → c61(U62'(z0))
U62'(mark(z0)) → c62(U62'(z0))
U122'(mark(z0)) → c63(U122'(z0))
U122'(ok(z0)) → c64(U122'(z0))
U121'(mark(z0), z1) → c65(U121'(z0, z1))
U121'(ok(z0), ok(z1)) → c66(U121'(z0, z1))
U21'(mark(z0), z1) → c67(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c68(U21'(z0, z1))
U131'(ok(z0), ok(z1), ok(z2)) → c69(U131'(z0, z1, z2))
U131'(mark(z0), z1, z2) → c70(U131'(z0, z1, z2))
U133'(ok(z0)) → c71(U133'(z0))
U133'(mark(z0)) → c72(U133'(z0))
TAKE(z0, mark(z1)) → c73(TAKE(z0, z1))
TAKE(mark(z0), z1) → c74(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c75(TAKE(z0, z1))
AND(mark(z0), z1) → c76(AND(z0, z1))
AND(ok(z0), ok(z1)) → c77(AND(z0, z1))
NATSFROM(mark(z0)) → c78(NATSFROM(z0))
NATSFROM(ok(z0)) → c79(NATSFROM(z0))
U171'(mark(z0), z1, z2) → c80(U171'(z0, z1, z2))
U171'(ok(z0), ok(z1), ok(z2)) → c81(U171'(z0, z1, z2))
ISPLNATKIND(ok(z0)) → c82(ISPLNATKIND(z0))
S(mark(z0)) → c83(S(z0))
S(ok(z0)) → c84(S(z0))
U41'(mark(z0), z1, z2) → c85(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c86(U41'(z0, z1, z2))
CONS(mark(z0), z1) → c87(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c88(CONS(z0, z1))
U91'(ok(z0), ok(z1)) → c89(U91'(z0, z1))
U91'(mark(z0), z1) → c90(U91'(z0, z1))
ISNATURAL(ok(z0)) → c91(ISNATURAL(z0))
U221'(mark(z0), z1, z2) → c92(U221'(z0, z1, z2))
U221'(ok(z0), ok(z1), ok(z2)) → c93(U221'(z0, z1, z2))
U101'(ok(z0), ok(z1), ok(z2)) → c94(U101'(z0, z1, z2))
U101'(mark(z0), z1, z2) → c95(U101'(z0, z1, z2))
U61'(ok(z0), ok(z1)) → c96(U61'(z0, z1))
U61'(mark(z0), z1) → c97(U61'(z0, z1))
SND(ok(z0)) → c98(SND(z0))
SND(mark(z0)) → c99(SND(z0))
U102'(mark(z0), z1) → c100(U102'(z0, z1))
U102'(ok(z0), ok(z1)) → c101(U102'(z0, z1))
U161'(mark(z0), z1) → c102(U161'(z0, z1))
U161'(ok(z0), ok(z1)) → c103(U161'(z0, z1))
U92'(mark(z0)) → c104(U92'(z0))
U92'(ok(z0)) → c105(U92'(z0))
ISPLNAT(ok(z0)) → c106(ISPLNAT(z0))
U141'(mark(z0), z1, z2) → c107(U141'(z0, z1, z2))
U141'(ok(z0), ok(z1), ok(z2)) → c108(U141'(z0, z1, z2))
U31'(ok(z0), ok(z1)) → c109(U31'(z0, z1))
U31'(mark(z0), z1) → c110(U31'(z0, z1))
U191'(ok(z0), ok(z1)) → c111(U191'(z0, z1))
U191'(mark(z0), z1) → c112(U191'(z0, z1))
U103'(ok(z0)) → c113(U103'(z0))
U103'(mark(z0)) → c114(U103'(z0))
U211'(ok(z0), ok(z1)) → c115(U211'(z0, z1))
U211'(mark(z0), z1) → c116(U211'(z0, z1))
U153'(ok(z0)) → c117(U153'(z0))
U153'(mark(z0)) → c118(U153'(z0))
U132'(ok(z0), ok(z1)) → c119(U132'(z0, z1))
U132'(mark(z0), z1) → c120(U132'(z0, z1))
U81'(ok(z0), ok(z1)) → c121(U81'(z0, z1))
U81'(mark(z0), z1) → c122(U81'(z0, z1))
ISLNAT(ok(z0)) → c123(ISLNAT(z0))
TOP(mark(z0)) → c1(TOP(proper(z0)))
S tuples:

U151'(ok(z0), ok(z1), ok(z2)) → c2(U151'(z0, z1, z2))
U151'(mark(z0), z1, z2) → c3(U151'(z0, z1, z2))
U43'(ok(z0)) → c4(U43'(z0))
U43'(mark(z0)) → c5(U43'(z0))
U202'(mark(z0), z1) → c6(U202'(z0, z1))
U202'(ok(z0), ok(z1)) → c7(U202'(z0, z1))
U143'(ok(z0)) → c8(U143'(z0))
U143'(mark(z0)) → c9(U143'(z0))
U52'(ok(z0), ok(z1)) → c10(U52'(z0, z1))
U52'(mark(z0), z1) → c11(U52'(z0, z1))
U51'(mark(z0), z1, z2) → c12(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c13(U51'(z0, z1, z2))
U152'(ok(z0), ok(z1)) → c14(U152'(z0, z1))
U152'(mark(z0), z1) → c15(U152'(z0, z1))
PAIR(mark(z0), z1) → c16(PAIR(z0, z1))
PAIR(ok(z0), ok(z1)) → c17(PAIR(z0, z1))
PAIR(z0, mark(z1)) → c18(PAIR(z0, z1))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
SEL(mark(z0), z1) → c20(SEL(z0, z1))
SEL(z0, mark(z1)) → c21(SEL(z0, z1))
U181'(ok(z0), ok(z1)) → c22(U181'(z0, z1))
U181'(mark(z0), z1) → c23(U181'(z0, z1))
U111'(ok(z0), ok(z1)) → c27(U111'(z0, z1))
U111'(mark(z0), z1) → c28(U111'(z0, z1))
ISLNATKIND(ok(z0)) → c29(ISLNATKIND(z0))
U11'(mark(z0), z1, z2) → c30(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c31(U11'(z0, z1, z2))
U53'(mark(z0)) → c32(U53'(z0))
U53'(ok(z0)) → c33(U53'(z0))
U72'(mark(z0)) → c34(U72'(z0))
U72'(ok(z0)) → c35(U72'(z0))
U201'(ok(z0), ok(z1), ok(z2), ok(z3)) → c36(U201'(z0, z1, z2, z3))
U201'(mark(z0), z1, z2, z3) → c37(U201'(z0, z1, z2, z3))
SPLITAT(z0, mark(z1)) → c38(SPLITAT(z0, z1))
SPLITAT(mark(z0), z1) → c39(SPLITAT(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c40(SPLITAT(z0, z1))
U112'(mark(z0)) → c41(U112'(z0))
U112'(ok(z0)) → c42(U112'(z0))
HEAD(mark(z0)) → c43(HEAD(z0))
HEAD(ok(z0)) → c44(HEAD(z0))
U82'(mark(z0)) → c45(U82'(z0))
U82'(ok(z0)) → c46(U82'(z0))
FST(mark(z0)) → c47(FST(z0))
FST(ok(z0)) → c48(FST(z0))
AFTERNTH(ok(z0), ok(z1)) → c49(AFTERNTH(z0, z1))
AFTERNTH(z0, mark(z1)) → c50(AFTERNTH(z0, z1))
AFTERNTH(mark(z0), z1) → c51(AFTERNTH(z0, z1))
TAIL(mark(z0)) → c52(TAIL(z0))
TAIL(ok(z0)) → c53(TAIL(z0))
U71'(ok(z0), ok(z1)) → c54(U71'(z0, z1))
U71'(mark(z0), z1) → c55(U71'(z0, z1))
ISNATURALKIND(ok(z0)) → c56(ISNATURALKIND(z0))
U42'(ok(z0), ok(z1)) → c57(U42'(z0, z1))
U42'(mark(z0), z1) → c58(U42'(z0, z1))
U142'(mark(z0), z1) → c59(U142'(z0, z1))
U142'(ok(z0), ok(z1)) → c60(U142'(z0, z1))
U62'(ok(z0)) → c61(U62'(z0))
U62'(mark(z0)) → c62(U62'(z0))
U122'(mark(z0)) → c63(U122'(z0))
U122'(ok(z0)) → c64(U122'(z0))
U121'(mark(z0), z1) → c65(U121'(z0, z1))
U121'(ok(z0), ok(z1)) → c66(U121'(z0, z1))
U21'(mark(z0), z1) → c67(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c68(U21'(z0, z1))
U131'(ok(z0), ok(z1), ok(z2)) → c69(U131'(z0, z1, z2))
U131'(mark(z0), z1, z2) → c70(U131'(z0, z1, z2))
U133'(ok(z0)) → c71(U133'(z0))
U133'(mark(z0)) → c72(U133'(z0))
TAKE(z0, mark(z1)) → c73(TAKE(z0, z1))
TAKE(mark(z0), z1) → c74(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c75(TAKE(z0, z1))
AND(mark(z0), z1) → c76(AND(z0, z1))
AND(ok(z0), ok(z1)) → c77(AND(z0, z1))
NATSFROM(mark(z0)) → c78(NATSFROM(z0))
NATSFROM(ok(z0)) → c79(NATSFROM(z0))
U171'(mark(z0), z1, z2) → c80(U171'(z0, z1, z2))
U171'(ok(z0), ok(z1), ok(z2)) → c81(U171'(z0, z1, z2))
ISPLNATKIND(ok(z0)) → c82(ISPLNATKIND(z0))
S(mark(z0)) → c83(S(z0))
S(ok(z0)) → c84(S(z0))
U41'(mark(z0), z1, z2) → c85(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c86(U41'(z0, z1, z2))
CONS(mark(z0), z1) → c87(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c88(CONS(z0, z1))
U91'(ok(z0), ok(z1)) → c89(U91'(z0, z1))
U91'(mark(z0), z1) → c90(U91'(z0, z1))
ISNATURAL(ok(z0)) → c91(ISNATURAL(z0))
U221'(mark(z0), z1, z2) → c92(U221'(z0, z1, z2))
U221'(ok(z0), ok(z1), ok(z2)) → c93(U221'(z0, z1, z2))
U101'(ok(z0), ok(z1), ok(z2)) → c94(U101'(z0, z1, z2))
U101'(mark(z0), z1, z2) → c95(U101'(z0, z1, z2))
U61'(ok(z0), ok(z1)) → c96(U61'(z0, z1))
U61'(mark(z0), z1) → c97(U61'(z0, z1))
SND(ok(z0)) → c98(SND(z0))
SND(mark(z0)) → c99(SND(z0))
U102'(mark(z0), z1) → c100(U102'(z0, z1))
U102'(ok(z0), ok(z1)) → c101(U102'(z0, z1))
U161'(mark(z0), z1) → c102(U161'(z0, z1))
U161'(ok(z0), ok(z1)) → c103(U161'(z0, z1))
U92'(mark(z0)) → c104(U92'(z0))
U92'(ok(z0)) → c105(U92'(z0))
ISPLNAT(ok(z0)) → c106(ISPLNAT(z0))
U141'(mark(z0), z1, z2) → c107(U141'(z0, z1, z2))
U141'(ok(z0), ok(z1), ok(z2)) → c108(U141'(z0, z1, z2))
U31'(ok(z0), ok(z1)) → c109(U31'(z0, z1))
U31'(mark(z0), z1) → c110(U31'(z0, z1))
U191'(ok(z0), ok(z1)) → c111(U191'(z0, z1))
U191'(mark(z0), z1) → c112(U191'(z0, z1))
U103'(ok(z0)) → c113(U103'(z0))
U103'(mark(z0)) → c114(U103'(z0))
U211'(ok(z0), ok(z1)) → c115(U211'(z0, z1))
U211'(mark(z0), z1) → c116(U211'(z0, z1))
U153'(ok(z0)) → c117(U153'(z0))
U153'(mark(z0)) → c118(U153'(z0))
U132'(ok(z0), ok(z1)) → c119(U132'(z0, z1))
U132'(mark(z0), z1) → c120(U132'(z0, z1))
U81'(ok(z0), ok(z1)) → c121(U81'(z0, z1))
U81'(mark(z0), z1) → c122(U81'(z0, z1))
ISLNAT(ok(z0)) → c123(ISLNAT(z0))
TOP(mark(z0)) → c1(TOP(proper(z0)))
K tuples:none
Defined Rule Symbols:

proper

Defined Pair Symbols:

U151', U43', U202', U143', U52', U51', U152', PAIR, SEL, U181', U111', ISLNATKIND, U11', U53', U72', U201', SPLITAT, U112', HEAD, U82', FST, AFTERNTH, TAIL, U71', ISNATURALKIND, U42', U142', U62', U122', U121', U21', U131', U133', TAKE, AND, NATSFROM, U171', ISPLNATKIND, S, U41', CONS, U91', ISNATURAL, U221', U101', U61', SND, U102', U161', U92', ISPLNAT, U141', U31', U191', U103', U211', U153', U132', U81', ISLNAT, TOP

Compound Symbols:

c2, c3, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c48, c49, c50, c51, c52, c53, c54, c55, c56, c57, c58, c59, c60, c61, c62, c63, c64, c65, c66, c67, c68, c69, c70, c71, c72, c73, c74, c75, c76, c77, c78, c79, c80, c81, c82, c83, c84, c85, c86, c87, c88, c89, c90, c91, c92, c93, c94, c95, c96, c97, c98, c99, c100, c101, c102, c103, c104, c105, c106, c107, c108, c109, c110, c111, c112, c113, c114, c115, c116, c117, c118, c119, c120, c121, c122, c123, c1

(13) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

TOP(mark(z0)) → c1(TOP(proper(z0)))
We considered the (Usable) Rules:

proper(tt) → ok(tt)
proper(nil) → ok(nil)
proper(0) → ok(0)
And the Tuples:

U151'(ok(z0), ok(z1), ok(z2)) → c2(U151'(z0, z1, z2))
U151'(mark(z0), z1, z2) → c3(U151'(z0, z1, z2))
U43'(ok(z0)) → c4(U43'(z0))
U43'(mark(z0)) → c5(U43'(z0))
U202'(mark(z0), z1) → c6(U202'(z0, z1))
U202'(ok(z0), ok(z1)) → c7(U202'(z0, z1))
U143'(ok(z0)) → c8(U143'(z0))
U143'(mark(z0)) → c9(U143'(z0))
U52'(ok(z0), ok(z1)) → c10(U52'(z0, z1))
U52'(mark(z0), z1) → c11(U52'(z0, z1))
U51'(mark(z0), z1, z2) → c12(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c13(U51'(z0, z1, z2))
U152'(ok(z0), ok(z1)) → c14(U152'(z0, z1))
U152'(mark(z0), z1) → c15(U152'(z0, z1))
PAIR(mark(z0), z1) → c16(PAIR(z0, z1))
PAIR(ok(z0), ok(z1)) → c17(PAIR(z0, z1))
PAIR(z0, mark(z1)) → c18(PAIR(z0, z1))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
SEL(mark(z0), z1) → c20(SEL(z0, z1))
SEL(z0, mark(z1)) → c21(SEL(z0, z1))
U181'(ok(z0), ok(z1)) → c22(U181'(z0, z1))
U181'(mark(z0), z1) → c23(U181'(z0, z1))
U111'(ok(z0), ok(z1)) → c27(U111'(z0, z1))
U111'(mark(z0), z1) → c28(U111'(z0, z1))
ISLNATKIND(ok(z0)) → c29(ISLNATKIND(z0))
U11'(mark(z0), z1, z2) → c30(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c31(U11'(z0, z1, z2))
U53'(mark(z0)) → c32(U53'(z0))
U53'(ok(z0)) → c33(U53'(z0))
U72'(mark(z0)) → c34(U72'(z0))
U72'(ok(z0)) → c35(U72'(z0))
U201'(ok(z0), ok(z1), ok(z2), ok(z3)) → c36(U201'(z0, z1, z2, z3))
U201'(mark(z0), z1, z2, z3) → c37(U201'(z0, z1, z2, z3))
SPLITAT(z0, mark(z1)) → c38(SPLITAT(z0, z1))
SPLITAT(mark(z0), z1) → c39(SPLITAT(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c40(SPLITAT(z0, z1))
U112'(mark(z0)) → c41(U112'(z0))
U112'(ok(z0)) → c42(U112'(z0))
HEAD(mark(z0)) → c43(HEAD(z0))
HEAD(ok(z0)) → c44(HEAD(z0))
U82'(mark(z0)) → c45(U82'(z0))
U82'(ok(z0)) → c46(U82'(z0))
FST(mark(z0)) → c47(FST(z0))
FST(ok(z0)) → c48(FST(z0))
AFTERNTH(ok(z0), ok(z1)) → c49(AFTERNTH(z0, z1))
AFTERNTH(z0, mark(z1)) → c50(AFTERNTH(z0, z1))
AFTERNTH(mark(z0), z1) → c51(AFTERNTH(z0, z1))
TAIL(mark(z0)) → c52(TAIL(z0))
TAIL(ok(z0)) → c53(TAIL(z0))
U71'(ok(z0), ok(z1)) → c54(U71'(z0, z1))
U71'(mark(z0), z1) → c55(U71'(z0, z1))
ISNATURALKIND(ok(z0)) → c56(ISNATURALKIND(z0))
U42'(ok(z0), ok(z1)) → c57(U42'(z0, z1))
U42'(mark(z0), z1) → c58(U42'(z0, z1))
U142'(mark(z0), z1) → c59(U142'(z0, z1))
U142'(ok(z0), ok(z1)) → c60(U142'(z0, z1))
U62'(ok(z0)) → c61(U62'(z0))
U62'(mark(z0)) → c62(U62'(z0))
U122'(mark(z0)) → c63(U122'(z0))
U122'(ok(z0)) → c64(U122'(z0))
U121'(mark(z0), z1) → c65(U121'(z0, z1))
U121'(ok(z0), ok(z1)) → c66(U121'(z0, z1))
U21'(mark(z0), z1) → c67(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c68(U21'(z0, z1))
U131'(ok(z0), ok(z1), ok(z2)) → c69(U131'(z0, z1, z2))
U131'(mark(z0), z1, z2) → c70(U131'(z0, z1, z2))
U133'(ok(z0)) → c71(U133'(z0))
U133'(mark(z0)) → c72(U133'(z0))
TAKE(z0, mark(z1)) → c73(TAKE(z0, z1))
TAKE(mark(z0), z1) → c74(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c75(TAKE(z0, z1))
AND(mark(z0), z1) → c76(AND(z0, z1))
AND(ok(z0), ok(z1)) → c77(AND(z0, z1))
NATSFROM(mark(z0)) → c78(NATSFROM(z0))
NATSFROM(ok(z0)) → c79(NATSFROM(z0))
U171'(mark(z0), z1, z2) → c80(U171'(z0, z1, z2))
U171'(ok(z0), ok(z1), ok(z2)) → c81(U171'(z0, z1, z2))
ISPLNATKIND(ok(z0)) → c82(ISPLNATKIND(z0))
S(mark(z0)) → c83(S(z0))
S(ok(z0)) → c84(S(z0))
U41'(mark(z0), z1, z2) → c85(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c86(U41'(z0, z1, z2))
CONS(mark(z0), z1) → c87(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c88(CONS(z0, z1))
U91'(ok(z0), ok(z1)) → c89(U91'(z0, z1))
U91'(mark(z0), z1) → c90(U91'(z0, z1))
ISNATURAL(ok(z0)) → c91(ISNATURAL(z0))
U221'(mark(z0), z1, z2) → c92(U221'(z0, z1, z2))
U221'(ok(z0), ok(z1), ok(z2)) → c93(U221'(z0, z1, z2))
U101'(ok(z0), ok(z1), ok(z2)) → c94(U101'(z0, z1, z2))
U101'(mark(z0), z1, z2) → c95(U101'(z0, z1, z2))
U61'(ok(z0), ok(z1)) → c96(U61'(z0, z1))
U61'(mark(z0), z1) → c97(U61'(z0, z1))
SND(ok(z0)) → c98(SND(z0))
SND(mark(z0)) → c99(SND(z0))
U102'(mark(z0), z1) → c100(U102'(z0, z1))
U102'(ok(z0), ok(z1)) → c101(U102'(z0, z1))
U161'(mark(z0), z1) → c102(U161'(z0, z1))
U161'(ok(z0), ok(z1)) → c103(U161'(z0, z1))
U92'(mark(z0)) → c104(U92'(z0))
U92'(ok(z0)) → c105(U92'(z0))
ISPLNAT(ok(z0)) → c106(ISPLNAT(z0))
U141'(mark(z0), z1, z2) → c107(U141'(z0, z1, z2))
U141'(ok(z0), ok(z1), ok(z2)) → c108(U141'(z0, z1, z2))
U31'(ok(z0), ok(z1)) → c109(U31'(z0, z1))
U31'(mark(z0), z1) → c110(U31'(z0, z1))
U191'(ok(z0), ok(z1)) → c111(U191'(z0, z1))
U191'(mark(z0), z1) → c112(U191'(z0, z1))
U103'(ok(z0)) → c113(U103'(z0))
U103'(mark(z0)) → c114(U103'(z0))
U211'(ok(z0), ok(z1)) → c115(U211'(z0, z1))
U211'(mark(z0), z1) → c116(U211'(z0, z1))
U153'(ok(z0)) → c117(U153'(z0))
U153'(mark(z0)) → c118(U153'(z0))
U132'(ok(z0), ok(z1)) → c119(U132'(z0, z1))
U132'(mark(z0), z1) → c120(U132'(z0, z1))
U81'(ok(z0), ok(z1)) → c121(U81'(z0, z1))
U81'(mark(z0), z1) → c122(U81'(z0, z1))
ISLNAT(ok(z0)) → c123(ISLNAT(z0))
TOP(mark(z0)) → c1(TOP(proper(z0)))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = 0   
POL(AFTERNTH(x1, x2)) = 0   
POL(AND(x1, x2)) = 0   
POL(CONS(x1, x2)) = 0   
POL(FST(x1)) = 0   
POL(HEAD(x1)) = 0   
POL(ISLNAT(x1)) = 0   
POL(ISLNATKIND(x1)) = 0   
POL(ISNATURAL(x1)) = 0   
POL(ISNATURALKIND(x1)) = 0   
POL(ISPLNAT(x1)) = 0   
POL(ISPLNATKIND(x1)) = 0   
POL(NATSFROM(x1)) = 0   
POL(PAIR(x1, x2)) = 0   
POL(S(x1)) = 0   
POL(SEL(x1, x2)) = 0   
POL(SND(x1)) = 0   
POL(SPLITAT(x1, x2)) = 0   
POL(TAIL(x1)) = 0   
POL(TAKE(x1, x2)) = 0   
POL(TOP(x1)) = x1   
POL(U101'(x1, x2, x3)) = 0   
POL(U102'(x1, x2)) = 0   
POL(U103'(x1)) = 0   
POL(U11'(x1, x2, x3)) = 0   
POL(U111'(x1, x2)) = 0   
POL(U112'(x1)) = 0   
POL(U121'(x1, x2)) = 0   
POL(U122'(x1)) = 0   
POL(U131'(x1, x2, x3)) = 0   
POL(U132'(x1, x2)) = 0   
POL(U133'(x1)) = 0   
POL(U141'(x1, x2, x3)) = 0   
POL(U142'(x1, x2)) = 0   
POL(U143'(x1)) = 0   
POL(U151'(x1, x2, x3)) = 0   
POL(U152'(x1, x2)) = 0   
POL(U153'(x1)) = 0   
POL(U161'(x1, x2)) = 0   
POL(U171'(x1, x2, x3)) = 0   
POL(U181'(x1, x2)) = 0   
POL(U191'(x1, x2)) = 0   
POL(U201'(x1, x2, x3, x4)) = 0   
POL(U202'(x1, x2)) = 0   
POL(U21'(x1, x2)) = 0   
POL(U211'(x1, x2)) = 0   
POL(U221'(x1, x2, x3)) = 0   
POL(U31'(x1, x2)) = 0   
POL(U41'(x1, x2, x3)) = 0   
POL(U42'(x1, x2)) = 0   
POL(U43'(x1)) = 0   
POL(U51'(x1, x2, x3)) = 0   
POL(U52'(x1, x2)) = 0   
POL(U53'(x1)) = 0   
POL(U61'(x1, x2)) = 0   
POL(U62'(x1)) = 0   
POL(U71'(x1, x2)) = 0   
POL(U72'(x1)) = 0   
POL(U81'(x1, x2)) = 0   
POL(U82'(x1)) = 0   
POL(U91'(x1, x2)) = 0   
POL(U92'(x1)) = 0   
POL(c1(x1)) = x1   
POL(c10(x1)) = x1   
POL(c100(x1)) = x1   
POL(c101(x1)) = x1   
POL(c102(x1)) = x1   
POL(c103(x1)) = x1   
POL(c104(x1)) = x1   
POL(c105(x1)) = x1   
POL(c106(x1)) = x1   
POL(c107(x1)) = x1   
POL(c108(x1)) = x1   
POL(c109(x1)) = x1   
POL(c11(x1)) = x1   
POL(c110(x1)) = x1   
POL(c111(x1)) = x1   
POL(c112(x1)) = x1   
POL(c113(x1)) = x1   
POL(c114(x1)) = x1   
POL(c115(x1)) = x1   
POL(c116(x1)) = x1   
POL(c117(x1)) = x1   
POL(c118(x1)) = x1   
POL(c119(x1)) = x1   
POL(c12(x1)) = x1   
POL(c120(x1)) = x1   
POL(c121(x1)) = x1   
POL(c122(x1)) = x1   
POL(c123(x1)) = x1   
POL(c13(x1)) = x1   
POL(c14(x1)) = x1   
POL(c15(x1)) = x1   
POL(c16(x1)) = x1   
POL(c17(x1)) = x1   
POL(c18(x1)) = x1   
POL(c19(x1)) = x1   
POL(c2(x1)) = x1   
POL(c20(x1)) = x1   
POL(c21(x1)) = x1   
POL(c22(x1)) = x1   
POL(c23(x1)) = x1   
POL(c27(x1)) = x1   
POL(c28(x1)) = x1   
POL(c29(x1)) = x1   
POL(c3(x1)) = x1   
POL(c30(x1)) = x1   
POL(c31(x1)) = x1   
POL(c32(x1)) = x1   
POL(c33(x1)) = x1   
POL(c34(x1)) = x1   
POL(c35(x1)) = x1   
POL(c36(x1)) = x1   
POL(c37(x1)) = x1   
POL(c38(x1)) = x1   
POL(c39(x1)) = x1   
POL(c4(x1)) = x1   
POL(c40(x1)) = x1   
POL(c41(x1)) = x1   
POL(c42(x1)) = x1   
POL(c43(x1)) = x1   
POL(c44(x1)) = x1   
POL(c45(x1)) = x1   
POL(c46(x1)) = x1   
POL(c47(x1)) = x1   
POL(c48(x1)) = x1   
POL(c49(x1)) = x1   
POL(c5(x1)) = x1   
POL(c50(x1)) = x1   
POL(c51(x1)) = x1   
POL(c52(x1)) = x1   
POL(c53(x1)) = x1   
POL(c54(x1)) = x1   
POL(c55(x1)) = x1   
POL(c56(x1)) = x1   
POL(c57(x1)) = x1   
POL(c58(x1)) = x1   
POL(c59(x1)) = x1   
POL(c6(x1)) = x1   
POL(c60(x1)) = x1   
POL(c61(x1)) = x1   
POL(c62(x1)) = x1   
POL(c63(x1)) = x1   
POL(c64(x1)) = x1   
POL(c65(x1)) = x1   
POL(c66(x1)) = x1   
POL(c67(x1)) = x1   
POL(c68(x1)) = x1   
POL(c69(x1)) = x1   
POL(c7(x1)) = x1   
POL(c70(x1)) = x1   
POL(c71(x1)) = x1   
POL(c72(x1)) = x1   
POL(c73(x1)) = x1   
POL(c74(x1)) = x1   
POL(c75(x1)) = x1   
POL(c76(x1)) = x1   
POL(c77(x1)) = x1   
POL(c78(x1)) = x1   
POL(c79(x1)) = x1   
POL(c8(x1)) = x1   
POL(c80(x1)) = x1   
POL(c81(x1)) = x1   
POL(c82(x1)) = x1   
POL(c83(x1)) = x1   
POL(c84(x1)) = x1   
POL(c85(x1)) = x1   
POL(c86(x1)) = x1   
POL(c87(x1)) = x1   
POL(c88(x1)) = x1   
POL(c89(x1)) = x1   
POL(c9(x1)) = x1   
POL(c90(x1)) = x1   
POL(c91(x1)) = x1   
POL(c92(x1)) = x1   
POL(c93(x1)) = x1   
POL(c94(x1)) = x1   
POL(c95(x1)) = x1   
POL(c96(x1)) = x1   
POL(c97(x1)) = x1   
POL(c98(x1)) = x1   
POL(c99(x1)) = x1   
POL(mark(x1)) = [1] + x1   
POL(nil) = 0   
POL(ok(x1)) = 0   
POL(proper(x1)) = 0   
POL(tt) = 0   

(14) Obligation:

Complexity Dependency Tuples Problem
Rules:

proper(tt) → ok(tt)
proper(nil) → ok(nil)
proper(0) → ok(0)
Tuples:

U151'(ok(z0), ok(z1), ok(z2)) → c2(U151'(z0, z1, z2))
U151'(mark(z0), z1, z2) → c3(U151'(z0, z1, z2))
U43'(ok(z0)) → c4(U43'(z0))
U43'(mark(z0)) → c5(U43'(z0))
U202'(mark(z0), z1) → c6(U202'(z0, z1))
U202'(ok(z0), ok(z1)) → c7(U202'(z0, z1))
U143'(ok(z0)) → c8(U143'(z0))
U143'(mark(z0)) → c9(U143'(z0))
U52'(ok(z0), ok(z1)) → c10(U52'(z0, z1))
U52'(mark(z0), z1) → c11(U52'(z0, z1))
U51'(mark(z0), z1, z2) → c12(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c13(U51'(z0, z1, z2))
U152'(ok(z0), ok(z1)) → c14(U152'(z0, z1))
U152'(mark(z0), z1) → c15(U152'(z0, z1))
PAIR(mark(z0), z1) → c16(PAIR(z0, z1))
PAIR(ok(z0), ok(z1)) → c17(PAIR(z0, z1))
PAIR(z0, mark(z1)) → c18(PAIR(z0, z1))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
SEL(mark(z0), z1) → c20(SEL(z0, z1))
SEL(z0, mark(z1)) → c21(SEL(z0, z1))
U181'(ok(z0), ok(z1)) → c22(U181'(z0, z1))
U181'(mark(z0), z1) → c23(U181'(z0, z1))
U111'(ok(z0), ok(z1)) → c27(U111'(z0, z1))
U111'(mark(z0), z1) → c28(U111'(z0, z1))
ISLNATKIND(ok(z0)) → c29(ISLNATKIND(z0))
U11'(mark(z0), z1, z2) → c30(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c31(U11'(z0, z1, z2))
U53'(mark(z0)) → c32(U53'(z0))
U53'(ok(z0)) → c33(U53'(z0))
U72'(mark(z0)) → c34(U72'(z0))
U72'(ok(z0)) → c35(U72'(z0))
U201'(ok(z0), ok(z1), ok(z2), ok(z3)) → c36(U201'(z0, z1, z2, z3))
U201'(mark(z0), z1, z2, z3) → c37(U201'(z0, z1, z2, z3))
SPLITAT(z0, mark(z1)) → c38(SPLITAT(z0, z1))
SPLITAT(mark(z0), z1) → c39(SPLITAT(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c40(SPLITAT(z0, z1))
U112'(mark(z0)) → c41(U112'(z0))
U112'(ok(z0)) → c42(U112'(z0))
HEAD(mark(z0)) → c43(HEAD(z0))
HEAD(ok(z0)) → c44(HEAD(z0))
U82'(mark(z0)) → c45(U82'(z0))
U82'(ok(z0)) → c46(U82'(z0))
FST(mark(z0)) → c47(FST(z0))
FST(ok(z0)) → c48(FST(z0))
AFTERNTH(ok(z0), ok(z1)) → c49(AFTERNTH(z0, z1))
AFTERNTH(z0, mark(z1)) → c50(AFTERNTH(z0, z1))
AFTERNTH(mark(z0), z1) → c51(AFTERNTH(z0, z1))
TAIL(mark(z0)) → c52(TAIL(z0))
TAIL(ok(z0)) → c53(TAIL(z0))
U71'(ok(z0), ok(z1)) → c54(U71'(z0, z1))
U71'(mark(z0), z1) → c55(U71'(z0, z1))
ISNATURALKIND(ok(z0)) → c56(ISNATURALKIND(z0))
U42'(ok(z0), ok(z1)) → c57(U42'(z0, z1))
U42'(mark(z0), z1) → c58(U42'(z0, z1))
U142'(mark(z0), z1) → c59(U142'(z0, z1))
U142'(ok(z0), ok(z1)) → c60(U142'(z0, z1))
U62'(ok(z0)) → c61(U62'(z0))
U62'(mark(z0)) → c62(U62'(z0))
U122'(mark(z0)) → c63(U122'(z0))
U122'(ok(z0)) → c64(U122'(z0))
U121'(mark(z0), z1) → c65(U121'(z0, z1))
U121'(ok(z0), ok(z1)) → c66(U121'(z0, z1))
U21'(mark(z0), z1) → c67(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c68(U21'(z0, z1))
U131'(ok(z0), ok(z1), ok(z2)) → c69(U131'(z0, z1, z2))
U131'(mark(z0), z1, z2) → c70(U131'(z0, z1, z2))
U133'(ok(z0)) → c71(U133'(z0))
U133'(mark(z0)) → c72(U133'(z0))
TAKE(z0, mark(z1)) → c73(TAKE(z0, z1))
TAKE(mark(z0), z1) → c74(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c75(TAKE(z0, z1))
AND(mark(z0), z1) → c76(AND(z0, z1))
AND(ok(z0), ok(z1)) → c77(AND(z0, z1))
NATSFROM(mark(z0)) → c78(NATSFROM(z0))
NATSFROM(ok(z0)) → c79(NATSFROM(z0))
U171'(mark(z0), z1, z2) → c80(U171'(z0, z1, z2))
U171'(ok(z0), ok(z1), ok(z2)) → c81(U171'(z0, z1, z2))
ISPLNATKIND(ok(z0)) → c82(ISPLNATKIND(z0))
S(mark(z0)) → c83(S(z0))
S(ok(z0)) → c84(S(z0))
U41'(mark(z0), z1, z2) → c85(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c86(U41'(z0, z1, z2))
CONS(mark(z0), z1) → c87(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c88(CONS(z0, z1))
U91'(ok(z0), ok(z1)) → c89(U91'(z0, z1))
U91'(mark(z0), z1) → c90(U91'(z0, z1))
ISNATURAL(ok(z0)) → c91(ISNATURAL(z0))
U221'(mark(z0), z1, z2) → c92(U221'(z0, z1, z2))
U221'(ok(z0), ok(z1), ok(z2)) → c93(U221'(z0, z1, z2))
U101'(ok(z0), ok(z1), ok(z2)) → c94(U101'(z0, z1, z2))
U101'(mark(z0), z1, z2) → c95(U101'(z0, z1, z2))
U61'(ok(z0), ok(z1)) → c96(U61'(z0, z1))
U61'(mark(z0), z1) → c97(U61'(z0, z1))
SND(ok(z0)) → c98(SND(z0))
SND(mark(z0)) → c99(SND(z0))
U102'(mark(z0), z1) → c100(U102'(z0, z1))
U102'(ok(z0), ok(z1)) → c101(U102'(z0, z1))
U161'(mark(z0), z1) → c102(U161'(z0, z1))
U161'(ok(z0), ok(z1)) → c103(U161'(z0, z1))
U92'(mark(z0)) → c104(U92'(z0))
U92'(ok(z0)) → c105(U92'(z0))
ISPLNAT(ok(z0)) → c106(ISPLNAT(z0))
U141'(mark(z0), z1, z2) → c107(U141'(z0, z1, z2))
U141'(ok(z0), ok(z1), ok(z2)) → c108(U141'(z0, z1, z2))
U31'(ok(z0), ok(z1)) → c109(U31'(z0, z1))
U31'(mark(z0), z1) → c110(U31'(z0, z1))
U191'(ok(z0), ok(z1)) → c111(U191'(z0, z1))
U191'(mark(z0), z1) → c112(U191'(z0, z1))
U103'(ok(z0)) → c113(U103'(z0))
U103'(mark(z0)) → c114(U103'(z0))
U211'(ok(z0), ok(z1)) → c115(U211'(z0, z1))
U211'(mark(z0), z1) → c116(U211'(z0, z1))
U153'(ok(z0)) → c117(U153'(z0))
U153'(mark(z0)) → c118(U153'(z0))
U132'(ok(z0), ok(z1)) → c119(U132'(z0, z1))
U132'(mark(z0), z1) → c120(U132'(z0, z1))
U81'(ok(z0), ok(z1)) → c121(U81'(z0, z1))
U81'(mark(z0), z1) → c122(U81'(z0, z1))
ISLNAT(ok(z0)) → c123(ISLNAT(z0))
TOP(mark(z0)) → c1(TOP(proper(z0)))
S tuples:

U151'(ok(z0), ok(z1), ok(z2)) → c2(U151'(z0, z1, z2))
U151'(mark(z0), z1, z2) → c3(U151'(z0, z1, z2))
U43'(ok(z0)) → c4(U43'(z0))
U43'(mark(z0)) → c5(U43'(z0))
U202'(mark(z0), z1) → c6(U202'(z0, z1))
U202'(ok(z0), ok(z1)) → c7(U202'(z0, z1))
U143'(ok(z0)) → c8(U143'(z0))
U143'(mark(z0)) → c9(U143'(z0))
U52'(ok(z0), ok(z1)) → c10(U52'(z0, z1))
U52'(mark(z0), z1) → c11(U52'(z0, z1))
U51'(mark(z0), z1, z2) → c12(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c13(U51'(z0, z1, z2))
U152'(ok(z0), ok(z1)) → c14(U152'(z0, z1))
U152'(mark(z0), z1) → c15(U152'(z0, z1))
PAIR(mark(z0), z1) → c16(PAIR(z0, z1))
PAIR(ok(z0), ok(z1)) → c17(PAIR(z0, z1))
PAIR(z0, mark(z1)) → c18(PAIR(z0, z1))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
SEL(mark(z0), z1) → c20(SEL(z0, z1))
SEL(z0, mark(z1)) → c21(SEL(z0, z1))
U181'(ok(z0), ok(z1)) → c22(U181'(z0, z1))
U181'(mark(z0), z1) → c23(U181'(z0, z1))
U111'(ok(z0), ok(z1)) → c27(U111'(z0, z1))
U111'(mark(z0), z1) → c28(U111'(z0, z1))
ISLNATKIND(ok(z0)) → c29(ISLNATKIND(z0))
U11'(mark(z0), z1, z2) → c30(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c31(U11'(z0, z1, z2))
U53'(mark(z0)) → c32(U53'(z0))
U53'(ok(z0)) → c33(U53'(z0))
U72'(mark(z0)) → c34(U72'(z0))
U72'(ok(z0)) → c35(U72'(z0))
U201'(ok(z0), ok(z1), ok(z2), ok(z3)) → c36(U201'(z0, z1, z2, z3))
U201'(mark(z0), z1, z2, z3) → c37(U201'(z0, z1, z2, z3))
SPLITAT(z0, mark(z1)) → c38(SPLITAT(z0, z1))
SPLITAT(mark(z0), z1) → c39(SPLITAT(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c40(SPLITAT(z0, z1))
U112'(mark(z0)) → c41(U112'(z0))
U112'(ok(z0)) → c42(U112'(z0))
HEAD(mark(z0)) → c43(HEAD(z0))
HEAD(ok(z0)) → c44(HEAD(z0))
U82'(mark(z0)) → c45(U82'(z0))
U82'(ok(z0)) → c46(U82'(z0))
FST(mark(z0)) → c47(FST(z0))
FST(ok(z0)) → c48(FST(z0))
AFTERNTH(ok(z0), ok(z1)) → c49(AFTERNTH(z0, z1))
AFTERNTH(z0, mark(z1)) → c50(AFTERNTH(z0, z1))
AFTERNTH(mark(z0), z1) → c51(AFTERNTH(z0, z1))
TAIL(mark(z0)) → c52(TAIL(z0))
TAIL(ok(z0)) → c53(TAIL(z0))
U71'(ok(z0), ok(z1)) → c54(U71'(z0, z1))
U71'(mark(z0), z1) → c55(U71'(z0, z1))
ISNATURALKIND(ok(z0)) → c56(ISNATURALKIND(z0))
U42'(ok(z0), ok(z1)) → c57(U42'(z0, z1))
U42'(mark(z0), z1) → c58(U42'(z0, z1))
U142'(mark(z0), z1) → c59(U142'(z0, z1))
U142'(ok(z0), ok(z1)) → c60(U142'(z0, z1))
U62'(ok(z0)) → c61(U62'(z0))
U62'(mark(z0)) → c62(U62'(z0))
U122'(mark(z0)) → c63(U122'(z0))
U122'(ok(z0)) → c64(U122'(z0))
U121'(mark(z0), z1) → c65(U121'(z0, z1))
U121'(ok(z0), ok(z1)) → c66(U121'(z0, z1))
U21'(mark(z0), z1) → c67(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c68(U21'(z0, z1))
U131'(ok(z0), ok(z1), ok(z2)) → c69(U131'(z0, z1, z2))
U131'(mark(z0), z1, z2) → c70(U131'(z0, z1, z2))
U133'(ok(z0)) → c71(U133'(z0))
U133'(mark(z0)) → c72(U133'(z0))
TAKE(z0, mark(z1)) → c73(TAKE(z0, z1))
TAKE(mark(z0), z1) → c74(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c75(TAKE(z0, z1))
AND(mark(z0), z1) → c76(AND(z0, z1))
AND(ok(z0), ok(z1)) → c77(AND(z0, z1))
NATSFROM(mark(z0)) → c78(NATSFROM(z0))
NATSFROM(ok(z0)) → c79(NATSFROM(z0))
U171'(mark(z0), z1, z2) → c80(U171'(z0, z1, z2))
U171'(ok(z0), ok(z1), ok(z2)) → c81(U171'(z0, z1, z2))
ISPLNATKIND(ok(z0)) → c82(ISPLNATKIND(z0))
S(mark(z0)) → c83(S(z0))
S(ok(z0)) → c84(S(z0))
U41'(mark(z0), z1, z2) → c85(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c86(U41'(z0, z1, z2))
CONS(mark(z0), z1) → c87(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c88(CONS(z0, z1))
U91'(ok(z0), ok(z1)) → c89(U91'(z0, z1))
U91'(mark(z0), z1) → c90(U91'(z0, z1))
ISNATURAL(ok(z0)) → c91(ISNATURAL(z0))
U221'(mark(z0), z1, z2) → c92(U221'(z0, z1, z2))
U221'(ok(z0), ok(z1), ok(z2)) → c93(U221'(z0, z1, z2))
U101'(ok(z0), ok(z1), ok(z2)) → c94(U101'(z0, z1, z2))
U101'(mark(z0), z1, z2) → c95(U101'(z0, z1, z2))
U61'(ok(z0), ok(z1)) → c96(U61'(z0, z1))
U61'(mark(z0), z1) → c97(U61'(z0, z1))
SND(ok(z0)) → c98(SND(z0))
SND(mark(z0)) → c99(SND(z0))
U102'(mark(z0), z1) → c100(U102'(z0, z1))
U102'(ok(z0), ok(z1)) → c101(U102'(z0, z1))
U161'(mark(z0), z1) → c102(U161'(z0, z1))
U161'(ok(z0), ok(z1)) → c103(U161'(z0, z1))
U92'(mark(z0)) → c104(U92'(z0))
U92'(ok(z0)) → c105(U92'(z0))
ISPLNAT(ok(z0)) → c106(ISPLNAT(z0))
U141'(mark(z0), z1, z2) → c107(U141'(z0, z1, z2))
U141'(ok(z0), ok(z1), ok(z2)) → c108(U141'(z0, z1, z2))
U31'(ok(z0), ok(z1)) → c109(U31'(z0, z1))
U31'(mark(z0), z1) → c110(U31'(z0, z1))
U191'(ok(z0), ok(z1)) → c111(U191'(z0, z1))
U191'(mark(z0), z1) → c112(U191'(z0, z1))
U103'(ok(z0)) → c113(U103'(z0))
U103'(mark(z0)) → c114(U103'(z0))
U211'(ok(z0), ok(z1)) → c115(U211'(z0, z1))
U211'(mark(z0), z1) → c116(U211'(z0, z1))
U153'(ok(z0)) → c117(U153'(z0))
U153'(mark(z0)) → c118(U153'(z0))
U132'(ok(z0), ok(z1)) → c119(U132'(z0, z1))
U132'(mark(z0), z1) → c120(U132'(z0, z1))
U81'(ok(z0), ok(z1)) → c121(U81'(z0, z1))
U81'(mark(z0), z1) → c122(U81'(z0, z1))
ISLNAT(ok(z0)) → c123(ISLNAT(z0))
K tuples:

TOP(mark(z0)) → c1(TOP(proper(z0)))
Defined Rule Symbols:

proper

Defined Pair Symbols:

U151', U43', U202', U143', U52', U51', U152', PAIR, SEL, U181', U111', ISLNATKIND, U11', U53', U72', U201', SPLITAT, U112', HEAD, U82', FST, AFTERNTH, TAIL, U71', ISNATURALKIND, U42', U142', U62', U122', U121', U21', U131', U133', TAKE, AND, NATSFROM, U171', ISPLNATKIND, S, U41', CONS, U91', ISNATURAL, U221', U101', U61', SND, U102', U161', U92', ISPLNAT, U141', U31', U191', U103', U211', U153', U132', U81', ISLNAT, TOP

Compound Symbols:

c2, c3, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c48, c49, c50, c51, c52, c53, c54, c55, c56, c57, c58, c59, c60, c61, c62, c63, c64, c65, c66, c67, c68, c69, c70, c71, c72, c73, c74, c75, c76, c77, c78, c79, c80, c81, c82, c83, c84, c85, c86, c87, c88, c89, c90, c91, c92, c93, c94, c95, c96, c97, c98, c99, c100, c101, c102, c103, c104, c105, c106, c107, c108, c109, c110, c111, c112, c113, c114, c115, c116, c117, c118, c119, c120, c121, c122, c123, c1

(15) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

AFTERNTH(mark(z0), z1) → c51(AFTERNTH(z0, z1))
U103'(mark(z0)) → c114(U103'(z0))
U81'(mark(z0), z1) → c122(U81'(z0, z1))
We considered the (Usable) Rules:

proper(tt) → ok(tt)
proper(nil) → ok(nil)
proper(0) → ok(0)
And the Tuples:

U151'(ok(z0), ok(z1), ok(z2)) → c2(U151'(z0, z1, z2))
U151'(mark(z0), z1, z2) → c3(U151'(z0, z1, z2))
U43'(ok(z0)) → c4(U43'(z0))
U43'(mark(z0)) → c5(U43'(z0))
U202'(mark(z0), z1) → c6(U202'(z0, z1))
U202'(ok(z0), ok(z1)) → c7(U202'(z0, z1))
U143'(ok(z0)) → c8(U143'(z0))
U143'(mark(z0)) → c9(U143'(z0))
U52'(ok(z0), ok(z1)) → c10(U52'(z0, z1))
U52'(mark(z0), z1) → c11(U52'(z0, z1))
U51'(mark(z0), z1, z2) → c12(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c13(U51'(z0, z1, z2))
U152'(ok(z0), ok(z1)) → c14(U152'(z0, z1))
U152'(mark(z0), z1) → c15(U152'(z0, z1))
PAIR(mark(z0), z1) → c16(PAIR(z0, z1))
PAIR(ok(z0), ok(z1)) → c17(PAIR(z0, z1))
PAIR(z0, mark(z1)) → c18(PAIR(z0, z1))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
SEL(mark(z0), z1) → c20(SEL(z0, z1))
SEL(z0, mark(z1)) → c21(SEL(z0, z1))
U181'(ok(z0), ok(z1)) → c22(U181'(z0, z1))
U181'(mark(z0), z1) → c23(U181'(z0, z1))
U111'(ok(z0), ok(z1)) → c27(U111'(z0, z1))
U111'(mark(z0), z1) → c28(U111'(z0, z1))
ISLNATKIND(ok(z0)) → c29(ISLNATKIND(z0))
U11'(mark(z0), z1, z2) → c30(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c31(U11'(z0, z1, z2))
U53'(mark(z0)) → c32(U53'(z0))
U53'(ok(z0)) → c33(U53'(z0))
U72'(mark(z0)) → c34(U72'(z0))
U72'(ok(z0)) → c35(U72'(z0))
U201'(ok(z0), ok(z1), ok(z2), ok(z3)) → c36(U201'(z0, z1, z2, z3))
U201'(mark(z0), z1, z2, z3) → c37(U201'(z0, z1, z2, z3))
SPLITAT(z0, mark(z1)) → c38(SPLITAT(z0, z1))
SPLITAT(mark(z0), z1) → c39(SPLITAT(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c40(SPLITAT(z0, z1))
U112'(mark(z0)) → c41(U112'(z0))
U112'(ok(z0)) → c42(U112'(z0))
HEAD(mark(z0)) → c43(HEAD(z0))
HEAD(ok(z0)) → c44(HEAD(z0))
U82'(mark(z0)) → c45(U82'(z0))
U82'(ok(z0)) → c46(U82'(z0))
FST(mark(z0)) → c47(FST(z0))
FST(ok(z0)) → c48(FST(z0))
AFTERNTH(ok(z0), ok(z1)) → c49(AFTERNTH(z0, z1))
AFTERNTH(z0, mark(z1)) → c50(AFTERNTH(z0, z1))
AFTERNTH(mark(z0), z1) → c51(AFTERNTH(z0, z1))
TAIL(mark(z0)) → c52(TAIL(z0))
TAIL(ok(z0)) → c53(TAIL(z0))
U71'(ok(z0), ok(z1)) → c54(U71'(z0, z1))
U71'(mark(z0), z1) → c55(U71'(z0, z1))
ISNATURALKIND(ok(z0)) → c56(ISNATURALKIND(z0))
U42'(ok(z0), ok(z1)) → c57(U42'(z0, z1))
U42'(mark(z0), z1) → c58(U42'(z0, z1))
U142'(mark(z0), z1) → c59(U142'(z0, z1))
U142'(ok(z0), ok(z1)) → c60(U142'(z0, z1))
U62'(ok(z0)) → c61(U62'(z0))
U62'(mark(z0)) → c62(U62'(z0))
U122'(mark(z0)) → c63(U122'(z0))
U122'(ok(z0)) → c64(U122'(z0))
U121'(mark(z0), z1) → c65(U121'(z0, z1))
U121'(ok(z0), ok(z1)) → c66(U121'(z0, z1))
U21'(mark(z0), z1) → c67(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c68(U21'(z0, z1))
U131'(ok(z0), ok(z1), ok(z2)) → c69(U131'(z0, z1, z2))
U131'(mark(z0), z1, z2) → c70(U131'(z0, z1, z2))
U133'(ok(z0)) → c71(U133'(z0))
U133'(mark(z0)) → c72(U133'(z0))
TAKE(z0, mark(z1)) → c73(TAKE(z0, z1))
TAKE(mark(z0), z1) → c74(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c75(TAKE(z0, z1))
AND(mark(z0), z1) → c76(AND(z0, z1))
AND(ok(z0), ok(z1)) → c77(AND(z0, z1))
NATSFROM(mark(z0)) → c78(NATSFROM(z0))
NATSFROM(ok(z0)) → c79(NATSFROM(z0))
U171'(mark(z0), z1, z2) → c80(U171'(z0, z1, z2))
U171'(ok(z0), ok(z1), ok(z2)) → c81(U171'(z0, z1, z2))
ISPLNATKIND(ok(z0)) → c82(ISPLNATKIND(z0))
S(mark(z0)) → c83(S(z0))
S(ok(z0)) → c84(S(z0))
U41'(mark(z0), z1, z2) → c85(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c86(U41'(z0, z1, z2))
CONS(mark(z0), z1) → c87(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c88(CONS(z0, z1))
U91'(ok(z0), ok(z1)) → c89(U91'(z0, z1))
U91'(mark(z0), z1) → c90(U91'(z0, z1))
ISNATURAL(ok(z0)) → c91(ISNATURAL(z0))
U221'(mark(z0), z1, z2) → c92(U221'(z0, z1, z2))
U221'(ok(z0), ok(z1), ok(z2)) → c93(U221'(z0, z1, z2))
U101'(ok(z0), ok(z1), ok(z2)) → c94(U101'(z0, z1, z2))
U101'(mark(z0), z1, z2) → c95(U101'(z0, z1, z2))
U61'(ok(z0), ok(z1)) → c96(U61'(z0, z1))
U61'(mark(z0), z1) → c97(U61'(z0, z1))
SND(ok(z0)) → c98(SND(z0))
SND(mark(z0)) → c99(SND(z0))
U102'(mark(z0), z1) → c100(U102'(z0, z1))
U102'(ok(z0), ok(z1)) → c101(U102'(z0, z1))
U161'(mark(z0), z1) → c102(U161'(z0, z1))
U161'(ok(z0), ok(z1)) → c103(U161'(z0, z1))
U92'(mark(z0)) → c104(U92'(z0))
U92'(ok(z0)) → c105(U92'(z0))
ISPLNAT(ok(z0)) → c106(ISPLNAT(z0))
U141'(mark(z0), z1, z2) → c107(U141'(z0, z1, z2))
U141'(ok(z0), ok(z1), ok(z2)) → c108(U141'(z0, z1, z2))
U31'(ok(z0), ok(z1)) → c109(U31'(z0, z1))
U31'(mark(z0), z1) → c110(U31'(z0, z1))
U191'(ok(z0), ok(z1)) → c111(U191'(z0, z1))
U191'(mark(z0), z1) → c112(U191'(z0, z1))
U103'(ok(z0)) → c113(U103'(z0))
U103'(mark(z0)) → c114(U103'(z0))
U211'(ok(z0), ok(z1)) → c115(U211'(z0, z1))
U211'(mark(z0), z1) → c116(U211'(z0, z1))
U153'(ok(z0)) → c117(U153'(z0))
U153'(mark(z0)) → c118(U153'(z0))
U132'(ok(z0), ok(z1)) → c119(U132'(z0, z1))
U132'(mark(z0), z1) → c120(U132'(z0, z1))
U81'(ok(z0), ok(z1)) → c121(U81'(z0, z1))
U81'(mark(z0), z1) → c122(U81'(z0, z1))
ISLNAT(ok(z0)) → c123(ISLNAT(z0))
TOP(mark(z0)) → c1(TOP(proper(z0)))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = 0   
POL(AFTERNTH(x1, x2)) = x1   
POL(AND(x1, x2)) = 0   
POL(CONS(x1, x2)) = 0   
POL(FST(x1)) = 0   
POL(HEAD(x1)) = 0   
POL(ISLNAT(x1)) = 0   
POL(ISLNATKIND(x1)) = 0   
POL(ISNATURAL(x1)) = 0   
POL(ISNATURALKIND(x1)) = 0   
POL(ISPLNAT(x1)) = 0   
POL(ISPLNATKIND(x1)) = 0   
POL(NATSFROM(x1)) = 0   
POL(PAIR(x1, x2)) = 0   
POL(S(x1)) = 0   
POL(SEL(x1, x2)) = 0   
POL(SND(x1)) = 0   
POL(SPLITAT(x1, x2)) = 0   
POL(TAIL(x1)) = 0   
POL(TAKE(x1, x2)) = 0   
POL(TOP(x1)) = x1   
POL(U101'(x1, x2, x3)) = 0   
POL(U102'(x1, x2)) = 0   
POL(U103'(x1)) = x1   
POL(U11'(x1, x2, x3)) = 0   
POL(U111'(x1, x2)) = 0   
POL(U112'(x1)) = 0   
POL(U121'(x1, x2)) = 0   
POL(U122'(x1)) = 0   
POL(U131'(x1, x2, x3)) = 0   
POL(U132'(x1, x2)) = 0   
POL(U133'(x1)) = 0   
POL(U141'(x1, x2, x3)) = 0   
POL(U142'(x1, x2)) = 0   
POL(U143'(x1)) = 0   
POL(U151'(x1, x2, x3)) = 0   
POL(U152'(x1, x2)) = 0   
POL(U153'(x1)) = 0   
POL(U161'(x1, x2)) = 0   
POL(U171'(x1, x2, x3)) = 0   
POL(U181'(x1, x2)) = 0   
POL(U191'(x1, x2)) = 0   
POL(U201'(x1, x2, x3, x4)) = 0   
POL(U202'(x1, x2)) = 0   
POL(U21'(x1, x2)) = 0   
POL(U211'(x1, x2)) = 0   
POL(U221'(x1, x2, x3)) = 0   
POL(U31'(x1, x2)) = 0   
POL(U41'(x1, x2, x3)) = 0   
POL(U42'(x1, x2)) = 0   
POL(U43'(x1)) = 0   
POL(U51'(x1, x2, x3)) = 0   
POL(U52'(x1, x2)) = 0   
POL(U53'(x1)) = 0   
POL(U61'(x1, x2)) = 0   
POL(U62'(x1)) = 0   
POL(U71'(x1, x2)) = 0   
POL(U72'(x1)) = 0   
POL(U81'(x1, x2)) = x1   
POL(U82'(x1)) = 0   
POL(U91'(x1, x2)) = 0   
POL(U92'(x1)) = 0   
POL(c1(x1)) = x1   
POL(c10(x1)) = x1   
POL(c100(x1)) = x1   
POL(c101(x1)) = x1   
POL(c102(x1)) = x1   
POL(c103(x1)) = x1   
POL(c104(x1)) = x1   
POL(c105(x1)) = x1   
POL(c106(x1)) = x1   
POL(c107(x1)) = x1   
POL(c108(x1)) = x1   
POL(c109(x1)) = x1   
POL(c11(x1)) = x1   
POL(c110(x1)) = x1   
POL(c111(x1)) = x1   
POL(c112(x1)) = x1   
POL(c113(x1)) = x1   
POL(c114(x1)) = x1   
POL(c115(x1)) = x1   
POL(c116(x1)) = x1   
POL(c117(x1)) = x1   
POL(c118(x1)) = x1   
POL(c119(x1)) = x1   
POL(c12(x1)) = x1   
POL(c120(x1)) = x1   
POL(c121(x1)) = x1   
POL(c122(x1)) = x1   
POL(c123(x1)) = x1   
POL(c13(x1)) = x1   
POL(c14(x1)) = x1   
POL(c15(x1)) = x1   
POL(c16(x1)) = x1   
POL(c17(x1)) = x1   
POL(c18(x1)) = x1   
POL(c19(x1)) = x1   
POL(c2(x1)) = x1   
POL(c20(x1)) = x1   
POL(c21(x1)) = x1   
POL(c22(x1)) = x1   
POL(c23(x1)) = x1   
POL(c27(x1)) = x1   
POL(c28(x1)) = x1   
POL(c29(x1)) = x1   
POL(c3(x1)) = x1   
POL(c30(x1)) = x1   
POL(c31(x1)) = x1   
POL(c32(x1)) = x1   
POL(c33(x1)) = x1   
POL(c34(x1)) = x1   
POL(c35(x1)) = x1   
POL(c36(x1)) = x1   
POL(c37(x1)) = x1   
POL(c38(x1)) = x1   
POL(c39(x1)) = x1   
POL(c4(x1)) = x1   
POL(c40(x1)) = x1   
POL(c41(x1)) = x1   
POL(c42(x1)) = x1   
POL(c43(x1)) = x1   
POL(c44(x1)) = x1   
POL(c45(x1)) = x1   
POL(c46(x1)) = x1   
POL(c47(x1)) = x1   
POL(c48(x1)) = x1   
POL(c49(x1)) = x1   
POL(c5(x1)) = x1   
POL(c50(x1)) = x1   
POL(c51(x1)) = x1   
POL(c52(x1)) = x1   
POL(c53(x1)) = x1   
POL(c54(x1)) = x1   
POL(c55(x1)) = x1   
POL(c56(x1)) = x1   
POL(c57(x1)) = x1   
POL(c58(x1)) = x1   
POL(c59(x1)) = x1   
POL(c6(x1)) = x1   
POL(c60(x1)) = x1   
POL(c61(x1)) = x1   
POL(c62(x1)) = x1   
POL(c63(x1)) = x1   
POL(c64(x1)) = x1   
POL(c65(x1)) = x1   
POL(c66(x1)) = x1   
POL(c67(x1)) = x1   
POL(c68(x1)) = x1   
POL(c69(x1)) = x1   
POL(c7(x1)) = x1   
POL(c70(x1)) = x1   
POL(c71(x1)) = x1   
POL(c72(x1)) = x1   
POL(c73(x1)) = x1   
POL(c74(x1)) = x1   
POL(c75(x1)) = x1   
POL(c76(x1)) = x1   
POL(c77(x1)) = x1   
POL(c78(x1)) = x1   
POL(c79(x1)) = x1   
POL(c8(x1)) = x1   
POL(c80(x1)) = x1   
POL(c81(x1)) = x1   
POL(c82(x1)) = x1   
POL(c83(x1)) = x1   
POL(c84(x1)) = x1   
POL(c85(x1)) = x1   
POL(c86(x1)) = x1   
POL(c87(x1)) = x1   
POL(c88(x1)) = x1   
POL(c89(x1)) = x1   
POL(c9(x1)) = x1   
POL(c90(x1)) = x1   
POL(c91(x1)) = x1   
POL(c92(x1)) = x1   
POL(c93(x1)) = x1   
POL(c94(x1)) = x1   
POL(c95(x1)) = x1   
POL(c96(x1)) = x1   
POL(c97(x1)) = x1   
POL(c98(x1)) = x1   
POL(c99(x1)) = x1   
POL(mark(x1)) = [1] + x1   
POL(nil) = 0   
POL(ok(x1)) = x1   
POL(proper(x1)) = 0   
POL(tt) = 0   

(16) Obligation:

Complexity Dependency Tuples Problem
Rules:

proper(tt) → ok(tt)
proper(nil) → ok(nil)
proper(0) → ok(0)
Tuples:

U151'(ok(z0), ok(z1), ok(z2)) → c2(U151'(z0, z1, z2))
U151'(mark(z0), z1, z2) → c3(U151'(z0, z1, z2))
U43'(ok(z0)) → c4(U43'(z0))
U43'(mark(z0)) → c5(U43'(z0))
U202'(mark(z0), z1) → c6(U202'(z0, z1))
U202'(ok(z0), ok(z1)) → c7(U202'(z0, z1))
U143'(ok(z0)) → c8(U143'(z0))
U143'(mark(z0)) → c9(U143'(z0))
U52'(ok(z0), ok(z1)) → c10(U52'(z0, z1))
U52'(mark(z0), z1) → c11(U52'(z0, z1))
U51'(mark(z0), z1, z2) → c12(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c13(U51'(z0, z1, z2))
U152'(ok(z0), ok(z1)) → c14(U152'(z0, z1))
U152'(mark(z0), z1) → c15(U152'(z0, z1))
PAIR(mark(z0), z1) → c16(PAIR(z0, z1))
PAIR(ok(z0), ok(z1)) → c17(PAIR(z0, z1))
PAIR(z0, mark(z1)) → c18(PAIR(z0, z1))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
SEL(mark(z0), z1) → c20(SEL(z0, z1))
SEL(z0, mark(z1)) → c21(SEL(z0, z1))
U181'(ok(z0), ok(z1)) → c22(U181'(z0, z1))
U181'(mark(z0), z1) → c23(U181'(z0, z1))
U111'(ok(z0), ok(z1)) → c27(U111'(z0, z1))
U111'(mark(z0), z1) → c28(U111'(z0, z1))
ISLNATKIND(ok(z0)) → c29(ISLNATKIND(z0))
U11'(mark(z0), z1, z2) → c30(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c31(U11'(z0, z1, z2))
U53'(mark(z0)) → c32(U53'(z0))
U53'(ok(z0)) → c33(U53'(z0))
U72'(mark(z0)) → c34(U72'(z0))
U72'(ok(z0)) → c35(U72'(z0))
U201'(ok(z0), ok(z1), ok(z2), ok(z3)) → c36(U201'(z0, z1, z2, z3))
U201'(mark(z0), z1, z2, z3) → c37(U201'(z0, z1, z2, z3))
SPLITAT(z0, mark(z1)) → c38(SPLITAT(z0, z1))
SPLITAT(mark(z0), z1) → c39(SPLITAT(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c40(SPLITAT(z0, z1))
U112'(mark(z0)) → c41(U112'(z0))
U112'(ok(z0)) → c42(U112'(z0))
HEAD(mark(z0)) → c43(HEAD(z0))
HEAD(ok(z0)) → c44(HEAD(z0))
U82'(mark(z0)) → c45(U82'(z0))
U82'(ok(z0)) → c46(U82'(z0))
FST(mark(z0)) → c47(FST(z0))
FST(ok(z0)) → c48(FST(z0))
AFTERNTH(ok(z0), ok(z1)) → c49(AFTERNTH(z0, z1))
AFTERNTH(z0, mark(z1)) → c50(AFTERNTH(z0, z1))
AFTERNTH(mark(z0), z1) → c51(AFTERNTH(z0, z1))
TAIL(mark(z0)) → c52(TAIL(z0))
TAIL(ok(z0)) → c53(TAIL(z0))
U71'(ok(z0), ok(z1)) → c54(U71'(z0, z1))
U71'(mark(z0), z1) → c55(U71'(z0, z1))
ISNATURALKIND(ok(z0)) → c56(ISNATURALKIND(z0))
U42'(ok(z0), ok(z1)) → c57(U42'(z0, z1))
U42'(mark(z0), z1) → c58(U42'(z0, z1))
U142'(mark(z0), z1) → c59(U142'(z0, z1))
U142'(ok(z0), ok(z1)) → c60(U142'(z0, z1))
U62'(ok(z0)) → c61(U62'(z0))
U62'(mark(z0)) → c62(U62'(z0))
U122'(mark(z0)) → c63(U122'(z0))
U122'(ok(z0)) → c64(U122'(z0))
U121'(mark(z0), z1) → c65(U121'(z0, z1))
U121'(ok(z0), ok(z1)) → c66(U121'(z0, z1))
U21'(mark(z0), z1) → c67(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c68(U21'(z0, z1))
U131'(ok(z0), ok(z1), ok(z2)) → c69(U131'(z0, z1, z2))
U131'(mark(z0), z1, z2) → c70(U131'(z0, z1, z2))
U133'(ok(z0)) → c71(U133'(z0))
U133'(mark(z0)) → c72(U133'(z0))
TAKE(z0, mark(z1)) → c73(TAKE(z0, z1))
TAKE(mark(z0), z1) → c74(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c75(TAKE(z0, z1))
AND(mark(z0), z1) → c76(AND(z0, z1))
AND(ok(z0), ok(z1)) → c77(AND(z0, z1))
NATSFROM(mark(z0)) → c78(NATSFROM(z0))
NATSFROM(ok(z0)) → c79(NATSFROM(z0))
U171'(mark(z0), z1, z2) → c80(U171'(z0, z1, z2))
U171'(ok(z0), ok(z1), ok(z2)) → c81(U171'(z0, z1, z2))
ISPLNATKIND(ok(z0)) → c82(ISPLNATKIND(z0))
S(mark(z0)) → c83(S(z0))
S(ok(z0)) → c84(S(z0))
U41'(mark(z0), z1, z2) → c85(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c86(U41'(z0, z1, z2))
CONS(mark(z0), z1) → c87(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c88(CONS(z0, z1))
U91'(ok(z0), ok(z1)) → c89(U91'(z0, z1))
U91'(mark(z0), z1) → c90(U91'(z0, z1))
ISNATURAL(ok(z0)) → c91(ISNATURAL(z0))
U221'(mark(z0), z1, z2) → c92(U221'(z0, z1, z2))
U221'(ok(z0), ok(z1), ok(z2)) → c93(U221'(z0, z1, z2))
U101'(ok(z0), ok(z1), ok(z2)) → c94(U101'(z0, z1, z2))
U101'(mark(z0), z1, z2) → c95(U101'(z0, z1, z2))
U61'(ok(z0), ok(z1)) → c96(U61'(z0, z1))
U61'(mark(z0), z1) → c97(U61'(z0, z1))
SND(ok(z0)) → c98(SND(z0))
SND(mark(z0)) → c99(SND(z0))
U102'(mark(z0), z1) → c100(U102'(z0, z1))
U102'(ok(z0), ok(z1)) → c101(U102'(z0, z1))
U161'(mark(z0), z1) → c102(U161'(z0, z1))
U161'(ok(z0), ok(z1)) → c103(U161'(z0, z1))
U92'(mark(z0)) → c104(U92'(z0))
U92'(ok(z0)) → c105(U92'(z0))
ISPLNAT(ok(z0)) → c106(ISPLNAT(z0))
U141'(mark(z0), z1, z2) → c107(U141'(z0, z1, z2))
U141'(ok(z0), ok(z1), ok(z2)) → c108(U141'(z0, z1, z2))
U31'(ok(z0), ok(z1)) → c109(U31'(z0, z1))
U31'(mark(z0), z1) → c110(U31'(z0, z1))
U191'(ok(z0), ok(z1)) → c111(U191'(z0, z1))
U191'(mark(z0), z1) → c112(U191'(z0, z1))
U103'(ok(z0)) → c113(U103'(z0))
U103'(mark(z0)) → c114(U103'(z0))
U211'(ok(z0), ok(z1)) → c115(U211'(z0, z1))
U211'(mark(z0), z1) → c116(U211'(z0, z1))
U153'(ok(z0)) → c117(U153'(z0))
U153'(mark(z0)) → c118(U153'(z0))
U132'(ok(z0), ok(z1)) → c119(U132'(z0, z1))
U132'(mark(z0), z1) → c120(U132'(z0, z1))
U81'(ok(z0), ok(z1)) → c121(U81'(z0, z1))
U81'(mark(z0), z1) → c122(U81'(z0, z1))
ISLNAT(ok(z0)) → c123(ISLNAT(z0))
TOP(mark(z0)) → c1(TOP(proper(z0)))
S tuples:

U151'(ok(z0), ok(z1), ok(z2)) → c2(U151'(z0, z1, z2))
U151'(mark(z0), z1, z2) → c3(U151'(z0, z1, z2))
U43'(ok(z0)) → c4(U43'(z0))
U43'(mark(z0)) → c5(U43'(z0))
U202'(mark(z0), z1) → c6(U202'(z0, z1))
U202'(ok(z0), ok(z1)) → c7(U202'(z0, z1))
U143'(ok(z0)) → c8(U143'(z0))
U143'(mark(z0)) → c9(U143'(z0))
U52'(ok(z0), ok(z1)) → c10(U52'(z0, z1))
U52'(mark(z0), z1) → c11(U52'(z0, z1))
U51'(mark(z0), z1, z2) → c12(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c13(U51'(z0, z1, z2))
U152'(ok(z0), ok(z1)) → c14(U152'(z0, z1))
U152'(mark(z0), z1) → c15(U152'(z0, z1))
PAIR(mark(z0), z1) → c16(PAIR(z0, z1))
PAIR(ok(z0), ok(z1)) → c17(PAIR(z0, z1))
PAIR(z0, mark(z1)) → c18(PAIR(z0, z1))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
SEL(mark(z0), z1) → c20(SEL(z0, z1))
SEL(z0, mark(z1)) → c21(SEL(z0, z1))
U181'(ok(z0), ok(z1)) → c22(U181'(z0, z1))
U181'(mark(z0), z1) → c23(U181'(z0, z1))
U111'(ok(z0), ok(z1)) → c27(U111'(z0, z1))
U111'(mark(z0), z1) → c28(U111'(z0, z1))
ISLNATKIND(ok(z0)) → c29(ISLNATKIND(z0))
U11'(mark(z0), z1, z2) → c30(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c31(U11'(z0, z1, z2))
U53'(mark(z0)) → c32(U53'(z0))
U53'(ok(z0)) → c33(U53'(z0))
U72'(mark(z0)) → c34(U72'(z0))
U72'(ok(z0)) → c35(U72'(z0))
U201'(ok(z0), ok(z1), ok(z2), ok(z3)) → c36(U201'(z0, z1, z2, z3))
U201'(mark(z0), z1, z2, z3) → c37(U201'(z0, z1, z2, z3))
SPLITAT(z0, mark(z1)) → c38(SPLITAT(z0, z1))
SPLITAT(mark(z0), z1) → c39(SPLITAT(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c40(SPLITAT(z0, z1))
U112'(mark(z0)) → c41(U112'(z0))
U112'(ok(z0)) → c42(U112'(z0))
HEAD(mark(z0)) → c43(HEAD(z0))
HEAD(ok(z0)) → c44(HEAD(z0))
U82'(mark(z0)) → c45(U82'(z0))
U82'(ok(z0)) → c46(U82'(z0))
FST(mark(z0)) → c47(FST(z0))
FST(ok(z0)) → c48(FST(z0))
AFTERNTH(ok(z0), ok(z1)) → c49(AFTERNTH(z0, z1))
AFTERNTH(z0, mark(z1)) → c50(AFTERNTH(z0, z1))
TAIL(mark(z0)) → c52(TAIL(z0))
TAIL(ok(z0)) → c53(TAIL(z0))
U71'(ok(z0), ok(z1)) → c54(U71'(z0, z1))
U71'(mark(z0), z1) → c55(U71'(z0, z1))
ISNATURALKIND(ok(z0)) → c56(ISNATURALKIND(z0))
U42'(ok(z0), ok(z1)) → c57(U42'(z0, z1))
U42'(mark(z0), z1) → c58(U42'(z0, z1))
U142'(mark(z0), z1) → c59(U142'(z0, z1))
U142'(ok(z0), ok(z1)) → c60(U142'(z0, z1))
U62'(ok(z0)) → c61(U62'(z0))
U62'(mark(z0)) → c62(U62'(z0))
U122'(mark(z0)) → c63(U122'(z0))
U122'(ok(z0)) → c64(U122'(z0))
U121'(mark(z0), z1) → c65(U121'(z0, z1))
U121'(ok(z0), ok(z1)) → c66(U121'(z0, z1))
U21'(mark(z0), z1) → c67(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c68(U21'(z0, z1))
U131'(ok(z0), ok(z1), ok(z2)) → c69(U131'(z0, z1, z2))
U131'(mark(z0), z1, z2) → c70(U131'(z0, z1, z2))
U133'(ok(z0)) → c71(U133'(z0))
U133'(mark(z0)) → c72(U133'(z0))
TAKE(z0, mark(z1)) → c73(TAKE(z0, z1))
TAKE(mark(z0), z1) → c74(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c75(TAKE(z0, z1))
AND(mark(z0), z1) → c76(AND(z0, z1))
AND(ok(z0), ok(z1)) → c77(AND(z0, z1))
NATSFROM(mark(z0)) → c78(NATSFROM(z0))
NATSFROM(ok(z0)) → c79(NATSFROM(z0))
U171'(mark(z0), z1, z2) → c80(U171'(z0, z1, z2))
U171'(ok(z0), ok(z1), ok(z2)) → c81(U171'(z0, z1, z2))
ISPLNATKIND(ok(z0)) → c82(ISPLNATKIND(z0))
S(mark(z0)) → c83(S(z0))
S(ok(z0)) → c84(S(z0))
U41'(mark(z0), z1, z2) → c85(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c86(U41'(z0, z1, z2))
CONS(mark(z0), z1) → c87(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c88(CONS(z0, z1))
U91'(ok(z0), ok(z1)) → c89(U91'(z0, z1))
U91'(mark(z0), z1) → c90(U91'(z0, z1))
ISNATURAL(ok(z0)) → c91(ISNATURAL(z0))
U221'(mark(z0), z1, z2) → c92(U221'(z0, z1, z2))
U221'(ok(z0), ok(z1), ok(z2)) → c93(U221'(z0, z1, z2))
U101'(ok(z0), ok(z1), ok(z2)) → c94(U101'(z0, z1, z2))
U101'(mark(z0), z1, z2) → c95(U101'(z0, z1, z2))
U61'(ok(z0), ok(z1)) → c96(U61'(z0, z1))
U61'(mark(z0), z1) → c97(U61'(z0, z1))
SND(ok(z0)) → c98(SND(z0))
SND(mark(z0)) → c99(SND(z0))
U102'(mark(z0), z1) → c100(U102'(z0, z1))
U102'(ok(z0), ok(z1)) → c101(U102'(z0, z1))
U161'(mark(z0), z1) → c102(U161'(z0, z1))
U161'(ok(z0), ok(z1)) → c103(U161'(z0, z1))
U92'(mark(z0)) → c104(U92'(z0))
U92'(ok(z0)) → c105(U92'(z0))
ISPLNAT(ok(z0)) → c106(ISPLNAT(z0))
U141'(mark(z0), z1, z2) → c107(U141'(z0, z1, z2))
U141'(ok(z0), ok(z1), ok(z2)) → c108(U141'(z0, z1, z2))
U31'(ok(z0), ok(z1)) → c109(U31'(z0, z1))
U31'(mark(z0), z1) → c110(U31'(z0, z1))
U191'(ok(z0), ok(z1)) → c111(U191'(z0, z1))
U191'(mark(z0), z1) → c112(U191'(z0, z1))
U103'(ok(z0)) → c113(U103'(z0))
U211'(ok(z0), ok(z1)) → c115(U211'(z0, z1))
U211'(mark(z0), z1) → c116(U211'(z0, z1))
U153'(ok(z0)) → c117(U153'(z0))
U153'(mark(z0)) → c118(U153'(z0))
U132'(ok(z0), ok(z1)) → c119(U132'(z0, z1))
U132'(mark(z0), z1) → c120(U132'(z0, z1))
U81'(ok(z0), ok(z1)) → c121(U81'(z0, z1))
ISLNAT(ok(z0)) → c123(ISLNAT(z0))
K tuples:

TOP(mark(z0)) → c1(TOP(proper(z0)))
AFTERNTH(mark(z0), z1) → c51(AFTERNTH(z0, z1))
U103'(mark(z0)) → c114(U103'(z0))
U81'(mark(z0), z1) → c122(U81'(z0, z1))
Defined Rule Symbols:

proper

Defined Pair Symbols:

U151', U43', U202', U143', U52', U51', U152', PAIR, SEL, U181', U111', ISLNATKIND, U11', U53', U72', U201', SPLITAT, U112', HEAD, U82', FST, AFTERNTH, TAIL, U71', ISNATURALKIND, U42', U142', U62', U122', U121', U21', U131', U133', TAKE, AND, NATSFROM, U171', ISPLNATKIND, S, U41', CONS, U91', ISNATURAL, U221', U101', U61', SND, U102', U161', U92', ISPLNAT, U141', U31', U191', U103', U211', U153', U132', U81', ISLNAT, TOP

Compound Symbols:

c2, c3, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c48, c49, c50, c51, c52, c53, c54, c55, c56, c57, c58, c59, c60, c61, c62, c63, c64, c65, c66, c67, c68, c69, c70, c71, c72, c73, c74, c75, c76, c77, c78, c79, c80, c81, c82, c83, c84, c85, c86, c87, c88, c89, c90, c91, c92, c93, c94, c95, c96, c97, c98, c99, c100, c101, c102, c103, c104, c105, c106, c107, c108, c109, c110, c111, c112, c113, c114, c115, c116, c117, c118, c119, c120, c121, c122, c123, c1

(17) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

U52'(mark(z0), z1) → c11(U52'(z0, z1))
U211'(mark(z0), z1) → c116(U211'(z0, z1))
We considered the (Usable) Rules:

proper(tt) → ok(tt)
proper(nil) → ok(nil)
proper(0) → ok(0)
And the Tuples:

U151'(ok(z0), ok(z1), ok(z2)) → c2(U151'(z0, z1, z2))
U151'(mark(z0), z1, z2) → c3(U151'(z0, z1, z2))
U43'(ok(z0)) → c4(U43'(z0))
U43'(mark(z0)) → c5(U43'(z0))
U202'(mark(z0), z1) → c6(U202'(z0, z1))
U202'(ok(z0), ok(z1)) → c7(U202'(z0, z1))
U143'(ok(z0)) → c8(U143'(z0))
U143'(mark(z0)) → c9(U143'(z0))
U52'(ok(z0), ok(z1)) → c10(U52'(z0, z1))
U52'(mark(z0), z1) → c11(U52'(z0, z1))
U51'(mark(z0), z1, z2) → c12(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c13(U51'(z0, z1, z2))
U152'(ok(z0), ok(z1)) → c14(U152'(z0, z1))
U152'(mark(z0), z1) → c15(U152'(z0, z1))
PAIR(mark(z0), z1) → c16(PAIR(z0, z1))
PAIR(ok(z0), ok(z1)) → c17(PAIR(z0, z1))
PAIR(z0, mark(z1)) → c18(PAIR(z0, z1))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
SEL(mark(z0), z1) → c20(SEL(z0, z1))
SEL(z0, mark(z1)) → c21(SEL(z0, z1))
U181'(ok(z0), ok(z1)) → c22(U181'(z0, z1))
U181'(mark(z0), z1) → c23(U181'(z0, z1))
U111'(ok(z0), ok(z1)) → c27(U111'(z0, z1))
U111'(mark(z0), z1) → c28(U111'(z0, z1))
ISLNATKIND(ok(z0)) → c29(ISLNATKIND(z0))
U11'(mark(z0), z1, z2) → c30(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c31(U11'(z0, z1, z2))
U53'(mark(z0)) → c32(U53'(z0))
U53'(ok(z0)) → c33(U53'(z0))
U72'(mark(z0)) → c34(U72'(z0))
U72'(ok(z0)) → c35(U72'(z0))
U201'(ok(z0), ok(z1), ok(z2), ok(z3)) → c36(U201'(z0, z1, z2, z3))
U201'(mark(z0), z1, z2, z3) → c37(U201'(z0, z1, z2, z3))
SPLITAT(z0, mark(z1)) → c38(SPLITAT(z0, z1))
SPLITAT(mark(z0), z1) → c39(SPLITAT(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c40(SPLITAT(z0, z1))
U112'(mark(z0)) → c41(U112'(z0))
U112'(ok(z0)) → c42(U112'(z0))
HEAD(mark(z0)) → c43(HEAD(z0))
HEAD(ok(z0)) → c44(HEAD(z0))
U82'(mark(z0)) → c45(U82'(z0))
U82'(ok(z0)) → c46(U82'(z0))
FST(mark(z0)) → c47(FST(z0))
FST(ok(z0)) → c48(FST(z0))
AFTERNTH(ok(z0), ok(z1)) → c49(AFTERNTH(z0, z1))
AFTERNTH(z0, mark(z1)) → c50(AFTERNTH(z0, z1))
AFTERNTH(mark(z0), z1) → c51(AFTERNTH(z0, z1))
TAIL(mark(z0)) → c52(TAIL(z0))
TAIL(ok(z0)) → c53(TAIL(z0))
U71'(ok(z0), ok(z1)) → c54(U71'(z0, z1))
U71'(mark(z0), z1) → c55(U71'(z0, z1))
ISNATURALKIND(ok(z0)) → c56(ISNATURALKIND(z0))
U42'(ok(z0), ok(z1)) → c57(U42'(z0, z1))
U42'(mark(z0), z1) → c58(U42'(z0, z1))
U142'(mark(z0), z1) → c59(U142'(z0, z1))
U142'(ok(z0), ok(z1)) → c60(U142'(z0, z1))
U62'(ok(z0)) → c61(U62'(z0))
U62'(mark(z0)) → c62(U62'(z0))
U122'(mark(z0)) → c63(U122'(z0))
U122'(ok(z0)) → c64(U122'(z0))
U121'(mark(z0), z1) → c65(U121'(z0, z1))
U121'(ok(z0), ok(z1)) → c66(U121'(z0, z1))
U21'(mark(z0), z1) → c67(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c68(U21'(z0, z1))
U131'(ok(z0), ok(z1), ok(z2)) → c69(U131'(z0, z1, z2))
U131'(mark(z0), z1, z2) → c70(U131'(z0, z1, z2))
U133'(ok(z0)) → c71(U133'(z0))
U133'(mark(z0)) → c72(U133'(z0))
TAKE(z0, mark(z1)) → c73(TAKE(z0, z1))
TAKE(mark(z0), z1) → c74(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c75(TAKE(z0, z1))
AND(mark(z0), z1) → c76(AND(z0, z1))
AND(ok(z0), ok(z1)) → c77(AND(z0, z1))
NATSFROM(mark(z0)) → c78(NATSFROM(z0))
NATSFROM(ok(z0)) → c79(NATSFROM(z0))
U171'(mark(z0), z1, z2) → c80(U171'(z0, z1, z2))
U171'(ok(z0), ok(z1), ok(z2)) → c81(U171'(z0, z1, z2))
ISPLNATKIND(ok(z0)) → c82(ISPLNATKIND(z0))
S(mark(z0)) → c83(S(z0))
S(ok(z0)) → c84(S(z0))
U41'(mark(z0), z1, z2) → c85(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c86(U41'(z0, z1, z2))
CONS(mark(z0), z1) → c87(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c88(CONS(z0, z1))
U91'(ok(z0), ok(z1)) → c89(U91'(z0, z1))
U91'(mark(z0), z1) → c90(U91'(z0, z1))
ISNATURAL(ok(z0)) → c91(ISNATURAL(z0))
U221'(mark(z0), z1, z2) → c92(U221'(z0, z1, z2))
U221'(ok(z0), ok(z1), ok(z2)) → c93(U221'(z0, z1, z2))
U101'(ok(z0), ok(z1), ok(z2)) → c94(U101'(z0, z1, z2))
U101'(mark(z0), z1, z2) → c95(U101'(z0, z1, z2))
U61'(ok(z0), ok(z1)) → c96(U61'(z0, z1))
U61'(mark(z0), z1) → c97(U61'(z0, z1))
SND(ok(z0)) → c98(SND(z0))
SND(mark(z0)) → c99(SND(z0))
U102'(mark(z0), z1) → c100(U102'(z0, z1))
U102'(ok(z0), ok(z1)) → c101(U102'(z0, z1))
U161'(mark(z0), z1) → c102(U161'(z0, z1))
U161'(ok(z0), ok(z1)) → c103(U161'(z0, z1))
U92'(mark(z0)) → c104(U92'(z0))
U92'(ok(z0)) → c105(U92'(z0))
ISPLNAT(ok(z0)) → c106(ISPLNAT(z0))
U141'(mark(z0), z1, z2) → c107(U141'(z0, z1, z2))
U141'(ok(z0), ok(z1), ok(z2)) → c108(U141'(z0, z1, z2))
U31'(ok(z0), ok(z1)) → c109(U31'(z0, z1))
U31'(mark(z0), z1) → c110(U31'(z0, z1))
U191'(ok(z0), ok(z1)) → c111(U191'(z0, z1))
U191'(mark(z0), z1) → c112(U191'(z0, z1))
U103'(ok(z0)) → c113(U103'(z0))
U103'(mark(z0)) → c114(U103'(z0))
U211'(ok(z0), ok(z1)) → c115(U211'(z0, z1))
U211'(mark(z0), z1) → c116(U211'(z0, z1))
U153'(ok(z0)) → c117(U153'(z0))
U153'(mark(z0)) → c118(U153'(z0))
U132'(ok(z0), ok(z1)) → c119(U132'(z0, z1))
U132'(mark(z0), z1) → c120(U132'(z0, z1))
U81'(ok(z0), ok(z1)) → c121(U81'(z0, z1))
U81'(mark(z0), z1) → c122(U81'(z0, z1))
ISLNAT(ok(z0)) → c123(ISLNAT(z0))
TOP(mark(z0)) → c1(TOP(proper(z0)))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = 0   
POL(AFTERNTH(x1, x2)) = 0   
POL(AND(x1, x2)) = 0   
POL(CONS(x1, x2)) = 0   
POL(FST(x1)) = 0   
POL(HEAD(x1)) = 0   
POL(ISLNAT(x1)) = 0   
POL(ISLNATKIND(x1)) = 0   
POL(ISNATURAL(x1)) = 0   
POL(ISNATURALKIND(x1)) = 0   
POL(ISPLNAT(x1)) = 0   
POL(ISPLNATKIND(x1)) = 0   
POL(NATSFROM(x1)) = 0   
POL(PAIR(x1, x2)) = 0   
POL(S(x1)) = 0   
POL(SEL(x1, x2)) = 0   
POL(SND(x1)) = 0   
POL(SPLITAT(x1, x2)) = 0   
POL(TAIL(x1)) = 0   
POL(TAKE(x1, x2)) = 0   
POL(TOP(x1)) = x1   
POL(U101'(x1, x2, x3)) = 0   
POL(U102'(x1, x2)) = 0   
POL(U103'(x1)) = 0   
POL(U11'(x1, x2, x3)) = 0   
POL(U111'(x1, x2)) = 0   
POL(U112'(x1)) = 0   
POL(U121'(x1, x2)) = 0   
POL(U122'(x1)) = 0   
POL(U131'(x1, x2, x3)) = 0   
POL(U132'(x1, x2)) = 0   
POL(U133'(x1)) = 0   
POL(U141'(x1, x2, x3)) = 0   
POL(U142'(x1, x2)) = 0   
POL(U143'(x1)) = 0   
POL(U151'(x1, x2, x3)) = 0   
POL(U152'(x1, x2)) = 0   
POL(U153'(x1)) = 0   
POL(U161'(x1, x2)) = 0   
POL(U171'(x1, x2, x3)) = 0   
POL(U181'(x1, x2)) = 0   
POL(U191'(x1, x2)) = 0   
POL(U201'(x1, x2, x3, x4)) = 0   
POL(U202'(x1, x2)) = 0   
POL(U21'(x1, x2)) = 0   
POL(U211'(x1, x2)) = x1   
POL(U221'(x1, x2, x3)) = 0   
POL(U31'(x1, x2)) = 0   
POL(U41'(x1, x2, x3)) = 0   
POL(U42'(x1, x2)) = 0   
POL(U43'(x1)) = 0   
POL(U51'(x1, x2, x3)) = 0   
POL(U52'(x1, x2)) = x1   
POL(U53'(x1)) = 0   
POL(U61'(x1, x2)) = 0   
POL(U62'(x1)) = 0   
POL(U71'(x1, x2)) = 0   
POL(U72'(x1)) = 0   
POL(U81'(x1, x2)) = 0   
POL(U82'(x1)) = 0   
POL(U91'(x1, x2)) = 0   
POL(U92'(x1)) = 0   
POL(c1(x1)) = x1   
POL(c10(x1)) = x1   
POL(c100(x1)) = x1   
POL(c101(x1)) = x1   
POL(c102(x1)) = x1   
POL(c103(x1)) = x1   
POL(c104(x1)) = x1   
POL(c105(x1)) = x1   
POL(c106(x1)) = x1   
POL(c107(x1)) = x1   
POL(c108(x1)) = x1   
POL(c109(x1)) = x1   
POL(c11(x1)) = x1   
POL(c110(x1)) = x1   
POL(c111(x1)) = x1   
POL(c112(x1)) = x1   
POL(c113(x1)) = x1   
POL(c114(x1)) = x1   
POL(c115(x1)) = x1   
POL(c116(x1)) = x1   
POL(c117(x1)) = x1   
POL(c118(x1)) = x1   
POL(c119(x1)) = x1   
POL(c12(x1)) = x1   
POL(c120(x1)) = x1   
POL(c121(x1)) = x1   
POL(c122(x1)) = x1   
POL(c123(x1)) = x1   
POL(c13(x1)) = x1   
POL(c14(x1)) = x1   
POL(c15(x1)) = x1   
POL(c16(x1)) = x1   
POL(c17(x1)) = x1   
POL(c18(x1)) = x1   
POL(c19(x1)) = x1   
POL(c2(x1)) = x1   
POL(c20(x1)) = x1   
POL(c21(x1)) = x1   
POL(c22(x1)) = x1   
POL(c23(x1)) = x1   
POL(c27(x1)) = x1   
POL(c28(x1)) = x1   
POL(c29(x1)) = x1   
POL(c3(x1)) = x1   
POL(c30(x1)) = x1   
POL(c31(x1)) = x1   
POL(c32(x1)) = x1   
POL(c33(x1)) = x1   
POL(c34(x1)) = x1   
POL(c35(x1)) = x1   
POL(c36(x1)) = x1   
POL(c37(x1)) = x1   
POL(c38(x1)) = x1   
POL(c39(x1)) = x1   
POL(c4(x1)) = x1   
POL(c40(x1)) = x1   
POL(c41(x1)) = x1   
POL(c42(x1)) = x1   
POL(c43(x1)) = x1   
POL(c44(x1)) = x1   
POL(c45(x1)) = x1   
POL(c46(x1)) = x1   
POL(c47(x1)) = x1   
POL(c48(x1)) = x1   
POL(c49(x1)) = x1   
POL(c5(x1)) = x1   
POL(c50(x1)) = x1   
POL(c51(x1)) = x1   
POL(c52(x1)) = x1   
POL(c53(x1)) = x1   
POL(c54(x1)) = x1   
POL(c55(x1)) = x1   
POL(c56(x1)) = x1   
POL(c57(x1)) = x1   
POL(c58(x1)) = x1   
POL(c59(x1)) = x1   
POL(c6(x1)) = x1   
POL(c60(x1)) = x1   
POL(c61(x1)) = x1   
POL(c62(x1)) = x1   
POL(c63(x1)) = x1   
POL(c64(x1)) = x1   
POL(c65(x1)) = x1   
POL(c66(x1)) = x1   
POL(c67(x1)) = x1   
POL(c68(x1)) = x1   
POL(c69(x1)) = x1   
POL(c7(x1)) = x1   
POL(c70(x1)) = x1   
POL(c71(x1)) = x1   
POL(c72(x1)) = x1   
POL(c73(x1)) = x1   
POL(c74(x1)) = x1   
POL(c75(x1)) = x1   
POL(c76(x1)) = x1   
POL(c77(x1)) = x1   
POL(c78(x1)) = x1   
POL(c79(x1)) = x1   
POL(c8(x1)) = x1   
POL(c80(x1)) = x1   
POL(c81(x1)) = x1   
POL(c82(x1)) = x1   
POL(c83(x1)) = x1   
POL(c84(x1)) = x1   
POL(c85(x1)) = x1   
POL(c86(x1)) = x1   
POL(c87(x1)) = x1   
POL(c88(x1)) = x1   
POL(c89(x1)) = x1   
POL(c9(x1)) = x1   
POL(c90(x1)) = x1   
POL(c91(x1)) = x1   
POL(c92(x1)) = x1   
POL(c93(x1)) = x1   
POL(c94(x1)) = x1   
POL(c95(x1)) = x1   
POL(c96(x1)) = x1   
POL(c97(x1)) = x1   
POL(c98(x1)) = x1   
POL(c99(x1)) = x1   
POL(mark(x1)) = [1] + x1   
POL(nil) = 0   
POL(ok(x1)) = x1   
POL(proper(x1)) = 0   
POL(tt) = 0   

(18) Obligation:

Complexity Dependency Tuples Problem
Rules:

proper(tt) → ok(tt)
proper(nil) → ok(nil)
proper(0) → ok(0)
Tuples:

U151'(ok(z0), ok(z1), ok(z2)) → c2(U151'(z0, z1, z2))
U151'(mark(z0), z1, z2) → c3(U151'(z0, z1, z2))
U43'(ok(z0)) → c4(U43'(z0))
U43'(mark(z0)) → c5(U43'(z0))
U202'(mark(z0), z1) → c6(U202'(z0, z1))
U202'(ok(z0), ok(z1)) → c7(U202'(z0, z1))
U143'(ok(z0)) → c8(U143'(z0))
U143'(mark(z0)) → c9(U143'(z0))
U52'(ok(z0), ok(z1)) → c10(U52'(z0, z1))
U52'(mark(z0), z1) → c11(U52'(z0, z1))
U51'(mark(z0), z1, z2) → c12(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c13(U51'(z0, z1, z2))
U152'(ok(z0), ok(z1)) → c14(U152'(z0, z1))
U152'(mark(z0), z1) → c15(U152'(z0, z1))
PAIR(mark(z0), z1) → c16(PAIR(z0, z1))
PAIR(ok(z0), ok(z1)) → c17(PAIR(z0, z1))
PAIR(z0, mark(z1)) → c18(PAIR(z0, z1))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
SEL(mark(z0), z1) → c20(SEL(z0, z1))
SEL(z0, mark(z1)) → c21(SEL(z0, z1))
U181'(ok(z0), ok(z1)) → c22(U181'(z0, z1))
U181'(mark(z0), z1) → c23(U181'(z0, z1))
U111'(ok(z0), ok(z1)) → c27(U111'(z0, z1))
U111'(mark(z0), z1) → c28(U111'(z0, z1))
ISLNATKIND(ok(z0)) → c29(ISLNATKIND(z0))
U11'(mark(z0), z1, z2) → c30(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c31(U11'(z0, z1, z2))
U53'(mark(z0)) → c32(U53'(z0))
U53'(ok(z0)) → c33(U53'(z0))
U72'(mark(z0)) → c34(U72'(z0))
U72'(ok(z0)) → c35(U72'(z0))
U201'(ok(z0), ok(z1), ok(z2), ok(z3)) → c36(U201'(z0, z1, z2, z3))
U201'(mark(z0), z1, z2, z3) → c37(U201'(z0, z1, z2, z3))
SPLITAT(z0, mark(z1)) → c38(SPLITAT(z0, z1))
SPLITAT(mark(z0), z1) → c39(SPLITAT(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c40(SPLITAT(z0, z1))
U112'(mark(z0)) → c41(U112'(z0))
U112'(ok(z0)) → c42(U112'(z0))
HEAD(mark(z0)) → c43(HEAD(z0))
HEAD(ok(z0)) → c44(HEAD(z0))
U82'(mark(z0)) → c45(U82'(z0))
U82'(ok(z0)) → c46(U82'(z0))
FST(mark(z0)) → c47(FST(z0))
FST(ok(z0)) → c48(FST(z0))
AFTERNTH(ok(z0), ok(z1)) → c49(AFTERNTH(z0, z1))
AFTERNTH(z0, mark(z1)) → c50(AFTERNTH(z0, z1))
AFTERNTH(mark(z0), z1) → c51(AFTERNTH(z0, z1))
TAIL(mark(z0)) → c52(TAIL(z0))
TAIL(ok(z0)) → c53(TAIL(z0))
U71'(ok(z0), ok(z1)) → c54(U71'(z0, z1))
U71'(mark(z0), z1) → c55(U71'(z0, z1))
ISNATURALKIND(ok(z0)) → c56(ISNATURALKIND(z0))
U42'(ok(z0), ok(z1)) → c57(U42'(z0, z1))
U42'(mark(z0), z1) → c58(U42'(z0, z1))
U142'(mark(z0), z1) → c59(U142'(z0, z1))
U142'(ok(z0), ok(z1)) → c60(U142'(z0, z1))
U62'(ok(z0)) → c61(U62'(z0))
U62'(mark(z0)) → c62(U62'(z0))
U122'(mark(z0)) → c63(U122'(z0))
U122'(ok(z0)) → c64(U122'(z0))
U121'(mark(z0), z1) → c65(U121'(z0, z1))
U121'(ok(z0), ok(z1)) → c66(U121'(z0, z1))
U21'(mark(z0), z1) → c67(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c68(U21'(z0, z1))
U131'(ok(z0), ok(z1), ok(z2)) → c69(U131'(z0, z1, z2))
U131'(mark(z0), z1, z2) → c70(U131'(z0, z1, z2))
U133'(ok(z0)) → c71(U133'(z0))
U133'(mark(z0)) → c72(U133'(z0))
TAKE(z0, mark(z1)) → c73(TAKE(z0, z1))
TAKE(mark(z0), z1) → c74(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c75(TAKE(z0, z1))
AND(mark(z0), z1) → c76(AND(z0, z1))
AND(ok(z0), ok(z1)) → c77(AND(z0, z1))
NATSFROM(mark(z0)) → c78(NATSFROM(z0))
NATSFROM(ok(z0)) → c79(NATSFROM(z0))
U171'(mark(z0), z1, z2) → c80(U171'(z0, z1, z2))
U171'(ok(z0), ok(z1), ok(z2)) → c81(U171'(z0, z1, z2))
ISPLNATKIND(ok(z0)) → c82(ISPLNATKIND(z0))
S(mark(z0)) → c83(S(z0))
S(ok(z0)) → c84(S(z0))
U41'(mark(z0), z1, z2) → c85(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c86(U41'(z0, z1, z2))
CONS(mark(z0), z1) → c87(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c88(CONS(z0, z1))
U91'(ok(z0), ok(z1)) → c89(U91'(z0, z1))
U91'(mark(z0), z1) → c90(U91'(z0, z1))
ISNATURAL(ok(z0)) → c91(ISNATURAL(z0))
U221'(mark(z0), z1, z2) → c92(U221'(z0, z1, z2))
U221'(ok(z0), ok(z1), ok(z2)) → c93(U221'(z0, z1, z2))
U101'(ok(z0), ok(z1), ok(z2)) → c94(U101'(z0, z1, z2))
U101'(mark(z0), z1, z2) → c95(U101'(z0, z1, z2))
U61'(ok(z0), ok(z1)) → c96(U61'(z0, z1))
U61'(mark(z0), z1) → c97(U61'(z0, z1))
SND(ok(z0)) → c98(SND(z0))
SND(mark(z0)) → c99(SND(z0))
U102'(mark(z0), z1) → c100(U102'(z0, z1))
U102'(ok(z0), ok(z1)) → c101(U102'(z0, z1))
U161'(mark(z0), z1) → c102(U161'(z0, z1))
U161'(ok(z0), ok(z1)) → c103(U161'(z0, z1))
U92'(mark(z0)) → c104(U92'(z0))
U92'(ok(z0)) → c105(U92'(z0))
ISPLNAT(ok(z0)) → c106(ISPLNAT(z0))
U141'(mark(z0), z1, z2) → c107(U141'(z0, z1, z2))
U141'(ok(z0), ok(z1), ok(z2)) → c108(U141'(z0, z1, z2))
U31'(ok(z0), ok(z1)) → c109(U31'(z0, z1))
U31'(mark(z0), z1) → c110(U31'(z0, z1))
U191'(ok(z0), ok(z1)) → c111(U191'(z0, z1))
U191'(mark(z0), z1) → c112(U191'(z0, z1))
U103'(ok(z0)) → c113(U103'(z0))
U103'(mark(z0)) → c114(U103'(z0))
U211'(ok(z0), ok(z1)) → c115(U211'(z0, z1))
U211'(mark(z0), z1) → c116(U211'(z0, z1))
U153'(ok(z0)) → c117(U153'(z0))
U153'(mark(z0)) → c118(U153'(z0))
U132'(ok(z0), ok(z1)) → c119(U132'(z0, z1))
U132'(mark(z0), z1) → c120(U132'(z0, z1))
U81'(ok(z0), ok(z1)) → c121(U81'(z0, z1))
U81'(mark(z0), z1) → c122(U81'(z0, z1))
ISLNAT(ok(z0)) → c123(ISLNAT(z0))
TOP(mark(z0)) → c1(TOP(proper(z0)))
S tuples:

U151'(ok(z0), ok(z1), ok(z2)) → c2(U151'(z0, z1, z2))
U151'(mark(z0), z1, z2) → c3(U151'(z0, z1, z2))
U43'(ok(z0)) → c4(U43'(z0))
U43'(mark(z0)) → c5(U43'(z0))
U202'(mark(z0), z1) → c6(U202'(z0, z1))
U202'(ok(z0), ok(z1)) → c7(U202'(z0, z1))
U143'(ok(z0)) → c8(U143'(z0))
U143'(mark(z0)) → c9(U143'(z0))
U52'(ok(z0), ok(z1)) → c10(U52'(z0, z1))
U51'(mark(z0), z1, z2) → c12(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c13(U51'(z0, z1, z2))
U152'(ok(z0), ok(z1)) → c14(U152'(z0, z1))
U152'(mark(z0), z1) → c15(U152'(z0, z1))
PAIR(mark(z0), z1) → c16(PAIR(z0, z1))
PAIR(ok(z0), ok(z1)) → c17(PAIR(z0, z1))
PAIR(z0, mark(z1)) → c18(PAIR(z0, z1))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
SEL(mark(z0), z1) → c20(SEL(z0, z1))
SEL(z0, mark(z1)) → c21(SEL(z0, z1))
U181'(ok(z0), ok(z1)) → c22(U181'(z0, z1))
U181'(mark(z0), z1) → c23(U181'(z0, z1))
U111'(ok(z0), ok(z1)) → c27(U111'(z0, z1))
U111'(mark(z0), z1) → c28(U111'(