MAYBE

Problem:
 a__U101(tt(),N,XS) -> a__fst(a__splitAt(mark(N),mark(XS)))
 a__U11(tt(),N,XS) -> a__snd(a__splitAt(mark(N),mark(XS)))
 a__U21(tt(),X) -> mark(X)
 a__U31(tt(),N) -> mark(N)
 a__U41(tt(),N) -> cons(mark(N),natsFrom(s(N)))
 a__U51(tt(),N,XS) -> a__head(a__afterNth(mark(N),mark(XS)))
 a__U61(tt(),Y) -> mark(Y)
 a__U71(tt(),XS) -> pair(nil(),mark(XS))
 a__U81(tt(),N,X,XS) -> a__U82(a__splitAt(mark(N),mark(XS)),X)
 a__U82(pair(YS,ZS),X) -> pair(cons(mark(X),YS),mark(ZS))
 a__U91(tt(),XS) -> mark(XS)
 a__afterNth(N,XS) -> a__U11(a__and(a__isNatural(N),isLNat(XS)),N,XS)
 a__and(tt(),X) -> mark(X)
 a__fst(pair(X,Y)) -> a__U21(a__and(a__isLNat(X),isLNat(Y)),X)
 a__head(cons(N,XS)) -> a__U31(a__and(a__isNatural(N),isLNat(XS)),N)
 a__isLNat(nil()) -> tt()
 a__isLNat(afterNth(V1,V2)) -> a__and(a__isNatural(V1),isLNat(V2))
 a__isLNat(cons(V1,V2)) -> a__and(a__isNatural(V1),isLNat(V2))
 a__isLNat(fst(V1)) -> a__isPLNat(V1)
 a__isLNat(natsFrom(V1)) -> a__isNatural(V1)
 a__isLNat(snd(V1)) -> a__isPLNat(V1)
 a__isLNat(tail(V1)) -> a__isLNat(V1)
 a__isLNat(take(V1,V2)) -> a__and(a__isNatural(V1),isLNat(V2))
 a__isNatural(0()) -> tt()
 a__isNatural(head(V1)) -> a__isLNat(V1)
 a__isNatural(s(V1)) -> a__isNatural(V1)
 a__isNatural(sel(V1,V2)) -> a__and(a__isNatural(V1),isLNat(V2))
 a__isPLNat(pair(V1,V2)) -> a__and(a__isLNat(V1),isLNat(V2))
 a__isPLNat(splitAt(V1,V2)) -> a__and(a__isNatural(V1),isLNat(V2))
 a__natsFrom(N) -> a__U41(a__isNatural(N),N)
 a__sel(N,XS) -> a__U51(a__and(a__isNatural(N),isLNat(XS)),N,XS)
 a__snd(pair(X,Y)) -> a__U61(a__and(a__isLNat(X),isLNat(Y)),Y)
 a__splitAt(0(),XS) -> a__U71(a__isLNat(XS),XS)
 a__splitAt(s(N),cons(X,XS)) -> a__U81(a__and(a__isNatural(N),and(isNatural(X),isLNat(XS))),N,X,XS)
 a__tail(cons(N,XS)) -> a__U91(a__and(a__isNatural(N),isLNat(XS)),XS)
 a__take(N,XS) -> a__U101(a__and(a__isNatural(N),isLNat(XS)),N,XS)
 mark(U101(X1,X2,X3)) -> a__U101(mark(X1),X2,X3)
 mark(fst(X)) -> a__fst(mark(X))
 mark(splitAt(X1,X2)) -> a__splitAt(mark(X1),mark(X2))
 mark(U11(X1,X2,X3)) -> a__U11(mark(X1),X2,X3)
 mark(snd(X)) -> a__snd(mark(X))
 mark(U21(X1,X2)) -> a__U21(mark(X1),X2)
 mark(U31(X1,X2)) -> a__U31(mark(X1),X2)
 mark(U41(X1,X2)) -> a__U41(mark(X1),X2)
 mark(natsFrom(X)) -> a__natsFrom(mark(X))
 mark(U51(X1,X2,X3)) -> a__U51(mark(X1),X2,X3)
 mark(head(X)) -> a__head(mark(X))
 mark(afterNth(X1,X2)) -> a__afterNth(mark(X1),mark(X2))
 mark(U61(X1,X2)) -> a__U61(mark(X1),X2)
 mark(U71(X1,X2)) -> a__U71(mark(X1),X2)
 mark(U81(X1,X2,X3,X4)) -> a__U81(mark(X1),X2,X3,X4)
 mark(U82(X1,X2)) -> a__U82(mark(X1),X2)
 mark(U91(X1,X2)) -> a__U91(mark(X1),X2)
 mark(and(X1,X2)) -> a__and(mark(X1),X2)
 mark(isNatural(X)) -> a__isNatural(X)
 mark(isLNat(X)) -> a__isLNat(X)
 mark(isPLNat(X)) -> a__isPLNat(X)
 mark(tail(X)) -> a__tail(mark(X))
 mark(take(X1,X2)) -> a__take(mark(X1),mark(X2))
 mark(sel(X1,X2)) -> a__sel(mark(X1),mark(X2))
 mark(tt()) -> tt()
 mark(cons(X1,X2)) -> cons(mark(X1),X2)
 mark(s(X)) -> s(mark(X))
 mark(pair(X1,X2)) -> pair(mark(X1),mark(X2))
 mark(nil()) -> nil()
 mark(0()) -> 0()
 a__U101(X1,X2,X3) -> U101(X1,X2,X3)
 a__fst(X) -> fst(X)
 a__splitAt(X1,X2) -> splitAt(X1,X2)
 a__U11(X1,X2,X3) -> U11(X1,X2,X3)
 a__snd(X) -> snd(X)
 a__U21(X1,X2) -> U21(X1,X2)
 a__U31(X1,X2) -> U31(X1,X2)
 a__U41(X1,X2) -> U41(X1,X2)
 a__natsFrom(X) -> natsFrom(X)
 a__U51(X1,X2,X3) -> U51(X1,X2,X3)
 a__head(X) -> head(X)
 a__afterNth(X1,X2) -> afterNth(X1,X2)
 a__U61(X1,X2) -> U61(X1,X2)
 a__U71(X1,X2) -> U71(X1,X2)
 a__U81(X1,X2,X3,X4) -> U81(X1,X2,X3,X4)
 a__U82(X1,X2) -> U82(X1,X2)
 a__U91(X1,X2) -> U91(X1,X2)
 a__and(X1,X2) -> and(X1,X2)
 a__isNatural(X) -> isNatural(X)
 a__isLNat(X) -> isLNat(X)
 a__isPLNat(X) -> isPLNat(X)
 a__tail(X) -> tail(X)
 a__take(X1,X2) -> take(X1,X2)
 a__sel(X1,X2) -> sel(X1,X2)

Proof:
 DP Processor:
  DPs:
   a__U101#(tt(),N,XS) -> mark#(XS)
   a__U101#(tt(),N,XS) -> mark#(N)
   a__U101#(tt(),N,XS) -> a__splitAt#(mark(N),mark(XS))
   a__U101#(tt(),N,XS) -> a__fst#(a__splitAt(mark(N),mark(XS)))
   a__U11#(tt(),N,XS) -> mark#(XS)
   a__U11#(tt(),N,XS) -> mark#(N)
   a__U11#(tt(),N,XS) -> a__splitAt#(mark(N),mark(XS))
   a__U11#(tt(),N,XS) -> a__snd#(a__splitAt(mark(N),mark(XS)))
   a__U21#(tt(),X) -> mark#(X)
   a__U31#(tt(),N) -> mark#(N)
   a__U41#(tt(),N) -> mark#(N)
   a__U51#(tt(),N,XS) -> mark#(XS)
   a__U51#(tt(),N,XS) -> mark#(N)
   a__U51#(tt(),N,XS) -> a__afterNth#(mark(N),mark(XS))
   a__U51#(tt(),N,XS) -> a__head#(a__afterNth(mark(N),mark(XS)))
   a__U61#(tt(),Y) -> mark#(Y)
   a__U71#(tt(),XS) -> mark#(XS)
   a__U81#(tt(),N,X,XS) -> mark#(XS)
   a__U81#(tt(),N,X,XS) -> mark#(N)
   a__U81#(tt(),N,X,XS) -> a__splitAt#(mark(N),mark(XS))
   a__U81#(tt(),N,X,XS) -> a__U82#(a__splitAt(mark(N),mark(XS)),X)
   a__U82#(pair(YS,ZS),X) -> mark#(ZS)
   a__U82#(pair(YS,ZS),X) -> mark#(X)
   a__U91#(tt(),XS) -> mark#(XS)
   a__afterNth#(N,XS) -> a__isNatural#(N)
   a__afterNth#(N,XS) -> a__and#(a__isNatural(N),isLNat(XS))
   a__afterNth#(N,XS) -> a__U11#(a__and(a__isNatural(N),isLNat(XS)),N,XS)
   a__and#(tt(),X) -> mark#(X)
   a__fst#(pair(X,Y)) -> a__isLNat#(X)
   a__fst#(pair(X,Y)) -> a__and#(a__isLNat(X),isLNat(Y))
   a__fst#(pair(X,Y)) -> a__U21#(a__and(a__isLNat(X),isLNat(Y)),X)
   a__head#(cons(N,XS)) -> a__isNatural#(N)
   a__head#(cons(N,XS)) -> a__and#(a__isNatural(N),isLNat(XS))
   a__head#(cons(N,XS)) -> a__U31#(a__and(a__isNatural(N),isLNat(XS)),N)
   a__isLNat#(afterNth(V1,V2)) -> a__isNatural#(V1)
   a__isLNat#(afterNth(V1,V2)) -> a__and#(a__isNatural(V1),isLNat(V2))
   a__isLNat#(cons(V1,V2)) -> a__isNatural#(V1)
   a__isLNat#(cons(V1,V2)) -> a__and#(a__isNatural(V1),isLNat(V2))
   a__isLNat#(fst(V1)) -> a__isPLNat#(V1)
   a__isLNat#(natsFrom(V1)) -> a__isNatural#(V1)
   a__isLNat#(snd(V1)) -> a__isPLNat#(V1)
   a__isLNat#(tail(V1)) -> a__isLNat#(V1)
   a__isLNat#(take(V1,V2)) -> a__isNatural#(V1)
   a__isLNat#(take(V1,V2)) -> a__and#(a__isNatural(V1),isLNat(V2))
   a__isNatural#(head(V1)) -> a__isLNat#(V1)
   a__isNatural#(s(V1)) -> a__isNatural#(V1)
   a__isNatural#(sel(V1,V2)) -> a__isNatural#(V1)
   a__isNatural#(sel(V1,V2)) -> a__and#(a__isNatural(V1),isLNat(V2))
   a__isPLNat#(pair(V1,V2)) -> a__isLNat#(V1)
   a__isPLNat#(pair(V1,V2)) -> a__and#(a__isLNat(V1),isLNat(V2))
   a__isPLNat#(splitAt(V1,V2)) -> a__isNatural#(V1)
   a__isPLNat#(splitAt(V1,V2)) -> a__and#(a__isNatural(V1),isLNat(V2))
   a__natsFrom#(N) -> a__isNatural#(N)
   a__natsFrom#(N) -> a__U41#(a__isNatural(N),N)
   a__sel#(N,XS) -> a__isNatural#(N)
   a__sel#(N,XS) -> a__and#(a__isNatural(N),isLNat(XS))
   a__sel#(N,XS) -> a__U51#(a__and(a__isNatural(N),isLNat(XS)),N,XS)
   a__snd#(pair(X,Y)) -> a__isLNat#(X)
   a__snd#(pair(X,Y)) -> a__and#(a__isLNat(X),isLNat(Y))
   a__snd#(pair(X,Y)) -> a__U61#(a__and(a__isLNat(X),isLNat(Y)),Y)
   a__splitAt#(0(),XS) -> a__isLNat#(XS)
   a__splitAt#(0(),XS) -> a__U71#(a__isLNat(XS),XS)
   a__splitAt#(s(N),cons(X,XS)) -> a__isNatural#(N)
   a__splitAt#(s(N),cons(X,XS)) -> a__and#(a__isNatural(N),and(isNatural(X),isLNat(XS)))
   a__splitAt#(s(N),cons(X,XS)) ->
   a__U81#(a__and(a__isNatural(N),and(isNatural(X),isLNat(XS))),N,X,XS)
   a__tail#(cons(N,XS)) -> a__isNatural#(N)
   a__tail#(cons(N,XS)) -> a__and#(a__isNatural(N),isLNat(XS))
   a__tail#(cons(N,XS)) -> a__U91#(a__and(a__isNatural(N),isLNat(XS)),XS)
   a__take#(N,XS) -> a__isNatural#(N)
   a__take#(N,XS) -> a__and#(a__isNatural(N),isLNat(XS))
   a__take#(N,XS) -> a__U101#(a__and(a__isNatural(N),isLNat(XS)),N,XS)
   mark#(U101(X1,X2,X3)) -> mark#(X1)
   mark#(U101(X1,X2,X3)) -> a__U101#(mark(X1),X2,X3)
   mark#(fst(X)) -> mark#(X)
   mark#(fst(X)) -> a__fst#(mark(X))
   mark#(splitAt(X1,X2)) -> mark#(X2)
   mark#(splitAt(X1,X2)) -> mark#(X1)
   mark#(splitAt(X1,X2)) -> a__splitAt#(mark(X1),mark(X2))
   mark#(U11(X1,X2,X3)) -> mark#(X1)
   mark#(U11(X1,X2,X3)) -> a__U11#(mark(X1),X2,X3)
   mark#(snd(X)) -> mark#(X)
   mark#(snd(X)) -> a__snd#(mark(X))
   mark#(U21(X1,X2)) -> mark#(X1)
   mark#(U21(X1,X2)) -> a__U21#(mark(X1),X2)
   mark#(U31(X1,X2)) -> mark#(X1)
   mark#(U31(X1,X2)) -> a__U31#(mark(X1),X2)
   mark#(U41(X1,X2)) -> mark#(X1)
   mark#(U41(X1,X2)) -> a__U41#(mark(X1),X2)
   mark#(natsFrom(X)) -> mark#(X)
   mark#(natsFrom(X)) -> a__natsFrom#(mark(X))
   mark#(U51(X1,X2,X3)) -> mark#(X1)
   mark#(U51(X1,X2,X3)) -> a__U51#(mark(X1),X2,X3)
   mark#(head(X)) -> mark#(X)
   mark#(head(X)) -> a__head#(mark(X))
   mark#(afterNth(X1,X2)) -> mark#(X2)
   mark#(afterNth(X1,X2)) -> mark#(X1)
   mark#(afterNth(X1,X2)) -> a__afterNth#(mark(X1),mark(X2))
   mark#(U61(X1,X2)) -> mark#(X1)
   mark#(U61(X1,X2)) -> a__U61#(mark(X1),X2)
   mark#(U71(X1,X2)) -> mark#(X1)
   mark#(U71(X1,X2)) -> a__U71#(mark(X1),X2)
   mark#(U81(X1,X2,X3,X4)) -> mark#(X1)
   mark#(U81(X1,X2,X3,X4)) -> a__U81#(mark(X1),X2,X3,X4)
   mark#(U82(X1,X2)) -> mark#(X1)
   mark#(U82(X1,X2)) -> a__U82#(mark(X1),X2)
   mark#(U91(X1,X2)) -> mark#(X1)
   mark#(U91(X1,X2)) -> a__U91#(mark(X1),X2)
   mark#(and(X1,X2)) -> mark#(X1)
   mark#(and(X1,X2)) -> a__and#(mark(X1),X2)
   mark#(isNatural(X)) -> a__isNatural#(X)
   mark#(isLNat(X)) -> a__isLNat#(X)
   mark#(isPLNat(X)) -> a__isPLNat#(X)
   mark#(tail(X)) -> mark#(X)
   mark#(tail(X)) -> a__tail#(mark(X))
   mark#(take(X1,X2)) -> mark#(X2)
   mark#(take(X1,X2)) -> mark#(X1)
   mark#(take(X1,X2)) -> a__take#(mark(X1),mark(X2))
   mark#(sel(X1,X2)) -> mark#(X2)
   mark#(sel(X1,X2)) -> mark#(X1)
   mark#(sel(X1,X2)) -> a__sel#(mark(X1),mark(X2))
   mark#(cons(X1,X2)) -> mark#(X1)
   mark#(s(X)) -> mark#(X)
   mark#(pair(X1,X2)) -> mark#(X2)
   mark#(pair(X1,X2)) -> mark#(X1)
  TRS:
   a__U101(tt(),N,XS) -> a__fst(a__splitAt(mark(N),mark(XS)))
   a__U11(tt(),N,XS) -> a__snd(a__splitAt(mark(N),mark(XS)))
   a__U21(tt(),X) -> mark(X)
   a__U31(tt(),N) -> mark(N)
   a__U41(tt(),N) -> cons(mark(N),natsFrom(s(N)))
   a__U51(tt(),N,XS) -> a__head(a__afterNth(mark(N),mark(XS)))
   a__U61(tt(),Y) -> mark(Y)
   a__U71(tt(),XS) -> pair(nil(),mark(XS))
   a__U81(tt(),N,X,XS) -> a__U82(a__splitAt(mark(N),mark(XS)),X)
   a__U82(pair(YS,ZS),X) -> pair(cons(mark(X),YS),mark(ZS))
   a__U91(tt(),XS) -> mark(XS)
   a__afterNth(N,XS) -> a__U11(a__and(a__isNatural(N),isLNat(XS)),N,XS)
   a__and(tt(),X) -> mark(X)
   a__fst(pair(X,Y)) -> a__U21(a__and(a__isLNat(X),isLNat(Y)),X)
   a__head(cons(N,XS)) -> a__U31(a__and(a__isNatural(N),isLNat(XS)),N)
   a__isLNat(nil()) -> tt()
   a__isLNat(afterNth(V1,V2)) -> a__and(a__isNatural(V1),isLNat(V2))
   a__isLNat(cons(V1,V2)) -> a__and(a__isNatural(V1),isLNat(V2))
   a__isLNat(fst(V1)) -> a__isPLNat(V1)
   a__isLNat(natsFrom(V1)) -> a__isNatural(V1)
   a__isLNat(snd(V1)) -> a__isPLNat(V1)
   a__isLNat(tail(V1)) -> a__isLNat(V1)
   a__isLNat(take(V1,V2)) -> a__and(a__isNatural(V1),isLNat(V2))
   a__isNatural(0()) -> tt()
   a__isNatural(head(V1)) -> a__isLNat(V1)
   a__isNatural(s(V1)) -> a__isNatural(V1)
   a__isNatural(sel(V1,V2)) -> a__and(a__isNatural(V1),isLNat(V2))
   a__isPLNat(pair(V1,V2)) -> a__and(a__isLNat(V1),isLNat(V2))
   a__isPLNat(splitAt(V1,V2)) -> a__and(a__isNatural(V1),isLNat(V2))
   a__natsFrom(N) -> a__U41(a__isNatural(N),N)
   a__sel(N,XS) -> a__U51(a__and(a__isNatural(N),isLNat(XS)),N,XS)
   a__snd(pair(X,Y)) -> a__U61(a__and(a__isLNat(X),isLNat(Y)),Y)
   a__splitAt(0(),XS) -> a__U71(a__isLNat(XS),XS)
   a__splitAt(s(N),cons(X,XS)) ->
   a__U81(a__and(a__isNatural(N),and(isNatural(X),isLNat(XS))),N,X,XS)
   a__tail(cons(N,XS)) -> a__U91(a__and(a__isNatural(N),isLNat(XS)),XS)
   a__take(N,XS) -> a__U101(a__and(a__isNatural(N),isLNat(XS)),N,XS)
   mark(U101(X1,X2,X3)) -> a__U101(mark(X1),X2,X3)
   mark(fst(X)) -> a__fst(mark(X))
   mark(splitAt(X1,X2)) -> a__splitAt(mark(X1),mark(X2))
   mark(U11(X1,X2,X3)) -> a__U11(mark(X1),X2,X3)
   mark(snd(X)) -> a__snd(mark(X))
   mark(U21(X1,X2)) -> a__U21(mark(X1),X2)
   mark(U31(X1,X2)) -> a__U31(mark(X1),X2)
   mark(U41(X1,X2)) -> a__U41(mark(X1),X2)
   mark(natsFrom(X)) -> a__natsFrom(mark(X))
   mark(U51(X1,X2,X3)) -> a__U51(mark(X1),X2,X3)
   mark(head(X)) -> a__head(mark(X))
   mark(afterNth(X1,X2)) -> a__afterNth(mark(X1),mark(X2))
   mark(U61(X1,X2)) -> a__U61(mark(X1),X2)
   mark(U71(X1,X2)) -> a__U71(mark(X1),X2)
   mark(U81(X1,X2,X3,X4)) -> a__U81(mark(X1),X2,X3,X4)
   mark(U82(X1,X2)) -> a__U82(mark(X1),X2)
   mark(U91(X1,X2)) -> a__U91(mark(X1),X2)
   mark(and(X1,X2)) -> a__and(mark(X1),X2)
   mark(isNatural(X)) -> a__isNatural(X)
   mark(isLNat(X)) -> a__isLNat(X)
   mark(isPLNat(X)) -> a__isPLNat(X)
   mark(tail(X)) -> a__tail(mark(X))
   mark(take(X1,X2)) -> a__take(mark(X1),mark(X2))
   mark(sel(X1,X2)) -> a__sel(mark(X1),mark(X2))
   mark(tt()) -> tt()
   mark(cons(X1,X2)) -> cons(mark(X1),X2)
   mark(s(X)) -> s(mark(X))
   mark(pair(X1,X2)) -> pair(mark(X1),mark(X2))
   mark(nil()) -> nil()
   mark(0()) -> 0()
   a__U101(X1,X2,X3) -> U101(X1,X2,X3)
   a__fst(X) -> fst(X)
   a__splitAt(X1,X2) -> splitAt(X1,X2)
   a__U11(X1,X2,X3) -> U11(X1,X2,X3)
   a__snd(X) -> snd(X)
   a__U21(X1,X2) -> U21(X1,X2)
   a__U31(X1,X2) -> U31(X1,X2)
   a__U41(X1,X2) -> U41(X1,X2)
   a__natsFrom(X) -> natsFrom(X)
   a__U51(X1,X2,X3) -> U51(X1,X2,X3)
   a__head(X) -> head(X)
   a__afterNth(X1,X2) -> afterNth(X1,X2)
   a__U61(X1,X2) -> U61(X1,X2)
   a__U71(X1,X2) -> U71(X1,X2)
   a__U81(X1,X2,X3,X4) -> U81(X1,X2,X3,X4)
   a__U82(X1,X2) -> U82(X1,X2)
   a__U91(X1,X2) -> U91(X1,X2)
   a__and(X1,X2) -> and(X1,X2)
   a__isNatural(X) -> isNatural(X)
   a__isLNat(X) -> isLNat(X)
   a__isPLNat(X) -> isPLNat(X)
   a__tail(X) -> tail(X)
   a__take(X1,X2) -> take(X1,X2)
   a__sel(X1,X2) -> sel(X1,X2)
  TDG Processor:
   DPs:
    a__U101#(tt(),N,XS) -> mark#(XS)
    a__U101#(tt(),N,XS) -> mark#(N)
    a__U101#(tt(),N,XS) -> a__splitAt#(mark(N),mark(XS))
    a__U101#(tt(),N,XS) -> a__fst#(a__splitAt(mark(N),mark(XS)))
    a__U11#(tt(),N,XS) -> mark#(XS)
    a__U11#(tt(),N,XS) -> mark#(N)
    a__U11#(tt(),N,XS) -> a__splitAt#(mark(N),mark(XS))
    a__U11#(tt(),N,XS) -> a__snd#(a__splitAt(mark(N),mark(XS)))
    a__U21#(tt(),X) -> mark#(X)
    a__U31#(tt(),N) -> mark#(N)
    a__U41#(tt(),N) -> mark#(N)
    a__U51#(tt(),N,XS) -> mark#(XS)
    a__U51#(tt(),N,XS) -> mark#(N)
    a__U51#(tt(),N,XS) -> a__afterNth#(mark(N),mark(XS))
    a__U51#(tt(),N,XS) -> a__head#(a__afterNth(mark(N),mark(XS)))
    a__U61#(tt(),Y) -> mark#(Y)
    a__U71#(tt(),XS) -> mark#(XS)
    a__U81#(tt(),N,X,XS) -> mark#(XS)
    a__U81#(tt(),N,X,XS) -> mark#(N)
    a__U81#(tt(),N,X,XS) -> a__splitAt#(mark(N),mark(XS))
    a__U81#(tt(),N,X,XS) -> a__U82#(a__splitAt(mark(N),mark(XS)),X)
    a__U82#(pair(YS,ZS),X) -> mark#(ZS)
    a__U82#(pair(YS,ZS),X) -> mark#(X)
    a__U91#(tt(),XS) -> mark#(XS)
    a__afterNth#(N,XS) -> a__isNatural#(N)
    a__afterNth#(N,XS) -> a__and#(a__isNatural(N),isLNat(XS))
    a__afterNth#(N,XS) -> a__U11#(a__and(a__isNatural(N),isLNat(XS)),N,XS)
    a__and#(tt(),X) -> mark#(X)
    a__fst#(pair(X,Y)) -> a__isLNat#(X)
    a__fst#(pair(X,Y)) -> a__and#(a__isLNat(X),isLNat(Y))
    a__fst#(pair(X,Y)) -> a__U21#(a__and(a__isLNat(X),isLNat(Y)),X)
    a__head#(cons(N,XS)) -> a__isNatural#(N)
    a__head#(cons(N,XS)) -> a__and#(a__isNatural(N),isLNat(XS))
    a__head#(cons(N,XS)) -> a__U31#(a__and(a__isNatural(N),isLNat(XS)),N)
    a__isLNat#(afterNth(V1,V2)) -> a__isNatural#(V1)
    a__isLNat#(afterNth(V1,V2)) -> a__and#(a__isNatural(V1),isLNat(V2))
    a__isLNat#(cons(V1,V2)) -> a__isNatural#(V1)
    a__isLNat#(cons(V1,V2)) -> a__and#(a__isNatural(V1),isLNat(V2))
    a__isLNat#(fst(V1)) -> a__isPLNat#(V1)
    a__isLNat#(natsFrom(V1)) -> a__isNatural#(V1)
    a__isLNat#(snd(V1)) -> a__isPLNat#(V1)
    a__isLNat#(tail(V1)) -> a__isLNat#(V1)
    a__isLNat#(take(V1,V2)) -> a__isNatural#(V1)
    a__isLNat#(take(V1,V2)) -> a__and#(a__isNatural(V1),isLNat(V2))
    a__isNatural#(head(V1)) -> a__isLNat#(V1)
    a__isNatural#(s(V1)) -> a__isNatural#(V1)
    a__isNatural#(sel(V1,V2)) -> a__isNatural#(V1)
    a__isNatural#(sel(V1,V2)) -> a__and#(a__isNatural(V1),isLNat(V2))
    a__isPLNat#(pair(V1,V2)) -> a__isLNat#(V1)
    a__isPLNat#(pair(V1,V2)) -> a__and#(a__isLNat(V1),isLNat(V2))
    a__isPLNat#(splitAt(V1,V2)) -> a__isNatural#(V1)
    a__isPLNat#(splitAt(V1,V2)) -> a__and#(a__isNatural(V1),isLNat(V2))
    a__natsFrom#(N) -> a__isNatural#(N)
    a__natsFrom#(N) -> a__U41#(a__isNatural(N),N)
    a__sel#(N,XS) -> a__isNatural#(N)
    a__sel#(N,XS) -> a__and#(a__isNatural(N),isLNat(XS))
    a__sel#(N,XS) -> a__U51#(a__and(a__isNatural(N),isLNat(XS)),N,XS)
    a__snd#(pair(X,Y)) -> a__isLNat#(X)
    a__snd#(pair(X,Y)) -> a__and#(a__isLNat(X),isLNat(Y))
    a__snd#(pair(X,Y)) -> a__U61#(a__and(a__isLNat(X),isLNat(Y)),Y)
    a__splitAt#(0(),XS) -> a__isLNat#(XS)
    a__splitAt#(0(),XS) -> a__U71#(a__isLNat(XS),XS)
    a__splitAt#(s(N),cons(X,XS)) -> a__isNatural#(N)
    a__splitAt#(s(N),cons(X,XS)) -> a__and#(a__isNatural(N),and(isNatural(X),isLNat(XS)))
    a__splitAt#(s(N),cons(X,XS)) ->
    a__U81#(a__and(a__isNatural(N),and(isNatural(X),isLNat(XS))),N,X,XS)
    a__tail#(cons(N,XS)) -> a__isNatural#(N)
    a__tail#(cons(N,XS)) -> a__and#(a__isNatural(N),isLNat(XS))
    a__tail#(cons(N,XS)) -> a__U91#(a__and(a__isNatural(N),isLNat(XS)),XS)
    a__take#(N,XS) -> a__isNatural#(N)
    a__take#(N,XS) -> a__and#(a__isNatural(N),isLNat(XS))
    a__take#(N,XS) -> a__U101#(a__and(a__isNatural(N),isLNat(XS)),N,XS)
    mark#(U101(X1,X2,X3)) -> mark#(X1)
    mark#(U101(X1,X2,X3)) -> a__U101#(mark(X1),X2,X3)
    mark#(fst(X)) -> mark#(X)
    mark#(fst(X)) -> a__fst#(mark(X))
    mark#(splitAt(X1,X2)) -> mark#(X2)
    mark#(splitAt(X1,X2)) -> mark#(X1)
    mark#(splitAt(X1,X2)) -> a__splitAt#(mark(X1),mark(X2))
    mark#(U11(X1,X2,X3)) -> mark#(X1)
    mark#(U11(X1,X2,X3)) -> a__U11#(mark(X1),X2,X3)
    mark#(snd(X)) -> mark#(X)
    mark#(snd(X)) -> a__snd#(mark(X))
    mark#(U21(X1,X2)) -> mark#(X1)
    mark#(U21(X1,X2)) -> a__U21#(mark(X1),X2)
    mark#(U31(X1,X2)) -> mark#(X1)
    mark#(U31(X1,X2)) -> a__U31#(mark(X1),X2)
    mark#(U41(X1,X2)) -> mark#(X1)
    mark#(U41(X1,X2)) -> a__U41#(mark(X1),X2)
    mark#(natsFrom(X)) -> mark#(X)
    mark#(natsFrom(X)) -> a__natsFrom#(mark(X))
    mark#(U51(X1,X2,X3)) -> mark#(X1)
    mark#(U51(X1,X2,X3)) -> a__U51#(mark(X1),X2,X3)
    mark#(head(X)) -> mark#(X)
    mark#(head(X)) -> a__head#(mark(X))
    mark#(afterNth(X1,X2)) -> mark#(X2)
    mark#(afterNth(X1,X2)) -> mark#(X1)
    mark#(afterNth(X1,X2)) -> a__afterNth#(mark(X1),mark(X2))
    mark#(U61(X1,X2)) -> mark#(X1)
    mark#(U61(X1,X2)) -> a__U61#(mark(X1),X2)
    mark#(U71(X1,X2)) -> mark#(X1)
    mark#(U71(X1,X2)) -> a__U71#(mark(X1),X2)
    mark#(U81(X1,X2,X3,X4)) -> mark#(X1)
    mark#(U81(X1,X2,X3,X4)) -> a__U81#(mark(X1),X2,X3,X4)
    mark#(U82(X1,X2)) -> mark#(X1)
    mark#(U82(X1,X2)) -> a__U82#(mark(X1),X2)
    mark#(U91(X1,X2)) -> mark#(X1)
    mark#(U91(X1,X2)) -> a__U91#(mark(X1),X2)
    mark#(and(X1,X2)) -> mark#(X1)
    mark#(and(X1,X2)) -> a__and#(mark(X1),X2)
    mark#(isNatural(X)) -> a__isNatural#(X)
    mark#(isLNat(X)) -> a__isLNat#(X)
    mark#(isPLNat(X)) -> a__isPLNat#(X)
    mark#(tail(X)) -> mark#(X)
    mark#(tail(X)) -> a__tail#(mark(X))
    mark#(take(X1,X2)) -> mark#(X2)
    mark#(take(X1,X2)) -> mark#(X1)
    mark#(take(X1,X2)) -> a__take#(mark(X1),mark(X2))
    mark#(sel(X1,X2)) -> mark#(X2)
    mark#(sel(X1,X2)) -> mark#(X1)
    mark#(sel(X1,X2)) -> a__sel#(mark(X1),mark(X2))
    mark#(cons(X1,X2)) -> mark#(X1)
    mark#(s(X)) -> mark#(X)
    mark#(pair(X1,X2)) -> mark#(X2)
    mark#(pair(X1,X2)) -> mark#(X1)
   TRS:
    a__U101(tt(),N,XS) -> a__fst(a__splitAt(mark(N),mark(XS)))
    a__U11(tt(),N,XS) -> a__snd(a__splitAt(mark(N),mark(XS)))
    a__U21(tt(),X) -> mark(X)
    a__U31(tt(),N) -> mark(N)
    a__U41(tt(),N) -> cons(mark(N),natsFrom(s(N)))
    a__U51(tt(),N,XS) -> a__head(a__afterNth(mark(N),mark(XS)))
    a__U61(tt(),Y) -> mark(Y)
    a__U71(tt(),XS) -> pair(nil(),mark(XS))
    a__U81(tt(),N,X,XS) -> a__U82(a__splitAt(mark(N),mark(XS)),X)
    a__U82(pair(YS,ZS),X) -> pair(cons(mark(X),YS),mark(ZS))
    a__U91(tt(),XS) -> mark(XS)
    a__afterNth(N,XS) -> a__U11(a__and(a__isNatural(N),isLNat(XS)),N,XS)
    a__and(tt(),X) -> mark(X)
    a__fst(pair(X,Y)) -> a__U21(a__and(a__isLNat(X),isLNat(Y)),X)
    a__head(cons(N,XS)) -> a__U31(a__and(a__isNatural(N),isLNat(XS)),N)
    a__isLNat(nil()) -> tt()
    a__isLNat(afterNth(V1,V2)) -> a__and(a__isNatural(V1),isLNat(V2))
    a__isLNat(cons(V1,V2)) -> a__and(a__isNatural(V1),isLNat(V2))
    a__isLNat(fst(V1)) -> a__isPLNat(V1)
    a__isLNat(natsFrom(V1)) -> a__isNatural(V1)
    a__isLNat(snd(V1)) -> a__isPLNat(V1)
    a__isLNat(tail(V1)) -> a__isLNat(V1)
    a__isLNat(take(V1,V2)) -> a__and(a__isNatural(V1),isLNat(V2))
    a__isNatural(0()) -> tt()
    a__isNatural(head(V1)) -> a__isLNat(V1)
    a__isNatural(s(V1)) -> a__isNatural(V1)
    a__isNatural(sel(V1,V2)) -> a__and(a__isNatural(V1),isLNat(V2))
    a__isPLNat(pair(V1,V2)) -> a__and(a__isLNat(V1),isLNat(V2))
    a__isPLNat(splitAt(V1,V2)) -> a__and(a__isNatural(V1),isLNat(V2))
    a__natsFrom(N) -> a__U41(a__isNatural(N),N)
    a__sel(N,XS) -> a__U51(a__and(a__isNatural(N),isLNat(XS)),N,XS)
    a__snd(pair(X,Y)) -> a__U61(a__and(a__isLNat(X),isLNat(Y)),Y)
    a__splitAt(0(),XS) -> a__U71(a__isLNat(XS),XS)
    a__splitAt(s(N),cons(X,XS)) ->
    a__U81(a__and(a__isNatural(N),and(isNatural(X),isLNat(XS))),N,X,XS)
    a__tail(cons(N,XS)) -> a__U91(a__and(a__isNatural(N),isLNat(XS)),XS)
    a__take(N,XS) -> a__U101(a__and(a__isNatural(N),isLNat(XS)),N,XS)
    mark(U101(X1,X2,X3)) -> a__U101(mark(X1),X2,X3)
    mark(fst(X)) -> a__fst(mark(X))
    mark(splitAt(X1,X2)) -> a__splitAt(mark(X1),mark(X2))
    mark(U11(X1,X2,X3)) -> a__U11(mark(X1),X2,X3)
    mark(snd(X)) -> a__snd(mark(X))
    mark(U21(X1,X2)) -> a__U21(mark(X1),X2)
    mark(U31(X1,X2)) -> a__U31(mark(X1),X2)
    mark(U41(X1,X2)) -> a__U41(mark(X1),X2)
    mark(natsFrom(X)) -> a__natsFrom(mark(X))
    mark(U51(X1,X2,X3)) -> a__U51(mark(X1),X2,X3)
    mark(head(X)) -> a__head(mark(X))
    mark(afterNth(X1,X2)) -> a__afterNth(mark(X1),mark(X2))
    mark(U61(X1,X2)) -> a__U61(mark(X1),X2)
    mark(U71(X1,X2)) -> a__U71(mark(X1),X2)
    mark(U81(X1,X2,X3,X4)) -> a__U81(mark(X1),X2,X3,X4)
    mark(U82(X1,X2)) -> a__U82(mark(X1),X2)
    mark(U91(X1,X2)) -> a__U91(mark(X1),X2)
    mark(and(X1,X2)) -> a__and(mark(X1),X2)
    mark(isNatural(X)) -> a__isNatural(X)
    mark(isLNat(X)) -> a__isLNat(X)
    mark(isPLNat(X)) -> a__isPLNat(X)
    mark(tail(X)) -> a__tail(mark(X))
    mark(take(X1,X2)) -> a__take(mark(X1),mark(X2))
    mark(sel(X1,X2)) -> a__sel(mark(X1),mark(X2))
    mark(tt()) -> tt()
    mark(cons(X1,X2)) -> cons(mark(X1),X2)
    mark(s(X)) -> s(mark(X))
    mark(pair(X1,X2)) -> pair(mark(X1),mark(X2))
    mark(nil()) -> nil()
    mark(0()) -> 0()
    a__U101(X1,X2,X3) -> U101(X1,X2,X3)
    a__fst(X) -> fst(X)
    a__splitAt(X1,X2) -> splitAt(X1,X2)
    a__U11(X1,X2,X3) -> U11(X1,X2,X3)
    a__snd(X) -> snd(X)
    a__U21(X1,X2) -> U21(X1,X2)
    a__U31(X1,X2) -> U31(X1,X2)
    a__U41(X1,X2) -> U41(X1,X2)
    a__natsFrom(X) -> natsFrom(X)
    a__U51(X1,X2,X3) -> U51(X1,X2,X3)
    a__head(X) -> head(X)
    a__afterNth(X1,X2) -> afterNth(X1,X2)
    a__U61(X1,X2) -> U61(X1,X2)
    a__U71(X1,X2) -> U71(X1,X2)
    a__U81(X1,X2,X3,X4) -> U81(X1,X2,X3,X4)
    a__U82(X1,X2) -> U82(X1,X2)
    a__U91(X1,X2) -> U91(X1,X2)
    a__and(X1,X2) -> and(X1,X2)
    a__isNatural(X) -> isNatural(X)
    a__isLNat(X) -> isLNat(X)
    a__isPLNat(X) -> isPLNat(X)
    a__tail(X) -> tail(X)
    a__take(X1,X2) -> take(X1,X2)
    a__sel(X1,X2) -> sel(X1,X2)
   graph:
    ...
   Open