MAYBE

Problem:
 active(U11(tt(),N,XS)) -> mark(U12(tt(),N,XS))
 active(U12(tt(),N,XS)) -> mark(snd(splitAt(N,XS)))
 active(U21(tt(),X)) -> mark(U22(tt(),X))
 active(U22(tt(),X)) -> mark(X)
 active(U31(tt(),N)) -> mark(U32(tt(),N))
 active(U32(tt(),N)) -> mark(N)
 active(U41(tt(),N,XS)) -> mark(U42(tt(),N,XS))
 active(U42(tt(),N,XS)) -> mark(head(afterNth(N,XS)))
 active(U51(tt(),Y)) -> mark(U52(tt(),Y))
 active(U52(tt(),Y)) -> mark(Y)
 active(U61(tt(),N,X,XS)) -> mark(U62(tt(),N,X,XS))
 active(U62(tt(),N,X,XS)) -> mark(U63(tt(),N,X,XS))
 active(U63(tt(),N,X,XS)) -> mark(U64(splitAt(N,XS),X))
 active(U64(pair(YS,ZS),X)) -> mark(pair(cons(X,YS),ZS))
 active(U71(tt(),XS)) -> mark(U72(tt(),XS))
 active(U72(tt(),XS)) -> mark(XS)
 active(U81(tt(),N,XS)) -> mark(U82(tt(),N,XS))
 active(U82(tt(),N,XS)) -> mark(fst(splitAt(N,XS)))
 active(afterNth(N,XS)) -> mark(U11(tt(),N,XS))
 active(fst(pair(X,Y))) -> mark(U21(tt(),X))
 active(head(cons(N,XS))) -> mark(U31(tt(),N))
 active(natsFrom(N)) -> mark(cons(N,natsFrom(s(N))))
 active(sel(N,XS)) -> mark(U41(tt(),N,XS))
 active(snd(pair(X,Y))) -> mark(U51(tt(),Y))
 active(splitAt(0(),XS)) -> mark(pair(nil(),XS))
 active(splitAt(s(N),cons(X,XS))) -> mark(U61(tt(),N,X,XS))
 active(tail(cons(N,XS))) -> mark(U71(tt(),XS))
 active(take(N,XS)) -> mark(U81(tt(),N,XS))
 active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3)
 active(U12(X1,X2,X3)) -> U12(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(U21(X1,X2)) -> U21(active(X1),X2)
 active(U22(X1,X2)) -> U22(active(X1),X2)
 active(U31(X1,X2)) -> U31(active(X1),X2)
 active(U32(X1,X2)) -> U32(active(X1),X2)
 active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3)
 active(U42(X1,X2,X3)) -> U42(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(U51(X1,X2)) -> U51(active(X1),X2)
 active(U52(X1,X2)) -> U52(active(X1),X2)
 active(U61(X1,X2,X3,X4)) -> U61(active(X1),X2,X3,X4)
 active(U62(X1,X2,X3,X4)) -> U62(active(X1),X2,X3,X4)
 active(U63(X1,X2,X3,X4)) -> U63(active(X1),X2,X3,X4)
 active(U64(X1,X2)) -> U64(active(X1),X2)
 active(pair(X1,X2)) -> pair(active(X1),X2)
 active(pair(X1,X2)) -> pair(X1,active(X2))
 active(cons(X1,X2)) -> cons(active(X1),X2)
 active(U71(X1,X2)) -> U71(active(X1),X2)
 active(U72(X1,X2)) -> U72(active(X1),X2)
 active(U81(X1,X2,X3)) -> U81(active(X1),X2,X3)
 active(U82(X1,X2,X3)) -> U82(active(X1),X2,X3)
 active(fst(X)) -> fst(active(X))
 active(natsFrom(X)) -> natsFrom(active(X))
 active(s(X)) -> s(active(X))
 active(sel(X1,X2)) -> sel(active(X1),X2)
 active(sel(X1,X2)) -> sel(X1,active(X2))
 active(tail(X)) -> tail(active(X))
 active(take(X1,X2)) -> take(active(X1),X2)
 active(take(X1,X2)) -> take(X1,active(X2))
 U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3))
 U12(mark(X1),X2,X3) -> mark(U12(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))
 U21(mark(X1),X2) -> mark(U21(X1,X2))
 U22(mark(X1),X2) -> mark(U22(X1,X2))
 U31(mark(X1),X2) -> mark(U31(X1,X2))
 U32(mark(X1),X2) -> mark(U32(X1,X2))
 U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3))
 U42(mark(X1),X2,X3) -> mark(U42(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))
 U51(mark(X1),X2) -> mark(U51(X1,X2))
 U52(mark(X1),X2) -> mark(U52(X1,X2))
 U61(mark(X1),X2,X3,X4) -> mark(U61(X1,X2,X3,X4))
 U62(mark(X1),X2,X3,X4) -> mark(U62(X1,X2,X3,X4))
 U63(mark(X1),X2,X3,X4) -> mark(U63(X1,X2,X3,X4))
 U64(mark(X1),X2) -> mark(U64(X1,X2))
 pair(mark(X1),X2) -> mark(pair(X1,X2))
 pair(X1,mark(X2)) -> mark(pair(X1,X2))
 cons(mark(X1),X2) -> mark(cons(X1,X2))
 U71(mark(X1),X2) -> mark(U71(X1,X2))
 U72(mark(X1),X2) -> mark(U72(X1,X2))
 U81(mark(X1),X2,X3) -> mark(U81(X1,X2,X3))
 U82(mark(X1),X2,X3) -> mark(U82(X1,X2,X3))
 fst(mark(X)) -> mark(fst(X))
 natsFrom(mark(X)) -> mark(natsFrom(X))
 s(mark(X)) -> mark(s(X))
 sel(mark(X1),X2) -> mark(sel(X1,X2))
 sel(X1,mark(X2)) -> mark(sel(X1,X2))
 tail(mark(X)) -> mark(tail(X))
 take(mark(X1),X2) -> mark(take(X1,X2))
 take(X1,mark(X2)) -> mark(take(X1,X2))
 proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3))
 proper(tt()) -> ok(tt())
 proper(U12(X1,X2,X3)) -> U12(proper(X1),proper(X2),proper(X3))
 proper(snd(X)) -> snd(proper(X))
 proper(splitAt(X1,X2)) -> splitAt(proper(X1),proper(X2))
 proper(U21(X1,X2)) -> U21(proper(X1),proper(X2))
 proper(U22(X1,X2)) -> U22(proper(X1),proper(X2))
 proper(U31(X1,X2)) -> U31(proper(X1),proper(X2))
 proper(U32(X1,X2)) -> U32(proper(X1),proper(X2))
 proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3))
 proper(U42(X1,X2,X3)) -> U42(proper(X1),proper(X2),proper(X3))
 proper(head(X)) -> head(proper(X))
 proper(afterNth(X1,X2)) -> afterNth(proper(X1),proper(X2))
 proper(U51(X1,X2)) -> U51(proper(X1),proper(X2))
 proper(U52(X1,X2)) -> U52(proper(X1),proper(X2))
 proper(U61(X1,X2,X3,X4)) -> U61(proper(X1),proper(X2),proper(X3),proper(X4))
 proper(U62(X1,X2,X3,X4)) -> U62(proper(X1),proper(X2),proper(X3),proper(X4))
 proper(U63(X1,X2,X3,X4)) -> U63(proper(X1),proper(X2),proper(X3),proper(X4))
 proper(U64(X1,X2)) -> U64(proper(X1),proper(X2))
 proper(pair(X1,X2)) -> pair(proper(X1),proper(X2))
 proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
 proper(U71(X1,X2)) -> U71(proper(X1),proper(X2))
 proper(U72(X1,X2)) -> U72(proper(X1),proper(X2))
 proper(U81(X1,X2,X3)) -> U81(proper(X1),proper(X2),proper(X3))
 proper(U82(X1,X2,X3)) -> U82(proper(X1),proper(X2),proper(X3))
 proper(fst(X)) -> fst(proper(X))
 proper(natsFrom(X)) -> natsFrom(proper(X))
 proper(s(X)) -> s(proper(X))
 proper(sel(X1,X2)) -> sel(proper(X1),proper(X2))
 proper(0()) -> ok(0())
 proper(nil()) -> ok(nil())
 proper(tail(X)) -> tail(proper(X))
 proper(take(X1,X2)) -> take(proper(X1),proper(X2))
 U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3))
 U12(ok(X1),ok(X2),ok(X3)) -> ok(U12(X1,X2,X3))
 snd(ok(X)) -> ok(snd(X))
 splitAt(ok(X1),ok(X2)) -> ok(splitAt(X1,X2))
 U21(ok(X1),ok(X2)) -> ok(U21(X1,X2))
 U22(ok(X1),ok(X2)) -> ok(U22(X1,X2))
 U31(ok(X1),ok(X2)) -> ok(U31(X1,X2))
 U32(ok(X1),ok(X2)) -> ok(U32(X1,X2))
 U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3))
 U42(ok(X1),ok(X2),ok(X3)) -> ok(U42(X1,X2,X3))
 head(ok(X)) -> ok(head(X))
 afterNth(ok(X1),ok(X2)) -> ok(afterNth(X1,X2))
 U51(ok(X1),ok(X2)) -> ok(U51(X1,X2))
 U52(ok(X1),ok(X2)) -> ok(U52(X1,X2))
 U61(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U61(X1,X2,X3,X4))
 U62(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U62(X1,X2,X3,X4))
 U63(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U63(X1,X2,X3,X4))
 U64(ok(X1),ok(X2)) -> ok(U64(X1,X2))
 pair(ok(X1),ok(X2)) -> ok(pair(X1,X2))
 cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
 U71(ok(X1),ok(X2)) -> ok(U71(X1,X2))
 U72(ok(X1),ok(X2)) -> ok(U72(X1,X2))
 U81(ok(X1),ok(X2),ok(X3)) -> ok(U81(X1,X2,X3))
 U82(ok(X1),ok(X2),ok(X3)) -> ok(U82(X1,X2,X3))
 fst(ok(X)) -> ok(fst(X))
 natsFrom(ok(X)) -> ok(natsFrom(X))
 s(ok(X)) -> ok(s(X))
 sel(ok(X1),ok(X2)) -> ok(sel(X1,X2))
 tail(ok(X)) -> ok(tail(X))
 take(ok(X1),ok(X2)) -> ok(take(X1,X2))
 top(mark(X)) -> top(proper(X))
 top(ok(X)) -> top(active(X))

Proof:
 DP Processor:
  DPs:
   active#(U11(tt(),N,XS)) -> U12#(tt(),N,XS)
   active#(U12(tt(),N,XS)) -> splitAt#(N,XS)
   active#(U12(tt(),N,XS)) -> snd#(splitAt(N,XS))
   active#(U21(tt(),X)) -> U22#(tt(),X)
   active#(U31(tt(),N)) -> U32#(tt(),N)
   active#(U41(tt(),N,XS)) -> U42#(tt(),N,XS)
   active#(U42(tt(),N,XS)) -> afterNth#(N,XS)
   active#(U42(tt(),N,XS)) -> head#(afterNth(N,XS))
   active#(U51(tt(),Y)) -> U52#(tt(),Y)
   active#(U61(tt(),N,X,XS)) -> U62#(tt(),N,X,XS)
   active#(U62(tt(),N,X,XS)) -> U63#(tt(),N,X,XS)
   active#(U63(tt(),N,X,XS)) -> splitAt#(N,XS)
   active#(U63(tt(),N,X,XS)) -> U64#(splitAt(N,XS),X)
   active#(U64(pair(YS,ZS),X)) -> cons#(X,YS)
   active#(U64(pair(YS,ZS),X)) -> pair#(cons(X,YS),ZS)
   active#(U71(tt(),XS)) -> U72#(tt(),XS)
   active#(U81(tt(),N,XS)) -> U82#(tt(),N,XS)
   active#(U82(tt(),N,XS)) -> splitAt#(N,XS)
   active#(U82(tt(),N,XS)) -> fst#(splitAt(N,XS))
   active#(afterNth(N,XS)) -> U11#(tt(),N,XS)
   active#(fst(pair(X,Y))) -> U21#(tt(),X)
   active#(head(cons(N,XS))) -> U31#(tt(),N)
   active#(natsFrom(N)) -> s#(N)
   active#(natsFrom(N)) -> natsFrom#(s(N))
   active#(natsFrom(N)) -> cons#(N,natsFrom(s(N)))
   active#(sel(N,XS)) -> U41#(tt(),N,XS)
   active#(snd(pair(X,Y))) -> U51#(tt(),Y)
   active#(splitAt(0(),XS)) -> pair#(nil(),XS)
   active#(splitAt(s(N),cons(X,XS))) -> U61#(tt(),N,X,XS)
   active#(tail(cons(N,XS))) -> U71#(tt(),XS)
   active#(take(N,XS)) -> U81#(tt(),N,XS)
   active#(U11(X1,X2,X3)) -> active#(X1)
   active#(U11(X1,X2,X3)) -> U11#(active(X1),X2,X3)
   active#(U12(X1,X2,X3)) -> active#(X1)
   active#(U12(X1,X2,X3)) -> U12#(active(X1),X2,X3)
   active#(snd(X)) -> active#(X)
   active#(snd(X)) -> snd#(active(X))
   active#(splitAt(X1,X2)) -> active#(X1)
   active#(splitAt(X1,X2)) -> splitAt#(active(X1),X2)
   active#(splitAt(X1,X2)) -> active#(X2)
   active#(splitAt(X1,X2)) -> splitAt#(X1,active(X2))
   active#(U21(X1,X2)) -> active#(X1)
   active#(U21(X1,X2)) -> U21#(active(X1),X2)
   active#(U22(X1,X2)) -> active#(X1)
   active#(U22(X1,X2)) -> U22#(active(X1),X2)
   active#(U31(X1,X2)) -> active#(X1)
   active#(U31(X1,X2)) -> U31#(active(X1),X2)
   active#(U32(X1,X2)) -> active#(X1)
   active#(U32(X1,X2)) -> U32#(active(X1),X2)
   active#(U41(X1,X2,X3)) -> active#(X1)
   active#(U41(X1,X2,X3)) -> U41#(active(X1),X2,X3)
   active#(U42(X1,X2,X3)) -> active#(X1)
   active#(U42(X1,X2,X3)) -> U42#(active(X1),X2,X3)
   active#(head(X)) -> active#(X)
   active#(head(X)) -> head#(active(X))
   active#(afterNth(X1,X2)) -> active#(X1)
   active#(afterNth(X1,X2)) -> afterNth#(active(X1),X2)
   active#(afterNth(X1,X2)) -> active#(X2)
   active#(afterNth(X1,X2)) -> afterNth#(X1,active(X2))
   active#(U51(X1,X2)) -> active#(X1)
   active#(U51(X1,X2)) -> U51#(active(X1),X2)
   active#(U52(X1,X2)) -> active#(X1)
   active#(U52(X1,X2)) -> U52#(active(X1),X2)
   active#(U61(X1,X2,X3,X4)) -> active#(X1)
   active#(U61(X1,X2,X3,X4)) -> U61#(active(X1),X2,X3,X4)
   active#(U62(X1,X2,X3,X4)) -> active#(X1)
   active#(U62(X1,X2,X3,X4)) -> U62#(active(X1),X2,X3,X4)
   active#(U63(X1,X2,X3,X4)) -> active#(X1)
   active#(U63(X1,X2,X3,X4)) -> U63#(active(X1),X2,X3,X4)
   active#(U64(X1,X2)) -> active#(X1)
   active#(U64(X1,X2)) -> U64#(active(X1),X2)
   active#(pair(X1,X2)) -> active#(X1)
   active#(pair(X1,X2)) -> pair#(active(X1),X2)
   active#(pair(X1,X2)) -> active#(X2)
   active#(pair(X1,X2)) -> pair#(X1,active(X2))
   active#(cons(X1,X2)) -> active#(X1)
   active#(cons(X1,X2)) -> cons#(active(X1),X2)
   active#(U71(X1,X2)) -> active#(X1)
   active#(U71(X1,X2)) -> U71#(active(X1),X2)
   active#(U72(X1,X2)) -> active#(X1)
   active#(U72(X1,X2)) -> U72#(active(X1),X2)
   active#(U81(X1,X2,X3)) -> active#(X1)
   active#(U81(X1,X2,X3)) -> U81#(active(X1),X2,X3)
   active#(U82(X1,X2,X3)) -> active#(X1)
   active#(U82(X1,X2,X3)) -> U82#(active(X1),X2,X3)
   active#(fst(X)) -> active#(X)
   active#(fst(X)) -> fst#(active(X))
   active#(natsFrom(X)) -> active#(X)
   active#(natsFrom(X)) -> natsFrom#(active(X))
   active#(s(X)) -> active#(X)
   active#(s(X)) -> s#(active(X))
   active#(sel(X1,X2)) -> active#(X1)
   active#(sel(X1,X2)) -> sel#(active(X1),X2)
   active#(sel(X1,X2)) -> active#(X2)
   active#(sel(X1,X2)) -> sel#(X1,active(X2))
   active#(tail(X)) -> active#(X)
   active#(tail(X)) -> tail#(active(X))
   active#(take(X1,X2)) -> active#(X1)
   active#(take(X1,X2)) -> take#(active(X1),X2)
   active#(take(X1,X2)) -> active#(X2)
   active#(take(X1,X2)) -> take#(X1,active(X2))
   U11#(mark(X1),X2,X3) -> U11#(X1,X2,X3)
   U12#(mark(X1),X2,X3) -> U12#(X1,X2,X3)
   snd#(mark(X)) -> snd#(X)
   splitAt#(mark(X1),X2) -> splitAt#(X1,X2)
   splitAt#(X1,mark(X2)) -> splitAt#(X1,X2)
   U21#(mark(X1),X2) -> U21#(X1,X2)
   U22#(mark(X1),X2) -> U22#(X1,X2)
   U31#(mark(X1),X2) -> U31#(X1,X2)
   U32#(mark(X1),X2) -> U32#(X1,X2)
   U41#(mark(X1),X2,X3) -> U41#(X1,X2,X3)
   U42#(mark(X1),X2,X3) -> U42#(X1,X2,X3)
   head#(mark(X)) -> head#(X)
   afterNth#(mark(X1),X2) -> afterNth#(X1,X2)
   afterNth#(X1,mark(X2)) -> afterNth#(X1,X2)
   U51#(mark(X1),X2) -> U51#(X1,X2)
   U52#(mark(X1),X2) -> U52#(X1,X2)
   U61#(mark(X1),X2,X3,X4) -> U61#(X1,X2,X3,X4)
   U62#(mark(X1),X2,X3,X4) -> U62#(X1,X2,X3,X4)
   U63#(mark(X1),X2,X3,X4) -> U63#(X1,X2,X3,X4)
   U64#(mark(X1),X2) -> U64#(X1,X2)
   pair#(mark(X1),X2) -> pair#(X1,X2)
   pair#(X1,mark(X2)) -> pair#(X1,X2)
   cons#(mark(X1),X2) -> cons#(X1,X2)
   U71#(mark(X1),X2) -> U71#(X1,X2)
   U72#(mark(X1),X2) -> U72#(X1,X2)
   U81#(mark(X1),X2,X3) -> U81#(X1,X2,X3)
   U82#(mark(X1),X2,X3) -> U82#(X1,X2,X3)
   fst#(mark(X)) -> fst#(X)
   natsFrom#(mark(X)) -> natsFrom#(X)
   s#(mark(X)) -> s#(X)
   sel#(mark(X1),X2) -> sel#(X1,X2)
   sel#(X1,mark(X2)) -> sel#(X1,X2)
   tail#(mark(X)) -> tail#(X)
   take#(mark(X1),X2) -> take#(X1,X2)
   take#(X1,mark(X2)) -> take#(X1,X2)
   proper#(U11(X1,X2,X3)) -> proper#(X3)
   proper#(U11(X1,X2,X3)) -> proper#(X2)
   proper#(U11(X1,X2,X3)) -> proper#(X1)
   proper#(U11(X1,X2,X3)) -> U11#(proper(X1),proper(X2),proper(X3))
   proper#(U12(X1,X2,X3)) -> proper#(X3)
   proper#(U12(X1,X2,X3)) -> proper#(X2)
   proper#(U12(X1,X2,X3)) -> proper#(X1)
   proper#(U12(X1,X2,X3)) -> U12#(proper(X1),proper(X2),proper(X3))
   proper#(snd(X)) -> proper#(X)
   proper#(snd(X)) -> snd#(proper(X))
   proper#(splitAt(X1,X2)) -> proper#(X2)
   proper#(splitAt(X1,X2)) -> proper#(X1)
   proper#(splitAt(X1,X2)) -> splitAt#(proper(X1),proper(X2))
   proper#(U21(X1,X2)) -> proper#(X2)
   proper#(U21(X1,X2)) -> proper#(X1)
   proper#(U21(X1,X2)) -> U21#(proper(X1),proper(X2))
   proper#(U22(X1,X2)) -> proper#(X2)
   proper#(U22(X1,X2)) -> proper#(X1)
   proper#(U22(X1,X2)) -> U22#(proper(X1),proper(X2))
   proper#(U31(X1,X2)) -> proper#(X2)
   proper#(U31(X1,X2)) -> proper#(X1)
   proper#(U31(X1,X2)) -> U31#(proper(X1),proper(X2))
   proper#(U32(X1,X2)) -> proper#(X2)
   proper#(U32(X1,X2)) -> proper#(X1)
   proper#(U32(X1,X2)) -> U32#(proper(X1),proper(X2))
   proper#(U41(X1,X2,X3)) -> proper#(X3)
   proper#(U41(X1,X2,X3)) -> proper#(X2)
   proper#(U41(X1,X2,X3)) -> proper#(X1)
   proper#(U41(X1,X2,X3)) -> U41#(proper(X1),proper(X2),proper(X3))
   proper#(U42(X1,X2,X3)) -> proper#(X3)
   proper#(U42(X1,X2,X3)) -> proper#(X2)
   proper#(U42(X1,X2,X3)) -> proper#(X1)
   proper#(U42(X1,X2,X3)) -> U42#(proper(X1),proper(X2),proper(X3))
   proper#(head(X)) -> proper#(X)
   proper#(head(X)) -> head#(proper(X))
   proper#(afterNth(X1,X2)) -> proper#(X2)
   proper#(afterNth(X1,X2)) -> proper#(X1)
   proper#(afterNth(X1,X2)) -> afterNth#(proper(X1),proper(X2))
   proper#(U51(X1,X2)) -> proper#(X2)
   proper#(U51(X1,X2)) -> proper#(X1)
   proper#(U51(X1,X2)) -> U51#(proper(X1),proper(X2))
   proper#(U52(X1,X2)) -> proper#(X2)
   proper#(U52(X1,X2)) -> proper#(X1)
   proper#(U52(X1,X2)) -> U52#(proper(X1),proper(X2))
   proper#(U61(X1,X2,X3,X4)) -> proper#(X4)
   proper#(U61(X1,X2,X3,X4)) -> proper#(X3)
   proper#(U61(X1,X2,X3,X4)) -> proper#(X2)
   proper#(U61(X1,X2,X3,X4)) -> proper#(X1)
   proper#(U61(X1,X2,X3,X4)) -> U61#(proper(X1),proper(X2),proper(X3),proper(X4))
   proper#(U62(X1,X2,X3,X4)) -> proper#(X4)
   proper#(U62(X1,X2,X3,X4)) -> proper#(X3)
   proper#(U62(X1,X2,X3,X4)) -> proper#(X2)
   proper#(U62(X1,X2,X3,X4)) -> proper#(X1)
   proper#(U62(X1,X2,X3,X4)) -> U62#(proper(X1),proper(X2),proper(X3),proper(X4))
   proper#(U63(X1,X2,X3,X4)) -> proper#(X4)
   proper#(U63(X1,X2,X3,X4)) -> proper#(X3)
   proper#(U63(X1,X2,X3,X4)) -> proper#(X2)
   proper#(U63(X1,X2,X3,X4)) -> proper#(X1)
   proper#(U63(X1,X2,X3,X4)) -> U63#(proper(X1),proper(X2),proper(X3),proper(X4))
   proper#(U64(X1,X2)) -> proper#(X2)
   proper#(U64(X1,X2)) -> proper#(X1)
   proper#(U64(X1,X2)) -> U64#(proper(X1),proper(X2))
   proper#(pair(X1,X2)) -> proper#(X2)
   proper#(pair(X1,X2)) -> proper#(X1)
   proper#(pair(X1,X2)) -> pair#(proper(X1),proper(X2))
   proper#(cons(X1,X2)) -> proper#(X2)
   proper#(cons(X1,X2)) -> proper#(X1)
   proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2))
   proper#(U71(X1,X2)) -> proper#(X2)
   proper#(U71(X1,X2)) -> proper#(X1)
   proper#(U71(X1,X2)) -> U71#(proper(X1),proper(X2))
   proper#(U72(X1,X2)) -> proper#(X2)
   proper#(U72(X1,X2)) -> proper#(X1)
   proper#(U72(X1,X2)) -> U72#(proper(X1),proper(X2))
   proper#(U81(X1,X2,X3)) -> proper#(X3)
   proper#(U81(X1,X2,X3)) -> proper#(X2)
   proper#(U81(X1,X2,X3)) -> proper#(X1)
   proper#(U81(X1,X2,X3)) -> U81#(proper(X1),proper(X2),proper(X3))
   proper#(U82(X1,X2,X3)) -> proper#(X3)
   proper#(U82(X1,X2,X3)) -> proper#(X2)
   proper#(U82(X1,X2,X3)) -> proper#(X1)
   proper#(U82(X1,X2,X3)) -> U82#(proper(X1),proper(X2),proper(X3))
   proper#(fst(X)) -> proper#(X)
   proper#(fst(X)) -> fst#(proper(X))
   proper#(natsFrom(X)) -> proper#(X)
   proper#(natsFrom(X)) -> natsFrom#(proper(X))
   proper#(s(X)) -> proper#(X)
   proper#(s(X)) -> s#(proper(X))
   proper#(sel(X1,X2)) -> proper#(X2)
   proper#(sel(X1,X2)) -> proper#(X1)
   proper#(sel(X1,X2)) -> sel#(proper(X1),proper(X2))
   proper#(tail(X)) -> proper#(X)
   proper#(tail(X)) -> tail#(proper(X))
   proper#(take(X1,X2)) -> proper#(X2)
   proper#(take(X1,X2)) -> proper#(X1)
   proper#(take(X1,X2)) -> take#(proper(X1),proper(X2))
   U11#(ok(X1),ok(X2),ok(X3)) -> U11#(X1,X2,X3)
   U12#(ok(X1),ok(X2),ok(X3)) -> U12#(X1,X2,X3)
   snd#(ok(X)) -> snd#(X)
   splitAt#(ok(X1),ok(X2)) -> splitAt#(X1,X2)
   U21#(ok(X1),ok(X2)) -> U21#(X1,X2)
   U22#(ok(X1),ok(X2)) -> U22#(X1,X2)
   U31#(ok(X1),ok(X2)) -> U31#(X1,X2)
   U32#(ok(X1),ok(X2)) -> U32#(X1,X2)
   U41#(ok(X1),ok(X2),ok(X3)) -> U41#(X1,X2,X3)
   U42#(ok(X1),ok(X2),ok(X3)) -> U42#(X1,X2,X3)
   head#(ok(X)) -> head#(X)
   afterNth#(ok(X1),ok(X2)) -> afterNth#(X1,X2)
   U51#(ok(X1),ok(X2)) -> U51#(X1,X2)
   U52#(ok(X1),ok(X2)) -> U52#(X1,X2)
   U61#(ok(X1),ok(X2),ok(X3),ok(X4)) -> U61#(X1,X2,X3,X4)
   U62#(ok(X1),ok(X2),ok(X3),ok(X4)) -> U62#(X1,X2,X3,X4)
   U63#(ok(X1),ok(X2),ok(X3),ok(X4)) -> U63#(X1,X2,X3,X4)
   U64#(ok(X1),ok(X2)) -> U64#(X1,X2)
   pair#(ok(X1),ok(X2)) -> pair#(X1,X2)
   cons#(ok(X1),ok(X2)) -> cons#(X1,X2)
   U71#(ok(X1),ok(X2)) -> U71#(X1,X2)
   U72#(ok(X1),ok(X2)) -> U72#(X1,X2)
   U81#(ok(X1),ok(X2),ok(X3)) -> U81#(X1,X2,X3)
   U82#(ok(X1),ok(X2),ok(X3)) -> U82#(X1,X2,X3)
   fst#(ok(X)) -> fst#(X)
   natsFrom#(ok(X)) -> natsFrom#(X)
   s#(ok(X)) -> s#(X)
   sel#(ok(X1),ok(X2)) -> sel#(X1,X2)
   tail#(ok(X)) -> tail#(X)
   take#(ok(X1),ok(X2)) -> take#(X1,X2)
   top#(mark(X)) -> proper#(X)
   top#(mark(X)) -> top#(proper(X))
   top#(ok(X)) -> active#(X)
   top#(ok(X)) -> top#(active(X))
  TRS:
   active(U11(tt(),N,XS)) -> mark(U12(tt(),N,XS))
   active(U12(tt(),N,XS)) -> mark(snd(splitAt(N,XS)))
   active(U21(tt(),X)) -> mark(U22(tt(),X))
   active(U22(tt(),X)) -> mark(X)
   active(U31(tt(),N)) -> mark(U32(tt(),N))
   active(U32(tt(),N)) -> mark(N)
   active(U41(tt(),N,XS)) -> mark(U42(tt(),N,XS))
   active(U42(tt(),N,XS)) -> mark(head(afterNth(N,XS)))
   active(U51(tt(),Y)) -> mark(U52(tt(),Y))
   active(U52(tt(),Y)) -> mark(Y)
   active(U61(tt(),N,X,XS)) -> mark(U62(tt(),N,X,XS))
   active(U62(tt(),N,X,XS)) -> mark(U63(tt(),N,X,XS))
   active(U63(tt(),N,X,XS)) -> mark(U64(splitAt(N,XS),X))
   active(U64(pair(YS,ZS),X)) -> mark(pair(cons(X,YS),ZS))
   active(U71(tt(),XS)) -> mark(U72(tt(),XS))
   active(U72(tt(),XS)) -> mark(XS)
   active(U81(tt(),N,XS)) -> mark(U82(tt(),N,XS))
   active(U82(tt(),N,XS)) -> mark(fst(splitAt(N,XS)))
   active(afterNth(N,XS)) -> mark(U11(tt(),N,XS))
   active(fst(pair(X,Y))) -> mark(U21(tt(),X))
   active(head(cons(N,XS))) -> mark(U31(tt(),N))
   active(natsFrom(N)) -> mark(cons(N,natsFrom(s(N))))
   active(sel(N,XS)) -> mark(U41(tt(),N,XS))
   active(snd(pair(X,Y))) -> mark(U51(tt(),Y))
   active(splitAt(0(),XS)) -> mark(pair(nil(),XS))
   active(splitAt(s(N),cons(X,XS))) -> mark(U61(tt(),N,X,XS))
   active(tail(cons(N,XS))) -> mark(U71(tt(),XS))
   active(take(N,XS)) -> mark(U81(tt(),N,XS))
   active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3)
   active(U12(X1,X2,X3)) -> U12(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(U21(X1,X2)) -> U21(active(X1),X2)
   active(U22(X1,X2)) -> U22(active(X1),X2)
   active(U31(X1,X2)) -> U31(active(X1),X2)
   active(U32(X1,X2)) -> U32(active(X1),X2)
   active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3)
   active(U42(X1,X2,X3)) -> U42(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(U51(X1,X2)) -> U51(active(X1),X2)
   active(U52(X1,X2)) -> U52(active(X1),X2)
   active(U61(X1,X2,X3,X4)) -> U61(active(X1),X2,X3,X4)
   active(U62(X1,X2,X3,X4)) -> U62(active(X1),X2,X3,X4)
   active(U63(X1,X2,X3,X4)) -> U63(active(X1),X2,X3,X4)
   active(U64(X1,X2)) -> U64(active(X1),X2)
   active(pair(X1,X2)) -> pair(active(X1),X2)
   active(pair(X1,X2)) -> pair(X1,active(X2))
   active(cons(X1,X2)) -> cons(active(X1),X2)
   active(U71(X1,X2)) -> U71(active(X1),X2)
   active(U72(X1,X2)) -> U72(active(X1),X2)
   active(U81(X1,X2,X3)) -> U81(active(X1),X2,X3)
   active(U82(X1,X2,X3)) -> U82(active(X1),X2,X3)
   active(fst(X)) -> fst(active(X))
   active(natsFrom(X)) -> natsFrom(active(X))
   active(s(X)) -> s(active(X))
   active(sel(X1,X2)) -> sel(active(X1),X2)
   active(sel(X1,X2)) -> sel(X1,active(X2))
   active(tail(X)) -> tail(active(X))
   active(take(X1,X2)) -> take(active(X1),X2)
   active(take(X1,X2)) -> take(X1,active(X2))
   U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3))
   U12(mark(X1),X2,X3) -> mark(U12(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))
   U21(mark(X1),X2) -> mark(U21(X1,X2))
   U22(mark(X1),X2) -> mark(U22(X1,X2))
   U31(mark(X1),X2) -> mark(U31(X1,X2))
   U32(mark(X1),X2) -> mark(U32(X1,X2))
   U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3))
   U42(mark(X1),X2,X3) -> mark(U42(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))
   U51(mark(X1),X2) -> mark(U51(X1,X2))
   U52(mark(X1),X2) -> mark(U52(X1,X2))
   U61(mark(X1),X2,X3,X4) -> mark(U61(X1,X2,X3,X4))
   U62(mark(X1),X2,X3,X4) -> mark(U62(X1,X2,X3,X4))
   U63(mark(X1),X2,X3,X4) -> mark(U63(X1,X2,X3,X4))
   U64(mark(X1),X2) -> mark(U64(X1,X2))
   pair(mark(X1),X2) -> mark(pair(X1,X2))
   pair(X1,mark(X2)) -> mark(pair(X1,X2))
   cons(mark(X1),X2) -> mark(cons(X1,X2))
   U71(mark(X1),X2) -> mark(U71(X1,X2))
   U72(mark(X1),X2) -> mark(U72(X1,X2))
   U81(mark(X1),X2,X3) -> mark(U81(X1,X2,X3))
   U82(mark(X1),X2,X3) -> mark(U82(X1,X2,X3))
   fst(mark(X)) -> mark(fst(X))
   natsFrom(mark(X)) -> mark(natsFrom(X))
   s(mark(X)) -> mark(s(X))
   sel(mark(X1),X2) -> mark(sel(X1,X2))
   sel(X1,mark(X2)) -> mark(sel(X1,X2))
   tail(mark(X)) -> mark(tail(X))
   take(mark(X1),X2) -> mark(take(X1,X2))
   take(X1,mark(X2)) -> mark(take(X1,X2))
   proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3))
   proper(tt()) -> ok(tt())
   proper(U12(X1,X2,X3)) -> U12(proper(X1),proper(X2),proper(X3))
   proper(snd(X)) -> snd(proper(X))
   proper(splitAt(X1,X2)) -> splitAt(proper(X1),proper(X2))
   proper(U21(X1,X2)) -> U21(proper(X1),proper(X2))
   proper(U22(X1,X2)) -> U22(proper(X1),proper(X2))
   proper(U31(X1,X2)) -> U31(proper(X1),proper(X2))
   proper(U32(X1,X2)) -> U32(proper(X1),proper(X2))
   proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3))
   proper(U42(X1,X2,X3)) -> U42(proper(X1),proper(X2),proper(X3))
   proper(head(X)) -> head(proper(X))
   proper(afterNth(X1,X2)) -> afterNth(proper(X1),proper(X2))
   proper(U51(X1,X2)) -> U51(proper(X1),proper(X2))
   proper(U52(X1,X2)) -> U52(proper(X1),proper(X2))
   proper(U61(X1,X2,X3,X4)) -> U61(proper(X1),proper(X2),proper(X3),proper(X4))
   proper(U62(X1,X2,X3,X4)) -> U62(proper(X1),proper(X2),proper(X3),proper(X4))
   proper(U63(X1,X2,X3,X4)) -> U63(proper(X1),proper(X2),proper(X3),proper(X4))
   proper(U64(X1,X2)) -> U64(proper(X1),proper(X2))
   proper(pair(X1,X2)) -> pair(proper(X1),proper(X2))
   proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
   proper(U71(X1,X2)) -> U71(proper(X1),proper(X2))
   proper(U72(X1,X2)) -> U72(proper(X1),proper(X2))
   proper(U81(X1,X2,X3)) -> U81(proper(X1),proper(X2),proper(X3))
   proper(U82(X1,X2,X3)) -> U82(proper(X1),proper(X2),proper(X3))
   proper(fst(X)) -> fst(proper(X))
   proper(natsFrom(X)) -> natsFrom(proper(X))
   proper(s(X)) -> s(proper(X))
   proper(sel(X1,X2)) -> sel(proper(X1),proper(X2))
   proper(0()) -> ok(0())
   proper(nil()) -> ok(nil())
   proper(tail(X)) -> tail(proper(X))
   proper(take(X1,X2)) -> take(proper(X1),proper(X2))
   U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3))
   U12(ok(X1),ok(X2),ok(X3)) -> ok(U12(X1,X2,X3))
   snd(ok(X)) -> ok(snd(X))
   splitAt(ok(X1),ok(X2)) -> ok(splitAt(X1,X2))
   U21(ok(X1),ok(X2)) -> ok(U21(X1,X2))
   U22(ok(X1),ok(X2)) -> ok(U22(X1,X2))
   U31(ok(X1),ok(X2)) -> ok(U31(X1,X2))
   U32(ok(X1),ok(X2)) -> ok(U32(X1,X2))
   U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3))
   U42(ok(X1),ok(X2),ok(X3)) -> ok(U42(X1,X2,X3))
   head(ok(X)) -> ok(head(X))
   afterNth(ok(X1),ok(X2)) -> ok(afterNth(X1,X2))
   U51(ok(X1),ok(X2)) -> ok(U51(X1,X2))
   U52(ok(X1),ok(X2)) -> ok(U52(X1,X2))
   U61(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U61(X1,X2,X3,X4))
   U62(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U62(X1,X2,X3,X4))
   U63(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U63(X1,X2,X3,X4))
   U64(ok(X1),ok(X2)) -> ok(U64(X1,X2))
   pair(ok(X1),ok(X2)) -> ok(pair(X1,X2))
   cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
   U71(ok(X1),ok(X2)) -> ok(U71(X1,X2))
   U72(ok(X1),ok(X2)) -> ok(U72(X1,X2))
   U81(ok(X1),ok(X2),ok(X3)) -> ok(U81(X1,X2,X3))
   U82(ok(X1),ok(X2),ok(X3)) -> ok(U82(X1,X2,X3))
   fst(ok(X)) -> ok(fst(X))
   natsFrom(ok(X)) -> ok(natsFrom(X))
   s(ok(X)) -> ok(s(X))
   sel(ok(X1),ok(X2)) -> ok(sel(X1,X2))
   tail(ok(X)) -> ok(tail(X))
   take(ok(X1),ok(X2)) -> ok(take(X1,X2))
   top(mark(X)) -> top(proper(X))
   top(ok(X)) -> top(active(X))
  Usable Rule Processor:
   DPs:
    active#(U11(tt(),N,XS)) -> U12#(tt(),N,XS)
    active#(U12(tt(),N,XS)) -> splitAt#(N,XS)
    active#(U12(tt(),N,XS)) -> snd#(splitAt(N,XS))
    active#(U21(tt(),X)) -> U22#(tt(),X)
    active#(U31(tt(),N)) -> U32#(tt(),N)
    active#(U41(tt(),N,XS)) -> U42#(tt(),N,XS)
    active#(U42(tt(),N,XS)) -> afterNth#(N,XS)
    active#(U42(tt(),N,XS)) -> head#(afterNth(N,XS))
    active#(U51(tt(),Y)) -> U52#(tt(),Y)
    active#(U61(tt(),N,X,XS)) -> U62#(tt(),N,X,XS)
    active#(U62(tt(),N,X,XS)) -> U63#(tt(),N,X,XS)
    active#(U63(tt(),N,X,XS)) -> splitAt#(N,XS)
    active#(U63(tt(),N,X,XS)) -> U64#(splitAt(N,XS),X)
    active#(U64(pair(YS,ZS),X)) -> cons#(X,YS)
    active#(U64(pair(YS,ZS),X)) -> pair#(cons(X,YS),ZS)
    active#(U71(tt(),XS)) -> U72#(tt(),XS)
    active#(U81(tt(),N,XS)) -> U82#(tt(),N,XS)
    active#(U82(tt(),N,XS)) -> splitAt#(N,XS)
    active#(U82(tt(),N,XS)) -> fst#(splitAt(N,XS))
    active#(afterNth(N,XS)) -> U11#(tt(),N,XS)
    active#(fst(pair(X,Y))) -> U21#(tt(),X)
    active#(head(cons(N,XS))) -> U31#(tt(),N)
    active#(natsFrom(N)) -> s#(N)
    active#(natsFrom(N)) -> natsFrom#(s(N))
    active#(natsFrom(N)) -> cons#(N,natsFrom(s(N)))
    active#(sel(N,XS)) -> U41#(tt(),N,XS)
    active#(snd(pair(X,Y))) -> U51#(tt(),Y)
    active#(splitAt(0(),XS)) -> pair#(nil(),XS)
    active#(splitAt(s(N),cons(X,XS))) -> U61#(tt(),N,X,XS)
    active#(tail(cons(N,XS))) -> U71#(tt(),XS)
    active#(take(N,XS)) -> U81#(tt(),N,XS)
    active#(U11(X1,X2,X3)) -> active#(X1)
    active#(U11(X1,X2,X3)) -> U11#(active(X1),X2,X3)
    active#(U12(X1,X2,X3)) -> active#(X1)
    active#(U12(X1,X2,X3)) -> U12#(active(X1),X2,X3)
    active#(snd(X)) -> active#(X)
    active#(snd(X)) -> snd#(active(X))
    active#(splitAt(X1,X2)) -> active#(X1)
    active#(splitAt(X1,X2)) -> splitAt#(active(X1),X2)
    active#(splitAt(X1,X2)) -> active#(X2)
    active#(splitAt(X1,X2)) -> splitAt#(X1,active(X2))
    active#(U21(X1,X2)) -> active#(X1)
    active#(U21(X1,X2)) -> U21#(active(X1),X2)
    active#(U22(X1,X2)) -> active#(X1)
    active#(U22(X1,X2)) -> U22#(active(X1),X2)
    active#(U31(X1,X2)) -> active#(X1)
    active#(U31(X1,X2)) -> U31#(active(X1),X2)
    active#(U32(X1,X2)) -> active#(X1)
    active#(U32(X1,X2)) -> U32#(active(X1),X2)
    active#(U41(X1,X2,X3)) -> active#(X1)
    active#(U41(X1,X2,X3)) -> U41#(active(X1),X2,X3)
    active#(U42(X1,X2,X3)) -> active#(X1)
    active#(U42(X1,X2,X3)) -> U42#(active(X1),X2,X3)
    active#(head(X)) -> active#(X)
    active#(head(X)) -> head#(active(X))
    active#(afterNth(X1,X2)) -> active#(X1)
    active#(afterNth(X1,X2)) -> afterNth#(active(X1),X2)
    active#(afterNth(X1,X2)) -> active#(X2)
    active#(afterNth(X1,X2)) -> afterNth#(X1,active(X2))
    active#(U51(X1,X2)) -> active#(X1)
    active#(U51(X1,X2)) -> U51#(active(X1),X2)
    active#(U52(X1,X2)) -> active#(X1)
    active#(U52(X1,X2)) -> U52#(active(X1),X2)
    active#(U61(X1,X2,X3,X4)) -> active#(X1)
    active#(U61(X1,X2,X3,X4)) -> U61#(active(X1),X2,X3,X4)
    active#(U62(X1,X2,X3,X4)) -> active#(X1)
    active#(U62(X1,X2,X3,X4)) -> U62#(active(X1),X2,X3,X4)
    active#(U63(X1,X2,X3,X4)) -> active#(X1)
    active#(U63(X1,X2,X3,X4)) -> U63#(active(X1),X2,X3,X4)
    active#(U64(X1,X2)) -> active#(X1)
    active#(U64(X1,X2)) -> U64#(active(X1),X2)
    active#(pair(X1,X2)) -> active#(X1)
    active#(pair(X1,X2)) -> pair#(active(X1),X2)
    active#(pair(X1,X2)) -> active#(X2)
    active#(pair(X1,X2)) -> pair#(X1,active(X2))
    active#(cons(X1,X2)) -> active#(X1)
    active#(cons(X1,X2)) -> cons#(active(X1),X2)
    active#(U71(X1,X2)) -> active#(X1)
    active#(U71(X1,X2)) -> U71#(active(X1),X2)
    active#(U72(X1,X2)) -> active#(X1)
    active#(U72(X1,X2)) -> U72#(active(X1),X2)
    active#(U81(X1,X2,X3)) -> active#(X1)
    active#(U81(X1,X2,X3)) -> U81#(active(X1),X2,X3)
    active#(U82(X1,X2,X3)) -> active#(X1)
    active#(U82(X1,X2,X3)) -> U82#(active(X1),X2,X3)
    active#(fst(X)) -> active#(X)
    active#(fst(X)) -> fst#(active(X))
    active#(natsFrom(X)) -> active#(X)
    active#(natsFrom(X)) -> natsFrom#(active(X))
    active#(s(X)) -> active#(X)
    active#(s(X)) -> s#(active(X))
    active#(sel(X1,X2)) -> active#(X1)
    active#(sel(X1,X2)) -> sel#(active(X1),X2)
    active#(sel(X1,X2)) -> active#(X2)
    active#(sel(X1,X2)) -> sel#(X1,active(X2))
    active#(tail(X)) -> active#(X)
    active#(tail(X)) -> tail#(active(X))
    active#(take(X1,X2)) -> active#(X1)
    active#(take(X1,X2)) -> take#(active(X1),X2)
    active#(take(X1,X2)) -> active#(X2)
    active#(take(X1,X2)) -> take#(X1,active(X2))
    U11#(mark(X1),X2,X3) -> U11#(X1,X2,X3)
    U12#(mark(X1),X2,X3) -> U12#(X1,X2,X3)
    snd#(mark(X)) -> snd#(X)
    splitAt#(mark(X1),X2) -> splitAt#(X1,X2)
    splitAt#(X1,mark(X2)) -> splitAt#(X1,X2)
    U21#(mark(X1),X2) -> U21#(X1,X2)
    U22#(mark(X1),X2) -> U22#(X1,X2)
    U31#(mark(X1),X2) -> U31#(X1,X2)
    U32#(mark(X1),X2) -> U32#(X1,X2)
    U41#(mark(X1),X2,X3) -> U41#(X1,X2,X3)
    U42#(mark(X1),X2,X3) -> U42#(X1,X2,X3)
    head#(mark(X)) -> head#(X)
    afterNth#(mark(X1),X2) -> afterNth#(X1,X2)
    afterNth#(X1,mark(X2)) -> afterNth#(X1,X2)
    U51#(mark(X1),X2) -> U51#(X1,X2)
    U52#(mark(X1),X2) -> U52#(X1,X2)
    U61#(mark(X1),X2,X3,X4) -> U61#(X1,X2,X3,X4)
    U62#(mark(X1),X2,X3,X4) -> U62#(X1,X2,X3,X4)
    U63#(mark(X1),X2,X3,X4) -> U63#(X1,X2,X3,X4)
    U64#(mark(X1),X2) -> U64#(X1,X2)
    pair#(mark(X1),X2) -> pair#(X1,X2)
    pair#(X1,mark(X2)) -> pair#(X1,X2)
    cons#(mark(X1),X2) -> cons#(X1,X2)
    U71#(mark(X1),X2) -> U71#(X1,X2)
    U72#(mark(X1),X2) -> U72#(X1,X2)
    U81#(mark(X1),X2,X3) -> U81#(X1,X2,X3)
    U82#(mark(X1),X2,X3) -> U82#(X1,X2,X3)
    fst#(mark(X)) -> fst#(X)
    natsFrom#(mark(X)) -> natsFrom#(X)
    s#(mark(X)) -> s#(X)
    sel#(mark(X1),X2) -> sel#(X1,X2)
    sel#(X1,mark(X2)) -> sel#(X1,X2)
    tail#(mark(X)) -> tail#(X)
    take#(mark(X1),X2) -> take#(X1,X2)
    take#(X1,mark(X2)) -> take#(X1,X2)
    proper#(U11(X1,X2,X3)) -> proper#(X3)
    proper#(U11(X1,X2,X3)) -> proper#(X2)
    proper#(U11(X1,X2,X3)) -> proper#(X1)
    proper#(U11(X1,X2,X3)) -> U11#(proper(X1),proper(X2),proper(X3))
    proper#(U12(X1,X2,X3)) -> proper#(X3)
    proper#(U12(X1,X2,X3)) -> proper#(X2)
    proper#(U12(X1,X2,X3)) -> proper#(X1)
    proper#(U12(X1,X2,X3)) -> U12#(proper(X1),proper(X2),proper(X3))
    proper#(snd(X)) -> proper#(X)
    proper#(snd(X)) -> snd#(proper(X))
    proper#(splitAt(X1,X2)) -> proper#(X2)
    proper#(splitAt(X1,X2)) -> proper#(X1)
    proper#(splitAt(X1,X2)) -> splitAt#(proper(X1),proper(X2))
    proper#(U21(X1,X2)) -> proper#(X2)
    proper#(U21(X1,X2)) -> proper#(X1)
    proper#(U21(X1,X2)) -> U21#(proper(X1),proper(X2))
    proper#(U22(X1,X2)) -> proper#(X2)
    proper#(U22(X1,X2)) -> proper#(X1)
    proper#(U22(X1,X2)) -> U22#(proper(X1),proper(X2))
    proper#(U31(X1,X2)) -> proper#(X2)
    proper#(U31(X1,X2)) -> proper#(X1)
    proper#(U31(X1,X2)) -> U31#(proper(X1),proper(X2))
    proper#(U32(X1,X2)) -> proper#(X2)
    proper#(U32(X1,X2)) -> proper#(X1)
    proper#(U32(X1,X2)) -> U32#(proper(X1),proper(X2))
    proper#(U41(X1,X2,X3)) -> proper#(X3)
    proper#(U41(X1,X2,X3)) -> proper#(X2)
    proper#(U41(X1,X2,X3)) -> proper#(X1)
    proper#(U41(X1,X2,X3)) -> U41#(proper(X1),proper(X2),proper(X3))
    proper#(U42(X1,X2,X3)) -> proper#(X3)
    proper#(U42(X1,X2,X3)) -> proper#(X2)
    proper#(U42(X1,X2,X3)) -> proper#(X1)
    proper#(U42(X1,X2,X3)) -> U42#(proper(X1),proper(X2),proper(X3))
    proper#(head(X)) -> proper#(X)
    proper#(head(X)) -> head#(proper(X))
    proper#(afterNth(X1,X2)) -> proper#(X2)
    proper#(afterNth(X1,X2)) -> proper#(X1)
    proper#(afterNth(X1,X2)) -> afterNth#(proper(X1),proper(X2))
    proper#(U51(X1,X2)) -> proper#(X2)
    proper#(U51(X1,X2)) -> proper#(X1)
    proper#(U51(X1,X2)) -> U51#(proper(X1),proper(X2))
    proper#(U52(X1,X2)) -> proper#(X2)
    proper#(U52(X1,X2)) -> proper#(X1)
    proper#(U52(X1,X2)) -> U52#(proper(X1),proper(X2))
    proper#(U61(X1,X2,X3,X4)) -> proper#(X4)
    proper#(U61(X1,X2,X3,X4)) -> proper#(X3)
    proper#(U61(X1,X2,X3,X4)) -> proper#(X2)
    proper#(U61(X1,X2,X3,X4)) -> proper#(X1)
    proper#(U61(X1,X2,X3,X4)) -> U61#(proper(X1),proper(X2),proper(X3),proper(X4))
    proper#(U62(X1,X2,X3,X4)) -> proper#(X4)
    proper#(U62(X1,X2,X3,X4)) -> proper#(X3)
    proper#(U62(X1,X2,X3,X4)) -> proper#(X2)
    proper#(U62(X1,X2,X3,X4)) -> proper#(X1)
    proper#(U62(X1,X2,X3,X4)) -> U62#(proper(X1),proper(X2),proper(X3),proper(X4))
    proper#(U63(X1,X2,X3,X4)) -> proper#(X4)
    proper#(U63(X1,X2,X3,X4)) -> proper#(X3)
    proper#(U63(X1,X2,X3,X4)) -> proper#(X2)
    proper#(U63(X1,X2,X3,X4)) -> proper#(X1)
    proper#(U63(X1,X2,X3,X4)) -> U63#(proper(X1),proper(X2),proper(X3),proper(X4))
    proper#(U64(X1,X2)) -> proper#(X2)
    proper#(U64(X1,X2)) -> proper#(X1)
    proper#(U64(X1,X2)) -> U64#(proper(X1),proper(X2))
    proper#(pair(X1,X2)) -> proper#(X2)
    proper#(pair(X1,X2)) -> proper#(X1)
    proper#(pair(X1,X2)) -> pair#(proper(X1),proper(X2))
    proper#(cons(X1,X2)) -> proper#(X2)
    proper#(cons(X1,X2)) -> proper#(X1)
    proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2))
    proper#(U71(X1,X2)) -> proper#(X2)
    proper#(U71(X1,X2)) -> proper#(X1)
    proper#(U71(X1,X2)) -> U71#(proper(X1),proper(X2))
    proper#(U72(X1,X2)) -> proper#(X2)
    proper#(U72(X1,X2)) -> proper#(X1)
    proper#(U72(X1,X2)) -> U72#(proper(X1),proper(X2))
    proper#(U81(X1,X2,X3)) -> proper#(X3)
    proper#(U81(X1,X2,X3)) -> proper#(X2)
    proper#(U81(X1,X2,X3)) -> proper#(X1)
    proper#(U81(X1,X2,X3)) -> U81#(proper(X1),proper(X2),proper(X3))
    proper#(U82(X1,X2,X3)) -> proper#(X3)
    proper#(U82(X1,X2,X3)) -> proper#(X2)
    proper#(U82(X1,X2,X3)) -> proper#(X1)
    proper#(U82(X1,X2,X3)) -> U82#(proper(X1),proper(X2),proper(X3))
    proper#(fst(X)) -> proper#(X)
    proper#(fst(X)) -> fst#(proper(X))
    proper#(natsFrom(X)) -> proper#(X)
    proper#(natsFrom(X)) -> natsFrom#(proper(X))
    proper#(s(X)) -> proper#(X)
    proper#(s(X)) -> s#(proper(X))
    proper#(sel(X1,X2)) -> proper#(X2)
    proper#(sel(X1,X2)) -> proper#(X1)
    proper#(sel(X1,X2)) -> sel#(proper(X1),proper(X2))
    proper#(tail(X)) -> proper#(X)
    proper#(tail(X)) -> tail#(proper(X))
    proper#(take(X1,X2)) -> proper#(X2)
    proper#(take(X1,X2)) -> proper#(X1)
    proper#(take(X1,X2)) -> take#(proper(X1),proper(X2))
    U11#(ok(X1),ok(X2),ok(X3)) -> U11#(X1,X2,X3)
    U12#(ok(X1),ok(X2),ok(X3)) -> U12#(X1,X2,X3)
    snd#(ok(X)) -> snd#(X)
    splitAt#(ok(X1),ok(X2)) -> splitAt#(X1,X2)
    U21#(ok(X1),ok(X2)) -> U21#(X1,X2)
    U22#(ok(X1),ok(X2)) -> U22#(X1,X2)
    U31#(ok(X1),ok(X2)) -> U31#(X1,X2)
    U32#(ok(X1),ok(X2)) -> U32#(X1,X2)
    U41#(ok(X1),ok(X2),ok(X3)) -> U41#(X1,X2,X3)
    U42#(ok(X1),ok(X2),ok(X3)) -> U42#(X1,X2,X3)
    head#(ok(X)) -> head#(X)
    afterNth#(ok(X1),ok(X2)) -> afterNth#(X1,X2)
    U51#(ok(X1),ok(X2)) -> U51#(X1,X2)
    U52#(ok(X1),ok(X2)) -> U52#(X1,X2)
    U61#(ok(X1),ok(X2),ok(X3),ok(X4)) -> U61#(X1,X2,X3,X4)
    U62#(ok(X1),ok(X2),ok(X3),ok(X4)) -> U62#(X1,X2,X3,X4)
    U63#(ok(X1),ok(X2),ok(X3),ok(X4)) -> U63#(X1,X2,X3,X4)
    U64#(ok(X1),ok(X2)) -> U64#(X1,X2)
    pair#(ok(X1),ok(X2)) -> pair#(X1,X2)
    cons#(ok(X1),ok(X2)) -> cons#(X1,X2)
    U71#(ok(X1),ok(X2)) -> U71#(X1,X2)
    U72#(ok(X1),ok(X2)) -> U72#(X1,X2)
    U81#(ok(X1),ok(X2),ok(X3)) -> U81#(X1,X2,X3)
    U82#(ok(X1),ok(X2),ok(X3)) -> U82#(X1,X2,X3)
    fst#(ok(X)) -> fst#(X)
    natsFrom#(ok(X)) -> natsFrom#(X)
    s#(ok(X)) -> s#(X)
    sel#(ok(X1),ok(X2)) -> sel#(X1,X2)
    tail#(ok(X)) -> tail#(X)
    take#(ok(X1),ok(X2)) -> take#(X1,X2)
    top#(mark(X)) -> proper#(X)
    top#(mark(X)) -> top#(proper(X))
    top#(ok(X)) -> active#(X)
    top#(ok(X)) -> top#(active(X))
   TRS:
    f71(x,y) -> x
    f71(x,y) -> y
    splitAt(mark(X1),X2) -> mark(splitAt(X1,X2))
    splitAt(X1,mark(X2)) -> mark(splitAt(X1,X2))
    splitAt(ok(X1),ok(X2)) -> ok(splitAt(X1,X2))
    afterNth(mark(X1),X2) -> mark(afterNth(X1,X2))
    afterNth(X1,mark(X2)) -> mark(afterNth(X1,X2))
    afterNth(ok(X1),ok(X2)) -> ok(afterNth(X1,X2))
    cons(mark(X1),X2) -> mark(cons(X1,X2))
    cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
    s(mark(X)) -> mark(s(X))
    s(ok(X)) -> ok(s(X))
    natsFrom(mark(X)) -> mark(natsFrom(X))
    natsFrom(ok(X)) -> ok(natsFrom(X))
    active(U11(tt(),N,XS)) -> mark(U12(tt(),N,XS))
    active(U12(tt(),N,XS)) -> mark(snd(splitAt(N,XS)))
    active(U21(tt(),X)) -> mark(U22(tt(),X))
    active(U22(tt(),X)) -> mark(X)
    active(U31(tt(),N)) -> mark(U32(tt(),N))
    active(U32(tt(),N)) -> mark(N)
    active(U41(tt(),N,XS)) -> mark(U42(tt(),N,XS))
    active(U42(tt(),N,XS)) -> mark(head(afterNth(N,XS)))
    active(U51(tt(),Y)) -> mark(U52(tt(),Y))
    active(U52(tt(),Y)) -> mark(Y)
    active(U61(tt(),N,X,XS)) -> mark(U62(tt(),N,X,XS))
    active(U62(tt(),N,X,XS)) -> mark(U63(tt(),N,X,XS))
    active(U63(tt(),N,X,XS)) -> mark(U64(splitAt(N,XS),X))
    active(U64(pair(YS,ZS),X)) -> mark(pair(cons(X,YS),ZS))
    active(U71(tt(),XS)) -> mark(U72(tt(),XS))
    active(U72(tt(),XS)) -> mark(XS)
    active(U81(tt(),N,XS)) -> mark(U82(tt(),N,XS))
    active(U82(tt(),N,XS)) -> mark(fst(splitAt(N,XS)))
    active(afterNth(N,XS)) -> mark(U11(tt(),N,XS))
    active(fst(pair(X,Y))) -> mark(U21(tt(),X))
    active(head(cons(N,XS))) -> mark(U31(tt(),N))
    active(natsFrom(N)) -> mark(cons(N,natsFrom(s(N))))
    active(sel(N,XS)) -> mark(U41(tt(),N,XS))
    active(snd(pair(X,Y))) -> mark(U51(tt(),Y))
    active(splitAt(0(),XS)) -> mark(pair(nil(),XS))
    active(splitAt(s(N),cons(X,XS))) -> mark(U61(tt(),N,X,XS))
    active(tail(cons(N,XS))) -> mark(U71(tt(),XS))
    active(take(N,XS)) -> mark(U81(tt(),N,XS))
    active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3)
    active(U12(X1,X2,X3)) -> U12(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(U21(X1,X2)) -> U21(active(X1),X2)
    active(U22(X1,X2)) -> U22(active(X1),X2)
    active(U31(X1,X2)) -> U31(active(X1),X2)
    active(U32(X1,X2)) -> U32(active(X1),X2)
    active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3)
    active(U42(X1,X2,X3)) -> U42(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(U51(X1,X2)) -> U51(active(X1),X2)
    active(U52(X1,X2)) -> U52(active(X1),X2)
    active(U61(X1,X2,X3,X4)) -> U61(active(X1),X2,X3,X4)
    active(U62(X1,X2,X3,X4)) -> U62(active(X1),X2,X3,X4)
    active(U63(X1,X2,X3,X4)) -> U63(active(X1),X2,X3,X4)
    active(U64(X1,X2)) -> U64(active(X1),X2)
    active(pair(X1,X2)) -> pair(active(X1),X2)
    active(pair(X1,X2)) -> pair(X1,active(X2))
    active(cons(X1,X2)) -> cons(active(X1),X2)
    active(U71(X1,X2)) -> U71(active(X1),X2)
    active(U72(X1,X2)) -> U72(active(X1),X2)
    active(U81(X1,X2,X3)) -> U81(active(X1),X2,X3)
    active(U82(X1,X2,X3)) -> U82(active(X1),X2,X3)
    active(fst(X)) -> fst(active(X))
    active(natsFrom(X)) -> natsFrom(active(X))
    active(s(X)) -> s(active(X))
    active(sel(X1,X2)) -> sel(active(X1),X2)
    active(sel(X1,X2)) -> sel(X1,active(X2))
    active(tail(X)) -> tail(active(X))
    active(take(X1,X2)) -> take(active(X1),X2)
    active(take(X1,X2)) -> take(X1,active(X2))
    snd(mark(X)) -> mark(snd(X))
    snd(ok(X)) -> ok(snd(X))
    head(mark(X)) -> mark(head(X))
    head(ok(X)) -> ok(head(X))
    U64(mark(X1),X2) -> mark(U64(X1,X2))
    U64(ok(X1),ok(X2)) -> ok(U64(X1,X2))
    pair(mark(X1),X2) -> mark(pair(X1,X2))
    pair(X1,mark(X2)) -> mark(pair(X1,X2))
    pair(ok(X1),ok(X2)) -> ok(pair(X1,X2))
    fst(mark(X)) -> mark(fst(X))
    fst(ok(X)) -> ok(fst(X))
    U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3))
    U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3))
    U12(mark(X1),X2,X3) -> mark(U12(X1,X2,X3))
    U12(ok(X1),ok(X2),ok(X3)) -> ok(U12(X1,X2,X3))
    U21(mark(X1),X2) -> mark(U21(X1,X2))
    U21(ok(X1),ok(X2)) -> ok(U21(X1,X2))
    U22(mark(X1),X2) -> mark(U22(X1,X2))
    U22(ok(X1),ok(X2)) -> ok(U22(X1,X2))
    U31(mark(X1),X2) -> mark(U31(X1,X2))
    U31(ok(X1),ok(X2)) -> ok(U31(X1,X2))
    U32(mark(X1),X2) -> mark(U32(X1,X2))
    U32(ok(X1),ok(X2)) -> ok(U32(X1,X2))
    U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3))
    U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3))
    U42(mark(X1),X2,X3) -> mark(U42(X1,X2,X3))
    U42(ok(X1),ok(X2),ok(X3)) -> ok(U42(X1,X2,X3))
    U51(mark(X1),X2) -> mark(U51(X1,X2))
    U51(ok(X1),ok(X2)) -> ok(U51(X1,X2))
    U52(mark(X1),X2) -> mark(U52(X1,X2))
    U52(ok(X1),ok(X2)) -> ok(U52(X1,X2))
    U61(mark(X1),X2,X3,X4) -> mark(U61(X1,X2,X3,X4))
    U61(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U61(X1,X2,X3,X4))
    U62(mark(X1),X2,X3,X4) -> mark(U62(X1,X2,X3,X4))
    U62(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U62(X1,X2,X3,X4))
    U63(mark(X1),X2,X3,X4) -> mark(U63(X1,X2,X3,X4))
    U63(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U63(X1,X2,X3,X4))
    U71(mark(X1),X2) -> mark(U71(X1,X2))
    U71(ok(X1),ok(X2)) -> ok(U71(X1,X2))
    U72(mark(X1),X2) -> mark(U72(X1,X2))
    U72(ok(X1),ok(X2)) -> ok(U72(X1,X2))
    U81(mark(X1),X2,X3) -> mark(U81(X1,X2,X3))
    U81(ok(X1),ok(X2),ok(X3)) -> ok(U81(X1,X2,X3))
    U82(mark(X1),X2,X3) -> mark(U82(X1,X2,X3))
    U82(ok(X1),ok(X2),ok(X3)) -> ok(U82(X1,X2,X3))
    sel(mark(X1),X2) -> mark(sel(X1,X2))
    sel(X1,mark(X2)) -> mark(sel(X1,X2))
    sel(ok(X1),ok(X2)) -> ok(sel(X1,X2))
    tail(mark(X)) -> mark(tail(X))
    tail(ok(X)) -> ok(tail(X))
    take(mark(X1),X2) -> mark(take(X1,X2))
    take(X1,mark(X2)) -> mark(take(X1,X2))
    take(ok(X1),ok(X2)) -> ok(take(X1,X2))
    proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3))
    proper(tt()) -> ok(tt())
    proper(U12(X1,X2,X3)) -> U12(proper(X1),proper(X2),proper(X3))
    proper(snd(X)) -> snd(proper(X))
    proper(splitAt(X1,X2)) -> splitAt(proper(X1),proper(X2))
    proper(U21(X1,X2)) -> U21(proper(X1),proper(X2))
    proper(U22(X1,X2)) -> U22(proper(X1),proper(X2))
    proper(U31(X1,X2)) -> U31(proper(X1),proper(X2))
    proper(U32(X1,X2)) -> U32(proper(X1),proper(X2))
    proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3))
    proper(U42(X1,X2,X3)) -> U42(proper(X1),proper(X2),proper(X3))
    proper(head(X)) -> head(proper(X))
    proper(afterNth(X1,X2)) -> afterNth(proper(X1),proper(X2))
    proper(U51(X1,X2)) -> U51(proper(X1),proper(X2))
    proper(U52(X1,X2)) -> U52(proper(X1),proper(X2))
    proper(U61(X1,X2,X3,X4)) -> U61(proper(X1),proper(X2),proper(X3),proper(X4))
    proper(U62(X1,X2,X3,X4)) -> U62(proper(X1),proper(X2),proper(X3),proper(X4))
    proper(U63(X1,X2,X3,X4)) -> U63(proper(X1),proper(X2),proper(X3),proper(X4))
    proper(U64(X1,X2)) -> U64(proper(X1),proper(X2))
    proper(pair(X1,X2)) -> pair(proper(X1),proper(X2))
    proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
    proper(U71(X1,X2)) -> U71(proper(X1),proper(X2))
    proper(U72(X1,X2)) -> U72(proper(X1),proper(X2))
    proper(U81(X1,X2,X3)) -> U81(proper(X1),proper(X2),proper(X3))
    proper(U82(X1,X2,X3)) -> U82(proper(X1),proper(X2),proper(X3))
    proper(fst(X)) -> fst(proper(X))
    proper(natsFrom(X)) -> natsFrom(proper(X))
    proper(s(X)) -> s(proper(X))
    proper(sel(X1,X2)) -> sel(proper(X1),proper(X2))
    proper(0()) -> ok(0())
    proper(nil()) -> ok(nil())
    proper(tail(X)) -> tail(proper(X))
    proper(take(X1,X2)) -> take(proper(X1),proper(X2))
   TDG Processor:
    DPs:
     active#(U11(tt(),N,XS)) -> U12#(tt(),N,XS)
     active#(U12(tt(),N,XS)) -> splitAt#(N,XS)
     active#(U12(tt(),N,XS)) -> snd#(splitAt(N,XS))
     active#(U21(tt(),X)) -> U22#(tt(),X)
     active#(U31(tt(),N)) -> U32#(tt(),N)
     active#(U41(tt(),N,XS)) -> U42#(tt(),N,XS)
     active#(U42(tt(),N,XS)) -> afterNth#(N,XS)
     active#(U42(tt(),N,XS)) -> head#(afterNth(N,XS))
     active#(U51(tt(),Y)) -> U52#(tt(),Y)
     active#(U61(tt(),N,X,XS)) -> U62#(tt(),N,X,XS)
     active#(U62(tt(),N,X,XS)) -> U63#(tt(),N,X,XS)
     active#(U63(tt(),N,X,XS)) -> splitAt#(N,XS)
     active#(U63(tt(),N,X,XS)) -> U64#(splitAt(N,XS),X)
     active#(U64(pair(YS,ZS),X)) -> cons#(X,YS)
     active#(U64(pair(YS,ZS),X)) -> pair#(cons(X,YS),ZS)
     active#(U71(tt(),XS)) -> U72#(tt(),XS)
     active#(U81(tt(),N,XS)) -> U82#(tt(),N,XS)
     active#(U82(tt(),N,XS)) -> splitAt#(N,XS)
     active#(U82(tt(),N,XS)) -> fst#(splitAt(N,XS))
     active#(afterNth(N,XS)) -> U11#(tt(),N,XS)
     active#(fst(pair(X,Y))) -> U21#(tt(),X)
     active#(head(cons(N,XS))) -> U31#(tt(),N)
     active#(natsFrom(N)) -> s#(N)
     active#(natsFrom(N)) -> natsFrom#(s(N))
     active#(natsFrom(N)) -> cons#(N,natsFrom(s(N)))
     active#(sel(N,XS)) -> U41#(tt(),N,XS)
     active#(snd(pair(X,Y))) -> U51#(tt(),Y)
     active#(splitAt(0(),XS)) -> pair#(nil(),XS)
     active#(splitAt(s(N),cons(X,XS))) -> U61#(tt(),N,X,XS)
     active#(tail(cons(N,XS))) -> U71#(tt(),XS)
     active#(take(N,XS)) -> U81#(tt(),N,XS)
     active#(U11(X1,X2,X3)) -> active#(X1)
     active#(U11(X1,X2,X3)) -> U11#(active(X1),X2,X3)
     active#(U12(X1,X2,X3)) -> active#(X1)
     active#(U12(X1,X2,X3)) -> U12#(active(X1),X2,X3)
     active#(snd(X)) -> active#(X)
     active#(snd(X)) -> snd#(active(X))
     active#(splitAt(X1,X2)) -> active#(X1)
     active#(splitAt(X1,X2)) -> splitAt#(active(X1),X2)
     active#(splitAt(X1,X2)) -> active#(X2)
     active#(splitAt(X1,X2)) -> splitAt#(X1,active(X2))
     active#(U21(X1,X2)) -> active#(X1)
     active#(U21(X1,X2)) -> U21#(active(X1),X2)
     active#(U22(X1,X2)) -> active#(X1)
     active#(U22(X1,X2)) -> U22#(active(X1),X2)
     active#(U31(X1,X2)) -> active#(X1)
     active#(U31(X1,X2)) -> U31#(active(X1),X2)
     active#(U32(X1,X2)) -> active#(X1)
     active#(U32(X1,X2)) -> U32#(active(X1),X2)
     active#(U41(X1,X2,X3)) -> active#(X1)
     active#(U41(X1,X2,X3)) -> U41#(active(X1),X2,X3)
     active#(U42(X1,X2,X3)) -> active#(X1)
     active#(U42(X1,X2,X3)) -> U42#(active(X1),X2,X3)
     active#(head(X)) -> active#(X)
     active#(head(X)) -> head#(active(X))
     active#(afterNth(X1,X2)) -> active#(X1)
     active#(afterNth(X1,X2)) -> afterNth#(active(X1),X2)
     active#(afterNth(X1,X2)) -> active#(X2)
     active#(afterNth(X1,X2)) -> afterNth#(X1,active(X2))
     active#(U51(X1,X2)) -> active#(X1)
     active#(U51(X1,X2)) -> U51#(active(X1),X2)
     active#(U52(X1,X2)) -> active#(X1)
     active#(U52(X1,X2)) -> U52#(active(X1),X2)
     active#(U61(X1,X2,X3,X4)) -> active#(X1)
     active#(U61(X1,X2,X3,X4)) -> U61#(active(X1),X2,X3,X4)
     active#(U62(X1,X2,X3,X4)) -> active#(X1)
     active#(U62(X1,X2,X3,X4)) -> U62#(active(X1),X2,X3,X4)
     active#(U63(X1,X2,X3,X4)) -> active#(X1)
     active#(U63(X1,X2,X3,X4)) -> U63#(active(X1),X2,X3,X4)
     active#(U64(X1,X2)) -> active#(X1)
     active#(U64(X1,X2)) -> U64#(active(X1),X2)
     active#(pair(X1,X2)) -> active#(X1)
     active#(pair(X1,X2)) -> pair#(active(X1),X2)
     active#(pair(X1,X2)) -> active#(X2)
     active#(pair(X1,X2)) -> pair#(X1,active(X2))
     active#(cons(X1,X2)) -> active#(X1)
     active#(cons(X1,X2)) -> cons#(active(X1),X2)
     active#(U71(X1,X2)) -> active#(X1)
     active#(U71(X1,X2)) -> U71#(active(X1),X2)
     active#(U72(X1,X2)) -> active#(X1)
     active#(U72(X1,X2)) -> U72#(active(X1),X2)
     active#(U81(X1,X2,X3)) -> active#(X1)
     active#(U81(X1,X2,X3)) -> U81#(active(X1),X2,X3)
     active#(U82(X1,X2,X3)) -> active#(X1)
     active#(U82(X1,X2,X3)) -> U82#(active(X1),X2,X3)
     active#(fst(X)) -> active#(X)
     active#(fst(X)) -> fst#(active(X))
     active#(natsFrom(X)) -> active#(X)
     active#(natsFrom(X)) -> natsFrom#(active(X))
     active#(s(X)) -> active#(X)
     active#(s(X)) -> s#(active(X))
     active#(sel(X1,X2)) -> active#(X1)
     active#(sel(X1,X2)) -> sel#(active(X1),X2)
     active#(sel(X1,X2)) -> active#(X2)
     active#(sel(X1,X2)) -> sel#(X1,active(X2))
     active#(tail(X)) -> active#(X)
     active#(tail(X)) -> tail#(active(X))
     active#(take(X1,X2)) -> active#(X1)
     active#(take(X1,X2)) -> take#(active(X1),X2)
     active#(take(X1,X2)) -> active#(X2)
     active#(take(X1,X2)) -> take#(X1,active(X2))
     U11#(mark(X1),X2,X3) -> U11#(X1,X2,X3)
     U12#(mark(X1),X2,X3) -> U12#(X1,X2,X3)
     snd#(mark(X)) -> snd#(X)
     splitAt#(mark(X1),X2) -> splitAt#(X1,X2)
     splitAt#(X1,mark(X2)) -> splitAt#(X1,X2)
     U21#(mark(X1),X2) -> U21#(X1,X2)
     U22#(mark(X1),X2) -> U22#(X1,X2)
     U31#(mark(X1),X2) -> U31#(X1,X2)
     U32#(mark(X1),X2) -> U32#(X1,X2)
     U41#(mark(X1),X2,X3) -> U41#(X1,X2,X3)
     U42#(mark(X1),X2,X3) -> U42#(X1,X2,X3)
     head#(mark(X)) -> head#(X)
     afterNth#(mark(X1),X2) -> afterNth#(X1,X2)
     afterNth#(X1,mark(X2)) -> afterNth#(X1,X2)
     U51#(mark(X1),X2) -> U51#(X1,X2)
     U52#(mark(X1),X2) -> U52#(X1,X2)
     U61#(mark(X1),X2,X3,X4) -> U61#(X1,X2,X3,X4)
     U62#(mark(X1),X2,X3,X4) -> U62#(X1,X2,X3,X4)
     U63#(mark(X1),X2,X3,X4) -> U63#(X1,X2,X3,X4)
     U64#(mark(X1),X2) -> U64#(X1,X2)
     pair#(mark(X1),X2) -> pair#(X1,X2)
     pair#(X1,mark(X2)) -> pair#(X1,X2)
     cons#(mark(X1),X2) -> cons#(X1,X2)
     U71#(mark(X1),X2) -> U71#(X1,X2)
     U72#(mark(X1),X2) -> U72#(X1,X2)
     U81#(mark(X1),X2,X3) -> U81#(X1,X2,X3)
     U82#(mark(X1),X2,X3) -> U82#(X1,X2,X3)
     fst#(mark(X)) -> fst#(X)
     natsFrom#(mark(X)) -> natsFrom#(X)
     s#(mark(X)) -> s#(X)
     sel#(mark(X1),X2) -> sel#(X1,X2)
     sel#(X1,mark(X2)) -> sel#(X1,X2)
     tail#(mark(X)) -> tail#(X)
     take#(mark(X1),X2) -> take#(X1,X2)
     take#(X1,mark(X2)) -> take#(X1,X2)
     proper#(U11(X1,X2,X3)) -> proper#(X3)
     proper#(U11(X1,X2,X3)) -> proper#(X2)
     proper#(U11(X1,X2,X3)) -> proper#(X1)
     proper#(U11(X1,X2,X3)) -> U11#(proper(X1),proper(X2),proper(X3))
     proper#(U12(X1,X2,X3)) -> proper#(X3)
     proper#(U12(X1,X2,X3)) -> proper#(X2)
     proper#(U12(X1,X2,X3)) -> proper#(X1)
     proper#(U12(X1,X2,X3)) -> U12#(proper(X1),proper(X2),proper(X3))
     proper#(snd(X)) -> proper#(X)
     proper#(snd(X)) -> snd#(proper(X))
     proper#(splitAt(X1,X2)) -> proper#(X2)
     proper#(splitAt(X1,X2)) -> proper#(X1)
     proper#(splitAt(X1,X2)) -> splitAt#(proper(X1),proper(X2))
     proper#(U21(X1,X2)) -> proper#(X2)
     proper#(U21(X1,X2)) -> proper#(X1)
     proper#(U21(X1,X2)) -> U21#(proper(X1),proper(X2))
     proper#(U22(X1,X2)) -> proper#(X2)
     proper#(U22(X1,X2)) -> proper#(X1)
     proper#(U22(X1,X2)) -> U22#(proper(X1),proper(X2))
     proper#(U31(X1,X2)) -> proper#(X2)
     proper#(U31(X1,X2)) -> proper#(X1)
     proper#(U31(X1,X2)) -> U31#(proper(X1),proper(X2))
     proper#(U32(X1,X2)) -> proper#(X2)
     proper#(U32(X1,X2)) -> proper#(X1)
     proper#(U32(X1,X2)) -> U32#(proper(X1),proper(X2))
     proper#(U41(X1,X2,X3)) -> proper#(X3)
     proper#(U41(X1,X2,X3)) -> proper#(X2)
     proper#(U41(X1,X2,X3)) -> proper#(X1)
     proper#(U41(X1,X2,X3)) -> U41#(proper(X1),proper(X2),proper(X3))
     proper#(U42(X1,X2,X3)) -> proper#(X3)
     proper#(U42(X1,X2,X3)) -> proper#(X2)
     proper#(U42(X1,X2,X3)) -> proper#(X1)
     proper#(U42(X1,X2,X3)) -> U42#(proper(X1),proper(X2),proper(X3))
     proper#(head(X)) -> proper#(X)
     proper#(head(X)) -> head#(proper(X))
     proper#(afterNth(X1,X2)) -> proper#(X2)
     proper#(afterNth(X1,X2)) -> proper#(X1)
     proper#(afterNth(X1,X2)) -> afterNth#(proper(X1),proper(X2))
     proper#(U51(X1,X2)) -> proper#(X2)
     proper#(U51(X1,X2)) -> proper#(X1)
     proper#(U51(X1,X2)) -> U51#(proper(X1),proper(X2))
     proper#(U52(X1,X2)) -> proper#(X2)
     proper#(U52(X1,X2)) -> proper#(X1)
     proper#(U52(X1,X2)) -> U52#(proper(X1),proper(X2))
     proper#(U61(X1,X2,X3,X4)) -> proper#(X4)
     proper#(U61(X1,X2,X3,X4)) -> proper#(X3)
     proper#(U61(X1,X2,X3,X4)) -> proper#(X2)
     proper#(U61(X1,X2,X3,X4)) -> proper#(X1)
     proper#(U61(X1,X2,X3,X4)) -> U61#(proper(X1),proper(X2),proper(X3),proper(X4))
     proper#(U62(X1,X2,X3,X4)) -> proper#(X4)
     proper#(U62(X1,X2,X3,X4)) -> proper#(X3)
     proper#(U62(X1,X2,X3,X4)) -> proper#(X2)
     proper#(U62(X1,X2,X3,X4)) -> proper#(X1)
     proper#(U62(X1,X2,X3,X4)) -> U62#(proper(X1),proper(X2),proper(X3),proper(X4))
     proper#(U63(X1,X2,X3,X4)) -> proper#(X4)
     proper#(U63(X1,X2,X3,X4)) -> proper#(X3)
     proper#(U63(X1,X2,X3,X4)) -> proper#(X2)
     proper#(U63(X1,X2,X3,X4)) -> proper#(X1)
     proper#(U63(X1,X2,X3,X4)) -> U63#(proper(X1),proper(X2),proper(X3),proper(X4))
     proper#(U64(X1,X2)) -> proper#(X2)
     proper#(U64(X1,X2)) -> proper#(X1)
     proper#(U64(X1,X2)) -> U64#(proper(X1),proper(X2))
     proper#(pair(X1,X2)) -> proper#(X2)
     proper#(pair(X1,X2)) -> proper#(X1)
     proper#(pair(X1,X2)) -> pair#(proper(X1),proper(X2))
     proper#(cons(X1,X2)) -> proper#(X2)
     proper#(cons(X1,X2)) -> proper#(X1)
     proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2))
     proper#(U71(X1,X2)) -> proper#(X2)
     proper#(U71(X1,X2)) -> proper#(X1)
     proper#(U71(X1,X2)) -> U71#(proper(X1),proper(X2))
     proper#(U72(X1,X2)) -> proper#(X2)
     proper#(U72(X1,X2)) -> proper#(X1)
     proper#(U72(X1,X2)) -> U72#(proper(X1),proper(X2))
     proper#(U81(X1,X2,X3)) -> proper#(X3)
     proper#(U81(X1,X2,X3)) -> proper#(X2)
     proper#(U81(X1,X2,X3)) -> proper#(X1)
     proper#(U81(X1,X2,X3)) -> U81#(proper(X1),proper(X2),proper(X3))
     proper#(U82(X1,X2,X3)) -> proper#(X3)
     proper#(U82(X1,X2,X3)) -> proper#(X2)
     proper#(U82(X1,X2,X3)) -> proper#(X1)
     proper#(U82(X1,X2,X3)) -> U82#(proper(X1),proper(X2),proper(X3))
     proper#(fst(X)) -> proper#(X)
     proper#(fst(X)) -> fst#(proper(X))
     proper#(natsFrom(X)) -> proper#(X)
     proper#(natsFrom(X)) -> natsFrom#(proper(X))
     proper#(s(X)) -> proper#(X)
     proper#(s(X)) -> s#(proper(X))
     proper#(sel(X1,X2)) -> proper#(X2)
     proper#(sel(X1,X2)) -> proper#(X1)
     proper#(sel(X1,X2)) -> sel#(proper(X1),proper(X2))
     proper#(tail(X)) -> proper#(X)
     proper#(tail(X)) -> tail#(proper(X))
     proper#(take(X1,X2)) -> proper#(X2)
     proper#(take(X1,X2)) -> proper#(X1)
     proper#(take(X1,X2)) -> take#(proper(X1),proper(X2))
     U11#(ok(X1),ok(X2),ok(X3)) -> U11#(X1,X2,X3)
     U12#(ok(X1),ok(X2),ok(X3)) -> U12#(X1,X2,X3)
     snd#(ok(X)) -> snd#(X)
     splitAt#(ok(X1),ok(X2)) -> splitAt#(X1,X2)
     U21#(ok(X1),ok(X2)) -> U21#(X1,X2)
     U22#(ok(X1),ok(X2)) -> U22#(X1,X2)
     U31#(ok(X1),ok(X2)) -> U31#(X1,X2)
     U32#(ok(X1),ok(X2)) -> U32#(X1,X2)
     U41#(ok(X1),ok(X2),ok(X3)) -> U41#(X1,X2,X3)
     U42#(ok(X1),ok(X2),ok(X3)) -> U42#(X1,X2,X3)
     head#(ok(X)) -> head#(X)
     afterNth#(ok(X1),ok(X2)) -> afterNth#(X1,X2)
     U51#(ok(X1),ok(X2)) -> U51#(X1,X2)
     U52#(ok(X1),ok(X2)) -> U52#(X1,X2)
     U61#(ok(X1),ok(X2),ok(X3),ok(X4)) -> U61#(X1,X2,X3,X4)
     U62#(ok(X1),ok(X2),ok(X3),ok(X4)) -> U62#(X1,X2,X3,X4)
     U63#(ok(X1),ok(X2),ok(X3),ok(X4)) -> U63#(X1,X2,X3,X4)
     U64#(ok(X1),ok(X2)) -> U64#(X1,X2)
     pair#(ok(X1),ok(X2)) -> pair#(X1,X2)
     cons#(ok(X1),ok(X2)) -> cons#(X1,X2)
     U71#(ok(X1),ok(X2)) -> U71#(X1,X2)
     U72#(ok(X1),ok(X2)) -> U72#(X1,X2)
     U81#(ok(X1),ok(X2),ok(X3)) -> U81#(X1,X2,X3)
     U82#(ok(X1),ok(X2),ok(X3)) -> U82#(X1,X2,X3)
     fst#(ok(X)) -> fst#(X)
     natsFrom#(ok(X)) -> natsFrom#(X)
     s#(ok(X)) -> s#(X)
     sel#(ok(X1),ok(X2)) -> sel#(X1,X2)
     tail#(ok(X)) -> tail#(X)
     take#(ok(X1),ok(X2)) -> take#(X1,X2)
     top#(mark(X)) -> proper#(X)
     top#(mark(X)) -> top#(proper(X))
     top#(ok(X)) -> active#(X)
     top#(ok(X)) -> top#(active(X))
    TRS:
     f71(x,y) -> x
     f71(x,y) -> y
     splitAt(mark(X1),X2) -> mark(splitAt(X1,X2))
     splitAt(X1,mark(X2)) -> mark(splitAt(X1,X2))
     splitAt(ok(X1),ok(X2)) -> ok(splitAt(X1,X2))
     afterNth(mark(X1),X2) -> mark(afterNth(X1,X2))
     afterNth(X1,mark(X2)) -> mark(afterNth(X1,X2))
     afterNth(ok(X1),ok(X2)) -> ok(afterNth(X1,X2))
     cons(mark(X1),X2) -> mark(cons(X1,X2))
     cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
     s(mark(X)) -> mark(s(X))
     s(ok(X)) -> ok(s(X))
     natsFrom(mark(X)) -> mark(natsFrom(X))
     natsFrom(ok(X)) -> ok(natsFrom(X))
     active(U11(tt(),N,XS)) -> mark(U12(tt(),N,XS))
     active(U12(tt(),N,XS)) -> mark(snd(splitAt(N,XS)))
     active(U21(tt(),X)) -> mark(U22(tt(),X))
     active(U22(tt(),X)) -> mark(X)
     active(U31(tt(),N)) -> mark(U32(tt(),N))
     active(U32(tt(),N)) -> mark(N)
     active(U41(tt(),N,XS)) -> mark(U42(tt(),N,XS))
     active(U42(tt(),N,XS)) -> mark(head(afterNth(N,XS)))
     active(U51(tt(),Y)) -> mark(U52(tt(),Y))
     active(U52(tt(),Y)) -> mark(Y)
     active(U61(tt(),N,X,XS)) -> mark(U62(tt(),N,X,XS))
     active(U62(tt(),N,X,XS)) -> mark(U63(tt(),N,X,XS))
     active(U63(tt(),N,X,XS)) -> mark(U64(splitAt(N,XS),X))
     active(U64(pair(YS,ZS),X)) -> mark(pair(cons(X,YS),ZS))
     active(U71(tt(),XS)) -> mark(U72(tt(),XS))
     active(U72(tt(),XS)) -> mark(XS)
     active(U81(tt(),N,XS)) -> mark(U82(tt(),N,XS))
     active(U82(tt(),N,XS)) -> mark(fst(splitAt(N,XS)))
     active(afterNth(N,XS)) -> mark(U11(tt(),N,XS))
     active(fst(pair(X,Y))) -> mark(U21(tt(),X))
     active(head(cons(N,XS))) -> mark(U31(tt(),N))
     active(natsFrom(N)) -> mark(cons(N,natsFrom(s(N))))
     active(sel(N,XS)) -> mark(U41(tt(),N,XS))
     active(snd(pair(X,Y))) -> mark(U51(tt(),Y))
     active(splitAt(0(),XS)) -> mark(pair(nil(),XS))
     active(splitAt(s(N),cons(X,XS))) -> mark(U61(tt(),N,X,XS))
     active(tail(cons(N,XS))) -> mark(U71(tt(),XS))
     active(take(N,XS)) -> mark(U81(tt(),N,XS))
     active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3)
     active(U12(X1,X2,X3)) -> U12(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(U21(X1,X2)) -> U21(active(X1),X2)
     active(U22(X1,X2)) -> U22(active(X1),X2)
     active(U31(X1,X2)) -> U31(active(X1),X2)
     active(U32(X1,X2)) -> U32(active(X1),X2)
     active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3)
     active(U42(X1,X2,X3)) -> U42(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(U51(X1,X2)) -> U51(active(X1),X2)
     active(U52(X1,X2)) -> U52(active(X1),X2)
     active(U61(X1,X2,X3,X4)) -> U61(active(X1),X2,X3,X4)
     active(U62(X1,X2,X3,X4)) -> U62(active(X1),X2,X3,X4)
     active(U63(X1,X2,X3,X4)) -> U63(active(X1),X2,X3,X4)
     active(U64(X1,X2)) -> U64(active(X1),X2)
     active(pair(X1,X2)) -> pair(active(X1),X2)
     active(pair(X1,X2)) -> pair(X1,active(X2))
     active(cons(X1,X2)) -> cons(active(X1),X2)
     active(U71(X1,X2)) -> U71(active(X1),X2)
     active(U72(X1,X2)) -> U72(active(X1),X2)
     active(U81(X1,X2,X3)) -> U81(active(X1),X2,X3)
     active(U82(X1,X2,X3)) -> U82(active(X1),X2,X3)
     active(fst(X)) -> fst(active(X))
     active(natsFrom(X)) -> natsFrom(active(X))
     active(s(X)) -> s(active(X))
     active(sel(X1,X2)) -> sel(active(X1),X2)
     active(sel(X1,X2)) -> sel(X1,active(X2))
     active(tail(X)) -> tail(active(X))
     active(take(X1,X2)) -> take(active(X1),X2)
     active(take(X1,X2)) -> take(X1,active(X2))
     snd(mark(X)) -> mark(snd(X))
     snd(ok(X)) -> ok(snd(X))
     head(mark(X)) -> mark(head(X))
     head(ok(X)) -> ok(head(X))
     U64(mark(X1),X2) -> mark(U64(X1,X2))
     U64(ok(X1),ok(X2)) -> ok(U64(X1,X2))
     pair(mark(X1),X2) -> mark(pair(X1,X2))
     pair(X1,mark(X2)) -> mark(pair(X1,X2))
     pair(ok(X1),ok(X2)) -> ok(pair(X1,X2))
     fst(mark(X)) -> mark(fst(X))
     fst(ok(X)) -> ok(fst(X))
     U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3))
     U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3))
     U12(mark(X1),X2,X3) -> mark(U12(X1,X2,X3))
     U12(ok(X1),ok(X2),ok(X3)) -> ok(U12(X1,X2,X3))
     U21(mark(X1),X2) -> mark(U21(X1,X2))
     U21(ok(X1),ok(X2)) -> ok(U21(X1,X2))
     U22(mark(X1),X2) -> mark(U22(X1,X2))
     U22(ok(X1),ok(X2)) -> ok(U22(X1,X2))
     U31(mark(X1),X2) -> mark(U31(X1,X2))
     U31(ok(X1),ok(X2)) -> ok(U31(X1,X2))
     U32(mark(X1),X2) -> mark(U32(X1,X2))
     U32(ok(X1),ok(X2)) -> ok(U32(X1,X2))
     U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3))
     U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3))
     U42(mark(X1),X2,X3) -> mark(U42(X1,X2,X3))
     U42(ok(X1),ok(X2),ok(X3)) -> ok(U42(X1,X2,X3))
     U51(mark(X1),X2) -> mark(U51(X1,X2))
     U51(ok(X1),ok(X2)) -> ok(U51(X1,X2))
     U52(mark(X1),X2) -> mark(U52(X1,X2))
     U52(ok(X1),ok(X2)) -> ok(U52(X1,X2))
     U61(mark(X1),X2,X3,X4) -> mark(U61(X1,X2,X3,X4))
     U61(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U61(X1,X2,X3,X4))
     U62(mark(X1),X2,X3,X4) -> mark(U62(X1,X2,X3,X4))
     U62(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U62(X1,X2,X3,X4))
     U63(mark(X1),X2,X3,X4) -> mark(U63(X1,X2,X3,X4))
     U63(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U63(X1,X2,X3,X4))
     U71(mark(X1),X2) -> mark(U71(X1,X2))
     U71(ok(X1),ok(X2)) -> ok(U71(X1,X2))
     U72(mark(X1),X2) -> mark(U72(X1,X2))
     U72(ok(X1),ok(X2)) -> ok(U72(X1,X2))
     U81(mark(X1),X2,X3) -> mark(U81(X1,X2,X3))
     U81(ok(X1),ok(X2),ok(X3)) -> ok(U81(X1,X2,X3))
     U82(mark(X1),X2,X3) -> mark(U82(X1,X2,X3))
     U82(ok(X1),ok(X2),ok(X3)) -> ok(U82(X1,X2,X3))
     sel(mark(X1),X2) -> mark(sel(X1,X2))
     sel(X1,mark(X2)) -> mark(sel(X1,X2))
     sel(ok(X1),ok(X2)) -> ok(sel(X1,X2))
     tail(mark(X)) -> mark(tail(X))
     tail(ok(X)) -> ok(tail(X))
     take(mark(X1),X2) -> mark(take(X1,X2))
     take(X1,mark(X2)) -> mark(take(X1,X2))
     take(ok(X1),ok(X2)) -> ok(take(X1,X2))
     proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3))
     proper(tt()) -> ok(tt())
     proper(U12(X1,X2,X3)) -> U12(proper(X1),proper(X2),proper(X3))
     proper(snd(X)) -> snd(proper(X))
     proper(splitAt(X1,X2)) -> splitAt(proper(X1),proper(X2))
     proper(U21(X1,X2)) -> U21(proper(X1),proper(X2))
     proper(U22(X1,X2)) -> U22(proper(X1),proper(X2))
     proper(U31(X1,X2)) -> U31(proper(X1),proper(X2))
     proper(U32(X1,X2)) -> U32(proper(X1),proper(X2))
     proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3))
     proper(U42(X1,X2,X3)) -> U42(proper(X1),proper(X2),proper(X3))
     proper(head(X)) -> head(proper(X))
     proper(afterNth(X1,X2)) -> afterNth(proper(X1),proper(X2))
     proper(U51(X1,X2)) -> U51(proper(X1),proper(X2))
     proper(U52(X1,X2)) -> U52(proper(X1),proper(X2))
     proper(U61(X1,X2,X3,X4)) -> U61(proper(X1),proper(X2),proper(X3),proper(X4))
     proper(U62(X1,X2,X3,X4)) -> U62(proper(X1),proper(X2),proper(X3),proper(X4))
     proper(U63(X1,X2,X3,X4)) -> U63(proper(X1),proper(X2),proper(X3),proper(X4))
     proper(U64(X1,X2)) -> U64(proper(X1),proper(X2))
     proper(pair(X1,X2)) -> pair(proper(X1),proper(X2))
     proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
     proper(U71(X1,X2)) -> U71(proper(X1),proper(X2))
     proper(U72(X1,X2)) -> U72(proper(X1),proper(X2))
     proper(U81(X1,X2,X3)) -> U81(proper(X1),proper(X2),proper(X3))
     proper(U82(X1,X2,X3)) -> U82(proper(X1),proper(X2),proper(X3))
     proper(fst(X)) -> fst(proper(X))
     proper(natsFrom(X)) -> natsFrom(proper(X))
     proper(s(X)) -> s(proper(X))
     proper(sel(X1,X2)) -> sel(proper(X1),proper(X2))
     proper(0()) -> ok(0())
     proper(nil()) -> ok(nil())
     proper(tail(X)) -> tail(proper(X))
     proper(take(X1,X2)) -> take(proper(X1),proper(X2))
    graph:
     ...
    Restore Modifier:
     DPs:
      active#(U11(tt(),N,XS)) -> U12#(tt(),N,XS)
      active#(U12(tt(),N,XS)) -> splitAt#(N,XS)
      active#(U12(tt(),N,XS)) -> snd#(splitAt(N,XS))
      active#(U21(tt(),X)) -> U22#(tt(),X)
      active#(U31(tt(),N)) -> U32#(tt(),N)
      active#(U41(tt(),N,XS)) -> U42#(tt(),N,XS)
      active#(U42(tt(),N,XS)) -> afterNth#(N,XS)
      active#(U42(tt(),N,XS)) -> head#(afterNth(N,XS))
      active#(U51(tt(),Y)) -> U52#(tt(),Y)
      active#(U61(tt(),N,X,XS)) -> U62#(tt(),N,X,XS)
      active#(U62(tt(),N,X,XS)) -> U63#(tt(),N,X,XS)
      active#(U63(tt(),N,X,XS)) -> splitAt#(N,XS)
      active#(U63(tt(),N,X,XS)) -> U64#(splitAt(N,XS),X)
      active#(U64(pair(YS,ZS),X)) -> cons#(X,YS)
      active#(U64(pair(YS,ZS),X)) -> pair#(cons(X,YS),ZS)
      active#(U71(tt(),XS)) -> U72#(tt(),XS)
      active#(U81(tt(),N,XS)) -> U82#(tt(),N,XS)
      active#(U82(tt(),N,XS)) -> splitAt#(N,XS)
      active#(U82(tt(),N,XS)) -> fst#(splitAt(N,XS))
      active#(afterNth(N,XS)) -> U11#(tt(),N,XS)
      active#(fst(pair(X,Y))) -> U21#(tt(),X)
      active#(head(cons(N,XS))) -> U31#(tt(),N)
      active#(natsFrom(N)) -> s#(N)
      active#(natsFrom(N)) -> natsFrom#(s(N))
      active#(natsFrom(N)) -> cons#(N,natsFrom(s(N)))
      active#(sel(N,XS)) -> U41#(tt(),N,XS)
      active#(snd(pair(X,Y))) -> U51#(tt(),Y)
      active#(splitAt(0(),XS)) -> pair#(nil(),XS)
      active#(splitAt(s(N),cons(X,XS))) -> U61#(tt(),N,X,XS)
      active#(tail(cons(N,XS))) -> U71#(tt(),XS)
      active#(take(N,XS)) -> U81#(tt(),N,XS)
      active#(U11(X1,X2,X3)) -> active#(X1)
      active#(U11(X1,X2,X3)) -> U11#(active(X1),X2,X3)
      active#(U12(X1,X2,X3)) -> active#(X1)
      active#(U12(X1,X2,X3)) -> U12#(active(X1),X2,X3)
      active#(snd(X)) -> active#(X)
      active#(snd(X)) -> snd#(active(X))
      active#(splitAt(X1,X2)) -> active#(X1)
      active#(splitAt(X1,X2)) -> splitAt#(active(X1),X2)
      active#(splitAt(X1,X2)) -> active#(X2)
      active#(splitAt(X1,X2)) -> splitAt#(X1,active(X2))
      active#(U21(X1,X2)) -> active#(X1)
      active#(U21(X1,X2)) -> U21#(active(X1),X2)
      active#(U22(X1,X2)) -> active#(X1)
      active#(U22(X1,X2)) -> U22#(active(X1),X2)
      active#(U31(X1,X2)) -> active#(X1)
      active#(U31(X1,X2)) -> U31#(active(X1),X2)
      active#(U32(X1,X2)) -> active#(X1)
      active#(U32(X1,X2)) -> U32#(active(X1),X2)
      active#(U41(X1,X2,X3)) -> active#(X1)
      active#(U41(X1,X2,X3)) -> U41#(active(X1),X2,X3)
      active#(U42(X1,X2,X3)) -> active#(X1)
      active#(U42(X1,X2,X3)) -> U42#(active(X1),X2,X3)
      active#(head(X)) -> active#(X)
      active#(head(X)) -> head#(active(X))
      active#(afterNth(X1,X2)) -> active#(X1)
      active#(afterNth(X1,X2)) -> afterNth#(active(X1),X2)
      active#(afterNth(X1,X2)) -> active#(X2)
      active#(afterNth(X1,X2)) -> afterNth#(X1,active(X2))
      active#(U51(X1,X2)) -> active#(X1)
      active#(U51(X1,X2)) -> U51#(active(X1),X2)
      active#(U52(X1,X2)) -> active#(X1)
      active#(U52(X1,X2)) -> U52#(active(X1),X2)
      active#(U61(X1,X2,X3,X4)) -> active#(X1)
      active#(U61(X1,X2,X3,X4)) -> U61#(active(X1),X2,X3,X4)
      active#(U62(X1,X2,X3,X4)) -> active#(X1)
      active#(U62(X1,X2,X3,X4)) -> U62#(active(X1),X2,X3,X4)
      active#(U63(X1,X2,X3,X4)) -> active#(X1)
      active#(U63(X1,X2,X3,X4)) -> U63#(active(X1),X2,X3,X4)
      active#(U64(X1,X2)) -> active#(X1)
      active#(U64(X1,X2)) -> U64#(active(X1),X2)
      active#(pair(X1,X2)) -> active#(X1)
      active#(pair(X1,X2)) -> pair#(active(X1),X2)
      active#(pair(X1,X2)) -> active#(X2)
      active#(pair(X1,X2)) -> pair#(X1,active(X2))
      active#(cons(X1,X2)) -> active#(X1)
      active#(cons(X1,X2)) -> cons#(active(X1),X2)
      active#(U71(X1,X2)) -> active#(X1)
      active#(U71(X1,X2)) -> U71#(active(X1),X2)
      active#(U72(X1,X2)) -> active#(X1)
      active#(U72(X1,X2)) -> U72#(active(X1),X2)
      active#(U81(X1,X2,X3)) -> active#(X1)
      active#(U81(X1,X2,X3)) -> U81#(active(X1),X2,X3)
      active#(U82(X1,X2,X3)) -> active#(X1)
      active#(U82(X1,X2,X3)) -> U82#(active(X1),X2,X3)
      active#(fst(X)) -> active#(X)
      active#(fst(X)) -> fst#(active(X))
      active#(natsFrom(X)) -> active#(X)
      active#(natsFrom(X)) -> natsFrom#(active(X))
      active#(s(X)) -> active#(X)
      active#(s(X)) -> s#(active(X))
      active#(sel(X1,X2)) -> active#(X1)
      active#(sel(X1,X2)) -> sel#(active(X1),X2)
      active#(sel(X1,X2)) -> active#(X2)
      active#(sel(X1,X2)) -> sel#(X1,active(X2))
      active#(tail(X)) -> active#(X)
      active#(tail(X)) -> tail#(active(X))
      active#(take(X1,X2)) -> active#(X1)
      active#(take(X1,X2)) -> take#(active(X1),X2)
      active#(take(X1,X2)) -> active#(X2)
      active#(take(X1,X2)) -> take#(X1,active(X2))
      U11#(mark(X1),X2,X3) -> U11#(X1,X2,X3)
      U12#(mark(X1),X2,X3) -> U12#(X1,X2,X3)
      snd#(mark(X)) -> snd#(X)
      splitAt#(mark(X1),X2) -> splitAt#(X1,X2)
      splitAt#(X1,mark(X2)) -> splitAt#(X1,X2)
      U21#(mark(X1),X2) -> U21#(X1,X2)
      U22#(mark(X1),X2) -> U22#(X1,X2)
      U31#(mark(X1),X2) -> U31#(X1,X2)
      U32#(mark(X1),X2) -> U32#(X1,X2)
      U41#(mark(X1),X2,X3) -> U41#(X1,X2,X3)
      U42#(mark(X1),X2,X3) -> U42#(X1,X2,X3)
      head#(mark(X)) -> head#(X)
      afterNth#(mark(X1),X2) -> afterNth#(X1,X2)
      afterNth#(X1,mark(X2)) -> afterNth#(X1,X2)
      U51#(mark(X1),X2) -> U51#(X1,X2)
      U52#(mark(X1),X2) -> U52#(X1,X2)
      U61#(mark(X1),X2,X3,X4) -> U61#(X1,X2,X3,X4)
      U62#(mark(X1),X2,X3,X4) -> U62#(X1,X2,X3,X4)
      U63#(mark(X1),X2,X3,X4) -> U63#(X1,X2,X3,X4)
      U64#(mark(X1),X2) -> U64#(X1,X2)
      pair#(mark(X1),X2) -> pair#(X1,X2)
      pair#(X1,mark(X2)) -> pair#(X1,X2)
      cons#(mark(X1),X2) -> cons#(X1,X2)
      U71#(mark(X1),X2) -> U71#(X1,X2)
      U72#(mark(X1),X2) -> U72#(X1,X2)
      U81#(mark(X1),X2,X3) -> U81#(X1,X2,X3)
      U82#(mark(X1),X2,X3) -> U82#(X1,X2,X3)
      fst#(mark(X)) -> fst#(X)
      natsFrom#(mark(X)) -> natsFrom#(X)
      s#(mark(X)) -> s#(X)
      sel#(mark(X1),X2) -> sel#(X1,X2)
      sel#(X1,mark(X2)) -> sel#(X1,X2)
      tail#(mark(X)) -> tail#(X)
      take#(mark(X1),X2) -> take#(X1,X2)
      take#(X1,mark(X2)) -> take#(X1,X2)
      proper#(U11(X1,X2,X3)) -> proper#(X3)
      proper#(U11(X1,X2,X3)) -> proper#(X2)
      proper#(U11(X1,X2,X3)) -> proper#(X1)
      proper#(U11(X1,X2,X3)) -> U11#(proper(X1),proper(X2),proper(X3))
      proper#(U12(X1,X2,X3)) -> proper#(X3)
      proper#(U12(X1,X2,X3)) -> proper#(X2)
      proper#(U12(X1,X2,X3)) -> proper#(X1)
      proper#(U12(X1,X2,X3)) -> U12#(proper(X1),proper(X2),proper(X3))
      proper#(snd(X)) -> proper#(X)
      proper#(snd(X)) -> snd#(proper(X))
      proper#(splitAt(X1,X2)) -> proper#(X2)
      proper#(splitAt(X1,X2)) -> proper#(X1)
      proper#(splitAt(X1,X2)) -> splitAt#(proper(X1),proper(X2))
      proper#(U21(X1,X2)) -> proper#(X2)
      proper#(U21(X1,X2)) -> proper#(X1)
      proper#(U21(X1,X2)) -> U21#(proper(X1),proper(X2))
      proper#(U22(X1,X2)) -> proper#(X2)
      proper#(U22(X1,X2)) -> proper#(X1)
      proper#(U22(X1,X2)) -> U22#(proper(X1),proper(X2))
      proper#(U31(X1,X2)) -> proper#(X2)
      proper#(U31(X1,X2)) -> proper#(X1)
      proper#(U31(X1,X2)) -> U31#(proper(X1),proper(X2))
      proper#(U32(X1,X2)) -> proper#(X2)
      proper#(U32(X1,X2)) -> proper#(X1)
      proper#(U32(X1,X2)) -> U32#(proper(X1),proper(X2))
      proper#(U41(X1,X2,X3)) -> proper#(X3)
      proper#(U41(X1,X2,X3)) -> proper#(X2)
      proper#(U41(X1,X2,X3)) -> proper#(X1)
      proper#(U41(X1,X2,X3)) -> U41#(proper(X1),proper(X2),proper(X3))
      proper#(U42(X1,X2,X3)) -> proper#(X3)
      proper#(U42(X1,X2,X3)) -> proper#(X2)
      proper#(U42(X1,X2,X3)) -> proper#(X1)
      proper#(U42(X1,X2,X3)) -> U42#(proper(X1),proper(X2),proper(X3))
      proper#(head(X)) -> proper#(X)
      proper#(head(X)) -> head#(proper(X))
      proper#(afterNth(X1,X2)) -> proper#(X2)
      proper#(afterNth(X1,X2)) -> proper#(X1)
      proper#(afterNth(X1,X2)) -> afterNth#(proper(X1),proper(X2))
      proper#(U51(X1,X2)) -> proper#(X2)
      proper#(U51(X1,X2)) -> proper#(X1)
      proper#(U51(X1,X2)) -> U51#(proper(X1),proper(X2))
      proper#(U52(X1,X2)) -> proper#(X2)
      proper#(U52(X1,X2)) -> proper#(X1)
      proper#(U52(X1,X2)) -> U52#(proper(X1),proper(X2))
      proper#(U61(X1,X2,X3,X4)) -> proper#(X4)
      proper#(U61(X1,X2,X3,X4)) -> proper#(X3)
      proper#(U61(X1,X2,X3,X4)) -> proper#(X2)
      proper#(U61(X1,X2,X3,X4)) -> proper#(X1)
      proper#(U61(X1,X2,X3,X4)) -> U61#(proper(X1),proper(X2),proper(X3),proper(X4))
      proper#(U62(X1,X2,X3,X4)) -> proper#(X4)
      proper#(U62(X1,X2,X3,X4)) -> proper#(X3)
      proper#(U62(X1,X2,X3,X4)) -> proper#(X2)
      proper#(U62(X1,X2,X3,X4)) -> proper#(X1)
      proper#(U62(X1,X2,X3,X4)) -> U62#(proper(X1),proper(X2),proper(X3),proper(X4))
      proper#(U63(X1,X2,X3,X4)) -> proper#(X4)
      proper#(U63(X1,X2,X3,X4)) -> proper#(X3)
      proper#(U63(X1,X2,X3,X4)) -> proper#(X2)
      proper#(U63(X1,X2,X3,X4)) -> proper#(X1)
      proper#(U63(X1,X2,X3,X4)) -> U63#(proper(X1),proper(X2),proper(X3),proper(X4))
      proper#(U64(X1,X2)) -> proper#(X2)
      proper#(U64(X1,X2)) -> proper#(X1)
      proper#(U64(X1,X2)) -> U64#(proper(X1),proper(X2))
      proper#(pair(X1,X2)) -> proper#(X2)
      proper#(pair(X1,X2)) -> proper#(X1)
      proper#(pair(X1,X2)) -> pair#(proper(X1),proper(X2))
      proper#(cons(X1,X2)) -> proper#(X2)
      proper#(cons(X1,X2)) -> proper#(X1)
      proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2))
      proper#(U71(X1,X2)) -> proper#(X2)
      proper#(U71(X1,X2)) -> proper#(X1)
      proper#(U71(X1,X2)) -> U71#(proper(X1),proper(X2))
      proper#(U72(X1,X2)) -> proper#(X2)
      proper#(U72(X1,X2)) -> proper#(X1)
      proper#(U72(X1,X2)) -> U72#(proper(X1),proper(X2))
      proper#(U81(X1,X2,X3)) -> proper#(X3)
      proper#(U81(X1,X2,X3)) -> proper#(X2)
      proper#(U81(X1,X2,X3)) -> proper#(X1)
      proper#(U81(X1,X2,X3)) -> U81#(proper(X1),proper(X2),proper(X3))
      proper#(U82(X1,X2,X3)) -> proper#(X3)
      proper#(U82(X1,X2,X3)) -> proper#(X2)
      proper#(U82(X1,X2,X3)) -> proper#(X1)
      proper#(U82(X1,X2,X3)) -> U82#(proper(X1),proper(X2),proper(X3))
      proper#(fst(X)) -> proper#(X)
      proper#(fst(X)) -> fst#(proper(X))
      proper#(natsFrom(X)) -> proper#(X)
      proper#(natsFrom(X)) -> natsFrom#(proper(X))
      proper#(s(X)) -> proper#(X)
      proper#(s(X)) -> s#(proper(X))
      proper#(sel(X1,X2)) -> proper#(X2)
      proper#(sel(X1,X2)) -> proper#(X1)
      proper#(sel(X1,X2)) -> sel#(proper(X1),proper(X2))
      proper#(tail(X)) -> proper#(X)
      proper#(tail(X)) -> tail#(proper(X))
      proper#(take(X1,X2)) -> proper#(X2)
      proper#(take(X1,X2)) -> proper#(X1)
      proper#(take(X1,X2)) -> take#(proper(X1),proper(X2))
      U11#(ok(X1),ok(X2),ok(X3)) -> U11#(X1,X2,X3)
      U12#(ok(X1),ok(X2),ok(X3)) -> U12#(X1,X2,X3)
      snd#(ok(X)) -> snd#(X)
      splitAt#(ok(X1),ok(X2)) -> splitAt#(X1,X2)
      U21#(ok(X1),ok(X2)) -> U21#(X1,X2)
      U22#(ok(X1),ok(X2)) -> U22#(X1,X2)
      U31#(ok(X1),ok(X2)) -> U31#(X1,X2)
      U32#(ok(X1),ok(X2)) -> U32#(X1,X2)
      U41#(ok(X1),ok(X2),ok(X3)) -> U41#(X1,X2,X3)
      U42#(ok(X1),ok(X2),ok(X3)) -> U42#(X1,X2,X3)
      head#(ok(X)) -> head#(X)
      afterNth#(ok(X1),ok(X2)) -> afterNth#(X1,X2)
      U51#(ok(X1),ok(X2)) -> U51#(X1,X2)
      U52#(ok(X1),ok(X2)) -> U52#(X1,X2)
      U61#(ok(X1),ok(X2),ok(X3),ok(X4)) -> U61#(X1,X2,X3,X4)
      U62#(ok(X1),ok(X2),ok(X3),ok(X4)) -> U62#(X1,X2,X3,X4)
      U63#(ok(X1),ok(X2),ok(X3),ok(X4)) -> U63#(X1,X2,X3,X4)
      U64#(ok(X1),ok(X2)) -> U64#(X1,X2)
      pair#(ok(X1),ok(X2)) -> pair#(X1,X2)
      cons#(ok(X1),ok(X2)) -> cons#(X1,X2)
      U71#(ok(X1),ok(X2)) -> U71#(X1,X2)
      U72#(ok(X1),ok(X2)) -> U72#(X1,X2)
      U81#(ok(X1),ok(X2),ok(X3)) -> U81#(X1,X2,X3)
      U82#(ok(X1),ok(X2),ok(X3)) -> U82#(X1,X2,X3)
      fst#(ok(X)) -> fst#(X)
      natsFrom#(ok(X)) -> natsFrom#(X)
      s#(ok(X)) -> s#(X)
      sel#(ok(X1),ok(X2)) -> sel#(X1,X2)
      tail#(ok(X)) -> tail#(X)
      take#(ok(X1),ok(X2)) -> take#(X1,X2)
      top#(mark(X)) -> proper#(X)
      top#(mark(X)) -> top#(proper(X))
      top#(ok(X)) -> active#(X)
      top#(ok(X)) -> top#(active(X))
     TRS:
      active(U11(tt(),N,XS)) -> mark(U12(tt(),N,XS))
      active(U12(tt(),N,XS)) -> mark(snd(splitAt(N,XS)))
      active(U21(tt(),X)) -> mark(U22(tt(),X))
      active(U22(tt(),X)) -> mark(X)
      active(U31(tt(),N)) -> mark(U32(tt(),N))
      active(U32(tt(),N)) -> mark(N)
      active(U41(tt(),N,XS)) -> mark(U42(tt(),N,XS))
      active(U42(tt(),N,XS)) -> mark(head(afterNth(N,XS)))
      active(U51(tt(),Y)) -> mark(U52(tt(),Y))
      active(U52(tt(),Y)) -> mark(Y)
      active(U61(tt(),N,X,XS)) -> mark(U62(tt(),N,X,XS))
      active(U62(tt(),N,X,XS)) -> mark(U63(tt(),N,X,XS))
      active(U63(tt(),N,X,XS)) -> mark(U64(splitAt(N,XS),X))
      active(U64(pair(YS,ZS),X)) -> mark(pair(cons(X,YS),ZS))
      active(U71(tt(),XS)) -> mark(U72(tt(),XS))
      active(U72(tt(),XS)) -> mark(XS)
      active(U81(tt(),N,XS)) -> mark(U82(tt(),N,XS))
      active(U82(tt(),N,XS)) -> mark(fst(splitAt(N,XS)))
      active(afterNth(N,XS)) -> mark(U11(tt(),N,XS))
      active(fst(pair(X,Y))) -> mark(U21(tt(),X))
      active(head(cons(N,XS))) -> mark(U31(tt(),N))
      active(natsFrom(N)) -> mark(cons(N,natsFrom(s(N))))
      active(sel(N,XS)) -> mark(U41(tt(),N,XS))
      active(snd(pair(X,Y))) -> mark(U51(tt(),Y))
      active(splitAt(0(),XS)) -> mark(pair(nil(),XS))
      active(splitAt(s(N),cons(X,XS))) -> mark(U61(tt(),N,X,XS))
      active(tail(cons(N,XS))) -> mark(U71(tt(),XS))
      active(take(N,XS)) -> mark(U81(tt(),N,XS))
      active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3)
      active(U12(X1,X2,X3)) -> U12(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(U21(X1,X2)) -> U21(active(X1),X2)
      active(U22(X1,X2)) -> U22(active(X1),X2)
      active(U31(X1,X2)) -> U31(active(X1),X2)
      active(U32(X1,X2)) -> U32(active(X1),X2)
      active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3)
      active(U42(X1,X2,X3)) -> U42(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(U51(X1,X2)) -> U51(active(X1),X2)
      active(U52(X1,X2)) -> U52(active(X1),X2)
      active(U61(X1,X2,X3,X4)) -> U61(active(X1),X2,X3,X4)
      active(U62(X1,X2,X3,X4)) -> U62(active(X1),X2,X3,X4)
      active(U63(X1,X2,X3,X4)) -> U63(active(X1),X2,X3,X4)
      active(U64(X1,X2)) -> U64(active(X1),X2)
      active(pair(X1,X2)) -> pair(active(X1),X2)
      active(pair(X1,X2)) -> pair(X1,active(X2))
      active(cons(X1,X2)) -> cons(active(X1),X2)
      active(U71(X1,X2)) -> U71(active(X1),X2)
      active(U72(X1,X2)) -> U72(active(X1),X2)
      active(U81(X1,X2,X3)) -> U81(active(X1),X2,X3)
      active(U82(X1,X2,X3)) -> U82(active(X1),X2,X3)
      active(fst(X)) -> fst(active(X))
      active(natsFrom(X)) -> natsFrom(active(X))
      active(s(X)) -> s(active(X))
      active(sel(X1,X2)) -> sel(active(X1),X2)
      active(sel(X1,X2)) -> sel(X1,active(X2))
      active(tail(X)) -> tail(active(X))
      active(take(X1,X2)) -> take(active(X1),X2)
      active(take(X1,X2)) -> take(X1,active(X2))
      U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3))
      U12(mark(X1),X2,X3) -> mark(U12(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))
      U21(mark(X1),X2) -> mark(U21(X1,X2))
      U22(mark(X1),X2) -> mark(U22(X1,X2))
      U31(mark(X1),X2) -> mark(U31(X1,X2))
      U32(mark(X1),X2) -> mark(U32(X1,X2))
      U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3))
      U42(mark(X1),X2,X3) -> mark(U42(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))
      U51(mark(X1),X2) -> mark(U51(X1,X2))
      U52(mark(X1),X2) -> mark(U52(X1,X2))
      U61(mark(X1),X2,X3,X4) -> mark(U61(X1,X2,X3,X4))
      U62(mark(X1),X2,X3,X4) -> mark(U62(X1,X2,X3,X4))
      U63(mark(X1),X2,X3,X4) -> mark(U63(X1,X2,X3,X4))
      U64(mark(X1),X2) -> mark(U64(X1,X2))
      pair(mark(X1),X2) -> mark(pair(X1,X2))
      pair(X1,mark(X2)) -> mark(pair(X1,X2))
      cons(mark(X1),X2) -> mark(cons(X1,X2))
      U71(mark(X1),X2) -> mark(U71(X1,X2))
      U72(mark(X1),X2) -> mark(U72(X1,X2))
      U81(mark(X1),X2,X3) -> mark(U81(X1,X2,X3))
      U82(mark(X1),X2,X3) -> mark(U82(X1,X2,X3))
      fst(mark(X)) -> mark(fst(X))
      natsFrom(mark(X)) -> mark(natsFrom(X))
      s(mark(X)) -> mark(s(X))
      sel(mark(X1),X2) -> mark(sel(X1,X2))
      sel(X1,mark(X2)) -> mark(sel(X1,X2))
      tail(mark(X)) -> mark(tail(X))
      take(mark(X1),X2) -> mark(take(X1,X2))
      take(X1,mark(X2)) -> mark(take(X1,X2))
      proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3))
      proper(tt()) -> ok(tt())
      proper(U12(X1,X2,X3)) -> U12(proper(X1),proper(X2),proper(X3))
      proper(snd(X)) -> snd(proper(X))
      proper(splitAt(X1,X2)) -> splitAt(proper(X1),proper(X2))
      proper(U21(X1,X2)) -> U21(proper(X1),proper(X2))
      proper(U22(X1,X2)) -> U22(proper(X1),proper(X2))
      proper(U31(X1,X2)) -> U31(proper(X1),proper(X2))
      proper(U32(X1,X2)) -> U32(proper(X1),proper(X2))
      proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3))
      proper(U42(X1,X2,X3)) -> U42(proper(X1),proper(X2),proper(X3))
      proper(head(X)) -> head(proper(X))
      proper(afterNth(X1,X2)) -> afterNth(proper(X1),proper(X2))
      proper(U51(X1,X2)) -> U51(proper(X1),proper(X2))
      proper(U52(X1,X2)) -> U52(proper(X1),proper(X2))
      proper(U61(X1,X2,X3,X4)) -> U61(proper(X1),proper(X2),proper(X3),proper(X4))
      proper(U62(X1,X2,X3,X4)) -> U62(proper(X1),proper(X2),proper(X3),proper(X4))
      proper(U63(X1,X2,X3,X4)) -> U63(proper(X1),proper(X2),proper(X3),proper(X4))
      proper(U64(X1,X2)) -> U64(proper(X1),proper(X2))
      proper(pair(X1,X2)) -> pair(proper(X1),proper(X2))
      proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
      proper(U71(X1,X2)) -> U71(proper(X1),proper(X2))
      proper(U72(X1,X2)) -> U72(proper(X1),proper(X2))
      proper(U81(X1,X2,X3)) -> U81(proper(X1),proper(X2),proper(X3))
      proper(U82(X1,X2,X3)) -> U82(proper(X1),proper(X2),proper(X3))
      proper(fst(X)) -> fst(proper(X))
      proper(natsFrom(X)) -> natsFrom(proper(X))
      proper(s(X)) -> s(proper(X))
      proper(sel(X1,X2)) -> sel(proper(X1),proper(X2))
      proper(0()) -> ok(0())
      proper(nil()) -> ok(nil())
      proper(tail(X)) -> tail(proper(X))
      proper(take(X1,X2)) -> take(proper(X1),proper(X2))
      U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3))
      U12(ok(X1),ok(X2),ok(X3)) -> ok(U12(X1,X2,X3))
      snd(ok(X)) -> ok(snd(X))
      splitAt(ok(X1),ok(X2)) -> ok(splitAt(X1,X2))
      U21(ok(X1),ok(X2)) -> ok(U21(X1,X2))
      U22(ok(X1),ok(X2)) -> ok(U22(X1,X2))
      U31(ok(X1),ok(X2)) -> ok(U31(X1,X2))
      U32(ok(X1),ok(X2)) -> ok(U32(X1,X2))
      U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3))
      U42(ok(X1),ok(X2),ok(X3)) -> ok(U42(X1,X2,X3))
      head(ok(X)) -> ok(head(X))
      afterNth(ok(X1),ok(X2)) -> ok(afterNth(X1,X2))
      U51(ok(X1),ok(X2)) -> ok(U51(X1,X2))
      U52(ok(X1),ok(X2)) -> ok(U52(X1,X2))
      U61(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U61(X1,X2,X3,X4))
      U62(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U62(X1,X2,X3,X4))
      U63(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U63(X1,X2,X3,X4))
      U64(ok(X1),ok(X2)) -> ok(U64(X1,X2))
      pair(ok(X1),ok(X2)) -> ok(pair(X1,X2))
      cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
      U71(ok(X1),ok(X2)) -> ok(U71(X1,X2))
      U72(ok(X1),ok(X2)) -> ok(U72(X1,X2))
      U81(ok(X1),ok(X2),ok(X3)) -> ok(U81(X1,X2,X3))
      U82(ok(X1),ok(X2),ok(X3)) -> ok(U82(X1,X2,X3))
      fst(ok(X)) -> ok(fst(X))
      natsFrom(ok(X)) -> ok(natsFrom(X))
      s(ok(X)) -> ok(s(X))
      sel(ok(X1),ok(X2)) -> ok(sel(X1,X2))
      tail(ok(X)) -> ok(tail(X))
      take(ok(X1),ok(X2)) -> ok(take(X1,X2))
      top(mark(X)) -> top(proper(X))
      top(ok(X)) -> top(active(X))
     SCC Processor:
      #sccs: 33
      #rules: 168
      #arcs: 10434/70756
      DPs:
       top#(ok(X)) -> top#(active(X))
       top#(mark(X)) -> top#(proper(X))
      TRS:
       active(U11(tt(),N,XS)) -> mark(U12(tt(),N,XS))
       active(U12(tt(),N,XS)) -> mark(snd(splitAt(N,XS)))
       active(U21(tt(),X)) -> mark(U22(tt(),X))
       active(U22(tt(),X)) -> mark(X)
       active(U31(tt(),N)) -> mark(U32(tt(),N))
       active(U32(tt(),N)) -> mark(N)
       active(U41(tt(),N,XS)) -> mark(U42(tt(),N,XS))
       active(U42(tt(),N,XS)) -> mark(head(afterNth(N,XS)))
       active(U51(tt(),Y)) -> mark(U52(tt(),Y))
       active(U52(tt(),Y)) -> mark(Y)
       active(U61(tt(),N,X,XS)) -> mark(U62(tt(),N,X,XS))
       active(U62(tt(),N,X,XS)) -> mark(U63(tt(),N,X,XS))
       active(U63(tt(),N,X,XS)) -> mark(U64(splitAt(N,XS),X))
       active(U64(pair(YS,ZS),X)) -> mark(pair(cons(X,YS),ZS))
       active(U71(tt(),XS)) -> mark(U72(tt(),XS))
       active(U72(tt(),XS)) -> mark(XS)
       active(U81(tt(),N,XS)) -> mark(U82(tt(),N,XS))
       active(U82(tt(),N,XS)) -> mark(fst(splitAt(N,XS)))
       active(afterNth(N,XS)) -> mark(U11(tt(),N,XS))
       active(fst(pair(X,Y))) -> mark(U21(tt(),X))
       active(head(cons(N,XS))) -> mark(U31(tt(),N))
       active(natsFrom(N)) -> mark(cons(N,natsFrom(s(N))))
       active(sel(N,XS)) -> mark(U41(tt(),N,XS))
       active(snd(pair(X,Y))) -> mark(U51(tt(),Y))
       active(splitAt(0(),XS)) -> mark(pair(nil(),XS))
       active(splitAt(s(N),cons(X,XS))) -> mark(U61(tt(),N,X,XS))
       active(tail(cons(N,XS))) -> mark(U71(tt(),XS))
       active(take(N,XS)) -> mark(U81(tt(),N,XS))
       active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3)
       active(U12(X1,X2,X3)) -> U12(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(U21(X1,X2)) -> U21(active(X1),X2)
       active(U22(X1,X2)) -> U22(active(X1),X2)
       active(U31(X1,X2)) -> U31(active(X1),X2)
       active(U32(X1,X2)) -> U32(active(X1),X2)
       active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3)
       active(U42(X1,X2,X3)) -> U42(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(U51(X1,X2)) -> U51(active(X1),X2)
       active(U52(X1,X2)) -> U52(active(X1),X2)
       active(U61(X1,X2,X3,X4)) -> U61(active(X1),X2,X3,X4)
       active(U62(X1,X2,X3,X4)) -> U62(active(X1),X2,X3,X4)
       active(U63(X1,X2,X3,X4)) -> U63(active(X1),X2,X3,X4)
       active(U64(X1,X2)) -> U64(active(X1),X2)
       active(pair(X1,X2)) -> pair(active(X1),X2)
       active(pair(X1,X2)) -> pair(X1,active(X2))
       active(cons(X1,X2)) -> cons(active(X1),X2)
       active(U71(X1,X2)) -> U71(active(X1),X2)
       active(U72(X1,X2)) -> U72(active(X1),X2)
       active(U81(X1,X2,X3)) -> U81(active(X1),X2,X3)
       active(U82(X1,X2,X3)) -> U82(active(X1),X2,X3)
       active(fst(X)) -> fst(active(X))
       active(natsFrom(X)) -> natsFrom(active(X))
       active(s(X)) -> s(active(X))
       active(sel(X1,X2)) -> sel(active(X1),X2)
       active(sel(X1,X2)) -> sel(X1,active(X2))
       active(tail(X)) -> tail(active(X))
       active(take(X1,X2)) -> take(active(X1),X2)
       active(take(X1,X2)) -> take(X1,active(X2))
       U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3))
       U12(mark(X1),X2,X3) -> mark(U12(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))
       U21(mark(X1),X2) -> mark(U21(X1,X2))
       U22(mark(X1),X2) -> mark(U22(X1,X2))
       U31(mark(X1),X2) -> mark(U31(X1,X2))
       U32(mark(X1),X2) -> mark(U32(X1,X2))
       U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3))
       U42(mark(X1),X2,X3) -> mark(U42(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))
       U51(mark(X1),X2) -> mark(U51(X1,X2))
       U52(mark(X1),X2) -> mark(U52(X1,X2))
       U61(mark(X1),X2,X3,X4) -> mark(U61(X1,X2,X3,X4))
       U62(mark(X1),X2,X3,X4) -> mark(U62(X1,X2,X3,X4))
       U63(mark(X1),X2,X3,X4) -> mark(U63(X1,X2,X3,X4))
       U64(mark(X1),X2) -> mark(U64(X1,X2))
       pair(mark(X1),X2) -> mark(pair(X1,X2))
       pair(X1,mark(X2)) -> mark(pair(X1,X2))
       cons(mark(X1),X2) -> mark(cons(X1,X2))
       U71(mark(X1),X2) -> mark(U71(X1,X2))
       U72(mark(X1),X2) -> mark(U72(X1,X2))
       U81(mark(X1),X2,X3) -> mark(U81(X1,X2,X3))
       U82(mark(X1),X2,X3) -> mark(U82(X1,X2,X3))
       fst(mark(X)) -> mark(fst(X))
       natsFrom(mark(X)) -> mark(natsFrom(X))
       s(mark(X)) -> mark(s(X))
       sel(mark(X1),X2) -> mark(sel(X1,X2))
       sel(X1,mark(X2)) -> mark(sel(X1,X2))
       tail(mark(X)) -> mark(tail(X))
       take(mark(X1),X2) -> mark(take(X1,X2))
       take(X1,mark(X2)) -> mark(take(X1,X2))
       proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3))
       proper(tt()) -> ok(tt())
       proper(U12(X1,X2,X3)) -> U12(proper(X1),proper(X2),proper(X3))
       proper(snd(X)) -> snd(proper(X))
       proper(splitAt(X1,X2)) -> splitAt(proper(X1),proper(X2))
       proper(U21(X1,X2)) -> U21(proper(X1),proper(X2))
       proper(U22(X1,X2)) -> U22(proper(X1),proper(X2))
       proper(U31(X1,X2)) -> U31(proper(X1),proper(X2))
       proper(U32(X1,X2)) -> U32(proper(X1),proper(X2))
       proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3))
       proper(U42(X1,X2,X3)) -> U42(proper(X1),proper(X2),proper(X3))
       proper(head(X)) -> head(proper(X))
       proper(afterNth(X1,X2)) -> afterNth(proper(X1),proper(X2))
       proper(U51(X1,X2)) -> U51(proper(X1),proper(X2))
       proper(U52(X1,X2)) -> U52(proper(X1),proper(X2))
       proper(U61(X1,X2,X3,X4)) -> U61(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(U62(X1,X2,X3,X4)) -> U62(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(U63(X1,X2,X3,X4)) -> U63(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(U64(X1,X2)) -> U64(proper(X1),proper(X2))
       proper(pair(X1,X2)) -> pair(proper(X1),proper(X2))
       proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
       proper(U71(X1,X2)) -> U71(proper(X1),proper(X2))
       proper(U72(X1,X2)) -> U72(proper(X1),proper(X2))
       proper(U81(X1,X2,X3)) -> U81(proper(X1),proper(X2),proper(X3))
       proper(U82(X1,X2,X3)) -> U82(proper(X1),proper(X2),proper(X3))
       proper(fst(X)) -> fst(proper(X))
       proper(natsFrom(X)) -> natsFrom(proper(X))
       proper(s(X)) -> s(proper(X))
       proper(sel(X1,X2)) -> sel(proper(X1),proper(X2))
       proper(0()) -> ok(0())
       proper(nil()) -> ok(nil())
       proper(tail(X)) -> tail(proper(X))
       proper(take(X1,X2)) -> take(proper(X1),proper(X2))
       U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3))
       U12(ok(X1),ok(X2),ok(X3)) -> ok(U12(X1,X2,X3))
       snd(ok(X)) -> ok(snd(X))
       splitAt(ok(X1),ok(X2)) -> ok(splitAt(X1,X2))
       U21(ok(X1),ok(X2)) -> ok(U21(X1,X2))
       U22(ok(X1),ok(X2)) -> ok(U22(X1,X2))
       U31(ok(X1),ok(X2)) -> ok(U31(X1,X2))
       U32(ok(X1),ok(X2)) -> ok(U32(X1,X2))
       U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3))
       U42(ok(X1),ok(X2),ok(X3)) -> ok(U42(X1,X2,X3))
       head(ok(X)) -> ok(head(X))
       afterNth(ok(X1),ok(X2)) -> ok(afterNth(X1,X2))
       U51(ok(X1),ok(X2)) -> ok(U51(X1,X2))
       U52(ok(X1),ok(X2)) -> ok(U52(X1,X2))
       U61(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U61(X1,X2,X3,X4))
       U62(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U62(X1,X2,X3,X4))
       U63(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U63(X1,X2,X3,X4))
       U64(ok(X1),ok(X2)) -> ok(U64(X1,X2))
       pair(ok(X1),ok(X2)) -> ok(pair(X1,X2))
       cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
       U71(ok(X1),ok(X2)) -> ok(U71(X1,X2))
       U72(ok(X1),ok(X2)) -> ok(U72(X1,X2))
       U81(ok(X1),ok(X2),ok(X3)) -> ok(U81(X1,X2,X3))
       U82(ok(X1),ok(X2),ok(X3)) -> ok(U82(X1,X2,X3))
       fst(ok(X)) -> ok(fst(X))
       natsFrom(ok(X)) -> ok(natsFrom(X))
       s(ok(X)) -> ok(s(X))
       sel(ok(X1),ok(X2)) -> ok(sel(X1,X2))
       tail(ok(X)) -> ok(tail(X))
       take(ok(X1),ok(X2)) -> ok(take(X1,X2))
       top(mark(X)) -> top(proper(X))
       top(ok(X)) -> top(active(X))
      Open
      
      DPs:
       active#(U11(X1,X2,X3)) -> active#(X1)
       active#(U12(X1,X2,X3)) -> active#(X1)
       active#(snd(X)) -> active#(X)
       active#(splitAt(X1,X2)) -> active#(X1)
       active#(splitAt(X1,X2)) -> active#(X2)
       active#(U21(X1,X2)) -> active#(X1)
       active#(U22(X1,X2)) -> active#(X1)
       active#(U31(X1,X2)) -> active#(X1)
       active#(U32(X1,X2)) -> active#(X1)
       active#(U41(X1,X2,X3)) -> active#(X1)
       active#(U42(X1,X2,X3)) -> active#(X1)
       active#(head(X)) -> active#(X)
       active#(afterNth(X1,X2)) -> active#(X1)
       active#(afterNth(X1,X2)) -> active#(X2)
       active#(U51(X1,X2)) -> active#(X1)
       active#(U52(X1,X2)) -> active#(X1)
       active#(U61(X1,X2,X3,X4)) -> active#(X1)
       active#(U62(X1,X2,X3,X4)) -> active#(X1)
       active#(U63(X1,X2,X3,X4)) -> active#(X1)
       active#(U64(X1,X2)) -> active#(X1)
       active#(pair(X1,X2)) -> active#(X1)
       active#(pair(X1,X2)) -> active#(X2)
       active#(cons(X1,X2)) -> active#(X1)
       active#(U71(X1,X2)) -> active#(X1)
       active#(U72(X1,X2)) -> active#(X1)
       active#(U81(X1,X2,X3)) -> active#(X1)
       active#(U82(X1,X2,X3)) -> active#(X1)
       active#(fst(X)) -> active#(X)
       active#(natsFrom(X)) -> active#(X)
       active#(s(X)) -> active#(X)
       active#(sel(X1,X2)) -> active#(X1)
       active#(sel(X1,X2)) -> active#(X2)
       active#(tail(X)) -> active#(X)
       active#(take(X1,X2)) -> active#(X1)
       active#(take(X1,X2)) -> active#(X2)
      TRS:
       active(U11(tt(),N,XS)) -> mark(U12(tt(),N,XS))
       active(U12(tt(),N,XS)) -> mark(snd(splitAt(N,XS)))
       active(U21(tt(),X)) -> mark(U22(tt(),X))
       active(U22(tt(),X)) -> mark(X)
       active(U31(tt(),N)) -> mark(U32(tt(),N))
       active(U32(tt(),N)) -> mark(N)
       active(U41(tt(),N,XS)) -> mark(U42(tt(),N,XS))
       active(U42(tt(),N,XS)) -> mark(head(afterNth(N,XS)))
       active(U51(tt(),Y)) -> mark(U52(tt(),Y))
       active(U52(tt(),Y)) -> mark(Y)
       active(U61(tt(),N,X,XS)) -> mark(U62(tt(),N,X,XS))
       active(U62(tt(),N,X,XS)) -> mark(U63(tt(),N,X,XS))
       active(U63(tt(),N,X,XS)) -> mark(U64(splitAt(N,XS),X))
       active(U64(pair(YS,ZS),X)) -> mark(pair(cons(X,YS),ZS))
       active(U71(tt(),XS)) -> mark(U72(tt(),XS))
       active(U72(tt(),XS)) -> mark(XS)
       active(U81(tt(),N,XS)) -> mark(U82(tt(),N,XS))
       active(U82(tt(),N,XS)) -> mark(fst(splitAt(N,XS)))
       active(afterNth(N,XS)) -> mark(U11(tt(),N,XS))
       active(fst(pair(X,Y))) -> mark(U21(tt(),X))
       active(head(cons(N,XS))) -> mark(U31(tt(),N))
       active(natsFrom(N)) -> mark(cons(N,natsFrom(s(N))))
       active(sel(N,XS)) -> mark(U41(tt(),N,XS))
       active(snd(pair(X,Y))) -> mark(U51(tt(),Y))
       active(splitAt(0(),XS)) -> mark(pair(nil(),XS))
       active(splitAt(s(N),cons(X,XS))) -> mark(U61(tt(),N,X,XS))
       active(tail(cons(N,XS))) -> mark(U71(tt(),XS))
       active(take(N,XS)) -> mark(U81(tt(),N,XS))
       active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3)
       active(U12(X1,X2,X3)) -> U12(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(U21(X1,X2)) -> U21(active(X1),X2)
       active(U22(X1,X2)) -> U22(active(X1),X2)
       active(U31(X1,X2)) -> U31(active(X1),X2)
       active(U32(X1,X2)) -> U32(active(X1),X2)
       active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3)
       active(U42(X1,X2,X3)) -> U42(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(U51(X1,X2)) -> U51(active(X1),X2)
       active(U52(X1,X2)) -> U52(active(X1),X2)
       active(U61(X1,X2,X3,X4)) -> U61(active(X1),X2,X3,X4)
       active(U62(X1,X2,X3,X4)) -> U62(active(X1),X2,X3,X4)
       active(U63(X1,X2,X3,X4)) -> U63(active(X1),X2,X3,X4)
       active(U64(X1,X2)) -> U64(active(X1),X2)
       active(pair(X1,X2)) -> pair(active(X1),X2)
       active(pair(X1,X2)) -> pair(X1,active(X2))
       active(cons(X1,X2)) -> cons(active(X1),X2)
       active(U71(X1,X2)) -> U71(active(X1),X2)
       active(U72(X1,X2)) -> U72(active(X1),X2)
       active(U81(X1,X2,X3)) -> U81(active(X1),X2,X3)
       active(U82(X1,X2,X3)) -> U82(active(X1),X2,X3)
       active(fst(X)) -> fst(active(X))
       active(natsFrom(X)) -> natsFrom(active(X))
       active(s(X)) -> s(active(X))
       active(sel(X1,X2)) -> sel(active(X1),X2)
       active(sel(X1,X2)) -> sel(X1,active(X2))
       active(tail(X)) -> tail(active(X))
       active(take(X1,X2)) -> take(active(X1),X2)
       active(take(X1,X2)) -> take(X1,active(X2))
       U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3))
       U12(mark(X1),X2,X3) -> mark(U12(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))
       U21(mark(X1),X2) -> mark(U21(X1,X2))
       U22(mark(X1),X2) -> mark(U22(X1,X2))
       U31(mark(X1),X2) -> mark(U31(X1,X2))
       U32(mark(X1),X2) -> mark(U32(X1,X2))
       U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3))
       U42(mark(X1),X2,X3) -> mark(U42(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))
       U51(mark(X1),X2) -> mark(U51(X1,X2))
       U52(mark(X1),X2) -> mark(U52(X1,X2))
       U61(mark(X1),X2,X3,X4) -> mark(U61(X1,X2,X3,X4))
       U62(mark(X1),X2,X3,X4) -> mark(U62(X1,X2,X3,X4))
       U63(mark(X1),X2,X3,X4) -> mark(U63(X1,X2,X3,X4))
       U64(mark(X1),X2) -> mark(U64(X1,X2))
       pair(mark(X1),X2) -> mark(pair(X1,X2))
       pair(X1,mark(X2)) -> mark(pair(X1,X2))
       cons(mark(X1),X2) -> mark(cons(X1,X2))
       U71(mark(X1),X2) -> mark(U71(X1,X2))
       U72(mark(X1),X2) -> mark(U72(X1,X2))
       U81(mark(X1),X2,X3) -> mark(U81(X1,X2,X3))
       U82(mark(X1),X2,X3) -> mark(U82(X1,X2,X3))
       fst(mark(X)) -> mark(fst(X))
       natsFrom(mark(X)) -> mark(natsFrom(X))
       s(mark(X)) -> mark(s(X))
       sel(mark(X1),X2) -> mark(sel(X1,X2))
       sel(X1,mark(X2)) -> mark(sel(X1,X2))
       tail(mark(X)) -> mark(tail(X))
       take(mark(X1),X2) -> mark(take(X1,X2))
       take(X1,mark(X2)) -> mark(take(X1,X2))
       proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3))
       proper(tt()) -> ok(tt())
       proper(U12(X1,X2,X3)) -> U12(proper(X1),proper(X2),proper(X3))
       proper(snd(X)) -> snd(proper(X))
       proper(splitAt(X1,X2)) -> splitAt(proper(X1),proper(X2))
       proper(U21(X1,X2)) -> U21(proper(X1),proper(X2))
       proper(U22(X1,X2)) -> U22(proper(X1),proper(X2))
       proper(U31(X1,X2)) -> U31(proper(X1),proper(X2))
       proper(U32(X1,X2)) -> U32(proper(X1),proper(X2))
       proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3))
       proper(U42(X1,X2,X3)) -> U42(proper(X1),proper(X2),proper(X3))
       proper(head(X)) -> head(proper(X))
       proper(afterNth(X1,X2)) -> afterNth(proper(X1),proper(X2))
       proper(U51(X1,X2)) -> U51(proper(X1),proper(X2))
       proper(U52(X1,X2)) -> U52(proper(X1),proper(X2))
       proper(U61(X1,X2,X3,X4)) -> U61(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(U62(X1,X2,X3,X4)) -> U62(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(U63(X1,X2,X3,X4)) -> U63(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(U64(X1,X2)) -> U64(proper(X1),proper(X2))
       proper(pair(X1,X2)) -> pair(proper(X1),proper(X2))
       proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
       proper(U71(X1,X2)) -> U71(proper(X1),proper(X2))
       proper(U72(X1,X2)) -> U72(proper(X1),proper(X2))
       proper(U81(X1,X2,X3)) -> U81(proper(X1),proper(X2),proper(X3))
       proper(U82(X1,X2,X3)) -> U82(proper(X1),proper(X2),proper(X3))
       proper(fst(X)) -> fst(proper(X))
       proper(natsFrom(X)) -> natsFrom(proper(X))
       proper(s(X)) -> s(proper(X))
       proper(sel(X1,X2)) -> sel(proper(X1),proper(X2))
       proper(0()) -> ok(0())
       proper(nil()) -> ok(nil())
       proper(tail(X)) -> tail(proper(X))
       proper(take(X1,X2)) -> take(proper(X1),proper(X2))
       U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3))
       U12(ok(X1),ok(X2),ok(X3)) -> ok(U12(X1,X2,X3))
       snd(ok(X)) -> ok(snd(X))
       splitAt(ok(X1),ok(X2)) -> ok(splitAt(X1,X2))
       U21(ok(X1),ok(X2)) -> ok(U21(X1,X2))
       U22(ok(X1),ok(X2)) -> ok(U22(X1,X2))
       U31(ok(X1),ok(X2)) -> ok(U31(X1,X2))
       U32(ok(X1),ok(X2)) -> ok(U32(X1,X2))
       U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3))
       U42(ok(X1),ok(X2),ok(X3)) -> ok(U42(X1,X2,X3))
       head(ok(X)) -> ok(head(X))
       afterNth(ok(X1),ok(X2)) -> ok(afterNth(X1,X2))
       U51(ok(X1),ok(X2)) -> ok(U51(X1,X2))
       U52(ok(X1),ok(X2)) -> ok(U52(X1,X2))
       U61(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U61(X1,X2,X3,X4))
       U62(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U62(X1,X2,X3,X4))
       U63(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U63(X1,X2,X3,X4))
       U64(ok(X1),ok(X2)) -> ok(U64(X1,X2))
       pair(ok(X1),ok(X2)) -> ok(pair(X1,X2))
       cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
       U71(ok(X1),ok(X2)) -> ok(U71(X1,X2))
       U72(ok(X1),ok(X2)) -> ok(U72(X1,X2))
       U81(ok(X1),ok(X2),ok(X3)) -> ok(U81(X1,X2,X3))
       U82(ok(X1),ok(X2),ok(X3)) -> ok(U82(X1,X2,X3))
       fst(ok(X)) -> ok(fst(X))
       natsFrom(ok(X)) -> ok(natsFrom(X))
       s(ok(X)) -> ok(s(X))
       sel(ok(X1),ok(X2)) -> ok(sel(X1,X2))
       tail(ok(X)) -> ok(tail(X))
       take(ok(X1),ok(X2)) -> ok(take(X1,X2))
       top(mark(X)) -> top(proper(X))
       top(ok(X)) -> top(active(X))
      Open
      
      DPs:
       proper#(U11(X1,X2,X3)) -> proper#(X3)
       proper#(U11(X1,X2,X3)) -> proper#(X2)
       proper#(U11(X1,X2,X3)) -> proper#(X1)
       proper#(U12(X1,X2,X3)) -> proper#(X3)
       proper#(U12(X1,X2,X3)) -> proper#(X2)
       proper#(U12(X1,X2,X3)) -> proper#(X1)
       proper#(snd(X)) -> proper#(X)
       proper#(splitAt(X1,X2)) -> proper#(X2)
       proper#(splitAt(X1,X2)) -> proper#(X1)
       proper#(U21(X1,X2)) -> proper#(X2)
       proper#(U21(X1,X2)) -> proper#(X1)
       proper#(U22(X1,X2)) -> proper#(X2)
       proper#(U22(X1,X2)) -> proper#(X1)
       proper#(U31(X1,X2)) -> proper#(X2)
       proper#(U31(X1,X2)) -> proper#(X1)
       proper#(U32(X1,X2)) -> proper#(X2)
       proper#(U32(X1,X2)) -> proper#(X1)
       proper#(U41(X1,X2,X3)) -> proper#(X3)
       proper#(U41(X1,X2,X3)) -> proper#(X2)
       proper#(U41(X1,X2,X3)) -> proper#(X1)
       proper#(U42(X1,X2,X3)) -> proper#(X3)
       proper#(U42(X1,X2,X3)) -> proper#(X2)
       proper#(U42(X1,X2,X3)) -> proper#(X1)
       proper#(head(X)) -> proper#(X)
       proper#(afterNth(X1,X2)) -> proper#(X2)
       proper#(afterNth(X1,X2)) -> proper#(X1)
       proper#(U51(X1,X2)) -> proper#(X2)
       proper#(U51(X1,X2)) -> proper#(X1)
       proper#(U52(X1,X2)) -> proper#(X2)
       proper#(U52(X1,X2)) -> proper#(X1)
       proper#(U61(X1,X2,X3,X4)) -> proper#(X4)
       proper#(U61(X1,X2,X3,X4)) -> proper#(X3)
       proper#(U61(X1,X2,X3,X4)) -> proper#(X2)
       proper#(U61(X1,X2,X3,X4)) -> proper#(X1)
       proper#(U62(X1,X2,X3,X4)) -> proper#(X4)
       proper#(U62(X1,X2,X3,X4)) -> proper#(X3)
       proper#(U62(X1,X2,X3,X4)) -> proper#(X2)
       proper#(U62(X1,X2,X3,X4)) -> proper#(X1)
       proper#(U63(X1,X2,X3,X4)) -> proper#(X4)
       proper#(U63(X1,X2,X3,X4)) -> proper#(X3)
       proper#(U63(X1,X2,X3,X4)) -> proper#(X2)
       proper#(U63(X1,X2,X3,X4)) -> proper#(X1)
       proper#(U64(X1,X2)) -> proper#(X2)
       proper#(U64(X1,X2)) -> proper#(X1)
       proper#(pair(X1,X2)) -> proper#(X2)
       proper#(pair(X1,X2)) -> proper#(X1)
       proper#(cons(X1,X2)) -> proper#(X2)
       proper#(cons(X1,X2)) -> proper#(X1)
       proper#(U71(X1,X2)) -> proper#(X2)
       proper#(U71(X1,X2)) -> proper#(X1)
       proper#(U72(X1,X2)) -> proper#(X2)
       proper#(U72(X1,X2)) -> proper#(X1)
       proper#(U81(X1,X2,X3)) -> proper#(X3)
       proper#(U81(X1,X2,X3)) -> proper#(X2)
       proper#(U81(X1,X2,X3)) -> proper#(X1)
       proper#(U82(X1,X2,X3)) -> proper#(X3)
       proper#(U82(X1,X2,X3)) -> proper#(X2)
       proper#(U82(X1,X2,X3)) -> proper#(X1)
       proper#(fst(X)) -> proper#(X)
       proper#(natsFrom(X)) -> proper#(X)
       proper#(s(X)) -> proper#(X)
       proper#(sel(X1,X2)) -> proper#(X2)
       proper#(sel(X1,X2)) -> proper#(X1)
       proper#(tail(X)) -> proper#(X)
       proper#(take(X1,X2)) -> proper#(X2)
       proper#(take(X1,X2)) -> proper#(X1)
      TRS:
       active(U11(tt(),N,XS)) -> mark(U12(tt(),N,XS))
       active(U12(tt(),N,XS)) -> mark(snd(splitAt(N,XS)))
       active(U21(tt(),X)) -> mark(U22(tt(),X))
       active(U22(tt(),X)) -> mark(X)
       active(U31(tt(),N)) -> mark(U32(tt(),N))
       active(U32(tt(),N)) -> mark(N)
       active(U41(tt(),N,XS)) -> mark(U42(tt(),N,XS))
       active(U42(tt(),N,XS)) -> mark(head(afterNth(N,XS)))
       active(U51(tt(),Y)) -> mark(U52(tt(),Y))
       active(U52(tt(),Y)) -> mark(Y)
       active(U61(tt(),N,X,XS)) -> mark(U62(tt(),N,X,XS))
       active(U62(tt(),N,X,XS)) -> mark(U63(tt(),N,X,XS))
       active(U63(tt(),N,X,XS)) -> mark(U64(splitAt(N,XS),X))
       active(U64(pair(YS,ZS),X)) -> mark(pair(cons(X,YS),ZS))
       active(U71(tt(),XS)) -> mark(U72(tt(),XS))
       active(U72(tt(),XS)) -> mark(XS)
       active(U81(tt(),N,XS)) -> mark(U82(tt(),N,XS))
       active(U82(tt(),N,XS)) -> mark(fst(splitAt(N,XS)))
       active(afterNth(N,XS)) -> mark(U11(tt(),N,XS))
       active(fst(pair(X,Y))) -> mark(U21(tt(),X))
       active(head(cons(N,XS))) -> mark(U31(tt(),N))
       active(natsFrom(N)) -> mark(cons(N,natsFrom(s(N))))
       active(sel(N,XS)) -> mark(U41(tt(),N,XS))
       active(snd(pair(X,Y))) -> mark(U51(tt(),Y))
       active(splitAt(0(),XS)) -> mark(pair(nil(),XS))
       active(splitAt(s(N),cons(X,XS))) -> mark(U61(tt(),N,X,XS))
       active(tail(cons(N,XS))) -> mark(U71(tt(),XS))
       active(take(N,XS)) -> mark(U81(tt(),N,XS))
       active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3)
       active(U12(X1,X2,X3)) -> U12(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(U21(X1,X2)) -> U21(active(X1),X2)
       active(U22(X1,X2)) -> U22(active(X1),X2)
       active(U31(X1,X2)) -> U31(active(X1),X2)
       active(U32(X1,X2)) -> U32(active(X1),X2)
       active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3)
       active(U42(X1,X2,X3)) -> U42(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(U51(X1,X2)) -> U51(active(X1),X2)
       active(U52(X1,X2)) -> U52(active(X1),X2)
       active(U61(X1,X2,X3,X4)) -> U61(active(X1),X2,X3,X4)
       active(U62(X1,X2,X3,X4)) -> U62(active(X1),X2,X3,X4)
       active(U63(X1,X2,X3,X4)) -> U63(active(X1),X2,X3,X4)
       active(U64(X1,X2)) -> U64(active(X1),X2)
       active(pair(X1,X2)) -> pair(active(X1),X2)
       active(pair(X1,X2)) -> pair(X1,active(X2))
       active(cons(X1,X2)) -> cons(active(X1),X2)
       active(U71(X1,X2)) -> U71(active(X1),X2)
       active(U72(X1,X2)) -> U72(active(X1),X2)
       active(U81(X1,X2,X3)) -> U81(active(X1),X2,X3)
       active(U82(X1,X2,X3)) -> U82(active(X1),X2,X3)
       active(fst(X)) -> fst(active(X))
       active(natsFrom(X)) -> natsFrom(active(X))
       active(s(X)) -> s(active(X))
       active(sel(X1,X2)) -> sel(active(X1),X2)
       active(sel(X1,X2)) -> sel(X1,active(X2))
       active(tail(X)) -> tail(active(X))
       active(take(X1,X2)) -> take(active(X1),X2)
       active(take(X1,X2)) -> take(X1,active(X2))
       U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3))
       U12(mark(X1),X2,X3) -> mark(U12(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))
       U21(mark(X1),X2) -> mark(U21(X1,X2))
       U22(mark(X1),X2) -> mark(U22(X1,X2))
       U31(mark(X1),X2) -> mark(U31(X1,X2))
       U32(mark(X1),X2) -> mark(U32(X1,X2))
       U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3))
       U42(mark(X1),X2,X3) -> mark(U42(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))
       U51(mark(X1),X2) -> mark(U51(X1,X2))
       U52(mark(X1),X2) -> mark(U52(X1,X2))
       U61(mark(X1),X2,X3,X4) -> mark(U61(X1,X2,X3,X4))
       U62(mark(X1),X2,X3,X4) -> mark(U62(X1,X2,X3,X4))
       U63(mark(X1),X2,X3,X4) -> mark(U63(X1,X2,X3,X4))
       U64(mark(X1),X2) -> mark(U64(X1,X2))
       pair(mark(X1),X2) -> mark(pair(X1,X2))
       pair(X1,mark(X2)) -> mark(pair(X1,X2))
       cons(mark(X1),X2) -> mark(cons(X1,X2))
       U71(mark(X1),X2) -> mark(U71(X1,X2))
       U72(mark(X1),X2) -> mark(U72(X1,X2))
       U81(mark(X1),X2,X3) -> mark(U81(X1,X2,X3))
       U82(mark(X1),X2,X3) -> mark(U82(X1,X2,X3))
       fst(mark(X)) -> mark(fst(X))
       natsFrom(mark(X)) -> mark(natsFrom(X))
       s(mark(X)) -> mark(s(X))
       sel(mark(X1),X2) -> mark(sel(X1,X2))
       sel(X1,mark(X2)) -> mark(sel(X1,X2))
       tail(mark(X)) -> mark(tail(X))
       take(mark(X1),X2) -> mark(take(X1,X2))
       take(X1,mark(X2)) -> mark(take(X1,X2))
       proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3))
       proper(tt()) -> ok(tt())
       proper(U12(X1,X2,X3)) -> U12(proper(X1),proper(X2),proper(X3))
       proper(snd(X)) -> snd(proper(X))
       proper(splitAt(X1,X2)) -> splitAt(proper(X1),proper(X2))
       proper(U21(X1,X2)) -> U21(proper(X1),proper(X2))
       proper(U22(X1,X2)) -> U22(proper(X1),proper(X2))
       proper(U31(X1,X2)) -> U31(proper(X1),proper(X2))
       proper(U32(X1,X2)) -> U32(proper(X1),proper(X2))
       proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3))
       proper(U42(X1,X2,X3)) -> U42(proper(X1),proper(X2),proper(X3))
       proper(head(X)) -> head(proper(X))
       proper(afterNth(X1,X2)) -> afterNth(proper(X1),proper(X2))
       proper(U51(X1,X2)) -> U51(proper(X1),proper(X2))
       proper(U52(X1,X2)) -> U52(proper(X1),proper(X2))
       proper(U61(X1,X2,X3,X4)) -> U61(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(U62(X1,X2,X3,X4)) -> U62(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(U63(X1,X2,X3,X4)) -> U63(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(U64(X1,X2)) -> U64(proper(X1),proper(X2))
       proper(pair(X1,X2)) -> pair(proper(X1),proper(X2))
       proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
       proper(U71(X1,X2)) -> U71(proper(X1),proper(X2))
       proper(U72(X1,X2)) -> U72(proper(X1),proper(X2))
       proper(U81(X1,X2,X3)) -> U81(proper(X1),proper(X2),proper(X3))
       proper(U82(X1,X2,X3)) -> U82(proper(X1),proper(X2),proper(X3))
       proper(fst(X)) -> fst(proper(X))
       proper(natsFrom(X)) -> natsFrom(proper(X))
       proper(s(X)) -> s(proper(X))
       proper(sel(X1,X2)) -> sel(proper(X1),proper(X2))
       proper(0()) -> ok(0())
       proper(nil()) -> ok(nil())
       proper(tail(X)) -> tail(proper(X))
       proper(take(X1,X2)) -> take(proper(X1),proper(X2))
       U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3))
       U12(ok(X1),ok(X2),ok(X3)) -> ok(U12(X1,X2,X3))
       snd(ok(X)) -> ok(snd(X))
       splitAt(ok(X1),ok(X2)) -> ok(splitAt(X1,X2))
       U21(ok(X1),ok(X2)) -> ok(U21(X1,X2))
       U22(ok(X1),ok(X2)) -> ok(U22(X1,X2))
       U31(ok(X1),ok(X2)) -> ok(U31(X1,X2))
       U32(ok(X1),ok(X2)) -> ok(U32(X1,X2))
       U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3))
       U42(ok(X1),ok(X2),ok(X3)) -> ok(U42(X1,X2,X3))
       head(ok(X)) -> ok(head(X))
       afterNth(ok(X1),ok(X2)) -> ok(afterNth(X1,X2))
       U51(ok(X1),ok(X2)) -> ok(U51(X1,X2))
       U52(ok(X1),ok(X2)) -> ok(U52(X1,X2))
       U61(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U61(X1,X2,X3,X4))
       U62(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U62(X1,X2,X3,X4))
       U63(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U63(X1,X2,X3,X4))
       U64(ok(X1),ok(X2)) -> ok(U64(X1,X2))
       pair(ok(X1),ok(X2)) -> ok(pair(X1,X2))
       cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
       U71(ok(X1),ok(X2)) -> ok(U71(X1,X2))
       U72(ok(X1),ok(X2)) -> ok(U72(X1,X2))
       U81(ok(X1),ok(X2),ok(X3)) -> ok(U81(X1,X2,X3))
       U82(ok(X1),ok(X2),ok(X3)) -> ok(U82(X1,X2,X3))
       fst(ok(X)) -> ok(fst(X))
       natsFrom(ok(X)) -> ok(natsFrom(X))
       s(ok(X)) -> ok(s(X))
       sel(ok(X1),ok(X2)) -> ok(sel(X1,X2))
       tail(ok(X)) -> ok(tail(X))
       take(ok(X1),ok(X2)) -> ok(take(X1,X2))
       top(mark(X)) -> top(proper(X))
       top(ok(X)) -> top(active(X))
      Open
      
      DPs:
       take#(mark(X1),X2) -> take#(X1,X2)
       take#(X1,mark(X2)) -> take#(X1,X2)
       take#(ok(X1),ok(X2)) -> take#(X1,X2)
      TRS:
       active(U11(tt(),N,XS)) -> mark(U12(tt(),N,XS))
       active(U12(tt(),N,XS)) -> mark(snd(splitAt(N,XS)))
       active(U21(tt(),X)) -> mark(U22(tt(),X))
       active(U22(tt(),X)) -> mark(X)
       active(U31(tt(),N)) -> mark(U32(tt(),N))
       active(U32(tt(),N)) -> mark(N)
       active(U41(tt(),N,XS)) -> mark(U42(tt(),N,XS))
       active(U42(tt(),N,XS)) -> mark(head(afterNth(N,XS)))
       active(U51(tt(),Y)) -> mark(U52(tt(),Y))
       active(U52(tt(),Y)) -> mark(Y)
       active(U61(tt(),N,X,XS)) -> mark(U62(tt(),N,X,XS))
       active(U62(tt(),N,X,XS)) -> mark(U63(tt(),N,X,XS))
       active(U63(tt(),N,X,XS)) -> mark(U64(splitAt(N,XS),X))
       active(U64(pair(YS,ZS),X)) -> mark(pair(cons(X,YS),ZS))
       active(U71(tt(),XS)) -> mark(U72(tt(),XS))
       active(U72(tt(),XS)) -> mark(XS)
       active(U81(tt(),N,XS)) -> mark(U82(tt(),N,XS))
       active(U82(tt(),N,XS)) -> mark(fst(splitAt(N,XS)))
       active(afterNth(N,XS)) -> mark(U11(tt(),N,XS))
       active(fst(pair(X,Y))) -> mark(U21(tt(),X))
       active(head(cons(N,XS))) -> mark(U31(tt(),N))
       active(natsFrom(N)) -> mark(cons(N,natsFrom(s(N))))
       active(sel(N,XS)) -> mark(U41(tt(),N,XS))
       active(snd(pair(X,Y))) -> mark(U51(tt(),Y))
       active(splitAt(0(),XS)) -> mark(pair(nil(),XS))
       active(splitAt(s(N),cons(X,XS))) -> mark(U61(tt(),N,X,XS))
       active(tail(cons(N,XS))) -> mark(U71(tt(),XS))
       active(take(N,XS)) -> mark(U81(tt(),N,XS))
       active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3)
       active(U12(X1,X2,X3)) -> U12(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(U21(X1,X2)) -> U21(active(X1),X2)
       active(U22(X1,X2)) -> U22(active(X1),X2)
       active(U31(X1,X2)) -> U31(active(X1),X2)
       active(U32(X1,X2)) -> U32(active(X1),X2)
       active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3)
       active(U42(X1,X2,X3)) -> U42(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(U51(X1,X2)) -> U51(active(X1),X2)
       active(U52(X1,X2)) -> U52(active(X1),X2)
       active(U61(X1,X2,X3,X4)) -> U61(active(X1),X2,X3,X4)
       active(U62(X1,X2,X3,X4)) -> U62(active(X1),X2,X3,X4)
       active(U63(X1,X2,X3,X4)) -> U63(active(X1),X2,X3,X4)
       active(U64(X1,X2)) -> U64(active(X1),X2)
       active(pair(X1,X2)) -> pair(active(X1),X2)
       active(pair(X1,X2)) -> pair(X1,active(X2))
       active(cons(X1,X2)) -> cons(active(X1),X2)
       active(U71(X1,X2)) -> U71(active(X1),X2)
       active(U72(X1,X2)) -> U72(active(X1),X2)
       active(U81(X1,X2,X3)) -> U81(active(X1),X2,X3)
       active(U82(X1,X2,X3)) -> U82(active(X1),X2,X3)
       active(fst(X)) -> fst(active(X))
       active(natsFrom(X)) -> natsFrom(active(X))
       active(s(X)) -> s(active(X))
       active(sel(X1,X2)) -> sel(active(X1),X2)
       active(sel(X1,X2)) -> sel(X1,active(X2))
       active(tail(X)) -> tail(active(X))
       active(take(X1,X2)) -> take(active(X1),X2)
       active(take(X1,X2)) -> take(X1,active(X2))
       U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3))
       U12(mark(X1),X2,X3) -> mark(U12(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))
       U21(mark(X1),X2) -> mark(U21(X1,X2))
       U22(mark(X1),X2) -> mark(U22(X1,X2))
       U31(mark(X1),X2) -> mark(U31(X1,X2))
       U32(mark(X1),X2) -> mark(U32(X1,X2))
       U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3))
       U42(mark(X1),X2,X3) -> mark(U42(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))
       U51(mark(X1),X2) -> mark(U51(X1,X2))
       U52(mark(X1),X2) -> mark(U52(X1,X2))
       U61(mark(X1),X2,X3,X4) -> mark(U61(X1,X2,X3,X4))
       U62(mark(X1),X2,X3,X4) -> mark(U62(X1,X2,X3,X4))
       U63(mark(X1),X2,X3,X4) -> mark(U63(X1,X2,X3,X4))
       U64(mark(X1),X2) -> mark(U64(X1,X2))
       pair(mark(X1),X2) -> mark(pair(X1,X2))
       pair(X1,mark(X2)) -> mark(pair(X1,X2))
       cons(mark(X1),X2) -> mark(cons(X1,X2))
       U71(mark(X1),X2) -> mark(U71(X1,X2))
       U72(mark(X1),X2) -> mark(U72(X1,X2))
       U81(mark(X1),X2,X3) -> mark(U81(X1,X2,X3))
       U82(mark(X1),X2,X3) -> mark(U82(X1,X2,X3))
       fst(mark(X)) -> mark(fst(X))
       natsFrom(mark(X)) -> mark(natsFrom(X))
       s(mark(X)) -> mark(s(X))
       sel(mark(X1),X2) -> mark(sel(X1,X2))
       sel(X1,mark(X2)) -> mark(sel(X1,X2))
       tail(mark(X)) -> mark(tail(X))
       take(mark(X1),X2) -> mark(take(X1,X2))
       take(X1,mark(X2)) -> mark(take(X1,X2))
       proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3))
       proper(tt()) -> ok(tt())
       proper(U12(X1,X2,X3)) -> U12(proper(X1),proper(X2),proper(X3))
       proper(snd(X)) -> snd(proper(X))
       proper(splitAt(X1,X2)) -> splitAt(proper(X1),proper(X2))
       proper(U21(X1,X2)) -> U21(proper(X1),proper(X2))
       proper(U22(X1,X2)) -> U22(proper(X1),proper(X2))
       proper(U31(X1,X2)) -> U31(proper(X1),proper(X2))
       proper(U32(X1,X2)) -> U32(proper(X1),proper(X2))
       proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3))
       proper(U42(X1,X2,X3)) -> U42(proper(X1),proper(X2),proper(X3))
       proper(head(X)) -> head(proper(X))
       proper(afterNth(X1,X2)) -> afterNth(proper(X1),proper(X2))
       proper(U51(X1,X2)) -> U51(proper(X1),proper(X2))
       proper(U52(X1,X2)) -> U52(proper(X1),proper(X2))
       proper(U61(X1,X2,X3,X4)) -> U61(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(U62(X1,X2,X3,X4)) -> U62(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(U63(X1,X2,X3,X4)) -> U63(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(U64(X1,X2)) -> U64(proper(X1),proper(X2))
       proper(pair(X1,X2)) -> pair(proper(X1),proper(X2))
       proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
       proper(U71(X1,X2)) -> U71(proper(X1),proper(X2))
       proper(U72(X1,X2)) -> U72(proper(X1),proper(X2))
       proper(U81(X1,X2,X3)) -> U81(proper(X1),proper(X2),proper(X3))
       proper(U82(X1,X2,X3)) -> U82(proper(X1),proper(X2),proper(X3))
       proper(fst(X)) -> fst(proper(X))
       proper(natsFrom(X)) -> natsFrom(proper(X))
       proper(s(X)) -> s(proper(X))
       proper(sel(X1,X2)) -> sel(proper(X1),proper(X2))
       proper(0()) -> ok(0())
       proper(nil()) -> ok(nil())
       proper(tail(X)) -> tail(proper(X))
       proper(take(X1,X2)) -> take(proper(X1),proper(X2))
       U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3))
       U12(ok(X1),ok(X2),ok(X3)) -> ok(U12(X1,X2,X3))
       snd(ok(X)) -> ok(snd(X))
       splitAt(ok(X1),ok(X2)) -> ok(splitAt(X1,X2))
       U21(ok(X1),ok(X2)) -> ok(U21(X1,X2))
       U22(ok(X1),ok(X2)) -> ok(U22(X1,X2))
       U31(ok(X1),ok(X2)) -> ok(U31(X1,X2))
       U32(ok(X1),ok(X2)) -> ok(U32(X1,X2))
       U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3))
       U42(ok(X1),ok(X2),ok(X3)) -> ok(U42(X1,X2,X3))
       head(ok(X)) -> ok(head(X))
       afterNth(ok(X1),ok(X2)) -> ok(afterNth(X1,X2))
       U51(ok(X1),ok(X2)) -> ok(U51(X1,X2))
       U52(ok(X1),ok(X2)) -> ok(U52(X1,X2))
       U61(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U61(X1,X2,X3,X4))
       U62(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U62(X1,X2,X3,X4))
       U63(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U63(X1,X2,X3,X4))
       U64(ok(X1),ok(X2)) -> ok(U64(X1,X2))
       pair(ok(X1),ok(X2)) -> ok(pair(X1,X2))
       cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
       U71(ok(X1),ok(X2)) -> ok(U71(X1,X2))
       U72(ok(X1),ok(X2)) -> ok(U72(X1,X2))
       U81(ok(X1),ok(X2),ok(X3)) -> ok(U81(X1,X2,X3))
       U82(ok(X1),ok(X2),ok(X3)) -> ok(U82(X1,X2,X3))
       fst(ok(X)) -> ok(fst(X))
       natsFrom(ok(X)) -> ok(natsFrom(X))
       s(ok(X)) -> ok(s(X))
       sel(ok(X1),ok(X2)) -> ok(sel(X1,X2))
       tail(ok(X)) -> ok(tail(X))
       take(ok(X1),ok(X2)) -> ok(take(X1,X2))
       top(mark(X)) -> top(proper(X))
       top(ok(X)) -> top(active(X))
      Open
      
      DPs:
       tail#(mark(X)) -> tail#(X)
       tail#(ok(X)) -> tail#(X)
      TRS:
       active(U11(tt(),N,XS)) -> mark(U12(tt(),N,XS))
       active(U12(tt(),N,XS)) -> mark(snd(splitAt(N,XS)))
       active(U21(tt(),X)) -> mark(U22(tt(),X))
       active(U22(tt(),X)) -> mark(X)
       active(U31(tt(),N)) -> mark(U32(tt(),N))
       active(U32(tt(),N)) -> mark(N)
       active(U41(tt(),N,XS)) -> mark(U42(tt(),N,XS))
       active(U42(tt(),N,XS)) -> mark(head(afterNth(N,XS)))
       active(U51(tt(),Y)) -> mark(U52(tt(),Y))
       active(U52(tt(),Y)) -> mark(Y)
       active(U61(tt(),N,X,XS)) -> mark(U62(tt(),N,X,XS))
       active(U62(tt(),N,X,XS)) -> mark(U63(tt(),N,X,XS))
       active(U63(tt(),N,X,XS)) -> mark(U64(splitAt(N,XS),X))
       active(U64(pair(YS,ZS),X)) -> mark(pair(cons(X,YS),ZS))
       active(U71(tt(),XS)) -> mark(U72(tt(),XS))
       active(U72(tt(),XS)) -> mark(XS)
       active(U81(tt(),N,XS)) -> mark(U82(tt(),N,XS))
       active(U82(tt(),N,XS)) -> mark(fst(splitAt(N,XS)))
       active(afterNth(N,XS)) -> mark(U11(tt(),N,XS))
       active(fst(pair(X,Y))) -> mark(U21(tt(),X))
       active(head(cons(N,XS))) -> mark(U31(tt(),N))
       active(natsFrom(N)) -> mark(cons(N,natsFrom(s(N))))
       active(sel(N,XS)) -> mark(U41(tt(),N,XS))
       active(snd(pair(X,Y))) -> mark(U51(tt(),Y))
       active(splitAt(0(),XS)) -> mark(pair(nil(),XS))
       active(splitAt(s(N),cons(X,XS))) -> mark(U61(tt(),N,X,XS))
       active(tail(cons(N,XS))) -> mark(U71(tt(),XS))
       active(take(N,XS)) -> mark(U81(tt(),N,XS))
       active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3)
       active(U12(X1,X2,X3)) -> U12(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(U21(X1,X2)) -> U21(active(X1),X2)
       active(U22(X1,X2)) -> U22(active(X1),X2)
       active(U31(X1,X2)) -> U31(active(X1),X2)
       active(U32(X1,X2)) -> U32(active(X1),X2)
       active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3)
       active(U42(X1,X2,X3)) -> U42(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(U51(X1,X2)) -> U51(active(X1),X2)
       active(U52(X1,X2)) -> U52(active(X1),X2)
       active(U61(X1,X2,X3,X4)) -> U61(active(X1),X2,X3,X4)
       active(U62(X1,X2,X3,X4)) -> U62(active(X1),X2,X3,X4)
       active(U63(X1,X2,X3,X4)) -> U63(active(X1),X2,X3,X4)
       active(U64(X1,X2)) -> U64(active(X1),X2)
       active(pair(X1,X2)) -> pair(active(X1),X2)
       active(pair(X1,X2)) -> pair(X1,active(X2))
       active(cons(X1,X2)) -> cons(active(X1),X2)
       active(U71(X1,X2)) -> U71(active(X1),X2)
       active(U72(X1,X2)) -> U72(active(X1),X2)
       active(U81(X1,X2,X3)) -> U81(active(X1),X2,X3)
       active(U82(X1,X2,X3)) -> U82(active(X1),X2,X3)
       active(fst(X)) -> fst(active(X))
       active(natsFrom(X)) -> natsFrom(active(X))
       active(s(X)) -> s(active(X))
       active(sel(X1,X2)) -> sel(active(X1),X2)
       active(sel(X1,X2)) -> sel(X1,active(X2))
       active(tail(X)) -> tail(active(X))
       active(take(X1,X2)) -> take(active(X1),X2)
       active(take(X1,X2)) -> take(X1,active(X2))
       U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3))
       U12(mark(X1),X2,X3) -> mark(U12(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))
       U21(mark(X1),X2) -> mark(U21(X1,X2))
       U22(mark(X1),X2) -> mark(U22(X1,X2))
       U31(mark(X1),X2) -> mark(U31(X1,X2))
       U32(mark(X1),X2) -> mark(U32(X1,X2))
       U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3))
       U42(mark(X1),X2,X3) -> mark(U42(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))
       U51(mark(X1),X2) -> mark(U51(X1,X2))
       U52(mark(X1),X2) -> mark(U52(X1,X2))
       U61(mark(X1),X2,X3,X4) -> mark(U61(X1,X2,X3,X4))
       U62(mark(X1),X2,X3,X4) -> mark(U62(X1,X2,X3,X4))
       U63(mark(X1),X2,X3,X4) -> mark(U63(X1,X2,X3,X4))
       U64(mark(X1),X2) -> mark(U64(X1,X2))
       pair(mark(X1),X2) -> mark(pair(X1,X2))
       pair(X1,mark(X2)) -> mark(pair(X1,X2))
       cons(mark(X1),X2) -> mark(cons(X1,X2))
       U71(mark(X1),X2) -> mark(U71(X1,X2))
       U72(mark(X1),X2) -> mark(U72(X1,X2))
       U81(mark(X1),X2,X3) -> mark(U81(X1,X2,X3))
       U82(mark(X1),X2,X3) -> mark(U82(X1,X2,X3))
       fst(mark(X)) -> mark(fst(X))
       natsFrom(mark(X)) -> mark(natsFrom(X))
       s(mark(X)) -> mark(s(X))
       sel(mark(X1),X2) -> mark(sel(X1,X2))
       sel(X1,mark(X2)) -> mark(sel(X1,X2))
       tail(mark(X)) -> mark(tail(X))
       take(mark(X1),X2) -> mark(take(X1,X2))
       take(X1,mark(X2)) -> mark(take(X1,X2))
       proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3))
       proper(tt()) -> ok(tt())
       proper(U12(X1,X2,X3)) -> U12(proper(X1),proper(X2),proper(X3))
       proper(snd(X)) -> snd(proper(X))
       proper(splitAt(X1,X2)) -> splitAt(proper(X1),proper(X2))
       proper(U21(X1,X2)) -> U21(proper(X1),proper(X2))
       proper(U22(X1,X2)) -> U22(proper(X1),proper(X2))
       proper(U31(X1,X2)) -> U31(proper(X1),proper(X2))
       proper(U32(X1,X2)) -> U32(proper(X1),proper(X2))
       proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3))
       proper(U42(X1,X2,X3)) -> U42(proper(X1),proper(X2),proper(X3))
       proper(head(X)) -> head(proper(X))
       proper(afterNth(X1,X2)) -> afterNth(proper(X1),proper(X2))
       proper(U51(X1,X2)) -> U51(proper(X1),proper(X2))
       proper(U52(X1,X2)) -> U52(proper(X1),proper(X2))
       proper(U61(X1,X2,X3,X4)) -> U61(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(U62(X1,X2,X3,X4)) -> U62(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(U63(X1,X2,X3,X4)) -> U63(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(U64(X1,X2)) -> U64(proper(X1),proper(X2))
       proper(pair(X1,X2)) -> pair(proper(X1),proper(X2))
       proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
       proper(U71(X1,X2)) -> U71(proper(X1),proper(X2))
       proper(U72(X1,X2)) -> U72(proper(X1),proper(X2))
       proper(U81(X1,X2,X3)) -> U81(proper(X1),proper(X2),proper(X3))
       proper(U82(X1,X2,X3)) -> U82(proper(X1),proper(X2),proper(X3))
       proper(fst(X)) -> fst(proper(X))
       proper(natsFrom(X)) -> natsFrom(proper(X))
       proper(s(X)) -> s(proper(X))
       proper(sel(X1,X2)) -> sel(proper(X1),proper(X2))
       proper(0()) -> ok(0())
       proper(nil()) -> ok(nil())
       proper(tail(X)) -> tail(proper(X))
       proper(take(X1,X2)) -> take(proper(X1),proper(X2))
       U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3))
       U12(ok(X1),ok(X2),ok(X3)) -> ok(U12(X1,X2,X3))
       snd(ok(X)) -> ok(snd(X))
       splitAt(ok(X1),ok(X2)) -> ok(splitAt(X1,X2))
       U21(ok(X1),ok(X2)) -> ok(U21(X1,X2))
       U22(ok(X1),ok(X2)) -> ok(U22(X1,X2))
       U31(ok(X1),ok(X2)) -> ok(U31(X1,X2))
       U32(ok(X1),ok(X2)) -> ok(U32(X1,X2))
       U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3))
       U42(ok(X1),ok(X2),ok(X3)) -> ok(U42(X1,X2,X3))
       head(ok(X)) -> ok(head(X))
       afterNth(ok(X1),ok(X2)) -> ok(afterNth(X1,X2))
       U51(ok(X1),ok(X2)) -> ok(U51(X1,X2))
       U52(ok(X1),ok(X2)) -> ok(U52(X1,X2))
       U61(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U61(X1,X2,X3,X4))
       U62(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U62(X1,X2,X3,X4))
       U63(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U63(X1,X2,X3,X4))
       U64(ok(X1),ok(X2)) -> ok(U64(X1,X2))
       pair(ok(X1),ok(X2)) -> ok(pair(X1,X2))
       cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
       U71(ok(X1),ok(X2)) -> ok(U71(X1,X2))
       U72(ok(X1),ok(X2)) -> ok(U72(X1,X2))
       U81(ok(X1),ok(X2),ok(X3)) -> ok(U81(X1,X2,X3))
       U82(ok(X1),ok(X2),ok(X3)) -> ok(U82(X1,X2,X3))
       fst(ok(X)) -> ok(fst(X))
       natsFrom(ok(X)) -> ok(natsFrom(X))
       s(ok(X)) -> ok(s(X))
       sel(ok(X1),ok(X2)) -> ok(sel(X1,X2))
       tail(ok(X)) -> ok(tail(X))
       take(ok(X1),ok(X2)) -> ok(take(X1,X2))
       top(mark(X)) -> top(proper(X))
       top(ok(X)) -> top(active(X))
      Open
      
      DPs:
       sel#(mark(X1),X2) -> sel#(X1,X2)
       sel#(X1,mark(X2)) -> sel#(X1,X2)
       sel#(ok(X1),ok(X2)) -> sel#(X1,X2)
      TRS:
       active(U11(tt(),N,XS)) -> mark(U12(tt(),N,XS))
       active(U12(tt(),N,XS)) -> mark(snd(splitAt(N,XS)))
       active(U21(tt(),X)) -> mark(U22(tt(),X))
       active(U22(tt(),X)) -> mark(X)
       active(U31(tt(),N)) -> mark(U32(tt(),N))
       active(U32(tt(),N)) -> mark(N)
       active(U41(tt(),N,XS)) -> mark(U42(tt(),N,XS))
       active(U42(tt(),N,XS)) -> mark(head(afterNth(N,XS)))
       active(U51(tt(),Y)) -> mark(U52(tt(),Y))
       active(U52(tt(),Y)) -> mark(Y)
       active(U61(tt(),N,X,XS)) -> mark(U62(tt(),N,X,XS))
       active(U62(tt(),N,X,XS)) -> mark(U63(tt(),N,X,XS))
       active(U63(tt(),N,X,XS)) -> mark(U64(splitAt(N,XS),X))
       active(U64(pair(YS,ZS),X)) -> mark(pair(cons(X,YS),ZS))
       active(U71(tt(),XS)) -> mark(U72(tt(),XS))
       active(U72(tt(),XS)) -> mark(XS)
       active(U81(tt(),N,XS)) -> mark(U82(tt(),N,XS))
       active(U82(tt(),N,XS)) -> mark(fst(splitAt(N,XS)))
       active(afterNth(N,XS)) -> mark(U11(tt(),N,XS))
       active(fst(pair(X,Y))) -> mark(U21(tt(),X))
       active(head(cons(N,XS))) -> mark(U31(tt(),N))
       active(natsFrom(N)) -> mark(cons(N,natsFrom(s(N))))
       active(sel(N,XS)) -> mark(U41(tt(),N,XS))
       active(snd(pair(X,Y))) -> mark(U51(tt(),Y))
       active(splitAt(0(),XS)) -> mark(pair(nil(),XS))
       active(splitAt(s(N),cons(X,XS))) -> mark(U61(tt(),N,X,XS))
       active(tail(cons(N,XS))) -> mark(U71(tt(),XS))
       active(take(N,XS)) -> mark(U81(tt(),N,XS))
       active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3)
       active(U12(X1,X2,X3)) -> U12(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(U21(X1,X2)) -> U21(active(X1),X2)
       active(U22(X1,X2)) -> U22(active(X1),X2)
       active(U31(X1,X2)) -> U31(active(X1),X2)
       active(U32(X1,X2)) -> U32(active(X1),X2)
       active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3)
       active(U42(X1,X2,X3)) -> U42(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(U51(X1,X2)) -> U51(active(X1),X2)
       active(U52(X1,X2)) -> U52(active(X1),X2)
       active(U61(X1,X2,X3,X4)) -> U61(active(X1),X2,X3,X4)
       active(U62(X1,X2,X3,X4)) -> U62(active(X1),X2,X3,X4)
       active(U63(X1,X2,X3,X4)) -> U63(active(X1),X2,X3,X4)
       active(U64(X1,X2)) -> U64(active(X1),X2)
       active(pair(X1,X2)) -> pair(active(X1),X2)
       active(pair(X1,X2)) -> pair(X1,active(X2))
       active(cons(X1,X2)) -> cons(active(X1),X2)
       active(U71(X1,X2)) -> U71(active(X1),X2)
       active(U72(X1,X2)) -> U72(active(X1),X2)
       active(U81(X1,X2,X3)) -> U81(active(X1),X2,X3)
       active(U82(X1,X2,X3)) -> U82(active(X1),X2,X3)
       active(fst(X)) -> fst(active(X))
       active(natsFrom(X)) -> natsFrom(active(X))
       active(s(X)) -> s(active(X))
       active(sel(X1,X2)) -> sel(active(X1),X2)
       active(sel(X1,X2)) -> sel(X1,active(X2))
       active(tail(X)) -> tail(active(X))
       active(take(X1,X2)) -> take(active(X1),X2)
       active(take(X1,X2)) -> take(X1,active(X2))
       U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3))
       U12(mark(X1),X2,X3) -> mark(U12(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))
       U21(mark(X1),X2) -> mark(U21(X1,X2))
       U22(mark(X1),X2) -> mark(U22(X1,X2))
       U31(mark(X1),X2) -> mark(U31(X1,X2))
       U32(mark(X1),X2) -> mark(U32(X1,X2))
       U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3))
       U42(mark(X1),X2,X3) -> mark(U42(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))
       U51(mark(X1),X2) -> mark(U51(X1,X2))
       U52(mark(X1),X2) -> mark(U52(X1,X2))
       U61(mark(X1),X2,X3,X4) -> mark(U61(X1,X2,X3,X4))
       U62(mark(X1),X2,X3,X4) -> mark(U62(X1,X2,X3,X4))
       U63(mark(X1),X2,X3,X4) -> mark(U63(X1,X2,X3,X4))
       U64(mark(X1),X2) -> mark(U64(X1,X2))
       pair(mark(X1),X2) -> mark(pair(X1,X2))
       pair(X1,mark(X2)) -> mark(pair(X1,X2))
       cons(mark(X1),X2) -> mark(cons(X1,X2))
       U71(mark(X1),X2) -> mark(U71(X1,X2))
       U72(mark(X1),X2) -> mark(U72(X1,X2))
       U81(mark(X1),X2,X3) -> mark(U81(X1,X2,X3))
       U82(mark(X1),X2,X3) -> mark(U82(X1,X2,X3))
       fst(mark(X)) -> mark(fst(X))
       natsFrom(mark(X)) -> mark(natsFrom(X))
       s(mark(X)) -> mark(s(X))
       sel(mark(X1),X2) -> mark(sel(X1,X2))
       sel(X1,mark(X2)) -> mark(sel(X1,X2))
       tail(mark(X)) -> mark(tail(X))
       take(mark(X1),X2) -> mark(take(X1,X2))
       take(X1,mark(X2)) -> mark(take(X1,X2))
       proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3))
       proper(tt()) -> ok(tt())
       proper(U12(X1,X2,X3)) -> U12(proper(X1),proper(X2),proper(X3))
       proper(snd(X)) -> snd(proper(X))
       proper(splitAt(X1,X2)) -> splitAt(proper(X1),proper(X2))
       proper(U21(X1,X2)) -> U21(proper(X1),proper(X2))
       proper(U22(X1,X2)) -> U22(proper(X1),proper(X2))
       proper(U31(X1,X2)) -> U31(proper(X1),proper(X2))
       proper(U32(X1,X2)) -> U32(proper(X1),proper(X2))
       proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3))
       proper(U42(X1,X2,X3)) -> U42(proper(X1),proper(X2),proper(X3))
       proper(head(X)) -> head(proper(X))
       proper(afterNth(X1,X2)) -> afterNth(proper(X1),proper(X2))
       proper(U51(X1,X2)) -> U51(proper(X1),proper(X2))
       proper(U52(X1,X2)) -> U52(proper(X1),proper(X2))
       proper(U61(X1,X2,X3,X4)) -> U61(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(U62(X1,X2,X3,X4)) -> U62(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(U63(X1,X2,X3,X4)) -> U63(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(U64(X1,X2)) -> U64(proper(X1),proper(X2))
       proper(pair(X1,X2)) -> pair(proper(X1),proper(X2))
       proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
       proper(U71(X1,X2)) -> U71(proper(X1),proper(X2))
       proper(U72(X1,X2)) -> U72(proper(X1),proper(X2))
       proper(U81(X1,X2,X3)) -> U81(proper(X1),proper(X2),proper(X3))
       proper(U82(X1,X2,X3)) -> U82(proper(X1),proper(X2),proper(X3))
       proper(fst(X)) -> fst(proper(X))
       proper(natsFrom(X)) -> natsFrom(proper(X))
       proper(s(X)) -> s(proper(X))
       proper(sel(X1,X2)) -> sel(proper(X1),proper(X2))
       proper(0()) -> ok(0())
       proper(nil()) -> ok(nil())
       proper(tail(X)) -> tail(proper(X))
       proper(take(X1,X2)) -> take(proper(X1),proper(X2))
       U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3))
       U12(ok(X1),ok(X2),ok(X3)) -> ok(U12(X1,X2,X3))
       snd(ok(X)) -> ok(snd(X))
       splitAt(ok(X1),ok(X2)) -> ok(splitAt(X1,X2))
       U21(ok(X1),ok(X2)) -> ok(U21(X1,X2))
       U22(ok(X1),ok(X2)) -> ok(U22(X1,X2))
       U31(ok(X1),ok(X2)) -> ok(U31(X1,X2))
       U32(ok(X1),ok(X2)) -> ok(U32(X1,X2))
       U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3))
       U42(ok(X1),ok(X2),ok(X3)) -> ok(U42(X1,X2,X3))
       head(ok(X)) -> ok(head(X))
       afterNth(ok(X1),ok(X2)) -> ok(afterNth(X1,X2))
       U51(ok(X1),ok(X2)) -> ok(U51(X1,X2))
       U52(ok(X1),ok(X2)) -> ok(U52(X1,X2))
       U61(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U61(X1,X2,X3,X4))
       U62(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U62(X1,X2,X3,X4))
       U63(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U63(X1,X2,X3,X4))
       U64(ok(X1),ok(X2)) -> ok(U64(X1,X2))
       pair(ok(X1),ok(X2)) -> ok(pair(X1,X2))
       cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
       U71(ok(X1),ok(X2)) -> ok(U71(X1,X2))
       U72(ok(X1),ok(X2)) -> ok(U72(X1,X2))
       U81(ok(X1),ok(X2),ok(X3)) -> ok(U81(X1,X2,X3))
       U82(ok(X1),ok(X2),ok(X3)) -> ok(U82(X1,X2,X3))
       fst(ok(X)) -> ok(fst(X))
       natsFrom(ok(X)) -> ok(natsFrom(X))
       s(ok(X)) -> ok(s(X))
       sel(ok(X1),ok(X2)) -> ok(sel(X1,X2))
       tail(ok(X)) -> ok(tail(X))
       take(ok(X1),ok(X2)) -> ok(take(X1,X2))
       top(mark(X)) -> top(proper(X))
       top(ok(X)) -> top(active(X))
      Open
      
      DPs:
       s#(mark(X)) -> s#(X)
       s#(ok(X)) -> s#(X)
      TRS:
       active(U11(tt(),N,XS)) -> mark(U12(tt(),N,XS))
       active(U12(tt(),N,XS)) -> mark(snd(splitAt(N,XS)))
       active(U21(tt(),X)) -> mark(U22(tt(),X))
       active(U22(tt(),X)) -> mark(X)
       active(U31(tt(),N)) -> mark(U32(tt(),N))
       active(U32(tt(),N)) -> mark(N)
       active(U41(tt(),N,XS)) -> mark(U42(tt(),N,XS))
       active(U42(tt(),N,XS)) -> mark(head(afterNth(N,XS)))
       active(U51(tt(),Y)) -> mark(U52(tt(),Y))
       active(U52(tt(),Y)) -> mark(Y)
       active(U61(tt(),N,X,XS)) -> mark(U62(tt(),N,X,XS))
       active(U62(tt(),N,X,XS)) -> mark(U63(tt(),N,X,XS))
       active(U63(tt(),N,X,XS)) -> mark(U64(splitAt(N,XS),X))
       active(U64(pair(YS,ZS),X)) -> mark(pair(cons(X,YS),ZS))
       active(U71(tt(),XS)) -> mark(U72(tt(),XS))
       active(U72(tt(),XS)) -> mark(XS)
       active(U81(tt(),N,XS)) -> mark(U82(tt(),N,XS))
       active(U82(tt(),N,XS)) -> mark(fst(splitAt(N,XS)))
       active(afterNth(N,XS)) -> mark(U11(tt(),N,XS))
       active(fst(pair(X,Y))) -> mark(U21(tt(),X))
       active(head(cons(N,XS))) -> mark(U31(tt(),N))
       active(natsFrom(N)) -> mark(cons(N,natsFrom(s(N))))
       active(sel(N,XS)) -> mark(U41(tt(),N,XS))
       active(snd(pair(X,Y))) -> mark(U51(tt(),Y))
       active(splitAt(0(),XS)) -> mark(pair(nil(),XS))
       active(splitAt(s(N),cons(X,XS))) -> mark(U61(tt(),N,X,XS))
       active(tail(cons(N,XS))) -> mark(U71(tt(),XS))
       active(take(N,XS)) -> mark(U81(tt(),N,XS))
       active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3)
       active(U12(X1,X2,X3)) -> U12(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(U21(X1,X2)) -> U21(active(X1),X2)
       active(U22(X1,X2)) -> U22(active(X1),X2)
       active(U31(X1,X2)) -> U31(active(X1),X2)
       active(U32(X1,X2)) -> U32(active(X1),X2)
       active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3)
       active(U42(X1,X2,X3)) -> U42(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(U51(X1,X2)) -> U51(active(X1),X2)
       active(U52(X1,X2)) -> U52(active(X1),X2)
       active(U61(X1,X2,X3,X4)) -> U61(active(X1),X2,X3,X4)
       active(U62(X1,X2,X3,X4)) -> U62(active(X1),X2,X3,X4)
       active(U63(X1,X2,X3,X4)) -> U63(active(X1),X2,X3,X4)
       active(U64(X1,X2)) -> U64(active(X1),X2)
       active(pair(X1,X2)) -> pair(active(X1),X2)
       active(pair(X1,X2)) -> pair(X1,active(X2))
       active(cons(X1,X2)) -> cons(active(X1),X2)
       active(U71(X1,X2)) -> U71(active(X1),X2)
       active(U72(X1,X2)) -> U72(active(X1),X2)
       active(U81(X1,X2,X3)) -> U81(active(X1),X2,X3)
       active(U82(X1,X2,X3)) -> U82(active(X1),X2,X3)
       active(fst(X)) -> fst(active(X))
       active(natsFrom(X)) -> natsFrom(active(X))
       active(s(X)) -> s(active(X))
       active(sel(X1,X2)) -> sel(active(X1),X2)
       active(sel(X1,X2)) -> sel(X1,active(X2))
       active(tail(X)) -> tail(active(X))
       active(take(X1,X2)) -> take(active(X1),X2)
       active(take(X1,X2)) -> take(X1,active(X2))
       U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3))
       U12(mark(X1),X2,X3) -> mark(U12(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))
       U21(mark(X1),X2) -> mark(U21(X1,X2))
       U22(mark(X1),X2) -> mark(U22(X1,X2))
       U31(mark(X1),X2) -> mark(U31(X1,X2))
       U32(mark(X1),X2) -> mark(U32(X1,X2))
       U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3))
       U42(mark(X1),X2,X3) -> mark(U42(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))
       U51(mark(X1),X2) -> mark(U51(X1,X2))
       U52(mark(X1),X2) -> mark(U52(X1,X2))
       U61(mark(X1),X2,X3,X4) -> mark(U61(X1,X2,X3,X4))
       U62(mark(X1),X2,X3,X4) -> mark(U62(X1,X2,X3,X4))
       U63(mark(X1),X2,X3,X4) -> mark(U63(X1,X2,X3,X4))
       U64(mark(X1),X2) -> mark(U64(X1,X2))
       pair(mark(X1),X2) -> mark(pair(X1,X2))
       pair(X1,mark(X2)) -> mark(pair(X1,X2))
       cons(mark(X1),X2) -> mark(cons(X1,X2))
       U71(mark(X1),X2) -> mark(U71(X1,X2))
       U72(mark(X1),X2) -> mark(U72(X1,X2))
       U81(mark(X1),X2,X3) -> mark(U81(X1,X2,X3))
       U82(mark(X1),X2,X3) -> mark(U82(X1,X2,X3))
       fst(mark(X)) -> mark(fst(X))
       natsFrom(mark(X)) -> mark(natsFrom(X))
       s(mark(X)) -> mark(s(X))
       sel(mark(X1),X2) -> mark(sel(X1,X2))
       sel(X1,mark(X2)) -> mark(sel(X1,X2))
       tail(mark(X)) -> mark(tail(X))
       take(mark(X1),X2) -> mark(take(X1,X2))
       take(X1,mark(X2)) -> mark(take(X1,X2))
       proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3))
       proper(tt()) -> ok(tt())
       proper(U12(X1,X2,X3)) -> U12(proper(X1),proper(X2),proper(X3))
       proper(snd(X)) -> snd(proper(X))
       proper(splitAt(X1,X2)) -> splitAt(proper(X1),proper(X2))
       proper(U21(X1,X2)) -> U21(proper(X1),proper(X2))
       proper(U22(X1,X2)) -> U22(proper(X1),proper(X2))
       proper(U31(X1,X2)) -> U31(proper(X1),proper(X2))
       proper(U32(X1,X2)) -> U32(proper(X1),proper(X2))
       proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3))
       proper(U42(X1,X2,X3)) -> U42(proper(X1),proper(X2),proper(X3))
       proper(head(X)) -> head(proper(X))
       proper(afterNth(X1,X2)) -> afterNth(proper(X1),proper(X2))
       proper(U51(X1,X2)) -> U51(proper(X1),proper(X2))
       proper(U52(X1,X2)) -> U52(proper(X1),proper(X2))
       proper(U61(X1,X2,X3,X4)) -> U61(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(U62(X1,X2,X3,X4)) -> U62(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(U63(X1,X2,X3,X4)) -> U63(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(U64(X1,X2)) -> U64(proper(X1),proper(X2))
       proper(pair(X1,X2)) -> pair(proper(X1),proper(X2))
       proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
       proper(U71(X1,X2)) -> U71(proper(X1),proper(X2))
       proper(U72(X1,X2)) -> U72(proper(X1),proper(X2))
       proper(U81(X1,X2,X3)) -> U81(proper(X1),proper(X2),proper(X3))
       proper(U82(X1,X2,X3)) -> U82(proper(X1),proper(X2),proper(X3))
       proper(fst(X)) -> fst(proper(X))
       proper(natsFrom(X)) -> natsFrom(proper(X))
       proper(s(X)) -> s(proper(X))
       proper(sel(X1,X2)) -> sel(proper(X1),proper(X2))
       proper(0()) -> ok(0())
       proper(nil()) -> ok(nil())
       proper(tail(X)) -> tail(proper(X))
       proper(take(X1,X2)) -> take(proper(X1),proper(X2))
       U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3))
       U12(ok(X1),ok(X2),ok(X3)) -> ok(U12(X1,X2,X3))
       snd(ok(X)) -> ok(snd(X))
       splitAt(ok(X1),ok(X2)) -> ok(splitAt(X1,X2))
       U21(ok(X1),ok(X2)) -> ok(U21(X1,X2))
       U22(ok(X1),ok(X2)) -> ok(U22(X1,X2))
       U31(ok(X1),ok(X2)) -> ok(U31(X1,X2))
       U32(ok(X1),ok(X2)) -> ok(U32(X1,X2))
       U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3))
       U42(ok(X1),ok(X2),ok(X3)) -> ok(U42(X1,X2,X3))
       head(ok(X)) -> ok(head(X))
       afterNth(ok(X1),ok(X2)) -> ok(afterNth(X1,X2))
       U51(ok(X1),ok(X2)) -> ok(U51(X1,X2))
       U52(ok(X1),ok(X2)) -> ok(U52(X1,X2))
       U61(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U61(X1,X2,X3,X4))
       U62(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U62(X1,X2,X3,X4))
       U63(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U63(X1,X2,X3,X4))
       U64(ok(X1),ok(X2)) -> ok(U64(X1,X2))
       pair(ok(X1),ok(X2)) -> ok(pair(X1,X2))
       cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
       U71(ok(X1),ok(X2)) -> ok(U71(X1,X2))
       U72(ok(X1),ok(X2)) -> ok(U72(X1,X2))
       U81(ok(X1),ok(X2),ok(X3)) -> ok(U81(X1,X2,X3))
       U82(ok(X1),ok(X2),ok(X3)) -> ok(U82(X1,X2,X3))
       fst(ok(X)) -> ok(fst(X))
       natsFrom(ok(X)) -> ok(natsFrom(X))
       s(ok(X)) -> ok(s(X))
       sel(ok(X1),ok(X2)) -> ok(sel(X1,X2))
       tail(ok(X)) -> ok(tail(X))
       take(ok(X1),ok(X2)) -> ok(take(X1,X2))
       top(mark(X)) -> top(proper(X))
       top(ok(X)) -> top(active(X))
      Open
      
      DPs:
       natsFrom#(mark(X)) -> natsFrom#(X)
       natsFrom#(ok(X)) -> natsFrom#(X)
      TRS:
       active(U11(tt(),N,XS)) -> mark(U12(tt(),N,XS))
       active(U12(tt(),N,XS)) -> mark(snd(splitAt(N,XS)))
       active(U21(tt(),X)) -> mark(U22(tt(),X))
       active(U22(tt(),X)) -> mark(X)
       active(U31(tt(),N)) -> mark(U32(tt(),N))
       active(U32(tt(),N)) -> mark(N)
       active(U41(tt(),N,XS)) -> mark(U42(tt(),N,XS))
       active(U42(tt(),N,XS)) -> mark(head(afterNth(N,XS)))
       active(U51(tt(),Y)) -> mark(U52(tt(),Y))
       active(U52(tt(),Y)) -> mark(Y)
       active(U61(tt(),N,X,XS)) -> mark(U62(tt(),N,X,XS))
       active(U62(tt(),N,X,XS)) -> mark(U63(tt(),N,X,XS))
       active(U63(tt(),N,X,XS)) -> mark(U64(splitAt(N,XS),X))
       active(U64(pair(YS,ZS),X)) -> mark(pair(cons(X,YS),ZS))
       active(U71(tt(),XS)) -> mark(U72(tt(),XS))
       active(U72(tt(),XS)) -> mark(XS)
       active(U81(tt(),N,XS)) -> mark(U82(tt(),N,XS))
       active(U82(tt(),N,XS)) -> mark(fst(splitAt(N,XS)))
       active(afterNth(N,XS)) -> mark(U11(tt(),N,XS))
       active(fst(pair(X,Y))) -> mark(U21(tt(),X))
       active(head(cons(N,XS))) -> mark(U31(tt(),N))
       active(natsFrom(N)) -> mark(cons(N,natsFrom(s(N))))
       active(sel(N,XS)) -> mark(U41(tt(),N,XS))
       active(snd(pair(X,Y))) -> mark(U51(tt(),Y))
       active(splitAt(0(),XS)) -> mark(pair(nil(),XS))
       active(splitAt(s(N),cons(X,XS))) -> mark(U61(tt(),N,X,XS))
       active(tail(cons(N,XS))) -> mark(U71(tt(),XS))
       active(take(N,XS)) -> mark(U81(tt(),N,XS))
       active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3)
       active(U12(X1,X2,X3)) -> U12(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(U21(X1,X2)) -> U21(active(X1),X2)
       active(U22(X1,X2)) -> U22(active(X1),X2)
       active(U31(X1,X2)) -> U31(active(X1),X2)
       active(U32(X1,X2)) -> U32(active(X1),X2)
       active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3)
       active(U42(X1,X2,X3)) -> U42(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(U51(X1,X2)) -> U51(active(X1),X2)
       active(U52(X1,X2)) -> U52(active(X1),X2)
       active(U61(X1,X2,X3,X4)) -> U61(active(X1),X2,X3,X4)
       active(U62(X1,X2,X3,X4)) -> U62(active(X1),X2,X3,X4)
       active(U63(X1,X2,X3,X4)) -> U63(active(X1),X2,X3,X4)
       active(U64(X1,X2)) -> U64(active(X1),X2)
       active(pair(X1,X2)) -> pair(active(X1),X2)
       active(pair(X1,X2)) -> pair(X1,active(X2))
       active(cons(X1,X2)) -> cons(active(X1),X2)
       active(U71(X1,X2)) -> U71(active(X1),X2)
       active(U72(X1,X2)) -> U72(active(X1),X2)
       active(U81(X1,X2,X3)) -> U81(active(X1),X2,X3)
       active(U82(X1,X2,X3)) -> U82(active(X1),X2,X3)
       active(fst(X)) -> fst(active(X))
       active(natsFrom(X)) -> natsFrom(active(X))
       active(s(X)) -> s(active(X))
       active(sel(X1,X2)) -> sel(active(X1),X2)
       active(sel(X1,X2)) -> sel(X1,active(X2))
       active(tail(X)) -> tail(active(X))
       active(take(X1,X2)) -> take(active(X1),X2)
       active(take(X1,X2)) -> take(X1,active(X2))
       U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3))
       U12(mark(X1),X2,X3) -> mark(U12(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))
       U21(mark(X1),X2) -> mark(U21(X1,X2))
       U22(mark(X1),X2) -> mark(U22(X1,X2))
       U31(mark(X1),X2) -> mark(U31(X1,X2))
       U32(mark(X1),X2) -> mark(U32(X1,X2))
       U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3))
       U42(mark(X1),X2,X3) -> mark(U42(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))
       U51(mark(X1),X2) -> mark(U51(X1,X2))
       U52(mark(X1),X2) -> mark(U52(X1,X2))
       U61(mark(X1),X2,X3,X4) -> mark(U61(X1,X2,X3,X4))
       U62(mark(X1),X2,X3,X4) -> mark(U62(X1,X2,X3,X4))
       U63(mark(X1),X2,X3,X4) -> mark(U63(X1,X2,X3,X4))
       U64(mark(X1),X2) -> mark(U64(X1,X2))
       pair(mark(X1),X2) -> mark(pair(X1,X2))
       pair(X1,mark(X2)) -> mark(pair(X1,X2))
       cons(mark(X1),X2) -> mark(cons(X1,X2))
       U71(mark(X1),X2) -> mark(U71(X1,X2))
       U72(mark(X1),X2) -> mark(U72(X1,X2))
       U81(mark(X1),X2,X3) -> mark(U81(X1,X2,X3))
       U82(mark(X1),X2,X3) -> mark(U82(X1,X2,X3))
       fst(mark(X)) -> mark(fst(X))
       natsFrom(mark(X)) -> mark(natsFrom(X))
       s(mark(X)) -> mark(s(X))
       sel(mark(X1),X2) -> mark(sel(X1,X2))
       sel(X1,mark(X2)) -> mark(sel(X1,X2))
       tail(mark(X)) -> mark(tail(X))
       take(mark(X1),X2) -> mark(take(X1,X2))
       take(X1,mark(X2)) -> mark(take(X1,X2))
       proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3))
       proper(tt()) -> ok(tt())
       proper(U12(X1,X2,X3)) -> U12(proper(X1),proper(X2),proper(X3))
       proper(snd(X)) -> snd(proper(X))
       proper(splitAt(X1,X2)) -> splitAt(proper(X1),proper(X2))
       proper(U21(X1,X2)) -> U21(proper(X1),proper(X2))
       proper(U22(X1,X2)) -> U22(proper(X1),proper(X2))
       proper(U31(X1,X2)) -> U31(proper(X1),proper(X2))
       proper(U32(X1,X2)) -> U32(proper(X1),proper(X2))
       proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3))
       proper(U42(X1,X2,X3)) -> U42(proper(X1),proper(X2),proper(X3))
       proper(head(X)) -> head(proper(X))
       proper(afterNth(X1,X2)) -> afterNth(proper(X1),proper(X2))
       proper(U51(X1,X2)) -> U51(proper(X1),proper(X2))
       proper(U52(X1,X2)) -> U52(proper(X1),proper(X2))
       proper(U61(X1,X2,X3,X4)) -> U61(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(U62(X1,X2,X3,X4)) -> U62(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(U63(X1,X2,X3,X4)) -> U63(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(U64(X1,X2)) -> U64(proper(X1),proper(X2))
       proper(pair(X1,X2)) -> pair(proper(X1),proper(X2))
       proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
       proper(U71(X1,X2)) -> U71(proper(X1),proper(X2))
       proper(U72(X1,X2)) -> U72(proper(X1),proper(X2))
       proper(U81(X1,X2,X3)) -> U81(proper(X1),proper(X2),proper(X3))
       proper(U82(X1,X2,X3)) -> U82(proper(X1),proper(X2),proper(X3))
       proper(fst(X)) -> fst(proper(X))
       proper(natsFrom(X)) -> natsFrom(proper(X))
       proper(s(X)) -> s(proper(X))
       proper(sel(X1,X2)) -> sel(proper(X1),proper(X2))
       proper(0()) -> ok(0())
       proper(nil()) -> ok(nil())
       proper(tail(X)) -> tail(proper(X))
       proper(take(X1,X2)) -> take(proper(X1),proper(X2))
       U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3))
       U12(ok(X1),ok(X2),ok(X3)) -> ok(U12(X1,X2,X3))
       snd(ok(X)) -> ok(snd(X))
       splitAt(ok(X1),ok(X2)) -> ok(splitAt(X1,X2))
       U21(ok(X1),ok(X2)) -> ok(U21(X1,X2))
       U22(ok(X1),ok(X2)) -> ok(U22(X1,X2))
       U31(ok(X1),ok(X2)) -> ok(U31(X1,X2))
       U32(ok(X1),ok(X2)) -> ok(U32(X1,X2))
       U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3))
       U42(ok(X1),ok(X2),ok(X3)) -> ok(U42(X1,X2,X3))
       head(ok(X)) -> ok(head(X))
       afterNth(ok(X1),ok(X2)) -> ok(afterNth(X1,X2))
       U51(ok(X1),ok(X2)) -> ok(U51(X1,X2))
       U52(ok(X1),ok(X2)) -> ok(U52(X1,X2))
       U61(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U61(X1,X2,X3,X4))
       U62(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U62(X1,X2,X3,X4))
       U63(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U63(X1,X2,X3,X4))
       U64(ok(X1),ok(X2)) -> ok(U64(X1,X2))
       pair(ok(X1),ok(X2)) -> ok(pair(X1,X2))
       cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
       U71(ok(X1),ok(X2)) -> ok(U71(X1,X2))
       U72(ok(X1),ok(X2)) -> ok(U72(X1,X2))
       U81(ok(X1),ok(X2),ok(X3)) -> ok(U81(X1,X2,X3))
       U82(ok(X1),ok(X2),ok(X3)) -> ok(U82(X1,X2,X3))
       fst(ok(X)) -> ok(fst(X))
       natsFrom(ok(X)) -> ok(natsFrom(X))
       s(ok(X)) -> ok(s(X))
       sel(ok(X1),ok(X2)) -> ok(sel(X1,X2))
       tail(ok(X)) -> ok(tail(X))
       take(ok(X1),ok(X2)) -> ok(take(X1,X2))
       top(mark(X)) -> top(proper(X))
       top(ok(X)) -> top(active(X))
      Open
      
      DPs:
       fst#(mark(X)) -> fst#(X)
       fst#(ok(X)) -> fst#(X)
      TRS:
       active(U11(tt(),N,XS)) -> mark(U12(tt(),N,XS))
       active(U12(tt(),N,XS)) -> mark(snd(splitAt(N,XS)))
       active(U21(tt(),X)) -> mark(U22(tt(),X))
       active(U22(tt(),X)) -> mark(X)
       active(U31(tt(),N)) -> mark(U32(tt(),N))
       active(U32(tt(),N)) -> mark(N)
       active(U41(tt(),N,XS)) -> mark(U42(tt(),N,XS))
       active(U42(tt(),N,XS)) -> mark(head(afterNth(N,XS)))
       active(U51(tt(),Y)) -> mark(U52(tt(),Y))
       active(U52(tt(),Y)) -> mark(Y)
       active(U61(tt(),N,X,XS)) -> mark(U62(tt(),N,X,XS))
       active(U62(tt(),N,X,XS)) -> mark(U63(tt(),N,X,XS))
       active(U63(tt(),N,X,XS)) -> mark(U64(splitAt(N,XS),X))
       active(U64(pair(YS,ZS),X)) -> mark(pair(cons(X,YS),ZS))
       active(U71(tt(),XS)) -> mark(U72(tt(),XS))
       active(U72(tt(),XS)) -> mark(XS)
       active(U81(tt(),N,XS)) -> mark(U82(tt(),N,XS))
       active(U82(tt(),N,XS)) -> mark(fst(splitAt(N,XS)))
       active(afterNth(N,XS)) -> mark(U11(tt(),N,XS))
       active(fst(pair(X,Y))) -> mark(U21(tt(),X))
       active(head(cons(N,XS))) -> mark(U31(tt(),N))
       active(natsFrom(N)) -> mark(cons(N,natsFrom(s(N))))
       active(sel(N,XS)) -> mark(U41(tt(),N,XS))
       active(snd(pair(X,Y))) -> mark(U51(tt(),Y))
       active(splitAt(0(),XS)) -> mark(pair(nil(),XS))
       active(splitAt(s(N),cons(X,XS))) -> mark(U61(tt(),N,X,XS))
       active(tail(cons(N,XS))) -> mark(U71(tt(),XS))
       active(take(N,XS)) -> mark(U81(tt(),N,XS))
       active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3)
       active(U12(X1,X2,X3)) -> U12(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(U21(X1,X2)) -> U21(active(X1),X2)
       active(U22(X1,X2)) -> U22(active(X1),X2)
       active(U31(X1,X2)) -> U31(active(X1),X2)
       active(U32(X1,X2)) -> U32(active(X1),X2)
       active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3)
       active(U42(X1,X2,X3)) -> U42(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(U51(X1,X2)) -> U51(active(X1),X2)
       active(U52(X1,X2)) -> U52(active(X1),X2)
       active(U61(X1,X2,X3,X4)) -> U61(active(X1),X2,X3,X4)
       active(U62(X1,X2,X3,X4)) -> U62(active(X1),X2,X3,X4)
       active(U63(X1,X2,X3,X4)) -> U63(active(X1),X2,X3,X4)
       active(U64(X1,X2)) -> U64(active(X1),X2)
       active(pair(X1,X2)) -> pair(active(X1),X2)
       active(pair(X1,X2)) -> pair(X1,active(X2))
       active(cons(X1,X2)) -> cons(active(X1),X2)
       active(U71(X1,X2)) -> U71(active(X1),X2)
       active(U72(X1,X2)) -> U72(active(X1),X2)
       active(U81(X1,X2,X3)) -> U81(active(X1),X2,X3)
       active(U82(X1,X2,X3)) -> U82(active(X1),X2,X3)
       active(fst(X)) -> fst(active(X))
       active(natsFrom(X)) -> natsFrom(active(X))
       active(s(X)) -> s(active(X))
       active(sel(X1,X2)) -> sel(active(X1),X2)
       active(sel(X1,X2)) -> sel(X1,active(X2))
       active(tail(X)) -> tail(active(X))
       active(take(X1,X2)) -> take(active(X1),X2)
       active(take(X1,X2)) -> take(X1,active(X2))
       U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3))
       U12(mark(X1),X2,X3) -> mark(U12(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))
       U21(mark(X1),X2) -> mark(U21(X1,X2))
       U22(mark(X1),X2) -> mark(U22(X1,X2))
       U31(mark(X1),X2) -> mark(U31(X1,X2))
       U32(mark(X1),X2) -> mark(U32(X1,X2))
       U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3))
       U42(mark(X1),X2,X3) -> mark(U42(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))
       U51(mark(X1),X2) -> mark(U51(X1,X2))
       U52(mark(X1),X2) -> mark(U52(X1,X2))
       U61(mark(X1),X2,X3,X4) -> mark(U61(X1,X2,X3,X4))
       U62(mark(X1),X2,X3,X4) -> mark(U62(X1,X2,X3,X4))
       U63(mark(X1),X2,X3,X4) -> mark(U63(X1,X2,X3,X4))
       U64(mark(X1),X2) -> mark(U64(X1,X2))
       pair(mark(X1),X2) -> mark(pair(X1,X2))
       pair(X1,mark(X2)) -> mark(pair(X1,X2))
       cons(mark(X1),X2) -> mark(cons(X1,X2))
       U71(mark(X1),X2) -> mark(U71(X1,X2))
       U72(mark(X1),X2) -> mark(U72(X1,X2))
       U81(mark(X1),X2,X3) -> mark(U81(X1,X2,X3))
       U82(mark(X1),X2,X3) -> mark(U82(X1,X2,X3))
       fst(mark(X)) -> mark(fst(X))
       natsFrom(mark(X)) -> mark(natsFrom(X))
       s(mark(X)) -> mark(s(X))
       sel(mark(X1),X2) -> mark(sel(X1,X2))
       sel(X1,mark(X2)) -> mark(sel(X1,X2))
       tail(mark(X)) -> mark(tail(X))
       take(mark(X1),X2) -> mark(take(X1,X2))
       take(X1,mark(X2)) -> mark(take(X1,X2))
       proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3))
       proper(tt()) -> ok(tt())
       proper(U12(X1,X2,X3)) -> U12(proper(X1),proper(X2),proper(X3))
       proper(snd(X)) -> snd(proper(X))
       proper(splitAt(X1,X2)) -> splitAt(proper(X1),proper(X2))
       proper(U21(X1,X2)) -> U21(proper(X1),proper(X2))
       proper(U22(X1,X2)) -> U22(proper(X1),proper(X2))
       proper(U31(X1,X2)) -> U31(proper(X1),proper(X2))
       proper(U32(X1,X2)) -> U32(proper(X1),proper(X2))
       proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3))
       proper(U42(X1,X2,X3)) -> U42(proper(X1),proper(X2),proper(X3))
       proper(head(X)) -> head(proper(X))
       proper(afterNth(X1,X2)) -> afterNth(proper(X1),proper(X2))
       proper(U51(X1,X2)) -> U51(proper(X1),proper(X2))
       proper(U52(X1,X2)) -> U52(proper(X1),proper(X2))
       proper(U61(X1,X2,X3,X4)) -> U61(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(U62(X1,X2,X3,X4)) -> U62(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(U63(X1,X2,X3,X4)) -> U63(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(U64(X1,X2)) -> U64(proper(X1),proper(X2))
       proper(pair(X1,X2)) -> pair(proper(X1),proper(X2))
       proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
       proper(U71(X1,X2)) -> U71(proper(X1),proper(X2))
       proper(U72(X1,X2)) -> U72(proper(X1),proper(X2))
       proper(U81(X1,X2,X3)) -> U81(proper(X1),proper(X2),proper(X3))
       proper(U82(X1,X2,X3)) -> U82(proper(X1),proper(X2),proper(X3))
       proper(fst(X)) -> fst(proper(X))
       proper(natsFrom(X)) -> natsFrom(proper(X))
       proper(s(X)) -> s(proper(X))
       proper(sel(X1,X2)) -> sel(proper(X1),proper(X2))
       proper(0()) -> ok(0())
       proper(nil()) -> ok(nil())
       proper(tail(X)) -> tail(proper(X))
       proper(take(X1,X2)) -> take(proper(X1),proper(X2))
       U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3))
       U12(ok(X1),ok(X2),ok(X3)) -> ok(U12(X1,X2,X3))
       snd(ok(X)) -> ok(snd(X))
       splitAt(ok(X1),ok(X2)) -> ok(splitAt(X1,X2))
       U21(ok(X1),ok(X2)) -> ok(U21(X1,X2))
       U22(ok(X1),ok(X2)) -> ok(U22(X1,X2))
       U31(ok(X1),ok(X2)) -> ok(U31(X1,X2))
       U32(ok(X1),ok(X2)) -> ok(U32(X1,X2))
       U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3))
       U42(ok(X1),ok(X2),ok(X3)) -> ok(U42(X1,X2,X3))
       head(ok(X)) -> ok(head(X))
       afterNth(ok(X1),ok(X2)) -> ok(afterNth(X1,X2))
       U51(ok(X1),ok(X2)) -> ok(U51(X1,X2))
       U52(ok(X1),ok(X2)) -> ok(U52(X1,X2))
       U61(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U61(X1,X2,X3,X4))
       U62(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U62(X1,X2,X3,X4))
       U63(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U63(X1,X2,X3,X4))
       U64(ok(X1),ok(X2)) -> ok(U64(X1,X2))
       pair(ok(X1),ok(X2)) -> ok(pair(X1,X2))
       cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
       U71(ok(X1),ok(X2)) -> ok(U71(X1,X2))
       U72(ok(X1),ok(X2)) -> ok(U72(X1,X2))
       U81(ok(X1),ok(X2),ok(X3)) -> ok(U81(X1,X2,X3))
       U82(ok(X1),ok(X2),ok(X3)) -> ok(U82(X1,X2,X3))
       fst(ok(X)) -> ok(fst(X))
       natsFrom(ok(X)) -> ok(natsFrom(X))
       s(ok(X)) -> ok(s(X))
       sel(ok(X1),ok(X2)) -> ok(sel(X1,X2))
       tail(ok(X)) -> ok(tail(X))
       take(ok(X1),ok(X2)) -> ok(take(X1,X2))
       top(mark(X)) -> top(proper(X))
       top(ok(X)) -> top(active(X))
      Open
      
      DPs:
       U82#(mark(X1),X2,X3) -> U82#(X1,X2,X3)
       U82#(ok(X1),ok(X2),ok(X3)) -> U82#(X1,X2,X3)
      TRS:
       active(U11(tt(),N,XS)) -> mark(U12(tt(),N,XS))
       active(U12(tt(),N,XS)) -> mark(snd(splitAt(N,XS)))
       active(U21(tt(),X)) -> mark(U22(tt(),X))
       active(U22(tt(),X)) -> mark(X)
       active(U31(tt(),N)) -> mark(U32(tt(),N))
       active(U32(tt(),N)) -> mark(N)
       active(U41(tt(),N,XS)) -> mark(U42(tt(),N,XS))
       active(U42(tt(),N,XS)) -> mark(head(afterNth(N,XS)))
       active(U51(tt(),Y)) -> mark(U52(tt(),Y))
       active(U52(tt(),Y)) -> mark(Y)
       active(U61(tt(),N,X,XS)) -> mark(U62(tt(),N,X,XS))
       active(U62(tt(),N,X,XS)) -> mark(U63(tt(),N,X,XS))
       active(U63(tt(),N,X,XS)) -> mark(U64(splitAt(N,XS),X))
       active(U64(pair(YS,ZS),X)) -> mark(pair(cons(X,YS),ZS))
       active(U71(tt(),XS)) -> mark(U72(tt(),XS))
       active(U72(tt(),XS)) -> mark(XS)
       active(U81(tt(),N,XS)) -> mark(U82(tt(),N,XS))
       active(U82(tt(),N,XS)) -> mark(fst(splitAt(N,XS)))
       active(afterNth(N,XS)) -> mark(U11(tt(),N,XS))
       active(fst(pair(X,Y))) -> mark(U21(tt(),X))
       active(head(cons(N,XS))) -> mark(U31(tt(),N))
       active(natsFrom(N)) -> mark(cons(N,natsFrom(s(N))))
       active(sel(N,XS)) -> mark(U41(tt(),N,XS))
       active(snd(pair(X,Y))) -> mark(U51(tt(),Y))
       active(splitAt(0(),XS)) -> mark(pair(nil(),XS))
       active(splitAt(s(N),cons(X,XS))) -> mark(U61(tt(),N,X,XS))
       active(tail(cons(N,XS))) -> mark(U71(tt(),XS))
       active(take(N,XS)) -> mark(U81(tt(),N,XS))
       active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3)
       active(U12(X1,X2,X3)) -> U12(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(U21(X1,X2)) -> U21(active(X1),X2)
       active(U22(X1,X2)) -> U22(active(X1),X2)
       active(U31(X1,X2)) -> U31(active(X1),X2)
       active(U32(X1,X2)) -> U32(active(X1),X2)
       active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3)
       active(U42(X1,X2,X3)) -> U42(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(U51(X1,X2)) -> U51(active(X1),X2)
       active(U52(X1,X2)) -> U52(active(X1),X2)
       active(U61(X1,X2,X3,X4)) -> U61(active(X1),X2,X3,X4)
       active(U62(X1,X2,X3,X4)) -> U62(active(X1),X2,X3,X4)
       active(U63(X1,X2,X3,X4)) -> U63(active(X1),X2,X3,X4)
       active(U64(X1,X2)) -> U64(active(X1),X2)
       active(pair(X1,X2)) -> pair(active(X1),X2)
       active(pair(X1,X2)) -> pair(X1,active(X2))
       active(cons(X1,X2)) -> cons(active(X1),X2)
       active(U71(X1,X2)) -> U71(active(X1),X2)
       active(U72(X1,X2)) -> U72(active(X1),X2)
       active(U81(X1,X2,X3)) -> U81(active(X1),X2,X3)
       active(U82(X1,X2,X3)) -> U82(active(X1),X2,X3)
       active(fst(X)) -> fst(active(X))
       active(natsFrom(X)) -> natsFrom(active(X))
       active(s(X)) -> s(active(X))
       active(sel(X1,X2)) -> sel(active(X1),X2)
       active(sel(X1,X2)) -> sel(X1,active(X2))
       active(tail(X)) -> tail(active(X))
       active(take(X1,X2)) -> take(active(X1),X2)
       active(take(X1,X2)) -> take(X1,active(X2))
       U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3))
       U12(mark(X1),X2,X3) -> mark(U12(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))
       U21(mark(X1),X2) -> mark(U21(X1,X2))
       U22(mark(X1),X2) -> mark(U22(X1,X2))
       U31(mark(X1),X2) -> mark(U31(X1,X2))
       U32(mark(X1),X2) -> mark(U32(X1,X2))
       U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3))
       U42(mark(X1),X2,X3) -> mark(U42(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))
       U51(mark(X1),X2) -> mark(U51(X1,X2))
       U52(mark(X1),X2) -> mark(U52(X1,X2))
       U61(mark(X1),X2,X3,X4) -> mark(U61(X1,X2,X3,X4))
       U62(mark(X1),X2,X3,X4) -> mark(U62(X1,X2,X3,X4))
       U63(mark(X1),X2,X3,X4) -> mark(U63(X1,X2,X3,X4))
       U64(mark(X1),X2) -> mark(U64(X1,X2))
       pair(mark(X1),X2) -> mark(pair(X1,X2))
       pair(X1,mark(X2)) -> mark(pair(X1,X2))
       cons(mark(X1),X2) -> mark(cons(X1,X2))
       U71(mark(X1),X2) -> mark(U71(X1,X2))
       U72(mark(X1),X2) -> mark(U72(X1,X2))
       U81(mark(X1),X2,X3) -> mark(U81(X1,X2,X3))
       U82(mark(X1),X2,X3) -> mark(U82(X1,X2,X3))
       fst(mark(X)) -> mark(fst(X))
       natsFrom(mark(X)) -> mark(natsFrom(X))
       s(mark(X)) -> mark(s(X))
       sel(mark(X1),X2) -> mark(sel(X1,X2))
       sel(X1,mark(X2)) -> mark(sel(X1,X2))
       tail(mark(X)) -> mark(tail(X))
       take(mark(X1),X2) -> mark(take(X1,X2))
       take(X1,mark(X2)) -> mark(take(X1,X2))
       proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3))
       proper(tt()) -> ok(tt())
       proper(U12(X1,X2,X3)) -> U12(proper(X1),proper(X2),proper(X3))
       proper(snd(X)) -> snd(proper(X))
       proper(splitAt(X1,X2)) -> splitAt(proper(X1),proper(X2))
       proper(U21(X1,X2)) -> U21(proper(X1),proper(X2))
       proper(U22(X1,X2)) -> U22(proper(X1),proper(X2))
       proper(U31(X1,X2)) -> U31(proper(X1),proper(X2))
       proper(U32(X1,X2)) -> U32(proper(X1),proper(X2))
       proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3))
       proper(U42(X1,X2,X3)) -> U42(proper(X1),proper(X2),proper(X3))
       proper(head(X)) -> head(proper(X))
       proper(afterNth(X1,X2)) -> afterNth(proper(X1),proper(X2))
       proper(U51(X1,X2)) -> U51(proper(X1),proper(X2))
       proper(U52(X1,X2)) -> U52(proper(X1),proper(X2))
       proper(U61(X1,X2,X3,X4)) -> U61(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(U62(X1,X2,X3,X4)) -> U62(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(U63(X1,X2,X3,X4)) -> U63(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(U64(X1,X2)) -> U64(proper(X1),proper(X2))
       proper(pair(X1,X2)) -> pair(proper(X1),proper(X2))
       proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
       proper(U71(X1,X2)) -> U71(proper(X1),proper(X2))
       proper(U72(X1,X2)) -> U72(proper(X1),proper(X2))
       proper(U81(X1,X2,X3)) -> U81(proper(X1),proper(X2),proper(X3))
       proper(U82(X1,X2,X3)) -> U82(proper(X1),proper(X2),proper(X3))
       proper(fst(X)) -> fst(proper(X))
       proper(natsFrom(X)) -> natsFrom(proper(X))
       proper(s(X)) -> s(proper(X))
       proper(sel(X1,X2)) -> sel(proper(X1),proper(X2))
       proper(0()) -> ok(0())
       proper(nil()) -> ok(nil())
       proper(tail(X)) -> tail(proper(X))
       proper(take(X1,X2)) -> take(proper(X1),proper(X2))
       U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3))
       U12(ok(X1),ok(X2),ok(X3)) -> ok(U12(X1,X2,X3))
       snd(ok(X)) -> ok(snd(X))
       splitAt(ok(X1),ok(X2)) -> ok(splitAt(X1,X2))
       U21(ok(X1),ok(X2)) -> ok(U21(X1,X2))
       U22(ok(X1),ok(X2)) -> ok(U22(X1,X2))
       U31(ok(X1),ok(X2)) -> ok(U31(X1,X2))
       U32(ok(X1),ok(X2)) -> ok(U32(X1,X2))
       U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3))
       U42(ok(X1),ok(X2),ok(X3)) -> ok(U42(X1,X2,X3))
       head(ok(X)) -> ok(head(X))
       afterNth(ok(X1),ok(X2)) -> ok(afterNth(X1,X2))
       U51(ok(X1),ok(X2)) -> ok(U51(X1,X2))
       U52(ok(X1),ok(X2)) -> ok(U52(X1,X2))
       U61(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U61(X1,X2,X3,X4))
       U62(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U62(X1,X2,X3,X4))
       U63(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U63(X1,X2,X3,X4))
       U64(ok(X1),ok(X2)) -> ok(U64(X1,X2))
       pair(ok(X1),ok(X2)) -> ok(pair(X1,X2))
       cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
       U71(ok(X1),ok(X2)) -> ok(U71(X1,X2))
       U72(ok(X1),ok(X2)) -> ok(U72(X1,X2))
       U81(ok(X1),ok(X2),ok(X3)) -> ok(U81(X1,X2,X3))
       U82(ok(X1),ok(X2),ok(X3)) -> ok(U82(X1,X2,X3))
       fst(ok(X)) -> ok(fst(X))
       natsFrom(ok(X)) -> ok(natsFrom(X))
       s(ok(X)) -> ok(s(X))
       sel(ok(X1),ok(X2)) -> ok(sel(X1,X2))
       tail(ok(X)) -> ok(tail(X))
       take(ok(X1),ok(X2)) -> ok(take(X1,X2))
       top(mark(X)) -> top(proper(X))
       top(ok(X)) -> top(active(X))
      Open
      
      DPs:
       U81#(mark(X1),X2,X3) -> U81#(X1,X2,X3)
       U81#(ok(X1),ok(X2),ok(X3)) -> U81#(X1,X2,X3)
      TRS:
       active(U11(tt(),N,XS)) -> mark(U12(tt(),N,XS))
       active(U12(tt(),N,XS)) -> mark(snd(splitAt(N,XS)))
       active(U21(tt(),X)) -> mark(U22(tt(),X))
       active(U22(tt(),X)) -> mark(X)
       active(U31(tt(),N)) -> mark(U32(tt(),N))
       active(U32(tt(),N)) -> mark(N)
       active(U41(tt(),N,XS)) -> mark(U42(tt(),N,XS))
       active(U42(tt(),N,XS)) -> mark(head(afterNth(N,XS)))
       active(U51(tt(),Y)) -> mark(U52(tt(),Y))
       active(U52(tt(),Y)) -> mark(Y)
       active(U61(tt(),N,X,XS)) -> mark(U62(tt(),N,X,XS))
       active(U62(tt(),N,X,XS)) -> mark(U63(tt(),N,X,XS))
       active(U63(tt(),N,X,XS)) -> mark(U64(splitAt(N,XS),X))
       active(U64(pair(YS,ZS),X)) -> mark(pair(cons(X,YS),ZS))
       active(U71(tt(),XS)) -> mark(U72(tt(),XS))
       active(U72(tt(),XS)) -> mark(XS)
       active(U81(tt(),N,XS)) -> mark(U82(tt(),N,XS))
       active(U82(tt(),N,XS)) -> mark(fst(splitAt(N,XS)))
       active(afterNth(N,XS)) -> mark(U11(tt(),N,XS))
       active(fst(pair(X,Y))) -> mark(U21(tt(),X))
       active(head(cons(N,XS))) -> mark(U31(tt(),N))
       active(natsFrom(N)) -> mark(cons(N,natsFrom(s(N))))
       active(sel(N,XS)) -> mark(U41(tt(),N,XS))
       active(snd(pair(X,Y))) -> mark(U51(tt(),Y))
       active(splitAt(0(),XS)) -> mark(pair(nil(),XS))
       active(splitAt(s(N),cons(X,XS))) -> mark(U61(tt(),N,X,XS))
       active(tail(cons(N,XS))) -> mark(U71(tt(),XS))
       active(take(N,XS)) -> mark(U81(tt(),N,XS))
       active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3)
       active(U12(X1,X2,X3)) -> U12(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(U21(X1,X2)) -> U21(active(X1),X2)
       active(U22(X1,X2)) -> U22(active(X1),X2)
       active(U31(X1,X2)) -> U31(active(X1),X2)
       active(U32(X1,X2)) -> U32(active(X1),X2)
       active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3)
       active(U42(X1,X2,X3)) -> U42(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(U51(X1,X2)) -> U51(active(X1),X2)
       active(U52(X1,X2)) -> U52(active(X1),X2)
       active(U61(X1,X2,X3,X4)) -> U61(active(X1),X2,X3,X4)
       active(U62(X1,X2,X3,X4)) -> U62(active(X1),X2,X3,X4)
       active(U63(X1,X2,X3,X4)) -> U63(active(X1),X2,X3,X4)
       active(U64(X1,X2)) -> U64(active(X1),X2)
       active(pair(X1,X2)) -> pair(active(X1),X2)
       active(pair(X1,X2)) -> pair(X1,active(X2))
       active(cons(X1,X2)) -> cons(active(X1),X2)
       active(U71(X1,X2)) -> U71(active(X1),X2)
       active(U72(X1,X2)) -> U72(active(X1),X2)
       active(U81(X1,X2,X3)) -> U81(active(X1),X2,X3)
       active(U82(X1,X2,X3)) -> U82(active(X1),X2,X3)
       active(fst(X)) -> fst(active(X))
       active(natsFrom(X)) -> natsFrom(active(X))
       active(s(X)) -> s(active(X))
       active(sel(X1,X2)) -> sel(active(X1),X2)
       active(sel(X1,X2)) -> sel(X1,active(X2))
       active(tail(X)) -> tail(active(X))
       active(take(X1,X2)) -> take(active(X1),X2)
       active(take(X1,X2)) -> take(X1,active(X2))
       U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3))
       U12(mark(X1),X2,X3) -> mark(U12(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))
       U21(mark(X1),X2) -> mark(U21(X1,X2))
       U22(mark(X1),X2) -> mark(U22(X1,X2))
       U31(mark(X1),X2) -> mark(U31(X1,X2))
       U32(mark(X1),X2) -> mark(U32(X1,X2))
       U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3))
       U42(mark(X1),X2,X3) -> mark(U42(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))
       U51(mark(X1),X2) -> mark(U51(X1,X2))
       U52(mark(X1),X2) -> mark(U52(X1,X2))
       U61(mark(X1),X2,X3,X4) -> mark(U61(X1,X2,X3,X4))
       U62(mark(X1),X2,X3,X4) -> mark(U62(X1,X2,X3,X4))
       U63(mark(X1),X2,X3,X4) -> mark(U63(X1,X2,X3,X4))
       U64(mark(X1),X2) -> mark(U64(X1,X2))
       pair(mark(X1),X2) -> mark(pair(X1,X2))
       pair(X1,mark(X2)) -> mark(pair(X1,X2))
       cons(mark(X1),X2) -> mark(cons(X1,X2))
       U71(mark(X1),X2) -> mark(U71(X1,X2))
       U72(mark(X1),X2) -> mark(U72(X1,X2))
       U81(mark(X1),X2,X3) -> mark(U81(X1,X2,X3))
       U82(mark(X1),X2,X3) -> mark(U82(X1,X2,X3))
       fst(mark(X)) -> mark(fst(X))
       natsFrom(mark(X)) -> mark(natsFrom(X))
       s(mark(X)) -> mark(s(X))
       sel(mark(X1),X2) -> mark(sel(X1,X2))
       sel(X1,mark(X2)) -> mark(sel(X1,X2))
       tail(mark(X)) -> mark(tail(X))
       take(mark(X1),X2) -> mark(take(X1,X2))
       take(X1,mark(X2)) -> mark(take(X1,X2))
       proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3))
       proper(tt()) -> ok(tt())
       proper(U12(X1,X2,X3)) -> U12(proper(X1),proper(X2),proper(X3))
       proper(snd(X)) -> snd(proper(X))
       proper(splitAt(X1,X2)) -> splitAt(proper(X1),proper(X2))
       proper(U21(X1,X2)) -> U21(proper(X1),proper(X2))
       proper(U22(X1,X2)) -> U22(proper(X1),proper(X2))
       proper(U31(X1,X2)) -> U31(proper(X1),proper(X2))
       proper(U32(X1,X2)) -> U32(proper(X1),proper(X2))
       proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3))
       proper(U42(X1,X2,X3)) -> U42(proper(X1),proper(X2),proper(X3))
       proper(head(X)) -> head(proper(X))
       proper(afterNth(X1,X2)) -> afterNth(proper(X1),proper(X2))
       proper(U51(X1,X2)) -> U51(proper(X1),proper(X2))
       proper(U52(X1,X2)) -> U52(proper(X1),proper(X2))
       proper(U61(X1,X2,X3,X4)) -> U61(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(U62(X1,X2,X3,X4)) -> U62(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(U63(X1,X2,X3,X4)) -> U63(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(U64(X1,X2)) -> U64(proper(X1),proper(X2))
       proper(pair(X1,X2)) -> pair(proper(X1),proper(X2))
       proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
       proper(U71(X1,X2)) -> U71(proper(X1),proper(X2))
       proper(U72(X1,X2)) -> U72(proper(X1),proper(X2))
       proper(U81(X1,X2,X3)) -> U81(proper(X1),proper(X2),proper(X3))
       proper(U82(X1,X2,X3)) -> U82(proper(X1),proper(X2),proper(X3))
       proper(fst(X)) -> fst(proper(X))
       proper(natsFrom(X)) -> natsFrom(proper(X))
       proper(s(X)) -> s(proper(X))
       proper(sel(X1,X2)) -> sel(proper(X1),proper(X2))
       proper(0()) -> ok(0())
       proper(nil()) -> ok(nil())
       proper(tail(X)) -> tail(proper(X))
       proper(take(X1,X2)) -> take(proper(X1),proper(X2))
       U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3))
       U12(ok(X1),ok(X2),ok(X3)) -> ok(U12(X1,X2,X3))
       snd(ok(X)) -> ok(snd(X))
       splitAt(ok(X1),ok(X2)) -> ok(splitAt(X1,X2))
       U21(ok(X1),ok(X2)) -> ok(U21(X1,X2))
       U22(ok(X1),ok(X2)) -> ok(U22(X1,X2))
       U31(ok(X1),ok(X2)) -> ok(U31(X1,X2))
       U32(ok(X1),ok(X2)) -> ok(U32(X1,X2))
       U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3))
       U42(ok(X1),ok(X2),ok(X3)) -> ok(U42(X1,X2,X3))
       head(ok(X)) -> ok(head(X))
       afterNth(ok(X1),ok(X2)) -> ok(afterNth(X1,X2))
       U51(ok(X1),ok(X2)) -> ok(U51(X1,X2))
       U52(ok(X1),ok(X2)) -> ok(U52(X1,X2))
       U61(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U61(X1,X2,X3,X4))
       U62(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U62(X1,X2,X3,X4))
       U63(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U63(X1,X2,X3,X4))
       U64(ok(X1),ok(X2)) -> ok(U64(X1,X2))
       pair(ok(X1),ok(X2)) -> ok(pair(X1,X2))
       cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
       U71(ok(X1),ok(X2)) -> ok(U71(X1,X2))
       U72(ok(X1),ok(X2)) -> ok(U72(X1,X2))
       U81(ok(X1),ok(X2),ok(X3)) -> ok(U81(X1,X2,X3))
       U82(ok(X1),ok(X2),ok(X3)) -> ok(U82(X1,X2,X3))
       fst(ok(X)) -> ok(fst(X))
       natsFrom(ok(X)) -> ok(natsFrom(X))
       s(ok(X)) -> ok(s(X))
       sel(ok(X1),ok(X2)) -> ok(sel(X1,X2))
       tail(ok(X)) -> ok(tail(X))
       take(ok(X1),ok(X2)) -> ok(take(X1,X2))
       top(mark(X)) -> top(proper(X))
       top(ok(X)) -> top(active(X))
      Open
      
      DPs:
       U72#(mark(X1),X2) -> U72#(X1,X2)
       U72#(ok(X1),ok(X2)) -> U72#(X1,X2)
      TRS:
       active(U11(tt(),N,XS)) -> mark(U12(tt(),N,XS))
       active(U12(tt(),N,XS)) -> mark(snd(splitAt(N,XS)))
       active(U21(tt(),X)) -> mark(U22(tt(),X))
       active(U22(tt(),X)) -> mark(X)
       active(U31(tt(),N)) -> mark(U32(tt(),N))
       active(U32(tt(),N)) -> mark(N)
       active(U41(tt(),N,XS)) -> mark(U42(tt(),N,XS))
       active(U42(tt(),N,XS)) -> mark(head(afterNth(N,XS)))
       active(U51(tt(),Y)) -> mark(U52(tt(),Y))
       active(U52(tt(),Y)) -> mark(Y)
       active(U61(tt(),N,X,XS)) -> mark(U62(tt(),N,X,XS))
       active(U62(tt(),N,X,XS)) -> mark(U63(tt(),N,X,XS))
       active(U63(tt(),N,X,XS)) -> mark(U64(splitAt(N,XS),X))
       active(U64(pair(YS,ZS),X)) -> mark(pair(cons(X,YS),ZS))
       active(U71(tt(),XS)) -> mark(U72(tt(),XS))
       active(U72(tt(),XS)) -> mark(XS)
       active(U81(tt(),N,XS)) -> mark(U82(tt(),N,XS))
       active(U82(tt(),N,XS)) -> mark(fst(splitAt(N,XS)))
       active(afterNth(N,XS)) -> mark(U11(tt(),N,XS))
       active(fst(pair(X,Y))) -> mark(U21(tt(),X))
       active(head(cons(N,XS))) -> mark(U31(tt(),N))
       active(natsFrom(N)) -> mark(cons(N,natsFrom(s(N))))
       active(sel(N,XS)) -> mark(U41(tt(),N,XS))
       active(snd(pair(X,Y))) -> mark(U51(tt(),Y))
       active(splitAt(0(),XS)) -> mark(pair(nil(),XS))
       active(splitAt(s(N),cons(X,XS))) -> mark(U61(tt(),N,X,XS))
       active(tail(cons(N,XS))) -> mark(U71(tt(),XS))
       active(take(N,XS)) -> mark(U81(tt(),N,XS))
       active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3)
       active(U12(X1,X2,X3)) -> U12(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(U21(X1,X2)) -> U21(active(X1),X2)
       active(U22(X1,X2)) -> U22(active(X1),X2)
       active(U31(X1,X2)) -> U31(active(X1),X2)
       active(U32(X1,X2)) -> U32(active(X1),X2)
       active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3)
       active(U42(X1,X2,X3)) -> U42(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(U51(X1,X2)) -> U51(active(X1),X2)
       active(U52(X1,X2)) -> U52(active(X1),X2)
       active(U61(X1,X2,X3,X4)) -> U61(active(X1),X2,X3,X4)
       active(U62(X1,X2,X3,X4)) -> U62(active(X1),X2,X3,X4)
       active(U63(X1,X2,X3,X4)) -> U63(active(X1),X2,X3,X4)
       active(U64(X1,X2)) -> U64(active(X1),X2)
       active(pair(X1,X2)) -> pair(active(X1),X2)
       active(pair(X1,X2)) -> pair(X1,active(X2))
       active(cons(X1,X2)) -> cons(active(X1),X2)
       active(U71(X1,X2)) -> U71(active(X1),X2)
       active(U72(X1,X2)) -> U72(active(X1),X2)
       active(U81(X1,X2,X3)) -> U81(active(X1),X2,X3)
       active(U82(X1,X2,X3)) -> U82(active(X1),X2,X3)
       active(fst(X)) -> fst(active(X))
       active(natsFrom(X)) -> natsFrom(active(X))
       active(s(X)) -> s(active(X))
       active(sel(X1,X2)) -> sel(active(X1),X2)
       active(sel(X1,X2)) -> sel(X1,active(X2))
       active(tail(X)) -> tail(active(X))
       active(take(X1,X2)) -> take(active(X1),X2)
       active(take(X1,X2)) -> take(X1,active(X2))
       U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3))
       U12(mark(X1),X2,X3) -> mark(U12(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))
       U21(mark(X1),X2) -> mark(U21(X1,X2))
       U22(mark(X1),X2) -> mark(U22(X1,X2))
       U31(mark(X1),X2) -> mark(U31(X1,X2))
       U32(mark(X1),X2) -> mark(U32(X1,X2))
       U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3))
       U42(mark(X1),X2,X3) -> mark(U42(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))
       U51(mark(X1),X2) -> mark(U51(X1,X2))
       U52(mark(X1),X2) -> mark(U52(X1,X2))
       U61(mark(X1),X2,X3,X4) -> mark(U61(X1,X2,X3,X4))
       U62(mark(X1),X2,X3,X4) -> mark(U62(X1,X2,X3,X4))
       U63(mark(X1),X2,X3,X4) -> mark(U63(X1,X2,X3,X4))
       U64(mark(X1),X2) -> mark(U64(X1,X2))
       pair(mark(X1),X2) -> mark(pair(X1,X2))
       pair(X1,mark(X2)) -> mark(pair(X1,X2))
       cons(mark(X1),X2) -> mark(cons(X1,X2))
       U71(mark(X1),X2) -> mark(U71(X1,X2))
       U72(mark(X1),X2) -> mark(U72(X1,X2))
       U81(mark(X1),X2,X3) -> mark(U81(X1,X2,X3))
       U82(mark(X1),X2,X3) -> mark(U82(X1,X2,X3))
       fst(mark(X)) -> mark(fst(X))
       natsFrom(mark(X)) -> mark(natsFrom(X))
       s(mark(X)) -> mark(s(X))
       sel(mark(X1),X2) -> mark(sel(X1,X2))
       sel(X1,mark(X2)) -> mark(sel(X1,X2))
       tail(mark(X)) -> mark(tail(X))
       take(mark(X1),X2) -> mark(take(X1,X2))
       take(X1,mark(X2)) -> mark(take(X1,X2))
       proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3))
       proper(tt()) -> ok(tt())
       proper(U12(X1,X2,X3)) -> U12(proper(X1),proper(X2),proper(X3))
       proper(snd(X)) -> snd(proper(X))
       proper(splitAt(X1,X2)) -> splitAt(proper(X1),proper(X2))
       proper(U21(X1,X2)) -> U21(proper(X1),proper(X2))
       proper(U22(X1,X2)) -> U22(proper(X1),proper(X2))
       proper(U31(X1,X2)) -> U31(proper(X1),proper(X2))
       proper(U32(X1,X2)) -> U32(proper(X1),proper(X2))
       proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3))
       proper(U42(X1,X2,X3)) -> U42(proper(X1),proper(X2),proper(X3))
       proper(head(X)) -> head(proper(X))
       proper(afterNth(X1,X2)) -> afterNth(proper(X1),proper(X2))
       proper(U51(X1,X2)) -> U51(proper(X1),proper(X2))
       proper(U52(X1,X2)) -> U52(proper(X1),proper(X2))
       proper(U61(X1,X2,X3,X4)) -> U61(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(U62(X1,X2,X3,X4)) -> U62(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(U63(X1,X2,X3,X4)) -> U63(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(U64(X1,X2)) -> U64(proper(X1),proper(X2))
       proper(pair(X1,X2)) -> pair(proper(X1),proper(X2))
       proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
       proper(U71(X1,X2)) -> U71(proper(X1),proper(X2))
       proper(U72(X1,X2)) -> U72(proper(X1),proper(X2))
       proper(U81(X1,X2,X3)) -> U81(proper(X1),proper(X2),proper(X3))
       proper(U82(X1,X2,X3)) -> U82(proper(X1),proper(X2),proper(X3))
       proper(fst(X)) -> fst(proper(X))
       proper(natsFrom(X)) -> natsFrom(proper(X))
       proper(s(X)) -> s(proper(X))
       proper(sel(X1,X2)) -> sel(proper(X1),proper(X2))
       proper(0()) -> ok(0())
       proper(nil()) -> ok(nil())
       proper(tail(X)) -> tail(proper(X))
       proper(take(X1,X2)) -> take(proper(X1),proper(X2))
       U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3))
       U12(ok(X1),ok(X2),ok(X3)) -> ok(U12(X1,X2,X3))
       snd(ok(X)) -> ok(snd(X))
       splitAt(ok(X1),ok(X2)) -> ok(splitAt(X1,X2))
       U21(ok(X1),ok(X2)) -> ok(U21(X1,X2))
       U22(ok(X1),ok(X2)) -> ok(U22(X1,X2))
       U31(ok(X1),ok(X2)) -> ok(U31(X1,X2))
       U32(ok(X1),ok(X2)) -> ok(U32(X1,X2))
       U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3))
       U42(ok(X1),ok(X2),ok(X3)) -> ok(U42(X1,X2,X3))
       head(ok(X)) -> ok(head(X))
       afterNth(ok(X1),ok(X2)) -> ok(afterNth(X1,X2))
       U51(ok(X1),ok(X2)) -> ok(U51(X1,X2))
       U52(ok(X1),ok(X2)) -> ok(U52(X1,X2))
       U61(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U61(X1,X2,X3,X4))
       U62(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U62(X1,X2,X3,X4))
       U63(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U63(X1,X2,X3,X4))
       U64(ok(X1),ok(X2)) -> ok(U64(X1,X2))
       pair(ok(X1),ok(X2)) -> ok(pair(X1,X2))
       cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
       U71(ok(X1),ok(X2)) -> ok(U71(X1,X2))
       U72(ok(X1),ok(X2)) -> ok(U72(X1,X2))
       U81(ok(X1),ok(X2),ok(X3)) -> ok(U81(X1,X2,X3))
       U82(ok(X1),ok(X2),ok(X3)) -> ok(U82(X1,X2,X3))
       fst(ok(X)) -> ok(fst(X))
       natsFrom(ok(X)) -> ok(natsFrom(X))
       s(ok(X)) -> ok(s(X))
       sel(ok(X1),ok(X2)) -> ok(sel(X1,X2))
       tail(ok(X)) -> ok(tail(X))
       take(ok(X1),ok(X2)) -> ok(take(X1,X2))
       top(mark(X)) -> top(proper(X))
       top(ok(X)) -> top(active(X))
      Open
      
      DPs:
       U71#(mark(X1),X2) -> U71#(X1,X2)
       U71#(ok(X1),ok(X2)) -> U71#(X1,X2)
      TRS:
       active(U11(tt(),N,XS)) -> mark(U12(tt(),N,XS))
       active(U12(tt(),N,XS)) -> mark(snd(splitAt(N,XS)))
       active(U21(tt(),X)) -> mark(U22(tt(),X))
       active(U22(tt(),X)) -> mark(X)
       active(U31(tt(),N)) -> mark(U32(tt(),N))
       active(U32(tt(),N)) -> mark(N)
       active(U41(tt(),N,XS)) -> mark(U42(tt(),N,XS))
       active(U42(tt(),N,XS)) -> mark(head(afterNth(N,XS)))
       active(U51(tt(),Y)) -> mark(U52(tt(),Y))
       active(U52(tt(),Y)) -> mark(Y)
       active(U61(tt(),N,X,XS)) -> mark(U62(tt(),N,X,XS))
       active(U62(tt(),N,X,XS)) -> mark(U63(tt(),N,X,XS))
       active(U63(tt(),N,X,XS)) -> mark(U64(splitAt(N,XS),X))
       active(U64(pair(YS,ZS),X)) -> mark(pair(cons(X,YS),ZS))
       active(U71(tt(),XS)) -> mark(U72(tt(),XS))
       active(U72(tt(),XS)) -> mark(XS)
       active(U81(tt(),N,XS)) -> mark(U82(tt(),N,XS))
       active(U82(tt(),N,XS)) -> mark(fst(splitAt(N,XS)))
       active(afterNth(N,XS)) -> mark(U11(tt(),N,XS))
       active(fst(pair(X,Y))) -> mark(U21(tt(),X))
       active(head(cons(N,XS))) -> mark(U31(tt(),N))
       active(natsFrom(N)) -> mark(cons(N,natsFrom(s(N))))
       active(sel(N,XS)) -> mark(U41(tt(),N,XS))
       active(snd(pair(X,Y))) -> mark(U51(tt(),Y))
       active(splitAt(0(),XS)) -> mark(pair(nil(),XS))
       active(splitAt(s(N),cons(X,XS))) -> mark(U61(tt(),N,X,XS))
       active(tail(cons(N,XS))) -> mark(U71(tt(),XS))
       active(take(N,XS)) -> mark(U81(tt(),N,XS))
       active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3)
       active(U12(X1,X2,X3)) -> U12(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(U21(X1,X2)) -> U21(active(X1),X2)
       active(U22(X1,X2)) -> U22(active(X1),X2)
       active(U31(X1,X2)) -> U31(active(X1),X2)
       active(U32(X1,X2)) -> U32(active(X1),X2)
       active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3)
       active(U42(X1,X2,X3)) -> U42(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(U51(X1,X2)) -> U51(active(X1),X2)
       active(U52(X1,X2)) -> U52(active(X1),X2)
       active(U61(X1,X2,X3,X4)) -> U61(active(X1),X2,X3,X4)
       active(U62(X1,X2,X3,X4)) -> U62(active(X1),X2,X3,X4)
       active(U63(X1,X2,X3,X4)) -> U63(active(X1),X2,X3,X4)
       active(U64(X1,X2)) -> U64(active(X1),X2)
       active(pair(X1,X2)) -> pair(active(X1),X2)
       active(pair(X1,X2)) -> pair(X1,active(X2))
       active(cons(X1,X2)) -> cons(active(X1),X2)
       active(U71(X1,X2)) -> U71(active(X1),X2)
       active(U72(X1,X2)) -> U72(active(X1),X2)
       active(U81(X1,X2,X3)) -> U81(active(X1),X2,X3)
       active(U82(X1,X2,X3)) -> U82(active(X1),X2,X3)
       active(fst(X)) -> fst(active(X))
       active(natsFrom(X)) -> natsFrom(active(X))
       active(s(X)) -> s(active(X))
       active(sel(X1,X2)) -> sel(active(X1),X2)
       active(sel(X1,X2)) -> sel(X1,active(X2))
       active(tail(X)) -> tail(active(X))
       active(take(X1,X2)) -> take(active(X1),X2)
       active(take(X1,X2)) -> take(X1,active(X2))
       U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3))
       U12(mark(X1),X2,X3) -> mark(U12(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))
       U21(mark(X1),X2) -> mark(U21(X1,X2))
       U22(mark(X1),X2) -> mark(U22(X1,X2))
       U31(mark(X1),X2) -> mark(U31(X1,X2))
       U32(mark(X1),X2) -> mark(U32(X1,X2))
       U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3))
       U42(mark(X1),X2,X3) -> mark(U42(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))
       U51(mark(X1),X2) -> mark(U51(X1,X2))
       U52(mark(X1),X2) -> mark(U52(X1,X2))
       U61(mark(X1),X2,X3,X4) -> mark(U61(X1,X2,X3,X4))
       U62(mark(X1),X2,X3,X4) -> mark(U62(X1,X2,X3,X4))
       U63(mark(X1),X2,X3,X4) -> mark(U63(X1,X2,X3,X4))
       U64(mark(X1),X2) -> mark(U64(X1,X2))
       pair(mark(X1),X2) -> mark(pair(X1,X2))
       pair(X1,mark(X2)) -> mark(pair(X1,X2))
       cons(mark(X1),X2) -> mark(cons(X1,X2))
       U71(mark(X1),X2) -> mark(U71(X1,X2))
       U72(mark(X1),X2) -> mark(U72(X1,X2))
       U81(mark(X1),X2,X3) -> mark(U81(X1,X2,X3))
       U82(mark(X1),X2,X3) -> mark(U82(X1,X2,X3))
       fst(mark(X)) -> mark(fst(X))
       natsFrom(mark(X)) -> mark(natsFrom(X))
       s(mark(X)) -> mark(s(X))
       sel(mark(X1),X2) -> mark(sel(X1,X2))
       sel(X1,mark(X2)) -> mark(sel(X1,X2))
       tail(mark(X)) -> mark(tail(X))
       take(mark(X1),X2) -> mark(take(X1,X2))
       take(X1,mark(X2)) -> mark(take(X1,X2))
       proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3))
       proper(tt()) -> ok(tt())
       proper(U12(X1,X2,X3)) -> U12(proper(X1),proper(X2),proper(X3))
       proper(snd(X)) -> snd(proper(X))
       proper(splitAt(X1,X2)) -> splitAt(proper(X1),proper(X2))
       proper(U21(X1,X2)) -> U21(proper(X1),proper(X2))
       proper(U22(X1,X2)) -> U22(proper(X1),proper(X2))
       proper(U31(X1,X2)) -> U31(proper(X1),proper(X2))
       proper(U32(X1,X2)) -> U32(proper(X1),proper(X2))
       proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3))
       proper(U42(X1,X2,X3)) -> U42(proper(X1),proper(X2),proper(X3))
       proper(head(X)) -> head(proper(X))
       proper(afterNth(X1,X2)) -> afterNth(proper(X1),proper(X2))
       proper(U51(X1,X2)) -> U51(proper(X1),proper(X2))
       proper(U52(X1,X2)) -> U52(proper(X1),proper(X2))
       proper(U61(X1,X2,X3,X4)) -> U61(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(U62(X1,X2,X3,X4)) -> U62(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(U63(X1,X2,X3,X4)) -> U63(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(U64(X1,X2)) -> U64(proper(X1),proper(X2))
       proper(pair(X1,X2)) -> pair(proper(X1),proper(X2))
       proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
       proper(U71(X1,X2)) -> U71(proper(X1),proper(X2))
       proper(U72(X1,X2)) -> U72(proper(X1),proper(X2))
       proper(U81(X1,X2,X3)) -> U81(proper(X1),proper(X2),proper(X3))
       proper(U82(X1,X2,X3)) -> U82(proper(X1),proper(X2),proper(X3))
       proper(fst(X)) -> fst(proper(X))
       proper(natsFrom(X)) -> natsFrom(proper(X))
       proper(s(X)) -> s(proper(X))
       proper(sel(X1,X2)) -> sel(proper(X1),proper(X2))
       proper(0()) -> ok(0())
       proper(nil()) -> ok(nil())
       proper(tail(X)) -> tail(proper(X))
       proper(take(X1,X2)) -> take(proper(X1),proper(X2))
       U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3))
       U12(ok(X1),ok(X2),ok(X3)) -> ok(U12(X1,X2,X3))
       snd(ok(X)) -> ok(snd(X))
       splitAt(ok(X1),ok(X2)) -> ok(splitAt(X1,X2))
       U21(ok(X1),ok(X2)) -> ok(U21(X1,X2))
       U22(ok(X1),ok(X2)) -> ok(U22(X1,X2))
       U31(ok(X1),ok(X2)) -> ok(U31(X1,X2))
       U32(ok(X1),ok(X2)) -> ok(U32(X1,X2))
       U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3))
       U42(ok(X1),ok(X2),ok(X3)) -> ok(U42(X1,X2,X3))
       head(ok(X)) -> ok(head(X))
       afterNth(ok(X1),ok(X2)) -> ok(afterNth(X1,X2))
       U51(ok(X1),ok(X2)) -> ok(U51(X1,X2))
       U52(ok(X1),ok(X2)) -> ok(U52(X1,X2))
       U61(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U61(X1,X2,X3,X4))
       U62(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U62(X1,X2,X3,X4))
       U63(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U63(X1,X2,X3,X4))
       U64(ok(X1),ok(X2)) -> ok(U64(X1,X2))
       pair(ok(X1),ok(X2)) -> ok(pair(X1,X2))
       cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
       U71(ok(X1),ok(X2)) -> ok(U71(X1,X2))
       U72(ok(X1),ok(X2)) -> ok(U72(X1,X2))
       U81(ok(X1),ok(X2),ok(X3)) -> ok(U81(X1,X2,X3))
       U82(ok(X1),ok(X2),ok(X3)) -> ok(U82(X1,X2,X3))
       fst(ok(X)) -> ok(fst(X))
       natsFrom(ok(X)) -> ok(natsFrom(X))
       s(ok(X)) -> ok(s(X))
       sel(ok(X1),ok(X2)) -> ok(sel(X1,X2))
       tail(ok(X)) -> ok(tail(X))
       take(ok(X1),ok(X2)) -> ok(take(X1,X2))
       top(mark(X)) -> top(proper(X))
       top(ok(X)) -> top(active(X))
      Open
      
      DPs:
       cons#(mark(X1),X2) -> cons#(X1,X2)
       cons#(ok(X1),ok(X2)) -> cons#(X1,X2)
      TRS:
       active(U11(tt(),N,XS)) -> mark(U12(tt(),N,XS))
       active(U12(tt(),N,XS)) -> mark(snd(splitAt(N,XS)))
       active(U21(tt(),X)) -> mark(U22(tt(),X))
       active(U22(tt(),X)) -> mark(X)
       active(U31(tt(),N)) -> mark(U32(tt(),N))
       active(U32(tt(),N)) -> mark(N)
       active(U41(tt(),N,XS)) -> mark(U42(tt(),N,XS))
       active(U42(tt(),N,XS)) -> mark(head(afterNth(N,XS)))
       active(U51(tt(),Y)) -> mark(U52(tt(),Y))
       active(U52(tt(),Y)) -> mark(Y)
       active(U61(tt(),N,X,XS)) -> mark(U62(tt(),N,X,XS))
       active(U62(tt(),N,X,XS)) -> mark(U63(tt(),N,X,XS))
       active(U63(tt(),N,X,XS)) -> mark(U64(splitAt(N,XS),X))
       active(U64(pair(YS,ZS),X)) -> mark(pair(cons(X,YS),ZS))
       active(U71(tt(),XS)) -> mark(U72(tt(),XS))
       active(U72(tt(),XS)) -> mark(XS)
       active(U81(tt(),N,XS)) -> mark(U82(tt(),N,XS))
       active(U82(tt(),N,XS)) -> mark(fst(splitAt(N,XS)))
       active(afterNth(N,XS)) -> mark(U11(tt(),N,XS))
       active(fst(pair(X,Y))) -> mark(U21(tt(),X))
       active(head(cons(N,XS))) -> mark(U31(tt(),N))
       active(natsFrom(N)) -> mark(cons(N,natsFrom(s(N))))
       active(sel(N,XS)) -> mark(U41(tt(),N,XS))
       active(snd(pair(X,Y))) -> mark(U51(tt(),Y))
       active(splitAt(0(),XS)) -> mark(pair(nil(),XS))
       active(splitAt(s(N),cons(X,XS))) -> mark(U61(tt(),N,X,XS))
       active(tail(cons(N,XS))) -> mark(U71(tt(),XS))
       active(take(N,XS)) -> mark(U81(tt(),N,XS))
       active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3)
       active(U12(X1,X2,X3)) -> U12(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(U21(X1,X2)) -> U21(active(X1),X2)
       active(U22(X1,X2)) -> U22(active(X1),X2)
       active(U31(X1,X2)) -> U31(active(X1),X2)
       active(U32(X1,X2)) -> U32(active(X1),X2)
       active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3)
       active(U42(X1,X2,X3)) -> U42(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(U51(X1,X2)) -> U51(active(X1),X2)
       active(U52(X1,X2)) -> U52(active(X1),X2)
       active(U61(X1,X2,X3,X4)) -> U61(active(X1),X2,X3,X4)
       active(U62(X1,X2,X3,X4)) -> U62(active(X1),X2,X3,X4)
       active(U63(X1,X2,X3,X4)) -> U63(active(X1),X2,X3,X4)
       active(U64(X1,X2)) -> U64(active(X1),X2)
       active(pair(X1,X2)) -> pair(active(X1),X2)
       active(pair(X1,X2)) -> pair(X1,active(X2))
       active(cons(X1,X2)) -> cons(active(X1),X2)
       active(U71(X1,X2)) -> U71(active(X1),X2)
       active(U72(X1,X2)) -> U72(active(X1),X2)
       active(U81(X1,X2,X3)) -> U81(active(X1),X2,X3)
       active(U82(X1,X2,X3)) -> U82(active(X1),X2,X3)
       active(fst(X)) -> fst(active(X))
       active(natsFrom(X)) -> natsFrom(active(X))
       active(s(X)) -> s(active(X))
       active(sel(X1,X2)) -> sel(active(X1),X2)
       active(sel(X1,X2)) -> sel(X1,active(X2))
       active(tail(X)) -> tail(active(X))
       active(take(X1,X2)) -> take(active(X1),X2)
       active(take(X1,X2)) -> take(X1,active(X2))
       U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3))
       U12(mark(X1),X2,X3) -> mark(U12(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))
       U21(mark(X1),X2) -> mark(U21(X1,X2))
       U22(mark(X1),X2) -> mark(U22(X1,X2))
       U31(mark(X1),X2) -> mark(U31(X1,X2))
       U32(mark(X1),X2) -> mark(U32(X1,X2))
       U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3))
       U42(mark(X1),X2,X3) -> mark(U42(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))
       U51(mark(X1),X2) -> mark(U51(X1,X2))
       U52(mark(X1),X2) -> mark(U52(X1,X2))
       U61(mark(X1),X2,X3,X4) -> mark(U61(X1,X2,X3,X4))
       U62(mark(X1),X2,X3,X4) -> mark(U62(X1,X2,X3,X4))
       U63(mark(X1),X2,X3,X4) -> mark(U63(X1,X2,X3,X4))
       U64(mark(X1),X2) -> mark(U64(X1,X2))
       pair(mark(X1),X2) -> mark(pair(X1,X2))
       pair(X1,mark(X2)) -> mark(pair(X1,X2))
       cons(mark(X1),X2) -> mark(cons(X1,X2))
       U71(mark(X1),X2) -> mark(U71(X1,X2))
       U72(mark(X1),X2) -> mark(U72(X1,X2))
       U81(mark(X1),X2,X3) -> mark(U81(X1,X2,X3))
       U82(mark(X1),X2,X3) -> mark(U82(X1,X2,X3))
       fst(mark(X)) -> mark(fst(X))
       natsFrom(mark(X)) -> mark(natsFrom(X))
       s(mark(X)) -> mark(s(X))
       sel(mark(X1),X2) -> mark(sel(X1,X2))
       sel(X1,mark(X2)) -> mark(sel(X1,X2))
       tail(mark(X)) -> mark(tail(X))
       take(mark(X1),X2) -> mark(take(X1,X2))
       take(X1,mark(X2)) -> mark(take(X1,X2))
       proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3))
       proper(tt()) -> ok(tt())
       proper(U12(X1,X2,X3)) -> U12(proper(X1),proper(X2),proper(X3))
       proper(snd(X)) -> snd(proper(X))
       proper(splitAt(X1,X2)) -> splitAt(proper(X1),proper(X2))
       proper(U21(X1,X2)) -> U21(proper(X1),proper(X2))
       proper(U22(X1,X2)) -> U22(proper(X1),proper(X2))
       proper(U31(X1,X2)) -> U31(proper(X1),proper(X2))
       proper(U32(X1,X2)) -> U32(proper(X1),proper(X2))
       proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3))
       proper(U42(X1,X2,X3)) -> U42(proper(X1),proper(X2),proper(X3))
       proper(head(X)) -> head(proper(X))
       proper(afterNth(X1,X2)) -> afterNth(proper(X1),proper(X2))
       proper(U51(X1,X2)) -> U51(proper(X1),proper(X2))
       proper(U52(X1,X2)) -> U52(proper(X1),proper(X2))
       proper(U61(X1,X2,X3,X4)) -> U61(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(U62(X1,X2,X3,X4)) -> U62(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(U63(X1,X2,X3,X4)) -> U63(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(U64(X1,X2)) -> U64(proper(X1),proper(X2))
       proper(pair(X1,X2)) -> pair(proper(X1),proper(X2))
       proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
       proper(U71(X1,X2)) -> U71(proper(X1),proper(X2))
       proper(U72(X1,X2)) -> U72(proper(X1),proper(X2))
       proper(U81(X1,X2,X3)) -> U81(proper(X1),proper(X2),proper(X3))
       proper(U82(X1,X2,X3)) -> U82(proper(X1),proper(X2),proper(X3))
       proper(fst(X)) -> fst(proper(X))
       proper(natsFrom(X)) -> natsFrom(proper(X))
       proper(s(X)) -> s(proper(X))
       proper(sel(X1,X2)) -> sel(proper(X1),proper(X2))
       proper(0()) -> ok(0())
       proper(nil()) -> ok(nil())
       proper(tail(X)) -> tail(proper(X))
       proper(take(X1,X2)) -> take(proper(X1),proper(X2))
       U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3))
       U12(ok(X1),ok(X2),ok(X3)) -> ok(U12(X1,X2,X3))
       snd(ok(X)) -> ok(snd(X))
       splitAt(ok(X1),ok(X2)) -> ok(splitAt(X1,X2))
       U21(ok(X1),ok(X2)) -> ok(U21(X1,X2))
       U22(ok(X1),ok(X2)) -> ok(U22(X1,X2))
       U31(ok(X1),ok(X2)) -> ok(U31(X1,X2))
       U32(ok(X1),ok(X2)) -> ok(U32(X1,X2))
       U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3))
       U42(ok(X1),ok(X2),ok(X3)) -> ok(U42(X1,X2,X3))
       head(ok(X)) -> ok(head(X))
       afterNth(ok(X1),ok(X2)) -> ok(afterNth(X1,X2))
       U51(ok(X1),ok(X2)) -> ok(U51(X1,X2))
       U52(ok(X1),ok(X2)) -> ok(U52(X1,X2))
       U61(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U61(X1,X2,X3,X4))
       U62(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U62(X1,X2,X3,X4))
       U63(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U63(X1,X2,X3,X4))
       U64(ok(X1),ok(X2)) -> ok(U64(X1,X2))
       pair(ok(X1),ok(X2)) -> ok(pair(X1,X2))
       cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
       U71(ok(X1),ok(X2)) -> ok(U71(X1,X2))
       U72(ok(X1),ok(X2)) -> ok(U72(X1,X2))
       U81(ok(X1),ok(X2),ok(X3)) -> ok(U81(X1,X2,X3))
       U82(ok(X1),ok(X2),ok(X3)) -> ok(U82(X1,X2,X3))
       fst(ok(X)) -> ok(fst(X))
       natsFrom(ok(X)) -> ok(natsFrom(X))
       s(ok(X)) -> ok(s(X))
       sel(ok(X1),ok(X2)) -> ok(sel(X1,X2))
       tail(ok(X)) -> ok(tail(X))
       take(ok(X1),ok(X2)) -> ok(take(X1,X2))
       top(mark(X)) -> top(proper(X))
       top(ok(X)) -> top(active(X))
      Open
      
      DPs:
       pair#(mark(X1),X2) -> pair#(X1,X2)
       pair#(X1,mark(X2)) -> pair#(X1,X2)
       pair#(ok(X1),ok(X2)) -> pair#(X1,X2)
      TRS:
       active(U11(tt(),N,XS)) -> mark(U12(tt(),N,XS))
       active(U12(tt(),N,XS)) -> mark(snd(splitAt(N,XS)))
       active(U21(tt(),X)) -> mark(U22(tt(),X))
       active(U22(tt(),X)) -> mark(X)
       active(U31(tt(),N)) -> mark(U32(tt(),N))
       active(U32(tt(),N)) -> mark(N)
       active(U41(tt(),N,XS)) -> mark(U42(tt(),N,XS))
       active(U42(tt(),N,XS)) -> mark(head(afterNth(N,XS)))
       active(U51(tt(),Y)) -> mark(U52(tt(),Y))
       active(U52(tt(),Y)) -> mark(Y)
       active(U61(tt(),N,X,XS)) -> mark(U62(tt(),N,X,XS))
       active(U62(tt(),N,X,XS)) -> mark(U63(tt(),N,X,XS))
       active(U63(tt(),N,X,XS)) -> mark(U64(splitAt(N,XS),X))
       active(U64(pair(YS,ZS),X)) -> mark(pair(cons(X,YS),ZS))
       active(U71(tt(),XS)) -> mark(U72(tt(),XS))
       active(U72(tt(),XS)) -> mark(XS)
       active(U81(tt(),N,XS)) -> mark(U82(tt(),N,XS))
       active(U82(tt(),N,XS)) -> mark(fst(splitAt(N,XS)))
       active(afterNth(N,XS)) -> mark(U11(tt(),N,XS))
       active(fst(pair(X,Y))) -> mark(U21(tt(),X))
       active(head(cons(N,XS))) -> mark(U31(tt(),N))
       active(natsFrom(N)) -> mark(cons(N,natsFrom(s(N))))
       active(sel(N,XS)) -> mark(U41(tt(),N,XS))
       active(snd(pair(X,Y))) -> mark(U51(tt(),Y))
       active(splitAt(0(),XS)) -> mark(pair(nil(),XS))
       active(splitAt(s(N),cons(X,XS))) -> mark(U61(tt(),N,X,XS))
       active(tail(cons(N,XS))) -> mark(U71(tt(),XS))
       active(take(N,XS)) -> mark(U81(tt(),N,XS))
       active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3)
       active(U12(X1,X2,X3)) -> U12(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(U21(X1,X2)) -> U21(active(X1),X2)
       active(U22(X1,X2)) -> U22(active(X1),X2)
       active(U31(X1,X2)) -> U31(active(X1),X2)
       active(U32(X1,X2)) -> U32(active(X1),X2)
       active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3)
       active(U42(X1,X2,X3)) -> U42(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(U51(X1,X2)) -> U51(active(X1),X2)
       active(U52(X1,X2)) -> U52(active(X1),X2)
       active(U61(X1,X2,X3,X4)) -> U61(active(X1),X2,X3,X4)
       active(U62(X1,X2,X3,X4)) -> U62(active(X1),X2,X3,X4)
       active(U63(X1,X2,X3,X4)) -> U63(active(X1),X2,X3,X4)
       active(U64(X1,X2)) -> U64(active(X1),X2)
       active(pair(X1,X2)) -> pair(active(X1),X2)
       active(pair(X1,X2)) -> pair(X1,active(X2))
       active(cons(X1,X2)) -> cons(active(X1),X2)
       active(U71(X1,X2)) -> U71(active(X1),X2)
       active(U72(X1,X2)) -> U72(active(X1),X2)
       active(U81(X1,X2,X3)) -> U81(active(X1),X2,X3)
       active(U82(X1,X2,X3)) -> U82(active(X1),X2,X3)
       active(fst(X)) -> fst(active(X))
       active(natsFrom(X)) -> natsFrom(active(X))
       active(s(X)) -> s(active(X))
       active(sel(X1,X2)) -> sel(active(X1),X2)
       active(sel(X1,X2)) -> sel(X1,active(X2))
       active(tail(X)) -> tail(active(X))
       active(take(X1,X2)) -> take(active(X1),X2)
       active(take(X1,X2)) -> take(X1,active(X2))
       U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3))
       U12(mark(X1),X2,X3) -> mark(U12(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))
       U21(mark(X1),X2) -> mark(U21(X1,X2))
       U22(mark(X1),X2) -> mark(U22(X1,X2))
       U31(mark(X1),X2) -> mark(U31(X1,X2))
       U32(mark(X1),X2) -> mark(U32(X1,X2))
       U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3))
       U42(mark(X1),X2,X3) -> mark(U42(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))
       U51(mark(X1),X2) -> mark(U51(X1,X2))
       U52(mark(X1),X2) -> mark(U52(X1,X2))
       U61(mark(X1),X2,X3,X4) -> mark(U61(X1,X2,X3,X4))
       U62(mark(X1),X2,X3,X4) -> mark(U62(X1,X2,X3,X4))
       U63(mark(X1),X2,X3,X4) -> mark(U63(X1,X2,X3,X4))
       U64(mark(X1),X2) -> mark(U64(X1,X2))
       pair(mark(X1),X2) -> mark(pair(X1,X2))
       pair(X1,mark(X2)) -> mark(pair(X1,X2))
       cons(mark(X1),X2) -> mark(cons(X1,X2))
       U71(mark(X1),X2) -> mark(U71(X1,X2))
       U72(mark(X1),X2) -> mark(U72(X1,X2))
       U81(mark(X1),X2,X3) -> mark(U81(X1,X2,X3))
       U82(mark(X1),X2,X3) -> mark(U82(X1,X2,X3))
       fst(mark(X)) -> mark(fst(X))
       natsFrom(mark(X)) -> mark(natsFrom(X))
       s(mark(X)) -> mark(s(X))
       sel(mark(X1),X2) -> mark(sel(X1,X2))
       sel(X1,mark(X2)) -> mark(sel(X1,X2))
       tail(mark(X)) -> mark(tail(X))
       take(mark(X1),X2) -> mark(take(X1,X2))
       take(X1,mark(X2)) -> mark(take(X1,X2))
       proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3))
       proper(tt()) -> ok(tt())
       proper(U12(X1,X2,X3)) -> U12(proper(X1),proper(X2),proper(X3))
       proper(snd(X)) -> snd(proper(X))
       proper(splitAt(X1,X2)) -> splitAt(proper(X1),proper(X2))
       proper(U21(X1,X2)) -> U21(proper(X1),proper(X2))
       proper(U22(X1,X2)) -> U22(proper(X1),proper(X2))
       proper(U31(X1,X2)) -> U31(proper(X1),proper(X2))
       proper(U32(X1,X2)) -> U32(proper(X1),proper(X2))
       proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3))
       proper(U42(X1,X2,X3)) -> U42(proper(X1),proper(X2),proper(X3))
       proper(head(X)) -> head(proper(X))
       proper(afterNth(X1,X2)) -> afterNth(proper(X1),proper(X2))
       proper(U51(X1,X2)) -> U51(proper(X1),proper(X2))
       proper(U52(X1,X2)) -> U52(proper(X1),proper(X2))
       proper(U61(X1,X2,X3,X4)) -> U61(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(U62(X1,X2,X3,X4)) -> U62(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(U63(X1,X2,X3,X4)) -> U63(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(U64(X1,X2)) -> U64(proper(X1),proper(X2))
       proper(pair(X1,X2)) -> pair(proper(X1),proper(X2))
       proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
       proper(U71(X1,X2)) -> U71(proper(X1),proper(X2))
       proper(U72(X1,X2)) -> U72(proper(X1),proper(X2))
       proper(U81(X1,X2,X3)) -> U81(proper(X1),proper(X2),proper(X3))
       proper(U82(X1,X2,X3)) -> U82(proper(X1),proper(X2),proper(X3))
       proper(fst(X)) -> fst(proper(X))
       proper(natsFrom(X)) -> natsFrom(proper(X))
       proper(s(X)) -> s(proper(X))
       proper(sel(X1,X2)) -> sel(proper(X1),proper(X2))
       proper(0()) -> ok(0())
       proper(nil()) -> ok(nil())
       proper(tail(X)) -> tail(proper(X))
       proper(take(X1,X2)) -> take(proper(X1),proper(X2))
       U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3))
       U12(ok(X1),ok(X2),ok(X3)) -> ok(U12(X1,X2,X3))
       snd(ok(X)) -> ok(snd(X))
       splitAt(ok(X1),ok(X2)) -> ok(splitAt(X1,X2))
       U21(ok(X1),ok(X2)) -> ok(U21(X1,X2))
       U22(ok(X1),ok(X2)) -> ok(U22(X1,X2))
       U31(ok(X1),ok(X2)) -> ok(U31(X1,X2))
       U32(ok(X1),ok(X2)) -> ok(U32(X1,X2))
       U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3))
       U42(ok(X1),ok(X2),ok(X3)) -> ok(U42(X1,X2,X3))
       head(ok(X)) -> ok(head(X))
       afterNth(ok(X1),ok(X2)) -> ok(afterNth(X1,X2))
       U51(ok(X1),ok(X2)) -> ok(U51(X1,X2))
       U52(ok(X1),ok(X2)) -> ok(U52(X1,X2))
       U61(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U61(X1,X2,X3,X4))
       U62(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U62(X1,X2,X3,X4))
       U63(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U63(X1,X2,X3,X4))
       U64(ok(X1),ok(X2)) -> ok(U64(X1,X2))
       pair(ok(X1),ok(X2)) -> ok(pair(X1,X2))
       cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
       U71(ok(X1),ok(X2)) -> ok(U71(X1,X2))
       U72(ok(X1),ok(X2)) -> ok(U72(X1,X2))
       U81(ok(X1),ok(X2),ok(X3)) -> ok(U81(X1,X2,X3))
       U82(ok(X1),ok(X2),ok(X3)) -> ok(U82(X1,X2,X3))
       fst(ok(X)) -> ok(fst(X))
       natsFrom(ok(X)) -> ok(natsFrom(X))
       s(ok(X)) -> ok(s(X))
       sel(ok(X1),ok(X2)) -> ok(sel(X1,X2))
       tail(ok(X)) -> ok(tail(X))
       take(ok(X1),ok(X2)) -> ok(take(X1,X2))
       top(mark(X)) -> top(proper(X))
       top(ok(X)) -> top(active(X))
      Open
      
      DPs:
       U64#(mark(X1),X2) -> U64#(X1,X2)
       U64#(ok(X1),ok(X2)) -> U64#(X1,X2)
      TRS:
       active(U11(tt(),N,XS)) -> mark(U12(tt(),N,XS))
       active(U12(tt(),N,XS)) -> mark(snd(splitAt(N,XS)))
       active(U21(tt(),X)) -> mark(U22(tt(),X))
       active(U22(tt(),X)) -> mark(X)
       active(U31(tt(),N)) -> mark(U32(tt(),N))
       active(U32(tt(),N)) -> mark(N)
       active(U41(tt(),N,XS)) -> mark(U42(tt(),N,XS))
       active(U42(tt(),N,XS)) -> mark(head(afterNth(N,XS)))
       active(U51(tt(),Y)) -> mark(U52(tt(),Y))
       active(U52(tt(),Y)) -> mark(Y)
       active(U61(tt(),N,X,XS)) -> mark(U62(tt(),N,X,XS))
       active(U62(tt(),N,X,XS)) -> mark(U63(tt(),N,X,XS))
       active(U63(tt(),N,X,XS)) -> mark(U64(splitAt(N,XS),X))
       active(U64(pair(YS,ZS),X)) -> mark(pair(cons(X,YS),ZS))
       active(U71(tt(),XS)) -> mark(U72(tt(),XS))
       active(U72(tt(),XS)) -> mark(XS)
       active(U81(tt(),N,XS)) -> mark(U82(tt(),N,XS))
       active(U82(tt(),N,XS)) -> mark(fst(splitAt(N,XS)))
       active(afterNth(N,XS)) -> mark(U11(tt(),N,XS))
       active(fst(pair(X,Y))) -> mark(U21(tt(),X))
       active(head(cons(N,XS))) -> mark(U31(tt(),N))
       active(natsFrom(N)) -> mark(cons(N,natsFrom(s(N))))
       active(sel(N,XS)) -> mark(U41(tt(),N,XS))
       active(snd(pair(X,Y))) -> mark(U51(tt(),Y))
       active(splitAt(0(),XS)) -> mark(pair(nil(),XS))
       active(splitAt(s(N),cons(X,XS))) -> mark(U61(tt(),N,X,XS))
       active(tail(cons(N,XS))) -> mark(U71(tt(),XS))
       active(take(N,XS)) -> mark(U81(tt(),N,XS))
       active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3)
       active(U12(X1,X2,X3)) -> U12(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(U21(X1,X2)) -> U21(active(X1),X2)
       active(U22(X1,X2)) -> U22(active(X1),X2)
       active(U31(X1,X2)) -> U31(active(X1),X2)
       active(U32(X1,X2)) -> U32(active(X1),X2)
       active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3)
       active(U42(X1,X2,X3)) -> U42(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(U51(X1,X2)) -> U51(active(X1),X2)
       active(U52(X1,X2)) -> U52(active(X1),X2)
       active(U61(X1,X2,X3,X4)) -> U61(active(X1),X2,X3,X4)
       active(U62(X1,X2,X3,X4)) -> U62(active(X1),X2,X3,X4)
       active(U63(X1,X2,X3,X4)) -> U63(active(X1),X2,X3,X4)
       active(U64(X1,X2)) -> U64(active(X1),X2)
       active(pair(X1,X2)) -> pair(active(X1),X2)
       active(pair(X1,X2)) -> pair(X1,active(X2))
       active(cons(X1,X2)) -> cons(active(X1),X2)
       active(U71(X1,X2)) -> U71(active(X1),X2)
       active(U72(X1,X2)) -> U72(active(X1),X2)
       active(U81(X1,X2,X3)) -> U81(active(X1),X2,X3)
       active(U82(X1,X2,X3)) -> U82(active(X1),X2,X3)
       active(fst(X)) -> fst(active(X))
       active(natsFrom(X)) -> natsFrom(active(X))
       active(s(X)) -> s(active(X))
       active(sel(X1,X2)) -> sel(active(X1),X2)
       active(sel(X1,X2)) -> sel(X1,active(X2))
       active(tail(X)) -> tail(active(X))
       active(take(X1,X2)) -> take(active(X1),X2)
       active(take(X1,X2)) -> take(X1,active(X2))
       U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3))
       U12(mark(X1),X2,X3) -> mark(U12(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))
       U21(mark(X1),X2) -> mark(U21(X1,X2))
       U22(mark(X1),X2) -> mark(U22(X1,X2))
       U31(mark(X1),X2) -> mark(U31(X1,X2))
       U32(mark(X1),X2) -> mark(U32(X1,X2))
       U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3))
       U42(mark(X1),X2,X3) -> mark(U42(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))
       U51(mark(X1),X2) -> mark(U51(X1,X2))
       U52(mark(X1),X2) -> mark(U52(X1,X2))
       U61(mark(X1),X2,X3,X4) -> mark(U61(X1,X2,X3,X4))
       U62(mark(X1),X2,X3,X4) -> mark(U62(X1,X2,X3,X4))
       U63(mark(X1),X2,X3,X4) -> mark(U63(X1,X2,X3,X4))
       U64(mark(X1),X2) -> mark(U64(X1,X2))
       pair(mark(X1),X2) -> mark(pair(X1,X2))
       pair(X1,mark(X2)) -> mark(pair(X1,X2))
       cons(mark(X1),X2) -> mark(cons(X1,X2))
       U71(mark(X1),X2) -> mark(U71(X1,X2))
       U72(mark(X1),X2) -> mark(U72(X1,X2))
       U81(mark(X1),X2,X3) -> mark(U81(X1,X2,X3))
       U82(mark(X1),X2,X3) -> mark(U82(X1,X2,X3))
       fst(mark(X)) -> mark(fst(X))
       natsFrom(mark(X)) -> mark(natsFrom(X))
       s(mark(X)) -> mark(s(X))
       sel(mark(X1),X2) -> mark(sel(X1,X2))
       sel(X1,mark(X2)) -> mark(sel(X1,X2))
       tail(mark(X)) -> mark(tail(X))
       take(mark(X1),X2) -> mark(take(X1,X2))
       take(X1,mark(X2)) -> mark(take(X1,X2))
       proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3))
       proper(tt()) -> ok(tt())
       proper(U12(X1,X2,X3)) -> U12(proper(X1),proper(X2),proper(X3))
       proper(snd(X)) -> snd(proper(X))
       proper(splitAt(X1,X2)) -> splitAt(proper(X1),proper(X2))
       proper(U21(X1,X2)) -> U21(proper(X1),proper(X2))
       proper(U22(X1,X2)) -> U22(proper(X1),proper(X2))
       proper(U31(X1,X2)) -> U31(proper(X1),proper(X2))
       proper(U32(X1,X2)) -> U32(proper(X1),proper(X2))
       proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3))
       proper(U42(X1,X2,X3)) -> U42(proper(X1),proper(X2),proper(X3))
       proper(head(X)) -> head(proper(X))
       proper(afterNth(X1,X2)) -> afterNth(proper(X1),proper(X2))
       proper(U51(X1,X2)) -> U51(proper(X1),proper(X2))
       proper(U52(X1,X2)) -> U52(proper(X1),proper(X2))
       proper(U61(X1,X2,X3,X4)) -> U61(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(U62(X1,X2,X3,X4)) -> U62(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(U63(X1,X2,X3,X4)) -> U63(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(U64(X1,X2)) -> U64(proper(X1),proper(X2))
       proper(pair(X1,X2)) -> pair(proper(X1),proper(X2))
       proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
       proper(U71(X1,X2)) -> U71(proper(X1),proper(X2))
       proper(U72(X1,X2)) -> U72(proper(X1),proper(X2))
       proper(U81(X1,X2,X3)) -> U81(proper(X1),proper(X2),proper(X3))
       proper(U82(X1,X2,X3)) -> U82(proper(X1),proper(X2),proper(X3))
       proper(fst(X)) -> fst(proper(X))
       proper(natsFrom(X)) -> natsFrom(proper(X))
       proper(s(X)) -> s(proper(X))
       proper(sel(X1,X2)) -> sel(proper(X1),proper(X2))
       proper(0()) -> ok(0())
       proper(nil()) -> ok(nil())
       proper(tail(X)) -> tail(proper(X))
       proper(take(X1,X2)) -> take(proper(X1),proper(X2))
       U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3))
       U12(ok(X1),ok(X2),ok(X3)) -> ok(U12(X1,X2,X3))
       snd(ok(X)) -> ok(snd(X))
       splitAt(ok(X1),ok(X2)) -> ok(splitAt(X1,X2))
       U21(ok(X1),ok(X2)) -> ok(U21(X1,X2))
       U22(ok(X1),ok(X2)) -> ok(U22(X1,X2))
       U31(ok(X1),ok(X2)) -> ok(U31(X1,X2))
       U32(ok(X1),ok(X2)) -> ok(U32(X1,X2))
       U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3))
       U42(ok(X1),ok(X2),ok(X3)) -> ok(U42(X1,X2,X3))
       head(ok(X)) -> ok(head(X))
       afterNth(ok(X1),ok(X2)) -> ok(afterNth(X1,X2))
       U51(ok(X1),ok(X2)) -> ok(U51(X1,X2))
       U52(ok(X1),ok(X2)) -> ok(U52(X1,X2))
       U61(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U61(X1,X2,X3,X4))
       U62(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U62(X1,X2,X3,X4))
       U63(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U63(X1,X2,X3,X4))
       U64(ok(X1),ok(X2)) -> ok(U64(X1,X2))
       pair(ok(X1),ok(X2)) -> ok(pair(X1,X2))
       cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
       U71(ok(X1),ok(X2)) -> ok(U71(X1,X2))
       U72(ok(X1),ok(X2)) -> ok(U72(X1,X2))
       U81(ok(X1),ok(X2),ok(X3)) -> ok(U81(X1,X2,X3))
       U82(ok(X1),ok(X2),ok(X3)) -> ok(U82(X1,X2,X3))
       fst(ok(X)) -> ok(fst(X))
       natsFrom(ok(X)) -> ok(natsFrom(X))
       s(ok(X)) -> ok(s(X))
       sel(ok(X1),ok(X2)) -> ok(sel(X1,X2))
       tail(ok(X)) -> ok(tail(X))
       take(ok(X1),ok(X2)) -> ok(take(X1,X2))
       top(mark(X)) -> top(proper(X))
       top(ok(X)) -> top(active(X))
      Open
      
      DPs:
       U63#(mark(X1),X2,X3,X4) -> U63#(X1,X2,X3,X4)
       U63#(ok(X1),ok(X2),ok(X3),ok(X4)) -> U63#(X1,X2,X3,X4)
      TRS:
       active(U11(tt(),N,XS)) -> mark(U12(tt(),N,XS))
       active(U12(tt(),N,XS)) -> mark(snd(splitAt(N,XS)))
       active(U21(tt(),X)) -> mark(U22(tt(),X))
       active(U22(tt(),X)) -> mark(X)
       active(U31(tt(),N)) -> mark(U32(tt(),N))
       active(U32(tt(),N)) -> mark(N)
       active(U41(tt(),N,XS)) -> mark(U42(tt(),N,XS))
       active(U42(tt(),N,XS)) -> mark(head(afterNth(N,XS)))
       active(U51(tt(),Y)) -> mark(U52(tt(),Y))
       active(U52(tt(),Y)) -> mark(Y)
       active(U61(tt(),N,X,XS)) -> mark(U62(tt(),N,X,XS))
       active(U62(tt(),N,X,XS)) -> mark(U63(tt(),N,X,XS))
       active(U63(tt(),N,X,XS)) -> mark(U64(splitAt(N,XS),X))
       active(U64(pair(YS,ZS),X)) -> mark(pair(cons(X,YS),ZS))
       active(U71(tt(),XS)) -> mark(U72(tt(),XS))
       active(U72(tt(),XS)) -> mark(XS)
       active(U81(tt(),N,XS)) -> mark(U82(tt(),N,XS))
       active(U82(tt(),N,XS)) -> mark(fst(splitAt(N,XS)))
       active(afterNth(N,XS)) -> mark(U11(tt(),N,XS))
       active(fst(pair(X,Y))) -> mark(U21(tt(),X))
       active(head(cons(N,XS))) -> mark(U31(tt(),N))
       active(natsFrom(N)) -> mark(cons(N,natsFrom(s(N))))
       active(sel(N,XS)) -> mark(U41(tt(),N,XS))
       active(snd(pair(X,Y))) -> mark(U51(tt(),Y))
       active(splitAt(0(),XS)) -> mark(pair(nil(),XS))
       active(splitAt(s(N),cons(X,XS))) -> mark(U61(tt(),N,X,XS))
       active(tail(cons(N,XS))) -> mark(U71(tt(),XS))
       active(take(N,XS)) -> mark(U81(tt(),N,XS))
       active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3)
       active(U12(X1,X2,X3)) -> U12(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(U21(X1,X2)) -> U21(active(X1),X2)
       active(U22(X1,X2)) -> U22(active(X1),X2)
       active(U31(X1,X2)) -> U31(active(X1),X2)
       active(U32(X1,X2)) -> U32(active(X1),X2)
       active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3)
       active(U42(X1,X2,X3)) -> U42(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(U51(X1,X2)) -> U51(active(X1),X2)
       active(U52(X1,X2)) -> U52(active(X1),X2)
       active(U61(X1,X2,X3,X4)) -> U61(active(X1),X2,X3,X4)
       active(U62(X1,X2,X3,X4)) -> U62(active(X1),X2,X3,X4)
       active(U63(X1,X2,X3,X4)) -> U63(active(X1),X2,X3,X4)
       active(U64(X1,X2)) -> U64(active(X1),X2)
       active(pair(X1,X2)) -> pair(active(X1),X2)
       active(pair(X1,X2)) -> pair(X1,active(X2))
       active(cons(X1,X2)) -> cons(active(X1),X2)
       active(U71(X1,X2)) -> U71(active(X1),X2)
       active(U72(X1,X2)) -> U72(active(X1),X2)
       active(U81(X1,X2,X3)) -> U81(active(X1),X2,X3)
       active(U82(X1,X2,X3)) -> U82(active(X1),X2,X3)
       active(fst(X)) -> fst(active(X))
       active(natsFrom(X)) -> natsFrom(active(X))
       active(s(X)) -> s(active(X))
       active(sel(X1,X2)) -> sel(active(X1),X2)
       active(sel(X1,X2)) -> sel(X1,active(X2))
       active(tail(X)) -> tail(active(X))
       active(take(X1,X2)) -> take(active(X1),X2)
       active(take(X1,X2)) -> take(X1,active(X2))
       U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3))
       U12(mark(X1),X2,X3) -> mark(U12(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))
       U21(mark(X1),X2) -> mark(U21(X1,X2))
       U22(mark(X1),X2) -> mark(U22(X1,X2))
       U31(mark(X1),X2) -> mark(U31(X1,X2))
       U32(mark(X1),X2) -> mark(U32(X1,X2))
       U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3))
       U42(mark(X1),X2,X3) -> mark(U42(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))
       U51(mark(X1),X2) -> mark(U51(X1,X2))
       U52(mark(X1),X2) -> mark(U52(X1,X2))
       U61(mark(X1),X2,X3,X4) -> mark(U61(X1,X2,X3,X4))
       U62(mark(X1),X2,X3,X4) -> mark(U62(X1,X2,X3,X4))
       U63(mark(X1),X2,X3,X4) -> mark(U63(X1,X2,X3,X4))
       U64(mark(X1),X2) -> mark(U64(X1,X2))
       pair(mark(X1),X2) -> mark(pair(X1,X2))
       pair(X1,mark(X2)) -> mark(pair(X1,X2))
       cons(mark(X1),X2) -> mark(cons(X1,X2))
       U71(mark(X1),X2) -> mark(U71(X1,X2))
       U72(mark(X1),X2) -> mark(U72(X1,X2))
       U81(mark(X1),X2,X3) -> mark(U81(X1,X2,X3))
       U82(mark(X1),X2,X3) -> mark(U82(X1,X2,X3))
       fst(mark(X)) -> mark(fst(X))
       natsFrom(mark(X)) -> mark(natsFrom(X))
       s(mark(X)) -> mark(s(X))
       sel(mark(X1),X2) -> mark(sel(X1,X2))
       sel(X1,mark(X2)) -> mark(sel(X1,X2))
       tail(mark(X)) -> mark(tail(X))
       take(mark(X1),X2) -> mark(take(X1,X2))
       take(X1,mark(X2)) -> mark(take(X1,X2))
       proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3))
       proper(tt()) -> ok(tt())
       proper(U12(X1,X2,X3)) -> U12(proper(X1),proper(X2),proper(X3))
       proper(snd(X)) -> snd(proper(X))
       proper(splitAt(X1,X2)) -> splitAt(proper(X1),proper(X2))
       proper(U21(X1,X2)) -> U21(proper(X1),proper(X2))
       proper(U22(X1,X2)) -> U22(proper(X1),proper(X2))
       proper(U31(X1,X2)) -> U31(proper(X1),proper(X2))
       proper(U32(X1,X2)) -> U32(proper(X1),proper(X2))
       proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3))
       proper(U42(X1,X2,X3)) -> U42(proper(X1),proper(X2),proper(X3))
       proper(head(X)) -> head(proper(X))
       proper(afterNth(X1,X2)) -> afterNth(proper(X1),proper(X2))
       proper(U51(X1,X2)) -> U51(proper(X1),proper(X2))
       proper(U52(X1,X2)) -> U52(proper(X1),proper(X2))
       proper(U61(X1,X2,X3,X4)) -> U61(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(U62(X1,X2,X3,X4)) -> U62(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(U63(X1,X2,X3,X4)) -> U63(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(U64(X1,X2)) -> U64(proper(X1),proper(X2))
       proper(pair(X1,X2)) -> pair(proper(X1),proper(X2))
       proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
       proper(U71(X1,X2)) -> U71(proper(X1),proper(X2))
       proper(U72(X1,X2)) -> U72(proper(X1),proper(X2))
       proper(U81(X1,X2,X3)) -> U81(proper(X1),proper(X2),proper(X3))
       proper(U82(X1,X2,X3)) -> U82(proper(X1),proper(X2),proper(X3))
       proper(fst(X)) -> fst(proper(X))
       proper(natsFrom(X)) -> natsFrom(proper(X))
       proper(s(X)) -> s(proper(X))
       proper(sel(X1,X2)) -> sel(proper(X1),proper(X2))
       proper(0()) -> ok(0())
       proper(nil()) -> ok(nil())
       proper(tail(X)) -> tail(proper(X))
       proper(take(X1,X2)) -> take(proper(X1),proper(X2))
       U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3))
       U12(ok(X1),ok(X2),ok(X3)) -> ok(U12(X1,X2,X3))
       snd(ok(X)) -> ok(snd(X))
       splitAt(ok(X1),ok(X2)) -> ok(splitAt(X1,X2))
       U21(ok(X1),ok(X2)) -> ok(U21(X1,X2))
       U22(ok(X1),ok(X2)) -> ok(U22(X1,X2))
       U31(ok(X1),ok(X2)) -> ok(U31(X1,X2))
       U32(ok(X1),ok(X2)) -> ok(U32(X1,X2))
       U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3))
       U42(ok(X1),ok(X2),ok(X3)) -> ok(U42(X1,X2,X3))
       head(ok(X)) -> ok(head(X))
       afterNth(ok(X1),ok(X2)) -> ok(afterNth(X1,X2))
       U51(ok(X1),ok(X2)) -> ok(U51(X1,X2))
       U52(ok(X1),ok(X2)) -> ok(U52(X1,X2))
       U61(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U61(X1,X2,X3,X4))
       U62(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U62(X1,X2,X3,X4))
       U63(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U63(X1,X2,X3,X4))
       U64(ok(X1),ok(X2)) -> ok(U64(X1,X2))
       pair(ok(X1),ok(X2)) -> ok(pair(X1,X2))
       cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
       U71(ok(X1),ok(X2)) -> ok(U71(X1,X2))
       U72(ok(X1),ok(X2)) -> ok(U72(X1,X2))
       U81(ok(X1),ok(X2),ok(X3)) -> ok(U81(X1,X2,X3))
       U82(ok(X1),ok(X2),ok(X3)) -> ok(U82(X1,X2,X3))
       fst(ok(X)) -> ok(fst(X))
       natsFrom(ok(X)) -> ok(natsFrom(X))
       s(ok(X)) -> ok(s(X))
       sel(ok(X1),ok(X2)) -> ok(sel(X1,X2))
       tail(ok(X)) -> ok(tail(X))
       take(ok(X1),ok(X2)) -> ok(take(X1,X2))
       top(mark(X)) -> top(proper(X))
       top(ok(X)) -> top(active(X))
      Open
      
      DPs:
       U62#(mark(X1),X2,X3,X4) -> U62#(X1,X2,X3,X4)
       U62#(ok(X1),ok(X2),ok(X3),ok(X4)) -> U62#(X1,X2,X3,X4)
      TRS:
       active(U11(tt(),N,XS)) -> mark(U12(tt(),N,XS))
       active(U12(tt(),N,XS)) -> mark(snd(splitAt(N,XS)))
       active(U21(tt(),X)) -> mark(U22(tt(),X))
       active(U22(tt(),X)) -> mark(X)
       active(U31(tt(),N)) -> mark(U32(tt(),N))
       active(U32(tt(),N)) -> mark(N)
       active(U41(tt(),N,XS)) -> mark(U42(tt(),N,XS))
       active(U42(tt(),N,XS)) -> mark(head(afterNth(N,XS)))
       active(U51(tt(),Y)) -> mark(U52(tt(),Y))
       active(U52(tt(),Y)) -> mark(Y)
       active(U61(tt(),N,X,XS)) -> mark(U62(tt(),N,X,XS))
       active(U62(tt(),N,X,XS)) -> mark(U63(tt(),N,X,XS))
       active(U63(tt(),N,X,XS)) -> mark(U64(splitAt(N,XS),X))
       active(U64(pair(YS,ZS),X)) -> mark(pair(cons(X,YS),ZS))
       active(U71(tt(),XS)) -> mark(U72(tt(),XS))
       active(U72(tt(),XS)) -> mark(XS)
       active(U81(tt(),N,XS)) -> mark(U82(tt(),N,XS))
       active(U82(tt(),N,XS)) -> mark(fst(splitAt(N,XS)))
       active(afterNth(N,XS)) -> mark(U11(tt(),N,XS))
       active(fst(pair(X,Y))) -> mark(U21(tt(),X))
       active(head(cons(N,XS))) -> mark(U31(tt(),N))
       active(natsFrom(N)) -> mark(cons(N,natsFrom(s(N))))
       active(sel(N,XS)) -> mark(U41(tt(),N,XS))
       active(snd(pair(X,Y))) -> mark(U51(tt(),Y))
       active(splitAt(0(),XS)) -> mark(pair(nil(),XS))
       active(splitAt(s(N),cons(X,XS))) -> mark(U61(tt(),N,X,XS))
       active(tail(cons(N,XS))) -> mark(U71(tt(),XS))
       active(take(N,XS)) -> mark(U81(tt(),N,XS))
       active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3)
       active(U12(X1,X2,X3)) -> U12(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(U21(X1,X2)) -> U21(active(X1),X2)
       active(U22(X1,X2)) -> U22(active(X1),X2)
       active(U31(X1,X2)) -> U31(active(X1),X2)
       active(U32(X1,X2)) -> U32(active(X1),X2)
       active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3)
       active(U42(X1,X2,X3)) -> U42(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(U51(X1,X2)) -> U51(active(X1),X2)
       active(U52(X1,X2)) -> U52(active(X1),X2)
       active(U61(X1,X2,X3,X4)) -> U61(active(X1),X2,X3,X4)
       active(U62(X1,X2,X3,X4)) -> U62(active(X1),X2,X3,X4)
       active(U63(X1,X2,X3,X4)) -> U63(active(X1),X2,X3,X4)
       active(U64(X1,X2)) -> U64(active(X1),X2)
       active(pair(X1,X2)) -> pair(active(X1),X2)
       active(pair(X1,X2)) -> pair(X1,active(X2))
       active(cons(X1,X2)) -> cons(active(X1),X2)
       active(U71(X1,X2)) -> U71(active(X1),X2)
       active(U72(X1,X2)) -> U72(active(X1),X2)
       active(U81(X1,X2,X3)) -> U81(active(X1),X2,X3)
       active(U82(X1,X2,X3)) -> U82(active(X1),X2,X3)
       active(fst(X)) -> fst(active(X))
       active(natsFrom(X)) -> natsFrom(active(X))
       active(s(X)) -> s(active(X))
       active(sel(X1,X2)) -> sel(active(X1),X2)
       active(sel(X1,X2)) -> sel(X1,active(X2))
       active(tail(X)) -> tail(active(X))
       active(take(X1,X2)) -> take(active(X1),X2)
       active(take(X1,X2)) -> take(X1,active(X2))
       U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3))
       U12(mark(X1),X2,X3) -> mark(U12(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))
       U21(mark(X1),X2) -> mark(U21(X1,X2))
       U22(mark(X1),X2) -> mark(U22(X1,X2))
       U31(mark(X1),X2) -> mark(U31(X1,X2))
       U32(mark(X1),X2) -> mark(U32(X1,X2))
       U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3))
       U42(mark(X1),X2,X3) -> mark(U42(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))
       U51(mark(X1),X2) -> mark(U51(X1,X2))
       U52(mark(X1),X2) -> mark(U52(X1,X2))
       U61(mark(X1),X2,X3,X4) -> mark(U61(X1,X2,X3,X4))
       U62(mark(X1),X2,X3,X4) -> mark(U62(X1,X2,X3,X4))
       U63(mark(X1),X2,X3,X4) -> mark(U63(X1,X2,X3,X4))
       U64(mark(X1),X2) -> mark(U64(X1,X2))
       pair(mark(X1),X2) -> mark(pair(X1,X2))
       pair(X1,mark(X2)) -> mark(pair(X1,X2))
       cons(mark(X1),X2) -> mark(cons(X1,X2))
       U71(mark(X1),X2) -> mark(U71(X1,X2))
       U72(mark(X1),X2) -> mark(U72(X1,X2))
       U81(mark(X1),X2,X3) -> mark(U81(X1,X2,X3))
       U82(mark(X1),X2,X3) -> mark(U82(X1,X2,X3))
       fst(mark(X)) -> mark(fst(X))
       natsFrom(mark(X)) -> mark(natsFrom(X))
       s(mark(X)) -> mark(s(X))
       sel(mark(X1),X2) -> mark(sel(X1,X2))
       sel(X1,mark(X2)) -> mark(sel(X1,X2))
       tail(mark(X)) -> mark(tail(X))
       take(mark(X1),X2) -> mark(take(X1,X2))
       take(X1,mark(X2)) -> mark(take(X1,X2))
       proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3))
       proper(tt()) -> ok(tt())
       proper(U12(X1,X2,X3)) -> U12(proper(X1),proper(X2),proper(X3))
       proper(snd(X)) -> snd(proper(X))
       proper(splitAt(X1,X2)) -> splitAt(proper(X1),proper(X2))
       proper(U21(X1,X2)) -> U21(proper(X1),proper(X2))
       proper(U22(X1,X2)) -> U22(proper(X1),proper(X2))
       proper(U31(X1,X2)) -> U31(proper(X1),proper(X2))
       proper(U32(X1,X2)) -> U32(proper(X1),proper(X2))
       proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3))
       proper(U42(X1,X2,X3)) -> U42(proper(X1),proper(X2),proper(X3))
       proper(head(X)) -> head(proper(X))
       proper(afterNth(X1,X2)) -> afterNth(proper(X1),proper(X2))
       proper(U51(X1,X2)) -> U51(proper(X1),proper(X2))
       proper(U52(X1,X2)) -> U52(proper(X1),proper(X2))
       proper(U61(X1,X2,X3,X4)) -> U61(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(U62(X1,X2,X3,X4)) -> U62(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(U63(X1,X2,X3,X4)) -> U63(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(U64(X1,X2)) -> U64(proper(X1),proper(X2))
       proper(pair(X1,X2)) -> pair(proper(X1),proper(X2))
       proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
       proper(U71(X1,X2)) -> U71(proper(X1),proper(X2))
       proper(U72(X1,X2)) -> U72(proper(X1),proper(X2))
       proper(U81(X1,X2,X3)) -> U81(proper(X1),proper(X2),proper(X3))
       proper(U82(X1,X2,X3)) -> U82(proper(X1),proper(X2),proper(X3))
       proper(fst(X)) -> fst(proper(X))
       proper(natsFrom(X)) -> natsFrom(proper(X))
       proper(s(X)) -> s(proper(X))
       proper(sel(X1,X2)) -> sel(proper(X1),proper(X2))
       proper(0()) -> ok(0())
       proper(nil()) -> ok(nil())
       proper(tail(X)) -> tail(proper(X))
       proper(take(X1,X2)) -> take(proper(X1),proper(X2))
       U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3))
       U12(ok(X1),ok(X2),ok(X3)) -> ok(U12(X1,X2,X3))
       snd(ok(X)) -> ok(snd(X))
       splitAt(ok(X1),ok(X2)) -> ok(splitAt(X1,X2))
       U21(ok(X1),ok(X2)) -> ok(U21(X1,X2))
       U22(ok(X1),ok(X2)) -> ok(U22(X1,X2))
       U31(ok(X1),ok(X2)) -> ok(U31(X1,X2))
       U32(ok(X1),ok(X2)) -> ok(U32(X1,X2))
       U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3))
       U42(ok(X1),ok(X2),ok(X3)) -> ok(U42(X1,X2,X3))
       head(ok(X)) -> ok(head(X))
       afterNth(ok(X1),ok(X2)) -> ok(afterNth(X1,X2))
       U51(ok(X1),ok(X2)) -> ok(U51(X1,X2))
       U52(ok(X1),ok(X2)) -> ok(U52(X1,X2))
       U61(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U61(X1,X2,X3,X4))
       U62(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U62(X1,X2,X3,X4))
       U63(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U63(X1,X2,X3,X4))
       U64(ok(X1),ok(X2)) -> ok(U64(X1,X2))
       pair(ok(X1),ok(X2)) -> ok(pair(X1,X2))
       cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
       U71(ok(X1),ok(X2)) -> ok(U71(X1,X2))
       U72(ok(X1),ok(X2)) -> ok(U72(X1,X2))
       U81(ok(X1),ok(X2),ok(X3)) -> ok(U81(X1,X2,X3))
       U82(ok(X1),ok(X2),ok(X3)) -> ok(U82(X1,X2,X3))
       fst(ok(X)) -> ok(fst(X))
       natsFrom(ok(X)) -> ok(natsFrom(X))
       s(ok(X)) -> ok(s(X))
       sel(ok(X1),ok(X2)) -> ok(sel(X1,X2))
       tail(ok(X)) -> ok(tail(X))
       take(ok(X1),ok(X2)) -> ok(take(X1,X2))
       top(mark(X)) -> top(proper(X))
       top(ok(X)) -> top(active(X))
      Open
      
      DPs:
       U61#(mark(X1),X2,X3,X4) -> U61#(X1,X2,X3,X4)
       U61#(ok(X1),ok(X2),ok(X3),ok(X4)) -> U61#(X1,X2,X3,X4)
      TRS:
       active(U11(tt(),N,XS)) -> mark(U12(tt(),N,XS))
       active(U12(tt(),N,XS)) -> mark(snd(splitAt(N,XS)))
       active(U21(tt(),X)) -> mark(U22(tt(),X))
       active(U22(tt(),X)) -> mark(X)
       active(U31(tt(),N)) -> mark(U32(tt(),N))
       active(U32(tt(),N)) -> mark(N)
       active(U41(tt(),N,XS)) -> mark(U42(tt(),N,XS))
       active(U42(tt(),N,XS)) -> mark(head(afterNth(N,XS)))
       active(U51(tt(),Y)) -> mark(U52(tt(),Y))
       active(U52(tt(),Y)) -> mark(Y)
       active(U61(tt(),N,X,XS)) -> mark(U62(tt(),N,X,XS))
       active(U62(tt(),N,X,XS)) -> mark(U63(tt(),N,X,XS))
       active(U63(tt(),N,X,XS)) -> mark(U64(splitAt(N,XS),X))
       active(U64(pair(YS,ZS),X)) -> mark(pair(cons(X,YS),ZS))
       active(U71(tt(),XS)) -> mark(U72(tt(),XS))
       active(U72(tt(),XS)) -> mark(XS)
       active(U81(tt(),N,XS)) -> mark(U82(tt(),N,XS))
       active(U82(tt(),N,XS)) -> mark(fst(splitAt(N,XS)))
       active(afterNth(N,XS)) -> mark(U11(tt(),N,XS))
       active(fst(pair(X,Y))) -> mark(U21(tt(),X))
       active(head(cons(N,XS))) -> mark(U31(tt(),N))
       active(natsFrom(N)) -> mark(cons(N,natsFrom(s(N))))
       active(sel(N,XS)) -> mark(U41(tt(),N,XS))
       active(snd(pair(X,Y))) -> mark(U51(tt(),Y))
       active(splitAt(0(),XS)) -> mark(pair(nil(),XS))
       active(splitAt(s(N),cons(X,XS))) -> mark(U61(tt(),N,X,XS))
       active(tail(cons(N,XS))) -> mark(U71(tt(),XS))
       active(take(N,XS)) -> mark(U81(tt(),N,XS))
       active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3)
       active(U12(X1,X2,X3)) -> U12(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(U21(X1,X2)) -> U21(active(X1),X2)
       active(U22(X1,X2)) -> U22(active(X1),X2)
       active(U31(X1,X2)) -> U31(active(X1),X2)
       active(U32(X1,X2)) -> U32(active(X1),X2)
       active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3)
       active(U42(X1,X2,X3)) -> U42(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(U51(X1,X2)) -> U51(active(X1),X2)
       active(U52(X1,X2)) -> U52(active(X1),X2)
       active(U61(X1,X2,X3,X4)) -> U61(active(X1),X2,X3,X4)
       active(U62(X1,X2,X3,X4)) -> U62(active(X1),X2,X3,X4)
       active(U63(X1,X2,X3,X4)) -> U63(active(X1),X2,X3,X4)
       active(U64(X1,X2)) -> U64(active(X1),X2)
       active(pair(X1,X2)) -> pair(active(X1),X2)
       active(pair(X1,X2)) -> pair(X1,active(X2))
       active(cons(X1,X2)) -> cons(active(X1),X2)
       active(U71(X1,X2)) -> U71(active(X1),X2)
       active(U72(X1,X2)) -> U72(active(X1),X2)
       active(U81(X1,X2,X3)) -> U81(active(X1),X2,X3)
       active(U82(X1,X2,X3)) -> U82(active(X1),X2,X3)
       active(fst(X)) -> fst(active(X))
       active(natsFrom(X)) -> natsFrom(active(X))
       active(s(X)) -> s(active(X))
       active(sel(X1,X2)) -> sel(active(X1),X2)
       active(sel(X1,X2)) -> sel(X1,active(X2))
       active(tail(X)) -> tail(active(X))
       active(take(X1,X2)) -> take(active(X1),X2)
       active(take(X1,X2)) -> take(X1,active(X2))
       U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3))
       U12(mark(X1),X2,X3) -> mark(U12(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))
       U21(mark(X1),X2) -> mark(U21(X1,X2))
       U22(mark(X1),X2) -> mark(U22(X1,X2))
       U31(mark(X1),X2) -> mark(U31(X1,X2))
       U32(mark(X1),X2) -> mark(U32(X1,X2))
       U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3))
       U42(mark(X1),X2,X3) -> mark(U42(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))
       U51(mark(X1),X2) -> mark(U51(X1,X2))
       U52(mark(X1),X2) -> mark(U52(X1,X2))
       U61(mark(X1),X2,X3,X4) -> mark(U61(X1,X2,X3,X4))
       U62(mark(X1),X2,X3,X4) -> mark(U62(X1,X2,X3,X4))
       U63(mark(X1),X2,X3,X4) -> mark(U63(X1,X2,X3,X4))
       U64(mark(X1),X2) -> mark(U64(X1,X2))
       pair(mark(X1),X2) -> mark(pair(X1,X2))
       pair(X1,mark(X2)) -> mark(pair(X1,X2))
       cons(mark(X1),X2) -> mark(cons(X1,X2))
       U71(mark(X1),X2) -> mark(U71(X1,X2))
       U72(mark(X1),X2) -> mark(U72(X1,X2))
       U81(mark(X1),X2,X3) -> mark(U81(X1,X2,X3))
       U82(mark(X1),X2,X3) -> mark(U82(X1,X2,X3))
       fst(mark(X)) -> mark(fst(X))
       natsFrom(mark(X)) -> mark(natsFrom(X))
       s(mark(X)) -> mark(s(X))
       sel(mark(X1),X2) -> mark(sel(X1,X2))
       sel(X1,mark(X2)) -> mark(sel(X1,X2))
       tail(mark(X)) -> mark(tail(X))
       take(mark(X1),X2) -> mark(take(X1,X2))
       take(X1,mark(X2)) -> mark(take(X1,X2))
       proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3))
       proper(tt()) -> ok(tt())
       proper(U12(X1,X2,X3)) -> U12(proper(X1),proper(X2),proper(X3))
       proper(snd(X)) -> snd(proper(X))
       proper(splitAt(X1,X2)) -> splitAt(proper(X1),proper(X2))
       proper(U21(X1,X2)) -> U21(proper(X1),proper(X2))
       proper(U22(X1,X2)) -> U22(proper(X1),proper(X2))
       proper(U31(X1,X2)) -> U31(proper(X1),proper(X2))
       proper(U32(X1,X2)) -> U32(proper(X1),proper(X2))
       proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3))
       proper(U42(X1,X2,X3)) -> U42(proper(X1),proper(X2),proper(X3))
       proper(head(X)) -> head(proper(X))
       proper(afterNth(X1,X2)) -> afterNth(proper(X1),proper(X2))
       proper(U51(X1,X2)) -> U51(proper(X1),proper(X2))
       proper(U52(X1,X2)) -> U52(proper(X1),proper(X2))
       proper(U61(X1,X2,X3,X4)) -> U61(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(U62(X1,X2,X3,X4)) -> U62(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(U63(X1,X2,X3,X4)) -> U63(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(U64(X1,X2)) -> U64(proper(X1),proper(X2))
       proper(pair(X1,X2)) -> pair(proper(X1),proper(X2))
       proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
       proper(U71(X1,X2)) -> U71(proper(X1),proper(X2))
       proper(U72(X1,X2)) -> U72(proper(X1),proper(X2))
       proper(U81(X1,X2,X3)) -> U81(proper(X1),proper(X2),proper(X3))
       proper(U82(X1,X2,X3)) -> U82(proper(X1),proper(X2),proper(X3))
       proper(fst(X)) -> fst(proper(X))
       proper(natsFrom(X)) -> natsFrom(proper(X))
       proper(s(X)) -> s(proper(X))
       proper(sel(X1,X2)) -> sel(proper(X1),proper(X2))
       proper(0()) -> ok(0())
       proper(nil()) -> ok(nil())
       proper(tail(X)) -> tail(proper(X))
       proper(take(X1,X2)) -> take(proper(X1),proper(X2))
       U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3))
       U12(ok(X1),ok(X2),ok(X3)) -> ok(U12(X1,X2,X3))
       snd(ok(X)) -> ok(snd(X))
       splitAt(ok(X1),ok(X2)) -> ok(splitAt(X1,X2))
       U21(ok(X1),ok(X2)) -> ok(U21(X1,X2))
       U22(ok(X1),ok(X2)) -> ok(U22(X1,X2))
       U31(ok(X1),ok(X2)) -> ok(U31(X1,X2))
       U32(ok(X1),ok(X2)) -> ok(U32(X1,X2))
       U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3))
       U42(ok(X1),ok(X2),ok(X3)) -> ok(U42(X1,X2,X3))
       head(ok(X)) -> ok(head(X))
       afterNth(ok(X1),ok(X2)) -> ok(afterNth(X1,X2))
       U51(ok(X1),ok(X2)) -> ok(U51(X1,X2))
       U52(ok(X1),ok(X2)) -> ok(U52(X1,X2))
       U61(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U61(X1,X2,X3,X4))
       U62(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U62(X1,X2,X3,X4))
       U63(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U63(X1,X2,X3,X4))
       U64(ok(X1),ok(X2)) -> ok(U64(X1,X2))
       pair(ok(X1),ok(X2)) -> ok(pair(X1,X2))
       cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
       U71(ok(X1),ok(X2)) -> ok(U71(X1,X2))
       U72(ok(X1),ok(X2)) -> ok(U72(X1,X2))
       U81(ok(X1),ok(X2),ok(X3)) -> ok(U81(X1,X2,X3))
       U82(ok(X1),ok(X2),ok(X3)) -> ok(U82(X1,X2,X3))
       fst(ok(X)) -> ok(fst(X))
       natsFrom(ok(X)) -> ok(natsFrom(X))
       s(ok(X)) -> ok(s(X))
       sel(ok(X1),ok(X2)) -> ok(sel(X1,X2))
       tail(ok(X)) -> ok(tail(X))
       take(ok(X1),ok(X2)) -> ok(take(X1,X2))
       top(mark(X)) -> top(proper(X))
       top(ok(X)) -> top(active(X))
      Open
      
      DPs:
       U52#(mark(X1),X2) -> U52#(X1,X2)
       U52#(ok(X1),ok(X2)) -> U52#(X1,X2)
      TRS:
       active(U11(tt(),N,XS)) -> mark(U12(tt(),N,XS))
       active(U12(tt(),N,XS)) -> mark(snd(splitAt(N,XS)))
       active(U21(tt(),X)) -> mark(U22(tt(),X))
       active(U22(tt(),X)) -> mark(X)
       active(U31(tt(),N)) -> mark(U32(tt(),N))
       active(U32(tt(),N)) -> mark(N)
       active(U41(tt(),N,XS)) -> mark(U42(tt(),N,XS))
       active(U42(tt(),N,XS)) -> mark(head(afterNth(N,XS)))
       active(U51(tt(),Y)) -> mark(U52(tt(),Y))
       active(U52(tt(),Y)) -> mark(Y)
       active(U61(tt(),N,X,XS)) -> mark(U62(tt(),N,X,XS))
       active(U62(tt(),N,X,XS)) -> mark(U63(tt(),N,X,XS))
       active(U63(tt(),N,X,XS)) -> mark(U64(splitAt(N,XS),X))
       active(U64(pair(YS,ZS),X)) -> mark(pair(cons(X,YS),ZS))
       active(U71(tt(),XS)) -> mark(U72(tt(),XS))
       active(U72(tt(),XS)) -> mark(XS)
       active(U81(tt(),N,XS)) -> mark(U82(tt(),N,XS))
       active(U82(tt(),N,XS)) -> mark(fst(splitAt(N,XS)))
       active(afterNth(N,XS)) -> mark(U11(tt(),N,XS))
       active(fst(pair(X,Y))) -> mark(U21(tt(),X))
       active(head(cons(N,XS))) -> mark(U31(tt(),N))
       active(natsFrom(N)) -> mark(cons(N,natsFrom(s(N))))
       active(sel(N,XS)) -> mark(U41(tt(),N,XS))
       active(snd(pair(X,Y))) -> mark(U51(tt(),Y))
       active(splitAt(0(),XS)) -> mark(pair(nil(),XS))
       active(splitAt(s(N),cons(X,XS))) -> mark(U61(tt(),N,X,XS))
       active(tail(cons(N,XS))) -> mark(U71(tt(),XS))
       active(take(N,XS)) -> mark(U81(tt(),N,XS))
       active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3)
       active(U12(X1,X2,X3)) -> U12(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(U21(X1,X2)) -> U21(active(X1),X2)
       active(U22(X1,X2)) -> U22(active(X1),X2)
       active(U31(X1,X2)) -> U31(active(X1),X2)
       active(U32(X1,X2)) -> U32(active(X1),X2)
       active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3)
       active(U42(X1,X2,X3)) -> U42(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(U51(X1,X2)) -> U51(active(X1),X2)
       active(U52(X1,X2)) -> U52(active(X1),X2)
       active(U61(X1,X2,X3,X4)) -> U61(active(X1),X2,X3,X4)
       active(U62(X1,X2,X3,X4)) -> U62(active(X1),X2,X3,X4)
       active(U63(X1,X2,X3,X4)) -> U63(active(X1),X2,X3,X4)
       active(U64(X1,X2)) -> U64(active(X1),X2)
       active(pair(X1,X2)) -> pair(active(X1),X2)
       active(pair(X1,X2)) -> pair(X1,active(X2))
       active(cons(X1,X2)) -> cons(active(X1),X2)
       active(U71(X1,X2)) -> U71(active(X1),X2)
       active(U72(X1,X2)) -> U72(active(X1),X2)
       active(U81(X1,X2,X3)) -> U81(active(X1),X2,X3)
       active(U82(X1,X2,X3)) -> U82(active(X1),X2,X3)
       active(fst(X)) -> fst(active(X))
       active(natsFrom(X)) -> natsFrom(active(X))
       active(s(X)) -> s(active(X))
       active(sel(X1,X2)) -> sel(active(X1),X2)
       active(sel(X1,X2)) -> sel(X1,active(X2))
       active(tail(X)) -> tail(active(X))
       active(take(X1,X2)) -> take(active(X1),X2)
       active(take(X1,X2)) -> take(X1,active(X2))
       U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3))
       U12(mark(X1),X2,X3) -> mark(U12(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))
       U21(mark(X1),X2) -> mark(U21(X1,X2))
       U22(mark(X1),X2) -> mark(U22(X1,X2))
       U31(mark(X1),X2) -> mark(U31(X1,X2))
       U32(mark(X1),X2) -> mark(U32(X1,X2))
       U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3))
       U42(mark(X1),X2,X3) -> mark(U42(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))
       U51(mark(X1),X2) -> mark(U51(X1,X2))
       U52(mark(X1),X2) -> mark(U52(X1,X2))
       U61(mark(X1),X2,X3,X4) -> mark(U61(X1,X2,X3,X4))
       U62(mark(X1),X2,X3,X4) -> mark(U62(X1,X2,X3,X4))
       U63(mark(X1),X2,X3,X4) -> mark(U63(X1,X2,X3,X4))
       U64(mark(X1),X2) -> mark(U64(X1,X2))
       pair(mark(X1),X2) -> mark(pair(X1,X2))
       pair(X1,mark(X2)) -> mark(pair(X1,X2))
       cons(mark(X1),X2) -> mark(cons(X1,X2))
       U71(mark(X1),X2) -> mark(U71(X1,X2))
       U72(mark(X1),X2) -> mark(U72(X1,X2))
       U81(mark(X1),X2,X3) -> mark(U81(X1,X2,X3))
       U82(mark(X1),X2,X3) -> mark(U82(X1,X2,X3))
       fst(mark(X)) -> mark(fst(X))
       natsFrom(mark(X)) -> mark(natsFrom(X))
       s(mark(X)) -> mark(s(X))
       sel(mark(X1),X2) -> mark(sel(X1,X2))
       sel(X1,mark(X2)) -> mark(sel(X1,X2))
       tail(mark(X)) -> mark(tail(X))
       take(mark(X1),X2) -> mark(take(X1,X2))
       take(X1,mark(X2)) -> mark(take(X1,X2))
       proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3))
       proper(tt()) -> ok(tt())
       proper(U12(X1,X2,X3)) -> U12(proper(X1),proper(X2),proper(X3))
       proper(snd(X)) -> snd(proper(X))
       proper(splitAt(X1,X2)) -> splitAt(proper(X1),proper(X2))
       proper(U21(X1,X2)) -> U21(proper(X1),proper(X2))
       proper(U22(X1,X2)) -> U22(proper(X1),proper(X2))
       proper(U31(X1,X2)) -> U31(proper(X1),proper(X2))
       proper(U32(X1,X2)) -> U32(proper(X1),proper(X2))
       proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3))
       proper(U42(X1,X2,X3)) -> U42(proper(X1),proper(X2),proper(X3))
       proper(head(X)) -> head(proper(X))
       proper(afterNth(X1,X2)) -> afterNth(proper(X1),proper(X2))
       proper(U51(X1,X2)) -> U51(proper(X1),proper(X2))
       proper(U52(X1,X2)) -> U52(proper(X1),proper(X2))
       proper(U61(X1,X2,X3,X4)) -> U61(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(U62(X1,X2,X3,X4)) -> U62(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(U63(X1,X2,X3,X4)) -> U63(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(U64(X1,X2)) -> U64(proper(X1),proper(X2))
       proper(pair(X1,X2)) -> pair(proper(X1),proper(X2))
       proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
       proper(U71(X1,X2)) -> U71(proper(X1),proper(X2))
       proper(U72(X1,X2)) -> U72(proper(X1),proper(X2))
       proper(U81(X1,X2,X3)) -> U81(proper(X1),proper(X2),proper(X3))
       proper(U82(X1,X2,X3)) -> U82(proper(X1),proper(X2),proper(X3))
       proper(fst(X)) -> fst(proper(X))
       proper(natsFrom(X)) -> natsFrom(proper(X))
       proper(s(X)) -> s(proper(X))
       proper(sel(X1,X2)) -> sel(proper(X1),proper(X2))
       proper(0()) -> ok(0())
       proper(nil()) -> ok(nil())
       proper(tail(X)) -> tail(proper(X))
       proper(take(X1,X2)) -> take(proper(X1),proper(X2))
       U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3))
       U12(ok(X1),ok(X2),ok(X3)) -> ok(U12(X1,X2,X3))
       snd(ok(X)) -> ok(snd(X))
       splitAt(ok(X1),ok(X2)) -> ok(splitAt(X1,X2))
       U21(ok(X1),ok(X2)) -> ok(U21(X1,X2))
       U22(ok(X1),ok(X2)) -> ok(U22(X1,X2))
       U31(ok(X1),ok(X2)) -> ok(U31(X1,X2))
       U32(ok(X1),ok(X2)) -> ok(U32(X1,X2))
       U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3))
       U42(ok(X1),ok(X2),ok(X3)) -> ok(U42(X1,X2,X3))
       head(ok(X)) -> ok(head(X))
       afterNth(ok(X1),ok(X2)) -> ok(afterNth(X1,X2))
       U51(ok(X1),ok(X2)) -> ok(U51(X1,X2))
       U52(ok(X1),ok(X2)) -> ok(U52(X1,X2))
       U61(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U61(X1,X2,X3,X4))
       U62(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U62(X1,X2,X3,X4))
       U63(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U63(X1,X2,X3,X4))
       U64(ok(X1),ok(X2)) -> ok(U64(X1,X2))
       pair(ok(X1),ok(X2)) -> ok(pair(X1,X2))
       cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
       U71(ok(X1),ok(X2)) -> ok(U71(X1,X2))
       U72(ok(X1),ok(X2)) -> ok(U72(X1,X2))
       U81(ok(X1),ok(X2),ok(X3)) -> ok(U81(X1,X2,X3))
       U82(ok(X1),ok(X2),ok(X3)) -> ok(U82(X1,X2,X3))
       fst(ok(X)) -> ok(fst(X))
       natsFrom(ok(X)) -> ok(natsFrom(X))
       s(ok(X)) -> ok(s(X))
       sel(ok(X1),ok(X2)) -> ok(sel(X1,X2))
       tail(ok(X)) -> ok(tail(X))
       take(ok(X1),ok(X2)) -> ok(take(X1,X2))
       top(mark(X)) -> top(proper(X))
       top(ok(X)) -> top(active(X))
      Open
      
      DPs:
       U51#(mark(X1),X2) -> U51#(X1,X2)
       U51#(ok(X1),ok(X2)) -> U51#(X1,X2)
      TRS:
       active(U11(tt(),N,XS)) -> mark(U12(tt(),N,XS))
       active(U12(tt(),N,XS)) -> mark(snd(splitAt(N,XS)))
       active(U21(tt(),X)) -> mark(U22(tt(),X))
       active(U22(tt(),X)) -> mark(X)
       active(U31(tt(),N)) -> mark(U32(tt(),N))
       active(U32(tt(),N)) -> mark(N)
       active(U41(tt(),N,XS)) -> mark(U42(tt(),N,XS))
       active(U42(tt(),N,XS)) -> mark(head(afterNth(N,XS)))
       active(U51(tt(),Y)) -> mark(U52(tt(),Y))
       active(U52(tt(),Y)) -> mark(Y)
       active(U61(tt(),N,X,XS)) -> mark(U62(tt(),N,X,XS))
       active(U62(tt(),N,X,XS)) -> mark(U63(tt(),N,X,XS))
       active(U63(tt(),N,X,XS)) -> mark(U64(splitAt(N,XS),X))
       active(U64(pair(YS,ZS),X)) -> mark(pair(cons(X,YS),ZS))
       active(U71(tt(),XS)) -> mark(U72(tt(),XS))
       active(U72(tt(),XS)) -> mark(XS)
       active(U81(tt(),N,XS)) -> mark(U82(tt(),N,XS))
       active(U82(tt(),N,XS)) -> mark(fst(splitAt(N,XS)))
       active(afterNth(N,XS)) -> mark(U11(tt(),N,XS))
       active(fst(pair(X,Y))) -> mark(U21(tt(),X))
       active(head(cons(N,XS))) -> mark(U31(tt(),N))
       active(natsFrom(N)) -> mark(cons(N,natsFrom(s(N))))
       active(sel(N,XS)) -> mark(U41(tt(),N,XS))
       active(snd(pair(X,Y))) -> mark(U51(tt(),Y))
       active(splitAt(0(),XS)) -> mark(pair(nil(),XS))
       active(splitAt(s(N),cons(X,XS))) -> mark(U61(tt(),N,X,XS))
       active(tail(cons(N,XS))) -> mark(U71(tt(),XS))
       active(take(N,XS)) -> mark(U81(tt(),N,XS))
       active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3)
       active(U12(X1,X2,X3)) -> U12(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(U21(X1,X2)) -> U21(active(X1),X2)
       active(U22(X1,X2)) -> U22(active(X1),X2)
       active(U31(X1,X2)) -> U31(active(X1),X2)
       active(U32(X1,X2)) -> U32(active(X1),X2)
       active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3)
       active(U42(X1,X2,X3)) -> U42(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(U51(X1,X2)) -> U51(active(X1),X2)
       active(U52(X1,X2)) -> U52(active(X1),X2)
       active(U61(X1,X2,X3,X4)) -> U61(active(X1),X2,X3,X4)
       active(U62(X1,X2,X3,X4)) -> U62(active(X1),X2,X3,X4)
       active(U63(X1,X2,X3,X4)) -> U63(active(X1),X2,X3,X4)
       active(U64(X1,X2)) -> U64(active(X1),X2)
       active(pair(X1,X2)) -> pair(active(X1),X2)
       active(pair(X1,X2)) -> pair(X1,active(X2))
       active(cons(X1,X2)) -> cons(active(X1),X2)
       active(U71(X1,X2)) -> U71(active(X1),X2)
       active(U72(X1,X2)) -> U72(active(X1),X2)
       active(U81(X1,X2,X3)) -> U81(active(X1),X2,X3)
       active(U82(X1,X2,X3)) -> U82(active(X1),X2,X3)
       active(fst(X)) -> fst(active(X))
       active(natsFrom(X)) -> natsFrom(active(X))
       active(s(X)) -> s(active(X))
       active(sel(X1,X2)) -> sel(active(X1),X2)
       active(sel(X1,X2)) -> sel(X1,active(X2))
       active(tail(X)) -> tail(active(X))
       active(take(X1,X2)) -> take(active(X1),X2)
       active(take(X1,X2)) -> take(X1,active(X2))
       U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3))
       U12(mark(X1),X2,X3) -> mark(U12(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))
       U21(mark(X1),X2) -> mark(U21(X1,X2))
       U22(mark(X1),X2) -> mark(U22(X1,X2))
       U31(mark(X1),X2) -> mark(U31(X1,X2))
       U32(mark(X1),X2) -> mark(U32(X1,X2))
       U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3))
       U42(mark(X1),X2,X3) -> mark(U42(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))
       U51(mark(X1),X2) -> mark(U51(X1,X2))
       U52(mark(X1),X2) -> mark(U52(X1,X2))
       U61(mark(X1),X2,X3,X4) -> mark(U61(X1,X2,X3,X4))
       U62(mark(X1),X2,X3,X4) -> mark(U62(X1,X2,X3,X4))
       U63(mark(X1),X2,X3,X4) -> mark(U63(X1,X2,X3,X4))
       U64(mark(X1),X2) -> mark(U64(X1,X2))
       pair(mark(X1),X2) -> mark(pair(X1,X2))
       pair(X1,mark(X2)) -> mark(pair(X1,X2))
       cons(mark(X1),X2) -> mark(cons(X1,X2))
       U71(mark(X1),X2) -> mark(U71(X1,X2))
       U72(mark(X1),X2) -> mark(U72(X1,X2))
       U81(mark(X1),X2,X3) -> mark(U81(X1,X2,X3))
       U82(mark(X1),X2,X3) -> mark(U82(X1,X2,X3))
       fst(mark(X)) -> mark(fst(X))
       natsFrom(mark(X)) -> mark(natsFrom(X))
       s(mark(X)) -> mark(s(X))
       sel(mark(X1),X2) -> mark(sel(X1,X2))
       sel(X1,mark(X2)) -> mark(sel(X1,X2))
       tail(mark(X)) -> mark(tail(X))
       take(mark(X1),X2) -> mark(take(X1,X2))
       take(X1,mark(X2)) -> mark(take(X1,X2))
       proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3))
       proper(tt()) -> ok(tt())
       proper(U12(X1,X2,X3)) -> U12(proper(X1),proper(X2),proper(X3))
       proper(snd(X)) -> snd(proper(X))
       proper(splitAt(X1,X2)) -> splitAt(proper(X1),proper(X2))
       proper(U21(X1,X2)) -> U21(proper(X1),proper(X2))
       proper(U22(X1,X2)) -> U22(proper(X1),proper(X2))
       proper(U31(X1,X2)) -> U31(proper(X1),proper(X2))
       proper(U32(X1,X2)) -> U32(proper(X1),proper(X2))
       proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3))
       proper(U42(X1,X2,X3)) -> U42(proper(X1),proper(X2),proper(X3))
       proper(head(X)) -> head(proper(X))
       proper(afterNth(X1,X2)) -> afterNth(proper(X1),proper(X2))
       proper(U51(X1,X2)) -> U51(proper(X1),proper(X2))
       proper(U52(X1,X2)) -> U52(proper(X1),proper(X2))
       proper(U61(X1,X2,X3,X4)) -> U61(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(U62(X1,X2,X3,X4)) -> U62(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(U63(X1,X2,X3,X4)) -> U63(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(U64(X1,X2)) -> U64(proper(X1),proper(X2))
       proper(pair(X1,X2)) -> pair(proper(X1),proper(X2))
       proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
       proper(U71(X1,X2)) -> U71(proper(X1),proper(X2))
       proper(U72(X1,X2)) -> U72(proper(X1),proper(X2))
       proper(U81(X1,X2,X3)) -> U81(proper(X1),proper(X2),proper(X3))
       proper(U82(X1,X2,X3)) -> U82(proper(X1),proper(X2),proper(X3))
       proper(fst(X)) -> fst(proper(X))
       proper(natsFrom(X)) -> natsFrom(proper(X))
       proper(s(X)) -> s(proper(X))
       proper(sel(X1,X2)) -> sel(proper(X1),proper(X2))
       proper(0()) -> ok(0())
       proper(nil()) -> ok(nil())
       proper(tail(X)) -> tail(proper(X))
       proper(take(X1,X2)) -> take(proper(X1),proper(X2))
       U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3))
       U12(ok(X1),ok(X2),ok(X3)) -> ok(U12(X1,X2,X3))
       snd(ok(X)) -> ok(snd(X))
       splitAt(ok(X1),ok(X2)) -> ok(splitAt(X1,X2))
       U21(ok(X1),ok(X2)) -> ok(U21(X1,X2))
       U22(ok(X1),ok(X2)) -> ok(U22(X1,X2))
       U31(ok(X1),ok(X2)) -> ok(U31(X1,X2))
       U32(ok(X1),ok(X2)) -> ok(U32(X1,X2))
       U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3))
       U42(ok(X1),ok(X2),ok(X3)) -> ok(U42(X1,X2,X3))
       head(ok(X)) -> ok(head(X))
       afterNth(ok(X1),ok(X2)) -> ok(afterNth(X1,X2))
       U51(ok(X1),ok(X2)) -> ok(U51(X1,X2))
       U52(ok(X1),ok(X2)) -> ok(U52(X1,X2))
       U61(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U61(X1,X2,X3,X4))
       U62(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U62(X1,X2,X3,X4))
       U63(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U63(X1,X2,X3,X4))
       U64(ok(X1),ok(X2)) -> ok(U64(X1,X2))
       pair(ok(X1),ok(X2)) -> ok(pair(X1,X2))
       cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
       U71(ok(X1),ok(X2)) -> ok(U71(X1,X2))
       U72(ok(X1),ok(X2)) -> ok(U72(X1,X2))
       U81(ok(X1),ok(X2),ok(X3)) -> ok(U81(X1,X2,X3))
       U82(ok(X1),ok(X2),ok(X3)) -> ok(U82(X1,X2,X3))
       fst(ok(X)) -> ok(fst(X))
       natsFrom(ok(X)) -> ok(natsFrom(X))
       s(ok(X)) -> ok(s(X))
       sel(ok(X1),ok(X2)) -> ok(sel(X1,X2))
       tail(ok(X)) -> ok(tail(X))
       take(ok(X1),ok(X2)) -> ok(take(X1,X2))
       top(mark(X)) -> top(proper(X))
       top(ok(X)) -> top(active(X))
      Open
      
      DPs:
       afterNth#(mark(X1),X2) -> afterNth#(X1,X2)
       afterNth#(X1,mark(X2)) -> afterNth#(X1,X2)
       afterNth#(ok(X1),ok(X2)) -> afterNth#(X1,X2)
      TRS:
       active(U11(tt(),N,XS)) -> mark(U12(tt(),N,XS))
       active(U12(tt(),N,XS)) -> mark(snd(splitAt(N,XS)))
       active(U21(tt(),X)) -> mark(U22(tt(),X))
       active(U22(tt(),X)) -> mark(X)
       active(U31(tt(),N)) -> mark(U32(tt(),N))
       active(U32(tt(),N)) -> mark(N)
       active(U41(tt(),N,XS)) -> mark(U42(tt(),N,XS))
       active(U42(tt(),N,XS)) -> mark(head(afterNth(N,XS)))
       active(U51(tt(),Y)) -> mark(U52(tt(),Y))
       active(U52(tt(),Y)) -> mark(Y)
       active(U61(tt(),N,X,XS)) -> mark(U62(tt(),N,X,XS))
       active(U62(tt(),N,X,XS)) -> mark(U63(tt(),N,X,XS))
       active(U63(tt(),N,X,XS)) -> mark(U64(splitAt(N,XS),X))
       active(U64(pair(YS,ZS),X)) -> mark(pair(cons(X,YS),ZS))
       active(U71(tt(),XS)) -> mark(U72(tt(),XS))
       active(U72(tt(),XS)) -> mark(XS)
       active(U81(tt(),N,XS)) -> mark(U82(tt(),N,XS))
       active(U82(tt(),N,XS)) -> mark(fst(splitAt(N,XS)))
       active(afterNth(N,XS)) -> mark(U11(tt(),N,XS))
       active(fst(pair(X,Y))) -> mark(U21(tt(),X))
       active(head(cons(N,XS))) -> mark(U31(tt(),N))
       active(natsFrom(N)) -> mark(cons(N,natsFrom(s(N))))
       active(sel(N,XS)) -> mark(U41(tt(),N,XS))
       active(snd(pair(X,Y))) -> mark(U51(tt(),Y))
       active(splitAt(0(),XS)) -> mark(pair(nil(),XS))
       active(splitAt(s(N),cons(X,XS))) -> mark(U61(tt(),N,X,XS))
       active(tail(cons(N,XS))) -> mark(U71(tt(),XS))
       active(take(N,XS)) -> mark(U81(tt(),N,XS))
       active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3)
       active(U12(X1,X2,X3)) -> U12(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(U21(X1,X2)) -> U21(active(X1),X2)
       active(U22(X1,X2)) -> U22(active(X1),X2)
       active(U31(X1,X2)) -> U31(active(X1),X2)
       active(U32(X1,X2)) -> U32(active(X1),X2)
       active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3)
       active(U42(X1,X2,X3)) -> U42(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(U51(X1,X2)) -> U51(active(X1),X2)
       active(U52(X1,X2)) -> U52(active(X1),X2)
       active(U61(X1,X2,X3,X4)) -> U61(active(X1),X2,X3,X4)
       active(U62(X1,X2,X3,X4)) -> U62(active(X1),X2,X3,X4)
       active(U63(X1,X2,X3,X4)) -> U63(active(X1),X2,X3,X4)
       active(U64(X1,X2)) -> U64(active(X1),X2)
       active(pair(X1,X2)) -> pair(active(X1),X2)
       active(pair(X1,X2)) -> pair(X1,active(X2))
       active(cons(X1,X2)) -> cons(active(X1),X2)
       active(U71(X1,X2)) -> U71(active(X1),X2)
       active(U72(X1,X2)) -> U72(active(X1),X2)
       active(U81(X1,X2,X3)) -> U81(active(X1),X2,X3)
       active(U82(X1,X2,X3)) -> U82(active(X1),X2,X3)
       active(fst(X)) -> fst(active(X))
       active(natsFrom(X)) -> natsFrom(active(X))
       active(s(X)) -> s(active(X))
       active(sel(X1,X2)) -> sel(active(X1),X2)
       active(sel(X1,X2)) -> sel(X1,active(X2))
       active(tail(X)) -> tail(active(X))
       active(take(X1,X2)) -> take(active(X1),X2)
       active(take(X1,X2)) -> take(X1,active(X2))
       U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3))
       U12(mark(X1),X2,X3) -> mark(U12(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))
       U21(mark(X1),X2) -> mark(U21(X1,X2))
       U22(mark(X1),X2) -> mark(U22(X1,X2))
       U31(mark(X1),X2) -> mark(U31(X1,X2))
       U32(mark(X1),X2) -> mark(U32(X1,X2))
       U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3))
       U42(mark(X1),X2,X3) -> mark(U42(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))
       U51(mark(X1),X2) -> mark(U51(X1,X2))
       U52(mark(X1),X2) -> mark(U52(X1,X2))
       U61(mark(X1),X2,X3,X4) -> mark(U61(X1,X2,X3,X4))
       U62(mark(X1),X2,X3,X4) -> mark(U62(X1,X2,X3,X4))
       U63(mark(X1),X2,X3,X4) -> mark(U63(X1,X2,X3,X4))
       U64(mark(X1),X2) -> mark(U64(X1,X2))
       pair(mark(X1),X2) -> mark(pair(X1,X2))
       pair(X1,mark(X2)) -> mark(pair(X1,X2))
       cons(mark(X1),X2) -> mark(cons(X1,X2))
       U71(mark(X1),X2) -> mark(U71(X1,X2))
       U72(mark(X1),X2) -> mark(U72(X1,X2))
       U81(mark(X1),X2,X3) -> mark(U81(X1,X2,X3))
       U82(mark(X1),X2,X3) -> mark(U82(X1,X2,X3))
       fst(mark(X)) -> mark(fst(X))
       natsFrom(mark(X)) -> mark(natsFrom(X))
       s(mark(X)) -> mark(s(X))
       sel(mark(X1),X2) -> mark(sel(X1,X2))
       sel(X1,mark(X2)) -> mark(sel(X1,X2))
       tail(mark(X)) -> mark(tail(X))
       take(mark(X1),X2) -> mark(take(X1,X2))
       take(X1,mark(X2)) -> mark(take(X1,X2))
       proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3))
       proper(tt()) -> ok(tt())
       proper(U12(X1,X2,X3)) -> U12(proper(X1),proper(X2),proper(X3))
       proper(snd(X)) -> snd(proper(X))
       proper(splitAt(X1,X2)) -> splitAt(proper(X1),proper(X2))
       proper(U21(X1,X2)) -> U21(proper(X1),proper(X2))
       proper(U22(X1,X2)) -> U22(proper(X1),proper(X2))
       proper(U31(X1,X2)) -> U31(proper(X1),proper(X2))
       proper(U32(X1,X2)) -> U32(proper(X1),proper(X2))
       proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3))
       proper(U42(X1,X2,X3)) -> U42(proper(X1),proper(X2),proper(X3))
       proper(head(X)) -> head(proper(X))
       proper(afterNth(X1,X2)) -> afterNth(proper(X1),proper(X2))
       proper(U51(X1,X2)) -> U51(proper(X1),proper(X2))
       proper(U52(X1,X2)) -> U52(proper(X1),proper(X2))
       proper(U61(X1,X2,X3,X4)) -> U61(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(U62(X1,X2,X3,X4)) -> U62(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(U63(X1,X2,X3,X4)) -> U63(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(U64(X1,X2)) -> U64(proper(X1),proper(X2))
       proper(pair(X1,X2)) -> pair(proper(X1),proper(X2))
       proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
       proper(U71(X1,X2)) -> U71(proper(X1),proper(X2))
       proper(U72(X1,X2)) -> U72(proper(X1),proper(X2))
       proper(U81(X1,X2,X3)) -> U81(proper(X1),proper(X2),proper(X3))
       proper(U82(X1,X2,X3)) -> U82(proper(X1),proper(X2),proper(X3))
       proper(fst(X)) -> fst(proper(X))
       proper(natsFrom(X)) -> natsFrom(proper(X))
       proper(s(X)) -> s(proper(X))
       proper(sel(X1,X2)) -> sel(proper(X1),proper(X2))
       proper(0()) -> ok(0())
       proper(nil()) -> ok(nil())
       proper(tail(X)) -> tail(proper(X))
       proper(take(X1,X2)) -> take(proper(X1),proper(X2))
       U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3))
       U12(ok(X1),ok(X2),ok(X3)) -> ok(U12(X1,X2,X3))
       snd(ok(X)) -> ok(snd(X))
       splitAt(ok(X1),ok(X2)) -> ok(splitAt(X1,X2))
       U21(ok(X1),ok(X2)) -> ok(U21(X1,X2))
       U22(ok(X1),ok(X2)) -> ok(U22(X1,X2))
       U31(ok(X1),ok(X2)) -> ok(U31(X1,X2))
       U32(ok(X1),ok(X2)) -> ok(U32(X1,X2))
       U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3))
       U42(ok(X1),ok(X2),ok(X3)) -> ok(U42(X1,X2,X3))
       head(ok(X)) -> ok(head(X))
       afterNth(ok(X1),ok(X2)) -> ok(afterNth(X1,X2))
       U51(ok(X1),ok(X2)) -> ok(U51(X1,X2))
       U52(ok(X1),ok(X2)) -> ok(U52(X1,X2))
       U61(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U61(X1,X2,X3,X4))
       U62(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U62(X1,X2,X3,X4))
       U63(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U63(X1,X2,X3,X4))
       U64(ok(X1),ok(X2)) -> ok(U64(X1,X2))
       pair(ok(X1),ok(X2)) -> ok(pair(X1,X2))
       cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
       U71(ok(X1),ok(X2)) -> ok(U71(X1,X2))
       U72(ok(X1),ok(X2)) -> ok(U72(X1,X2))
       U81(ok(X1),ok(X2),ok(X3)) -> ok(U81(X1,X2,X3))
       U82(ok(X1),ok(X2),ok(X3)) -> ok(U82(X1,X2,X3))
       fst(ok(X)) -> ok(fst(X))
       natsFrom(ok(X)) -> ok(natsFrom(X))
       s(ok(X)) -> ok(s(X))
       sel(ok(X1),ok(X2)) -> ok(sel(X1,X2))
       tail(ok(X)) -> ok(tail(X))
       take(ok(X1),ok(X2)) -> ok(take(X1,X2))
       top(mark(X)) -> top(proper(X))
       top(ok(X)) -> top(active(X))
      Open
      
      DPs:
       head#(mark(X)) -> head#(X)
       head#(ok(X)) -> head#(X)
      TRS:
       active(U11(tt(),N,XS)) -> mark(U12(tt(),N,XS))
       active(U12(tt(),N,XS)) -> mark(snd(splitAt(N,XS)))
       active(U21(tt(),X)) -> mark(U22(tt(),X))
       active(U22(tt(),X)) -> mark(X)
       active(U31(tt(),N)) -> mark(U32(tt(),N))
       active(U32(tt(),N)) -> mark(N)
       active(U41(tt(),N,XS)) -> mark(U42(tt(),N,XS))
       active(U42(tt(),N,XS)) -> mark(head(afterNth(N,XS)))
       active(U51(tt(),Y)) -> mark(U52(tt(),Y))
       active(U52(tt(),Y)) -> mark(Y)
       active(U61(tt(),N,X,XS)) -> mark(U62(tt(),N,X,XS))
       active(U62(tt(),N,X,XS)) -> mark(U63(tt(),N,X,XS))
       active(U63(tt(),N,X,XS)) -> mark(U64(splitAt(N,XS),X))
       active(U64(pair(YS,ZS),X)) -> mark(pair(cons(X,YS),ZS))
       active(U71(tt(),XS)) -> mark(U72(tt(),XS))
       active(U72(tt(),XS)) -> mark(XS)
       active(U81(tt(),N,XS)) -> mark(U82(tt(),N,XS))
       active(U82(tt(),N,XS)) -> mark(fst(splitAt(N,XS)))
       active(afterNth(N,XS)) -> mark(U11(tt(),N,XS))
       active(fst(pair(X,Y))) -> mark(U21(tt(),X))
       active(head(cons(N,XS))) -> mark(U31(tt(),N))
       active(natsFrom(N)) -> mark(cons(N,natsFrom(s(N))))
       active(sel(N,XS)) -> mark(U41(tt(),N,XS))
       active(snd(pair(X,Y))) -> mark(U51(tt(),Y))
       active(splitAt(0(),XS)) -> mark(pair(nil(),XS))
       active(splitAt(s(N),cons(X,XS))) -> mark(U61(tt(),N,X,XS))
       active(tail(cons(N,XS))) -> mark(U71(tt(),XS))
       active(take(N,XS)) -> mark(U81(tt(),N,XS))
       active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3)
       active(U12(X1,X2,X3)) -> U12(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(U21(X1,X2)) -> U21(active(X1),X2)
       active(U22(X1,X2)) -> U22(active(X1),X2)
       active(U31(X1,X2)) -> U31(active(X1),X2)
       active(U32(X1,X2)) -> U32(active(X1),X2)
       active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3)
       active(U42(X1,X2,X3)) -> U42(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(U51(X1,X2)) -> U51(active(X1),X2)
       active(U52(X1,X2)) -> U52(active(X1),X2)
       active(U61(X1,X2,X3,X4)) -> U61(active(X1),X2,X3,X4)
       active(U62(X1,X2,X3,X4)) -> U62(active(X1),X2,X3,X4)
       active(U63(X1,X2,X3,X4)) -> U63(active(X1),X2,X3,X4)
       active(U64(X1,X2)) -> U64(active(X1),X2)
       active(pair(X1,X2)) -> pair(active(X1),X2)
       active(pair(X1,X2)) -> pair(X1,active(X2))
       active(cons(X1,X2)) -> cons(active(X1),X2)
       active(U71(X1,X2)) -> U71(active(X1),X2)
       active(U72(X1,X2)) -> U72(active(X1),X2)
       active(U81(X1,X2,X3)) -> U81(active(X1),X2,X3)
       active(U82(X1,X2,X3)) -> U82(active(X1),X2,X3)
       active(fst(X)) -> fst(active(X))
       active(natsFrom(X)) -> natsFrom(active(X))
       active(s(X)) -> s(active(X))
       active(sel(X1,X2)) -> sel(active(X1),X2)
       active(sel(X1,X2)) -> sel(X1,active(X2))
       active(tail(X)) -> tail(active(X))
       active(take(X1,X2)) -> take(active(X1),X2)
       active(take(X1,X2)) -> take(X1,active(X2))
       U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3))
       U12(mark(X1),X2,X3) -> mark(U12(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))
       U21(mark(X1),X2) -> mark(U21(X1,X2))
       U22(mark(X1),X2) -> mark(U22(X1,X2))
       U31(mark(X1),X2) -> mark(U31(X1,X2))
       U32(mark(X1),X2) -> mark(U32(X1,X2))
       U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3))
       U42(mark(X1),X2,X3) -> mark(U42(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))
       U51(mark(X1),X2) -> mark(U51(X1,X2))
       U52(mark(X1),X2) -> mark(U52(X1,X2))
       U61(mark(X1),X2,X3,X4) -> mark(U61(X1,X2,X3,X4))
       U62(mark(X1),X2,X3,X4) -> mark(U62(X1,X2,X3,X4))
       U63(mark(X1),X2,X3,X4) -> mark(U63(X1,X2,X3,X4))
       U64(mark(X1),X2) -> mark(U64(X1,X2))
       pair(mark(X1),X2) -> mark(pair(X1,X2))
       pair(X1,mark(X2)) -> mark(pair(X1,X2))
       cons(mark(X1),X2) -> mark(cons(X1,X2))
       U71(mark(X1),X2) -> mark(U71(X1,X2))
       U72(mark(X1),X2) -> mark(U72(X1,X2))
       U81(mark(X1),X2,X3) -> mark(U81(X1,X2,X3))
       U82(mark(X1),X2,X3) -> mark(U82(X1,X2,X3))
       fst(mark(X)) -> mark(fst(X))
       natsFrom(mark(X)) -> mark(natsFrom(X))
       s(mark(X)) -> mark(s(X))
       sel(mark(X1),X2) -> mark(sel(X1,X2))
       sel(X1,mark(X2)) -> mark(sel(X1,X2))
       tail(mark(X)) -> mark(tail(X))
       take(mark(X1),X2) -> mark(take(X1,X2))
       take(X1,mark(X2)) -> mark(take(X1,X2))
       proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3))
       proper(tt()) -> ok(tt())
       proper(U12(X1,X2,X3)) -> U12(proper(X1),proper(X2),proper(X3))
       proper(snd(X)) -> snd(proper(X))
       proper(splitAt(X1,X2)) -> splitAt(proper(X1),proper(X2))
       proper(U21(X1,X2)) -> U21(proper(X1),proper(X2))
       proper(U22(X1,X2)) -> U22(proper(X1),proper(X2))
       proper(U31(X1,X2)) -> U31(proper(X1),proper(X2))
       proper(U32(X1,X2)) -> U32(proper(X1),proper(X2))
       proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3))
       proper(U42(X1,X2,X3)) -> U42(proper(X1),proper(X2),proper(X3))
       proper(head(X)) -> head(proper(X))
       proper(afterNth(X1,X2)) -> afterNth(proper(X1),proper(X2))
       proper(U51(X1,X2)) -> U51(proper(X1),proper(X2))
       proper(U52(X1,X2)) -> U52(proper(X1),proper(X2))
       proper(U61(X1,X2,X3,X4)) -> U61(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(U62(X1,X2,X3,X4)) -> U62(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(U63(X1,X2,X3,X4)) -> U63(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(U64(X1,X2)) -> U64(proper(X1),proper(X2))
       proper(pair(X1,X2)) -> pair(proper(X1),proper(X2))
       proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
       proper(U71(X1,X2)) -> U71(proper(X1),proper(X2))
       proper(U72(X1,X2)) -> U72(proper(X1),proper(X2))
       proper(U81(X1,X2,X3)) -> U81(proper(X1),proper(X2),proper(X3))
       proper(U82(X1,X2,X3)) -> U82(proper(X1),proper(X2),proper(X3))
       proper(fst(X)) -> fst(proper(X))
       proper(natsFrom(X)) -> natsFrom(proper(X))
       proper(s(X)) -> s(proper(X))
       proper(sel(X1,X2)) -> sel(proper(X1),proper(X2))
       proper(0()) -> ok(0())
       proper(nil()) -> ok(nil())
       proper(tail(X)) -> tail(proper(X))
       proper(take(X1,X2)) -> take(proper(X1),proper(X2))
       U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3))
       U12(ok(X1),ok(X2),ok(X3)) -> ok(U12(X1,X2,X3))
       snd(ok(X)) -> ok(snd(X))
       splitAt(ok(X1),ok(X2)) -> ok(splitAt(X1,X2))
       U21(ok(X1),ok(X2)) -> ok(U21(X1,X2))
       U22(ok(X1),ok(X2)) -> ok(U22(X1,X2))
       U31(ok(X1),ok(X2)) -> ok(U31(X1,X2))
       U32(ok(X1),ok(X2)) -> ok(U32(X1,X2))
       U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3))
       U42(ok(X1),ok(X2),ok(X3)) -> ok(U42(X1,X2,X3))
       head(ok(X)) -> ok(head(X))
       afterNth(ok(X1),ok(X2)) -> ok(afterNth(X1,X2))
       U51(ok(X1),ok(X2)) -> ok(U51(X1,X2))
       U52(ok(X1),ok(X2)) -> ok(U52(X1,X2))
       U61(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U61(X1,X2,X3,X4))
       U62(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U62(X1,X2,X3,X4))
       U63(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U63(X1,X2,X3,X4))
       U64(ok(X1),ok(X2)) -> ok(U64(X1,X2))
       pair(ok(X1),ok(X2)) -> ok(pair(X1,X2))
       cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
       U71(ok(X1),ok(X2)) -> ok(U71(X1,X2))
       U72(ok(X1),ok(X2)) -> ok(U72(X1,X2))
       U81(ok(X1),ok(X2),ok(X3)) -> ok(U81(X1,X2,X3))
       U82(ok(X1),ok(X2),ok(X3)) -> ok(U82(X1,X2,X3))
       fst(ok(X)) -> ok(fst(X))
       natsFrom(ok(X)) -> ok(natsFrom(X))
       s(ok(X)) -> ok(s(X))
       sel(ok(X1),ok(X2)) -> ok(sel(X1,X2))
       tail(ok(X)) -> ok(tail(X))
       take(ok(X1),ok(X2)) -> ok(take(X1,X2))
       top(mark(X)) -> top(proper(X))
       top(ok(X)) -> top(active(X))
      Open
      
      DPs:
       U42#(mark(X1),X2,X3) -> U42#(X1,X2,X3)
       U42#(ok(X1),ok(X2),ok(X3)) -> U42#(X1,X2,X3)
      TRS:
       active(U11(tt(),N,XS)) -> mark(U12(tt(),N,XS))
       active(U12(tt(),N,XS)) -> mark(snd(splitAt(N,XS)))
       active(U21(tt(),X)) -> mark(U22(tt(),X))
       active(U22(tt(),X)) -> mark(X)
       active(U31(tt(),N)) -> mark(U32(tt(),N))
       active(U32(tt(),N)) -> mark(N)
       active(U41(tt(),N,XS)) -> mark(U42(tt(),N,XS))
       active(U42(tt(),N,XS)) -> mark(head(afterNth(N,XS)))
       active(U51(tt(),Y)) -> mark(U52(tt(),Y))
       active(U52(tt(),Y)) -> mark(Y)
       active(U61(tt(),N,X,XS)) -> mark(U62(tt(),N,X,XS))
       active(U62(tt(),N,X,XS)) -> mark(U63(tt(),N,X,XS))
       active(U63(tt(),N,X,XS)) -> mark(U64(splitAt(N,XS),X))
       active(U64(pair(YS,ZS),X)) -> mark(pair(cons(X,YS),ZS))
       active(U71(tt(),XS)) -> mark(U72(tt(),XS))
       active(U72(tt(),XS)) -> mark(XS)
       active(U81(tt(),N,XS)) -> mark(U82(tt(),N,XS))
       active(U82(tt(),N,XS)) -> mark(fst(splitAt(N,XS)))
       active(afterNth(N,XS)) -> mark(U11(tt(),N,XS))
       active(fst(pair(X,Y))) -> mark(U21(tt(),X))
       active(head(cons(N,XS))) -> mark(U31(tt(),N))
       active(natsFrom(N)) -> mark(cons(N,natsFrom(s(N))))
       active(sel(N,XS)) -> mark(U41(tt(),N,XS))
       active(snd(pair(X,Y))) -> mark(U51(tt(),Y))
       active(splitAt(0(),XS)) -> mark(pair(nil(),XS))
       active(splitAt(s(N),cons(X,XS))) -> mark(U61(tt(),N,X,XS))
       active(tail(cons(N,XS))) -> mark(U71(tt(),XS))
       active(take(N,XS)) -> mark(U81(tt(),N,XS))
       active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3)
       active(U12(X1,X2,X3)) -> U12(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(U21(X1,X2)) -> U21(active(X1),X2)
       active(U22(X1,X2)) -> U22(active(X1),X2)
       active(U31(X1,X2)) -> U31(active(X1),X2)
       active(U32(X1,X2)) -> U32(active(X1),X2)
       active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3)
       active(U42(X1,X2,X3)) -> U42(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(U51(X1,X2)) -> U51(active(X1),X2)
       active(U52(X1,X2)) -> U52(active(X1),X2)
       active(U61(X1,X2,X3,X4)) -> U61(active(X1),X2,X3,X4)
       active(U62(X1,X2,X3,X4)) -> U62(active(X1),X2,X3,X4)
       active(U63(X1,X2,X3,X4)) -> U63(active(X1),X2,X3,X4)
       active(U64(X1,X2)) -> U64(active(X1),X2)
       active(pair(X1,X2)) -> pair(active(X1),X2)
       active(pair(X1,X2)) -> pair(X1,active(X2))
       active(cons(X1,X2)) -> cons(active(X1),X2)
       active(U71(X1,X2)) -> U71(active(X1),X2)
       active(U72(X1,X2)) -> U72(active(X1),X2)
       active(U81(X1,X2,X3)) -> U81(active(X1),X2,X3)
       active(U82(X1,X2,X3)) -> U82(active(X1),X2,X3)
       active(fst(X)) -> fst(active(X))
       active(natsFrom(X)) -> natsFrom(active(X))
       active(s(X)) -> s(active(X))
       active(sel(X1,X2)) -> sel(active(X1),X2)
       active(sel(X1,X2)) -> sel(X1,active(X2))
       active(tail(X)) -> tail(active(X))
       active(take(X1,X2)) -> take(active(X1),X2)
       active(take(X1,X2)) -> take(X1,active(X2))
       U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3))
       U12(mark(X1),X2,X3) -> mark(U12(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))
       U21(mark(X1),X2) -> mark(U21(X1,X2))
       U22(mark(X1),X2) -> mark(U22(X1,X2))
       U31(mark(X1),X2) -> mark(U31(X1,X2))
       U32(mark(X1),X2) -> mark(U32(X1,X2))
       U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3))
       U42(mark(X1),X2,X3) -> mark(U42(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))
       U51(mark(X1),X2) -> mark(U51(X1,X2))
       U52(mark(X1),X2) -> mark(U52(X1,X2))
       U61(mark(X1),X2,X3,X4) -> mark(U61(X1,X2,X3,X4))
       U62(mark(X1),X2,X3,X4) -> mark(U62(X1,X2,X3,X4))
       U63(mark(X1),X2,X3,X4) -> mark(U63(X1,X2,X3,X4))
       U64(mark(X1),X2) -> mark(U64(X1,X2))
       pair(mark(X1),X2) -> mark(pair(X1,X2))
       pair(X1,mark(X2)) -> mark(pair(X1,X2))
       cons(mark(X1),X2) -> mark(cons(X1,X2))
       U71(mark(X1),X2) -> mark(U71(X1,X2))
       U72(mark(X1),X2) -> mark(U72(X1,X2))
       U81(mark(X1),X2,X3) -> mark(U81(X1,X2,X3))
       U82(mark(X1),X2,X3) -> mark(U82(X1,X2,X3))
       fst(mark(X)) -> mark(fst(X))
       natsFrom(mark(X)) -> mark(natsFrom(X))
       s(mark(X)) -> mark(s(X))
       sel(mark(X1),X2) -> mark(sel(X1,X2))
       sel(X1,mark(X2)) -> mark(sel(X1,X2))
       tail(mark(X)) -> mark(tail(X))
       take(mark(X1),X2) -> mark(take(X1,X2))
       take(X1,mark(X2)) -> mark(take(X1,X2))
       proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3))
       proper(tt()) -> ok(tt())
       proper(U12(X1,X2,X3)) -> U12(proper(X1),proper(X2),proper(X3))
       proper(snd(X)) -> snd(proper(X))
       proper(splitAt(X1,X2)) -> splitAt(proper(X1),proper(X2))
       proper(U21(X1,X2)) -> U21(proper(X1),proper(X2))
       proper(U22(X1,X2)) -> U22(proper(X1),proper(X2))
       proper(U31(X1,X2)) -> U31(proper(X1),proper(X2))
       proper(U32(X1,X2)) -> U32(proper(X1),proper(X2))
       proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3))
       proper(U42(X1,X2,X3)) -> U42(proper(X1),proper(X2),proper(X3))
       proper(head(X)) -> head(proper(X))
       proper(afterNth(X1,X2)) -> afterNth(proper(X1),proper(X2))
       proper(U51(X1,X2)) -> U51(proper(X1),proper(X2))
       proper(U52(X1,X2)) -> U52(proper(X1),proper(X2))
       proper(U61(X1,X2,X3,X4)) -> U61(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(U62(X1,X2,X3,X4)) -> U62(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(U63(X1,X2,X3,X4)) -> U63(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(U64(X1,X2)) -> U64(proper(X1),proper(X2))
       proper(pair(X1,X2)) -> pair(proper(X1),proper(X2))
       proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
       proper(U71(X1,X2)) -> U71(proper(X1),proper(X2))
       proper(U72(X1,X2)) -> U72(proper(X1),proper(X2))
       proper(U81(X1,X2,X3)) -> U81(proper(X1),proper(X2),proper(X3))
       proper(U82(X1,X2,X3)) -> U82(proper(X1),proper(X2),proper(X3))
       proper(fst(X)) -> fst(proper(X))
       proper(natsFrom(X)) -> natsFrom(proper(X))
       proper(s(X)) -> s(proper(X))
       proper(sel(X1,X2)) -> sel(proper(X1),proper(X2))
       proper(0()) -> ok(0())
       proper(nil()) -> ok(nil())
       proper(tail(X)) -> tail(proper(X))
       proper(take(X1,X2)) -> take(proper(X1),proper(X2))
       U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3))
       U12(ok(X1),ok(X2),ok(X3)) -> ok(U12(X1,X2,X3))
       snd(ok(X)) -> ok(snd(X))
       splitAt(ok(X1),ok(X2)) -> ok(splitAt(X1,X2))
       U21(ok(X1),ok(X2)) -> ok(U21(X1,X2))
       U22(ok(X1),ok(X2)) -> ok(U22(X1,X2))
       U31(ok(X1),ok(X2)) -> ok(U31(X1,X2))
       U32(ok(X1),ok(X2)) -> ok(U32(X1,X2))
       U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3))
       U42(ok(X1),ok(X2),ok(X3)) -> ok(U42(X1,X2,X3))
       head(ok(X)) -> ok(head(X))
       afterNth(ok(X1),ok(X2)) -> ok(afterNth(X1,X2))
       U51(ok(X1),ok(X2)) -> ok(U51(X1,X2))
       U52(ok(X1),ok(X2)) -> ok(U52(X1,X2))
       U61(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U61(X1,X2,X3,X4))
       U62(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U62(X1,X2,X3,X4))
       U63(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U63(X1,X2,X3,X4))
       U64(ok(X1),ok(X2)) -> ok(U64(X1,X2))
       pair(ok(X1),ok(X2)) -> ok(pair(X1,X2))
       cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
       U71(ok(X1),ok(X2)) -> ok(U71(X1,X2))
       U72(ok(X1),ok(X2)) -> ok(U72(X1,X2))
       U81(ok(X1),ok(X2),ok(X3)) -> ok(U81(X1,X2,X3))
       U82(ok(X1),ok(X2),ok(X3)) -> ok(U82(X1,X2,X3))
       fst(ok(X)) -> ok(fst(X))
       natsFrom(ok(X)) -> ok(natsFrom(X))
       s(ok(X)) -> ok(s(X))
       sel(ok(X1),ok(X2)) -> ok(sel(X1,X2))
       tail(ok(X)) -> ok(tail(X))
       take(ok(X1),ok(X2)) -> ok(take(X1,X2))
       top(mark(X)) -> top(proper(X))
       top(ok(X)) -> top(active(X))
      Open
      
      DPs:
       U41#(mark(X1),X2,X3) -> U41#(X1,X2,X3)
       U41#(ok(X1),ok(X2),ok(X3)) -> U41#(X1,X2,X3)
      TRS:
       active(U11(tt(),N,XS)) -> mark(U12(tt(),N,XS))
       active(U12(tt(),N,XS)) -> mark(snd(splitAt(N,XS)))
       active(U21(tt(),X)) -> mark(U22(tt(),X))
       active(U22(tt(),X)) -> mark(X)
       active(U31(tt(),N)) -> mark(U32(tt(),N))
       active(U32(tt(),N)) -> mark(N)
       active(U41(tt(),N,XS)) -> mark(U42(tt(),N,XS))
       active(U42(tt(),N,XS)) -> mark(head(afterNth(N,XS)))
       active(U51(tt(),Y)) -> mark(U52(tt(),Y))
       active(U52(tt(),Y)) -> mark(Y)
       active(U61(tt(),N,X,XS)) -> mark(U62(tt(),N,X,XS))
       active(U62(tt(),N,X,XS)) -> mark(U63(tt(),N,X,XS))
       active(U63(tt(),N,X,XS)) -> mark(U64(splitAt(N,XS),X))
       active(U64(pair(YS,ZS),X)) -> mark(pair(cons(X,YS),ZS))
       active(U71(tt(),XS)) -> mark(U72(tt(),XS))
       active(U72(tt(),XS)) -> mark(XS)
       active(U81(tt(),N,XS)) -> mark(U82(tt(),N,XS))
       active(U82(tt(),N,XS)) -> mark(fst(splitAt(N,XS)))
       active(afterNth(N,XS)) -> mark(U11(tt(),N,XS))
       active(fst(pair(X,Y))) -> mark(U21(tt(),X))
       active(head(cons(N,XS))) -> mark(U31(tt(),N))
       active(natsFrom(N)) -> mark(cons(N,natsFrom(s(N))))
       active(sel(N,XS)) -> mark(U41(tt(),N,XS))
       active(snd(pair(X,Y))) -> mark(U51(tt(),Y))
       active(splitAt(0(),XS)) -> mark(pair(nil(),XS))
       active(splitAt(s(N),cons(X,XS))) -> mark(U61(tt(),N,X,XS))
       active(tail(cons(N,XS))) -> mark(U71(tt(),XS))
       active(take(N,XS)) -> mark(U81(tt(),N,XS))
       active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3)
       active(U12(X1,X2,X3)) -> U12(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(U21(X1,X2)) -> U21(active(X1),X2)
       active(U22(X1,X2)) -> U22(active(X1),X2)
       active(U31(X1,X2)) -> U31(active(X1),X2)
       active(U32(X1,X2)) -> U32(active(X1),X2)
       active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3)
       active(U42(X1,X2,X3)) -> U42(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(U51(X1,X2)) -> U51(active(X1),X2)
       active(U52(X1,X2)) -> U52(active(X1),X2)
       active(U61(X1,X2,X3,X4)) -> U61(active(X1),X2,X3,X4)
       active(U62(X1,X2,X3,X4)) -> U62(active(X1),X2,X3,X4)
       active(U63(X1,X2,X3,X4)) -> U63(active(X1),X2,X3,X4)
       active(U64(X1,X2)) -> U64(active(X1),X2)
       active(pair(X1,X2)) -> pair(active(X1),X2)
       active(pair(X1,X2)) -> pair(X1,active(X2))
       active(cons(X1,X2)) -> cons(active(X1),X2)
       active(U71(X1,X2)) -> U71(active(X1),X2)
       active(U72(X1,X2)) -> U72(active(X1),X2)
       active(U81(X1,X2,X3)) -> U81(active(X1),X2,X3)
       active(U82(X1,X2,X3)) -> U82(active(X1),X2,X3)
       active(fst(X)) -> fst(active(X))
       active(natsFrom(X)) -> natsFrom(active(X))
       active(s(X)) -> s(active(X))
       active(sel(X1,X2)) -> sel(active(X1),X2)
       active(sel(X1,X2)) -> sel(X1,active(X2))
       active(tail(X)) -> tail(active(X))
       active(take(X1,X2)) -> take(active(X1),X2)
       active(take(X1,X2)) -> take(X1,active(X2))
       U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3))
       U12(mark(X1),X2,X3) -> mark(U12(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))
       U21(mark(X1),X2) -> mark(U21(X1,X2))
       U22(mark(X1),X2) -> mark(U22(X1,X2))
       U31(mark(X1),X2) -> mark(U31(X1,X2))
       U32(mark(X1),X2) -> mark(U32(X1,X2))
       U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3))
       U42(mark(X1),X2,X3) -> mark(U42(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))
       U51(mark(X1),X2) -> mark(U51(X1,X2))
       U52(mark(X1),X2) -> mark(U52(X1,X2))
       U61(mark(X1),X2,X3,X4) -> mark(U61(X1,X2,X3,X4))
       U62(mark(X1),X2,X3,X4) -> mark(U62(X1,X2,X3,X4))
       U63(mark(X1),X2,X3,X4) -> mark(U63(X1,X2,X3,X4))
       U64(mark(X1),X2) -> mark(U64(X1,X2))
       pair(mark(X1),X2) -> mark(pair(X1,X2))
       pair(X1,mark(X2)) -> mark(pair(X1,X2))
       cons(mark(X1),X2) -> mark(cons(X1,X2))
       U71(mark(X1),X2) -> mark(U71(X1,X2))
       U72(mark(X1),X2) -> mark(U72(X1,X2))
       U81(mark(X1),X2,X3) -> mark(U81(X1,X2,X3))
       U82(mark(X1),X2,X3) -> mark(U82(X1,X2,X3))
       fst(mark(X)) -> mark(fst(X))
       natsFrom(mark(X)) -> mark(natsFrom(X))
       s(mark(X)) -> mark(s(X))
       sel(mark(X1),X2) -> mark(sel(X1,X2))
       sel(X1,mark(X2)) -> mark(sel(X1,X2))
       tail(mark(X)) -> mark(tail(X))
       take(mark(X1),X2) -> mark(take(X1,X2))
       take(X1,mark(X2)) -> mark(take(X1,X2))
       proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3))
       proper(tt()) -> ok(tt())
       proper(U12(X1,X2,X3)) -> U12(proper(X1),proper(X2),proper(X3))
       proper(snd(X)) -> snd(proper(X))
       proper(splitAt(X1,X2)) -> splitAt(proper(X1),proper(X2))
       proper(U21(X1,X2)) -> U21(proper(X1),proper(X2))
       proper(U22(X1,X2)) -> U22(proper(X1),proper(X2))
       proper(U31(X1,X2)) -> U31(proper(X1),proper(X2))
       proper(U32(X1,X2)) -> U32(proper(X1),proper(X2))
       proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3))
       proper(U42(X1,X2,X3)) -> U42(proper(X1),proper(X2),proper(X3))
       proper(head(X)) -> head(proper(X))
       proper(afterNth(X1,X2)) -> afterNth(proper(X1),proper(X2))
       proper(U51(X1,X2)) -> U51(proper(X1),proper(X2))
       proper(U52(X1,X2)) -> U52(proper(X1),proper(X2))
       proper(U61(X1,X2,X3,X4)) -> U61(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(U62(X1,X2,X3,X4)) -> U62(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(U63(X1,X2,X3,X4)) -> U63(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(U64(X1,X2)) -> U64(proper(X1),proper(X2))
       proper(pair(X1,X2)) -> pair(proper(X1),proper(X2))
       proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
       proper(U71(X1,X2)) -> U71(proper(X1),proper(X2))
       proper(U72(X1,X2)) -> U72(proper(X1),proper(X2))
       proper(U81(X1,X2,X3)) -> U81(proper(X1),proper(X2),proper(X3))
       proper(U82(X1,X2,X3)) -> U82(proper(X1),proper(X2),proper(X3))
       proper(fst(X)) -> fst(proper(X))
       proper(natsFrom(X)) -> natsFrom(proper(X))
       proper(s(X)) -> s(proper(X))
       proper(sel(X1,X2)) -> sel(proper(X1),proper(X2))
       proper(0()) -> ok(0())
       proper(nil()) -> ok(nil())
       proper(tail(X)) -> tail(proper(X))
       proper(take(X1,X2)) -> take(proper(X1),proper(X2))
       U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3))
       U12(ok(X1),ok(X2),ok(X3)) -> ok(U12(X1,X2,X3))
       snd(ok(X)) -> ok(snd(X))
       splitAt(ok(X1),ok(X2)) -> ok(splitAt(X1,X2))
       U21(ok(X1),ok(X2)) -> ok(U21(X1,X2))
       U22(ok(X1),ok(X2)) -> ok(U22(X1,X2))
       U31(ok(X1),ok(X2)) -> ok(U31(X1,X2))
       U32(ok(X1),ok(X2)) -> ok(U32(X1,X2))
       U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3))
       U42(ok(X1),ok(X2),ok(X3)) -> ok(U42(X1,X2,X3))
       head(ok(X)) -> ok(head(X))
       afterNth(ok(X1),ok(X2)) -> ok(afterNth(X1,X2))
       U51(ok(X1),ok(X2)) -> ok(U51(X1,X2))
       U52(ok(X1),ok(X2)) -> ok(U52(X1,X2))
       U61(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U61(X1,X2,X3,X4))
       U62(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U62(X1,X2,X3,X4))
       U63(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U63(X1,X2,X3,X4))
       U64(ok(X1),ok(X2)) -> ok(U64(X1,X2))
       pair(ok(X1),ok(X2)) -> ok(pair(X1,X2))
       cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
       U71(ok(X1),ok(X2)) -> ok(U71(X1,X2))
       U72(ok(X1),ok(X2)) -> ok(U72(X1,X2))
       U81(ok(X1),ok(X2),ok(X3)) -> ok(U81(X1,X2,X3))
       U82(ok(X1),ok(X2),ok(X3)) -> ok(U82(X1,X2,X3))
       fst(ok(X)) -> ok(fst(X))
       natsFrom(ok(X)) -> ok(natsFrom(X))
       s(ok(X)) -> ok(s(X))
       sel(ok(X1),ok(X2)) -> ok(sel(X1,X2))
       tail(ok(X)) -> ok(tail(X))
       take(ok(X1),ok(X2)) -> ok(take(X1,X2))
       top(mark(X)) -> top(proper(X))
       top(ok(X)) -> top(active(X))
      Open
      
      DPs:
       U32#(mark(X1),X2) -> U32#(X1,X2)
       U32#(ok(X1),ok(X2)) -> U32#(X1,X2)
      TRS:
       active(U11(tt(),N,XS)) -> mark(U12(tt(),N,XS))
       active(U12(tt(),N,XS)) -> mark(snd(splitAt(N,XS)))
       active(U21(tt(),X)) -> mark(U22(tt(),X))
       active(U22(tt(),X)) -> mark(X)
       active(U31(tt(),N)) -> mark(U32(tt(),N))
       active(U32(tt(),N)) -> mark(N)
       active(U41(tt(),N,XS)) -> mark(U42(tt(),N,XS))
       active(U42(tt(),N,XS)) -> mark(head(afterNth(N,XS)))
       active(U51(tt(),Y)) -> mark(U52(tt(),Y))
       active(U52(tt(),Y)) -> mark(Y)
       active(U61(tt(),N,X,XS)) -> mark(U62(tt(),N,X,XS))
       active(U62(tt(),N,X,XS)) -> mark(U63(tt(),N,X,XS))
       active(U63(tt(),N,X,XS)) -> mark(U64(splitAt(N,XS),X))
       active(U64(pair(YS,ZS),X)) -> mark(pair(cons(X,YS),ZS))
       active(U71(tt(),XS)) -> mark(U72(tt(),XS))
       active(U72(tt(),XS)) -> mark(XS)
       active(U81(tt(),N,XS)) -> mark(U82(tt(),N,XS))
       active(U82(tt(),N,XS)) -> mark(fst(splitAt(N,XS)))
       active(afterNth(N,XS)) -> mark(U11(tt(),N,XS))
       active(fst(pair(X,Y))) -> mark(U21(tt(),X))
       active(head(cons(N,XS))) -> mark(U31(tt(),N))
       active(natsFrom(N)) -> mark(cons(N,natsFrom(s(N))))
       active(sel(N,XS)) -> mark(U41(tt(),N,XS))
       active(snd(pair(X,Y))) -> mark(U51(tt(),Y))
       active(splitAt(0(),XS)) -> mark(pair(nil(),XS))
       active(splitAt(s(N),cons(X,XS))) -> mark(U61(tt(),N,X,XS))
       active(tail(cons(N,XS))) -> mark(U71(tt(),XS))
       active(take(N,XS)) -> mark(U81(tt(),N,XS))
       active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3)
       active(U12(X1,X2,X3)) -> U12(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(U21(X1,X2)) -> U21(active(X1),X2)
       active(U22(X1,X2)) -> U22(active(X1),X2)
       active(U31(X1,X2)) -> U31(active(X1),X2)
       active(U32(X1,X2)) -> U32(active(X1),X2)
       active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3)
       active(U42(X1,X2,X3)) -> U42(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(U51(X1,X2)) -> U51(active(X1),X2)
       active(U52(X1,X2)) -> U52(active(X1),X2)
       active(U61(X1,X2,X3,X4)) -> U61(active(X1),X2,X3,X4)
       active(U62(X1,X2,X3,X4)) -> U62(active(X1),X2,X3,X4)
       active(U63(X1,X2,X3,X4)) -> U63(active(X1),X2,X3,X4)
       active(U64(X1,X2)) -> U64(active(X1),X2)
       active(pair(X1,X2)) -> pair(active(X1),X2)
       active(pair(X1,X2)) -> pair(X1,active(X2))
       active(cons(X1,X2)) -> cons(active(X1),X2)
       active(U71(X1,X2)) -> U71(active(X1),X2)
       active(U72(X1,X2)) -> U72(active(X1),X2)
       active(U81(X1,X2,X3)) -> U81(active(X1),X2,X3)
       active(U82(X1,X2,X3)) -> U82(active(X1),X2,X3)
       active(fst(X)) -> fst(active(X))
       active(natsFrom(X)) -> natsFrom(active(X))
       active(s(X)) -> s(active(X))
       active(sel(X1,X2)) -> sel(active(X1),X2)
       active(sel(X1,X2)) -> sel(X1,active(X2))
       active(tail(X)) -> tail(active(X))
       active(take(X1,X2)) -> take(active(X1),X2)
       active(take(X1,X2)) -> take(X1,active(X2))
       U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3))
       U12(mark(X1),X2,X3) -> mark(U12(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))
       U21(mark(X1),X2) -> mark(U21(X1,X2))
       U22(mark(X1),X2) -> mark(U22(X1,X2))
       U31(mark(X1),X2) -> mark(U31(X1,X2))
       U32(mark(X1),X2) -> mark(U32(X1,X2))
       U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3))
       U42(mark(X1),X2,X3) -> mark(U42(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))
       U51(mark(X1),X2) -> mark(U51(X1,X2))
       U52(mark(X1),X2) -> mark(U52(X1,X2))
       U61(mark(X1),X2,X3,X4) -> mark(U61(X1,X2,X3,X4))
       U62(mark(X1),X2,X3,X4) -> mark(U62(X1,X2,X3,X4))
       U63(mark(X1),X2,X3,X4) -> mark(U63(X1,X2,X3,X4))
       U64(mark(X1),X2) -> mark(U64(X1,X2))
       pair(mark(X1),X2) -> mark(pair(X1,X2))
       pair(X1,mark(X2)) -> mark(pair(X1,X2))
       cons(mark(X1),X2) -> mark(cons(X1,X2))
       U71(mark(X1),X2) -> mark(U71(X1,X2))
       U72(mark(X1),X2) -> mark(U72(X1,X2))
       U81(mark(X1),X2,X3) -> mark(U81(X1,X2,X3))
       U82(mark(X1),X2,X3) -> mark(U82(X1,X2,X3))
       fst(mark(X)) -> mark(fst(X))
       natsFrom(mark(X)) -> mark(natsFrom(X))
       s(mark(X)) -> mark(s(X))
       sel(mark(X1),X2) -> mark(sel(X1,X2))
       sel(X1,mark(X2)) -> mark(sel(X1,X2))
       tail(mark(X)) -> mark(tail(X))
       take(mark(X1),X2) -> mark(take(X1,X2))
       take(X1,mark(X2)) -> mark(take(X1,X2))
       proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3))
       proper(tt()) -> ok(tt())
       proper(U12(X1,X2,X3)) -> U12(proper(X1),proper(X2),proper(X3))
       proper(snd(X)) -> snd(proper(X))
       proper(splitAt(X1,X2)) -> splitAt(proper(X1),proper(X2))
       proper(U21(X1,X2)) -> U21(proper(X1),proper(X2))
       proper(U22(X1,X2)) -> U22(proper(X1),proper(X2))
       proper(U31(X1,X2)) -> U31(proper(X1),proper(X2))
       proper(U32(X1,X2)) -> U32(proper(X1),proper(X2))
       proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3))
       proper(U42(X1,X2,X3)) -> U42(proper(X1),proper(X2),proper(X3))
       proper(head(X)) -> head(proper(X))
       proper(afterNth(X1,X2)) -> afterNth(proper(X1),proper(X2))
       proper(U51(X1,X2)) -> U51(proper(X1),proper(X2))
       proper(U52(X1,X2)) -> U52(proper(X1),proper(X2))
       proper(U61(X1,X2,X3,X4)) -> U61(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(U62(X1,X2,X3,X4)) -> U62(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(U63(X1,X2,X3,X4)) -> U63(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(U64(X1,X2)) -> U64(proper(X1),proper(X2))
       proper(pair(X1,X2)) -> pair(proper(X1),proper(X2))
       proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
       proper(U71(X1,X2)) -> U71(proper(X1),proper(X2))
       proper(U72(X1,X2)) -> U72(proper(X1),proper(X2))
       proper(U81(X1,X2,X3)) -> U81(proper(X1),proper(X2),proper(X3))
       proper(U82(X1,X2,X3)) -> U82(proper(X1),proper(X2),proper(X3))
       proper(fst(X)) -> fst(proper(X))
       proper(natsFrom(X)) -> natsFrom(proper(X))
       proper(s(X)) -> s(proper(X))
       proper(sel(X1,X2)) -> sel(proper(X1),proper(X2))
       proper(0()) -> ok(0())
       proper(nil()) -> ok(nil())
       proper(tail(X)) -> tail(proper(X))
       proper(take(X1,X2)) -> take(proper(X1),proper(X2))
       U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3))
       U12(ok(X1),ok(X2),ok(X3)) -> ok(U12(X1,X2,X3))
       snd(ok(X)) -> ok(snd(X))
       splitAt(ok(X1),ok(X2)) -> ok(splitAt(X1,X2))
       U21(ok(X1),ok(X2)) -> ok(U21(X1,X2))
       U22(ok(X1),ok(X2)) -> ok(U22(X1,X2))
       U31(ok(X1),ok(X2)) -> ok(U31(X1,X2))
       U32(ok(X1),ok(X2)) -> ok(U32(X1,X2))
       U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3))
       U42(ok(X1),ok(X2),ok(X3)) -> ok(U42(X1,X2,X3))
       head(ok(X)) -> ok(head(X))
       afterNth(ok(X1),ok(X2)) -> ok(afterNth(X1,X2))
       U51(ok(X1),ok(X2)) -> ok(U51(X1,X2))
       U52(ok(X1),ok(X2)) -> ok(U52(X1,X2))
       U61(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U61(X1,X2,X3,X4))
       U62(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U62(X1,X2,X3,X4))
       U63(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U63(X1,X2,X3,X4))
       U64(ok(X1),ok(X2)) -> ok(U64(X1,X2))
       pair(ok(X1),ok(X2)) -> ok(pair(X1,X2))
       cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
       U71(ok(X1),ok(X2)) -> ok(U71(X1,X2))
       U72(ok(X1),ok(X2)) -> ok(U72(X1,X2))
       U81(ok(X1),ok(X2),ok(X3)) -> ok(U81(X1,X2,X3))
       U82(ok(X1),ok(X2),ok(X3)) -> ok(U82(X1,X2,X3))
       fst(ok(X)) -> ok(fst(X))
       natsFrom(ok(X)) -> ok(natsFrom(X))
       s(ok(X)) -> ok(s(X))
       sel(ok(X1),ok(X2)) -> ok(sel(X1,X2))
       tail(ok(X)) -> ok(tail(X))
       take(ok(X1),ok(X2)) -> ok(take(X1,X2))
       top(mark(X)) -> top(proper(X))
       top(ok(X)) -> top(active(X))
      Open
      
      DPs:
       U31#(mark(X1),X2) -> U31#(X1,X2)
       U31#(ok(X1),ok(X2)) -> U31#(X1,X2)
      TRS:
       active(U11(tt(),N,XS)) -> mark(U12(tt(),N,XS))
       active(U12(tt(),N,XS)) -> mark(snd(splitAt(N,XS)))
       active(U21(tt(),X)) -> mark(U22(tt(),X))
       active(U22(tt(),X)) -> mark(X)
       active(U31(tt(),N)) -> mark(U32(tt(),N))
       active(U32(tt(),N)) -> mark(N)
       active(U41(tt(),N,XS)) -> mark(U42(tt(),N,XS))
       active(U42(tt(),N,XS)) -> mark(head(afterNth(N,XS)))
       active(U51(tt(),Y)) -> mark(U52(tt(),Y))
       active(U52(tt(),Y)) -> mark(Y)
       active(U61(tt(),N,X,XS)) -> mark(U62(tt(),N,X,XS))
       active(U62(tt(),N,X,XS)) -> mark(U63(tt(),N,X,XS))
       active(U63(tt(),N,X,XS)) -> mark(U64(splitAt(N,XS),X))
       active(U64(pair(YS,ZS),X)) -> mark(pair(cons(X,YS),ZS))
       active(U71(tt(),XS)) -> mark(U72(tt(),XS))
       active(U72(tt(),XS)) -> mark(XS)
       active(U81(tt(),N,XS)) -> mark(U82(tt(),N,XS))
       active(U82(tt(),N,XS)) -> mark(fst(splitAt(N,XS)))
       active(afterNth(N,XS)) -> mark(U11(tt(),N,XS))
       active(fst(pair(X,Y))) -> mark(U21(tt(),X))
       active(head(cons(N,XS))) -> mark(U31(tt(),N))
       active(natsFrom(N)) -> mark(cons(N,natsFrom(s(N))))
       active(sel(N,XS)) -> mark(U41(tt(),N,XS))
       active(snd(pair(X,Y))) -> mark(U51(tt(),Y))
       active(splitAt(0(),XS)) -> mark(pair(nil(),XS))
       active(splitAt(s(N),cons(X,XS))) -> mark(U61(tt(),N,X,XS))
       active(tail(cons(N,XS))) -> mark(U71(tt(),XS))
       active(take(N,XS)) -> mark(U81(tt(),N,XS))
       active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3)
       active(U12(X1,X2,X3)) -> U12(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(U21(X1,X2)) -> U21(active(X1),X2)
       active(U22(X1,X2)) -> U22(active(X1),X2)
       active(U31(X1,X2)) -> U31(active(X1),X2)
       active(U32(X1,X2)) -> U32(active(X1),X2)
       active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3)
       active(U42(X1,X2,X3)) -> U42(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(U51(X1,X2)) -> U51(active(X1),X2)
       active(U52(X1,X2)) -> U52(active(X1),X2)
       active(U61(X1,X2,X3,X4)) -> U61(active(X1),X2,X3,X4)
       active(U62(X1,X2,X3,X4)) -> U62(active(X1),X2,X3,X4)
       active(U63(X1,X2,X3,X4)) -> U63(active(X1),X2,X3,X4)
       active(U64(X1,X2)) -> U64(active(X1),X2)
       active(pair(X1,X2)) -> pair(active(X1),X2)
       active(pair(X1,X2)) -> pair(X1,active(X2))
       active(cons(X1,X2)) -> cons(active(X1),X2)
       active(U71(X1,X2)) -> U71(active(X1),X2)
       active(U72(X1,X2)) -> U72(active(X1),X2)
       active(U81(X1,X2,X3)) -> U81(active(X1),X2,X3)
       active(U82(X1,X2,X3)) -> U82(active(X1),X2,X3)
       active(fst(X)) -> fst(active(X))
       active(natsFrom(X)) -> natsFrom(active(X))
       active(s(X)) -> s(active(X))
       active(sel(X1,X2)) -> sel(active(X1),X2)
       active(sel(X1,X2)) -> sel(X1,active(X2))
       active(tail(X)) -> tail(active(X))
       active(take(X1,X2)) -> take(active(X1),X2)
       active(take(X1,X2)) -> take(X1,active(X2))
       U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3))
       U12(mark(X1),X2,X3) -> mark(U12(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))
       U21(mark(X1),X2) -> mark(U21(X1,X2))
       U22(mark(X1),X2) -> mark(U22(X1,X2))
       U31(mark(X1),X2) -> mark(U31(X1,X2))
       U32(mark(X1),X2) -> mark(U32(X1,X2))
       U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3))
       U42(mark(X1),X2,X3) -> mark(U42(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))
       U51(mark(X1),X2) -> mark(U51(X1,X2))
       U52(mark(X1),X2) -> mark(U52(X1,X2))
       U61(mark(X1),X2,X3,X4) -> mark(U61(X1,X2,X3,X4))
       U62(mark(X1),X2,X3,X4) -> mark(U62(X1,X2,X3,X4))
       U63(mark(X1),X2,X3,X4) -> mark(U63(X1,X2,X3,X4))
       U64(mark(X1),X2) -> mark(U64(X1,X2))
       pair(mark(X1),X2) -> mark(pair(X1,X2))
       pair(X1,mark(X2)) -> mark(pair(X1,X2))
       cons(mark(X1),X2) -> mark(cons(X1,X2))
       U71(mark(X1),X2) -> mark(U71(X1,X2))
       U72(mark(X1),X2) -> mark(U72(X1,X2))
       U81(mark(X1),X2,X3) -> mark(U81(X1,X2,X3))
       U82(mark(X1),X2,X3) -> mark(U82(X1,X2,X3))
       fst(mark(X)) -> mark(fst(X))
       natsFrom(mark(X)) -> mark(natsFrom(X))
       s(mark(X)) -> mark(s(X))
       sel(mark(X1),X2) -> mark(sel(X1,X2))
       sel(X1,mark(X2)) -> mark(sel(X1,X2))
       tail(mark(X)) -> mark(tail(X))
       take(mark(X1),X2) -> mark(take(X1,X2))
       take(X1,mark(X2)) -> mark(take(X1,X2))
       proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3))
       proper(tt()) -> ok(tt())
       proper(U12(X1,X2,X3)) -> U12(proper(X1),proper(X2),proper(X3))
       proper(snd(X)) -> snd(proper(X))
       proper(splitAt(X1,X2)) -> splitAt(proper(X1),proper(X2))
       proper(U21(X1,X2)) -> U21(proper(X1),proper(X2))
       proper(U22(X1,X2)) -> U22(proper(X1),proper(X2))
       proper(U31(X1,X2)) -> U31(proper(X1),proper(X2))
       proper(U32(X1,X2)) -> U32(proper(X1),proper(X2))
       proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3))
       proper(U42(X1,X2,X3)) -> U42(proper(X1),proper(X2),proper(X3))
       proper(head(X)) -> head(proper(X))
       proper(afterNth(X1,X2)) -> afterNth(proper(X1),proper(X2))
       proper(U51(X1,X2)) -> U51(proper(X1),proper(X2))
       proper(U52(X1,X2)) -> U52(proper(X1),proper(X2))
       proper(U61(X1,X2,X3,X4)) -> U61(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(U62(X1,X2,X3,X4)) -> U62(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(U63(X1,X2,X3,X4)) -> U63(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(U64(X1,X2)) -> U64(proper(X1),proper(X2))
       proper(pair(X1,X2)) -> pair(proper(X1),proper(X2))
       proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
       proper(U71(X1,X2)) -> U71(proper(X1),proper(X2))
       proper(U72(X1,X2)) -> U72(proper(X1),proper(X2))
       proper(U81(X1,X2,X3)) -> U81(proper(X1),proper(X2),proper(X3))
       proper(U82(X1,X2,X3)) -> U82(proper(X1),proper(X2),proper(X3))
       proper(fst(X)) -> fst(proper(X))
       proper(natsFrom(X)) -> natsFrom(proper(X))
       proper(s(X)) -> s(proper(X))
       proper(sel(X1,X2)) -> sel(proper(X1),proper(X2))
       proper(0()) -> ok(0())
       proper(nil()) -> ok(nil())
       proper(tail(X)) -> tail(proper(X))
       proper(take(X1,X2)) -> take(proper(X1),proper(X2))
       U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3))
       U12(ok(X1),ok(X2),ok(X3)) -> ok(U12(X1,X2,X3))
       snd(ok(X)) -> ok(snd(X))
       splitAt(ok(X1),ok(X2)) -> ok(splitAt(X1,X2))
       U21(ok(X1),ok(X2)) -> ok(U21(X1,X2))
       U22(ok(X1),ok(X2)) -> ok(U22(X1,X2))
       U31(ok(X1),ok(X2)) -> ok(U31(X1,X2))
       U32(ok(X1),ok(X2)) -> ok(U32(X1,X2))
       U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3))
       U42(ok(X1),ok(X2),ok(X3)) -> ok(U42(X1,X2,X3))
       head(ok(X)) -> ok(head(X))
       afterNth(ok(X1),ok(X2)) -> ok(afterNth(X1,X2))
       U51(ok(X1),ok(X2)) -> ok(U51(X1,X2))
       U52(ok(X1),ok(X2)) -> ok(U52(X1,X2))
       U61(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U61(X1,X2,X3,X4))
       U62(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U62(X1,X2,X3,X4))
       U63(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U63(X1,X2,X3,X4))
       U64(ok(X1),ok(X2)) -> ok(U64(X1,X2))
       pair(ok(X1),ok(X2)) -> ok(pair(X1,X2))
       cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
       U71(ok(X1),ok(X2)) -> ok(U71(X1,X2))
       U72(ok(X1),ok(X2)) -> ok(U72(X1,X2))
       U81(ok(X1),ok(X2),ok(X3)) -> ok(U81(X1,X2,X3))
       U82(ok(X1),ok(X2),ok(X3)) -> ok(U82(X1,X2,X3))
       fst(ok(X)) -> ok(fst(X))
       natsFrom(ok(X)) -> ok(natsFrom(X))
       s(ok(X)) -> ok(s(X))
       sel(ok(X1),ok(X2)) -> ok(sel(X1,X2))
       tail(ok(X)) -> ok(tail(X))
       take(ok(X1),ok(X2)) -> ok(take(X1,X2))
       top(mark(X)) -> top(proper(X))
       top(ok(X)) -> top(active(X))
      Open
      
      DPs:
       U22#(mark(X1),X2) -> U22#(X1,X2)
       U22#(ok(X1),ok(X2)) -> U22#(X1,X2)
      TRS:
       active(U11(tt(),N,XS)) -> mark(U12(tt(),N,XS))
       active(U12(tt(),N,XS)) -> mark(snd(splitAt(N,XS)))
       active(U21(tt(),X)) -> mark(U22(tt(),X))
       active(U22(tt(),X)) -> mark(X)
       active(U31(tt(),N)) -> mark(U32(tt(),N))
       active(U32(tt(),N)) -> mark(N)
       active(U41(tt(),N,XS)) -> mark(U42(tt(),N,XS))
       active(U42(tt(),N,XS)) -> mark(head(afterNth(N,XS)))
       active(U51(tt(),Y)) -> mark(U52(tt(),Y))
       active(U52(tt(),Y)) -> mark(Y)
       active(U61(tt(),N,X,XS)) -> mark(U62(tt(),N,X,XS))
       active(U62(tt(),N,X,XS)) -> mark(U63(tt(),N,X,XS))
       active(U63(tt(),N,X,XS)) -> mark(U64(splitAt(N,XS),X))
       active(U64(pair(YS,ZS),X)) -> mark(pair(cons(X,YS),ZS))
       active(U71(tt(),XS)) -> mark(U72(tt(),XS))
       active(U72(tt(),XS)) -> mark(XS)
       active(U81(tt(),N,XS)) -> mark(U82(tt(),N,XS))
       active(U82(tt(),N,XS)) -> mark(fst(splitAt(N,XS)))
       active(afterNth(N,XS)) -> mark(U11(tt(),N,XS))
       active(fst(pair(X,Y))) -> mark(U21(tt(),X))
       active(head(cons(N,XS))) -> mark(U31(tt(),N))
       active(natsFrom(N)) -> mark(cons(N,natsFrom(s(N))))
       active(sel(N,XS)) -> mark(U41(tt(),N,XS))
       active(snd(pair(X,Y))) -> mark(U51(tt(),Y))
       active(splitAt(0(),XS)) -> mark(pair(nil(),XS))
       active(splitAt(s(N),cons(X,XS))) -> mark(U61(tt(),N,X,XS))
       active(tail(cons(N,XS))) -> mark(U71(tt(),XS))
       active(take(N,XS)) -> mark(U81(tt(),N,XS))
       active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3)
       active(U12(X1,X2,X3)) -> U12(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(U21(X1,X2)) -> U21(active(X1),X2)
       active(U22(X1,X2)) -> U22(active(X1),X2)
       active(U31(X1,X2)) -> U31(active(X1),X2)
       active(U32(X1,X2)) -> U32(active(X1),X2)
       active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3)
       active(U42(X1,X2,X3)) -> U42(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(U51(X1,X2)) -> U51(active(X1),X2)
       active(U52(X1,X2)) -> U52(active(X1),X2)
       active(U61(X1,X2,X3,X4)) -> U61(active(X1),X2,X3,X4)
       active(U62(X1,X2,X3,X4)) -> U62(active(X1),X2,X3,X4)
       active(U63(X1,X2,X3,X4)) -> U63(active(X1),X2,X3,X4)
       active(U64(X1,X2)) -> U64(active(X1),X2)
       active(pair(X1,X2)) -> pair(active(X1),X2)
       active(pair(X1,X2)) -> pair(X1,active(X2))
       active(cons(X1,X2)) -> cons(active(X1),X2)
       active(U71(X1,X2)) -> U71(active(X1),X2)
       active(U72(X1,X2)) -> U72(active(X1),X2)
       active(U81(X1,X2,X3)) -> U81(active(X1),X2,X3)
       active(U82(X1,X2,X3)) -> U82(active(X1),X2,X3)
       active(fst(X)) -> fst(active(X))
       active(natsFrom(X)) -> natsFrom(active(X))
       active(s(X)) -> s(active(X))
       active(sel(X1,X2)) -> sel(active(X1),X2)
       active(sel(X1,X2)) -> sel(X1,active(X2))
       active(tail(X)) -> tail(active(X))
       active(take(X1,X2)) -> take(active(X1),X2)
       active(take(X1,X2)) -> take(X1,active(X2))
       U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3))
       U12(mark(X1),X2,X3) -> mark(U12(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))
       U21(mark(X1),X2) -> mark(U21(X1,X2))
       U22(mark(X1),X2) -> mark(U22(X1,X2))
       U31(mark(X1),X2) -> mark(U31(X1,X2))
       U32(mark(X1),X2) -> mark(U32(X1,X2))
       U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3))
       U42(mark(X1),X2,X3) -> mark(U42(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))
       U51(mark(X1),X2) -> mark(U51(X1,X2))
       U52(mark(X1),X2) -> mark(U52(X1,X2))
       U61(mark(X1),X2,X3,X4) -> mark(U61(X1,X2,X3,X4))
       U62(mark(X1),X2,X3,X4) -> mark(U62(X1,X2,X3,X4))
       U63(mark(X1),X2,X3,X4) -> mark(U63(X1,X2,X3,X4))
       U64(mark(X1),X2) -> mark(U64(X1,X2))
       pair(mark(X1),X2) -> mark(pair(X1,X2))
       pair(X1,mark(X2)) -> mark(pair(X1,X2))
       cons(mark(X1),X2) -> mark(cons(X1,X2))
       U71(mark(X1),X2) -> mark(U71(X1,X2))
       U72(mark(X1),X2) -> mark(U72(X1,X2))
       U81(mark(X1),X2,X3) -> mark(U81(X1,X2,X3))
       U82(mark(X1),X2,X3) -> mark(U82(X1,X2,X3))
       fst(mark(X)) -> mark(fst(X))
       natsFrom(mark(X)) -> mark(natsFrom(X))
       s(mark(X)) -> mark(s(X))
       sel(mark(X1),X2) -> mark(sel(X1,X2))
       sel(X1,mark(X2)) -> mark(sel(X1,X2))
       tail(mark(X)) -> mark(tail(X))
       take(mark(X1),X2) -> mark(take(X1,X2))
       take(X1,mark(X2)) -> mark(take(X1,X2))
       proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3))
       proper(tt()) -> ok(tt())
       proper(U12(X1,X2,X3)) -> U12(proper(X1),proper(X2),proper(X3))
       proper(snd(X)) -> snd(proper(X))
       proper(splitAt(X1,X2)) -> splitAt(proper(X1),proper(X2))
       proper(U21(X1,X2)) -> U21(proper(X1),proper(X2))
       proper(U22(X1,X2)) -> U22(proper(X1),proper(X2))
       proper(U31(X1,X2)) -> U31(proper(X1),proper(X2))
       proper(U32(X1,X2)) -> U32(proper(X1),proper(X2))
       proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3))
       proper(U42(X1,X2,X3)) -> U42(proper(X1),proper(X2),proper(X3))
       proper(head(X)) -> head(proper(X))
       proper(afterNth(X1,X2)) -> afterNth(proper(X1),proper(X2))
       proper(U51(X1,X2)) -> U51(proper(X1),proper(X2))
       proper(U52(X1,X2)) -> U52(proper(X1),proper(X2))
       proper(U61(X1,X2,X3,X4)) -> U61(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(U62(X1,X2,X3,X4)) -> U62(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(U63(X1,X2,X3,X4)) -> U63(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(U64(X1,X2)) -> U64(proper(X1),proper(X2))
       proper(pair(X1,X2)) -> pair(proper(X1),proper(X2))
       proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
       proper(U71(X1,X2)) -> U71(proper(X1),proper(X2))
       proper(U72(X1,X2)) -> U72(proper(X1),proper(X2))
       proper(U81(X1,X2,X3)) -> U81(proper(X1),proper(X2),proper(X3))
       proper(U82(X1,X2,X3)) -> U82(proper(X1),proper(X2),proper(X3))
       proper(fst(X)) -> fst(proper(X))
       proper(natsFrom(X)) -> natsFrom(proper(X))
       proper(s(X)) -> s(proper(X))
       proper(sel(X1,X2)) -> sel(proper(X1),proper(X2))
       proper(0()) -> ok(0())
       proper(nil()) -> ok(nil())
       proper(tail(X)) -> tail(proper(X))
       proper(take(X1,X2)) -> take(proper(X1),proper(X2))
       U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3))
       U12(ok(X1),ok(X2),ok(X3)) -> ok(U12(X1,X2,X3))
       snd(ok(X)) -> ok(snd(X))
       splitAt(ok(X1),ok(X2)) -> ok(splitAt(X1,X2))
       U21(ok(X1),ok(X2)) -> ok(U21(X1,X2))
       U22(ok(X1),ok(X2)) -> ok(U22(X1,X2))
       U31(ok(X1),ok(X2)) -> ok(U31(X1,X2))
       U32(ok(X1),ok(X2)) -> ok(U32(X1,X2))
       U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3))
       U42(ok(X1),ok(X2),ok(X3)) -> ok(U42(X1,X2,X3))
       head(ok(X)) -> ok(head(X))
       afterNth(ok(X1),ok(X2)) -> ok(afterNth(X1,X2))
       U51(ok(X1),ok(X2)) -> ok(U51(X1,X2))
       U52(ok(X1),ok(X2)) -> ok(U52(X1,X2))
       U61(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U61(X1,X2,X3,X4))
       U62(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U62(X1,X2,X3,X4))
       U63(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U63(X1,X2,X3,X4))
       U64(ok(X1),ok(X2)) -> ok(U64(X1,X2))
       pair(ok(X1),ok(X2)) -> ok(pair(X1,X2))
       cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
       U71(ok(X1),ok(X2)) -> ok(U71(X1,X2))
       U72(ok(X1),ok(X2)) -> ok(U72(X1,X2))
       U81(ok(X1),ok(X2),ok(X3)) -> ok(U81(X1,X2,X3))
       U82(ok(X1),ok(X2),ok(X3)) -> ok(U82(X1,X2,X3))
       fst(ok(X)) -> ok(fst(X))
       natsFrom(ok(X)) -> ok(natsFrom(X))
       s(ok(X)) -> ok(s(X))
       sel(ok(X1),ok(X2)) -> ok(sel(X1,X2))
       tail(ok(X)) -> ok(tail(X))
       take(ok(X1),ok(X2)) -> ok(take(X1,X2))
       top(mark(X)) -> top(proper(X))
       top(ok(X)) -> top(active(X))
      Open
      
      DPs:
       U21#(mark(X1),X2) -> U21#(X1,X2)
       U21#(ok(X1),ok(X2)) -> U21#(X1,X2)
      TRS:
       active(U11(tt(),N,XS)) -> mark(U12(tt(),N,XS))
       active(U12(tt(),N,XS)) -> mark(snd(splitAt(N,XS)))
       active(U21(tt(),X)) -> mark(U22(tt(),X))
       active(U22(tt(),X)) -> mark(X)
       active(U31(tt(),N)) -> mark(U32(tt(),N))
       active(U32(tt(),N)) -> mark(N)
       active(U41(tt(),N,XS)) -> mark(U42(tt(),N,XS))
       active(U42(tt(),N,XS)) -> mark(head(afterNth(N,XS)))
       active(U51(tt(),Y)) -> mark(U52(tt(),Y))
       active(U52(tt(),Y)) -> mark(Y)
       active(U61(tt(),N,X,XS)) -> mark(U62(tt(),N,X,XS))
       active(U62(tt(),N,X,XS)) -> mark(U63(tt(),N,X,XS))
       active(U63(tt(),N,X,XS)) -> mark(U64(splitAt(N,XS),X))
       active(U64(pair(YS,ZS),X)) -> mark(pair(cons(X,YS),ZS))
       active(U71(tt(),XS)) -> mark(U72(tt(),XS))
       active(U72(tt(),XS)) -> mark(XS)
       active(U81(tt(),N,XS)) -> mark(U82(tt(),N,XS))
       active(U82(tt(),N,XS)) -> mark(fst(splitAt(N,XS)))
       active(afterNth(N,XS)) -> mark(U11(tt(),N,XS))
       active(fst(pair(X,Y))) -> mark(U21(tt(),X))
       active(head(cons(N,XS))) -> mark(U31(tt(),N))
       active(natsFrom(N)) -> mark(cons(N,natsFrom(s(N))))
       active(sel(N,XS)) -> mark(U41(tt(),N,XS))
       active(snd(pair(X,Y))) -> mark(U51(tt(),Y))
       active(splitAt(0(),XS)) -> mark(pair(nil(),XS))
       active(splitAt(s(N),cons(X,XS))) -> mark(U61(tt(),N,X,XS))
       active(tail(cons(N,XS))) -> mark(U71(tt(),XS))
       active(take(N,XS)) -> mark(U81(tt(),N,XS))
       active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3)
       active(U12(X1,X2,X3)) -> U12(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(U21(X1,X2)) -> U21(active(X1),X2)
       active(U22(X1,X2)) -> U22(active(X1),X2)
       active(U31(X1,X2)) -> U31(active(X1),X2)
       active(U32(X1,X2)) -> U32(active(X1),X2)
       active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3)
       active(U42(X1,X2,X3)) -> U42(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(U51(X1,X2)) -> U51(active(X1),X2)
       active(U52(X1,X2)) -> U52(active(X1),X2)
       active(U61(X1,X2,X3,X4)) -> U61(active(X1),X2,X3,X4)
       active(U62(X1,X2,X3,X4)) -> U62(active(X1),X2,X3,X4)
       active(U63(X1,X2,X3,X4)) -> U63(active(X1),X2,X3,X4)
       active(U64(X1,X2)) -> U64(active(X1),X2)
       active(pair(X1,X2)) -> pair(active(X1),X2)
       active(pair(X1,X2)) -> pair(X1,active(X2))
       active(cons(X1,X2)) -> cons(active(X1),X2)
       active(U71(X1,X2)) -> U71(active(X1),X2)
       active(U72(X1,X2)) -> U72(active(X1),X2)
       active(U81(X1,X2,X3)) -> U81(active(X1),X2,X3)
       active(U82(X1,X2,X3)) -> U82(active(X1),X2,X3)
       active(fst(X)) -> fst(active(X))
       active(natsFrom(X)) -> natsFrom(active(X))
       active(s(X)) -> s(active(X))
       active(sel(X1,X2)) -> sel(active(X1),X2)
       active(sel(X1,X2)) -> sel(X1,active(X2))
       active(tail(X)) -> tail(active(X))
       active(take(X1,X2)) -> take(active(X1),X2)
       active(take(X1,X2)) -> take(X1,active(X2))
       U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3))
       U12(mark(X1),X2,X3) -> mark(U12(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))
       U21(mark(X1),X2) -> mark(U21(X1,X2))
       U22(mark(X1),X2) -> mark(U22(X1,X2))
       U31(mark(X1),X2) -> mark(U31(X1,X2))
       U32(mark(X1),X2) -> mark(U32(X1,X2))
       U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3))
       U42(mark(X1),X2,X3) -> mark(U42(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))
       U51(mark(X1),X2) -> mark(U51(X1,X2))
       U52(mark(X1),X2) -> mark(U52(X1,X2))
       U61(mark(X1),X2,X3,X4) -> mark(U61(X1,X2,X3,X4))
       U62(mark(X1),X2,X3,X4) -> mark(U62(X1,X2,X3,X4))
       U63(mark(X1),X2,X3,X4) -> mark(U63(X1,X2,X3,X4))
       U64(mark(X1),X2) -> mark(U64(X1,X2))
       pair(mark(X1),X2) -> mark(pair(X1,X2))
       pair(X1,mark(X2)) -> mark(pair(X1,X2))
       cons(mark(X1),X2) -> mark(cons(X1,X2))
       U71(mark(X1),X2) -> mark(U71(X1,X2))
       U72(mark(X1),X2) -> mark(U72(X1,X2))
       U81(mark(X1),X2,X3) -> mark(U81(X1,X2,X3))
       U82(mark(X1),X2,X3) -> mark(U82(X1,X2,X3))
       fst(mark(X)) -> mark(fst(X))
       natsFrom(mark(X)) -> mark(natsFrom(X))
       s(mark(X)) -> mark(s(X))
       sel(mark(X1),X2) -> mark(sel(X1,X2))
       sel(X1,mark(X2)) -> mark(sel(X1,X2))
       tail(mark(X)) -> mark(tail(X))
       take(mark(X1),X2) -> mark(take(X1,X2))
       take(X1,mark(X2)) -> mark(take(X1,X2))
       proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3))
       proper(tt()) -> ok(tt())
       proper(U12(X1,X2,X3)) -> U12(proper(X1),proper(X2),proper(X3))
       proper(snd(X)) -> snd(proper(X))
       proper(splitAt(X1,X2)) -> splitAt(proper(X1),proper(X2))
       proper(U21(X1,X2)) -> U21(proper(X1),proper(X2))
       proper(U22(X1,X2)) -> U22(proper(X1),proper(X2))
       proper(U31(X1,X2)) -> U31(proper(X1),proper(X2))
       proper(U32(X1,X2)) -> U32(proper(X1),proper(X2))
       proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3))
       proper(U42(X1,X2,X3)) -> U42(proper(X1),proper(X2),proper(X3))
       proper(head(X)) -> head(proper(X))
       proper(afterNth(X1,X2)) -> afterNth(proper(X1),proper(X2))
       proper(U51(X1,X2)) -> U51(proper(X1),proper(X2))
       proper(U52(X1,X2)) -> U52(proper(X1),proper(X2))
       proper(U61(X1,X2,X3,X4)) -> U61(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(U62(X1,X2,X3,X4)) -> U62(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(U63(X1,X2,X3,X4)) -> U63(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(U64(X1,X2)) -> U64(proper(X1),proper(X2))
       proper(pair(X1,X2)) -> pair(proper(X1),proper(X2))
       proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
       proper(U71(X1,X2)) -> U71(proper(X1),proper(X2))
       proper(U72(X1,X2)) -> U72(proper(X1),proper(X2))
       proper(U81(X1,X2,X3)) -> U81(proper(X1),proper(X2),proper(X3))
       proper(U82(X1,X2,X3)) -> U82(proper(X1),proper(X2),proper(X3))
       proper(fst(X)) -> fst(proper(X))
       proper(natsFrom(X)) -> natsFrom(proper(X))
       proper(s(X)) -> s(proper(X))
       proper(sel(X1,X2)) -> sel(proper(X1),proper(X2))
       proper(0()) -> ok(0())
       proper(nil()) -> ok(nil())
       proper(tail(X)) -> tail(proper(X))
       proper(take(X1,X2)) -> take(proper(X1),proper(X2))
       U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3))
       U12(ok(X1),ok(X2),ok(X3)) -> ok(U12(X1,X2,X3))
       snd(ok(X)) -> ok(snd(X))
       splitAt(ok(X1),ok(X2)) -> ok(splitAt(X1,X2))
       U21(ok(X1),ok(X2)) -> ok(U21(X1,X2))
       U22(ok(X1),ok(X2)) -> ok(U22(X1,X2))
       U31(ok(X1),ok(X2)) -> ok(U31(X1,X2))
       U32(ok(X1),ok(X2)) -> ok(U32(X1,X2))
       U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3))
       U42(ok(X1),ok(X2),ok(X3)) -> ok(U42(X1,X2,X3))
       head(ok(X)) -> ok(head(X))
       afterNth(ok(X1),ok(X2)) -> ok(afterNth(X1,X2))
       U51(ok(X1),ok(X2)) -> ok(U51(X1,X2))
       U52(ok(X1),ok(X2)) -> ok(U52(X1,X2))
       U61(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U61(X1,X2,X3,X4))
       U62(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U62(X1,X2,X3,X4))
       U63(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U63(X1,X2,X3,X4))
       U64(ok(X1),ok(X2)) -> ok(U64(X1,X2))
       pair(ok(X1),ok(X2)) -> ok(pair(X1,X2))
       cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
       U71(ok(X1),ok(X2)) -> ok(U71(X1,X2))
       U72(ok(X1),ok(X2)) -> ok(U72(X1,X2))
       U81(ok(X1),ok(X2),ok(X3)) -> ok(U81(X1,X2,X3))
       U82(ok(X1),ok(X2),ok(X3)) -> ok(U82(X1,X2,X3))
       fst(ok(X)) -> ok(fst(X))
       natsFrom(ok(X)) -> ok(natsFrom(X))
       s(ok(X)) -> ok(s(X))
       sel(ok(X1),ok(X2)) -> ok(sel(X1,X2))
       tail(ok(X)) -> ok(tail(X))
       take(ok(X1),ok(X2)) -> ok(take(X1,X2))
       top(mark(X)) -> top(proper(X))
       top(ok(X)) -> top(active(X))
      Open
      
      DPs:
       splitAt#(mark(X1),X2) -> splitAt#(X1,X2)
       splitAt#(X1,mark(X2)) -> splitAt#(X1,X2)
       splitAt#(ok(X1),ok(X2)) -> splitAt#(X1,X2)
      TRS:
       active(U11(tt(),N,XS)) -> mark(U12(tt(),N,XS))
       active(U12(tt(),N,XS)) -> mark(snd(splitAt(N,XS)))
       active(U21(tt(),X)) -> mark(U22(tt(),X))
       active(U22(tt(),X)) -> mark(X)
       active(U31(tt(),N)) -> mark(U32(tt(),N))
       active(U32(tt(),N)) -> mark(N)
       active(U41(tt(),N,XS)) -> mark(U42(tt(),N,XS))
       active(U42(tt(),N,XS)) -> mark(head(afterNth(N,XS)))
       active(U51(tt(),Y)) -> mark(U52(tt(),Y))
       active(U52(tt(),Y)) -> mark(Y)
       active(U61(tt(),N,X,XS)) -> mark(U62(tt(),N,X,XS))
       active(U62(tt(),N,X,XS)) -> mark(U63(tt(),N,X,XS))
       active(U63(tt(),N,X,XS)) -> mark(U64(splitAt(N,XS),X))
       active(U64(pair(YS,ZS),X)) -> mark(pair(cons(X,YS),ZS))
       active(U71(tt(),XS)) -> mark(U72(tt(),XS))
       active(U72(tt(),XS)) -> mark(XS)
       active(U81(tt(),N,XS)) -> mark(U82(tt(),N,XS))
       active(U82(tt(),N,XS)) -> mark(fst(splitAt(N,XS)))
       active(afterNth(N,XS)) -> mark(U11(tt(),N,XS))
       active(fst(pair(X,Y))) -> mark(U21(tt(),X))
       active(head(cons(N,XS))) -> mark(U31(tt(),N))
       active(natsFrom(N)) -> mark(cons(N,natsFrom(s(N))))
       active(sel(N,XS)) -> mark(U41(tt(),N,XS))
       active(snd(pair(X,Y))) -> mark(U51(tt(),Y))
       active(splitAt(0(),XS)) -> mark(pair(nil(),XS))
       active(splitAt(s(N),cons(X,XS))) -> mark(U61(tt(),N,X,XS))
       active(tail(cons(N,XS))) -> mark(U71(tt(),XS))
       active(take(N,XS)) -> mark(U81(tt(),N,XS))
       active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3)
       active(U12(X1,X2,X3)) -> U12(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(U21(X1,X2)) -> U21(active(X1),X2)
       active(U22(X1,X2)) -> U22(active(X1),X2)
       active(U31(X1,X2)) -> U31(active(X1),X2)
       active(U32(X1,X2)) -> U32(active(X1),X2)
       active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3)
       active(U42(X1,X2,X3)) -> U42(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(U51(X1,X2)) -> U51(active(X1),X2)
       active(U52(X1,X2)) -> U52(active(X1),X2)
       active(U61(X1,X2,X3,X4)) -> U61(active(X1),X2,X3,X4)
       active(U62(X1,X2,X3,X4)) -> U62(active(X1),X2,X3,X4)
       active(U63(X1,X2,X3,X4)) -> U63(active(X1),X2,X3,X4)
       active(U64(X1,X2)) -> U64(active(X1),X2)
       active(pair(X1,X2)) -> pair(active(X1),X2)
       active(pair(X1,X2)) -> pair(X1,active(X2))
       active(cons(X1,X2)) -> cons(active(X1),X2)
       active(U71(X1,X2)) -> U71(active(X1),X2)
       active(U72(X1,X2)) -> U72(active(X1),X2)
       active(U81(X1,X2,X3)) -> U81(active(X1),X2,X3)
       active(U82(X1,X2,X3)) -> U82(active(X1),X2,X3)
       active(fst(X)) -> fst(active(X))
       active(natsFrom(X)) -> natsFrom(active(X))
       active(s(X)) -> s(active(X))
       active(sel(X1,X2)) -> sel(active(X1),X2)
       active(sel(X1,X2)) -> sel(X1,active(X2))
       active(tail(X)) -> tail(active(X))
       active(take(X1,X2)) -> take(active(X1),X2)
       active(take(X1,X2)) -> take(X1,active(X2))
       U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3))
       U12(mark(X1),X2,X3) -> mark(U12(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))
       U21(mark(X1),X2) -> mark(U21(X1,X2))
       U22(mark(X1),X2) -> mark(U22(X1,X2))
       U31(mark(X1),X2) -> mark(U31(X1,X2))
       U32(mark(X1),X2) -> mark(U32(X1,X2))
       U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3))
       U42(mark(X1),X2,X3) -> mark(U42(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))
       U51(mark(X1),X2) -> mark(U51(X1,X2))
       U52(mark(X1),X2) -> mark(U52(X1,X2))
       U61(mark(X1),X2,X3,X4) -> mark(U61(X1,X2,X3,X4))
       U62(mark(X1),X2,X3,X4) -> mark(U62(X1,X2,X3,X4))
       U63(mark(X1),X2,X3,X4) -> mark(U63(X1,X2,X3,X4))
       U64(mark(X1),X2) -> mark(U64(X1,X2))
       pair(mark(X1),X2) -> mark(pair(X1,X2))
       pair(X1,mark(X2)) -> mark(pair(X1,X2))
       cons(mark(X1),X2) -> mark(cons(X1,X2))
       U71(mark(X1),X2) -> mark(U71(X1,X2))
       U72(mark(X1),X2) -> mark(U72(X1,X2))
       U81(mark(X1),X2,X3) -> mark(U81(X1,X2,X3))
       U82(mark(X1),X2,X3) -> mark(U82(X1,X2,X3))
       fst(mark(X)) -> mark(fst(X))
       natsFrom(mark(X)) -> mark(natsFrom(X))
       s(mark(X)) -> mark(s(X))
       sel(mark(X1),X2) -> mark(sel(X1,X2))
       sel(X1,mark(X2)) -> mark(sel(X1,X2))
       tail(mark(X)) -> mark(tail(X))
       take(mark(X1),X2) -> mark(take(X1,X2))
       take(X1,mark(X2)) -> mark(take(X1,X2))
       proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3))
       proper(tt()) -> ok(tt())
       proper(U12(X1,X2,X3)) -> U12(proper(X1),proper(X2),proper(X3))
       proper(snd(X)) -> snd(proper(X))
       proper(splitAt(X1,X2)) -> splitAt(proper(X1),proper(X2))
       proper(U21(X1,X2)) -> U21(proper(X1),proper(X2))
       proper(U22(X1,X2)) -> U22(proper(X1),proper(X2))
       proper(U31(X1,X2)) -> U31(proper(X1),proper(X2))
       proper(U32(X1,X2)) -> U32(proper(X1),proper(X2))
       proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3))
       proper(U42(X1,X2,X3)) -> U42(proper(X1),proper(X2),proper(X3))
       proper(head(X)) -> head(proper(X))
       proper(afterNth(X1,X2)) -> afterNth(proper(X1),proper(X2))
       proper(U51(X1,X2)) -> U51(proper(X1),proper(X2))
       proper(U52(X1,X2)) -> U52(proper(X1),proper(X2))
       proper(U61(X1,X2,X3,X4)) -> U61(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(U62(X1,X2,X3,X4)) -> U62(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(U63(X1,X2,X3,X4)) -> U63(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(U64(X1,X2)) -> U64(proper(X1),proper(X2))
       proper(pair(X1,X2)) -> pair(proper(X1),proper(X2))
       proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
       proper(U71(X1,X2)) -> U71(proper(X1),proper(X2))
       proper(U72(X1,X2)) -> U72(proper(X1),proper(X2))
       proper(U81(X1,X2,X3)) -> U81(proper(X1),proper(X2),proper(X3))
       proper(U82(X1,X2,X3)) -> U82(proper(X1),proper(X2),proper(X3))
       proper(fst(X)) -> fst(proper(X))
       proper(natsFrom(X)) -> natsFrom(proper(X))
       proper(s(X)) -> s(proper(X))
       proper(sel(X1,X2)) -> sel(proper(X1),proper(X2))
       proper(0()) -> ok(0())
       proper(nil()) -> ok(nil())
       proper(tail(X)) -> tail(proper(X))
       proper(take(X1,X2)) -> take(proper(X1),proper(X2))
       U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3))
       U12(ok(X1),ok(X2),ok(X3)) -> ok(U12(X1,X2,X3))
       snd(ok(X)) -> ok(snd(X))
       splitAt(ok(X1),ok(X2)) -> ok(splitAt(X1,X2))
       U21(ok(X1),ok(X2)) -> ok(U21(X1,X2))
       U22(ok(X1),ok(X2)) -> ok(U22(X1,X2))
       U31(ok(X1),ok(X2)) -> ok(U31(X1,X2))
       U32(ok(X1),ok(X2)) -> ok(U32(X1,X2))
       U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3))
       U42(ok(X1),ok(X2),ok(X3)) -> ok(U42(X1,X2,X3))
       head(ok(X)) -> ok(head(X))
       afterNth(ok(X1),ok(X2)) -> ok(afterNth(X1,X2))
       U51(ok(X1),ok(X2)) -> ok(U51(X1,X2))
       U52(ok(X1),ok(X2)) -> ok(U52(X1,X2))
       U61(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U61(X1,X2,X3,X4))
       U62(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U62(X1,X2,X3,X4))
       U63(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U63(X1,X2,X3,X4))
       U64(ok(X1),ok(X2)) -> ok(U64(X1,X2))
       pair(ok(X1),ok(X2)) -> ok(pair(X1,X2))
       cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
       U71(ok(X1),ok(X2)) -> ok(U71(X1,X2))
       U72(ok(X1),ok(X2)) -> ok(U72(X1,X2))
       U81(ok(X1),ok(X2),ok(X3)) -> ok(U81(X1,X2,X3))
       U82(ok(X1),ok(X2),ok(X3)) -> ok(U82(X1,X2,X3))
       fst(ok(X)) -> ok(fst(X))
       natsFrom(ok(X)) -> ok(natsFrom(X))
       s(ok(X)) -> ok(s(X))
       sel(ok(X1),ok(X2)) -> ok(sel(X1,X2))
       tail(ok(X)) -> ok(tail(X))
       take(ok(X1),ok(X2)) -> ok(take(X1,X2))
       top(mark(X)) -> top(proper(X))
       top(ok(X)) -> top(active(X))
      Open
      
      DPs:
       snd#(mark(X)) -> snd#(X)
       snd#(ok(X)) -> snd#(X)
      TRS:
       active(U11(tt(),N,XS)) -> mark(U12(tt(),N,XS))
       active(U12(tt(),N,XS)) -> mark(snd(splitAt(N,XS)))
       active(U21(tt(),X)) -> mark(U22(tt(),X))
       active(U22(tt(),X)) -> mark(X)
       active(U31(tt(),N)) -> mark(U32(tt(),N))
       active(U32(tt(),N)) -> mark(N)
       active(U41(tt(),N,XS)) -> mark(U42(tt(),N,XS))
       active(U42(tt(),N,XS)) -> mark(head(afterNth(N,XS)))
       active(U51(tt(),Y)) -> mark(U52(tt(),Y))
       active(U52(tt(),Y)) -> mark(Y)
       active(U61(tt(),N,X,XS)) -> mark(U62(tt(),N,X,XS))
       active(U62(tt(),N,X,XS)) -> mark(U63(tt(),N,X,XS))
       active(U63(tt(),N,X,XS)) -> mark(U64(splitAt(N,XS),X))
       active(U64(pair(YS,ZS),X)) -> mark(pair(cons(X,YS),ZS))
       active(U71(tt(),XS)) -> mark(U72(tt(),XS))
       active(U72(tt(),XS)) -> mark(XS)
       active(U81(tt(),N,XS)) -> mark(U82(tt(),N,XS))
       active(U82(tt(),N,XS)) -> mark(fst(splitAt(N,XS)))
       active(afterNth(N,XS)) -> mark(U11(tt(),N,XS))
       active(fst(pair(X,Y))) -> mark(U21(tt(),X))
       active(head(cons(N,XS))) -> mark(U31(tt(),N))
       active(natsFrom(N)) -> mark(cons(N,natsFrom(s(N))))
       active(sel(N,XS)) -> mark(U41(tt(),N,XS))
       active(snd(pair(X,Y))) -> mark(U51(tt(),Y))
       active(splitAt(0(),XS)) -> mark(pair(nil(),XS))
       active(splitAt(s(N),cons(X,XS))) -> mark(U61(tt(),N,X,XS))
       active(tail(cons(N,XS))) -> mark(U71(tt(),XS))
       active(take(N,XS)) -> mark(U81(tt(),N,XS))
       active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3)
       active(U12(X1,X2,X3)) -> U12(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(U21(X1,X2)) -> U21(active(X1),X2)
       active(U22(X1,X2)) -> U22(active(X1),X2)
       active(U31(X1,X2)) -> U31(active(X1),X2)
       active(U32(X1,X2)) -> U32(active(X1),X2)
       active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3)
       active(U42(X1,X2,X3)) -> U42(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(U51(X1,X2)) -> U51(active(X1),X2)
       active(U52(X1,X2)) -> U52(active(X1),X2)
       active(U61(X1,X2,X3,X4)) -> U61(active(X1),X2,X3,X4)
       active(U62(X1,X2,X3,X4)) -> U62(active(X1),X2,X3,X4)
       active(U63(X1,X2,X3,X4)) -> U63(active(X1),X2,X3,X4)
       active(U64(X1,X2)) -> U64(active(X1),X2)
       active(pair(X1,X2)) -> pair(active(X1),X2)
       active(pair(X1,X2)) -> pair(X1,active(X2))
       active(cons(X1,X2)) -> cons(active(X1),X2)
       active(U71(X1,X2)) -> U71(active(X1),X2)
       active(U72(X1,X2)) -> U72(active(X1),X2)
       active(U81(X1,X2,X3)) -> U81(active(X1),X2,X3)
       active(U82(X1,X2,X3)) -> U82(active(X1),X2,X3)
       active(fst(X)) -> fst(active(X))
       active(natsFrom(X)) -> natsFrom(active(X))
       active(s(X)) -> s(active(X))
       active(sel(X1,X2)) -> sel(active(X1),X2)
       active(sel(X1,X2)) -> sel(X1,active(X2))
       active(tail(X)) -> tail(active(X))
       active(take(X1,X2)) -> take(active(X1),X2)
       active(take(X1,X2)) -> take(X1,active(X2))
       U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3))
       U12(mark(X1),X2,X3) -> mark(U12(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))
       U21(mark(X1),X2) -> mark(U21(X1,X2))
       U22(mark(X1),X2) -> mark(U22(X1,X2))
       U31(mark(X1),X2) -> mark(U31(X1,X2))
       U32(mark(X1),X2) -> mark(U32(X1,X2))
       U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3))
       U42(mark(X1),X2,X3) -> mark(U42(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))
       U51(mark(X1),X2) -> mark(U51(X1,X2))
       U52(mark(X1),X2) -> mark(U52(X1,X2))
       U61(mark(X1),X2,X3,X4) -> mark(U61(X1,X2,X3,X4))
       U62(mark(X1),X2,X3,X4) -> mark(U62(X1,X2,X3,X4))
       U63(mark(X1),X2,X3,X4) -> mark(U63(X1,X2,X3,X4))
       U64(mark(X1),X2) -> mark(U64(X1,X2))
       pair(mark(X1),X2) -> mark(pair(X1,X2))
       pair(X1,mark(X2)) -> mark(pair(X1,X2))
       cons(mark(X1),X2) -> mark(cons(X1,X2))
       U71(mark(X1),X2) -> mark(U71(X1,X2))
       U72(mark(X1),X2) -> mark(U72(X1,X2))
       U81(mark(X1),X2,X3) -> mark(U81(X1,X2,X3))
       U82(mark(X1),X2,X3) -> mark(U82(X1,X2,X3))
       fst(mark(X)) -> mark(fst(X))
       natsFrom(mark(X)) -> mark(natsFrom(X))
       s(mark(X)) -> mark(s(X))
       sel(mark(X1),X2) -> mark(sel(X1,X2))
       sel(X1,mark(X2)) -> mark(sel(X1,X2))
       tail(mark(X)) -> mark(tail(X))
       take(mark(X1),X2) -> mark(take(X1,X2))
       take(X1,mark(X2)) -> mark(take(X1,X2))
       proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3))
       proper(tt()) -> ok(tt())
       proper(U12(X1,X2,X3)) -> U12(proper(X1),proper(X2),proper(X3))
       proper(snd(X)) -> snd(proper(X))
       proper(splitAt(X1,X2)) -> splitAt(proper(X1),proper(X2))
       proper(U21(X1,X2)) -> U21(proper(X1),proper(X2))
       proper(U22(X1,X2)) -> U22(proper(X1),proper(X2))
       proper(U31(X1,X2)) -> U31(proper(X1),proper(X2))
       proper(U32(X1,X2)) -> U32(proper(X1),proper(X2))
       proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3))
       proper(U42(X1,X2,X3)) -> U42(proper(X1),proper(X2),proper(X3))
       proper(head(X)) -> head(proper(X))
       proper(afterNth(X1,X2)) -> afterNth(proper(X1),proper(X2))
       proper(U51(X1,X2)) -> U51(proper(X1),proper(X2))
       proper(U52(X1,X2)) -> U52(proper(X1),proper(X2))
       proper(U61(X1,X2,X3,X4)) -> U61(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(U62(X1,X2,X3,X4)) -> U62(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(U63(X1,X2,X3,X4)) -> U63(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(U64(X1,X2)) -> U64(proper(X1),proper(X2))
       proper(pair(X1,X2)) -> pair(proper(X1),proper(X2))
       proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
       proper(U71(X1,X2)) -> U71(proper(X1),proper(X2))
       proper(U72(X1,X2)) -> U72(proper(X1),proper(X2))
       proper(U81(X1,X2,X3)) -> U81(proper(X1),proper(X2),proper(X3))
       proper(U82(X1,X2,X3)) -> U82(proper(X1),proper(X2),proper(X3))
       proper(fst(X)) -> fst(proper(X))
       proper(natsFrom(X)) -> natsFrom(proper(X))
       proper(s(X)) -> s(proper(X))
       proper(sel(X1,X2)) -> sel(proper(X1),proper(X2))
       proper(0()) -> ok(0())
       proper(nil()) -> ok(nil())
       proper(tail(X)) -> tail(proper(X))
       proper(take(X1,X2)) -> take(proper(X1),proper(X2))
       U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3))
       U12(ok(X1),ok(X2),ok(X3)) -> ok(U12(X1,X2,X3))
       snd(ok(X)) -> ok(snd(X))
       splitAt(ok(X1),ok(X2)) -> ok(splitAt(X1,X2))
       U21(ok(X1),ok(X2)) -> ok(U21(X1,X2))
       U22(ok(X1),ok(X2)) -> ok(U22(X1,X2))
       U31(ok(X1),ok(X2)) -> ok(U31(X1,X2))
       U32(ok(X1),ok(X2)) -> ok(U32(X1,X2))
       U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3))
       U42(ok(X1),ok(X2),ok(X3)) -> ok(U42(X1,X2,X3))
       head(ok(X)) -> ok(head(X))
       afterNth(ok(X1),ok(X2)) -> ok(afterNth(X1,X2))
       U51(ok(X1),ok(X2)) -> ok(U51(X1,X2))
       U52(ok(X1),ok(X2)) -> ok(U52(X1,X2))
       U61(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U61(X1,X2,X3,X4))
       U62(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U62(X1,X2,X3,X4))
       U63(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U63(X1,X2,X3,X4))
       U64(ok(X1),ok(X2)) -> ok(U64(X1,X2))
       pair(ok(X1),ok(X2)) -> ok(pair(X1,X2))
       cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
       U71(ok(X1),ok(X2)) -> ok(U71(X1,X2))
       U72(ok(X1),ok(X2)) -> ok(U72(X1,X2))
       U81(ok(X1),ok(X2),ok(X3)) -> ok(U81(X1,X2,X3))
       U82(ok(X1),ok(X2),ok(X3)) -> ok(U82(X1,X2,X3))
       fst(ok(X)) -> ok(fst(X))
       natsFrom(ok(X)) -> ok(natsFrom(X))
       s(ok(X)) -> ok(s(X))
       sel(ok(X1),ok(X2)) -> ok(sel(X1,X2))
       tail(ok(X)) -> ok(tail(X))
       take(ok(X1),ok(X2)) -> ok(take(X1,X2))
       top(mark(X)) -> top(proper(X))
       top(ok(X)) -> top(active(X))
      Open
      
      DPs:
       U12#(mark(X1),X2,X3) -> U12#(X1,X2,X3)
       U12#(ok(X1),ok(X2),ok(X3)) -> U12#(X1,X2,X3)
      TRS:
       active(U11(tt(),N,XS)) -> mark(U12(tt(),N,XS))
       active(U12(tt(),N,XS)) -> mark(snd(splitAt(N,XS)))
       active(U21(tt(),X)) -> mark(U22(tt(),X))
       active(U22(tt(),X)) -> mark(X)
       active(U31(tt(),N)) -> mark(U32(tt(),N))
       active(U32(tt(),N)) -> mark(N)
       active(U41(tt(),N,XS)) -> mark(U42(tt(),N,XS))
       active(U42(tt(),N,XS)) -> mark(head(afterNth(N,XS)))
       active(U51(tt(),Y)) -> mark(U52(tt(),Y))
       active(U52(tt(),Y)) -> mark(Y)
       active(U61(tt(),N,X,XS)) -> mark(U62(tt(),N,X,XS))
       active(U62(tt(),N,X,XS)) -> mark(U63(tt(),N,X,XS))
       active(U63(tt(),N,X,XS)) -> mark(U64(splitAt(N,XS),X))
       active(U64(pair(YS,ZS),X)) -> mark(pair(cons(X,YS),ZS))
       active(U71(tt(),XS)) -> mark(U72(tt(),XS))
       active(U72(tt(),XS)) -> mark(XS)
       active(U81(tt(),N,XS)) -> mark(U82(tt(),N,XS))
       active(U82(tt(),N,XS)) -> mark(fst(splitAt(N,XS)))
       active(afterNth(N,XS)) -> mark(U11(tt(),N,XS))
       active(fst(pair(X,Y))) -> mark(U21(tt(),X))
       active(head(cons(N,XS))) -> mark(U31(tt(),N))
       active(natsFrom(N)) -> mark(cons(N,natsFrom(s(N))))
       active(sel(N,XS)) -> mark(U41(tt(),N,XS))
       active(snd(pair(X,Y))) -> mark(U51(tt(),Y))
       active(splitAt(0(),XS)) -> mark(pair(nil(),XS))
       active(splitAt(s(N),cons(X,XS))) -> mark(U61(tt(),N,X,XS))
       active(tail(cons(N,XS))) -> mark(U71(tt(),XS))
       active(take(N,XS)) -> mark(U81(tt(),N,XS))
       active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3)
       active(U12(X1,X2,X3)) -> U12(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(U21(X1,X2)) -> U21(active(X1),X2)
       active(U22(X1,X2)) -> U22(active(X1),X2)
       active(U31(X1,X2)) -> U31(active(X1),X2)
       active(U32(X1,X2)) -> U32(active(X1),X2)
       active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3)
       active(U42(X1,X2,X3)) -> U42(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(U51(X1,X2)) -> U51(active(X1),X2)
       active(U52(X1,X2)) -> U52(active(X1),X2)
       active(U61(X1,X2,X3,X4)) -> U61(active(X1),X2,X3,X4)
       active(U62(X1,X2,X3,X4)) -> U62(active(X1),X2,X3,X4)
       active(U63(X1,X2,X3,X4)) -> U63(active(X1),X2,X3,X4)
       active(U64(X1,X2)) -> U64(active(X1),X2)
       active(pair(X1,X2)) -> pair(active(X1),X2)
       active(pair(X1,X2)) -> pair(X1,active(X2))
       active(cons(X1,X2)) -> cons(active(X1),X2)
       active(U71(X1,X2)) -> U71(active(X1),X2)
       active(U72(X1,X2)) -> U72(active(X1),X2)
       active(U81(X1,X2,X3)) -> U81(active(X1),X2,X3)
       active(U82(X1,X2,X3)) -> U82(active(X1),X2,X3)
       active(fst(X)) -> fst(active(X))
       active(natsFrom(X)) -> natsFrom(active(X))
       active(s(X)) -> s(active(X))
       active(sel(X1,X2)) -> sel(active(X1),X2)
       active(sel(X1,X2)) -> sel(X1,active(X2))
       active(tail(X)) -> tail(active(X))
       active(take(X1,X2)) -> take(active(X1),X2)
       active(take(X1,X2)) -> take(X1,active(X2))
       U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3))
       U12(mark(X1),X2,X3) -> mark(U12(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))
       U21(mark(X1),X2) -> mark(U21(X1,X2))
       U22(mark(X1),X2) -> mark(U22(X1,X2))
       U31(mark(X1),X2) -> mark(U31(X1,X2))
       U32(mark(X1),X2) -> mark(U32(X1,X2))
       U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3))
       U42(mark(X1),X2,X3) -> mark(U42(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))
       U51(mark(X1),X2) -> mark(U51(X1,X2))
       U52(mark(X1),X2) -> mark(U52(X1,X2))
       U61(mark(X1),X2,X3,X4) -> mark(U61(X1,X2,X3,X4))
       U62(mark(X1),X2,X3,X4) -> mark(U62(X1,X2,X3,X4))
       U63(mark(X1),X2,X3,X4) -> mark(U63(X1,X2,X3,X4))
       U64(mark(X1),X2) -> mark(U64(X1,X2))
       pair(mark(X1),X2) -> mark(pair(X1,X2))
       pair(X1,mark(X2)) -> mark(pair(X1,X2))
       cons(mark(X1),X2) -> mark(cons(X1,X2))
       U71(mark(X1),X2) -> mark(U71(X1,X2))
       U72(mark(X1),X2) -> mark(U72(X1,X2))
       U81(mark(X1),X2,X3) -> mark(U81(X1,X2,X3))
       U82(mark(X1),X2,X3) -> mark(U82(X1,X2,X3))
       fst(mark(X)) -> mark(fst(X))
       natsFrom(mark(X)) -> mark(natsFrom(X))
       s(mark(X)) -> mark(s(X))
       sel(mark(X1),X2) -> mark(sel(X1,X2))
       sel(X1,mark(X2)) -> mark(sel(X1,X2))
       tail(mark(X)) -> mark(tail(X))
       take(mark(X1),X2) -> mark(take(X1,X2))
       take(X1,mark(X2)) -> mark(take(X1,X2))
       proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3))
       proper(tt()) -> ok(tt())
       proper(U12(X1,X2,X3)) -> U12(proper(X1),proper(X2),proper(X3))
       proper(snd(X)) -> snd(proper(X))
       proper(splitAt(X1,X2)) -> splitAt(proper(X1),proper(X2))
       proper(U21(X1,X2)) -> U21(proper(X1),proper(X2))
       proper(U22(X1,X2)) -> U22(proper(X1),proper(X2))
       proper(U31(X1,X2)) -> U31(proper(X1),proper(X2))
       proper(U32(X1,X2)) -> U32(proper(X1),proper(X2))
       proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3))
       proper(U42(X1,X2,X3)) -> U42(proper(X1),proper(X2),proper(X3))
       proper(head(X)) -> head(proper(X))
       proper(afterNth(X1,X2)) -> afterNth(proper(X1),proper(X2))
       proper(U51(X1,X2)) -> U51(proper(X1),proper(X2))
       proper(U52(X1,X2)) -> U52(proper(X1),proper(X2))
       proper(U61(X1,X2,X3,X4)) -> U61(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(U62(X1,X2,X3,X4)) -> U62(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(U63(X1,X2,X3,X4)) -> U63(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(U64(X1,X2)) -> U64(proper(X1),proper(X2))
       proper(pair(X1,X2)) -> pair(proper(X1),proper(X2))
       proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
       proper(U71(X1,X2)) -> U71(proper(X1),proper(X2))
       proper(U72(X1,X2)) -> U72(proper(X1),proper(X2))
       proper(U81(X1,X2,X3)) -> U81(proper(X1),proper(X2),proper(X3))
       proper(U82(X1,X2,X3)) -> U82(proper(X1),proper(X2),proper(X3))
       proper(fst(X)) -> fst(proper(X))
       proper(natsFrom(X)) -> natsFrom(proper(X))
       proper(s(X)) -> s(proper(X))
       proper(sel(X1,X2)) -> sel(proper(X1),proper(X2))
       proper(0()) -> ok(0())
       proper(nil()) -> ok(nil())
       proper(tail(X)) -> tail(proper(X))
       proper(take(X1,X2)) -> take(proper(X1),proper(X2))
       U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3))
       U12(ok(X1),ok(X2),ok(X3)) -> ok(U12(X1,X2,X3))
       snd(ok(X)) -> ok(snd(X))
       splitAt(ok(X1),ok(X2)) -> ok(splitAt(X1,X2))
       U21(ok(X1),ok(X2)) -> ok(U21(X1,X2))
       U22(ok(X1),ok(X2)) -> ok(U22(X1,X2))
       U31(ok(X1),ok(X2)) -> ok(U31(X1,X2))
       U32(ok(X1),ok(X2)) -> ok(U32(X1,X2))
       U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3))
       U42(ok(X1),ok(X2),ok(X3)) -> ok(U42(X1,X2,X3))
       head(ok(X)) -> ok(head(X))
       afterNth(ok(X1),ok(X2)) -> ok(afterNth(X1,X2))
       U51(ok(X1),ok(X2)) -> ok(U51(X1,X2))
       U52(ok(X1),ok(X2)) -> ok(U52(X1,X2))
       U61(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U61(X1,X2,X3,X4))
       U62(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U62(X1,X2,X3,X4))
       U63(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U63(X1,X2,X3,X4))
       U64(ok(X1),ok(X2)) -> ok(U64(X1,X2))
       pair(ok(X1),ok(X2)) -> ok(pair(X1,X2))
       cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
       U71(ok(X1),ok(X2)) -> ok(U71(X1,X2))
       U72(ok(X1),ok(X2)) -> ok(U72(X1,X2))
       U81(ok(X1),ok(X2),ok(X3)) -> ok(U81(X1,X2,X3))
       U82(ok(X1),ok(X2),ok(X3)) -> ok(U82(X1,X2,X3))
       fst(ok(X)) -> ok(fst(X))
       natsFrom(ok(X)) -> ok(natsFrom(X))
       s(ok(X)) -> ok(s(X))
       sel(ok(X1),ok(X2)) -> ok(sel(X1,X2))
       tail(ok(X)) -> ok(tail(X))
       take(ok(X1),ok(X2)) -> ok(take(X1,X2))
       top(mark(X)) -> top(proper(X))
       top(ok(X)) -> top(active(X))
      Open
      
      DPs:
       U11#(mark(X1),X2,X3) -> U11#(X1,X2,X3)
       U11#(ok(X1),ok(X2),ok(X3)) -> U11#(X1,X2,X3)
      TRS:
       active(U11(tt(),N,XS)) -> mark(U12(tt(),N,XS))
       active(U12(tt(),N,XS)) -> mark(snd(splitAt(N,XS)))
       active(U21(tt(),X)) -> mark(U22(tt(),X))
       active(U22(tt(),X)) -> mark(X)
       active(U31(tt(),N)) -> mark(U32(tt(),N))
       active(U32(tt(),N)) -> mark(N)
       active(U41(tt(),N,XS)) -> mark(U42(tt(),N,XS))
       active(U42(tt(),N,XS)) -> mark(head(afterNth(N,XS)))
       active(U51(tt(),Y)) -> mark(U52(tt(),Y))
       active(U52(tt(),Y)) -> mark(Y)
       active(U61(tt(),N,X,XS)) -> mark(U62(tt(),N,X,XS))
       active(U62(tt(),N,X,XS)) -> mark(U63(tt(),N,X,XS))
       active(U63(tt(),N,X,XS)) -> mark(U64(splitAt(N,XS),X))
       active(U64(pair(YS,ZS),X)) -> mark(pair(cons(X,YS),ZS))
       active(U71(tt(),XS)) -> mark(U72(tt(),XS))
       active(U72(tt(),XS)) -> mark(XS)
       active(U81(tt(),N,XS)) -> mark(U82(tt(),N,XS))
       active(U82(tt(),N,XS)) -> mark(fst(splitAt(N,XS)))
       active(afterNth(N,XS)) -> mark(U11(tt(),N,XS))
       active(fst(pair(X,Y))) -> mark(U21(tt(),X))
       active(head(cons(N,XS))) -> mark(U31(tt(),N))
       active(natsFrom(N)) -> mark(cons(N,natsFrom(s(N))))
       active(sel(N,XS)) -> mark(U41(tt(),N,XS))
       active(snd(pair(X,Y))) -> mark(U51(tt(),Y))
       active(splitAt(0(),XS)) -> mark(pair(nil(),XS))
       active(splitAt(s(N),cons(X,XS))) -> mark(U61(tt(),N,X,XS))
       active(tail(cons(N,XS))) -> mark(U71(tt(),XS))
       active(take(N,XS)) -> mark(U81(tt(),N,XS))
       active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3)
       active(U12(X1,X2,X3)) -> U12(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(U21(X1,X2)) -> U21(active(X1),X2)
       active(U22(X1,X2)) -> U22(active(X1),X2)
       active(U31(X1,X2)) -> U31(active(X1),X2)
       active(U32(X1,X2)) -> U32(active(X1),X2)
       active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3)
       active(U42(X1,X2,X3)) -> U42(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(U51(X1,X2)) -> U51(active(X1),X2)
       active(U52(X1,X2)) -> U52(active(X1),X2)
       active(U61(X1,X2,X3,X4)) -> U61(active(X1),X2,X3,X4)
       active(U62(X1,X2,X3,X4)) -> U62(active(X1),X2,X3,X4)
       active(U63(X1,X2,X3,X4)) -> U63(active(X1),X2,X3,X4)
       active(U64(X1,X2)) -> U64(active(X1),X2)
       active(pair(X1,X2)) -> pair(active(X1),X2)
       active(pair(X1,X2)) -> pair(X1,active(X2))
       active(cons(X1,X2)) -> cons(active(X1),X2)
       active(U71(X1,X2)) -> U71(active(X1),X2)
       active(U72(X1,X2)) -> U72(active(X1),X2)
       active(U81(X1,X2,X3)) -> U81(active(X1),X2,X3)
       active(U82(X1,X2,X3)) -> U82(active(X1),X2,X3)
       active(fst(X)) -> fst(active(X))
       active(natsFrom(X)) -> natsFrom(active(X))
       active(s(X)) -> s(active(X))
       active(sel(X1,X2)) -> sel(active(X1),X2)
       active(sel(X1,X2)) -> sel(X1,active(X2))
       active(tail(X)) -> tail(active(X))
       active(take(X1,X2)) -> take(active(X1),X2)
       active(take(X1,X2)) -> take(X1,active(X2))
       U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3))
       U12(mark(X1),X2,X3) -> mark(U12(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))
       U21(mark(X1),X2) -> mark(U21(X1,X2))
       U22(mark(X1),X2) -> mark(U22(X1,X2))
       U31(mark(X1),X2) -> mark(U31(X1,X2))
       U32(mark(X1),X2) -> mark(U32(X1,X2))
       U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3))
       U42(mark(X1),X2,X3) -> mark(U42(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))
       U51(mark(X1),X2) -> mark(U51(X1,X2))
       U52(mark(X1),X2) -> mark(U52(X1,X2))
       U61(mark(X1),X2,X3,X4) -> mark(U61(X1,X2,X3,X4))
       U62(mark(X1),X2,X3,X4) -> mark(U62(X1,X2,X3,X4))
       U63(mark(X1),X2,X3,X4) -> mark(U63(X1,X2,X3,X4))
       U64(mark(X1),X2) -> mark(U64(X1,X2))
       pair(mark(X1),X2) -> mark(pair(X1,X2))
       pair(X1,mark(X2)) -> mark(pair(X1,X2))
       cons(mark(X1),X2) -> mark(cons(X1,X2))
       U71(mark(X1),X2) -> mark(U71(X1,X2))
       U72(mark(X1),X2) -> mark(U72(X1,X2))
       U81(mark(X1),X2,X3) -> mark(U81(X1,X2,X3))
       U82(mark(X1),X2,X3) -> mark(U82(X1,X2,X3))
       fst(mark(X)) -> mark(fst(X))
       natsFrom(mark(X)) -> mark(natsFrom(X))
       s(mark(X)) -> mark(s(X))
       sel(mark(X1),X2) -> mark(sel(X1,X2))
       sel(X1,mark(X2)) -> mark(sel(X1,X2))
       tail(mark(X)) -> mark(tail(X))
       take(mark(X1),X2) -> mark(take(X1,X2))
       take(X1,mark(X2)) -> mark(take(X1,X2))
       proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3))
       proper(tt()) -> ok(tt())
       proper(U12(X1,X2,X3)) -> U12(proper(X1),proper(X2),proper(X3))
       proper(snd(X)) -> snd(proper(X))
       proper(splitAt(X1,X2)) -> splitAt(proper(X1),proper(X2))
       proper(U21(X1,X2)) -> U21(proper(X1),proper(X2))
       proper(U22(X1,X2)) -> U22(proper(X1),proper(X2))
       proper(U31(X1,X2)) -> U31(proper(X1),proper(X2))
       proper(U32(X1,X2)) -> U32(proper(X1),proper(X2))
       proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3))
       proper(U42(X1,X2,X3)) -> U42(proper(X1),proper(X2),proper(X3))
       proper(head(X)) -> head(proper(X))
       proper(afterNth(X1,X2)) -> afterNth(proper(X1),proper(X2))
       proper(U51(X1,X2)) -> U51(proper(X1),proper(X2))
       proper(U52(X1,X2)) -> U52(proper(X1),proper(X2))
       proper(U61(X1,X2,X3,X4)) -> U61(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(U62(X1,X2,X3,X4)) -> U62(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(U63(X1,X2,X3,X4)) -> U63(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(U64(X1,X2)) -> U64(proper(X1),proper(X2))
       proper(pair(X1,X2)) -> pair(proper(X1),proper(X2))
       proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
       proper(U71(X1,X2)) -> U71(proper(X1),proper(X2))
       proper(U72(X1,X2)) -> U72(proper(X1),proper(X2))
       proper(U81(X1,X2,X3)) -> U81(proper(X1),proper(X2),proper(X3))
       proper(U82(X1,X2,X3)) -> U82(proper(X1),proper(X2),proper(X3))
       proper(fst(X)) -> fst(proper(X))
       proper(natsFrom(X)) -> natsFrom(proper(X))
       proper(s(X)) -> s(proper(X))
       proper(sel(X1,X2)) -> sel(proper(X1),proper(X2))
       proper(0()) -> ok(0())
       proper(nil()) -> ok(nil())
       proper(tail(X)) -> tail(proper(X))
       proper(take(X1,X2)) -> take(proper(X1),proper(X2))
       U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3))
       U12(ok(X1),ok(X2),ok(X3)) -> ok(U12(X1,X2,X3))
       snd(ok(X)) -> ok(snd(X))
       splitAt(ok(X1),ok(X2)) -> ok(splitAt(X1,X2))
       U21(ok(X1),ok(X2)) -> ok(U21(X1,X2))
       U22(ok(X1),ok(X2)) -> ok(U22(X1,X2))
       U31(ok(X1),ok(X2)) -> ok(U31(X1,X2))
       U32(ok(X1),ok(X2)) -> ok(U32(X1,X2))
       U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3))
       U42(ok(X1),ok(X2),ok(X3)) -> ok(U42(X1,X2,X3))
       head(ok(X)) -> ok(head(X))
       afterNth(ok(X1),ok(X2)) -> ok(afterNth(X1,X2))
       U51(ok(X1),ok(X2)) -> ok(U51(X1,X2))
       U52(ok(X1),ok(X2)) -> ok(U52(X1,X2))
       U61(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U61(X1,X2,X3,X4))
       U62(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U62(X1,X2,X3,X4))
       U63(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U63(X1,X2,X3,X4))
       U64(ok(X1),ok(X2)) -> ok(U64(X1,X2))
       pair(ok(X1),ok(X2)) -> ok(pair(X1,X2))
       cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
       U71(ok(X1),ok(X2)) -> ok(U71(X1,X2))
       U72(ok(X1),ok(X2)) -> ok(U72(X1,X2))
       U81(ok(X1),ok(X2),ok(X3)) -> ok(U81(X1,X2,X3))
       U82(ok(X1),ok(X2),ok(X3)) -> ok(U82(X1,X2,X3))
       fst(ok(X)) -> ok(fst(X))
       natsFrom(ok(X)) -> ok(natsFrom(X))
       s(ok(X)) -> ok(s(X))
       sel(ok(X1),ok(X2)) -> ok(sel(X1,X2))
       tail(ok(X)) -> ok(tail(X))
       take(ok(X1),ok(X2)) -> ok(take(X1,X2))
       top(mark(X)) -> top(proper(X))
       top(ok(X)) -> top(active(X))
      Open