MAYBE

Problem:
 active(zeros()) -> mark(cons(0(),zeros()))
 active(U11(tt())) -> mark(tt())
 active(U21(tt())) -> mark(tt())
 active(U31(tt())) -> mark(tt())
 active(U41(tt(),V2)) -> mark(U42(isNatIList(V2)))
 active(U42(tt())) -> mark(tt())
 active(U51(tt(),V2)) -> mark(U52(isNatList(V2)))
 active(U52(tt())) -> mark(tt())
 active(U61(tt(),V2)) -> mark(U62(isNatIList(V2)))
 active(U62(tt())) -> mark(tt())
 active(U71(tt(),L,N)) -> mark(U72(isNat(N),L))
 active(U72(tt(),L)) -> mark(s(length(L)))
 active(U81(tt())) -> mark(nil())
 active(U91(tt(),IL,M,N)) -> mark(U92(isNat(M),IL,M,N))
 active(U92(tt(),IL,M,N)) -> mark(U93(isNat(N),IL,M,N))
 active(U93(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
 active(isNat(0())) -> mark(tt())
 active(isNat(length(V1))) -> mark(U11(isNatList(V1)))
 active(isNat(s(V1))) -> mark(U21(isNat(V1)))
 active(isNatIList(V)) -> mark(U31(isNatList(V)))
 active(isNatIList(zeros())) -> mark(tt())
 active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2))
 active(isNatList(nil())) -> mark(tt())
 active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2))
 active(isNatList(take(V1,V2))) -> mark(U61(isNat(V1),V2))
 active(length(nil())) -> mark(0())
 active(length(cons(N,L))) -> mark(U71(isNatList(L),L,N))
 active(take(0(),IL)) -> mark(U81(isNatIList(IL)))
 active(take(s(M),cons(N,IL))) -> mark(U91(isNatIList(IL),IL,M,N))
 active(cons(X1,X2)) -> cons(active(X1),X2)
 active(U11(X)) -> U11(active(X))
 active(U21(X)) -> U21(active(X))
 active(U31(X)) -> U31(active(X))
 active(U41(X1,X2)) -> U41(active(X1),X2)
 active(U42(X)) -> U42(active(X))
 active(U51(X1,X2)) -> U51(active(X1),X2)
 active(U52(X)) -> U52(active(X))
 active(U61(X1,X2)) -> U61(active(X1),X2)
 active(U62(X)) -> U62(active(X))
 active(U71(X1,X2,X3)) -> U71(active(X1),X2,X3)
 active(U72(X1,X2)) -> U72(active(X1),X2)
 active(s(X)) -> s(active(X))
 active(length(X)) -> length(active(X))
 active(U81(X)) -> U81(active(X))
 active(U91(X1,X2,X3,X4)) -> U91(active(X1),X2,X3,X4)
 active(U92(X1,X2,X3,X4)) -> U92(active(X1),X2,X3,X4)
 active(U93(X1,X2,X3,X4)) -> U93(active(X1),X2,X3,X4)
 active(take(X1,X2)) -> take(active(X1),X2)
 active(take(X1,X2)) -> take(X1,active(X2))
 cons(mark(X1),X2) -> mark(cons(X1,X2))
 U11(mark(X)) -> mark(U11(X))
 U21(mark(X)) -> mark(U21(X))
 U31(mark(X)) -> mark(U31(X))
 U41(mark(X1),X2) -> mark(U41(X1,X2))
 U42(mark(X)) -> mark(U42(X))
 U51(mark(X1),X2) -> mark(U51(X1,X2))
 U52(mark(X)) -> mark(U52(X))
 U61(mark(X1),X2) -> mark(U61(X1,X2))
 U62(mark(X)) -> mark(U62(X))
 U71(mark(X1),X2,X3) -> mark(U71(X1,X2,X3))
 U72(mark(X1),X2) -> mark(U72(X1,X2))
 s(mark(X)) -> mark(s(X))
 length(mark(X)) -> mark(length(X))
 U81(mark(X)) -> mark(U81(X))
 U91(mark(X1),X2,X3,X4) -> mark(U91(X1,X2,X3,X4))
 U92(mark(X1),X2,X3,X4) -> mark(U92(X1,X2,X3,X4))
 U93(mark(X1),X2,X3,X4) -> mark(U93(X1,X2,X3,X4))
 take(mark(X1),X2) -> mark(take(X1,X2))
 take(X1,mark(X2)) -> mark(take(X1,X2))
 proper(zeros()) -> ok(zeros())
 proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
 proper(0()) -> ok(0())
 proper(U11(X)) -> U11(proper(X))
 proper(tt()) -> ok(tt())
 proper(U21(X)) -> U21(proper(X))
 proper(U31(X)) -> U31(proper(X))
 proper(U41(X1,X2)) -> U41(proper(X1),proper(X2))
 proper(U42(X)) -> U42(proper(X))
 proper(isNatIList(X)) -> isNatIList(proper(X))
 proper(U51(X1,X2)) -> U51(proper(X1),proper(X2))
 proper(U52(X)) -> U52(proper(X))
 proper(isNatList(X)) -> isNatList(proper(X))
 proper(U61(X1,X2)) -> U61(proper(X1),proper(X2))
 proper(U62(X)) -> U62(proper(X))
 proper(U71(X1,X2,X3)) -> U71(proper(X1),proper(X2),proper(X3))
 proper(U72(X1,X2)) -> U72(proper(X1),proper(X2))
 proper(isNat(X)) -> isNat(proper(X))
 proper(s(X)) -> s(proper(X))
 proper(length(X)) -> length(proper(X))
 proper(U81(X)) -> U81(proper(X))
 proper(nil()) -> ok(nil())
 proper(U91(X1,X2,X3,X4)) -> U91(proper(X1),proper(X2),proper(X3),proper(X4))
 proper(U92(X1,X2,X3,X4)) -> U92(proper(X1),proper(X2),proper(X3),proper(X4))
 proper(U93(X1,X2,X3,X4)) -> U93(proper(X1),proper(X2),proper(X3),proper(X4))
 proper(take(X1,X2)) -> take(proper(X1),proper(X2))
 cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
 U11(ok(X)) -> ok(U11(X))
 U21(ok(X)) -> ok(U21(X))
 U31(ok(X)) -> ok(U31(X))
 U41(ok(X1),ok(X2)) -> ok(U41(X1,X2))
 U42(ok(X)) -> ok(U42(X))
 isNatIList(ok(X)) -> ok(isNatIList(X))
 U51(ok(X1),ok(X2)) -> ok(U51(X1,X2))
 U52(ok(X)) -> ok(U52(X))
 isNatList(ok(X)) -> ok(isNatList(X))
 U61(ok(X1),ok(X2)) -> ok(U61(X1,X2))
 U62(ok(X)) -> ok(U62(X))
 U71(ok(X1),ok(X2),ok(X3)) -> ok(U71(X1,X2,X3))
 U72(ok(X1),ok(X2)) -> ok(U72(X1,X2))
 isNat(ok(X)) -> ok(isNat(X))
 s(ok(X)) -> ok(s(X))
 length(ok(X)) -> ok(length(X))
 U81(ok(X)) -> ok(U81(X))
 U91(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U91(X1,X2,X3,X4))
 U92(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U92(X1,X2,X3,X4))
 U93(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U93(X1,X2,X3,X4))
 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#(zeros()) -> cons#(0(),zeros())
   active#(U41(tt(),V2)) -> isNatIList#(V2)
   active#(U41(tt(),V2)) -> U42#(isNatIList(V2))
   active#(U51(tt(),V2)) -> isNatList#(V2)
   active#(U51(tt(),V2)) -> U52#(isNatList(V2))
   active#(U61(tt(),V2)) -> isNatIList#(V2)
   active#(U61(tt(),V2)) -> U62#(isNatIList(V2))
   active#(U71(tt(),L,N)) -> isNat#(N)
   active#(U71(tt(),L,N)) -> U72#(isNat(N),L)
   active#(U72(tt(),L)) -> length#(L)
   active#(U72(tt(),L)) -> s#(length(L))
   active#(U91(tt(),IL,M,N)) -> isNat#(M)
   active#(U91(tt(),IL,M,N)) -> U92#(isNat(M),IL,M,N)
   active#(U92(tt(),IL,M,N)) -> isNat#(N)
   active#(U92(tt(),IL,M,N)) -> U93#(isNat(N),IL,M,N)
   active#(U93(tt(),IL,M,N)) -> take#(M,IL)
   active#(U93(tt(),IL,M,N)) -> cons#(N,take(M,IL))
   active#(isNat(length(V1))) -> isNatList#(V1)
   active#(isNat(length(V1))) -> U11#(isNatList(V1))
   active#(isNat(s(V1))) -> isNat#(V1)
   active#(isNat(s(V1))) -> U21#(isNat(V1))
   active#(isNatIList(V)) -> isNatList#(V)
   active#(isNatIList(V)) -> U31#(isNatList(V))
   active#(isNatIList(cons(V1,V2))) -> isNat#(V1)
   active#(isNatIList(cons(V1,V2))) -> U41#(isNat(V1),V2)
   active#(isNatList(cons(V1,V2))) -> isNat#(V1)
   active#(isNatList(cons(V1,V2))) -> U51#(isNat(V1),V2)
   active#(isNatList(take(V1,V2))) -> isNat#(V1)
   active#(isNatList(take(V1,V2))) -> U61#(isNat(V1),V2)
   active#(length(cons(N,L))) -> isNatList#(L)
   active#(length(cons(N,L))) -> U71#(isNatList(L),L,N)
   active#(take(0(),IL)) -> isNatIList#(IL)
   active#(take(0(),IL)) -> U81#(isNatIList(IL))
   active#(take(s(M),cons(N,IL))) -> isNatIList#(IL)
   active#(take(s(M),cons(N,IL))) -> U91#(isNatIList(IL),IL,M,N)
   active#(cons(X1,X2)) -> active#(X1)
   active#(cons(X1,X2)) -> cons#(active(X1),X2)
   active#(U11(X)) -> active#(X)
   active#(U11(X)) -> U11#(active(X))
   active#(U21(X)) -> active#(X)
   active#(U21(X)) -> U21#(active(X))
   active#(U31(X)) -> active#(X)
   active#(U31(X)) -> U31#(active(X))
   active#(U41(X1,X2)) -> active#(X1)
   active#(U41(X1,X2)) -> U41#(active(X1),X2)
   active#(U42(X)) -> active#(X)
   active#(U42(X)) -> U42#(active(X))
   active#(U51(X1,X2)) -> active#(X1)
   active#(U51(X1,X2)) -> U51#(active(X1),X2)
   active#(U52(X)) -> active#(X)
   active#(U52(X)) -> U52#(active(X))
   active#(U61(X1,X2)) -> active#(X1)
   active#(U61(X1,X2)) -> U61#(active(X1),X2)
   active#(U62(X)) -> active#(X)
   active#(U62(X)) -> U62#(active(X))
   active#(U71(X1,X2,X3)) -> active#(X1)
   active#(U71(X1,X2,X3)) -> U71#(active(X1),X2,X3)
   active#(U72(X1,X2)) -> active#(X1)
   active#(U72(X1,X2)) -> U72#(active(X1),X2)
   active#(s(X)) -> active#(X)
   active#(s(X)) -> s#(active(X))
   active#(length(X)) -> active#(X)
   active#(length(X)) -> length#(active(X))
   active#(U81(X)) -> active#(X)
   active#(U81(X)) -> U81#(active(X))
   active#(U91(X1,X2,X3,X4)) -> active#(X1)
   active#(U91(X1,X2,X3,X4)) -> U91#(active(X1),X2,X3,X4)
   active#(U92(X1,X2,X3,X4)) -> active#(X1)
   active#(U92(X1,X2,X3,X4)) -> U92#(active(X1),X2,X3,X4)
   active#(U93(X1,X2,X3,X4)) -> active#(X1)
   active#(U93(X1,X2,X3,X4)) -> U93#(active(X1),X2,X3,X4)
   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))
   cons#(mark(X1),X2) -> cons#(X1,X2)
   U11#(mark(X)) -> U11#(X)
   U21#(mark(X)) -> U21#(X)
   U31#(mark(X)) -> U31#(X)
   U41#(mark(X1),X2) -> U41#(X1,X2)
   U42#(mark(X)) -> U42#(X)
   U51#(mark(X1),X2) -> U51#(X1,X2)
   U52#(mark(X)) -> U52#(X)
   U61#(mark(X1),X2) -> U61#(X1,X2)
   U62#(mark(X)) -> U62#(X)
   U71#(mark(X1),X2,X3) -> U71#(X1,X2,X3)
   U72#(mark(X1),X2) -> U72#(X1,X2)
   s#(mark(X)) -> s#(X)
   length#(mark(X)) -> length#(X)
   U81#(mark(X)) -> U81#(X)
   U91#(mark(X1),X2,X3,X4) -> U91#(X1,X2,X3,X4)
   U92#(mark(X1),X2,X3,X4) -> U92#(X1,X2,X3,X4)
   U93#(mark(X1),X2,X3,X4) -> U93#(X1,X2,X3,X4)
   take#(mark(X1),X2) -> take#(X1,X2)
   take#(X1,mark(X2)) -> take#(X1,X2)
   proper#(cons(X1,X2)) -> proper#(X2)
   proper#(cons(X1,X2)) -> proper#(X1)
   proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2))
   proper#(U11(X)) -> proper#(X)
   proper#(U11(X)) -> U11#(proper(X))
   proper#(U21(X)) -> proper#(X)
   proper#(U21(X)) -> U21#(proper(X))
   proper#(U31(X)) -> proper#(X)
   proper#(U31(X)) -> U31#(proper(X))
   proper#(U41(X1,X2)) -> proper#(X2)
   proper#(U41(X1,X2)) -> proper#(X1)
   proper#(U41(X1,X2)) -> U41#(proper(X1),proper(X2))
   proper#(U42(X)) -> proper#(X)
   proper#(U42(X)) -> U42#(proper(X))
   proper#(isNatIList(X)) -> proper#(X)
   proper#(isNatIList(X)) -> isNatIList#(proper(X))
   proper#(U51(X1,X2)) -> proper#(X2)
   proper#(U51(X1,X2)) -> proper#(X1)
   proper#(U51(X1,X2)) -> U51#(proper(X1),proper(X2))
   proper#(U52(X)) -> proper#(X)
   proper#(U52(X)) -> U52#(proper(X))
   proper#(isNatList(X)) -> proper#(X)
   proper#(isNatList(X)) -> isNatList#(proper(X))
   proper#(U61(X1,X2)) -> proper#(X2)
   proper#(U61(X1,X2)) -> proper#(X1)
   proper#(U61(X1,X2)) -> U61#(proper(X1),proper(X2))
   proper#(U62(X)) -> proper#(X)
   proper#(U62(X)) -> U62#(proper(X))
   proper#(U71(X1,X2,X3)) -> proper#(X3)
   proper#(U71(X1,X2,X3)) -> proper#(X2)
   proper#(U71(X1,X2,X3)) -> proper#(X1)
   proper#(U71(X1,X2,X3)) -> U71#(proper(X1),proper(X2),proper(X3))
   proper#(U72(X1,X2)) -> proper#(X2)
   proper#(U72(X1,X2)) -> proper#(X1)
   proper#(U72(X1,X2)) -> U72#(proper(X1),proper(X2))
   proper#(isNat(X)) -> proper#(X)
   proper#(isNat(X)) -> isNat#(proper(X))
   proper#(s(X)) -> proper#(X)
   proper#(s(X)) -> s#(proper(X))
   proper#(length(X)) -> proper#(X)
   proper#(length(X)) -> length#(proper(X))
   proper#(U81(X)) -> proper#(X)
   proper#(U81(X)) -> U81#(proper(X))
   proper#(U91(X1,X2,X3,X4)) -> proper#(X4)
   proper#(U91(X1,X2,X3,X4)) -> proper#(X3)
   proper#(U91(X1,X2,X3,X4)) -> proper#(X2)
   proper#(U91(X1,X2,X3,X4)) -> proper#(X1)
   proper#(U91(X1,X2,X3,X4)) -> U91#(proper(X1),proper(X2),proper(X3),proper(X4))
   proper#(U92(X1,X2,X3,X4)) -> proper#(X4)
   proper#(U92(X1,X2,X3,X4)) -> proper#(X3)
   proper#(U92(X1,X2,X3,X4)) -> proper#(X2)
   proper#(U92(X1,X2,X3,X4)) -> proper#(X1)
   proper#(U92(X1,X2,X3,X4)) -> U92#(proper(X1),proper(X2),proper(X3),proper(X4))
   proper#(U93(X1,X2,X3,X4)) -> proper#(X4)
   proper#(U93(X1,X2,X3,X4)) -> proper#(X3)
   proper#(U93(X1,X2,X3,X4)) -> proper#(X2)
   proper#(U93(X1,X2,X3,X4)) -> proper#(X1)
   proper#(U93(X1,X2,X3,X4)) -> U93#(proper(X1),proper(X2),proper(X3),proper(X4))
   proper#(take(X1,X2)) -> proper#(X2)
   proper#(take(X1,X2)) -> proper#(X1)
   proper#(take(X1,X2)) -> take#(proper(X1),proper(X2))
   cons#(ok(X1),ok(X2)) -> cons#(X1,X2)
   U11#(ok(X)) -> U11#(X)
   U21#(ok(X)) -> U21#(X)
   U31#(ok(X)) -> U31#(X)
   U41#(ok(X1),ok(X2)) -> U41#(X1,X2)
   U42#(ok(X)) -> U42#(X)
   isNatIList#(ok(X)) -> isNatIList#(X)
   U51#(ok(X1),ok(X2)) -> U51#(X1,X2)
   U52#(ok(X)) -> U52#(X)
   isNatList#(ok(X)) -> isNatList#(X)
   U61#(ok(X1),ok(X2)) -> U61#(X1,X2)
   U62#(ok(X)) -> U62#(X)
   U71#(ok(X1),ok(X2),ok(X3)) -> U71#(X1,X2,X3)
   U72#(ok(X1),ok(X2)) -> U72#(X1,X2)
   isNat#(ok(X)) -> isNat#(X)
   s#(ok(X)) -> s#(X)
   length#(ok(X)) -> length#(X)
   U81#(ok(X)) -> U81#(X)
   U91#(ok(X1),ok(X2),ok(X3),ok(X4)) -> U91#(X1,X2,X3,X4)
   U92#(ok(X1),ok(X2),ok(X3),ok(X4)) -> U92#(X1,X2,X3,X4)
   U93#(ok(X1),ok(X2),ok(X3),ok(X4)) -> U93#(X1,X2,X3,X4)
   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(zeros()) -> mark(cons(0(),zeros()))
   active(U11(tt())) -> mark(tt())
   active(U21(tt())) -> mark(tt())
   active(U31(tt())) -> mark(tt())
   active(U41(tt(),V2)) -> mark(U42(isNatIList(V2)))
   active(U42(tt())) -> mark(tt())
   active(U51(tt(),V2)) -> mark(U52(isNatList(V2)))
   active(U52(tt())) -> mark(tt())
   active(U61(tt(),V2)) -> mark(U62(isNatIList(V2)))
   active(U62(tt())) -> mark(tt())
   active(U71(tt(),L,N)) -> mark(U72(isNat(N),L))
   active(U72(tt(),L)) -> mark(s(length(L)))
   active(U81(tt())) -> mark(nil())
   active(U91(tt(),IL,M,N)) -> mark(U92(isNat(M),IL,M,N))
   active(U92(tt(),IL,M,N)) -> mark(U93(isNat(N),IL,M,N))
   active(U93(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
   active(isNat(0())) -> mark(tt())
   active(isNat(length(V1))) -> mark(U11(isNatList(V1)))
   active(isNat(s(V1))) -> mark(U21(isNat(V1)))
   active(isNatIList(V)) -> mark(U31(isNatList(V)))
   active(isNatIList(zeros())) -> mark(tt())
   active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2))
   active(isNatList(nil())) -> mark(tt())
   active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2))
   active(isNatList(take(V1,V2))) -> mark(U61(isNat(V1),V2))
   active(length(nil())) -> mark(0())
   active(length(cons(N,L))) -> mark(U71(isNatList(L),L,N))
   active(take(0(),IL)) -> mark(U81(isNatIList(IL)))
   active(take(s(M),cons(N,IL))) -> mark(U91(isNatIList(IL),IL,M,N))
   active(cons(X1,X2)) -> cons(active(X1),X2)
   active(U11(X)) -> U11(active(X))
   active(U21(X)) -> U21(active(X))
   active(U31(X)) -> U31(active(X))
   active(U41(X1,X2)) -> U41(active(X1),X2)
   active(U42(X)) -> U42(active(X))
   active(U51(X1,X2)) -> U51(active(X1),X2)
   active(U52(X)) -> U52(active(X))
   active(U61(X1,X2)) -> U61(active(X1),X2)
   active(U62(X)) -> U62(active(X))
   active(U71(X1,X2,X3)) -> U71(active(X1),X2,X3)
   active(U72(X1,X2)) -> U72(active(X1),X2)
   active(s(X)) -> s(active(X))
   active(length(X)) -> length(active(X))
   active(U81(X)) -> U81(active(X))
   active(U91(X1,X2,X3,X4)) -> U91(active(X1),X2,X3,X4)
   active(U92(X1,X2,X3,X4)) -> U92(active(X1),X2,X3,X4)
   active(U93(X1,X2,X3,X4)) -> U93(active(X1),X2,X3,X4)
   active(take(X1,X2)) -> take(active(X1),X2)
   active(take(X1,X2)) -> take(X1,active(X2))
   cons(mark(X1),X2) -> mark(cons(X1,X2))
   U11(mark(X)) -> mark(U11(X))
   U21(mark(X)) -> mark(U21(X))
   U31(mark(X)) -> mark(U31(X))
   U41(mark(X1),X2) -> mark(U41(X1,X2))
   U42(mark(X)) -> mark(U42(X))
   U51(mark(X1),X2) -> mark(U51(X1,X2))
   U52(mark(X)) -> mark(U52(X))
   U61(mark(X1),X2) -> mark(U61(X1,X2))
   U62(mark(X)) -> mark(U62(X))
   U71(mark(X1),X2,X3) -> mark(U71(X1,X2,X3))
   U72(mark(X1),X2) -> mark(U72(X1,X2))
   s(mark(X)) -> mark(s(X))
   length(mark(X)) -> mark(length(X))
   U81(mark(X)) -> mark(U81(X))
   U91(mark(X1),X2,X3,X4) -> mark(U91(X1,X2,X3,X4))
   U92(mark(X1),X2,X3,X4) -> mark(U92(X1,X2,X3,X4))
   U93(mark(X1),X2,X3,X4) -> mark(U93(X1,X2,X3,X4))
   take(mark(X1),X2) -> mark(take(X1,X2))
   take(X1,mark(X2)) -> mark(take(X1,X2))
   proper(zeros()) -> ok(zeros())
   proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
   proper(0()) -> ok(0())
   proper(U11(X)) -> U11(proper(X))
   proper(tt()) -> ok(tt())
   proper(U21(X)) -> U21(proper(X))
   proper(U31(X)) -> U31(proper(X))
   proper(U41(X1,X2)) -> U41(proper(X1),proper(X2))
   proper(U42(X)) -> U42(proper(X))
   proper(isNatIList(X)) -> isNatIList(proper(X))
   proper(U51(X1,X2)) -> U51(proper(X1),proper(X2))
   proper(U52(X)) -> U52(proper(X))
   proper(isNatList(X)) -> isNatList(proper(X))
   proper(U61(X1,X2)) -> U61(proper(X1),proper(X2))
   proper(U62(X)) -> U62(proper(X))
   proper(U71(X1,X2,X3)) -> U71(proper(X1),proper(X2),proper(X3))
   proper(U72(X1,X2)) -> U72(proper(X1),proper(X2))
   proper(isNat(X)) -> isNat(proper(X))
   proper(s(X)) -> s(proper(X))
   proper(length(X)) -> length(proper(X))
   proper(U81(X)) -> U81(proper(X))
   proper(nil()) -> ok(nil())
   proper(U91(X1,X2,X3,X4)) -> U91(proper(X1),proper(X2),proper(X3),proper(X4))
   proper(U92(X1,X2,X3,X4)) -> U92(proper(X1),proper(X2),proper(X3),proper(X4))
   proper(U93(X1,X2,X3,X4)) -> U93(proper(X1),proper(X2),proper(X3),proper(X4))
   proper(take(X1,X2)) -> take(proper(X1),proper(X2))
   cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
   U11(ok(X)) -> ok(U11(X))
   U21(ok(X)) -> ok(U21(X))
   U31(ok(X)) -> ok(U31(X))
   U41(ok(X1),ok(X2)) -> ok(U41(X1,X2))
   U42(ok(X)) -> ok(U42(X))
   isNatIList(ok(X)) -> ok(isNatIList(X))
   U51(ok(X1),ok(X2)) -> ok(U51(X1,X2))
   U52(ok(X)) -> ok(U52(X))
   isNatList(ok(X)) -> ok(isNatList(X))
   U61(ok(X1),ok(X2)) -> ok(U61(X1,X2))
   U62(ok(X)) -> ok(U62(X))
   U71(ok(X1),ok(X2),ok(X3)) -> ok(U71(X1,X2,X3))
   U72(ok(X1),ok(X2)) -> ok(U72(X1,X2))
   isNat(ok(X)) -> ok(isNat(X))
   s(ok(X)) -> ok(s(X))
   length(ok(X)) -> ok(length(X))
   U81(ok(X)) -> ok(U81(X))
   U91(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U91(X1,X2,X3,X4))
   U92(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U92(X1,X2,X3,X4))
   U93(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U93(X1,X2,X3,X4))
   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#(zeros()) -> cons#(0(),zeros())
    active#(U41(tt(),V2)) -> isNatIList#(V2)
    active#(U41(tt(),V2)) -> U42#(isNatIList(V2))
    active#(U51(tt(),V2)) -> isNatList#(V2)
    active#(U51(tt(),V2)) -> U52#(isNatList(V2))
    active#(U61(tt(),V2)) -> isNatIList#(V2)
    active#(U61(tt(),V2)) -> U62#(isNatIList(V2))
    active#(U71(tt(),L,N)) -> isNat#(N)
    active#(U71(tt(),L,N)) -> U72#(isNat(N),L)
    active#(U72(tt(),L)) -> length#(L)
    active#(U72(tt(),L)) -> s#(length(L))
    active#(U91(tt(),IL,M,N)) -> isNat#(M)
    active#(U91(tt(),IL,M,N)) -> U92#(isNat(M),IL,M,N)
    active#(U92(tt(),IL,M,N)) -> isNat#(N)
    active#(U92(tt(),IL,M,N)) -> U93#(isNat(N),IL,M,N)
    active#(U93(tt(),IL,M,N)) -> take#(M,IL)
    active#(U93(tt(),IL,M,N)) -> cons#(N,take(M,IL))
    active#(isNat(length(V1))) -> isNatList#(V1)
    active#(isNat(length(V1))) -> U11#(isNatList(V1))
    active#(isNat(s(V1))) -> isNat#(V1)
    active#(isNat(s(V1))) -> U21#(isNat(V1))
    active#(isNatIList(V)) -> isNatList#(V)
    active#(isNatIList(V)) -> U31#(isNatList(V))
    active#(isNatIList(cons(V1,V2))) -> isNat#(V1)
    active#(isNatIList(cons(V1,V2))) -> U41#(isNat(V1),V2)
    active#(isNatList(cons(V1,V2))) -> isNat#(V1)
    active#(isNatList(cons(V1,V2))) -> U51#(isNat(V1),V2)
    active#(isNatList(take(V1,V2))) -> isNat#(V1)
    active#(isNatList(take(V1,V2))) -> U61#(isNat(V1),V2)
    active#(length(cons(N,L))) -> isNatList#(L)
    active#(length(cons(N,L))) -> U71#(isNatList(L),L,N)
    active#(take(0(),IL)) -> isNatIList#(IL)
    active#(take(0(),IL)) -> U81#(isNatIList(IL))
    active#(take(s(M),cons(N,IL))) -> isNatIList#(IL)
    active#(take(s(M),cons(N,IL))) -> U91#(isNatIList(IL),IL,M,N)
    active#(cons(X1,X2)) -> active#(X1)
    active#(cons(X1,X2)) -> cons#(active(X1),X2)
    active#(U11(X)) -> active#(X)
    active#(U11(X)) -> U11#(active(X))
    active#(U21(X)) -> active#(X)
    active#(U21(X)) -> U21#(active(X))
    active#(U31(X)) -> active#(X)
    active#(U31(X)) -> U31#(active(X))
    active#(U41(X1,X2)) -> active#(X1)
    active#(U41(X1,X2)) -> U41#(active(X1),X2)
    active#(U42(X)) -> active#(X)
    active#(U42(X)) -> U42#(active(X))
    active#(U51(X1,X2)) -> active#(X1)
    active#(U51(X1,X2)) -> U51#(active(X1),X2)
    active#(U52(X)) -> active#(X)
    active#(U52(X)) -> U52#(active(X))
    active#(U61(X1,X2)) -> active#(X1)
    active#(U61(X1,X2)) -> U61#(active(X1),X2)
    active#(U62(X)) -> active#(X)
    active#(U62(X)) -> U62#(active(X))
    active#(U71(X1,X2,X3)) -> active#(X1)
    active#(U71(X1,X2,X3)) -> U71#(active(X1),X2,X3)
    active#(U72(X1,X2)) -> active#(X1)
    active#(U72(X1,X2)) -> U72#(active(X1),X2)
    active#(s(X)) -> active#(X)
    active#(s(X)) -> s#(active(X))
    active#(length(X)) -> active#(X)
    active#(length(X)) -> length#(active(X))
    active#(U81(X)) -> active#(X)
    active#(U81(X)) -> U81#(active(X))
    active#(U91(X1,X2,X3,X4)) -> active#(X1)
    active#(U91(X1,X2,X3,X4)) -> U91#(active(X1),X2,X3,X4)
    active#(U92(X1,X2,X3,X4)) -> active#(X1)
    active#(U92(X1,X2,X3,X4)) -> U92#(active(X1),X2,X3,X4)
    active#(U93(X1,X2,X3,X4)) -> active#(X1)
    active#(U93(X1,X2,X3,X4)) -> U93#(active(X1),X2,X3,X4)
    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))
    cons#(mark(X1),X2) -> cons#(X1,X2)
    U11#(mark(X)) -> U11#(X)
    U21#(mark(X)) -> U21#(X)
    U31#(mark(X)) -> U31#(X)
    U41#(mark(X1),X2) -> U41#(X1,X2)
    U42#(mark(X)) -> U42#(X)
    U51#(mark(X1),X2) -> U51#(X1,X2)
    U52#(mark(X)) -> U52#(X)
    U61#(mark(X1),X2) -> U61#(X1,X2)
    U62#(mark(X)) -> U62#(X)
    U71#(mark(X1),X2,X3) -> U71#(X1,X2,X3)
    U72#(mark(X1),X2) -> U72#(X1,X2)
    s#(mark(X)) -> s#(X)
    length#(mark(X)) -> length#(X)
    U81#(mark(X)) -> U81#(X)
    U91#(mark(X1),X2,X3,X4) -> U91#(X1,X2,X3,X4)
    U92#(mark(X1),X2,X3,X4) -> U92#(X1,X2,X3,X4)
    U93#(mark(X1),X2,X3,X4) -> U93#(X1,X2,X3,X4)
    take#(mark(X1),X2) -> take#(X1,X2)
    take#(X1,mark(X2)) -> take#(X1,X2)
    proper#(cons(X1,X2)) -> proper#(X2)
    proper#(cons(X1,X2)) -> proper#(X1)
    proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2))
    proper#(U11(X)) -> proper#(X)
    proper#(U11(X)) -> U11#(proper(X))
    proper#(U21(X)) -> proper#(X)
    proper#(U21(X)) -> U21#(proper(X))
    proper#(U31(X)) -> proper#(X)
    proper#(U31(X)) -> U31#(proper(X))
    proper#(U41(X1,X2)) -> proper#(X2)
    proper#(U41(X1,X2)) -> proper#(X1)
    proper#(U41(X1,X2)) -> U41#(proper(X1),proper(X2))
    proper#(U42(X)) -> proper#(X)
    proper#(U42(X)) -> U42#(proper(X))
    proper#(isNatIList(X)) -> proper#(X)
    proper#(isNatIList(X)) -> isNatIList#(proper(X))
    proper#(U51(X1,X2)) -> proper#(X2)
    proper#(U51(X1,X2)) -> proper#(X1)
    proper#(U51(X1,X2)) -> U51#(proper(X1),proper(X2))
    proper#(U52(X)) -> proper#(X)
    proper#(U52(X)) -> U52#(proper(X))
    proper#(isNatList(X)) -> proper#(X)
    proper#(isNatList(X)) -> isNatList#(proper(X))
    proper#(U61(X1,X2)) -> proper#(X2)
    proper#(U61(X1,X2)) -> proper#(X1)
    proper#(U61(X1,X2)) -> U61#(proper(X1),proper(X2))
    proper#(U62(X)) -> proper#(X)
    proper#(U62(X)) -> U62#(proper(X))
    proper#(U71(X1,X2,X3)) -> proper#(X3)
    proper#(U71(X1,X2,X3)) -> proper#(X2)
    proper#(U71(X1,X2,X3)) -> proper#(X1)
    proper#(U71(X1,X2,X3)) -> U71#(proper(X1),proper(X2),proper(X3))
    proper#(U72(X1,X2)) -> proper#(X2)
    proper#(U72(X1,X2)) -> proper#(X1)
    proper#(U72(X1,X2)) -> U72#(proper(X1),proper(X2))
    proper#(isNat(X)) -> proper#(X)
    proper#(isNat(X)) -> isNat#(proper(X))
    proper#(s(X)) -> proper#(X)
    proper#(s(X)) -> s#(proper(X))
    proper#(length(X)) -> proper#(X)
    proper#(length(X)) -> length#(proper(X))
    proper#(U81(X)) -> proper#(X)
    proper#(U81(X)) -> U81#(proper(X))
    proper#(U91(X1,X2,X3,X4)) -> proper#(X4)
    proper#(U91(X1,X2,X3,X4)) -> proper#(X3)
    proper#(U91(X1,X2,X3,X4)) -> proper#(X2)
    proper#(U91(X1,X2,X3,X4)) -> proper#(X1)
    proper#(U91(X1,X2,X3,X4)) -> U91#(proper(X1),proper(X2),proper(X3),proper(X4))
    proper#(U92(X1,X2,X3,X4)) -> proper#(X4)
    proper#(U92(X1,X2,X3,X4)) -> proper#(X3)
    proper#(U92(X1,X2,X3,X4)) -> proper#(X2)
    proper#(U92(X1,X2,X3,X4)) -> proper#(X1)
    proper#(U92(X1,X2,X3,X4)) -> U92#(proper(X1),proper(X2),proper(X3),proper(X4))
    proper#(U93(X1,X2,X3,X4)) -> proper#(X4)
    proper#(U93(X1,X2,X3,X4)) -> proper#(X3)
    proper#(U93(X1,X2,X3,X4)) -> proper#(X2)
    proper#(U93(X1,X2,X3,X4)) -> proper#(X1)
    proper#(U93(X1,X2,X3,X4)) -> U93#(proper(X1),proper(X2),proper(X3),proper(X4))
    proper#(take(X1,X2)) -> proper#(X2)
    proper#(take(X1,X2)) -> proper#(X1)
    proper#(take(X1,X2)) -> take#(proper(X1),proper(X2))
    cons#(ok(X1),ok(X2)) -> cons#(X1,X2)
    U11#(ok(X)) -> U11#(X)
    U21#(ok(X)) -> U21#(X)
    U31#(ok(X)) -> U31#(X)
    U41#(ok(X1),ok(X2)) -> U41#(X1,X2)
    U42#(ok(X)) -> U42#(X)
    isNatIList#(ok(X)) -> isNatIList#(X)
    U51#(ok(X1),ok(X2)) -> U51#(X1,X2)
    U52#(ok(X)) -> U52#(X)
    isNatList#(ok(X)) -> isNatList#(X)
    U61#(ok(X1),ok(X2)) -> U61#(X1,X2)
    U62#(ok(X)) -> U62#(X)
    U71#(ok(X1),ok(X2),ok(X3)) -> U71#(X1,X2,X3)
    U72#(ok(X1),ok(X2)) -> U72#(X1,X2)
    isNat#(ok(X)) -> isNat#(X)
    s#(ok(X)) -> s#(X)
    length#(ok(X)) -> length#(X)
    U81#(ok(X)) -> U81#(X)
    U91#(ok(X1),ok(X2),ok(X3),ok(X4)) -> U91#(X1,X2,X3,X4)
    U92#(ok(X1),ok(X2),ok(X3),ok(X4)) -> U92#(X1,X2,X3,X4)
    U93#(ok(X1),ok(X2),ok(X3),ok(X4)) -> U93#(X1,X2,X3,X4)
    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:
    f56(x,y) -> x
    f56(x,y) -> y
    isNatIList(ok(X)) -> ok(isNatIList(X))
    isNatList(ok(X)) -> ok(isNatList(X))
    isNat(ok(X)) -> ok(isNat(X))
    length(mark(X)) -> mark(length(X))
    length(ok(X)) -> ok(length(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))
    active(zeros()) -> mark(cons(0(),zeros()))
    active(U11(tt())) -> mark(tt())
    active(U21(tt())) -> mark(tt())
    active(U31(tt())) -> mark(tt())
    active(U41(tt(),V2)) -> mark(U42(isNatIList(V2)))
    active(U42(tt())) -> mark(tt())
    active(U51(tt(),V2)) -> mark(U52(isNatList(V2)))
    active(U52(tt())) -> mark(tt())
    active(U61(tt(),V2)) -> mark(U62(isNatIList(V2)))
    active(U62(tt())) -> mark(tt())
    active(U71(tt(),L,N)) -> mark(U72(isNat(N),L))
    active(U72(tt(),L)) -> mark(s(length(L)))
    active(U81(tt())) -> mark(nil())
    active(U91(tt(),IL,M,N)) -> mark(U92(isNat(M),IL,M,N))
    active(U92(tt(),IL,M,N)) -> mark(U93(isNat(N),IL,M,N))
    active(U93(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
    active(isNat(0())) -> mark(tt())
    active(isNat(length(V1))) -> mark(U11(isNatList(V1)))
    active(isNat(s(V1))) -> mark(U21(isNat(V1)))
    active(isNatIList(V)) -> mark(U31(isNatList(V)))
    active(isNatIList(zeros())) -> mark(tt())
    active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2))
    active(isNatList(nil())) -> mark(tt())
    active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2))
    active(isNatList(take(V1,V2))) -> mark(U61(isNat(V1),V2))
    active(length(nil())) -> mark(0())
    active(length(cons(N,L))) -> mark(U71(isNatList(L),L,N))
    active(take(0(),IL)) -> mark(U81(isNatIList(IL)))
    active(take(s(M),cons(N,IL))) -> mark(U91(isNatIList(IL),IL,M,N))
    active(cons(X1,X2)) -> cons(active(X1),X2)
    active(U11(X)) -> U11(active(X))
    active(U21(X)) -> U21(active(X))
    active(U31(X)) -> U31(active(X))
    active(U41(X1,X2)) -> U41(active(X1),X2)
    active(U42(X)) -> U42(active(X))
    active(U51(X1,X2)) -> U51(active(X1),X2)
    active(U52(X)) -> U52(active(X))
    active(U61(X1,X2)) -> U61(active(X1),X2)
    active(U62(X)) -> U62(active(X))
    active(U71(X1,X2,X3)) -> U71(active(X1),X2,X3)
    active(U72(X1,X2)) -> U72(active(X1),X2)
    active(s(X)) -> s(active(X))
    active(length(X)) -> length(active(X))
    active(U81(X)) -> U81(active(X))
    active(U91(X1,X2,X3,X4)) -> U91(active(X1),X2,X3,X4)
    active(U92(X1,X2,X3,X4)) -> U92(active(X1),X2,X3,X4)
    active(U93(X1,X2,X3,X4)) -> U93(active(X1),X2,X3,X4)
    active(take(X1,X2)) -> take(active(X1),X2)
    active(take(X1,X2)) -> take(X1,active(X2))
    U42(mark(X)) -> mark(U42(X))
    U42(ok(X)) -> ok(U42(X))
    U52(mark(X)) -> mark(U52(X))
    U52(ok(X)) -> ok(U52(X))
    U62(mark(X)) -> mark(U62(X))
    U62(ok(X)) -> ok(U62(X))
    U72(mark(X1),X2) -> mark(U72(X1,X2))
    U72(ok(X1),ok(X2)) -> ok(U72(X1,X2))
    s(mark(X)) -> mark(s(X))
    s(ok(X)) -> ok(s(X))
    U92(mark(X1),X2,X3,X4) -> mark(U92(X1,X2,X3,X4))
    U92(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U92(X1,X2,X3,X4))
    U93(mark(X1),X2,X3,X4) -> mark(U93(X1,X2,X3,X4))
    U93(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U93(X1,X2,X3,X4))
    cons(mark(X1),X2) -> mark(cons(X1,X2))
    cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
    U11(mark(X)) -> mark(U11(X))
    U11(ok(X)) -> ok(U11(X))
    U21(mark(X)) -> mark(U21(X))
    U21(ok(X)) -> ok(U21(X))
    U31(mark(X)) -> mark(U31(X))
    U31(ok(X)) -> ok(U31(X))
    U41(mark(X1),X2) -> mark(U41(X1,X2))
    U41(ok(X1),ok(X2)) -> ok(U41(X1,X2))
    U51(mark(X1),X2) -> mark(U51(X1,X2))
    U51(ok(X1),ok(X2)) -> ok(U51(X1,X2))
    U61(mark(X1),X2) -> mark(U61(X1,X2))
    U61(ok(X1),ok(X2)) -> ok(U61(X1,X2))
    U71(mark(X1),X2,X3) -> mark(U71(X1,X2,X3))
    U71(ok(X1),ok(X2),ok(X3)) -> ok(U71(X1,X2,X3))
    U81(mark(X)) -> mark(U81(X))
    U81(ok(X)) -> ok(U81(X))
    U91(mark(X1),X2,X3,X4) -> mark(U91(X1,X2,X3,X4))
    U91(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U91(X1,X2,X3,X4))
    proper(zeros()) -> ok(zeros())
    proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
    proper(0()) -> ok(0())
    proper(U11(X)) -> U11(proper(X))
    proper(tt()) -> ok(tt())
    proper(U21(X)) -> U21(proper(X))
    proper(U31(X)) -> U31(proper(X))
    proper(U41(X1,X2)) -> U41(proper(X1),proper(X2))
    proper(U42(X)) -> U42(proper(X))
    proper(isNatIList(X)) -> isNatIList(proper(X))
    proper(U51(X1,X2)) -> U51(proper(X1),proper(X2))
    proper(U52(X)) -> U52(proper(X))
    proper(isNatList(X)) -> isNatList(proper(X))
    proper(U61(X1,X2)) -> U61(proper(X1),proper(X2))
    proper(U62(X)) -> U62(proper(X))
    proper(U71(X1,X2,X3)) -> U71(proper(X1),proper(X2),proper(X3))
    proper(U72(X1,X2)) -> U72(proper(X1),proper(X2))
    proper(isNat(X)) -> isNat(proper(X))
    proper(s(X)) -> s(proper(X))
    proper(length(X)) -> length(proper(X))
    proper(U81(X)) -> U81(proper(X))
    proper(nil()) -> ok(nil())
    proper(U91(X1,X2,X3,X4)) -> U91(proper(X1),proper(X2),proper(X3),proper(X4))
    proper(U92(X1,X2,X3,X4)) -> U92(proper(X1),proper(X2),proper(X3),proper(X4))
    proper(U93(X1,X2,X3,X4)) -> U93(proper(X1),proper(X2),proper(X3),proper(X4))
    proper(take(X1,X2)) -> take(proper(X1),proper(X2))
   CDG Processor:
    DPs:
     active#(zeros()) -> cons#(0(),zeros())
     active#(U41(tt(),V2)) -> isNatIList#(V2)
     active#(U41(tt(),V2)) -> U42#(isNatIList(V2))
     active#(U51(tt(),V2)) -> isNatList#(V2)
     active#(U51(tt(),V2)) -> U52#(isNatList(V2))
     active#(U61(tt(),V2)) -> isNatIList#(V2)
     active#(U61(tt(),V2)) -> U62#(isNatIList(V2))
     active#(U71(tt(),L,N)) -> isNat#(N)
     active#(U71(tt(),L,N)) -> U72#(isNat(N),L)
     active#(U72(tt(),L)) -> length#(L)
     active#(U72(tt(),L)) -> s#(length(L))
     active#(U91(tt(),IL,M,N)) -> isNat#(M)
     active#(U91(tt(),IL,M,N)) -> U92#(isNat(M),IL,M,N)
     active#(U92(tt(),IL,M,N)) -> isNat#(N)
     active#(U92(tt(),IL,M,N)) -> U93#(isNat(N),IL,M,N)
     active#(U93(tt(),IL,M,N)) -> take#(M,IL)
     active#(U93(tt(),IL,M,N)) -> cons#(N,take(M,IL))
     active#(isNat(length(V1))) -> isNatList#(V1)
     active#(isNat(length(V1))) -> U11#(isNatList(V1))
     active#(isNat(s(V1))) -> isNat#(V1)
     active#(isNat(s(V1))) -> U21#(isNat(V1))
     active#(isNatIList(V)) -> isNatList#(V)
     active#(isNatIList(V)) -> U31#(isNatList(V))
     active#(isNatIList(cons(V1,V2))) -> isNat#(V1)
     active#(isNatIList(cons(V1,V2))) -> U41#(isNat(V1),V2)
     active#(isNatList(cons(V1,V2))) -> isNat#(V1)
     active#(isNatList(cons(V1,V2))) -> U51#(isNat(V1),V2)
     active#(isNatList(take(V1,V2))) -> isNat#(V1)
     active#(isNatList(take(V1,V2))) -> U61#(isNat(V1),V2)
     active#(length(cons(N,L))) -> isNatList#(L)
     active#(length(cons(N,L))) -> U71#(isNatList(L),L,N)
     active#(take(0(),IL)) -> isNatIList#(IL)
     active#(take(0(),IL)) -> U81#(isNatIList(IL))
     active#(take(s(M),cons(N,IL))) -> isNatIList#(IL)
     active#(take(s(M),cons(N,IL))) -> U91#(isNatIList(IL),IL,M,N)
     active#(cons(X1,X2)) -> active#(X1)
     active#(cons(X1,X2)) -> cons#(active(X1),X2)
     active#(U11(X)) -> active#(X)
     active#(U11(X)) -> U11#(active(X))
     active#(U21(X)) -> active#(X)
     active#(U21(X)) -> U21#(active(X))
     active#(U31(X)) -> active#(X)
     active#(U31(X)) -> U31#(active(X))
     active#(U41(X1,X2)) -> active#(X1)
     active#(U41(X1,X2)) -> U41#(active(X1),X2)
     active#(U42(X)) -> active#(X)
     active#(U42(X)) -> U42#(active(X))
     active#(U51(X1,X2)) -> active#(X1)
     active#(U51(X1,X2)) -> U51#(active(X1),X2)
     active#(U52(X)) -> active#(X)
     active#(U52(X)) -> U52#(active(X))
     active#(U61(X1,X2)) -> active#(X1)
     active#(U61(X1,X2)) -> U61#(active(X1),X2)
     active#(U62(X)) -> active#(X)
     active#(U62(X)) -> U62#(active(X))
     active#(U71(X1,X2,X3)) -> active#(X1)
     active#(U71(X1,X2,X3)) -> U71#(active(X1),X2,X3)
     active#(U72(X1,X2)) -> active#(X1)
     active#(U72(X1,X2)) -> U72#(active(X1),X2)
     active#(s(X)) -> active#(X)
     active#(s(X)) -> s#(active(X))
     active#(length(X)) -> active#(X)
     active#(length(X)) -> length#(active(X))
     active#(U81(X)) -> active#(X)
     active#(U81(X)) -> U81#(active(X))
     active#(U91(X1,X2,X3,X4)) -> active#(X1)
     active#(U91(X1,X2,X3,X4)) -> U91#(active(X1),X2,X3,X4)
     active#(U92(X1,X2,X3,X4)) -> active#(X1)
     active#(U92(X1,X2,X3,X4)) -> U92#(active(X1),X2,X3,X4)
     active#(U93(X1,X2,X3,X4)) -> active#(X1)
     active#(U93(X1,X2,X3,X4)) -> U93#(active(X1),X2,X3,X4)
     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))
     cons#(mark(X1),X2) -> cons#(X1,X2)
     U11#(mark(X)) -> U11#(X)
     U21#(mark(X)) -> U21#(X)
     U31#(mark(X)) -> U31#(X)
     U41#(mark(X1),X2) -> U41#(X1,X2)
     U42#(mark(X)) -> U42#(X)
     U51#(mark(X1),X2) -> U51#(X1,X2)
     U52#(mark(X)) -> U52#(X)
     U61#(mark(X1),X2) -> U61#(X1,X2)
     U62#(mark(X)) -> U62#(X)
     U71#(mark(X1),X2,X3) -> U71#(X1,X2,X3)
     U72#(mark(X1),X2) -> U72#(X1,X2)
     s#(mark(X)) -> s#(X)
     length#(mark(X)) -> length#(X)
     U81#(mark(X)) -> U81#(X)
     U91#(mark(X1),X2,X3,X4) -> U91#(X1,X2,X3,X4)
     U92#(mark(X1),X2,X3,X4) -> U92#(X1,X2,X3,X4)
     U93#(mark(X1),X2,X3,X4) -> U93#(X1,X2,X3,X4)
     take#(mark(X1),X2) -> take#(X1,X2)
     take#(X1,mark(X2)) -> take#(X1,X2)
     proper#(cons(X1,X2)) -> proper#(X2)
     proper#(cons(X1,X2)) -> proper#(X1)
     proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2))
     proper#(U11(X)) -> proper#(X)
     proper#(U11(X)) -> U11#(proper(X))
     proper#(U21(X)) -> proper#(X)
     proper#(U21(X)) -> U21#(proper(X))
     proper#(U31(X)) -> proper#(X)
     proper#(U31(X)) -> U31#(proper(X))
     proper#(U41(X1,X2)) -> proper#(X2)
     proper#(U41(X1,X2)) -> proper#(X1)
     proper#(U41(X1,X2)) -> U41#(proper(X1),proper(X2))
     proper#(U42(X)) -> proper#(X)
     proper#(U42(X)) -> U42#(proper(X))
     proper#(isNatIList(X)) -> proper#(X)
     proper#(isNatIList(X)) -> isNatIList#(proper(X))
     proper#(U51(X1,X2)) -> proper#(X2)
     proper#(U51(X1,X2)) -> proper#(X1)
     proper#(U51(X1,X2)) -> U51#(proper(X1),proper(X2))
     proper#(U52(X)) -> proper#(X)
     proper#(U52(X)) -> U52#(proper(X))
     proper#(isNatList(X)) -> proper#(X)
     proper#(isNatList(X)) -> isNatList#(proper(X))
     proper#(U61(X1,X2)) -> proper#(X2)
     proper#(U61(X1,X2)) -> proper#(X1)
     proper#(U61(X1,X2)) -> U61#(proper(X1),proper(X2))
     proper#(U62(X)) -> proper#(X)
     proper#(U62(X)) -> U62#(proper(X))
     proper#(U71(X1,X2,X3)) -> proper#(X3)
     proper#(U71(X1,X2,X3)) -> proper#(X2)
     proper#(U71(X1,X2,X3)) -> proper#(X1)
     proper#(U71(X1,X2,X3)) -> U71#(proper(X1),proper(X2),proper(X3))
     proper#(U72(X1,X2)) -> proper#(X2)
     proper#(U72(X1,X2)) -> proper#(X1)
     proper#(U72(X1,X2)) -> U72#(proper(X1),proper(X2))
     proper#(isNat(X)) -> proper#(X)
     proper#(isNat(X)) -> isNat#(proper(X))
     proper#(s(X)) -> proper#(X)
     proper#(s(X)) -> s#(proper(X))
     proper#(length(X)) -> proper#(X)
     proper#(length(X)) -> length#(proper(X))
     proper#(U81(X)) -> proper#(X)
     proper#(U81(X)) -> U81#(proper(X))
     proper#(U91(X1,X2,X3,X4)) -> proper#(X4)
     proper#(U91(X1,X2,X3,X4)) -> proper#(X3)
     proper#(U91(X1,X2,X3,X4)) -> proper#(X2)
     proper#(U91(X1,X2,X3,X4)) -> proper#(X1)
     proper#(U91(X1,X2,X3,X4)) -> U91#(proper(X1),proper(X2),proper(X3),proper(X4))
     proper#(U92(X1,X2,X3,X4)) -> proper#(X4)
     proper#(U92(X1,X2,X3,X4)) -> proper#(X3)
     proper#(U92(X1,X2,X3,X4)) -> proper#(X2)
     proper#(U92(X1,X2,X3,X4)) -> proper#(X1)
     proper#(U92(X1,X2,X3,X4)) -> U92#(proper(X1),proper(X2),proper(X3),proper(X4))
     proper#(U93(X1,X2,X3,X4)) -> proper#(X4)
     proper#(U93(X1,X2,X3,X4)) -> proper#(X3)
     proper#(U93(X1,X2,X3,X4)) -> proper#(X2)
     proper#(U93(X1,X2,X3,X4)) -> proper#(X1)
     proper#(U93(X1,X2,X3,X4)) -> U93#(proper(X1),proper(X2),proper(X3),proper(X4))
     proper#(take(X1,X2)) -> proper#(X2)
     proper#(take(X1,X2)) -> proper#(X1)
     proper#(take(X1,X2)) -> take#(proper(X1),proper(X2))
     cons#(ok(X1),ok(X2)) -> cons#(X1,X2)
     U11#(ok(X)) -> U11#(X)
     U21#(ok(X)) -> U21#(X)
     U31#(ok(X)) -> U31#(X)
     U41#(ok(X1),ok(X2)) -> U41#(X1,X2)
     U42#(ok(X)) -> U42#(X)
     isNatIList#(ok(X)) -> isNatIList#(X)
     U51#(ok(X1),ok(X2)) -> U51#(X1,X2)
     U52#(ok(X)) -> U52#(X)
     isNatList#(ok(X)) -> isNatList#(X)
     U61#(ok(X1),ok(X2)) -> U61#(X1,X2)
     U62#(ok(X)) -> U62#(X)
     U71#(ok(X1),ok(X2),ok(X3)) -> U71#(X1,X2,X3)
     U72#(ok(X1),ok(X2)) -> U72#(X1,X2)
     isNat#(ok(X)) -> isNat#(X)
     s#(ok(X)) -> s#(X)
     length#(ok(X)) -> length#(X)
     U81#(ok(X)) -> U81#(X)
     U91#(ok(X1),ok(X2),ok(X3),ok(X4)) -> U91#(X1,X2,X3,X4)
     U92#(ok(X1),ok(X2),ok(X3),ok(X4)) -> U92#(X1,X2,X3,X4)
     U93#(ok(X1),ok(X2),ok(X3),ok(X4)) -> U93#(X1,X2,X3,X4)
     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:
     f56(x,y) -> x
     f56(x,y) -> y
     isNatIList(ok(X)) -> ok(isNatIList(X))
     isNatList(ok(X)) -> ok(isNatList(X))
     isNat(ok(X)) -> ok(isNat(X))
     length(mark(X)) -> mark(length(X))
     length(ok(X)) -> ok(length(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))
     active(zeros()) -> mark(cons(0(),zeros()))
     active(U11(tt())) -> mark(tt())
     active(U21(tt())) -> mark(tt())
     active(U31(tt())) -> mark(tt())
     active(U41(tt(),V2)) -> mark(U42(isNatIList(V2)))
     active(U42(tt())) -> mark(tt())
     active(U51(tt(),V2)) -> mark(U52(isNatList(V2)))
     active(U52(tt())) -> mark(tt())
     active(U61(tt(),V2)) -> mark(U62(isNatIList(V2)))
     active(U62(tt())) -> mark(tt())
     active(U71(tt(),L,N)) -> mark(U72(isNat(N),L))
     active(U72(tt(),L)) -> mark(s(length(L)))
     active(U81(tt())) -> mark(nil())
     active(U91(tt(),IL,M,N)) -> mark(U92(isNat(M),IL,M,N))
     active(U92(tt(),IL,M,N)) -> mark(U93(isNat(N),IL,M,N))
     active(U93(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
     active(isNat(0())) -> mark(tt())
     active(isNat(length(V1))) -> mark(U11(isNatList(V1)))
     active(isNat(s(V1))) -> mark(U21(isNat(V1)))
     active(isNatIList(V)) -> mark(U31(isNatList(V)))
     active(isNatIList(zeros())) -> mark(tt())
     active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2))
     active(isNatList(nil())) -> mark(tt())
     active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2))
     active(isNatList(take(V1,V2))) -> mark(U61(isNat(V1),V2))
     active(length(nil())) -> mark(0())
     active(length(cons(N,L))) -> mark(U71(isNatList(L),L,N))
     active(take(0(),IL)) -> mark(U81(isNatIList(IL)))
     active(take(s(M),cons(N,IL))) -> mark(U91(isNatIList(IL),IL,M,N))
     active(cons(X1,X2)) -> cons(active(X1),X2)
     active(U11(X)) -> U11(active(X))
     active(U21(X)) -> U21(active(X))
     active(U31(X)) -> U31(active(X))
     active(U41(X1,X2)) -> U41(active(X1),X2)
     active(U42(X)) -> U42(active(X))
     active(U51(X1,X2)) -> U51(active(X1),X2)
     active(U52(X)) -> U52(active(X))
     active(U61(X1,X2)) -> U61(active(X1),X2)
     active(U62(X)) -> U62(active(X))
     active(U71(X1,X2,X3)) -> U71(active(X1),X2,X3)
     active(U72(X1,X2)) -> U72(active(X1),X2)
     active(s(X)) -> s(active(X))
     active(length(X)) -> length(active(X))
     active(U81(X)) -> U81(active(X))
     active(U91(X1,X2,X3,X4)) -> U91(active(X1),X2,X3,X4)
     active(U92(X1,X2,X3,X4)) -> U92(active(X1),X2,X3,X4)
     active(U93(X1,X2,X3,X4)) -> U93(active(X1),X2,X3,X4)
     active(take(X1,X2)) -> take(active(X1),X2)
     active(take(X1,X2)) -> take(X1,active(X2))
     U42(mark(X)) -> mark(U42(X))
     U42(ok(X)) -> ok(U42(X))
     U52(mark(X)) -> mark(U52(X))
     U52(ok(X)) -> ok(U52(X))
     U62(mark(X)) -> mark(U62(X))
     U62(ok(X)) -> ok(U62(X))
     U72(mark(X1),X2) -> mark(U72(X1,X2))
     U72(ok(X1),ok(X2)) -> ok(U72(X1,X2))
     s(mark(X)) -> mark(s(X))
     s(ok(X)) -> ok(s(X))
     U92(mark(X1),X2,X3,X4) -> mark(U92(X1,X2,X3,X4))
     U92(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U92(X1,X2,X3,X4))
     U93(mark(X1),X2,X3,X4) -> mark(U93(X1,X2,X3,X4))
     U93(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U93(X1,X2,X3,X4))
     cons(mark(X1),X2) -> mark(cons(X1,X2))
     cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
     U11(mark(X)) -> mark(U11(X))
     U11(ok(X)) -> ok(U11(X))
     U21(mark(X)) -> mark(U21(X))
     U21(ok(X)) -> ok(U21(X))
     U31(mark(X)) -> mark(U31(X))
     U31(ok(X)) -> ok(U31(X))
     U41(mark(X1),X2) -> mark(U41(X1,X2))
     U41(ok(X1),ok(X2)) -> ok(U41(X1,X2))
     U51(mark(X1),X2) -> mark(U51(X1,X2))
     U51(ok(X1),ok(X2)) -> ok(U51(X1,X2))
     U61(mark(X1),X2) -> mark(U61(X1,X2))
     U61(ok(X1),ok(X2)) -> ok(U61(X1,X2))
     U71(mark(X1),X2,X3) -> mark(U71(X1,X2,X3))
     U71(ok(X1),ok(X2),ok(X3)) -> ok(U71(X1,X2,X3))
     U81(mark(X)) -> mark(U81(X))
     U81(ok(X)) -> ok(U81(X))
     U91(mark(X1),X2,X3,X4) -> mark(U91(X1,X2,X3,X4))
     U91(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U91(X1,X2,X3,X4))
     proper(zeros()) -> ok(zeros())
     proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
     proper(0()) -> ok(0())
     proper(U11(X)) -> U11(proper(X))
     proper(tt()) -> ok(tt())
     proper(U21(X)) -> U21(proper(X))
     proper(U31(X)) -> U31(proper(X))
     proper(U41(X1,X2)) -> U41(proper(X1),proper(X2))
     proper(U42(X)) -> U42(proper(X))
     proper(isNatIList(X)) -> isNatIList(proper(X))
     proper(U51(X1,X2)) -> U51(proper(X1),proper(X2))
     proper(U52(X)) -> U52(proper(X))
     proper(isNatList(X)) -> isNatList(proper(X))
     proper(U61(X1,X2)) -> U61(proper(X1),proper(X2))
     proper(U62(X)) -> U62(proper(X))
     proper(U71(X1,X2,X3)) -> U71(proper(X1),proper(X2),proper(X3))
     proper(U72(X1,X2)) -> U72(proper(X1),proper(X2))
     proper(isNat(X)) -> isNat(proper(X))
     proper(s(X)) -> s(proper(X))
     proper(length(X)) -> length(proper(X))
     proper(U81(X)) -> U81(proper(X))
     proper(nil()) -> ok(nil())
     proper(U91(X1,X2,X3,X4)) -> U91(proper(X1),proper(X2),proper(X3),proper(X4))
     proper(U92(X1,X2,X3,X4)) -> U92(proper(X1),proper(X2),proper(X3),proper(X4))
     proper(U93(X1,X2,X3,X4)) -> U93(proper(X1),proper(X2),proper(X3),proper(X4))
     proper(take(X1,X2)) -> take(proper(X1),proper(X2))
    graph:
     ...
    Restore Modifier:
     DPs:
      active#(zeros()) -> cons#(0(),zeros())
      active#(U41(tt(),V2)) -> isNatIList#(V2)
      active#(U41(tt(),V2)) -> U42#(isNatIList(V2))
      active#(U51(tt(),V2)) -> isNatList#(V2)
      active#(U51(tt(),V2)) -> U52#(isNatList(V2))
      active#(U61(tt(),V2)) -> isNatIList#(V2)
      active#(U61(tt(),V2)) -> U62#(isNatIList(V2))
      active#(U71(tt(),L,N)) -> isNat#(N)
      active#(U71(tt(),L,N)) -> U72#(isNat(N),L)
      active#(U72(tt(),L)) -> length#(L)
      active#(U72(tt(),L)) -> s#(length(L))
      active#(U91(tt(),IL,M,N)) -> isNat#(M)
      active#(U91(tt(),IL,M,N)) -> U92#(isNat(M),IL,M,N)
      active#(U92(tt(),IL,M,N)) -> isNat#(N)
      active#(U92(tt(),IL,M,N)) -> U93#(isNat(N),IL,M,N)
      active#(U93(tt(),IL,M,N)) -> take#(M,IL)
      active#(U93(tt(),IL,M,N)) -> cons#(N,take(M,IL))
      active#(isNat(length(V1))) -> isNatList#(V1)
      active#(isNat(length(V1))) -> U11#(isNatList(V1))
      active#(isNat(s(V1))) -> isNat#(V1)
      active#(isNat(s(V1))) -> U21#(isNat(V1))
      active#(isNatIList(V)) -> isNatList#(V)
      active#(isNatIList(V)) -> U31#(isNatList(V))
      active#(isNatIList(cons(V1,V2))) -> isNat#(V1)
      active#(isNatIList(cons(V1,V2))) -> U41#(isNat(V1),V2)
      active#(isNatList(cons(V1,V2))) -> isNat#(V1)
      active#(isNatList(cons(V1,V2))) -> U51#(isNat(V1),V2)
      active#(isNatList(take(V1,V2))) -> isNat#(V1)
      active#(isNatList(take(V1,V2))) -> U61#(isNat(V1),V2)
      active#(length(cons(N,L))) -> isNatList#(L)
      active#(length(cons(N,L))) -> U71#(isNatList(L),L,N)
      active#(take(0(),IL)) -> isNatIList#(IL)
      active#(take(0(),IL)) -> U81#(isNatIList(IL))
      active#(take(s(M),cons(N,IL))) -> isNatIList#(IL)
      active#(take(s(M),cons(N,IL))) -> U91#(isNatIList(IL),IL,M,N)
      active#(cons(X1,X2)) -> active#(X1)
      active#(cons(X1,X2)) -> cons#(active(X1),X2)
      active#(U11(X)) -> active#(X)
      active#(U11(X)) -> U11#(active(X))
      active#(U21(X)) -> active#(X)
      active#(U21(X)) -> U21#(active(X))
      active#(U31(X)) -> active#(X)
      active#(U31(X)) -> U31#(active(X))
      active#(U41(X1,X2)) -> active#(X1)
      active#(U41(X1,X2)) -> U41#(active(X1),X2)
      active#(U42(X)) -> active#(X)
      active#(U42(X)) -> U42#(active(X))
      active#(U51(X1,X2)) -> active#(X1)
      active#(U51(X1,X2)) -> U51#(active(X1),X2)
      active#(U52(X)) -> active#(X)
      active#(U52(X)) -> U52#(active(X))
      active#(U61(X1,X2)) -> active#(X1)
      active#(U61(X1,X2)) -> U61#(active(X1),X2)
      active#(U62(X)) -> active#(X)
      active#(U62(X)) -> U62#(active(X))
      active#(U71(X1,X2,X3)) -> active#(X1)
      active#(U71(X1,X2,X3)) -> U71#(active(X1),X2,X3)
      active#(U72(X1,X2)) -> active#(X1)
      active#(U72(X1,X2)) -> U72#(active(X1),X2)
      active#(s(X)) -> active#(X)
      active#(s(X)) -> s#(active(X))
      active#(length(X)) -> active#(X)
      active#(length(X)) -> length#(active(X))
      active#(U81(X)) -> active#(X)
      active#(U81(X)) -> U81#(active(X))
      active#(U91(X1,X2,X3,X4)) -> active#(X1)
      active#(U91(X1,X2,X3,X4)) -> U91#(active(X1),X2,X3,X4)
      active#(U92(X1,X2,X3,X4)) -> active#(X1)
      active#(U92(X1,X2,X3,X4)) -> U92#(active(X1),X2,X3,X4)
      active#(U93(X1,X2,X3,X4)) -> active#(X1)
      active#(U93(X1,X2,X3,X4)) -> U93#(active(X1),X2,X3,X4)
      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))
      cons#(mark(X1),X2) -> cons#(X1,X2)
      U11#(mark(X)) -> U11#(X)
      U21#(mark(X)) -> U21#(X)
      U31#(mark(X)) -> U31#(X)
      U41#(mark(X1),X2) -> U41#(X1,X2)
      U42#(mark(X)) -> U42#(X)
      U51#(mark(X1),X2) -> U51#(X1,X2)
      U52#(mark(X)) -> U52#(X)
      U61#(mark(X1),X2) -> U61#(X1,X2)
      U62#(mark(X)) -> U62#(X)
      U71#(mark(X1),X2,X3) -> U71#(X1,X2,X3)
      U72#(mark(X1),X2) -> U72#(X1,X2)
      s#(mark(X)) -> s#(X)
      length#(mark(X)) -> length#(X)
      U81#(mark(X)) -> U81#(X)
      U91#(mark(X1),X2,X3,X4) -> U91#(X1,X2,X3,X4)
      U92#(mark(X1),X2,X3,X4) -> U92#(X1,X2,X3,X4)
      U93#(mark(X1),X2,X3,X4) -> U93#(X1,X2,X3,X4)
      take#(mark(X1),X2) -> take#(X1,X2)
      take#(X1,mark(X2)) -> take#(X1,X2)
      proper#(cons(X1,X2)) -> proper#(X2)
      proper#(cons(X1,X2)) -> proper#(X1)
      proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2))
      proper#(U11(X)) -> proper#(X)
      proper#(U11(X)) -> U11#(proper(X))
      proper#(U21(X)) -> proper#(X)
      proper#(U21(X)) -> U21#(proper(X))
      proper#(U31(X)) -> proper#(X)
      proper#(U31(X)) -> U31#(proper(X))
      proper#(U41(X1,X2)) -> proper#(X2)
      proper#(U41(X1,X2)) -> proper#(X1)
      proper#(U41(X1,X2)) -> U41#(proper(X1),proper(X2))
      proper#(U42(X)) -> proper#(X)
      proper#(U42(X)) -> U42#(proper(X))
      proper#(isNatIList(X)) -> proper#(X)
      proper#(isNatIList(X)) -> isNatIList#(proper(X))
      proper#(U51(X1,X2)) -> proper#(X2)
      proper#(U51(X1,X2)) -> proper#(X1)
      proper#(U51(X1,X2)) -> U51#(proper(X1),proper(X2))
      proper#(U52(X)) -> proper#(X)
      proper#(U52(X)) -> U52#(proper(X))
      proper#(isNatList(X)) -> proper#(X)
      proper#(isNatList(X)) -> isNatList#(proper(X))
      proper#(U61(X1,X2)) -> proper#(X2)
      proper#(U61(X1,X2)) -> proper#(X1)
      proper#(U61(X1,X2)) -> U61#(proper(X1),proper(X2))
      proper#(U62(X)) -> proper#(X)
      proper#(U62(X)) -> U62#(proper(X))
      proper#(U71(X1,X2,X3)) -> proper#(X3)
      proper#(U71(X1,X2,X3)) -> proper#(X2)
      proper#(U71(X1,X2,X3)) -> proper#(X1)
      proper#(U71(X1,X2,X3)) -> U71#(proper(X1),proper(X2),proper(X3))
      proper#(U72(X1,X2)) -> proper#(X2)
      proper#(U72(X1,X2)) -> proper#(X1)
      proper#(U72(X1,X2)) -> U72#(proper(X1),proper(X2))
      proper#(isNat(X)) -> proper#(X)
      proper#(isNat(X)) -> isNat#(proper(X))
      proper#(s(X)) -> proper#(X)
      proper#(s(X)) -> s#(proper(X))
      proper#(length(X)) -> proper#(X)
      proper#(length(X)) -> length#(proper(X))
      proper#(U81(X)) -> proper#(X)
      proper#(U81(X)) -> U81#(proper(X))
      proper#(U91(X1,X2,X3,X4)) -> proper#(X4)
      proper#(U91(X1,X2,X3,X4)) -> proper#(X3)
      proper#(U91(X1,X2,X3,X4)) -> proper#(X2)
      proper#(U91(X1,X2,X3,X4)) -> proper#(X1)
      proper#(U91(X1,X2,X3,X4)) -> U91#(proper(X1),proper(X2),proper(X3),proper(X4))
      proper#(U92(X1,X2,X3,X4)) -> proper#(X4)
      proper#(U92(X1,X2,X3,X4)) -> proper#(X3)
      proper#(U92(X1,X2,X3,X4)) -> proper#(X2)
      proper#(U92(X1,X2,X3,X4)) -> proper#(X1)
      proper#(U92(X1,X2,X3,X4)) -> U92#(proper(X1),proper(X2),proper(X3),proper(X4))
      proper#(U93(X1,X2,X3,X4)) -> proper#(X4)
      proper#(U93(X1,X2,X3,X4)) -> proper#(X3)
      proper#(U93(X1,X2,X3,X4)) -> proper#(X2)
      proper#(U93(X1,X2,X3,X4)) -> proper#(X1)
      proper#(U93(X1,X2,X3,X4)) -> U93#(proper(X1),proper(X2),proper(X3),proper(X4))
      proper#(take(X1,X2)) -> proper#(X2)
      proper#(take(X1,X2)) -> proper#(X1)
      proper#(take(X1,X2)) -> take#(proper(X1),proper(X2))
      cons#(ok(X1),ok(X2)) -> cons#(X1,X2)
      U11#(ok(X)) -> U11#(X)
      U21#(ok(X)) -> U21#(X)
      U31#(ok(X)) -> U31#(X)
      U41#(ok(X1),ok(X2)) -> U41#(X1,X2)
      U42#(ok(X)) -> U42#(X)
      isNatIList#(ok(X)) -> isNatIList#(X)
      U51#(ok(X1),ok(X2)) -> U51#(X1,X2)
      U52#(ok(X)) -> U52#(X)
      isNatList#(ok(X)) -> isNatList#(X)
      U61#(ok(X1),ok(X2)) -> U61#(X1,X2)
      U62#(ok(X)) -> U62#(X)
      U71#(ok(X1),ok(X2),ok(X3)) -> U71#(X1,X2,X3)
      U72#(ok(X1),ok(X2)) -> U72#(X1,X2)
      isNat#(ok(X)) -> isNat#(X)
      s#(ok(X)) -> s#(X)
      length#(ok(X)) -> length#(X)
      U81#(ok(X)) -> U81#(X)
      U91#(ok(X1),ok(X2),ok(X3),ok(X4)) -> U91#(X1,X2,X3,X4)
      U92#(ok(X1),ok(X2),ok(X3),ok(X4)) -> U92#(X1,X2,X3,X4)
      U93#(ok(X1),ok(X2),ok(X3),ok(X4)) -> U93#(X1,X2,X3,X4)
      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(zeros()) -> mark(cons(0(),zeros()))
      active(U11(tt())) -> mark(tt())
      active(U21(tt())) -> mark(tt())
      active(U31(tt())) -> mark(tt())
      active(U41(tt(),V2)) -> mark(U42(isNatIList(V2)))
      active(U42(tt())) -> mark(tt())
      active(U51(tt(),V2)) -> mark(U52(isNatList(V2)))
      active(U52(tt())) -> mark(tt())
      active(U61(tt(),V2)) -> mark(U62(isNatIList(V2)))
      active(U62(tt())) -> mark(tt())
      active(U71(tt(),L,N)) -> mark(U72(isNat(N),L))
      active(U72(tt(),L)) -> mark(s(length(L)))
      active(U81(tt())) -> mark(nil())
      active(U91(tt(),IL,M,N)) -> mark(U92(isNat(M),IL,M,N))
      active(U92(tt(),IL,M,N)) -> mark(U93(isNat(N),IL,M,N))
      active(U93(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
      active(isNat(0())) -> mark(tt())
      active(isNat(length(V1))) -> mark(U11(isNatList(V1)))
      active(isNat(s(V1))) -> mark(U21(isNat(V1)))
      active(isNatIList(V)) -> mark(U31(isNatList(V)))
      active(isNatIList(zeros())) -> mark(tt())
      active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2))
      active(isNatList(nil())) -> mark(tt())
      active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2))
      active(isNatList(take(V1,V2))) -> mark(U61(isNat(V1),V2))
      active(length(nil())) -> mark(0())
      active(length(cons(N,L))) -> mark(U71(isNatList(L),L,N))
      active(take(0(),IL)) -> mark(U81(isNatIList(IL)))
      active(take(s(M),cons(N,IL))) -> mark(U91(isNatIList(IL),IL,M,N))
      active(cons(X1,X2)) -> cons(active(X1),X2)
      active(U11(X)) -> U11(active(X))
      active(U21(X)) -> U21(active(X))
      active(U31(X)) -> U31(active(X))
      active(U41(X1,X2)) -> U41(active(X1),X2)
      active(U42(X)) -> U42(active(X))
      active(U51(X1,X2)) -> U51(active(X1),X2)
      active(U52(X)) -> U52(active(X))
      active(U61(X1,X2)) -> U61(active(X1),X2)
      active(U62(X)) -> U62(active(X))
      active(U71(X1,X2,X3)) -> U71(active(X1),X2,X3)
      active(U72(X1,X2)) -> U72(active(X1),X2)
      active(s(X)) -> s(active(X))
      active(length(X)) -> length(active(X))
      active(U81(X)) -> U81(active(X))
      active(U91(X1,X2,X3,X4)) -> U91(active(X1),X2,X3,X4)
      active(U92(X1,X2,X3,X4)) -> U92(active(X1),X2,X3,X4)
      active(U93(X1,X2,X3,X4)) -> U93(active(X1),X2,X3,X4)
      active(take(X1,X2)) -> take(active(X1),X2)
      active(take(X1,X2)) -> take(X1,active(X2))
      cons(mark(X1),X2) -> mark(cons(X1,X2))
      U11(mark(X)) -> mark(U11(X))
      U21(mark(X)) -> mark(U21(X))
      U31(mark(X)) -> mark(U31(X))
      U41(mark(X1),X2) -> mark(U41(X1,X2))
      U42(mark(X)) -> mark(U42(X))
      U51(mark(X1),X2) -> mark(U51(X1,X2))
      U52(mark(X)) -> mark(U52(X))
      U61(mark(X1),X2) -> mark(U61(X1,X2))
      U62(mark(X)) -> mark(U62(X))
      U71(mark(X1),X2,X3) -> mark(U71(X1,X2,X3))
      U72(mark(X1),X2) -> mark(U72(X1,X2))
      s(mark(X)) -> mark(s(X))
      length(mark(X)) -> mark(length(X))
      U81(mark(X)) -> mark(U81(X))
      U91(mark(X1),X2,X3,X4) -> mark(U91(X1,X2,X3,X4))
      U92(mark(X1),X2,X3,X4) -> mark(U92(X1,X2,X3,X4))
      U93(mark(X1),X2,X3,X4) -> mark(U93(X1,X2,X3,X4))
      take(mark(X1),X2) -> mark(take(X1,X2))
      take(X1,mark(X2)) -> mark(take(X1,X2))
      proper(zeros()) -> ok(zeros())
      proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
      proper(0()) -> ok(0())
      proper(U11(X)) -> U11(proper(X))
      proper(tt()) -> ok(tt())
      proper(U21(X)) -> U21(proper(X))
      proper(U31(X)) -> U31(proper(X))
      proper(U41(X1,X2)) -> U41(proper(X1),proper(X2))
      proper(U42(X)) -> U42(proper(X))
      proper(isNatIList(X)) -> isNatIList(proper(X))
      proper(U51(X1,X2)) -> U51(proper(X1),proper(X2))
      proper(U52(X)) -> U52(proper(X))
      proper(isNatList(X)) -> isNatList(proper(X))
      proper(U61(X1,X2)) -> U61(proper(X1),proper(X2))
      proper(U62(X)) -> U62(proper(X))
      proper(U71(X1,X2,X3)) -> U71(proper(X1),proper(X2),proper(X3))
      proper(U72(X1,X2)) -> U72(proper(X1),proper(X2))
      proper(isNat(X)) -> isNat(proper(X))
      proper(s(X)) -> s(proper(X))
      proper(length(X)) -> length(proper(X))
      proper(U81(X)) -> U81(proper(X))
      proper(nil()) -> ok(nil())
      proper(U91(X1,X2,X3,X4)) -> U91(proper(X1),proper(X2),proper(X3),proper(X4))
      proper(U92(X1,X2,X3,X4)) -> U92(proper(X1),proper(X2),proper(X3),proper(X4))
      proper(U93(X1,X2,X3,X4)) -> U93(proper(X1),proper(X2),proper(X3),proper(X4))
      proper(take(X1,X2)) -> take(proper(X1),proper(X2))
      cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
      U11(ok(X)) -> ok(U11(X))
      U21(ok(X)) -> ok(U21(X))
      U31(ok(X)) -> ok(U31(X))
      U41(ok(X1),ok(X2)) -> ok(U41(X1,X2))
      U42(ok(X)) -> ok(U42(X))
      isNatIList(ok(X)) -> ok(isNatIList(X))
      U51(ok(X1),ok(X2)) -> ok(U51(X1,X2))
      U52(ok(X)) -> ok(U52(X))
      isNatList(ok(X)) -> ok(isNatList(X))
      U61(ok(X1),ok(X2)) -> ok(U61(X1,X2))
      U62(ok(X)) -> ok(U62(X))
      U71(ok(X1),ok(X2),ok(X3)) -> ok(U71(X1,X2,X3))
      U72(ok(X1),ok(X2)) -> ok(U72(X1,X2))
      isNat(ok(X)) -> ok(isNat(X))
      s(ok(X)) -> ok(s(X))
      length(ok(X)) -> ok(length(X))
      U81(ok(X)) -> ok(U81(X))
      U91(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U91(X1,X2,X3,X4))
      U92(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U92(X1,X2,X3,X4))
      U93(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U93(X1,X2,X3,X4))
      take(ok(X1),ok(X2)) -> ok(take(X1,X2))
      top(mark(X)) -> top(proper(X))
      top(ok(X)) -> top(active(X))
     SCC Processor:
      #sccs: 25
      #rules: 103
      #arcs: 4230/33124
      DPs:
       top#(ok(X)) -> top#(active(X))
       top#(mark(X)) -> top#(proper(X))
      TRS:
       active(zeros()) -> mark(cons(0(),zeros()))
       active(U11(tt())) -> mark(tt())
       active(U21(tt())) -> mark(tt())
       active(U31(tt())) -> mark(tt())
       active(U41(tt(),V2)) -> mark(U42(isNatIList(V2)))
       active(U42(tt())) -> mark(tt())
       active(U51(tt(),V2)) -> mark(U52(isNatList(V2)))
       active(U52(tt())) -> mark(tt())
       active(U61(tt(),V2)) -> mark(U62(isNatIList(V2)))
       active(U62(tt())) -> mark(tt())
       active(U71(tt(),L,N)) -> mark(U72(isNat(N),L))
       active(U72(tt(),L)) -> mark(s(length(L)))
       active(U81(tt())) -> mark(nil())
       active(U91(tt(),IL,M,N)) -> mark(U92(isNat(M),IL,M,N))
       active(U92(tt(),IL,M,N)) -> mark(U93(isNat(N),IL,M,N))
       active(U93(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
       active(isNat(0())) -> mark(tt())
       active(isNat(length(V1))) -> mark(U11(isNatList(V1)))
       active(isNat(s(V1))) -> mark(U21(isNat(V1)))
       active(isNatIList(V)) -> mark(U31(isNatList(V)))
       active(isNatIList(zeros())) -> mark(tt())
       active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2))
       active(isNatList(nil())) -> mark(tt())
       active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2))
       active(isNatList(take(V1,V2))) -> mark(U61(isNat(V1),V2))
       active(length(nil())) -> mark(0())
       active(length(cons(N,L))) -> mark(U71(isNatList(L),L,N))
       active(take(0(),IL)) -> mark(U81(isNatIList(IL)))
       active(take(s(M),cons(N,IL))) -> mark(U91(isNatIList(IL),IL,M,N))
       active(cons(X1,X2)) -> cons(active(X1),X2)
       active(U11(X)) -> U11(active(X))
       active(U21(X)) -> U21(active(X))
       active(U31(X)) -> U31(active(X))
       active(U41(X1,X2)) -> U41(active(X1),X2)
       active(U42(X)) -> U42(active(X))
       active(U51(X1,X2)) -> U51(active(X1),X2)
       active(U52(X)) -> U52(active(X))
       active(U61(X1,X2)) -> U61(active(X1),X2)
       active(U62(X)) -> U62(active(X))
       active(U71(X1,X2,X3)) -> U71(active(X1),X2,X3)
       active(U72(X1,X2)) -> U72(active(X1),X2)
       active(s(X)) -> s(active(X))
       active(length(X)) -> length(active(X))
       active(U81(X)) -> U81(active(X))
       active(U91(X1,X2,X3,X4)) -> U91(active(X1),X2,X3,X4)
       active(U92(X1,X2,X3,X4)) -> U92(active(X1),X2,X3,X4)
       active(U93(X1,X2,X3,X4)) -> U93(active(X1),X2,X3,X4)
       active(take(X1,X2)) -> take(active(X1),X2)
       active(take(X1,X2)) -> take(X1,active(X2))
       cons(mark(X1),X2) -> mark(cons(X1,X2))
       U11(mark(X)) -> mark(U11(X))
       U21(mark(X)) -> mark(U21(X))
       U31(mark(X)) -> mark(U31(X))
       U41(mark(X1),X2) -> mark(U41(X1,X2))
       U42(mark(X)) -> mark(U42(X))
       U51(mark(X1),X2) -> mark(U51(X1,X2))
       U52(mark(X)) -> mark(U52(X))
       U61(mark(X1),X2) -> mark(U61(X1,X2))
       U62(mark(X)) -> mark(U62(X))
       U71(mark(X1),X2,X3) -> mark(U71(X1,X2,X3))
       U72(mark(X1),X2) -> mark(U72(X1,X2))
       s(mark(X)) -> mark(s(X))
       length(mark(X)) -> mark(length(X))
       U81(mark(X)) -> mark(U81(X))
       U91(mark(X1),X2,X3,X4) -> mark(U91(X1,X2,X3,X4))
       U92(mark(X1),X2,X3,X4) -> mark(U92(X1,X2,X3,X4))
       U93(mark(X1),X2,X3,X4) -> mark(U93(X1,X2,X3,X4))
       take(mark(X1),X2) -> mark(take(X1,X2))
       take(X1,mark(X2)) -> mark(take(X1,X2))
       proper(zeros()) -> ok(zeros())
       proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
       proper(0()) -> ok(0())
       proper(U11(X)) -> U11(proper(X))
       proper(tt()) -> ok(tt())
       proper(U21(X)) -> U21(proper(X))
       proper(U31(X)) -> U31(proper(X))
       proper(U41(X1,X2)) -> U41(proper(X1),proper(X2))
       proper(U42(X)) -> U42(proper(X))
       proper(isNatIList(X)) -> isNatIList(proper(X))
       proper(U51(X1,X2)) -> U51(proper(X1),proper(X2))
       proper(U52(X)) -> U52(proper(X))
       proper(isNatList(X)) -> isNatList(proper(X))
       proper(U61(X1,X2)) -> U61(proper(X1),proper(X2))
       proper(U62(X)) -> U62(proper(X))
       proper(U71(X1,X2,X3)) -> U71(proper(X1),proper(X2),proper(X3))
       proper(U72(X1,X2)) -> U72(proper(X1),proper(X2))
       proper(isNat(X)) -> isNat(proper(X))
       proper(s(X)) -> s(proper(X))
       proper(length(X)) -> length(proper(X))
       proper(U81(X)) -> U81(proper(X))
       proper(nil()) -> ok(nil())
       proper(U91(X1,X2,X3,X4)) -> U91(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(U92(X1,X2,X3,X4)) -> U92(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(U93(X1,X2,X3,X4)) -> U93(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(take(X1,X2)) -> take(proper(X1),proper(X2))
       cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
       U11(ok(X)) -> ok(U11(X))
       U21(ok(X)) -> ok(U21(X))
       U31(ok(X)) -> ok(U31(X))
       U41(ok(X1),ok(X2)) -> ok(U41(X1,X2))
       U42(ok(X)) -> ok(U42(X))
       isNatIList(ok(X)) -> ok(isNatIList(X))
       U51(ok(X1),ok(X2)) -> ok(U51(X1,X2))
       U52(ok(X)) -> ok(U52(X))
       isNatList(ok(X)) -> ok(isNatList(X))
       U61(ok(X1),ok(X2)) -> ok(U61(X1,X2))
       U62(ok(X)) -> ok(U62(X))
       U71(ok(X1),ok(X2),ok(X3)) -> ok(U71(X1,X2,X3))
       U72(ok(X1),ok(X2)) -> ok(U72(X1,X2))
       isNat(ok(X)) -> ok(isNat(X))
       s(ok(X)) -> ok(s(X))
       length(ok(X)) -> ok(length(X))
       U81(ok(X)) -> ok(U81(X))
       U91(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U91(X1,X2,X3,X4))
       U92(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U92(X1,X2,X3,X4))
       U93(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U93(X1,X2,X3,X4))
       take(ok(X1),ok(X2)) -> ok(take(X1,X2))
       top(mark(X)) -> top(proper(X))
       top(ok(X)) -> top(active(X))
      Open
      
      DPs:
       proper#(take(X1,X2)) -> proper#(X1)
       proper#(take(X1,X2)) -> proper#(X2)
       proper#(U93(X1,X2,X3,X4)) -> proper#(X1)
       proper#(U93(X1,X2,X3,X4)) -> proper#(X2)
       proper#(U93(X1,X2,X3,X4)) -> proper#(X3)
       proper#(U93(X1,X2,X3,X4)) -> proper#(X4)
       proper#(U92(X1,X2,X3,X4)) -> proper#(X1)
       proper#(U92(X1,X2,X3,X4)) -> proper#(X2)
       proper#(U92(X1,X2,X3,X4)) -> proper#(X3)
       proper#(U92(X1,X2,X3,X4)) -> proper#(X4)
       proper#(U91(X1,X2,X3,X4)) -> proper#(X1)
       proper#(U91(X1,X2,X3,X4)) -> proper#(X2)
       proper#(U91(X1,X2,X3,X4)) -> proper#(X3)
       proper#(U91(X1,X2,X3,X4)) -> proper#(X4)
       proper#(U81(X)) -> proper#(X)
       proper#(length(X)) -> proper#(X)
       proper#(s(X)) -> proper#(X)
       proper#(isNat(X)) -> proper#(X)
       proper#(U72(X1,X2)) -> proper#(X1)
       proper#(U72(X1,X2)) -> proper#(X2)
       proper#(U71(X1,X2,X3)) -> proper#(X1)
       proper#(U71(X1,X2,X3)) -> proper#(X2)
       proper#(U71(X1,X2,X3)) -> proper#(X3)
       proper#(U62(X)) -> proper#(X)
       proper#(U61(X1,X2)) -> proper#(X1)
       proper#(U61(X1,X2)) -> proper#(X2)
       proper#(isNatList(X)) -> proper#(X)
       proper#(U52(X)) -> proper#(X)
       proper#(U51(X1,X2)) -> proper#(X1)
       proper#(U51(X1,X2)) -> proper#(X2)
       proper#(isNatIList(X)) -> proper#(X)
       proper#(U42(X)) -> proper#(X)
       proper#(U41(X1,X2)) -> proper#(X1)
       proper#(U41(X1,X2)) -> proper#(X2)
       proper#(U31(X)) -> proper#(X)
       proper#(U21(X)) -> proper#(X)
       proper#(U11(X)) -> proper#(X)
       proper#(cons(X1,X2)) -> proper#(X1)
       proper#(cons(X1,X2)) -> proper#(X2)
      TRS:
       active(zeros()) -> mark(cons(0(),zeros()))
       active(U11(tt())) -> mark(tt())
       active(U21(tt())) -> mark(tt())
       active(U31(tt())) -> mark(tt())
       active(U41(tt(),V2)) -> mark(U42(isNatIList(V2)))
       active(U42(tt())) -> mark(tt())
       active(U51(tt(),V2)) -> mark(U52(isNatList(V2)))
       active(U52(tt())) -> mark(tt())
       active(U61(tt(),V2)) -> mark(U62(isNatIList(V2)))
       active(U62(tt())) -> mark(tt())
       active(U71(tt(),L,N)) -> mark(U72(isNat(N),L))
       active(U72(tt(),L)) -> mark(s(length(L)))
       active(U81(tt())) -> mark(nil())
       active(U91(tt(),IL,M,N)) -> mark(U92(isNat(M),IL,M,N))
       active(U92(tt(),IL,M,N)) -> mark(U93(isNat(N),IL,M,N))
       active(U93(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
       active(isNat(0())) -> mark(tt())
       active(isNat(length(V1))) -> mark(U11(isNatList(V1)))
       active(isNat(s(V1))) -> mark(U21(isNat(V1)))
       active(isNatIList(V)) -> mark(U31(isNatList(V)))
       active(isNatIList(zeros())) -> mark(tt())
       active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2))
       active(isNatList(nil())) -> mark(tt())
       active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2))
       active(isNatList(take(V1,V2))) -> mark(U61(isNat(V1),V2))
       active(length(nil())) -> mark(0())
       active(length(cons(N,L))) -> mark(U71(isNatList(L),L,N))
       active(take(0(),IL)) -> mark(U81(isNatIList(IL)))
       active(take(s(M),cons(N,IL))) -> mark(U91(isNatIList(IL),IL,M,N))
       active(cons(X1,X2)) -> cons(active(X1),X2)
       active(U11(X)) -> U11(active(X))
       active(U21(X)) -> U21(active(X))
       active(U31(X)) -> U31(active(X))
       active(U41(X1,X2)) -> U41(active(X1),X2)
       active(U42(X)) -> U42(active(X))
       active(U51(X1,X2)) -> U51(active(X1),X2)
       active(U52(X)) -> U52(active(X))
       active(U61(X1,X2)) -> U61(active(X1),X2)
       active(U62(X)) -> U62(active(X))
       active(U71(X1,X2,X3)) -> U71(active(X1),X2,X3)
       active(U72(X1,X2)) -> U72(active(X1),X2)
       active(s(X)) -> s(active(X))
       active(length(X)) -> length(active(X))
       active(U81(X)) -> U81(active(X))
       active(U91(X1,X2,X3,X4)) -> U91(active(X1),X2,X3,X4)
       active(U92(X1,X2,X3,X4)) -> U92(active(X1),X2,X3,X4)
       active(U93(X1,X2,X3,X4)) -> U93(active(X1),X2,X3,X4)
       active(take(X1,X2)) -> take(active(X1),X2)
       active(take(X1,X2)) -> take(X1,active(X2))
       cons(mark(X1),X2) -> mark(cons(X1,X2))
       U11(mark(X)) -> mark(U11(X))
       U21(mark(X)) -> mark(U21(X))
       U31(mark(X)) -> mark(U31(X))
       U41(mark(X1),X2) -> mark(U41(X1,X2))
       U42(mark(X)) -> mark(U42(X))
       U51(mark(X1),X2) -> mark(U51(X1,X2))
       U52(mark(X)) -> mark(U52(X))
       U61(mark(X1),X2) -> mark(U61(X1,X2))
       U62(mark(X)) -> mark(U62(X))
       U71(mark(X1),X2,X3) -> mark(U71(X1,X2,X3))
       U72(mark(X1),X2) -> mark(U72(X1,X2))
       s(mark(X)) -> mark(s(X))
       length(mark(X)) -> mark(length(X))
       U81(mark(X)) -> mark(U81(X))
       U91(mark(X1),X2,X3,X4) -> mark(U91(X1,X2,X3,X4))
       U92(mark(X1),X2,X3,X4) -> mark(U92(X1,X2,X3,X4))
       U93(mark(X1),X2,X3,X4) -> mark(U93(X1,X2,X3,X4))
       take(mark(X1),X2) -> mark(take(X1,X2))
       take(X1,mark(X2)) -> mark(take(X1,X2))
       proper(zeros()) -> ok(zeros())
       proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
       proper(0()) -> ok(0())
       proper(U11(X)) -> U11(proper(X))
       proper(tt()) -> ok(tt())
       proper(U21(X)) -> U21(proper(X))
       proper(U31(X)) -> U31(proper(X))
       proper(U41(X1,X2)) -> U41(proper(X1),proper(X2))
       proper(U42(X)) -> U42(proper(X))
       proper(isNatIList(X)) -> isNatIList(proper(X))
       proper(U51(X1,X2)) -> U51(proper(X1),proper(X2))
       proper(U52(X)) -> U52(proper(X))
       proper(isNatList(X)) -> isNatList(proper(X))
       proper(U61(X1,X2)) -> U61(proper(X1),proper(X2))
       proper(U62(X)) -> U62(proper(X))
       proper(U71(X1,X2,X3)) -> U71(proper(X1),proper(X2),proper(X3))
       proper(U72(X1,X2)) -> U72(proper(X1),proper(X2))
       proper(isNat(X)) -> isNat(proper(X))
       proper(s(X)) -> s(proper(X))
       proper(length(X)) -> length(proper(X))
       proper(U81(X)) -> U81(proper(X))
       proper(nil()) -> ok(nil())
       proper(U91(X1,X2,X3,X4)) -> U91(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(U92(X1,X2,X3,X4)) -> U92(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(U93(X1,X2,X3,X4)) -> U93(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(take(X1,X2)) -> take(proper(X1),proper(X2))
       cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
       U11(ok(X)) -> ok(U11(X))
       U21(ok(X)) -> ok(U21(X))
       U31(ok(X)) -> ok(U31(X))
       U41(ok(X1),ok(X2)) -> ok(U41(X1,X2))
       U42(ok(X)) -> ok(U42(X))
       isNatIList(ok(X)) -> ok(isNatIList(X))
       U51(ok(X1),ok(X2)) -> ok(U51(X1,X2))
       U52(ok(X)) -> ok(U52(X))
       isNatList(ok(X)) -> ok(isNatList(X))
       U61(ok(X1),ok(X2)) -> ok(U61(X1,X2))
       U62(ok(X)) -> ok(U62(X))
       U71(ok(X1),ok(X2),ok(X3)) -> ok(U71(X1,X2,X3))
       U72(ok(X1),ok(X2)) -> ok(U72(X1,X2))
       isNat(ok(X)) -> ok(isNat(X))
       s(ok(X)) -> ok(s(X))
       length(ok(X)) -> ok(length(X))
       U81(ok(X)) -> ok(U81(X))
       U91(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U91(X1,X2,X3,X4))
       U92(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U92(X1,X2,X3,X4))
       U93(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U93(X1,X2,X3,X4))
       take(ok(X1),ok(X2)) -> ok(take(X1,X2))
       top(mark(X)) -> top(proper(X))
       top(ok(X)) -> top(active(X))
      Open
      
      DPs:
       active#(take(X1,X2)) -> active#(X2)
       active#(take(X1,X2)) -> active#(X1)
       active#(U93(X1,X2,X3,X4)) -> active#(X1)
       active#(U92(X1,X2,X3,X4)) -> active#(X1)
       active#(U91(X1,X2,X3,X4)) -> active#(X1)
       active#(U81(X)) -> active#(X)
       active#(length(X)) -> active#(X)
       active#(s(X)) -> active#(X)
       active#(U72(X1,X2)) -> active#(X1)
       active#(U71(X1,X2,X3)) -> active#(X1)
       active#(U62(X)) -> active#(X)
       active#(U61(X1,X2)) -> active#(X1)
       active#(U52(X)) -> active#(X)
       active#(U51(X1,X2)) -> active#(X1)
       active#(U42(X)) -> active#(X)
       active#(U41(X1,X2)) -> active#(X1)
       active#(U31(X)) -> active#(X)
       active#(U21(X)) -> active#(X)
       active#(U11(X)) -> active#(X)
       active#(cons(X1,X2)) -> active#(X1)
      TRS:
       active(zeros()) -> mark(cons(0(),zeros()))
       active(U11(tt())) -> mark(tt())
       active(U21(tt())) -> mark(tt())
       active(U31(tt())) -> mark(tt())
       active(U41(tt(),V2)) -> mark(U42(isNatIList(V2)))
       active(U42(tt())) -> mark(tt())
       active(U51(tt(),V2)) -> mark(U52(isNatList(V2)))
       active(U52(tt())) -> mark(tt())
       active(U61(tt(),V2)) -> mark(U62(isNatIList(V2)))
       active(U62(tt())) -> mark(tt())
       active(U71(tt(),L,N)) -> mark(U72(isNat(N),L))
       active(U72(tt(),L)) -> mark(s(length(L)))
       active(U81(tt())) -> mark(nil())
       active(U91(tt(),IL,M,N)) -> mark(U92(isNat(M),IL,M,N))
       active(U92(tt(),IL,M,N)) -> mark(U93(isNat(N),IL,M,N))
       active(U93(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
       active(isNat(0())) -> mark(tt())
       active(isNat(length(V1))) -> mark(U11(isNatList(V1)))
       active(isNat(s(V1))) -> mark(U21(isNat(V1)))
       active(isNatIList(V)) -> mark(U31(isNatList(V)))
       active(isNatIList(zeros())) -> mark(tt())
       active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2))
       active(isNatList(nil())) -> mark(tt())
       active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2))
       active(isNatList(take(V1,V2))) -> mark(U61(isNat(V1),V2))
       active(length(nil())) -> mark(0())
       active(length(cons(N,L))) -> mark(U71(isNatList(L),L,N))
       active(take(0(),IL)) -> mark(U81(isNatIList(IL)))
       active(take(s(M),cons(N,IL))) -> mark(U91(isNatIList(IL),IL,M,N))
       active(cons(X1,X2)) -> cons(active(X1),X2)
       active(U11(X)) -> U11(active(X))
       active(U21(X)) -> U21(active(X))
       active(U31(X)) -> U31(active(X))
       active(U41(X1,X2)) -> U41(active(X1),X2)
       active(U42(X)) -> U42(active(X))
       active(U51(X1,X2)) -> U51(active(X1),X2)
       active(U52(X)) -> U52(active(X))
       active(U61(X1,X2)) -> U61(active(X1),X2)
       active(U62(X)) -> U62(active(X))
       active(U71(X1,X2,X3)) -> U71(active(X1),X2,X3)
       active(U72(X1,X2)) -> U72(active(X1),X2)
       active(s(X)) -> s(active(X))
       active(length(X)) -> length(active(X))
       active(U81(X)) -> U81(active(X))
       active(U91(X1,X2,X3,X4)) -> U91(active(X1),X2,X3,X4)
       active(U92(X1,X2,X3,X4)) -> U92(active(X1),X2,X3,X4)
       active(U93(X1,X2,X3,X4)) -> U93(active(X1),X2,X3,X4)
       active(take(X1,X2)) -> take(active(X1),X2)
       active(take(X1,X2)) -> take(X1,active(X2))
       cons(mark(X1),X2) -> mark(cons(X1,X2))
       U11(mark(X)) -> mark(U11(X))
       U21(mark(X)) -> mark(U21(X))
       U31(mark(X)) -> mark(U31(X))
       U41(mark(X1),X2) -> mark(U41(X1,X2))
       U42(mark(X)) -> mark(U42(X))
       U51(mark(X1),X2) -> mark(U51(X1,X2))
       U52(mark(X)) -> mark(U52(X))
       U61(mark(X1),X2) -> mark(U61(X1,X2))
       U62(mark(X)) -> mark(U62(X))
       U71(mark(X1),X2,X3) -> mark(U71(X1,X2,X3))
       U72(mark(X1),X2) -> mark(U72(X1,X2))
       s(mark(X)) -> mark(s(X))
       length(mark(X)) -> mark(length(X))
       U81(mark(X)) -> mark(U81(X))
       U91(mark(X1),X2,X3,X4) -> mark(U91(X1,X2,X3,X4))
       U92(mark(X1),X2,X3,X4) -> mark(U92(X1,X2,X3,X4))
       U93(mark(X1),X2,X3,X4) -> mark(U93(X1,X2,X3,X4))
       take(mark(X1),X2) -> mark(take(X1,X2))
       take(X1,mark(X2)) -> mark(take(X1,X2))
       proper(zeros()) -> ok(zeros())
       proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
       proper(0()) -> ok(0())
       proper(U11(X)) -> U11(proper(X))
       proper(tt()) -> ok(tt())
       proper(U21(X)) -> U21(proper(X))
       proper(U31(X)) -> U31(proper(X))
       proper(U41(X1,X2)) -> U41(proper(X1),proper(X2))
       proper(U42(X)) -> U42(proper(X))
       proper(isNatIList(X)) -> isNatIList(proper(X))
       proper(U51(X1,X2)) -> U51(proper(X1),proper(X2))
       proper(U52(X)) -> U52(proper(X))
       proper(isNatList(X)) -> isNatList(proper(X))
       proper(U61(X1,X2)) -> U61(proper(X1),proper(X2))
       proper(U62(X)) -> U62(proper(X))
       proper(U71(X1,X2,X3)) -> U71(proper(X1),proper(X2),proper(X3))
       proper(U72(X1,X2)) -> U72(proper(X1),proper(X2))
       proper(isNat(X)) -> isNat(proper(X))
       proper(s(X)) -> s(proper(X))
       proper(length(X)) -> length(proper(X))
       proper(U81(X)) -> U81(proper(X))
       proper(nil()) -> ok(nil())
       proper(U91(X1,X2,X3,X4)) -> U91(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(U92(X1,X2,X3,X4)) -> U92(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(U93(X1,X2,X3,X4)) -> U93(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(take(X1,X2)) -> take(proper(X1),proper(X2))
       cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
       U11(ok(X)) -> ok(U11(X))
       U21(ok(X)) -> ok(U21(X))
       U31(ok(X)) -> ok(U31(X))
       U41(ok(X1),ok(X2)) -> ok(U41(X1,X2))
       U42(ok(X)) -> ok(U42(X))
       isNatIList(ok(X)) -> ok(isNatIList(X))
       U51(ok(X1),ok(X2)) -> ok(U51(X1,X2))
       U52(ok(X)) -> ok(U52(X))
       isNatList(ok(X)) -> ok(isNatList(X))
       U61(ok(X1),ok(X2)) -> ok(U61(X1,X2))
       U62(ok(X)) -> ok(U62(X))
       U71(ok(X1),ok(X2),ok(X3)) -> ok(U71(X1,X2,X3))
       U72(ok(X1),ok(X2)) -> ok(U72(X1,X2))
       isNat(ok(X)) -> ok(isNat(X))
       s(ok(X)) -> ok(s(X))
       length(ok(X)) -> ok(length(X))
       U81(ok(X)) -> ok(U81(X))
       U91(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U91(X1,X2,X3,X4))
       U92(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U92(X1,X2,X3,X4))
       U93(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U93(X1,X2,X3,X4))
       take(ok(X1),ok(X2)) -> ok(take(X1,X2))
       top(mark(X)) -> top(proper(X))
       top(ok(X)) -> top(active(X))
      Open
      
      DPs:
       isNat#(ok(X)) -> isNat#(X)
      TRS:
       active(zeros()) -> mark(cons(0(),zeros()))
       active(U11(tt())) -> mark(tt())
       active(U21(tt())) -> mark(tt())
       active(U31(tt())) -> mark(tt())
       active(U41(tt(),V2)) -> mark(U42(isNatIList(V2)))
       active(U42(tt())) -> mark(tt())
       active(U51(tt(),V2)) -> mark(U52(isNatList(V2)))
       active(U52(tt())) -> mark(tt())
       active(U61(tt(),V2)) -> mark(U62(isNatIList(V2)))
       active(U62(tt())) -> mark(tt())
       active(U71(tt(),L,N)) -> mark(U72(isNat(N),L))
       active(U72(tt(),L)) -> mark(s(length(L)))
       active(U81(tt())) -> mark(nil())
       active(U91(tt(),IL,M,N)) -> mark(U92(isNat(M),IL,M,N))
       active(U92(tt(),IL,M,N)) -> mark(U93(isNat(N),IL,M,N))
       active(U93(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
       active(isNat(0())) -> mark(tt())
       active(isNat(length(V1))) -> mark(U11(isNatList(V1)))
       active(isNat(s(V1))) -> mark(U21(isNat(V1)))
       active(isNatIList(V)) -> mark(U31(isNatList(V)))
       active(isNatIList(zeros())) -> mark(tt())
       active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2))
       active(isNatList(nil())) -> mark(tt())
       active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2))
       active(isNatList(take(V1,V2))) -> mark(U61(isNat(V1),V2))
       active(length(nil())) -> mark(0())
       active(length(cons(N,L))) -> mark(U71(isNatList(L),L,N))
       active(take(0(),IL)) -> mark(U81(isNatIList(IL)))
       active(take(s(M),cons(N,IL))) -> mark(U91(isNatIList(IL),IL,M,N))
       active(cons(X1,X2)) -> cons(active(X1),X2)
       active(U11(X)) -> U11(active(X))
       active(U21(X)) -> U21(active(X))
       active(U31(X)) -> U31(active(X))
       active(U41(X1,X2)) -> U41(active(X1),X2)
       active(U42(X)) -> U42(active(X))
       active(U51(X1,X2)) -> U51(active(X1),X2)
       active(U52(X)) -> U52(active(X))
       active(U61(X1,X2)) -> U61(active(X1),X2)
       active(U62(X)) -> U62(active(X))
       active(U71(X1,X2,X3)) -> U71(active(X1),X2,X3)
       active(U72(X1,X2)) -> U72(active(X1),X2)
       active(s(X)) -> s(active(X))
       active(length(X)) -> length(active(X))
       active(U81(X)) -> U81(active(X))
       active(U91(X1,X2,X3,X4)) -> U91(active(X1),X2,X3,X4)
       active(U92(X1,X2,X3,X4)) -> U92(active(X1),X2,X3,X4)
       active(U93(X1,X2,X3,X4)) -> U93(active(X1),X2,X3,X4)
       active(take(X1,X2)) -> take(active(X1),X2)
       active(take(X1,X2)) -> take(X1,active(X2))
       cons(mark(X1),X2) -> mark(cons(X1,X2))
       U11(mark(X)) -> mark(U11(X))
       U21(mark(X)) -> mark(U21(X))
       U31(mark(X)) -> mark(U31(X))
       U41(mark(X1),X2) -> mark(U41(X1,X2))
       U42(mark(X)) -> mark(U42(X))
       U51(mark(X1),X2) -> mark(U51(X1,X2))
       U52(mark(X)) -> mark(U52(X))
       U61(mark(X1),X2) -> mark(U61(X1,X2))
       U62(mark(X)) -> mark(U62(X))
       U71(mark(X1),X2,X3) -> mark(U71(X1,X2,X3))
       U72(mark(X1),X2) -> mark(U72(X1,X2))
       s(mark(X)) -> mark(s(X))
       length(mark(X)) -> mark(length(X))
       U81(mark(X)) -> mark(U81(X))
       U91(mark(X1),X2,X3,X4) -> mark(U91(X1,X2,X3,X4))
       U92(mark(X1),X2,X3,X4) -> mark(U92(X1,X2,X3,X4))
       U93(mark(X1),X2,X3,X4) -> mark(U93(X1,X2,X3,X4))
       take(mark(X1),X2) -> mark(take(X1,X2))
       take(X1,mark(X2)) -> mark(take(X1,X2))
       proper(zeros()) -> ok(zeros())
       proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
       proper(0()) -> ok(0())
       proper(U11(X)) -> U11(proper(X))
       proper(tt()) -> ok(tt())
       proper(U21(X)) -> U21(proper(X))
       proper(U31(X)) -> U31(proper(X))
       proper(U41(X1,X2)) -> U41(proper(X1),proper(X2))
       proper(U42(X)) -> U42(proper(X))
       proper(isNatIList(X)) -> isNatIList(proper(X))
       proper(U51(X1,X2)) -> U51(proper(X1),proper(X2))
       proper(U52(X)) -> U52(proper(X))
       proper(isNatList(X)) -> isNatList(proper(X))
       proper(U61(X1,X2)) -> U61(proper(X1),proper(X2))
       proper(U62(X)) -> U62(proper(X))
       proper(U71(X1,X2,X3)) -> U71(proper(X1),proper(X2),proper(X3))
       proper(U72(X1,X2)) -> U72(proper(X1),proper(X2))
       proper(isNat(X)) -> isNat(proper(X))
       proper(s(X)) -> s(proper(X))
       proper(length(X)) -> length(proper(X))
       proper(U81(X)) -> U81(proper(X))
       proper(nil()) -> ok(nil())
       proper(U91(X1,X2,X3,X4)) -> U91(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(U92(X1,X2,X3,X4)) -> U92(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(U93(X1,X2,X3,X4)) -> U93(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(take(X1,X2)) -> take(proper(X1),proper(X2))
       cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
       U11(ok(X)) -> ok(U11(X))
       U21(ok(X)) -> ok(U21(X))
       U31(ok(X)) -> ok(U31(X))
       U41(ok(X1),ok(X2)) -> ok(U41(X1,X2))
       U42(ok(X)) -> ok(U42(X))
       isNatIList(ok(X)) -> ok(isNatIList(X))
       U51(ok(X1),ok(X2)) -> ok(U51(X1,X2))
       U52(ok(X)) -> ok(U52(X))
       isNatList(ok(X)) -> ok(isNatList(X))
       U61(ok(X1),ok(X2)) -> ok(U61(X1,X2))
       U62(ok(X)) -> ok(U62(X))
       U71(ok(X1),ok(X2),ok(X3)) -> ok(U71(X1,X2,X3))
       U72(ok(X1),ok(X2)) -> ok(U72(X1,X2))
       isNat(ok(X)) -> ok(isNat(X))
       s(ok(X)) -> ok(s(X))
       length(ok(X)) -> ok(length(X))
       U81(ok(X)) -> ok(U81(X))
       U91(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U91(X1,X2,X3,X4))
       U92(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U92(X1,X2,X3,X4))
       U93(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U93(X1,X2,X3,X4))
       take(ok(X1),ok(X2)) -> ok(take(X1,X2))
       top(mark(X)) -> top(proper(X))
       top(ok(X)) -> top(active(X))
      Open
      
      DPs:
       isNatList#(ok(X)) -> isNatList#(X)
      TRS:
       active(zeros()) -> mark(cons(0(),zeros()))
       active(U11(tt())) -> mark(tt())
       active(U21(tt())) -> mark(tt())
       active(U31(tt())) -> mark(tt())
       active(U41(tt(),V2)) -> mark(U42(isNatIList(V2)))
       active(U42(tt())) -> mark(tt())
       active(U51(tt(),V2)) -> mark(U52(isNatList(V2)))
       active(U52(tt())) -> mark(tt())
       active(U61(tt(),V2)) -> mark(U62(isNatIList(V2)))
       active(U62(tt())) -> mark(tt())
       active(U71(tt(),L,N)) -> mark(U72(isNat(N),L))
       active(U72(tt(),L)) -> mark(s(length(L)))
       active(U81(tt())) -> mark(nil())
       active(U91(tt(),IL,M,N)) -> mark(U92(isNat(M),IL,M,N))
       active(U92(tt(),IL,M,N)) -> mark(U93(isNat(N),IL,M,N))
       active(U93(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
       active(isNat(0())) -> mark(tt())
       active(isNat(length(V1))) -> mark(U11(isNatList(V1)))
       active(isNat(s(V1))) -> mark(U21(isNat(V1)))
       active(isNatIList(V)) -> mark(U31(isNatList(V)))
       active(isNatIList(zeros())) -> mark(tt())
       active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2))
       active(isNatList(nil())) -> mark(tt())
       active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2))
       active(isNatList(take(V1,V2))) -> mark(U61(isNat(V1),V2))
       active(length(nil())) -> mark(0())
       active(length(cons(N,L))) -> mark(U71(isNatList(L),L,N))
       active(take(0(),IL)) -> mark(U81(isNatIList(IL)))
       active(take(s(M),cons(N,IL))) -> mark(U91(isNatIList(IL),IL,M,N))
       active(cons(X1,X2)) -> cons(active(X1),X2)
       active(U11(X)) -> U11(active(X))
       active(U21(X)) -> U21(active(X))
       active(U31(X)) -> U31(active(X))
       active(U41(X1,X2)) -> U41(active(X1),X2)
       active(U42(X)) -> U42(active(X))
       active(U51(X1,X2)) -> U51(active(X1),X2)
       active(U52(X)) -> U52(active(X))
       active(U61(X1,X2)) -> U61(active(X1),X2)
       active(U62(X)) -> U62(active(X))
       active(U71(X1,X2,X3)) -> U71(active(X1),X2,X3)
       active(U72(X1,X2)) -> U72(active(X1),X2)
       active(s(X)) -> s(active(X))
       active(length(X)) -> length(active(X))
       active(U81(X)) -> U81(active(X))
       active(U91(X1,X2,X3,X4)) -> U91(active(X1),X2,X3,X4)
       active(U92(X1,X2,X3,X4)) -> U92(active(X1),X2,X3,X4)
       active(U93(X1,X2,X3,X4)) -> U93(active(X1),X2,X3,X4)
       active(take(X1,X2)) -> take(active(X1),X2)
       active(take(X1,X2)) -> take(X1,active(X2))
       cons(mark(X1),X2) -> mark(cons(X1,X2))
       U11(mark(X)) -> mark(U11(X))
       U21(mark(X)) -> mark(U21(X))
       U31(mark(X)) -> mark(U31(X))
       U41(mark(X1),X2) -> mark(U41(X1,X2))
       U42(mark(X)) -> mark(U42(X))
       U51(mark(X1),X2) -> mark(U51(X1,X2))
       U52(mark(X)) -> mark(U52(X))
       U61(mark(X1),X2) -> mark(U61(X1,X2))
       U62(mark(X)) -> mark(U62(X))
       U71(mark(X1),X2,X3) -> mark(U71(X1,X2,X3))
       U72(mark(X1),X2) -> mark(U72(X1,X2))
       s(mark(X)) -> mark(s(X))
       length(mark(X)) -> mark(length(X))
       U81(mark(X)) -> mark(U81(X))
       U91(mark(X1),X2,X3,X4) -> mark(U91(X1,X2,X3,X4))
       U92(mark(X1),X2,X3,X4) -> mark(U92(X1,X2,X3,X4))
       U93(mark(X1),X2,X3,X4) -> mark(U93(X1,X2,X3,X4))
       take(mark(X1),X2) -> mark(take(X1,X2))
       take(X1,mark(X2)) -> mark(take(X1,X2))
       proper(zeros()) -> ok(zeros())
       proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
       proper(0()) -> ok(0())
       proper(U11(X)) -> U11(proper(X))
       proper(tt()) -> ok(tt())
       proper(U21(X)) -> U21(proper(X))
       proper(U31(X)) -> U31(proper(X))
       proper(U41(X1,X2)) -> U41(proper(X1),proper(X2))
       proper(U42(X)) -> U42(proper(X))
       proper(isNatIList(X)) -> isNatIList(proper(X))
       proper(U51(X1,X2)) -> U51(proper(X1),proper(X2))
       proper(U52(X)) -> U52(proper(X))
       proper(isNatList(X)) -> isNatList(proper(X))
       proper(U61(X1,X2)) -> U61(proper(X1),proper(X2))
       proper(U62(X)) -> U62(proper(X))
       proper(U71(X1,X2,X3)) -> U71(proper(X1),proper(X2),proper(X3))
       proper(U72(X1,X2)) -> U72(proper(X1),proper(X2))
       proper(isNat(X)) -> isNat(proper(X))
       proper(s(X)) -> s(proper(X))
       proper(length(X)) -> length(proper(X))
       proper(U81(X)) -> U81(proper(X))
       proper(nil()) -> ok(nil())
       proper(U91(X1,X2,X3,X4)) -> U91(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(U92(X1,X2,X3,X4)) -> U92(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(U93(X1,X2,X3,X4)) -> U93(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(take(X1,X2)) -> take(proper(X1),proper(X2))
       cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
       U11(ok(X)) -> ok(U11(X))
       U21(ok(X)) -> ok(U21(X))
       U31(ok(X)) -> ok(U31(X))
       U41(ok(X1),ok(X2)) -> ok(U41(X1,X2))
       U42(ok(X)) -> ok(U42(X))
       isNatIList(ok(X)) -> ok(isNatIList(X))
       U51(ok(X1),ok(X2)) -> ok(U51(X1,X2))
       U52(ok(X)) -> ok(U52(X))
       isNatList(ok(X)) -> ok(isNatList(X))
       U61(ok(X1),ok(X2)) -> ok(U61(X1,X2))
       U62(ok(X)) -> ok(U62(X))
       U71(ok(X1),ok(X2),ok(X3)) -> ok(U71(X1,X2,X3))
       U72(ok(X1),ok(X2)) -> ok(U72(X1,X2))
       isNat(ok(X)) -> ok(isNat(X))
       s(ok(X)) -> ok(s(X))
       length(ok(X)) -> ok(length(X))
       U81(ok(X)) -> ok(U81(X))
       U91(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U91(X1,X2,X3,X4))
       U92(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U92(X1,X2,X3,X4))
       U93(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U93(X1,X2,X3,X4))
       take(ok(X1),ok(X2)) -> ok(take(X1,X2))
       top(mark(X)) -> top(proper(X))
       top(ok(X)) -> top(active(X))
      Open
      
      DPs:
       isNatIList#(ok(X)) -> isNatIList#(X)
      TRS:
       active(zeros()) -> mark(cons(0(),zeros()))
       active(U11(tt())) -> mark(tt())
       active(U21(tt())) -> mark(tt())
       active(U31(tt())) -> mark(tt())
       active(U41(tt(),V2)) -> mark(U42(isNatIList(V2)))
       active(U42(tt())) -> mark(tt())
       active(U51(tt(),V2)) -> mark(U52(isNatList(V2)))
       active(U52(tt())) -> mark(tt())
       active(U61(tt(),V2)) -> mark(U62(isNatIList(V2)))
       active(U62(tt())) -> mark(tt())
       active(U71(tt(),L,N)) -> mark(U72(isNat(N),L))
       active(U72(tt(),L)) -> mark(s(length(L)))
       active(U81(tt())) -> mark(nil())
       active(U91(tt(),IL,M,N)) -> mark(U92(isNat(M),IL,M,N))
       active(U92(tt(),IL,M,N)) -> mark(U93(isNat(N),IL,M,N))
       active(U93(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
       active(isNat(0())) -> mark(tt())
       active(isNat(length(V1))) -> mark(U11(isNatList(V1)))
       active(isNat(s(V1))) -> mark(U21(isNat(V1)))
       active(isNatIList(V)) -> mark(U31(isNatList(V)))
       active(isNatIList(zeros())) -> mark(tt())
       active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2))
       active(isNatList(nil())) -> mark(tt())
       active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2))
       active(isNatList(take(V1,V2))) -> mark(U61(isNat(V1),V2))
       active(length(nil())) -> mark(0())
       active(length(cons(N,L))) -> mark(U71(isNatList(L),L,N))
       active(take(0(),IL)) -> mark(U81(isNatIList(IL)))
       active(take(s(M),cons(N,IL))) -> mark(U91(isNatIList(IL),IL,M,N))
       active(cons(X1,X2)) -> cons(active(X1),X2)
       active(U11(X)) -> U11(active(X))
       active(U21(X)) -> U21(active(X))
       active(U31(X)) -> U31(active(X))
       active(U41(X1,X2)) -> U41(active(X1),X2)
       active(U42(X)) -> U42(active(X))
       active(U51(X1,X2)) -> U51(active(X1),X2)
       active(U52(X)) -> U52(active(X))
       active(U61(X1,X2)) -> U61(active(X1),X2)
       active(U62(X)) -> U62(active(X))
       active(U71(X1,X2,X3)) -> U71(active(X1),X2,X3)
       active(U72(X1,X2)) -> U72(active(X1),X2)
       active(s(X)) -> s(active(X))
       active(length(X)) -> length(active(X))
       active(U81(X)) -> U81(active(X))
       active(U91(X1,X2,X3,X4)) -> U91(active(X1),X2,X3,X4)
       active(U92(X1,X2,X3,X4)) -> U92(active(X1),X2,X3,X4)
       active(U93(X1,X2,X3,X4)) -> U93(active(X1),X2,X3,X4)
       active(take(X1,X2)) -> take(active(X1),X2)
       active(take(X1,X2)) -> take(X1,active(X2))
       cons(mark(X1),X2) -> mark(cons(X1,X2))
       U11(mark(X)) -> mark(U11(X))
       U21(mark(X)) -> mark(U21(X))
       U31(mark(X)) -> mark(U31(X))
       U41(mark(X1),X2) -> mark(U41(X1,X2))
       U42(mark(X)) -> mark(U42(X))
       U51(mark(X1),X2) -> mark(U51(X1,X2))
       U52(mark(X)) -> mark(U52(X))
       U61(mark(X1),X2) -> mark(U61(X1,X2))
       U62(mark(X)) -> mark(U62(X))
       U71(mark(X1),X2,X3) -> mark(U71(X1,X2,X3))
       U72(mark(X1),X2) -> mark(U72(X1,X2))
       s(mark(X)) -> mark(s(X))
       length(mark(X)) -> mark(length(X))
       U81(mark(X)) -> mark(U81(X))
       U91(mark(X1),X2,X3,X4) -> mark(U91(X1,X2,X3,X4))
       U92(mark(X1),X2,X3,X4) -> mark(U92(X1,X2,X3,X4))
       U93(mark(X1),X2,X3,X4) -> mark(U93(X1,X2,X3,X4))
       take(mark(X1),X2) -> mark(take(X1,X2))
       take(X1,mark(X2)) -> mark(take(X1,X2))
       proper(zeros()) -> ok(zeros())
       proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
       proper(0()) -> ok(0())
       proper(U11(X)) -> U11(proper(X))
       proper(tt()) -> ok(tt())
       proper(U21(X)) -> U21(proper(X))
       proper(U31(X)) -> U31(proper(X))
       proper(U41(X1,X2)) -> U41(proper(X1),proper(X2))
       proper(U42(X)) -> U42(proper(X))
       proper(isNatIList(X)) -> isNatIList(proper(X))
       proper(U51(X1,X2)) -> U51(proper(X1),proper(X2))
       proper(U52(X)) -> U52(proper(X))
       proper(isNatList(X)) -> isNatList(proper(X))
       proper(U61(X1,X2)) -> U61(proper(X1),proper(X2))
       proper(U62(X)) -> U62(proper(X))
       proper(U71(X1,X2,X3)) -> U71(proper(X1),proper(X2),proper(X3))
       proper(U72(X1,X2)) -> U72(proper(X1),proper(X2))
       proper(isNat(X)) -> isNat(proper(X))
       proper(s(X)) -> s(proper(X))
       proper(length(X)) -> length(proper(X))
       proper(U81(X)) -> U81(proper(X))
       proper(nil()) -> ok(nil())
       proper(U91(X1,X2,X3,X4)) -> U91(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(U92(X1,X2,X3,X4)) -> U92(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(U93(X1,X2,X3,X4)) -> U93(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(take(X1,X2)) -> take(proper(X1),proper(X2))
       cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
       U11(ok(X)) -> ok(U11(X))
       U21(ok(X)) -> ok(U21(X))
       U31(ok(X)) -> ok(U31(X))
       U41(ok(X1),ok(X2)) -> ok(U41(X1,X2))
       U42(ok(X)) -> ok(U42(X))
       isNatIList(ok(X)) -> ok(isNatIList(X))
       U51(ok(X1),ok(X2)) -> ok(U51(X1,X2))
       U52(ok(X)) -> ok(U52(X))
       isNatList(ok(X)) -> ok(isNatList(X))
       U61(ok(X1),ok(X2)) -> ok(U61(X1,X2))
       U62(ok(X)) -> ok(U62(X))
       U71(ok(X1),ok(X2),ok(X3)) -> ok(U71(X1,X2,X3))
       U72(ok(X1),ok(X2)) -> ok(U72(X1,X2))
       isNat(ok(X)) -> ok(isNat(X))
       s(ok(X)) -> ok(s(X))
       length(ok(X)) -> ok(length(X))
       U81(ok(X)) -> ok(U81(X))
       U91(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U91(X1,X2,X3,X4))
       U92(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U92(X1,X2,X3,X4))
       U93(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U93(X1,X2,X3,X4))
       take(ok(X1),ok(X2)) -> ok(take(X1,X2))
       top(mark(X)) -> top(proper(X))
       top(ok(X)) -> top(active(X))
      Open
      
      DPs:
       cons#(ok(X1),ok(X2)) -> cons#(X1,X2)
       cons#(mark(X1),X2) -> cons#(X1,X2)
      TRS:
       active(zeros()) -> mark(cons(0(),zeros()))
       active(U11(tt())) -> mark(tt())
       active(U21(tt())) -> mark(tt())
       active(U31(tt())) -> mark(tt())
       active(U41(tt(),V2)) -> mark(U42(isNatIList(V2)))
       active(U42(tt())) -> mark(tt())
       active(U51(tt(),V2)) -> mark(U52(isNatList(V2)))
       active(U52(tt())) -> mark(tt())
       active(U61(tt(),V2)) -> mark(U62(isNatIList(V2)))
       active(U62(tt())) -> mark(tt())
       active(U71(tt(),L,N)) -> mark(U72(isNat(N),L))
       active(U72(tt(),L)) -> mark(s(length(L)))
       active(U81(tt())) -> mark(nil())
       active(U91(tt(),IL,M,N)) -> mark(U92(isNat(M),IL,M,N))
       active(U92(tt(),IL,M,N)) -> mark(U93(isNat(N),IL,M,N))
       active(U93(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
       active(isNat(0())) -> mark(tt())
       active(isNat(length(V1))) -> mark(U11(isNatList(V1)))
       active(isNat(s(V1))) -> mark(U21(isNat(V1)))
       active(isNatIList(V)) -> mark(U31(isNatList(V)))
       active(isNatIList(zeros())) -> mark(tt())
       active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2))
       active(isNatList(nil())) -> mark(tt())
       active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2))
       active(isNatList(take(V1,V2))) -> mark(U61(isNat(V1),V2))
       active(length(nil())) -> mark(0())
       active(length(cons(N,L))) -> mark(U71(isNatList(L),L,N))
       active(take(0(),IL)) -> mark(U81(isNatIList(IL)))
       active(take(s(M),cons(N,IL))) -> mark(U91(isNatIList(IL),IL,M,N))
       active(cons(X1,X2)) -> cons(active(X1),X2)
       active(U11(X)) -> U11(active(X))
       active(U21(X)) -> U21(active(X))
       active(U31(X)) -> U31(active(X))
       active(U41(X1,X2)) -> U41(active(X1),X2)
       active(U42(X)) -> U42(active(X))
       active(U51(X1,X2)) -> U51(active(X1),X2)
       active(U52(X)) -> U52(active(X))
       active(U61(X1,X2)) -> U61(active(X1),X2)
       active(U62(X)) -> U62(active(X))
       active(U71(X1,X2,X3)) -> U71(active(X1),X2,X3)
       active(U72(X1,X2)) -> U72(active(X1),X2)
       active(s(X)) -> s(active(X))
       active(length(X)) -> length(active(X))
       active(U81(X)) -> U81(active(X))
       active(U91(X1,X2,X3,X4)) -> U91(active(X1),X2,X3,X4)
       active(U92(X1,X2,X3,X4)) -> U92(active(X1),X2,X3,X4)
       active(U93(X1,X2,X3,X4)) -> U93(active(X1),X2,X3,X4)
       active(take(X1,X2)) -> take(active(X1),X2)
       active(take(X1,X2)) -> take(X1,active(X2))
       cons(mark(X1),X2) -> mark(cons(X1,X2))
       U11(mark(X)) -> mark(U11(X))
       U21(mark(X)) -> mark(U21(X))
       U31(mark(X)) -> mark(U31(X))
       U41(mark(X1),X2) -> mark(U41(X1,X2))
       U42(mark(X)) -> mark(U42(X))
       U51(mark(X1),X2) -> mark(U51(X1,X2))
       U52(mark(X)) -> mark(U52(X))
       U61(mark(X1),X2) -> mark(U61(X1,X2))
       U62(mark(X)) -> mark(U62(X))
       U71(mark(X1),X2,X3) -> mark(U71(X1,X2,X3))
       U72(mark(X1),X2) -> mark(U72(X1,X2))
       s(mark(X)) -> mark(s(X))
       length(mark(X)) -> mark(length(X))
       U81(mark(X)) -> mark(U81(X))
       U91(mark(X1),X2,X3,X4) -> mark(U91(X1,X2,X3,X4))
       U92(mark(X1),X2,X3,X4) -> mark(U92(X1,X2,X3,X4))
       U93(mark(X1),X2,X3,X4) -> mark(U93(X1,X2,X3,X4))
       take(mark(X1),X2) -> mark(take(X1,X2))
       take(X1,mark(X2)) -> mark(take(X1,X2))
       proper(zeros()) -> ok(zeros())
       proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
       proper(0()) -> ok(0())
       proper(U11(X)) -> U11(proper(X))
       proper(tt()) -> ok(tt())
       proper(U21(X)) -> U21(proper(X))
       proper(U31(X)) -> U31(proper(X))
       proper(U41(X1,X2)) -> U41(proper(X1),proper(X2))
       proper(U42(X)) -> U42(proper(X))
       proper(isNatIList(X)) -> isNatIList(proper(X))
       proper(U51(X1,X2)) -> U51(proper(X1),proper(X2))
       proper(U52(X)) -> U52(proper(X))
       proper(isNatList(X)) -> isNatList(proper(X))
       proper(U61(X1,X2)) -> U61(proper(X1),proper(X2))
       proper(U62(X)) -> U62(proper(X))
       proper(U71(X1,X2,X3)) -> U71(proper(X1),proper(X2),proper(X3))
       proper(U72(X1,X2)) -> U72(proper(X1),proper(X2))
       proper(isNat(X)) -> isNat(proper(X))
       proper(s(X)) -> s(proper(X))
       proper(length(X)) -> length(proper(X))
       proper(U81(X)) -> U81(proper(X))
       proper(nil()) -> ok(nil())
       proper(U91(X1,X2,X3,X4)) -> U91(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(U92(X1,X2,X3,X4)) -> U92(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(U93(X1,X2,X3,X4)) -> U93(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(take(X1,X2)) -> take(proper(X1),proper(X2))
       cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
       U11(ok(X)) -> ok(U11(X))
       U21(ok(X)) -> ok(U21(X))
       U31(ok(X)) -> ok(U31(X))
       U41(ok(X1),ok(X2)) -> ok(U41(X1,X2))
       U42(ok(X)) -> ok(U42(X))
       isNatIList(ok(X)) -> ok(isNatIList(X))
       U51(ok(X1),ok(X2)) -> ok(U51(X1,X2))
       U52(ok(X)) -> ok(U52(X))
       isNatList(ok(X)) -> ok(isNatList(X))
       U61(ok(X1),ok(X2)) -> ok(U61(X1,X2))
       U62(ok(X)) -> ok(U62(X))
       U71(ok(X1),ok(X2),ok(X3)) -> ok(U71(X1,X2,X3))
       U72(ok(X1),ok(X2)) -> ok(U72(X1,X2))
       isNat(ok(X)) -> ok(isNat(X))
       s(ok(X)) -> ok(s(X))
       length(ok(X)) -> ok(length(X))
       U81(ok(X)) -> ok(U81(X))
       U91(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U91(X1,X2,X3,X4))
       U92(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U92(X1,X2,X3,X4))
       U93(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U93(X1,X2,X3,X4))
       take(ok(X1),ok(X2)) -> ok(take(X1,X2))
       top(mark(X)) -> top(proper(X))
       top(ok(X)) -> top(active(X))
      Open
      
      DPs:
       U11#(ok(X)) -> U11#(X)
       U11#(mark(X)) -> U11#(X)
      TRS:
       active(zeros()) -> mark(cons(0(),zeros()))
       active(U11(tt())) -> mark(tt())
       active(U21(tt())) -> mark(tt())
       active(U31(tt())) -> mark(tt())
       active(U41(tt(),V2)) -> mark(U42(isNatIList(V2)))
       active(U42(tt())) -> mark(tt())
       active(U51(tt(),V2)) -> mark(U52(isNatList(V2)))
       active(U52(tt())) -> mark(tt())
       active(U61(tt(),V2)) -> mark(U62(isNatIList(V2)))
       active(U62(tt())) -> mark(tt())
       active(U71(tt(),L,N)) -> mark(U72(isNat(N),L))
       active(U72(tt(),L)) -> mark(s(length(L)))
       active(U81(tt())) -> mark(nil())
       active(U91(tt(),IL,M,N)) -> mark(U92(isNat(M),IL,M,N))
       active(U92(tt(),IL,M,N)) -> mark(U93(isNat(N),IL,M,N))
       active(U93(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
       active(isNat(0())) -> mark(tt())
       active(isNat(length(V1))) -> mark(U11(isNatList(V1)))
       active(isNat(s(V1))) -> mark(U21(isNat(V1)))
       active(isNatIList(V)) -> mark(U31(isNatList(V)))
       active(isNatIList(zeros())) -> mark(tt())
       active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2))
       active(isNatList(nil())) -> mark(tt())
       active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2))
       active(isNatList(take(V1,V2))) -> mark(U61(isNat(V1),V2))
       active(length(nil())) -> mark(0())
       active(length(cons(N,L))) -> mark(U71(isNatList(L),L,N))
       active(take(0(),IL)) -> mark(U81(isNatIList(IL)))
       active(take(s(M),cons(N,IL))) -> mark(U91(isNatIList(IL),IL,M,N))
       active(cons(X1,X2)) -> cons(active(X1),X2)
       active(U11(X)) -> U11(active(X))
       active(U21(X)) -> U21(active(X))
       active(U31(X)) -> U31(active(X))
       active(U41(X1,X2)) -> U41(active(X1),X2)
       active(U42(X)) -> U42(active(X))
       active(U51(X1,X2)) -> U51(active(X1),X2)
       active(U52(X)) -> U52(active(X))
       active(U61(X1,X2)) -> U61(active(X1),X2)
       active(U62(X)) -> U62(active(X))
       active(U71(X1,X2,X3)) -> U71(active(X1),X2,X3)
       active(U72(X1,X2)) -> U72(active(X1),X2)
       active(s(X)) -> s(active(X))
       active(length(X)) -> length(active(X))
       active(U81(X)) -> U81(active(X))
       active(U91(X1,X2,X3,X4)) -> U91(active(X1),X2,X3,X4)
       active(U92(X1,X2,X3,X4)) -> U92(active(X1),X2,X3,X4)
       active(U93(X1,X2,X3,X4)) -> U93(active(X1),X2,X3,X4)
       active(take(X1,X2)) -> take(active(X1),X2)
       active(take(X1,X2)) -> take(X1,active(X2))
       cons(mark(X1),X2) -> mark(cons(X1,X2))
       U11(mark(X)) -> mark(U11(X))
       U21(mark(X)) -> mark(U21(X))
       U31(mark(X)) -> mark(U31(X))
       U41(mark(X1),X2) -> mark(U41(X1,X2))
       U42(mark(X)) -> mark(U42(X))
       U51(mark(X1),X2) -> mark(U51(X1,X2))
       U52(mark(X)) -> mark(U52(X))
       U61(mark(X1),X2) -> mark(U61(X1,X2))
       U62(mark(X)) -> mark(U62(X))
       U71(mark(X1),X2,X3) -> mark(U71(X1,X2,X3))
       U72(mark(X1),X2) -> mark(U72(X1,X2))
       s(mark(X)) -> mark(s(X))
       length(mark(X)) -> mark(length(X))
       U81(mark(X)) -> mark(U81(X))
       U91(mark(X1),X2,X3,X4) -> mark(U91(X1,X2,X3,X4))
       U92(mark(X1),X2,X3,X4) -> mark(U92(X1,X2,X3,X4))
       U93(mark(X1),X2,X3,X4) -> mark(U93(X1,X2,X3,X4))
       take(mark(X1),X2) -> mark(take(X1,X2))
       take(X1,mark(X2)) -> mark(take(X1,X2))
       proper(zeros()) -> ok(zeros())
       proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
       proper(0()) -> ok(0())
       proper(U11(X)) -> U11(proper(X))
       proper(tt()) -> ok(tt())
       proper(U21(X)) -> U21(proper(X))
       proper(U31(X)) -> U31(proper(X))
       proper(U41(X1,X2)) -> U41(proper(X1),proper(X2))
       proper(U42(X)) -> U42(proper(X))
       proper(isNatIList(X)) -> isNatIList(proper(X))
       proper(U51(X1,X2)) -> U51(proper(X1),proper(X2))
       proper(U52(X)) -> U52(proper(X))
       proper(isNatList(X)) -> isNatList(proper(X))
       proper(U61(X1,X2)) -> U61(proper(X1),proper(X2))
       proper(U62(X)) -> U62(proper(X))
       proper(U71(X1,X2,X3)) -> U71(proper(X1),proper(X2),proper(X3))
       proper(U72(X1,X2)) -> U72(proper(X1),proper(X2))
       proper(isNat(X)) -> isNat(proper(X))
       proper(s(X)) -> s(proper(X))
       proper(length(X)) -> length(proper(X))
       proper(U81(X)) -> U81(proper(X))
       proper(nil()) -> ok(nil())
       proper(U91(X1,X2,X3,X4)) -> U91(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(U92(X1,X2,X3,X4)) -> U92(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(U93(X1,X2,X3,X4)) -> U93(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(take(X1,X2)) -> take(proper(X1),proper(X2))
       cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
       U11(ok(X)) -> ok(U11(X))
       U21(ok(X)) -> ok(U21(X))
       U31(ok(X)) -> ok(U31(X))
       U41(ok(X1),ok(X2)) -> ok(U41(X1,X2))
       U42(ok(X)) -> ok(U42(X))
       isNatIList(ok(X)) -> ok(isNatIList(X))
       U51(ok(X1),ok(X2)) -> ok(U51(X1,X2))
       U52(ok(X)) -> ok(U52(X))
       isNatList(ok(X)) -> ok(isNatList(X))
       U61(ok(X1),ok(X2)) -> ok(U61(X1,X2))
       U62(ok(X)) -> ok(U62(X))
       U71(ok(X1),ok(X2),ok(X3)) -> ok(U71(X1,X2,X3))
       U72(ok(X1),ok(X2)) -> ok(U72(X1,X2))
       isNat(ok(X)) -> ok(isNat(X))
       s(ok(X)) -> ok(s(X))
       length(ok(X)) -> ok(length(X))
       U81(ok(X)) -> ok(U81(X))
       U91(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U91(X1,X2,X3,X4))
       U92(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U92(X1,X2,X3,X4))
       U93(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U93(X1,X2,X3,X4))
       take(ok(X1),ok(X2)) -> ok(take(X1,X2))
       top(mark(X)) -> top(proper(X))
       top(ok(X)) -> top(active(X))
      Open
      
      DPs:
       U21#(ok(X)) -> U21#(X)
       U21#(mark(X)) -> U21#(X)
      TRS:
       active(zeros()) -> mark(cons(0(),zeros()))
       active(U11(tt())) -> mark(tt())
       active(U21(tt())) -> mark(tt())
       active(U31(tt())) -> mark(tt())
       active(U41(tt(),V2)) -> mark(U42(isNatIList(V2)))
       active(U42(tt())) -> mark(tt())
       active(U51(tt(),V2)) -> mark(U52(isNatList(V2)))
       active(U52(tt())) -> mark(tt())
       active(U61(tt(),V2)) -> mark(U62(isNatIList(V2)))
       active(U62(tt())) -> mark(tt())
       active(U71(tt(),L,N)) -> mark(U72(isNat(N),L))
       active(U72(tt(),L)) -> mark(s(length(L)))
       active(U81(tt())) -> mark(nil())
       active(U91(tt(),IL,M,N)) -> mark(U92(isNat(M),IL,M,N))
       active(U92(tt(),IL,M,N)) -> mark(U93(isNat(N),IL,M,N))
       active(U93(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
       active(isNat(0())) -> mark(tt())
       active(isNat(length(V1))) -> mark(U11(isNatList(V1)))
       active(isNat(s(V1))) -> mark(U21(isNat(V1)))
       active(isNatIList(V)) -> mark(U31(isNatList(V)))
       active(isNatIList(zeros())) -> mark(tt())
       active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2))
       active(isNatList(nil())) -> mark(tt())
       active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2))
       active(isNatList(take(V1,V2))) -> mark(U61(isNat(V1),V2))
       active(length(nil())) -> mark(0())
       active(length(cons(N,L))) -> mark(U71(isNatList(L),L,N))
       active(take(0(),IL)) -> mark(U81(isNatIList(IL)))
       active(take(s(M),cons(N,IL))) -> mark(U91(isNatIList(IL),IL,M,N))
       active(cons(X1,X2)) -> cons(active(X1),X2)
       active(U11(X)) -> U11(active(X))
       active(U21(X)) -> U21(active(X))
       active(U31(X)) -> U31(active(X))
       active(U41(X1,X2)) -> U41(active(X1),X2)
       active(U42(X)) -> U42(active(X))
       active(U51(X1,X2)) -> U51(active(X1),X2)
       active(U52(X)) -> U52(active(X))
       active(U61(X1,X2)) -> U61(active(X1),X2)
       active(U62(X)) -> U62(active(X))
       active(U71(X1,X2,X3)) -> U71(active(X1),X2,X3)
       active(U72(X1,X2)) -> U72(active(X1),X2)
       active(s(X)) -> s(active(X))
       active(length(X)) -> length(active(X))
       active(U81(X)) -> U81(active(X))
       active(U91(X1,X2,X3,X4)) -> U91(active(X1),X2,X3,X4)
       active(U92(X1,X2,X3,X4)) -> U92(active(X1),X2,X3,X4)
       active(U93(X1,X2,X3,X4)) -> U93(active(X1),X2,X3,X4)
       active(take(X1,X2)) -> take(active(X1),X2)
       active(take(X1,X2)) -> take(X1,active(X2))
       cons(mark(X1),X2) -> mark(cons(X1,X2))
       U11(mark(X)) -> mark(U11(X))
       U21(mark(X)) -> mark(U21(X))
       U31(mark(X)) -> mark(U31(X))
       U41(mark(X1),X2) -> mark(U41(X1,X2))
       U42(mark(X)) -> mark(U42(X))
       U51(mark(X1),X2) -> mark(U51(X1,X2))
       U52(mark(X)) -> mark(U52(X))
       U61(mark(X1),X2) -> mark(U61(X1,X2))
       U62(mark(X)) -> mark(U62(X))
       U71(mark(X1),X2,X3) -> mark(U71(X1,X2,X3))
       U72(mark(X1),X2) -> mark(U72(X1,X2))
       s(mark(X)) -> mark(s(X))
       length(mark(X)) -> mark(length(X))
       U81(mark(X)) -> mark(U81(X))
       U91(mark(X1),X2,X3,X4) -> mark(U91(X1,X2,X3,X4))
       U92(mark(X1),X2,X3,X4) -> mark(U92(X1,X2,X3,X4))
       U93(mark(X1),X2,X3,X4) -> mark(U93(X1,X2,X3,X4))
       take(mark(X1),X2) -> mark(take(X1,X2))
       take(X1,mark(X2)) -> mark(take(X1,X2))
       proper(zeros()) -> ok(zeros())
       proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
       proper(0()) -> ok(0())
       proper(U11(X)) -> U11(proper(X))
       proper(tt()) -> ok(tt())
       proper(U21(X)) -> U21(proper(X))
       proper(U31(X)) -> U31(proper(X))
       proper(U41(X1,X2)) -> U41(proper(X1),proper(X2))
       proper(U42(X)) -> U42(proper(X))
       proper(isNatIList(X)) -> isNatIList(proper(X))
       proper(U51(X1,X2)) -> U51(proper(X1),proper(X2))
       proper(U52(X)) -> U52(proper(X))
       proper(isNatList(X)) -> isNatList(proper(X))
       proper(U61(X1,X2)) -> U61(proper(X1),proper(X2))
       proper(U62(X)) -> U62(proper(X))
       proper(U71(X1,X2,X3)) -> U71(proper(X1),proper(X2),proper(X3))
       proper(U72(X1,X2)) -> U72(proper(X1),proper(X2))
       proper(isNat(X)) -> isNat(proper(X))
       proper(s(X)) -> s(proper(X))
       proper(length(X)) -> length(proper(X))
       proper(U81(X)) -> U81(proper(X))
       proper(nil()) -> ok(nil())
       proper(U91(X1,X2,X3,X4)) -> U91(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(U92(X1,X2,X3,X4)) -> U92(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(U93(X1,X2,X3,X4)) -> U93(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(take(X1,X2)) -> take(proper(X1),proper(X2))
       cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
       U11(ok(X)) -> ok(U11(X))
       U21(ok(X)) -> ok(U21(X))
       U31(ok(X)) -> ok(U31(X))
       U41(ok(X1),ok(X2)) -> ok(U41(X1,X2))
       U42(ok(X)) -> ok(U42(X))
       isNatIList(ok(X)) -> ok(isNatIList(X))
       U51(ok(X1),ok(X2)) -> ok(U51(X1,X2))
       U52(ok(X)) -> ok(U52(X))
       isNatList(ok(X)) -> ok(isNatList(X))
       U61(ok(X1),ok(X2)) -> ok(U61(X1,X2))
       U62(ok(X)) -> ok(U62(X))
       U71(ok(X1),ok(X2),ok(X3)) -> ok(U71(X1,X2,X3))
       U72(ok(X1),ok(X2)) -> ok(U72(X1,X2))
       isNat(ok(X)) -> ok(isNat(X))
       s(ok(X)) -> ok(s(X))
       length(ok(X)) -> ok(length(X))
       U81(ok(X)) -> ok(U81(X))
       U91(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U91(X1,X2,X3,X4))
       U92(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U92(X1,X2,X3,X4))
       U93(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U93(X1,X2,X3,X4))
       take(ok(X1),ok(X2)) -> ok(take(X1,X2))
       top(mark(X)) -> top(proper(X))
       top(ok(X)) -> top(active(X))
      Open
      
      DPs:
       U31#(ok(X)) -> U31#(X)
       U31#(mark(X)) -> U31#(X)
      TRS:
       active(zeros()) -> mark(cons(0(),zeros()))
       active(U11(tt())) -> mark(tt())
       active(U21(tt())) -> mark(tt())
       active(U31(tt())) -> mark(tt())
       active(U41(tt(),V2)) -> mark(U42(isNatIList(V2)))
       active(U42(tt())) -> mark(tt())
       active(U51(tt(),V2)) -> mark(U52(isNatList(V2)))
       active(U52(tt())) -> mark(tt())
       active(U61(tt(),V2)) -> mark(U62(isNatIList(V2)))
       active(U62(tt())) -> mark(tt())
       active(U71(tt(),L,N)) -> mark(U72(isNat(N),L))
       active(U72(tt(),L)) -> mark(s(length(L)))
       active(U81(tt())) -> mark(nil())
       active(U91(tt(),IL,M,N)) -> mark(U92(isNat(M),IL,M,N))
       active(U92(tt(),IL,M,N)) -> mark(U93(isNat(N),IL,M,N))
       active(U93(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
       active(isNat(0())) -> mark(tt())
       active(isNat(length(V1))) -> mark(U11(isNatList(V1)))
       active(isNat(s(V1))) -> mark(U21(isNat(V1)))
       active(isNatIList(V)) -> mark(U31(isNatList(V)))
       active(isNatIList(zeros())) -> mark(tt())
       active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2))
       active(isNatList(nil())) -> mark(tt())
       active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2))
       active(isNatList(take(V1,V2))) -> mark(U61(isNat(V1),V2))
       active(length(nil())) -> mark(0())
       active(length(cons(N,L))) -> mark(U71(isNatList(L),L,N))
       active(take(0(),IL)) -> mark(U81(isNatIList(IL)))
       active(take(s(M),cons(N,IL))) -> mark(U91(isNatIList(IL),IL,M,N))
       active(cons(X1,X2)) -> cons(active(X1),X2)
       active(U11(X)) -> U11(active(X))
       active(U21(X)) -> U21(active(X))
       active(U31(X)) -> U31(active(X))
       active(U41(X1,X2)) -> U41(active(X1),X2)
       active(U42(X)) -> U42(active(X))
       active(U51(X1,X2)) -> U51(active(X1),X2)
       active(U52(X)) -> U52(active(X))
       active(U61(X1,X2)) -> U61(active(X1),X2)
       active(U62(X)) -> U62(active(X))
       active(U71(X1,X2,X3)) -> U71(active(X1),X2,X3)
       active(U72(X1,X2)) -> U72(active(X1),X2)
       active(s(X)) -> s(active(X))
       active(length(X)) -> length(active(X))
       active(U81(X)) -> U81(active(X))
       active(U91(X1,X2,X3,X4)) -> U91(active(X1),X2,X3,X4)
       active(U92(X1,X2,X3,X4)) -> U92(active(X1),X2,X3,X4)
       active(U93(X1,X2,X3,X4)) -> U93(active(X1),X2,X3,X4)
       active(take(X1,X2)) -> take(active(X1),X2)
       active(take(X1,X2)) -> take(X1,active(X2))
       cons(mark(X1),X2) -> mark(cons(X1,X2))
       U11(mark(X)) -> mark(U11(X))
       U21(mark(X)) -> mark(U21(X))
       U31(mark(X)) -> mark(U31(X))
       U41(mark(X1),X2) -> mark(U41(X1,X2))
       U42(mark(X)) -> mark(U42(X))
       U51(mark(X1),X2) -> mark(U51(X1,X2))
       U52(mark(X)) -> mark(U52(X))
       U61(mark(X1),X2) -> mark(U61(X1,X2))
       U62(mark(X)) -> mark(U62(X))
       U71(mark(X1),X2,X3) -> mark(U71(X1,X2,X3))
       U72(mark(X1),X2) -> mark(U72(X1,X2))
       s(mark(X)) -> mark(s(X))
       length(mark(X)) -> mark(length(X))
       U81(mark(X)) -> mark(U81(X))
       U91(mark(X1),X2,X3,X4) -> mark(U91(X1,X2,X3,X4))
       U92(mark(X1),X2,X3,X4) -> mark(U92(X1,X2,X3,X4))
       U93(mark(X1),X2,X3,X4) -> mark(U93(X1,X2,X3,X4))
       take(mark(X1),X2) -> mark(take(X1,X2))
       take(X1,mark(X2)) -> mark(take(X1,X2))
       proper(zeros()) -> ok(zeros())
       proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
       proper(0()) -> ok(0())
       proper(U11(X)) -> U11(proper(X))
       proper(tt()) -> ok(tt())
       proper(U21(X)) -> U21(proper(X))
       proper(U31(X)) -> U31(proper(X))
       proper(U41(X1,X2)) -> U41(proper(X1),proper(X2))
       proper(U42(X)) -> U42(proper(X))
       proper(isNatIList(X)) -> isNatIList(proper(X))
       proper(U51(X1,X2)) -> U51(proper(X1),proper(X2))
       proper(U52(X)) -> U52(proper(X))
       proper(isNatList(X)) -> isNatList(proper(X))
       proper(U61(X1,X2)) -> U61(proper(X1),proper(X2))
       proper(U62(X)) -> U62(proper(X))
       proper(U71(X1,X2,X3)) -> U71(proper(X1),proper(X2),proper(X3))
       proper(U72(X1,X2)) -> U72(proper(X1),proper(X2))
       proper(isNat(X)) -> isNat(proper(X))
       proper(s(X)) -> s(proper(X))
       proper(length(X)) -> length(proper(X))
       proper(U81(X)) -> U81(proper(X))
       proper(nil()) -> ok(nil())
       proper(U91(X1,X2,X3,X4)) -> U91(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(U92(X1,X2,X3,X4)) -> U92(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(U93(X1,X2,X3,X4)) -> U93(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(take(X1,X2)) -> take(proper(X1),proper(X2))
       cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
       U11(ok(X)) -> ok(U11(X))
       U21(ok(X)) -> ok(U21(X))
       U31(ok(X)) -> ok(U31(X))
       U41(ok(X1),ok(X2)) -> ok(U41(X1,X2))
       U42(ok(X)) -> ok(U42(X))
       isNatIList(ok(X)) -> ok(isNatIList(X))
       U51(ok(X1),ok(X2)) -> ok(U51(X1,X2))
       U52(ok(X)) -> ok(U52(X))
       isNatList(ok(X)) -> ok(isNatList(X))
       U61(ok(X1),ok(X2)) -> ok(U61(X1,X2))
       U62(ok(X)) -> ok(U62(X))
       U71(ok(X1),ok(X2),ok(X3)) -> ok(U71(X1,X2,X3))
       U72(ok(X1),ok(X2)) -> ok(U72(X1,X2))
       isNat(ok(X)) -> ok(isNat(X))
       s(ok(X)) -> ok(s(X))
       length(ok(X)) -> ok(length(X))
       U81(ok(X)) -> ok(U81(X))
       U91(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U91(X1,X2,X3,X4))
       U92(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U92(X1,X2,X3,X4))
       U93(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U93(X1,X2,X3,X4))
       take(ok(X1),ok(X2)) -> ok(take(X1,X2))
       top(mark(X)) -> top(proper(X))
       top(ok(X)) -> top(active(X))
      Open
      
      DPs:
       U41#(ok(X1),ok(X2)) -> U41#(X1,X2)
       U41#(mark(X1),X2) -> U41#(X1,X2)
      TRS:
       active(zeros()) -> mark(cons(0(),zeros()))
       active(U11(tt())) -> mark(tt())
       active(U21(tt())) -> mark(tt())
       active(U31(tt())) -> mark(tt())
       active(U41(tt(),V2)) -> mark(U42(isNatIList(V2)))
       active(U42(tt())) -> mark(tt())
       active(U51(tt(),V2)) -> mark(U52(isNatList(V2)))
       active(U52(tt())) -> mark(tt())
       active(U61(tt(),V2)) -> mark(U62(isNatIList(V2)))
       active(U62(tt())) -> mark(tt())
       active(U71(tt(),L,N)) -> mark(U72(isNat(N),L))
       active(U72(tt(),L)) -> mark(s(length(L)))
       active(U81(tt())) -> mark(nil())
       active(U91(tt(),IL,M,N)) -> mark(U92(isNat(M),IL,M,N))
       active(U92(tt(),IL,M,N)) -> mark(U93(isNat(N),IL,M,N))
       active(U93(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
       active(isNat(0())) -> mark(tt())
       active(isNat(length(V1))) -> mark(U11(isNatList(V1)))
       active(isNat(s(V1))) -> mark(U21(isNat(V1)))
       active(isNatIList(V)) -> mark(U31(isNatList(V)))
       active(isNatIList(zeros())) -> mark(tt())
       active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2))
       active(isNatList(nil())) -> mark(tt())
       active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2))
       active(isNatList(take(V1,V2))) -> mark(U61(isNat(V1),V2))
       active(length(nil())) -> mark(0())
       active(length(cons(N,L))) -> mark(U71(isNatList(L),L,N))
       active(take(0(),IL)) -> mark(U81(isNatIList(IL)))
       active(take(s(M),cons(N,IL))) -> mark(U91(isNatIList(IL),IL,M,N))
       active(cons(X1,X2)) -> cons(active(X1),X2)
       active(U11(X)) -> U11(active(X))
       active(U21(X)) -> U21(active(X))
       active(U31(X)) -> U31(active(X))
       active(U41(X1,X2)) -> U41(active(X1),X2)
       active(U42(X)) -> U42(active(X))
       active(U51(X1,X2)) -> U51(active(X1),X2)
       active(U52(X)) -> U52(active(X))
       active(U61(X1,X2)) -> U61(active(X1),X2)
       active(U62(X)) -> U62(active(X))
       active(U71(X1,X2,X3)) -> U71(active(X1),X2,X3)
       active(U72(X1,X2)) -> U72(active(X1),X2)
       active(s(X)) -> s(active(X))
       active(length(X)) -> length(active(X))
       active(U81(X)) -> U81(active(X))
       active(U91(X1,X2,X3,X4)) -> U91(active(X1),X2,X3,X4)
       active(U92(X1,X2,X3,X4)) -> U92(active(X1),X2,X3,X4)
       active(U93(X1,X2,X3,X4)) -> U93(active(X1),X2,X3,X4)
       active(take(X1,X2)) -> take(active(X1),X2)
       active(take(X1,X2)) -> take(X1,active(X2))
       cons(mark(X1),X2) -> mark(cons(X1,X2))
       U11(mark(X)) -> mark(U11(X))
       U21(mark(X)) -> mark(U21(X))
       U31(mark(X)) -> mark(U31(X))
       U41(mark(X1),X2) -> mark(U41(X1,X2))
       U42(mark(X)) -> mark(U42(X))
       U51(mark(X1),X2) -> mark(U51(X1,X2))
       U52(mark(X)) -> mark(U52(X))
       U61(mark(X1),X2) -> mark(U61(X1,X2))
       U62(mark(X)) -> mark(U62(X))
       U71(mark(X1),X2,X3) -> mark(U71(X1,X2,X3))
       U72(mark(X1),X2) -> mark(U72(X1,X2))
       s(mark(X)) -> mark(s(X))
       length(mark(X)) -> mark(length(X))
       U81(mark(X)) -> mark(U81(X))
       U91(mark(X1),X2,X3,X4) -> mark(U91(X1,X2,X3,X4))
       U92(mark(X1),X2,X3,X4) -> mark(U92(X1,X2,X3,X4))
       U93(mark(X1),X2,X3,X4) -> mark(U93(X1,X2,X3,X4))
       take(mark(X1),X2) -> mark(take(X1,X2))
       take(X1,mark(X2)) -> mark(take(X1,X2))
       proper(zeros()) -> ok(zeros())
       proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
       proper(0()) -> ok(0())
       proper(U11(X)) -> U11(proper(X))
       proper(tt()) -> ok(tt())
       proper(U21(X)) -> U21(proper(X))
       proper(U31(X)) -> U31(proper(X))
       proper(U41(X1,X2)) -> U41(proper(X1),proper(X2))
       proper(U42(X)) -> U42(proper(X))
       proper(isNatIList(X)) -> isNatIList(proper(X))
       proper(U51(X1,X2)) -> U51(proper(X1),proper(X2))
       proper(U52(X)) -> U52(proper(X))
       proper(isNatList(X)) -> isNatList(proper(X))
       proper(U61(X1,X2)) -> U61(proper(X1),proper(X2))
       proper(U62(X)) -> U62(proper(X))
       proper(U71(X1,X2,X3)) -> U71(proper(X1),proper(X2),proper(X3))
       proper(U72(X1,X2)) -> U72(proper(X1),proper(X2))
       proper(isNat(X)) -> isNat(proper(X))
       proper(s(X)) -> s(proper(X))
       proper(length(X)) -> length(proper(X))
       proper(U81(X)) -> U81(proper(X))
       proper(nil()) -> ok(nil())
       proper(U91(X1,X2,X3,X4)) -> U91(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(U92(X1,X2,X3,X4)) -> U92(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(U93(X1,X2,X3,X4)) -> U93(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(take(X1,X2)) -> take(proper(X1),proper(X2))
       cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
       U11(ok(X)) -> ok(U11(X))
       U21(ok(X)) -> ok(U21(X))
       U31(ok(X)) -> ok(U31(X))
       U41(ok(X1),ok(X2)) -> ok(U41(X1,X2))
       U42(ok(X)) -> ok(U42(X))
       isNatIList(ok(X)) -> ok(isNatIList(X))
       U51(ok(X1),ok(X2)) -> ok(U51(X1,X2))
       U52(ok(X)) -> ok(U52(X))
       isNatList(ok(X)) -> ok(isNatList(X))
       U61(ok(X1),ok(X2)) -> ok(U61(X1,X2))
       U62(ok(X)) -> ok(U62(X))
       U71(ok(X1),ok(X2),ok(X3)) -> ok(U71(X1,X2,X3))
       U72(ok(X1),ok(X2)) -> ok(U72(X1,X2))
       isNat(ok(X)) -> ok(isNat(X))
       s(ok(X)) -> ok(s(X))
       length(ok(X)) -> ok(length(X))
       U81(ok(X)) -> ok(U81(X))
       U91(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U91(X1,X2,X3,X4))
       U92(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U92(X1,X2,X3,X4))
       U93(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U93(X1,X2,X3,X4))
       take(ok(X1),ok(X2)) -> ok(take(X1,X2))
       top(mark(X)) -> top(proper(X))
       top(ok(X)) -> top(active(X))
      Open
      
      DPs:
       U42#(ok(X)) -> U42#(X)
       U42#(mark(X)) -> U42#(X)
      TRS:
       active(zeros()) -> mark(cons(0(),zeros()))
       active(U11(tt())) -> mark(tt())
       active(U21(tt())) -> mark(tt())
       active(U31(tt())) -> mark(tt())
       active(U41(tt(),V2)) -> mark(U42(isNatIList(V2)))
       active(U42(tt())) -> mark(tt())
       active(U51(tt(),V2)) -> mark(U52(isNatList(V2)))
       active(U52(tt())) -> mark(tt())
       active(U61(tt(),V2)) -> mark(U62(isNatIList(V2)))
       active(U62(tt())) -> mark(tt())
       active(U71(tt(),L,N)) -> mark(U72(isNat(N),L))
       active(U72(tt(),L)) -> mark(s(length(L)))
       active(U81(tt())) -> mark(nil())
       active(U91(tt(),IL,M,N)) -> mark(U92(isNat(M),IL,M,N))
       active(U92(tt(),IL,M,N)) -> mark(U93(isNat(N),IL,M,N))
       active(U93(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
       active(isNat(0())) -> mark(tt())
       active(isNat(length(V1))) -> mark(U11(isNatList(V1)))
       active(isNat(s(V1))) -> mark(U21(isNat(V1)))
       active(isNatIList(V)) -> mark(U31(isNatList(V)))
       active(isNatIList(zeros())) -> mark(tt())
       active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2))
       active(isNatList(nil())) -> mark(tt())
       active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2))
       active(isNatList(take(V1,V2))) -> mark(U61(isNat(V1),V2))
       active(length(nil())) -> mark(0())
       active(length(cons(N,L))) -> mark(U71(isNatList(L),L,N))
       active(take(0(),IL)) -> mark(U81(isNatIList(IL)))
       active(take(s(M),cons(N,IL))) -> mark(U91(isNatIList(IL),IL,M,N))
       active(cons(X1,X2)) -> cons(active(X1),X2)
       active(U11(X)) -> U11(active(X))
       active(U21(X)) -> U21(active(X))
       active(U31(X)) -> U31(active(X))
       active(U41(X1,X2)) -> U41(active(X1),X2)
       active(U42(X)) -> U42(active(X))
       active(U51(X1,X2)) -> U51(active(X1),X2)
       active(U52(X)) -> U52(active(X))
       active(U61(X1,X2)) -> U61(active(X1),X2)
       active(U62(X)) -> U62(active(X))
       active(U71(X1,X2,X3)) -> U71(active(X1),X2,X3)
       active(U72(X1,X2)) -> U72(active(X1),X2)
       active(s(X)) -> s(active(X))
       active(length(X)) -> length(active(X))
       active(U81(X)) -> U81(active(X))
       active(U91(X1,X2,X3,X4)) -> U91(active(X1),X2,X3,X4)
       active(U92(X1,X2,X3,X4)) -> U92(active(X1),X2,X3,X4)
       active(U93(X1,X2,X3,X4)) -> U93(active(X1),X2,X3,X4)
       active(take(X1,X2)) -> take(active(X1),X2)
       active(take(X1,X2)) -> take(X1,active(X2))
       cons(mark(X1),X2) -> mark(cons(X1,X2))
       U11(mark(X)) -> mark(U11(X))
       U21(mark(X)) -> mark(U21(X))
       U31(mark(X)) -> mark(U31(X))
       U41(mark(X1),X2) -> mark(U41(X1,X2))
       U42(mark(X)) -> mark(U42(X))
       U51(mark(X1),X2) -> mark(U51(X1,X2))
       U52(mark(X)) -> mark(U52(X))
       U61(mark(X1),X2) -> mark(U61(X1,X2))
       U62(mark(X)) -> mark(U62(X))
       U71(mark(X1),X2,X3) -> mark(U71(X1,X2,X3))
       U72(mark(X1),X2) -> mark(U72(X1,X2))
       s(mark(X)) -> mark(s(X))
       length(mark(X)) -> mark(length(X))
       U81(mark(X)) -> mark(U81(X))
       U91(mark(X1),X2,X3,X4) -> mark(U91(X1,X2,X3,X4))
       U92(mark(X1),X2,X3,X4) -> mark(U92(X1,X2,X3,X4))
       U93(mark(X1),X2,X3,X4) -> mark(U93(X1,X2,X3,X4))
       take(mark(X1),X2) -> mark(take(X1,X2))
       take(X1,mark(X2)) -> mark(take(X1,X2))
       proper(zeros()) -> ok(zeros())
       proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
       proper(0()) -> ok(0())
       proper(U11(X)) -> U11(proper(X))
       proper(tt()) -> ok(tt())
       proper(U21(X)) -> U21(proper(X))
       proper(U31(X)) -> U31(proper(X))
       proper(U41(X1,X2)) -> U41(proper(X1),proper(X2))
       proper(U42(X)) -> U42(proper(X))
       proper(isNatIList(X)) -> isNatIList(proper(X))
       proper(U51(X1,X2)) -> U51(proper(X1),proper(X2))
       proper(U52(X)) -> U52(proper(X))
       proper(isNatList(X)) -> isNatList(proper(X))
       proper(U61(X1,X2)) -> U61(proper(X1),proper(X2))
       proper(U62(X)) -> U62(proper(X))
       proper(U71(X1,X2,X3)) -> U71(proper(X1),proper(X2),proper(X3))
       proper(U72(X1,X2)) -> U72(proper(X1),proper(X2))
       proper(isNat(X)) -> isNat(proper(X))
       proper(s(X)) -> s(proper(X))
       proper(length(X)) -> length(proper(X))
       proper(U81(X)) -> U81(proper(X))
       proper(nil()) -> ok(nil())
       proper(U91(X1,X2,X3,X4)) -> U91(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(U92(X1,X2,X3,X4)) -> U92(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(U93(X1,X2,X3,X4)) -> U93(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(take(X1,X2)) -> take(proper(X1),proper(X2))
       cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
       U11(ok(X)) -> ok(U11(X))
       U21(ok(X)) -> ok(U21(X))
       U31(ok(X)) -> ok(U31(X))
       U41(ok(X1),ok(X2)) -> ok(U41(X1,X2))
       U42(ok(X)) -> ok(U42(X))
       isNatIList(ok(X)) -> ok(isNatIList(X))
       U51(ok(X1),ok(X2)) -> ok(U51(X1,X2))
       U52(ok(X)) -> ok(U52(X))
       isNatList(ok(X)) -> ok(isNatList(X))
       U61(ok(X1),ok(X2)) -> ok(U61(X1,X2))
       U62(ok(X)) -> ok(U62(X))
       U71(ok(X1),ok(X2),ok(X3)) -> ok(U71(X1,X2,X3))
       U72(ok(X1),ok(X2)) -> ok(U72(X1,X2))
       isNat(ok(X)) -> ok(isNat(X))
       s(ok(X)) -> ok(s(X))
       length(ok(X)) -> ok(length(X))
       U81(ok(X)) -> ok(U81(X))
       U91(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U91(X1,X2,X3,X4))
       U92(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U92(X1,X2,X3,X4))
       U93(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U93(X1,X2,X3,X4))
       take(ok(X1),ok(X2)) -> ok(take(X1,X2))
       top(mark(X)) -> top(proper(X))
       top(ok(X)) -> top(active(X))
      Open
      
      DPs:
       U51#(ok(X1),ok(X2)) -> U51#(X1,X2)
       U51#(mark(X1),X2) -> U51#(X1,X2)
      TRS:
       active(zeros()) -> mark(cons(0(),zeros()))
       active(U11(tt())) -> mark(tt())
       active(U21(tt())) -> mark(tt())
       active(U31(tt())) -> mark(tt())
       active(U41(tt(),V2)) -> mark(U42(isNatIList(V2)))
       active(U42(tt())) -> mark(tt())
       active(U51(tt(),V2)) -> mark(U52(isNatList(V2)))
       active(U52(tt())) -> mark(tt())
       active(U61(tt(),V2)) -> mark(U62(isNatIList(V2)))
       active(U62(tt())) -> mark(tt())
       active(U71(tt(),L,N)) -> mark(U72(isNat(N),L))
       active(U72(tt(),L)) -> mark(s(length(L)))
       active(U81(tt())) -> mark(nil())
       active(U91(tt(),IL,M,N)) -> mark(U92(isNat(M),IL,M,N))
       active(U92(tt(),IL,M,N)) -> mark(U93(isNat(N),IL,M,N))
       active(U93(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
       active(isNat(0())) -> mark(tt())
       active(isNat(length(V1))) -> mark(U11(isNatList(V1)))
       active(isNat(s(V1))) -> mark(U21(isNat(V1)))
       active(isNatIList(V)) -> mark(U31(isNatList(V)))
       active(isNatIList(zeros())) -> mark(tt())
       active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2))
       active(isNatList(nil())) -> mark(tt())
       active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2))
       active(isNatList(take(V1,V2))) -> mark(U61(isNat(V1),V2))
       active(length(nil())) -> mark(0())
       active(length(cons(N,L))) -> mark(U71(isNatList(L),L,N))
       active(take(0(),IL)) -> mark(U81(isNatIList(IL)))
       active(take(s(M),cons(N,IL))) -> mark(U91(isNatIList(IL),IL,M,N))
       active(cons(X1,X2)) -> cons(active(X1),X2)
       active(U11(X)) -> U11(active(X))
       active(U21(X)) -> U21(active(X))
       active(U31(X)) -> U31(active(X))
       active(U41(X1,X2)) -> U41(active(X1),X2)
       active(U42(X)) -> U42(active(X))
       active(U51(X1,X2)) -> U51(active(X1),X2)
       active(U52(X)) -> U52(active(X))
       active(U61(X1,X2)) -> U61(active(X1),X2)
       active(U62(X)) -> U62(active(X))
       active(U71(X1,X2,X3)) -> U71(active(X1),X2,X3)
       active(U72(X1,X2)) -> U72(active(X1),X2)
       active(s(X)) -> s(active(X))
       active(length(X)) -> length(active(X))
       active(U81(X)) -> U81(active(X))
       active(U91(X1,X2,X3,X4)) -> U91(active(X1),X2,X3,X4)
       active(U92(X1,X2,X3,X4)) -> U92(active(X1),X2,X3,X4)
       active(U93(X1,X2,X3,X4)) -> U93(active(X1),X2,X3,X4)
       active(take(X1,X2)) -> take(active(X1),X2)
       active(take(X1,X2)) -> take(X1,active(X2))
       cons(mark(X1),X2) -> mark(cons(X1,X2))
       U11(mark(X)) -> mark(U11(X))
       U21(mark(X)) -> mark(U21(X))
       U31(mark(X)) -> mark(U31(X))
       U41(mark(X1),X2) -> mark(U41(X1,X2))
       U42(mark(X)) -> mark(U42(X))
       U51(mark(X1),X2) -> mark(U51(X1,X2))
       U52(mark(X)) -> mark(U52(X))
       U61(mark(X1),X2) -> mark(U61(X1,X2))
       U62(mark(X)) -> mark(U62(X))
       U71(mark(X1),X2,X3) -> mark(U71(X1,X2,X3))
       U72(mark(X1),X2) -> mark(U72(X1,X2))
       s(mark(X)) -> mark(s(X))
       length(mark(X)) -> mark(length(X))
       U81(mark(X)) -> mark(U81(X))
       U91(mark(X1),X2,X3,X4) -> mark(U91(X1,X2,X3,X4))
       U92(mark(X1),X2,X3,X4) -> mark(U92(X1,X2,X3,X4))
       U93(mark(X1),X2,X3,X4) -> mark(U93(X1,X2,X3,X4))
       take(mark(X1),X2) -> mark(take(X1,X2))
       take(X1,mark(X2)) -> mark(take(X1,X2))
       proper(zeros()) -> ok(zeros())
       proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
       proper(0()) -> ok(0())
       proper(U11(X)) -> U11(proper(X))
       proper(tt()) -> ok(tt())
       proper(U21(X)) -> U21(proper(X))
       proper(U31(X)) -> U31(proper(X))
       proper(U41(X1,X2)) -> U41(proper(X1),proper(X2))
       proper(U42(X)) -> U42(proper(X))
       proper(isNatIList(X)) -> isNatIList(proper(X))
       proper(U51(X1,X2)) -> U51(proper(X1),proper(X2))
       proper(U52(X)) -> U52(proper(X))
       proper(isNatList(X)) -> isNatList(proper(X))
       proper(U61(X1,X2)) -> U61(proper(X1),proper(X2))
       proper(U62(X)) -> U62(proper(X))
       proper(U71(X1,X2,X3)) -> U71(proper(X1),proper(X2),proper(X3))
       proper(U72(X1,X2)) -> U72(proper(X1),proper(X2))
       proper(isNat(X)) -> isNat(proper(X))
       proper(s(X)) -> s(proper(X))
       proper(length(X)) -> length(proper(X))
       proper(U81(X)) -> U81(proper(X))
       proper(nil()) -> ok(nil())
       proper(U91(X1,X2,X3,X4)) -> U91(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(U92(X1,X2,X3,X4)) -> U92(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(U93(X1,X2,X3,X4)) -> U93(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(take(X1,X2)) -> take(proper(X1),proper(X2))
       cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
       U11(ok(X)) -> ok(U11(X))
       U21(ok(X)) -> ok(U21(X))
       U31(ok(X)) -> ok(U31(X))
       U41(ok(X1),ok(X2)) -> ok(U41(X1,X2))
       U42(ok(X)) -> ok(U42(X))
       isNatIList(ok(X)) -> ok(isNatIList(X))
       U51(ok(X1),ok(X2)) -> ok(U51(X1,X2))
       U52(ok(X)) -> ok(U52(X))
       isNatList(ok(X)) -> ok(isNatList(X))
       U61(ok(X1),ok(X2)) -> ok(U61(X1,X2))
       U62(ok(X)) -> ok(U62(X))
       U71(ok(X1),ok(X2),ok(X3)) -> ok(U71(X1,X2,X3))
       U72(ok(X1),ok(X2)) -> ok(U72(X1,X2))
       isNat(ok(X)) -> ok(isNat(X))
       s(ok(X)) -> ok(s(X))
       length(ok(X)) -> ok(length(X))
       U81(ok(X)) -> ok(U81(X))
       U91(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U91(X1,X2,X3,X4))
       U92(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U92(X1,X2,X3,X4))
       U93(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U93(X1,X2,X3,X4))
       take(ok(X1),ok(X2)) -> ok(take(X1,X2))
       top(mark(X)) -> top(proper(X))
       top(ok(X)) -> top(active(X))
      Open
      
      DPs:
       U52#(ok(X)) -> U52#(X)
       U52#(mark(X)) -> U52#(X)
      TRS:
       active(zeros()) -> mark(cons(0(),zeros()))
       active(U11(tt())) -> mark(tt())
       active(U21(tt())) -> mark(tt())
       active(U31(tt())) -> mark(tt())
       active(U41(tt(),V2)) -> mark(U42(isNatIList(V2)))
       active(U42(tt())) -> mark(tt())
       active(U51(tt(),V2)) -> mark(U52(isNatList(V2)))
       active(U52(tt())) -> mark(tt())
       active(U61(tt(),V2)) -> mark(U62(isNatIList(V2)))
       active(U62(tt())) -> mark(tt())
       active(U71(tt(),L,N)) -> mark(U72(isNat(N),L))
       active(U72(tt(),L)) -> mark(s(length(L)))
       active(U81(tt())) -> mark(nil())
       active(U91(tt(),IL,M,N)) -> mark(U92(isNat(M),IL,M,N))
       active(U92(tt(),IL,M,N)) -> mark(U93(isNat(N),IL,M,N))
       active(U93(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
       active(isNat(0())) -> mark(tt())
       active(isNat(length(V1))) -> mark(U11(isNatList(V1)))
       active(isNat(s(V1))) -> mark(U21(isNat(V1)))
       active(isNatIList(V)) -> mark(U31(isNatList(V)))
       active(isNatIList(zeros())) -> mark(tt())
       active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2))
       active(isNatList(nil())) -> mark(tt())
       active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2))
       active(isNatList(take(V1,V2))) -> mark(U61(isNat(V1),V2))
       active(length(nil())) -> mark(0())
       active(length(cons(N,L))) -> mark(U71(isNatList(L),L,N))
       active(take(0(),IL)) -> mark(U81(isNatIList(IL)))
       active(take(s(M),cons(N,IL))) -> mark(U91(isNatIList(IL),IL,M,N))
       active(cons(X1,X2)) -> cons(active(X1),X2)
       active(U11(X)) -> U11(active(X))
       active(U21(X)) -> U21(active(X))
       active(U31(X)) -> U31(active(X))
       active(U41(X1,X2)) -> U41(active(X1),X2)
       active(U42(X)) -> U42(active(X))
       active(U51(X1,X2)) -> U51(active(X1),X2)
       active(U52(X)) -> U52(active(X))
       active(U61(X1,X2)) -> U61(active(X1),X2)
       active(U62(X)) -> U62(active(X))
       active(U71(X1,X2,X3)) -> U71(active(X1),X2,X3)
       active(U72(X1,X2)) -> U72(active(X1),X2)
       active(s(X)) -> s(active(X))
       active(length(X)) -> length(active(X))
       active(U81(X)) -> U81(active(X))
       active(U91(X1,X2,X3,X4)) -> U91(active(X1),X2,X3,X4)
       active(U92(X1,X2,X3,X4)) -> U92(active(X1),X2,X3,X4)
       active(U93(X1,X2,X3,X4)) -> U93(active(X1),X2,X3,X4)
       active(take(X1,X2)) -> take(active(X1),X2)
       active(take(X1,X2)) -> take(X1,active(X2))
       cons(mark(X1),X2) -> mark(cons(X1,X2))
       U11(mark(X)) -> mark(U11(X))
       U21(mark(X)) -> mark(U21(X))
       U31(mark(X)) -> mark(U31(X))
       U41(mark(X1),X2) -> mark(U41(X1,X2))
       U42(mark(X)) -> mark(U42(X))
       U51(mark(X1),X2) -> mark(U51(X1,X2))
       U52(mark(X)) -> mark(U52(X))
       U61(mark(X1),X2) -> mark(U61(X1,X2))
       U62(mark(X)) -> mark(U62(X))
       U71(mark(X1),X2,X3) -> mark(U71(X1,X2,X3))
       U72(mark(X1),X2) -> mark(U72(X1,X2))
       s(mark(X)) -> mark(s(X))
       length(mark(X)) -> mark(length(X))
       U81(mark(X)) -> mark(U81(X))
       U91(mark(X1),X2,X3,X4) -> mark(U91(X1,X2,X3,X4))
       U92(mark(X1),X2,X3,X4) -> mark(U92(X1,X2,X3,X4))
       U93(mark(X1),X2,X3,X4) -> mark(U93(X1,X2,X3,X4))
       take(mark(X1),X2) -> mark(take(X1,X2))
       take(X1,mark(X2)) -> mark(take(X1,X2))
       proper(zeros()) -> ok(zeros())
       proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
       proper(0()) -> ok(0())
       proper(U11(X)) -> U11(proper(X))
       proper(tt()) -> ok(tt())
       proper(U21(X)) -> U21(proper(X))
       proper(U31(X)) -> U31(proper(X))
       proper(U41(X1,X2)) -> U41(proper(X1),proper(X2))
       proper(U42(X)) -> U42(proper(X))
       proper(isNatIList(X)) -> isNatIList(proper(X))
       proper(U51(X1,X2)) -> U51(proper(X1),proper(X2))
       proper(U52(X)) -> U52(proper(X))
       proper(isNatList(X)) -> isNatList(proper(X))
       proper(U61(X1,X2)) -> U61(proper(X1),proper(X2))
       proper(U62(X)) -> U62(proper(X))
       proper(U71(X1,X2,X3)) -> U71(proper(X1),proper(X2),proper(X3))
       proper(U72(X1,X2)) -> U72(proper(X1),proper(X2))
       proper(isNat(X)) -> isNat(proper(X))
       proper(s(X)) -> s(proper(X))
       proper(length(X)) -> length(proper(X))
       proper(U81(X)) -> U81(proper(X))
       proper(nil()) -> ok(nil())
       proper(U91(X1,X2,X3,X4)) -> U91(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(U92(X1,X2,X3,X4)) -> U92(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(U93(X1,X2,X3,X4)) -> U93(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(take(X1,X2)) -> take(proper(X1),proper(X2))
       cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
       U11(ok(X)) -> ok(U11(X))
       U21(ok(X)) -> ok(U21(X))
       U31(ok(X)) -> ok(U31(X))
       U41(ok(X1),ok(X2)) -> ok(U41(X1,X2))
       U42(ok(X)) -> ok(U42(X))
       isNatIList(ok(X)) -> ok(isNatIList(X))
       U51(ok(X1),ok(X2)) -> ok(U51(X1,X2))
       U52(ok(X)) -> ok(U52(X))
       isNatList(ok(X)) -> ok(isNatList(X))
       U61(ok(X1),ok(X2)) -> ok(U61(X1,X2))
       U62(ok(X)) -> ok(U62(X))
       U71(ok(X1),ok(X2),ok(X3)) -> ok(U71(X1,X2,X3))
       U72(ok(X1),ok(X2)) -> ok(U72(X1,X2))
       isNat(ok(X)) -> ok(isNat(X))
       s(ok(X)) -> ok(s(X))
       length(ok(X)) -> ok(length(X))
       U81(ok(X)) -> ok(U81(X))
       U91(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U91(X1,X2,X3,X4))
       U92(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U92(X1,X2,X3,X4))
       U93(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U93(X1,X2,X3,X4))
       take(ok(X1),ok(X2)) -> ok(take(X1,X2))
       top(mark(X)) -> top(proper(X))
       top(ok(X)) -> top(active(X))
      Open
      
      DPs:
       U61#(ok(X1),ok(X2)) -> U61#(X1,X2)
       U61#(mark(X1),X2) -> U61#(X1,X2)
      TRS:
       active(zeros()) -> mark(cons(0(),zeros()))
       active(U11(tt())) -> mark(tt())
       active(U21(tt())) -> mark(tt())
       active(U31(tt())) -> mark(tt())
       active(U41(tt(),V2)) -> mark(U42(isNatIList(V2)))
       active(U42(tt())) -> mark(tt())
       active(U51(tt(),V2)) -> mark(U52(isNatList(V2)))
       active(U52(tt())) -> mark(tt())
       active(U61(tt(),V2)) -> mark(U62(isNatIList(V2)))
       active(U62(tt())) -> mark(tt())
       active(U71(tt(),L,N)) -> mark(U72(isNat(N),L))
       active(U72(tt(),L)) -> mark(s(length(L)))
       active(U81(tt())) -> mark(nil())
       active(U91(tt(),IL,M,N)) -> mark(U92(isNat(M),IL,M,N))
       active(U92(tt(),IL,M,N)) -> mark(U93(isNat(N),IL,M,N))
       active(U93(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
       active(isNat(0())) -> mark(tt())
       active(isNat(length(V1))) -> mark(U11(isNatList(V1)))
       active(isNat(s(V1))) -> mark(U21(isNat(V1)))
       active(isNatIList(V)) -> mark(U31(isNatList(V)))
       active(isNatIList(zeros())) -> mark(tt())
       active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2))
       active(isNatList(nil())) -> mark(tt())
       active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2))
       active(isNatList(take(V1,V2))) -> mark(U61(isNat(V1),V2))
       active(length(nil())) -> mark(0())
       active(length(cons(N,L))) -> mark(U71(isNatList(L),L,N))
       active(take(0(),IL)) -> mark(U81(isNatIList(IL)))
       active(take(s(M),cons(N,IL))) -> mark(U91(isNatIList(IL),IL,M,N))
       active(cons(X1,X2)) -> cons(active(X1),X2)
       active(U11(X)) -> U11(active(X))
       active(U21(X)) -> U21(active(X))
       active(U31(X)) -> U31(active(X))
       active(U41(X1,X2)) -> U41(active(X1),X2)
       active(U42(X)) -> U42(active(X))
       active(U51(X1,X2)) -> U51(active(X1),X2)
       active(U52(X)) -> U52(active(X))
       active(U61(X1,X2)) -> U61(active(X1),X2)
       active(U62(X)) -> U62(active(X))
       active(U71(X1,X2,X3)) -> U71(active(X1),X2,X3)
       active(U72(X1,X2)) -> U72(active(X1),X2)
       active(s(X)) -> s(active(X))
       active(length(X)) -> length(active(X))
       active(U81(X)) -> U81(active(X))
       active(U91(X1,X2,X3,X4)) -> U91(active(X1),X2,X3,X4)
       active(U92(X1,X2,X3,X4)) -> U92(active(X1),X2,X3,X4)
       active(U93(X1,X2,X3,X4)) -> U93(active(X1),X2,X3,X4)
       active(take(X1,X2)) -> take(active(X1),X2)
       active(take(X1,X2)) -> take(X1,active(X2))
       cons(mark(X1),X2) -> mark(cons(X1,X2))
       U11(mark(X)) -> mark(U11(X))
       U21(mark(X)) -> mark(U21(X))
       U31(mark(X)) -> mark(U31(X))
       U41(mark(X1),X2) -> mark(U41(X1,X2))
       U42(mark(X)) -> mark(U42(X))
       U51(mark(X1),X2) -> mark(U51(X1,X2))
       U52(mark(X)) -> mark(U52(X))
       U61(mark(X1),X2) -> mark(U61(X1,X2))
       U62(mark(X)) -> mark(U62(X))
       U71(mark(X1),X2,X3) -> mark(U71(X1,X2,X3))
       U72(mark(X1),X2) -> mark(U72(X1,X2))
       s(mark(X)) -> mark(s(X))
       length(mark(X)) -> mark(length(X))
       U81(mark(X)) -> mark(U81(X))
       U91(mark(X1),X2,X3,X4) -> mark(U91(X1,X2,X3,X4))
       U92(mark(X1),X2,X3,X4) -> mark(U92(X1,X2,X3,X4))
       U93(mark(X1),X2,X3,X4) -> mark(U93(X1,X2,X3,X4))
       take(mark(X1),X2) -> mark(take(X1,X2))
       take(X1,mark(X2)) -> mark(take(X1,X2))
       proper(zeros()) -> ok(zeros())
       proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
       proper(0()) -> ok(0())
       proper(U11(X)) -> U11(proper(X))
       proper(tt()) -> ok(tt())
       proper(U21(X)) -> U21(proper(X))
       proper(U31(X)) -> U31(proper(X))
       proper(U41(X1,X2)) -> U41(proper(X1),proper(X2))
       proper(U42(X)) -> U42(proper(X))
       proper(isNatIList(X)) -> isNatIList(proper(X))
       proper(U51(X1,X2)) -> U51(proper(X1),proper(X2))
       proper(U52(X)) -> U52(proper(X))
       proper(isNatList(X)) -> isNatList(proper(X))
       proper(U61(X1,X2)) -> U61(proper(X1),proper(X2))
       proper(U62(X)) -> U62(proper(X))
       proper(U71(X1,X2,X3)) -> U71(proper(X1),proper(X2),proper(X3))
       proper(U72(X1,X2)) -> U72(proper(X1),proper(X2))
       proper(isNat(X)) -> isNat(proper(X))
       proper(s(X)) -> s(proper(X))
       proper(length(X)) -> length(proper(X))
       proper(U81(X)) -> U81(proper(X))
       proper(nil()) -> ok(nil())
       proper(U91(X1,X2,X3,X4)) -> U91(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(U92(X1,X2,X3,X4)) -> U92(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(U93(X1,X2,X3,X4)) -> U93(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(take(X1,X2)) -> take(proper(X1),proper(X2))
       cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
       U11(ok(X)) -> ok(U11(X))
       U21(ok(X)) -> ok(U21(X))
       U31(ok(X)) -> ok(U31(X))
       U41(ok(X1),ok(X2)) -> ok(U41(X1,X2))
       U42(ok(X)) -> ok(U42(X))
       isNatIList(ok(X)) -> ok(isNatIList(X))
       U51(ok(X1),ok(X2)) -> ok(U51(X1,X2))
       U52(ok(X)) -> ok(U52(X))
       isNatList(ok(X)) -> ok(isNatList(X))
       U61(ok(X1),ok(X2)) -> ok(U61(X1,X2))
       U62(ok(X)) -> ok(U62(X))
       U71(ok(X1),ok(X2),ok(X3)) -> ok(U71(X1,X2,X3))
       U72(ok(X1),ok(X2)) -> ok(U72(X1,X2))
       isNat(ok(X)) -> ok(isNat(X))
       s(ok(X)) -> ok(s(X))
       length(ok(X)) -> ok(length(X))
       U81(ok(X)) -> ok(U81(X))
       U91(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U91(X1,X2,X3,X4))
       U92(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U92(X1,X2,X3,X4))
       U93(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U93(X1,X2,X3,X4))
       take(ok(X1),ok(X2)) -> ok(take(X1,X2))
       top(mark(X)) -> top(proper(X))
       top(ok(X)) -> top(active(X))
      Open
      
      DPs:
       U62#(ok(X)) -> U62#(X)
       U62#(mark(X)) -> U62#(X)
      TRS:
       active(zeros()) -> mark(cons(0(),zeros()))
       active(U11(tt())) -> mark(tt())
       active(U21(tt())) -> mark(tt())
       active(U31(tt())) -> mark(tt())
       active(U41(tt(),V2)) -> mark(U42(isNatIList(V2)))
       active(U42(tt())) -> mark(tt())
       active(U51(tt(),V2)) -> mark(U52(isNatList(V2)))
       active(U52(tt())) -> mark(tt())
       active(U61(tt(),V2)) -> mark(U62(isNatIList(V2)))
       active(U62(tt())) -> mark(tt())
       active(U71(tt(),L,N)) -> mark(U72(isNat(N),L))
       active(U72(tt(),L)) -> mark(s(length(L)))
       active(U81(tt())) -> mark(nil())
       active(U91(tt(),IL,M,N)) -> mark(U92(isNat(M),IL,M,N))
       active(U92(tt(),IL,M,N)) -> mark(U93(isNat(N),IL,M,N))
       active(U93(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
       active(isNat(0())) -> mark(tt())
       active(isNat(length(V1))) -> mark(U11(isNatList(V1)))
       active(isNat(s(V1))) -> mark(U21(isNat(V1)))
       active(isNatIList(V)) -> mark(U31(isNatList(V)))
       active(isNatIList(zeros())) -> mark(tt())
       active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2))
       active(isNatList(nil())) -> mark(tt())
       active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2))
       active(isNatList(take(V1,V2))) -> mark(U61(isNat(V1),V2))
       active(length(nil())) -> mark(0())
       active(length(cons(N,L))) -> mark(U71(isNatList(L),L,N))
       active(take(0(),IL)) -> mark(U81(isNatIList(IL)))
       active(take(s(M),cons(N,IL))) -> mark(U91(isNatIList(IL),IL,M,N))
       active(cons(X1,X2)) -> cons(active(X1),X2)
       active(U11(X)) -> U11(active(X))
       active(U21(X)) -> U21(active(X))
       active(U31(X)) -> U31(active(X))
       active(U41(X1,X2)) -> U41(active(X1),X2)
       active(U42(X)) -> U42(active(X))
       active(U51(X1,X2)) -> U51(active(X1),X2)
       active(U52(X)) -> U52(active(X))
       active(U61(X1,X2)) -> U61(active(X1),X2)
       active(U62(X)) -> U62(active(X))
       active(U71(X1,X2,X3)) -> U71(active(X1),X2,X3)
       active(U72(X1,X2)) -> U72(active(X1),X2)
       active(s(X)) -> s(active(X))
       active(length(X)) -> length(active(X))
       active(U81(X)) -> U81(active(X))
       active(U91(X1,X2,X3,X4)) -> U91(active(X1),X2,X3,X4)
       active(U92(X1,X2,X3,X4)) -> U92(active(X1),X2,X3,X4)
       active(U93(X1,X2,X3,X4)) -> U93(active(X1),X2,X3,X4)
       active(take(X1,X2)) -> take(active(X1),X2)
       active(take(X1,X2)) -> take(X1,active(X2))
       cons(mark(X1),X2) -> mark(cons(X1,X2))
       U11(mark(X)) -> mark(U11(X))
       U21(mark(X)) -> mark(U21(X))
       U31(mark(X)) -> mark(U31(X))
       U41(mark(X1),X2) -> mark(U41(X1,X2))
       U42(mark(X)) -> mark(U42(X))
       U51(mark(X1),X2) -> mark(U51(X1,X2))
       U52(mark(X)) -> mark(U52(X))
       U61(mark(X1),X2) -> mark(U61(X1,X2))
       U62(mark(X)) -> mark(U62(X))
       U71(mark(X1),X2,X3) -> mark(U71(X1,X2,X3))
       U72(mark(X1),X2) -> mark(U72(X1,X2))
       s(mark(X)) -> mark(s(X))
       length(mark(X)) -> mark(length(X))
       U81(mark(X)) -> mark(U81(X))
       U91(mark(X1),X2,X3,X4) -> mark(U91(X1,X2,X3,X4))
       U92(mark(X1),X2,X3,X4) -> mark(U92(X1,X2,X3,X4))
       U93(mark(X1),X2,X3,X4) -> mark(U93(X1,X2,X3,X4))
       take(mark(X1),X2) -> mark(take(X1,X2))
       take(X1,mark(X2)) -> mark(take(X1,X2))
       proper(zeros()) -> ok(zeros())
       proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
       proper(0()) -> ok(0())
       proper(U11(X)) -> U11(proper(X))
       proper(tt()) -> ok(tt())
       proper(U21(X)) -> U21(proper(X))
       proper(U31(X)) -> U31(proper(X))
       proper(U41(X1,X2)) -> U41(proper(X1),proper(X2))
       proper(U42(X)) -> U42(proper(X))
       proper(isNatIList(X)) -> isNatIList(proper(X))
       proper(U51(X1,X2)) -> U51(proper(X1),proper(X2))
       proper(U52(X)) -> U52(proper(X))
       proper(isNatList(X)) -> isNatList(proper(X))
       proper(U61(X1,X2)) -> U61(proper(X1),proper(X2))
       proper(U62(X)) -> U62(proper(X))
       proper(U71(X1,X2,X3)) -> U71(proper(X1),proper(X2),proper(X3))
       proper(U72(X1,X2)) -> U72(proper(X1),proper(X2))
       proper(isNat(X)) -> isNat(proper(X))
       proper(s(X)) -> s(proper(X))
       proper(length(X)) -> length(proper(X))
       proper(U81(X)) -> U81(proper(X))
       proper(nil()) -> ok(nil())
       proper(U91(X1,X2,X3,X4)) -> U91(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(U92(X1,X2,X3,X4)) -> U92(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(U93(X1,X2,X3,X4)) -> U93(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(take(X1,X2)) -> take(proper(X1),proper(X2))
       cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
       U11(ok(X)) -> ok(U11(X))
       U21(ok(X)) -> ok(U21(X))
       U31(ok(X)) -> ok(U31(X))
       U41(ok(X1),ok(X2)) -> ok(U41(X1,X2))
       U42(ok(X)) -> ok(U42(X))
       isNatIList(ok(X)) -> ok(isNatIList(X))
       U51(ok(X1),ok(X2)) -> ok(U51(X1,X2))
       U52(ok(X)) -> ok(U52(X))
       isNatList(ok(X)) -> ok(isNatList(X))
       U61(ok(X1),ok(X2)) -> ok(U61(X1,X2))
       U62(ok(X)) -> ok(U62(X))
       U71(ok(X1),ok(X2),ok(X3)) -> ok(U71(X1,X2,X3))
       U72(ok(X1),ok(X2)) -> ok(U72(X1,X2))
       isNat(ok(X)) -> ok(isNat(X))
       s(ok(X)) -> ok(s(X))
       length(ok(X)) -> ok(length(X))
       U81(ok(X)) -> ok(U81(X))
       U91(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U91(X1,X2,X3,X4))
       U92(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U92(X1,X2,X3,X4))
       U93(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U93(X1,X2,X3,X4))
       take(ok(X1),ok(X2)) -> ok(take(X1,X2))
       top(mark(X)) -> top(proper(X))
       top(ok(X)) -> top(active(X))
      Open
      
      DPs:
       U71#(ok(X1),ok(X2),ok(X3)) -> U71#(X1,X2,X3)
       U71#(mark(X1),X2,X3) -> U71#(X1,X2,X3)
      TRS:
       active(zeros()) -> mark(cons(0(),zeros()))
       active(U11(tt())) -> mark(tt())
       active(U21(tt())) -> mark(tt())
       active(U31(tt())) -> mark(tt())
       active(U41(tt(),V2)) -> mark(U42(isNatIList(V2)))
       active(U42(tt())) -> mark(tt())
       active(U51(tt(),V2)) -> mark(U52(isNatList(V2)))
       active(U52(tt())) -> mark(tt())
       active(U61(tt(),V2)) -> mark(U62(isNatIList(V2)))
       active(U62(tt())) -> mark(tt())
       active(U71(tt(),L,N)) -> mark(U72(isNat(N),L))
       active(U72(tt(),L)) -> mark(s(length(L)))
       active(U81(tt())) -> mark(nil())
       active(U91(tt(),IL,M,N)) -> mark(U92(isNat(M),IL,M,N))
       active(U92(tt(),IL,M,N)) -> mark(U93(isNat(N),IL,M,N))
       active(U93(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
       active(isNat(0())) -> mark(tt())
       active(isNat(length(V1))) -> mark(U11(isNatList(V1)))
       active(isNat(s(V1))) -> mark(U21(isNat(V1)))
       active(isNatIList(V)) -> mark(U31(isNatList(V)))
       active(isNatIList(zeros())) -> mark(tt())
       active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2))
       active(isNatList(nil())) -> mark(tt())
       active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2))
       active(isNatList(take(V1,V2))) -> mark(U61(isNat(V1),V2))
       active(length(nil())) -> mark(0())
       active(length(cons(N,L))) -> mark(U71(isNatList(L),L,N))
       active(take(0(),IL)) -> mark(U81(isNatIList(IL)))
       active(take(s(M),cons(N,IL))) -> mark(U91(isNatIList(IL),IL,M,N))
       active(cons(X1,X2)) -> cons(active(X1),X2)
       active(U11(X)) -> U11(active(X))
       active(U21(X)) -> U21(active(X))
       active(U31(X)) -> U31(active(X))
       active(U41(X1,X2)) -> U41(active(X1),X2)
       active(U42(X)) -> U42(active(X))
       active(U51(X1,X2)) -> U51(active(X1),X2)
       active(U52(X)) -> U52(active(X))
       active(U61(X1,X2)) -> U61(active(X1),X2)
       active(U62(X)) -> U62(active(X))
       active(U71(X1,X2,X3)) -> U71(active(X1),X2,X3)
       active(U72(X1,X2)) -> U72(active(X1),X2)
       active(s(X)) -> s(active(X))
       active(length(X)) -> length(active(X))
       active(U81(X)) -> U81(active(X))
       active(U91(X1,X2,X3,X4)) -> U91(active(X1),X2,X3,X4)
       active(U92(X1,X2,X3,X4)) -> U92(active(X1),X2,X3,X4)
       active(U93(X1,X2,X3,X4)) -> U93(active(X1),X2,X3,X4)
       active(take(X1,X2)) -> take(active(X1),X2)
       active(take(X1,X2)) -> take(X1,active(X2))
       cons(mark(X1),X2) -> mark(cons(X1,X2))
       U11(mark(X)) -> mark(U11(X))
       U21(mark(X)) -> mark(U21(X))
       U31(mark(X)) -> mark(U31(X))
       U41(mark(X1),X2) -> mark(U41(X1,X2))
       U42(mark(X)) -> mark(U42(X))
       U51(mark(X1),X2) -> mark(U51(X1,X2))
       U52(mark(X)) -> mark(U52(X))
       U61(mark(X1),X2) -> mark(U61(X1,X2))
       U62(mark(X)) -> mark(U62(X))
       U71(mark(X1),X2,X3) -> mark(U71(X1,X2,X3))
       U72(mark(X1),X2) -> mark(U72(X1,X2))
       s(mark(X)) -> mark(s(X))
       length(mark(X)) -> mark(length(X))
       U81(mark(X)) -> mark(U81(X))
       U91(mark(X1),X2,X3,X4) -> mark(U91(X1,X2,X3,X4))
       U92(mark(X1),X2,X3,X4) -> mark(U92(X1,X2,X3,X4))
       U93(mark(X1),X2,X3,X4) -> mark(U93(X1,X2,X3,X4))
       take(mark(X1),X2) -> mark(take(X1,X2))
       take(X1,mark(X2)) -> mark(take(X1,X2))
       proper(zeros()) -> ok(zeros())
       proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
       proper(0()) -> ok(0())
       proper(U11(X)) -> U11(proper(X))
       proper(tt()) -> ok(tt())
       proper(U21(X)) -> U21(proper(X))
       proper(U31(X)) -> U31(proper(X))
       proper(U41(X1,X2)) -> U41(proper(X1),proper(X2))
       proper(U42(X)) -> U42(proper(X))
       proper(isNatIList(X)) -> isNatIList(proper(X))
       proper(U51(X1,X2)) -> U51(proper(X1),proper(X2))
       proper(U52(X)) -> U52(proper(X))
       proper(isNatList(X)) -> isNatList(proper(X))
       proper(U61(X1,X2)) -> U61(proper(X1),proper(X2))
       proper(U62(X)) -> U62(proper(X))
       proper(U71(X1,X2,X3)) -> U71(proper(X1),proper(X2),proper(X3))
       proper(U72(X1,X2)) -> U72(proper(X1),proper(X2))
       proper(isNat(X)) -> isNat(proper(X))
       proper(s(X)) -> s(proper(X))
       proper(length(X)) -> length(proper(X))
       proper(U81(X)) -> U81(proper(X))
       proper(nil()) -> ok(nil())
       proper(U91(X1,X2,X3,X4)) -> U91(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(U92(X1,X2,X3,X4)) -> U92(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(U93(X1,X2,X3,X4)) -> U93(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(take(X1,X2)) -> take(proper(X1),proper(X2))
       cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
       U11(ok(X)) -> ok(U11(X))
       U21(ok(X)) -> ok(U21(X))
       U31(ok(X)) -> ok(U31(X))
       U41(ok(X1),ok(X2)) -> ok(U41(X1,X2))
       U42(ok(X)) -> ok(U42(X))
       isNatIList(ok(X)) -> ok(isNatIList(X))
       U51(ok(X1),ok(X2)) -> ok(U51(X1,X2))
       U52(ok(X)) -> ok(U52(X))
       isNatList(ok(X)) -> ok(isNatList(X))
       U61(ok(X1),ok(X2)) -> ok(U61(X1,X2))
       U62(ok(X)) -> ok(U62(X))
       U71(ok(X1),ok(X2),ok(X3)) -> ok(U71(X1,X2,X3))
       U72(ok(X1),ok(X2)) -> ok(U72(X1,X2))
       isNat(ok(X)) -> ok(isNat(X))
       s(ok(X)) -> ok(s(X))
       length(ok(X)) -> ok(length(X))
       U81(ok(X)) -> ok(U81(X))
       U91(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U91(X1,X2,X3,X4))
       U92(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U92(X1,X2,X3,X4))
       U93(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U93(X1,X2,X3,X4))
       take(ok(X1),ok(X2)) -> ok(take(X1,X2))
       top(mark(X)) -> top(proper(X))
       top(ok(X)) -> top(active(X))
      Open
      
      DPs:
       U72#(ok(X1),ok(X2)) -> U72#(X1,X2)
       U72#(mark(X1),X2) -> U72#(X1,X2)
      TRS:
       active(zeros()) -> mark(cons(0(),zeros()))
       active(U11(tt())) -> mark(tt())
       active(U21(tt())) -> mark(tt())
       active(U31(tt())) -> mark(tt())
       active(U41(tt(),V2)) -> mark(U42(isNatIList(V2)))
       active(U42(tt())) -> mark(tt())
       active(U51(tt(),V2)) -> mark(U52(isNatList(V2)))
       active(U52(tt())) -> mark(tt())
       active(U61(tt(),V2)) -> mark(U62(isNatIList(V2)))
       active(U62(tt())) -> mark(tt())
       active(U71(tt(),L,N)) -> mark(U72(isNat(N),L))
       active(U72(tt(),L)) -> mark(s(length(L)))
       active(U81(tt())) -> mark(nil())
       active(U91(tt(),IL,M,N)) -> mark(U92(isNat(M),IL,M,N))
       active(U92(tt(),IL,M,N)) -> mark(U93(isNat(N),IL,M,N))
       active(U93(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
       active(isNat(0())) -> mark(tt())
       active(isNat(length(V1))) -> mark(U11(isNatList(V1)))
       active(isNat(s(V1))) -> mark(U21(isNat(V1)))
       active(isNatIList(V)) -> mark(U31(isNatList(V)))
       active(isNatIList(zeros())) -> mark(tt())
       active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2))
       active(isNatList(nil())) -> mark(tt())
       active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2))
       active(isNatList(take(V1,V2))) -> mark(U61(isNat(V1),V2))
       active(length(nil())) -> mark(0())
       active(length(cons(N,L))) -> mark(U71(isNatList(L),L,N))
       active(take(0(),IL)) -> mark(U81(isNatIList(IL)))
       active(take(s(M),cons(N,IL))) -> mark(U91(isNatIList(IL),IL,M,N))
       active(cons(X1,X2)) -> cons(active(X1),X2)
       active(U11(X)) -> U11(active(X))
       active(U21(X)) -> U21(active(X))
       active(U31(X)) -> U31(active(X))
       active(U41(X1,X2)) -> U41(active(X1),X2)
       active(U42(X)) -> U42(active(X))
       active(U51(X1,X2)) -> U51(active(X1),X2)
       active(U52(X)) -> U52(active(X))
       active(U61(X1,X2)) -> U61(active(X1),X2)
       active(U62(X)) -> U62(active(X))
       active(U71(X1,X2,X3)) -> U71(active(X1),X2,X3)
       active(U72(X1,X2)) -> U72(active(X1),X2)
       active(s(X)) -> s(active(X))
       active(length(X)) -> length(active(X))
       active(U81(X)) -> U81(active(X))
       active(U91(X1,X2,X3,X4)) -> U91(active(X1),X2,X3,X4)
       active(U92(X1,X2,X3,X4)) -> U92(active(X1),X2,X3,X4)
       active(U93(X1,X2,X3,X4)) -> U93(active(X1),X2,X3,X4)
       active(take(X1,X2)) -> take(active(X1),X2)
       active(take(X1,X2)) -> take(X1,active(X2))
       cons(mark(X1),X2) -> mark(cons(X1,X2))
       U11(mark(X)) -> mark(U11(X))
       U21(mark(X)) -> mark(U21(X))
       U31(mark(X)) -> mark(U31(X))
       U41(mark(X1),X2) -> mark(U41(X1,X2))
       U42(mark(X)) -> mark(U42(X))
       U51(mark(X1),X2) -> mark(U51(X1,X2))
       U52(mark(X)) -> mark(U52(X))
       U61(mark(X1),X2) -> mark(U61(X1,X2))
       U62(mark(X)) -> mark(U62(X))
       U71(mark(X1),X2,X3) -> mark(U71(X1,X2,X3))
       U72(mark(X1),X2) -> mark(U72(X1,X2))
       s(mark(X)) -> mark(s(X))
       length(mark(X)) -> mark(length(X))
       U81(mark(X)) -> mark(U81(X))
       U91(mark(X1),X2,X3,X4) -> mark(U91(X1,X2,X3,X4))
       U92(mark(X1),X2,X3,X4) -> mark(U92(X1,X2,X3,X4))
       U93(mark(X1),X2,X3,X4) -> mark(U93(X1,X2,X3,X4))
       take(mark(X1),X2) -> mark(take(X1,X2))
       take(X1,mark(X2)) -> mark(take(X1,X2))
       proper(zeros()) -> ok(zeros())
       proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
       proper(0()) -> ok(0())
       proper(U11(X)) -> U11(proper(X))
       proper(tt()) -> ok(tt())
       proper(U21(X)) -> U21(proper(X))
       proper(U31(X)) -> U31(proper(X))
       proper(U41(X1,X2)) -> U41(proper(X1),proper(X2))
       proper(U42(X)) -> U42(proper(X))
       proper(isNatIList(X)) -> isNatIList(proper(X))
       proper(U51(X1,X2)) -> U51(proper(X1),proper(X2))
       proper(U52(X)) -> U52(proper(X))
       proper(isNatList(X)) -> isNatList(proper(X))
       proper(U61(X1,X2)) -> U61(proper(X1),proper(X2))
       proper(U62(X)) -> U62(proper(X))
       proper(U71(X1,X2,X3)) -> U71(proper(X1),proper(X2),proper(X3))
       proper(U72(X1,X2)) -> U72(proper(X1),proper(X2))
       proper(isNat(X)) -> isNat(proper(X))
       proper(s(X)) -> s(proper(X))
       proper(length(X)) -> length(proper(X))
       proper(U81(X)) -> U81(proper(X))
       proper(nil()) -> ok(nil())
       proper(U91(X1,X2,X3,X4)) -> U91(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(U92(X1,X2,X3,X4)) -> U92(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(U93(X1,X2,X3,X4)) -> U93(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(take(X1,X2)) -> take(proper(X1),proper(X2))
       cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
       U11(ok(X)) -> ok(U11(X))
       U21(ok(X)) -> ok(U21(X))
       U31(ok(X)) -> ok(U31(X))
       U41(ok(X1),ok(X2)) -> ok(U41(X1,X2))
       U42(ok(X)) -> ok(U42(X))
       isNatIList(ok(X)) -> ok(isNatIList(X))
       U51(ok(X1),ok(X2)) -> ok(U51(X1,X2))
       U52(ok(X)) -> ok(U52(X))
       isNatList(ok(X)) -> ok(isNatList(X))
       U61(ok(X1),ok(X2)) -> ok(U61(X1,X2))
       U62(ok(X)) -> ok(U62(X))
       U71(ok(X1),ok(X2),ok(X3)) -> ok(U71(X1,X2,X3))
       U72(ok(X1),ok(X2)) -> ok(U72(X1,X2))
       isNat(ok(X)) -> ok(isNat(X))
       s(ok(X)) -> ok(s(X))
       length(ok(X)) -> ok(length(X))
       U81(ok(X)) -> ok(U81(X))
       U91(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U91(X1,X2,X3,X4))
       U92(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U92(X1,X2,X3,X4))
       U93(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U93(X1,X2,X3,X4))
       take(ok(X1),ok(X2)) -> ok(take(X1,X2))
       top(mark(X)) -> top(proper(X))
       top(ok(X)) -> top(active(X))
      Open
      
      DPs:
       s#(ok(X)) -> s#(X)
       s#(mark(X)) -> s#(X)
      TRS:
       active(zeros()) -> mark(cons(0(),zeros()))
       active(U11(tt())) -> mark(tt())
       active(U21(tt())) -> mark(tt())
       active(U31(tt())) -> mark(tt())
       active(U41(tt(),V2)) -> mark(U42(isNatIList(V2)))
       active(U42(tt())) -> mark(tt())
       active(U51(tt(),V2)) -> mark(U52(isNatList(V2)))
       active(U52(tt())) -> mark(tt())
       active(U61(tt(),V2)) -> mark(U62(isNatIList(V2)))
       active(U62(tt())) -> mark(tt())
       active(U71(tt(),L,N)) -> mark(U72(isNat(N),L))
       active(U72(tt(),L)) -> mark(s(length(L)))
       active(U81(tt())) -> mark(nil())
       active(U91(tt(),IL,M,N)) -> mark(U92(isNat(M),IL,M,N))
       active(U92(tt(),IL,M,N)) -> mark(U93(isNat(N),IL,M,N))
       active(U93(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
       active(isNat(0())) -> mark(tt())
       active(isNat(length(V1))) -> mark(U11(isNatList(V1)))
       active(isNat(s(V1))) -> mark(U21(isNat(V1)))
       active(isNatIList(V)) -> mark(U31(isNatList(V)))
       active(isNatIList(zeros())) -> mark(tt())
       active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2))
       active(isNatList(nil())) -> mark(tt())
       active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2))
       active(isNatList(take(V1,V2))) -> mark(U61(isNat(V1),V2))
       active(length(nil())) -> mark(0())
       active(length(cons(N,L))) -> mark(U71(isNatList(L),L,N))
       active(take(0(),IL)) -> mark(U81(isNatIList(IL)))
       active(take(s(M),cons(N,IL))) -> mark(U91(isNatIList(IL),IL,M,N))
       active(cons(X1,X2)) -> cons(active(X1),X2)
       active(U11(X)) -> U11(active(X))
       active(U21(X)) -> U21(active(X))
       active(U31(X)) -> U31(active(X))
       active(U41(X1,X2)) -> U41(active(X1),X2)
       active(U42(X)) -> U42(active(X))
       active(U51(X1,X2)) -> U51(active(X1),X2)
       active(U52(X)) -> U52(active(X))
       active(U61(X1,X2)) -> U61(active(X1),X2)
       active(U62(X)) -> U62(active(X))
       active(U71(X1,X2,X3)) -> U71(active(X1),X2,X3)
       active(U72(X1,X2)) -> U72(active(X1),X2)
       active(s(X)) -> s(active(X))
       active(length(X)) -> length(active(X))
       active(U81(X)) -> U81(active(X))
       active(U91(X1,X2,X3,X4)) -> U91(active(X1),X2,X3,X4)
       active(U92(X1,X2,X3,X4)) -> U92(active(X1),X2,X3,X4)
       active(U93(X1,X2,X3,X4)) -> U93(active(X1),X2,X3,X4)
       active(take(X1,X2)) -> take(active(X1),X2)
       active(take(X1,X2)) -> take(X1,active(X2))
       cons(mark(X1),X2) -> mark(cons(X1,X2))
       U11(mark(X)) -> mark(U11(X))
       U21(mark(X)) -> mark(U21(X))
       U31(mark(X)) -> mark(U31(X))
       U41(mark(X1),X2) -> mark(U41(X1,X2))
       U42(mark(X)) -> mark(U42(X))
       U51(mark(X1),X2) -> mark(U51(X1,X2))
       U52(mark(X)) -> mark(U52(X))
       U61(mark(X1),X2) -> mark(U61(X1,X2))
       U62(mark(X)) -> mark(U62(X))
       U71(mark(X1),X2,X3) -> mark(U71(X1,X2,X3))
       U72(mark(X1),X2) -> mark(U72(X1,X2))
       s(mark(X)) -> mark(s(X))
       length(mark(X)) -> mark(length(X))
       U81(mark(X)) -> mark(U81(X))
       U91(mark(X1),X2,X3,X4) -> mark(U91(X1,X2,X3,X4))
       U92(mark(X1),X2,X3,X4) -> mark(U92(X1,X2,X3,X4))
       U93(mark(X1),X2,X3,X4) -> mark(U93(X1,X2,X3,X4))
       take(mark(X1),X2) -> mark(take(X1,X2))
       take(X1,mark(X2)) -> mark(take(X1,X2))
       proper(zeros()) -> ok(zeros())
       proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
       proper(0()) -> ok(0())
       proper(U11(X)) -> U11(proper(X))
       proper(tt()) -> ok(tt())
       proper(U21(X)) -> U21(proper(X))
       proper(U31(X)) -> U31(proper(X))
       proper(U41(X1,X2)) -> U41(proper(X1),proper(X2))
       proper(U42(X)) -> U42(proper(X))
       proper(isNatIList(X)) -> isNatIList(proper(X))
       proper(U51(X1,X2)) -> U51(proper(X1),proper(X2))
       proper(U52(X)) -> U52(proper(X))
       proper(isNatList(X)) -> isNatList(proper(X))
       proper(U61(X1,X2)) -> U61(proper(X1),proper(X2))
       proper(U62(X)) -> U62(proper(X))
       proper(U71(X1,X2,X3)) -> U71(proper(X1),proper(X2),proper(X3))
       proper(U72(X1,X2)) -> U72(proper(X1),proper(X2))
       proper(isNat(X)) -> isNat(proper(X))
       proper(s(X)) -> s(proper(X))
       proper(length(X)) -> length(proper(X))
       proper(U81(X)) -> U81(proper(X))
       proper(nil()) -> ok(nil())
       proper(U91(X1,X2,X3,X4)) -> U91(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(U92(X1,X2,X3,X4)) -> U92(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(U93(X1,X2,X3,X4)) -> U93(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(take(X1,X2)) -> take(proper(X1),proper(X2))
       cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
       U11(ok(X)) -> ok(U11(X))
       U21(ok(X)) -> ok(U21(X))
       U31(ok(X)) -> ok(U31(X))
       U41(ok(X1),ok(X2)) -> ok(U41(X1,X2))
       U42(ok(X)) -> ok(U42(X))
       isNatIList(ok(X)) -> ok(isNatIList(X))
       U51(ok(X1),ok(X2)) -> ok(U51(X1,X2))
       U52(ok(X)) -> ok(U52(X))
       isNatList(ok(X)) -> ok(isNatList(X))
       U61(ok(X1),ok(X2)) -> ok(U61(X1,X2))
       U62(ok(X)) -> ok(U62(X))
       U71(ok(X1),ok(X2),ok(X3)) -> ok(U71(X1,X2,X3))
       U72(ok(X1),ok(X2)) -> ok(U72(X1,X2))
       isNat(ok(X)) -> ok(isNat(X))
       s(ok(X)) -> ok(s(X))
       length(ok(X)) -> ok(length(X))
       U81(ok(X)) -> ok(U81(X))
       U91(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U91(X1,X2,X3,X4))
       U92(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U92(X1,X2,X3,X4))
       U93(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U93(X1,X2,X3,X4))
       take(ok(X1),ok(X2)) -> ok(take(X1,X2))
       top(mark(X)) -> top(proper(X))
       top(ok(X)) -> top(active(X))
      Open
      
      DPs:
       length#(ok(X)) -> length#(X)
       length#(mark(X)) -> length#(X)
      TRS:
       active(zeros()) -> mark(cons(0(),zeros()))
       active(U11(tt())) -> mark(tt())
       active(U21(tt())) -> mark(tt())
       active(U31(tt())) -> mark(tt())
       active(U41(tt(),V2)) -> mark(U42(isNatIList(V2)))
       active(U42(tt())) -> mark(tt())
       active(U51(tt(),V2)) -> mark(U52(isNatList(V2)))
       active(U52(tt())) -> mark(tt())
       active(U61(tt(),V2)) -> mark(U62(isNatIList(V2)))
       active(U62(tt())) -> mark(tt())
       active(U71(tt(),L,N)) -> mark(U72(isNat(N),L))
       active(U72(tt(),L)) -> mark(s(length(L)))
       active(U81(tt())) -> mark(nil())
       active(U91(tt(),IL,M,N)) -> mark(U92(isNat(M),IL,M,N))
       active(U92(tt(),IL,M,N)) -> mark(U93(isNat(N),IL,M,N))
       active(U93(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
       active(isNat(0())) -> mark(tt())
       active(isNat(length(V1))) -> mark(U11(isNatList(V1)))
       active(isNat(s(V1))) -> mark(U21(isNat(V1)))
       active(isNatIList(V)) -> mark(U31(isNatList(V)))
       active(isNatIList(zeros())) -> mark(tt())
       active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2))
       active(isNatList(nil())) -> mark(tt())
       active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2))
       active(isNatList(take(V1,V2))) -> mark(U61(isNat(V1),V2))
       active(length(nil())) -> mark(0())
       active(length(cons(N,L))) -> mark(U71(isNatList(L),L,N))
       active(take(0(),IL)) -> mark(U81(isNatIList(IL)))
       active(take(s(M),cons(N,IL))) -> mark(U91(isNatIList(IL),IL,M,N))
       active(cons(X1,X2)) -> cons(active(X1),X2)
       active(U11(X)) -> U11(active(X))
       active(U21(X)) -> U21(active(X))
       active(U31(X)) -> U31(active(X))
       active(U41(X1,X2)) -> U41(active(X1),X2)
       active(U42(X)) -> U42(active(X))
       active(U51(X1,X2)) -> U51(active(X1),X2)
       active(U52(X)) -> U52(active(X))
       active(U61(X1,X2)) -> U61(active(X1),X2)
       active(U62(X)) -> U62(active(X))
       active(U71(X1,X2,X3)) -> U71(active(X1),X2,X3)
       active(U72(X1,X2)) -> U72(active(X1),X2)
       active(s(X)) -> s(active(X))
       active(length(X)) -> length(active(X))
       active(U81(X)) -> U81(active(X))
       active(U91(X1,X2,X3,X4)) -> U91(active(X1),X2,X3,X4)
       active(U92(X1,X2,X3,X4)) -> U92(active(X1),X2,X3,X4)
       active(U93(X1,X2,X3,X4)) -> U93(active(X1),X2,X3,X4)
       active(take(X1,X2)) -> take(active(X1),X2)
       active(take(X1,X2)) -> take(X1,active(X2))
       cons(mark(X1),X2) -> mark(cons(X1,X2))
       U11(mark(X)) -> mark(U11(X))
       U21(mark(X)) -> mark(U21(X))
       U31(mark(X)) -> mark(U31(X))
       U41(mark(X1),X2) -> mark(U41(X1,X2))
       U42(mark(X)) -> mark(U42(X))
       U51(mark(X1),X2) -> mark(U51(X1,X2))
       U52(mark(X)) -> mark(U52(X))
       U61(mark(X1),X2) -> mark(U61(X1,X2))
       U62(mark(X)) -> mark(U62(X))
       U71(mark(X1),X2,X3) -> mark(U71(X1,X2,X3))
       U72(mark(X1),X2) -> mark(U72(X1,X2))
       s(mark(X)) -> mark(s(X))
       length(mark(X)) -> mark(length(X))
       U81(mark(X)) -> mark(U81(X))
       U91(mark(X1),X2,X3,X4) -> mark(U91(X1,X2,X3,X4))
       U92(mark(X1),X2,X3,X4) -> mark(U92(X1,X2,X3,X4))
       U93(mark(X1),X2,X3,X4) -> mark(U93(X1,X2,X3,X4))
       take(mark(X1),X2) -> mark(take(X1,X2))
       take(X1,mark(X2)) -> mark(take(X1,X2))
       proper(zeros()) -> ok(zeros())
       proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
       proper(0()) -> ok(0())
       proper(U11(X)) -> U11(proper(X))
       proper(tt()) -> ok(tt())
       proper(U21(X)) -> U21(proper(X))
       proper(U31(X)) -> U31(proper(X))
       proper(U41(X1,X2)) -> U41(proper(X1),proper(X2))
       proper(U42(X)) -> U42(proper(X))
       proper(isNatIList(X)) -> isNatIList(proper(X))
       proper(U51(X1,X2)) -> U51(proper(X1),proper(X2))
       proper(U52(X)) -> U52(proper(X))
       proper(isNatList(X)) -> isNatList(proper(X))
       proper(U61(X1,X2)) -> U61(proper(X1),proper(X2))
       proper(U62(X)) -> U62(proper(X))
       proper(U71(X1,X2,X3)) -> U71(proper(X1),proper(X2),proper(X3))
       proper(U72(X1,X2)) -> U72(proper(X1),proper(X2))
       proper(isNat(X)) -> isNat(proper(X))
       proper(s(X)) -> s(proper(X))
       proper(length(X)) -> length(proper(X))
       proper(U81(X)) -> U81(proper(X))
       proper(nil()) -> ok(nil())
       proper(U91(X1,X2,X3,X4)) -> U91(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(U92(X1,X2,X3,X4)) -> U92(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(U93(X1,X2,X3,X4)) -> U93(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(take(X1,X2)) -> take(proper(X1),proper(X2))
       cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
       U11(ok(X)) -> ok(U11(X))
       U21(ok(X)) -> ok(U21(X))
       U31(ok(X)) -> ok(U31(X))
       U41(ok(X1),ok(X2)) -> ok(U41(X1,X2))
       U42(ok(X)) -> ok(U42(X))
       isNatIList(ok(X)) -> ok(isNatIList(X))
       U51(ok(X1),ok(X2)) -> ok(U51(X1,X2))
       U52(ok(X)) -> ok(U52(X))
       isNatList(ok(X)) -> ok(isNatList(X))
       U61(ok(X1),ok(X2)) -> ok(U61(X1,X2))
       U62(ok(X)) -> ok(U62(X))
       U71(ok(X1),ok(X2),ok(X3)) -> ok(U71(X1,X2,X3))
       U72(ok(X1),ok(X2)) -> ok(U72(X1,X2))
       isNat(ok(X)) -> ok(isNat(X))
       s(ok(X)) -> ok(s(X))
       length(ok(X)) -> ok(length(X))
       U81(ok(X)) -> ok(U81(X))
       U91(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U91(X1,X2,X3,X4))
       U92(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U92(X1,X2,X3,X4))
       U93(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U93(X1,X2,X3,X4))
       take(ok(X1),ok(X2)) -> ok(take(X1,X2))
       top(mark(X)) -> top(proper(X))
       top(ok(X)) -> top(active(X))
      Open
      
      DPs:
       U81#(ok(X)) -> U81#(X)
       U81#(mark(X)) -> U81#(X)
      TRS:
       active(zeros()) -> mark(cons(0(),zeros()))
       active(U11(tt())) -> mark(tt())
       active(U21(tt())) -> mark(tt())
       active(U31(tt())) -> mark(tt())
       active(U41(tt(),V2)) -> mark(U42(isNatIList(V2)))
       active(U42(tt())) -> mark(tt())
       active(U51(tt(),V2)) -> mark(U52(isNatList(V2)))
       active(U52(tt())) -> mark(tt())
       active(U61(tt(),V2)) -> mark(U62(isNatIList(V2)))
       active(U62(tt())) -> mark(tt())
       active(U71(tt(),L,N)) -> mark(U72(isNat(N),L))
       active(U72(tt(),L)) -> mark(s(length(L)))
       active(U81(tt())) -> mark(nil())
       active(U91(tt(),IL,M,N)) -> mark(U92(isNat(M),IL,M,N))
       active(U92(tt(),IL,M,N)) -> mark(U93(isNat(N),IL,M,N))
       active(U93(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
       active(isNat(0())) -> mark(tt())
       active(isNat(length(V1))) -> mark(U11(isNatList(V1)))
       active(isNat(s(V1))) -> mark(U21(isNat(V1)))
       active(isNatIList(V)) -> mark(U31(isNatList(V)))
       active(isNatIList(zeros())) -> mark(tt())
       active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2))
       active(isNatList(nil())) -> mark(tt())
       active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2))
       active(isNatList(take(V1,V2))) -> mark(U61(isNat(V1),V2))
       active(length(nil())) -> mark(0())
       active(length(cons(N,L))) -> mark(U71(isNatList(L),L,N))
       active(take(0(),IL)) -> mark(U81(isNatIList(IL)))
       active(take(s(M),cons(N,IL))) -> mark(U91(isNatIList(IL),IL,M,N))
       active(cons(X1,X2)) -> cons(active(X1),X2)
       active(U11(X)) -> U11(active(X))
       active(U21(X)) -> U21(active(X))
       active(U31(X)) -> U31(active(X))
       active(U41(X1,X2)) -> U41(active(X1),X2)
       active(U42(X)) -> U42(active(X))
       active(U51(X1,X2)) -> U51(active(X1),X2)
       active(U52(X)) -> U52(active(X))
       active(U61(X1,X2)) -> U61(active(X1),X2)
       active(U62(X)) -> U62(active(X))
       active(U71(X1,X2,X3)) -> U71(active(X1),X2,X3)
       active(U72(X1,X2)) -> U72(active(X1),X2)
       active(s(X)) -> s(active(X))
       active(length(X)) -> length(active(X))
       active(U81(X)) -> U81(active(X))
       active(U91(X1,X2,X3,X4)) -> U91(active(X1),X2,X3,X4)
       active(U92(X1,X2,X3,X4)) -> U92(active(X1),X2,X3,X4)
       active(U93(X1,X2,X3,X4)) -> U93(active(X1),X2,X3,X4)
       active(take(X1,X2)) -> take(active(X1),X2)
       active(take(X1,X2)) -> take(X1,active(X2))
       cons(mark(X1),X2) -> mark(cons(X1,X2))
       U11(mark(X)) -> mark(U11(X))
       U21(mark(X)) -> mark(U21(X))
       U31(mark(X)) -> mark(U31(X))
       U41(mark(X1),X2) -> mark(U41(X1,X2))
       U42(mark(X)) -> mark(U42(X))
       U51(mark(X1),X2) -> mark(U51(X1,X2))
       U52(mark(X)) -> mark(U52(X))
       U61(mark(X1),X2) -> mark(U61(X1,X2))
       U62(mark(X)) -> mark(U62(X))
       U71(mark(X1),X2,X3) -> mark(U71(X1,X2,X3))
       U72(mark(X1),X2) -> mark(U72(X1,X2))
       s(mark(X)) -> mark(s(X))
       length(mark(X)) -> mark(length(X))
       U81(mark(X)) -> mark(U81(X))
       U91(mark(X1),X2,X3,X4) -> mark(U91(X1,X2,X3,X4))
       U92(mark(X1),X2,X3,X4) -> mark(U92(X1,X2,X3,X4))
       U93(mark(X1),X2,X3,X4) -> mark(U93(X1,X2,X3,X4))
       take(mark(X1),X2) -> mark(take(X1,X2))
       take(X1,mark(X2)) -> mark(take(X1,X2))
       proper(zeros()) -> ok(zeros())
       proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
       proper(0()) -> ok(0())
       proper(U11(X)) -> U11(proper(X))
       proper(tt()) -> ok(tt())
       proper(U21(X)) -> U21(proper(X))
       proper(U31(X)) -> U31(proper(X))
       proper(U41(X1,X2)) -> U41(proper(X1),proper(X2))
       proper(U42(X)) -> U42(proper(X))
       proper(isNatIList(X)) -> isNatIList(proper(X))
       proper(U51(X1,X2)) -> U51(proper(X1),proper(X2))
       proper(U52(X)) -> U52(proper(X))
       proper(isNatList(X)) -> isNatList(proper(X))
       proper(U61(X1,X2)) -> U61(proper(X1),proper(X2))
       proper(U62(X)) -> U62(proper(X))
       proper(U71(X1,X2,X3)) -> U71(proper(X1),proper(X2),proper(X3))
       proper(U72(X1,X2)) -> U72(proper(X1),proper(X2))
       proper(isNat(X)) -> isNat(proper(X))
       proper(s(X)) -> s(proper(X))
       proper(length(X)) -> length(proper(X))
       proper(U81(X)) -> U81(proper(X))
       proper(nil()) -> ok(nil())
       proper(U91(X1,X2,X3,X4)) -> U91(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(U92(X1,X2,X3,X4)) -> U92(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(U93(X1,X2,X3,X4)) -> U93(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(take(X1,X2)) -> take(proper(X1),proper(X2))
       cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
       U11(ok(X)) -> ok(U11(X))
       U21(ok(X)) -> ok(U21(X))
       U31(ok(X)) -> ok(U31(X))
       U41(ok(X1),ok(X2)) -> ok(U41(X1,X2))
       U42(ok(X)) -> ok(U42(X))
       isNatIList(ok(X)) -> ok(isNatIList(X))
       U51(ok(X1),ok(X2)) -> ok(U51(X1,X2))
       U52(ok(X)) -> ok(U52(X))
       isNatList(ok(X)) -> ok(isNatList(X))
       U61(ok(X1),ok(X2)) -> ok(U61(X1,X2))
       U62(ok(X)) -> ok(U62(X))
       U71(ok(X1),ok(X2),ok(X3)) -> ok(U71(X1,X2,X3))
       U72(ok(X1),ok(X2)) -> ok(U72(X1,X2))
       isNat(ok(X)) -> ok(isNat(X))
       s(ok(X)) -> ok(s(X))
       length(ok(X)) -> ok(length(X))
       U81(ok(X)) -> ok(U81(X))
       U91(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U91(X1,X2,X3,X4))
       U92(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U92(X1,X2,X3,X4))
       U93(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U93(X1,X2,X3,X4))
       take(ok(X1),ok(X2)) -> ok(take(X1,X2))
       top(mark(X)) -> top(proper(X))
       top(ok(X)) -> top(active(X))
      Open
      
      DPs:
       U91#(ok(X1),ok(X2),ok(X3),ok(X4)) -> U91#(X1,X2,X3,X4)
       U91#(mark(X1),X2,X3,X4) -> U91#(X1,X2,X3,X4)
      TRS:
       active(zeros()) -> mark(cons(0(),zeros()))
       active(U11(tt())) -> mark(tt())
       active(U21(tt())) -> mark(tt())
       active(U31(tt())) -> mark(tt())
       active(U41(tt(),V2)) -> mark(U42(isNatIList(V2)))
       active(U42(tt())) -> mark(tt())
       active(U51(tt(),V2)) -> mark(U52(isNatList(V2)))
       active(U52(tt())) -> mark(tt())
       active(U61(tt(),V2)) -> mark(U62(isNatIList(V2)))
       active(U62(tt())) -> mark(tt())
       active(U71(tt(),L,N)) -> mark(U72(isNat(N),L))
       active(U72(tt(),L)) -> mark(s(length(L)))
       active(U81(tt())) -> mark(nil())
       active(U91(tt(),IL,M,N)) -> mark(U92(isNat(M),IL,M,N))
       active(U92(tt(),IL,M,N)) -> mark(U93(isNat(N),IL,M,N))
       active(U93(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
       active(isNat(0())) -> mark(tt())
       active(isNat(length(V1))) -> mark(U11(isNatList(V1)))
       active(isNat(s(V1))) -> mark(U21(isNat(V1)))
       active(isNatIList(V)) -> mark(U31(isNatList(V)))
       active(isNatIList(zeros())) -> mark(tt())
       active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2))
       active(isNatList(nil())) -> mark(tt())
       active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2))
       active(isNatList(take(V1,V2))) -> mark(U61(isNat(V1),V2))
       active(length(nil())) -> mark(0())
       active(length(cons(N,L))) -> mark(U71(isNatList(L),L,N))
       active(take(0(),IL)) -> mark(U81(isNatIList(IL)))
       active(take(s(M),cons(N,IL))) -> mark(U91(isNatIList(IL),IL,M,N))
       active(cons(X1,X2)) -> cons(active(X1),X2)
       active(U11(X)) -> U11(active(X))
       active(U21(X)) -> U21(active(X))
       active(U31(X)) -> U31(active(X))
       active(U41(X1,X2)) -> U41(active(X1),X2)
       active(U42(X)) -> U42(active(X))
       active(U51(X1,X2)) -> U51(active(X1),X2)
       active(U52(X)) -> U52(active(X))
       active(U61(X1,X2)) -> U61(active(X1),X2)
       active(U62(X)) -> U62(active(X))
       active(U71(X1,X2,X3)) -> U71(active(X1),X2,X3)
       active(U72(X1,X2)) -> U72(active(X1),X2)
       active(s(X)) -> s(active(X))
       active(length(X)) -> length(active(X))
       active(U81(X)) -> U81(active(X))
       active(U91(X1,X2,X3,X4)) -> U91(active(X1),X2,X3,X4)
       active(U92(X1,X2,X3,X4)) -> U92(active(X1),X2,X3,X4)
       active(U93(X1,X2,X3,X4)) -> U93(active(X1),X2,X3,X4)
       active(take(X1,X2)) -> take(active(X1),X2)
       active(take(X1,X2)) -> take(X1,active(X2))
       cons(mark(X1),X2) -> mark(cons(X1,X2))
       U11(mark(X)) -> mark(U11(X))
       U21(mark(X)) -> mark(U21(X))
       U31(mark(X)) -> mark(U31(X))
       U41(mark(X1),X2) -> mark(U41(X1,X2))
       U42(mark(X)) -> mark(U42(X))
       U51(mark(X1),X2) -> mark(U51(X1,X2))
       U52(mark(X)) -> mark(U52(X))
       U61(mark(X1),X2) -> mark(U61(X1,X2))
       U62(mark(X)) -> mark(U62(X))
       U71(mark(X1),X2,X3) -> mark(U71(X1,X2,X3))
       U72(mark(X1),X2) -> mark(U72(X1,X2))
       s(mark(X)) -> mark(s(X))
       length(mark(X)) -> mark(length(X))
       U81(mark(X)) -> mark(U81(X))
       U91(mark(X1),X2,X3,X4) -> mark(U91(X1,X2,X3,X4))
       U92(mark(X1),X2,X3,X4) -> mark(U92(X1,X2,X3,X4))
       U93(mark(X1),X2,X3,X4) -> mark(U93(X1,X2,X3,X4))
       take(mark(X1),X2) -> mark(take(X1,X2))
       take(X1,mark(X2)) -> mark(take(X1,X2))
       proper(zeros()) -> ok(zeros())
       proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
       proper(0()) -> ok(0())
       proper(U11(X)) -> U11(proper(X))
       proper(tt()) -> ok(tt())
       proper(U21(X)) -> U21(proper(X))
       proper(U31(X)) -> U31(proper(X))
       proper(U41(X1,X2)) -> U41(proper(X1),proper(X2))
       proper(U42(X)) -> U42(proper(X))
       proper(isNatIList(X)) -> isNatIList(proper(X))
       proper(U51(X1,X2)) -> U51(proper(X1),proper(X2))
       proper(U52(X)) -> U52(proper(X))
       proper(isNatList(X)) -> isNatList(proper(X))
       proper(U61(X1,X2)) -> U61(proper(X1),proper(X2))
       proper(U62(X)) -> U62(proper(X))
       proper(U71(X1,X2,X3)) -> U71(proper(X1),proper(X2),proper(X3))
       proper(U72(X1,X2)) -> U72(proper(X1),proper(X2))
       proper(isNat(X)) -> isNat(proper(X))
       proper(s(X)) -> s(proper(X))
       proper(length(X)) -> length(proper(X))
       proper(U81(X)) -> U81(proper(X))
       proper(nil()) -> ok(nil())
       proper(U91(X1,X2,X3,X4)) -> U91(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(U92(X1,X2,X3,X4)) -> U92(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(U93(X1,X2,X3,X4)) -> U93(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(take(X1,X2)) -> take(proper(X1),proper(X2))
       cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
       U11(ok(X)) -> ok(U11(X))
       U21(ok(X)) -> ok(U21(X))
       U31(ok(X)) -> ok(U31(X))
       U41(ok(X1),ok(X2)) -> ok(U41(X1,X2))
       U42(ok(X)) -> ok(U42(X))
       isNatIList(ok(X)) -> ok(isNatIList(X))
       U51(ok(X1),ok(X2)) -> ok(U51(X1,X2))
       U52(ok(X)) -> ok(U52(X))
       isNatList(ok(X)) -> ok(isNatList(X))
       U61(ok(X1),ok(X2)) -> ok(U61(X1,X2))
       U62(ok(X)) -> ok(U62(X))
       U71(ok(X1),ok(X2),ok(X3)) -> ok(U71(X1,X2,X3))
       U72(ok(X1),ok(X2)) -> ok(U72(X1,X2))
       isNat(ok(X)) -> ok(isNat(X))
       s(ok(X)) -> ok(s(X))
       length(ok(X)) -> ok(length(X))
       U81(ok(X)) -> ok(U81(X))
       U91(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U91(X1,X2,X3,X4))
       U92(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U92(X1,X2,X3,X4))
       U93(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U93(X1,X2,X3,X4))
       take(ok(X1),ok(X2)) -> ok(take(X1,X2))
       top(mark(X)) -> top(proper(X))
       top(ok(X)) -> top(active(X))
      Open
      
      DPs:
       U92#(ok(X1),ok(X2),ok(X3),ok(X4)) -> U92#(X1,X2,X3,X4)
       U92#(mark(X1),X2,X3,X4) -> U92#(X1,X2,X3,X4)
      TRS:
       active(zeros()) -> mark(cons(0(),zeros()))
       active(U11(tt())) -> mark(tt())
       active(U21(tt())) -> mark(tt())
       active(U31(tt())) -> mark(tt())
       active(U41(tt(),V2)) -> mark(U42(isNatIList(V2)))
       active(U42(tt())) -> mark(tt())
       active(U51(tt(),V2)) -> mark(U52(isNatList(V2)))
       active(U52(tt())) -> mark(tt())
       active(U61(tt(),V2)) -> mark(U62(isNatIList(V2)))
       active(U62(tt())) -> mark(tt())
       active(U71(tt(),L,N)) -> mark(U72(isNat(N),L))
       active(U72(tt(),L)) -> mark(s(length(L)))
       active(U81(tt())) -> mark(nil())
       active(U91(tt(),IL,M,N)) -> mark(U92(isNat(M),IL,M,N))
       active(U92(tt(),IL,M,N)) -> mark(U93(isNat(N),IL,M,N))
       active(U93(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
       active(isNat(0())) -> mark(tt())
       active(isNat(length(V1))) -> mark(U11(isNatList(V1)))
       active(isNat(s(V1))) -> mark(U21(isNat(V1)))
       active(isNatIList(V)) -> mark(U31(isNatList(V)))
       active(isNatIList(zeros())) -> mark(tt())
       active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2))
       active(isNatList(nil())) -> mark(tt())
       active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2))
       active(isNatList(take(V1,V2))) -> mark(U61(isNat(V1),V2))
       active(length(nil())) -> mark(0())
       active(length(cons(N,L))) -> mark(U71(isNatList(L),L,N))
       active(take(0(),IL)) -> mark(U81(isNatIList(IL)))
       active(take(s(M),cons(N,IL))) -> mark(U91(isNatIList(IL),IL,M,N))
       active(cons(X1,X2)) -> cons(active(X1),X2)
       active(U11(X)) -> U11(active(X))
       active(U21(X)) -> U21(active(X))
       active(U31(X)) -> U31(active(X))
       active(U41(X1,X2)) -> U41(active(X1),X2)
       active(U42(X)) -> U42(active(X))
       active(U51(X1,X2)) -> U51(active(X1),X2)
       active(U52(X)) -> U52(active(X))
       active(U61(X1,X2)) -> U61(active(X1),X2)
       active(U62(X)) -> U62(active(X))
       active(U71(X1,X2,X3)) -> U71(active(X1),X2,X3)
       active(U72(X1,X2)) -> U72(active(X1),X2)
       active(s(X)) -> s(active(X))
       active(length(X)) -> length(active(X))
       active(U81(X)) -> U81(active(X))
       active(U91(X1,X2,X3,X4)) -> U91(active(X1),X2,X3,X4)
       active(U92(X1,X2,X3,X4)) -> U92(active(X1),X2,X3,X4)
       active(U93(X1,X2,X3,X4)) -> U93(active(X1),X2,X3,X4)
       active(take(X1,X2)) -> take(active(X1),X2)
       active(take(X1,X2)) -> take(X1,active(X2))
       cons(mark(X1),X2) -> mark(cons(X1,X2))
       U11(mark(X)) -> mark(U11(X))
       U21(mark(X)) -> mark(U21(X))
       U31(mark(X)) -> mark(U31(X))
       U41(mark(X1),X2) -> mark(U41(X1,X2))
       U42(mark(X)) -> mark(U42(X))
       U51(mark(X1),X2) -> mark(U51(X1,X2))
       U52(mark(X)) -> mark(U52(X))
       U61(mark(X1),X2) -> mark(U61(X1,X2))
       U62(mark(X)) -> mark(U62(X))
       U71(mark(X1),X2,X3) -> mark(U71(X1,X2,X3))
       U72(mark(X1),X2) -> mark(U72(X1,X2))
       s(mark(X)) -> mark(s(X))
       length(mark(X)) -> mark(length(X))
       U81(mark(X)) -> mark(U81(X))
       U91(mark(X1),X2,X3,X4) -> mark(U91(X1,X2,X3,X4))
       U92(mark(X1),X2,X3,X4) -> mark(U92(X1,X2,X3,X4))
       U93(mark(X1),X2,X3,X4) -> mark(U93(X1,X2,X3,X4))
       take(mark(X1),X2) -> mark(take(X1,X2))
       take(X1,mark(X2)) -> mark(take(X1,X2))
       proper(zeros()) -> ok(zeros())
       proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
       proper(0()) -> ok(0())
       proper(U11(X)) -> U11(proper(X))
       proper(tt()) -> ok(tt())
       proper(U21(X)) -> U21(proper(X))
       proper(U31(X)) -> U31(proper(X))
       proper(U41(X1,X2)) -> U41(proper(X1),proper(X2))
       proper(U42(X)) -> U42(proper(X))
       proper(isNatIList(X)) -> isNatIList(proper(X))
       proper(U51(X1,X2)) -> U51(proper(X1),proper(X2))
       proper(U52(X)) -> U52(proper(X))
       proper(isNatList(X)) -> isNatList(proper(X))
       proper(U61(X1,X2)) -> U61(proper(X1),proper(X2))
       proper(U62(X)) -> U62(proper(X))
       proper(U71(X1,X2,X3)) -> U71(proper(X1),proper(X2),proper(X3))
       proper(U72(X1,X2)) -> U72(proper(X1),proper(X2))
       proper(isNat(X)) -> isNat(proper(X))
       proper(s(X)) -> s(proper(X))
       proper(length(X)) -> length(proper(X))
       proper(U81(X)) -> U81(proper(X))
       proper(nil()) -> ok(nil())
       proper(U91(X1,X2,X3,X4)) -> U91(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(U92(X1,X2,X3,X4)) -> U92(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(U93(X1,X2,X3,X4)) -> U93(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(take(X1,X2)) -> take(proper(X1),proper(X2))
       cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
       U11(ok(X)) -> ok(U11(X))
       U21(ok(X)) -> ok(U21(X))
       U31(ok(X)) -> ok(U31(X))
       U41(ok(X1),ok(X2)) -> ok(U41(X1,X2))
       U42(ok(X)) -> ok(U42(X))
       isNatIList(ok(X)) -> ok(isNatIList(X))
       U51(ok(X1),ok(X2)) -> ok(U51(X1,X2))
       U52(ok(X)) -> ok(U52(X))
       isNatList(ok(X)) -> ok(isNatList(X))
       U61(ok(X1),ok(X2)) -> ok(U61(X1,X2))
       U62(ok(X)) -> ok(U62(X))
       U71(ok(X1),ok(X2),ok(X3)) -> ok(U71(X1,X2,X3))
       U72(ok(X1),ok(X2)) -> ok(U72(X1,X2))
       isNat(ok(X)) -> ok(isNat(X))
       s(ok(X)) -> ok(s(X))
       length(ok(X)) -> ok(length(X))
       U81(ok(X)) -> ok(U81(X))
       U91(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U91(X1,X2,X3,X4))
       U92(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U92(X1,X2,X3,X4))
       U93(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U93(X1,X2,X3,X4))
       take(ok(X1),ok(X2)) -> ok(take(X1,X2))
       top(mark(X)) -> top(proper(X))
       top(ok(X)) -> top(active(X))
      Open
      
      DPs:
       U93#(ok(X1),ok(X2),ok(X3),ok(X4)) -> U93#(X1,X2,X3,X4)
       U93#(mark(X1),X2,X3,X4) -> U93#(X1,X2,X3,X4)
      TRS:
       active(zeros()) -> mark(cons(0(),zeros()))
       active(U11(tt())) -> mark(tt())
       active(U21(tt())) -> mark(tt())
       active(U31(tt())) -> mark(tt())
       active(U41(tt(),V2)) -> mark(U42(isNatIList(V2)))
       active(U42(tt())) -> mark(tt())
       active(U51(tt(),V2)) -> mark(U52(isNatList(V2)))
       active(U52(tt())) -> mark(tt())
       active(U61(tt(),V2)) -> mark(U62(isNatIList(V2)))
       active(U62(tt())) -> mark(tt())
       active(U71(tt(),L,N)) -> mark(U72(isNat(N),L))
       active(U72(tt(),L)) -> mark(s(length(L)))
       active(U81(tt())) -> mark(nil())
       active(U91(tt(),IL,M,N)) -> mark(U92(isNat(M),IL,M,N))
       active(U92(tt(),IL,M,N)) -> mark(U93(isNat(N),IL,M,N))
       active(U93(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
       active(isNat(0())) -> mark(tt())
       active(isNat(length(V1))) -> mark(U11(isNatList(V1)))
       active(isNat(s(V1))) -> mark(U21(isNat(V1)))
       active(isNatIList(V)) -> mark(U31(isNatList(V)))
       active(isNatIList(zeros())) -> mark(tt())
       active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2))
       active(isNatList(nil())) -> mark(tt())
       active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2))
       active(isNatList(take(V1,V2))) -> mark(U61(isNat(V1),V2))
       active(length(nil())) -> mark(0())
       active(length(cons(N,L))) -> mark(U71(isNatList(L),L,N))
       active(take(0(),IL)) -> mark(U81(isNatIList(IL)))
       active(take(s(M),cons(N,IL))) -> mark(U91(isNatIList(IL),IL,M,N))
       active(cons(X1,X2)) -> cons(active(X1),X2)
       active(U11(X)) -> U11(active(X))
       active(U21(X)) -> U21(active(X))
       active(U31(X)) -> U31(active(X))
       active(U41(X1,X2)) -> U41(active(X1),X2)
       active(U42(X)) -> U42(active(X))
       active(U51(X1,X2)) -> U51(active(X1),X2)
       active(U52(X)) -> U52(active(X))
       active(U61(X1,X2)) -> U61(active(X1),X2)
       active(U62(X)) -> U62(active(X))
       active(U71(X1,X2,X3)) -> U71(active(X1),X2,X3)
       active(U72(X1,X2)) -> U72(active(X1),X2)
       active(s(X)) -> s(active(X))
       active(length(X)) -> length(active(X))
       active(U81(X)) -> U81(active(X))
       active(U91(X1,X2,X3,X4)) -> U91(active(X1),X2,X3,X4)
       active(U92(X1,X2,X3,X4)) -> U92(active(X1),X2,X3,X4)
       active(U93(X1,X2,X3,X4)) -> U93(active(X1),X2,X3,X4)
       active(take(X1,X2)) -> take(active(X1),X2)
       active(take(X1,X2)) -> take(X1,active(X2))
       cons(mark(X1),X2) -> mark(cons(X1,X2))
       U11(mark(X)) -> mark(U11(X))
       U21(mark(X)) -> mark(U21(X))
       U31(mark(X)) -> mark(U31(X))
       U41(mark(X1),X2) -> mark(U41(X1,X2))
       U42(mark(X)) -> mark(U42(X))
       U51(mark(X1),X2) -> mark(U51(X1,X2))
       U52(mark(X)) -> mark(U52(X))
       U61(mark(X1),X2) -> mark(U61(X1,X2))
       U62(mark(X)) -> mark(U62(X))
       U71(mark(X1),X2,X3) -> mark(U71(X1,X2,X3))
       U72(mark(X1),X2) -> mark(U72(X1,X2))
       s(mark(X)) -> mark(s(X))
       length(mark(X)) -> mark(length(X))
       U81(mark(X)) -> mark(U81(X))
       U91(mark(X1),X2,X3,X4) -> mark(U91(X1,X2,X3,X4))
       U92(mark(X1),X2,X3,X4) -> mark(U92(X1,X2,X3,X4))
       U93(mark(X1),X2,X3,X4) -> mark(U93(X1,X2,X3,X4))
       take(mark(X1),X2) -> mark(take(X1,X2))
       take(X1,mark(X2)) -> mark(take(X1,X2))
       proper(zeros()) -> ok(zeros())
       proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
       proper(0()) -> ok(0())
       proper(U11(X)) -> U11(proper(X))
       proper(tt()) -> ok(tt())
       proper(U21(X)) -> U21(proper(X))
       proper(U31(X)) -> U31(proper(X))
       proper(U41(X1,X2)) -> U41(proper(X1),proper(X2))
       proper(U42(X)) -> U42(proper(X))
       proper(isNatIList(X)) -> isNatIList(proper(X))
       proper(U51(X1,X2)) -> U51(proper(X1),proper(X2))
       proper(U52(X)) -> U52(proper(X))
       proper(isNatList(X)) -> isNatList(proper(X))
       proper(U61(X1,X2)) -> U61(proper(X1),proper(X2))
       proper(U62(X)) -> U62(proper(X))
       proper(U71(X1,X2,X3)) -> U71(proper(X1),proper(X2),proper(X3))
       proper(U72(X1,X2)) -> U72(proper(X1),proper(X2))
       proper(isNat(X)) -> isNat(proper(X))
       proper(s(X)) -> s(proper(X))
       proper(length(X)) -> length(proper(X))
       proper(U81(X)) -> U81(proper(X))
       proper(nil()) -> ok(nil())
       proper(U91(X1,X2,X3,X4)) -> U91(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(U92(X1,X2,X3,X4)) -> U92(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(U93(X1,X2,X3,X4)) -> U93(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(take(X1,X2)) -> take(proper(X1),proper(X2))
       cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
       U11(ok(X)) -> ok(U11(X))
       U21(ok(X)) -> ok(U21(X))
       U31(ok(X)) -> ok(U31(X))
       U41(ok(X1),ok(X2)) -> ok(U41(X1,X2))
       U42(ok(X)) -> ok(U42(X))
       isNatIList(ok(X)) -> ok(isNatIList(X))
       U51(ok(X1),ok(X2)) -> ok(U51(X1,X2))
       U52(ok(X)) -> ok(U52(X))
       isNatList(ok(X)) -> ok(isNatList(X))
       U61(ok(X1),ok(X2)) -> ok(U61(X1,X2))
       U62(ok(X)) -> ok(U62(X))
       U71(ok(X1),ok(X2),ok(X3)) -> ok(U71(X1,X2,X3))
       U72(ok(X1),ok(X2)) -> ok(U72(X1,X2))
       isNat(ok(X)) -> ok(isNat(X))
       s(ok(X)) -> ok(s(X))
       length(ok(X)) -> ok(length(X))
       U81(ok(X)) -> ok(U81(X))
       U91(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U91(X1,X2,X3,X4))
       U92(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U92(X1,X2,X3,X4))
       U93(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U93(X1,X2,X3,X4))
       take(ok(X1),ok(X2)) -> ok(take(X1,X2))
       top(mark(X)) -> top(proper(X))
       top(ok(X)) -> top(active(X))
      Open
      
      DPs:
       take#(ok(X1),ok(X2)) -> take#(X1,X2)
       take#(X1,mark(X2)) -> take#(X1,X2)
       take#(mark(X1),X2) -> take#(X1,X2)
      TRS:
       active(zeros()) -> mark(cons(0(),zeros()))
       active(U11(tt())) -> mark(tt())
       active(U21(tt())) -> mark(tt())
       active(U31(tt())) -> mark(tt())
       active(U41(tt(),V2)) -> mark(U42(isNatIList(V2)))
       active(U42(tt())) -> mark(tt())
       active(U51(tt(),V2)) -> mark(U52(isNatList(V2)))
       active(U52(tt())) -> mark(tt())
       active(U61(tt(),V2)) -> mark(U62(isNatIList(V2)))
       active(U62(tt())) -> mark(tt())
       active(U71(tt(),L,N)) -> mark(U72(isNat(N),L))
       active(U72(tt(),L)) -> mark(s(length(L)))
       active(U81(tt())) -> mark(nil())
       active(U91(tt(),IL,M,N)) -> mark(U92(isNat(M),IL,M,N))
       active(U92(tt(),IL,M,N)) -> mark(U93(isNat(N),IL,M,N))
       active(U93(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
       active(isNat(0())) -> mark(tt())
       active(isNat(length(V1))) -> mark(U11(isNatList(V1)))
       active(isNat(s(V1))) -> mark(U21(isNat(V1)))
       active(isNatIList(V)) -> mark(U31(isNatList(V)))
       active(isNatIList(zeros())) -> mark(tt())
       active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2))
       active(isNatList(nil())) -> mark(tt())
       active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2))
       active(isNatList(take(V1,V2))) -> mark(U61(isNat(V1),V2))
       active(length(nil())) -> mark(0())
       active(length(cons(N,L))) -> mark(U71(isNatList(L),L,N))
       active(take(0(),IL)) -> mark(U81(isNatIList(IL)))
       active(take(s(M),cons(N,IL))) -> mark(U91(isNatIList(IL),IL,M,N))
       active(cons(X1,X2)) -> cons(active(X1),X2)
       active(U11(X)) -> U11(active(X))
       active(U21(X)) -> U21(active(X))
       active(U31(X)) -> U31(active(X))
       active(U41(X1,X2)) -> U41(active(X1),X2)
       active(U42(X)) -> U42(active(X))
       active(U51(X1,X2)) -> U51(active(X1),X2)
       active(U52(X)) -> U52(active(X))
       active(U61(X1,X2)) -> U61(active(X1),X2)
       active(U62(X)) -> U62(active(X))
       active(U71(X1,X2,X3)) -> U71(active(X1),X2,X3)
       active(U72(X1,X2)) -> U72(active(X1),X2)
       active(s(X)) -> s(active(X))
       active(length(X)) -> length(active(X))
       active(U81(X)) -> U81(active(X))
       active(U91(X1,X2,X3,X4)) -> U91(active(X1),X2,X3,X4)
       active(U92(X1,X2,X3,X4)) -> U92(active(X1),X2,X3,X4)
       active(U93(X1,X2,X3,X4)) -> U93(active(X1),X2,X3,X4)
       active(take(X1,X2)) -> take(active(X1),X2)
       active(take(X1,X2)) -> take(X1,active(X2))
       cons(mark(X1),X2) -> mark(cons(X1,X2))
       U11(mark(X)) -> mark(U11(X))
       U21(mark(X)) -> mark(U21(X))
       U31(mark(X)) -> mark(U31(X))
       U41(mark(X1),X2) -> mark(U41(X1,X2))
       U42(mark(X)) -> mark(U42(X))
       U51(mark(X1),X2) -> mark(U51(X1,X2))
       U52(mark(X)) -> mark(U52(X))
       U61(mark(X1),X2) -> mark(U61(X1,X2))
       U62(mark(X)) -> mark(U62(X))
       U71(mark(X1),X2,X3) -> mark(U71(X1,X2,X3))
       U72(mark(X1),X2) -> mark(U72(X1,X2))
       s(mark(X)) -> mark(s(X))
       length(mark(X)) -> mark(length(X))
       U81(mark(X)) -> mark(U81(X))
       U91(mark(X1),X2,X3,X4) -> mark(U91(X1,X2,X3,X4))
       U92(mark(X1),X2,X3,X4) -> mark(U92(X1,X2,X3,X4))
       U93(mark(X1),X2,X3,X4) -> mark(U93(X1,X2,X3,X4))
       take(mark(X1),X2) -> mark(take(X1,X2))
       take(X1,mark(X2)) -> mark(take(X1,X2))
       proper(zeros()) -> ok(zeros())
       proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
       proper(0()) -> ok(0())
       proper(U11(X)) -> U11(proper(X))
       proper(tt()) -> ok(tt())
       proper(U21(X)) -> U21(proper(X))
       proper(U31(X)) -> U31(proper(X))
       proper(U41(X1,X2)) -> U41(proper(X1),proper(X2))
       proper(U42(X)) -> U42(proper(X))
       proper(isNatIList(X)) -> isNatIList(proper(X))
       proper(U51(X1,X2)) -> U51(proper(X1),proper(X2))
       proper(U52(X)) -> U52(proper(X))
       proper(isNatList(X)) -> isNatList(proper(X))
       proper(U61(X1,X2)) -> U61(proper(X1),proper(X2))
       proper(U62(X)) -> U62(proper(X))
       proper(U71(X1,X2,X3)) -> U71(proper(X1),proper(X2),proper(X3))
       proper(U72(X1,X2)) -> U72(proper(X1),proper(X2))
       proper(isNat(X)) -> isNat(proper(X))
       proper(s(X)) -> s(proper(X))
       proper(length(X)) -> length(proper(X))
       proper(U81(X)) -> U81(proper(X))
       proper(nil()) -> ok(nil())
       proper(U91(X1,X2,X3,X4)) -> U91(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(U92(X1,X2,X3,X4)) -> U92(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(U93(X1,X2,X3,X4)) -> U93(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(take(X1,X2)) -> take(proper(X1),proper(X2))
       cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
       U11(ok(X)) -> ok(U11(X))
       U21(ok(X)) -> ok(U21(X))
       U31(ok(X)) -> ok(U31(X))
       U41(ok(X1),ok(X2)) -> ok(U41(X1,X2))
       U42(ok(X)) -> ok(U42(X))
       isNatIList(ok(X)) -> ok(isNatIList(X))
       U51(ok(X1),ok(X2)) -> ok(U51(X1,X2))
       U52(ok(X)) -> ok(U52(X))
       isNatList(ok(X)) -> ok(isNatList(X))
       U61(ok(X1),ok(X2)) -> ok(U61(X1,X2))
       U62(ok(X)) -> ok(U62(X))
       U71(ok(X1),ok(X2),ok(X3)) -> ok(U71(X1,X2,X3))
       U72(ok(X1),ok(X2)) -> ok(U72(X1,X2))
       isNat(ok(X)) -> ok(isNat(X))
       s(ok(X)) -> ok(s(X))
       length(ok(X)) -> ok(length(X))
       U81(ok(X)) -> ok(U81(X))
       U91(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U91(X1,X2,X3,X4))
       U92(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U92(X1,X2,X3,X4))
       U93(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U93(X1,X2,X3,X4))
       take(ok(X1),ok(X2)) -> ok(take(X1,X2))
       top(mark(X)) -> top(proper(X))
       top(ok(X)) -> top(active(X))
      Open