MAYBE Problem: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),L)) -> mark(s(length(L))) active(U21(tt())) -> mark(nil()) active(U31(tt(),IL,M,N)) -> mark(cons(N,take(M,IL))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(length(V1))) -> mark(isNatList(V1)) active(isNat(s(V1))) -> mark(isNat(V1)) active(isNatIList(V)) -> mark(isNatList(V)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(and(isNat(V1),isNatIList(V2))) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(and(isNat(V1),isNatList(V2))) active(isNatList(take(V1,V2))) -> mark(and(isNat(V1),isNatIList(V2))) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U11(and(isNatList(L),isNat(N)),L)) active(take(0(),IL)) -> mark(U21(isNatIList(IL))) active(take(s(M),cons(N,IL))) -> mark(U31(and(isNatIList(IL),and(isNat(M),isNat(N))),IL,M,N)) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(nil()) -> active(nil()) mark(U31(X1,X2,X3,X4)) -> active(U31(mark(X1),X2,X3,X4)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(isNatIList(X)) -> active(isNatIList(X)) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) U21(mark(X)) -> U21(X) U21(active(X)) -> U21(X) U31(mark(X1),X2,X3,X4) -> U31(X1,X2,X3,X4) U31(X1,mark(X2),X3,X4) -> U31(X1,X2,X3,X4) U31(X1,X2,mark(X3),X4) -> U31(X1,X2,X3,X4) U31(X1,X2,X3,mark(X4)) -> U31(X1,X2,X3,X4) U31(active(X1),X2,X3,X4) -> U31(X1,X2,X3,X4) U31(X1,active(X2),X3,X4) -> U31(X1,X2,X3,X4) U31(X1,X2,active(X3),X4) -> U31(X1,X2,X3,X4) U31(X1,X2,X3,active(X4)) -> U31(X1,X2,X3,X4) take(mark(X1),X2) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) take(active(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) Proof: DP Processor: DPs: active#(zeros()) -> cons#(0(),zeros()) active#(zeros()) -> mark#(cons(0(),zeros())) active#(U11(tt(),L)) -> length#(L) active#(U11(tt(),L)) -> s#(length(L)) active#(U11(tt(),L)) -> mark#(s(length(L))) active#(U21(tt())) -> mark#(nil()) active#(U31(tt(),IL,M,N)) -> take#(M,IL) active#(U31(tt(),IL,M,N)) -> cons#(N,take(M,IL)) active#(U31(tt(),IL,M,N)) -> mark#(cons(N,take(M,IL))) active#(and(tt(),X)) -> mark#(X) active#(isNat(0())) -> mark#(tt()) active#(isNat(length(V1))) -> isNatList#(V1) active#(isNat(length(V1))) -> mark#(isNatList(V1)) active#(isNat(s(V1))) -> isNat#(V1) active#(isNat(s(V1))) -> mark#(isNat(V1)) active#(isNatIList(V)) -> isNatList#(V) active#(isNatIList(V)) -> mark#(isNatList(V)) active#(isNatIList(zeros())) -> mark#(tt()) active#(isNatIList(cons(V1,V2))) -> isNatIList#(V2) active#(isNatIList(cons(V1,V2))) -> isNat#(V1) active#(isNatIList(cons(V1,V2))) -> and#(isNat(V1),isNatIList(V2)) active#(isNatIList(cons(V1,V2))) -> mark#(and(isNat(V1),isNatIList(V2))) active#(isNatList(nil())) -> mark#(tt()) active#(isNatList(cons(V1,V2))) -> isNatList#(V2) active#(isNatList(cons(V1,V2))) -> isNat#(V1) active#(isNatList(cons(V1,V2))) -> and#(isNat(V1),isNatList(V2)) active#(isNatList(cons(V1,V2))) -> mark#(and(isNat(V1),isNatList(V2))) active#(isNatList(take(V1,V2))) -> isNatIList#(V2) active#(isNatList(take(V1,V2))) -> isNat#(V1) active#(isNatList(take(V1,V2))) -> and#(isNat(V1),isNatIList(V2)) active#(isNatList(take(V1,V2))) -> mark#(and(isNat(V1),isNatIList(V2))) active#(length(nil())) -> mark#(0()) active#(length(cons(N,L))) -> isNat#(N) active#(length(cons(N,L))) -> isNatList#(L) active#(length(cons(N,L))) -> and#(isNatList(L),isNat(N)) active#(length(cons(N,L))) -> U11#(and(isNatList(L),isNat(N)),L) active#(length(cons(N,L))) -> mark#(U11(and(isNatList(L),isNat(N)),L)) active#(take(0(),IL)) -> isNatIList#(IL) active#(take(0(),IL)) -> U21#(isNatIList(IL)) active#(take(0(),IL)) -> mark#(U21(isNatIList(IL))) active#(take(s(M),cons(N,IL))) -> isNat#(N) active#(take(s(M),cons(N,IL))) -> isNat#(M) active#(take(s(M),cons(N,IL))) -> and#(isNat(M),isNat(N)) active#(take(s(M),cons(N,IL))) -> isNatIList#(IL) active#(take(s(M),cons(N,IL))) -> and#(isNatIList(IL),and(isNat(M),isNat(N))) active#(take(s(M),cons(N,IL))) -> U31#(and(isNatIList(IL),and(isNat(M),isNat(N))),IL,M,N) active#(take(s(M),cons(N,IL))) -> mark#(U31(and(isNatIList(IL),and(isNat(M),isNat(N))),IL,M,N)) mark#(zeros()) -> active#(zeros()) mark#(cons(X1,X2)) -> mark#(X1) mark#(cons(X1,X2)) -> cons#(mark(X1),X2) mark#(cons(X1,X2)) -> active#(cons(mark(X1),X2)) mark#(0()) -> active#(0()) mark#(U11(X1,X2)) -> mark#(X1) mark#(U11(X1,X2)) -> U11#(mark(X1),X2) mark#(U11(X1,X2)) -> active#(U11(mark(X1),X2)) mark#(tt()) -> active#(tt()) mark#(s(X)) -> mark#(X) mark#(s(X)) -> s#(mark(X)) mark#(s(X)) -> active#(s(mark(X))) mark#(length(X)) -> mark#(X) mark#(length(X)) -> length#(mark(X)) mark#(length(X)) -> active#(length(mark(X))) mark#(U21(X)) -> mark#(X) mark#(U21(X)) -> U21#(mark(X)) mark#(U21(X)) -> active#(U21(mark(X))) mark#(nil()) -> active#(nil()) mark#(U31(X1,X2,X3,X4)) -> mark#(X1) mark#(U31(X1,X2,X3,X4)) -> U31#(mark(X1),X2,X3,X4) mark#(U31(X1,X2,X3,X4)) -> active#(U31(mark(X1),X2,X3,X4)) mark#(take(X1,X2)) -> mark#(X2) mark#(take(X1,X2)) -> mark#(X1) mark#(take(X1,X2)) -> take#(mark(X1),mark(X2)) mark#(take(X1,X2)) -> active#(take(mark(X1),mark(X2))) mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> and#(mark(X1),X2) mark#(and(X1,X2)) -> active#(and(mark(X1),X2)) mark#(isNat(X)) -> active#(isNat(X)) mark#(isNatList(X)) -> active#(isNatList(X)) mark#(isNatIList(X)) -> active#(isNatIList(X)) cons#(mark(X1),X2) -> cons#(X1,X2) cons#(X1,mark(X2)) -> cons#(X1,X2) cons#(active(X1),X2) -> cons#(X1,X2) cons#(X1,active(X2)) -> cons#(X1,X2) U11#(mark(X1),X2) -> U11#(X1,X2) U11#(X1,mark(X2)) -> U11#(X1,X2) U11#(active(X1),X2) -> U11#(X1,X2) U11#(X1,active(X2)) -> U11#(X1,X2) s#(mark(X)) -> s#(X) s#(active(X)) -> s#(X) length#(mark(X)) -> length#(X) length#(active(X)) -> length#(X) U21#(mark(X)) -> U21#(X) U21#(active(X)) -> U21#(X) U31#(mark(X1),X2,X3,X4) -> U31#(X1,X2,X3,X4) U31#(X1,mark(X2),X3,X4) -> U31#(X1,X2,X3,X4) U31#(X1,X2,mark(X3),X4) -> U31#(X1,X2,X3,X4) U31#(X1,X2,X3,mark(X4)) -> U31#(X1,X2,X3,X4) U31#(active(X1),X2,X3,X4) -> U31#(X1,X2,X3,X4) U31#(X1,active(X2),X3,X4) -> U31#(X1,X2,X3,X4) U31#(X1,X2,active(X3),X4) -> U31#(X1,X2,X3,X4) U31#(X1,X2,X3,active(X4)) -> U31#(X1,X2,X3,X4) take#(mark(X1),X2) -> take#(X1,X2) take#(X1,mark(X2)) -> take#(X1,X2) take#(active(X1),X2) -> take#(X1,X2) take#(X1,active(X2)) -> take#(X1,X2) and#(mark(X1),X2) -> and#(X1,X2) and#(X1,mark(X2)) -> and#(X1,X2) and#(active(X1),X2) -> and#(X1,X2) and#(X1,active(X2)) -> and#(X1,X2) isNat#(mark(X)) -> isNat#(X) isNat#(active(X)) -> isNat#(X) isNatList#(mark(X)) -> isNatList#(X) isNatList#(active(X)) -> isNatList#(X) isNatIList#(mark(X)) -> isNatIList#(X) isNatIList#(active(X)) -> isNatIList#(X) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),L)) -> mark(s(length(L))) active(U21(tt())) -> mark(nil()) active(U31(tt(),IL,M,N)) -> mark(cons(N,take(M,IL))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(length(V1))) -> mark(isNatList(V1)) active(isNat(s(V1))) -> mark(isNat(V1)) active(isNatIList(V)) -> mark(isNatList(V)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(and(isNat(V1),isNatIList(V2))) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(and(isNat(V1),isNatList(V2))) active(isNatList(take(V1,V2))) -> mark(and(isNat(V1),isNatIList(V2))) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U11(and(isNatList(L),isNat(N)),L)) active(take(0(),IL)) -> mark(U21(isNatIList(IL))) active(take(s(M),cons(N,IL))) -> mark(U31(and(isNatIList(IL),and(isNat(M),isNat(N))),IL,M,N)) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(nil()) -> active(nil()) mark(U31(X1,X2,X3,X4)) -> active(U31(mark(X1),X2,X3,X4)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(isNatIList(X)) -> active(isNatIList(X)) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) U21(mark(X)) -> U21(X) U21(active(X)) -> U21(X) U31(mark(X1),X2,X3,X4) -> U31(X1,X2,X3,X4) U31(X1,mark(X2),X3,X4) -> U31(X1,X2,X3,X4) U31(X1,X2,mark(X3),X4) -> U31(X1,X2,X3,X4) U31(X1,X2,X3,mark(X4)) -> U31(X1,X2,X3,X4) U31(active(X1),X2,X3,X4) -> U31(X1,X2,X3,X4) U31(X1,active(X2),X3,X4) -> U31(X1,X2,X3,X4) U31(X1,X2,active(X3),X4) -> U31(X1,X2,X3,X4) U31(X1,X2,X3,active(X4)) -> U31(X1,X2,X3,X4) take(mark(X1),X2) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) take(active(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) EDG Processor: DPs: active#(zeros()) -> cons#(0(),zeros()) active#(zeros()) -> mark#(cons(0(),zeros())) active#(U11(tt(),L)) -> length#(L) active#(U11(tt(),L)) -> s#(length(L)) active#(U11(tt(),L)) -> mark#(s(length(L))) active#(U21(tt())) -> mark#(nil()) active#(U31(tt(),IL,M,N)) -> take#(M,IL) active#(U31(tt(),IL,M,N)) -> cons#(N,take(M,IL)) active#(U31(tt(),IL,M,N)) -> mark#(cons(N,take(M,IL))) active#(and(tt(),X)) -> mark#(X) active#(isNat(0())) -> mark#(tt()) active#(isNat(length(V1))) -> isNatList#(V1) active#(isNat(length(V1))) -> mark#(isNatList(V1)) active#(isNat(s(V1))) -> isNat#(V1) active#(isNat(s(V1))) -> mark#(isNat(V1)) active#(isNatIList(V)) -> isNatList#(V) active#(isNatIList(V)) -> mark#(isNatList(V)) active#(isNatIList(zeros())) -> mark#(tt()) active#(isNatIList(cons(V1,V2))) -> isNatIList#(V2) active#(isNatIList(cons(V1,V2))) -> isNat#(V1) active#(isNatIList(cons(V1,V2))) -> and#(isNat(V1),isNatIList(V2)) active#(isNatIList(cons(V1,V2))) -> mark#(and(isNat(V1),isNatIList(V2))) active#(isNatList(nil())) -> mark#(tt()) active#(isNatList(cons(V1,V2))) -> isNatList#(V2) active#(isNatList(cons(V1,V2))) -> isNat#(V1) active#(isNatList(cons(V1,V2))) -> and#(isNat(V1),isNatList(V2)) active#(isNatList(cons(V1,V2))) -> mark#(and(isNat(V1),isNatList(V2))) active#(isNatList(take(V1,V2))) -> isNatIList#(V2) active#(isNatList(take(V1,V2))) -> isNat#(V1) active#(isNatList(take(V1,V2))) -> and#(isNat(V1),isNatIList(V2)) active#(isNatList(take(V1,V2))) -> mark#(and(isNat(V1),isNatIList(V2))) active#(length(nil())) -> mark#(0()) active#(length(cons(N,L))) -> isNat#(N) active#(length(cons(N,L))) -> isNatList#(L) active#(length(cons(N,L))) -> and#(isNatList(L),isNat(N)) active#(length(cons(N,L))) -> U11#(and(isNatList(L),isNat(N)),L) active#(length(cons(N,L))) -> mark#(U11(and(isNatList(L),isNat(N)),L)) active#(take(0(),IL)) -> isNatIList#(IL) active#(take(0(),IL)) -> U21#(isNatIList(IL)) active#(take(0(),IL)) -> mark#(U21(isNatIList(IL))) active#(take(s(M),cons(N,IL))) -> isNat#(N) active#(take(s(M),cons(N,IL))) -> isNat#(M) active#(take(s(M),cons(N,IL))) -> and#(isNat(M),isNat(N)) active#(take(s(M),cons(N,IL))) -> isNatIList#(IL) active#(take(s(M),cons(N,IL))) -> and#(isNatIList(IL),and(isNat(M),isNat(N))) active#(take(s(M),cons(N,IL))) -> U31#(and(isNatIList(IL),and(isNat(M),isNat(N))),IL,M,N) active#(take(s(M),cons(N,IL))) -> mark#(U31(and(isNatIList(IL),and(isNat(M),isNat(N))),IL,M,N)) mark#(zeros()) -> active#(zeros()) mark#(cons(X1,X2)) -> mark#(X1) mark#(cons(X1,X2)) -> cons#(mark(X1),X2) mark#(cons(X1,X2)) -> active#(cons(mark(X1),X2)) mark#(0()) -> active#(0()) mark#(U11(X1,X2)) -> mark#(X1) mark#(U11(X1,X2)) -> U11#(mark(X1),X2) mark#(U11(X1,X2)) -> active#(U11(mark(X1),X2)) mark#(tt()) -> active#(tt()) mark#(s(X)) -> mark#(X) mark#(s(X)) -> s#(mark(X)) mark#(s(X)) -> active#(s(mark(X))) mark#(length(X)) -> mark#(X) mark#(length(X)) -> length#(mark(X)) mark#(length(X)) -> active#(length(mark(X))) mark#(U21(X)) -> mark#(X) mark#(U21(X)) -> U21#(mark(X)) mark#(U21(X)) -> active#(U21(mark(X))) mark#(nil()) -> active#(nil()) mark#(U31(X1,X2,X3,X4)) -> mark#(X1) mark#(U31(X1,X2,X3,X4)) -> U31#(mark(X1),X2,X3,X4) mark#(U31(X1,X2,X3,X4)) -> active#(U31(mark(X1),X2,X3,X4)) mark#(take(X1,X2)) -> mark#(X2) mark#(take(X1,X2)) -> mark#(X1) mark#(take(X1,X2)) -> take#(mark(X1),mark(X2)) mark#(take(X1,X2)) -> active#(take(mark(X1),mark(X2))) mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> and#(mark(X1),X2) mark#(and(X1,X2)) -> active#(and(mark(X1),X2)) mark#(isNat(X)) -> active#(isNat(X)) mark#(isNatList(X)) -> active#(isNatList(X)) mark#(isNatIList(X)) -> active#(isNatIList(X)) cons#(mark(X1),X2) -> cons#(X1,X2) cons#(X1,mark(X2)) -> cons#(X1,X2) cons#(active(X1),X2) -> cons#(X1,X2) cons#(X1,active(X2)) -> cons#(X1,X2) U11#(mark(X1),X2) -> U11#(X1,X2) U11#(X1,mark(X2)) -> U11#(X1,X2) U11#(active(X1),X2) -> U11#(X1,X2) U11#(X1,active(X2)) -> U11#(X1,X2) s#(mark(X)) -> s#(X) s#(active(X)) -> s#(X) length#(mark(X)) -> length#(X) length#(active(X)) -> length#(X) U21#(mark(X)) -> U21#(X) U21#(active(X)) -> U21#(X) U31#(mark(X1),X2,X3,X4) -> U31#(X1,X2,X3,X4) U31#(X1,mark(X2),X3,X4) -> U31#(X1,X2,X3,X4) U31#(X1,X2,mark(X3),X4) -> U31#(X1,X2,X3,X4) U31#(X1,X2,X3,mark(X4)) -> U31#(X1,X2,X3,X4) U31#(active(X1),X2,X3,X4) -> U31#(X1,X2,X3,X4) U31#(X1,active(X2),X3,X4) -> U31#(X1,X2,X3,X4) U31#(X1,X2,active(X3),X4) -> U31#(X1,X2,X3,X4) U31#(X1,X2,X3,active(X4)) -> U31#(X1,X2,X3,X4) take#(mark(X1),X2) -> take#(X1,X2) take#(X1,mark(X2)) -> take#(X1,X2) take#(active(X1),X2) -> take#(X1,X2) take#(X1,active(X2)) -> take#(X1,X2) and#(mark(X1),X2) -> and#(X1,X2) and#(X1,mark(X2)) -> and#(X1,X2) and#(active(X1),X2) -> and#(X1,X2) and#(X1,active(X2)) -> and#(X1,X2) isNat#(mark(X)) -> isNat#(X) isNat#(active(X)) -> isNat#(X) isNatList#(mark(X)) -> isNatList#(X) isNatList#(active(X)) -> isNatList#(X) isNatIList#(mark(X)) -> isNatIList#(X) isNatIList#(active(X)) -> isNatIList#(X) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),L)) -> mark(s(length(L))) active(U21(tt())) -> mark(nil()) active(U31(tt(),IL,M,N)) -> mark(cons(N,take(M,IL))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(length(V1))) -> mark(isNatList(V1)) active(isNat(s(V1))) -> mark(isNat(V1)) active(isNatIList(V)) -> mark(isNatList(V)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(and(isNat(V1),isNatIList(V2))) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(and(isNat(V1),isNatList(V2))) active(isNatList(take(V1,V2))) -> mark(and(isNat(V1),isNatIList(V2))) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U11(and(isNatList(L),isNat(N)),L)) active(take(0(),IL)) -> mark(U21(isNatIList(IL))) active(take(s(M),cons(N,IL))) -> mark(U31(and(isNatIList(IL),and(isNat(M),isNat(N))),IL,M,N)) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(nil()) -> active(nil()) mark(U31(X1,X2,X3,X4)) -> active(U31(mark(X1),X2,X3,X4)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(isNatIList(X)) -> active(isNatIList(X)) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) U21(mark(X)) -> U21(X) U21(active(X)) -> U21(X) U31(mark(X1),X2,X3,X4) -> U31(X1,X2,X3,X4) U31(X1,mark(X2),X3,X4) -> U31(X1,X2,X3,X4) U31(X1,X2,mark(X3),X4) -> U31(X1,X2,X3,X4) U31(X1,X2,X3,mark(X4)) -> U31(X1,X2,X3,X4) U31(active(X1),X2,X3,X4) -> U31(X1,X2,X3,X4) U31(X1,active(X2),X3,X4) -> U31(X1,X2,X3,X4) U31(X1,X2,active(X3),X4) -> U31(X1,X2,X3,X4) U31(X1,X2,X3,active(X4)) -> U31(X1,X2,X3,X4) take(mark(X1),X2) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) take(active(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) graph: ... SCC Processor: #sccs: 12 #rules: 70 #arcs: 1223/13225 DPs: mark#(isNatIList(X)) -> active#(isNatIList(X)) active#(isNatIList(cons(V1,V2))) -> mark#(and(isNat(V1),isNatIList(V2))) mark#(isNatList(X)) -> active#(isNatList(X)) active#(isNatList(take(V1,V2))) -> mark#(and(isNat(V1),isNatIList(V2))) mark#(isNat(X)) -> active#(isNat(X)) active#(isNat(s(V1))) -> mark#(isNat(V1)) mark#(and(X1,X2)) -> active#(and(mark(X1),X2)) active#(take(s(M),cons(N,IL))) -> mark#(U31(and(isNatIList(IL),and(isNat(M),isNat(N))),IL,M,N)) mark#(and(X1,X2)) -> mark#(X1) mark#(take(X1,X2)) -> active#(take(mark(X1),mark(X2))) active#(take(0(),IL)) -> mark#(U21(isNatIList(IL))) mark#(take(X1,X2)) -> mark#(X1) mark#(take(X1,X2)) -> mark#(X2) mark#(U31(X1,X2,X3,X4)) -> active#(U31(mark(X1),X2,X3,X4)) active#(length(cons(N,L))) -> mark#(U11(and(isNatList(L),isNat(N)),L)) mark#(U31(X1,X2,X3,X4)) -> mark#(X1) mark#(U21(X)) -> active#(U21(mark(X))) active#(isNatList(cons(V1,V2))) -> mark#(and(isNat(V1),isNatList(V2))) mark#(U21(X)) -> mark#(X) mark#(length(X)) -> active#(length(mark(X))) active#(isNatIList(V)) -> mark#(isNatList(V)) mark#(length(X)) -> mark#(X) mark#(s(X)) -> active#(s(mark(X))) active#(isNat(length(V1))) -> mark#(isNatList(V1)) mark#(s(X)) -> mark#(X) mark#(U11(X1,X2)) -> active#(U11(mark(X1),X2)) active#(and(tt(),X)) -> mark#(X) mark#(U11(X1,X2)) -> mark#(X1) mark#(cons(X1,X2)) -> active#(cons(mark(X1),X2)) active#(U31(tt(),IL,M,N)) -> mark#(cons(N,take(M,IL))) mark#(cons(X1,X2)) -> mark#(X1) mark#(zeros()) -> active#(zeros()) active#(zeros()) -> mark#(cons(0(),zeros())) active#(U11(tt(),L)) -> mark#(s(length(L))) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),L)) -> mark(s(length(L))) active(U21(tt())) -> mark(nil()) active(U31(tt(),IL,M,N)) -> mark(cons(N,take(M,IL))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(length(V1))) -> mark(isNatList(V1)) active(isNat(s(V1))) -> mark(isNat(V1)) active(isNatIList(V)) -> mark(isNatList(V)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(and(isNat(V1),isNatIList(V2))) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(and(isNat(V1),isNatList(V2))) active(isNatList(take(V1,V2))) -> mark(and(isNat(V1),isNatIList(V2))) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U11(and(isNatList(L),isNat(N)),L)) active(take(0(),IL)) -> mark(U21(isNatIList(IL))) active(take(s(M),cons(N,IL))) -> mark(U31(and(isNatIList(IL),and(isNat(M),isNat(N))),IL,M,N)) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(nil()) -> active(nil()) mark(U31(X1,X2,X3,X4)) -> active(U31(mark(X1),X2,X3,X4)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(isNatIList(X)) -> active(isNatIList(X)) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) U21(mark(X)) -> U21(X) U21(active(X)) -> U21(X) U31(mark(X1),X2,X3,X4) -> U31(X1,X2,X3,X4) U31(X1,mark(X2),X3,X4) -> U31(X1,X2,X3,X4) U31(X1,X2,mark(X3),X4) -> U31(X1,X2,X3,X4) U31(X1,X2,X3,mark(X4)) -> U31(X1,X2,X3,X4) U31(active(X1),X2,X3,X4) -> U31(X1,X2,X3,X4) U31(X1,active(X2),X3,X4) -> U31(X1,X2,X3,X4) U31(X1,X2,active(X3),X4) -> U31(X1,X2,X3,X4) U31(X1,X2,X3,active(X4)) -> U31(X1,X2,X3,X4) take(mark(X1),X2) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) take(active(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) Open DPs: cons#(active(X1),X2) -> cons#(X1,X2) cons#(X1,active(X2)) -> cons#(X1,X2) cons#(X1,mark(X2)) -> cons#(X1,X2) cons#(mark(X1),X2) -> cons#(X1,X2) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),L)) -> mark(s(length(L))) active(U21(tt())) -> mark(nil()) active(U31(tt(),IL,M,N)) -> mark(cons(N,take(M,IL))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(length(V1))) -> mark(isNatList(V1)) active(isNat(s(V1))) -> mark(isNat(V1)) active(isNatIList(V)) -> mark(isNatList(V)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(and(isNat(V1),isNatIList(V2))) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(and(isNat(V1),isNatList(V2))) active(isNatList(take(V1,V2))) -> mark(and(isNat(V1),isNatIList(V2))) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U11(and(isNatList(L),isNat(N)),L)) active(take(0(),IL)) -> mark(U21(isNatIList(IL))) active(take(s(M),cons(N,IL))) -> mark(U31(and(isNatIList(IL),and(isNat(M),isNat(N))),IL,M,N)) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(nil()) -> active(nil()) mark(U31(X1,X2,X3,X4)) -> active(U31(mark(X1),X2,X3,X4)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(isNatIList(X)) -> active(isNatIList(X)) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) U21(mark(X)) -> U21(X) U21(active(X)) -> U21(X) U31(mark(X1),X2,X3,X4) -> U31(X1,X2,X3,X4) U31(X1,mark(X2),X3,X4) -> U31(X1,X2,X3,X4) U31(X1,X2,mark(X3),X4) -> U31(X1,X2,X3,X4) U31(X1,X2,X3,mark(X4)) -> U31(X1,X2,X3,X4) U31(active(X1),X2,X3,X4) -> U31(X1,X2,X3,X4) U31(X1,active(X2),X3,X4) -> U31(X1,X2,X3,X4) U31(X1,X2,active(X3),X4) -> U31(X1,X2,X3,X4) U31(X1,X2,X3,active(X4)) -> U31(X1,X2,X3,X4) take(mark(X1),X2) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) take(active(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) Open DPs: length#(mark(X)) -> length#(X) length#(active(X)) -> length#(X) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),L)) -> mark(s(length(L))) active(U21(tt())) -> mark(nil()) active(U31(tt(),IL,M,N)) -> mark(cons(N,take(M,IL))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(length(V1))) -> mark(isNatList(V1)) active(isNat(s(V1))) -> mark(isNat(V1)) active(isNatIList(V)) -> mark(isNatList(V)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(and(isNat(V1),isNatIList(V2))) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(and(isNat(V1),isNatList(V2))) active(isNatList(take(V1,V2))) -> mark(and(isNat(V1),isNatIList(V2))) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U11(and(isNatList(L),isNat(N)),L)) active(take(0(),IL)) -> mark(U21(isNatIList(IL))) active(take(s(M),cons(N,IL))) -> mark(U31(and(isNatIList(IL),and(isNat(M),isNat(N))),IL,M,N)) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(nil()) -> active(nil()) mark(U31(X1,X2,X3,X4)) -> active(U31(mark(X1),X2,X3,X4)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(isNatIList(X)) -> active(isNatIList(X)) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) U21(mark(X)) -> U21(X) U21(active(X)) -> U21(X) U31(mark(X1),X2,X3,X4) -> U31(X1,X2,X3,X4) U31(X1,mark(X2),X3,X4) -> U31(X1,X2,X3,X4) U31(X1,X2,mark(X3),X4) -> U31(X1,X2,X3,X4) U31(X1,X2,X3,mark(X4)) -> U31(X1,X2,X3,X4) U31(active(X1),X2,X3,X4) -> U31(X1,X2,X3,X4) U31(X1,active(X2),X3,X4) -> U31(X1,X2,X3,X4) U31(X1,X2,active(X3),X4) -> U31(X1,X2,X3,X4) U31(X1,X2,X3,active(X4)) -> U31(X1,X2,X3,X4) take(mark(X1),X2) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) take(active(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) Open DPs: s#(mark(X)) -> s#(X) s#(active(X)) -> s#(X) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),L)) -> mark(s(length(L))) active(U21(tt())) -> mark(nil()) active(U31(tt(),IL,M,N)) -> mark(cons(N,take(M,IL))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(length(V1))) -> mark(isNatList(V1)) active(isNat(s(V1))) -> mark(isNat(V1)) active(isNatIList(V)) -> mark(isNatList(V)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(and(isNat(V1),isNatIList(V2))) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(and(isNat(V1),isNatList(V2))) active(isNatList(take(V1,V2))) -> mark(and(isNat(V1),isNatIList(V2))) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U11(and(isNatList(L),isNat(N)),L)) active(take(0(),IL)) -> mark(U21(isNatIList(IL))) active(take(s(M),cons(N,IL))) -> mark(U31(and(isNatIList(IL),and(isNat(M),isNat(N))),IL,M,N)) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(nil()) -> active(nil()) mark(U31(X1,X2,X3,X4)) -> active(U31(mark(X1),X2,X3,X4)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(isNatIList(X)) -> active(isNatIList(X)) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) U21(mark(X)) -> U21(X) U21(active(X)) -> U21(X) U31(mark(X1),X2,X3,X4) -> U31(X1,X2,X3,X4) U31(X1,mark(X2),X3,X4) -> U31(X1,X2,X3,X4) U31(X1,X2,mark(X3),X4) -> U31(X1,X2,X3,X4) U31(X1,X2,X3,mark(X4)) -> U31(X1,X2,X3,X4) U31(active(X1),X2,X3,X4) -> U31(X1,X2,X3,X4) U31(X1,active(X2),X3,X4) -> U31(X1,X2,X3,X4) U31(X1,X2,active(X3),X4) -> U31(X1,X2,X3,X4) U31(X1,X2,X3,active(X4)) -> U31(X1,X2,X3,X4) take(mark(X1),X2) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) take(active(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) Open DPs: take#(mark(X1),X2) -> take#(X1,X2) take#(X1,active(X2)) -> take#(X1,X2) take#(active(X1),X2) -> take#(X1,X2) take#(X1,mark(X2)) -> take#(X1,X2) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),L)) -> mark(s(length(L))) active(U21(tt())) -> mark(nil()) active(U31(tt(),IL,M,N)) -> mark(cons(N,take(M,IL))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(length(V1))) -> mark(isNatList(V1)) active(isNat(s(V1))) -> mark(isNat(V1)) active(isNatIList(V)) -> mark(isNatList(V)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(and(isNat(V1),isNatIList(V2))) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(and(isNat(V1),isNatList(V2))) active(isNatList(take(V1,V2))) -> mark(and(isNat(V1),isNatIList(V2))) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U11(and(isNatList(L),isNat(N)),L)) active(take(0(),IL)) -> mark(U21(isNatIList(IL))) active(take(s(M),cons(N,IL))) -> mark(U31(and(isNatIList(IL),and(isNat(M),isNat(N))),IL,M,N)) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(nil()) -> active(nil()) mark(U31(X1,X2,X3,X4)) -> active(U31(mark(X1),X2,X3,X4)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(isNatIList(X)) -> active(isNatIList(X)) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) U21(mark(X)) -> U21(X) U21(active(X)) -> U21(X) U31(mark(X1),X2,X3,X4) -> U31(X1,X2,X3,X4) U31(X1,mark(X2),X3,X4) -> U31(X1,X2,X3,X4) U31(X1,X2,mark(X3),X4) -> U31(X1,X2,X3,X4) U31(X1,X2,X3,mark(X4)) -> U31(X1,X2,X3,X4) U31(active(X1),X2,X3,X4) -> U31(X1,X2,X3,X4) U31(X1,active(X2),X3,X4) -> U31(X1,X2,X3,X4) U31(X1,X2,active(X3),X4) -> U31(X1,X2,X3,X4) U31(X1,X2,X3,active(X4)) -> U31(X1,X2,X3,X4) take(mark(X1),X2) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) take(active(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) Open DPs: isNatList#(mark(X)) -> isNatList#(X) isNatList#(active(X)) -> isNatList#(X) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),L)) -> mark(s(length(L))) active(U21(tt())) -> mark(nil()) active(U31(tt(),IL,M,N)) -> mark(cons(N,take(M,IL))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(length(V1))) -> mark(isNatList(V1)) active(isNat(s(V1))) -> mark(isNat(V1)) active(isNatIList(V)) -> mark(isNatList(V)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(and(isNat(V1),isNatIList(V2))) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(and(isNat(V1),isNatList(V2))) active(isNatList(take(V1,V2))) -> mark(and(isNat(V1),isNatIList(V2))) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U11(and(isNatList(L),isNat(N)),L)) active(take(0(),IL)) -> mark(U21(isNatIList(IL))) active(take(s(M),cons(N,IL))) -> mark(U31(and(isNatIList(IL),and(isNat(M),isNat(N))),IL,M,N)) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(nil()) -> active(nil()) mark(U31(X1,X2,X3,X4)) -> active(U31(mark(X1),X2,X3,X4)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(isNatIList(X)) -> active(isNatIList(X)) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) U21(mark(X)) -> U21(X) U21(active(X)) -> U21(X) U31(mark(X1),X2,X3,X4) -> U31(X1,X2,X3,X4) U31(X1,mark(X2),X3,X4) -> U31(X1,X2,X3,X4) U31(X1,X2,mark(X3),X4) -> U31(X1,X2,X3,X4) U31(X1,X2,X3,mark(X4)) -> U31(X1,X2,X3,X4) U31(active(X1),X2,X3,X4) -> U31(X1,X2,X3,X4) U31(X1,active(X2),X3,X4) -> U31(X1,X2,X3,X4) U31(X1,X2,active(X3),X4) -> U31(X1,X2,X3,X4) U31(X1,X2,X3,active(X4)) -> U31(X1,X2,X3,X4) take(mark(X1),X2) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) take(active(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) Open DPs: isNat#(mark(X)) -> isNat#(X) isNat#(active(X)) -> isNat#(X) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),L)) -> mark(s(length(L))) active(U21(tt())) -> mark(nil()) active(U31(tt(),IL,M,N)) -> mark(cons(N,take(M,IL))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(length(V1))) -> mark(isNatList(V1)) active(isNat(s(V1))) -> mark(isNat(V1)) active(isNatIList(V)) -> mark(isNatList(V)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(and(isNat(V1),isNatIList(V2))) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(and(isNat(V1),isNatList(V2))) active(isNatList(take(V1,V2))) -> mark(and(isNat(V1),isNatIList(V2))) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U11(and(isNatList(L),isNat(N)),L)) active(take(0(),IL)) -> mark(U21(isNatIList(IL))) active(take(s(M),cons(N,IL))) -> mark(U31(and(isNatIList(IL),and(isNat(M),isNat(N))),IL,M,N)) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(nil()) -> active(nil()) mark(U31(X1,X2,X3,X4)) -> active(U31(mark(X1),X2,X3,X4)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(isNatIList(X)) -> active(isNatIList(X)) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) U21(mark(X)) -> U21(X) U21(active(X)) -> U21(X) U31(mark(X1),X2,X3,X4) -> U31(X1,X2,X3,X4) U31(X1,mark(X2),X3,X4) -> U31(X1,X2,X3,X4) U31(X1,X2,mark(X3),X4) -> U31(X1,X2,X3,X4) U31(X1,X2,X3,mark(X4)) -> U31(X1,X2,X3,X4) U31(active(X1),X2,X3,X4) -> U31(X1,X2,X3,X4) U31(X1,active(X2),X3,X4) -> U31(X1,X2,X3,X4) U31(X1,X2,active(X3),X4) -> U31(X1,X2,X3,X4) U31(X1,X2,X3,active(X4)) -> U31(X1,X2,X3,X4) take(mark(X1),X2) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) take(active(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) Open DPs: isNatIList#(mark(X)) -> isNatIList#(X) isNatIList#(active(X)) -> isNatIList#(X) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),L)) -> mark(s(length(L))) active(U21(tt())) -> mark(nil()) active(U31(tt(),IL,M,N)) -> mark(cons(N,take(M,IL))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(length(V1))) -> mark(isNatList(V1)) active(isNat(s(V1))) -> mark(isNat(V1)) active(isNatIList(V)) -> mark(isNatList(V)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(and(isNat(V1),isNatIList(V2))) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(and(isNat(V1),isNatList(V2))) active(isNatList(take(V1,V2))) -> mark(and(isNat(V1),isNatIList(V2))) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U11(and(isNatList(L),isNat(N)),L)) active(take(0(),IL)) -> mark(U21(isNatIList(IL))) active(take(s(M),cons(N,IL))) -> mark(U31(and(isNatIList(IL),and(isNat(M),isNat(N))),IL,M,N)) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(nil()) -> active(nil()) mark(U31(X1,X2,X3,X4)) -> active(U31(mark(X1),X2,X3,X4)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(isNatIList(X)) -> active(isNatIList(X)) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) U21(mark(X)) -> U21(X) U21(active(X)) -> U21(X) U31(mark(X1),X2,X3,X4) -> U31(X1,X2,X3,X4) U31(X1,mark(X2),X3,X4) -> U31(X1,X2,X3,X4) U31(X1,X2,mark(X3),X4) -> U31(X1,X2,X3,X4) U31(X1,X2,X3,mark(X4)) -> U31(X1,X2,X3,X4) U31(active(X1),X2,X3,X4) -> U31(X1,X2,X3,X4) U31(X1,active(X2),X3,X4) -> U31(X1,X2,X3,X4) U31(X1,X2,active(X3),X4) -> U31(X1,X2,X3,X4) U31(X1,X2,X3,active(X4)) -> U31(X1,X2,X3,X4) take(mark(X1),X2) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) take(active(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) Open DPs: and#(mark(X1),X2) -> and#(X1,X2) and#(X1,active(X2)) -> and#(X1,X2) and#(active(X1),X2) -> and#(X1,X2) and#(X1,mark(X2)) -> and#(X1,X2) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),L)) -> mark(s(length(L))) active(U21(tt())) -> mark(nil()) active(U31(tt(),IL,M,N)) -> mark(cons(N,take(M,IL))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(length(V1))) -> mark(isNatList(V1)) active(isNat(s(V1))) -> mark(isNat(V1)) active(isNatIList(V)) -> mark(isNatList(V)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(and(isNat(V1),isNatIList(V2))) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(and(isNat(V1),isNatList(V2))) active(isNatList(take(V1,V2))) -> mark(and(isNat(V1),isNatIList(V2))) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U11(and(isNatList(L),isNat(N)),L)) active(take(0(),IL)) -> mark(U21(isNatIList(IL))) active(take(s(M),cons(N,IL))) -> mark(U31(and(isNatIList(IL),and(isNat(M),isNat(N))),IL,M,N)) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(nil()) -> active(nil()) mark(U31(X1,X2,X3,X4)) -> active(U31(mark(X1),X2,X3,X4)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(isNatIList(X)) -> active(isNatIList(X)) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) U21(mark(X)) -> U21(X) U21(active(X)) -> U21(X) U31(mark(X1),X2,X3,X4) -> U31(X1,X2,X3,X4) U31(X1,mark(X2),X3,X4) -> U31(X1,X2,X3,X4) U31(X1,X2,mark(X3),X4) -> U31(X1,X2,X3,X4) U31(X1,X2,X3,mark(X4)) -> U31(X1,X2,X3,X4) U31(active(X1),X2,X3,X4) -> U31(X1,X2,X3,X4) U31(X1,active(X2),X3,X4) -> U31(X1,X2,X3,X4) U31(X1,X2,active(X3),X4) -> U31(X1,X2,X3,X4) U31(X1,X2,X3,active(X4)) -> U31(X1,X2,X3,X4) take(mark(X1),X2) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) take(active(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) Open DPs: U11#(mark(X1),X2) -> U11#(X1,X2) U11#(X1,active(X2)) -> U11#(X1,X2) U11#(active(X1),X2) -> U11#(X1,X2) U11#(X1,mark(X2)) -> U11#(X1,X2) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),L)) -> mark(s(length(L))) active(U21(tt())) -> mark(nil()) active(U31(tt(),IL,M,N)) -> mark(cons(N,take(M,IL))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(length(V1))) -> mark(isNatList(V1)) active(isNat(s(V1))) -> mark(isNat(V1)) active(isNatIList(V)) -> mark(isNatList(V)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(and(isNat(V1),isNatIList(V2))) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(and(isNat(V1),isNatList(V2))) active(isNatList(take(V1,V2))) -> mark(and(isNat(V1),isNatIList(V2))) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U11(and(isNatList(L),isNat(N)),L)) active(take(0(),IL)) -> mark(U21(isNatIList(IL))) active(take(s(M),cons(N,IL))) -> mark(U31(and(isNatIList(IL),and(isNat(M),isNat(N))),IL,M,N)) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(nil()) -> active(nil()) mark(U31(X1,X2,X3,X4)) -> active(U31(mark(X1),X2,X3,X4)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(isNatIList(X)) -> active(isNatIList(X)) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) U21(mark(X)) -> U21(X) U21(active(X)) -> U21(X) U31(mark(X1),X2,X3,X4) -> U31(X1,X2,X3,X4) U31(X1,mark(X2),X3,X4) -> U31(X1,X2,X3,X4) U31(X1,X2,mark(X3),X4) -> U31(X1,X2,X3,X4) U31(X1,X2,X3,mark(X4)) -> U31(X1,X2,X3,X4) U31(active(X1),X2,X3,X4) -> U31(X1,X2,X3,X4) U31(X1,active(X2),X3,X4) -> U31(X1,X2,X3,X4) U31(X1,X2,active(X3),X4) -> U31(X1,X2,X3,X4) U31(X1,X2,X3,active(X4)) -> U31(X1,X2,X3,X4) take(mark(X1),X2) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) take(active(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) Open DPs: U21#(mark(X)) -> U21#(X) U21#(active(X)) -> U21#(X) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),L)) -> mark(s(length(L))) active(U21(tt())) -> mark(nil()) active(U31(tt(),IL,M,N)) -> mark(cons(N,take(M,IL))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(length(V1))) -> mark(isNatList(V1)) active(isNat(s(V1))) -> mark(isNat(V1)) active(isNatIList(V)) -> mark(isNatList(V)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(and(isNat(V1),isNatIList(V2))) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(and(isNat(V1),isNatList(V2))) active(isNatList(take(V1,V2))) -> mark(and(isNat(V1),isNatIList(V2))) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U11(and(isNatList(L),isNat(N)),L)) active(take(0(),IL)) -> mark(U21(isNatIList(IL))) active(take(s(M),cons(N,IL))) -> mark(U31(and(isNatIList(IL),and(isNat(M),isNat(N))),IL,M,N)) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(nil()) -> active(nil()) mark(U31(X1,X2,X3,X4)) -> active(U31(mark(X1),X2,X3,X4)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(isNatIList(X)) -> active(isNatIList(X)) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) U21(mark(X)) -> U21(X) U21(active(X)) -> U21(X) U31(mark(X1),X2,X3,X4) -> U31(X1,X2,X3,X4) U31(X1,mark(X2),X3,X4) -> U31(X1,X2,X3,X4) U31(X1,X2,mark(X3),X4) -> U31(X1,X2,X3,X4) U31(X1,X2,X3,mark(X4)) -> U31(X1,X2,X3,X4) U31(active(X1),X2,X3,X4) -> U31(X1,X2,X3,X4) U31(X1,active(X2),X3,X4) -> U31(X1,X2,X3,X4) U31(X1,X2,active(X3),X4) -> U31(X1,X2,X3,X4) U31(X1,X2,X3,active(X4)) -> U31(X1,X2,X3,X4) take(mark(X1),X2) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) take(active(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) Open DPs: U31#(mark(X1),X2,X3,X4) -> U31#(X1,X2,X3,X4) U31#(X1,X2,X3,active(X4)) -> U31#(X1,X2,X3,X4) U31#(X1,X2,active(X3),X4) -> U31#(X1,X2,X3,X4) U31#(X1,active(X2),X3,X4) -> U31#(X1,X2,X3,X4) U31#(active(X1),X2,X3,X4) -> U31#(X1,X2,X3,X4) U31#(X1,X2,X3,mark(X4)) -> U31#(X1,X2,X3,X4) U31#(X1,X2,mark(X3),X4) -> U31#(X1,X2,X3,X4) U31#(X1,mark(X2),X3,X4) -> U31#(X1,X2,X3,X4) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),L)) -> mark(s(length(L))) active(U21(tt())) -> mark(nil()) active(U31(tt(),IL,M,N)) -> mark(cons(N,take(M,IL))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(length(V1))) -> mark(isNatList(V1)) active(isNat(s(V1))) -> mark(isNat(V1)) active(isNatIList(V)) -> mark(isNatList(V)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(and(isNat(V1),isNatIList(V2))) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(and(isNat(V1),isNatList(V2))) active(isNatList(take(V1,V2))) -> mark(and(isNat(V1),isNatIList(V2))) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U11(and(isNatList(L),isNat(N)),L)) active(take(0(),IL)) -> mark(U21(isNatIList(IL))) active(take(s(M),cons(N,IL))) -> mark(U31(and(isNatIList(IL),and(isNat(M),isNat(N))),IL,M,N)) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(nil()) -> active(nil()) mark(U31(X1,X2,X3,X4)) -> active(U31(mark(X1),X2,X3,X4)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(isNatIList(X)) -> active(isNatIList(X)) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) U21(mark(X)) -> U21(X) U21(active(X)) -> U21(X) U31(mark(X1),X2,X3,X4) -> U31(X1,X2,X3,X4) U31(X1,mark(X2),X3,X4) -> U31(X1,X2,X3,X4) U31(X1,X2,mark(X3),X4) -> U31(X1,X2,X3,X4) U31(X1,X2,X3,mark(X4)) -> U31(X1,X2,X3,X4) U31(active(X1),X2,X3,X4) -> U31(X1,X2,X3,X4) U31(X1,active(X2),X3,X4) -> U31(X1,X2,X3,X4) U31(X1,X2,active(X3),X4) -> U31(X1,X2,X3,X4) U31(X1,X2,X3,active(X4)) -> U31(X1,X2,X3,X4) take(mark(X1),X2) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) take(active(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) Open