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)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X1,X2)) -> U11(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(U21(X)) -> U21(active(X)) active(U31(X1,X2,X3,X4)) -> U31(active(X1),X2,X3,X4) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) active(and(X1,X2)) -> and(active(X1),X2) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X1),X2) -> mark(U11(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) U21(mark(X)) -> mark(U21(X)) U31(mark(X1),X2,X3,X4) -> mark(U31(X1,X2,X3,X4)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(U21(X)) -> U21(proper(X)) proper(nil()) -> ok(nil()) proper(U31(X1,X2,X3,X4)) -> U31(proper(X1),proper(X2),proper(X3),proper(X4)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) U21(ok(X)) -> ok(U21(X)) U31(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U31(X1,X2,X3,X4)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) isNatList(ok(X)) -> ok(isNatList(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Proof: DP Processor: DPs: active#(zeros()) -> cons#(0(),zeros()) active#(U11(tt(),L)) -> length#(L) active#(U11(tt(),L)) -> s#(length(L)) active#(U31(tt(),IL,M,N)) -> take#(M,IL) active#(U31(tt(),IL,M,N)) -> cons#(N,take(M,IL)) active#(isNat(length(V1))) -> isNatList#(V1) active#(isNat(s(V1))) -> isNat#(V1) active#(isNatIList(V)) -> isNatList#(V) 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#(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(take(V1,V2))) -> isNatIList#(V2) active#(isNatList(take(V1,V2))) -> isNat#(V1) active#(isNatList(take(V1,V2))) -> and#(isNat(V1),isNatIList(V2)) 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#(take(0(),IL)) -> isNatIList#(IL) active#(take(0(),IL)) -> 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#(cons(X1,X2)) -> active#(X1) active#(cons(X1,X2)) -> cons#(active(X1),X2) active#(U11(X1,X2)) -> active#(X1) active#(U11(X1,X2)) -> U11#(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#(U21(X)) -> active#(X) active#(U21(X)) -> U21#(active(X)) active#(U31(X1,X2,X3,X4)) -> active#(X1) active#(U31(X1,X2,X3,X4)) -> U31#(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)) active#(and(X1,X2)) -> active#(X1) active#(and(X1,X2)) -> and#(active(X1),X2) cons#(mark(X1),X2) -> cons#(X1,X2) U11#(mark(X1),X2) -> U11#(X1,X2) s#(mark(X)) -> s#(X) length#(mark(X)) -> length#(X) U21#(mark(X)) -> U21#(X) U31#(mark(X1),X2,X3,X4) -> U31#(X1,X2,X3,X4) take#(mark(X1),X2) -> take#(X1,X2) take#(X1,mark(X2)) -> take#(X1,X2) and#(mark(X1),X2) -> and#(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(X1,X2)) -> proper#(X2) proper#(U11(X1,X2)) -> proper#(X1) proper#(U11(X1,X2)) -> U11#(proper(X1),proper(X2)) proper#(s(X)) -> proper#(X) proper#(s(X)) -> s#(proper(X)) proper#(length(X)) -> proper#(X) proper#(length(X)) -> length#(proper(X)) proper#(U21(X)) -> proper#(X) proper#(U21(X)) -> U21#(proper(X)) proper#(U31(X1,X2,X3,X4)) -> proper#(X4) proper#(U31(X1,X2,X3,X4)) -> proper#(X3) proper#(U31(X1,X2,X3,X4)) -> proper#(X2) proper#(U31(X1,X2,X3,X4)) -> proper#(X1) proper#(U31(X1,X2,X3,X4)) -> U31#(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)) proper#(and(X1,X2)) -> proper#(X2) proper#(and(X1,X2)) -> proper#(X1) proper#(and(X1,X2)) -> and#(proper(X1),proper(X2)) proper#(isNat(X)) -> proper#(X) proper#(isNat(X)) -> isNat#(proper(X)) proper#(isNatList(X)) -> proper#(X) proper#(isNatList(X)) -> isNatList#(proper(X)) proper#(isNatIList(X)) -> proper#(X) proper#(isNatIList(X)) -> isNatIList#(proper(X)) cons#(ok(X1),ok(X2)) -> cons#(X1,X2) U11#(ok(X1),ok(X2)) -> U11#(X1,X2) s#(ok(X)) -> s#(X) length#(ok(X)) -> length#(X) U21#(ok(X)) -> U21#(X) U31#(ok(X1),ok(X2),ok(X3),ok(X4)) -> U31#(X1,X2,X3,X4) take#(ok(X1),ok(X2)) -> take#(X1,X2) and#(ok(X1),ok(X2)) -> and#(X1,X2) isNat#(ok(X)) -> isNat#(X) isNatList#(ok(X)) -> isNatList#(X) isNatIList#(ok(X)) -> isNatIList#(X) 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(),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)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X1,X2)) -> U11(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(U21(X)) -> U21(active(X)) active(U31(X1,X2,X3,X4)) -> U31(active(X1),X2,X3,X4) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) active(and(X1,X2)) -> and(active(X1),X2) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X1),X2) -> mark(U11(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) U21(mark(X)) -> mark(U21(X)) U31(mark(X1),X2,X3,X4) -> mark(U31(X1,X2,X3,X4)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(U21(X)) -> U21(proper(X)) proper(nil()) -> ok(nil()) proper(U31(X1,X2,X3,X4)) -> U31(proper(X1),proper(X2),proper(X3),proper(X4)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) U21(ok(X)) -> ok(U21(X)) U31(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U31(X1,X2,X3,X4)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) isNatList(ok(X)) -> ok(isNatList(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) TDG Processor: DPs: active#(zeros()) -> cons#(0(),zeros()) active#(U11(tt(),L)) -> length#(L) active#(U11(tt(),L)) -> s#(length(L)) active#(U31(tt(),IL,M,N)) -> take#(M,IL) active#(U31(tt(),IL,M,N)) -> cons#(N,take(M,IL)) active#(isNat(length(V1))) -> isNatList#(V1) active#(isNat(s(V1))) -> isNat#(V1) active#(isNatIList(V)) -> isNatList#(V) 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#(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(take(V1,V2))) -> isNatIList#(V2) active#(isNatList(take(V1,V2))) -> isNat#(V1) active#(isNatList(take(V1,V2))) -> and#(isNat(V1),isNatIList(V2)) 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#(take(0(),IL)) -> isNatIList#(IL) active#(take(0(),IL)) -> 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#(cons(X1,X2)) -> active#(X1) active#(cons(X1,X2)) -> cons#(active(X1),X2) active#(U11(X1,X2)) -> active#(X1) active#(U11(X1,X2)) -> U11#(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#(U21(X)) -> active#(X) active#(U21(X)) -> U21#(active(X)) active#(U31(X1,X2,X3,X4)) -> active#(X1) active#(U31(X1,X2,X3,X4)) -> U31#(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)) active#(and(X1,X2)) -> active#(X1) active#(and(X1,X2)) -> and#(active(X1),X2) cons#(mark(X1),X2) -> cons#(X1,X2) U11#(mark(X1),X2) -> U11#(X1,X2) s#(mark(X)) -> s#(X) length#(mark(X)) -> length#(X) U21#(mark(X)) -> U21#(X) U31#(mark(X1),X2,X3,X4) -> U31#(X1,X2,X3,X4) take#(mark(X1),X2) -> take#(X1,X2) take#(X1,mark(X2)) -> take#(X1,X2) and#(mark(X1),X2) -> and#(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(X1,X2)) -> proper#(X2) proper#(U11(X1,X2)) -> proper#(X1) proper#(U11(X1,X2)) -> U11#(proper(X1),proper(X2)) proper#(s(X)) -> proper#(X) proper#(s(X)) -> s#(proper(X)) proper#(length(X)) -> proper#(X) proper#(length(X)) -> length#(proper(X)) proper#(U21(X)) -> proper#(X) proper#(U21(X)) -> U21#(proper(X)) proper#(U31(X1,X2,X3,X4)) -> proper#(X4) proper#(U31(X1,X2,X3,X4)) -> proper#(X3) proper#(U31(X1,X2,X3,X4)) -> proper#(X2) proper#(U31(X1,X2,X3,X4)) -> proper#(X1) proper#(U31(X1,X2,X3,X4)) -> U31#(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)) proper#(and(X1,X2)) -> proper#(X2) proper#(and(X1,X2)) -> proper#(X1) proper#(and(X1,X2)) -> and#(proper(X1),proper(X2)) proper#(isNat(X)) -> proper#(X) proper#(isNat(X)) -> isNat#(proper(X)) proper#(isNatList(X)) -> proper#(X) proper#(isNatList(X)) -> isNatList#(proper(X)) proper#(isNatIList(X)) -> proper#(X) proper#(isNatIList(X)) -> isNatIList#(proper(X)) cons#(ok(X1),ok(X2)) -> cons#(X1,X2) U11#(ok(X1),ok(X2)) -> U11#(X1,X2) s#(ok(X)) -> s#(X) length#(ok(X)) -> length#(X) U21#(ok(X)) -> U21#(X) U31#(ok(X1),ok(X2),ok(X3),ok(X4)) -> U31#(X1,X2,X3,X4) take#(ok(X1),ok(X2)) -> take#(X1,X2) and#(ok(X1),ok(X2)) -> and#(X1,X2) isNat#(ok(X)) -> isNat#(X) isNatList#(ok(X)) -> isNatList#(X) isNatIList#(ok(X)) -> isNatIList#(X) 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(),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)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X1,X2)) -> U11(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(U21(X)) -> U21(active(X)) active(U31(X1,X2,X3,X4)) -> U31(active(X1),X2,X3,X4) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) active(and(X1,X2)) -> and(active(X1),X2) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X1),X2) -> mark(U11(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) U21(mark(X)) -> mark(U21(X)) U31(mark(X1),X2,X3,X4) -> mark(U31(X1,X2,X3,X4)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(U21(X)) -> U21(proper(X)) proper(nil()) -> ok(nil()) proper(U31(X1,X2,X3,X4)) -> U31(proper(X1),proper(X2),proper(X3),proper(X4)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) U21(ok(X)) -> ok(U21(X)) U31(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U31(X1,X2,X3,X4)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) isNatList(ok(X)) -> ok(isNatList(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) graph: ... SCC Processor: #sccs: 14 #rules: 49 #arcs: 1153/10000 DPs: top#(ok(X)) -> top#(active(X)) top#(mark(X)) -> top#(proper(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)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X1,X2)) -> U11(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(U21(X)) -> U21(active(X)) active(U31(X1,X2,X3,X4)) -> U31(active(X1),X2,X3,X4) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) active(and(X1,X2)) -> and(active(X1),X2) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X1),X2) -> mark(U11(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) U21(mark(X)) -> mark(U21(X)) U31(mark(X1),X2,X3,X4) -> mark(U31(X1,X2,X3,X4)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(U21(X)) -> U21(proper(X)) proper(nil()) -> ok(nil()) proper(U31(X1,X2,X3,X4)) -> U31(proper(X1),proper(X2),proper(X3),proper(X4)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) U21(ok(X)) -> ok(U21(X)) U31(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U31(X1,X2,X3,X4)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) isNatList(ok(X)) -> ok(isNatList(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Open DPs: active#(cons(X1,X2)) -> active#(X1) active#(U11(X1,X2)) -> active#(X1) active#(s(X)) -> active#(X) active#(length(X)) -> active#(X) active#(U21(X)) -> active#(X) active#(U31(X1,X2,X3,X4)) -> active#(X1) active#(take(X1,X2)) -> active#(X1) active#(take(X1,X2)) -> active#(X2) active#(and(X1,X2)) -> active#(X1) 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)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X1,X2)) -> U11(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(U21(X)) -> U21(active(X)) active(U31(X1,X2,X3,X4)) -> U31(active(X1),X2,X3,X4) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) active(and(X1,X2)) -> and(active(X1),X2) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X1),X2) -> mark(U11(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) U21(mark(X)) -> mark(U21(X)) U31(mark(X1),X2,X3,X4) -> mark(U31(X1,X2,X3,X4)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(U21(X)) -> U21(proper(X)) proper(nil()) -> ok(nil()) proper(U31(X1,X2,X3,X4)) -> U31(proper(X1),proper(X2),proper(X3),proper(X4)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) U21(ok(X)) -> ok(U21(X)) U31(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U31(X1,X2,X3,X4)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) isNatList(ok(X)) -> ok(isNatList(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Subterm Criterion Processor: simple projection: pi(active#) = 0 problem: DPs: 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)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X1,X2)) -> U11(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(U21(X)) -> U21(active(X)) active(U31(X1,X2,X3,X4)) -> U31(active(X1),X2,X3,X4) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) active(and(X1,X2)) -> and(active(X1),X2) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X1),X2) -> mark(U11(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) U21(mark(X)) -> mark(U21(X)) U31(mark(X1),X2,X3,X4) -> mark(U31(X1,X2,X3,X4)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(U21(X)) -> U21(proper(X)) proper(nil()) -> ok(nil()) proper(U31(X1,X2,X3,X4)) -> U31(proper(X1),proper(X2),proper(X3),proper(X4)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) U21(ok(X)) -> ok(U21(X)) U31(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U31(X1,X2,X3,X4)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) isNatList(ok(X)) -> ok(isNatList(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed DPs: proper#(cons(X1,X2)) -> proper#(X2) proper#(cons(X1,X2)) -> proper#(X1) proper#(U11(X1,X2)) -> proper#(X2) proper#(U11(X1,X2)) -> proper#(X1) proper#(s(X)) -> proper#(X) proper#(length(X)) -> proper#(X) proper#(U21(X)) -> proper#(X) proper#(U31(X1,X2,X3,X4)) -> proper#(X4) proper#(U31(X1,X2,X3,X4)) -> proper#(X3) proper#(U31(X1,X2,X3,X4)) -> proper#(X2) proper#(U31(X1,X2,X3,X4)) -> proper#(X1) proper#(take(X1,X2)) -> proper#(X2) proper#(take(X1,X2)) -> proper#(X1) proper#(and(X1,X2)) -> proper#(X2) proper#(and(X1,X2)) -> proper#(X1) proper#(isNat(X)) -> proper#(X) proper#(isNatList(X)) -> proper#(X) proper#(isNatIList(X)) -> proper#(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)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X1,X2)) -> U11(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(U21(X)) -> U21(active(X)) active(U31(X1,X2,X3,X4)) -> U31(active(X1),X2,X3,X4) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) active(and(X1,X2)) -> and(active(X1),X2) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X1),X2) -> mark(U11(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) U21(mark(X)) -> mark(U21(X)) U31(mark(X1),X2,X3,X4) -> mark(U31(X1,X2,X3,X4)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(U21(X)) -> U21(proper(X)) proper(nil()) -> ok(nil()) proper(U31(X1,X2,X3,X4)) -> U31(proper(X1),proper(X2),proper(X3),proper(X4)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) U21(ok(X)) -> ok(U21(X)) U31(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U31(X1,X2,X3,X4)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) isNatList(ok(X)) -> ok(isNatList(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Subterm Criterion Processor: simple projection: pi(proper#) = 0 problem: DPs: 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)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X1,X2)) -> U11(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(U21(X)) -> U21(active(X)) active(U31(X1,X2,X3,X4)) -> U31(active(X1),X2,X3,X4) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) active(and(X1,X2)) -> and(active(X1),X2) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X1),X2) -> mark(U11(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) U21(mark(X)) -> mark(U21(X)) U31(mark(X1),X2,X3,X4) -> mark(U31(X1,X2,X3,X4)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(U21(X)) -> U21(proper(X)) proper(nil()) -> ok(nil()) proper(U31(X1,X2,X3,X4)) -> U31(proper(X1),proper(X2),proper(X3),proper(X4)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) U21(ok(X)) -> ok(U21(X)) U31(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U31(X1,X2,X3,X4)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) isNatList(ok(X)) -> ok(isNatList(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed DPs: isNatIList#(ok(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)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X1,X2)) -> U11(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(U21(X)) -> U21(active(X)) active(U31(X1,X2,X3,X4)) -> U31(active(X1),X2,X3,X4) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) active(and(X1,X2)) -> and(active(X1),X2) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X1),X2) -> mark(U11(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) U21(mark(X)) -> mark(U21(X)) U31(mark(X1),X2,X3,X4) -> mark(U31(X1,X2,X3,X4)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(U21(X)) -> U21(proper(X)) proper(nil()) -> ok(nil()) proper(U31(X1,X2,X3,X4)) -> U31(proper(X1),proper(X2),proper(X3),proper(X4)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) U21(ok(X)) -> ok(U21(X)) U31(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U31(X1,X2,X3,X4)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) isNatList(ok(X)) -> ok(isNatList(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Subterm Criterion Processor: simple projection: pi(isNatIList#) = 0 problem: DPs: 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)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X1,X2)) -> U11(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(U21(X)) -> U21(active(X)) active(U31(X1,X2,X3,X4)) -> U31(active(X1),X2,X3,X4) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) active(and(X1,X2)) -> and(active(X1),X2) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X1),X2) -> mark(U11(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) U21(mark(X)) -> mark(U21(X)) U31(mark(X1),X2,X3,X4) -> mark(U31(X1,X2,X3,X4)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(U21(X)) -> U21(proper(X)) proper(nil()) -> ok(nil()) proper(U31(X1,X2,X3,X4)) -> U31(proper(X1),proper(X2),proper(X3),proper(X4)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) U21(ok(X)) -> ok(U21(X)) U31(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U31(X1,X2,X3,X4)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) isNatList(ok(X)) -> ok(isNatList(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed DPs: isNatList#(ok(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)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X1,X2)) -> U11(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(U21(X)) -> U21(active(X)) active(U31(X1,X2,X3,X4)) -> U31(active(X1),X2,X3,X4) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) active(and(X1,X2)) -> and(active(X1),X2) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X1),X2) -> mark(U11(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) U21(mark(X)) -> mark(U21(X)) U31(mark(X1),X2,X3,X4) -> mark(U31(X1,X2,X3,X4)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(U21(X)) -> U21(proper(X)) proper(nil()) -> ok(nil()) proper(U31(X1,X2,X3,X4)) -> U31(proper(X1),proper(X2),proper(X3),proper(X4)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) U21(ok(X)) -> ok(U21(X)) U31(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U31(X1,X2,X3,X4)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) isNatList(ok(X)) -> ok(isNatList(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Subterm Criterion Processor: simple projection: pi(isNatList#) = 0 problem: DPs: 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)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X1,X2)) -> U11(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(U21(X)) -> U21(active(X)) active(U31(X1,X2,X3,X4)) -> U31(active(X1),X2,X3,X4) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) active(and(X1,X2)) -> and(active(X1),X2) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X1),X2) -> mark(U11(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) U21(mark(X)) -> mark(U21(X)) U31(mark(X1),X2,X3,X4) -> mark(U31(X1,X2,X3,X4)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(U21(X)) -> U21(proper(X)) proper(nil()) -> ok(nil()) proper(U31(X1,X2,X3,X4)) -> U31(proper(X1),proper(X2),proper(X3),proper(X4)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) U21(ok(X)) -> ok(U21(X)) U31(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U31(X1,X2,X3,X4)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) isNatList(ok(X)) -> ok(isNatList(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed DPs: isNat#(ok(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)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X1,X2)) -> U11(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(U21(X)) -> U21(active(X)) active(U31(X1,X2,X3,X4)) -> U31(active(X1),X2,X3,X4) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) active(and(X1,X2)) -> and(active(X1),X2) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X1),X2) -> mark(U11(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) U21(mark(X)) -> mark(U21(X)) U31(mark(X1),X2,X3,X4) -> mark(U31(X1,X2,X3,X4)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(U21(X)) -> U21(proper(X)) proper(nil()) -> ok(nil()) proper(U31(X1,X2,X3,X4)) -> U31(proper(X1),proper(X2),proper(X3),proper(X4)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) U21(ok(X)) -> ok(U21(X)) U31(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U31(X1,X2,X3,X4)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) isNatList(ok(X)) -> ok(isNatList(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Subterm Criterion Processor: simple projection: pi(isNat#) = 0 problem: DPs: 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)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X1,X2)) -> U11(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(U21(X)) -> U21(active(X)) active(U31(X1,X2,X3,X4)) -> U31(active(X1),X2,X3,X4) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) active(and(X1,X2)) -> and(active(X1),X2) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X1),X2) -> mark(U11(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) U21(mark(X)) -> mark(U21(X)) U31(mark(X1),X2,X3,X4) -> mark(U31(X1,X2,X3,X4)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(U21(X)) -> U21(proper(X)) proper(nil()) -> ok(nil()) proper(U31(X1,X2,X3,X4)) -> U31(proper(X1),proper(X2),proper(X3),proper(X4)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) U21(ok(X)) -> ok(U21(X)) U31(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U31(X1,X2,X3,X4)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) isNatList(ok(X)) -> ok(isNatList(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed DPs: and#(mark(X1),X2) -> and#(X1,X2) and#(ok(X1),ok(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)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X1,X2)) -> U11(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(U21(X)) -> U21(active(X)) active(U31(X1,X2,X3,X4)) -> U31(active(X1),X2,X3,X4) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) active(and(X1,X2)) -> and(active(X1),X2) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X1),X2) -> mark(U11(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) U21(mark(X)) -> mark(U21(X)) U31(mark(X1),X2,X3,X4) -> mark(U31(X1,X2,X3,X4)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(U21(X)) -> U21(proper(X)) proper(nil()) -> ok(nil()) proper(U31(X1,X2,X3,X4)) -> U31(proper(X1),proper(X2),proper(X3),proper(X4)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) U21(ok(X)) -> ok(U21(X)) U31(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U31(X1,X2,X3,X4)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) isNatList(ok(X)) -> ok(isNatList(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Subterm Criterion Processor: simple projection: pi(and#) = 1 problem: DPs: and#(mark(X1),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)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X1,X2)) -> U11(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(U21(X)) -> U21(active(X)) active(U31(X1,X2,X3,X4)) -> U31(active(X1),X2,X3,X4) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) active(and(X1,X2)) -> and(active(X1),X2) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X1),X2) -> mark(U11(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) U21(mark(X)) -> mark(U21(X)) U31(mark(X1),X2,X3,X4) -> mark(U31(X1,X2,X3,X4)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(U21(X)) -> U21(proper(X)) proper(nil()) -> ok(nil()) proper(U31(X1,X2,X3,X4)) -> U31(proper(X1),proper(X2),proper(X3),proper(X4)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) U21(ok(X)) -> ok(U21(X)) U31(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U31(X1,X2,X3,X4)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) isNatList(ok(X)) -> ok(isNatList(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Subterm Criterion Processor: simple projection: pi(and#) = 0 problem: DPs: 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)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X1,X2)) -> U11(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(U21(X)) -> U21(active(X)) active(U31(X1,X2,X3,X4)) -> U31(active(X1),X2,X3,X4) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) active(and(X1,X2)) -> and(active(X1),X2) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X1),X2) -> mark(U11(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) U21(mark(X)) -> mark(U21(X)) U31(mark(X1),X2,X3,X4) -> mark(U31(X1,X2,X3,X4)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(U21(X)) -> U21(proper(X)) proper(nil()) -> ok(nil()) proper(U31(X1,X2,X3,X4)) -> U31(proper(X1),proper(X2),proper(X3),proper(X4)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) U21(ok(X)) -> ok(U21(X)) U31(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U31(X1,X2,X3,X4)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) isNatList(ok(X)) -> ok(isNatList(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed DPs: take#(mark(X1),X2) -> take#(X1,X2) take#(X1,mark(X2)) -> take#(X1,X2) take#(ok(X1),ok(X2)) -> take#(X1,X2) TRS: active(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)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X1,X2)) -> U11(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(U21(X)) -> U21(active(X)) active(U31(X1,X2,X3,X4)) -> U31(active(X1),X2,X3,X4) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) active(and(X1,X2)) -> and(active(X1),X2) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X1),X2) -> mark(U11(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) U21(mark(X)) -> mark(U21(X)) U31(mark(X1),X2,X3,X4) -> mark(U31(X1,X2,X3,X4)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(U21(X)) -> U21(proper(X)) proper(nil()) -> ok(nil()) proper(U31(X1,X2,X3,X4)) -> U31(proper(X1),proper(X2),proper(X3),proper(X4)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) U21(ok(X)) -> ok(U21(X)) U31(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U31(X1,X2,X3,X4)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) isNatList(ok(X)) -> ok(isNatList(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Subterm Criterion Processor: simple projection: pi(take#) = 1 problem: DPs: take#(mark(X1),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)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X1,X2)) -> U11(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(U21(X)) -> U21(active(X)) active(U31(X1,X2,X3,X4)) -> U31(active(X1),X2,X3,X4) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) active(and(X1,X2)) -> and(active(X1),X2) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X1),X2) -> mark(U11(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) U21(mark(X)) -> mark(U21(X)) U31(mark(X1),X2,X3,X4) -> mark(U31(X1,X2,X3,X4)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(U21(X)) -> U21(proper(X)) proper(nil()) -> ok(nil()) proper(U31(X1,X2,X3,X4)) -> U31(proper(X1),proper(X2),proper(X3),proper(X4)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) U21(ok(X)) -> ok(U21(X)) U31(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U31(X1,X2,X3,X4)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) isNatList(ok(X)) -> ok(isNatList(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Subterm Criterion Processor: simple projection: pi(take#) = 0 problem: DPs: 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)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X1,X2)) -> U11(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(U21(X)) -> U21(active(X)) active(U31(X1,X2,X3,X4)) -> U31(active(X1),X2,X3,X4) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) active(and(X1,X2)) -> and(active(X1),X2) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X1),X2) -> mark(U11(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) U21(mark(X)) -> mark(U21(X)) U31(mark(X1),X2,X3,X4) -> mark(U31(X1,X2,X3,X4)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(U21(X)) -> U21(proper(X)) proper(nil()) -> ok(nil()) proper(U31(X1,X2,X3,X4)) -> U31(proper(X1),proper(X2),proper(X3),proper(X4)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) U21(ok(X)) -> ok(U21(X)) U31(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U31(X1,X2,X3,X4)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) isNatList(ok(X)) -> ok(isNatList(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed DPs: U31#(mark(X1),X2,X3,X4) -> U31#(X1,X2,X3,X4) U31#(ok(X1),ok(X2),ok(X3),ok(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)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X1,X2)) -> U11(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(U21(X)) -> U21(active(X)) active(U31(X1,X2,X3,X4)) -> U31(active(X1),X2,X3,X4) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) active(and(X1,X2)) -> and(active(X1),X2) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X1),X2) -> mark(U11(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) U21(mark(X)) -> mark(U21(X)) U31(mark(X1),X2,X3,X4) -> mark(U31(X1,X2,X3,X4)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(U21(X)) -> U21(proper(X)) proper(nil()) -> ok(nil()) proper(U31(X1,X2,X3,X4)) -> U31(proper(X1),proper(X2),proper(X3),proper(X4)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) U21(ok(X)) -> ok(U21(X)) U31(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U31(X1,X2,X3,X4)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) isNatList(ok(X)) -> ok(isNatList(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Subterm Criterion Processor: simple projection: pi(U31#) = 1 problem: DPs: U31#(mark(X1),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)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X1,X2)) -> U11(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(U21(X)) -> U21(active(X)) active(U31(X1,X2,X3,X4)) -> U31(active(X1),X2,X3,X4) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) active(and(X1,X2)) -> and(active(X1),X2) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X1),X2) -> mark(U11(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) U21(mark(X)) -> mark(U21(X)) U31(mark(X1),X2,X3,X4) -> mark(U31(X1,X2,X3,X4)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(U21(X)) -> U21(proper(X)) proper(nil()) -> ok(nil()) proper(U31(X1,X2,X3,X4)) -> U31(proper(X1),proper(X2),proper(X3),proper(X4)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) U21(ok(X)) -> ok(U21(X)) U31(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U31(X1,X2,X3,X4)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) isNatList(ok(X)) -> ok(isNatList(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Subterm Criterion Processor: simple projection: pi(U31#) = 0 problem: DPs: 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)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X1,X2)) -> U11(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(U21(X)) -> U21(active(X)) active(U31(X1,X2,X3,X4)) -> U31(active(X1),X2,X3,X4) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) active(and(X1,X2)) -> and(active(X1),X2) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X1),X2) -> mark(U11(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) U21(mark(X)) -> mark(U21(X)) U31(mark(X1),X2,X3,X4) -> mark(U31(X1,X2,X3,X4)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(U21(X)) -> U21(proper(X)) proper(nil()) -> ok(nil()) proper(U31(X1,X2,X3,X4)) -> U31(proper(X1),proper(X2),proper(X3),proper(X4)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) U21(ok(X)) -> ok(U21(X)) U31(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U31(X1,X2,X3,X4)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) isNatList(ok(X)) -> ok(isNatList(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed DPs: U21#(mark(X)) -> U21#(X) U21#(ok(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)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X1,X2)) -> U11(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(U21(X)) -> U21(active(X)) active(U31(X1,X2,X3,X4)) -> U31(active(X1),X2,X3,X4) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) active(and(X1,X2)) -> and(active(X1),X2) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X1),X2) -> mark(U11(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) U21(mark(X)) -> mark(U21(X)) U31(mark(X1),X2,X3,X4) -> mark(U31(X1,X2,X3,X4)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(U21(X)) -> U21(proper(X)) proper(nil()) -> ok(nil()) proper(U31(X1,X2,X3,X4)) -> U31(proper(X1),proper(X2),proper(X3),proper(X4)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) U21(ok(X)) -> ok(U21(X)) U31(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U31(X1,X2,X3,X4)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) isNatList(ok(X)) -> ok(isNatList(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Subterm Criterion Processor: simple projection: pi(U21#) = 0 problem: DPs: 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)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X1,X2)) -> U11(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(U21(X)) -> U21(active(X)) active(U31(X1,X2,X3,X4)) -> U31(active(X1),X2,X3,X4) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) active(and(X1,X2)) -> and(active(X1),X2) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X1),X2) -> mark(U11(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) U21(mark(X)) -> mark(U21(X)) U31(mark(X1),X2,X3,X4) -> mark(U31(X1,X2,X3,X4)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(U21(X)) -> U21(proper(X)) proper(nil()) -> ok(nil()) proper(U31(X1,X2,X3,X4)) -> U31(proper(X1),proper(X2),proper(X3),proper(X4)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) U21(ok(X)) -> ok(U21(X)) U31(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U31(X1,X2,X3,X4)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) isNatList(ok(X)) -> ok(isNatList(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed DPs: length#(mark(X)) -> length#(X) length#(ok(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)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X1,X2)) -> U11(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(U21(X)) -> U21(active(X)) active(U31(X1,X2,X3,X4)) -> U31(active(X1),X2,X3,X4) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) active(and(X1,X2)) -> and(active(X1),X2) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X1),X2) -> mark(U11(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) U21(mark(X)) -> mark(U21(X)) U31(mark(X1),X2,X3,X4) -> mark(U31(X1,X2,X3,X4)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(U21(X)) -> U21(proper(X)) proper(nil()) -> ok(nil()) proper(U31(X1,X2,X3,X4)) -> U31(proper(X1),proper(X2),proper(X3),proper(X4)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) U21(ok(X)) -> ok(U21(X)) U31(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U31(X1,X2,X3,X4)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) isNatList(ok(X)) -> ok(isNatList(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Subterm Criterion Processor: simple projection: pi(length#) = 0 problem: DPs: 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)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X1,X2)) -> U11(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(U21(X)) -> U21(active(X)) active(U31(X1,X2,X3,X4)) -> U31(active(X1),X2,X3,X4) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) active(and(X1,X2)) -> and(active(X1),X2) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X1),X2) -> mark(U11(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) U21(mark(X)) -> mark(U21(X)) U31(mark(X1),X2,X3,X4) -> mark(U31(X1,X2,X3,X4)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(U21(X)) -> U21(proper(X)) proper(nil()) -> ok(nil()) proper(U31(X1,X2,X3,X4)) -> U31(proper(X1),proper(X2),proper(X3),proper(X4)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) U21(ok(X)) -> ok(U21(X)) U31(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U31(X1,X2,X3,X4)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) isNatList(ok(X)) -> ok(isNatList(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed DPs: s#(mark(X)) -> s#(X) s#(ok(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)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X1,X2)) -> U11(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(U21(X)) -> U21(active(X)) active(U31(X1,X2,X3,X4)) -> U31(active(X1),X2,X3,X4) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) active(and(X1,X2)) -> and(active(X1),X2) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X1),X2) -> mark(U11(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) U21(mark(X)) -> mark(U21(X)) U31(mark(X1),X2,X3,X4) -> mark(U31(X1,X2,X3,X4)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(U21(X)) -> U21(proper(X)) proper(nil()) -> ok(nil()) proper(U31(X1,X2,X3,X4)) -> U31(proper(X1),proper(X2),proper(X3),proper(X4)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) U21(ok(X)) -> ok(U21(X)) U31(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U31(X1,X2,X3,X4)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) isNatList(ok(X)) -> ok(isNatList(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Subterm Criterion Processor: simple projection: pi(s#) = 0 problem: DPs: 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)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X1,X2)) -> U11(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(U21(X)) -> U21(active(X)) active(U31(X1,X2,X3,X4)) -> U31(active(X1),X2,X3,X4) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) active(and(X1,X2)) -> and(active(X1),X2) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X1),X2) -> mark(U11(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) U21(mark(X)) -> mark(U21(X)) U31(mark(X1),X2,X3,X4) -> mark(U31(X1,X2,X3,X4)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(U21(X)) -> U21(proper(X)) proper(nil()) -> ok(nil()) proper(U31(X1,X2,X3,X4)) -> U31(proper(X1),proper(X2),proper(X3),proper(X4)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) U21(ok(X)) -> ok(U21(X)) U31(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U31(X1,X2,X3,X4)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) isNatList(ok(X)) -> ok(isNatList(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed DPs: U11#(mark(X1),X2) -> U11#(X1,X2) U11#(ok(X1),ok(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)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X1,X2)) -> U11(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(U21(X)) -> U21(active(X)) active(U31(X1,X2,X3,X4)) -> U31(active(X1),X2,X3,X4) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) active(and(X1,X2)) -> and(active(X1),X2) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X1),X2) -> mark(U11(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) U21(mark(X)) -> mark(U21(X)) U31(mark(X1),X2,X3,X4) -> mark(U31(X1,X2,X3,X4)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(U21(X)) -> U21(proper(X)) proper(nil()) -> ok(nil()) proper(U31(X1,X2,X3,X4)) -> U31(proper(X1),proper(X2),proper(X3),proper(X4)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) U21(ok(X)) -> ok(U21(X)) U31(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U31(X1,X2,X3,X4)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) isNatList(ok(X)) -> ok(isNatList(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Subterm Criterion Processor: simple projection: pi(U11#) = 1 problem: DPs: U11#(mark(X1),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)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X1,X2)) -> U11(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(U21(X)) -> U21(active(X)) active(U31(X1,X2,X3,X4)) -> U31(active(X1),X2,X3,X4) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) active(and(X1,X2)) -> and(active(X1),X2) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X1),X2) -> mark(U11(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) U21(mark(X)) -> mark(U21(X)) U31(mark(X1),X2,X3,X4) -> mark(U31(X1,X2,X3,X4)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(U21(X)) -> U21(proper(X)) proper(nil()) -> ok(nil()) proper(U31(X1,X2,X3,X4)) -> U31(proper(X1),proper(X2),proper(X3),proper(X4)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) U21(ok(X)) -> ok(U21(X)) U31(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U31(X1,X2,X3,X4)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) isNatList(ok(X)) -> ok(isNatList(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Subterm Criterion Processor: simple projection: pi(U11#) = 0 problem: DPs: 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)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X1,X2)) -> U11(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(U21(X)) -> U21(active(X)) active(U31(X1,X2,X3,X4)) -> U31(active(X1),X2,X3,X4) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) active(and(X1,X2)) -> and(active(X1),X2) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X1),X2) -> mark(U11(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) U21(mark(X)) -> mark(U21(X)) U31(mark(X1),X2,X3,X4) -> mark(U31(X1,X2,X3,X4)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(U21(X)) -> U21(proper(X)) proper(nil()) -> ok(nil()) proper(U31(X1,X2,X3,X4)) -> U31(proper(X1),proper(X2),proper(X3),proper(X4)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) U21(ok(X)) -> ok(U21(X)) U31(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U31(X1,X2,X3,X4)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) isNatList(ok(X)) -> ok(isNatList(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed DPs: cons#(mark(X1),X2) -> cons#(X1,X2) cons#(ok(X1),ok(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)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X1,X2)) -> U11(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(U21(X)) -> U21(active(X)) active(U31(X1,X2,X3,X4)) -> U31(active(X1),X2,X3,X4) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) active(and(X1,X2)) -> and(active(X1),X2) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X1),X2) -> mark(U11(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) U21(mark(X)) -> mark(U21(X)) U31(mark(X1),X2,X3,X4) -> mark(U31(X1,X2,X3,X4)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(U21(X)) -> U21(proper(X)) proper(nil()) -> ok(nil()) proper(U31(X1,X2,X3,X4)) -> U31(proper(X1),proper(X2),proper(X3),proper(X4)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) U21(ok(X)) -> ok(U21(X)) U31(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U31(X1,X2,X3,X4)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) isNatList(ok(X)) -> ok(isNatList(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Subterm Criterion Processor: simple projection: pi(cons#) = 1 problem: DPs: 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)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X1,X2)) -> U11(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(U21(X)) -> U21(active(X)) active(U31(X1,X2,X3,X4)) -> U31(active(X1),X2,X3,X4) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) active(and(X1,X2)) -> and(active(X1),X2) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X1),X2) -> mark(U11(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) U21(mark(X)) -> mark(U21(X)) U31(mark(X1),X2,X3,X4) -> mark(U31(X1,X2,X3,X4)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(U21(X)) -> U21(proper(X)) proper(nil()) -> ok(nil()) proper(U31(X1,X2,X3,X4)) -> U31(proper(X1),proper(X2),proper(X3),proper(X4)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) U21(ok(X)) -> ok(U21(X)) U31(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U31(X1,X2,X3,X4)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) isNatList(ok(X)) -> ok(isNatList(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Subterm Criterion Processor: simple projection: pi(cons#) = 0 problem: DPs: 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)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X1,X2)) -> U11(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(U21(X)) -> U21(active(X)) active(U31(X1,X2,X3,X4)) -> U31(active(X1),X2,X3,X4) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) active(and(X1,X2)) -> and(active(X1),X2) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X1),X2) -> mark(U11(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) U21(mark(X)) -> mark(U21(X)) U31(mark(X1),X2,X3,X4) -> mark(U31(X1,X2,X3,X4)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(U21(X)) -> U21(proper(X)) proper(nil()) -> ok(nil()) proper(U31(X1,X2,X3,X4)) -> U31(proper(X1),proper(X2),proper(X3),proper(X4)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) U21(ok(X)) -> ok(U21(X)) U31(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U31(X1,X2,X3,X4)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) isNatList(ok(X)) -> ok(isNatList(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed