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)) EDG 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: 1144/10000 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: proper#(isNatIList(X)) -> proper#(X) proper#(isNatList(X)) -> proper#(X) proper#(isNat(X)) -> proper#(X) proper#(and(X1,X2)) -> proper#(X1) proper#(and(X1,X2)) -> proper#(X2) proper#(take(X1,X2)) -> proper#(X1) proper#(take(X1,X2)) -> proper#(X2) proper#(U31(X1,X2,X3,X4)) -> proper#(X1) proper#(U31(X1,X2,X3,X4)) -> proper#(X2) proper#(U31(X1,X2,X3,X4)) -> proper#(X3) proper#(U31(X1,X2,X3,X4)) -> proper#(X4) proper#(U21(X)) -> proper#(X) proper#(length(X)) -> proper#(X) proper#(s(X)) -> proper#(X) proper#(U11(X1,X2)) -> proper#(X1) proper#(U11(X1,X2)) -> proper#(X2) proper#(cons(X1,X2)) -> proper#(X1) proper#(cons(X1,X2)) -> proper#(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)) Matrix Interpretation Processor: dimension: 1 interpretation: [proper#](x0) = x0 + 1, [top](x0) = 0, [ok](x0) = 0, [proper](x0) = x0, [isNatIList](x0) = x0 + 1, [isNatList](x0) = x0 + 1, [isNat](x0) = x0 + 1, [and](x0, x1) = x0 + x1, [take](x0, x1) = x0 + x1, [U31](x0, x1, x2, x3) = x0 + x1 + x2 + x3, [nil] = 0, [U21](x0) = x0 + 1, [s](x0) = x0 + 1, [length](x0) = x0 + 1, [U11](x0, x1) = x0 + x1 + 1, [tt] = 1, [mark](x0) = 0, [cons](x0, x1) = x0 + x1, [0] = 1, [active](x0) = x0, [zeros] = 0 orientation: proper#(isNatIList(X)) = X + 2 >= X + 1 = proper#(X) proper#(isNatList(X)) = X + 2 >= X + 1 = proper#(X) proper#(isNat(X)) = X + 2 >= X + 1 = proper#(X) proper#(and(X1,X2)) = X1 + X2 + 1 >= X1 + 1 = proper#(X1) proper#(and(X1,X2)) = X1 + X2 + 1 >= X2 + 1 = proper#(X2) proper#(take(X1,X2)) = X1 + X2 + 1 >= X1 + 1 = proper#(X1) proper#(take(X1,X2)) = X1 + X2 + 1 >= X2 + 1 = proper#(X2) proper#(U31(X1,X2,X3,X4)) = X1 + X2 + X3 + X4 + 1 >= X1 + 1 = proper#(X1) proper#(U31(X1,X2,X3,X4)) = X1 + X2 + X3 + X4 + 1 >= X2 + 1 = proper#(X2) proper#(U31(X1,X2,X3,X4)) = X1 + X2 + X3 + X4 + 1 >= X3 + 1 = proper#(X3) proper#(U31(X1,X2,X3,X4)) = X1 + X2 + X3 + X4 + 1 >= X4 + 1 = proper#(X4) proper#(U21(X)) = X + 2 >= X + 1 = proper#(X) proper#(length(X)) = X + 2 >= X + 1 = proper#(X) proper#(s(X)) = X + 2 >= X + 1 = proper#(X) proper#(U11(X1,X2)) = X1 + X2 + 2 >= X1 + 1 = proper#(X1) proper#(U11(X1,X2)) = X1 + X2 + 2 >= X2 + 1 = proper#(X2) proper#(cons(X1,X2)) = X1 + X2 + 1 >= X1 + 1 = proper#(X1) proper#(cons(X1,X2)) = X1 + X2 + 1 >= X2 + 1 = proper#(X2) active(zeros()) = 0 >= 0 = mark(cons(0(),zeros())) active(U11(tt(),L)) = L + 2 >= 0 = mark(s(length(L))) active(U21(tt())) = 2 >= 0 = mark(nil()) active(U31(tt(),IL,M,N)) = IL + M + N + 1 >= 0 = mark(cons(N,take(M,IL))) active(and(tt(),X)) = X + 1 >= 0 = mark(X) active(isNat(0())) = 2 >= 0 = mark(tt()) active(isNat(length(V1))) = V1 + 2 >= 0 = mark(isNatList(V1)) active(isNat(s(V1))) = V1 + 2 >= 0 = mark(isNat(V1)) active(isNatIList(V)) = V + 1 >= 0 = mark(isNatList(V)) active(isNatIList(zeros())) = 1 >= 0 = mark(tt()) active(isNatIList(cons(V1,V2))) = V1 + V2 + 1 >= 0 = mark(and(isNat(V1),isNatIList(V2))) active(isNatList(nil())) = 1 >= 0 = mark(tt()) active(isNatList(cons(V1,V2))) = V1 + V2 + 1 >= 0 = mark(and(isNat(V1),isNatList(V2))) active(isNatList(take(V1,V2))) = V1 + V2 + 1 >= 0 = mark(and(isNat(V1),isNatIList(V2))) active(length(nil())) = 1 >= 0 = mark(0()) active(length(cons(N,L))) = L + N + 1 >= 0 = mark(U11(and(isNatList(L),isNat(N)),L)) active(take(0(),IL)) = IL + 1 >= 0 = mark(U21(isNatIList(IL))) active(take(s(M),cons(N,IL))) = IL + M + N + 1 >= 0 = mark(U31(and(isNatIList(IL),and(isNat(M),isNat(N))),IL,M,N)) active(cons(X1,X2)) = X1 + X2 >= X1 + X2 = cons(active(X1),X2) active(U11(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = U11(active(X1),X2) active(s(X)) = X + 1 >= X + 1 = s(active(X)) active(length(X)) = X + 1 >= X + 1 = length(active(X)) active(U21(X)) = X + 1 >= X + 1 = U21(active(X)) active(U31(X1,X2,X3,X4)) = X1 + X2 + X3 + X4 >= X1 + X2 + X3 + X4 = U31(active(X1),X2,X3,X4) active(take(X1,X2)) = X1 + X2 >= X1 + X2 = take(active(X1),X2) active(take(X1,X2)) = X1 + X2 >= X1 + X2 = take(X1,active(X2)) active(and(X1,X2)) = X1 + X2 >= X1 + X2 = and(active(X1),X2) cons(mark(X1),X2) = X2 >= 0 = mark(cons(X1,X2)) U11(mark(X1),X2) = X2 + 1 >= 0 = mark(U11(X1,X2)) s(mark(X)) = 1 >= 0 = mark(s(X)) length(mark(X)) = 1 >= 0 = mark(length(X)) U21(mark(X)) = 1 >= 0 = mark(U21(X)) U31(mark(X1),X2,X3,X4) = X2 + X3 + X4 >= 0 = mark(U31(X1,X2,X3,X4)) take(mark(X1),X2) = X2 >= 0 = mark(take(X1,X2)) take(X1,mark(X2)) = X1 >= 0 = mark(take(X1,X2)) and(mark(X1),X2) = X2 >= 0 = mark(and(X1,X2)) proper(zeros()) = 0 >= 0 = ok(zeros()) proper(cons(X1,X2)) = X1 + X2 >= X1 + X2 = cons(proper(X1),proper(X2)) proper(0()) = 1 >= 0 = ok(0()) proper(U11(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = U11(proper(X1),proper(X2)) proper(tt()) = 1 >= 0 = ok(tt()) proper(s(X)) = X + 1 >= X + 1 = s(proper(X)) proper(length(X)) = X + 1 >= X + 1 = length(proper(X)) proper(U21(X)) = X + 1 >= X + 1 = U21(proper(X)) proper(nil()) = 0 >= 0 = ok(nil()) proper(U31(X1,X2,X3,X4)) = X1 + X2 + X3 + X4 >= X1 + X2 + X3 + X4 = U31(proper(X1),proper(X2),proper(X3),proper(X4)) proper(take(X1,X2)) = X1 + X2 >= X1 + X2 = take(proper(X1),proper(X2)) proper(and(X1,X2)) = X1 + X2 >= X1 + X2 = and(proper(X1),proper(X2)) proper(isNat(X)) = X + 1 >= X + 1 = isNat(proper(X)) proper(isNatList(X)) = X + 1 >= X + 1 = isNatList(proper(X)) proper(isNatIList(X)) = X + 1 >= X + 1 = isNatIList(proper(X)) cons(ok(X1),ok(X2)) = 0 >= 0 = ok(cons(X1,X2)) U11(ok(X1),ok(X2)) = 1 >= 0 = ok(U11(X1,X2)) s(ok(X)) = 1 >= 0 = ok(s(X)) length(ok(X)) = 1 >= 0 = ok(length(X)) U21(ok(X)) = 1 >= 0 = ok(U21(X)) U31(ok(X1),ok(X2),ok(X3),ok(X4)) = 0 >= 0 = ok(U31(X1,X2,X3,X4)) take(ok(X1),ok(X2)) = 0 >= 0 = ok(take(X1,X2)) and(ok(X1),ok(X2)) = 0 >= 0 = ok(and(X1,X2)) isNat(ok(X)) = 1 >= 0 = ok(isNat(X)) isNatList(ok(X)) = 1 >= 0 = ok(isNatList(X)) isNatIList(ok(X)) = 1 >= 0 = ok(isNatIList(X)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) problem: DPs: proper#(and(X1,X2)) -> proper#(X1) proper#(and(X1,X2)) -> proper#(X2) proper#(take(X1,X2)) -> proper#(X1) proper#(take(X1,X2)) -> proper#(X2) proper#(U31(X1,X2,X3,X4)) -> proper#(X1) proper#(U31(X1,X2,X3,X4)) -> proper#(X2) proper#(U31(X1,X2,X3,X4)) -> proper#(X3) proper#(U31(X1,X2,X3,X4)) -> proper#(X4) proper#(cons(X1,X2)) -> proper#(X1) proper#(cons(X1,X2)) -> proper#(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)) Matrix Interpretation Processor: dimension: 1 interpretation: [proper#](x0) = x0 + 1, [top](x0) = 0, [ok](x0) = x0, [proper](x0) = x0, [isNatIList](x0) = 0, [isNatList](x0) = 0, [isNat](x0) = 0, [and](x0, x1) = x0 + x1 + 1, [take](x0, x1) = x0 + x1 + 1, [U31](x0, x1, x2, x3) = x0 + x1 + x2 + x3 + 1, [nil] = 1, [U21](x0) = 0, [s](x0) = x0, [length](x0) = 1, [U11](x0, x1) = 0, [tt] = 0, [mark](x0) = x0, [cons](x0, x1) = x0 + x1 + 1, [0] = 0, [active](x0) = x0 + 1, [zeros] = 0 orientation: proper#(and(X1,X2)) = X1 + X2 + 2 >= X1 + 1 = proper#(X1) proper#(and(X1,X2)) = X1 + X2 + 2 >= X2 + 1 = proper#(X2) proper#(take(X1,X2)) = X1 + X2 + 2 >= X1 + 1 = proper#(X1) proper#(take(X1,X2)) = X1 + X2 + 2 >= X2 + 1 = proper#(X2) proper#(U31(X1,X2,X3,X4)) = X1 + X2 + X3 + X4 + 2 >= X1 + 1 = proper#(X1) proper#(U31(X1,X2,X3,X4)) = X1 + X2 + X3 + X4 + 2 >= X2 + 1 = proper#(X2) proper#(U31(X1,X2,X3,X4)) = X1 + X2 + X3 + X4 + 2 >= X3 + 1 = proper#(X3) proper#(U31(X1,X2,X3,X4)) = X1 + X2 + X3 + X4 + 2 >= X4 + 1 = proper#(X4) proper#(cons(X1,X2)) = X1 + X2 + 2 >= X1 + 1 = proper#(X1) proper#(cons(X1,X2)) = X1 + X2 + 2 >= X2 + 1 = proper#(X2) active(zeros()) = 1 >= 1 = mark(cons(0(),zeros())) active(U11(tt(),L)) = 1 >= 1 = mark(s(length(L))) active(U21(tt())) = 1 >= 1 = mark(nil()) active(U31(tt(),IL,M,N)) = IL + M + N + 2 >= IL + M + N + 2 = mark(cons(N,take(M,IL))) active(and(tt(),X)) = X + 2 >= X = mark(X) active(isNat(0())) = 1 >= 0 = mark(tt()) active(isNat(length(V1))) = 1 >= 0 = mark(isNatList(V1)) active(isNat(s(V1))) = 1 >= 0 = mark(isNat(V1)) active(isNatIList(V)) = 1 >= 0 = mark(isNatList(V)) active(isNatIList(zeros())) = 1 >= 0 = mark(tt()) active(isNatIList(cons(V1,V2))) = 1 >= 1 = mark(and(isNat(V1),isNatIList(V2))) active(isNatList(nil())) = 1 >= 0 = mark(tt()) active(isNatList(cons(V1,V2))) = 1 >= 1 = mark(and(isNat(V1),isNatList(V2))) active(isNatList(take(V1,V2))) = 1 >= 1 = mark(and(isNat(V1),isNatIList(V2))) active(length(nil())) = 2 >= 0 = mark(0()) active(length(cons(N,L))) = 2 >= 0 = mark(U11(and(isNatList(L),isNat(N)),L)) active(take(0(),IL)) = IL + 2 >= 0 = mark(U21(isNatIList(IL))) active(take(s(M),cons(N,IL))) = IL + M + N + 3 >= IL + M + N + 3 = mark(U31(and(isNatIList(IL),and(isNat(M),isNat(N))),IL,M,N)) active(cons(X1,X2)) = X1 + X2 + 2 >= X1 + X2 + 2 = cons(active(X1),X2) active(U11(X1,X2)) = 1 >= 0 = U11(active(X1),X2) active(s(X)) = X + 1 >= X + 1 = s(active(X)) active(length(X)) = 2 >= 1 = length(active(X)) active(U21(X)) = 1 >= 0 = U21(active(X)) active(U31(X1,X2,X3,X4)) = X1 + X2 + X3 + X4 + 2 >= X1 + X2 + X3 + X4 + 2 = U31(active(X1),X2,X3,X4) active(take(X1,X2)) = X1 + X2 + 2 >= X1 + X2 + 2 = take(active(X1),X2) active(take(X1,X2)) = X1 + X2 + 2 >= X1 + X2 + 2 = take(X1,active(X2)) active(and(X1,X2)) = X1 + X2 + 2 >= X1 + X2 + 2 = and(active(X1),X2) cons(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(cons(X1,X2)) U11(mark(X1),X2) = 0 >= 0 = mark(U11(X1,X2)) s(mark(X)) = X >= X = mark(s(X)) length(mark(X)) = 1 >= 1 = mark(length(X)) U21(mark(X)) = 0 >= 0 = mark(U21(X)) U31(mark(X1),X2,X3,X4) = X1 + X2 + X3 + X4 + 1 >= X1 + X2 + X3 + X4 + 1 = mark(U31(X1,X2,X3,X4)) take(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(take(X1,X2)) take(X1,mark(X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(take(X1,X2)) and(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(and(X1,X2)) proper(zeros()) = 0 >= 0 = ok(zeros()) proper(cons(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = cons(proper(X1),proper(X2)) proper(0()) = 0 >= 0 = ok(0()) proper(U11(X1,X2)) = 0 >= 0 = U11(proper(X1),proper(X2)) proper(tt()) = 0 >= 0 = ok(tt()) proper(s(X)) = X >= X = s(proper(X)) proper(length(X)) = 1 >= 1 = length(proper(X)) proper(U21(X)) = 0 >= 0 = U21(proper(X)) proper(nil()) = 1 >= 1 = ok(nil()) proper(U31(X1,X2,X3,X4)) = X1 + X2 + X3 + X4 + 1 >= X1 + X2 + X3 + X4 + 1 = U31(proper(X1),proper(X2),proper(X3),proper(X4)) proper(take(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = take(proper(X1),proper(X2)) proper(and(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = and(proper(X1),proper(X2)) proper(isNat(X)) = 0 >= 0 = isNat(proper(X)) proper(isNatList(X)) = 0 >= 0 = isNatList(proper(X)) proper(isNatIList(X)) = 0 >= 0 = isNatIList(proper(X)) cons(ok(X1),ok(X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = ok(cons(X1,X2)) U11(ok(X1),ok(X2)) = 0 >= 0 = ok(U11(X1,X2)) s(ok(X)) = X >= X = ok(s(X)) length(ok(X)) = 1 >= 1 = ok(length(X)) U21(ok(X)) = 0 >= 0 = ok(U21(X)) U31(ok(X1),ok(X2),ok(X3),ok(X4)) = X1 + X2 + X3 + X4 + 1 >= X1 + X2 + X3 + X4 + 1 = ok(U31(X1,X2,X3,X4)) take(ok(X1),ok(X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = ok(take(X1,X2)) and(ok(X1),ok(X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = ok(and(X1,X2)) isNat(ok(X)) = 0 >= 0 = ok(isNat(X)) isNatList(ok(X)) = 0 >= 0 = ok(isNatList(X)) isNatIList(ok(X)) = 0 >= 0 = ok(isNatIList(X)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) 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: active#(and(X1,X2)) -> active#(X1) active#(take(X1,X2)) -> active#(X2) active#(take(X1,X2)) -> active#(X1) active#(U31(X1,X2,X3,X4)) -> active#(X1) active#(U21(X)) -> active#(X) active#(length(X)) -> active#(X) active#(s(X)) -> active#(X) active#(U11(X1,X2)) -> active#(X1) active#(cons(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)) Matrix Interpretation Processor: dimension: 1 interpretation: [active#](x0) = x0 + 1, [top](x0) = 0, [ok](x0) = x0, [proper](x0) = x0, [isNatIList](x0) = 1, [isNatList](x0) = 1, [isNat](x0) = 1, [and](x0, x1) = x0 + 1, [take](x0, x1) = x0 + x1 + 1, [U31](x0, x1, x2, x3) = x0 + 1, [nil] = 0, [U21](x0) = x0 + 1, [s](x0) = x0 + 1, [length](x0) = x0 + 1, [U11](x0, x1) = x0 + 1, [tt] = 0, [mark](x0) = 1, [cons](x0, x1) = x0 + 1, [0] = 0, [active](x0) = x0, [zeros] = 1 orientation: active#(and(X1,X2)) = X1 + 2 >= X1 + 1 = active#(X1) active#(take(X1,X2)) = X1 + X2 + 2 >= X2 + 1 = active#(X2) active#(take(X1,X2)) = X1 + X2 + 2 >= X1 + 1 = active#(X1) active#(U31(X1,X2,X3,X4)) = X1 + 2 >= X1 + 1 = active#(X1) active#(U21(X)) = X + 2 >= X + 1 = active#(X) active#(length(X)) = X + 2 >= X + 1 = active#(X) active#(s(X)) = X + 2 >= X + 1 = active#(X) active#(U11(X1,X2)) = X1 + 2 >= X1 + 1 = active#(X1) active#(cons(X1,X2)) = X1 + 2 >= X1 + 1 = active#(X1) active(zeros()) = 1 >= 1 = mark(cons(0(),zeros())) active(U11(tt(),L)) = 1 >= 1 = mark(s(length(L))) active(U21(tt())) = 1 >= 1 = mark(nil()) active(U31(tt(),IL,M,N)) = 1 >= 1 = mark(cons(N,take(M,IL))) active(and(tt(),X)) = 1 >= 1 = mark(X) active(isNat(0())) = 1 >= 1 = mark(tt()) active(isNat(length(V1))) = 1 >= 1 = mark(isNatList(V1)) active(isNat(s(V1))) = 1 >= 1 = mark(isNat(V1)) active(isNatIList(V)) = 1 >= 1 = mark(isNatList(V)) active(isNatIList(zeros())) = 1 >= 1 = mark(tt()) active(isNatIList(cons(V1,V2))) = 1 >= 1 = mark(and(isNat(V1),isNatIList(V2))) active(isNatList(nil())) = 1 >= 1 = mark(tt()) active(isNatList(cons(V1,V2))) = 1 >= 1 = mark(and(isNat(V1),isNatList(V2))) active(isNatList(take(V1,V2))) = 1 >= 1 = mark(and(isNat(V1),isNatIList(V2))) active(length(nil())) = 1 >= 1 = mark(0()) active(length(cons(N,L))) = N + 2 >= 1 = mark(U11(and(isNatList(L),isNat(N)),L)) active(take(0(),IL)) = IL + 1 >= 1 = mark(U21(isNatIList(IL))) active(take(s(M),cons(N,IL))) = M + N + 3 >= 1 = mark(U31(and(isNatIList(IL),and(isNat(M),isNat(N))),IL,M,N)) active(cons(X1,X2)) = X1 + 1 >= X1 + 1 = cons(active(X1),X2) active(U11(X1,X2)) = X1 + 1 >= X1 + 1 = U11(active(X1),X2) active(s(X)) = X + 1 >= X + 1 = s(active(X)) active(length(X)) = X + 1 >= X + 1 = length(active(X)) active(U21(X)) = X + 1 >= X + 1 = U21(active(X)) active(U31(X1,X2,X3,X4)) = X1 + 1 >= X1 + 1 = U31(active(X1),X2,X3,X4) active(take(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = take(active(X1),X2) active(take(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = take(X1,active(X2)) active(and(X1,X2)) = X1 + 1 >= X1 + 1 = and(active(X1),X2) cons(mark(X1),X2) = 2 >= 1 = mark(cons(X1,X2)) U11(mark(X1),X2) = 2 >= 1 = mark(U11(X1,X2)) s(mark(X)) = 2 >= 1 = mark(s(X)) length(mark(X)) = 2 >= 1 = mark(length(X)) U21(mark(X)) = 2 >= 1 = mark(U21(X)) U31(mark(X1),X2,X3,X4) = 2 >= 1 = mark(U31(X1,X2,X3,X4)) take(mark(X1),X2) = X2 + 2 >= 1 = mark(take(X1,X2)) take(X1,mark(X2)) = X1 + 2 >= 1 = mark(take(X1,X2)) and(mark(X1),X2) = 2 >= 1 = mark(and(X1,X2)) proper(zeros()) = 1 >= 1 = ok(zeros()) proper(cons(X1,X2)) = X1 + 1 >= X1 + 1 = cons(proper(X1),proper(X2)) proper(0()) = 0 >= 0 = ok(0()) proper(U11(X1,X2)) = X1 + 1 >= X1 + 1 = U11(proper(X1),proper(X2)) proper(tt()) = 0 >= 0 = ok(tt()) proper(s(X)) = X + 1 >= X + 1 = s(proper(X)) proper(length(X)) = X + 1 >= X + 1 = length(proper(X)) proper(U21(X)) = X + 1 >= X + 1 = U21(proper(X)) proper(nil()) = 0 >= 0 = ok(nil()) proper(U31(X1,X2,X3,X4)) = X1 + 1 >= X1 + 1 = U31(proper(X1),proper(X2),proper(X3),proper(X4)) proper(take(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = take(proper(X1),proper(X2)) proper(and(X1,X2)) = X1 + 1 >= X1 + 1 = and(proper(X1),proper(X2)) proper(isNat(X)) = 1 >= 1 = isNat(proper(X)) proper(isNatList(X)) = 1 >= 1 = isNatList(proper(X)) proper(isNatIList(X)) = 1 >= 1 = isNatIList(proper(X)) cons(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(cons(X1,X2)) U11(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(U11(X1,X2)) s(ok(X)) = X + 1 >= X + 1 = ok(s(X)) length(ok(X)) = X + 1 >= X + 1 = ok(length(X)) U21(ok(X)) = X + 1 >= X + 1 = ok(U21(X)) U31(ok(X1),ok(X2),ok(X3),ok(X4)) = X1 + 1 >= X1 + 1 = ok(U31(X1,X2,X3,X4)) take(ok(X1),ok(X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = ok(take(X1,X2)) and(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(and(X1,X2)) isNat(ok(X)) = 1 >= 1 = ok(isNat(X)) isNatList(ok(X)) = 1 >= 1 = ok(isNatList(X)) isNatIList(ok(X)) = 1 >= 1 = ok(isNatIList(X)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) 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)) Matrix Interpretation Processor: dimension: 1 interpretation: [isNatList#](x0) = x0 + 1, [top](x0) = 0, [ok](x0) = x0 + 1, [proper](x0) = 1, [isNatIList](x0) = x0, [isNatList](x0) = x0, [isNat](x0) = x0, [and](x0, x1) = x1, [take](x0, x1) = x0, [U31](x0, x1, x2, x3) = x3, [nil] = 0, [U21](x0) = x0, [s](x0) = x0, [length](x0) = x0, [U11](x0, x1) = x0, [tt] = 0, [mark](x0) = 0, [cons](x0, x1) = x1, [0] = 0, [active](x0) = x0, [zeros] = 0 orientation: isNatList#(ok(X)) = X + 2 >= X + 1 = isNatList#(X) active(zeros()) = 0 >= 0 = mark(cons(0(),zeros())) active(U11(tt(),L)) = 0 >= 0 = mark(s(length(L))) active(U21(tt())) = 0 >= 0 = mark(nil()) active(U31(tt(),IL,M,N)) = N >= 0 = mark(cons(N,take(M,IL))) active(and(tt(),X)) = X >= 0 = mark(X) active(isNat(0())) = 0 >= 0 = mark(tt()) active(isNat(length(V1))) = V1 >= 0 = mark(isNatList(V1)) active(isNat(s(V1))) = V1 >= 0 = mark(isNat(V1)) active(isNatIList(V)) = V >= 0 = mark(isNatList(V)) active(isNatIList(zeros())) = 0 >= 0 = mark(tt()) active(isNatIList(cons(V1,V2))) = V2 >= 0 = mark(and(isNat(V1),isNatIList(V2))) active(isNatList(nil())) = 0 >= 0 = mark(tt()) active(isNatList(cons(V1,V2))) = V2 >= 0 = mark(and(isNat(V1),isNatList(V2))) active(isNatList(take(V1,V2))) = V1 >= 0 = mark(and(isNat(V1),isNatIList(V2))) active(length(nil())) = 0 >= 0 = mark(0()) active(length(cons(N,L))) = L >= 0 = mark(U11(and(isNatList(L),isNat(N)),L)) active(take(0(),IL)) = 0 >= 0 = mark(U21(isNatIList(IL))) active(take(s(M),cons(N,IL))) = M >= 0 = mark(U31(and(isNatIList(IL),and(isNat(M),isNat(N))),IL,M,N)) active(cons(X1,X2)) = X2 >= X2 = cons(active(X1),X2) active(U11(X1,X2)) = X1 >= X1 = U11(active(X1),X2) active(s(X)) = X >= X = s(active(X)) active(length(X)) = X >= X = length(active(X)) active(U21(X)) = X >= X = U21(active(X)) active(U31(X1,X2,X3,X4)) = X4 >= X4 = U31(active(X1),X2,X3,X4) active(take(X1,X2)) = X1 >= X1 = take(active(X1),X2) active(take(X1,X2)) = X1 >= X1 = take(X1,active(X2)) active(and(X1,X2)) = X2 >= X2 = and(active(X1),X2) cons(mark(X1),X2) = X2 >= 0 = mark(cons(X1,X2)) U11(mark(X1),X2) = 0 >= 0 = mark(U11(X1,X2)) s(mark(X)) = 0 >= 0 = mark(s(X)) length(mark(X)) = 0 >= 0 = mark(length(X)) U21(mark(X)) = 0 >= 0 = mark(U21(X)) U31(mark(X1),X2,X3,X4) = X4 >= 0 = mark(U31(X1,X2,X3,X4)) take(mark(X1),X2) = 0 >= 0 = mark(take(X1,X2)) take(X1,mark(X2)) = X1 >= 0 = mark(take(X1,X2)) and(mark(X1),X2) = X2 >= 0 = mark(and(X1,X2)) proper(zeros()) = 1 >= 1 = ok(zeros()) proper(cons(X1,X2)) = 1 >= 1 = cons(proper(X1),proper(X2)) proper(0()) = 1 >= 1 = ok(0()) proper(U11(X1,X2)) = 1 >= 1 = U11(proper(X1),proper(X2)) proper(tt()) = 1 >= 1 = ok(tt()) proper(s(X)) = 1 >= 1 = s(proper(X)) proper(length(X)) = 1 >= 1 = length(proper(X)) proper(U21(X)) = 1 >= 1 = U21(proper(X)) proper(nil()) = 1 >= 1 = ok(nil()) proper(U31(X1,X2,X3,X4)) = 1 >= 1 = U31(proper(X1),proper(X2),proper(X3),proper(X4)) proper(take(X1,X2)) = 1 >= 1 = take(proper(X1),proper(X2)) proper(and(X1,X2)) = 1 >= 1 = and(proper(X1),proper(X2)) proper(isNat(X)) = 1 >= 1 = isNat(proper(X)) proper(isNatList(X)) = 1 >= 1 = isNatList(proper(X)) proper(isNatIList(X)) = 1 >= 1 = isNatIList(proper(X)) cons(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(cons(X1,X2)) U11(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(U11(X1,X2)) s(ok(X)) = X + 1 >= X + 1 = ok(s(X)) length(ok(X)) = X + 1 >= X + 1 = ok(length(X)) U21(ok(X)) = X + 1 >= X + 1 = ok(U21(X)) U31(ok(X1),ok(X2),ok(X3),ok(X4)) = X4 + 1 >= X4 + 1 = ok(U31(X1,X2,X3,X4)) take(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(take(X1,X2)) and(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(and(X1,X2)) isNat(ok(X)) = X + 1 >= X + 1 = ok(isNat(X)) isNatList(ok(X)) = X + 1 >= X + 1 = ok(isNatList(X)) isNatIList(ok(X)) = X + 1 >= X + 1 = ok(isNatIList(X)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) 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)) Matrix Interpretation Processor: dimension: 1 interpretation: [isNat#](x0) = x0 + 1, [top](x0) = 0, [ok](x0) = x0 + 1, [proper](x0) = 1, [isNatIList](x0) = x0, [isNatList](x0) = x0, [isNat](x0) = x0, [and](x0, x1) = x1, [take](x0, x1) = x0, [U31](x0, x1, x2, x3) = x3, [nil] = 0, [U21](x0) = x0, [s](x0) = x0, [length](x0) = x0, [U11](x0, x1) = x0, [tt] = 0, [mark](x0) = 0, [cons](x0, x1) = x1, [0] = 0, [active](x0) = x0, [zeros] = 0 orientation: isNat#(ok(X)) = X + 2 >= X + 1 = isNat#(X) active(zeros()) = 0 >= 0 = mark(cons(0(),zeros())) active(U11(tt(),L)) = 0 >= 0 = mark(s(length(L))) active(U21(tt())) = 0 >= 0 = mark(nil()) active(U31(tt(),IL,M,N)) = N >= 0 = mark(cons(N,take(M,IL))) active(and(tt(),X)) = X >= 0 = mark(X) active(isNat(0())) = 0 >= 0 = mark(tt()) active(isNat(length(V1))) = V1 >= 0 = mark(isNatList(V1)) active(isNat(s(V1))) = V1 >= 0 = mark(isNat(V1)) active(isNatIList(V)) = V >= 0 = mark(isNatList(V)) active(isNatIList(zeros())) = 0 >= 0 = mark(tt()) active(isNatIList(cons(V1,V2))) = V2 >= 0 = mark(and(isNat(V1),isNatIList(V2))) active(isNatList(nil())) = 0 >= 0 = mark(tt()) active(isNatList(cons(V1,V2))) = V2 >= 0 = mark(and(isNat(V1),isNatList(V2))) active(isNatList(take(V1,V2))) = V1 >= 0 = mark(and(isNat(V1),isNatIList(V2))) active(length(nil())) = 0 >= 0 = mark(0()) active(length(cons(N,L))) = L >= 0 = mark(U11(and(isNatList(L),isNat(N)),L)) active(take(0(),IL)) = 0 >= 0 = mark(U21(isNatIList(IL))) active(take(s(M),cons(N,IL))) = M >= 0 = mark(U31(and(isNatIList(IL),and(isNat(M),isNat(N))),IL,M,N)) active(cons(X1,X2)) = X2 >= X2 = cons(active(X1),X2) active(U11(X1,X2)) = X1 >= X1 = U11(active(X1),X2) active(s(X)) = X >= X = s(active(X)) active(length(X)) = X >= X = length(active(X)) active(U21(X)) = X >= X = U21(active(X)) active(U31(X1,X2,X3,X4)) = X4 >= X4 = U31(active(X1),X2,X3,X4) active(take(X1,X2)) = X1 >= X1 = take(active(X1),X2) active(take(X1,X2)) = X1 >= X1 = take(X1,active(X2)) active(and(X1,X2)) = X2 >= X2 = and(active(X1),X2) cons(mark(X1),X2) = X2 >= 0 = mark(cons(X1,X2)) U11(mark(X1),X2) = 0 >= 0 = mark(U11(X1,X2)) s(mark(X)) = 0 >= 0 = mark(s(X)) length(mark(X)) = 0 >= 0 = mark(length(X)) U21(mark(X)) = 0 >= 0 = mark(U21(X)) U31(mark(X1),X2,X3,X4) = X4 >= 0 = mark(U31(X1,X2,X3,X4)) take(mark(X1),X2) = 0 >= 0 = mark(take(X1,X2)) take(X1,mark(X2)) = X1 >= 0 = mark(take(X1,X2)) and(mark(X1),X2) = X2 >= 0 = mark(and(X1,X2)) proper(zeros()) = 1 >= 1 = ok(zeros()) proper(cons(X1,X2)) = 1 >= 1 = cons(proper(X1),proper(X2)) proper(0()) = 1 >= 1 = ok(0()) proper(U11(X1,X2)) = 1 >= 1 = U11(proper(X1),proper(X2)) proper(tt()) = 1 >= 1 = ok(tt()) proper(s(X)) = 1 >= 1 = s(proper(X)) proper(length(X)) = 1 >= 1 = length(proper(X)) proper(U21(X)) = 1 >= 1 = U21(proper(X)) proper(nil()) = 1 >= 1 = ok(nil()) proper(U31(X1,X2,X3,X4)) = 1 >= 1 = U31(proper(X1),proper(X2),proper(X3),proper(X4)) proper(take(X1,X2)) = 1 >= 1 = take(proper(X1),proper(X2)) proper(and(X1,X2)) = 1 >= 1 = and(proper(X1),proper(X2)) proper(isNat(X)) = 1 >= 1 = isNat(proper(X)) proper(isNatList(X)) = 1 >= 1 = isNatList(proper(X)) proper(isNatIList(X)) = 1 >= 1 = isNatIList(proper(X)) cons(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(cons(X1,X2)) U11(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(U11(X1,X2)) s(ok(X)) = X + 1 >= X + 1 = ok(s(X)) length(ok(X)) = X + 1 >= X + 1 = ok(length(X)) U21(ok(X)) = X + 1 >= X + 1 = ok(U21(X)) U31(ok(X1),ok(X2),ok(X3),ok(X4)) = X4 + 1 >= X4 + 1 = ok(U31(X1,X2,X3,X4)) take(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(take(X1,X2)) and(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(and(X1,X2)) isNat(ok(X)) = X + 1 >= X + 1 = ok(isNat(X)) isNatList(ok(X)) = X + 1 >= X + 1 = ok(isNatList(X)) isNatIList(ok(X)) = X + 1 >= X + 1 = ok(isNatIList(X)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) 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)) Matrix Interpretation Processor: dimension: 1 interpretation: [isNatIList#](x0) = x0 + 1, [top](x0) = 0, [ok](x0) = x0 + 1, [proper](x0) = 1, [isNatIList](x0) = x0, [isNatList](x0) = x0, [isNat](x0) = x0, [and](x0, x1) = x1, [take](x0, x1) = x0, [U31](x0, x1, x2, x3) = x3, [nil] = 0, [U21](x0) = x0, [s](x0) = x0, [length](x0) = x0, [U11](x0, x1) = x0, [tt] = 0, [mark](x0) = 0, [cons](x0, x1) = x1, [0] = 0, [active](x0) = x0, [zeros] = 0 orientation: isNatIList#(ok(X)) = X + 2 >= X + 1 = isNatIList#(X) active(zeros()) = 0 >= 0 = mark(cons(0(),zeros())) active(U11(tt(),L)) = 0 >= 0 = mark(s(length(L))) active(U21(tt())) = 0 >= 0 = mark(nil()) active(U31(tt(),IL,M,N)) = N >= 0 = mark(cons(N,take(M,IL))) active(and(tt(),X)) = X >= 0 = mark(X) active(isNat(0())) = 0 >= 0 = mark(tt()) active(isNat(length(V1))) = V1 >= 0 = mark(isNatList(V1)) active(isNat(s(V1))) = V1 >= 0 = mark(isNat(V1)) active(isNatIList(V)) = V >= 0 = mark(isNatList(V)) active(isNatIList(zeros())) = 0 >= 0 = mark(tt()) active(isNatIList(cons(V1,V2))) = V2 >= 0 = mark(and(isNat(V1),isNatIList(V2))) active(isNatList(nil())) = 0 >= 0 = mark(tt()) active(isNatList(cons(V1,V2))) = V2 >= 0 = mark(and(isNat(V1),isNatList(V2))) active(isNatList(take(V1,V2))) = V1 >= 0 = mark(and(isNat(V1),isNatIList(V2))) active(length(nil())) = 0 >= 0 = mark(0()) active(length(cons(N,L))) = L >= 0 = mark(U11(and(isNatList(L),isNat(N)),L)) active(take(0(),IL)) = 0 >= 0 = mark(U21(isNatIList(IL))) active(take(s(M),cons(N,IL))) = M >= 0 = mark(U31(and(isNatIList(IL),and(isNat(M),isNat(N))),IL,M,N)) active(cons(X1,X2)) = X2 >= X2 = cons(active(X1),X2) active(U11(X1,X2)) = X1 >= X1 = U11(active(X1),X2) active(s(X)) = X >= X = s(active(X)) active(length(X)) = X >= X = length(active(X)) active(U21(X)) = X >= X = U21(active(X)) active(U31(X1,X2,X3,X4)) = X4 >= X4 = U31(active(X1),X2,X3,X4) active(take(X1,X2)) = X1 >= X1 = take(active(X1),X2) active(take(X1,X2)) = X1 >= X1 = take(X1,active(X2)) active(and(X1,X2)) = X2 >= X2 = and(active(X1),X2) cons(mark(X1),X2) = X2 >= 0 = mark(cons(X1,X2)) U11(mark(X1),X2) = 0 >= 0 = mark(U11(X1,X2)) s(mark(X)) = 0 >= 0 = mark(s(X)) length(mark(X)) = 0 >= 0 = mark(length(X)) U21(mark(X)) = 0 >= 0 = mark(U21(X)) U31(mark(X1),X2,X3,X4) = X4 >= 0 = mark(U31(X1,X2,X3,X4)) take(mark(X1),X2) = 0 >= 0 = mark(take(X1,X2)) take(X1,mark(X2)) = X1 >= 0 = mark(take(X1,X2)) and(mark(X1),X2) = X2 >= 0 = mark(and(X1,X2)) proper(zeros()) = 1 >= 1 = ok(zeros()) proper(cons(X1,X2)) = 1 >= 1 = cons(proper(X1),proper(X2)) proper(0()) = 1 >= 1 = ok(0()) proper(U11(X1,X2)) = 1 >= 1 = U11(proper(X1),proper(X2)) proper(tt()) = 1 >= 1 = ok(tt()) proper(s(X)) = 1 >= 1 = s(proper(X)) proper(length(X)) = 1 >= 1 = length(proper(X)) proper(U21(X)) = 1 >= 1 = U21(proper(X)) proper(nil()) = 1 >= 1 = ok(nil()) proper(U31(X1,X2,X3,X4)) = 1 >= 1 = U31(proper(X1),proper(X2),proper(X3),proper(X4)) proper(take(X1,X2)) = 1 >= 1 = take(proper(X1),proper(X2)) proper(and(X1,X2)) = 1 >= 1 = and(proper(X1),proper(X2)) proper(isNat(X)) = 1 >= 1 = isNat(proper(X)) proper(isNatList(X)) = 1 >= 1 = isNatList(proper(X)) proper(isNatIList(X)) = 1 >= 1 = isNatIList(proper(X)) cons(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(cons(X1,X2)) U11(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(U11(X1,X2)) s(ok(X)) = X + 1 >= X + 1 = ok(s(X)) length(ok(X)) = X + 1 >= X + 1 = ok(length(X)) U21(ok(X)) = X + 1 >= X + 1 = ok(U21(X)) U31(ok(X1),ok(X2),ok(X3),ok(X4)) = X4 + 1 >= X4 + 1 = ok(U31(X1,X2,X3,X4)) take(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(take(X1,X2)) and(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(and(X1,X2)) isNat(ok(X)) = X + 1 >= X + 1 = ok(isNat(X)) isNatList(ok(X)) = X + 1 >= X + 1 = ok(isNatList(X)) isNatIList(ok(X)) = X + 1 >= X + 1 = ok(isNatIList(X)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) 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#(ok(X1),ok(X2)) -> cons#(X1,X2) cons#(mark(X1),X2) -> cons#(X1,X2) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),L)) -> mark(s(length(L))) active(U21(tt())) -> mark(nil()) active(U31(tt(),IL,M,N)) -> mark(cons(N,take(M,IL))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(length(V1))) -> mark(isNatList(V1)) active(isNat(s(V1))) -> mark(isNat(V1)) active(isNatIList(V)) -> mark(isNatList(V)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(and(isNat(V1),isNatIList(V2))) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(and(isNat(V1),isNatList(V2))) active(isNatList(take(V1,V2))) -> mark(and(isNat(V1),isNatIList(V2))) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U11(and(isNatList(L),isNat(N)),L)) active(take(0(),IL)) -> mark(U21(isNatIList(IL))) active(take(s(M),cons(N,IL))) -> mark(U31(and(isNatIList(IL),and(isNat(M),isNat(N))),IL,M,N)) 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)) Matrix Interpretation Processor: dimension: 1 interpretation: [cons#](x0, x1) = x1 + 1, [top](x0) = 0, [ok](x0) = x0 + 1, [proper](x0) = x0 + 1, [isNatIList](x0) = x0, [isNatList](x0) = x0, [isNat](x0) = x0, [and](x0, x1) = x0, [take](x0, x1) = x1, [U31](x0, x1, x2, x3) = x3, [nil] = 0, [U21](x0) = x0, [s](x0) = x0, [length](x0) = x0, [U11](x0, x1) = x1, [tt] = 0, [mark](x0) = 0, [cons](x0, x1) = x0, [0] = 0, [active](x0) = x0, [zeros] = 1 orientation: cons#(ok(X1),ok(X2)) = X2 + 2 >= X2 + 1 = cons#(X1,X2) cons#(mark(X1),X2) = X2 + 1 >= X2 + 1 = cons#(X1,X2) active(zeros()) = 1 >= 0 = mark(cons(0(),zeros())) active(U11(tt(),L)) = L >= 0 = mark(s(length(L))) active(U21(tt())) = 0 >= 0 = mark(nil()) active(U31(tt(),IL,M,N)) = N >= 0 = mark(cons(N,take(M,IL))) active(and(tt(),X)) = 0 >= 0 = mark(X) active(isNat(0())) = 0 >= 0 = mark(tt()) active(isNat(length(V1))) = V1 >= 0 = mark(isNatList(V1)) active(isNat(s(V1))) = V1 >= 0 = mark(isNat(V1)) active(isNatIList(V)) = V >= 0 = mark(isNatList(V)) active(isNatIList(zeros())) = 1 >= 0 = mark(tt()) active(isNatIList(cons(V1,V2))) = V1 >= 0 = mark(and(isNat(V1),isNatIList(V2))) active(isNatList(nil())) = 0 >= 0 = mark(tt()) active(isNatList(cons(V1,V2))) = V1 >= 0 = mark(and(isNat(V1),isNatList(V2))) active(isNatList(take(V1,V2))) = V2 >= 0 = mark(and(isNat(V1),isNatIList(V2))) active(length(nil())) = 0 >= 0 = mark(0()) active(length(cons(N,L))) = N >= 0 = mark(U11(and(isNatList(L),isNat(N)),L)) active(take(0(),IL)) = IL >= 0 = mark(U21(isNatIList(IL))) active(take(s(M),cons(N,IL))) = N >= 0 = mark(U31(and(isNatIList(IL),and(isNat(M),isNat(N))),IL,M,N)) active(cons(X1,X2)) = X1 >= X1 = cons(active(X1),X2) active(U11(X1,X2)) = X2 >= X2 = U11(active(X1),X2) active(s(X)) = X >= X = s(active(X)) active(length(X)) = X >= X = length(active(X)) active(U21(X)) = X >= X = U21(active(X)) active(U31(X1,X2,X3,X4)) = X4 >= X4 = U31(active(X1),X2,X3,X4) active(take(X1,X2)) = X2 >= X2 = take(active(X1),X2) active(take(X1,X2)) = X2 >= X2 = take(X1,active(X2)) active(and(X1,X2)) = X1 >= X1 = and(active(X1),X2) cons(mark(X1),X2) = 0 >= 0 = mark(cons(X1,X2)) U11(mark(X1),X2) = X2 >= 0 = mark(U11(X1,X2)) s(mark(X)) = 0 >= 0 = mark(s(X)) length(mark(X)) = 0 >= 0 = mark(length(X)) U21(mark(X)) = 0 >= 0 = mark(U21(X)) U31(mark(X1),X2,X3,X4) = X4 >= 0 = mark(U31(X1,X2,X3,X4)) take(mark(X1),X2) = X2 >= 0 = mark(take(X1,X2)) take(X1,mark(X2)) = 0 >= 0 = mark(take(X1,X2)) and(mark(X1),X2) = 0 >= 0 = mark(and(X1,X2)) proper(zeros()) = 2 >= 2 = ok(zeros()) proper(cons(X1,X2)) = X1 + 1 >= X1 + 1 = cons(proper(X1),proper(X2)) proper(0()) = 1 >= 1 = ok(0()) proper(U11(X1,X2)) = X2 + 1 >= X2 + 1 = U11(proper(X1),proper(X2)) proper(tt()) = 1 >= 1 = ok(tt()) proper(s(X)) = X + 1 >= X + 1 = s(proper(X)) proper(length(X)) = X + 1 >= X + 1 = length(proper(X)) proper(U21(X)) = X + 1 >= X + 1 = U21(proper(X)) proper(nil()) = 1 >= 1 = ok(nil()) proper(U31(X1,X2,X3,X4)) = X4 + 1 >= X4 + 1 = U31(proper(X1),proper(X2),proper(X3),proper(X4)) proper(take(X1,X2)) = X2 + 1 >= X2 + 1 = take(proper(X1),proper(X2)) proper(and(X1,X2)) = X1 + 1 >= X1 + 1 = and(proper(X1),proper(X2)) proper(isNat(X)) = X + 1 >= X + 1 = isNat(proper(X)) proper(isNatList(X)) = X + 1 >= X + 1 = isNatList(proper(X)) proper(isNatIList(X)) = X + 1 >= X + 1 = isNatIList(proper(X)) cons(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(cons(X1,X2)) U11(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(U11(X1,X2)) s(ok(X)) = X + 1 >= X + 1 = ok(s(X)) length(ok(X)) = X + 1 >= X + 1 = ok(length(X)) U21(ok(X)) = X + 1 >= X + 1 = ok(U21(X)) U31(ok(X1),ok(X2),ok(X3),ok(X4)) = X4 + 1 >= X4 + 1 = ok(U31(X1,X2,X3,X4)) take(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(take(X1,X2)) and(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(and(X1,X2)) isNat(ok(X)) = X + 1 >= X + 1 = ok(isNat(X)) isNatList(ok(X)) = X + 1 >= X + 1 = ok(isNatList(X)) isNatIList(ok(X)) = X + 1 >= X + 1 = ok(isNatIList(X)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) 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)) Matrix Interpretation Processor: dimension: 1 interpretation: [cons#](x0, x1) = x0 + 1, [top](x0) = 0, [ok](x0) = 0, [proper](x0) = 0, [isNatIList](x0) = 0, [isNatList](x0) = 0, [isNat](x0) = 0, [and](x0, x1) = x0 + x1, [take](x0, x1) = x0 + x1, [U31](x0, x1, x2, x3) = x0 + x1 + x2 + x3, [nil] = 0, [U21](x0) = x0, [s](x0) = x0, [length](x0) = x0, [U11](x0, x1) = x0 + x1, [tt] = 0, [mark](x0) = x0 + 1, [cons](x0, x1) = x0 + x1, [0] = 0, [active](x0) = x0 + 1, [zeros] = 0 orientation: cons#(mark(X1),X2) = X1 + 2 >= X1 + 1 = cons#(X1,X2) active(zeros()) = 1 >= 1 = mark(cons(0(),zeros())) active(U11(tt(),L)) = L + 1 >= L + 1 = mark(s(length(L))) active(U21(tt())) = 1 >= 1 = mark(nil()) active(U31(tt(),IL,M,N)) = IL + M + N + 1 >= IL + M + N + 1 = mark(cons(N,take(M,IL))) active(and(tt(),X)) = X + 1 >= X + 1 = mark(X) active(isNat(0())) = 1 >= 1 = mark(tt()) active(isNat(length(V1))) = 1 >= 1 = mark(isNatList(V1)) active(isNat(s(V1))) = 1 >= 1 = mark(isNat(V1)) active(isNatIList(V)) = 1 >= 1 = mark(isNatList(V)) active(isNatIList(zeros())) = 1 >= 1 = mark(tt()) active(isNatIList(cons(V1,V2))) = 1 >= 1 = mark(and(isNat(V1),isNatIList(V2))) active(isNatList(nil())) = 1 >= 1 = mark(tt()) active(isNatList(cons(V1,V2))) = 1 >= 1 = mark(and(isNat(V1),isNatList(V2))) active(isNatList(take(V1,V2))) = 1 >= 1 = mark(and(isNat(V1),isNatIList(V2))) active(length(nil())) = 1 >= 1 = mark(0()) active(length(cons(N,L))) = L + N + 1 >= L + 1 = mark(U11(and(isNatList(L),isNat(N)),L)) active(take(0(),IL)) = IL + 1 >= 1 = mark(U21(isNatIList(IL))) active(take(s(M),cons(N,IL))) = IL + M + N + 1 >= IL + M + N + 1 = mark(U31(and(isNatIList(IL),and(isNat(M),isNat(N))),IL,M,N)) active(cons(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = cons(active(X1),X2) active(U11(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = U11(active(X1),X2) active(s(X)) = X + 1 >= X + 1 = s(active(X)) active(length(X)) = X + 1 >= X + 1 = length(active(X)) active(U21(X)) = X + 1 >= X + 1 = U21(active(X)) active(U31(X1,X2,X3,X4)) = X1 + X2 + X3 + X4 + 1 >= X1 + X2 + X3 + X4 + 1 = U31(active(X1),X2,X3,X4) active(take(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = take(active(X1),X2) active(take(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = take(X1,active(X2)) active(and(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = and(active(X1),X2) cons(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(cons(X1,X2)) U11(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(U11(X1,X2)) s(mark(X)) = X + 1 >= X + 1 = mark(s(X)) length(mark(X)) = X + 1 >= X + 1 = mark(length(X)) U21(mark(X)) = X + 1 >= X + 1 = mark(U21(X)) U31(mark(X1),X2,X3,X4) = X1 + X2 + X3 + X4 + 1 >= X1 + X2 + X3 + X4 + 1 = mark(U31(X1,X2,X3,X4)) take(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(take(X1,X2)) take(X1,mark(X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(take(X1,X2)) and(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(and(X1,X2)) proper(zeros()) = 0 >= 0 = ok(zeros()) proper(cons(X1,X2)) = 0 >= 0 = cons(proper(X1),proper(X2)) proper(0()) = 0 >= 0 = ok(0()) proper(U11(X1,X2)) = 0 >= 0 = U11(proper(X1),proper(X2)) proper(tt()) = 0 >= 0 = ok(tt()) proper(s(X)) = 0 >= 0 = s(proper(X)) proper(length(X)) = 0 >= 0 = length(proper(X)) proper(U21(X)) = 0 >= 0 = U21(proper(X)) proper(nil()) = 0 >= 0 = ok(nil()) proper(U31(X1,X2,X3,X4)) = 0 >= 0 = U31(proper(X1),proper(X2),proper(X3),proper(X4)) proper(take(X1,X2)) = 0 >= 0 = take(proper(X1),proper(X2)) proper(and(X1,X2)) = 0 >= 0 = and(proper(X1),proper(X2)) proper(isNat(X)) = 0 >= 0 = isNat(proper(X)) proper(isNatList(X)) = 0 >= 0 = isNatList(proper(X)) proper(isNatIList(X)) = 0 >= 0 = isNatIList(proper(X)) cons(ok(X1),ok(X2)) = 0 >= 0 = ok(cons(X1,X2)) U11(ok(X1),ok(X2)) = 0 >= 0 = ok(U11(X1,X2)) s(ok(X)) = 0 >= 0 = ok(s(X)) length(ok(X)) = 0 >= 0 = ok(length(X)) U21(ok(X)) = 0 >= 0 = ok(U21(X)) U31(ok(X1),ok(X2),ok(X3),ok(X4)) = 0 >= 0 = ok(U31(X1,X2,X3,X4)) take(ok(X1),ok(X2)) = 0 >= 0 = ok(take(X1,X2)) and(ok(X1),ok(X2)) = 0 >= 0 = ok(and(X1,X2)) isNat(ok(X)) = 0 >= 0 = ok(isNat(X)) isNatList(ok(X)) = 0 >= 0 = ok(isNatList(X)) isNatIList(ok(X)) = 0 >= 0 = ok(isNatIList(X)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) 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#(ok(X1),ok(X2)) -> U11#(X1,X2) 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)) Matrix Interpretation Processor: dimension: 1 interpretation: [U11#](x0, x1) = x1 + 1, [top](x0) = 0, [ok](x0) = x0 + 1, [proper](x0) = x0 + 1, [isNatIList](x0) = x0, [isNatList](x0) = x0, [isNat](x0) = x0, [and](x0, x1) = x0, [take](x0, x1) = x1, [U31](x0, x1, x2, x3) = x3, [nil] = 0, [U21](x0) = x0, [s](x0) = x0, [length](x0) = x0, [U11](x0, x1) = x1, [tt] = 0, [mark](x0) = 0, [cons](x0, x1) = x0, [0] = 0, [active](x0) = x0, [zeros] = 1 orientation: U11#(ok(X1),ok(X2)) = X2 + 2 >= X2 + 1 = U11#(X1,X2) U11#(mark(X1),X2) = X2 + 1 >= X2 + 1 = U11#(X1,X2) active(zeros()) = 1 >= 0 = mark(cons(0(),zeros())) active(U11(tt(),L)) = L >= 0 = mark(s(length(L))) active(U21(tt())) = 0 >= 0 = mark(nil()) active(U31(tt(),IL,M,N)) = N >= 0 = mark(cons(N,take(M,IL))) active(and(tt(),X)) = 0 >= 0 = mark(X) active(isNat(0())) = 0 >= 0 = mark(tt()) active(isNat(length(V1))) = V1 >= 0 = mark(isNatList(V1)) active(isNat(s(V1))) = V1 >= 0 = mark(isNat(V1)) active(isNatIList(V)) = V >= 0 = mark(isNatList(V)) active(isNatIList(zeros())) = 1 >= 0 = mark(tt()) active(isNatIList(cons(V1,V2))) = V1 >= 0 = mark(and(isNat(V1),isNatIList(V2))) active(isNatList(nil())) = 0 >= 0 = mark(tt()) active(isNatList(cons(V1,V2))) = V1 >= 0 = mark(and(isNat(V1),isNatList(V2))) active(isNatList(take(V1,V2))) = V2 >= 0 = mark(and(isNat(V1),isNatIList(V2))) active(length(nil())) = 0 >= 0 = mark(0()) active(length(cons(N,L))) = N >= 0 = mark(U11(and(isNatList(L),isNat(N)),L)) active(take(0(),IL)) = IL >= 0 = mark(U21(isNatIList(IL))) active(take(s(M),cons(N,IL))) = N >= 0 = mark(U31(and(isNatIList(IL),and(isNat(M),isNat(N))),IL,M,N)) active(cons(X1,X2)) = X1 >= X1 = cons(active(X1),X2) active(U11(X1,X2)) = X2 >= X2 = U11(active(X1),X2) active(s(X)) = X >= X = s(active(X)) active(length(X)) = X >= X = length(active(X)) active(U21(X)) = X >= X = U21(active(X)) active(U31(X1,X2,X3,X4)) = X4 >= X4 = U31(active(X1),X2,X3,X4) active(take(X1,X2)) = X2 >= X2 = take(active(X1),X2) active(take(X1,X2)) = X2 >= X2 = take(X1,active(X2)) active(and(X1,X2)) = X1 >= X1 = and(active(X1),X2) cons(mark(X1),X2) = 0 >= 0 = mark(cons(X1,X2)) U11(mark(X1),X2) = X2 >= 0 = mark(U11(X1,X2)) s(mark(X)) = 0 >= 0 = mark(s(X)) length(mark(X)) = 0 >= 0 = mark(length(X)) U21(mark(X)) = 0 >= 0 = mark(U21(X)) U31(mark(X1),X2,X3,X4) = X4 >= 0 = mark(U31(X1,X2,X3,X4)) take(mark(X1),X2) = X2 >= 0 = mark(take(X1,X2)) take(X1,mark(X2)) = 0 >= 0 = mark(take(X1,X2)) and(mark(X1),X2) = 0 >= 0 = mark(and(X1,X2)) proper(zeros()) = 2 >= 2 = ok(zeros()) proper(cons(X1,X2)) = X1 + 1 >= X1 + 1 = cons(proper(X1),proper(X2)) proper(0()) = 1 >= 1 = ok(0()) proper(U11(X1,X2)) = X2 + 1 >= X2 + 1 = U11(proper(X1),proper(X2)) proper(tt()) = 1 >= 1 = ok(tt()) proper(s(X)) = X + 1 >= X + 1 = s(proper(X)) proper(length(X)) = X + 1 >= X + 1 = length(proper(X)) proper(U21(X)) = X + 1 >= X + 1 = U21(proper(X)) proper(nil()) = 1 >= 1 = ok(nil()) proper(U31(X1,X2,X3,X4)) = X4 + 1 >= X4 + 1 = U31(proper(X1),proper(X2),proper(X3),proper(X4)) proper(take(X1,X2)) = X2 + 1 >= X2 + 1 = take(proper(X1),proper(X2)) proper(and(X1,X2)) = X1 + 1 >= X1 + 1 = and(proper(X1),proper(X2)) proper(isNat(X)) = X + 1 >= X + 1 = isNat(proper(X)) proper(isNatList(X)) = X + 1 >= X + 1 = isNatList(proper(X)) proper(isNatIList(X)) = X + 1 >= X + 1 = isNatIList(proper(X)) cons(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(cons(X1,X2)) U11(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(U11(X1,X2)) s(ok(X)) = X + 1 >= X + 1 = ok(s(X)) length(ok(X)) = X + 1 >= X + 1 = ok(length(X)) U21(ok(X)) = X + 1 >= X + 1 = ok(U21(X)) U31(ok(X1),ok(X2),ok(X3),ok(X4)) = X4 + 1 >= X4 + 1 = ok(U31(X1,X2,X3,X4)) take(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(take(X1,X2)) and(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(and(X1,X2)) isNat(ok(X)) = X + 1 >= X + 1 = ok(isNat(X)) isNatList(ok(X)) = X + 1 >= X + 1 = ok(isNatList(X)) isNatIList(ok(X)) = X + 1 >= X + 1 = ok(isNatIList(X)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) 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)) Matrix Interpretation Processor: dimension: 1 interpretation: [U11#](x0, x1) = x0 + 1, [top](x0) = 0, [ok](x0) = 0, [proper](x0) = 0, [isNatIList](x0) = 0, [isNatList](x0) = 0, [isNat](x0) = 0, [and](x0, x1) = x0 + x1, [take](x0, x1) = x0 + x1, [U31](x0, x1, x2, x3) = x0 + x1 + x2 + x3, [nil] = 0, [U21](x0) = x0, [s](x0) = x0, [length](x0) = x0, [U11](x0, x1) = x0 + x1, [tt] = 0, [mark](x0) = x0 + 1, [cons](x0, x1) = x0 + x1, [0] = 0, [active](x0) = x0 + 1, [zeros] = 0 orientation: U11#(mark(X1),X2) = X1 + 2 >= X1 + 1 = U11#(X1,X2) active(zeros()) = 1 >= 1 = mark(cons(0(),zeros())) active(U11(tt(),L)) = L + 1 >= L + 1 = mark(s(length(L))) active(U21(tt())) = 1 >= 1 = mark(nil()) active(U31(tt(),IL,M,N)) = IL + M + N + 1 >= IL + M + N + 1 = mark(cons(N,take(M,IL))) active(and(tt(),X)) = X + 1 >= X + 1 = mark(X) active(isNat(0())) = 1 >= 1 = mark(tt()) active(isNat(length(V1))) = 1 >= 1 = mark(isNatList(V1)) active(isNat(s(V1))) = 1 >= 1 = mark(isNat(V1)) active(isNatIList(V)) = 1 >= 1 = mark(isNatList(V)) active(isNatIList(zeros())) = 1 >= 1 = mark(tt()) active(isNatIList(cons(V1,V2))) = 1 >= 1 = mark(and(isNat(V1),isNatIList(V2))) active(isNatList(nil())) = 1 >= 1 = mark(tt()) active(isNatList(cons(V1,V2))) = 1 >= 1 = mark(and(isNat(V1),isNatList(V2))) active(isNatList(take(V1,V2))) = 1 >= 1 = mark(and(isNat(V1),isNatIList(V2))) active(length(nil())) = 1 >= 1 = mark(0()) active(length(cons(N,L))) = L + N + 1 >= L + 1 = mark(U11(and(isNatList(L),isNat(N)),L)) active(take(0(),IL)) = IL + 1 >= 1 = mark(U21(isNatIList(IL))) active(take(s(M),cons(N,IL))) = IL + M + N + 1 >= IL + M + N + 1 = mark(U31(and(isNatIList(IL),and(isNat(M),isNat(N))),IL,M,N)) active(cons(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = cons(active(X1),X2) active(U11(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = U11(active(X1),X2) active(s(X)) = X + 1 >= X + 1 = s(active(X)) active(length(X)) = X + 1 >= X + 1 = length(active(X)) active(U21(X)) = X + 1 >= X + 1 = U21(active(X)) active(U31(X1,X2,X3,X4)) = X1 + X2 + X3 + X4 + 1 >= X1 + X2 + X3 + X4 + 1 = U31(active(X1),X2,X3,X4) active(take(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = take(active(X1),X2) active(take(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = take(X1,active(X2)) active(and(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = and(active(X1),X2) cons(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(cons(X1,X2)) U11(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(U11(X1,X2)) s(mark(X)) = X + 1 >= X + 1 = mark(s(X)) length(mark(X)) = X + 1 >= X + 1 = mark(length(X)) U21(mark(X)) = X + 1 >= X + 1 = mark(U21(X)) U31(mark(X1),X2,X3,X4) = X1 + X2 + X3 + X4 + 1 >= X1 + X2 + X3 + X4 + 1 = mark(U31(X1,X2,X3,X4)) take(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(take(X1,X2)) take(X1,mark(X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(take(X1,X2)) and(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(and(X1,X2)) proper(zeros()) = 0 >= 0 = ok(zeros()) proper(cons(X1,X2)) = 0 >= 0 = cons(proper(X1),proper(X2)) proper(0()) = 0 >= 0 = ok(0()) proper(U11(X1,X2)) = 0 >= 0 = U11(proper(X1),proper(X2)) proper(tt()) = 0 >= 0 = ok(tt()) proper(s(X)) = 0 >= 0 = s(proper(X)) proper(length(X)) = 0 >= 0 = length(proper(X)) proper(U21(X)) = 0 >= 0 = U21(proper(X)) proper(nil()) = 0 >= 0 = ok(nil()) proper(U31(X1,X2,X3,X4)) = 0 >= 0 = U31(proper(X1),proper(X2),proper(X3),proper(X4)) proper(take(X1,X2)) = 0 >= 0 = take(proper(X1),proper(X2)) proper(and(X1,X2)) = 0 >= 0 = and(proper(X1),proper(X2)) proper(isNat(X)) = 0 >= 0 = isNat(proper(X)) proper(isNatList(X)) = 0 >= 0 = isNatList(proper(X)) proper(isNatIList(X)) = 0 >= 0 = isNatIList(proper(X)) cons(ok(X1),ok(X2)) = 0 >= 0 = ok(cons(X1,X2)) U11(ok(X1),ok(X2)) = 0 >= 0 = ok(U11(X1,X2)) s(ok(X)) = 0 >= 0 = ok(s(X)) length(ok(X)) = 0 >= 0 = ok(length(X)) U21(ok(X)) = 0 >= 0 = ok(U21(X)) U31(ok(X1),ok(X2),ok(X3),ok(X4)) = 0 >= 0 = ok(U31(X1,X2,X3,X4)) take(ok(X1),ok(X2)) = 0 >= 0 = ok(take(X1,X2)) and(ok(X1),ok(X2)) = 0 >= 0 = ok(and(X1,X2)) isNat(ok(X)) = 0 >= 0 = ok(isNat(X)) isNatList(ok(X)) = 0 >= 0 = ok(isNatList(X)) isNatIList(ok(X)) = 0 >= 0 = ok(isNatIList(X)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) 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#(ok(X)) -> s#(X) s#(mark(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)) Matrix Interpretation Processor: dimension: 1 interpretation: [s#](x0) = x0 + 1, [top](x0) = 0, [ok](x0) = x0 + 1, [proper](x0) = x0 + 1, [isNatIList](x0) = x0, [isNatList](x0) = x0, [isNat](x0) = x0, [and](x0, x1) = x1, [take](x0, x1) = x1 + 1, [U31](x0, x1, x2, x3) = x1 + 1, [nil] = 0, [U21](x0) = x0, [s](x0) = x0, [length](x0) = x0, [U11](x0, x1) = x1, [tt] = 0, [mark](x0) = x0, [cons](x0, x1) = x1, [0] = 0, [active](x0) = x0, [zeros] = 0 orientation: s#(ok(X)) = X + 2 >= X + 1 = s#(X) s#(mark(X)) = X + 1 >= X + 1 = s#(X) active(zeros()) = 0 >= 0 = mark(cons(0(),zeros())) active(U11(tt(),L)) = L >= L = mark(s(length(L))) active(U21(tt())) = 0 >= 0 = mark(nil()) active(U31(tt(),IL,M,N)) = IL + 1 >= IL + 1 = mark(cons(N,take(M,IL))) active(and(tt(),X)) = X >= X = mark(X) active(isNat(0())) = 0 >= 0 = mark(tt()) active(isNat(length(V1))) = V1 >= V1 = mark(isNatList(V1)) active(isNat(s(V1))) = V1 >= V1 = mark(isNat(V1)) active(isNatIList(V)) = V >= V = mark(isNatList(V)) active(isNatIList(zeros())) = 0 >= 0 = mark(tt()) active(isNatIList(cons(V1,V2))) = V2 >= V2 = mark(and(isNat(V1),isNatIList(V2))) active(isNatList(nil())) = 0 >= 0 = mark(tt()) active(isNatList(cons(V1,V2))) = V2 >= V2 = mark(and(isNat(V1),isNatList(V2))) active(isNatList(take(V1,V2))) = V2 + 1 >= V2 = mark(and(isNat(V1),isNatIList(V2))) active(length(nil())) = 0 >= 0 = mark(0()) active(length(cons(N,L))) = L >= L = mark(U11(and(isNatList(L),isNat(N)),L)) active(take(0(),IL)) = IL + 1 >= IL = mark(U21(isNatIList(IL))) active(take(s(M),cons(N,IL))) = IL + 1 >= IL + 1 = mark(U31(and(isNatIList(IL),and(isNat(M),isNat(N))),IL,M,N)) active(cons(X1,X2)) = X2 >= X2 = cons(active(X1),X2) active(U11(X1,X2)) = X2 >= X2 = U11(active(X1),X2) active(s(X)) = X >= X = s(active(X)) active(length(X)) = X >= X = length(active(X)) active(U21(X)) = X >= X = U21(active(X)) active(U31(X1,X2,X3,X4)) = X2 + 1 >= X2 + 1 = U31(active(X1),X2,X3,X4) active(take(X1,X2)) = X2 + 1 >= X2 + 1 = take(active(X1),X2) active(take(X1,X2)) = X2 + 1 >= X2 + 1 = take(X1,active(X2)) active(and(X1,X2)) = X2 >= X2 = and(active(X1),X2) cons(mark(X1),X2) = X2 >= X2 = mark(cons(X1,X2)) U11(mark(X1),X2) = X2 >= X2 = mark(U11(X1,X2)) s(mark(X)) = X >= X = mark(s(X)) length(mark(X)) = X >= X = mark(length(X)) U21(mark(X)) = X >= X = mark(U21(X)) U31(mark(X1),X2,X3,X4) = X2 + 1 >= X2 + 1 = mark(U31(X1,X2,X3,X4)) take(mark(X1),X2) = X2 + 1 >= X2 + 1 = mark(take(X1,X2)) take(X1,mark(X2)) = X2 + 1 >= X2 + 1 = mark(take(X1,X2)) and(mark(X1),X2) = X2 >= X2 = mark(and(X1,X2)) proper(zeros()) = 1 >= 1 = ok(zeros()) proper(cons(X1,X2)) = X2 + 1 >= X2 + 1 = cons(proper(X1),proper(X2)) proper(0()) = 1 >= 1 = ok(0()) proper(U11(X1,X2)) = X2 + 1 >= X2 + 1 = U11(proper(X1),proper(X2)) proper(tt()) = 1 >= 1 = ok(tt()) proper(s(X)) = X + 1 >= X + 1 = s(proper(X)) proper(length(X)) = X + 1 >= X + 1 = length(proper(X)) proper(U21(X)) = X + 1 >= X + 1 = U21(proper(X)) proper(nil()) = 1 >= 1 = ok(nil()) proper(U31(X1,X2,X3,X4)) = X2 + 2 >= X2 + 2 = U31(proper(X1),proper(X2),proper(X3),proper(X4)) proper(take(X1,X2)) = X2 + 2 >= X2 + 2 = take(proper(X1),proper(X2)) proper(and(X1,X2)) = X2 + 1 >= X2 + 1 = and(proper(X1),proper(X2)) proper(isNat(X)) = X + 1 >= X + 1 = isNat(proper(X)) proper(isNatList(X)) = X + 1 >= X + 1 = isNatList(proper(X)) proper(isNatIList(X)) = X + 1 >= X + 1 = isNatIList(proper(X)) cons(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(cons(X1,X2)) U11(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(U11(X1,X2)) s(ok(X)) = X + 1 >= X + 1 = ok(s(X)) length(ok(X)) = X + 1 >= X + 1 = ok(length(X)) U21(ok(X)) = X + 1 >= X + 1 = ok(U21(X)) U31(ok(X1),ok(X2),ok(X3),ok(X4)) = X2 + 2 >= X2 + 2 = ok(U31(X1,X2,X3,X4)) take(ok(X1),ok(X2)) = X2 + 2 >= X2 + 2 = ok(take(X1,X2)) and(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(and(X1,X2)) isNat(ok(X)) = X + 1 >= X + 1 = ok(isNat(X)) isNatList(ok(X)) = X + 1 >= X + 1 = ok(isNatList(X)) isNatIList(ok(X)) = X + 1 >= X + 1 = ok(isNatIList(X)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) problem: DPs: s#(mark(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)) Matrix Interpretation Processor: dimension: 1 interpretation: [s#](x0) = x0 + 1, [top](x0) = 1, [ok](x0) = 0, [proper](x0) = 0, [isNatIList](x0) = 0, [isNatList](x0) = 0, [isNat](x0) = 0, [and](x0, x1) = x0 + x1, [take](x0, x1) = x0 + x1, [U31](x0, x1, x2, x3) = x0 + x1 + x2 + x3, [nil] = 0, [U21](x0) = x0, [s](x0) = x0, [length](x0) = x0, [U11](x0, x1) = x0 + x1, [tt] = 0, [mark](x0) = x0 + 1, [cons](x0, x1) = x0 + x1, [0] = 0, [active](x0) = x0 + 1, [zeros] = 0 orientation: s#(mark(X)) = X + 2 >= X + 1 = s#(X) active(zeros()) = 1 >= 1 = mark(cons(0(),zeros())) active(U11(tt(),L)) = L + 1 >= L + 1 = mark(s(length(L))) active(U21(tt())) = 1 >= 1 = mark(nil()) active(U31(tt(),IL,M,N)) = IL + M + N + 1 >= IL + M + N + 1 = mark(cons(N,take(M,IL))) active(and(tt(),X)) = X + 1 >= X + 1 = mark(X) active(isNat(0())) = 1 >= 1 = mark(tt()) active(isNat(length(V1))) = 1 >= 1 = mark(isNatList(V1)) active(isNat(s(V1))) = 1 >= 1 = mark(isNat(V1)) active(isNatIList(V)) = 1 >= 1 = mark(isNatList(V)) active(isNatIList(zeros())) = 1 >= 1 = mark(tt()) active(isNatIList(cons(V1,V2))) = 1 >= 1 = mark(and(isNat(V1),isNatIList(V2))) active(isNatList(nil())) = 1 >= 1 = mark(tt()) active(isNatList(cons(V1,V2))) = 1 >= 1 = mark(and(isNat(V1),isNatList(V2))) active(isNatList(take(V1,V2))) = 1 >= 1 = mark(and(isNat(V1),isNatIList(V2))) active(length(nil())) = 1 >= 1 = mark(0()) active(length(cons(N,L))) = L + N + 1 >= L + 1 = mark(U11(and(isNatList(L),isNat(N)),L)) active(take(0(),IL)) = IL + 1 >= 1 = mark(U21(isNatIList(IL))) active(take(s(M),cons(N,IL))) = IL + M + N + 1 >= IL + M + N + 1 = mark(U31(and(isNatIList(IL),and(isNat(M),isNat(N))),IL,M,N)) active(cons(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = cons(active(X1),X2) active(U11(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = U11(active(X1),X2) active(s(X)) = X + 1 >= X + 1 = s(active(X)) active(length(X)) = X + 1 >= X + 1 = length(active(X)) active(U21(X)) = X + 1 >= X + 1 = U21(active(X)) active(U31(X1,X2,X3,X4)) = X1 + X2 + X3 + X4 + 1 >= X1 + X2 + X3 + X4 + 1 = U31(active(X1),X2,X3,X4) active(take(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = take(active(X1),X2) active(take(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = take(X1,active(X2)) active(and(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = and(active(X1),X2) cons(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(cons(X1,X2)) U11(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(U11(X1,X2)) s(mark(X)) = X + 1 >= X + 1 = mark(s(X)) length(mark(X)) = X + 1 >= X + 1 = mark(length(X)) U21(mark(X)) = X + 1 >= X + 1 = mark(U21(X)) U31(mark(X1),X2,X3,X4) = X1 + X2 + X3 + X4 + 1 >= X1 + X2 + X3 + X4 + 1 = mark(U31(X1,X2,X3,X4)) take(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(take(X1,X2)) take(X1,mark(X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(take(X1,X2)) and(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(and(X1,X2)) proper(zeros()) = 0 >= 0 = ok(zeros()) proper(cons(X1,X2)) = 0 >= 0 = cons(proper(X1),proper(X2)) proper(0()) = 0 >= 0 = ok(0()) proper(U11(X1,X2)) = 0 >= 0 = U11(proper(X1),proper(X2)) proper(tt()) = 0 >= 0 = ok(tt()) proper(s(X)) = 0 >= 0 = s(proper(X)) proper(length(X)) = 0 >= 0 = length(proper(X)) proper(U21(X)) = 0 >= 0 = U21(proper(X)) proper(nil()) = 0 >= 0 = ok(nil()) proper(U31(X1,X2,X3,X4)) = 0 >= 0 = U31(proper(X1),proper(X2),proper(X3),proper(X4)) proper(take(X1,X2)) = 0 >= 0 = take(proper(X1),proper(X2)) proper(and(X1,X2)) = 0 >= 0 = and(proper(X1),proper(X2)) proper(isNat(X)) = 0 >= 0 = isNat(proper(X)) proper(isNatList(X)) = 0 >= 0 = isNatList(proper(X)) proper(isNatIList(X)) = 0 >= 0 = isNatIList(proper(X)) cons(ok(X1),ok(X2)) = 0 >= 0 = ok(cons(X1,X2)) U11(ok(X1),ok(X2)) = 0 >= 0 = ok(U11(X1,X2)) s(ok(X)) = 0 >= 0 = ok(s(X)) length(ok(X)) = 0 >= 0 = ok(length(X)) U21(ok(X)) = 0 >= 0 = ok(U21(X)) U31(ok(X1),ok(X2),ok(X3),ok(X4)) = 0 >= 0 = ok(U31(X1,X2,X3,X4)) take(ok(X1),ok(X2)) = 0 >= 0 = ok(take(X1,X2)) and(ok(X1),ok(X2)) = 0 >= 0 = ok(and(X1,X2)) isNat(ok(X)) = 0 >= 0 = ok(isNat(X)) isNatList(ok(X)) = 0 >= 0 = ok(isNatList(X)) isNatIList(ok(X)) = 0 >= 0 = ok(isNatIList(X)) top(mark(X)) = 1 >= 1 = top(proper(X)) top(ok(X)) = 1 >= 1 = top(active(X)) 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#(ok(X)) -> length#(X) length#(mark(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)) Matrix Interpretation Processor: dimension: 1 interpretation: [length#](x0) = x0 + 1, [top](x0) = 0, [ok](x0) = x0 + 1, [proper](x0) = x0 + 1, [isNatIList](x0) = x0, [isNatList](x0) = x0, [isNat](x0) = x0, [and](x0, x1) = x1, [take](x0, x1) = x1 + 1, [U31](x0, x1, x2, x3) = x1 + 1, [nil] = 0, [U21](x0) = x0, [s](x0) = x0, [length](x0) = x0, [U11](x0, x1) = x1, [tt] = 0, [mark](x0) = x0, [cons](x0, x1) = x1, [0] = 0, [active](x0) = x0, [zeros] = 0 orientation: length#(ok(X)) = X + 2 >= X + 1 = length#(X) length#(mark(X)) = X + 1 >= X + 1 = length#(X) active(zeros()) = 0 >= 0 = mark(cons(0(),zeros())) active(U11(tt(),L)) = L >= L = mark(s(length(L))) active(U21(tt())) = 0 >= 0 = mark(nil()) active(U31(tt(),IL,M,N)) = IL + 1 >= IL + 1 = mark(cons(N,take(M,IL))) active(and(tt(),X)) = X >= X = mark(X) active(isNat(0())) = 0 >= 0 = mark(tt()) active(isNat(length(V1))) = V1 >= V1 = mark(isNatList(V1)) active(isNat(s(V1))) = V1 >= V1 = mark(isNat(V1)) active(isNatIList(V)) = V >= V = mark(isNatList(V)) active(isNatIList(zeros())) = 0 >= 0 = mark(tt()) active(isNatIList(cons(V1,V2))) = V2 >= V2 = mark(and(isNat(V1),isNatIList(V2))) active(isNatList(nil())) = 0 >= 0 = mark(tt()) active(isNatList(cons(V1,V2))) = V2 >= V2 = mark(and(isNat(V1),isNatList(V2))) active(isNatList(take(V1,V2))) = V2 + 1 >= V2 = mark(and(isNat(V1),isNatIList(V2))) active(length(nil())) = 0 >= 0 = mark(0()) active(length(cons(N,L))) = L >= L = mark(U11(and(isNatList(L),isNat(N)),L)) active(take(0(),IL)) = IL + 1 >= IL = mark(U21(isNatIList(IL))) active(take(s(M),cons(N,IL))) = IL + 1 >= IL + 1 = mark(U31(and(isNatIList(IL),and(isNat(M),isNat(N))),IL,M,N)) active(cons(X1,X2)) = X2 >= X2 = cons(active(X1),X2) active(U11(X1,X2)) = X2 >= X2 = U11(active(X1),X2) active(s(X)) = X >= X = s(active(X)) active(length(X)) = X >= X = length(active(X)) active(U21(X)) = X >= X = U21(active(X)) active(U31(X1,X2,X3,X4)) = X2 + 1 >= X2 + 1 = U31(active(X1),X2,X3,X4) active(take(X1,X2)) = X2 + 1 >= X2 + 1 = take(active(X1),X2) active(take(X1,X2)) = X2 + 1 >= X2 + 1 = take(X1,active(X2)) active(and(X1,X2)) = X2 >= X2 = and(active(X1),X2) cons(mark(X1),X2) = X2 >= X2 = mark(cons(X1,X2)) U11(mark(X1),X2) = X2 >= X2 = mark(U11(X1,X2)) s(mark(X)) = X >= X = mark(s(X)) length(mark(X)) = X >= X = mark(length(X)) U21(mark(X)) = X >= X = mark(U21(X)) U31(mark(X1),X2,X3,X4) = X2 + 1 >= X2 + 1 = mark(U31(X1,X2,X3,X4)) take(mark(X1),X2) = X2 + 1 >= X2 + 1 = mark(take(X1,X2)) take(X1,mark(X2)) = X2 + 1 >= X2 + 1 = mark(take(X1,X2)) and(mark(X1),X2) = X2 >= X2 = mark(and(X1,X2)) proper(zeros()) = 1 >= 1 = ok(zeros()) proper(cons(X1,X2)) = X2 + 1 >= X2 + 1 = cons(proper(X1),proper(X2)) proper(0()) = 1 >= 1 = ok(0()) proper(U11(X1,X2)) = X2 + 1 >= X2 + 1 = U11(proper(X1),proper(X2)) proper(tt()) = 1 >= 1 = ok(tt()) proper(s(X)) = X + 1 >= X + 1 = s(proper(X)) proper(length(X)) = X + 1 >= X + 1 = length(proper(X)) proper(U21(X)) = X + 1 >= X + 1 = U21(proper(X)) proper(nil()) = 1 >= 1 = ok(nil()) proper(U31(X1,X2,X3,X4)) = X2 + 2 >= X2 + 2 = U31(proper(X1),proper(X2),proper(X3),proper(X4)) proper(take(X1,X2)) = X2 + 2 >= X2 + 2 = take(proper(X1),proper(X2)) proper(and(X1,X2)) = X2 + 1 >= X2 + 1 = and(proper(X1),proper(X2)) proper(isNat(X)) = X + 1 >= X + 1 = isNat(proper(X)) proper(isNatList(X)) = X + 1 >= X + 1 = isNatList(proper(X)) proper(isNatIList(X)) = X + 1 >= X + 1 = isNatIList(proper(X)) cons(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(cons(X1,X2)) U11(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(U11(X1,X2)) s(ok(X)) = X + 1 >= X + 1 = ok(s(X)) length(ok(X)) = X + 1 >= X + 1 = ok(length(X)) U21(ok(X)) = X + 1 >= X + 1 = ok(U21(X)) U31(ok(X1),ok(X2),ok(X3),ok(X4)) = X2 + 2 >= X2 + 2 = ok(U31(X1,X2,X3,X4)) take(ok(X1),ok(X2)) = X2 + 2 >= X2 + 2 = ok(take(X1,X2)) and(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(and(X1,X2)) isNat(ok(X)) = X + 1 >= X + 1 = ok(isNat(X)) isNatList(ok(X)) = X + 1 >= X + 1 = ok(isNatList(X)) isNatIList(ok(X)) = X + 1 >= X + 1 = ok(isNatIList(X)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) problem: DPs: length#(mark(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)) Matrix Interpretation Processor: dimension: 1 interpretation: [length#](x0) = x0 + 1, [top](x0) = 1, [ok](x0) = 0, [proper](x0) = 0, [isNatIList](x0) = 0, [isNatList](x0) = 0, [isNat](x0) = 0, [and](x0, x1) = x0 + x1, [take](x0, x1) = x0 + x1, [U31](x0, x1, x2, x3) = x0 + x1 + x2 + x3, [nil] = 0, [U21](x0) = x0, [s](x0) = x0, [length](x0) = x0, [U11](x0, x1) = x0 + x1, [tt] = 0, [mark](x0) = x0 + 1, [cons](x0, x1) = x0 + x1, [0] = 0, [active](x0) = x0 + 1, [zeros] = 0 orientation: length#(mark(X)) = X + 2 >= X + 1 = length#(X) active(zeros()) = 1 >= 1 = mark(cons(0(),zeros())) active(U11(tt(),L)) = L + 1 >= L + 1 = mark(s(length(L))) active(U21(tt())) = 1 >= 1 = mark(nil()) active(U31(tt(),IL,M,N)) = IL + M + N + 1 >= IL + M + N + 1 = mark(cons(N,take(M,IL))) active(and(tt(),X)) = X + 1 >= X + 1 = mark(X) active(isNat(0())) = 1 >= 1 = mark(tt()) active(isNat(length(V1))) = 1 >= 1 = mark(isNatList(V1)) active(isNat(s(V1))) = 1 >= 1 = mark(isNat(V1)) active(isNatIList(V)) = 1 >= 1 = mark(isNatList(V)) active(isNatIList(zeros())) = 1 >= 1 = mark(tt()) active(isNatIList(cons(V1,V2))) = 1 >= 1 = mark(and(isNat(V1),isNatIList(V2))) active(isNatList(nil())) = 1 >= 1 = mark(tt()) active(isNatList(cons(V1,V2))) = 1 >= 1 = mark(and(isNat(V1),isNatList(V2))) active(isNatList(take(V1,V2))) = 1 >= 1 = mark(and(isNat(V1),isNatIList(V2))) active(length(nil())) = 1 >= 1 = mark(0()) active(length(cons(N,L))) = L + N + 1 >= L + 1 = mark(U11(and(isNatList(L),isNat(N)),L)) active(take(0(),IL)) = IL + 1 >= 1 = mark(U21(isNatIList(IL))) active(take(s(M),cons(N,IL))) = IL + M + N + 1 >= IL + M + N + 1 = mark(U31(and(isNatIList(IL),and(isNat(M),isNat(N))),IL,M,N)) active(cons(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = cons(active(X1),X2) active(U11(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = U11(active(X1),X2) active(s(X)) = X + 1 >= X + 1 = s(active(X)) active(length(X)) = X + 1 >= X + 1 = length(active(X)) active(U21(X)) = X + 1 >= X + 1 = U21(active(X)) active(U31(X1,X2,X3,X4)) = X1 + X2 + X3 + X4 + 1 >= X1 + X2 + X3 + X4 + 1 = U31(active(X1),X2,X3,X4) active(take(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = take(active(X1),X2) active(take(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = take(X1,active(X2)) active(and(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = and(active(X1),X2) cons(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(cons(X1,X2)) U11(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(U11(X1,X2)) s(mark(X)) = X + 1 >= X + 1 = mark(s(X)) length(mark(X)) = X + 1 >= X + 1 = mark(length(X)) U21(mark(X)) = X + 1 >= X + 1 = mark(U21(X)) U31(mark(X1),X2,X3,X4) = X1 + X2 + X3 + X4 + 1 >= X1 + X2 + X3 + X4 + 1 = mark(U31(X1,X2,X3,X4)) take(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(take(X1,X2)) take(X1,mark(X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(take(X1,X2)) and(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(and(X1,X2)) proper(zeros()) = 0 >= 0 = ok(zeros()) proper(cons(X1,X2)) = 0 >= 0 = cons(proper(X1),proper(X2)) proper(0()) = 0 >= 0 = ok(0()) proper(U11(X1,X2)) = 0 >= 0 = U11(proper(X1),proper(X2)) proper(tt()) = 0 >= 0 = ok(tt()) proper(s(X)) = 0 >= 0 = s(proper(X)) proper(length(X)) = 0 >= 0 = length(proper(X)) proper(U21(X)) = 0 >= 0 = U21(proper(X)) proper(nil()) = 0 >= 0 = ok(nil()) proper(U31(X1,X2,X3,X4)) = 0 >= 0 = U31(proper(X1),proper(X2),proper(X3),proper(X4)) proper(take(X1,X2)) = 0 >= 0 = take(proper(X1),proper(X2)) proper(and(X1,X2)) = 0 >= 0 = and(proper(X1),proper(X2)) proper(isNat(X)) = 0 >= 0 = isNat(proper(X)) proper(isNatList(X)) = 0 >= 0 = isNatList(proper(X)) proper(isNatIList(X)) = 0 >= 0 = isNatIList(proper(X)) cons(ok(X1),ok(X2)) = 0 >= 0 = ok(cons(X1,X2)) U11(ok(X1),ok(X2)) = 0 >= 0 = ok(U11(X1,X2)) s(ok(X)) = 0 >= 0 = ok(s(X)) length(ok(X)) = 0 >= 0 = ok(length(X)) U21(ok(X)) = 0 >= 0 = ok(U21(X)) U31(ok(X1),ok(X2),ok(X3),ok(X4)) = 0 >= 0 = ok(U31(X1,X2,X3,X4)) take(ok(X1),ok(X2)) = 0 >= 0 = ok(take(X1,X2)) and(ok(X1),ok(X2)) = 0 >= 0 = ok(and(X1,X2)) isNat(ok(X)) = 0 >= 0 = ok(isNat(X)) isNatList(ok(X)) = 0 >= 0 = ok(isNatList(X)) isNatIList(ok(X)) = 0 >= 0 = ok(isNatIList(X)) top(mark(X)) = 1 >= 1 = top(proper(X)) top(ok(X)) = 1 >= 1 = top(active(X)) 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#(ok(X)) -> U21#(X) U21#(mark(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)) Matrix Interpretation Processor: dimension: 1 interpretation: [U21#](x0) = x0 + 1, [top](x0) = 0, [ok](x0) = x0 + 1, [proper](x0) = x0 + 1, [isNatIList](x0) = x0, [isNatList](x0) = x0, [isNat](x0) = x0, [and](x0, x1) = x1, [take](x0, x1) = x1 + 1, [U31](x0, x1, x2, x3) = x1 + 1, [nil] = 0, [U21](x0) = x0, [s](x0) = x0, [length](x0) = x0, [U11](x0, x1) = x1, [tt] = 0, [mark](x0) = x0, [cons](x0, x1) = x1, [0] = 0, [active](x0) = x0, [zeros] = 0 orientation: U21#(ok(X)) = X + 2 >= X + 1 = U21#(X) U21#(mark(X)) = X + 1 >= X + 1 = U21#(X) active(zeros()) = 0 >= 0 = mark(cons(0(),zeros())) active(U11(tt(),L)) = L >= L = mark(s(length(L))) active(U21(tt())) = 0 >= 0 = mark(nil()) active(U31(tt(),IL,M,N)) = IL + 1 >= IL + 1 = mark(cons(N,take(M,IL))) active(and(tt(),X)) = X >= X = mark(X) active(isNat(0())) = 0 >= 0 = mark(tt()) active(isNat(length(V1))) = V1 >= V1 = mark(isNatList(V1)) active(isNat(s(V1))) = V1 >= V1 = mark(isNat(V1)) active(isNatIList(V)) = V >= V = mark(isNatList(V)) active(isNatIList(zeros())) = 0 >= 0 = mark(tt()) active(isNatIList(cons(V1,V2))) = V2 >= V2 = mark(and(isNat(V1),isNatIList(V2))) active(isNatList(nil())) = 0 >= 0 = mark(tt()) active(isNatList(cons(V1,V2))) = V2 >= V2 = mark(and(isNat(V1),isNatList(V2))) active(isNatList(take(V1,V2))) = V2 + 1 >= V2 = mark(and(isNat(V1),isNatIList(V2))) active(length(nil())) = 0 >= 0 = mark(0()) active(length(cons(N,L))) = L >= L = mark(U11(and(isNatList(L),isNat(N)),L)) active(take(0(),IL)) = IL + 1 >= IL = mark(U21(isNatIList(IL))) active(take(s(M),cons(N,IL))) = IL + 1 >= IL + 1 = mark(U31(and(isNatIList(IL),and(isNat(M),isNat(N))),IL,M,N)) active(cons(X1,X2)) = X2 >= X2 = cons(active(X1),X2) active(U11(X1,X2)) = X2 >= X2 = U11(active(X1),X2) active(s(X)) = X >= X = s(active(X)) active(length(X)) = X >= X = length(active(X)) active(U21(X)) = X >= X = U21(active(X)) active(U31(X1,X2,X3,X4)) = X2 + 1 >= X2 + 1 = U31(active(X1),X2,X3,X4) active(take(X1,X2)) = X2 + 1 >= X2 + 1 = take(active(X1),X2) active(take(X1,X2)) = X2 + 1 >= X2 + 1 = take(X1,active(X2)) active(and(X1,X2)) = X2 >= X2 = and(active(X1),X2) cons(mark(X1),X2) = X2 >= X2 = mark(cons(X1,X2)) U11(mark(X1),X2) = X2 >= X2 = mark(U11(X1,X2)) s(mark(X)) = X >= X = mark(s(X)) length(mark(X)) = X >= X = mark(length(X)) U21(mark(X)) = X >= X = mark(U21(X)) U31(mark(X1),X2,X3,X4) = X2 + 1 >= X2 + 1 = mark(U31(X1,X2,X3,X4)) take(mark(X1),X2) = X2 + 1 >= X2 + 1 = mark(take(X1,X2)) take(X1,mark(X2)) = X2 + 1 >= X2 + 1 = mark(take(X1,X2)) and(mark(X1),X2) = X2 >= X2 = mark(and(X1,X2)) proper(zeros()) = 1 >= 1 = ok(zeros()) proper(cons(X1,X2)) = X2 + 1 >= X2 + 1 = cons(proper(X1),proper(X2)) proper(0()) = 1 >= 1 = ok(0()) proper(U11(X1,X2)) = X2 + 1 >= X2 + 1 = U11(proper(X1),proper(X2)) proper(tt()) = 1 >= 1 = ok(tt()) proper(s(X)) = X + 1 >= X + 1 = s(proper(X)) proper(length(X)) = X + 1 >= X + 1 = length(proper(X)) proper(U21(X)) = X + 1 >= X + 1 = U21(proper(X)) proper(nil()) = 1 >= 1 = ok(nil()) proper(U31(X1,X2,X3,X4)) = X2 + 2 >= X2 + 2 = U31(proper(X1),proper(X2),proper(X3),proper(X4)) proper(take(X1,X2)) = X2 + 2 >= X2 + 2 = take(proper(X1),proper(X2)) proper(and(X1,X2)) = X2 + 1 >= X2 + 1 = and(proper(X1),proper(X2)) proper(isNat(X)) = X + 1 >= X + 1 = isNat(proper(X)) proper(isNatList(X)) = X + 1 >= X + 1 = isNatList(proper(X)) proper(isNatIList(X)) = X + 1 >= X + 1 = isNatIList(proper(X)) cons(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(cons(X1,X2)) U11(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(U11(X1,X2)) s(ok(X)) = X + 1 >= X + 1 = ok(s(X)) length(ok(X)) = X + 1 >= X + 1 = ok(length(X)) U21(ok(X)) = X + 1 >= X + 1 = ok(U21(X)) U31(ok(X1),ok(X2),ok(X3),ok(X4)) = X2 + 2 >= X2 + 2 = ok(U31(X1,X2,X3,X4)) take(ok(X1),ok(X2)) = X2 + 2 >= X2 + 2 = ok(take(X1,X2)) and(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(and(X1,X2)) isNat(ok(X)) = X + 1 >= X + 1 = ok(isNat(X)) isNatList(ok(X)) = X + 1 >= X + 1 = ok(isNatList(X)) isNatIList(ok(X)) = X + 1 >= X + 1 = ok(isNatIList(X)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) problem: DPs: U21#(mark(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)) Matrix Interpretation Processor: dimension: 1 interpretation: [U21#](x0) = x0 + 1, [top](x0) = 1, [ok](x0) = 0, [proper](x0) = 0, [isNatIList](x0) = 0, [isNatList](x0) = 0, [isNat](x0) = 0, [and](x0, x1) = x0 + x1, [take](x0, x1) = x0 + x1, [U31](x0, x1, x2, x3) = x0 + x1 + x2 + x3, [nil] = 0, [U21](x0) = x0, [s](x0) = x0, [length](x0) = x0, [U11](x0, x1) = x0 + x1, [tt] = 0, [mark](x0) = x0 + 1, [cons](x0, x1) = x0 + x1, [0] = 0, [active](x0) = x0 + 1, [zeros] = 0 orientation: U21#(mark(X)) = X + 2 >= X + 1 = U21#(X) active(zeros()) = 1 >= 1 = mark(cons(0(),zeros())) active(U11(tt(),L)) = L + 1 >= L + 1 = mark(s(length(L))) active(U21(tt())) = 1 >= 1 = mark(nil()) active(U31(tt(),IL,M,N)) = IL + M + N + 1 >= IL + M + N + 1 = mark(cons(N,take(M,IL))) active(and(tt(),X)) = X + 1 >= X + 1 = mark(X) active(isNat(0())) = 1 >= 1 = mark(tt()) active(isNat(length(V1))) = 1 >= 1 = mark(isNatList(V1)) active(isNat(s(V1))) = 1 >= 1 = mark(isNat(V1)) active(isNatIList(V)) = 1 >= 1 = mark(isNatList(V)) active(isNatIList(zeros())) = 1 >= 1 = mark(tt()) active(isNatIList(cons(V1,V2))) = 1 >= 1 = mark(and(isNat(V1),isNatIList(V2))) active(isNatList(nil())) = 1 >= 1 = mark(tt()) active(isNatList(cons(V1,V2))) = 1 >= 1 = mark(and(isNat(V1),isNatList(V2))) active(isNatList(take(V1,V2))) = 1 >= 1 = mark(and(isNat(V1),isNatIList(V2))) active(length(nil())) = 1 >= 1 = mark(0()) active(length(cons(N,L))) = L + N + 1 >= L + 1 = mark(U11(and(isNatList(L),isNat(N)),L)) active(take(0(),IL)) = IL + 1 >= 1 = mark(U21(isNatIList(IL))) active(take(s(M),cons(N,IL))) = IL + M + N + 1 >= IL + M + N + 1 = mark(U31(and(isNatIList(IL),and(isNat(M),isNat(N))),IL,M,N)) active(cons(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = cons(active(X1),X2) active(U11(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = U11(active(X1),X2) active(s(X)) = X + 1 >= X + 1 = s(active(X)) active(length(X)) = X + 1 >= X + 1 = length(active(X)) active(U21(X)) = X + 1 >= X + 1 = U21(active(X)) active(U31(X1,X2,X3,X4)) = X1 + X2 + X3 + X4 + 1 >= X1 + X2 + X3 + X4 + 1 = U31(active(X1),X2,X3,X4) active(take(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = take(active(X1),X2) active(take(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = take(X1,active(X2)) active(and(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = and(active(X1),X2) cons(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(cons(X1,X2)) U11(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(U11(X1,X2)) s(mark(X)) = X + 1 >= X + 1 = mark(s(X)) length(mark(X)) = X + 1 >= X + 1 = mark(length(X)) U21(mark(X)) = X + 1 >= X + 1 = mark(U21(X)) U31(mark(X1),X2,X3,X4) = X1 + X2 + X3 + X4 + 1 >= X1 + X2 + X3 + X4 + 1 = mark(U31(X1,X2,X3,X4)) take(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(take(X1,X2)) take(X1,mark(X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(take(X1,X2)) and(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(and(X1,X2)) proper(zeros()) = 0 >= 0 = ok(zeros()) proper(cons(X1,X2)) = 0 >= 0 = cons(proper(X1),proper(X2)) proper(0()) = 0 >= 0 = ok(0()) proper(U11(X1,X2)) = 0 >= 0 = U11(proper(X1),proper(X2)) proper(tt()) = 0 >= 0 = ok(tt()) proper(s(X)) = 0 >= 0 = s(proper(X)) proper(length(X)) = 0 >= 0 = length(proper(X)) proper(U21(X)) = 0 >= 0 = U21(proper(X)) proper(nil()) = 0 >= 0 = ok(nil()) proper(U31(X1,X2,X3,X4)) = 0 >= 0 = U31(proper(X1),proper(X2),proper(X3),proper(X4)) proper(take(X1,X2)) = 0 >= 0 = take(proper(X1),proper(X2)) proper(and(X1,X2)) = 0 >= 0 = and(proper(X1),proper(X2)) proper(isNat(X)) = 0 >= 0 = isNat(proper(X)) proper(isNatList(X)) = 0 >= 0 = isNatList(proper(X)) proper(isNatIList(X)) = 0 >= 0 = isNatIList(proper(X)) cons(ok(X1),ok(X2)) = 0 >= 0 = ok(cons(X1,X2)) U11(ok(X1),ok(X2)) = 0 >= 0 = ok(U11(X1,X2)) s(ok(X)) = 0 >= 0 = ok(s(X)) length(ok(X)) = 0 >= 0 = ok(length(X)) U21(ok(X)) = 0 >= 0 = ok(U21(X)) U31(ok(X1),ok(X2),ok(X3),ok(X4)) = 0 >= 0 = ok(U31(X1,X2,X3,X4)) take(ok(X1),ok(X2)) = 0 >= 0 = ok(take(X1,X2)) and(ok(X1),ok(X2)) = 0 >= 0 = ok(and(X1,X2)) isNat(ok(X)) = 0 >= 0 = ok(isNat(X)) isNatList(ok(X)) = 0 >= 0 = ok(isNatList(X)) isNatIList(ok(X)) = 0 >= 0 = ok(isNatIList(X)) top(mark(X)) = 1 >= 1 = top(proper(X)) top(ok(X)) = 1 >= 1 = top(active(X)) 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#(ok(X1),ok(X2),ok(X3),ok(X4)) -> U31#(X1,X2,X3,X4) 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)) Matrix Interpretation Processor: dimension: 1 interpretation: [U31#](x0, x1, x2, x3) = x1 + 1, [top](x0) = 0, [ok](x0) = x0 + 1, [proper](x0) = x0 + 1, [isNatIList](x0) = x0, [isNatList](x0) = x0, [isNat](x0) = x0, [and](x0, x1) = x0, [take](x0, x1) = x0, [U31](x0, x1, x2, x3) = x2, [nil] = 0, [U21](x0) = x0, [s](x0) = x0, [length](x0) = x0, [U11](x0, x1) = x0 + 1, [tt] = 0, [mark](x0) = 0, [cons](x0, x1) = x0, [0] = 1, [active](x0) = x0, [zeros] = 0 orientation: U31#(ok(X1),ok(X2),ok(X3),ok(X4)) = X2 + 2 >= X2 + 1 = U31#(X1,X2,X3,X4) U31#(mark(X1),X2,X3,X4) = X2 + 1 >= X2 + 1 = U31#(X1,X2,X3,X4) active(zeros()) = 0 >= 0 = mark(cons(0(),zeros())) active(U11(tt(),L)) = 1 >= 0 = mark(s(length(L))) active(U21(tt())) = 0 >= 0 = mark(nil()) active(U31(tt(),IL,M,N)) = M >= 0 = mark(cons(N,take(M,IL))) active(and(tt(),X)) = 0 >= 0 = mark(X) active(isNat(0())) = 1 >= 0 = mark(tt()) active(isNat(length(V1))) = V1 >= 0 = mark(isNatList(V1)) active(isNat(s(V1))) = V1 >= 0 = mark(isNat(V1)) active(isNatIList(V)) = V >= 0 = mark(isNatList(V)) active(isNatIList(zeros())) = 0 >= 0 = mark(tt()) active(isNatIList(cons(V1,V2))) = V1 >= 0 = mark(and(isNat(V1),isNatIList(V2))) active(isNatList(nil())) = 0 >= 0 = mark(tt()) active(isNatList(cons(V1,V2))) = V1 >= 0 = mark(and(isNat(V1),isNatList(V2))) active(isNatList(take(V1,V2))) = V1 >= 0 = mark(and(isNat(V1),isNatIList(V2))) active(length(nil())) = 0 >= 0 = mark(0()) active(length(cons(N,L))) = N >= 0 = mark(U11(and(isNatList(L),isNat(N)),L)) active(take(0(),IL)) = 1 >= 0 = mark(U21(isNatIList(IL))) active(take(s(M),cons(N,IL))) = M >= 0 = mark(U31(and(isNatIList(IL),and(isNat(M),isNat(N))),IL,M,N)) active(cons(X1,X2)) = X1 >= X1 = cons(active(X1),X2) active(U11(X1,X2)) = X1 + 1 >= X1 + 1 = U11(active(X1),X2) active(s(X)) = X >= X = s(active(X)) active(length(X)) = X >= X = length(active(X)) active(U21(X)) = X >= X = U21(active(X)) active(U31(X1,X2,X3,X4)) = X3 >= X3 = U31(active(X1),X2,X3,X4) active(take(X1,X2)) = X1 >= X1 = take(active(X1),X2) active(take(X1,X2)) = X1 >= X1 = take(X1,active(X2)) active(and(X1,X2)) = X1 >= X1 = and(active(X1),X2) cons(mark(X1),X2) = 0 >= 0 = mark(cons(X1,X2)) U11(mark(X1),X2) = 1 >= 0 = mark(U11(X1,X2)) s(mark(X)) = 0 >= 0 = mark(s(X)) length(mark(X)) = 0 >= 0 = mark(length(X)) U21(mark(X)) = 0 >= 0 = mark(U21(X)) U31(mark(X1),X2,X3,X4) = X3 >= 0 = mark(U31(X1,X2,X3,X4)) take(mark(X1),X2) = 0 >= 0 = mark(take(X1,X2)) take(X1,mark(X2)) = X1 >= 0 = mark(take(X1,X2)) and(mark(X1),X2) = 0 >= 0 = mark(and(X1,X2)) proper(zeros()) = 1 >= 1 = ok(zeros()) proper(cons(X1,X2)) = X1 + 1 >= X1 + 1 = cons(proper(X1),proper(X2)) proper(0()) = 2 >= 2 = ok(0()) proper(U11(X1,X2)) = X1 + 2 >= X1 + 2 = U11(proper(X1),proper(X2)) proper(tt()) = 1 >= 1 = ok(tt()) proper(s(X)) = X + 1 >= X + 1 = s(proper(X)) proper(length(X)) = X + 1 >= X + 1 = length(proper(X)) proper(U21(X)) = X + 1 >= X + 1 = U21(proper(X)) proper(nil()) = 1 >= 1 = ok(nil()) proper(U31(X1,X2,X3,X4)) = X3 + 1 >= X3 + 1 = U31(proper(X1),proper(X2),proper(X3),proper(X4)) proper(take(X1,X2)) = X1 + 1 >= X1 + 1 = take(proper(X1),proper(X2)) proper(and(X1,X2)) = X1 + 1 >= X1 + 1 = and(proper(X1),proper(X2)) proper(isNat(X)) = X + 1 >= X + 1 = isNat(proper(X)) proper(isNatList(X)) = X + 1 >= X + 1 = isNatList(proper(X)) proper(isNatIList(X)) = X + 1 >= X + 1 = isNatIList(proper(X)) cons(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(cons(X1,X2)) U11(ok(X1),ok(X2)) = X1 + 2 >= X1 + 2 = ok(U11(X1,X2)) s(ok(X)) = X + 1 >= X + 1 = ok(s(X)) length(ok(X)) = X + 1 >= X + 1 = ok(length(X)) U21(ok(X)) = X + 1 >= X + 1 = ok(U21(X)) U31(ok(X1),ok(X2),ok(X3),ok(X4)) = X3 + 1 >= X3 + 1 = ok(U31(X1,X2,X3,X4)) take(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(take(X1,X2)) and(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(and(X1,X2)) isNat(ok(X)) = X + 1 >= X + 1 = ok(isNat(X)) isNatList(ok(X)) = X + 1 >= X + 1 = ok(isNatList(X)) isNatIList(ok(X)) = X + 1 >= X + 1 = ok(isNatIList(X)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) 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)) Matrix Interpretation Processor: dimension: 1 interpretation: [U31#](x0, x1, x2, x3) = x0, [top](x0) = 1, [ok](x0) = 0, [proper](x0) = 0, [isNatIList](x0) = 0, [isNatList](x0) = 0, [isNat](x0) = 0, [and](x0, x1) = x0 + x1, [take](x0, x1) = x0 + x1, [U31](x0, x1, x2, x3) = x0 + x1 + x2 + x3, [nil] = 0, [U21](x0) = x0, [s](x0) = x0, [length](x0) = x0, [U11](x0, x1) = x0 + x1, [tt] = 0, [mark](x0) = x0 + 1, [cons](x0, x1) = x0 + x1, [0] = 0, [active](x0) = x0 + 1, [zeros] = 0 orientation: U31#(mark(X1),X2,X3,X4) = X1 + 1 >= X1 = U31#(X1,X2,X3,X4) active(zeros()) = 1 >= 1 = mark(cons(0(),zeros())) active(U11(tt(),L)) = L + 1 >= L + 1 = mark(s(length(L))) active(U21(tt())) = 1 >= 1 = mark(nil()) active(U31(tt(),IL,M,N)) = IL + M + N + 1 >= IL + M + N + 1 = mark(cons(N,take(M,IL))) active(and(tt(),X)) = X + 1 >= X + 1 = mark(X) active(isNat(0())) = 1 >= 1 = mark(tt()) active(isNat(length(V1))) = 1 >= 1 = mark(isNatList(V1)) active(isNat(s(V1))) = 1 >= 1 = mark(isNat(V1)) active(isNatIList(V)) = 1 >= 1 = mark(isNatList(V)) active(isNatIList(zeros())) = 1 >= 1 = mark(tt()) active(isNatIList(cons(V1,V2))) = 1 >= 1 = mark(and(isNat(V1),isNatIList(V2))) active(isNatList(nil())) = 1 >= 1 = mark(tt()) active(isNatList(cons(V1,V2))) = 1 >= 1 = mark(and(isNat(V1),isNatList(V2))) active(isNatList(take(V1,V2))) = 1 >= 1 = mark(and(isNat(V1),isNatIList(V2))) active(length(nil())) = 1 >= 1 = mark(0()) active(length(cons(N,L))) = L + N + 1 >= L + 1 = mark(U11(and(isNatList(L),isNat(N)),L)) active(take(0(),IL)) = IL + 1 >= 1 = mark(U21(isNatIList(IL))) active(take(s(M),cons(N,IL))) = IL + M + N + 1 >= IL + M + N + 1 = mark(U31(and(isNatIList(IL),and(isNat(M),isNat(N))),IL,M,N)) active(cons(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = cons(active(X1),X2) active(U11(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = U11(active(X1),X2) active(s(X)) = X + 1 >= X + 1 = s(active(X)) active(length(X)) = X + 1 >= X + 1 = length(active(X)) active(U21(X)) = X + 1 >= X + 1 = U21(active(X)) active(U31(X1,X2,X3,X4)) = X1 + X2 + X3 + X4 + 1 >= X1 + X2 + X3 + X4 + 1 = U31(active(X1),X2,X3,X4) active(take(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = take(active(X1),X2) active(take(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = take(X1,active(X2)) active(and(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = and(active(X1),X2) cons(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(cons(X1,X2)) U11(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(U11(X1,X2)) s(mark(X)) = X + 1 >= X + 1 = mark(s(X)) length(mark(X)) = X + 1 >= X + 1 = mark(length(X)) U21(mark(X)) = X + 1 >= X + 1 = mark(U21(X)) U31(mark(X1),X2,X3,X4) = X1 + X2 + X3 + X4 + 1 >= X1 + X2 + X3 + X4 + 1 = mark(U31(X1,X2,X3,X4)) take(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(take(X1,X2)) take(X1,mark(X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(take(X1,X2)) and(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(and(X1,X2)) proper(zeros()) = 0 >= 0 = ok(zeros()) proper(cons(X1,X2)) = 0 >= 0 = cons(proper(X1),proper(X2)) proper(0()) = 0 >= 0 = ok(0()) proper(U11(X1,X2)) = 0 >= 0 = U11(proper(X1),proper(X2)) proper(tt()) = 0 >= 0 = ok(tt()) proper(s(X)) = 0 >= 0 = s(proper(X)) proper(length(X)) = 0 >= 0 = length(proper(X)) proper(U21(X)) = 0 >= 0 = U21(proper(X)) proper(nil()) = 0 >= 0 = ok(nil()) proper(U31(X1,X2,X3,X4)) = 0 >= 0 = U31(proper(X1),proper(X2),proper(X3),proper(X4)) proper(take(X1,X2)) = 0 >= 0 = take(proper(X1),proper(X2)) proper(and(X1,X2)) = 0 >= 0 = and(proper(X1),proper(X2)) proper(isNat(X)) = 0 >= 0 = isNat(proper(X)) proper(isNatList(X)) = 0 >= 0 = isNatList(proper(X)) proper(isNatIList(X)) = 0 >= 0 = isNatIList(proper(X)) cons(ok(X1),ok(X2)) = 0 >= 0 = ok(cons(X1,X2)) U11(ok(X1),ok(X2)) = 0 >= 0 = ok(U11(X1,X2)) s(ok(X)) = 0 >= 0 = ok(s(X)) length(ok(X)) = 0 >= 0 = ok(length(X)) U21(ok(X)) = 0 >= 0 = ok(U21(X)) U31(ok(X1),ok(X2),ok(X3),ok(X4)) = 0 >= 0 = ok(U31(X1,X2,X3,X4)) take(ok(X1),ok(X2)) = 0 >= 0 = ok(take(X1,X2)) and(ok(X1),ok(X2)) = 0 >= 0 = ok(and(X1,X2)) isNat(ok(X)) = 0 >= 0 = ok(isNat(X)) isNatList(ok(X)) = 0 >= 0 = ok(isNatList(X)) isNatIList(ok(X)) = 0 >= 0 = ok(isNatIList(X)) top(mark(X)) = 1 >= 1 = top(proper(X)) top(ok(X)) = 1 >= 1 = top(active(X)) 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#(ok(X1),ok(X2)) -> take#(X1,X2) take#(X1,mark(X2)) -> take#(X1,X2) take#(mark(X1),X2) -> take#(X1,X2) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),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)) Matrix Interpretation Processor: dimension: 1 interpretation: [take#](x0, x1) = x1, [top](x0) = 0, [ok](x0) = x0 + 1, [proper](x0) = x0 + 1, [isNatIList](x0) = x0 + 1, [isNatList](x0) = x0, [isNat](x0) = x0, [and](x0, x1) = x1, [take](x0, x1) = x1, [U31](x0, x1, x2, x3) = x1 + 1, [nil] = 1, [U21](x0) = x0, [s](x0) = x0, [length](x0) = x0, [U11](x0, x1) = x1, [tt] = 0, [mark](x0) = x0, [cons](x0, x1) = x1, [0] = 1, [active](x0) = x0 + 1, [zeros] = 0 orientation: take#(ok(X1),ok(X2)) = X2 + 1 >= X2 = take#(X1,X2) take#(X1,mark(X2)) = X2 >= X2 = take#(X1,X2) take#(mark(X1),X2) = X2 >= X2 = take#(X1,X2) active(zeros()) = 1 >= 0 = mark(cons(0(),zeros())) active(U11(tt(),L)) = L + 1 >= L = mark(s(length(L))) active(U21(tt())) = 1 >= 1 = mark(nil()) active(U31(tt(),IL,M,N)) = IL + 2 >= IL = mark(cons(N,take(M,IL))) active(and(tt(),X)) = X + 1 >= X = mark(X) active(isNat(0())) = 2 >= 0 = mark(tt()) active(isNat(length(V1))) = V1 + 1 >= V1 = mark(isNatList(V1)) active(isNat(s(V1))) = V1 + 1 >= V1 = mark(isNat(V1)) active(isNatIList(V)) = V + 2 >= V = mark(isNatList(V)) active(isNatIList(zeros())) = 2 >= 0 = mark(tt()) active(isNatIList(cons(V1,V2))) = V2 + 2 >= V2 + 1 = mark(and(isNat(V1),isNatIList(V2))) active(isNatList(nil())) = 2 >= 0 = mark(tt()) active(isNatList(cons(V1,V2))) = V2 + 1 >= V2 = mark(and(isNat(V1),isNatList(V2))) active(isNatList(take(V1,V2))) = V2 + 1 >= V2 + 1 = mark(and(isNat(V1),isNatIList(V2))) active(length(nil())) = 2 >= 1 = mark(0()) active(length(cons(N,L))) = L + 1 >= L = mark(U11(and(isNatList(L),isNat(N)),L)) active(take(0(),IL)) = IL + 1 >= IL + 1 = mark(U21(isNatIList(IL))) active(take(s(M),cons(N,IL))) = IL + 1 >= IL + 1 = mark(U31(and(isNatIList(IL),and(isNat(M),isNat(N))),IL,M,N)) active(cons(X1,X2)) = X2 + 1 >= X2 = cons(active(X1),X2) active(U11(X1,X2)) = X2 + 1 >= X2 = U11(active(X1),X2) active(s(X)) = X + 1 >= X + 1 = s(active(X)) active(length(X)) = X + 1 >= X + 1 = length(active(X)) active(U21(X)) = X + 1 >= X + 1 = U21(active(X)) active(U31(X1,X2,X3,X4)) = X2 + 2 >= X2 + 1 = U31(active(X1),X2,X3,X4) active(take(X1,X2)) = X2 + 1 >= X2 = take(active(X1),X2) active(take(X1,X2)) = X2 + 1 >= X2 + 1 = take(X1,active(X2)) active(and(X1,X2)) = X2 + 1 >= X2 = and(active(X1),X2) cons(mark(X1),X2) = X2 >= X2 = mark(cons(X1,X2)) U11(mark(X1),X2) = X2 >= X2 = mark(U11(X1,X2)) s(mark(X)) = X >= X = mark(s(X)) length(mark(X)) = X >= X = mark(length(X)) U21(mark(X)) = X >= X = mark(U21(X)) U31(mark(X1),X2,X3,X4) = X2 + 1 >= X2 + 1 = mark(U31(X1,X2,X3,X4)) take(mark(X1),X2) = X2 >= X2 = mark(take(X1,X2)) take(X1,mark(X2)) = X2 >= X2 = mark(take(X1,X2)) and(mark(X1),X2) = X2 >= X2 = mark(and(X1,X2)) proper(zeros()) = 1 >= 1 = ok(zeros()) proper(cons(X1,X2)) = X2 + 1 >= X2 + 1 = cons(proper(X1),proper(X2)) proper(0()) = 2 >= 2 = ok(0()) proper(U11(X1,X2)) = X2 + 1 >= X2 + 1 = U11(proper(X1),proper(X2)) proper(tt()) = 1 >= 1 = ok(tt()) proper(s(X)) = X + 1 >= X + 1 = s(proper(X)) proper(length(X)) = X + 1 >= X + 1 = length(proper(X)) proper(U21(X)) = X + 1 >= X + 1 = U21(proper(X)) proper(nil()) = 2 >= 2 = ok(nil()) proper(U31(X1,X2,X3,X4)) = X2 + 2 >= X2 + 2 = U31(proper(X1),proper(X2),proper(X3),proper(X4)) proper(take(X1,X2)) = X2 + 1 >= X2 + 1 = take(proper(X1),proper(X2)) proper(and(X1,X2)) = X2 + 1 >= X2 + 1 = and(proper(X1),proper(X2)) proper(isNat(X)) = X + 1 >= X + 1 = isNat(proper(X)) proper(isNatList(X)) = X + 1 >= X + 1 = isNatList(proper(X)) proper(isNatIList(X)) = X + 2 >= X + 2 = isNatIList(proper(X)) cons(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(cons(X1,X2)) U11(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(U11(X1,X2)) s(ok(X)) = X + 1 >= X + 1 = ok(s(X)) length(ok(X)) = X + 1 >= X + 1 = ok(length(X)) U21(ok(X)) = X + 1 >= X + 1 = ok(U21(X)) U31(ok(X1),ok(X2),ok(X3),ok(X4)) = X2 + 2 >= X2 + 2 = ok(U31(X1,X2,X3,X4)) take(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(take(X1,X2)) and(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(and(X1,X2)) isNat(ok(X)) = X + 1 >= X + 1 = ok(isNat(X)) isNatList(ok(X)) = X + 1 >= X + 1 = ok(isNatList(X)) isNatIList(ok(X)) = X + 2 >= X + 2 = ok(isNatIList(X)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) problem: DPs: take#(X1,mark(X2)) -> take#(X1,X2) 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)) Matrix Interpretation Processor: dimension: 1 interpretation: [take#](x0, x1) = x0, [top](x0) = 0, [ok](x0) = 0, [proper](x0) = x0, [isNatIList](x0) = 0, [isNatList](x0) = 0, [isNat](x0) = 0, [and](x0, x1) = x0 + x1, [take](x0, x1) = x0 + x1 + 1, [U31](x0, x1, x2, x3) = x0 + x1 + x2 + x3 + 1, [nil] = 0, [U21](x0) = x0 + 1, [s](x0) = x0, [length](x0) = x0, [U11](x0, x1) = x0 + x1, [tt] = 0, [mark](x0) = x0 + 1, [cons](x0, x1) = x0 + x1, [0] = 0, [active](x0) = x0 + 1, [zeros] = 0 orientation: take#(X1,mark(X2)) = X1 >= X1 = take#(X1,X2) take#(mark(X1),X2) = X1 + 1 >= X1 = take#(X1,X2) active(zeros()) = 1 >= 1 = mark(cons(0(),zeros())) active(U11(tt(),L)) = L + 1 >= L + 1 = mark(s(length(L))) active(U21(tt())) = 2 >= 1 = mark(nil()) active(U31(tt(),IL,M,N)) = IL + M + N + 2 >= IL + M + N + 2 = mark(cons(N,take(M,IL))) active(and(tt(),X)) = X + 1 >= X + 1 = mark(X) active(isNat(0())) = 1 >= 1 = mark(tt()) active(isNat(length(V1))) = 1 >= 1 = mark(isNatList(V1)) active(isNat(s(V1))) = 1 >= 1 = mark(isNat(V1)) active(isNatIList(V)) = 1 >= 1 = mark(isNatList(V)) active(isNatIList(zeros())) = 1 >= 1 = mark(tt()) active(isNatIList(cons(V1,V2))) = 1 >= 1 = mark(and(isNat(V1),isNatIList(V2))) active(isNatList(nil())) = 1 >= 1 = mark(tt()) active(isNatList(cons(V1,V2))) = 1 >= 1 = mark(and(isNat(V1),isNatList(V2))) active(isNatList(take(V1,V2))) = 1 >= 1 = mark(and(isNat(V1),isNatIList(V2))) active(length(nil())) = 1 >= 1 = mark(0()) active(length(cons(N,L))) = L + N + 1 >= L + 1 = mark(U11(and(isNatList(L),isNat(N)),L)) active(take(0(),IL)) = IL + 2 >= 2 = mark(U21(isNatIList(IL))) active(take(s(M),cons(N,IL))) = IL + M + N + 2 >= IL + M + N + 2 = mark(U31(and(isNatIList(IL),and(isNat(M),isNat(N))),IL,M,N)) active(cons(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = cons(active(X1),X2) active(U11(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = U11(active(X1),X2) active(s(X)) = X + 1 >= X + 1 = s(active(X)) active(length(X)) = X + 1 >= X + 1 = length(active(X)) active(U21(X)) = X + 2 >= X + 2 = U21(active(X)) active(U31(X1,X2,X3,X4)) = X1 + X2 + X3 + X4 + 2 >= X1 + X2 + X3 + X4 + 2 = U31(active(X1),X2,X3,X4) active(take(X1,X2)) = X1 + X2 + 2 >= X1 + X2 + 2 = take(active(X1),X2) active(take(X1,X2)) = X1 + X2 + 2 >= X1 + X2 + 2 = take(X1,active(X2)) active(and(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = and(active(X1),X2) cons(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(cons(X1,X2)) U11(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(U11(X1,X2)) s(mark(X)) = X + 1 >= X + 1 = mark(s(X)) length(mark(X)) = X + 1 >= X + 1 = mark(length(X)) U21(mark(X)) = X + 2 >= X + 2 = mark(U21(X)) U31(mark(X1),X2,X3,X4) = X1 + X2 + X3 + X4 + 2 >= X1 + X2 + X3 + X4 + 2 = mark(U31(X1,X2,X3,X4)) take(mark(X1),X2) = X1 + X2 + 2 >= X1 + X2 + 2 = mark(take(X1,X2)) take(X1,mark(X2)) = X1 + X2 + 2 >= X1 + X2 + 2 = mark(take(X1,X2)) and(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(and(X1,X2)) proper(zeros()) = 0 >= 0 = ok(zeros()) proper(cons(X1,X2)) = X1 + X2 >= X1 + X2 = cons(proper(X1),proper(X2)) proper(0()) = 0 >= 0 = ok(0()) proper(U11(X1,X2)) = X1 + X2 >= X1 + X2 = U11(proper(X1),proper(X2)) proper(tt()) = 0 >= 0 = ok(tt()) proper(s(X)) = X >= X = s(proper(X)) proper(length(X)) = X >= X = length(proper(X)) proper(U21(X)) = X + 1 >= X + 1 = U21(proper(X)) proper(nil()) = 0 >= 0 = ok(nil()) proper(U31(X1,X2,X3,X4)) = X1 + X2 + X3 + X4 + 1 >= X1 + X2 + X3 + X4 + 1 = U31(proper(X1),proper(X2),proper(X3),proper(X4)) proper(take(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = take(proper(X1),proper(X2)) proper(and(X1,X2)) = X1 + X2 >= X1 + X2 = and(proper(X1),proper(X2)) proper(isNat(X)) = 0 >= 0 = isNat(proper(X)) proper(isNatList(X)) = 0 >= 0 = isNatList(proper(X)) proper(isNatIList(X)) = 0 >= 0 = isNatIList(proper(X)) cons(ok(X1),ok(X2)) = 0 >= 0 = ok(cons(X1,X2)) U11(ok(X1),ok(X2)) = 0 >= 0 = ok(U11(X1,X2)) s(ok(X)) = 0 >= 0 = ok(s(X)) length(ok(X)) = 0 >= 0 = ok(length(X)) U21(ok(X)) = 1 >= 0 = ok(U21(X)) U31(ok(X1),ok(X2),ok(X3),ok(X4)) = 1 >= 0 = ok(U31(X1,X2,X3,X4)) take(ok(X1),ok(X2)) = 1 >= 0 = ok(take(X1,X2)) and(ok(X1),ok(X2)) = 0 >= 0 = ok(and(X1,X2)) isNat(ok(X)) = 0 >= 0 = ok(isNat(X)) isNatList(ok(X)) = 0 >= 0 = ok(isNatList(X)) isNatIList(ok(X)) = 0 >= 0 = ok(isNatIList(X)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) problem: DPs: take#(X1,mark(X2)) -> take#(X1,X2) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),L)) -> mark(s(length(L))) active(U21(tt())) -> mark(nil()) active(U31(tt(),IL,M,N)) -> mark(cons(N,take(M,IL))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(length(V1))) -> mark(isNatList(V1)) active(isNat(s(V1))) -> mark(isNat(V1)) active(isNatIList(V)) -> mark(isNatList(V)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(and(isNat(V1),isNatIList(V2))) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(and(isNat(V1),isNatList(V2))) active(isNatList(take(V1,V2))) -> mark(and(isNat(V1),isNatIList(V2))) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U11(and(isNatList(L),isNat(N)),L)) active(take(0(),IL)) -> mark(U21(isNatIList(IL))) active(take(s(M),cons(N,IL))) -> mark(U31(and(isNatIList(IL),and(isNat(M),isNat(N))),IL,M,N)) 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)) Matrix Interpretation Processor: dimension: 1 interpretation: [take#](x0, x1) = x1 + 1, [top](x0) = 0, [ok](x0) = 0, [proper](x0) = 0, [isNatIList](x0) = 0, [isNatList](x0) = 0, [isNat](x0) = 0, [and](x0, x1) = x0 + x1, [take](x0, x1) = x0 + x1, [U31](x0, x1, x2, x3) = x0 + x1 + x2 + x3, [nil] = 0, [U21](x0) = x0, [s](x0) = x0, [length](x0) = x0, [U11](x0, x1) = x0 + x1, [tt] = 0, [mark](x0) = x0 + 1, [cons](x0, x1) = x0 + x1, [0] = 0, [active](x0) = x0 + 1, [zeros] = 0 orientation: take#(X1,mark(X2)) = X2 + 2 >= X2 + 1 = take#(X1,X2) active(zeros()) = 1 >= 1 = mark(cons(0(),zeros())) active(U11(tt(),L)) = L + 1 >= L + 1 = mark(s(length(L))) active(U21(tt())) = 1 >= 1 = mark(nil()) active(U31(tt(),IL,M,N)) = IL + M + N + 1 >= IL + M + N + 1 = mark(cons(N,take(M,IL))) active(and(tt(),X)) = X + 1 >= X + 1 = mark(X) active(isNat(0())) = 1 >= 1 = mark(tt()) active(isNat(length(V1))) = 1 >= 1 = mark(isNatList(V1)) active(isNat(s(V1))) = 1 >= 1 = mark(isNat(V1)) active(isNatIList(V)) = 1 >= 1 = mark(isNatList(V)) active(isNatIList(zeros())) = 1 >= 1 = mark(tt()) active(isNatIList(cons(V1,V2))) = 1 >= 1 = mark(and(isNat(V1),isNatIList(V2))) active(isNatList(nil())) = 1 >= 1 = mark(tt()) active(isNatList(cons(V1,V2))) = 1 >= 1 = mark(and(isNat(V1),isNatList(V2))) active(isNatList(take(V1,V2))) = 1 >= 1 = mark(and(isNat(V1),isNatIList(V2))) active(length(nil())) = 1 >= 1 = mark(0()) active(length(cons(N,L))) = L + N + 1 >= L + 1 = mark(U11(and(isNatList(L),isNat(N)),L)) active(take(0(),IL)) = IL + 1 >= 1 = mark(U21(isNatIList(IL))) active(take(s(M),cons(N,IL))) = IL + M + N + 1 >= IL + M + N + 1 = mark(U31(and(isNatIList(IL),and(isNat(M),isNat(N))),IL,M,N)) active(cons(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = cons(active(X1),X2) active(U11(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = U11(active(X1),X2) active(s(X)) = X + 1 >= X + 1 = s(active(X)) active(length(X)) = X + 1 >= X + 1 = length(active(X)) active(U21(X)) = X + 1 >= X + 1 = U21(active(X)) active(U31(X1,X2,X3,X4)) = X1 + X2 + X3 + X4 + 1 >= X1 + X2 + X3 + X4 + 1 = U31(active(X1),X2,X3,X4) active(take(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = take(active(X1),X2) active(take(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = take(X1,active(X2)) active(and(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = and(active(X1),X2) cons(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(cons(X1,X2)) U11(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(U11(X1,X2)) s(mark(X)) = X + 1 >= X + 1 = mark(s(X)) length(mark(X)) = X + 1 >= X + 1 = mark(length(X)) U21(mark(X)) = X + 1 >= X + 1 = mark(U21(X)) U31(mark(X1),X2,X3,X4) = X1 + X2 + X3 + X4 + 1 >= X1 + X2 + X3 + X4 + 1 = mark(U31(X1,X2,X3,X4)) take(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(take(X1,X2)) take(X1,mark(X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(take(X1,X2)) and(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(and(X1,X2)) proper(zeros()) = 0 >= 0 = ok(zeros()) proper(cons(X1,X2)) = 0 >= 0 = cons(proper(X1),proper(X2)) proper(0()) = 0 >= 0 = ok(0()) proper(U11(X1,X2)) = 0 >= 0 = U11(proper(X1),proper(X2)) proper(tt()) = 0 >= 0 = ok(tt()) proper(s(X)) = 0 >= 0 = s(proper(X)) proper(length(X)) = 0 >= 0 = length(proper(X)) proper(U21(X)) = 0 >= 0 = U21(proper(X)) proper(nil()) = 0 >= 0 = ok(nil()) proper(U31(X1,X2,X3,X4)) = 0 >= 0 = U31(proper(X1),proper(X2),proper(X3),proper(X4)) proper(take(X1,X2)) = 0 >= 0 = take(proper(X1),proper(X2)) proper(and(X1,X2)) = 0 >= 0 = and(proper(X1),proper(X2)) proper(isNat(X)) = 0 >= 0 = isNat(proper(X)) proper(isNatList(X)) = 0 >= 0 = isNatList(proper(X)) proper(isNatIList(X)) = 0 >= 0 = isNatIList(proper(X)) cons(ok(X1),ok(X2)) = 0 >= 0 = ok(cons(X1,X2)) U11(ok(X1),ok(X2)) = 0 >= 0 = ok(U11(X1,X2)) s(ok(X)) = 0 >= 0 = ok(s(X)) length(ok(X)) = 0 >= 0 = ok(length(X)) U21(ok(X)) = 0 >= 0 = ok(U21(X)) U31(ok(X1),ok(X2),ok(X3),ok(X4)) = 0 >= 0 = ok(U31(X1,X2,X3,X4)) take(ok(X1),ok(X2)) = 0 >= 0 = ok(take(X1,X2)) and(ok(X1),ok(X2)) = 0 >= 0 = ok(and(X1,X2)) isNat(ok(X)) = 0 >= 0 = ok(isNat(X)) isNatList(ok(X)) = 0 >= 0 = ok(isNatList(X)) isNatIList(ok(X)) = 0 >= 0 = ok(isNatIList(X)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) 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#(ok(X1),ok(X2)) -> and#(X1,X2) 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)) Matrix Interpretation Processor: dimension: 1 interpretation: [and#](x0, x1) = x1 + 1, [top](x0) = 0, [ok](x0) = x0 + 1, [proper](x0) = x0 + 1, [isNatIList](x0) = x0, [isNatList](x0) = x0, [isNat](x0) = x0, [and](x0, x1) = x0, [take](x0, x1) = x1, [U31](x0, x1, x2, x3) = x3, [nil] = 0, [U21](x0) = x0, [s](x0) = x0, [length](x0) = x0, [U11](x0, x1) = x1, [tt] = 0, [mark](x0) = 0, [cons](x0, x1) = x0, [0] = 0, [active](x0) = x0, [zeros] = 1 orientation: and#(ok(X1),ok(X2)) = X2 + 2 >= X2 + 1 = and#(X1,X2) and#(mark(X1),X2) = X2 + 1 >= X2 + 1 = and#(X1,X2) active(zeros()) = 1 >= 0 = mark(cons(0(),zeros())) active(U11(tt(),L)) = L >= 0 = mark(s(length(L))) active(U21(tt())) = 0 >= 0 = mark(nil()) active(U31(tt(),IL,M,N)) = N >= 0 = mark(cons(N,take(M,IL))) active(and(tt(),X)) = 0 >= 0 = mark(X) active(isNat(0())) = 0 >= 0 = mark(tt()) active(isNat(length(V1))) = V1 >= 0 = mark(isNatList(V1)) active(isNat(s(V1))) = V1 >= 0 = mark(isNat(V1)) active(isNatIList(V)) = V >= 0 = mark(isNatList(V)) active(isNatIList(zeros())) = 1 >= 0 = mark(tt()) active(isNatIList(cons(V1,V2))) = V1 >= 0 = mark(and(isNat(V1),isNatIList(V2))) active(isNatList(nil())) = 0 >= 0 = mark(tt()) active(isNatList(cons(V1,V2))) = V1 >= 0 = mark(and(isNat(V1),isNatList(V2))) active(isNatList(take(V1,V2))) = V2 >= 0 = mark(and(isNat(V1),isNatIList(V2))) active(length(nil())) = 0 >= 0 = mark(0()) active(length(cons(N,L))) = N >= 0 = mark(U11(and(isNatList(L),isNat(N)),L)) active(take(0(),IL)) = IL >= 0 = mark(U21(isNatIList(IL))) active(take(s(M),cons(N,IL))) = N >= 0 = mark(U31(and(isNatIList(IL),and(isNat(M),isNat(N))),IL,M,N)) active(cons(X1,X2)) = X1 >= X1 = cons(active(X1),X2) active(U11(X1,X2)) = X2 >= X2 = U11(active(X1),X2) active(s(X)) = X >= X = s(active(X)) active(length(X)) = X >= X = length(active(X)) active(U21(X)) = X >= X = U21(active(X)) active(U31(X1,X2,X3,X4)) = X4 >= X4 = U31(active(X1),X2,X3,X4) active(take(X1,X2)) = X2 >= X2 = take(active(X1),X2) active(take(X1,X2)) = X2 >= X2 = take(X1,active(X2)) active(and(X1,X2)) = X1 >= X1 = and(active(X1),X2) cons(mark(X1),X2) = 0 >= 0 = mark(cons(X1,X2)) U11(mark(X1),X2) = X2 >= 0 = mark(U11(X1,X2)) s(mark(X)) = 0 >= 0 = mark(s(X)) length(mark(X)) = 0 >= 0 = mark(length(X)) U21(mark(X)) = 0 >= 0 = mark(U21(X)) U31(mark(X1),X2,X3,X4) = X4 >= 0 = mark(U31(X1,X2,X3,X4)) take(mark(X1),X2) = X2 >= 0 = mark(take(X1,X2)) take(X1,mark(X2)) = 0 >= 0 = mark(take(X1,X2)) and(mark(X1),X2) = 0 >= 0 = mark(and(X1,X2)) proper(zeros()) = 2 >= 2 = ok(zeros()) proper(cons(X1,X2)) = X1 + 1 >= X1 + 1 = cons(proper(X1),proper(X2)) proper(0()) = 1 >= 1 = ok(0()) proper(U11(X1,X2)) = X2 + 1 >= X2 + 1 = U11(proper(X1),proper(X2)) proper(tt()) = 1 >= 1 = ok(tt()) proper(s(X)) = X + 1 >= X + 1 = s(proper(X)) proper(length(X)) = X + 1 >= X + 1 = length(proper(X)) proper(U21(X)) = X + 1 >= X + 1 = U21(proper(X)) proper(nil()) = 1 >= 1 = ok(nil()) proper(U31(X1,X2,X3,X4)) = X4 + 1 >= X4 + 1 = U31(proper(X1),proper(X2),proper(X3),proper(X4)) proper(take(X1,X2)) = X2 + 1 >= X2 + 1 = take(proper(X1),proper(X2)) proper(and(X1,X2)) = X1 + 1 >= X1 + 1 = and(proper(X1),proper(X2)) proper(isNat(X)) = X + 1 >= X + 1 = isNat(proper(X)) proper(isNatList(X)) = X + 1 >= X + 1 = isNatList(proper(X)) proper(isNatIList(X)) = X + 1 >= X + 1 = isNatIList(proper(X)) cons(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(cons(X1,X2)) U11(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(U11(X1,X2)) s(ok(X)) = X + 1 >= X + 1 = ok(s(X)) length(ok(X)) = X + 1 >= X + 1 = ok(length(X)) U21(ok(X)) = X + 1 >= X + 1 = ok(U21(X)) U31(ok(X1),ok(X2),ok(X3),ok(X4)) = X4 + 1 >= X4 + 1 = ok(U31(X1,X2,X3,X4)) take(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(take(X1,X2)) and(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(and(X1,X2)) isNat(ok(X)) = X + 1 >= X + 1 = ok(isNat(X)) isNatList(ok(X)) = X + 1 >= X + 1 = ok(isNatList(X)) isNatIList(ok(X)) = X + 1 >= X + 1 = ok(isNatIList(X)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) 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)) Matrix Interpretation Processor: dimension: 1 interpretation: [and#](x0, x1) = x0 + 1, [top](x0) = 0, [ok](x0) = 0, [proper](x0) = 0, [isNatIList](x0) = 0, [isNatList](x0) = 0, [isNat](x0) = 0, [and](x0, x1) = x0 + x1, [take](x0, x1) = x0 + x1, [U31](x0, x1, x2, x3) = x0 + x1 + x2 + x3, [nil] = 0, [U21](x0) = x0, [s](x0) = x0, [length](x0) = x0, [U11](x0, x1) = x0 + x1, [tt] = 0, [mark](x0) = x0 + 1, [cons](x0, x1) = x0 + x1, [0] = 0, [active](x0) = x0 + 1, [zeros] = 0 orientation: and#(mark(X1),X2) = X1 + 2 >= X1 + 1 = and#(X1,X2) active(zeros()) = 1 >= 1 = mark(cons(0(),zeros())) active(U11(tt(),L)) = L + 1 >= L + 1 = mark(s(length(L))) active(U21(tt())) = 1 >= 1 = mark(nil()) active(U31(tt(),IL,M,N)) = IL + M + N + 1 >= IL + M + N + 1 = mark(cons(N,take(M,IL))) active(and(tt(),X)) = X + 1 >= X + 1 = mark(X) active(isNat(0())) = 1 >= 1 = mark(tt()) active(isNat(length(V1))) = 1 >= 1 = mark(isNatList(V1)) active(isNat(s(V1))) = 1 >= 1 = mark(isNat(V1)) active(isNatIList(V)) = 1 >= 1 = mark(isNatList(V)) active(isNatIList(zeros())) = 1 >= 1 = mark(tt()) active(isNatIList(cons(V1,V2))) = 1 >= 1 = mark(and(isNat(V1),isNatIList(V2))) active(isNatList(nil())) = 1 >= 1 = mark(tt()) active(isNatList(cons(V1,V2))) = 1 >= 1 = mark(and(isNat(V1),isNatList(V2))) active(isNatList(take(V1,V2))) = 1 >= 1 = mark(and(isNat(V1),isNatIList(V2))) active(length(nil())) = 1 >= 1 = mark(0()) active(length(cons(N,L))) = L + N + 1 >= L + 1 = mark(U11(and(isNatList(L),isNat(N)),L)) active(take(0(),IL)) = IL + 1 >= 1 = mark(U21(isNatIList(IL))) active(take(s(M),cons(N,IL))) = IL + M + N + 1 >= IL + M + N + 1 = mark(U31(and(isNatIList(IL),and(isNat(M),isNat(N))),IL,M,N)) active(cons(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = cons(active(X1),X2) active(U11(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = U11(active(X1),X2) active(s(X)) = X + 1 >= X + 1 = s(active(X)) active(length(X)) = X + 1 >= X + 1 = length(active(X)) active(U21(X)) = X + 1 >= X + 1 = U21(active(X)) active(U31(X1,X2,X3,X4)) = X1 + X2 + X3 + X4 + 1 >= X1 + X2 + X3 + X4 + 1 = U31(active(X1),X2,X3,X4) active(take(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = take(active(X1),X2) active(take(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = take(X1,active(X2)) active(and(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = and(active(X1),X2) cons(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(cons(X1,X2)) U11(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(U11(X1,X2)) s(mark(X)) = X + 1 >= X + 1 = mark(s(X)) length(mark(X)) = X + 1 >= X + 1 = mark(length(X)) U21(mark(X)) = X + 1 >= X + 1 = mark(U21(X)) U31(mark(X1),X2,X3,X4) = X1 + X2 + X3 + X4 + 1 >= X1 + X2 + X3 + X4 + 1 = mark(U31(X1,X2,X3,X4)) take(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(take(X1,X2)) take(X1,mark(X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(take(X1,X2)) and(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(and(X1,X2)) proper(zeros()) = 0 >= 0 = ok(zeros()) proper(cons(X1,X2)) = 0 >= 0 = cons(proper(X1),proper(X2)) proper(0()) = 0 >= 0 = ok(0()) proper(U11(X1,X2)) = 0 >= 0 = U11(proper(X1),proper(X2)) proper(tt()) = 0 >= 0 = ok(tt()) proper(s(X)) = 0 >= 0 = s(proper(X)) proper(length(X)) = 0 >= 0 = length(proper(X)) proper(U21(X)) = 0 >= 0 = U21(proper(X)) proper(nil()) = 0 >= 0 = ok(nil()) proper(U31(X1,X2,X3,X4)) = 0 >= 0 = U31(proper(X1),proper(X2),proper(X3),proper(X4)) proper(take(X1,X2)) = 0 >= 0 = take(proper(X1),proper(X2)) proper(and(X1,X2)) = 0 >= 0 = and(proper(X1),proper(X2)) proper(isNat(X)) = 0 >= 0 = isNat(proper(X)) proper(isNatList(X)) = 0 >= 0 = isNatList(proper(X)) proper(isNatIList(X)) = 0 >= 0 = isNatIList(proper(X)) cons(ok(X1),ok(X2)) = 0 >= 0 = ok(cons(X1,X2)) U11(ok(X1),ok(X2)) = 0 >= 0 = ok(U11(X1,X2)) s(ok(X)) = 0 >= 0 = ok(s(X)) length(ok(X)) = 0 >= 0 = ok(length(X)) U21(ok(X)) = 0 >= 0 = ok(U21(X)) U31(ok(X1),ok(X2),ok(X3),ok(X4)) = 0 >= 0 = ok(U31(X1,X2,X3,X4)) take(ok(X1),ok(X2)) = 0 >= 0 = ok(take(X1,X2)) and(ok(X1),ok(X2)) = 0 >= 0 = ok(and(X1,X2)) isNat(ok(X)) = 0 >= 0 = ok(isNat(X)) isNatList(ok(X)) = 0 >= 0 = ok(isNatList(X)) isNatIList(ok(X)) = 0 >= 0 = ok(isNatIList(X)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) 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