MAYBE Problem: active(and(tt(),T)) -> mark(T) active(isNatIList(IL)) -> mark(isNatList(IL)) active(isNat(0())) -> mark(tt()) active(isNat(s(N))) -> mark(isNat(N)) active(isNat(length(L))) -> mark(isNatList(L)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(N,L))) -> mark(and(isNat(N),isNatList(L))) active(isNatList(take(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(zeros()) -> mark(cons(0(),zeros())) active(take(0(),IL)) -> mark(uTake1(isNatIList(IL))) active(uTake1(tt())) -> mark(nil()) active(take(s(M),cons(N,IL))) -> mark(uTake2(and(isNat(M),and(isNat(N),isNatIList(IL))),M,N,IL)) active(uTake2(tt(),M,N,IL)) -> mark(cons(N,take(M,IL))) active(length(cons(N,L))) -> mark(uLength(and(isNat(N),isNatList(L)),L)) active(uLength(tt(),L)) -> mark(s(length(L))) active(and(X1,X2)) -> and(active(X1),X2) active(and(X1,X2)) -> and(X1,active(X2)) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) active(uTake1(X)) -> uTake1(active(X)) active(uTake2(X1,X2,X3,X4)) -> uTake2(active(X1),X2,X3,X4) active(uLength(X1,X2)) -> uLength(active(X1),X2) and(mark(X1),X2) -> mark(and(X1,X2)) and(X1,mark(X2)) -> mark(and(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) uTake1(mark(X)) -> mark(uTake1(X)) uTake2(mark(X1),X2,X3,X4) -> mark(uTake2(X1,X2,X3,X4)) uLength(mark(X1),X2) -> mark(uLength(X1,X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(isNat(X)) -> isNat(proper(X)) proper(0()) -> ok(0()) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(nil()) -> ok(nil()) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) proper(uTake1(X)) -> uTake1(proper(X)) proper(uTake2(X1,X2,X3,X4)) -> uTake2(proper(X1),proper(X2),proper(X3),proper(X4)) proper(uLength(X1,X2)) -> uLength(proper(X1),proper(X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatIList(ok(X)) -> ok(isNatIList(X)) isNatList(ok(X)) -> ok(isNatList(X)) isNat(ok(X)) -> ok(isNat(X)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) uTake1(ok(X)) -> ok(uTake1(X)) uTake2(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(uTake2(X1,X2,X3,X4)) uLength(ok(X1),ok(X2)) -> ok(uLength(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Proof: DP Processor: DPs: active#(isNatIList(IL)) -> isNatList#(IL) active#(isNat(s(N))) -> isNat#(N) active#(isNat(length(L))) -> isNatList#(L) active#(isNatIList(cons(N,IL))) -> isNatIList#(IL) active#(isNatIList(cons(N,IL))) -> isNat#(N) active#(isNatIList(cons(N,IL))) -> and#(isNat(N),isNatIList(IL)) active#(isNatList(cons(N,L))) -> isNatList#(L) active#(isNatList(cons(N,L))) -> isNat#(N) active#(isNatList(cons(N,L))) -> and#(isNat(N),isNatList(L)) active#(isNatList(take(N,IL))) -> isNatIList#(IL) active#(isNatList(take(N,IL))) -> isNat#(N) active#(isNatList(take(N,IL))) -> and#(isNat(N),isNatIList(IL)) active#(zeros()) -> cons#(0(),zeros()) active#(take(0(),IL)) -> isNatIList#(IL) active#(take(0(),IL)) -> uTake1#(isNatIList(IL)) active#(take(s(M),cons(N,IL))) -> isNatIList#(IL) active#(take(s(M),cons(N,IL))) -> isNat#(N) active#(take(s(M),cons(N,IL))) -> and#(isNat(N),isNatIList(IL)) active#(take(s(M),cons(N,IL))) -> isNat#(M) active#(take(s(M),cons(N,IL))) -> and#(isNat(M),and(isNat(N),isNatIList(IL))) active#(take(s(M),cons(N,IL))) -> uTake2#(and(isNat(M),and(isNat(N),isNatIList(IL))),M,N,IL) active#(uTake2(tt(),M,N,IL)) -> take#(M,IL) active#(uTake2(tt(),M,N,IL)) -> cons#(N,take(M,IL)) active#(length(cons(N,L))) -> isNatList#(L) active#(length(cons(N,L))) -> isNat#(N) active#(length(cons(N,L))) -> and#(isNat(N),isNatList(L)) active#(length(cons(N,L))) -> uLength#(and(isNat(N),isNatList(L)),L) active#(uLength(tt(),L)) -> length#(L) active#(uLength(tt(),L)) -> s#(length(L)) active#(and(X1,X2)) -> active#(X1) active#(and(X1,X2)) -> and#(active(X1),X2) active#(and(X1,X2)) -> active#(X2) active#(and(X1,X2)) -> and#(X1,active(X2)) active#(s(X)) -> active#(X) active#(s(X)) -> s#(active(X)) active#(length(X)) -> active#(X) active#(length(X)) -> length#(active(X)) active#(cons(X1,X2)) -> active#(X1) active#(cons(X1,X2)) -> cons#(active(X1),X2) 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#(uTake1(X)) -> active#(X) active#(uTake1(X)) -> uTake1#(active(X)) active#(uTake2(X1,X2,X3,X4)) -> active#(X1) active#(uTake2(X1,X2,X3,X4)) -> uTake2#(active(X1),X2,X3,X4) active#(uLength(X1,X2)) -> active#(X1) active#(uLength(X1,X2)) -> uLength#(active(X1),X2) and#(mark(X1),X2) -> and#(X1,X2) and#(X1,mark(X2)) -> and#(X1,X2) s#(mark(X)) -> s#(X) length#(mark(X)) -> length#(X) cons#(mark(X1),X2) -> cons#(X1,X2) take#(mark(X1),X2) -> take#(X1,X2) take#(X1,mark(X2)) -> take#(X1,X2) uTake1#(mark(X)) -> uTake1#(X) uTake2#(mark(X1),X2,X3,X4) -> uTake2#(X1,X2,X3,X4) uLength#(mark(X1),X2) -> uLength#(X1,X2) proper#(and(X1,X2)) -> proper#(X2) proper#(and(X1,X2)) -> proper#(X1) proper#(and(X1,X2)) -> and#(proper(X1),proper(X2)) proper#(isNatIList(X)) -> proper#(X) proper#(isNatIList(X)) -> isNatIList#(proper(X)) proper#(isNatList(X)) -> proper#(X) proper#(isNatList(X)) -> isNatList#(proper(X)) proper#(isNat(X)) -> proper#(X) proper#(isNat(X)) -> isNat#(proper(X)) proper#(s(X)) -> proper#(X) proper#(s(X)) -> s#(proper(X)) proper#(length(X)) -> proper#(X) proper#(length(X)) -> length#(proper(X)) proper#(cons(X1,X2)) -> proper#(X2) proper#(cons(X1,X2)) -> proper#(X1) proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2)) proper#(take(X1,X2)) -> proper#(X2) proper#(take(X1,X2)) -> proper#(X1) proper#(take(X1,X2)) -> take#(proper(X1),proper(X2)) proper#(uTake1(X)) -> proper#(X) proper#(uTake1(X)) -> uTake1#(proper(X)) proper#(uTake2(X1,X2,X3,X4)) -> proper#(X4) proper#(uTake2(X1,X2,X3,X4)) -> proper#(X3) proper#(uTake2(X1,X2,X3,X4)) -> proper#(X2) proper#(uTake2(X1,X2,X3,X4)) -> proper#(X1) proper#(uTake2(X1,X2,X3,X4)) -> uTake2#(proper(X1),proper(X2),proper(X3),proper(X4)) proper#(uLength(X1,X2)) -> proper#(X2) proper#(uLength(X1,X2)) -> proper#(X1) proper#(uLength(X1,X2)) -> uLength#(proper(X1),proper(X2)) and#(ok(X1),ok(X2)) -> and#(X1,X2) isNatIList#(ok(X)) -> isNatIList#(X) isNatList#(ok(X)) -> isNatList#(X) isNat#(ok(X)) -> isNat#(X) s#(ok(X)) -> s#(X) length#(ok(X)) -> length#(X) cons#(ok(X1),ok(X2)) -> cons#(X1,X2) take#(ok(X1),ok(X2)) -> take#(X1,X2) uTake1#(ok(X)) -> uTake1#(X) uTake2#(ok(X1),ok(X2),ok(X3),ok(X4)) -> uTake2#(X1,X2,X3,X4) uLength#(ok(X1),ok(X2)) -> uLength#(X1,X2) top#(mark(X)) -> proper#(X) top#(mark(X)) -> top#(proper(X)) top#(ok(X)) -> active#(X) top#(ok(X)) -> top#(active(X)) TRS: active(and(tt(),T)) -> mark(T) active(isNatIList(IL)) -> mark(isNatList(IL)) active(isNat(0())) -> mark(tt()) active(isNat(s(N))) -> mark(isNat(N)) active(isNat(length(L))) -> mark(isNatList(L)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(N,L))) -> mark(and(isNat(N),isNatList(L))) active(isNatList(take(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(zeros()) -> mark(cons(0(),zeros())) active(take(0(),IL)) -> mark(uTake1(isNatIList(IL))) active(uTake1(tt())) -> mark(nil()) active(take(s(M),cons(N,IL))) -> mark(uTake2(and(isNat(M),and(isNat(N),isNatIList(IL))),M,N,IL)) active(uTake2(tt(),M,N,IL)) -> mark(cons(N,take(M,IL))) active(length(cons(N,L))) -> mark(uLength(and(isNat(N),isNatList(L)),L)) active(uLength(tt(),L)) -> mark(s(length(L))) active(and(X1,X2)) -> and(active(X1),X2) active(and(X1,X2)) -> and(X1,active(X2)) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) active(uTake1(X)) -> uTake1(active(X)) active(uTake2(X1,X2,X3,X4)) -> uTake2(active(X1),X2,X3,X4) active(uLength(X1,X2)) -> uLength(active(X1),X2) and(mark(X1),X2) -> mark(and(X1,X2)) and(X1,mark(X2)) -> mark(and(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) uTake1(mark(X)) -> mark(uTake1(X)) uTake2(mark(X1),X2,X3,X4) -> mark(uTake2(X1,X2,X3,X4)) uLength(mark(X1),X2) -> mark(uLength(X1,X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(isNat(X)) -> isNat(proper(X)) proper(0()) -> ok(0()) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(nil()) -> ok(nil()) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) proper(uTake1(X)) -> uTake1(proper(X)) proper(uTake2(X1,X2,X3,X4)) -> uTake2(proper(X1),proper(X2),proper(X3),proper(X4)) proper(uLength(X1,X2)) -> uLength(proper(X1),proper(X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatIList(ok(X)) -> ok(isNatIList(X)) isNatList(ok(X)) -> ok(isNatList(X)) isNat(ok(X)) -> ok(isNat(X)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) uTake1(ok(X)) -> ok(uTake1(X)) uTake2(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(uTake2(X1,X2,X3,X4)) uLength(ok(X1),ok(X2)) -> ok(uLength(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) TDG Processor: DPs: active#(isNatIList(IL)) -> isNatList#(IL) active#(isNat(s(N))) -> isNat#(N) active#(isNat(length(L))) -> isNatList#(L) active#(isNatIList(cons(N,IL))) -> isNatIList#(IL) active#(isNatIList(cons(N,IL))) -> isNat#(N) active#(isNatIList(cons(N,IL))) -> and#(isNat(N),isNatIList(IL)) active#(isNatList(cons(N,L))) -> isNatList#(L) active#(isNatList(cons(N,L))) -> isNat#(N) active#(isNatList(cons(N,L))) -> and#(isNat(N),isNatList(L)) active#(isNatList(take(N,IL))) -> isNatIList#(IL) active#(isNatList(take(N,IL))) -> isNat#(N) active#(isNatList(take(N,IL))) -> and#(isNat(N),isNatIList(IL)) active#(zeros()) -> cons#(0(),zeros()) active#(take(0(),IL)) -> isNatIList#(IL) active#(take(0(),IL)) -> uTake1#(isNatIList(IL)) active#(take(s(M),cons(N,IL))) -> isNatIList#(IL) active#(take(s(M),cons(N,IL))) -> isNat#(N) active#(take(s(M),cons(N,IL))) -> and#(isNat(N),isNatIList(IL)) active#(take(s(M),cons(N,IL))) -> isNat#(M) active#(take(s(M),cons(N,IL))) -> and#(isNat(M),and(isNat(N),isNatIList(IL))) active#(take(s(M),cons(N,IL))) -> uTake2#(and(isNat(M),and(isNat(N),isNatIList(IL))),M,N,IL) active#(uTake2(tt(),M,N,IL)) -> take#(M,IL) active#(uTake2(tt(),M,N,IL)) -> cons#(N,take(M,IL)) active#(length(cons(N,L))) -> isNatList#(L) active#(length(cons(N,L))) -> isNat#(N) active#(length(cons(N,L))) -> and#(isNat(N),isNatList(L)) active#(length(cons(N,L))) -> uLength#(and(isNat(N),isNatList(L)),L) active#(uLength(tt(),L)) -> length#(L) active#(uLength(tt(),L)) -> s#(length(L)) active#(and(X1,X2)) -> active#(X1) active#(and(X1,X2)) -> and#(active(X1),X2) active#(and(X1,X2)) -> active#(X2) active#(and(X1,X2)) -> and#(X1,active(X2)) active#(s(X)) -> active#(X) active#(s(X)) -> s#(active(X)) active#(length(X)) -> active#(X) active#(length(X)) -> length#(active(X)) active#(cons(X1,X2)) -> active#(X1) active#(cons(X1,X2)) -> cons#(active(X1),X2) 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#(uTake1(X)) -> active#(X) active#(uTake1(X)) -> uTake1#(active(X)) active#(uTake2(X1,X2,X3,X4)) -> active#(X1) active#(uTake2(X1,X2,X3,X4)) -> uTake2#(active(X1),X2,X3,X4) active#(uLength(X1,X2)) -> active#(X1) active#(uLength(X1,X2)) -> uLength#(active(X1),X2) and#(mark(X1),X2) -> and#(X1,X2) and#(X1,mark(X2)) -> and#(X1,X2) s#(mark(X)) -> s#(X) length#(mark(X)) -> length#(X) cons#(mark(X1),X2) -> cons#(X1,X2) take#(mark(X1),X2) -> take#(X1,X2) take#(X1,mark(X2)) -> take#(X1,X2) uTake1#(mark(X)) -> uTake1#(X) uTake2#(mark(X1),X2,X3,X4) -> uTake2#(X1,X2,X3,X4) uLength#(mark(X1),X2) -> uLength#(X1,X2) proper#(and(X1,X2)) -> proper#(X2) proper#(and(X1,X2)) -> proper#(X1) proper#(and(X1,X2)) -> and#(proper(X1),proper(X2)) proper#(isNatIList(X)) -> proper#(X) proper#(isNatIList(X)) -> isNatIList#(proper(X)) proper#(isNatList(X)) -> proper#(X) proper#(isNatList(X)) -> isNatList#(proper(X)) proper#(isNat(X)) -> proper#(X) proper#(isNat(X)) -> isNat#(proper(X)) proper#(s(X)) -> proper#(X) proper#(s(X)) -> s#(proper(X)) proper#(length(X)) -> proper#(X) proper#(length(X)) -> length#(proper(X)) proper#(cons(X1,X2)) -> proper#(X2) proper#(cons(X1,X2)) -> proper#(X1) proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2)) proper#(take(X1,X2)) -> proper#(X2) proper#(take(X1,X2)) -> proper#(X1) proper#(take(X1,X2)) -> take#(proper(X1),proper(X2)) proper#(uTake1(X)) -> proper#(X) proper#(uTake1(X)) -> uTake1#(proper(X)) proper#(uTake2(X1,X2,X3,X4)) -> proper#(X4) proper#(uTake2(X1,X2,X3,X4)) -> proper#(X3) proper#(uTake2(X1,X2,X3,X4)) -> proper#(X2) proper#(uTake2(X1,X2,X3,X4)) -> proper#(X1) proper#(uTake2(X1,X2,X3,X4)) -> uTake2#(proper(X1),proper(X2),proper(X3),proper(X4)) proper#(uLength(X1,X2)) -> proper#(X2) proper#(uLength(X1,X2)) -> proper#(X1) proper#(uLength(X1,X2)) -> uLength#(proper(X1),proper(X2)) and#(ok(X1),ok(X2)) -> and#(X1,X2) isNatIList#(ok(X)) -> isNatIList#(X) isNatList#(ok(X)) -> isNatList#(X) isNat#(ok(X)) -> isNat#(X) s#(ok(X)) -> s#(X) length#(ok(X)) -> length#(X) cons#(ok(X1),ok(X2)) -> cons#(X1,X2) take#(ok(X1),ok(X2)) -> take#(X1,X2) uTake1#(ok(X)) -> uTake1#(X) uTake2#(ok(X1),ok(X2),ok(X3),ok(X4)) -> uTake2#(X1,X2,X3,X4) uLength#(ok(X1),ok(X2)) -> uLength#(X1,X2) top#(mark(X)) -> proper#(X) top#(mark(X)) -> top#(proper(X)) top#(ok(X)) -> active#(X) top#(ok(X)) -> top#(active(X)) TRS: active(and(tt(),T)) -> mark(T) active(isNatIList(IL)) -> mark(isNatList(IL)) active(isNat(0())) -> mark(tt()) active(isNat(s(N))) -> mark(isNat(N)) active(isNat(length(L))) -> mark(isNatList(L)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(N,L))) -> mark(and(isNat(N),isNatList(L))) active(isNatList(take(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(zeros()) -> mark(cons(0(),zeros())) active(take(0(),IL)) -> mark(uTake1(isNatIList(IL))) active(uTake1(tt())) -> mark(nil()) active(take(s(M),cons(N,IL))) -> mark(uTake2(and(isNat(M),and(isNat(N),isNatIList(IL))),M,N,IL)) active(uTake2(tt(),M,N,IL)) -> mark(cons(N,take(M,IL))) active(length(cons(N,L))) -> mark(uLength(and(isNat(N),isNatList(L)),L)) active(uLength(tt(),L)) -> mark(s(length(L))) active(and(X1,X2)) -> and(active(X1),X2) active(and(X1,X2)) -> and(X1,active(X2)) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) active(uTake1(X)) -> uTake1(active(X)) active(uTake2(X1,X2,X3,X4)) -> uTake2(active(X1),X2,X3,X4) active(uLength(X1,X2)) -> uLength(active(X1),X2) and(mark(X1),X2) -> mark(and(X1,X2)) and(X1,mark(X2)) -> mark(and(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) uTake1(mark(X)) -> mark(uTake1(X)) uTake2(mark(X1),X2,X3,X4) -> mark(uTake2(X1,X2,X3,X4)) uLength(mark(X1),X2) -> mark(uLength(X1,X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(isNat(X)) -> isNat(proper(X)) proper(0()) -> ok(0()) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(nil()) -> ok(nil()) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) proper(uTake1(X)) -> uTake1(proper(X)) proper(uTake2(X1,X2,X3,X4)) -> uTake2(proper(X1),proper(X2),proper(X3),proper(X4)) proper(uLength(X1,X2)) -> uLength(proper(X1),proper(X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatIList(ok(X)) -> ok(isNatIList(X)) isNatList(ok(X)) -> ok(isNatList(X)) isNat(ok(X)) -> ok(isNat(X)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) uTake1(ok(X)) -> ok(uTake1(X)) uTake2(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(uTake2(X1,X2,X3,X4)) uLength(ok(X1),ok(X2)) -> ok(uLength(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) graph: ... SCC Processor: #sccs: 14 #rules: 51 #arcs: 1238/10609 DPs: top#(ok(X)) -> top#(active(X)) top#(mark(X)) -> top#(proper(X)) TRS: active(and(tt(),T)) -> mark(T) active(isNatIList(IL)) -> mark(isNatList(IL)) active(isNat(0())) -> mark(tt()) active(isNat(s(N))) -> mark(isNat(N)) active(isNat(length(L))) -> mark(isNatList(L)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(N,L))) -> mark(and(isNat(N),isNatList(L))) active(isNatList(take(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(zeros()) -> mark(cons(0(),zeros())) active(take(0(),IL)) -> mark(uTake1(isNatIList(IL))) active(uTake1(tt())) -> mark(nil()) active(take(s(M),cons(N,IL))) -> mark(uTake2(and(isNat(M),and(isNat(N),isNatIList(IL))),M,N,IL)) active(uTake2(tt(),M,N,IL)) -> mark(cons(N,take(M,IL))) active(length(cons(N,L))) -> mark(uLength(and(isNat(N),isNatList(L)),L)) active(uLength(tt(),L)) -> mark(s(length(L))) active(and(X1,X2)) -> and(active(X1),X2) active(and(X1,X2)) -> and(X1,active(X2)) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) active(uTake1(X)) -> uTake1(active(X)) active(uTake2(X1,X2,X3,X4)) -> uTake2(active(X1),X2,X3,X4) active(uLength(X1,X2)) -> uLength(active(X1),X2) and(mark(X1),X2) -> mark(and(X1,X2)) and(X1,mark(X2)) -> mark(and(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) uTake1(mark(X)) -> mark(uTake1(X)) uTake2(mark(X1),X2,X3,X4) -> mark(uTake2(X1,X2,X3,X4)) uLength(mark(X1),X2) -> mark(uLength(X1,X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(isNat(X)) -> isNat(proper(X)) proper(0()) -> ok(0()) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(nil()) -> ok(nil()) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) proper(uTake1(X)) -> uTake1(proper(X)) proper(uTake2(X1,X2,X3,X4)) -> uTake2(proper(X1),proper(X2),proper(X3),proper(X4)) proper(uLength(X1,X2)) -> uLength(proper(X1),proper(X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatIList(ok(X)) -> ok(isNatIList(X)) isNatList(ok(X)) -> ok(isNatList(X)) isNat(ok(X)) -> ok(isNat(X)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) uTake1(ok(X)) -> ok(uTake1(X)) uTake2(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(uTake2(X1,X2,X3,X4)) uLength(ok(X1),ok(X2)) -> ok(uLength(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Open DPs: active#(and(X1,X2)) -> active#(X1) active#(and(X1,X2)) -> active#(X2) active#(s(X)) -> active#(X) active#(length(X)) -> active#(X) active#(cons(X1,X2)) -> active#(X1) active#(take(X1,X2)) -> active#(X1) active#(take(X1,X2)) -> active#(X2) active#(uTake1(X)) -> active#(X) active#(uTake2(X1,X2,X3,X4)) -> active#(X1) active#(uLength(X1,X2)) -> active#(X1) TRS: active(and(tt(),T)) -> mark(T) active(isNatIList(IL)) -> mark(isNatList(IL)) active(isNat(0())) -> mark(tt()) active(isNat(s(N))) -> mark(isNat(N)) active(isNat(length(L))) -> mark(isNatList(L)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(N,L))) -> mark(and(isNat(N),isNatList(L))) active(isNatList(take(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(zeros()) -> mark(cons(0(),zeros())) active(take(0(),IL)) -> mark(uTake1(isNatIList(IL))) active(uTake1(tt())) -> mark(nil()) active(take(s(M),cons(N,IL))) -> mark(uTake2(and(isNat(M),and(isNat(N),isNatIList(IL))),M,N,IL)) active(uTake2(tt(),M,N,IL)) -> mark(cons(N,take(M,IL))) active(length(cons(N,L))) -> mark(uLength(and(isNat(N),isNatList(L)),L)) active(uLength(tt(),L)) -> mark(s(length(L))) active(and(X1,X2)) -> and(active(X1),X2) active(and(X1,X2)) -> and(X1,active(X2)) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) active(uTake1(X)) -> uTake1(active(X)) active(uTake2(X1,X2,X3,X4)) -> uTake2(active(X1),X2,X3,X4) active(uLength(X1,X2)) -> uLength(active(X1),X2) and(mark(X1),X2) -> mark(and(X1,X2)) and(X1,mark(X2)) -> mark(and(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) uTake1(mark(X)) -> mark(uTake1(X)) uTake2(mark(X1),X2,X3,X4) -> mark(uTake2(X1,X2,X3,X4)) uLength(mark(X1),X2) -> mark(uLength(X1,X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(isNat(X)) -> isNat(proper(X)) proper(0()) -> ok(0()) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(nil()) -> ok(nil()) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) proper(uTake1(X)) -> uTake1(proper(X)) proper(uTake2(X1,X2,X3,X4)) -> uTake2(proper(X1),proper(X2),proper(X3),proper(X4)) proper(uLength(X1,X2)) -> uLength(proper(X1),proper(X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatIList(ok(X)) -> ok(isNatIList(X)) isNatList(ok(X)) -> ok(isNatList(X)) isNat(ok(X)) -> ok(isNat(X)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) uTake1(ok(X)) -> ok(uTake1(X)) uTake2(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(uTake2(X1,X2,X3,X4)) uLength(ok(X1),ok(X2)) -> ok(uLength(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dimension: 1 interpretation: [active#](x0) = x0, [top](x0) = 0, [ok](x0) = 0, [proper](x0) = x0, [uLength](x0, x1) = x0, [uTake2](x0, x1, x2, x3) = x0, [uTake1](x0) = x0, [take](x0, x1) = x0 + x1, [nil] = 0, [cons](x0, x1) = x0 + 1, [zeros] = 0, [length](x0) = x0, [s](x0) = x0, [isNat](x0) = 0, [0] = 0, [isNatList](x0) = 0, [isNatIList](x0) = 0, [mark](x0) = 0, [active](x0) = x0, [and](x0, x1) = x0 + x1, [tt] = 0 orientation: active#(and(X1,X2)) = X1 + X2 >= X1 = active#(X1) active#(and(X1,X2)) = X1 + X2 >= X2 = active#(X2) active#(s(X)) = X >= X = active#(X) active#(length(X)) = X >= X = active#(X) active#(cons(X1,X2)) = X1 + 1 >= X1 = active#(X1) active#(take(X1,X2)) = X1 + X2 >= X1 = active#(X1) active#(take(X1,X2)) = X1 + X2 >= X2 = active#(X2) active#(uTake1(X)) = X >= X = active#(X) active#(uTake2(X1,X2,X3,X4)) = X1 >= X1 = active#(X1) active#(uLength(X1,X2)) = X1 >= X1 = active#(X1) active(and(tt(),T)) = T >= 0 = mark(T) active(isNatIList(IL)) = 0 >= 0 = mark(isNatList(IL)) active(isNat(0())) = 0 >= 0 = mark(tt()) active(isNat(s(N))) = 0 >= 0 = mark(isNat(N)) active(isNat(length(L))) = 0 >= 0 = mark(isNatList(L)) active(isNatIList(zeros())) = 0 >= 0 = mark(tt()) active(isNatIList(cons(N,IL))) = 0 >= 0 = mark(and(isNat(N),isNatIList(IL))) active(isNatList(nil())) = 0 >= 0 = mark(tt()) active(isNatList(cons(N,L))) = 0 >= 0 = mark(and(isNat(N),isNatList(L))) active(isNatList(take(N,IL))) = 0 >= 0 = mark(and(isNat(N),isNatIList(IL))) active(zeros()) = 0 >= 0 = mark(cons(0(),zeros())) active(take(0(),IL)) = IL >= 0 = mark(uTake1(isNatIList(IL))) active(uTake1(tt())) = 0 >= 0 = mark(nil()) active(take(s(M),cons(N,IL))) = M + N + 1 >= 0 = mark(uTake2(and(isNat(M),and(isNat(N),isNatIList(IL))),M,N,IL)) active(uTake2(tt(),M,N,IL)) = 0 >= 0 = mark(cons(N,take(M,IL))) active(length(cons(N,L))) = N + 1 >= 0 = mark(uLength(and(isNat(N),isNatList(L)),L)) active(uLength(tt(),L)) = 0 >= 0 = mark(s(length(L))) active(and(X1,X2)) = X1 + X2 >= X1 + X2 = and(active(X1),X2) active(and(X1,X2)) = X1 + X2 >= X1 + X2 = and(X1,active(X2)) active(s(X)) = X >= X = s(active(X)) active(length(X)) = X >= X = length(active(X)) active(cons(X1,X2)) = X1 + 1 >= X1 + 1 = cons(active(X1),X2) 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(uTake1(X)) = X >= X = uTake1(active(X)) active(uTake2(X1,X2,X3,X4)) = X1 >= X1 = uTake2(active(X1),X2,X3,X4) active(uLength(X1,X2)) = X1 >= X1 = uLength(active(X1),X2) and(mark(X1),X2) = X2 >= 0 = mark(and(X1,X2)) and(X1,mark(X2)) = X1 >= 0 = mark(and(X1,X2)) s(mark(X)) = 0 >= 0 = mark(s(X)) length(mark(X)) = 0 >= 0 = mark(length(X)) cons(mark(X1),X2) = 1 >= 0 = mark(cons(X1,X2)) take(mark(X1),X2) = X2 >= 0 = mark(take(X1,X2)) take(X1,mark(X2)) = X1 >= 0 = mark(take(X1,X2)) uTake1(mark(X)) = 0 >= 0 = mark(uTake1(X)) uTake2(mark(X1),X2,X3,X4) = 0 >= 0 = mark(uTake2(X1,X2,X3,X4)) uLength(mark(X1),X2) = 0 >= 0 = mark(uLength(X1,X2)) proper(and(X1,X2)) = X1 + X2 >= X1 + X2 = and(proper(X1),proper(X2)) proper(tt()) = 0 >= 0 = ok(tt()) proper(isNatIList(X)) = 0 >= 0 = isNatIList(proper(X)) proper(isNatList(X)) = 0 >= 0 = isNatList(proper(X)) proper(isNat(X)) = 0 >= 0 = isNat(proper(X)) proper(0()) = 0 >= 0 = ok(0()) proper(s(X)) = X >= X = s(proper(X)) proper(length(X)) = X >= X = length(proper(X)) proper(zeros()) = 0 >= 0 = ok(zeros()) proper(cons(X1,X2)) = X1 + 1 >= X1 + 1 = cons(proper(X1),proper(X2)) proper(nil()) = 0 >= 0 = ok(nil()) proper(take(X1,X2)) = X1 + X2 >= X1 + X2 = take(proper(X1),proper(X2)) proper(uTake1(X)) = X >= X = uTake1(proper(X)) proper(uTake2(X1,X2,X3,X4)) = X1 >= X1 = uTake2(proper(X1),proper(X2),proper(X3),proper(X4)) proper(uLength(X1,X2)) = X1 >= X1 = uLength(proper(X1),proper(X2)) and(ok(X1),ok(X2)) = 0 >= 0 = ok(and(X1,X2)) isNatIList(ok(X)) = 0 >= 0 = ok(isNatIList(X)) isNatList(ok(X)) = 0 >= 0 = ok(isNatList(X)) isNat(ok(X)) = 0 >= 0 = ok(isNat(X)) s(ok(X)) = 0 >= 0 = ok(s(X)) length(ok(X)) = 0 >= 0 = ok(length(X)) cons(ok(X1),ok(X2)) = 1 >= 0 = ok(cons(X1,X2)) take(ok(X1),ok(X2)) = 0 >= 0 = ok(take(X1,X2)) uTake1(ok(X)) = 0 >= 0 = ok(uTake1(X)) uTake2(ok(X1),ok(X2),ok(X3),ok(X4)) = 0 >= 0 = ok(uTake2(X1,X2,X3,X4)) uLength(ok(X1),ok(X2)) = 0 >= 0 = ok(uLength(X1,X2)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) problem: DPs: active#(and(X1,X2)) -> active#(X1) active#(and(X1,X2)) -> active#(X2) active#(s(X)) -> active#(X) active#(length(X)) -> active#(X) active#(take(X1,X2)) -> active#(X1) active#(take(X1,X2)) -> active#(X2) active#(uTake1(X)) -> active#(X) active#(uTake2(X1,X2,X3,X4)) -> active#(X1) active#(uLength(X1,X2)) -> active#(X1) TRS: active(and(tt(),T)) -> mark(T) active(isNatIList(IL)) -> mark(isNatList(IL)) active(isNat(0())) -> mark(tt()) active(isNat(s(N))) -> mark(isNat(N)) active(isNat(length(L))) -> mark(isNatList(L)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(N,L))) -> mark(and(isNat(N),isNatList(L))) active(isNatList(take(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(zeros()) -> mark(cons(0(),zeros())) active(take(0(),IL)) -> mark(uTake1(isNatIList(IL))) active(uTake1(tt())) -> mark(nil()) active(take(s(M),cons(N,IL))) -> mark(uTake2(and(isNat(M),and(isNat(N),isNatIList(IL))),M,N,IL)) active(uTake2(tt(),M,N,IL)) -> mark(cons(N,take(M,IL))) active(length(cons(N,L))) -> mark(uLength(and(isNat(N),isNatList(L)),L)) active(uLength(tt(),L)) -> mark(s(length(L))) active(and(X1,X2)) -> and(active(X1),X2) active(and(X1,X2)) -> and(X1,active(X2)) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) active(uTake1(X)) -> uTake1(active(X)) active(uTake2(X1,X2,X3,X4)) -> uTake2(active(X1),X2,X3,X4) active(uLength(X1,X2)) -> uLength(active(X1),X2) and(mark(X1),X2) -> mark(and(X1,X2)) and(X1,mark(X2)) -> mark(and(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) uTake1(mark(X)) -> mark(uTake1(X)) uTake2(mark(X1),X2,X3,X4) -> mark(uTake2(X1,X2,X3,X4)) uLength(mark(X1),X2) -> mark(uLength(X1,X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(isNat(X)) -> isNat(proper(X)) proper(0()) -> ok(0()) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(nil()) -> ok(nil()) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) proper(uTake1(X)) -> uTake1(proper(X)) proper(uTake2(X1,X2,X3,X4)) -> uTake2(proper(X1),proper(X2),proper(X3),proper(X4)) proper(uLength(X1,X2)) -> uLength(proper(X1),proper(X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatIList(ok(X)) -> ok(isNatIList(X)) isNatList(ok(X)) -> ok(isNatList(X)) isNat(ok(X)) -> ok(isNat(X)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) uTake1(ok(X)) -> ok(uTake1(X)) uTake2(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(uTake2(X1,X2,X3,X4)) uLength(ok(X1),ok(X2)) -> ok(uLength(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dimension: 1 interpretation: [active#](x0) = x0, [top](x0) = 0, [ok](x0) = x0, [proper](x0) = x0, [uLength](x0, x1) = x0, [uTake2](x0, x1, x2, x3) = x0, [uTake1](x0) = x0 + 1, [take](x0, x1) = x0 + x1, [nil] = 0, [cons](x0, x1) = x0 + 1, [zeros] = 0, [length](x0) = x0, [s](x0) = x0, [isNat](x0) = 0, [0] = 1, [isNatList](x0) = x0, [isNatIList](x0) = x0, [mark](x0) = 0, [active](x0) = x0, [and](x0, x1) = x0 + x1, [tt] = 1 orientation: active#(and(X1,X2)) = X1 + X2 >= X1 = active#(X1) active#(and(X1,X2)) = X1 + X2 >= X2 = active#(X2) active#(s(X)) = X >= X = active#(X) active#(length(X)) = X >= X = active#(X) active#(take(X1,X2)) = X1 + X2 >= X1 = active#(X1) active#(take(X1,X2)) = X1 + X2 >= X2 = active#(X2) active#(uTake1(X)) = X + 1 >= X = active#(X) active#(uTake2(X1,X2,X3,X4)) = X1 >= X1 = active#(X1) active#(uLength(X1,X2)) = X1 >= X1 = active#(X1) active(and(tt(),T)) = T + 1 >= 0 = mark(T) active(isNatIList(IL)) = IL >= 0 = mark(isNatList(IL)) active(isNat(0())) = 0 >= 0 = mark(tt()) active(isNat(s(N))) = 0 >= 0 = mark(isNat(N)) active(isNat(length(L))) = 0 >= 0 = mark(isNatList(L)) active(isNatIList(zeros())) = 0 >= 0 = mark(tt()) active(isNatIList(cons(N,IL))) = N + 1 >= 0 = mark(and(isNat(N),isNatIList(IL))) active(isNatList(nil())) = 0 >= 0 = mark(tt()) active(isNatList(cons(N,L))) = N + 1 >= 0 = mark(and(isNat(N),isNatList(L))) active(isNatList(take(N,IL))) = IL + N >= 0 = mark(and(isNat(N),isNatIList(IL))) active(zeros()) = 0 >= 0 = mark(cons(0(),zeros())) active(take(0(),IL)) = IL + 1 >= 0 = mark(uTake1(isNatIList(IL))) active(uTake1(tt())) = 2 >= 0 = mark(nil()) active(take(s(M),cons(N,IL))) = M + N + 1 >= 0 = mark(uTake2(and(isNat(M),and(isNat(N),isNatIList(IL))),M,N,IL)) active(uTake2(tt(),M,N,IL)) = 1 >= 0 = mark(cons(N,take(M,IL))) active(length(cons(N,L))) = N + 1 >= 0 = mark(uLength(and(isNat(N),isNatList(L)),L)) active(uLength(tt(),L)) = 1 >= 0 = mark(s(length(L))) active(and(X1,X2)) = X1 + X2 >= X1 + X2 = and(active(X1),X2) active(and(X1,X2)) = X1 + X2 >= X1 + X2 = and(X1,active(X2)) active(s(X)) = X >= X = s(active(X)) active(length(X)) = X >= X = length(active(X)) active(cons(X1,X2)) = X1 + 1 >= X1 + 1 = cons(active(X1),X2) 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(uTake1(X)) = X + 1 >= X + 1 = uTake1(active(X)) active(uTake2(X1,X2,X3,X4)) = X1 >= X1 = uTake2(active(X1),X2,X3,X4) active(uLength(X1,X2)) = X1 >= X1 = uLength(active(X1),X2) and(mark(X1),X2) = X2 >= 0 = mark(and(X1,X2)) and(X1,mark(X2)) = X1 >= 0 = mark(and(X1,X2)) s(mark(X)) = 0 >= 0 = mark(s(X)) length(mark(X)) = 0 >= 0 = mark(length(X)) cons(mark(X1),X2) = 1 >= 0 = mark(cons(X1,X2)) take(mark(X1),X2) = X2 >= 0 = mark(take(X1,X2)) take(X1,mark(X2)) = X1 >= 0 = mark(take(X1,X2)) uTake1(mark(X)) = 1 >= 0 = mark(uTake1(X)) uTake2(mark(X1),X2,X3,X4) = 0 >= 0 = mark(uTake2(X1,X2,X3,X4)) uLength(mark(X1),X2) = 0 >= 0 = mark(uLength(X1,X2)) proper(and(X1,X2)) = X1 + X2 >= X1 + X2 = and(proper(X1),proper(X2)) proper(tt()) = 1 >= 1 = ok(tt()) proper(isNatIList(X)) = X >= X = isNatIList(proper(X)) proper(isNatList(X)) = X >= X = isNatList(proper(X)) proper(isNat(X)) = 0 >= 0 = isNat(proper(X)) proper(0()) = 1 >= 1 = ok(0()) proper(s(X)) = X >= X = s(proper(X)) proper(length(X)) = X >= X = length(proper(X)) proper(zeros()) = 0 >= 0 = ok(zeros()) proper(cons(X1,X2)) = X1 + 1 >= X1 + 1 = cons(proper(X1),proper(X2)) proper(nil()) = 0 >= 0 = ok(nil()) proper(take(X1,X2)) = X1 + X2 >= X1 + X2 = take(proper(X1),proper(X2)) proper(uTake1(X)) = X + 1 >= X + 1 = uTake1(proper(X)) proper(uTake2(X1,X2,X3,X4)) = X1 >= X1 = uTake2(proper(X1),proper(X2),proper(X3),proper(X4)) proper(uLength(X1,X2)) = X1 >= X1 = uLength(proper(X1),proper(X2)) and(ok(X1),ok(X2)) = X1 + X2 >= X1 + X2 = ok(and(X1,X2)) isNatIList(ok(X)) = X >= X = ok(isNatIList(X)) isNatList(ok(X)) = X >= X = ok(isNatList(X)) isNat(ok(X)) = 0 >= 0 = ok(isNat(X)) s(ok(X)) = X >= X = ok(s(X)) length(ok(X)) = X >= X = ok(length(X)) cons(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(cons(X1,X2)) take(ok(X1),ok(X2)) = X1 + X2 >= X1 + X2 = ok(take(X1,X2)) uTake1(ok(X)) = X + 1 >= X + 1 = ok(uTake1(X)) uTake2(ok(X1),ok(X2),ok(X3),ok(X4)) = X1 >= X1 = ok(uTake2(X1,X2,X3,X4)) uLength(ok(X1),ok(X2)) = X1 >= X1 = ok(uLength(X1,X2)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) problem: DPs: active#(and(X1,X2)) -> active#(X1) active#(and(X1,X2)) -> active#(X2) active#(s(X)) -> active#(X) active#(length(X)) -> active#(X) active#(take(X1,X2)) -> active#(X1) active#(take(X1,X2)) -> active#(X2) active#(uTake2(X1,X2,X3,X4)) -> active#(X1) active#(uLength(X1,X2)) -> active#(X1) TRS: active(and(tt(),T)) -> mark(T) active(isNatIList(IL)) -> mark(isNatList(IL)) active(isNat(0())) -> mark(tt()) active(isNat(s(N))) -> mark(isNat(N)) active(isNat(length(L))) -> mark(isNatList(L)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(N,L))) -> mark(and(isNat(N),isNatList(L))) active(isNatList(take(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(zeros()) -> mark(cons(0(),zeros())) active(take(0(),IL)) -> mark(uTake1(isNatIList(IL))) active(uTake1(tt())) -> mark(nil()) active(take(s(M),cons(N,IL))) -> mark(uTake2(and(isNat(M),and(isNat(N),isNatIList(IL))),M,N,IL)) active(uTake2(tt(),M,N,IL)) -> mark(cons(N,take(M,IL))) active(length(cons(N,L))) -> mark(uLength(and(isNat(N),isNatList(L)),L)) active(uLength(tt(),L)) -> mark(s(length(L))) active(and(X1,X2)) -> and(active(X1),X2) active(and(X1,X2)) -> and(X1,active(X2)) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) active(uTake1(X)) -> uTake1(active(X)) active(uTake2(X1,X2,X3,X4)) -> uTake2(active(X1),X2,X3,X4) active(uLength(X1,X2)) -> uLength(active(X1),X2) and(mark(X1),X2) -> mark(and(X1,X2)) and(X1,mark(X2)) -> mark(and(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) uTake1(mark(X)) -> mark(uTake1(X)) uTake2(mark(X1),X2,X3,X4) -> mark(uTake2(X1,X2,X3,X4)) uLength(mark(X1),X2) -> mark(uLength(X1,X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(isNat(X)) -> isNat(proper(X)) proper(0()) -> ok(0()) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(nil()) -> ok(nil()) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) proper(uTake1(X)) -> uTake1(proper(X)) proper(uTake2(X1,X2,X3,X4)) -> uTake2(proper(X1),proper(X2),proper(X3),proper(X4)) proper(uLength(X1,X2)) -> uLength(proper(X1),proper(X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatIList(ok(X)) -> ok(isNatIList(X)) isNatList(ok(X)) -> ok(isNatList(X)) isNat(ok(X)) -> ok(isNat(X)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) uTake1(ok(X)) -> ok(uTake1(X)) uTake2(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(uTake2(X1,X2,X3,X4)) uLength(ok(X1),ok(X2)) -> ok(uLength(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dimension: 1 interpretation: [active#](x0) = x0 + 1, [top](x0) = 1, [ok](x0) = x0, [proper](x0) = x0, [uLength](x0, x1) = x0, [uTake2](x0, x1, x2, x3) = x0, [uTake1](x0) = 0, [take](x0, x1) = x0 + x1, [nil] = 1, [cons](x0, x1) = 1, [zeros] = 0, [length](x0) = x0, [s](x0) = x0 + 1, [isNat](x0) = x0, [0] = 1, [isNatList](x0) = 0, [isNatIList](x0) = 0, [mark](x0) = 0, [active](x0) = x0, [and](x0, x1) = x0 + x1 + 1, [tt] = 0 orientation: active#(and(X1,X2)) = X1 + X2 + 2 >= X1 + 1 = active#(X1) active#(and(X1,X2)) = X1 + X2 + 2 >= X2 + 1 = active#(X2) active#(s(X)) = X + 2 >= X + 1 = active#(X) active#(length(X)) = X + 1 >= X + 1 = active#(X) active#(take(X1,X2)) = X1 + X2 + 1 >= X1 + 1 = active#(X1) active#(take(X1,X2)) = X1 + X2 + 1 >= X2 + 1 = active#(X2) active#(uTake2(X1,X2,X3,X4)) = X1 + 1 >= X1 + 1 = active#(X1) active#(uLength(X1,X2)) = X1 + 1 >= X1 + 1 = active#(X1) active(and(tt(),T)) = T + 1 >= 0 = mark(T) active(isNatIList(IL)) = 0 >= 0 = mark(isNatList(IL)) active(isNat(0())) = 1 >= 0 = mark(tt()) active(isNat(s(N))) = N + 1 >= 0 = mark(isNat(N)) active(isNat(length(L))) = L >= 0 = mark(isNatList(L)) active(isNatIList(zeros())) = 0 >= 0 = mark(tt()) active(isNatIList(cons(N,IL))) = 0 >= 0 = mark(and(isNat(N),isNatIList(IL))) active(isNatList(nil())) = 0 >= 0 = mark(tt()) active(isNatList(cons(N,L))) = 0 >= 0 = mark(and(isNat(N),isNatList(L))) active(isNatList(take(N,IL))) = 0 >= 0 = mark(and(isNat(N),isNatIList(IL))) active(zeros()) = 0 >= 0 = mark(cons(0(),zeros())) active(take(0(),IL)) = IL + 1 >= 0 = mark(uTake1(isNatIList(IL))) active(uTake1(tt())) = 0 >= 0 = mark(nil()) active(take(s(M),cons(N,IL))) = M + 2 >= 0 = mark(uTake2(and(isNat(M),and(isNat(N),isNatIList(IL))),M,N,IL)) active(uTake2(tt(),M,N,IL)) = 0 >= 0 = mark(cons(N,take(M,IL))) active(length(cons(N,L))) = 1 >= 0 = mark(uLength(and(isNat(N),isNatList(L)),L)) active(uLength(tt(),L)) = 0 >= 0 = mark(s(length(L))) active(and(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = and(active(X1),X2) active(and(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = and(X1,active(X2)) active(s(X)) = X + 1 >= X + 1 = s(active(X)) active(length(X)) = X >= X = length(active(X)) active(cons(X1,X2)) = 1 >= 1 = cons(active(X1),X2) 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(uTake1(X)) = 0 >= 0 = uTake1(active(X)) active(uTake2(X1,X2,X3,X4)) = X1 >= X1 = uTake2(active(X1),X2,X3,X4) active(uLength(X1,X2)) = X1 >= X1 = uLength(active(X1),X2) and(mark(X1),X2) = X2 + 1 >= 0 = mark(and(X1,X2)) and(X1,mark(X2)) = X1 + 1 >= 0 = mark(and(X1,X2)) s(mark(X)) = 1 >= 0 = mark(s(X)) length(mark(X)) = 0 >= 0 = mark(length(X)) cons(mark(X1),X2) = 1 >= 0 = mark(cons(X1,X2)) take(mark(X1),X2) = X2 >= 0 = mark(take(X1,X2)) take(X1,mark(X2)) = X1 >= 0 = mark(take(X1,X2)) uTake1(mark(X)) = 0 >= 0 = mark(uTake1(X)) uTake2(mark(X1),X2,X3,X4) = 0 >= 0 = mark(uTake2(X1,X2,X3,X4)) uLength(mark(X1),X2) = 0 >= 0 = mark(uLength(X1,X2)) proper(and(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = and(proper(X1),proper(X2)) proper(tt()) = 0 >= 0 = ok(tt()) proper(isNatIList(X)) = 0 >= 0 = isNatIList(proper(X)) proper(isNatList(X)) = 0 >= 0 = isNatList(proper(X)) proper(isNat(X)) = X >= X = isNat(proper(X)) proper(0()) = 1 >= 1 = ok(0()) proper(s(X)) = X + 1 >= X + 1 = s(proper(X)) proper(length(X)) = X >= X = length(proper(X)) proper(zeros()) = 0 >= 0 = ok(zeros()) proper(cons(X1,X2)) = 1 >= 1 = cons(proper(X1),proper(X2)) proper(nil()) = 1 >= 1 = ok(nil()) proper(take(X1,X2)) = X1 + X2 >= X1 + X2 = take(proper(X1),proper(X2)) proper(uTake1(X)) = 0 >= 0 = uTake1(proper(X)) proper(uTake2(X1,X2,X3,X4)) = X1 >= X1 = uTake2(proper(X1),proper(X2),proper(X3),proper(X4)) proper(uLength(X1,X2)) = X1 >= X1 = uLength(proper(X1),proper(X2)) and(ok(X1),ok(X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = ok(and(X1,X2)) isNatIList(ok(X)) = 0 >= 0 = ok(isNatIList(X)) isNatList(ok(X)) = 0 >= 0 = ok(isNatList(X)) isNat(ok(X)) = X >= X = ok(isNat(X)) s(ok(X)) = X + 1 >= X + 1 = ok(s(X)) length(ok(X)) = X >= X = ok(length(X)) cons(ok(X1),ok(X2)) = 1 >= 1 = ok(cons(X1,X2)) take(ok(X1),ok(X2)) = X1 + X2 >= X1 + X2 = ok(take(X1,X2)) uTake1(ok(X)) = 0 >= 0 = ok(uTake1(X)) uTake2(ok(X1),ok(X2),ok(X3),ok(X4)) = X1 >= X1 = ok(uTake2(X1,X2,X3,X4)) uLength(ok(X1),ok(X2)) = X1 >= X1 = ok(uLength(X1,X2)) top(mark(X)) = 1 >= 1 = top(proper(X)) top(ok(X)) = 1 >= 1 = top(active(X)) problem: DPs: active#(length(X)) -> active#(X) active#(take(X1,X2)) -> active#(X1) active#(take(X1,X2)) -> active#(X2) active#(uTake2(X1,X2,X3,X4)) -> active#(X1) active#(uLength(X1,X2)) -> active#(X1) TRS: active(and(tt(),T)) -> mark(T) active(isNatIList(IL)) -> mark(isNatList(IL)) active(isNat(0())) -> mark(tt()) active(isNat(s(N))) -> mark(isNat(N)) active(isNat(length(L))) -> mark(isNatList(L)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(N,L))) -> mark(and(isNat(N),isNatList(L))) active(isNatList(take(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(zeros()) -> mark(cons(0(),zeros())) active(take(0(),IL)) -> mark(uTake1(isNatIList(IL))) active(uTake1(tt())) -> mark(nil()) active(take(s(M),cons(N,IL))) -> mark(uTake2(and(isNat(M),and(isNat(N),isNatIList(IL))),M,N,IL)) active(uTake2(tt(),M,N,IL)) -> mark(cons(N,take(M,IL))) active(length(cons(N,L))) -> mark(uLength(and(isNat(N),isNatList(L)),L)) active(uLength(tt(),L)) -> mark(s(length(L))) active(and(X1,X2)) -> and(active(X1),X2) active(and(X1,X2)) -> and(X1,active(X2)) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) active(uTake1(X)) -> uTake1(active(X)) active(uTake2(X1,X2,X3,X4)) -> uTake2(active(X1),X2,X3,X4) active(uLength(X1,X2)) -> uLength(active(X1),X2) and(mark(X1),X2) -> mark(and(X1,X2)) and(X1,mark(X2)) -> mark(and(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) uTake1(mark(X)) -> mark(uTake1(X)) uTake2(mark(X1),X2,X3,X4) -> mark(uTake2(X1,X2,X3,X4)) uLength(mark(X1),X2) -> mark(uLength(X1,X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(isNat(X)) -> isNat(proper(X)) proper(0()) -> ok(0()) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(nil()) -> ok(nil()) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) proper(uTake1(X)) -> uTake1(proper(X)) proper(uTake2(X1,X2,X3,X4)) -> uTake2(proper(X1),proper(X2),proper(X3),proper(X4)) proper(uLength(X1,X2)) -> uLength(proper(X1),proper(X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatIList(ok(X)) -> ok(isNatIList(X)) isNatList(ok(X)) -> ok(isNatList(X)) isNat(ok(X)) -> ok(isNat(X)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) uTake1(ok(X)) -> ok(uTake1(X)) uTake2(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(uTake2(X1,X2,X3,X4)) uLength(ok(X1),ok(X2)) -> ok(uLength(X1,X2)) 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, [uLength](x0, x1) = x0, [uTake2](x0, x1, x2, x3) = x0, [uTake1](x0) = 0, [take](x0, x1) = x0 + x1 + 1, [nil] = 1, [cons](x0, x1) = 1, [zeros] = 0, [length](x0) = x0 + 1, [s](x0) = 0, [isNat](x0) = 0, [0] = 0, [isNatList](x0) = x0 + 1, [isNatIList](x0) = 1, [mark](x0) = 0, [active](x0) = x0, [and](x0, x1) = 0, [tt] = 0 orientation: active#(length(X)) = X + 2 >= X + 1 = active#(X) active#(take(X1,X2)) = X1 + X2 + 2 >= X1 + 1 = active#(X1) active#(take(X1,X2)) = X1 + X2 + 2 >= X2 + 1 = active#(X2) active#(uTake2(X1,X2,X3,X4)) = X1 + 1 >= X1 + 1 = active#(X1) active#(uLength(X1,X2)) = X1 + 1 >= X1 + 1 = active#(X1) active(and(tt(),T)) = 0 >= 0 = mark(T) active(isNatIList(IL)) = 1 >= 0 = mark(isNatList(IL)) active(isNat(0())) = 0 >= 0 = mark(tt()) active(isNat(s(N))) = 0 >= 0 = mark(isNat(N)) active(isNat(length(L))) = 0 >= 0 = mark(isNatList(L)) active(isNatIList(zeros())) = 1 >= 0 = mark(tt()) active(isNatIList(cons(N,IL))) = 1 >= 0 = mark(and(isNat(N),isNatIList(IL))) active(isNatList(nil())) = 2 >= 0 = mark(tt()) active(isNatList(cons(N,L))) = 2 >= 0 = mark(and(isNat(N),isNatList(L))) active(isNatList(take(N,IL))) = IL + N + 2 >= 0 = mark(and(isNat(N),isNatIList(IL))) active(zeros()) = 0 >= 0 = mark(cons(0(),zeros())) active(take(0(),IL)) = IL + 1 >= 0 = mark(uTake1(isNatIList(IL))) active(uTake1(tt())) = 0 >= 0 = mark(nil()) active(take(s(M),cons(N,IL))) = 2 >= 0 = mark(uTake2(and(isNat(M),and(isNat(N),isNatIList(IL))),M,N,IL)) active(uTake2(tt(),M,N,IL)) = 0 >= 0 = mark(cons(N,take(M,IL))) active(length(cons(N,L))) = 2 >= 0 = mark(uLength(and(isNat(N),isNatList(L)),L)) active(uLength(tt(),L)) = 0 >= 0 = mark(s(length(L))) active(and(X1,X2)) = 0 >= 0 = and(active(X1),X2) active(and(X1,X2)) = 0 >= 0 = and(X1,active(X2)) active(s(X)) = 0 >= 0 = s(active(X)) active(length(X)) = X + 1 >= X + 1 = length(active(X)) active(cons(X1,X2)) = 1 >= 1 = cons(active(X1),X2) 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(uTake1(X)) = 0 >= 0 = uTake1(active(X)) active(uTake2(X1,X2,X3,X4)) = X1 >= X1 = uTake2(active(X1),X2,X3,X4) active(uLength(X1,X2)) = X1 >= X1 = uLength(active(X1),X2) and(mark(X1),X2) = 0 >= 0 = mark(and(X1,X2)) and(X1,mark(X2)) = 0 >= 0 = mark(and(X1,X2)) s(mark(X)) = 0 >= 0 = mark(s(X)) length(mark(X)) = 1 >= 0 = mark(length(X)) cons(mark(X1),X2) = 1 >= 0 = mark(cons(X1,X2)) take(mark(X1),X2) = X2 + 1 >= 0 = mark(take(X1,X2)) take(X1,mark(X2)) = X1 + 1 >= 0 = mark(take(X1,X2)) uTake1(mark(X)) = 0 >= 0 = mark(uTake1(X)) uTake2(mark(X1),X2,X3,X4) = 0 >= 0 = mark(uTake2(X1,X2,X3,X4)) uLength(mark(X1),X2) = 0 >= 0 = mark(uLength(X1,X2)) proper(and(X1,X2)) = 0 >= 0 = and(proper(X1),proper(X2)) proper(tt()) = 0 >= 0 = ok(tt()) proper(isNatIList(X)) = 1 >= 1 = isNatIList(proper(X)) proper(isNatList(X)) = X + 1 >= X + 1 = isNatList(proper(X)) proper(isNat(X)) = 0 >= 0 = isNat(proper(X)) proper(0()) = 0 >= 0 = ok(0()) proper(s(X)) = 0 >= 0 = s(proper(X)) proper(length(X)) = X + 1 >= X + 1 = length(proper(X)) proper(zeros()) = 0 >= 0 = ok(zeros()) proper(cons(X1,X2)) = 1 >= 1 = cons(proper(X1),proper(X2)) proper(nil()) = 1 >= 1 = ok(nil()) proper(take(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = take(proper(X1),proper(X2)) proper(uTake1(X)) = 0 >= 0 = uTake1(proper(X)) proper(uTake2(X1,X2,X3,X4)) = X1 >= X1 = uTake2(proper(X1),proper(X2),proper(X3),proper(X4)) proper(uLength(X1,X2)) = X1 >= X1 = uLength(proper(X1),proper(X2)) and(ok(X1),ok(X2)) = 0 >= 0 = ok(and(X1,X2)) isNatIList(ok(X)) = 1 >= 1 = ok(isNatIList(X)) isNatList(ok(X)) = X + 1 >= X + 1 = ok(isNatList(X)) isNat(ok(X)) = 0 >= 0 = ok(isNat(X)) s(ok(X)) = 0 >= 0 = ok(s(X)) length(ok(X)) = X + 1 >= X + 1 = ok(length(X)) cons(ok(X1),ok(X2)) = 1 >= 1 = ok(cons(X1,X2)) take(ok(X1),ok(X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = ok(take(X1,X2)) uTake1(ok(X)) = 0 >= 0 = ok(uTake1(X)) uTake2(ok(X1),ok(X2),ok(X3),ok(X4)) = X1 >= X1 = ok(uTake2(X1,X2,X3,X4)) uLength(ok(X1),ok(X2)) = X1 >= X1 = ok(uLength(X1,X2)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) problem: DPs: active#(uTake2(X1,X2,X3,X4)) -> active#(X1) active#(uLength(X1,X2)) -> active#(X1) TRS: active(and(tt(),T)) -> mark(T) active(isNatIList(IL)) -> mark(isNatList(IL)) active(isNat(0())) -> mark(tt()) active(isNat(s(N))) -> mark(isNat(N)) active(isNat(length(L))) -> mark(isNatList(L)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(N,L))) -> mark(and(isNat(N),isNatList(L))) active(isNatList(take(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(zeros()) -> mark(cons(0(),zeros())) active(take(0(),IL)) -> mark(uTake1(isNatIList(IL))) active(uTake1(tt())) -> mark(nil()) active(take(s(M),cons(N,IL))) -> mark(uTake2(and(isNat(M),and(isNat(N),isNatIList(IL))),M,N,IL)) active(uTake2(tt(),M,N,IL)) -> mark(cons(N,take(M,IL))) active(length(cons(N,L))) -> mark(uLength(and(isNat(N),isNatList(L)),L)) active(uLength(tt(),L)) -> mark(s(length(L))) active(and(X1,X2)) -> and(active(X1),X2) active(and(X1,X2)) -> and(X1,active(X2)) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) active(uTake1(X)) -> uTake1(active(X)) active(uTake2(X1,X2,X3,X4)) -> uTake2(active(X1),X2,X3,X4) active(uLength(X1,X2)) -> uLength(active(X1),X2) and(mark(X1),X2) -> mark(and(X1,X2)) and(X1,mark(X2)) -> mark(and(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) uTake1(mark(X)) -> mark(uTake1(X)) uTake2(mark(X1),X2,X3,X4) -> mark(uTake2(X1,X2,X3,X4)) uLength(mark(X1),X2) -> mark(uLength(X1,X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(isNat(X)) -> isNat(proper(X)) proper(0()) -> ok(0()) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(nil()) -> ok(nil()) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) proper(uTake1(X)) -> uTake1(proper(X)) proper(uTake2(X1,X2,X3,X4)) -> uTake2(proper(X1),proper(X2),proper(X3),proper(X4)) proper(uLength(X1,X2)) -> uLength(proper(X1),proper(X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatIList(ok(X)) -> ok(isNatIList(X)) isNatList(ok(X)) -> ok(isNatList(X)) isNat(ok(X)) -> ok(isNat(X)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) uTake1(ok(X)) -> ok(uTake1(X)) uTake2(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(uTake2(X1,X2,X3,X4)) uLength(ok(X1),ok(X2)) -> ok(uLength(X1,X2)) 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 + 1, [uLength](x0, x1) = x0 + 1, [uTake2](x0, x1, x2, x3) = x0 + 1, [uTake1](x0) = 0, [take](x0, x1) = x0, [nil] = 0, [cons](x0, x1) = 1, [zeros] = 1, [length](x0) = 1, [s](x0) = 1, [isNat](x0) = 0, [0] = 0, [isNatList](x0) = 0, [isNatIList](x0) = 0, [mark](x0) = x0, [active](x0) = x0, [and](x0, x1) = x1, [tt] = 0 orientation: active#(uTake2(X1,X2,X3,X4)) = X1 + 2 >= X1 + 1 = active#(X1) active#(uLength(X1,X2)) = X1 + 2 >= X1 + 1 = active#(X1) active(and(tt(),T)) = T >= T = mark(T) active(isNatIList(IL)) = 0 >= 0 = mark(isNatList(IL)) active(isNat(0())) = 0 >= 0 = mark(tt()) active(isNat(s(N))) = 0 >= 0 = mark(isNat(N)) active(isNat(length(L))) = 0 >= 0 = mark(isNatList(L)) active(isNatIList(zeros())) = 0 >= 0 = mark(tt()) active(isNatIList(cons(N,IL))) = 0 >= 0 = mark(and(isNat(N),isNatIList(IL))) active(isNatList(nil())) = 0 >= 0 = mark(tt()) active(isNatList(cons(N,L))) = 0 >= 0 = mark(and(isNat(N),isNatList(L))) active(isNatList(take(N,IL))) = 0 >= 0 = mark(and(isNat(N),isNatIList(IL))) active(zeros()) = 1 >= 1 = mark(cons(0(),zeros())) active(take(0(),IL)) = 0 >= 0 = mark(uTake1(isNatIList(IL))) active(uTake1(tt())) = 0 >= 0 = mark(nil()) active(take(s(M),cons(N,IL))) = 1 >= 1 = mark(uTake2(and(isNat(M),and(isNat(N),isNatIList(IL))),M,N,IL)) active(uTake2(tt(),M,N,IL)) = 1 >= 1 = mark(cons(N,take(M,IL))) active(length(cons(N,L))) = 1 >= 1 = mark(uLength(and(isNat(N),isNatList(L)),L)) active(uLength(tt(),L)) = 1 >= 1 = mark(s(length(L))) active(and(X1,X2)) = X2 >= X2 = and(active(X1),X2) active(and(X1,X2)) = X2 >= X2 = and(X1,active(X2)) active(s(X)) = 1 >= 1 = s(active(X)) active(length(X)) = 1 >= 1 = length(active(X)) active(cons(X1,X2)) = 1 >= 1 = cons(active(X1),X2) active(take(X1,X2)) = X1 >= X1 = take(active(X1),X2) active(take(X1,X2)) = X1 >= X1 = take(X1,active(X2)) active(uTake1(X)) = 0 >= 0 = uTake1(active(X)) active(uTake2(X1,X2,X3,X4)) = X1 + 1 >= X1 + 1 = uTake2(active(X1),X2,X3,X4) active(uLength(X1,X2)) = X1 + 1 >= X1 + 1 = uLength(active(X1),X2) and(mark(X1),X2) = X2 >= X2 = mark(and(X1,X2)) and(X1,mark(X2)) = X2 >= X2 = mark(and(X1,X2)) s(mark(X)) = 1 >= 1 = mark(s(X)) length(mark(X)) = 1 >= 1 = mark(length(X)) cons(mark(X1),X2) = 1 >= 1 = mark(cons(X1,X2)) take(mark(X1),X2) = X1 >= X1 = mark(take(X1,X2)) take(X1,mark(X2)) = X1 >= X1 = mark(take(X1,X2)) uTake1(mark(X)) = 0 >= 0 = mark(uTake1(X)) uTake2(mark(X1),X2,X3,X4) = X1 + 1 >= X1 + 1 = mark(uTake2(X1,X2,X3,X4)) uLength(mark(X1),X2) = X1 + 1 >= X1 + 1 = mark(uLength(X1,X2)) proper(and(X1,X2)) = X2 + 1 >= X2 + 1 = and(proper(X1),proper(X2)) proper(tt()) = 1 >= 0 = ok(tt()) proper(isNatIList(X)) = 1 >= 0 = isNatIList(proper(X)) proper(isNatList(X)) = 1 >= 0 = isNatList(proper(X)) proper(isNat(X)) = 1 >= 0 = isNat(proper(X)) proper(0()) = 1 >= 0 = ok(0()) proper(s(X)) = 2 >= 1 = s(proper(X)) proper(length(X)) = 2 >= 1 = length(proper(X)) proper(zeros()) = 2 >= 1 = ok(zeros()) proper(cons(X1,X2)) = 2 >= 1 = cons(proper(X1),proper(X2)) proper(nil()) = 1 >= 0 = ok(nil()) proper(take(X1,X2)) = X1 + 1 >= X1 + 1 = take(proper(X1),proper(X2)) proper(uTake1(X)) = 1 >= 0 = uTake1(proper(X)) proper(uTake2(X1,X2,X3,X4)) = X1 + 2 >= X1 + 2 = uTake2(proper(X1),proper(X2),proper(X3),proper(X4)) proper(uLength(X1,X2)) = X1 + 2 >= X1 + 2 = uLength(proper(X1),proper(X2)) and(ok(X1),ok(X2)) = X2 >= X2 = ok(and(X1,X2)) isNatIList(ok(X)) = 0 >= 0 = ok(isNatIList(X)) isNatList(ok(X)) = 0 >= 0 = ok(isNatList(X)) isNat(ok(X)) = 0 >= 0 = ok(isNat(X)) s(ok(X)) = 1 >= 1 = ok(s(X)) length(ok(X)) = 1 >= 1 = ok(length(X)) cons(ok(X1),ok(X2)) = 1 >= 1 = ok(cons(X1,X2)) take(ok(X1),ok(X2)) = X1 >= X1 = ok(take(X1,X2)) uTake1(ok(X)) = 0 >= 0 = ok(uTake1(X)) uTake2(ok(X1),ok(X2),ok(X3),ok(X4)) = X1 + 1 >= X1 + 1 = ok(uTake2(X1,X2,X3,X4)) uLength(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(uLength(X1,X2)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) problem: DPs: TRS: active(and(tt(),T)) -> mark(T) active(isNatIList(IL)) -> mark(isNatList(IL)) active(isNat(0())) -> mark(tt()) active(isNat(s(N))) -> mark(isNat(N)) active(isNat(length(L))) -> mark(isNatList(L)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(N,L))) -> mark(and(isNat(N),isNatList(L))) active(isNatList(take(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(zeros()) -> mark(cons(0(),zeros())) active(take(0(),IL)) -> mark(uTake1(isNatIList(IL))) active(uTake1(tt())) -> mark(nil()) active(take(s(M),cons(N,IL))) -> mark(uTake2(and(isNat(M),and(isNat(N),isNatIList(IL))),M,N,IL)) active(uTake2(tt(),M,N,IL)) -> mark(cons(N,take(M,IL))) active(length(cons(N,L))) -> mark(uLength(and(isNat(N),isNatList(L)),L)) active(uLength(tt(),L)) -> mark(s(length(L))) active(and(X1,X2)) -> and(active(X1),X2) active(and(X1,X2)) -> and(X1,active(X2)) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) active(uTake1(X)) -> uTake1(active(X)) active(uTake2(X1,X2,X3,X4)) -> uTake2(active(X1),X2,X3,X4) active(uLength(X1,X2)) -> uLength(active(X1),X2) and(mark(X1),X2) -> mark(and(X1,X2)) and(X1,mark(X2)) -> mark(and(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) uTake1(mark(X)) -> mark(uTake1(X)) uTake2(mark(X1),X2,X3,X4) -> mark(uTake2(X1,X2,X3,X4)) uLength(mark(X1),X2) -> mark(uLength(X1,X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(isNat(X)) -> isNat(proper(X)) proper(0()) -> ok(0()) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(nil()) -> ok(nil()) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) proper(uTake1(X)) -> uTake1(proper(X)) proper(uTake2(X1,X2,X3,X4)) -> uTake2(proper(X1),proper(X2),proper(X3),proper(X4)) proper(uLength(X1,X2)) -> uLength(proper(X1),proper(X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatIList(ok(X)) -> ok(isNatIList(X)) isNatList(ok(X)) -> ok(isNatList(X)) isNat(ok(X)) -> ok(isNat(X)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) uTake1(ok(X)) -> ok(uTake1(X)) uTake2(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(uTake2(X1,X2,X3,X4)) uLength(ok(X1),ok(X2)) -> ok(uLength(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed DPs: proper#(and(X1,X2)) -> proper#(X2) proper#(and(X1,X2)) -> proper#(X1) proper#(isNatIList(X)) -> proper#(X) proper#(isNatList(X)) -> proper#(X) proper#(isNat(X)) -> proper#(X) proper#(s(X)) -> proper#(X) proper#(length(X)) -> proper#(X) proper#(cons(X1,X2)) -> proper#(X2) proper#(cons(X1,X2)) -> proper#(X1) proper#(take(X1,X2)) -> proper#(X2) proper#(take(X1,X2)) -> proper#(X1) proper#(uTake1(X)) -> proper#(X) proper#(uTake2(X1,X2,X3,X4)) -> proper#(X4) proper#(uTake2(X1,X2,X3,X4)) -> proper#(X3) proper#(uTake2(X1,X2,X3,X4)) -> proper#(X2) proper#(uTake2(X1,X2,X3,X4)) -> proper#(X1) proper#(uLength(X1,X2)) -> proper#(X2) proper#(uLength(X1,X2)) -> proper#(X1) TRS: active(and(tt(),T)) -> mark(T) active(isNatIList(IL)) -> mark(isNatList(IL)) active(isNat(0())) -> mark(tt()) active(isNat(s(N))) -> mark(isNat(N)) active(isNat(length(L))) -> mark(isNatList(L)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(N,L))) -> mark(and(isNat(N),isNatList(L))) active(isNatList(take(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(zeros()) -> mark(cons(0(),zeros())) active(take(0(),IL)) -> mark(uTake1(isNatIList(IL))) active(uTake1(tt())) -> mark(nil()) active(take(s(M),cons(N,IL))) -> mark(uTake2(and(isNat(M),and(isNat(N),isNatIList(IL))),M,N,IL)) active(uTake2(tt(),M,N,IL)) -> mark(cons(N,take(M,IL))) active(length(cons(N,L))) -> mark(uLength(and(isNat(N),isNatList(L)),L)) active(uLength(tt(),L)) -> mark(s(length(L))) active(and(X1,X2)) -> and(active(X1),X2) active(and(X1,X2)) -> and(X1,active(X2)) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) active(uTake1(X)) -> uTake1(active(X)) active(uTake2(X1,X2,X3,X4)) -> uTake2(active(X1),X2,X3,X4) active(uLength(X1,X2)) -> uLength(active(X1),X2) and(mark(X1),X2) -> mark(and(X1,X2)) and(X1,mark(X2)) -> mark(and(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) uTake1(mark(X)) -> mark(uTake1(X)) uTake2(mark(X1),X2,X3,X4) -> mark(uTake2(X1,X2,X3,X4)) uLength(mark(X1),X2) -> mark(uLength(X1,X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(isNat(X)) -> isNat(proper(X)) proper(0()) -> ok(0()) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(nil()) -> ok(nil()) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) proper(uTake1(X)) -> uTake1(proper(X)) proper(uTake2(X1,X2,X3,X4)) -> uTake2(proper(X1),proper(X2),proper(X3),proper(X4)) proper(uLength(X1,X2)) -> uLength(proper(X1),proper(X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatIList(ok(X)) -> ok(isNatIList(X)) isNatList(ok(X)) -> ok(isNatList(X)) isNat(ok(X)) -> ok(isNat(X)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) uTake1(ok(X)) -> ok(uTake1(X)) uTake2(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(uTake2(X1,X2,X3,X4)) uLength(ok(X1),ok(X2)) -> ok(uLength(X1,X2)) 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, [uLength](x0, x1) = x0 + x1, [uTake2](x0, x1, x2, x3) = x0 + x1 + x2 + x3, [uTake1](x0) = x0, [take](x0, x1) = x0 + x1, [nil] = 0, [cons](x0, x1) = x0 + x1, [zeros] = 0, [length](x0) = x0, [s](x0) = x0, [isNat](x0) = x0, [0] = 0, [isNatList](x0) = x0, [isNatIList](x0) = x0, [mark](x0) = 0, [active](x0) = x0, [and](x0, x1) = x0 + x1 + 1, [tt] = 0 orientation: proper#(and(X1,X2)) = X1 + X2 + 2 >= X2 + 1 = proper#(X2) proper#(and(X1,X2)) = X1 + X2 + 2 >= X1 + 1 = proper#(X1) proper#(isNatIList(X)) = X + 1 >= X + 1 = proper#(X) proper#(isNatList(X)) = X + 1 >= X + 1 = proper#(X) proper#(isNat(X)) = X + 1 >= X + 1 = proper#(X) proper#(s(X)) = X + 1 >= X + 1 = proper#(X) proper#(length(X)) = X + 1 >= X + 1 = proper#(X) proper#(cons(X1,X2)) = X1 + X2 + 1 >= X2 + 1 = proper#(X2) proper#(cons(X1,X2)) = X1 + X2 + 1 >= X1 + 1 = proper#(X1) proper#(take(X1,X2)) = X1 + X2 + 1 >= X2 + 1 = proper#(X2) proper#(take(X1,X2)) = X1 + X2 + 1 >= X1 + 1 = proper#(X1) proper#(uTake1(X)) = X + 1 >= X + 1 = proper#(X) proper#(uTake2(X1,X2,X3,X4)) = X1 + X2 + X3 + X4 + 1 >= X4 + 1 = proper#(X4) proper#(uTake2(X1,X2,X3,X4)) = X1 + X2 + X3 + X4 + 1 >= X3 + 1 = proper#(X3) proper#(uTake2(X1,X2,X3,X4)) = X1 + X2 + X3 + X4 + 1 >= X2 + 1 = proper#(X2) proper#(uTake2(X1,X2,X3,X4)) = X1 + X2 + X3 + X4 + 1 >= X1 + 1 = proper#(X1) proper#(uLength(X1,X2)) = X1 + X2 + 1 >= X2 + 1 = proper#(X2) proper#(uLength(X1,X2)) = X1 + X2 + 1 >= X1 + 1 = proper#(X1) active(and(tt(),T)) = T + 1 >= 0 = mark(T) active(isNatIList(IL)) = IL >= 0 = mark(isNatList(IL)) active(isNat(0())) = 0 >= 0 = mark(tt()) active(isNat(s(N))) = N >= 0 = mark(isNat(N)) active(isNat(length(L))) = L >= 0 = mark(isNatList(L)) active(isNatIList(zeros())) = 0 >= 0 = mark(tt()) active(isNatIList(cons(N,IL))) = IL + N >= 0 = mark(and(isNat(N),isNatIList(IL))) active(isNatList(nil())) = 0 >= 0 = mark(tt()) active(isNatList(cons(N,L))) = L + N >= 0 = mark(and(isNat(N),isNatList(L))) active(isNatList(take(N,IL))) = IL + N >= 0 = mark(and(isNat(N),isNatIList(IL))) active(zeros()) = 0 >= 0 = mark(cons(0(),zeros())) active(take(0(),IL)) = IL >= 0 = mark(uTake1(isNatIList(IL))) active(uTake1(tt())) = 0 >= 0 = mark(nil()) active(take(s(M),cons(N,IL))) = IL + M + N >= 0 = mark(uTake2(and(isNat(M),and(isNat(N),isNatIList(IL))),M,N,IL)) active(uTake2(tt(),M,N,IL)) = IL + M + N >= 0 = mark(cons(N,take(M,IL))) active(length(cons(N,L))) = L + N >= 0 = mark(uLength(and(isNat(N),isNatList(L)),L)) active(uLength(tt(),L)) = L >= 0 = mark(s(length(L))) active(and(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = and(active(X1),X2) active(and(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = and(X1,active(X2)) active(s(X)) = X >= X = s(active(X)) active(length(X)) = X >= X = length(active(X)) active(cons(X1,X2)) = X1 + X2 >= X1 + X2 = cons(active(X1),X2) 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(uTake1(X)) = X >= X = uTake1(active(X)) active(uTake2(X1,X2,X3,X4)) = X1 + X2 + X3 + X4 >= X1 + X2 + X3 + X4 = uTake2(active(X1),X2,X3,X4) active(uLength(X1,X2)) = X1 + X2 >= X1 + X2 = uLength(active(X1),X2) and(mark(X1),X2) = X2 + 1 >= 0 = mark(and(X1,X2)) and(X1,mark(X2)) = X1 + 1 >= 0 = mark(and(X1,X2)) s(mark(X)) = 0 >= 0 = mark(s(X)) length(mark(X)) = 0 >= 0 = mark(length(X)) cons(mark(X1),X2) = X2 >= 0 = mark(cons(X1,X2)) take(mark(X1),X2) = X2 >= 0 = mark(take(X1,X2)) take(X1,mark(X2)) = X1 >= 0 = mark(take(X1,X2)) uTake1(mark(X)) = 0 >= 0 = mark(uTake1(X)) uTake2(mark(X1),X2,X3,X4) = X2 + X3 + X4 >= 0 = mark(uTake2(X1,X2,X3,X4)) uLength(mark(X1),X2) = X2 >= 0 = mark(uLength(X1,X2)) proper(and(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = and(proper(X1),proper(X2)) proper(tt()) = 0 >= 0 = ok(tt()) proper(isNatIList(X)) = X >= X = isNatIList(proper(X)) proper(isNatList(X)) = X >= X = isNatList(proper(X)) proper(isNat(X)) = X >= X = isNat(proper(X)) proper(0()) = 0 >= 0 = ok(0()) proper(s(X)) = X >= X = s(proper(X)) proper(length(X)) = X >= X = length(proper(X)) proper(zeros()) = 0 >= 0 = ok(zeros()) proper(cons(X1,X2)) = X1 + X2 >= X1 + X2 = cons(proper(X1),proper(X2)) proper(nil()) = 0 >= 0 = ok(nil()) proper(take(X1,X2)) = X1 + X2 >= X1 + X2 = take(proper(X1),proper(X2)) proper(uTake1(X)) = X >= X = uTake1(proper(X)) proper(uTake2(X1,X2,X3,X4)) = X1 + X2 + X3 + X4 >= X1 + X2 + X3 + X4 = uTake2(proper(X1),proper(X2),proper(X3),proper(X4)) proper(uLength(X1,X2)) = X1 + X2 >= X1 + X2 = uLength(proper(X1),proper(X2)) and(ok(X1),ok(X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = ok(and(X1,X2)) isNatIList(ok(X)) = X >= X = ok(isNatIList(X)) isNatList(ok(X)) = X >= X = ok(isNatList(X)) isNat(ok(X)) = X >= X = ok(isNat(X)) s(ok(X)) = X >= X = ok(s(X)) length(ok(X)) = X >= X = ok(length(X)) cons(ok(X1),ok(X2)) = X1 + X2 >= X1 + X2 = ok(cons(X1,X2)) take(ok(X1),ok(X2)) = X1 + X2 >= X1 + X2 = ok(take(X1,X2)) uTake1(ok(X)) = X >= X = ok(uTake1(X)) uTake2(ok(X1),ok(X2),ok(X3),ok(X4)) = X1 + X2 + X3 + X4 >= X1 + X2 + X3 + X4 = ok(uTake2(X1,X2,X3,X4)) uLength(ok(X1),ok(X2)) = X1 + X2 >= X1 + X2 = ok(uLength(X1,X2)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) problem: DPs: proper#(isNatIList(X)) -> proper#(X) proper#(isNatList(X)) -> proper#(X) proper#(isNat(X)) -> proper#(X) proper#(s(X)) -> proper#(X) proper#(length(X)) -> proper#(X) proper#(cons(X1,X2)) -> proper#(X2) proper#(cons(X1,X2)) -> proper#(X1) proper#(take(X1,X2)) -> proper#(X2) proper#(take(X1,X2)) -> proper#(X1) proper#(uTake1(X)) -> proper#(X) proper#(uTake2(X1,X2,X3,X4)) -> proper#(X4) proper#(uTake2(X1,X2,X3,X4)) -> proper#(X3) proper#(uTake2(X1,X2,X3,X4)) -> proper#(X2) proper#(uTake2(X1,X2,X3,X4)) -> proper#(X1) proper#(uLength(X1,X2)) -> proper#(X2) proper#(uLength(X1,X2)) -> proper#(X1) TRS: active(and(tt(),T)) -> mark(T) active(isNatIList(IL)) -> mark(isNatList(IL)) active(isNat(0())) -> mark(tt()) active(isNat(s(N))) -> mark(isNat(N)) active(isNat(length(L))) -> mark(isNatList(L)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(N,L))) -> mark(and(isNat(N),isNatList(L))) active(isNatList(take(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(zeros()) -> mark(cons(0(),zeros())) active(take(0(),IL)) -> mark(uTake1(isNatIList(IL))) active(uTake1(tt())) -> mark(nil()) active(take(s(M),cons(N,IL))) -> mark(uTake2(and(isNat(M),and(isNat(N),isNatIList(IL))),M,N,IL)) active(uTake2(tt(),M,N,IL)) -> mark(cons(N,take(M,IL))) active(length(cons(N,L))) -> mark(uLength(and(isNat(N),isNatList(L)),L)) active(uLength(tt(),L)) -> mark(s(length(L))) active(and(X1,X2)) -> and(active(X1),X2) active(and(X1,X2)) -> and(X1,active(X2)) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) active(uTake1(X)) -> uTake1(active(X)) active(uTake2(X1,X2,X3,X4)) -> uTake2(active(X1),X2,X3,X4) active(uLength(X1,X2)) -> uLength(active(X1),X2) and(mark(X1),X2) -> mark(and(X1,X2)) and(X1,mark(X2)) -> mark(and(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) uTake1(mark(X)) -> mark(uTake1(X)) uTake2(mark(X1),X2,X3,X4) -> mark(uTake2(X1,X2,X3,X4)) uLength(mark(X1),X2) -> mark(uLength(X1,X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(isNat(X)) -> isNat(proper(X)) proper(0()) -> ok(0()) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(nil()) -> ok(nil()) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) proper(uTake1(X)) -> uTake1(proper(X)) proper(uTake2(X1,X2,X3,X4)) -> uTake2(proper(X1),proper(X2),proper(X3),proper(X4)) proper(uLength(X1,X2)) -> uLength(proper(X1),proper(X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatIList(ok(X)) -> ok(isNatIList(X)) isNatList(ok(X)) -> ok(isNatList(X)) isNat(ok(X)) -> ok(isNat(X)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) uTake1(ok(X)) -> ok(uTake1(X)) uTake2(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(uTake2(X1,X2,X3,X4)) uLength(ok(X1),ok(X2)) -> ok(uLength(X1,X2)) 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, [uLength](x0, x1) = x0 + x1 + 1, [uTake2](x0, x1, x2, x3) = x0 + x1 + x2 + x3 + 1, [uTake1](x0) = x0 + 1, [take](x0, x1) = x0 + x1 + 1, [nil] = 0, [cons](x0, x1) = x0 + x1 + 1, [zeros] = 1, [length](x0) = x0 + 1, [s](x0) = x0 + 1, [isNat](x0) = x0 + 1, [0] = 0, [isNatList](x0) = x0 + 1, [isNatIList](x0) = x0 + 1, [mark](x0) = 1, [active](x0) = x0, [and](x0, x1) = 1, [tt] = 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#(s(X)) = X + 2 >= X + 1 = proper#(X) proper#(length(X)) = X + 2 >= X + 1 = proper#(X) proper#(cons(X1,X2)) = X1 + X2 + 2 >= X2 + 1 = proper#(X2) proper#(cons(X1,X2)) = X1 + X2 + 2 >= X1 + 1 = proper#(X1) proper#(take(X1,X2)) = X1 + X2 + 2 >= X2 + 1 = proper#(X2) proper#(take(X1,X2)) = X1 + X2 + 2 >= X1 + 1 = proper#(X1) proper#(uTake1(X)) = X + 2 >= X + 1 = proper#(X) proper#(uTake2(X1,X2,X3,X4)) = X1 + X2 + X3 + X4 + 2 >= X4 + 1 = proper#(X4) proper#(uTake2(X1,X2,X3,X4)) = X1 + X2 + X3 + X4 + 2 >= X3 + 1 = proper#(X3) proper#(uTake2(X1,X2,X3,X4)) = X1 + X2 + X3 + X4 + 2 >= X2 + 1 = proper#(X2) proper#(uTake2(X1,X2,X3,X4)) = X1 + X2 + X3 + X4 + 2 >= X1 + 1 = proper#(X1) proper#(uLength(X1,X2)) = X1 + X2 + 2 >= X2 + 1 = proper#(X2) proper#(uLength(X1,X2)) = X1 + X2 + 2 >= X1 + 1 = proper#(X1) active(and(tt(),T)) = 1 >= 1 = mark(T) active(isNatIList(IL)) = IL + 1 >= 1 = mark(isNatList(IL)) active(isNat(0())) = 1 >= 1 = mark(tt()) active(isNat(s(N))) = N + 2 >= 1 = mark(isNat(N)) active(isNat(length(L))) = L + 2 >= 1 = mark(isNatList(L)) active(isNatIList(zeros())) = 2 >= 1 = mark(tt()) active(isNatIList(cons(N,IL))) = IL + N + 2 >= 1 = mark(and(isNat(N),isNatIList(IL))) active(isNatList(nil())) = 1 >= 1 = mark(tt()) active(isNatList(cons(N,L))) = L + N + 2 >= 1 = mark(and(isNat(N),isNatList(L))) active(isNatList(take(N,IL))) = IL + N + 2 >= 1 = mark(and(isNat(N),isNatIList(IL))) active(zeros()) = 1 >= 1 = mark(cons(0(),zeros())) active(take(0(),IL)) = IL + 1 >= 1 = mark(uTake1(isNatIList(IL))) active(uTake1(tt())) = 1 >= 1 = mark(nil()) active(take(s(M),cons(N,IL))) = IL + M + N + 3 >= 1 = mark(uTake2(and(isNat(M),and(isNat(N),isNatIList(IL))),M,N,IL)) active(uTake2(tt(),M,N,IL)) = IL + M + N + 1 >= 1 = mark(cons(N,take(M,IL))) active(length(cons(N,L))) = L + N + 2 >= 1 = mark(uLength(and(isNat(N),isNatList(L)),L)) active(uLength(tt(),L)) = L + 1 >= 1 = mark(s(length(L))) active(and(X1,X2)) = 1 >= 1 = and(active(X1),X2) active(and(X1,X2)) = 1 >= 1 = and(X1,active(X2)) active(s(X)) = X + 1 >= X + 1 = s(active(X)) active(length(X)) = X + 1 >= X + 1 = length(active(X)) active(cons(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = cons(active(X1),X2) 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(uTake1(X)) = X + 1 >= X + 1 = uTake1(active(X)) active(uTake2(X1,X2,X3,X4)) = X1 + X2 + X3 + X4 + 1 >= X1 + X2 + X3 + X4 + 1 = uTake2(active(X1),X2,X3,X4) active(uLength(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = uLength(active(X1),X2) and(mark(X1),X2) = 1 >= 1 = mark(and(X1,X2)) and(X1,mark(X2)) = 1 >= 1 = mark(and(X1,X2)) s(mark(X)) = 2 >= 1 = mark(s(X)) length(mark(X)) = 2 >= 1 = mark(length(X)) cons(mark(X1),X2) = X2 + 2 >= 1 = mark(cons(X1,X2)) take(mark(X1),X2) = X2 + 2 >= 1 = mark(take(X1,X2)) take(X1,mark(X2)) = X1 + 2 >= 1 = mark(take(X1,X2)) uTake1(mark(X)) = 2 >= 1 = mark(uTake1(X)) uTake2(mark(X1),X2,X3,X4) = X2 + X3 + X4 + 2 >= 1 = mark(uTake2(X1,X2,X3,X4)) uLength(mark(X1),X2) = X2 + 2 >= 1 = mark(uLength(X1,X2)) proper(and(X1,X2)) = 1 >= 1 = and(proper(X1),proper(X2)) proper(tt()) = 0 >= 0 = ok(tt()) proper(isNatIList(X)) = X + 1 >= X + 1 = isNatIList(proper(X)) proper(isNatList(X)) = X + 1 >= X + 1 = isNatList(proper(X)) proper(isNat(X)) = X + 1 >= X + 1 = isNat(proper(X)) proper(0()) = 0 >= 0 = ok(0()) proper(s(X)) = X + 1 >= X + 1 = s(proper(X)) proper(length(X)) = X + 1 >= X + 1 = length(proper(X)) proper(zeros()) = 1 >= 1 = ok(zeros()) proper(cons(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = cons(proper(X1),proper(X2)) proper(nil()) = 0 >= 0 = ok(nil()) proper(take(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = take(proper(X1),proper(X2)) proper(uTake1(X)) = X + 1 >= X + 1 = uTake1(proper(X)) proper(uTake2(X1,X2,X3,X4)) = X1 + X2 + X3 + X4 + 1 >= X1 + X2 + X3 + X4 + 1 = uTake2(proper(X1),proper(X2),proper(X3),proper(X4)) proper(uLength(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = uLength(proper(X1),proper(X2)) and(ok(X1),ok(X2)) = 1 >= 1 = ok(and(X1,X2)) isNatIList(ok(X)) = X + 1 >= X + 1 = ok(isNatIList(X)) isNatList(ok(X)) = X + 1 >= X + 1 = ok(isNatList(X)) isNat(ok(X)) = X + 1 >= X + 1 = ok(isNat(X)) s(ok(X)) = X + 1 >= X + 1 = ok(s(X)) length(ok(X)) = X + 1 >= X + 1 = ok(length(X)) cons(ok(X1),ok(X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = ok(cons(X1,X2)) take(ok(X1),ok(X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = ok(take(X1,X2)) uTake1(ok(X)) = X + 1 >= X + 1 = ok(uTake1(X)) uTake2(ok(X1),ok(X2),ok(X3),ok(X4)) = X1 + X2 + X3 + X4 + 1 >= X1 + X2 + X3 + X4 + 1 = ok(uTake2(X1,X2,X3,X4)) uLength(ok(X1),ok(X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = ok(uLength(X1,X2)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) problem: DPs: TRS: active(and(tt(),T)) -> mark(T) active(isNatIList(IL)) -> mark(isNatList(IL)) active(isNat(0())) -> mark(tt()) active(isNat(s(N))) -> mark(isNat(N)) active(isNat(length(L))) -> mark(isNatList(L)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(N,L))) -> mark(and(isNat(N),isNatList(L))) active(isNatList(take(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(zeros()) -> mark(cons(0(),zeros())) active(take(0(),IL)) -> mark(uTake1(isNatIList(IL))) active(uTake1(tt())) -> mark(nil()) active(take(s(M),cons(N,IL))) -> mark(uTake2(and(isNat(M),and(isNat(N),isNatIList(IL))),M,N,IL)) active(uTake2(tt(),M,N,IL)) -> mark(cons(N,take(M,IL))) active(length(cons(N,L))) -> mark(uLength(and(isNat(N),isNatList(L)),L)) active(uLength(tt(),L)) -> mark(s(length(L))) active(and(X1,X2)) -> and(active(X1),X2) active(and(X1,X2)) -> and(X1,active(X2)) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) active(uTake1(X)) -> uTake1(active(X)) active(uTake2(X1,X2,X3,X4)) -> uTake2(active(X1),X2,X3,X4) active(uLength(X1,X2)) -> uLength(active(X1),X2) and(mark(X1),X2) -> mark(and(X1,X2)) and(X1,mark(X2)) -> mark(and(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) uTake1(mark(X)) -> mark(uTake1(X)) uTake2(mark(X1),X2,X3,X4) -> mark(uTake2(X1,X2,X3,X4)) uLength(mark(X1),X2) -> mark(uLength(X1,X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(isNat(X)) -> isNat(proper(X)) proper(0()) -> ok(0()) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(nil()) -> ok(nil()) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) proper(uTake1(X)) -> uTake1(proper(X)) proper(uTake2(X1,X2,X3,X4)) -> uTake2(proper(X1),proper(X2),proper(X3),proper(X4)) proper(uLength(X1,X2)) -> uLength(proper(X1),proper(X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatIList(ok(X)) -> ok(isNatIList(X)) isNatList(ok(X)) -> ok(isNatList(X)) isNat(ok(X)) -> ok(isNat(X)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) uTake1(ok(X)) -> ok(uTake1(X)) uTake2(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(uTake2(X1,X2,X3,X4)) uLength(ok(X1),ok(X2)) -> ok(uLength(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed DPs: uLength#(mark(X1),X2) -> uLength#(X1,X2) uLength#(ok(X1),ok(X2)) -> uLength#(X1,X2) TRS: active(and(tt(),T)) -> mark(T) active(isNatIList(IL)) -> mark(isNatList(IL)) active(isNat(0())) -> mark(tt()) active(isNat(s(N))) -> mark(isNat(N)) active(isNat(length(L))) -> mark(isNatList(L)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(N,L))) -> mark(and(isNat(N),isNatList(L))) active(isNatList(take(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(zeros()) -> mark(cons(0(),zeros())) active(take(0(),IL)) -> mark(uTake1(isNatIList(IL))) active(uTake1(tt())) -> mark(nil()) active(take(s(M),cons(N,IL))) -> mark(uTake2(and(isNat(M),and(isNat(N),isNatIList(IL))),M,N,IL)) active(uTake2(tt(),M,N,IL)) -> mark(cons(N,take(M,IL))) active(length(cons(N,L))) -> mark(uLength(and(isNat(N),isNatList(L)),L)) active(uLength(tt(),L)) -> mark(s(length(L))) active(and(X1,X2)) -> and(active(X1),X2) active(and(X1,X2)) -> and(X1,active(X2)) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) active(uTake1(X)) -> uTake1(active(X)) active(uTake2(X1,X2,X3,X4)) -> uTake2(active(X1),X2,X3,X4) active(uLength(X1,X2)) -> uLength(active(X1),X2) and(mark(X1),X2) -> mark(and(X1,X2)) and(X1,mark(X2)) -> mark(and(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) uTake1(mark(X)) -> mark(uTake1(X)) uTake2(mark(X1),X2,X3,X4) -> mark(uTake2(X1,X2,X3,X4)) uLength(mark(X1),X2) -> mark(uLength(X1,X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(isNat(X)) -> isNat(proper(X)) proper(0()) -> ok(0()) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(nil()) -> ok(nil()) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) proper(uTake1(X)) -> uTake1(proper(X)) proper(uTake2(X1,X2,X3,X4)) -> uTake2(proper(X1),proper(X2),proper(X3),proper(X4)) proper(uLength(X1,X2)) -> uLength(proper(X1),proper(X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatIList(ok(X)) -> ok(isNatIList(X)) isNatList(ok(X)) -> ok(isNatList(X)) isNat(ok(X)) -> ok(isNat(X)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) uTake1(ok(X)) -> ok(uTake1(X)) uTake2(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(uTake2(X1,X2,X3,X4)) uLength(ok(X1),ok(X2)) -> ok(uLength(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dimension: 1 interpretation: [uLength#](x0, x1) = x1, [top](x0) = 1, [ok](x0) = x0 + 1, [proper](x0) = 1, [uLength](x0, x1) = x1, [uTake2](x0, x1, x2, x3) = x1, [uTake1](x0) = x0, [take](x0, x1) = x0, [nil] = 0, [cons](x0, x1) = x0, [zeros] = 0, [length](x0) = x0, [s](x0) = x0, [isNat](x0) = x0, [0] = 0, [isNatList](x0) = x0, [isNatIList](x0) = x0, [mark](x0) = 0, [active](x0) = x0, [and](x0, x1) = x1, [tt] = 0 orientation: uLength#(mark(X1),X2) = X2 >= X2 = uLength#(X1,X2) uLength#(ok(X1),ok(X2)) = X2 + 1 >= X2 = uLength#(X1,X2) active(and(tt(),T)) = T >= 0 = mark(T) active(isNatIList(IL)) = IL >= 0 = mark(isNatList(IL)) active(isNat(0())) = 0 >= 0 = mark(tt()) active(isNat(s(N))) = N >= 0 = mark(isNat(N)) active(isNat(length(L))) = L >= 0 = mark(isNatList(L)) active(isNatIList(zeros())) = 0 >= 0 = mark(tt()) active(isNatIList(cons(N,IL))) = N >= 0 = mark(and(isNat(N),isNatIList(IL))) active(isNatList(nil())) = 0 >= 0 = mark(tt()) active(isNatList(cons(N,L))) = N >= 0 = mark(and(isNat(N),isNatList(L))) active(isNatList(take(N,IL))) = N >= 0 = mark(and(isNat(N),isNatIList(IL))) active(zeros()) = 0 >= 0 = mark(cons(0(),zeros())) active(take(0(),IL)) = 0 >= 0 = mark(uTake1(isNatIList(IL))) active(uTake1(tt())) = 0 >= 0 = mark(nil()) active(take(s(M),cons(N,IL))) = M >= 0 = mark(uTake2(and(isNat(M),and(isNat(N),isNatIList(IL))),M,N,IL)) active(uTake2(tt(),M,N,IL)) = M >= 0 = mark(cons(N,take(M,IL))) active(length(cons(N,L))) = N >= 0 = mark(uLength(and(isNat(N),isNatList(L)),L)) active(uLength(tt(),L)) = L >= 0 = mark(s(length(L))) active(and(X1,X2)) = X2 >= X2 = and(active(X1),X2) active(and(X1,X2)) = X2 >= X2 = and(X1,active(X2)) active(s(X)) = X >= X = s(active(X)) active(length(X)) = X >= X = length(active(X)) active(cons(X1,X2)) = X1 >= X1 = cons(active(X1),X2) active(take(X1,X2)) = X1 >= X1 = take(active(X1),X2) active(take(X1,X2)) = X1 >= X1 = take(X1,active(X2)) active(uTake1(X)) = X >= X = uTake1(active(X)) active(uTake2(X1,X2,X3,X4)) = X2 >= X2 = uTake2(active(X1),X2,X3,X4) active(uLength(X1,X2)) = X2 >= X2 = uLength(active(X1),X2) and(mark(X1),X2) = X2 >= 0 = mark(and(X1,X2)) and(X1,mark(X2)) = 0 >= 0 = mark(and(X1,X2)) s(mark(X)) = 0 >= 0 = mark(s(X)) length(mark(X)) = 0 >= 0 = mark(length(X)) cons(mark(X1),X2) = 0 >= 0 = mark(cons(X1,X2)) take(mark(X1),X2) = 0 >= 0 = mark(take(X1,X2)) take(X1,mark(X2)) = X1 >= 0 = mark(take(X1,X2)) uTake1(mark(X)) = 0 >= 0 = mark(uTake1(X)) uTake2(mark(X1),X2,X3,X4) = X2 >= 0 = mark(uTake2(X1,X2,X3,X4)) uLength(mark(X1),X2) = X2 >= 0 = mark(uLength(X1,X2)) proper(and(X1,X2)) = 1 >= 1 = and(proper(X1),proper(X2)) proper(tt()) = 1 >= 1 = ok(tt()) proper(isNatIList(X)) = 1 >= 1 = isNatIList(proper(X)) proper(isNatList(X)) = 1 >= 1 = isNatList(proper(X)) proper(isNat(X)) = 1 >= 1 = isNat(proper(X)) proper(0()) = 1 >= 1 = ok(0()) proper(s(X)) = 1 >= 1 = s(proper(X)) proper(length(X)) = 1 >= 1 = length(proper(X)) proper(zeros()) = 1 >= 1 = ok(zeros()) proper(cons(X1,X2)) = 1 >= 1 = cons(proper(X1),proper(X2)) proper(nil()) = 1 >= 1 = ok(nil()) proper(take(X1,X2)) = 1 >= 1 = take(proper(X1),proper(X2)) proper(uTake1(X)) = 1 >= 1 = uTake1(proper(X)) proper(uTake2(X1,X2,X3,X4)) = 1 >= 1 = uTake2(proper(X1),proper(X2),proper(X3),proper(X4)) proper(uLength(X1,X2)) = 1 >= 1 = uLength(proper(X1),proper(X2)) and(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(and(X1,X2)) isNatIList(ok(X)) = X + 1 >= X + 1 = ok(isNatIList(X)) isNatList(ok(X)) = X + 1 >= X + 1 = ok(isNatList(X)) isNat(ok(X)) = X + 1 >= X + 1 = ok(isNat(X)) s(ok(X)) = X + 1 >= X + 1 = ok(s(X)) length(ok(X)) = X + 1 >= X + 1 = ok(length(X)) cons(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(cons(X1,X2)) take(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(take(X1,X2)) uTake1(ok(X)) = X + 1 >= X + 1 = ok(uTake1(X)) uTake2(ok(X1),ok(X2),ok(X3),ok(X4)) = X2 + 1 >= X2 + 1 = ok(uTake2(X1,X2,X3,X4)) uLength(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(uLength(X1,X2)) top(mark(X)) = 1 >= 1 = top(proper(X)) top(ok(X)) = 1 >= 1 = top(active(X)) problem: DPs: uLength#(mark(X1),X2) -> uLength#(X1,X2) TRS: active(and(tt(),T)) -> mark(T) active(isNatIList(IL)) -> mark(isNatList(IL)) active(isNat(0())) -> mark(tt()) active(isNat(s(N))) -> mark(isNat(N)) active(isNat(length(L))) -> mark(isNatList(L)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(N,L))) -> mark(and(isNat(N),isNatList(L))) active(isNatList(take(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(zeros()) -> mark(cons(0(),zeros())) active(take(0(),IL)) -> mark(uTake1(isNatIList(IL))) active(uTake1(tt())) -> mark(nil()) active(take(s(M),cons(N,IL))) -> mark(uTake2(and(isNat(M),and(isNat(N),isNatIList(IL))),M,N,IL)) active(uTake2(tt(),M,N,IL)) -> mark(cons(N,take(M,IL))) active(length(cons(N,L))) -> mark(uLength(and(isNat(N),isNatList(L)),L)) active(uLength(tt(),L)) -> mark(s(length(L))) active(and(X1,X2)) -> and(active(X1),X2) active(and(X1,X2)) -> and(X1,active(X2)) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) active(uTake1(X)) -> uTake1(active(X)) active(uTake2(X1,X2,X3,X4)) -> uTake2(active(X1),X2,X3,X4) active(uLength(X1,X2)) -> uLength(active(X1),X2) and(mark(X1),X2) -> mark(and(X1,X2)) and(X1,mark(X2)) -> mark(and(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) uTake1(mark(X)) -> mark(uTake1(X)) uTake2(mark(X1),X2,X3,X4) -> mark(uTake2(X1,X2,X3,X4)) uLength(mark(X1),X2) -> mark(uLength(X1,X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(isNat(X)) -> isNat(proper(X)) proper(0()) -> ok(0()) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(nil()) -> ok(nil()) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) proper(uTake1(X)) -> uTake1(proper(X)) proper(uTake2(X1,X2,X3,X4)) -> uTake2(proper(X1),proper(X2),proper(X3),proper(X4)) proper(uLength(X1,X2)) -> uLength(proper(X1),proper(X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatIList(ok(X)) -> ok(isNatIList(X)) isNatList(ok(X)) -> ok(isNatList(X)) isNat(ok(X)) -> ok(isNat(X)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) uTake1(ok(X)) -> ok(uTake1(X)) uTake2(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(uTake2(X1,X2,X3,X4)) uLength(ok(X1),ok(X2)) -> ok(uLength(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dimension: 1 interpretation: [uLength#](x0, x1) = x0 + 1, [top](x0) = 0, [ok](x0) = 0, [proper](x0) = 0, [uLength](x0, x1) = x0 + x1, [uTake2](x0, x1, x2, x3) = x0 + x1 + x2 + x3, [uTake1](x0) = x0, [take](x0, x1) = x0 + x1, [nil] = 0, [cons](x0, x1) = x0 + x1, [zeros] = 1, [length](x0) = x0, [s](x0) = x0, [isNat](x0) = 0, [0] = 0, [isNatList](x0) = 0, [isNatIList](x0) = 0, [mark](x0) = x0 + 1, [active](x0) = x0 + 1, [and](x0, x1) = x0 + x1, [tt] = 0 orientation: uLength#(mark(X1),X2) = X1 + 2 >= X1 + 1 = uLength#(X1,X2) active(and(tt(),T)) = T + 1 >= T + 1 = mark(T) active(isNatIList(IL)) = 1 >= 1 = mark(isNatList(IL)) active(isNat(0())) = 1 >= 1 = mark(tt()) active(isNat(s(N))) = 1 >= 1 = mark(isNat(N)) active(isNat(length(L))) = 1 >= 1 = mark(isNatList(L)) active(isNatIList(zeros())) = 1 >= 1 = mark(tt()) active(isNatIList(cons(N,IL))) = 1 >= 1 = mark(and(isNat(N),isNatIList(IL))) active(isNatList(nil())) = 1 >= 1 = mark(tt()) active(isNatList(cons(N,L))) = 1 >= 1 = mark(and(isNat(N),isNatList(L))) active(isNatList(take(N,IL))) = 1 >= 1 = mark(and(isNat(N),isNatIList(IL))) active(zeros()) = 2 >= 2 = mark(cons(0(),zeros())) active(take(0(),IL)) = IL + 1 >= 1 = mark(uTake1(isNatIList(IL))) active(uTake1(tt())) = 1 >= 1 = mark(nil()) active(take(s(M),cons(N,IL))) = IL + M + N + 1 >= IL + M + N + 1 = mark(uTake2(and(isNat(M),and(isNat(N),isNatIList(IL))),M,N,IL)) active(uTake2(tt(),M,N,IL)) = IL + M + N + 1 >= IL + M + N + 1 = mark(cons(N,take(M,IL))) active(length(cons(N,L))) = L + N + 1 >= L + 1 = mark(uLength(and(isNat(N),isNatList(L)),L)) active(uLength(tt(),L)) = L + 1 >= L + 1 = mark(s(length(L))) active(and(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = and(active(X1),X2) active(and(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = and(X1,active(X2)) active(s(X)) = X + 1 >= X + 1 = s(active(X)) active(length(X)) = X + 1 >= X + 1 = length(active(X)) active(cons(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = cons(active(X1),X2) 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(uTake1(X)) = X + 1 >= X + 1 = uTake1(active(X)) active(uTake2(X1,X2,X3,X4)) = X1 + X2 + X3 + X4 + 1 >= X1 + X2 + X3 + X4 + 1 = uTake2(active(X1),X2,X3,X4) active(uLength(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = uLength(active(X1),X2) and(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(and(X1,X2)) and(X1,mark(X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(and(X1,X2)) s(mark(X)) = X + 1 >= X + 1 = mark(s(X)) length(mark(X)) = X + 1 >= X + 1 = mark(length(X)) cons(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(cons(X1,X2)) 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)) uTake1(mark(X)) = X + 1 >= X + 1 = mark(uTake1(X)) uTake2(mark(X1),X2,X3,X4) = X1 + X2 + X3 + X4 + 1 >= X1 + X2 + X3 + X4 + 1 = mark(uTake2(X1,X2,X3,X4)) uLength(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(uLength(X1,X2)) proper(and(X1,X2)) = 0 >= 0 = and(proper(X1),proper(X2)) proper(tt()) = 0 >= 0 = ok(tt()) proper(isNatIList(X)) = 0 >= 0 = isNatIList(proper(X)) proper(isNatList(X)) = 0 >= 0 = isNatList(proper(X)) proper(isNat(X)) = 0 >= 0 = isNat(proper(X)) proper(0()) = 0 >= 0 = ok(0()) proper(s(X)) = 0 >= 0 = s(proper(X)) proper(length(X)) = 0 >= 0 = length(proper(X)) proper(zeros()) = 0 >= 0 = ok(zeros()) proper(cons(X1,X2)) = 0 >= 0 = cons(proper(X1),proper(X2)) proper(nil()) = 0 >= 0 = ok(nil()) proper(take(X1,X2)) = 0 >= 0 = take(proper(X1),proper(X2)) proper(uTake1(X)) = 0 >= 0 = uTake1(proper(X)) proper(uTake2(X1,X2,X3,X4)) = 0 >= 0 = uTake2(proper(X1),proper(X2),proper(X3),proper(X4)) proper(uLength(X1,X2)) = 0 >= 0 = uLength(proper(X1),proper(X2)) and(ok(X1),ok(X2)) = 0 >= 0 = ok(and(X1,X2)) isNatIList(ok(X)) = 0 >= 0 = ok(isNatIList(X)) isNatList(ok(X)) = 0 >= 0 = ok(isNatList(X)) isNat(ok(X)) = 0 >= 0 = ok(isNat(X)) s(ok(X)) = 0 >= 0 = ok(s(X)) length(ok(X)) = 0 >= 0 = ok(length(X)) cons(ok(X1),ok(X2)) = 0 >= 0 = ok(cons(X1,X2)) take(ok(X1),ok(X2)) = 0 >= 0 = ok(take(X1,X2)) uTake1(ok(X)) = 0 >= 0 = ok(uTake1(X)) uTake2(ok(X1),ok(X2),ok(X3),ok(X4)) = 0 >= 0 = ok(uTake2(X1,X2,X3,X4)) uLength(ok(X1),ok(X2)) = 0 >= 0 = ok(uLength(X1,X2)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) problem: DPs: TRS: active(and(tt(),T)) -> mark(T) active(isNatIList(IL)) -> mark(isNatList(IL)) active(isNat(0())) -> mark(tt()) active(isNat(s(N))) -> mark(isNat(N)) active(isNat(length(L))) -> mark(isNatList(L)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(N,L))) -> mark(and(isNat(N),isNatList(L))) active(isNatList(take(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(zeros()) -> mark(cons(0(),zeros())) active(take(0(),IL)) -> mark(uTake1(isNatIList(IL))) active(uTake1(tt())) -> mark(nil()) active(take(s(M),cons(N,IL))) -> mark(uTake2(and(isNat(M),and(isNat(N),isNatIList(IL))),M,N,IL)) active(uTake2(tt(),M,N,IL)) -> mark(cons(N,take(M,IL))) active(length(cons(N,L))) -> mark(uLength(and(isNat(N),isNatList(L)),L)) active(uLength(tt(),L)) -> mark(s(length(L))) active(and(X1,X2)) -> and(active(X1),X2) active(and(X1,X2)) -> and(X1,active(X2)) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) active(uTake1(X)) -> uTake1(active(X)) active(uTake2(X1,X2,X3,X4)) -> uTake2(active(X1),X2,X3,X4) active(uLength(X1,X2)) -> uLength(active(X1),X2) and(mark(X1),X2) -> mark(and(X1,X2)) and(X1,mark(X2)) -> mark(and(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) uTake1(mark(X)) -> mark(uTake1(X)) uTake2(mark(X1),X2,X3,X4) -> mark(uTake2(X1,X2,X3,X4)) uLength(mark(X1),X2) -> mark(uLength(X1,X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(isNat(X)) -> isNat(proper(X)) proper(0()) -> ok(0()) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(nil()) -> ok(nil()) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) proper(uTake1(X)) -> uTake1(proper(X)) proper(uTake2(X1,X2,X3,X4)) -> uTake2(proper(X1),proper(X2),proper(X3),proper(X4)) proper(uLength(X1,X2)) -> uLength(proper(X1),proper(X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatIList(ok(X)) -> ok(isNatIList(X)) isNatList(ok(X)) -> ok(isNatList(X)) isNat(ok(X)) -> ok(isNat(X)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) uTake1(ok(X)) -> ok(uTake1(X)) uTake2(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(uTake2(X1,X2,X3,X4)) uLength(ok(X1),ok(X2)) -> ok(uLength(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed DPs: uTake2#(mark(X1),X2,X3,X4) -> uTake2#(X1,X2,X3,X4) uTake2#(ok(X1),ok(X2),ok(X3),ok(X4)) -> uTake2#(X1,X2,X3,X4) TRS: active(and(tt(),T)) -> mark(T) active(isNatIList(IL)) -> mark(isNatList(IL)) active(isNat(0())) -> mark(tt()) active(isNat(s(N))) -> mark(isNat(N)) active(isNat(length(L))) -> mark(isNatList(L)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(N,L))) -> mark(and(isNat(N),isNatList(L))) active(isNatList(take(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(zeros()) -> mark(cons(0(),zeros())) active(take(0(),IL)) -> mark(uTake1(isNatIList(IL))) active(uTake1(tt())) -> mark(nil()) active(take(s(M),cons(N,IL))) -> mark(uTake2(and(isNat(M),and(isNat(N),isNatIList(IL))),M,N,IL)) active(uTake2(tt(),M,N,IL)) -> mark(cons(N,take(M,IL))) active(length(cons(N,L))) -> mark(uLength(and(isNat(N),isNatList(L)),L)) active(uLength(tt(),L)) -> mark(s(length(L))) active(and(X1,X2)) -> and(active(X1),X2) active(and(X1,X2)) -> and(X1,active(X2)) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) active(uTake1(X)) -> uTake1(active(X)) active(uTake2(X1,X2,X3,X4)) -> uTake2(active(X1),X2,X3,X4) active(uLength(X1,X2)) -> uLength(active(X1),X2) and(mark(X1),X2) -> mark(and(X1,X2)) and(X1,mark(X2)) -> mark(and(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) uTake1(mark(X)) -> mark(uTake1(X)) uTake2(mark(X1),X2,X3,X4) -> mark(uTake2(X1,X2,X3,X4)) uLength(mark(X1),X2) -> mark(uLength(X1,X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(isNat(X)) -> isNat(proper(X)) proper(0()) -> ok(0()) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(nil()) -> ok(nil()) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) proper(uTake1(X)) -> uTake1(proper(X)) proper(uTake2(X1,X2,X3,X4)) -> uTake2(proper(X1),proper(X2),proper(X3),proper(X4)) proper(uLength(X1,X2)) -> uLength(proper(X1),proper(X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatIList(ok(X)) -> ok(isNatIList(X)) isNatList(ok(X)) -> ok(isNatList(X)) isNat(ok(X)) -> ok(isNat(X)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) uTake1(ok(X)) -> ok(uTake1(X)) uTake2(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(uTake2(X1,X2,X3,X4)) uLength(ok(X1),ok(X2)) -> ok(uLength(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dimension: 1 interpretation: [uTake2#](x0, x1, x2, x3) = x1 + x2 + x3, [top](x0) = 0, [ok](x0) = x0 + 1, [proper](x0) = x0 + 1, [uLength](x0, x1) = x0, [uTake2](x0, x1, x2, x3) = x0, [uTake1](x0) = x0, [take](x0, x1) = x0, [nil] = 0, [cons](x0, x1) = x0, [zeros] = 1, [length](x0) = x0, [s](x0) = x0, [isNat](x0) = x0, [0] = 1, [isNatList](x0) = x0 + 1, [isNatIList](x0) = x0, [mark](x0) = 0, [active](x0) = x0, [and](x0, x1) = x0 + 1, [tt] = 0 orientation: uTake2#(mark(X1),X2,X3,X4) = X2 + X3 + X4 >= X2 + X3 + X4 = uTake2#(X1,X2,X3,X4) uTake2#(ok(X1),ok(X2),ok(X3),ok(X4)) = X2 + X3 + X4 + 3 >= X2 + X3 + X4 = uTake2#(X1,X2,X3,X4) active(and(tt(),T)) = 1 >= 0 = mark(T) active(isNatIList(IL)) = IL >= 0 = mark(isNatList(IL)) active(isNat(0())) = 1 >= 0 = mark(tt()) active(isNat(s(N))) = N >= 0 = mark(isNat(N)) active(isNat(length(L))) = L >= 0 = mark(isNatList(L)) active(isNatIList(zeros())) = 1 >= 0 = mark(tt()) active(isNatIList(cons(N,IL))) = N >= 0 = mark(and(isNat(N),isNatIList(IL))) active(isNatList(nil())) = 1 >= 0 = mark(tt()) active(isNatList(cons(N,L))) = N + 1 >= 0 = mark(and(isNat(N),isNatList(L))) active(isNatList(take(N,IL))) = N + 1 >= 0 = mark(and(isNat(N),isNatIList(IL))) active(zeros()) = 1 >= 0 = mark(cons(0(),zeros())) active(take(0(),IL)) = 1 >= 0 = mark(uTake1(isNatIList(IL))) active(uTake1(tt())) = 0 >= 0 = mark(nil()) active(take(s(M),cons(N,IL))) = M >= 0 = mark(uTake2(and(isNat(M),and(isNat(N),isNatIList(IL))),M,N,IL)) active(uTake2(tt(),M,N,IL)) = 0 >= 0 = mark(cons(N,take(M,IL))) active(length(cons(N,L))) = N >= 0 = mark(uLength(and(isNat(N),isNatList(L)),L)) active(uLength(tt(),L)) = 0 >= 0 = mark(s(length(L))) active(and(X1,X2)) = X1 + 1 >= X1 + 1 = and(active(X1),X2) active(and(X1,X2)) = X1 + 1 >= X1 + 1 = and(X1,active(X2)) active(s(X)) = X >= X = s(active(X)) active(length(X)) = X >= X = length(active(X)) active(cons(X1,X2)) = X1 >= X1 = cons(active(X1),X2) active(take(X1,X2)) = X1 >= X1 = take(active(X1),X2) active(take(X1,X2)) = X1 >= X1 = take(X1,active(X2)) active(uTake1(X)) = X >= X = uTake1(active(X)) active(uTake2(X1,X2,X3,X4)) = X1 >= X1 = uTake2(active(X1),X2,X3,X4) active(uLength(X1,X2)) = X1 >= X1 = uLength(active(X1),X2) and(mark(X1),X2) = 1 >= 0 = mark(and(X1,X2)) and(X1,mark(X2)) = X1 + 1 >= 0 = mark(and(X1,X2)) s(mark(X)) = 0 >= 0 = mark(s(X)) length(mark(X)) = 0 >= 0 = mark(length(X)) cons(mark(X1),X2) = 0 >= 0 = mark(cons(X1,X2)) take(mark(X1),X2) = 0 >= 0 = mark(take(X1,X2)) take(X1,mark(X2)) = X1 >= 0 = mark(take(X1,X2)) uTake1(mark(X)) = 0 >= 0 = mark(uTake1(X)) uTake2(mark(X1),X2,X3,X4) = 0 >= 0 = mark(uTake2(X1,X2,X3,X4)) uLength(mark(X1),X2) = 0 >= 0 = mark(uLength(X1,X2)) proper(and(X1,X2)) = X1 + 2 >= X1 + 2 = and(proper(X1),proper(X2)) proper(tt()) = 1 >= 1 = ok(tt()) proper(isNatIList(X)) = X + 1 >= X + 1 = isNatIList(proper(X)) proper(isNatList(X)) = X + 2 >= X + 2 = isNatList(proper(X)) proper(isNat(X)) = X + 1 >= X + 1 = isNat(proper(X)) proper(0()) = 2 >= 2 = ok(0()) proper(s(X)) = X + 1 >= X + 1 = s(proper(X)) proper(length(X)) = X + 1 >= X + 1 = length(proper(X)) proper(zeros()) = 2 >= 2 = ok(zeros()) proper(cons(X1,X2)) = X1 + 1 >= X1 + 1 = cons(proper(X1),proper(X2)) proper(nil()) = 1 >= 1 = ok(nil()) proper(take(X1,X2)) = X1 + 1 >= X1 + 1 = take(proper(X1),proper(X2)) proper(uTake1(X)) = X + 1 >= X + 1 = uTake1(proper(X)) proper(uTake2(X1,X2,X3,X4)) = X1 + 1 >= X1 + 1 = uTake2(proper(X1),proper(X2),proper(X3),proper(X4)) proper(uLength(X1,X2)) = X1 + 1 >= X1 + 1 = uLength(proper(X1),proper(X2)) and(ok(X1),ok(X2)) = X1 + 2 >= X1 + 2 = ok(and(X1,X2)) isNatIList(ok(X)) = X + 1 >= X + 1 = ok(isNatIList(X)) isNatList(ok(X)) = X + 2 >= X + 2 = ok(isNatList(X)) isNat(ok(X)) = X + 1 >= X + 1 = ok(isNat(X)) s(ok(X)) = X + 1 >= X + 1 = ok(s(X)) length(ok(X)) = X + 1 >= X + 1 = ok(length(X)) cons(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(cons(X1,X2)) take(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(take(X1,X2)) uTake1(ok(X)) = X + 1 >= X + 1 = ok(uTake1(X)) uTake2(ok(X1),ok(X2),ok(X3),ok(X4)) = X1 + 1 >= X1 + 1 = ok(uTake2(X1,X2,X3,X4)) uLength(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(uLength(X1,X2)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) problem: DPs: uTake2#(mark(X1),X2,X3,X4) -> uTake2#(X1,X2,X3,X4) TRS: active(and(tt(),T)) -> mark(T) active(isNatIList(IL)) -> mark(isNatList(IL)) active(isNat(0())) -> mark(tt()) active(isNat(s(N))) -> mark(isNat(N)) active(isNat(length(L))) -> mark(isNatList(L)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(N,L))) -> mark(and(isNat(N),isNatList(L))) active(isNatList(take(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(zeros()) -> mark(cons(0(),zeros())) active(take(0(),IL)) -> mark(uTake1(isNatIList(IL))) active(uTake1(tt())) -> mark(nil()) active(take(s(M),cons(N,IL))) -> mark(uTake2(and(isNat(M),and(isNat(N),isNatIList(IL))),M,N,IL)) active(uTake2(tt(),M,N,IL)) -> mark(cons(N,take(M,IL))) active(length(cons(N,L))) -> mark(uLength(and(isNat(N),isNatList(L)),L)) active(uLength(tt(),L)) -> mark(s(length(L))) active(and(X1,X2)) -> and(active(X1),X2) active(and(X1,X2)) -> and(X1,active(X2)) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) active(uTake1(X)) -> uTake1(active(X)) active(uTake2(X1,X2,X3,X4)) -> uTake2(active(X1),X2,X3,X4) active(uLength(X1,X2)) -> uLength(active(X1),X2) and(mark(X1),X2) -> mark(and(X1,X2)) and(X1,mark(X2)) -> mark(and(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) uTake1(mark(X)) -> mark(uTake1(X)) uTake2(mark(X1),X2,X3,X4) -> mark(uTake2(X1,X2,X3,X4)) uLength(mark(X1),X2) -> mark(uLength(X1,X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(isNat(X)) -> isNat(proper(X)) proper(0()) -> ok(0()) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(nil()) -> ok(nil()) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) proper(uTake1(X)) -> uTake1(proper(X)) proper(uTake2(X1,X2,X3,X4)) -> uTake2(proper(X1),proper(X2),proper(X3),proper(X4)) proper(uLength(X1,X2)) -> uLength(proper(X1),proper(X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatIList(ok(X)) -> ok(isNatIList(X)) isNatList(ok(X)) -> ok(isNatList(X)) isNat(ok(X)) -> ok(isNat(X)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) uTake1(ok(X)) -> ok(uTake1(X)) uTake2(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(uTake2(X1,X2,X3,X4)) uLength(ok(X1),ok(X2)) -> ok(uLength(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dimension: 1 interpretation: [uTake2#](x0, x1, x2, x3) = x0 + 1, [top](x0) = 0, [ok](x0) = 0, [proper](x0) = x0, [uLength](x0, x1) = x0 + x1, [uTake2](x0, x1, x2, x3) = x0 + x1 + x2 + x3 + 1, [uTake1](x0) = x0, [take](x0, x1) = x0 + x1 + 1, [nil] = 0, [cons](x0, x1) = x0 + x1, [zeros] = 0, [length](x0) = x0, [s](x0) = x0, [isNat](x0) = 0, [0] = 0, [isNatList](x0) = 0, [isNatIList](x0) = 0, [mark](x0) = x0 + 1, [active](x0) = x0 + 1, [and](x0, x1) = x0 + x1, [tt] = 0 orientation: uTake2#(mark(X1),X2,X3,X4) = X1 + 2 >= X1 + 1 = uTake2#(X1,X2,X3,X4) active(and(tt(),T)) = T + 1 >= T + 1 = mark(T) active(isNatIList(IL)) = 1 >= 1 = mark(isNatList(IL)) active(isNat(0())) = 1 >= 1 = mark(tt()) active(isNat(s(N))) = 1 >= 1 = mark(isNat(N)) active(isNat(length(L))) = 1 >= 1 = mark(isNatList(L)) active(isNatIList(zeros())) = 1 >= 1 = mark(tt()) active(isNatIList(cons(N,IL))) = 1 >= 1 = mark(and(isNat(N),isNatIList(IL))) active(isNatList(nil())) = 1 >= 1 = mark(tt()) active(isNatList(cons(N,L))) = 1 >= 1 = mark(and(isNat(N),isNatList(L))) active(isNatList(take(N,IL))) = 1 >= 1 = mark(and(isNat(N),isNatIList(IL))) active(zeros()) = 1 >= 1 = mark(cons(0(),zeros())) active(take(0(),IL)) = IL + 2 >= 1 = mark(uTake1(isNatIList(IL))) active(uTake1(tt())) = 1 >= 1 = mark(nil()) active(take(s(M),cons(N,IL))) = IL + M + N + 2 >= IL + M + N + 2 = mark(uTake2(and(isNat(M),and(isNat(N),isNatIList(IL))),M,N,IL)) active(uTake2(tt(),M,N,IL)) = IL + M + N + 2 >= IL + M + N + 2 = mark(cons(N,take(M,IL))) active(length(cons(N,L))) = L + N + 1 >= L + 1 = mark(uLength(and(isNat(N),isNatList(L)),L)) active(uLength(tt(),L)) = L + 1 >= L + 1 = mark(s(length(L))) active(and(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = and(active(X1),X2) active(and(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = and(X1,active(X2)) active(s(X)) = X + 1 >= X + 1 = s(active(X)) active(length(X)) = X + 1 >= X + 1 = length(active(X)) active(cons(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = cons(active(X1),X2) 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(uTake1(X)) = X + 1 >= X + 1 = uTake1(active(X)) active(uTake2(X1,X2,X3,X4)) = X1 + X2 + X3 + X4 + 2 >= X1 + X2 + X3 + X4 + 2 = uTake2(active(X1),X2,X3,X4) active(uLength(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = uLength(active(X1),X2) and(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(and(X1,X2)) and(X1,mark(X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(and(X1,X2)) s(mark(X)) = X + 1 >= X + 1 = mark(s(X)) length(mark(X)) = X + 1 >= X + 1 = mark(length(X)) cons(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(cons(X1,X2)) 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)) uTake1(mark(X)) = X + 1 >= X + 1 = mark(uTake1(X)) uTake2(mark(X1),X2,X3,X4) = X1 + X2 + X3 + X4 + 2 >= X1 + X2 + X3 + X4 + 2 = mark(uTake2(X1,X2,X3,X4)) uLength(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(uLength(X1,X2)) proper(and(X1,X2)) = X1 + X2 >= X1 + X2 = and(proper(X1),proper(X2)) proper(tt()) = 0 >= 0 = ok(tt()) proper(isNatIList(X)) = 0 >= 0 = isNatIList(proper(X)) proper(isNatList(X)) = 0 >= 0 = isNatList(proper(X)) proper(isNat(X)) = 0 >= 0 = isNat(proper(X)) proper(0()) = 0 >= 0 = ok(0()) proper(s(X)) = X >= X = s(proper(X)) proper(length(X)) = X >= X = length(proper(X)) proper(zeros()) = 0 >= 0 = ok(zeros()) proper(cons(X1,X2)) = X1 + X2 >= X1 + X2 = cons(proper(X1),proper(X2)) proper(nil()) = 0 >= 0 = ok(nil()) proper(take(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = take(proper(X1),proper(X2)) proper(uTake1(X)) = X >= X = uTake1(proper(X)) proper(uTake2(X1,X2,X3,X4)) = X1 + X2 + X3 + X4 + 1 >= X1 + X2 + X3 + X4 + 1 = uTake2(proper(X1),proper(X2),proper(X3),proper(X4)) proper(uLength(X1,X2)) = X1 + X2 >= X1 + X2 = uLength(proper(X1),proper(X2)) and(ok(X1),ok(X2)) = 0 >= 0 = ok(and(X1,X2)) isNatIList(ok(X)) = 0 >= 0 = ok(isNatIList(X)) isNatList(ok(X)) = 0 >= 0 = ok(isNatList(X)) isNat(ok(X)) = 0 >= 0 = ok(isNat(X)) s(ok(X)) = 0 >= 0 = ok(s(X)) length(ok(X)) = 0 >= 0 = ok(length(X)) cons(ok(X1),ok(X2)) = 0 >= 0 = ok(cons(X1,X2)) take(ok(X1),ok(X2)) = 1 >= 0 = ok(take(X1,X2)) uTake1(ok(X)) = 0 >= 0 = ok(uTake1(X)) uTake2(ok(X1),ok(X2),ok(X3),ok(X4)) = 1 >= 0 = ok(uTake2(X1,X2,X3,X4)) uLength(ok(X1),ok(X2)) = 0 >= 0 = ok(uLength(X1,X2)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) problem: DPs: TRS: active(and(tt(),T)) -> mark(T) active(isNatIList(IL)) -> mark(isNatList(IL)) active(isNat(0())) -> mark(tt()) active(isNat(s(N))) -> mark(isNat(N)) active(isNat(length(L))) -> mark(isNatList(L)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(N,L))) -> mark(and(isNat(N),isNatList(L))) active(isNatList(take(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(zeros()) -> mark(cons(0(),zeros())) active(take(0(),IL)) -> mark(uTake1(isNatIList(IL))) active(uTake1(tt())) -> mark(nil()) active(take(s(M),cons(N,IL))) -> mark(uTake2(and(isNat(M),and(isNat(N),isNatIList(IL))),M,N,IL)) active(uTake2(tt(),M,N,IL)) -> mark(cons(N,take(M,IL))) active(length(cons(N,L))) -> mark(uLength(and(isNat(N),isNatList(L)),L)) active(uLength(tt(),L)) -> mark(s(length(L))) active(and(X1,X2)) -> and(active(X1),X2) active(and(X1,X2)) -> and(X1,active(X2)) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) active(uTake1(X)) -> uTake1(active(X)) active(uTake2(X1,X2,X3,X4)) -> uTake2(active(X1),X2,X3,X4) active(uLength(X1,X2)) -> uLength(active(X1),X2) and(mark(X1),X2) -> mark(and(X1,X2)) and(X1,mark(X2)) -> mark(and(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) uTake1(mark(X)) -> mark(uTake1(X)) uTake2(mark(X1),X2,X3,X4) -> mark(uTake2(X1,X2,X3,X4)) uLength(mark(X1),X2) -> mark(uLength(X1,X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(isNat(X)) -> isNat(proper(X)) proper(0()) -> ok(0()) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(nil()) -> ok(nil()) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) proper(uTake1(X)) -> uTake1(proper(X)) proper(uTake2(X1,X2,X3,X4)) -> uTake2(proper(X1),proper(X2),proper(X3),proper(X4)) proper(uLength(X1,X2)) -> uLength(proper(X1),proper(X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatIList(ok(X)) -> ok(isNatIList(X)) isNatList(ok(X)) -> ok(isNatList(X)) isNat(ok(X)) -> ok(isNat(X)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) uTake1(ok(X)) -> ok(uTake1(X)) uTake2(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(uTake2(X1,X2,X3,X4)) uLength(ok(X1),ok(X2)) -> ok(uLength(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed DPs: uTake1#(mark(X)) -> uTake1#(X) uTake1#(ok(X)) -> uTake1#(X) TRS: active(and(tt(),T)) -> mark(T) active(isNatIList(IL)) -> mark(isNatList(IL)) active(isNat(0())) -> mark(tt()) active(isNat(s(N))) -> mark(isNat(N)) active(isNat(length(L))) -> mark(isNatList(L)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(N,L))) -> mark(and(isNat(N),isNatList(L))) active(isNatList(take(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(zeros()) -> mark(cons(0(),zeros())) active(take(0(),IL)) -> mark(uTake1(isNatIList(IL))) active(uTake1(tt())) -> mark(nil()) active(take(s(M),cons(N,IL))) -> mark(uTake2(and(isNat(M),and(isNat(N),isNatIList(IL))),M,N,IL)) active(uTake2(tt(),M,N,IL)) -> mark(cons(N,take(M,IL))) active(length(cons(N,L))) -> mark(uLength(and(isNat(N),isNatList(L)),L)) active(uLength(tt(),L)) -> mark(s(length(L))) active(and(X1,X2)) -> and(active(X1),X2) active(and(X1,X2)) -> and(X1,active(X2)) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) active(uTake1(X)) -> uTake1(active(X)) active(uTake2(X1,X2,X3,X4)) -> uTake2(active(X1),X2,X3,X4) active(uLength(X1,X2)) -> uLength(active(X1),X2) and(mark(X1),X2) -> mark(and(X1,X2)) and(X1,mark(X2)) -> mark(and(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) uTake1(mark(X)) -> mark(uTake1(X)) uTake2(mark(X1),X2,X3,X4) -> mark(uTake2(X1,X2,X3,X4)) uLength(mark(X1),X2) -> mark(uLength(X1,X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(isNat(X)) -> isNat(proper(X)) proper(0()) -> ok(0()) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(nil()) -> ok(nil()) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) proper(uTake1(X)) -> uTake1(proper(X)) proper(uTake2(X1,X2,X3,X4)) -> uTake2(proper(X1),proper(X2),proper(X3),proper(X4)) proper(uLength(X1,X2)) -> uLength(proper(X1),proper(X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatIList(ok(X)) -> ok(isNatIList(X)) isNatList(ok(X)) -> ok(isNatList(X)) isNat(ok(X)) -> ok(isNat(X)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) uTake1(ok(X)) -> ok(uTake1(X)) uTake2(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(uTake2(X1,X2,X3,X4)) uLength(ok(X1),ok(X2)) -> ok(uLength(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dimension: 1 interpretation: [uTake1#](x0) = x0, [top](x0) = 0, [ok](x0) = x0 + 1, [proper](x0) = x0 + 1, [uLength](x0, x1) = x1, [uTake2](x0, x1, x2, x3) = x3 + 1, [uTake1](x0) = x0, [take](x0, x1) = x1 + 1, [nil] = 0, [cons](x0, x1) = x1, [zeros] = 0, [length](x0) = x0, [s](x0) = x0, [isNat](x0) = x0, [0] = 0, [isNatList](x0) = x0, [isNatIList](x0) = x0 + 1, [mark](x0) = x0, [active](x0) = x0, [and](x0, x1) = x1, [tt] = 0 orientation: uTake1#(mark(X)) = X >= X = uTake1#(X) uTake1#(ok(X)) = X + 1 >= X = uTake1#(X) active(and(tt(),T)) = T >= T = mark(T) active(isNatIList(IL)) = IL + 1 >= IL = mark(isNatList(IL)) active(isNat(0())) = 0 >= 0 = mark(tt()) active(isNat(s(N))) = N >= N = mark(isNat(N)) active(isNat(length(L))) = L >= L = mark(isNatList(L)) active(isNatIList(zeros())) = 1 >= 0 = mark(tt()) active(isNatIList(cons(N,IL))) = IL + 1 >= IL + 1 = mark(and(isNat(N),isNatIList(IL))) active(isNatList(nil())) = 0 >= 0 = mark(tt()) active(isNatList(cons(N,L))) = L >= L = mark(and(isNat(N),isNatList(L))) active(isNatList(take(N,IL))) = IL + 1 >= IL + 1 = mark(and(isNat(N),isNatIList(IL))) active(zeros()) = 0 >= 0 = mark(cons(0(),zeros())) active(take(0(),IL)) = IL + 1 >= IL + 1 = mark(uTake1(isNatIList(IL))) active(uTake1(tt())) = 0 >= 0 = mark(nil()) active(take(s(M),cons(N,IL))) = IL + 1 >= IL + 1 = mark(uTake2(and(isNat(M),and(isNat(N),isNatIList(IL))),M,N,IL)) active(uTake2(tt(),M,N,IL)) = IL + 1 >= IL + 1 = mark(cons(N,take(M,IL))) active(length(cons(N,L))) = L >= L = mark(uLength(and(isNat(N),isNatList(L)),L)) active(uLength(tt(),L)) = L >= L = mark(s(length(L))) active(and(X1,X2)) = X2 >= X2 = and(active(X1),X2) active(and(X1,X2)) = X2 >= X2 = and(X1,active(X2)) active(s(X)) = X >= X = s(active(X)) active(length(X)) = X >= X = length(active(X)) active(cons(X1,X2)) = X2 >= X2 = cons(active(X1),X2) 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(uTake1(X)) = X >= X = uTake1(active(X)) active(uTake2(X1,X2,X3,X4)) = X4 + 1 >= X4 + 1 = uTake2(active(X1),X2,X3,X4) active(uLength(X1,X2)) = X2 >= X2 = uLength(active(X1),X2) and(mark(X1),X2) = X2 >= X2 = mark(and(X1,X2)) and(X1,mark(X2)) = X2 >= X2 = mark(and(X1,X2)) s(mark(X)) = X >= X = mark(s(X)) length(mark(X)) = X >= X = mark(length(X)) cons(mark(X1),X2) = X2 >= X2 = mark(cons(X1,X2)) take(mark(X1),X2) = X2 + 1 >= X2 + 1 = mark(take(X1,X2)) take(X1,mark(X2)) = X2 + 1 >= X2 + 1 = mark(take(X1,X2)) uTake1(mark(X)) = X >= X = mark(uTake1(X)) uTake2(mark(X1),X2,X3,X4) = X4 + 1 >= X4 + 1 = mark(uTake2(X1,X2,X3,X4)) uLength(mark(X1),X2) = X2 >= X2 = mark(uLength(X1,X2)) proper(and(X1,X2)) = X2 + 1 >= X2 + 1 = and(proper(X1),proper(X2)) proper(tt()) = 1 >= 1 = ok(tt()) proper(isNatIList(X)) = X + 2 >= X + 2 = isNatIList(proper(X)) proper(isNatList(X)) = X + 1 >= X + 1 = isNatList(proper(X)) proper(isNat(X)) = X + 1 >= X + 1 = isNat(proper(X)) proper(0()) = 1 >= 1 = ok(0()) proper(s(X)) = X + 1 >= X + 1 = s(proper(X)) proper(length(X)) = X + 1 >= X + 1 = length(proper(X)) proper(zeros()) = 1 >= 1 = ok(zeros()) proper(cons(X1,X2)) = X2 + 1 >= X2 + 1 = cons(proper(X1),proper(X2)) proper(nil()) = 1 >= 1 = ok(nil()) proper(take(X1,X2)) = X2 + 2 >= X2 + 2 = take(proper(X1),proper(X2)) proper(uTake1(X)) = X + 1 >= X + 1 = uTake1(proper(X)) proper(uTake2(X1,X2,X3,X4)) = X4 + 2 >= X4 + 2 = uTake2(proper(X1),proper(X2),proper(X3),proper(X4)) proper(uLength(X1,X2)) = X2 + 1 >= X2 + 1 = uLength(proper(X1),proper(X2)) and(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(and(X1,X2)) isNatIList(ok(X)) = X + 2 >= X + 2 = ok(isNatIList(X)) isNatList(ok(X)) = X + 1 >= X + 1 = ok(isNatList(X)) isNat(ok(X)) = X + 1 >= X + 1 = ok(isNat(X)) s(ok(X)) = X + 1 >= X + 1 = ok(s(X)) length(ok(X)) = X + 1 >= X + 1 = ok(length(X)) cons(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(cons(X1,X2)) take(ok(X1),ok(X2)) = X2 + 2 >= X2 + 2 = ok(take(X1,X2)) uTake1(ok(X)) = X + 1 >= X + 1 = ok(uTake1(X)) uTake2(ok(X1),ok(X2),ok(X3),ok(X4)) = X4 + 2 >= X4 + 2 = ok(uTake2(X1,X2,X3,X4)) uLength(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(uLength(X1,X2)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) problem: DPs: uTake1#(mark(X)) -> uTake1#(X) TRS: active(and(tt(),T)) -> mark(T) active(isNatIList(IL)) -> mark(isNatList(IL)) active(isNat(0())) -> mark(tt()) active(isNat(s(N))) -> mark(isNat(N)) active(isNat(length(L))) -> mark(isNatList(L)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(N,L))) -> mark(and(isNat(N),isNatList(L))) active(isNatList(take(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(zeros()) -> mark(cons(0(),zeros())) active(take(0(),IL)) -> mark(uTake1(isNatIList(IL))) active(uTake1(tt())) -> mark(nil()) active(take(s(M),cons(N,IL))) -> mark(uTake2(and(isNat(M),and(isNat(N),isNatIList(IL))),M,N,IL)) active(uTake2(tt(),M,N,IL)) -> mark(cons(N,take(M,IL))) active(length(cons(N,L))) -> mark(uLength(and(isNat(N),isNatList(L)),L)) active(uLength(tt(),L)) -> mark(s(length(L))) active(and(X1,X2)) -> and(active(X1),X2) active(and(X1,X2)) -> and(X1,active(X2)) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) active(uTake1(X)) -> uTake1(active(X)) active(uTake2(X1,X2,X3,X4)) -> uTake2(active(X1),X2,X3,X4) active(uLength(X1,X2)) -> uLength(active(X1),X2) and(mark(X1),X2) -> mark(and(X1,X2)) and(X1,mark(X2)) -> mark(and(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) uTake1(mark(X)) -> mark(uTake1(X)) uTake2(mark(X1),X2,X3,X4) -> mark(uTake2(X1,X2,X3,X4)) uLength(mark(X1),X2) -> mark(uLength(X1,X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(isNat(X)) -> isNat(proper(X)) proper(0()) -> ok(0()) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(nil()) -> ok(nil()) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) proper(uTake1(X)) -> uTake1(proper(X)) proper(uTake2(X1,X2,X3,X4)) -> uTake2(proper(X1),proper(X2),proper(X3),proper(X4)) proper(uLength(X1,X2)) -> uLength(proper(X1),proper(X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatIList(ok(X)) -> ok(isNatIList(X)) isNatList(ok(X)) -> ok(isNatList(X)) isNat(ok(X)) -> ok(isNat(X)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) uTake1(ok(X)) -> ok(uTake1(X)) uTake2(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(uTake2(X1,X2,X3,X4)) uLength(ok(X1),ok(X2)) -> ok(uLength(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dimension: 1 interpretation: [uTake1#](x0) = x0, [top](x0) = 1, [ok](x0) = 0, [proper](x0) = 0, [uLength](x0, x1) = x0 + x1, [uTake2](x0, x1, x2, x3) = x0 + x1 + x2 + x3, [uTake1](x0) = x0, [take](x0, x1) = x0 + x1, [nil] = 0, [cons](x0, x1) = x0 + x1, [zeros] = 1, [length](x0) = x0, [s](x0) = x0, [isNat](x0) = 0, [0] = 0, [isNatList](x0) = 0, [isNatIList](x0) = 0, [mark](x0) = x0 + 1, [active](x0) = x0 + 1, [and](x0, x1) = x0 + x1, [tt] = 0 orientation: uTake1#(mark(X)) = X + 1 >= X = uTake1#(X) active(and(tt(),T)) = T + 1 >= T + 1 = mark(T) active(isNatIList(IL)) = 1 >= 1 = mark(isNatList(IL)) active(isNat(0())) = 1 >= 1 = mark(tt()) active(isNat(s(N))) = 1 >= 1 = mark(isNat(N)) active(isNat(length(L))) = 1 >= 1 = mark(isNatList(L)) active(isNatIList(zeros())) = 1 >= 1 = mark(tt()) active(isNatIList(cons(N,IL))) = 1 >= 1 = mark(and(isNat(N),isNatIList(IL))) active(isNatList(nil())) = 1 >= 1 = mark(tt()) active(isNatList(cons(N,L))) = 1 >= 1 = mark(and(isNat(N),isNatList(L))) active(isNatList(take(N,IL))) = 1 >= 1 = mark(and(isNat(N),isNatIList(IL))) active(zeros()) = 2 >= 2 = mark(cons(0(),zeros())) active(take(0(),IL)) = IL + 1 >= 1 = mark(uTake1(isNatIList(IL))) active(uTake1(tt())) = 1 >= 1 = mark(nil()) active(take(s(M),cons(N,IL))) = IL + M + N + 1 >= IL + M + N + 1 = mark(uTake2(and(isNat(M),and(isNat(N),isNatIList(IL))),M,N,IL)) active(uTake2(tt(),M,N,IL)) = IL + M + N + 1 >= IL + M + N + 1 = mark(cons(N,take(M,IL))) active(length(cons(N,L))) = L + N + 1 >= L + 1 = mark(uLength(and(isNat(N),isNatList(L)),L)) active(uLength(tt(),L)) = L + 1 >= L + 1 = mark(s(length(L))) active(and(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = and(active(X1),X2) active(and(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = and(X1,active(X2)) active(s(X)) = X + 1 >= X + 1 = s(active(X)) active(length(X)) = X + 1 >= X + 1 = length(active(X)) active(cons(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = cons(active(X1),X2) 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(uTake1(X)) = X + 1 >= X + 1 = uTake1(active(X)) active(uTake2(X1,X2,X3,X4)) = X1 + X2 + X3 + X4 + 1 >= X1 + X2 + X3 + X4 + 1 = uTake2(active(X1),X2,X3,X4) active(uLength(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = uLength(active(X1),X2) and(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(and(X1,X2)) and(X1,mark(X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(and(X1,X2)) s(mark(X)) = X + 1 >= X + 1 = mark(s(X)) length(mark(X)) = X + 1 >= X + 1 = mark(length(X)) cons(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(cons(X1,X2)) 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)) uTake1(mark(X)) = X + 1 >= X + 1 = mark(uTake1(X)) uTake2(mark(X1),X2,X3,X4) = X1 + X2 + X3 + X4 + 1 >= X1 + X2 + X3 + X4 + 1 = mark(uTake2(X1,X2,X3,X4)) uLength(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(uLength(X1,X2)) proper(and(X1,X2)) = 0 >= 0 = and(proper(X1),proper(X2)) proper(tt()) = 0 >= 0 = ok(tt()) proper(isNatIList(X)) = 0 >= 0 = isNatIList(proper(X)) proper(isNatList(X)) = 0 >= 0 = isNatList(proper(X)) proper(isNat(X)) = 0 >= 0 = isNat(proper(X)) proper(0()) = 0 >= 0 = ok(0()) proper(s(X)) = 0 >= 0 = s(proper(X)) proper(length(X)) = 0 >= 0 = length(proper(X)) proper(zeros()) = 0 >= 0 = ok(zeros()) proper(cons(X1,X2)) = 0 >= 0 = cons(proper(X1),proper(X2)) proper(nil()) = 0 >= 0 = ok(nil()) proper(take(X1,X2)) = 0 >= 0 = take(proper(X1),proper(X2)) proper(uTake1(X)) = 0 >= 0 = uTake1(proper(X)) proper(uTake2(X1,X2,X3,X4)) = 0 >= 0 = uTake2(proper(X1),proper(X2),proper(X3),proper(X4)) proper(uLength(X1,X2)) = 0 >= 0 = uLength(proper(X1),proper(X2)) and(ok(X1),ok(X2)) = 0 >= 0 = ok(and(X1,X2)) isNatIList(ok(X)) = 0 >= 0 = ok(isNatIList(X)) isNatList(ok(X)) = 0 >= 0 = ok(isNatList(X)) isNat(ok(X)) = 0 >= 0 = ok(isNat(X)) s(ok(X)) = 0 >= 0 = ok(s(X)) length(ok(X)) = 0 >= 0 = ok(length(X)) cons(ok(X1),ok(X2)) = 0 >= 0 = ok(cons(X1,X2)) take(ok(X1),ok(X2)) = 0 >= 0 = ok(take(X1,X2)) uTake1(ok(X)) = 0 >= 0 = ok(uTake1(X)) uTake2(ok(X1),ok(X2),ok(X3),ok(X4)) = 0 >= 0 = ok(uTake2(X1,X2,X3,X4)) uLength(ok(X1),ok(X2)) = 0 >= 0 = ok(uLength(X1,X2)) top(mark(X)) = 1 >= 1 = top(proper(X)) top(ok(X)) = 1 >= 1 = top(active(X)) problem: DPs: TRS: active(and(tt(),T)) -> mark(T) active(isNatIList(IL)) -> mark(isNatList(IL)) active(isNat(0())) -> mark(tt()) active(isNat(s(N))) -> mark(isNat(N)) active(isNat(length(L))) -> mark(isNatList(L)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(N,L))) -> mark(and(isNat(N),isNatList(L))) active(isNatList(take(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(zeros()) -> mark(cons(0(),zeros())) active(take(0(),IL)) -> mark(uTake1(isNatIList(IL))) active(uTake1(tt())) -> mark(nil()) active(take(s(M),cons(N,IL))) -> mark(uTake2(and(isNat(M),and(isNat(N),isNatIList(IL))),M,N,IL)) active(uTake2(tt(),M,N,IL)) -> mark(cons(N,take(M,IL))) active(length(cons(N,L))) -> mark(uLength(and(isNat(N),isNatList(L)),L)) active(uLength(tt(),L)) -> mark(s(length(L))) active(and(X1,X2)) -> and(active(X1),X2) active(and(X1,X2)) -> and(X1,active(X2)) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) active(uTake1(X)) -> uTake1(active(X)) active(uTake2(X1,X2,X3,X4)) -> uTake2(active(X1),X2,X3,X4) active(uLength(X1,X2)) -> uLength(active(X1),X2) and(mark(X1),X2) -> mark(and(X1,X2)) and(X1,mark(X2)) -> mark(and(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) uTake1(mark(X)) -> mark(uTake1(X)) uTake2(mark(X1),X2,X3,X4) -> mark(uTake2(X1,X2,X3,X4)) uLength(mark(X1),X2) -> mark(uLength(X1,X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(isNat(X)) -> isNat(proper(X)) proper(0()) -> ok(0()) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(nil()) -> ok(nil()) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) proper(uTake1(X)) -> uTake1(proper(X)) proper(uTake2(X1,X2,X3,X4)) -> uTake2(proper(X1),proper(X2),proper(X3),proper(X4)) proper(uLength(X1,X2)) -> uLength(proper(X1),proper(X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatIList(ok(X)) -> ok(isNatIList(X)) isNatList(ok(X)) -> ok(isNatList(X)) isNat(ok(X)) -> ok(isNat(X)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) uTake1(ok(X)) -> ok(uTake1(X)) uTake2(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(uTake2(X1,X2,X3,X4)) uLength(ok(X1),ok(X2)) -> ok(uLength(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed DPs: take#(mark(X1),X2) -> take#(X1,X2) take#(X1,mark(X2)) -> take#(X1,X2) take#(ok(X1),ok(X2)) -> take#(X1,X2) TRS: active(and(tt(),T)) -> mark(T) active(isNatIList(IL)) -> mark(isNatList(IL)) active(isNat(0())) -> mark(tt()) active(isNat(s(N))) -> mark(isNat(N)) active(isNat(length(L))) -> mark(isNatList(L)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(N,L))) -> mark(and(isNat(N),isNatList(L))) active(isNatList(take(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(zeros()) -> mark(cons(0(),zeros())) active(take(0(),IL)) -> mark(uTake1(isNatIList(IL))) active(uTake1(tt())) -> mark(nil()) active(take(s(M),cons(N,IL))) -> mark(uTake2(and(isNat(M),and(isNat(N),isNatIList(IL))),M,N,IL)) active(uTake2(tt(),M,N,IL)) -> mark(cons(N,take(M,IL))) active(length(cons(N,L))) -> mark(uLength(and(isNat(N),isNatList(L)),L)) active(uLength(tt(),L)) -> mark(s(length(L))) active(and(X1,X2)) -> and(active(X1),X2) active(and(X1,X2)) -> and(X1,active(X2)) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) active(uTake1(X)) -> uTake1(active(X)) active(uTake2(X1,X2,X3,X4)) -> uTake2(active(X1),X2,X3,X4) active(uLength(X1,X2)) -> uLength(active(X1),X2) and(mark(X1),X2) -> mark(and(X1,X2)) and(X1,mark(X2)) -> mark(and(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) uTake1(mark(X)) -> mark(uTake1(X)) uTake2(mark(X1),X2,X3,X4) -> mark(uTake2(X1,X2,X3,X4)) uLength(mark(X1),X2) -> mark(uLength(X1,X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(isNat(X)) -> isNat(proper(X)) proper(0()) -> ok(0()) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(nil()) -> ok(nil()) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) proper(uTake1(X)) -> uTake1(proper(X)) proper(uTake2(X1,X2,X3,X4)) -> uTake2(proper(X1),proper(X2),proper(X3),proper(X4)) proper(uLength(X1,X2)) -> uLength(proper(X1),proper(X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatIList(ok(X)) -> ok(isNatIList(X)) isNatList(ok(X)) -> ok(isNatList(X)) isNat(ok(X)) -> ok(isNat(X)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) uTake1(ok(X)) -> ok(uTake1(X)) uTake2(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(uTake2(X1,X2,X3,X4)) uLength(ok(X1),ok(X2)) -> ok(uLength(X1,X2)) 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) = x0 + 1, [proper](x0) = 1, [uLength](x0, x1) = x1, [uTake2](x0, x1, x2, x3) = x3, [uTake1](x0) = x0, [take](x0, x1) = x1, [nil] = 0, [cons](x0, x1) = x1, [zeros] = 0, [length](x0) = x0, [s](x0) = x0, [isNat](x0) = x0, [0] = 0, [isNatList](x0) = x0, [isNatIList](x0) = x0, [mark](x0) = x0, [active](x0) = x0, [and](x0, x1) = x1, [tt] = 0 orientation: take#(mark(X1),X2) = X2 + 1 >= X2 + 1 = take#(X1,X2) take#(X1,mark(X2)) = X2 + 1 >= X2 + 1 = take#(X1,X2) take#(ok(X1),ok(X2)) = X2 + 2 >= X2 + 1 = take#(X1,X2) active(and(tt(),T)) = T >= T = mark(T) active(isNatIList(IL)) = IL >= IL = mark(isNatList(IL)) active(isNat(0())) = 0 >= 0 = mark(tt()) active(isNat(s(N))) = N >= N = mark(isNat(N)) active(isNat(length(L))) = L >= L = mark(isNatList(L)) active(isNatIList(zeros())) = 0 >= 0 = mark(tt()) active(isNatIList(cons(N,IL))) = IL >= IL = mark(and(isNat(N),isNatIList(IL))) active(isNatList(nil())) = 0 >= 0 = mark(tt()) active(isNatList(cons(N,L))) = L >= L = mark(and(isNat(N),isNatList(L))) active(isNatList(take(N,IL))) = IL >= IL = mark(and(isNat(N),isNatIList(IL))) active(zeros()) = 0 >= 0 = mark(cons(0(),zeros())) active(take(0(),IL)) = IL >= IL = mark(uTake1(isNatIList(IL))) active(uTake1(tt())) = 0 >= 0 = mark(nil()) active(take(s(M),cons(N,IL))) = IL >= IL = mark(uTake2(and(isNat(M),and(isNat(N),isNatIList(IL))),M,N,IL)) active(uTake2(tt(),M,N,IL)) = IL >= IL = mark(cons(N,take(M,IL))) active(length(cons(N,L))) = L >= L = mark(uLength(and(isNat(N),isNatList(L)),L)) active(uLength(tt(),L)) = L >= L = mark(s(length(L))) active(and(X1,X2)) = X2 >= X2 = and(active(X1),X2) active(and(X1,X2)) = X2 >= X2 = and(X1,active(X2)) active(s(X)) = X >= X = s(active(X)) active(length(X)) = X >= X = length(active(X)) active(cons(X1,X2)) = X2 >= X2 = cons(active(X1),X2) active(take(X1,X2)) = X2 >= X2 = take(active(X1),X2) active(take(X1,X2)) = X2 >= X2 = take(X1,active(X2)) active(uTake1(X)) = X >= X = uTake1(active(X)) active(uTake2(X1,X2,X3,X4)) = X4 >= X4 = uTake2(active(X1),X2,X3,X4) active(uLength(X1,X2)) = X2 >= X2 = uLength(active(X1),X2) and(mark(X1),X2) = X2 >= X2 = mark(and(X1,X2)) and(X1,mark(X2)) = X2 >= X2 = mark(and(X1,X2)) s(mark(X)) = X >= X = mark(s(X)) length(mark(X)) = X >= X = mark(length(X)) cons(mark(X1),X2) = X2 >= X2 = mark(cons(X1,X2)) take(mark(X1),X2) = X2 >= X2 = mark(take(X1,X2)) take(X1,mark(X2)) = X2 >= X2 = mark(take(X1,X2)) uTake1(mark(X)) = X >= X = mark(uTake1(X)) uTake2(mark(X1),X2,X3,X4) = X4 >= X4 = mark(uTake2(X1,X2,X3,X4)) uLength(mark(X1),X2) = X2 >= X2 = mark(uLength(X1,X2)) proper(and(X1,X2)) = 1 >= 1 = and(proper(X1),proper(X2)) proper(tt()) = 1 >= 1 = ok(tt()) proper(isNatIList(X)) = 1 >= 1 = isNatIList(proper(X)) proper(isNatList(X)) = 1 >= 1 = isNatList(proper(X)) proper(isNat(X)) = 1 >= 1 = isNat(proper(X)) proper(0()) = 1 >= 1 = ok(0()) proper(s(X)) = 1 >= 1 = s(proper(X)) proper(length(X)) = 1 >= 1 = length(proper(X)) proper(zeros()) = 1 >= 1 = ok(zeros()) proper(cons(X1,X2)) = 1 >= 1 = cons(proper(X1),proper(X2)) proper(nil()) = 1 >= 1 = ok(nil()) proper(take(X1,X2)) = 1 >= 1 = take(proper(X1),proper(X2)) proper(uTake1(X)) = 1 >= 1 = uTake1(proper(X)) proper(uTake2(X1,X2,X3,X4)) = 1 >= 1 = uTake2(proper(X1),proper(X2),proper(X3),proper(X4)) proper(uLength(X1,X2)) = 1 >= 1 = uLength(proper(X1),proper(X2)) and(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(and(X1,X2)) isNatIList(ok(X)) = X + 1 >= X + 1 = ok(isNatIList(X)) isNatList(ok(X)) = X + 1 >= X + 1 = ok(isNatList(X)) isNat(ok(X)) = X + 1 >= X + 1 = ok(isNat(X)) s(ok(X)) = X + 1 >= X + 1 = ok(s(X)) length(ok(X)) = X + 1 >= X + 1 = ok(length(X)) cons(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(cons(X1,X2)) take(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(take(X1,X2)) uTake1(ok(X)) = X + 1 >= X + 1 = ok(uTake1(X)) uTake2(ok(X1),ok(X2),ok(X3),ok(X4)) = X4 + 1 >= X4 + 1 = ok(uTake2(X1,X2,X3,X4)) uLength(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(uLength(X1,X2)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) problem: DPs: take#(mark(X1),X2) -> take#(X1,X2) take#(X1,mark(X2)) -> take#(X1,X2) TRS: active(and(tt(),T)) -> mark(T) active(isNatIList(IL)) -> mark(isNatList(IL)) active(isNat(0())) -> mark(tt()) active(isNat(s(N))) -> mark(isNat(N)) active(isNat(length(L))) -> mark(isNatList(L)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(N,L))) -> mark(and(isNat(N),isNatList(L))) active(isNatList(take(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(zeros()) -> mark(cons(0(),zeros())) active(take(0(),IL)) -> mark(uTake1(isNatIList(IL))) active(uTake1(tt())) -> mark(nil()) active(take(s(M),cons(N,IL))) -> mark(uTake2(and(isNat(M),and(isNat(N),isNatIList(IL))),M,N,IL)) active(uTake2(tt(),M,N,IL)) -> mark(cons(N,take(M,IL))) active(length(cons(N,L))) -> mark(uLength(and(isNat(N),isNatList(L)),L)) active(uLength(tt(),L)) -> mark(s(length(L))) active(and(X1,X2)) -> and(active(X1),X2) active(and(X1,X2)) -> and(X1,active(X2)) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) active(uTake1(X)) -> uTake1(active(X)) active(uTake2(X1,X2,X3,X4)) -> uTake2(active(X1),X2,X3,X4) active(uLength(X1,X2)) -> uLength(active(X1),X2) and(mark(X1),X2) -> mark(and(X1,X2)) and(X1,mark(X2)) -> mark(and(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) uTake1(mark(X)) -> mark(uTake1(X)) uTake2(mark(X1),X2,X3,X4) -> mark(uTake2(X1,X2,X3,X4)) uLength(mark(X1),X2) -> mark(uLength(X1,X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(isNat(X)) -> isNat(proper(X)) proper(0()) -> ok(0()) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(nil()) -> ok(nil()) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) proper(uTake1(X)) -> uTake1(proper(X)) proper(uTake2(X1,X2,X3,X4)) -> uTake2(proper(X1),proper(X2),proper(X3),proper(X4)) proper(uLength(X1,X2)) -> uLength(proper(X1),proper(X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatIList(ok(X)) -> ok(isNatIList(X)) isNatList(ok(X)) -> ok(isNatList(X)) isNat(ok(X)) -> ok(isNat(X)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) uTake1(ok(X)) -> ok(uTake1(X)) uTake2(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(uTake2(X1,X2,X3,X4)) uLength(ok(X1),ok(X2)) -> ok(uLength(X1,X2)) 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) = 0, [uLength](x0, x1) = x0 + x1, [uTake2](x0, x1, x2, x3) = x0 + x1 + x2 + x3, [uTake1](x0) = x0, [take](x0, x1) = x0 + x1, [nil] = 0, [cons](x0, x1) = x0 + x1, [zeros] = 0, [length](x0) = x0, [s](x0) = x0, [isNat](x0) = 0, [0] = 0, [isNatList](x0) = 0, [isNatIList](x0) = 0, [mark](x0) = x0 + 1, [active](x0) = x0 + 1, [and](x0, x1) = x0 + x1, [tt] = 0 orientation: take#(mark(X1),X2) = X1 + 1 >= X1 = take#(X1,X2) take#(X1,mark(X2)) = X1 >= X1 = take#(X1,X2) active(and(tt(),T)) = T + 1 >= T + 1 = mark(T) active(isNatIList(IL)) = 1 >= 1 = mark(isNatList(IL)) active(isNat(0())) = 1 >= 1 = mark(tt()) active(isNat(s(N))) = 1 >= 1 = mark(isNat(N)) active(isNat(length(L))) = 1 >= 1 = mark(isNatList(L)) active(isNatIList(zeros())) = 1 >= 1 = mark(tt()) active(isNatIList(cons(N,IL))) = 1 >= 1 = mark(and(isNat(N),isNatIList(IL))) active(isNatList(nil())) = 1 >= 1 = mark(tt()) active(isNatList(cons(N,L))) = 1 >= 1 = mark(and(isNat(N),isNatList(L))) active(isNatList(take(N,IL))) = 1 >= 1 = mark(and(isNat(N),isNatIList(IL))) active(zeros()) = 1 >= 1 = mark(cons(0(),zeros())) active(take(0(),IL)) = IL + 1 >= 1 = mark(uTake1(isNatIList(IL))) active(uTake1(tt())) = 1 >= 1 = mark(nil()) active(take(s(M),cons(N,IL))) = IL + M + N + 1 >= IL + M + N + 1 = mark(uTake2(and(isNat(M),and(isNat(N),isNatIList(IL))),M,N,IL)) active(uTake2(tt(),M,N,IL)) = IL + M + N + 1 >= IL + M + N + 1 = mark(cons(N,take(M,IL))) active(length(cons(N,L))) = L + N + 1 >= L + 1 = mark(uLength(and(isNat(N),isNatList(L)),L)) active(uLength(tt(),L)) = L + 1 >= L + 1 = mark(s(length(L))) active(and(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = and(active(X1),X2) active(and(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = and(X1,active(X2)) active(s(X)) = X + 1 >= X + 1 = s(active(X)) active(length(X)) = X + 1 >= X + 1 = length(active(X)) active(cons(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = cons(active(X1),X2) 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(uTake1(X)) = X + 1 >= X + 1 = uTake1(active(X)) active(uTake2(X1,X2,X3,X4)) = X1 + X2 + X3 + X4 + 1 >= X1 + X2 + X3 + X4 + 1 = uTake2(active(X1),X2,X3,X4) active(uLength(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = uLength(active(X1),X2) and(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(and(X1,X2)) and(X1,mark(X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(and(X1,X2)) s(mark(X)) = X + 1 >= X + 1 = mark(s(X)) length(mark(X)) = X + 1 >= X + 1 = mark(length(X)) cons(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(cons(X1,X2)) 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)) uTake1(mark(X)) = X + 1 >= X + 1 = mark(uTake1(X)) uTake2(mark(X1),X2,X3,X4) = X1 + X2 + X3 + X4 + 1 >= X1 + X2 + X3 + X4 + 1 = mark(uTake2(X1,X2,X3,X4)) uLength(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(uLength(X1,X2)) proper(and(X1,X2)) = 0 >= 0 = and(proper(X1),proper(X2)) proper(tt()) = 0 >= 0 = ok(tt()) proper(isNatIList(X)) = 0 >= 0 = isNatIList(proper(X)) proper(isNatList(X)) = 0 >= 0 = isNatList(proper(X)) proper(isNat(X)) = 0 >= 0 = isNat(proper(X)) proper(0()) = 0 >= 0 = ok(0()) proper(s(X)) = 0 >= 0 = s(proper(X)) proper(length(X)) = 0 >= 0 = length(proper(X)) proper(zeros()) = 0 >= 0 = ok(zeros()) proper(cons(X1,X2)) = 0 >= 0 = cons(proper(X1),proper(X2)) proper(nil()) = 0 >= 0 = ok(nil()) proper(take(X1,X2)) = 0 >= 0 = take(proper(X1),proper(X2)) proper(uTake1(X)) = 0 >= 0 = uTake1(proper(X)) proper(uTake2(X1,X2,X3,X4)) = 0 >= 0 = uTake2(proper(X1),proper(X2),proper(X3),proper(X4)) proper(uLength(X1,X2)) = 0 >= 0 = uLength(proper(X1),proper(X2)) and(ok(X1),ok(X2)) = 0 >= 0 = ok(and(X1,X2)) isNatIList(ok(X)) = 0 >= 0 = ok(isNatIList(X)) isNatList(ok(X)) = 0 >= 0 = ok(isNatList(X)) isNat(ok(X)) = 0 >= 0 = ok(isNat(X)) s(ok(X)) = 0 >= 0 = ok(s(X)) length(ok(X)) = 0 >= 0 = ok(length(X)) cons(ok(X1),ok(X2)) = 0 >= 0 = ok(cons(X1,X2)) take(ok(X1),ok(X2)) = 0 >= 0 = ok(take(X1,X2)) uTake1(ok(X)) = 0 >= 0 = ok(uTake1(X)) uTake2(ok(X1),ok(X2),ok(X3),ok(X4)) = 0 >= 0 = ok(uTake2(X1,X2,X3,X4)) uLength(ok(X1),ok(X2)) = 0 >= 0 = ok(uLength(X1,X2)) 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(and(tt(),T)) -> mark(T) active(isNatIList(IL)) -> mark(isNatList(IL)) active(isNat(0())) -> mark(tt()) active(isNat(s(N))) -> mark(isNat(N)) active(isNat(length(L))) -> mark(isNatList(L)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(N,L))) -> mark(and(isNat(N),isNatList(L))) active(isNatList(take(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(zeros()) -> mark(cons(0(),zeros())) active(take(0(),IL)) -> mark(uTake1(isNatIList(IL))) active(uTake1(tt())) -> mark(nil()) active(take(s(M),cons(N,IL))) -> mark(uTake2(and(isNat(M),and(isNat(N),isNatIList(IL))),M,N,IL)) active(uTake2(tt(),M,N,IL)) -> mark(cons(N,take(M,IL))) active(length(cons(N,L))) -> mark(uLength(and(isNat(N),isNatList(L)),L)) active(uLength(tt(),L)) -> mark(s(length(L))) active(and(X1,X2)) -> and(active(X1),X2) active(and(X1,X2)) -> and(X1,active(X2)) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) active(uTake1(X)) -> uTake1(active(X)) active(uTake2(X1,X2,X3,X4)) -> uTake2(active(X1),X2,X3,X4) active(uLength(X1,X2)) -> uLength(active(X1),X2) and(mark(X1),X2) -> mark(and(X1,X2)) and(X1,mark(X2)) -> mark(and(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) uTake1(mark(X)) -> mark(uTake1(X)) uTake2(mark(X1),X2,X3,X4) -> mark(uTake2(X1,X2,X3,X4)) uLength(mark(X1),X2) -> mark(uLength(X1,X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(isNat(X)) -> isNat(proper(X)) proper(0()) -> ok(0()) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(nil()) -> ok(nil()) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) proper(uTake1(X)) -> uTake1(proper(X)) proper(uTake2(X1,X2,X3,X4)) -> uTake2(proper(X1),proper(X2),proper(X3),proper(X4)) proper(uLength(X1,X2)) -> uLength(proper(X1),proper(X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatIList(ok(X)) -> ok(isNatIList(X)) isNatList(ok(X)) -> ok(isNatList(X)) isNat(ok(X)) -> ok(isNat(X)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) uTake1(ok(X)) -> ok(uTake1(X)) uTake2(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(uTake2(X1,X2,X3,X4)) uLength(ok(X1),ok(X2)) -> ok(uLength(X1,X2)) 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, [uLength](x0, x1) = x0 + x1, [uTake2](x0, x1, x2, x3) = x0 + x1 + x2 + x3, [uTake1](x0) = x0, [take](x0, x1) = x0 + x1, [nil] = 0, [cons](x0, x1) = x0 + x1, [zeros] = 1, [length](x0) = x0, [s](x0) = x0, [isNat](x0) = 0, [0] = 0, [isNatList](x0) = 0, [isNatIList](x0) = 0, [mark](x0) = x0 + 1, [active](x0) = x0 + 1, [and](x0, x1) = x0 + x1, [tt] = 0 orientation: take#(X1,mark(X2)) = X2 + 2 >= X2 + 1 = take#(X1,X2) active(and(tt(),T)) = T + 1 >= T + 1 = mark(T) active(isNatIList(IL)) = 1 >= 1 = mark(isNatList(IL)) active(isNat(0())) = 1 >= 1 = mark(tt()) active(isNat(s(N))) = 1 >= 1 = mark(isNat(N)) active(isNat(length(L))) = 1 >= 1 = mark(isNatList(L)) active(isNatIList(zeros())) = 1 >= 1 = mark(tt()) active(isNatIList(cons(N,IL))) = 1 >= 1 = mark(and(isNat(N),isNatIList(IL))) active(isNatList(nil())) = 1 >= 1 = mark(tt()) active(isNatList(cons(N,L))) = 1 >= 1 = mark(and(isNat(N),isNatList(L))) active(isNatList(take(N,IL))) = 1 >= 1 = mark(and(isNat(N),isNatIList(IL))) active(zeros()) = 2 >= 2 = mark(cons(0(),zeros())) active(take(0(),IL)) = IL + 1 >= 1 = mark(uTake1(isNatIList(IL))) active(uTake1(tt())) = 1 >= 1 = mark(nil()) active(take(s(M),cons(N,IL))) = IL + M + N + 1 >= IL + M + N + 1 = mark(uTake2(and(isNat(M),and(isNat(N),isNatIList(IL))),M,N,IL)) active(uTake2(tt(),M,N,IL)) = IL + M + N + 1 >= IL + M + N + 1 = mark(cons(N,take(M,IL))) active(length(cons(N,L))) = L + N + 1 >= L + 1 = mark(uLength(and(isNat(N),isNatList(L)),L)) active(uLength(tt(),L)) = L + 1 >= L + 1 = mark(s(length(L))) active(and(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = and(active(X1),X2) active(and(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = and(X1,active(X2)) active(s(X)) = X + 1 >= X + 1 = s(active(X)) active(length(X)) = X + 1 >= X + 1 = length(active(X)) active(cons(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = cons(active(X1),X2) 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(uTake1(X)) = X + 1 >= X + 1 = uTake1(active(X)) active(uTake2(X1,X2,X3,X4)) = X1 + X2 + X3 + X4 + 1 >= X1 + X2 + X3 + X4 + 1 = uTake2(active(X1),X2,X3,X4) active(uLength(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = uLength(active(X1),X2) and(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(and(X1,X2)) and(X1,mark(X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(and(X1,X2)) s(mark(X)) = X + 1 >= X + 1 = mark(s(X)) length(mark(X)) = X + 1 >= X + 1 = mark(length(X)) cons(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(cons(X1,X2)) 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)) uTake1(mark(X)) = X + 1 >= X + 1 = mark(uTake1(X)) uTake2(mark(X1),X2,X3,X4) = X1 + X2 + X3 + X4 + 1 >= X1 + X2 + X3 + X4 + 1 = mark(uTake2(X1,X2,X3,X4)) uLength(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(uLength(X1,X2)) proper(and(X1,X2)) = 0 >= 0 = and(proper(X1),proper(X2)) proper(tt()) = 0 >= 0 = ok(tt()) proper(isNatIList(X)) = 0 >= 0 = isNatIList(proper(X)) proper(isNatList(X)) = 0 >= 0 = isNatList(proper(X)) proper(isNat(X)) = 0 >= 0 = isNat(proper(X)) proper(0()) = 0 >= 0 = ok(0()) proper(s(X)) = 0 >= 0 = s(proper(X)) proper(length(X)) = 0 >= 0 = length(proper(X)) proper(zeros()) = 0 >= 0 = ok(zeros()) proper(cons(X1,X2)) = 0 >= 0 = cons(proper(X1),proper(X2)) proper(nil()) = 0 >= 0 = ok(nil()) proper(take(X1,X2)) = 0 >= 0 = take(proper(X1),proper(X2)) proper(uTake1(X)) = 0 >= 0 = uTake1(proper(X)) proper(uTake2(X1,X2,X3,X4)) = 0 >= 0 = uTake2(proper(X1),proper(X2),proper(X3),proper(X4)) proper(uLength(X1,X2)) = 0 >= 0 = uLength(proper(X1),proper(X2)) and(ok(X1),ok(X2)) = 0 >= 0 = ok(and(X1,X2)) isNatIList(ok(X)) = 0 >= 0 = ok(isNatIList(X)) isNatList(ok(X)) = 0 >= 0 = ok(isNatList(X)) isNat(ok(X)) = 0 >= 0 = ok(isNat(X)) s(ok(X)) = 0 >= 0 = ok(s(X)) length(ok(X)) = 0 >= 0 = ok(length(X)) cons(ok(X1),ok(X2)) = 0 >= 0 = ok(cons(X1,X2)) take(ok(X1),ok(X2)) = 0 >= 0 = ok(take(X1,X2)) uTake1(ok(X)) = 0 >= 0 = ok(uTake1(X)) uTake2(ok(X1),ok(X2),ok(X3),ok(X4)) = 0 >= 0 = ok(uTake2(X1,X2,X3,X4)) uLength(ok(X1),ok(X2)) = 0 >= 0 = ok(uLength(X1,X2)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) problem: DPs: TRS: active(and(tt(),T)) -> mark(T) active(isNatIList(IL)) -> mark(isNatList(IL)) active(isNat(0())) -> mark(tt()) active(isNat(s(N))) -> mark(isNat(N)) active(isNat(length(L))) -> mark(isNatList(L)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(N,L))) -> mark(and(isNat(N),isNatList(L))) active(isNatList(take(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(zeros()) -> mark(cons(0(),zeros())) active(take(0(),IL)) -> mark(uTake1(isNatIList(IL))) active(uTake1(tt())) -> mark(nil()) active(take(s(M),cons(N,IL))) -> mark(uTake2(and(isNat(M),and(isNat(N),isNatIList(IL))),M,N,IL)) active(uTake2(tt(),M,N,IL)) -> mark(cons(N,take(M,IL))) active(length(cons(N,L))) -> mark(uLength(and(isNat(N),isNatList(L)),L)) active(uLength(tt(),L)) -> mark(s(length(L))) active(and(X1,X2)) -> and(active(X1),X2) active(and(X1,X2)) -> and(X1,active(X2)) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) active(uTake1(X)) -> uTake1(active(X)) active(uTake2(X1,X2,X3,X4)) -> uTake2(active(X1),X2,X3,X4) active(uLength(X1,X2)) -> uLength(active(X1),X2) and(mark(X1),X2) -> mark(and(X1,X2)) and(X1,mark(X2)) -> mark(and(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) uTake1(mark(X)) -> mark(uTake1(X)) uTake2(mark(X1),X2,X3,X4) -> mark(uTake2(X1,X2,X3,X4)) uLength(mark(X1),X2) -> mark(uLength(X1,X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(isNat(X)) -> isNat(proper(X)) proper(0()) -> ok(0()) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(nil()) -> ok(nil()) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) proper(uTake1(X)) -> uTake1(proper(X)) proper(uTake2(X1,X2,X3,X4)) -> uTake2(proper(X1),proper(X2),proper(X3),proper(X4)) proper(uLength(X1,X2)) -> uLength(proper(X1),proper(X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatIList(ok(X)) -> ok(isNatIList(X)) isNatList(ok(X)) -> ok(isNatList(X)) isNat(ok(X)) -> ok(isNat(X)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) uTake1(ok(X)) -> ok(uTake1(X)) uTake2(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(uTake2(X1,X2,X3,X4)) uLength(ok(X1),ok(X2)) -> ok(uLength(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed DPs: cons#(mark(X1),X2) -> cons#(X1,X2) cons#(ok(X1),ok(X2)) -> cons#(X1,X2) TRS: active(and(tt(),T)) -> mark(T) active(isNatIList(IL)) -> mark(isNatList(IL)) active(isNat(0())) -> mark(tt()) active(isNat(s(N))) -> mark(isNat(N)) active(isNat(length(L))) -> mark(isNatList(L)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(N,L))) -> mark(and(isNat(N),isNatList(L))) active(isNatList(take(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(zeros()) -> mark(cons(0(),zeros())) active(take(0(),IL)) -> mark(uTake1(isNatIList(IL))) active(uTake1(tt())) -> mark(nil()) active(take(s(M),cons(N,IL))) -> mark(uTake2(and(isNat(M),and(isNat(N),isNatIList(IL))),M,N,IL)) active(uTake2(tt(),M,N,IL)) -> mark(cons(N,take(M,IL))) active(length(cons(N,L))) -> mark(uLength(and(isNat(N),isNatList(L)),L)) active(uLength(tt(),L)) -> mark(s(length(L))) active(and(X1,X2)) -> and(active(X1),X2) active(and(X1,X2)) -> and(X1,active(X2)) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) active(uTake1(X)) -> uTake1(active(X)) active(uTake2(X1,X2,X3,X4)) -> uTake2(active(X1),X2,X3,X4) active(uLength(X1,X2)) -> uLength(active(X1),X2) and(mark(X1),X2) -> mark(and(X1,X2)) and(X1,mark(X2)) -> mark(and(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) uTake1(mark(X)) -> mark(uTake1(X)) uTake2(mark(X1),X2,X3,X4) -> mark(uTake2(X1,X2,X3,X4)) uLength(mark(X1),X2) -> mark(uLength(X1,X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(isNat(X)) -> isNat(proper(X)) proper(0()) -> ok(0()) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(nil()) -> ok(nil()) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) proper(uTake1(X)) -> uTake1(proper(X)) proper(uTake2(X1,X2,X3,X4)) -> uTake2(proper(X1),proper(X2),proper(X3),proper(X4)) proper(uLength(X1,X2)) -> uLength(proper(X1),proper(X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatIList(ok(X)) -> ok(isNatIList(X)) isNatList(ok(X)) -> ok(isNatList(X)) isNat(ok(X)) -> ok(isNat(X)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) uTake1(ok(X)) -> ok(uTake1(X)) uTake2(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(uTake2(X1,X2,X3,X4)) uLength(ok(X1),ok(X2)) -> ok(uLength(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dimension: 1 interpretation: [cons#](x0, x1) = x1, [top](x0) = 1, [ok](x0) = x0 + 1, [proper](x0) = 1, [uLength](x0, x1) = x1, [uTake2](x0, x1, x2, x3) = x1, [uTake1](x0) = x0, [take](x0, x1) = x0, [nil] = 0, [cons](x0, x1) = x0, [zeros] = 0, [length](x0) = x0, [s](x0) = x0, [isNat](x0) = x0, [0] = 0, [isNatList](x0) = x0, [isNatIList](x0) = x0, [mark](x0) = 0, [active](x0) = x0, [and](x0, x1) = x1, [tt] = 0 orientation: cons#(mark(X1),X2) = X2 >= X2 = cons#(X1,X2) cons#(ok(X1),ok(X2)) = X2 + 1 >= X2 = cons#(X1,X2) active(and(tt(),T)) = T >= 0 = mark(T) active(isNatIList(IL)) = IL >= 0 = mark(isNatList(IL)) active(isNat(0())) = 0 >= 0 = mark(tt()) active(isNat(s(N))) = N >= 0 = mark(isNat(N)) active(isNat(length(L))) = L >= 0 = mark(isNatList(L)) active(isNatIList(zeros())) = 0 >= 0 = mark(tt()) active(isNatIList(cons(N,IL))) = N >= 0 = mark(and(isNat(N),isNatIList(IL))) active(isNatList(nil())) = 0 >= 0 = mark(tt()) active(isNatList(cons(N,L))) = N >= 0 = mark(and(isNat(N),isNatList(L))) active(isNatList(take(N,IL))) = N >= 0 = mark(and(isNat(N),isNatIList(IL))) active(zeros()) = 0 >= 0 = mark(cons(0(),zeros())) active(take(0(),IL)) = 0 >= 0 = mark(uTake1(isNatIList(IL))) active(uTake1(tt())) = 0 >= 0 = mark(nil()) active(take(s(M),cons(N,IL))) = M >= 0 = mark(uTake2(and(isNat(M),and(isNat(N),isNatIList(IL))),M,N,IL)) active(uTake2(tt(),M,N,IL)) = M >= 0 = mark(cons(N,take(M,IL))) active(length(cons(N,L))) = N >= 0 = mark(uLength(and(isNat(N),isNatList(L)),L)) active(uLength(tt(),L)) = L >= 0 = mark(s(length(L))) active(and(X1,X2)) = X2 >= X2 = and(active(X1),X2) active(and(X1,X2)) = X2 >= X2 = and(X1,active(X2)) active(s(X)) = X >= X = s(active(X)) active(length(X)) = X >= X = length(active(X)) active(cons(X1,X2)) = X1 >= X1 = cons(active(X1),X2) active(take(X1,X2)) = X1 >= X1 = take(active(X1),X2) active(take(X1,X2)) = X1 >= X1 = take(X1,active(X2)) active(uTake1(X)) = X >= X = uTake1(active(X)) active(uTake2(X1,X2,X3,X4)) = X2 >= X2 = uTake2(active(X1),X2,X3,X4) active(uLength(X1,X2)) = X2 >= X2 = uLength(active(X1),X2) and(mark(X1),X2) = X2 >= 0 = mark(and(X1,X2)) and(X1,mark(X2)) = 0 >= 0 = mark(and(X1,X2)) s(mark(X)) = 0 >= 0 = mark(s(X)) length(mark(X)) = 0 >= 0 = mark(length(X)) cons(mark(X1),X2) = 0 >= 0 = mark(cons(X1,X2)) take(mark(X1),X2) = 0 >= 0 = mark(take(X1,X2)) take(X1,mark(X2)) = X1 >= 0 = mark(take(X1,X2)) uTake1(mark(X)) = 0 >= 0 = mark(uTake1(X)) uTake2(mark(X1),X2,X3,X4) = X2 >= 0 = mark(uTake2(X1,X2,X3,X4)) uLength(mark(X1),X2) = X2 >= 0 = mark(uLength(X1,X2)) proper(and(X1,X2)) = 1 >= 1 = and(proper(X1),proper(X2)) proper(tt()) = 1 >= 1 = ok(tt()) proper(isNatIList(X)) = 1 >= 1 = isNatIList(proper(X)) proper(isNatList(X)) = 1 >= 1 = isNatList(proper(X)) proper(isNat(X)) = 1 >= 1 = isNat(proper(X)) proper(0()) = 1 >= 1 = ok(0()) proper(s(X)) = 1 >= 1 = s(proper(X)) proper(length(X)) = 1 >= 1 = length(proper(X)) proper(zeros()) = 1 >= 1 = ok(zeros()) proper(cons(X1,X2)) = 1 >= 1 = cons(proper(X1),proper(X2)) proper(nil()) = 1 >= 1 = ok(nil()) proper(take(X1,X2)) = 1 >= 1 = take(proper(X1),proper(X2)) proper(uTake1(X)) = 1 >= 1 = uTake1(proper(X)) proper(uTake2(X1,X2,X3,X4)) = 1 >= 1 = uTake2(proper(X1),proper(X2),proper(X3),proper(X4)) proper(uLength(X1,X2)) = 1 >= 1 = uLength(proper(X1),proper(X2)) and(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(and(X1,X2)) isNatIList(ok(X)) = X + 1 >= X + 1 = ok(isNatIList(X)) isNatList(ok(X)) = X + 1 >= X + 1 = ok(isNatList(X)) isNat(ok(X)) = X + 1 >= X + 1 = ok(isNat(X)) s(ok(X)) = X + 1 >= X + 1 = ok(s(X)) length(ok(X)) = X + 1 >= X + 1 = ok(length(X)) cons(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(cons(X1,X2)) take(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(take(X1,X2)) uTake1(ok(X)) = X + 1 >= X + 1 = ok(uTake1(X)) uTake2(ok(X1),ok(X2),ok(X3),ok(X4)) = X2 + 1 >= X2 + 1 = ok(uTake2(X1,X2,X3,X4)) uLength(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(uLength(X1,X2)) top(mark(X)) = 1 >= 1 = top(proper(X)) top(ok(X)) = 1 >= 1 = top(active(X)) problem: DPs: cons#(mark(X1),X2) -> cons#(X1,X2) TRS: active(and(tt(),T)) -> mark(T) active(isNatIList(IL)) -> mark(isNatList(IL)) active(isNat(0())) -> mark(tt()) active(isNat(s(N))) -> mark(isNat(N)) active(isNat(length(L))) -> mark(isNatList(L)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(N,L))) -> mark(and(isNat(N),isNatList(L))) active(isNatList(take(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(zeros()) -> mark(cons(0(),zeros())) active(take(0(),IL)) -> mark(uTake1(isNatIList(IL))) active(uTake1(tt())) -> mark(nil()) active(take(s(M),cons(N,IL))) -> mark(uTake2(and(isNat(M),and(isNat(N),isNatIList(IL))),M,N,IL)) active(uTake2(tt(),M,N,IL)) -> mark(cons(N,take(M,IL))) active(length(cons(N,L))) -> mark(uLength(and(isNat(N),isNatList(L)),L)) active(uLength(tt(),L)) -> mark(s(length(L))) active(and(X1,X2)) -> and(active(X1),X2) active(and(X1,X2)) -> and(X1,active(X2)) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) active(uTake1(X)) -> uTake1(active(X)) active(uTake2(X1,X2,X3,X4)) -> uTake2(active(X1),X2,X3,X4) active(uLength(X1,X2)) -> uLength(active(X1),X2) and(mark(X1),X2) -> mark(and(X1,X2)) and(X1,mark(X2)) -> mark(and(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) uTake1(mark(X)) -> mark(uTake1(X)) uTake2(mark(X1),X2,X3,X4) -> mark(uTake2(X1,X2,X3,X4)) uLength(mark(X1),X2) -> mark(uLength(X1,X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(isNat(X)) -> isNat(proper(X)) proper(0()) -> ok(0()) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(nil()) -> ok(nil()) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) proper(uTake1(X)) -> uTake1(proper(X)) proper(uTake2(X1,X2,X3,X4)) -> uTake2(proper(X1),proper(X2),proper(X3),proper(X4)) proper(uLength(X1,X2)) -> uLength(proper(X1),proper(X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatIList(ok(X)) -> ok(isNatIList(X)) isNatList(ok(X)) -> ok(isNatList(X)) isNat(ok(X)) -> ok(isNat(X)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) uTake1(ok(X)) -> ok(uTake1(X)) uTake2(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(uTake2(X1,X2,X3,X4)) uLength(ok(X1),ok(X2)) -> ok(uLength(X1,X2)) 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, [uLength](x0, x1) = x0 + x1, [uTake2](x0, x1, x2, x3) = x0 + x1 + x2 + x3, [uTake1](x0) = x0, [take](x0, x1) = x0 + x1, [nil] = 0, [cons](x0, x1) = x0 + x1, [zeros] = 1, [length](x0) = x0, [s](x0) = x0, [isNat](x0) = 0, [0] = 0, [isNatList](x0) = 0, [isNatIList](x0) = 0, [mark](x0) = x0 + 1, [active](x0) = x0 + 1, [and](x0, x1) = x0 + x1, [tt] = 0 orientation: cons#(mark(X1),X2) = X1 + 2 >= X1 + 1 = cons#(X1,X2) active(and(tt(),T)) = T + 1 >= T + 1 = mark(T) active(isNatIList(IL)) = 1 >= 1 = mark(isNatList(IL)) active(isNat(0())) = 1 >= 1 = mark(tt()) active(isNat(s(N))) = 1 >= 1 = mark(isNat(N)) active(isNat(length(L))) = 1 >= 1 = mark(isNatList(L)) active(isNatIList(zeros())) = 1 >= 1 = mark(tt()) active(isNatIList(cons(N,IL))) = 1 >= 1 = mark(and(isNat(N),isNatIList(IL))) active(isNatList(nil())) = 1 >= 1 = mark(tt()) active(isNatList(cons(N,L))) = 1 >= 1 = mark(and(isNat(N),isNatList(L))) active(isNatList(take(N,IL))) = 1 >= 1 = mark(and(isNat(N),isNatIList(IL))) active(zeros()) = 2 >= 2 = mark(cons(0(),zeros())) active(take(0(),IL)) = IL + 1 >= 1 = mark(uTake1(isNatIList(IL))) active(uTake1(tt())) = 1 >= 1 = mark(nil()) active(take(s(M),cons(N,IL))) = IL + M + N + 1 >= IL + M + N + 1 = mark(uTake2(and(isNat(M),and(isNat(N),isNatIList(IL))),M,N,IL)) active(uTake2(tt(),M,N,IL)) = IL + M + N + 1 >= IL + M + N + 1 = mark(cons(N,take(M,IL))) active(length(cons(N,L))) = L + N + 1 >= L + 1 = mark(uLength(and(isNat(N),isNatList(L)),L)) active(uLength(tt(),L)) = L + 1 >= L + 1 = mark(s(length(L))) active(and(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = and(active(X1),X2) active(and(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = and(X1,active(X2)) active(s(X)) = X + 1 >= X + 1 = s(active(X)) active(length(X)) = X + 1 >= X + 1 = length(active(X)) active(cons(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = cons(active(X1),X2) 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(uTake1(X)) = X + 1 >= X + 1 = uTake1(active(X)) active(uTake2(X1,X2,X3,X4)) = X1 + X2 + X3 + X4 + 1 >= X1 + X2 + X3 + X4 + 1 = uTake2(active(X1),X2,X3,X4) active(uLength(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = uLength(active(X1),X2) and(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(and(X1,X2)) and(X1,mark(X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(and(X1,X2)) s(mark(X)) = X + 1 >= X + 1 = mark(s(X)) length(mark(X)) = X + 1 >= X + 1 = mark(length(X)) cons(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(cons(X1,X2)) 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)) uTake1(mark(X)) = X + 1 >= X + 1 = mark(uTake1(X)) uTake2(mark(X1),X2,X3,X4) = X1 + X2 + X3 + X4 + 1 >= X1 + X2 + X3 + X4 + 1 = mark(uTake2(X1,X2,X3,X4)) uLength(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(uLength(X1,X2)) proper(and(X1,X2)) = 0 >= 0 = and(proper(X1),proper(X2)) proper(tt()) = 0 >= 0 = ok(tt()) proper(isNatIList(X)) = 0 >= 0 = isNatIList(proper(X)) proper(isNatList(X)) = 0 >= 0 = isNatList(proper(X)) proper(isNat(X)) = 0 >= 0 = isNat(proper(X)) proper(0()) = 0 >= 0 = ok(0()) proper(s(X)) = 0 >= 0 = s(proper(X)) proper(length(X)) = 0 >= 0 = length(proper(X)) proper(zeros()) = 0 >= 0 = ok(zeros()) proper(cons(X1,X2)) = 0 >= 0 = cons(proper(X1),proper(X2)) proper(nil()) = 0 >= 0 = ok(nil()) proper(take(X1,X2)) = 0 >= 0 = take(proper(X1),proper(X2)) proper(uTake1(X)) = 0 >= 0 = uTake1(proper(X)) proper(uTake2(X1,X2,X3,X4)) = 0 >= 0 = uTake2(proper(X1),proper(X2),proper(X3),proper(X4)) proper(uLength(X1,X2)) = 0 >= 0 = uLength(proper(X1),proper(X2)) and(ok(X1),ok(X2)) = 0 >= 0 = ok(and(X1,X2)) isNatIList(ok(X)) = 0 >= 0 = ok(isNatIList(X)) isNatList(ok(X)) = 0 >= 0 = ok(isNatList(X)) isNat(ok(X)) = 0 >= 0 = ok(isNat(X)) s(ok(X)) = 0 >= 0 = ok(s(X)) length(ok(X)) = 0 >= 0 = ok(length(X)) cons(ok(X1),ok(X2)) = 0 >= 0 = ok(cons(X1,X2)) take(ok(X1),ok(X2)) = 0 >= 0 = ok(take(X1,X2)) uTake1(ok(X)) = 0 >= 0 = ok(uTake1(X)) uTake2(ok(X1),ok(X2),ok(X3),ok(X4)) = 0 >= 0 = ok(uTake2(X1,X2,X3,X4)) uLength(ok(X1),ok(X2)) = 0 >= 0 = ok(uLength(X1,X2)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) problem: DPs: TRS: active(and(tt(),T)) -> mark(T) active(isNatIList(IL)) -> mark(isNatList(IL)) active(isNat(0())) -> mark(tt()) active(isNat(s(N))) -> mark(isNat(N)) active(isNat(length(L))) -> mark(isNatList(L)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(N,L))) -> mark(and(isNat(N),isNatList(L))) active(isNatList(take(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(zeros()) -> mark(cons(0(),zeros())) active(take(0(),IL)) -> mark(uTake1(isNatIList(IL))) active(uTake1(tt())) -> mark(nil()) active(take(s(M),cons(N,IL))) -> mark(uTake2(and(isNat(M),and(isNat(N),isNatIList(IL))),M,N,IL)) active(uTake2(tt(),M,N,IL)) -> mark(cons(N,take(M,IL))) active(length(cons(N,L))) -> mark(uLength(and(isNat(N),isNatList(L)),L)) active(uLength(tt(),L)) -> mark(s(length(L))) active(and(X1,X2)) -> and(active(X1),X2) active(and(X1,X2)) -> and(X1,active(X2)) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) active(uTake1(X)) -> uTake1(active(X)) active(uTake2(X1,X2,X3,X4)) -> uTake2(active(X1),X2,X3,X4) active(uLength(X1,X2)) -> uLength(active(X1),X2) and(mark(X1),X2) -> mark(and(X1,X2)) and(X1,mark(X2)) -> mark(and(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) uTake1(mark(X)) -> mark(uTake1(X)) uTake2(mark(X1),X2,X3,X4) -> mark(uTake2(X1,X2,X3,X4)) uLength(mark(X1),X2) -> mark(uLength(X1,X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(isNat(X)) -> isNat(proper(X)) proper(0()) -> ok(0()) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(nil()) -> ok(nil()) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) proper(uTake1(X)) -> uTake1(proper(X)) proper(uTake2(X1,X2,X3,X4)) -> uTake2(proper(X1),proper(X2),proper(X3),proper(X4)) proper(uLength(X1,X2)) -> uLength(proper(X1),proper(X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatIList(ok(X)) -> ok(isNatIList(X)) isNatList(ok(X)) -> ok(isNatList(X)) isNat(ok(X)) -> ok(isNat(X)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) uTake1(ok(X)) -> ok(uTake1(X)) uTake2(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(uTake2(X1,X2,X3,X4)) uLength(ok(X1),ok(X2)) -> ok(uLength(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed DPs: length#(mark(X)) -> length#(X) length#(ok(X)) -> length#(X) TRS: active(and(tt(),T)) -> mark(T) active(isNatIList(IL)) -> mark(isNatList(IL)) active(isNat(0())) -> mark(tt()) active(isNat(s(N))) -> mark(isNat(N)) active(isNat(length(L))) -> mark(isNatList(L)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(N,L))) -> mark(and(isNat(N),isNatList(L))) active(isNatList(take(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(zeros()) -> mark(cons(0(),zeros())) active(take(0(),IL)) -> mark(uTake1(isNatIList(IL))) active(uTake1(tt())) -> mark(nil()) active(take(s(M),cons(N,IL))) -> mark(uTake2(and(isNat(M),and(isNat(N),isNatIList(IL))),M,N,IL)) active(uTake2(tt(),M,N,IL)) -> mark(cons(N,take(M,IL))) active(length(cons(N,L))) -> mark(uLength(and(isNat(N),isNatList(L)),L)) active(uLength(tt(),L)) -> mark(s(length(L))) active(and(X1,X2)) -> and(active(X1),X2) active(and(X1,X2)) -> and(X1,active(X2)) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) active(uTake1(X)) -> uTake1(active(X)) active(uTake2(X1,X2,X3,X4)) -> uTake2(active(X1),X2,X3,X4) active(uLength(X1,X2)) -> uLength(active(X1),X2) and(mark(X1),X2) -> mark(and(X1,X2)) and(X1,mark(X2)) -> mark(and(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) uTake1(mark(X)) -> mark(uTake1(X)) uTake2(mark(X1),X2,X3,X4) -> mark(uTake2(X1,X2,X3,X4)) uLength(mark(X1),X2) -> mark(uLength(X1,X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(isNat(X)) -> isNat(proper(X)) proper(0()) -> ok(0()) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(nil()) -> ok(nil()) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) proper(uTake1(X)) -> uTake1(proper(X)) proper(uTake2(X1,X2,X3,X4)) -> uTake2(proper(X1),proper(X2),proper(X3),proper(X4)) proper(uLength(X1,X2)) -> uLength(proper(X1),proper(X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatIList(ok(X)) -> ok(isNatIList(X)) isNatList(ok(X)) -> ok(isNatList(X)) isNat(ok(X)) -> ok(isNat(X)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) uTake1(ok(X)) -> ok(uTake1(X)) uTake2(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(uTake2(X1,X2,X3,X4)) uLength(ok(X1),ok(X2)) -> ok(uLength(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dimension: 1 interpretation: [length#](x0) = x0, [top](x0) = 0, [ok](x0) = x0 + 1, [proper](x0) = x0 + 1, [uLength](x0, x1) = x1, [uTake2](x0, x1, x2, x3) = x3 + 1, [uTake1](x0) = x0, [take](x0, x1) = x1 + 1, [nil] = 0, [cons](x0, x1) = x1, [zeros] = 0, [length](x0) = x0, [s](x0) = x0, [isNat](x0) = x0, [0] = 0, [isNatList](x0) = x0, [isNatIList](x0) = x0 + 1, [mark](x0) = x0, [active](x0) = x0, [and](x0, x1) = x1, [tt] = 0 orientation: length#(mark(X)) = X >= X = length#(X) length#(ok(X)) = X + 1 >= X = length#(X) active(and(tt(),T)) = T >= T = mark(T) active(isNatIList(IL)) = IL + 1 >= IL = mark(isNatList(IL)) active(isNat(0())) = 0 >= 0 = mark(tt()) active(isNat(s(N))) = N >= N = mark(isNat(N)) active(isNat(length(L))) = L >= L = mark(isNatList(L)) active(isNatIList(zeros())) = 1 >= 0 = mark(tt()) active(isNatIList(cons(N,IL))) = IL + 1 >= IL + 1 = mark(and(isNat(N),isNatIList(IL))) active(isNatList(nil())) = 0 >= 0 = mark(tt()) active(isNatList(cons(N,L))) = L >= L = mark(and(isNat(N),isNatList(L))) active(isNatList(take(N,IL))) = IL + 1 >= IL + 1 = mark(and(isNat(N),isNatIList(IL))) active(zeros()) = 0 >= 0 = mark(cons(0(),zeros())) active(take(0(),IL)) = IL + 1 >= IL + 1 = mark(uTake1(isNatIList(IL))) active(uTake1(tt())) = 0 >= 0 = mark(nil()) active(take(s(M),cons(N,IL))) = IL + 1 >= IL + 1 = mark(uTake2(and(isNat(M),and(isNat(N),isNatIList(IL))),M,N,IL)) active(uTake2(tt(),M,N,IL)) = IL + 1 >= IL + 1 = mark(cons(N,take(M,IL))) active(length(cons(N,L))) = L >= L = mark(uLength(and(isNat(N),isNatList(L)),L)) active(uLength(tt(),L)) = L >= L = mark(s(length(L))) active(and(X1,X2)) = X2 >= X2 = and(active(X1),X2) active(and(X1,X2)) = X2 >= X2 = and(X1,active(X2)) active(s(X)) = X >= X = s(active(X)) active(length(X)) = X >= X = length(active(X)) active(cons(X1,X2)) = X2 >= X2 = cons(active(X1),X2) 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(uTake1(X)) = X >= X = uTake1(active(X)) active(uTake2(X1,X2,X3,X4)) = X4 + 1 >= X4 + 1 = uTake2(active(X1),X2,X3,X4) active(uLength(X1,X2)) = X2 >= X2 = uLength(active(X1),X2) and(mark(X1),X2) = X2 >= X2 = mark(and(X1,X2)) and(X1,mark(X2)) = X2 >= X2 = mark(and(X1,X2)) s(mark(X)) = X >= X = mark(s(X)) length(mark(X)) = X >= X = mark(length(X)) cons(mark(X1),X2) = X2 >= X2 = mark(cons(X1,X2)) take(mark(X1),X2) = X2 + 1 >= X2 + 1 = mark(take(X1,X2)) take(X1,mark(X2)) = X2 + 1 >= X2 + 1 = mark(take(X1,X2)) uTake1(mark(X)) = X >= X = mark(uTake1(X)) uTake2(mark(X1),X2,X3,X4) = X4 + 1 >= X4 + 1 = mark(uTake2(X1,X2,X3,X4)) uLength(mark(X1),X2) = X2 >= X2 = mark(uLength(X1,X2)) proper(and(X1,X2)) = X2 + 1 >= X2 + 1 = and(proper(X1),proper(X2)) proper(tt()) = 1 >= 1 = ok(tt()) proper(isNatIList(X)) = X + 2 >= X + 2 = isNatIList(proper(X)) proper(isNatList(X)) = X + 1 >= X + 1 = isNatList(proper(X)) proper(isNat(X)) = X + 1 >= X + 1 = isNat(proper(X)) proper(0()) = 1 >= 1 = ok(0()) proper(s(X)) = X + 1 >= X + 1 = s(proper(X)) proper(length(X)) = X + 1 >= X + 1 = length(proper(X)) proper(zeros()) = 1 >= 1 = ok(zeros()) proper(cons(X1,X2)) = X2 + 1 >= X2 + 1 = cons(proper(X1),proper(X2)) proper(nil()) = 1 >= 1 = ok(nil()) proper(take(X1,X2)) = X2 + 2 >= X2 + 2 = take(proper(X1),proper(X2)) proper(uTake1(X)) = X + 1 >= X + 1 = uTake1(proper(X)) proper(uTake2(X1,X2,X3,X4)) = X4 + 2 >= X4 + 2 = uTake2(proper(X1),proper(X2),proper(X3),proper(X4)) proper(uLength(X1,X2)) = X2 + 1 >= X2 + 1 = uLength(proper(X1),proper(X2)) and(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(and(X1,X2)) isNatIList(ok(X)) = X + 2 >= X + 2 = ok(isNatIList(X)) isNatList(ok(X)) = X + 1 >= X + 1 = ok(isNatList(X)) isNat(ok(X)) = X + 1 >= X + 1 = ok(isNat(X)) s(ok(X)) = X + 1 >= X + 1 = ok(s(X)) length(ok(X)) = X + 1 >= X + 1 = ok(length(X)) cons(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(cons(X1,X2)) take(ok(X1),ok(X2)) = X2 + 2 >= X2 + 2 = ok(take(X1,X2)) uTake1(ok(X)) = X + 1 >= X + 1 = ok(uTake1(X)) uTake2(ok(X1),ok(X2),ok(X3),ok(X4)) = X4 + 2 >= X4 + 2 = ok(uTake2(X1,X2,X3,X4)) uLength(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(uLength(X1,X2)) 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(and(tt(),T)) -> mark(T) active(isNatIList(IL)) -> mark(isNatList(IL)) active(isNat(0())) -> mark(tt()) active(isNat(s(N))) -> mark(isNat(N)) active(isNat(length(L))) -> mark(isNatList(L)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(N,L))) -> mark(and(isNat(N),isNatList(L))) active(isNatList(take(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(zeros()) -> mark(cons(0(),zeros())) active(take(0(),IL)) -> mark(uTake1(isNatIList(IL))) active(uTake1(tt())) -> mark(nil()) active(take(s(M),cons(N,IL))) -> mark(uTake2(and(isNat(M),and(isNat(N),isNatIList(IL))),M,N,IL)) active(uTake2(tt(),M,N,IL)) -> mark(cons(N,take(M,IL))) active(length(cons(N,L))) -> mark(uLength(and(isNat(N),isNatList(L)),L)) active(uLength(tt(),L)) -> mark(s(length(L))) active(and(X1,X2)) -> and(active(X1),X2) active(and(X1,X2)) -> and(X1,active(X2)) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) active(uTake1(X)) -> uTake1(active(X)) active(uTake2(X1,X2,X3,X4)) -> uTake2(active(X1),X2,X3,X4) active(uLength(X1,X2)) -> uLength(active(X1),X2) and(mark(X1),X2) -> mark(and(X1,X2)) and(X1,mark(X2)) -> mark(and(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) uTake1(mark(X)) -> mark(uTake1(X)) uTake2(mark(X1),X2,X3,X4) -> mark(uTake2(X1,X2,X3,X4)) uLength(mark(X1),X2) -> mark(uLength(X1,X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(isNat(X)) -> isNat(proper(X)) proper(0()) -> ok(0()) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(nil()) -> ok(nil()) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) proper(uTake1(X)) -> uTake1(proper(X)) proper(uTake2(X1,X2,X3,X4)) -> uTake2(proper(X1),proper(X2),proper(X3),proper(X4)) proper(uLength(X1,X2)) -> uLength(proper(X1),proper(X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatIList(ok(X)) -> ok(isNatIList(X)) isNatList(ok(X)) -> ok(isNatList(X)) isNat(ok(X)) -> ok(isNat(X)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) uTake1(ok(X)) -> ok(uTake1(X)) uTake2(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(uTake2(X1,X2,X3,X4)) uLength(ok(X1),ok(X2)) -> ok(uLength(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dimension: 1 interpretation: [length#](x0) = x0, [top](x0) = 1, [ok](x0) = 0, [proper](x0) = 0, [uLength](x0, x1) = x0 + x1, [uTake2](x0, x1, x2, x3) = x0 + x1 + x2 + x3, [uTake1](x0) = x0, [take](x0, x1) = x0 + x1, [nil] = 0, [cons](x0, x1) = x0 + x1, [zeros] = 1, [length](x0) = x0, [s](x0) = x0, [isNat](x0) = 0, [0] = 0, [isNatList](x0) = 0, [isNatIList](x0) = 0, [mark](x0) = x0 + 1, [active](x0) = x0 + 1, [and](x0, x1) = x0 + x1, [tt] = 0 orientation: length#(mark(X)) = X + 1 >= X = length#(X) active(and(tt(),T)) = T + 1 >= T + 1 = mark(T) active(isNatIList(IL)) = 1 >= 1 = mark(isNatList(IL)) active(isNat(0())) = 1 >= 1 = mark(tt()) active(isNat(s(N))) = 1 >= 1 = mark(isNat(N)) active(isNat(length(L))) = 1 >= 1 = mark(isNatList(L)) active(isNatIList(zeros())) = 1 >= 1 = mark(tt()) active(isNatIList(cons(N,IL))) = 1 >= 1 = mark(and(isNat(N),isNatIList(IL))) active(isNatList(nil())) = 1 >= 1 = mark(tt()) active(isNatList(cons(N,L))) = 1 >= 1 = mark(and(isNat(N),isNatList(L))) active(isNatList(take(N,IL))) = 1 >= 1 = mark(and(isNat(N),isNatIList(IL))) active(zeros()) = 2 >= 2 = mark(cons(0(),zeros())) active(take(0(),IL)) = IL + 1 >= 1 = mark(uTake1(isNatIList(IL))) active(uTake1(tt())) = 1 >= 1 = mark(nil()) active(take(s(M),cons(N,IL))) = IL + M + N + 1 >= IL + M + N + 1 = mark(uTake2(and(isNat(M),and(isNat(N),isNatIList(IL))),M,N,IL)) active(uTake2(tt(),M,N,IL)) = IL + M + N + 1 >= IL + M + N + 1 = mark(cons(N,take(M,IL))) active(length(cons(N,L))) = L + N + 1 >= L + 1 = mark(uLength(and(isNat(N),isNatList(L)),L)) active(uLength(tt(),L)) = L + 1 >= L + 1 = mark(s(length(L))) active(and(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = and(active(X1),X2) active(and(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = and(X1,active(X2)) active(s(X)) = X + 1 >= X + 1 = s(active(X)) active(length(X)) = X + 1 >= X + 1 = length(active(X)) active(cons(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = cons(active(X1),X2) 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(uTake1(X)) = X + 1 >= X + 1 = uTake1(active(X)) active(uTake2(X1,X2,X3,X4)) = X1 + X2 + X3 + X4 + 1 >= X1 + X2 + X3 + X4 + 1 = uTake2(active(X1),X2,X3,X4) active(uLength(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = uLength(active(X1),X2) and(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(and(X1,X2)) and(X1,mark(X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(and(X1,X2)) s(mark(X)) = X + 1 >= X + 1 = mark(s(X)) length(mark(X)) = X + 1 >= X + 1 = mark(length(X)) cons(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(cons(X1,X2)) 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)) uTake1(mark(X)) = X + 1 >= X + 1 = mark(uTake1(X)) uTake2(mark(X1),X2,X3,X4) = X1 + X2 + X3 + X4 + 1 >= X1 + X2 + X3 + X4 + 1 = mark(uTake2(X1,X2,X3,X4)) uLength(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(uLength(X1,X2)) proper(and(X1,X2)) = 0 >= 0 = and(proper(X1),proper(X2)) proper(tt()) = 0 >= 0 = ok(tt()) proper(isNatIList(X)) = 0 >= 0 = isNatIList(proper(X)) proper(isNatList(X)) = 0 >= 0 = isNatList(proper(X)) proper(isNat(X)) = 0 >= 0 = isNat(proper(X)) proper(0()) = 0 >= 0 = ok(0()) proper(s(X)) = 0 >= 0 = s(proper(X)) proper(length(X)) = 0 >= 0 = length(proper(X)) proper(zeros()) = 0 >= 0 = ok(zeros()) proper(cons(X1,X2)) = 0 >= 0 = cons(proper(X1),proper(X2)) proper(nil()) = 0 >= 0 = ok(nil()) proper(take(X1,X2)) = 0 >= 0 = take(proper(X1),proper(X2)) proper(uTake1(X)) = 0 >= 0 = uTake1(proper(X)) proper(uTake2(X1,X2,X3,X4)) = 0 >= 0 = uTake2(proper(X1),proper(X2),proper(X3),proper(X4)) proper(uLength(X1,X2)) = 0 >= 0 = uLength(proper(X1),proper(X2)) and(ok(X1),ok(X2)) = 0 >= 0 = ok(and(X1,X2)) isNatIList(ok(X)) = 0 >= 0 = ok(isNatIList(X)) isNatList(ok(X)) = 0 >= 0 = ok(isNatList(X)) isNat(ok(X)) = 0 >= 0 = ok(isNat(X)) s(ok(X)) = 0 >= 0 = ok(s(X)) length(ok(X)) = 0 >= 0 = ok(length(X)) cons(ok(X1),ok(X2)) = 0 >= 0 = ok(cons(X1,X2)) take(ok(X1),ok(X2)) = 0 >= 0 = ok(take(X1,X2)) uTake1(ok(X)) = 0 >= 0 = ok(uTake1(X)) uTake2(ok(X1),ok(X2),ok(X3),ok(X4)) = 0 >= 0 = ok(uTake2(X1,X2,X3,X4)) uLength(ok(X1),ok(X2)) = 0 >= 0 = ok(uLength(X1,X2)) top(mark(X)) = 1 >= 1 = top(proper(X)) top(ok(X)) = 1 >= 1 = top(active(X)) problem: DPs: TRS: active(and(tt(),T)) -> mark(T) active(isNatIList(IL)) -> mark(isNatList(IL)) active(isNat(0())) -> mark(tt()) active(isNat(s(N))) -> mark(isNat(N)) active(isNat(length(L))) -> mark(isNatList(L)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(N,L))) -> mark(and(isNat(N),isNatList(L))) active(isNatList(take(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(zeros()) -> mark(cons(0(),zeros())) active(take(0(),IL)) -> mark(uTake1(isNatIList(IL))) active(uTake1(tt())) -> mark(nil()) active(take(s(M),cons(N,IL))) -> mark(uTake2(and(isNat(M),and(isNat(N),isNatIList(IL))),M,N,IL)) active(uTake2(tt(),M,N,IL)) -> mark(cons(N,take(M,IL))) active(length(cons(N,L))) -> mark(uLength(and(isNat(N),isNatList(L)),L)) active(uLength(tt(),L)) -> mark(s(length(L))) active(and(X1,X2)) -> and(active(X1),X2) active(and(X1,X2)) -> and(X1,active(X2)) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) active(uTake1(X)) -> uTake1(active(X)) active(uTake2(X1,X2,X3,X4)) -> uTake2(active(X1),X2,X3,X4) active(uLength(X1,X2)) -> uLength(active(X1),X2) and(mark(X1),X2) -> mark(and(X1,X2)) and(X1,mark(X2)) -> mark(and(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) uTake1(mark(X)) -> mark(uTake1(X)) uTake2(mark(X1),X2,X3,X4) -> mark(uTake2(X1,X2,X3,X4)) uLength(mark(X1),X2) -> mark(uLength(X1,X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(isNat(X)) -> isNat(proper(X)) proper(0()) -> ok(0()) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(nil()) -> ok(nil()) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) proper(uTake1(X)) -> uTake1(proper(X)) proper(uTake2(X1,X2,X3,X4)) -> uTake2(proper(X1),proper(X2),proper(X3),proper(X4)) proper(uLength(X1,X2)) -> uLength(proper(X1),proper(X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatIList(ok(X)) -> ok(isNatIList(X)) isNatList(ok(X)) -> ok(isNatList(X)) isNat(ok(X)) -> ok(isNat(X)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) uTake1(ok(X)) -> ok(uTake1(X)) uTake2(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(uTake2(X1,X2,X3,X4)) uLength(ok(X1),ok(X2)) -> ok(uLength(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed DPs: s#(mark(X)) -> s#(X) s#(ok(X)) -> s#(X) TRS: active(and(tt(),T)) -> mark(T) active(isNatIList(IL)) -> mark(isNatList(IL)) active(isNat(0())) -> mark(tt()) active(isNat(s(N))) -> mark(isNat(N)) active(isNat(length(L))) -> mark(isNatList(L)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(N,L))) -> mark(and(isNat(N),isNatList(L))) active(isNatList(take(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(zeros()) -> mark(cons(0(),zeros())) active(take(0(),IL)) -> mark(uTake1(isNatIList(IL))) active(uTake1(tt())) -> mark(nil()) active(take(s(M),cons(N,IL))) -> mark(uTake2(and(isNat(M),and(isNat(N),isNatIList(IL))),M,N,IL)) active(uTake2(tt(),M,N,IL)) -> mark(cons(N,take(M,IL))) active(length(cons(N,L))) -> mark(uLength(and(isNat(N),isNatList(L)),L)) active(uLength(tt(),L)) -> mark(s(length(L))) active(and(X1,X2)) -> and(active(X1),X2) active(and(X1,X2)) -> and(X1,active(X2)) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) active(uTake1(X)) -> uTake1(active(X)) active(uTake2(X1,X2,X3,X4)) -> uTake2(active(X1),X2,X3,X4) active(uLength(X1,X2)) -> uLength(active(X1),X2) and(mark(X1),X2) -> mark(and(X1,X2)) and(X1,mark(X2)) -> mark(and(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) uTake1(mark(X)) -> mark(uTake1(X)) uTake2(mark(X1),X2,X3,X4) -> mark(uTake2(X1,X2,X3,X4)) uLength(mark(X1),X2) -> mark(uLength(X1,X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(isNat(X)) -> isNat(proper(X)) proper(0()) -> ok(0()) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(nil()) -> ok(nil()) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) proper(uTake1(X)) -> uTake1(proper(X)) proper(uTake2(X1,X2,X3,X4)) -> uTake2(proper(X1),proper(X2),proper(X3),proper(X4)) proper(uLength(X1,X2)) -> uLength(proper(X1),proper(X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatIList(ok(X)) -> ok(isNatIList(X)) isNatList(ok(X)) -> ok(isNatList(X)) isNat(ok(X)) -> ok(isNat(X)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) uTake1(ok(X)) -> ok(uTake1(X)) uTake2(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(uTake2(X1,X2,X3,X4)) uLength(ok(X1),ok(X2)) -> ok(uLength(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dimension: 1 interpretation: [s#](x0) = x0, [top](x0) = 0, [ok](x0) = x0 + 1, [proper](x0) = x0 + 1, [uLength](x0, x1) = x1, [uTake2](x0, x1, x2, x3) = x3 + 1, [uTake1](x0) = x0, [take](x0, x1) = x1 + 1, [nil] = 0, [cons](x0, x1) = x1, [zeros] = 0, [length](x0) = x0, [s](x0) = x0, [isNat](x0) = x0, [0] = 0, [isNatList](x0) = x0, [isNatIList](x0) = x0 + 1, [mark](x0) = x0, [active](x0) = x0, [and](x0, x1) = x1, [tt] = 0 orientation: s#(mark(X)) = X >= X = s#(X) s#(ok(X)) = X + 1 >= X = s#(X) active(and(tt(),T)) = T >= T = mark(T) active(isNatIList(IL)) = IL + 1 >= IL = mark(isNatList(IL)) active(isNat(0())) = 0 >= 0 = mark(tt()) active(isNat(s(N))) = N >= N = mark(isNat(N)) active(isNat(length(L))) = L >= L = mark(isNatList(L)) active(isNatIList(zeros())) = 1 >= 0 = mark(tt()) active(isNatIList(cons(N,IL))) = IL + 1 >= IL + 1 = mark(and(isNat(N),isNatIList(IL))) active(isNatList(nil())) = 0 >= 0 = mark(tt()) active(isNatList(cons(N,L))) = L >= L = mark(and(isNat(N),isNatList(L))) active(isNatList(take(N,IL))) = IL + 1 >= IL + 1 = mark(and(isNat(N),isNatIList(IL))) active(zeros()) = 0 >= 0 = mark(cons(0(),zeros())) active(take(0(),IL)) = IL + 1 >= IL + 1 = mark(uTake1(isNatIList(IL))) active(uTake1(tt())) = 0 >= 0 = mark(nil()) active(take(s(M),cons(N,IL))) = IL + 1 >= IL + 1 = mark(uTake2(and(isNat(M),and(isNat(N),isNatIList(IL))),M,N,IL)) active(uTake2(tt(),M,N,IL)) = IL + 1 >= IL + 1 = mark(cons(N,take(M,IL))) active(length(cons(N,L))) = L >= L = mark(uLength(and(isNat(N),isNatList(L)),L)) active(uLength(tt(),L)) = L >= L = mark(s(length(L))) active(and(X1,X2)) = X2 >= X2 = and(active(X1),X2) active(and(X1,X2)) = X2 >= X2 = and(X1,active(X2)) active(s(X)) = X >= X = s(active(X)) active(length(X)) = X >= X = length(active(X)) active(cons(X1,X2)) = X2 >= X2 = cons(active(X1),X2) 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(uTake1(X)) = X >= X = uTake1(active(X)) active(uTake2(X1,X2,X3,X4)) = X4 + 1 >= X4 + 1 = uTake2(active(X1),X2,X3,X4) active(uLength(X1,X2)) = X2 >= X2 = uLength(active(X1),X2) and(mark(X1),X2) = X2 >= X2 = mark(and(X1,X2)) and(X1,mark(X2)) = X2 >= X2 = mark(and(X1,X2)) s(mark(X)) = X >= X = mark(s(X)) length(mark(X)) = X >= X = mark(length(X)) cons(mark(X1),X2) = X2 >= X2 = mark(cons(X1,X2)) take(mark(X1),X2) = X2 + 1 >= X2 + 1 = mark(take(X1,X2)) take(X1,mark(X2)) = X2 + 1 >= X2 + 1 = mark(take(X1,X2)) uTake1(mark(X)) = X >= X = mark(uTake1(X)) uTake2(mark(X1),X2,X3,X4) = X4 + 1 >= X4 + 1 = mark(uTake2(X1,X2,X3,X4)) uLength(mark(X1),X2) = X2 >= X2 = mark(uLength(X1,X2)) proper(and(X1,X2)) = X2 + 1 >= X2 + 1 = and(proper(X1),proper(X2)) proper(tt()) = 1 >= 1 = ok(tt()) proper(isNatIList(X)) = X + 2 >= X + 2 = isNatIList(proper(X)) proper(isNatList(X)) = X + 1 >= X + 1 = isNatList(proper(X)) proper(isNat(X)) = X + 1 >= X + 1 = isNat(proper(X)) proper(0()) = 1 >= 1 = ok(0()) proper(s(X)) = X + 1 >= X + 1 = s(proper(X)) proper(length(X)) = X + 1 >= X + 1 = length(proper(X)) proper(zeros()) = 1 >= 1 = ok(zeros()) proper(cons(X1,X2)) = X2 + 1 >= X2 + 1 = cons(proper(X1),proper(X2)) proper(nil()) = 1 >= 1 = ok(nil()) proper(take(X1,X2)) = X2 + 2 >= X2 + 2 = take(proper(X1),proper(X2)) proper(uTake1(X)) = X + 1 >= X + 1 = uTake1(proper(X)) proper(uTake2(X1,X2,X3,X4)) = X4 + 2 >= X4 + 2 = uTake2(proper(X1),proper(X2),proper(X3),proper(X4)) proper(uLength(X1,X2)) = X2 + 1 >= X2 + 1 = uLength(proper(X1),proper(X2)) and(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(and(X1,X2)) isNatIList(ok(X)) = X + 2 >= X + 2 = ok(isNatIList(X)) isNatList(ok(X)) = X + 1 >= X + 1 = ok(isNatList(X)) isNat(ok(X)) = X + 1 >= X + 1 = ok(isNat(X)) s(ok(X)) = X + 1 >= X + 1 = ok(s(X)) length(ok(X)) = X + 1 >= X + 1 = ok(length(X)) cons(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(cons(X1,X2)) take(ok(X1),ok(X2)) = X2 + 2 >= X2 + 2 = ok(take(X1,X2)) uTake1(ok(X)) = X + 1 >= X + 1 = ok(uTake1(X)) uTake2(ok(X1),ok(X2),ok(X3),ok(X4)) = X4 + 2 >= X4 + 2 = ok(uTake2(X1,X2,X3,X4)) uLength(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(uLength(X1,X2)) 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(and(tt(),T)) -> mark(T) active(isNatIList(IL)) -> mark(isNatList(IL)) active(isNat(0())) -> mark(tt()) active(isNat(s(N))) -> mark(isNat(N)) active(isNat(length(L))) -> mark(isNatList(L)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(N,L))) -> mark(and(isNat(N),isNatList(L))) active(isNatList(take(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(zeros()) -> mark(cons(0(),zeros())) active(take(0(),IL)) -> mark(uTake1(isNatIList(IL))) active(uTake1(tt())) -> mark(nil()) active(take(s(M),cons(N,IL))) -> mark(uTake2(and(isNat(M),and(isNat(N),isNatIList(IL))),M,N,IL)) active(uTake2(tt(),M,N,IL)) -> mark(cons(N,take(M,IL))) active(length(cons(N,L))) -> mark(uLength(and(isNat(N),isNatList(L)),L)) active(uLength(tt(),L)) -> mark(s(length(L))) active(and(X1,X2)) -> and(active(X1),X2) active(and(X1,X2)) -> and(X1,active(X2)) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) active(uTake1(X)) -> uTake1(active(X)) active(uTake2(X1,X2,X3,X4)) -> uTake2(active(X1),X2,X3,X4) active(uLength(X1,X2)) -> uLength(active(X1),X2) and(mark(X1),X2) -> mark(and(X1,X2)) and(X1,mark(X2)) -> mark(and(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) uTake1(mark(X)) -> mark(uTake1(X)) uTake2(mark(X1),X2,X3,X4) -> mark(uTake2(X1,X2,X3,X4)) uLength(mark(X1),X2) -> mark(uLength(X1,X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(isNat(X)) -> isNat(proper(X)) proper(0()) -> ok(0()) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(nil()) -> ok(nil()) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) proper(uTake1(X)) -> uTake1(proper(X)) proper(uTake2(X1,X2,X3,X4)) -> uTake2(proper(X1),proper(X2),proper(X3),proper(X4)) proper(uLength(X1,X2)) -> uLength(proper(X1),proper(X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatIList(ok(X)) -> ok(isNatIList(X)) isNatList(ok(X)) -> ok(isNatList(X)) isNat(ok(X)) -> ok(isNat(X)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) uTake1(ok(X)) -> ok(uTake1(X)) uTake2(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(uTake2(X1,X2,X3,X4)) uLength(ok(X1),ok(X2)) -> ok(uLength(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dimension: 1 interpretation: [s#](x0) = x0, [top](x0) = 1, [ok](x0) = 0, [proper](x0) = 0, [uLength](x0, x1) = x0 + x1, [uTake2](x0, x1, x2, x3) = x0 + x1 + x2 + x3, [uTake1](x0) = x0, [take](x0, x1) = x0 + x1, [nil] = 0, [cons](x0, x1) = x0 + x1, [zeros] = 1, [length](x0) = x0, [s](x0) = x0, [isNat](x0) = 0, [0] = 0, [isNatList](x0) = 0, [isNatIList](x0) = 0, [mark](x0) = x0 + 1, [active](x0) = x0 + 1, [and](x0, x1) = x0 + x1, [tt] = 0 orientation: s#(mark(X)) = X + 1 >= X = s#(X) active(and(tt(),T)) = T + 1 >= T + 1 = mark(T) active(isNatIList(IL)) = 1 >= 1 = mark(isNatList(IL)) active(isNat(0())) = 1 >= 1 = mark(tt()) active(isNat(s(N))) = 1 >= 1 = mark(isNat(N)) active(isNat(length(L))) = 1 >= 1 = mark(isNatList(L)) active(isNatIList(zeros())) = 1 >= 1 = mark(tt()) active(isNatIList(cons(N,IL))) = 1 >= 1 = mark(and(isNat(N),isNatIList(IL))) active(isNatList(nil())) = 1 >= 1 = mark(tt()) active(isNatList(cons(N,L))) = 1 >= 1 = mark(and(isNat(N),isNatList(L))) active(isNatList(take(N,IL))) = 1 >= 1 = mark(and(isNat(N),isNatIList(IL))) active(zeros()) = 2 >= 2 = mark(cons(0(),zeros())) active(take(0(),IL)) = IL + 1 >= 1 = mark(uTake1(isNatIList(IL))) active(uTake1(tt())) = 1 >= 1 = mark(nil()) active(take(s(M),cons(N,IL))) = IL + M + N + 1 >= IL + M + N + 1 = mark(uTake2(and(isNat(M),and(isNat(N),isNatIList(IL))),M,N,IL)) active(uTake2(tt(),M,N,IL)) = IL + M + N + 1 >= IL + M + N + 1 = mark(cons(N,take(M,IL))) active(length(cons(N,L))) = L + N + 1 >= L + 1 = mark(uLength(and(isNat(N),isNatList(L)),L)) active(uLength(tt(),L)) = L + 1 >= L + 1 = mark(s(length(L))) active(and(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = and(active(X1),X2) active(and(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = and(X1,active(X2)) active(s(X)) = X + 1 >= X + 1 = s(active(X)) active(length(X)) = X + 1 >= X + 1 = length(active(X)) active(cons(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = cons(active(X1),X2) 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(uTake1(X)) = X + 1 >= X + 1 = uTake1(active(X)) active(uTake2(X1,X2,X3,X4)) = X1 + X2 + X3 + X4 + 1 >= X1 + X2 + X3 + X4 + 1 = uTake2(active(X1),X2,X3,X4) active(uLength(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = uLength(active(X1),X2) and(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(and(X1,X2)) and(X1,mark(X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(and(X1,X2)) s(mark(X)) = X + 1 >= X + 1 = mark(s(X)) length(mark(X)) = X + 1 >= X + 1 = mark(length(X)) cons(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(cons(X1,X2)) 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)) uTake1(mark(X)) = X + 1 >= X + 1 = mark(uTake1(X)) uTake2(mark(X1),X2,X3,X4) = X1 + X2 + X3 + X4 + 1 >= X1 + X2 + X3 + X4 + 1 = mark(uTake2(X1,X2,X3,X4)) uLength(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(uLength(X1,X2)) proper(and(X1,X2)) = 0 >= 0 = and(proper(X1),proper(X2)) proper(tt()) = 0 >= 0 = ok(tt()) proper(isNatIList(X)) = 0 >= 0 = isNatIList(proper(X)) proper(isNatList(X)) = 0 >= 0 = isNatList(proper(X)) proper(isNat(X)) = 0 >= 0 = isNat(proper(X)) proper(0()) = 0 >= 0 = ok(0()) proper(s(X)) = 0 >= 0 = s(proper(X)) proper(length(X)) = 0 >= 0 = length(proper(X)) proper(zeros()) = 0 >= 0 = ok(zeros()) proper(cons(X1,X2)) = 0 >= 0 = cons(proper(X1),proper(X2)) proper(nil()) = 0 >= 0 = ok(nil()) proper(take(X1,X2)) = 0 >= 0 = take(proper(X1),proper(X2)) proper(uTake1(X)) = 0 >= 0 = uTake1(proper(X)) proper(uTake2(X1,X2,X3,X4)) = 0 >= 0 = uTake2(proper(X1),proper(X2),proper(X3),proper(X4)) proper(uLength(X1,X2)) = 0 >= 0 = uLength(proper(X1),proper(X2)) and(ok(X1),ok(X2)) = 0 >= 0 = ok(and(X1,X2)) isNatIList(ok(X)) = 0 >= 0 = ok(isNatIList(X)) isNatList(ok(X)) = 0 >= 0 = ok(isNatList(X)) isNat(ok(X)) = 0 >= 0 = ok(isNat(X)) s(ok(X)) = 0 >= 0 = ok(s(X)) length(ok(X)) = 0 >= 0 = ok(length(X)) cons(ok(X1),ok(X2)) = 0 >= 0 = ok(cons(X1,X2)) take(ok(X1),ok(X2)) = 0 >= 0 = ok(take(X1,X2)) uTake1(ok(X)) = 0 >= 0 = ok(uTake1(X)) uTake2(ok(X1),ok(X2),ok(X3),ok(X4)) = 0 >= 0 = ok(uTake2(X1,X2,X3,X4)) uLength(ok(X1),ok(X2)) = 0 >= 0 = ok(uLength(X1,X2)) top(mark(X)) = 1 >= 1 = top(proper(X)) top(ok(X)) = 1 >= 1 = top(active(X)) problem: DPs: TRS: active(and(tt(),T)) -> mark(T) active(isNatIList(IL)) -> mark(isNatList(IL)) active(isNat(0())) -> mark(tt()) active(isNat(s(N))) -> mark(isNat(N)) active(isNat(length(L))) -> mark(isNatList(L)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(N,L))) -> mark(and(isNat(N),isNatList(L))) active(isNatList(take(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(zeros()) -> mark(cons(0(),zeros())) active(take(0(),IL)) -> mark(uTake1(isNatIList(IL))) active(uTake1(tt())) -> mark(nil()) active(take(s(M),cons(N,IL))) -> mark(uTake2(and(isNat(M),and(isNat(N),isNatIList(IL))),M,N,IL)) active(uTake2(tt(),M,N,IL)) -> mark(cons(N,take(M,IL))) active(length(cons(N,L))) -> mark(uLength(and(isNat(N),isNatList(L)),L)) active(uLength(tt(),L)) -> mark(s(length(L))) active(and(X1,X2)) -> and(active(X1),X2) active(and(X1,X2)) -> and(X1,active(X2)) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) active(uTake1(X)) -> uTake1(active(X)) active(uTake2(X1,X2,X3,X4)) -> uTake2(active(X1),X2,X3,X4) active(uLength(X1,X2)) -> uLength(active(X1),X2) and(mark(X1),X2) -> mark(and(X1,X2)) and(X1,mark(X2)) -> mark(and(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) uTake1(mark(X)) -> mark(uTake1(X)) uTake2(mark(X1),X2,X3,X4) -> mark(uTake2(X1,X2,X3,X4)) uLength(mark(X1),X2) -> mark(uLength(X1,X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(isNat(X)) -> isNat(proper(X)) proper(0()) -> ok(0()) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(nil()) -> ok(nil()) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) proper(uTake1(X)) -> uTake1(proper(X)) proper(uTake2(X1,X2,X3,X4)) -> uTake2(proper(X1),proper(X2),proper(X3),proper(X4)) proper(uLength(X1,X2)) -> uLength(proper(X1),proper(X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatIList(ok(X)) -> ok(isNatIList(X)) isNatList(ok(X)) -> ok(isNatList(X)) isNat(ok(X)) -> ok(isNat(X)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) uTake1(ok(X)) -> ok(uTake1(X)) uTake2(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(uTake2(X1,X2,X3,X4)) uLength(ok(X1),ok(X2)) -> ok(uLength(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed DPs: isNat#(ok(X)) -> isNat#(X) TRS: active(and(tt(),T)) -> mark(T) active(isNatIList(IL)) -> mark(isNatList(IL)) active(isNat(0())) -> mark(tt()) active(isNat(s(N))) -> mark(isNat(N)) active(isNat(length(L))) -> mark(isNatList(L)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(N,L))) -> mark(and(isNat(N),isNatList(L))) active(isNatList(take(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(zeros()) -> mark(cons(0(),zeros())) active(take(0(),IL)) -> mark(uTake1(isNatIList(IL))) active(uTake1(tt())) -> mark(nil()) active(take(s(M),cons(N,IL))) -> mark(uTake2(and(isNat(M),and(isNat(N),isNatIList(IL))),M,N,IL)) active(uTake2(tt(),M,N,IL)) -> mark(cons(N,take(M,IL))) active(length(cons(N,L))) -> mark(uLength(and(isNat(N),isNatList(L)),L)) active(uLength(tt(),L)) -> mark(s(length(L))) active(and(X1,X2)) -> and(active(X1),X2) active(and(X1,X2)) -> and(X1,active(X2)) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) active(uTake1(X)) -> uTake1(active(X)) active(uTake2(X1,X2,X3,X4)) -> uTake2(active(X1),X2,X3,X4) active(uLength(X1,X2)) -> uLength(active(X1),X2) and(mark(X1),X2) -> mark(and(X1,X2)) and(X1,mark(X2)) -> mark(and(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) uTake1(mark(X)) -> mark(uTake1(X)) uTake2(mark(X1),X2,X3,X4) -> mark(uTake2(X1,X2,X3,X4)) uLength(mark(X1),X2) -> mark(uLength(X1,X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(isNat(X)) -> isNat(proper(X)) proper(0()) -> ok(0()) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(nil()) -> ok(nil()) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) proper(uTake1(X)) -> uTake1(proper(X)) proper(uTake2(X1,X2,X3,X4)) -> uTake2(proper(X1),proper(X2),proper(X3),proper(X4)) proper(uLength(X1,X2)) -> uLength(proper(X1),proper(X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatIList(ok(X)) -> ok(isNatIList(X)) isNatList(ok(X)) -> ok(isNatList(X)) isNat(ok(X)) -> ok(isNat(X)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) uTake1(ok(X)) -> ok(uTake1(X)) uTake2(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(uTake2(X1,X2,X3,X4)) uLength(ok(X1),ok(X2)) -> ok(uLength(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dimension: 1 interpretation: [isNat#](x0) = x0 + 1, [top](x0) = 1, [ok](x0) = x0 + 1, [proper](x0) = x0 + 1, [uLength](x0, x1) = x0 + 1, [uTake2](x0, x1, x2, x3) = x3, [uTake1](x0) = x0, [take](x0, x1) = x1, [nil] = 0, [cons](x0, x1) = x0, [zeros] = 0, [length](x0) = x0, [s](x0) = x0, [isNat](x0) = x0, [0] = 0, [isNatList](x0) = x0, [isNatIList](x0) = x0 + 1, [mark](x0) = 0, [active](x0) = x0, [and](x0, x1) = x1, [tt] = 0 orientation: isNat#(ok(X)) = X + 2 >= X + 1 = isNat#(X) active(and(tt(),T)) = T >= 0 = mark(T) active(isNatIList(IL)) = IL + 1 >= 0 = mark(isNatList(IL)) active(isNat(0())) = 0 >= 0 = mark(tt()) active(isNat(s(N))) = N >= 0 = mark(isNat(N)) active(isNat(length(L))) = L >= 0 = mark(isNatList(L)) active(isNatIList(zeros())) = 1 >= 0 = mark(tt()) active(isNatIList(cons(N,IL))) = N + 1 >= 0 = mark(and(isNat(N),isNatIList(IL))) active(isNatList(nil())) = 0 >= 0 = mark(tt()) active(isNatList(cons(N,L))) = N >= 0 = mark(and(isNat(N),isNatList(L))) active(isNatList(take(N,IL))) = IL >= 0 = mark(and(isNat(N),isNatIList(IL))) active(zeros()) = 0 >= 0 = mark(cons(0(),zeros())) active(take(0(),IL)) = IL >= 0 = mark(uTake1(isNatIList(IL))) active(uTake1(tt())) = 0 >= 0 = mark(nil()) active(take(s(M),cons(N,IL))) = N >= 0 = mark(uTake2(and(isNat(M),and(isNat(N),isNatIList(IL))),M,N,IL)) active(uTake2(tt(),M,N,IL)) = IL >= 0 = mark(cons(N,take(M,IL))) active(length(cons(N,L))) = N >= 0 = mark(uLength(and(isNat(N),isNatList(L)),L)) active(uLength(tt(),L)) = 1 >= 0 = mark(s(length(L))) active(and(X1,X2)) = X2 >= X2 = and(active(X1),X2) active(and(X1,X2)) = X2 >= X2 = and(X1,active(X2)) active(s(X)) = X >= X = s(active(X)) active(length(X)) = X >= X = length(active(X)) active(cons(X1,X2)) = X1 >= X1 = cons(active(X1),X2) active(take(X1,X2)) = X2 >= X2 = take(active(X1),X2) active(take(X1,X2)) = X2 >= X2 = take(X1,active(X2)) active(uTake1(X)) = X >= X = uTake1(active(X)) active(uTake2(X1,X2,X3,X4)) = X4 >= X4 = uTake2(active(X1),X2,X3,X4) active(uLength(X1,X2)) = X1 + 1 >= X1 + 1 = uLength(active(X1),X2) and(mark(X1),X2) = X2 >= 0 = mark(and(X1,X2)) and(X1,mark(X2)) = 0 >= 0 = mark(and(X1,X2)) s(mark(X)) = 0 >= 0 = mark(s(X)) length(mark(X)) = 0 >= 0 = mark(length(X)) cons(mark(X1),X2) = 0 >= 0 = mark(cons(X1,X2)) take(mark(X1),X2) = X2 >= 0 = mark(take(X1,X2)) take(X1,mark(X2)) = 0 >= 0 = mark(take(X1,X2)) uTake1(mark(X)) = 0 >= 0 = mark(uTake1(X)) uTake2(mark(X1),X2,X3,X4) = X4 >= 0 = mark(uTake2(X1,X2,X3,X4)) uLength(mark(X1),X2) = 1 >= 0 = mark(uLength(X1,X2)) proper(and(X1,X2)) = X2 + 1 >= X2 + 1 = and(proper(X1),proper(X2)) proper(tt()) = 1 >= 1 = ok(tt()) proper(isNatIList(X)) = X + 2 >= X + 2 = isNatIList(proper(X)) proper(isNatList(X)) = X + 1 >= X + 1 = isNatList(proper(X)) proper(isNat(X)) = X + 1 >= X + 1 = isNat(proper(X)) proper(0()) = 1 >= 1 = ok(0()) proper(s(X)) = X + 1 >= X + 1 = s(proper(X)) proper(length(X)) = X + 1 >= X + 1 = length(proper(X)) proper(zeros()) = 1 >= 1 = ok(zeros()) proper(cons(X1,X2)) = X1 + 1 >= X1 + 1 = cons(proper(X1),proper(X2)) proper(nil()) = 1 >= 1 = ok(nil()) proper(take(X1,X2)) = X2 + 1 >= X2 + 1 = take(proper(X1),proper(X2)) proper(uTake1(X)) = X + 1 >= X + 1 = uTake1(proper(X)) proper(uTake2(X1,X2,X3,X4)) = X4 + 1 >= X4 + 1 = uTake2(proper(X1),proper(X2),proper(X3),proper(X4)) proper(uLength(X1,X2)) = X1 + 2 >= X1 + 2 = uLength(proper(X1),proper(X2)) and(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(and(X1,X2)) isNatIList(ok(X)) = X + 2 >= X + 2 = ok(isNatIList(X)) isNatList(ok(X)) = X + 1 >= X + 1 = ok(isNatList(X)) isNat(ok(X)) = X + 1 >= X + 1 = ok(isNat(X)) s(ok(X)) = X + 1 >= X + 1 = ok(s(X)) length(ok(X)) = X + 1 >= X + 1 = ok(length(X)) cons(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(cons(X1,X2)) take(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(take(X1,X2)) uTake1(ok(X)) = X + 1 >= X + 1 = ok(uTake1(X)) uTake2(ok(X1),ok(X2),ok(X3),ok(X4)) = X4 + 1 >= X4 + 1 = ok(uTake2(X1,X2,X3,X4)) uLength(ok(X1),ok(X2)) = X1 + 2 >= X1 + 2 = ok(uLength(X1,X2)) top(mark(X)) = 1 >= 1 = top(proper(X)) top(ok(X)) = 1 >= 1 = top(active(X)) problem: DPs: TRS: active(and(tt(),T)) -> mark(T) active(isNatIList(IL)) -> mark(isNatList(IL)) active(isNat(0())) -> mark(tt()) active(isNat(s(N))) -> mark(isNat(N)) active(isNat(length(L))) -> mark(isNatList(L)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(N,L))) -> mark(and(isNat(N),isNatList(L))) active(isNatList(take(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(zeros()) -> mark(cons(0(),zeros())) active(take(0(),IL)) -> mark(uTake1(isNatIList(IL))) active(uTake1(tt())) -> mark(nil()) active(take(s(M),cons(N,IL))) -> mark(uTake2(and(isNat(M),and(isNat(N),isNatIList(IL))),M,N,IL)) active(uTake2(tt(),M,N,IL)) -> mark(cons(N,take(M,IL))) active(length(cons(N,L))) -> mark(uLength(and(isNat(N),isNatList(L)),L)) active(uLength(tt(),L)) -> mark(s(length(L))) active(and(X1,X2)) -> and(active(X1),X2) active(and(X1,X2)) -> and(X1,active(X2)) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) active(uTake1(X)) -> uTake1(active(X)) active(uTake2(X1,X2,X3,X4)) -> uTake2(active(X1),X2,X3,X4) active(uLength(X1,X2)) -> uLength(active(X1),X2) and(mark(X1),X2) -> mark(and(X1,X2)) and(X1,mark(X2)) -> mark(and(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) uTake1(mark(X)) -> mark(uTake1(X)) uTake2(mark(X1),X2,X3,X4) -> mark(uTake2(X1,X2,X3,X4)) uLength(mark(X1),X2) -> mark(uLength(X1,X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(isNat(X)) -> isNat(proper(X)) proper(0()) -> ok(0()) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(nil()) -> ok(nil()) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) proper(uTake1(X)) -> uTake1(proper(X)) proper(uTake2(X1,X2,X3,X4)) -> uTake2(proper(X1),proper(X2),proper(X3),proper(X4)) proper(uLength(X1,X2)) -> uLength(proper(X1),proper(X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatIList(ok(X)) -> ok(isNatIList(X)) isNatList(ok(X)) -> ok(isNatList(X)) isNat(ok(X)) -> ok(isNat(X)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) uTake1(ok(X)) -> ok(uTake1(X)) uTake2(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(uTake2(X1,X2,X3,X4)) uLength(ok(X1),ok(X2)) -> ok(uLength(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed DPs: isNatList#(ok(X)) -> isNatList#(X) TRS: active(and(tt(),T)) -> mark(T) active(isNatIList(IL)) -> mark(isNatList(IL)) active(isNat(0())) -> mark(tt()) active(isNat(s(N))) -> mark(isNat(N)) active(isNat(length(L))) -> mark(isNatList(L)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(N,L))) -> mark(and(isNat(N),isNatList(L))) active(isNatList(take(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(zeros()) -> mark(cons(0(),zeros())) active(take(0(),IL)) -> mark(uTake1(isNatIList(IL))) active(uTake1(tt())) -> mark(nil()) active(take(s(M),cons(N,IL))) -> mark(uTake2(and(isNat(M),and(isNat(N),isNatIList(IL))),M,N,IL)) active(uTake2(tt(),M,N,IL)) -> mark(cons(N,take(M,IL))) active(length(cons(N,L))) -> mark(uLength(and(isNat(N),isNatList(L)),L)) active(uLength(tt(),L)) -> mark(s(length(L))) active(and(X1,X2)) -> and(active(X1),X2) active(and(X1,X2)) -> and(X1,active(X2)) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) active(uTake1(X)) -> uTake1(active(X)) active(uTake2(X1,X2,X3,X4)) -> uTake2(active(X1),X2,X3,X4) active(uLength(X1,X2)) -> uLength(active(X1),X2) and(mark(X1),X2) -> mark(and(X1,X2)) and(X1,mark(X2)) -> mark(and(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) uTake1(mark(X)) -> mark(uTake1(X)) uTake2(mark(X1),X2,X3,X4) -> mark(uTake2(X1,X2,X3,X4)) uLength(mark(X1),X2) -> mark(uLength(X1,X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(isNat(X)) -> isNat(proper(X)) proper(0()) -> ok(0()) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(nil()) -> ok(nil()) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) proper(uTake1(X)) -> uTake1(proper(X)) proper(uTake2(X1,X2,X3,X4)) -> uTake2(proper(X1),proper(X2),proper(X3),proper(X4)) proper(uLength(X1,X2)) -> uLength(proper(X1),proper(X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatIList(ok(X)) -> ok(isNatIList(X)) isNatList(ok(X)) -> ok(isNatList(X)) isNat(ok(X)) -> ok(isNat(X)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) uTake1(ok(X)) -> ok(uTake1(X)) uTake2(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(uTake2(X1,X2,X3,X4)) uLength(ok(X1),ok(X2)) -> ok(uLength(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dimension: 1 interpretation: [isNatList#](x0) = x0 + 1, [top](x0) = 1, [ok](x0) = x0 + 1, [proper](x0) = x0 + 1, [uLength](x0, x1) = x0 + 1, [uTake2](x0, x1, x2, x3) = x3, [uTake1](x0) = x0, [take](x0, x1) = x1, [nil] = 0, [cons](x0, x1) = x0, [zeros] = 0, [length](x0) = x0, [s](x0) = x0, [isNat](x0) = x0, [0] = 0, [isNatList](x0) = x0, [isNatIList](x0) = x0 + 1, [mark](x0) = 0, [active](x0) = x0, [and](x0, x1) = x1, [tt] = 0 orientation: isNatList#(ok(X)) = X + 2 >= X + 1 = isNatList#(X) active(and(tt(),T)) = T >= 0 = mark(T) active(isNatIList(IL)) = IL + 1 >= 0 = mark(isNatList(IL)) active(isNat(0())) = 0 >= 0 = mark(tt()) active(isNat(s(N))) = N >= 0 = mark(isNat(N)) active(isNat(length(L))) = L >= 0 = mark(isNatList(L)) active(isNatIList(zeros())) = 1 >= 0 = mark(tt()) active(isNatIList(cons(N,IL))) = N + 1 >= 0 = mark(and(isNat(N),isNatIList(IL))) active(isNatList(nil())) = 0 >= 0 = mark(tt()) active(isNatList(cons(N,L))) = N >= 0 = mark(and(isNat(N),isNatList(L))) active(isNatList(take(N,IL))) = IL >= 0 = mark(and(isNat(N),isNatIList(IL))) active(zeros()) = 0 >= 0 = mark(cons(0(),zeros())) active(take(0(),IL)) = IL >= 0 = mark(uTake1(isNatIList(IL))) active(uTake1(tt())) = 0 >= 0 = mark(nil()) active(take(s(M),cons(N,IL))) = N >= 0 = mark(uTake2(and(isNat(M),and(isNat(N),isNatIList(IL))),M,N,IL)) active(uTake2(tt(),M,N,IL)) = IL >= 0 = mark(cons(N,take(M,IL))) active(length(cons(N,L))) = N >= 0 = mark(uLength(and(isNat(N),isNatList(L)),L)) active(uLength(tt(),L)) = 1 >= 0 = mark(s(length(L))) active(and(X1,X2)) = X2 >= X2 = and(active(X1),X2) active(and(X1,X2)) = X2 >= X2 = and(X1,active(X2)) active(s(X)) = X >= X = s(active(X)) active(length(X)) = X >= X = length(active(X)) active(cons(X1,X2)) = X1 >= X1 = cons(active(X1),X2) active(take(X1,X2)) = X2 >= X2 = take(active(X1),X2) active(take(X1,X2)) = X2 >= X2 = take(X1,active(X2)) active(uTake1(X)) = X >= X = uTake1(active(X)) active(uTake2(X1,X2,X3,X4)) = X4 >= X4 = uTake2(active(X1),X2,X3,X4) active(uLength(X1,X2)) = X1 + 1 >= X1 + 1 = uLength(active(X1),X2) and(mark(X1),X2) = X2 >= 0 = mark(and(X1,X2)) and(X1,mark(X2)) = 0 >= 0 = mark(and(X1,X2)) s(mark(X)) = 0 >= 0 = mark(s(X)) length(mark(X)) = 0 >= 0 = mark(length(X)) cons(mark(X1),X2) = 0 >= 0 = mark(cons(X1,X2)) take(mark(X1),X2) = X2 >= 0 = mark(take(X1,X2)) take(X1,mark(X2)) = 0 >= 0 = mark(take(X1,X2)) uTake1(mark(X)) = 0 >= 0 = mark(uTake1(X)) uTake2(mark(X1),X2,X3,X4) = X4 >= 0 = mark(uTake2(X1,X2,X3,X4)) uLength(mark(X1),X2) = 1 >= 0 = mark(uLength(X1,X2)) proper(and(X1,X2)) = X2 + 1 >= X2 + 1 = and(proper(X1),proper(X2)) proper(tt()) = 1 >= 1 = ok(tt()) proper(isNatIList(X)) = X + 2 >= X + 2 = isNatIList(proper(X)) proper(isNatList(X)) = X + 1 >= X + 1 = isNatList(proper(X)) proper(isNat(X)) = X + 1 >= X + 1 = isNat(proper(X)) proper(0()) = 1 >= 1 = ok(0()) proper(s(X)) = X + 1 >= X + 1 = s(proper(X)) proper(length(X)) = X + 1 >= X + 1 = length(proper(X)) proper(zeros()) = 1 >= 1 = ok(zeros()) proper(cons(X1,X2)) = X1 + 1 >= X1 + 1 = cons(proper(X1),proper(X2)) proper(nil()) = 1 >= 1 = ok(nil()) proper(take(X1,X2)) = X2 + 1 >= X2 + 1 = take(proper(X1),proper(X2)) proper(uTake1(X)) = X + 1 >= X + 1 = uTake1(proper(X)) proper(uTake2(X1,X2,X3,X4)) = X4 + 1 >= X4 + 1 = uTake2(proper(X1),proper(X2),proper(X3),proper(X4)) proper(uLength(X1,X2)) = X1 + 2 >= X1 + 2 = uLength(proper(X1),proper(X2)) and(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(and(X1,X2)) isNatIList(ok(X)) = X + 2 >= X + 2 = ok(isNatIList(X)) isNatList(ok(X)) = X + 1 >= X + 1 = ok(isNatList(X)) isNat(ok(X)) = X + 1 >= X + 1 = ok(isNat(X)) s(ok(X)) = X + 1 >= X + 1 = ok(s(X)) length(ok(X)) = X + 1 >= X + 1 = ok(length(X)) cons(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(cons(X1,X2)) take(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(take(X1,X2)) uTake1(ok(X)) = X + 1 >= X + 1 = ok(uTake1(X)) uTake2(ok(X1),ok(X2),ok(X3),ok(X4)) = X4 + 1 >= X4 + 1 = ok(uTake2(X1,X2,X3,X4)) uLength(ok(X1),ok(X2)) = X1 + 2 >= X1 + 2 = ok(uLength(X1,X2)) top(mark(X)) = 1 >= 1 = top(proper(X)) top(ok(X)) = 1 >= 1 = top(active(X)) problem: DPs: TRS: active(and(tt(),T)) -> mark(T) active(isNatIList(IL)) -> mark(isNatList(IL)) active(isNat(0())) -> mark(tt()) active(isNat(s(N))) -> mark(isNat(N)) active(isNat(length(L))) -> mark(isNatList(L)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(N,L))) -> mark(and(isNat(N),isNatList(L))) active(isNatList(take(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(zeros()) -> mark(cons(0(),zeros())) active(take(0(),IL)) -> mark(uTake1(isNatIList(IL))) active(uTake1(tt())) -> mark(nil()) active(take(s(M),cons(N,IL))) -> mark(uTake2(and(isNat(M),and(isNat(N),isNatIList(IL))),M,N,IL)) active(uTake2(tt(),M,N,IL)) -> mark(cons(N,take(M,IL))) active(length(cons(N,L))) -> mark(uLength(and(isNat(N),isNatList(L)),L)) active(uLength(tt(),L)) -> mark(s(length(L))) active(and(X1,X2)) -> and(active(X1),X2) active(and(X1,X2)) -> and(X1,active(X2)) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) active(uTake1(X)) -> uTake1(active(X)) active(uTake2(X1,X2,X3,X4)) -> uTake2(active(X1),X2,X3,X4) active(uLength(X1,X2)) -> uLength(active(X1),X2) and(mark(X1),X2) -> mark(and(X1,X2)) and(X1,mark(X2)) -> mark(and(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) uTake1(mark(X)) -> mark(uTake1(X)) uTake2(mark(X1),X2,X3,X4) -> mark(uTake2(X1,X2,X3,X4)) uLength(mark(X1),X2) -> mark(uLength(X1,X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(isNat(X)) -> isNat(proper(X)) proper(0()) -> ok(0()) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(nil()) -> ok(nil()) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) proper(uTake1(X)) -> uTake1(proper(X)) proper(uTake2(X1,X2,X3,X4)) -> uTake2(proper(X1),proper(X2),proper(X3),proper(X4)) proper(uLength(X1,X2)) -> uLength(proper(X1),proper(X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatIList(ok(X)) -> ok(isNatIList(X)) isNatList(ok(X)) -> ok(isNatList(X)) isNat(ok(X)) -> ok(isNat(X)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) uTake1(ok(X)) -> ok(uTake1(X)) uTake2(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(uTake2(X1,X2,X3,X4)) uLength(ok(X1),ok(X2)) -> ok(uLength(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed DPs: isNatIList#(ok(X)) -> isNatIList#(X) TRS: active(and(tt(),T)) -> mark(T) active(isNatIList(IL)) -> mark(isNatList(IL)) active(isNat(0())) -> mark(tt()) active(isNat(s(N))) -> mark(isNat(N)) active(isNat(length(L))) -> mark(isNatList(L)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(N,L))) -> mark(and(isNat(N),isNatList(L))) active(isNatList(take(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(zeros()) -> mark(cons(0(),zeros())) active(take(0(),IL)) -> mark(uTake1(isNatIList(IL))) active(uTake1(tt())) -> mark(nil()) active(take(s(M),cons(N,IL))) -> mark(uTake2(and(isNat(M),and(isNat(N),isNatIList(IL))),M,N,IL)) active(uTake2(tt(),M,N,IL)) -> mark(cons(N,take(M,IL))) active(length(cons(N,L))) -> mark(uLength(and(isNat(N),isNatList(L)),L)) active(uLength(tt(),L)) -> mark(s(length(L))) active(and(X1,X2)) -> and(active(X1),X2) active(and(X1,X2)) -> and(X1,active(X2)) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) active(uTake1(X)) -> uTake1(active(X)) active(uTake2(X1,X2,X3,X4)) -> uTake2(active(X1),X2,X3,X4) active(uLength(X1,X2)) -> uLength(active(X1),X2) and(mark(X1),X2) -> mark(and(X1,X2)) and(X1,mark(X2)) -> mark(and(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) uTake1(mark(X)) -> mark(uTake1(X)) uTake2(mark(X1),X2,X3,X4) -> mark(uTake2(X1,X2,X3,X4)) uLength(mark(X1),X2) -> mark(uLength(X1,X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(isNat(X)) -> isNat(proper(X)) proper(0()) -> ok(0()) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(nil()) -> ok(nil()) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) proper(uTake1(X)) -> uTake1(proper(X)) proper(uTake2(X1,X2,X3,X4)) -> uTake2(proper(X1),proper(X2),proper(X3),proper(X4)) proper(uLength(X1,X2)) -> uLength(proper(X1),proper(X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatIList(ok(X)) -> ok(isNatIList(X)) isNatList(ok(X)) -> ok(isNatList(X)) isNat(ok(X)) -> ok(isNat(X)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) uTake1(ok(X)) -> ok(uTake1(X)) uTake2(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(uTake2(X1,X2,X3,X4)) uLength(ok(X1),ok(X2)) -> ok(uLength(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dimension: 1 interpretation: [isNatIList#](x0) = x0 + 1, [top](x0) = 1, [ok](x0) = x0 + 1, [proper](x0) = x0 + 1, [uLength](x0, x1) = x0 + 1, [uTake2](x0, x1, x2, x3) = x3, [uTake1](x0) = x0, [take](x0, x1) = x1, [nil] = 0, [cons](x0, x1) = x0, [zeros] = 0, [length](x0) = x0, [s](x0) = x0, [isNat](x0) = x0, [0] = 0, [isNatList](x0) = x0, [isNatIList](x0) = x0 + 1, [mark](x0) = 0, [active](x0) = x0, [and](x0, x1) = x1, [tt] = 0 orientation: isNatIList#(ok(X)) = X + 2 >= X + 1 = isNatIList#(X) active(and(tt(),T)) = T >= 0 = mark(T) active(isNatIList(IL)) = IL + 1 >= 0 = mark(isNatList(IL)) active(isNat(0())) = 0 >= 0 = mark(tt()) active(isNat(s(N))) = N >= 0 = mark(isNat(N)) active(isNat(length(L))) = L >= 0 = mark(isNatList(L)) active(isNatIList(zeros())) = 1 >= 0 = mark(tt()) active(isNatIList(cons(N,IL))) = N + 1 >= 0 = mark(and(isNat(N),isNatIList(IL))) active(isNatList(nil())) = 0 >= 0 = mark(tt()) active(isNatList(cons(N,L))) = N >= 0 = mark(and(isNat(N),isNatList(L))) active(isNatList(take(N,IL))) = IL >= 0 = mark(and(isNat(N),isNatIList(IL))) active(zeros()) = 0 >= 0 = mark(cons(0(),zeros())) active(take(0(),IL)) = IL >= 0 = mark(uTake1(isNatIList(IL))) active(uTake1(tt())) = 0 >= 0 = mark(nil()) active(take(s(M),cons(N,IL))) = N >= 0 = mark(uTake2(and(isNat(M),and(isNat(N),isNatIList(IL))),M,N,IL)) active(uTake2(tt(),M,N,IL)) = IL >= 0 = mark(cons(N,take(M,IL))) active(length(cons(N,L))) = N >= 0 = mark(uLength(and(isNat(N),isNatList(L)),L)) active(uLength(tt(),L)) = 1 >= 0 = mark(s(length(L))) active(and(X1,X2)) = X2 >= X2 = and(active(X1),X2) active(and(X1,X2)) = X2 >= X2 = and(X1,active(X2)) active(s(X)) = X >= X = s(active(X)) active(length(X)) = X >= X = length(active(X)) active(cons(X1,X2)) = X1 >= X1 = cons(active(X1),X2) active(take(X1,X2)) = X2 >= X2 = take(active(X1),X2) active(take(X1,X2)) = X2 >= X2 = take(X1,active(X2)) active(uTake1(X)) = X >= X = uTake1(active(X)) active(uTake2(X1,X2,X3,X4)) = X4 >= X4 = uTake2(active(X1),X2,X3,X4) active(uLength(X1,X2)) = X1 + 1 >= X1 + 1 = uLength(active(X1),X2) and(mark(X1),X2) = X2 >= 0 = mark(and(X1,X2)) and(X1,mark(X2)) = 0 >= 0 = mark(and(X1,X2)) s(mark(X)) = 0 >= 0 = mark(s(X)) length(mark(X)) = 0 >= 0 = mark(length(X)) cons(mark(X1),X2) = 0 >= 0 = mark(cons(X1,X2)) take(mark(X1),X2) = X2 >= 0 = mark(take(X1,X2)) take(X1,mark(X2)) = 0 >= 0 = mark(take(X1,X2)) uTake1(mark(X)) = 0 >= 0 = mark(uTake1(X)) uTake2(mark(X1),X2,X3,X4) = X4 >= 0 = mark(uTake2(X1,X2,X3,X4)) uLength(mark(X1),X2) = 1 >= 0 = mark(uLength(X1,X2)) proper(and(X1,X2)) = X2 + 1 >= X2 + 1 = and(proper(X1),proper(X2)) proper(tt()) = 1 >= 1 = ok(tt()) proper(isNatIList(X)) = X + 2 >= X + 2 = isNatIList(proper(X)) proper(isNatList(X)) = X + 1 >= X + 1 = isNatList(proper(X)) proper(isNat(X)) = X + 1 >= X + 1 = isNat(proper(X)) proper(0()) = 1 >= 1 = ok(0()) proper(s(X)) = X + 1 >= X + 1 = s(proper(X)) proper(length(X)) = X + 1 >= X + 1 = length(proper(X)) proper(zeros()) = 1 >= 1 = ok(zeros()) proper(cons(X1,X2)) = X1 + 1 >= X1 + 1 = cons(proper(X1),proper(X2)) proper(nil()) = 1 >= 1 = ok(nil()) proper(take(X1,X2)) = X2 + 1 >= X2 + 1 = take(proper(X1),proper(X2)) proper(uTake1(X)) = X + 1 >= X + 1 = uTake1(proper(X)) proper(uTake2(X1,X2,X3,X4)) = X4 + 1 >= X4 + 1 = uTake2(proper(X1),proper(X2),proper(X3),proper(X4)) proper(uLength(X1,X2)) = X1 + 2 >= X1 + 2 = uLength(proper(X1),proper(X2)) and(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(and(X1,X2)) isNatIList(ok(X)) = X + 2 >= X + 2 = ok(isNatIList(X)) isNatList(ok(X)) = X + 1 >= X + 1 = ok(isNatList(X)) isNat(ok(X)) = X + 1 >= X + 1 = ok(isNat(X)) s(ok(X)) = X + 1 >= X + 1 = ok(s(X)) length(ok(X)) = X + 1 >= X + 1 = ok(length(X)) cons(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(cons(X1,X2)) take(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(take(X1,X2)) uTake1(ok(X)) = X + 1 >= X + 1 = ok(uTake1(X)) uTake2(ok(X1),ok(X2),ok(X3),ok(X4)) = X4 + 1 >= X4 + 1 = ok(uTake2(X1,X2,X3,X4)) uLength(ok(X1),ok(X2)) = X1 + 2 >= X1 + 2 = ok(uLength(X1,X2)) top(mark(X)) = 1 >= 1 = top(proper(X)) top(ok(X)) = 1 >= 1 = top(active(X)) problem: DPs: TRS: active(and(tt(),T)) -> mark(T) active(isNatIList(IL)) -> mark(isNatList(IL)) active(isNat(0())) -> mark(tt()) active(isNat(s(N))) -> mark(isNat(N)) active(isNat(length(L))) -> mark(isNatList(L)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(N,L))) -> mark(and(isNat(N),isNatList(L))) active(isNatList(take(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(zeros()) -> mark(cons(0(),zeros())) active(take(0(),IL)) -> mark(uTake1(isNatIList(IL))) active(uTake1(tt())) -> mark(nil()) active(take(s(M),cons(N,IL))) -> mark(uTake2(and(isNat(M),and(isNat(N),isNatIList(IL))),M,N,IL)) active(uTake2(tt(),M,N,IL)) -> mark(cons(N,take(M,IL))) active(length(cons(N,L))) -> mark(uLength(and(isNat(N),isNatList(L)),L)) active(uLength(tt(),L)) -> mark(s(length(L))) active(and(X1,X2)) -> and(active(X1),X2) active(and(X1,X2)) -> and(X1,active(X2)) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) active(uTake1(X)) -> uTake1(active(X)) active(uTake2(X1,X2,X3,X4)) -> uTake2(active(X1),X2,X3,X4) active(uLength(X1,X2)) -> uLength(active(X1),X2) and(mark(X1),X2) -> mark(and(X1,X2)) and(X1,mark(X2)) -> mark(and(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) uTake1(mark(X)) -> mark(uTake1(X)) uTake2(mark(X1),X2,X3,X4) -> mark(uTake2(X1,X2,X3,X4)) uLength(mark(X1),X2) -> mark(uLength(X1,X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(isNat(X)) -> isNat(proper(X)) proper(0()) -> ok(0()) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(nil()) -> ok(nil()) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) proper(uTake1(X)) -> uTake1(proper(X)) proper(uTake2(X1,X2,X3,X4)) -> uTake2(proper(X1),proper(X2),proper(X3),proper(X4)) proper(uLength(X1,X2)) -> uLength(proper(X1),proper(X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatIList(ok(X)) -> ok(isNatIList(X)) isNatList(ok(X)) -> ok(isNatList(X)) isNat(ok(X)) -> ok(isNat(X)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) uTake1(ok(X)) -> ok(uTake1(X)) uTake2(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(uTake2(X1,X2,X3,X4)) uLength(ok(X1),ok(X2)) -> ok(uLength(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed DPs: and#(mark(X1),X2) -> and#(X1,X2) and#(X1,mark(X2)) -> and#(X1,X2) and#(ok(X1),ok(X2)) -> and#(X1,X2) TRS: active(and(tt(),T)) -> mark(T) active(isNatIList(IL)) -> mark(isNatList(IL)) active(isNat(0())) -> mark(tt()) active(isNat(s(N))) -> mark(isNat(N)) active(isNat(length(L))) -> mark(isNatList(L)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(N,L))) -> mark(and(isNat(N),isNatList(L))) active(isNatList(take(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(zeros()) -> mark(cons(0(),zeros())) active(take(0(),IL)) -> mark(uTake1(isNatIList(IL))) active(uTake1(tt())) -> mark(nil()) active(take(s(M),cons(N,IL))) -> mark(uTake2(and(isNat(M),and(isNat(N),isNatIList(IL))),M,N,IL)) active(uTake2(tt(),M,N,IL)) -> mark(cons(N,take(M,IL))) active(length(cons(N,L))) -> mark(uLength(and(isNat(N),isNatList(L)),L)) active(uLength(tt(),L)) -> mark(s(length(L))) active(and(X1,X2)) -> and(active(X1),X2) active(and(X1,X2)) -> and(X1,active(X2)) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) active(uTake1(X)) -> uTake1(active(X)) active(uTake2(X1,X2,X3,X4)) -> uTake2(active(X1),X2,X3,X4) active(uLength(X1,X2)) -> uLength(active(X1),X2) and(mark(X1),X2) -> mark(and(X1,X2)) and(X1,mark(X2)) -> mark(and(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) uTake1(mark(X)) -> mark(uTake1(X)) uTake2(mark(X1),X2,X3,X4) -> mark(uTake2(X1,X2,X3,X4)) uLength(mark(X1),X2) -> mark(uLength(X1,X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(isNat(X)) -> isNat(proper(X)) proper(0()) -> ok(0()) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(nil()) -> ok(nil()) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) proper(uTake1(X)) -> uTake1(proper(X)) proper(uTake2(X1,X2,X3,X4)) -> uTake2(proper(X1),proper(X2),proper(X3),proper(X4)) proper(uLength(X1,X2)) -> uLength(proper(X1),proper(X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatIList(ok(X)) -> ok(isNatIList(X)) isNatList(ok(X)) -> ok(isNatList(X)) isNat(ok(X)) -> ok(isNat(X)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) uTake1(ok(X)) -> ok(uTake1(X)) uTake2(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(uTake2(X1,X2,X3,X4)) uLength(ok(X1),ok(X2)) -> ok(uLength(X1,X2)) 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) = 1, [uLength](x0, x1) = x1, [uTake2](x0, x1, x2, x3) = x3, [uTake1](x0) = x0, [take](x0, x1) = x1, [nil] = 0, [cons](x0, x1) = x1, [zeros] = 0, [length](x0) = x0, [s](x0) = x0, [isNat](x0) = x0, [0] = 0, [isNatList](x0) = x0, [isNatIList](x0) = x0, [mark](x0) = x0, [active](x0) = x0, [and](x0, x1) = x1, [tt] = 0 orientation: and#(mark(X1),X2) = X2 + 1 >= X2 + 1 = and#(X1,X2) and#(X1,mark(X2)) = X2 + 1 >= X2 + 1 = and#(X1,X2) and#(ok(X1),ok(X2)) = X2 + 2 >= X2 + 1 = and#(X1,X2) active(and(tt(),T)) = T >= T = mark(T) active(isNatIList(IL)) = IL >= IL = mark(isNatList(IL)) active(isNat(0())) = 0 >= 0 = mark(tt()) active(isNat(s(N))) = N >= N = mark(isNat(N)) active(isNat(length(L))) = L >= L = mark(isNatList(L)) active(isNatIList(zeros())) = 0 >= 0 = mark(tt()) active(isNatIList(cons(N,IL))) = IL >= IL = mark(and(isNat(N),isNatIList(IL))) active(isNatList(nil())) = 0 >= 0 = mark(tt()) active(isNatList(cons(N,L))) = L >= L = mark(and(isNat(N),isNatList(L))) active(isNatList(take(N,IL))) = IL >= IL = mark(and(isNat(N),isNatIList(IL))) active(zeros()) = 0 >= 0 = mark(cons(0(),zeros())) active(take(0(),IL)) = IL >= IL = mark(uTake1(isNatIList(IL))) active(uTake1(tt())) = 0 >= 0 = mark(nil()) active(take(s(M),cons(N,IL))) = IL >= IL = mark(uTake2(and(isNat(M),and(isNat(N),isNatIList(IL))),M,N,IL)) active(uTake2(tt(),M,N,IL)) = IL >= IL = mark(cons(N,take(M,IL))) active(length(cons(N,L))) = L >= L = mark(uLength(and(isNat(N),isNatList(L)),L)) active(uLength(tt(),L)) = L >= L = mark(s(length(L))) active(and(X1,X2)) = X2 >= X2 = and(active(X1),X2) active(and(X1,X2)) = X2 >= X2 = and(X1,active(X2)) active(s(X)) = X >= X = s(active(X)) active(length(X)) = X >= X = length(active(X)) active(cons(X1,X2)) = X2 >= X2 = cons(active(X1),X2) active(take(X1,X2)) = X2 >= X2 = take(active(X1),X2) active(take(X1,X2)) = X2 >= X2 = take(X1,active(X2)) active(uTake1(X)) = X >= X = uTake1(active(X)) active(uTake2(X1,X2,X3,X4)) = X4 >= X4 = uTake2(active(X1),X2,X3,X4) active(uLength(X1,X2)) = X2 >= X2 = uLength(active(X1),X2) and(mark(X1),X2) = X2 >= X2 = mark(and(X1,X2)) and(X1,mark(X2)) = X2 >= X2 = mark(and(X1,X2)) s(mark(X)) = X >= X = mark(s(X)) length(mark(X)) = X >= X = mark(length(X)) cons(mark(X1),X2) = X2 >= X2 = mark(cons(X1,X2)) take(mark(X1),X2) = X2 >= X2 = mark(take(X1,X2)) take(X1,mark(X2)) = X2 >= X2 = mark(take(X1,X2)) uTake1(mark(X)) = X >= X = mark(uTake1(X)) uTake2(mark(X1),X2,X3,X4) = X4 >= X4 = mark(uTake2(X1,X2,X3,X4)) uLength(mark(X1),X2) = X2 >= X2 = mark(uLength(X1,X2)) proper(and(X1,X2)) = 1 >= 1 = and(proper(X1),proper(X2)) proper(tt()) = 1 >= 1 = ok(tt()) proper(isNatIList(X)) = 1 >= 1 = isNatIList(proper(X)) proper(isNatList(X)) = 1 >= 1 = isNatList(proper(X)) proper(isNat(X)) = 1 >= 1 = isNat(proper(X)) proper(0()) = 1 >= 1 = ok(0()) proper(s(X)) = 1 >= 1 = s(proper(X)) proper(length(X)) = 1 >= 1 = length(proper(X)) proper(zeros()) = 1 >= 1 = ok(zeros()) proper(cons(X1,X2)) = 1 >= 1 = cons(proper(X1),proper(X2)) proper(nil()) = 1 >= 1 = ok(nil()) proper(take(X1,X2)) = 1 >= 1 = take(proper(X1),proper(X2)) proper(uTake1(X)) = 1 >= 1 = uTake1(proper(X)) proper(uTake2(X1,X2,X3,X4)) = 1 >= 1 = uTake2(proper(X1),proper(X2),proper(X3),proper(X4)) proper(uLength(X1,X2)) = 1 >= 1 = uLength(proper(X1),proper(X2)) and(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(and(X1,X2)) isNatIList(ok(X)) = X + 1 >= X + 1 = ok(isNatIList(X)) isNatList(ok(X)) = X + 1 >= X + 1 = ok(isNatList(X)) isNat(ok(X)) = X + 1 >= X + 1 = ok(isNat(X)) s(ok(X)) = X + 1 >= X + 1 = ok(s(X)) length(ok(X)) = X + 1 >= X + 1 = ok(length(X)) cons(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(cons(X1,X2)) take(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(take(X1,X2)) uTake1(ok(X)) = X + 1 >= X + 1 = ok(uTake1(X)) uTake2(ok(X1),ok(X2),ok(X3),ok(X4)) = X4 + 1 >= X4 + 1 = ok(uTake2(X1,X2,X3,X4)) uLength(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(uLength(X1,X2)) 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) and#(X1,mark(X2)) -> and#(X1,X2) TRS: active(and(tt(),T)) -> mark(T) active(isNatIList(IL)) -> mark(isNatList(IL)) active(isNat(0())) -> mark(tt()) active(isNat(s(N))) -> mark(isNat(N)) active(isNat(length(L))) -> mark(isNatList(L)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(N,L))) -> mark(and(isNat(N),isNatList(L))) active(isNatList(take(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(zeros()) -> mark(cons(0(),zeros())) active(take(0(),IL)) -> mark(uTake1(isNatIList(IL))) active(uTake1(tt())) -> mark(nil()) active(take(s(M),cons(N,IL))) -> mark(uTake2(and(isNat(M),and(isNat(N),isNatIList(IL))),M,N,IL)) active(uTake2(tt(),M,N,IL)) -> mark(cons(N,take(M,IL))) active(length(cons(N,L))) -> mark(uLength(and(isNat(N),isNatList(L)),L)) active(uLength(tt(),L)) -> mark(s(length(L))) active(and(X1,X2)) -> and(active(X1),X2) active(and(X1,X2)) -> and(X1,active(X2)) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) active(uTake1(X)) -> uTake1(active(X)) active(uTake2(X1,X2,X3,X4)) -> uTake2(active(X1),X2,X3,X4) active(uLength(X1,X2)) -> uLength(active(X1),X2) and(mark(X1),X2) -> mark(and(X1,X2)) and(X1,mark(X2)) -> mark(and(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) uTake1(mark(X)) -> mark(uTake1(X)) uTake2(mark(X1),X2,X3,X4) -> mark(uTake2(X1,X2,X3,X4)) uLength(mark(X1),X2) -> mark(uLength(X1,X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(isNat(X)) -> isNat(proper(X)) proper(0()) -> ok(0()) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(nil()) -> ok(nil()) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) proper(uTake1(X)) -> uTake1(proper(X)) proper(uTake2(X1,X2,X3,X4)) -> uTake2(proper(X1),proper(X2),proper(X3),proper(X4)) proper(uLength(X1,X2)) -> uLength(proper(X1),proper(X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatIList(ok(X)) -> ok(isNatIList(X)) isNatList(ok(X)) -> ok(isNatList(X)) isNat(ok(X)) -> ok(isNat(X)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) uTake1(ok(X)) -> ok(uTake1(X)) uTake2(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(uTake2(X1,X2,X3,X4)) uLength(ok(X1),ok(X2)) -> ok(uLength(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dimension: 1 interpretation: [and#](x0, x1) = x0, [top](x0) = 0, [ok](x0) = 0, [proper](x0) = 0, [uLength](x0, x1) = x0 + x1, [uTake2](x0, x1, x2, x3) = x0 + x1 + x2 + x3, [uTake1](x0) = x0, [take](x0, x1) = x0 + x1, [nil] = 0, [cons](x0, x1) = x0 + x1, [zeros] = 0, [length](x0) = x0, [s](x0) = x0, [isNat](x0) = 0, [0] = 0, [isNatList](x0) = 0, [isNatIList](x0) = 0, [mark](x0) = x0 + 1, [active](x0) = x0 + 1, [and](x0, x1) = x0 + x1, [tt] = 0 orientation: and#(mark(X1),X2) = X1 + 1 >= X1 = and#(X1,X2) and#(X1,mark(X2)) = X1 >= X1 = and#(X1,X2) active(and(tt(),T)) = T + 1 >= T + 1 = mark(T) active(isNatIList(IL)) = 1 >= 1 = mark(isNatList(IL)) active(isNat(0())) = 1 >= 1 = mark(tt()) active(isNat(s(N))) = 1 >= 1 = mark(isNat(N)) active(isNat(length(L))) = 1 >= 1 = mark(isNatList(L)) active(isNatIList(zeros())) = 1 >= 1 = mark(tt()) active(isNatIList(cons(N,IL))) = 1 >= 1 = mark(and(isNat(N),isNatIList(IL))) active(isNatList(nil())) = 1 >= 1 = mark(tt()) active(isNatList(cons(N,L))) = 1 >= 1 = mark(and(isNat(N),isNatList(L))) active(isNatList(take(N,IL))) = 1 >= 1 = mark(and(isNat(N),isNatIList(IL))) active(zeros()) = 1 >= 1 = mark(cons(0(),zeros())) active(take(0(),IL)) = IL + 1 >= 1 = mark(uTake1(isNatIList(IL))) active(uTake1(tt())) = 1 >= 1 = mark(nil()) active(take(s(M),cons(N,IL))) = IL + M + N + 1 >= IL + M + N + 1 = mark(uTake2(and(isNat(M),and(isNat(N),isNatIList(IL))),M,N,IL)) active(uTake2(tt(),M,N,IL)) = IL + M + N + 1 >= IL + M + N + 1 = mark(cons(N,take(M,IL))) active(length(cons(N,L))) = L + N + 1 >= L + 1 = mark(uLength(and(isNat(N),isNatList(L)),L)) active(uLength(tt(),L)) = L + 1 >= L + 1 = mark(s(length(L))) active(and(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = and(active(X1),X2) active(and(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = and(X1,active(X2)) active(s(X)) = X + 1 >= X + 1 = s(active(X)) active(length(X)) = X + 1 >= X + 1 = length(active(X)) active(cons(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = cons(active(X1),X2) 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(uTake1(X)) = X + 1 >= X + 1 = uTake1(active(X)) active(uTake2(X1,X2,X3,X4)) = X1 + X2 + X3 + X4 + 1 >= X1 + X2 + X3 + X4 + 1 = uTake2(active(X1),X2,X3,X4) active(uLength(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = uLength(active(X1),X2) and(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(and(X1,X2)) and(X1,mark(X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(and(X1,X2)) s(mark(X)) = X + 1 >= X + 1 = mark(s(X)) length(mark(X)) = X + 1 >= X + 1 = mark(length(X)) cons(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(cons(X1,X2)) 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)) uTake1(mark(X)) = X + 1 >= X + 1 = mark(uTake1(X)) uTake2(mark(X1),X2,X3,X4) = X1 + X2 + X3 + X4 + 1 >= X1 + X2 + X3 + X4 + 1 = mark(uTake2(X1,X2,X3,X4)) uLength(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(uLength(X1,X2)) proper(and(X1,X2)) = 0 >= 0 = and(proper(X1),proper(X2)) proper(tt()) = 0 >= 0 = ok(tt()) proper(isNatIList(X)) = 0 >= 0 = isNatIList(proper(X)) proper(isNatList(X)) = 0 >= 0 = isNatList(proper(X)) proper(isNat(X)) = 0 >= 0 = isNat(proper(X)) proper(0()) = 0 >= 0 = ok(0()) proper(s(X)) = 0 >= 0 = s(proper(X)) proper(length(X)) = 0 >= 0 = length(proper(X)) proper(zeros()) = 0 >= 0 = ok(zeros()) proper(cons(X1,X2)) = 0 >= 0 = cons(proper(X1),proper(X2)) proper(nil()) = 0 >= 0 = ok(nil()) proper(take(X1,X2)) = 0 >= 0 = take(proper(X1),proper(X2)) proper(uTake1(X)) = 0 >= 0 = uTake1(proper(X)) proper(uTake2(X1,X2,X3,X4)) = 0 >= 0 = uTake2(proper(X1),proper(X2),proper(X3),proper(X4)) proper(uLength(X1,X2)) = 0 >= 0 = uLength(proper(X1),proper(X2)) and(ok(X1),ok(X2)) = 0 >= 0 = ok(and(X1,X2)) isNatIList(ok(X)) = 0 >= 0 = ok(isNatIList(X)) isNatList(ok(X)) = 0 >= 0 = ok(isNatList(X)) isNat(ok(X)) = 0 >= 0 = ok(isNat(X)) s(ok(X)) = 0 >= 0 = ok(s(X)) length(ok(X)) = 0 >= 0 = ok(length(X)) cons(ok(X1),ok(X2)) = 0 >= 0 = ok(cons(X1,X2)) take(ok(X1),ok(X2)) = 0 >= 0 = ok(take(X1,X2)) uTake1(ok(X)) = 0 >= 0 = ok(uTake1(X)) uTake2(ok(X1),ok(X2),ok(X3),ok(X4)) = 0 >= 0 = ok(uTake2(X1,X2,X3,X4)) uLength(ok(X1),ok(X2)) = 0 >= 0 = ok(uLength(X1,X2)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) problem: DPs: and#(X1,mark(X2)) -> and#(X1,X2) TRS: active(and(tt(),T)) -> mark(T) active(isNatIList(IL)) -> mark(isNatList(IL)) active(isNat(0())) -> mark(tt()) active(isNat(s(N))) -> mark(isNat(N)) active(isNat(length(L))) -> mark(isNatList(L)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(N,L))) -> mark(and(isNat(N),isNatList(L))) active(isNatList(take(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(zeros()) -> mark(cons(0(),zeros())) active(take(0(),IL)) -> mark(uTake1(isNatIList(IL))) active(uTake1(tt())) -> mark(nil()) active(take(s(M),cons(N,IL))) -> mark(uTake2(and(isNat(M),and(isNat(N),isNatIList(IL))),M,N,IL)) active(uTake2(tt(),M,N,IL)) -> mark(cons(N,take(M,IL))) active(length(cons(N,L))) -> mark(uLength(and(isNat(N),isNatList(L)),L)) active(uLength(tt(),L)) -> mark(s(length(L))) active(and(X1,X2)) -> and(active(X1),X2) active(and(X1,X2)) -> and(X1,active(X2)) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) active(uTake1(X)) -> uTake1(active(X)) active(uTake2(X1,X2,X3,X4)) -> uTake2(active(X1),X2,X3,X4) active(uLength(X1,X2)) -> uLength(active(X1),X2) and(mark(X1),X2) -> mark(and(X1,X2)) and(X1,mark(X2)) -> mark(and(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) uTake1(mark(X)) -> mark(uTake1(X)) uTake2(mark(X1),X2,X3,X4) -> mark(uTake2(X1,X2,X3,X4)) uLength(mark(X1),X2) -> mark(uLength(X1,X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(isNat(X)) -> isNat(proper(X)) proper(0()) -> ok(0()) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(nil()) -> ok(nil()) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) proper(uTake1(X)) -> uTake1(proper(X)) proper(uTake2(X1,X2,X3,X4)) -> uTake2(proper(X1),proper(X2),proper(X3),proper(X4)) proper(uLength(X1,X2)) -> uLength(proper(X1),proper(X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatIList(ok(X)) -> ok(isNatIList(X)) isNatList(ok(X)) -> ok(isNatList(X)) isNat(ok(X)) -> ok(isNat(X)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) uTake1(ok(X)) -> ok(uTake1(X)) uTake2(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(uTake2(X1,X2,X3,X4)) uLength(ok(X1),ok(X2)) -> ok(uLength(X1,X2)) 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) = 0, [proper](x0) = 0, [uLength](x0, x1) = x0 + x1, [uTake2](x0, x1, x2, x3) = x0 + x1 + x2 + x3, [uTake1](x0) = x0, [take](x0, x1) = x0 + x1, [nil] = 0, [cons](x0, x1) = x0 + x1, [zeros] = 1, [length](x0) = x0, [s](x0) = x0, [isNat](x0) = 0, [0] = 0, [isNatList](x0) = 0, [isNatIList](x0) = 0, [mark](x0) = x0 + 1, [active](x0) = x0 + 1, [and](x0, x1) = x0 + x1, [tt] = 0 orientation: and#(X1,mark(X2)) = X2 + 2 >= X2 + 1 = and#(X1,X2) active(and(tt(),T)) = T + 1 >= T + 1 = mark(T) active(isNatIList(IL)) = 1 >= 1 = mark(isNatList(IL)) active(isNat(0())) = 1 >= 1 = mark(tt()) active(isNat(s(N))) = 1 >= 1 = mark(isNat(N)) active(isNat(length(L))) = 1 >= 1 = mark(isNatList(L)) active(isNatIList(zeros())) = 1 >= 1 = mark(tt()) active(isNatIList(cons(N,IL))) = 1 >= 1 = mark(and(isNat(N),isNatIList(IL))) active(isNatList(nil())) = 1 >= 1 = mark(tt()) active(isNatList(cons(N,L))) = 1 >= 1 = mark(and(isNat(N),isNatList(L))) active(isNatList(take(N,IL))) = 1 >= 1 = mark(and(isNat(N),isNatIList(IL))) active(zeros()) = 2 >= 2 = mark(cons(0(),zeros())) active(take(0(),IL)) = IL + 1 >= 1 = mark(uTake1(isNatIList(IL))) active(uTake1(tt())) = 1 >= 1 = mark(nil()) active(take(s(M),cons(N,IL))) = IL + M + N + 1 >= IL + M + N + 1 = mark(uTake2(and(isNat(M),and(isNat(N),isNatIList(IL))),M,N,IL)) active(uTake2(tt(),M,N,IL)) = IL + M + N + 1 >= IL + M + N + 1 = mark(cons(N,take(M,IL))) active(length(cons(N,L))) = L + N + 1 >= L + 1 = mark(uLength(and(isNat(N),isNatList(L)),L)) active(uLength(tt(),L)) = L + 1 >= L + 1 = mark(s(length(L))) active(and(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = and(active(X1),X2) active(and(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = and(X1,active(X2)) active(s(X)) = X + 1 >= X + 1 = s(active(X)) active(length(X)) = X + 1 >= X + 1 = length(active(X)) active(cons(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = cons(active(X1),X2) 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(uTake1(X)) = X + 1 >= X + 1 = uTake1(active(X)) active(uTake2(X1,X2,X3,X4)) = X1 + X2 + X3 + X4 + 1 >= X1 + X2 + X3 + X4 + 1 = uTake2(active(X1),X2,X3,X4) active(uLength(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = uLength(active(X1),X2) and(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(and(X1,X2)) and(X1,mark(X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(and(X1,X2)) s(mark(X)) = X + 1 >= X + 1 = mark(s(X)) length(mark(X)) = X + 1 >= X + 1 = mark(length(X)) cons(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(cons(X1,X2)) 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)) uTake1(mark(X)) = X + 1 >= X + 1 = mark(uTake1(X)) uTake2(mark(X1),X2,X3,X4) = X1 + X2 + X3 + X4 + 1 >= X1 + X2 + X3 + X4 + 1 = mark(uTake2(X1,X2,X3,X4)) uLength(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(uLength(X1,X2)) proper(and(X1,X2)) = 0 >= 0 = and(proper(X1),proper(X2)) proper(tt()) = 0 >= 0 = ok(tt()) proper(isNatIList(X)) = 0 >= 0 = isNatIList(proper(X)) proper(isNatList(X)) = 0 >= 0 = isNatList(proper(X)) proper(isNat(X)) = 0 >= 0 = isNat(proper(X)) proper(0()) = 0 >= 0 = ok(0()) proper(s(X)) = 0 >= 0 = s(proper(X)) proper(length(X)) = 0 >= 0 = length(proper(X)) proper(zeros()) = 0 >= 0 = ok(zeros()) proper(cons(X1,X2)) = 0 >= 0 = cons(proper(X1),proper(X2)) proper(nil()) = 0 >= 0 = ok(nil()) proper(take(X1,X2)) = 0 >= 0 = take(proper(X1),proper(X2)) proper(uTake1(X)) = 0 >= 0 = uTake1(proper(X)) proper(uTake2(X1,X2,X3,X4)) = 0 >= 0 = uTake2(proper(X1),proper(X2),proper(X3),proper(X4)) proper(uLength(X1,X2)) = 0 >= 0 = uLength(proper(X1),proper(X2)) and(ok(X1),ok(X2)) = 0 >= 0 = ok(and(X1,X2)) isNatIList(ok(X)) = 0 >= 0 = ok(isNatIList(X)) isNatList(ok(X)) = 0 >= 0 = ok(isNatList(X)) isNat(ok(X)) = 0 >= 0 = ok(isNat(X)) s(ok(X)) = 0 >= 0 = ok(s(X)) length(ok(X)) = 0 >= 0 = ok(length(X)) cons(ok(X1),ok(X2)) = 0 >= 0 = ok(cons(X1,X2)) take(ok(X1),ok(X2)) = 0 >= 0 = ok(take(X1,X2)) uTake1(ok(X)) = 0 >= 0 = ok(uTake1(X)) uTake2(ok(X1),ok(X2),ok(X3),ok(X4)) = 0 >= 0 = ok(uTake2(X1,X2,X3,X4)) uLength(ok(X1),ok(X2)) = 0 >= 0 = ok(uLength(X1,X2)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) problem: DPs: TRS: active(and(tt(),T)) -> mark(T) active(isNatIList(IL)) -> mark(isNatList(IL)) active(isNat(0())) -> mark(tt()) active(isNat(s(N))) -> mark(isNat(N)) active(isNat(length(L))) -> mark(isNatList(L)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(N,L))) -> mark(and(isNat(N),isNatList(L))) active(isNatList(take(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(zeros()) -> mark(cons(0(),zeros())) active(take(0(),IL)) -> mark(uTake1(isNatIList(IL))) active(uTake1(tt())) -> mark(nil()) active(take(s(M),cons(N,IL))) -> mark(uTake2(and(isNat(M),and(isNat(N),isNatIList(IL))),M,N,IL)) active(uTake2(tt(),M,N,IL)) -> mark(cons(N,take(M,IL))) active(length(cons(N,L))) -> mark(uLength(and(isNat(N),isNatList(L)),L)) active(uLength(tt(),L)) -> mark(s(length(L))) active(and(X1,X2)) -> and(active(X1),X2) active(and(X1,X2)) -> and(X1,active(X2)) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) active(uTake1(X)) -> uTake1(active(X)) active(uTake2(X1,X2,X3,X4)) -> uTake2(active(X1),X2,X3,X4) active(uLength(X1,X2)) -> uLength(active(X1),X2) and(mark(X1),X2) -> mark(and(X1,X2)) and(X1,mark(X2)) -> mark(and(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) uTake1(mark(X)) -> mark(uTake1(X)) uTake2(mark(X1),X2,X3,X4) -> mark(uTake2(X1,X2,X3,X4)) uLength(mark(X1),X2) -> mark(uLength(X1,X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(isNat(X)) -> isNat(proper(X)) proper(0()) -> ok(0()) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(nil()) -> ok(nil()) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) proper(uTake1(X)) -> uTake1(proper(X)) proper(uTake2(X1,X2,X3,X4)) -> uTake2(proper(X1),proper(X2),proper(X3),proper(X4)) proper(uLength(X1,X2)) -> uLength(proper(X1),proper(X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatIList(ok(X)) -> ok(isNatIList(X)) isNatList(ok(X)) -> ok(isNatList(X)) isNat(ok(X)) -> ok(isNat(X)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) uTake1(ok(X)) -> ok(uTake1(X)) uTake2(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(uTake2(X1,X2,X3,X4)) uLength(ok(X1),ok(X2)) -> ok(uLength(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed