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)) Subterm Criterion Processor: simple projection: pi(active#) = 0 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)) Subterm Criterion Processor: simple projection: pi(proper#) = 0 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)) Subterm Criterion Processor: simple projection: pi(uLength#) = 1 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)) Subterm Criterion Processor: simple projection: pi(uLength#) = 0 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)) Subterm Criterion Processor: simple projection: pi(uTake2#) = 1 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)) Subterm Criterion Processor: simple projection: pi(uTake2#) = 0 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)) Subterm Criterion Processor: simple projection: pi(uTake1#) = 0 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)) Subterm Criterion Processor: simple projection: pi(take#) = 1 problem: DPs: take#(mark(X1),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)) Subterm Criterion Processor: simple projection: pi(take#) = 0 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)) Subterm Criterion Processor: simple projection: pi(cons#) = 1 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)) Subterm Criterion Processor: simple projection: pi(cons#) = 0 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)) Subterm Criterion Processor: simple projection: pi(length#) = 0 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)) Subterm Criterion Processor: simple projection: pi(s#) = 0 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)) Subterm Criterion Processor: simple projection: pi(isNat#) = 0 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)) Subterm Criterion Processor: simple projection: pi(isNatList#) = 0 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)) Subterm Criterion Processor: simple projection: pi(isNatIList#) = 0 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)) Subterm Criterion Processor: simple projection: pi(and#) = 1 problem: DPs: and#(mark(X1),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)) Subterm Criterion Processor: simple projection: pi(and#) = 0 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