MAYBE Problem: active(zeros()) -> mark(cons(0(),zeros())) active(and(tt(),X)) -> mark(X) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(s(length(L))) active(take(0(),IL)) -> mark(nil()) active(take(s(M),cons(N,IL))) -> mark(cons(N,take(M,IL))) active(cons(X1,X2)) -> cons(active(X1),X2) active(and(X1,X2)) -> and(active(X1),X2) active(length(X)) -> length(active(X)) active(s(X)) -> s(active(X)) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) cons(mark(X1),X2) -> mark(cons(X1,X2)) and(mark(X1),X2) -> mark(and(X1,X2)) length(mark(X)) -> mark(length(X)) s(mark(X)) -> mark(s(X)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(length(X)) -> length(proper(X)) proper(nil()) -> ok(nil()) proper(s(X)) -> s(proper(X)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) length(ok(X)) -> ok(length(X)) s(ok(X)) -> ok(s(X)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Proof: DP Processor: DPs: active#(zeros()) -> cons#(0(),zeros()) active#(length(cons(N,L))) -> length#(L) active#(length(cons(N,L))) -> s#(length(L)) active#(take(s(M),cons(N,IL))) -> take#(M,IL) active#(take(s(M),cons(N,IL))) -> cons#(N,take(M,IL)) active#(cons(X1,X2)) -> active#(X1) active#(cons(X1,X2)) -> cons#(active(X1),X2) active#(and(X1,X2)) -> active#(X1) active#(and(X1,X2)) -> and#(active(X1),X2) active#(length(X)) -> active#(X) active#(length(X)) -> length#(active(X)) active#(s(X)) -> active#(X) active#(s(X)) -> s#(active(X)) 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)) cons#(mark(X1),X2) -> cons#(X1,X2) and#(mark(X1),X2) -> and#(X1,X2) length#(mark(X)) -> length#(X) s#(mark(X)) -> s#(X) take#(mark(X1),X2) -> take#(X1,X2) take#(X1,mark(X2)) -> take#(X1,X2) proper#(cons(X1,X2)) -> proper#(X2) proper#(cons(X1,X2)) -> proper#(X1) proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2)) proper#(and(X1,X2)) -> proper#(X2) proper#(and(X1,X2)) -> proper#(X1) proper#(and(X1,X2)) -> and#(proper(X1),proper(X2)) proper#(length(X)) -> proper#(X) proper#(length(X)) -> length#(proper(X)) proper#(s(X)) -> proper#(X) proper#(s(X)) -> s#(proper(X)) proper#(take(X1,X2)) -> proper#(X2) proper#(take(X1,X2)) -> proper#(X1) proper#(take(X1,X2)) -> take#(proper(X1),proper(X2)) cons#(ok(X1),ok(X2)) -> cons#(X1,X2) and#(ok(X1),ok(X2)) -> and#(X1,X2) length#(ok(X)) -> length#(X) s#(ok(X)) -> s#(X) take#(ok(X1),ok(X2)) -> take#(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(zeros()) -> mark(cons(0(),zeros())) active(and(tt(),X)) -> mark(X) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(s(length(L))) active(take(0(),IL)) -> mark(nil()) active(take(s(M),cons(N,IL))) -> mark(cons(N,take(M,IL))) active(cons(X1,X2)) -> cons(active(X1),X2) active(and(X1,X2)) -> and(active(X1),X2) active(length(X)) -> length(active(X)) active(s(X)) -> s(active(X)) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) cons(mark(X1),X2) -> mark(cons(X1,X2)) and(mark(X1),X2) -> mark(and(X1,X2)) length(mark(X)) -> mark(length(X)) s(mark(X)) -> mark(s(X)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(length(X)) -> length(proper(X)) proper(nil()) -> ok(nil()) proper(s(X)) -> s(proper(X)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) length(ok(X)) -> ok(length(X)) s(ok(X)) -> ok(s(X)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) TDG Processor: DPs: active#(zeros()) -> cons#(0(),zeros()) active#(length(cons(N,L))) -> length#(L) active#(length(cons(N,L))) -> s#(length(L)) active#(take(s(M),cons(N,IL))) -> take#(M,IL) active#(take(s(M),cons(N,IL))) -> cons#(N,take(M,IL)) active#(cons(X1,X2)) -> active#(X1) active#(cons(X1,X2)) -> cons#(active(X1),X2) active#(and(X1,X2)) -> active#(X1) active#(and(X1,X2)) -> and#(active(X1),X2) active#(length(X)) -> active#(X) active#(length(X)) -> length#(active(X)) active#(s(X)) -> active#(X) active#(s(X)) -> s#(active(X)) 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)) cons#(mark(X1),X2) -> cons#(X1,X2) and#(mark(X1),X2) -> and#(X1,X2) length#(mark(X)) -> length#(X) s#(mark(X)) -> s#(X) take#(mark(X1),X2) -> take#(X1,X2) take#(X1,mark(X2)) -> take#(X1,X2) proper#(cons(X1,X2)) -> proper#(X2) proper#(cons(X1,X2)) -> proper#(X1) proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2)) proper#(and(X1,X2)) -> proper#(X2) proper#(and(X1,X2)) -> proper#(X1) proper#(and(X1,X2)) -> and#(proper(X1),proper(X2)) proper#(length(X)) -> proper#(X) proper#(length(X)) -> length#(proper(X)) proper#(s(X)) -> proper#(X) proper#(s(X)) -> s#(proper(X)) proper#(take(X1,X2)) -> proper#(X2) proper#(take(X1,X2)) -> proper#(X1) proper#(take(X1,X2)) -> take#(proper(X1),proper(X2)) cons#(ok(X1),ok(X2)) -> cons#(X1,X2) and#(ok(X1),ok(X2)) -> and#(X1,X2) length#(ok(X)) -> length#(X) s#(ok(X)) -> s#(X) take#(ok(X1),ok(X2)) -> take#(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(zeros()) -> mark(cons(0(),zeros())) active(and(tt(),X)) -> mark(X) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(s(length(L))) active(take(0(),IL)) -> mark(nil()) active(take(s(M),cons(N,IL))) -> mark(cons(N,take(M,IL))) active(cons(X1,X2)) -> cons(active(X1),X2) active(and(X1,X2)) -> and(active(X1),X2) active(length(X)) -> length(active(X)) active(s(X)) -> s(active(X)) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) cons(mark(X1),X2) -> mark(cons(X1,X2)) and(mark(X1),X2) -> mark(and(X1,X2)) length(mark(X)) -> mark(length(X)) s(mark(X)) -> mark(s(X)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(length(X)) -> length(proper(X)) proper(nil()) -> ok(nil()) proper(s(X)) -> s(proper(X)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) length(ok(X)) -> ok(length(X)) s(ok(X)) -> ok(s(X)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) graph: top#(ok(X)) -> top#(active(X)) -> top#(ok(X)) -> top#(active(X)) top#(ok(X)) -> top#(active(X)) -> top#(ok(X)) -> active#(X) top#(ok(X)) -> top#(active(X)) -> top#(mark(X)) -> top#(proper(X)) top#(ok(X)) -> top#(active(X)) -> top#(mark(X)) -> proper#(X) top#(ok(X)) -> active#(X) -> active#(take(X1,X2)) -> take#(X1,active(X2)) top#(ok(X)) -> active#(X) -> active#(take(X1,X2)) -> active#(X2) top#(ok(X)) -> active#(X) -> active#(take(X1,X2)) -> take#(active(X1),X2) top#(ok(X)) -> active#(X) -> active#(take(X1,X2)) -> active#(X1) top#(ok(X)) -> active#(X) -> active#(s(X)) -> s#(active(X)) top#(ok(X)) -> active#(X) -> active#(s(X)) -> active#(X) top#(ok(X)) -> active#(X) -> active#(length(X)) -> length#(active(X)) top#(ok(X)) -> active#(X) -> active#(length(X)) -> active#(X) top#(ok(X)) -> active#(X) -> active#(and(X1,X2)) -> and#(active(X1),X2) top#(ok(X)) -> active#(X) -> active#(and(X1,X2)) -> active#(X1) top#(ok(X)) -> active#(X) -> active#(cons(X1,X2)) -> cons#(active(X1),X2) top#(ok(X)) -> active#(X) -> active#(cons(X1,X2)) -> active#(X1) top#(ok(X)) -> active#(X) -> active#(take(s(M),cons(N,IL))) -> cons#(N,take(M,IL)) top#(ok(X)) -> active#(X) -> active#(take(s(M),cons(N,IL))) -> take#(M,IL) top#(ok(X)) -> active#(X) -> active#(length(cons(N,L))) -> s#(length(L)) top#(ok(X)) -> active#(X) -> active#(length(cons(N,L))) -> length#(L) top#(ok(X)) -> active#(X) -> active#(zeros()) -> cons#(0(),zeros()) top#(mark(X)) -> top#(proper(X)) -> top#(ok(X)) -> top#(active(X)) top#(mark(X)) -> top#(proper(X)) -> top#(ok(X)) -> active#(X) top#(mark(X)) -> top#(proper(X)) -> top#(mark(X)) -> top#(proper(X)) top#(mark(X)) -> top#(proper(X)) -> top#(mark(X)) -> proper#(X) top#(mark(X)) -> proper#(X) -> proper#(take(X1,X2)) -> take#(proper(X1),proper(X2)) top#(mark(X)) -> proper#(X) -> proper#(take(X1,X2)) -> proper#(X1) top#(mark(X)) -> proper#(X) -> proper#(take(X1,X2)) -> proper#(X2) top#(mark(X)) -> proper#(X) -> proper#(s(X)) -> s#(proper(X)) top#(mark(X)) -> proper#(X) -> proper#(s(X)) -> proper#(X) top#(mark(X)) -> proper#(X) -> proper#(length(X)) -> length#(proper(X)) top#(mark(X)) -> proper#(X) -> proper#(length(X)) -> proper#(X) top#(mark(X)) -> proper#(X) -> proper#(and(X1,X2)) -> and#(proper(X1),proper(X2)) top#(mark(X)) -> proper#(X) -> proper#(and(X1,X2)) -> proper#(X1) top#(mark(X)) -> proper#(X) -> proper#(and(X1,X2)) -> proper#(X2) top#(mark(X)) -> proper#(X) -> proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2)) top#(mark(X)) -> proper#(X) -> proper#(cons(X1,X2)) -> proper#(X1) top#(mark(X)) -> proper#(X) -> proper#(cons(X1,X2)) -> proper#(X2) proper#(take(X1,X2)) -> proper#(X2) -> proper#(take(X1,X2)) -> take#(proper(X1),proper(X2)) proper#(take(X1,X2)) -> proper#(X2) -> proper#(take(X1,X2)) -> proper#(X1) proper#(take(X1,X2)) -> proper#(X2) -> proper#(take(X1,X2)) -> proper#(X2) proper#(take(X1,X2)) -> proper#(X2) -> proper#(s(X)) -> s#(proper(X)) proper#(take(X1,X2)) -> proper#(X2) -> proper#(s(X)) -> proper#(X) proper#(take(X1,X2)) -> proper#(X2) -> proper#(length(X)) -> length#(proper(X)) proper#(take(X1,X2)) -> proper#(X2) -> proper#(length(X)) -> proper#(X) proper#(take(X1,X2)) -> proper#(X2) -> proper#(and(X1,X2)) -> and#(proper(X1),proper(X2)) proper#(take(X1,X2)) -> proper#(X2) -> proper#(and(X1,X2)) -> proper#(X1) proper#(take(X1,X2)) -> proper#(X2) -> proper#(and(X1,X2)) -> proper#(X2) proper#(take(X1,X2)) -> proper#(X2) -> proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2)) proper#(take(X1,X2)) -> proper#(X2) -> proper#(cons(X1,X2)) -> proper#(X1) proper#(take(X1,X2)) -> proper#(X2) -> proper#(cons(X1,X2)) -> proper#(X2) proper#(take(X1,X2)) -> proper#(X1) -> proper#(take(X1,X2)) -> take#(proper(X1),proper(X2)) proper#(take(X1,X2)) -> proper#(X1) -> proper#(take(X1,X2)) -> proper#(X1) proper#(take(X1,X2)) -> proper#(X1) -> proper#(take(X1,X2)) -> proper#(X2) proper#(take(X1,X2)) -> proper#(X1) -> proper#(s(X)) -> s#(proper(X)) proper#(take(X1,X2)) -> proper#(X1) -> proper#(s(X)) -> proper#(X) proper#(take(X1,X2)) -> proper#(X1) -> proper#(length(X)) -> length#(proper(X)) proper#(take(X1,X2)) -> proper#(X1) -> proper#(length(X)) -> proper#(X) proper#(take(X1,X2)) -> proper#(X1) -> proper#(and(X1,X2)) -> and#(proper(X1),proper(X2)) proper#(take(X1,X2)) -> proper#(X1) -> proper#(and(X1,X2)) -> proper#(X1) proper#(take(X1,X2)) -> proper#(X1) -> proper#(and(X1,X2)) -> proper#(X2) proper#(take(X1,X2)) -> proper#(X1) -> proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2)) proper#(take(X1,X2)) -> proper#(X1) -> proper#(cons(X1,X2)) -> proper#(X1) proper#(take(X1,X2)) -> proper#(X1) -> proper#(cons(X1,X2)) -> proper#(X2) proper#(take(X1,X2)) -> take#(proper(X1),proper(X2)) -> take#(ok(X1),ok(X2)) -> take#(X1,X2) proper#(take(X1,X2)) -> take#(proper(X1),proper(X2)) -> take#(X1,mark(X2)) -> take#(X1,X2) proper#(take(X1,X2)) -> take#(proper(X1),proper(X2)) -> take#(mark(X1),X2) -> take#(X1,X2) proper#(s(X)) -> proper#(X) -> proper#(take(X1,X2)) -> take#(proper(X1),proper(X2)) proper#(s(X)) -> proper#(X) -> proper#(take(X1,X2)) -> proper#(X1) proper#(s(X)) -> proper#(X) -> proper#(take(X1,X2)) -> proper#(X2) proper#(s(X)) -> proper#(X) -> proper#(s(X)) -> s#(proper(X)) proper#(s(X)) -> proper#(X) -> proper#(s(X)) -> proper#(X) proper#(s(X)) -> proper#(X) -> proper#(length(X)) -> length#(proper(X)) proper#(s(X)) -> proper#(X) -> proper#(length(X)) -> proper#(X) proper#(s(X)) -> proper#(X) -> proper#(and(X1,X2)) -> and#(proper(X1),proper(X2)) proper#(s(X)) -> proper#(X) -> proper#(and(X1,X2)) -> proper#(X1) proper#(s(X)) -> proper#(X) -> proper#(and(X1,X2)) -> proper#(X2) proper#(s(X)) -> proper#(X) -> proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2)) proper#(s(X)) -> proper#(X) -> proper#(cons(X1,X2)) -> proper#(X1) proper#(s(X)) -> proper#(X) -> proper#(cons(X1,X2)) -> proper#(X2) proper#(s(X)) -> s#(proper(X)) -> s#(ok(X)) -> s#(X) proper#(s(X)) -> s#(proper(X)) -> s#(mark(X)) -> s#(X) proper#(length(X)) -> proper#(X) -> proper#(take(X1,X2)) -> take#(proper(X1),proper(X2)) proper#(length(X)) -> proper#(X) -> proper#(take(X1,X2)) -> proper#(X1) proper#(length(X)) -> proper#(X) -> proper#(take(X1,X2)) -> proper#(X2) proper#(length(X)) -> proper#(X) -> proper#(s(X)) -> s#(proper(X)) proper#(length(X)) -> proper#(X) -> proper#(s(X)) -> proper#(X) proper#(length(X)) -> proper#(X) -> proper#(length(X)) -> length#(proper(X)) proper#(length(X)) -> proper#(X) -> proper#(length(X)) -> proper#(X) proper#(length(X)) -> proper#(X) -> proper#(and(X1,X2)) -> and#(proper(X1),proper(X2)) proper#(length(X)) -> proper#(X) -> proper#(and(X1,X2)) -> proper#(X1) proper#(length(X)) -> proper#(X) -> proper#(and(X1,X2)) -> proper#(X2) proper#(length(X)) -> proper#(X) -> proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2)) proper#(length(X)) -> proper#(X) -> proper#(cons(X1,X2)) -> proper#(X1) proper#(length(X)) -> proper#(X) -> proper#(cons(X1,X2)) -> proper#(X2) proper#(length(X)) -> length#(proper(X)) -> length#(ok(X)) -> length#(X) proper#(length(X)) -> length#(proper(X)) -> length#(mark(X)) -> length#(X) proper#(and(X1,X2)) -> proper#(X2) -> proper#(take(X1,X2)) -> take#(proper(X1),proper(X2)) proper#(and(X1,X2)) -> proper#(X2) -> proper#(take(X1,X2)) -> proper#(X1) proper#(and(X1,X2)) -> proper#(X2) -> proper#(take(X1,X2)) -> proper#(X2) proper#(and(X1,X2)) -> proper#(X2) -> proper#(s(X)) -> s#(proper(X)) proper#(and(X1,X2)) -> proper#(X2) -> proper#(s(X)) -> proper#(X) proper#(and(X1,X2)) -> proper#(X2) -> proper#(length(X)) -> length#(proper(X)) proper#(and(X1,X2)) -> proper#(X2) -> proper#(length(X)) -> proper#(X) proper#(and(X1,X2)) -> proper#(X2) -> proper#(and(X1,X2)) -> and#(proper(X1),proper(X2)) proper#(and(X1,X2)) -> proper#(X2) -> proper#(and(X1,X2)) -> proper#(X1) proper#(and(X1,X2)) -> proper#(X2) -> proper#(and(X1,X2)) -> proper#(X2) proper#(and(X1,X2)) -> proper#(X2) -> proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2)) proper#(and(X1,X2)) -> proper#(X2) -> proper#(cons(X1,X2)) -> proper#(X1) proper#(and(X1,X2)) -> proper#(X2) -> proper#(cons(X1,X2)) -> proper#(X2) proper#(and(X1,X2)) -> proper#(X1) -> proper#(take(X1,X2)) -> take#(proper(X1),proper(X2)) proper#(and(X1,X2)) -> proper#(X1) -> proper#(take(X1,X2)) -> proper#(X1) proper#(and(X1,X2)) -> proper#(X1) -> proper#(take(X1,X2)) -> proper#(X2) proper#(and(X1,X2)) -> proper#(X1) -> proper#(s(X)) -> s#(proper(X)) proper#(and(X1,X2)) -> proper#(X1) -> proper#(s(X)) -> proper#(X) proper#(and(X1,X2)) -> proper#(X1) -> proper#(length(X)) -> length#(proper(X)) proper#(and(X1,X2)) -> proper#(X1) -> proper#(length(X)) -> proper#(X) proper#(and(X1,X2)) -> proper#(X1) -> proper#(and(X1,X2)) -> and#(proper(X1),proper(X2)) proper#(and(X1,X2)) -> proper#(X1) -> proper#(and(X1,X2)) -> proper#(X1) proper#(and(X1,X2)) -> proper#(X1) -> proper#(and(X1,X2)) -> proper#(X2) proper#(and(X1,X2)) -> proper#(X1) -> proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2)) proper#(and(X1,X2)) -> proper#(X1) -> proper#(cons(X1,X2)) -> proper#(X1) proper#(and(X1,X2)) -> proper#(X1) -> proper#(cons(X1,X2)) -> proper#(X2) proper#(and(X1,X2)) -> and#(proper(X1),proper(X2)) -> and#(ok(X1),ok(X2)) -> and#(X1,X2) proper#(and(X1,X2)) -> and#(proper(X1),proper(X2)) -> and#(mark(X1),X2) -> and#(X1,X2) proper#(cons(X1,X2)) -> proper#(X2) -> proper#(take(X1,X2)) -> take#(proper(X1),proper(X2)) proper#(cons(X1,X2)) -> proper#(X2) -> proper#(take(X1,X2)) -> proper#(X1) proper#(cons(X1,X2)) -> proper#(X2) -> proper#(take(X1,X2)) -> proper#(X2) proper#(cons(X1,X2)) -> proper#(X2) -> proper#(s(X)) -> s#(proper(X)) proper#(cons(X1,X2)) -> proper#(X2) -> proper#(s(X)) -> proper#(X) proper#(cons(X1,X2)) -> proper#(X2) -> proper#(length(X)) -> length#(proper(X)) proper#(cons(X1,X2)) -> proper#(X2) -> proper#(length(X)) -> proper#(X) proper#(cons(X1,X2)) -> proper#(X2) -> proper#(and(X1,X2)) -> and#(proper(X1),proper(X2)) proper#(cons(X1,X2)) -> proper#(X2) -> proper#(and(X1,X2)) -> proper#(X1) proper#(cons(X1,X2)) -> proper#(X2) -> proper#(and(X1,X2)) -> proper#(X2) proper#(cons(X1,X2)) -> proper#(X2) -> proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2)) proper#(cons(X1,X2)) -> proper#(X2) -> proper#(cons(X1,X2)) -> proper#(X1) proper#(cons(X1,X2)) -> proper#(X2) -> proper#(cons(X1,X2)) -> proper#(X2) proper#(cons(X1,X2)) -> proper#(X1) -> proper#(take(X1,X2)) -> take#(proper(X1),proper(X2)) proper#(cons(X1,X2)) -> proper#(X1) -> proper#(take(X1,X2)) -> proper#(X1) proper#(cons(X1,X2)) -> proper#(X1) -> proper#(take(X1,X2)) -> proper#(X2) proper#(cons(X1,X2)) -> proper#(X1) -> proper#(s(X)) -> s#(proper(X)) proper#(cons(X1,X2)) -> proper#(X1) -> proper#(s(X)) -> proper#(X) proper#(cons(X1,X2)) -> proper#(X1) -> proper#(length(X)) -> length#(proper(X)) proper#(cons(X1,X2)) -> proper#(X1) -> proper#(length(X)) -> proper#(X) proper#(cons(X1,X2)) -> proper#(X1) -> proper#(and(X1,X2)) -> and#(proper(X1),proper(X2)) proper#(cons(X1,X2)) -> proper#(X1) -> proper#(and(X1,X2)) -> proper#(X1) proper#(cons(X1,X2)) -> proper#(X1) -> proper#(and(X1,X2)) -> proper#(X2) proper#(cons(X1,X2)) -> proper#(X1) -> proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2)) proper#(cons(X1,X2)) -> proper#(X1) -> proper#(cons(X1,X2)) -> proper#(X1) proper#(cons(X1,X2)) -> proper#(X1) -> proper#(cons(X1,X2)) -> proper#(X2) proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2)) -> cons#(ok(X1),ok(X2)) -> cons#(X1,X2) proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2)) -> cons#(mark(X1),X2) -> cons#(X1,X2) and#(ok(X1),ok(X2)) -> and#(X1,X2) -> and#(ok(X1),ok(X2)) -> and#(X1,X2) and#(ok(X1),ok(X2)) -> and#(X1,X2) -> and#(mark(X1),X2) -> and#(X1,X2) and#(mark(X1),X2) -> and#(X1,X2) -> and#(ok(X1),ok(X2)) -> and#(X1,X2) and#(mark(X1),X2) -> and#(X1,X2) -> and#(mark(X1),X2) -> and#(X1,X2) take#(ok(X1),ok(X2)) -> take#(X1,X2) -> take#(ok(X1),ok(X2)) -> take#(X1,X2) take#(ok(X1),ok(X2)) -> take#(X1,X2) -> take#(X1,mark(X2)) -> take#(X1,X2) take#(ok(X1),ok(X2)) -> take#(X1,X2) -> take#(mark(X1),X2) -> take#(X1,X2) take#(mark(X1),X2) -> take#(X1,X2) -> take#(ok(X1),ok(X2)) -> take#(X1,X2) take#(mark(X1),X2) -> take#(X1,X2) -> take#(X1,mark(X2)) -> take#(X1,X2) take#(mark(X1),X2) -> take#(X1,X2) -> take#(mark(X1),X2) -> take#(X1,X2) take#(X1,mark(X2)) -> take#(X1,X2) -> take#(ok(X1),ok(X2)) -> take#(X1,X2) take#(X1,mark(X2)) -> take#(X1,X2) -> take#(X1,mark(X2)) -> take#(X1,X2) take#(X1,mark(X2)) -> take#(X1,X2) -> take#(mark(X1),X2) -> take#(X1,X2) s#(ok(X)) -> s#(X) -> s#(ok(X)) -> s#(X) s#(ok(X)) -> s#(X) -> s#(mark(X)) -> s#(X) s#(mark(X)) -> s#(X) -> s#(ok(X)) -> s#(X) s#(mark(X)) -> s#(X) -> s#(mark(X)) -> s#(X) length#(ok(X)) -> length#(X) -> length#(ok(X)) -> length#(X) length#(ok(X)) -> length#(X) -> length#(mark(X)) -> length#(X) length#(mark(X)) -> length#(X) -> length#(ok(X)) -> length#(X) length#(mark(X)) -> length#(X) -> length#(mark(X)) -> length#(X) cons#(ok(X1),ok(X2)) -> cons#(X1,X2) -> cons#(ok(X1),ok(X2)) -> cons#(X1,X2) cons#(ok(X1),ok(X2)) -> cons#(X1,X2) -> cons#(mark(X1),X2) -> cons#(X1,X2) cons#(mark(X1),X2) -> cons#(X1,X2) -> cons#(ok(X1),ok(X2)) -> cons#(X1,X2) cons#(mark(X1),X2) -> cons#(X1,X2) -> cons#(mark(X1),X2) -> cons#(X1,X2) active#(take(s(M),cons(N,IL))) -> take#(M,IL) -> take#(ok(X1),ok(X2)) -> take#(X1,X2) active#(take(s(M),cons(N,IL))) -> take#(M,IL) -> take#(X1,mark(X2)) -> take#(X1,X2) active#(take(s(M),cons(N,IL))) -> take#(M,IL) -> take#(mark(X1),X2) -> take#(X1,X2) active#(take(s(M),cons(N,IL))) -> cons#(N,take(M,IL)) -> cons#(ok(X1),ok(X2)) -> cons#(X1,X2) active#(take(s(M),cons(N,IL))) -> cons#(N,take(M,IL)) -> cons#(mark(X1),X2) -> cons#(X1,X2) active#(take(X1,X2)) -> take#(active(X1),X2) -> take#(ok(X1),ok(X2)) -> take#(X1,X2) active#(take(X1,X2)) -> take#(active(X1),X2) -> take#(X1,mark(X2)) -> take#(X1,X2) active#(take(X1,X2)) -> take#(active(X1),X2) -> take#(mark(X1),X2) -> take#(X1,X2) active#(take(X1,X2)) -> take#(X1,active(X2)) -> take#(ok(X1),ok(X2)) -> take#(X1,X2) active#(take(X1,X2)) -> take#(X1,active(X2)) -> take#(X1,mark(X2)) -> take#(X1,X2) active#(take(X1,X2)) -> take#(X1,active(X2)) -> take#(mark(X1),X2) -> take#(X1,X2) active#(take(X1,X2)) -> active#(X2) -> active#(take(X1,X2)) -> take#(X1,active(X2)) active#(take(X1,X2)) -> active#(X2) -> active#(take(X1,X2)) -> active#(X2) active#(take(X1,X2)) -> active#(X2) -> active#(take(X1,X2)) -> take#(active(X1),X2) active#(take(X1,X2)) -> active#(X2) -> active#(take(X1,X2)) -> active#(X1) active#(take(X1,X2)) -> active#(X2) -> active#(s(X)) -> s#(active(X)) active#(take(X1,X2)) -> active#(X2) -> active#(s(X)) -> active#(X) active#(take(X1,X2)) -> active#(X2) -> active#(length(X)) -> length#(active(X)) active#(take(X1,X2)) -> active#(X2) -> active#(length(X)) -> active#(X) active#(take(X1,X2)) -> active#(X2) -> active#(and(X1,X2)) -> and#(active(X1),X2) active#(take(X1,X2)) -> active#(X2) -> active#(and(X1,X2)) -> active#(X1) active#(take(X1,X2)) -> active#(X2) -> active#(cons(X1,X2)) -> cons#(active(X1),X2) active#(take(X1,X2)) -> active#(X2) -> active#(cons(X1,X2)) -> active#(X1) active#(take(X1,X2)) -> active#(X2) -> active#(take(s(M),cons(N,IL))) -> cons#(N,take(M,IL)) active#(take(X1,X2)) -> active#(X2) -> active#(take(s(M),cons(N,IL))) -> take#(M,IL) active#(take(X1,X2)) -> active#(X2) -> active#(length(cons(N,L))) -> s#(length(L)) active#(take(X1,X2)) -> active#(X2) -> active#(length(cons(N,L))) -> length#(L) active#(take(X1,X2)) -> active#(X2) -> active#(zeros()) -> cons#(0(),zeros()) active#(take(X1,X2)) -> active#(X1) -> active#(take(X1,X2)) -> take#(X1,active(X2)) active#(take(X1,X2)) -> active#(X1) -> active#(take(X1,X2)) -> active#(X2) active#(take(X1,X2)) -> active#(X1) -> active#(take(X1,X2)) -> take#(active(X1),X2) active#(take(X1,X2)) -> active#(X1) -> active#(take(X1,X2)) -> active#(X1) active#(take(X1,X2)) -> active#(X1) -> active#(s(X)) -> s#(active(X)) active#(take(X1,X2)) -> active#(X1) -> active#(s(X)) -> active#(X) active#(take(X1,X2)) -> active#(X1) -> active#(length(X)) -> length#(active(X)) active#(take(X1,X2)) -> active#(X1) -> active#(length(X)) -> active#(X) active#(take(X1,X2)) -> active#(X1) -> active#(and(X1,X2)) -> and#(active(X1),X2) active#(take(X1,X2)) -> active#(X1) -> active#(and(X1,X2)) -> active#(X1) active#(take(X1,X2)) -> active#(X1) -> active#(cons(X1,X2)) -> cons#(active(X1),X2) active#(take(X1,X2)) -> active#(X1) -> active#(cons(X1,X2)) -> active#(X1) active#(take(X1,X2)) -> active#(X1) -> active#(take(s(M),cons(N,IL))) -> cons#(N,take(M,IL)) active#(take(X1,X2)) -> active#(X1) -> active#(take(s(M),cons(N,IL))) -> take#(M,IL) active#(take(X1,X2)) -> active#(X1) -> active#(length(cons(N,L))) -> s#(length(L)) active#(take(X1,X2)) -> active#(X1) -> active#(length(cons(N,L))) -> length#(L) active#(take(X1,X2)) -> active#(X1) -> active#(zeros()) -> cons#(0(),zeros()) active#(s(X)) -> s#(active(X)) -> s#(ok(X)) -> s#(X) active#(s(X)) -> s#(active(X)) -> s#(mark(X)) -> s#(X) active#(s(X)) -> active#(X) -> active#(take(X1,X2)) -> take#(X1,active(X2)) active#(s(X)) -> active#(X) -> active#(take(X1,X2)) -> active#(X2) active#(s(X)) -> active#(X) -> active#(take(X1,X2)) -> take#(active(X1),X2) active#(s(X)) -> active#(X) -> active#(take(X1,X2)) -> active#(X1) active#(s(X)) -> active#(X) -> active#(s(X)) -> s#(active(X)) active#(s(X)) -> active#(X) -> active#(s(X)) -> active#(X) active#(s(X)) -> active#(X) -> active#(length(X)) -> length#(active(X)) active#(s(X)) -> active#(X) -> active#(length(X)) -> active#(X) active#(s(X)) -> active#(X) -> active#(and(X1,X2)) -> and#(active(X1),X2) active#(s(X)) -> active#(X) -> active#(and(X1,X2)) -> active#(X1) active#(s(X)) -> active#(X) -> active#(cons(X1,X2)) -> cons#(active(X1),X2) active#(s(X)) -> active#(X) -> active#(cons(X1,X2)) -> active#(X1) active#(s(X)) -> active#(X) -> active#(take(s(M),cons(N,IL))) -> cons#(N,take(M,IL)) active#(s(X)) -> active#(X) -> active#(take(s(M),cons(N,IL))) -> take#(M,IL) active#(s(X)) -> active#(X) -> active#(length(cons(N,L))) -> s#(length(L)) active#(s(X)) -> active#(X) -> active#(length(cons(N,L))) -> length#(L) active#(s(X)) -> active#(X) -> active#(zeros()) -> cons#(0(),zeros()) active#(length(cons(N,L))) -> s#(length(L)) -> s#(ok(X)) -> s#(X) active#(length(cons(N,L))) -> s#(length(L)) -> s#(mark(X)) -> s#(X) active#(length(cons(N,L))) -> length#(L) -> length#(ok(X)) -> length#(X) active#(length(cons(N,L))) -> length#(L) -> length#(mark(X)) -> length#(X) active#(length(X)) -> length#(active(X)) -> length#(ok(X)) -> length#(X) active#(length(X)) -> length#(active(X)) -> length#(mark(X)) -> length#(X) active#(length(X)) -> active#(X) -> active#(take(X1,X2)) -> take#(X1,active(X2)) active#(length(X)) -> active#(X) -> active#(take(X1,X2)) -> active#(X2) active#(length(X)) -> active#(X) -> active#(take(X1,X2)) -> take#(active(X1),X2) active#(length(X)) -> active#(X) -> active#(take(X1,X2)) -> active#(X1) active#(length(X)) -> active#(X) -> active#(s(X)) -> s#(active(X)) active#(length(X)) -> active#(X) -> active#(s(X)) -> active#(X) active#(length(X)) -> active#(X) -> active#(length(X)) -> length#(active(X)) active#(length(X)) -> active#(X) -> active#(length(X)) -> active#(X) active#(length(X)) -> active#(X) -> active#(and(X1,X2)) -> and#(active(X1),X2) active#(length(X)) -> active#(X) -> active#(and(X1,X2)) -> active#(X1) active#(length(X)) -> active#(X) -> active#(cons(X1,X2)) -> cons#(active(X1),X2) active#(length(X)) -> active#(X) -> active#(cons(X1,X2)) -> active#(X1) active#(length(X)) -> active#(X) -> active#(take(s(M),cons(N,IL))) -> cons#(N,take(M,IL)) active#(length(X)) -> active#(X) -> active#(take(s(M),cons(N,IL))) -> take#(M,IL) active#(length(X)) -> active#(X) -> active#(length(cons(N,L))) -> s#(length(L)) active#(length(X)) -> active#(X) -> active#(length(cons(N,L))) -> length#(L) active#(length(X)) -> active#(X) -> active#(zeros()) -> cons#(0(),zeros()) active#(and(X1,X2)) -> and#(active(X1),X2) -> and#(ok(X1),ok(X2)) -> and#(X1,X2) active#(and(X1,X2)) -> and#(active(X1),X2) -> and#(mark(X1),X2) -> and#(X1,X2) active#(and(X1,X2)) -> active#(X1) -> active#(take(X1,X2)) -> take#(X1,active(X2)) active#(and(X1,X2)) -> active#(X1) -> active#(take(X1,X2)) -> active#(X2) active#(and(X1,X2)) -> active#(X1) -> active#(take(X1,X2)) -> take#(active(X1),X2) active#(and(X1,X2)) -> active#(X1) -> active#(take(X1,X2)) -> active#(X1) active#(and(X1,X2)) -> active#(X1) -> active#(s(X)) -> s#(active(X)) active#(and(X1,X2)) -> active#(X1) -> active#(s(X)) -> active#(X) active#(and(X1,X2)) -> active#(X1) -> active#(length(X)) -> length#(active(X)) active#(and(X1,X2)) -> active#(X1) -> active#(length(X)) -> active#(X) active#(and(X1,X2)) -> active#(X1) -> active#(and(X1,X2)) -> and#(active(X1),X2) active#(and(X1,X2)) -> active#(X1) -> active#(and(X1,X2)) -> active#(X1) active#(and(X1,X2)) -> active#(X1) -> active#(cons(X1,X2)) -> cons#(active(X1),X2) active#(and(X1,X2)) -> active#(X1) -> active#(cons(X1,X2)) -> active#(X1) active#(and(X1,X2)) -> active#(X1) -> active#(take(s(M),cons(N,IL))) -> cons#(N,take(M,IL)) active#(and(X1,X2)) -> active#(X1) -> active#(take(s(M),cons(N,IL))) -> take#(M,IL) active#(and(X1,X2)) -> active#(X1) -> active#(length(cons(N,L))) -> s#(length(L)) active#(and(X1,X2)) -> active#(X1) -> active#(length(cons(N,L))) -> length#(L) active#(and(X1,X2)) -> active#(X1) -> active#(zeros()) -> cons#(0(),zeros()) active#(cons(X1,X2)) -> cons#(active(X1),X2) -> cons#(ok(X1),ok(X2)) -> cons#(X1,X2) active#(cons(X1,X2)) -> cons#(active(X1),X2) -> cons#(mark(X1),X2) -> cons#(X1,X2) active#(cons(X1,X2)) -> active#(X1) -> active#(take(X1,X2)) -> take#(X1,active(X2)) active#(cons(X1,X2)) -> active#(X1) -> active#(take(X1,X2)) -> active#(X2) active#(cons(X1,X2)) -> active#(X1) -> active#(take(X1,X2)) -> take#(active(X1),X2) active#(cons(X1,X2)) -> active#(X1) -> active#(take(X1,X2)) -> active#(X1) active#(cons(X1,X2)) -> active#(X1) -> active#(s(X)) -> s#(active(X)) active#(cons(X1,X2)) -> active#(X1) -> active#(s(X)) -> active#(X) active#(cons(X1,X2)) -> active#(X1) -> active#(length(X)) -> length#(active(X)) active#(cons(X1,X2)) -> active#(X1) -> active#(length(X)) -> active#(X) active#(cons(X1,X2)) -> active#(X1) -> active#(and(X1,X2)) -> and#(active(X1),X2) active#(cons(X1,X2)) -> active#(X1) -> active#(and(X1,X2)) -> active#(X1) active#(cons(X1,X2)) -> active#(X1) -> active#(cons(X1,X2)) -> cons#(active(X1),X2) active#(cons(X1,X2)) -> active#(X1) -> active#(cons(X1,X2)) -> active#(X1) active#(cons(X1,X2)) -> active#(X1) -> active#(take(s(M),cons(N,IL))) -> cons#(N,take(M,IL)) active#(cons(X1,X2)) -> active#(X1) -> active#(take(s(M),cons(N,IL))) -> take#(M,IL) active#(cons(X1,X2)) -> active#(X1) -> active#(length(cons(N,L))) -> s#(length(L)) active#(cons(X1,X2)) -> active#(X1) -> active#(length(cons(N,L))) -> length#(L) active#(cons(X1,X2)) -> active#(X1) -> active#(zeros()) -> cons#(0(),zeros()) active#(zeros()) -> cons#(0(),zeros()) -> cons#(ok(X1),ok(X2)) -> cons#(X1,X2) active#(zeros()) -> cons#(0(),zeros()) -> cons#(mark(X1),X2) -> cons#(X1,X2) EDG Processor: DPs: active#(zeros()) -> cons#(0(),zeros()) active#(length(cons(N,L))) -> length#(L) active#(length(cons(N,L))) -> s#(length(L)) active#(take(s(M),cons(N,IL))) -> take#(M,IL) active#(take(s(M),cons(N,IL))) -> cons#(N,take(M,IL)) active#(cons(X1,X2)) -> active#(X1) active#(cons(X1,X2)) -> cons#(active(X1),X2) active#(and(X1,X2)) -> active#(X1) active#(and(X1,X2)) -> and#(active(X1),X2) active#(length(X)) -> active#(X) active#(length(X)) -> length#(active(X)) active#(s(X)) -> active#(X) active#(s(X)) -> s#(active(X)) 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)) cons#(mark(X1),X2) -> cons#(X1,X2) and#(mark(X1),X2) -> and#(X1,X2) length#(mark(X)) -> length#(X) s#(mark(X)) -> s#(X) take#(mark(X1),X2) -> take#(X1,X2) take#(X1,mark(X2)) -> take#(X1,X2) proper#(cons(X1,X2)) -> proper#(X2) proper#(cons(X1,X2)) -> proper#(X1) proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2)) proper#(and(X1,X2)) -> proper#(X2) proper#(and(X1,X2)) -> proper#(X1) proper#(and(X1,X2)) -> and#(proper(X1),proper(X2)) proper#(length(X)) -> proper#(X) proper#(length(X)) -> length#(proper(X)) proper#(s(X)) -> proper#(X) proper#(s(X)) -> s#(proper(X)) proper#(take(X1,X2)) -> proper#(X2) proper#(take(X1,X2)) -> proper#(X1) proper#(take(X1,X2)) -> take#(proper(X1),proper(X2)) cons#(ok(X1),ok(X2)) -> cons#(X1,X2) and#(ok(X1),ok(X2)) -> and#(X1,X2) length#(ok(X)) -> length#(X) s#(ok(X)) -> s#(X) take#(ok(X1),ok(X2)) -> take#(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(zeros()) -> mark(cons(0(),zeros())) active(and(tt(),X)) -> mark(X) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(s(length(L))) active(take(0(),IL)) -> mark(nil()) active(take(s(M),cons(N,IL))) -> mark(cons(N,take(M,IL))) active(cons(X1,X2)) -> cons(active(X1),X2) active(and(X1,X2)) -> and(active(X1),X2) active(length(X)) -> length(active(X)) active(s(X)) -> s(active(X)) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) cons(mark(X1),X2) -> mark(cons(X1,X2)) and(mark(X1),X2) -> mark(and(X1,X2)) length(mark(X)) -> mark(length(X)) s(mark(X)) -> mark(s(X)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(length(X)) -> length(proper(X)) proper(nil()) -> ok(nil()) proper(s(X)) -> s(proper(X)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) length(ok(X)) -> ok(length(X)) s(ok(X)) -> ok(s(X)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) graph: top#(ok(X)) -> top#(active(X)) -> top#(mark(X)) -> proper#(X) top#(ok(X)) -> top#(active(X)) -> top#(mark(X)) -> top#(proper(X)) top#(ok(X)) -> top#(active(X)) -> top#(ok(X)) -> active#(X) top#(ok(X)) -> top#(active(X)) -> top#(ok(X)) -> top#(active(X)) top#(ok(X)) -> active#(X) -> active#(zeros()) -> cons#(0(),zeros()) top#(ok(X)) -> active#(X) -> active#(length(cons(N,L))) -> length#(L) top#(ok(X)) -> active#(X) -> active#(length(cons(N,L))) -> s#(length(L)) top#(ok(X)) -> active#(X) -> active#(take(s(M),cons(N,IL))) -> take#(M,IL) top#(ok(X)) -> active#(X) -> active#(take(s(M),cons(N,IL))) -> cons#(N,take(M,IL)) top#(ok(X)) -> active#(X) -> active#(cons(X1,X2)) -> active#(X1) top#(ok(X)) -> active#(X) -> active#(cons(X1,X2)) -> cons#(active(X1),X2) top#(ok(X)) -> active#(X) -> active#(and(X1,X2)) -> active#(X1) top#(ok(X)) -> active#(X) -> active#(and(X1,X2)) -> and#(active(X1),X2) top#(ok(X)) -> active#(X) -> active#(length(X)) -> active#(X) top#(ok(X)) -> active#(X) -> active#(length(X)) -> length#(active(X)) top#(ok(X)) -> active#(X) -> active#(s(X)) -> active#(X) top#(ok(X)) -> active#(X) -> active#(s(X)) -> s#(active(X)) top#(ok(X)) -> active#(X) -> active#(take(X1,X2)) -> active#(X1) top#(ok(X)) -> active#(X) -> active#(take(X1,X2)) -> take#(active(X1),X2) top#(ok(X)) -> active#(X) -> active#(take(X1,X2)) -> active#(X2) top#(ok(X)) -> active#(X) -> active#(take(X1,X2)) -> take#(X1,active(X2)) top#(mark(X)) -> top#(proper(X)) -> top#(mark(X)) -> proper#(X) top#(mark(X)) -> top#(proper(X)) -> top#(mark(X)) -> top#(proper(X)) top#(mark(X)) -> top#(proper(X)) -> top#(ok(X)) -> active#(X) top#(mark(X)) -> top#(proper(X)) -> top#(ok(X)) -> top#(active(X)) top#(mark(X)) -> proper#(X) -> proper#(cons(X1,X2)) -> proper#(X2) top#(mark(X)) -> proper#(X) -> proper#(cons(X1,X2)) -> proper#(X1) top#(mark(X)) -> proper#(X) -> proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2)) top#(mark(X)) -> proper#(X) -> proper#(and(X1,X2)) -> proper#(X2) top#(mark(X)) -> proper#(X) -> proper#(and(X1,X2)) -> proper#(X1) top#(mark(X)) -> proper#(X) -> proper#(and(X1,X2)) -> and#(proper(X1),proper(X2)) top#(mark(X)) -> proper#(X) -> proper#(length(X)) -> proper#(X) top#(mark(X)) -> proper#(X) -> proper#(length(X)) -> length#(proper(X)) top#(mark(X)) -> proper#(X) -> proper#(s(X)) -> proper#(X) top#(mark(X)) -> proper#(X) -> proper#(s(X)) -> s#(proper(X)) top#(mark(X)) -> proper#(X) -> proper#(take(X1,X2)) -> proper#(X2) top#(mark(X)) -> proper#(X) -> proper#(take(X1,X2)) -> proper#(X1) top#(mark(X)) -> proper#(X) -> proper#(take(X1,X2)) -> take#(proper(X1),proper(X2)) proper#(take(X1,X2)) -> proper#(X2) -> proper#(cons(X1,X2)) -> proper#(X2) proper#(take(X1,X2)) -> proper#(X2) -> proper#(cons(X1,X2)) -> proper#(X1) proper#(take(X1,X2)) -> proper#(X2) -> proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2)) proper#(take(X1,X2)) -> proper#(X2) -> proper#(and(X1,X2)) -> proper#(X2) proper#(take(X1,X2)) -> proper#(X2) -> proper#(and(X1,X2)) -> proper#(X1) proper#(take(X1,X2)) -> proper#(X2) -> proper#(and(X1,X2)) -> and#(proper(X1),proper(X2)) proper#(take(X1,X2)) -> proper#(X2) -> proper#(length(X)) -> proper#(X) proper#(take(X1,X2)) -> proper#(X2) -> proper#(length(X)) -> length#(proper(X)) proper#(take(X1,X2)) -> proper#(X2) -> proper#(s(X)) -> proper#(X) proper#(take(X1,X2)) -> proper#(X2) -> proper#(s(X)) -> s#(proper(X)) proper#(take(X1,X2)) -> proper#(X2) -> proper#(take(X1,X2)) -> proper#(X2) proper#(take(X1,X2)) -> proper#(X2) -> proper#(take(X1,X2)) -> proper#(X1) proper#(take(X1,X2)) -> proper#(X2) -> proper#(take(X1,X2)) -> take#(proper(X1),proper(X2)) proper#(take(X1,X2)) -> proper#(X1) -> proper#(cons(X1,X2)) -> proper#(X2) proper#(take(X1,X2)) -> proper#(X1) -> proper#(cons(X1,X2)) -> proper#(X1) proper#(take(X1,X2)) -> proper#(X1) -> proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2)) proper#(take(X1,X2)) -> proper#(X1) -> proper#(and(X1,X2)) -> proper#(X2) proper#(take(X1,X2)) -> proper#(X1) -> proper#(and(X1,X2)) -> proper#(X1) proper#(take(X1,X2)) -> proper#(X1) -> proper#(and(X1,X2)) -> and#(proper(X1),proper(X2)) proper#(take(X1,X2)) -> proper#(X1) -> proper#(length(X)) -> proper#(X) proper#(take(X1,X2)) -> proper#(X1) -> proper#(length(X)) -> length#(proper(X)) proper#(take(X1,X2)) -> proper#(X1) -> proper#(s(X)) -> proper#(X) proper#(take(X1,X2)) -> proper#(X1) -> proper#(s(X)) -> s#(proper(X)) proper#(take(X1,X2)) -> proper#(X1) -> proper#(take(X1,X2)) -> proper#(X2) proper#(take(X1,X2)) -> proper#(X1) -> proper#(take(X1,X2)) -> proper#(X1) proper#(take(X1,X2)) -> proper#(X1) -> proper#(take(X1,X2)) -> take#(proper(X1),proper(X2)) proper#(take(X1,X2)) -> take#(proper(X1),proper(X2)) -> take#(mark(X1),X2) -> take#(X1,X2) proper#(take(X1,X2)) -> take#(proper(X1),proper(X2)) -> take#(X1,mark(X2)) -> take#(X1,X2) proper#(take(X1,X2)) -> take#(proper(X1),proper(X2)) -> take#(ok(X1),ok(X2)) -> take#(X1,X2) proper#(s(X)) -> proper#(X) -> proper#(cons(X1,X2)) -> proper#(X2) proper#(s(X)) -> proper#(X) -> proper#(cons(X1,X2)) -> proper#(X1) proper#(s(X)) -> proper#(X) -> proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2)) proper#(s(X)) -> proper#(X) -> proper#(and(X1,X2)) -> proper#(X2) proper#(s(X)) -> proper#(X) -> proper#(and(X1,X2)) -> proper#(X1) proper#(s(X)) -> proper#(X) -> proper#(and(X1,X2)) -> and#(proper(X1),proper(X2)) proper#(s(X)) -> proper#(X) -> proper#(length(X)) -> proper#(X) proper#(s(X)) -> proper#(X) -> proper#(length(X)) -> length#(proper(X)) proper#(s(X)) -> proper#(X) -> proper#(s(X)) -> proper#(X) proper#(s(X)) -> proper#(X) -> proper#(s(X)) -> s#(proper(X)) proper#(s(X)) -> proper#(X) -> proper#(take(X1,X2)) -> proper#(X2) proper#(s(X)) -> proper#(X) -> proper#(take(X1,X2)) -> proper#(X1) proper#(s(X)) -> proper#(X) -> proper#(take(X1,X2)) -> take#(proper(X1),proper(X2)) proper#(s(X)) -> s#(proper(X)) -> s#(mark(X)) -> s#(X) proper#(s(X)) -> s#(proper(X)) -> s#(ok(X)) -> s#(X) proper#(length(X)) -> proper#(X) -> proper#(cons(X1,X2)) -> proper#(X2) proper#(length(X)) -> proper#(X) -> proper#(cons(X1,X2)) -> proper#(X1) proper#(length(X)) -> proper#(X) -> proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2)) proper#(length(X)) -> proper#(X) -> proper#(and(X1,X2)) -> proper#(X2) proper#(length(X)) -> proper#(X) -> proper#(and(X1,X2)) -> proper#(X1) proper#(length(X)) -> proper#(X) -> proper#(and(X1,X2)) -> and#(proper(X1),proper(X2)) proper#(length(X)) -> proper#(X) -> proper#(length(X)) -> proper#(X) proper#(length(X)) -> proper#(X) -> proper#(length(X)) -> length#(proper(X)) proper#(length(X)) -> proper#(X) -> proper#(s(X)) -> proper#(X) proper#(length(X)) -> proper#(X) -> proper#(s(X)) -> s#(proper(X)) proper#(length(X)) -> proper#(X) -> proper#(take(X1,X2)) -> proper#(X2) proper#(length(X)) -> proper#(X) -> proper#(take(X1,X2)) -> proper#(X1) proper#(length(X)) -> proper#(X) -> proper#(take(X1,X2)) -> take#(proper(X1),proper(X2)) proper#(length(X)) -> length#(proper(X)) -> length#(mark(X)) -> length#(X) proper#(length(X)) -> length#(proper(X)) -> length#(ok(X)) -> length#(X) proper#(and(X1,X2)) -> proper#(X2) -> proper#(cons(X1,X2)) -> proper#(X2) proper#(and(X1,X2)) -> proper#(X2) -> proper#(cons(X1,X2)) -> proper#(X1) proper#(and(X1,X2)) -> proper#(X2) -> proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2)) proper#(and(X1,X2)) -> proper#(X2) -> proper#(and(X1,X2)) -> proper#(X2) proper#(and(X1,X2)) -> proper#(X2) -> proper#(and(X1,X2)) -> proper#(X1) proper#(and(X1,X2)) -> proper#(X2) -> proper#(and(X1,X2)) -> and#(proper(X1),proper(X2)) proper#(and(X1,X2)) -> proper#(X2) -> proper#(length(X)) -> proper#(X) proper#(and(X1,X2)) -> proper#(X2) -> proper#(length(X)) -> length#(proper(X)) proper#(and(X1,X2)) -> proper#(X2) -> proper#(s(X)) -> proper#(X) proper#(and(X1,X2)) -> proper#(X2) -> proper#(s(X)) -> s#(proper(X)) proper#(and(X1,X2)) -> proper#(X2) -> proper#(take(X1,X2)) -> proper#(X2) proper#(and(X1,X2)) -> proper#(X2) -> proper#(take(X1,X2)) -> proper#(X1) proper#(and(X1,X2)) -> proper#(X2) -> proper#(take(X1,X2)) -> take#(proper(X1),proper(X2)) proper#(and(X1,X2)) -> proper#(X1) -> proper#(cons(X1,X2)) -> proper#(X2) proper#(and(X1,X2)) -> proper#(X1) -> proper#(cons(X1,X2)) -> proper#(X1) proper#(and(X1,X2)) -> proper#(X1) -> proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2)) proper#(and(X1,X2)) -> proper#(X1) -> proper#(and(X1,X2)) -> proper#(X2) proper#(and(X1,X2)) -> proper#(X1) -> proper#(and(X1,X2)) -> proper#(X1) proper#(and(X1,X2)) -> proper#(X1) -> proper#(and(X1,X2)) -> and#(proper(X1),proper(X2)) proper#(and(X1,X2)) -> proper#(X1) -> proper#(length(X)) -> proper#(X) proper#(and(X1,X2)) -> proper#(X1) -> proper#(length(X)) -> length#(proper(X)) proper#(and(X1,X2)) -> proper#(X1) -> proper#(s(X)) -> proper#(X) proper#(and(X1,X2)) -> proper#(X1) -> proper#(s(X)) -> s#(proper(X)) proper#(and(X1,X2)) -> proper#(X1) -> proper#(take(X1,X2)) -> proper#(X2) proper#(and(X1,X2)) -> proper#(X1) -> proper#(take(X1,X2)) -> proper#(X1) proper#(and(X1,X2)) -> proper#(X1) -> proper#(take(X1,X2)) -> take#(proper(X1),proper(X2)) proper#(and(X1,X2)) -> and#(proper(X1),proper(X2)) -> and#(mark(X1),X2) -> and#(X1,X2) proper#(and(X1,X2)) -> and#(proper(X1),proper(X2)) -> and#(ok(X1),ok(X2)) -> and#(X1,X2) proper#(cons(X1,X2)) -> proper#(X2) -> proper#(cons(X1,X2)) -> proper#(X2) proper#(cons(X1,X2)) -> proper#(X2) -> proper#(cons(X1,X2)) -> proper#(X1) proper#(cons(X1,X2)) -> proper#(X2) -> proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2)) proper#(cons(X1,X2)) -> proper#(X2) -> proper#(and(X1,X2)) -> proper#(X2) proper#(cons(X1,X2)) -> proper#(X2) -> proper#(and(X1,X2)) -> proper#(X1) proper#(cons(X1,X2)) -> proper#(X2) -> proper#(and(X1,X2)) -> and#(proper(X1),proper(X2)) proper#(cons(X1,X2)) -> proper#(X2) -> proper#(length(X)) -> proper#(X) proper#(cons(X1,X2)) -> proper#(X2) -> proper#(length(X)) -> length#(proper(X)) proper#(cons(X1,X2)) -> proper#(X2) -> proper#(s(X)) -> proper#(X) proper#(cons(X1,X2)) -> proper#(X2) -> proper#(s(X)) -> s#(proper(X)) proper#(cons(X1,X2)) -> proper#(X2) -> proper#(take(X1,X2)) -> proper#(X2) proper#(cons(X1,X2)) -> proper#(X2) -> proper#(take(X1,X2)) -> proper#(X1) proper#(cons(X1,X2)) -> proper#(X2) -> proper#(take(X1,X2)) -> take#(proper(X1),proper(X2)) proper#(cons(X1,X2)) -> proper#(X1) -> proper#(cons(X1,X2)) -> proper#(X2) proper#(cons(X1,X2)) -> proper#(X1) -> proper#(cons(X1,X2)) -> proper#(X1) proper#(cons(X1,X2)) -> proper#(X1) -> proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2)) proper#(cons(X1,X2)) -> proper#(X1) -> proper#(and(X1,X2)) -> proper#(X2) proper#(cons(X1,X2)) -> proper#(X1) -> proper#(and(X1,X2)) -> proper#(X1) proper#(cons(X1,X2)) -> proper#(X1) -> proper#(and(X1,X2)) -> and#(proper(X1),proper(X2)) proper#(cons(X1,X2)) -> proper#(X1) -> proper#(length(X)) -> proper#(X) proper#(cons(X1,X2)) -> proper#(X1) -> proper#(length(X)) -> length#(proper(X)) proper#(cons(X1,X2)) -> proper#(X1) -> proper#(s(X)) -> proper#(X) proper#(cons(X1,X2)) -> proper#(X1) -> proper#(s(X)) -> s#(proper(X)) proper#(cons(X1,X2)) -> proper#(X1) -> proper#(take(X1,X2)) -> proper#(X2) proper#(cons(X1,X2)) -> proper#(X1) -> proper#(take(X1,X2)) -> proper#(X1) proper#(cons(X1,X2)) -> proper#(X1) -> proper#(take(X1,X2)) -> take#(proper(X1),proper(X2)) proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2)) -> cons#(mark(X1),X2) -> cons#(X1,X2) proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2)) -> cons#(ok(X1),ok(X2)) -> cons#(X1,X2) and#(ok(X1),ok(X2)) -> and#(X1,X2) -> and#(mark(X1),X2) -> and#(X1,X2) and#(ok(X1),ok(X2)) -> and#(X1,X2) -> and#(ok(X1),ok(X2)) -> and#(X1,X2) and#(mark(X1),X2) -> and#(X1,X2) -> and#(mark(X1),X2) -> and#(X1,X2) and#(mark(X1),X2) -> and#(X1,X2) -> and#(ok(X1),ok(X2)) -> and#(X1,X2) take#(ok(X1),ok(X2)) -> take#(X1,X2) -> take#(mark(X1),X2) -> take#(X1,X2) take#(ok(X1),ok(X2)) -> take#(X1,X2) -> take#(X1,mark(X2)) -> take#(X1,X2) take#(ok(X1),ok(X2)) -> take#(X1,X2) -> take#(ok(X1),ok(X2)) -> take#(X1,X2) take#(mark(X1),X2) -> take#(X1,X2) -> take#(mark(X1),X2) -> take#(X1,X2) take#(mark(X1),X2) -> take#(X1,X2) -> take#(X1,mark(X2)) -> take#(X1,X2) take#(mark(X1),X2) -> take#(X1,X2) -> take#(ok(X1),ok(X2)) -> take#(X1,X2) take#(X1,mark(X2)) -> take#(X1,X2) -> take#(mark(X1),X2) -> take#(X1,X2) take#(X1,mark(X2)) -> take#(X1,X2) -> take#(X1,mark(X2)) -> take#(X1,X2) take#(X1,mark(X2)) -> take#(X1,X2) -> take#(ok(X1),ok(X2)) -> take#(X1,X2) s#(ok(X)) -> s#(X) -> s#(mark(X)) -> s#(X) s#(ok(X)) -> s#(X) -> s#(ok(X)) -> s#(X) s#(mark(X)) -> s#(X) -> s#(mark(X)) -> s#(X) s#(mark(X)) -> s#(X) -> s#(ok(X)) -> s#(X) length#(ok(X)) -> length#(X) -> length#(mark(X)) -> length#(X) length#(ok(X)) -> length#(X) -> length#(ok(X)) -> length#(X) length#(mark(X)) -> length#(X) -> length#(mark(X)) -> length#(X) length#(mark(X)) -> length#(X) -> length#(ok(X)) -> length#(X) cons#(ok(X1),ok(X2)) -> cons#(X1,X2) -> cons#(mark(X1),X2) -> cons#(X1,X2) cons#(ok(X1),ok(X2)) -> cons#(X1,X2) -> cons#(ok(X1),ok(X2)) -> cons#(X1,X2) cons#(mark(X1),X2) -> cons#(X1,X2) -> cons#(mark(X1),X2) -> cons#(X1,X2) cons#(mark(X1),X2) -> cons#(X1,X2) -> cons#(ok(X1),ok(X2)) -> cons#(X1,X2) active#(take(s(M),cons(N,IL))) -> take#(M,IL) -> take#(X1,mark(X2)) -> take#(X1,X2) active#(take(s(M),cons(N,IL))) -> cons#(N,take(M,IL)) -> cons#(ok(X1),ok(X2)) -> cons#(X1,X2) active#(take(X1,X2)) -> take#(active(X1),X2) -> take#(mark(X1),X2) -> take#(X1,X2) active#(take(X1,X2)) -> take#(active(X1),X2) -> take#(ok(X1),ok(X2)) -> take#(X1,X2) active#(take(X1,X2)) -> take#(X1,active(X2)) -> take#(X1,mark(X2)) -> take#(X1,X2) active#(take(X1,X2)) -> take#(X1,active(X2)) -> take#(ok(X1),ok(X2)) -> take#(X1,X2) active#(take(X1,X2)) -> active#(X2) -> active#(zeros()) -> cons#(0(),zeros()) active#(take(X1,X2)) -> active#(X2) -> active#(length(cons(N,L))) -> length#(L) active#(take(X1,X2)) -> active#(X2) -> active#(length(cons(N,L))) -> s#(length(L)) active#(take(X1,X2)) -> active#(X2) -> active#(take(s(M),cons(N,IL))) -> take#(M,IL) active#(take(X1,X2)) -> active#(X2) -> active#(take(s(M),cons(N,IL))) -> cons#(N,take(M,IL)) active#(take(X1,X2)) -> active#(X2) -> active#(cons(X1,X2)) -> active#(X1) active#(take(X1,X2)) -> active#(X2) -> active#(cons(X1,X2)) -> cons#(active(X1),X2) active#(take(X1,X2)) -> active#(X2) -> active#(and(X1,X2)) -> active#(X1) active#(take(X1,X2)) -> active#(X2) -> active#(and(X1,X2)) -> and#(active(X1),X2) active#(take(X1,X2)) -> active#(X2) -> active#(length(X)) -> active#(X) active#(take(X1,X2)) -> active#(X2) -> active#(length(X)) -> length#(active(X)) active#(take(X1,X2)) -> active#(X2) -> active#(s(X)) -> active#(X) active#(take(X1,X2)) -> active#(X2) -> active#(s(X)) -> s#(active(X)) active#(take(X1,X2)) -> active#(X2) -> active#(take(X1,X2)) -> active#(X1) active#(take(X1,X2)) -> active#(X2) -> active#(take(X1,X2)) -> take#(active(X1),X2) active#(take(X1,X2)) -> active#(X2) -> active#(take(X1,X2)) -> active#(X2) active#(take(X1,X2)) -> active#(X2) -> active#(take(X1,X2)) -> take#(X1,active(X2)) active#(take(X1,X2)) -> active#(X1) -> active#(zeros()) -> cons#(0(),zeros()) active#(take(X1,X2)) -> active#(X1) -> active#(length(cons(N,L))) -> length#(L) active#(take(X1,X2)) -> active#(X1) -> active#(length(cons(N,L))) -> s#(length(L)) active#(take(X1,X2)) -> active#(X1) -> active#(take(s(M),cons(N,IL))) -> take#(M,IL) active#(take(X1,X2)) -> active#(X1) -> active#(take(s(M),cons(N,IL))) -> cons#(N,take(M,IL)) active#(take(X1,X2)) -> active#(X1) -> active#(cons(X1,X2)) -> active#(X1) active#(take(X1,X2)) -> active#(X1) -> active#(cons(X1,X2)) -> cons#(active(X1),X2) active#(take(X1,X2)) -> active#(X1) -> active#(and(X1,X2)) -> active#(X1) active#(take(X1,X2)) -> active#(X1) -> active#(and(X1,X2)) -> and#(active(X1),X2) active#(take(X1,X2)) -> active#(X1) -> active#(length(X)) -> active#(X) active#(take(X1,X2)) -> active#(X1) -> active#(length(X)) -> length#(active(X)) active#(take(X1,X2)) -> active#(X1) -> active#(s(X)) -> active#(X) active#(take(X1,X2)) -> active#(X1) -> active#(s(X)) -> s#(active(X)) active#(take(X1,X2)) -> active#(X1) -> active#(take(X1,X2)) -> active#(X1) active#(take(X1,X2)) -> active#(X1) -> active#(take(X1,X2)) -> take#(active(X1),X2) active#(take(X1,X2)) -> active#(X1) -> active#(take(X1,X2)) -> active#(X2) active#(take(X1,X2)) -> active#(X1) -> active#(take(X1,X2)) -> take#(X1,active(X2)) active#(s(X)) -> s#(active(X)) -> s#(mark(X)) -> s#(X) active#(s(X)) -> s#(active(X)) -> s#(ok(X)) -> s#(X) active#(s(X)) -> active#(X) -> active#(zeros()) -> cons#(0(),zeros()) active#(s(X)) -> active#(X) -> active#(length(cons(N,L))) -> length#(L) active#(s(X)) -> active#(X) -> active#(length(cons(N,L))) -> s#(length(L)) active#(s(X)) -> active#(X) -> active#(take(s(M),cons(N,IL))) -> take#(M,IL) active#(s(X)) -> active#(X) -> active#(take(s(M),cons(N,IL))) -> cons#(N,take(M,IL)) active#(s(X)) -> active#(X) -> active#(cons(X1,X2)) -> active#(X1) active#(s(X)) -> active#(X) -> active#(cons(X1,X2)) -> cons#(active(X1),X2) active#(s(X)) -> active#(X) -> active#(and(X1,X2)) -> active#(X1) active#(s(X)) -> active#(X) -> active#(and(X1,X2)) -> and#(active(X1),X2) active#(s(X)) -> active#(X) -> active#(length(X)) -> active#(X) active#(s(X)) -> active#(X) -> active#(length(X)) -> length#(active(X)) active#(s(X)) -> active#(X) -> active#(s(X)) -> active#(X) active#(s(X)) -> active#(X) -> active#(s(X)) -> s#(active(X)) active#(s(X)) -> active#(X) -> active#(take(X1,X2)) -> active#(X1) active#(s(X)) -> active#(X) -> active#(take(X1,X2)) -> take#(active(X1),X2) active#(s(X)) -> active#(X) -> active#(take(X1,X2)) -> active#(X2) active#(s(X)) -> active#(X) -> active#(take(X1,X2)) -> take#(X1,active(X2)) active#(length(cons(N,L))) -> s#(length(L)) -> s#(mark(X)) -> s#(X) active#(length(cons(N,L))) -> s#(length(L)) -> s#(ok(X)) -> s#(X) active#(length(cons(N,L))) -> length#(L) -> length#(mark(X)) -> length#(X) active#(length(cons(N,L))) -> length#(L) -> length#(ok(X)) -> length#(X) active#(length(X)) -> length#(active(X)) -> length#(mark(X)) -> length#(X) active#(length(X)) -> length#(active(X)) -> length#(ok(X)) -> length#(X) active#(length(X)) -> active#(X) -> active#(zeros()) -> cons#(0(),zeros()) active#(length(X)) -> active#(X) -> active#(length(cons(N,L))) -> length#(L) active#(length(X)) -> active#(X) -> active#(length(cons(N,L))) -> s#(length(L)) active#(length(X)) -> active#(X) -> active#(take(s(M),cons(N,IL))) -> take#(M,IL) active#(length(X)) -> active#(X) -> active#(take(s(M),cons(N,IL))) -> cons#(N,take(M,IL)) active#(length(X)) -> active#(X) -> active#(cons(X1,X2)) -> active#(X1) active#(length(X)) -> active#(X) -> active#(cons(X1,X2)) -> cons#(active(X1),X2) active#(length(X)) -> active#(X) -> active#(and(X1,X2)) -> active#(X1) active#(length(X)) -> active#(X) -> active#(and(X1,X2)) -> and#(active(X1),X2) active#(length(X)) -> active#(X) -> active#(length(X)) -> active#(X) active#(length(X)) -> active#(X) -> active#(length(X)) -> length#(active(X)) active#(length(X)) -> active#(X) -> active#(s(X)) -> active#(X) active#(length(X)) -> active#(X) -> active#(s(X)) -> s#(active(X)) active#(length(X)) -> active#(X) -> active#(take(X1,X2)) -> active#(X1) active#(length(X)) -> active#(X) -> active#(take(X1,X2)) -> take#(active(X1),X2) active#(length(X)) -> active#(X) -> active#(take(X1,X2)) -> active#(X2) active#(length(X)) -> active#(X) -> active#(take(X1,X2)) -> take#(X1,active(X2)) active#(and(X1,X2)) -> and#(active(X1),X2) -> and#(mark(X1),X2) -> and#(X1,X2) active#(and(X1,X2)) -> and#(active(X1),X2) -> and#(ok(X1),ok(X2)) -> and#(X1,X2) active#(and(X1,X2)) -> active#(X1) -> active#(zeros()) -> cons#(0(),zeros()) active#(and(X1,X2)) -> active#(X1) -> active#(length(cons(N,L))) -> length#(L) active#(and(X1,X2)) -> active#(X1) -> active#(length(cons(N,L))) -> s#(length(L)) active#(and(X1,X2)) -> active#(X1) -> active#(take(s(M),cons(N,IL))) -> take#(M,IL) active#(and(X1,X2)) -> active#(X1) -> active#(take(s(M),cons(N,IL))) -> cons#(N,take(M,IL)) active#(and(X1,X2)) -> active#(X1) -> active#(cons(X1,X2)) -> active#(X1) active#(and(X1,X2)) -> active#(X1) -> active#(cons(X1,X2)) -> cons#(active(X1),X2) active#(and(X1,X2)) -> active#(X1) -> active#(and(X1,X2)) -> active#(X1) active#(and(X1,X2)) -> active#(X1) -> active#(and(X1,X2)) -> and#(active(X1),X2) active#(and(X1,X2)) -> active#(X1) -> active#(length(X)) -> active#(X) active#(and(X1,X2)) -> active#(X1) -> active#(length(X)) -> length#(active(X)) active#(and(X1,X2)) -> active#(X1) -> active#(s(X)) -> active#(X) active#(and(X1,X2)) -> active#(X1) -> active#(s(X)) -> s#(active(X)) active#(and(X1,X2)) -> active#(X1) -> active#(take(X1,X2)) -> active#(X1) active#(and(X1,X2)) -> active#(X1) -> active#(take(X1,X2)) -> take#(active(X1),X2) active#(and(X1,X2)) -> active#(X1) -> active#(take(X1,X2)) -> active#(X2) active#(and(X1,X2)) -> active#(X1) -> active#(take(X1,X2)) -> take#(X1,active(X2)) active#(cons(X1,X2)) -> cons#(active(X1),X2) -> cons#(mark(X1),X2) -> cons#(X1,X2) active#(cons(X1,X2)) -> cons#(active(X1),X2) -> cons#(ok(X1),ok(X2)) -> cons#(X1,X2) active#(cons(X1,X2)) -> active#(X1) -> active#(zeros()) -> cons#(0(),zeros()) active#(cons(X1,X2)) -> active#(X1) -> active#(length(cons(N,L))) -> length#(L) active#(cons(X1,X2)) -> active#(X1) -> active#(length(cons(N,L))) -> s#(length(L)) active#(cons(X1,X2)) -> active#(X1) -> active#(take(s(M),cons(N,IL))) -> take#(M,IL) active#(cons(X1,X2)) -> active#(X1) -> active#(take(s(M),cons(N,IL))) -> cons#(N,take(M,IL)) active#(cons(X1,X2)) -> active#(X1) -> active#(cons(X1,X2)) -> active#(X1) active#(cons(X1,X2)) -> active#(X1) -> active#(cons(X1,X2)) -> cons#(active(X1),X2) active#(cons(X1,X2)) -> active#(X1) -> active#(and(X1,X2)) -> active#(X1) active#(cons(X1,X2)) -> active#(X1) -> active#(and(X1,X2)) -> and#(active(X1),X2) active#(cons(X1,X2)) -> active#(X1) -> active#(length(X)) -> active#(X) active#(cons(X1,X2)) -> active#(X1) -> active#(length(X)) -> length#(active(X)) active#(cons(X1,X2)) -> active#(X1) -> active#(s(X)) -> active#(X) active#(cons(X1,X2)) -> active#(X1) -> active#(s(X)) -> s#(active(X)) active#(cons(X1,X2)) -> active#(X1) -> active#(take(X1,X2)) -> active#(X1) active#(cons(X1,X2)) -> active#(X1) -> active#(take(X1,X2)) -> take#(active(X1),X2) active#(cons(X1,X2)) -> active#(X1) -> active#(take(X1,X2)) -> active#(X2) active#(cons(X1,X2)) -> active#(X1) -> active#(take(X1,X2)) -> take#(X1,active(X2)) SCC Processor: #sccs: 8 #rules: 27 #arcs: 298/2025 DPs: top#(ok(X)) -> top#(active(X)) top#(mark(X)) -> top#(proper(X)) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(and(tt(),X)) -> mark(X) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(s(length(L))) active(take(0(),IL)) -> mark(nil()) active(take(s(M),cons(N,IL))) -> mark(cons(N,take(M,IL))) active(cons(X1,X2)) -> cons(active(X1),X2) active(and(X1,X2)) -> and(active(X1),X2) active(length(X)) -> length(active(X)) active(s(X)) -> s(active(X)) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) cons(mark(X1),X2) -> mark(cons(X1,X2)) and(mark(X1),X2) -> mark(and(X1,X2)) length(mark(X)) -> mark(length(X)) s(mark(X)) -> mark(s(X)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(length(X)) -> length(proper(X)) proper(nil()) -> ok(nil()) proper(s(X)) -> s(proper(X)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) length(ok(X)) -> ok(length(X)) s(ok(X)) -> ok(s(X)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Open DPs: proper#(take(X1,X2)) -> proper#(X1) proper#(take(X1,X2)) -> proper#(X2) proper#(s(X)) -> proper#(X) proper#(length(X)) -> proper#(X) proper#(and(X1,X2)) -> proper#(X1) proper#(and(X1,X2)) -> proper#(X2) proper#(cons(X1,X2)) -> proper#(X1) proper#(cons(X1,X2)) -> proper#(X2) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(and(tt(),X)) -> mark(X) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(s(length(L))) active(take(0(),IL)) -> mark(nil()) active(take(s(M),cons(N,IL))) -> mark(cons(N,take(M,IL))) active(cons(X1,X2)) -> cons(active(X1),X2) active(and(X1,X2)) -> and(active(X1),X2) active(length(X)) -> length(active(X)) active(s(X)) -> s(active(X)) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) cons(mark(X1),X2) -> mark(cons(X1,X2)) and(mark(X1),X2) -> mark(and(X1,X2)) length(mark(X)) -> mark(length(X)) s(mark(X)) -> mark(s(X)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(length(X)) -> length(proper(X)) proper(nil()) -> ok(nil()) proper(s(X)) -> s(proper(X)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) length(ok(X)) -> ok(length(X)) s(ok(X)) -> ok(s(X)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dimension: 1 interpretation: [proper#](x0) = x0 + 1, [top](x0) = 0, [ok](x0) = x0, [proper](x0) = x0, [take](x0, x1) = x0 + x1 + 1, [s](x0) = x0 + 1, [length](x0) = x0 + 1, [nil] = 0, [and](x0, x1) = x0 + x1 + 1, [tt] = 1, [mark](x0) = 1, [cons](x0, x1) = x0 + x1, [0] = 0, [active](x0) = x0, [zeros] = 1 orientation: proper#(take(X1,X2)) = X1 + X2 + 2 >= X1 + 1 = proper#(X1) proper#(take(X1,X2)) = X1 + X2 + 2 >= X2 + 1 = proper#(X2) proper#(s(X)) = X + 2 >= X + 1 = proper#(X) proper#(length(X)) = X + 2 >= X + 1 = proper#(X) proper#(and(X1,X2)) = X1 + X2 + 2 >= X1 + 1 = proper#(X1) proper#(and(X1,X2)) = X1 + X2 + 2 >= X2 + 1 = proper#(X2) proper#(cons(X1,X2)) = X1 + X2 + 1 >= X1 + 1 = proper#(X1) proper#(cons(X1,X2)) = X1 + X2 + 1 >= X2 + 1 = proper#(X2) active(zeros()) = 1 >= 1 = mark(cons(0(),zeros())) active(and(tt(),X)) = X + 2 >= 1 = mark(X) active(length(nil())) = 1 >= 1 = mark(0()) active(length(cons(N,L))) = L + N + 1 >= 1 = mark(s(length(L))) active(take(0(),IL)) = IL + 1 >= 1 = mark(nil()) active(take(s(M),cons(N,IL))) = IL + M + N + 2 >= 1 = mark(cons(N,take(M,IL))) active(cons(X1,X2)) = X1 + X2 >= X1 + X2 = cons(active(X1),X2) active(and(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = and(active(X1),X2) active(length(X)) = X + 1 >= X + 1 = length(active(X)) active(s(X)) = X + 1 >= X + 1 = s(active(X)) active(take(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = take(active(X1),X2) active(take(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = take(X1,active(X2)) cons(mark(X1),X2) = X2 + 1 >= 1 = mark(cons(X1,X2)) and(mark(X1),X2) = X2 + 2 >= 1 = mark(and(X1,X2)) length(mark(X)) = 2 >= 1 = mark(length(X)) s(mark(X)) = 2 >= 1 = mark(s(X)) take(mark(X1),X2) = X2 + 2 >= 1 = mark(take(X1,X2)) take(X1,mark(X2)) = X1 + 2 >= 1 = mark(take(X1,X2)) proper(zeros()) = 1 >= 1 = ok(zeros()) proper(cons(X1,X2)) = X1 + X2 >= X1 + X2 = cons(proper(X1),proper(X2)) proper(0()) = 0 >= 0 = ok(0()) proper(and(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = and(proper(X1),proper(X2)) proper(tt()) = 1 >= 1 = ok(tt()) proper(length(X)) = X + 1 >= X + 1 = length(proper(X)) proper(nil()) = 0 >= 0 = ok(nil()) proper(s(X)) = X + 1 >= X + 1 = s(proper(X)) proper(take(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = take(proper(X1),proper(X2)) cons(ok(X1),ok(X2)) = X1 + X2 >= X1 + X2 = ok(cons(X1,X2)) and(ok(X1),ok(X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = ok(and(X1,X2)) length(ok(X)) = X + 1 >= X + 1 = ok(length(X)) s(ok(X)) = X + 1 >= X + 1 = ok(s(X)) take(ok(X1),ok(X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = ok(take(X1,X2)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) problem: DPs: proper#(cons(X1,X2)) -> proper#(X1) proper#(cons(X1,X2)) -> proper#(X2) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(and(tt(),X)) -> mark(X) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(s(length(L))) active(take(0(),IL)) -> mark(nil()) active(take(s(M),cons(N,IL))) -> mark(cons(N,take(M,IL))) active(cons(X1,X2)) -> cons(active(X1),X2) active(and(X1,X2)) -> and(active(X1),X2) active(length(X)) -> length(active(X)) active(s(X)) -> s(active(X)) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) cons(mark(X1),X2) -> mark(cons(X1,X2)) and(mark(X1),X2) -> mark(and(X1,X2)) length(mark(X)) -> mark(length(X)) s(mark(X)) -> mark(s(X)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(length(X)) -> length(proper(X)) proper(nil()) -> ok(nil()) proper(s(X)) -> s(proper(X)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) length(ok(X)) -> ok(length(X)) s(ok(X)) -> ok(s(X)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dimension: 1 interpretation: [proper#](x0) = x0 + 1, [top](x0) = 0, [ok](x0) = x0, [proper](x0) = x0, [take](x0, x1) = 0, [s](x0) = 0, [length](x0) = 0, [nil] = 0, [and](x0, x1) = 0, [tt] = 1, [mark](x0) = 0, [cons](x0, x1) = x0 + x1 + 1, [0] = 1, [active](x0) = x0, [zeros] = 0 orientation: proper#(cons(X1,X2)) = X1 + X2 + 2 >= X1 + 1 = proper#(X1) proper#(cons(X1,X2)) = X1 + X2 + 2 >= X2 + 1 = proper#(X2) active(zeros()) = 0 >= 0 = mark(cons(0(),zeros())) active(and(tt(),X)) = 0 >= 0 = mark(X) active(length(nil())) = 0 >= 0 = mark(0()) active(length(cons(N,L))) = 0 >= 0 = mark(s(length(L))) active(take(0(),IL)) = 0 >= 0 = mark(nil()) active(take(s(M),cons(N,IL))) = 0 >= 0 = mark(cons(N,take(M,IL))) active(cons(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = cons(active(X1),X2) active(and(X1,X2)) = 0 >= 0 = and(active(X1),X2) active(length(X)) = 0 >= 0 = length(active(X)) active(s(X)) = 0 >= 0 = s(active(X)) active(take(X1,X2)) = 0 >= 0 = take(active(X1),X2) active(take(X1,X2)) = 0 >= 0 = take(X1,active(X2)) cons(mark(X1),X2) = X2 + 1 >= 0 = mark(cons(X1,X2)) and(mark(X1),X2) = 0 >= 0 = mark(and(X1,X2)) length(mark(X)) = 0 >= 0 = mark(length(X)) s(mark(X)) = 0 >= 0 = mark(s(X)) take(mark(X1),X2) = 0 >= 0 = mark(take(X1,X2)) take(X1,mark(X2)) = 0 >= 0 = mark(take(X1,X2)) proper(zeros()) = 0 >= 0 = ok(zeros()) proper(cons(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = cons(proper(X1),proper(X2)) proper(0()) = 1 >= 1 = ok(0()) proper(and(X1,X2)) = 0 >= 0 = and(proper(X1),proper(X2)) proper(tt()) = 1 >= 1 = ok(tt()) proper(length(X)) = 0 >= 0 = length(proper(X)) proper(nil()) = 0 >= 0 = ok(nil()) proper(s(X)) = 0 >= 0 = s(proper(X)) proper(take(X1,X2)) = 0 >= 0 = take(proper(X1),proper(X2)) cons(ok(X1),ok(X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = ok(cons(X1,X2)) and(ok(X1),ok(X2)) = 0 >= 0 = ok(and(X1,X2)) length(ok(X)) = 0 >= 0 = ok(length(X)) s(ok(X)) = 0 >= 0 = ok(s(X)) take(ok(X1),ok(X2)) = 0 >= 0 = ok(take(X1,X2)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) problem: DPs: TRS: active(zeros()) -> mark(cons(0(),zeros())) active(and(tt(),X)) -> mark(X) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(s(length(L))) active(take(0(),IL)) -> mark(nil()) active(take(s(M),cons(N,IL))) -> mark(cons(N,take(M,IL))) active(cons(X1,X2)) -> cons(active(X1),X2) active(and(X1,X2)) -> and(active(X1),X2) active(length(X)) -> length(active(X)) active(s(X)) -> s(active(X)) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) cons(mark(X1),X2) -> mark(cons(X1,X2)) and(mark(X1),X2) -> mark(and(X1,X2)) length(mark(X)) -> mark(length(X)) s(mark(X)) -> mark(s(X)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(length(X)) -> length(proper(X)) proper(nil()) -> ok(nil()) proper(s(X)) -> s(proper(X)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) length(ok(X)) -> ok(length(X)) s(ok(X)) -> ok(s(X)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed DPs: active#(take(X1,X2)) -> active#(X2) active#(take(X1,X2)) -> active#(X1) active#(s(X)) -> active#(X) active#(length(X)) -> active#(X) active#(and(X1,X2)) -> active#(X1) active#(cons(X1,X2)) -> active#(X1) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(and(tt(),X)) -> mark(X) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(s(length(L))) active(take(0(),IL)) -> mark(nil()) active(take(s(M),cons(N,IL))) -> mark(cons(N,take(M,IL))) active(cons(X1,X2)) -> cons(active(X1),X2) active(and(X1,X2)) -> and(active(X1),X2) active(length(X)) -> length(active(X)) active(s(X)) -> s(active(X)) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) cons(mark(X1),X2) -> mark(cons(X1,X2)) and(mark(X1),X2) -> mark(and(X1,X2)) length(mark(X)) -> mark(length(X)) s(mark(X)) -> mark(s(X)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(length(X)) -> length(proper(X)) proper(nil()) -> ok(nil()) proper(s(X)) -> s(proper(X)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) length(ok(X)) -> ok(length(X)) s(ok(X)) -> ok(s(X)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dimension: 1 interpretation: [active#](x0) = x0 + 1, [top](x0) = 0, [ok](x0) = x0, [proper](x0) = x0, [take](x0, x1) = x0 + x1 + 1, [s](x0) = x0 + 1, [length](x0) = x0 + 1, [nil] = 0, [and](x0, x1) = x0 + 1, [tt] = 0, [mark](x0) = 1, [cons](x0, x1) = x0, [0] = 1, [active](x0) = x0, [zeros] = 1 orientation: active#(take(X1,X2)) = X1 + X2 + 2 >= X2 + 1 = active#(X2) active#(take(X1,X2)) = X1 + X2 + 2 >= X1 + 1 = active#(X1) active#(s(X)) = X + 2 >= X + 1 = active#(X) active#(length(X)) = X + 2 >= X + 1 = active#(X) active#(and(X1,X2)) = X1 + 2 >= X1 + 1 = active#(X1) active#(cons(X1,X2)) = X1 + 1 >= X1 + 1 = active#(X1) active(zeros()) = 1 >= 1 = mark(cons(0(),zeros())) active(and(tt(),X)) = 1 >= 1 = mark(X) active(length(nil())) = 1 >= 1 = mark(0()) active(length(cons(N,L))) = N + 1 >= 1 = mark(s(length(L))) active(take(0(),IL)) = IL + 2 >= 1 = mark(nil()) active(take(s(M),cons(N,IL))) = M + N + 2 >= 1 = mark(cons(N,take(M,IL))) active(cons(X1,X2)) = X1 >= X1 = cons(active(X1),X2) active(and(X1,X2)) = X1 + 1 >= X1 + 1 = and(active(X1),X2) active(length(X)) = X + 1 >= X + 1 = length(active(X)) active(s(X)) = X + 1 >= X + 1 = s(active(X)) active(take(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = take(active(X1),X2) active(take(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = take(X1,active(X2)) cons(mark(X1),X2) = 1 >= 1 = mark(cons(X1,X2)) and(mark(X1),X2) = 2 >= 1 = mark(and(X1,X2)) length(mark(X)) = 2 >= 1 = mark(length(X)) s(mark(X)) = 2 >= 1 = mark(s(X)) take(mark(X1),X2) = X2 + 2 >= 1 = mark(take(X1,X2)) take(X1,mark(X2)) = X1 + 2 >= 1 = mark(take(X1,X2)) proper(zeros()) = 1 >= 1 = ok(zeros()) proper(cons(X1,X2)) = X1 >= X1 = cons(proper(X1),proper(X2)) proper(0()) = 1 >= 1 = ok(0()) proper(and(X1,X2)) = X1 + 1 >= X1 + 1 = and(proper(X1),proper(X2)) proper(tt()) = 0 >= 0 = ok(tt()) proper(length(X)) = X + 1 >= X + 1 = length(proper(X)) proper(nil()) = 0 >= 0 = ok(nil()) proper(s(X)) = X + 1 >= X + 1 = s(proper(X)) proper(take(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = take(proper(X1),proper(X2)) cons(ok(X1),ok(X2)) = X1 >= X1 = ok(cons(X1,X2)) and(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(and(X1,X2)) length(ok(X)) = X + 1 >= X + 1 = ok(length(X)) s(ok(X)) = X + 1 >= X + 1 = ok(s(X)) take(ok(X1),ok(X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = ok(take(X1,X2)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) problem: DPs: active#(cons(X1,X2)) -> active#(X1) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(and(tt(),X)) -> mark(X) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(s(length(L))) active(take(0(),IL)) -> mark(nil()) active(take(s(M),cons(N,IL))) -> mark(cons(N,take(M,IL))) active(cons(X1,X2)) -> cons(active(X1),X2) active(and(X1,X2)) -> and(active(X1),X2) active(length(X)) -> length(active(X)) active(s(X)) -> s(active(X)) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) cons(mark(X1),X2) -> mark(cons(X1,X2)) and(mark(X1),X2) -> mark(and(X1,X2)) length(mark(X)) -> mark(length(X)) s(mark(X)) -> mark(s(X)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(length(X)) -> length(proper(X)) proper(nil()) -> ok(nil()) proper(s(X)) -> s(proper(X)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) length(ok(X)) -> ok(length(X)) s(ok(X)) -> ok(s(X)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dimension: 1 interpretation: [active#](x0) = x0 + 1, [top](x0) = 0, [ok](x0) = x0, [proper](x0) = x0, [take](x0, x1) = 0, [s](x0) = 0, [length](x0) = 0, [nil] = 0, [and](x0, x1) = 0, [tt] = 0, [mark](x0) = 0, [cons](x0, x1) = x0 + 1, [0] = 0, [active](x0) = x0, [zeros] = 0 orientation: active#(cons(X1,X2)) = X1 + 2 >= X1 + 1 = active#(X1) active(zeros()) = 0 >= 0 = mark(cons(0(),zeros())) active(and(tt(),X)) = 0 >= 0 = mark(X) active(length(nil())) = 0 >= 0 = mark(0()) active(length(cons(N,L))) = 0 >= 0 = mark(s(length(L))) active(take(0(),IL)) = 0 >= 0 = mark(nil()) active(take(s(M),cons(N,IL))) = 0 >= 0 = mark(cons(N,take(M,IL))) active(cons(X1,X2)) = X1 + 1 >= X1 + 1 = cons(active(X1),X2) active(and(X1,X2)) = 0 >= 0 = and(active(X1),X2) active(length(X)) = 0 >= 0 = length(active(X)) active(s(X)) = 0 >= 0 = s(active(X)) active(take(X1,X2)) = 0 >= 0 = take(active(X1),X2) active(take(X1,X2)) = 0 >= 0 = take(X1,active(X2)) cons(mark(X1),X2) = 1 >= 0 = mark(cons(X1,X2)) and(mark(X1),X2) = 0 >= 0 = mark(and(X1,X2)) length(mark(X)) = 0 >= 0 = mark(length(X)) s(mark(X)) = 0 >= 0 = mark(s(X)) take(mark(X1),X2) = 0 >= 0 = mark(take(X1,X2)) take(X1,mark(X2)) = 0 >= 0 = mark(take(X1,X2)) proper(zeros()) = 0 >= 0 = ok(zeros()) proper(cons(X1,X2)) = X1 + 1 >= X1 + 1 = cons(proper(X1),proper(X2)) proper(0()) = 0 >= 0 = ok(0()) proper(and(X1,X2)) = 0 >= 0 = and(proper(X1),proper(X2)) proper(tt()) = 0 >= 0 = ok(tt()) proper(length(X)) = 0 >= 0 = length(proper(X)) proper(nil()) = 0 >= 0 = ok(nil()) proper(s(X)) = 0 >= 0 = s(proper(X)) proper(take(X1,X2)) = 0 >= 0 = take(proper(X1),proper(X2)) cons(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(cons(X1,X2)) and(ok(X1),ok(X2)) = 0 >= 0 = ok(and(X1,X2)) length(ok(X)) = 0 >= 0 = ok(length(X)) s(ok(X)) = 0 >= 0 = ok(s(X)) take(ok(X1),ok(X2)) = 0 >= 0 = ok(take(X1,X2)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) problem: DPs: TRS: active(zeros()) -> mark(cons(0(),zeros())) active(and(tt(),X)) -> mark(X) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(s(length(L))) active(take(0(),IL)) -> mark(nil()) active(take(s(M),cons(N,IL))) -> mark(cons(N,take(M,IL))) active(cons(X1,X2)) -> cons(active(X1),X2) active(and(X1,X2)) -> and(active(X1),X2) active(length(X)) -> length(active(X)) active(s(X)) -> s(active(X)) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) cons(mark(X1),X2) -> mark(cons(X1,X2)) and(mark(X1),X2) -> mark(and(X1,X2)) length(mark(X)) -> mark(length(X)) s(mark(X)) -> mark(s(X)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(length(X)) -> length(proper(X)) proper(nil()) -> ok(nil()) proper(s(X)) -> s(proper(X)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) length(ok(X)) -> ok(length(X)) s(ok(X)) -> ok(s(X)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed DPs: cons#(ok(X1),ok(X2)) -> cons#(X1,X2) cons#(mark(X1),X2) -> cons#(X1,X2) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(and(tt(),X)) -> mark(X) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(s(length(L))) active(take(0(),IL)) -> mark(nil()) active(take(s(M),cons(N,IL))) -> mark(cons(N,take(M,IL))) active(cons(X1,X2)) -> cons(active(X1),X2) active(and(X1,X2)) -> and(active(X1),X2) active(length(X)) -> length(active(X)) active(s(X)) -> s(active(X)) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) cons(mark(X1),X2) -> mark(cons(X1,X2)) and(mark(X1),X2) -> mark(and(X1,X2)) length(mark(X)) -> mark(length(X)) s(mark(X)) -> mark(s(X)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(length(X)) -> length(proper(X)) proper(nil()) -> ok(nil()) proper(s(X)) -> s(proper(X)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) length(ok(X)) -> ok(length(X)) s(ok(X)) -> ok(s(X)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dimension: 1 interpretation: [cons#](x0, x1) = x0 + 1, [top](x0) = 1, [ok](x0) = x0, [proper](x0) = x0, [take](x0, x1) = x0 + x1, [s](x0) = x0, [length](x0) = x0, [nil] = 0, [and](x0, x1) = x0 + x1 + 1, [tt] = 0, [mark](x0) = x0 + 1, [cons](x0, x1) = x0 + x1, [0] = 0, [active](x0) = x0 + 1, [zeros] = 1 orientation: cons#(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = cons#(X1,X2) cons#(mark(X1),X2) = X1 + 2 >= X1 + 1 = cons#(X1,X2) active(zeros()) = 2 >= 2 = mark(cons(0(),zeros())) active(and(tt(),X)) = X + 2 >= X + 1 = mark(X) active(length(nil())) = 1 >= 1 = mark(0()) active(length(cons(N,L))) = L + N + 1 >= L + 1 = mark(s(length(L))) active(take(0(),IL)) = IL + 1 >= 1 = mark(nil()) active(take(s(M),cons(N,IL))) = IL + M + N + 1 >= IL + M + N + 1 = mark(cons(N,take(M,IL))) active(cons(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = cons(active(X1),X2) active(and(X1,X2)) = X1 + X2 + 2 >= X1 + X2 + 2 = and(active(X1),X2) active(length(X)) = X + 1 >= X + 1 = length(active(X)) active(s(X)) = X + 1 >= X + 1 = s(active(X)) active(take(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = take(active(X1),X2) active(take(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = take(X1,active(X2)) cons(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(cons(X1,X2)) and(mark(X1),X2) = X1 + X2 + 2 >= X1 + X2 + 2 = mark(and(X1,X2)) length(mark(X)) = X + 1 >= X + 1 = mark(length(X)) s(mark(X)) = X + 1 >= X + 1 = mark(s(X)) take(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(take(X1,X2)) take(X1,mark(X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(take(X1,X2)) proper(zeros()) = 1 >= 1 = ok(zeros()) proper(cons(X1,X2)) = X1 + X2 >= X1 + X2 = cons(proper(X1),proper(X2)) proper(0()) = 0 >= 0 = ok(0()) proper(and(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = and(proper(X1),proper(X2)) proper(tt()) = 0 >= 0 = ok(tt()) proper(length(X)) = X >= X = length(proper(X)) proper(nil()) = 0 >= 0 = ok(nil()) proper(s(X)) = X >= X = s(proper(X)) proper(take(X1,X2)) = X1 + X2 >= X1 + X2 = take(proper(X1),proper(X2)) cons(ok(X1),ok(X2)) = X1 + X2 >= X1 + X2 = ok(cons(X1,X2)) and(ok(X1),ok(X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = ok(and(X1,X2)) length(ok(X)) = X >= X = ok(length(X)) s(ok(X)) = X >= X = ok(s(X)) take(ok(X1),ok(X2)) = X1 + X2 >= X1 + X2 = ok(take(X1,X2)) top(mark(X)) = 1 >= 1 = top(proper(X)) top(ok(X)) = 1 >= 1 = top(active(X)) problem: DPs: cons#(ok(X1),ok(X2)) -> cons#(X1,X2) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(and(tt(),X)) -> mark(X) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(s(length(L))) active(take(0(),IL)) -> mark(nil()) active(take(s(M),cons(N,IL))) -> mark(cons(N,take(M,IL))) active(cons(X1,X2)) -> cons(active(X1),X2) active(and(X1,X2)) -> and(active(X1),X2) active(length(X)) -> length(active(X)) active(s(X)) -> s(active(X)) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) cons(mark(X1),X2) -> mark(cons(X1,X2)) and(mark(X1),X2) -> mark(and(X1,X2)) length(mark(X)) -> mark(length(X)) s(mark(X)) -> mark(s(X)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(length(X)) -> length(proper(X)) proper(nil()) -> ok(nil()) proper(s(X)) -> s(proper(X)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) length(ok(X)) -> ok(length(X)) s(ok(X)) -> ok(s(X)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dimension: 1 interpretation: [cons#](x0, x1) = x1 + 1, [top](x0) = 0, [ok](x0) = x0 + 1, [proper](x0) = x0 + 1, [take](x0, x1) = x0, [s](x0) = x0 + 1, [length](x0) = x0, [nil] = 0, [and](x0, x1) = x0 + 1, [tt] = 0, [mark](x0) = 0, [cons](x0, x1) = x0, [0] = 0, [active](x0) = x0, [zeros] = 0 orientation: cons#(ok(X1),ok(X2)) = X2 + 2 >= X2 + 1 = cons#(X1,X2) active(zeros()) = 0 >= 0 = mark(cons(0(),zeros())) active(and(tt(),X)) = 1 >= 0 = mark(X) active(length(nil())) = 0 >= 0 = mark(0()) active(length(cons(N,L))) = N >= 0 = mark(s(length(L))) active(take(0(),IL)) = 0 >= 0 = mark(nil()) active(take(s(M),cons(N,IL))) = M + 1 >= 0 = mark(cons(N,take(M,IL))) active(cons(X1,X2)) = X1 >= X1 = cons(active(X1),X2) active(and(X1,X2)) = X1 + 1 >= X1 + 1 = and(active(X1),X2) active(length(X)) = X >= X = length(active(X)) active(s(X)) = X + 1 >= X + 1 = s(active(X)) active(take(X1,X2)) = X1 >= X1 = take(active(X1),X2) active(take(X1,X2)) = X1 >= X1 = take(X1,active(X2)) cons(mark(X1),X2) = 0 >= 0 = mark(cons(X1,X2)) and(mark(X1),X2) = 1 >= 0 = mark(and(X1,X2)) length(mark(X)) = 0 >= 0 = mark(length(X)) s(mark(X)) = 1 >= 0 = mark(s(X)) take(mark(X1),X2) = 0 >= 0 = mark(take(X1,X2)) take(X1,mark(X2)) = X1 >= 0 = mark(take(X1,X2)) proper(zeros()) = 1 >= 1 = ok(zeros()) proper(cons(X1,X2)) = X1 + 1 >= X1 + 1 = cons(proper(X1),proper(X2)) proper(0()) = 1 >= 1 = ok(0()) proper(and(X1,X2)) = X1 + 2 >= X1 + 2 = and(proper(X1),proper(X2)) proper(tt()) = 1 >= 1 = ok(tt()) proper(length(X)) = X + 1 >= X + 1 = length(proper(X)) proper(nil()) = 1 >= 1 = ok(nil()) proper(s(X)) = X + 2 >= X + 2 = s(proper(X)) proper(take(X1,X2)) = X1 + 1 >= X1 + 1 = take(proper(X1),proper(X2)) cons(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(cons(X1,X2)) and(ok(X1),ok(X2)) = X1 + 2 >= X1 + 2 = ok(and(X1,X2)) length(ok(X)) = X + 1 >= X + 1 = ok(length(X)) s(ok(X)) = X + 2 >= X + 2 = ok(s(X)) take(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(take(X1,X2)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) problem: DPs: TRS: active(zeros()) -> mark(cons(0(),zeros())) active(and(tt(),X)) -> mark(X) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(s(length(L))) active(take(0(),IL)) -> mark(nil()) active(take(s(M),cons(N,IL))) -> mark(cons(N,take(M,IL))) active(cons(X1,X2)) -> cons(active(X1),X2) active(and(X1,X2)) -> and(active(X1),X2) active(length(X)) -> length(active(X)) active(s(X)) -> s(active(X)) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) cons(mark(X1),X2) -> mark(cons(X1,X2)) and(mark(X1),X2) -> mark(and(X1,X2)) length(mark(X)) -> mark(length(X)) s(mark(X)) -> mark(s(X)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(length(X)) -> length(proper(X)) proper(nil()) -> ok(nil()) proper(s(X)) -> s(proper(X)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) length(ok(X)) -> ok(length(X)) s(ok(X)) -> ok(s(X)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed DPs: and#(ok(X1),ok(X2)) -> and#(X1,X2) and#(mark(X1),X2) -> and#(X1,X2) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(and(tt(),X)) -> mark(X) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(s(length(L))) active(take(0(),IL)) -> mark(nil()) active(take(s(M),cons(N,IL))) -> mark(cons(N,take(M,IL))) active(cons(X1,X2)) -> cons(active(X1),X2) active(and(X1,X2)) -> and(active(X1),X2) active(length(X)) -> length(active(X)) active(s(X)) -> s(active(X)) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) cons(mark(X1),X2) -> mark(cons(X1,X2)) and(mark(X1),X2) -> mark(and(X1,X2)) length(mark(X)) -> mark(length(X)) s(mark(X)) -> mark(s(X)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(length(X)) -> length(proper(X)) proper(nil()) -> ok(nil()) proper(s(X)) -> s(proper(X)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) length(ok(X)) -> ok(length(X)) s(ok(X)) -> ok(s(X)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dimension: 1 interpretation: [and#](x0, x1) = x0 + 1, [top](x0) = 1, [ok](x0) = x0, [proper](x0) = x0, [take](x0, x1) = x0 + x1, [s](x0) = x0, [length](x0) = x0, [nil] = 0, [and](x0, x1) = x0 + x1 + 1, [tt] = 0, [mark](x0) = x0 + 1, [cons](x0, x1) = x0 + x1, [0] = 0, [active](x0) = x0 + 1, [zeros] = 1 orientation: and#(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = and#(X1,X2) and#(mark(X1),X2) = X1 + 2 >= X1 + 1 = and#(X1,X2) active(zeros()) = 2 >= 2 = mark(cons(0(),zeros())) active(and(tt(),X)) = X + 2 >= X + 1 = mark(X) active(length(nil())) = 1 >= 1 = mark(0()) active(length(cons(N,L))) = L + N + 1 >= L + 1 = mark(s(length(L))) active(take(0(),IL)) = IL + 1 >= 1 = mark(nil()) active(take(s(M),cons(N,IL))) = IL + M + N + 1 >= IL + M + N + 1 = mark(cons(N,take(M,IL))) active(cons(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = cons(active(X1),X2) active(and(X1,X2)) = X1 + X2 + 2 >= X1 + X2 + 2 = and(active(X1),X2) active(length(X)) = X + 1 >= X + 1 = length(active(X)) active(s(X)) = X + 1 >= X + 1 = s(active(X)) active(take(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = take(active(X1),X2) active(take(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = take(X1,active(X2)) cons(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(cons(X1,X2)) and(mark(X1),X2) = X1 + X2 + 2 >= X1 + X2 + 2 = mark(and(X1,X2)) length(mark(X)) = X + 1 >= X + 1 = mark(length(X)) s(mark(X)) = X + 1 >= X + 1 = mark(s(X)) take(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(take(X1,X2)) take(X1,mark(X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(take(X1,X2)) proper(zeros()) = 1 >= 1 = ok(zeros()) proper(cons(X1,X2)) = X1 + X2 >= X1 + X2 = cons(proper(X1),proper(X2)) proper(0()) = 0 >= 0 = ok(0()) proper(and(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = and(proper(X1),proper(X2)) proper(tt()) = 0 >= 0 = ok(tt()) proper(length(X)) = X >= X = length(proper(X)) proper(nil()) = 0 >= 0 = ok(nil()) proper(s(X)) = X >= X = s(proper(X)) proper(take(X1,X2)) = X1 + X2 >= X1 + X2 = take(proper(X1),proper(X2)) cons(ok(X1),ok(X2)) = X1 + X2 >= X1 + X2 = ok(cons(X1,X2)) and(ok(X1),ok(X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = ok(and(X1,X2)) length(ok(X)) = X >= X = ok(length(X)) s(ok(X)) = X >= X = ok(s(X)) take(ok(X1),ok(X2)) = X1 + X2 >= X1 + X2 = ok(take(X1,X2)) top(mark(X)) = 1 >= 1 = top(proper(X)) top(ok(X)) = 1 >= 1 = top(active(X)) problem: DPs: and#(ok(X1),ok(X2)) -> and#(X1,X2) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(and(tt(),X)) -> mark(X) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(s(length(L))) active(take(0(),IL)) -> mark(nil()) active(take(s(M),cons(N,IL))) -> mark(cons(N,take(M,IL))) active(cons(X1,X2)) -> cons(active(X1),X2) active(and(X1,X2)) -> and(active(X1),X2) active(length(X)) -> length(active(X)) active(s(X)) -> s(active(X)) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) cons(mark(X1),X2) -> mark(cons(X1,X2)) and(mark(X1),X2) -> mark(and(X1,X2)) length(mark(X)) -> mark(length(X)) s(mark(X)) -> mark(s(X)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(length(X)) -> length(proper(X)) proper(nil()) -> ok(nil()) proper(s(X)) -> s(proper(X)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) length(ok(X)) -> ok(length(X)) s(ok(X)) -> ok(s(X)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dimension: 1 interpretation: [and#](x0, x1) = x1 + 1, [top](x0) = 0, [ok](x0) = x0 + 1, [proper](x0) = x0 + 1, [take](x0, x1) = x0, [s](x0) = x0 + 1, [length](x0) = x0, [nil] = 0, [and](x0, x1) = x0 + 1, [tt] = 0, [mark](x0) = 0, [cons](x0, x1) = x0, [0] = 0, [active](x0) = x0, [zeros] = 0 orientation: and#(ok(X1),ok(X2)) = X2 + 2 >= X2 + 1 = and#(X1,X2) active(zeros()) = 0 >= 0 = mark(cons(0(),zeros())) active(and(tt(),X)) = 1 >= 0 = mark(X) active(length(nil())) = 0 >= 0 = mark(0()) active(length(cons(N,L))) = N >= 0 = mark(s(length(L))) active(take(0(),IL)) = 0 >= 0 = mark(nil()) active(take(s(M),cons(N,IL))) = M + 1 >= 0 = mark(cons(N,take(M,IL))) active(cons(X1,X2)) = X1 >= X1 = cons(active(X1),X2) active(and(X1,X2)) = X1 + 1 >= X1 + 1 = and(active(X1),X2) active(length(X)) = X >= X = length(active(X)) active(s(X)) = X + 1 >= X + 1 = s(active(X)) active(take(X1,X2)) = X1 >= X1 = take(active(X1),X2) active(take(X1,X2)) = X1 >= X1 = take(X1,active(X2)) cons(mark(X1),X2) = 0 >= 0 = mark(cons(X1,X2)) and(mark(X1),X2) = 1 >= 0 = mark(and(X1,X2)) length(mark(X)) = 0 >= 0 = mark(length(X)) s(mark(X)) = 1 >= 0 = mark(s(X)) take(mark(X1),X2) = 0 >= 0 = mark(take(X1,X2)) take(X1,mark(X2)) = X1 >= 0 = mark(take(X1,X2)) proper(zeros()) = 1 >= 1 = ok(zeros()) proper(cons(X1,X2)) = X1 + 1 >= X1 + 1 = cons(proper(X1),proper(X2)) proper(0()) = 1 >= 1 = ok(0()) proper(and(X1,X2)) = X1 + 2 >= X1 + 2 = and(proper(X1),proper(X2)) proper(tt()) = 1 >= 1 = ok(tt()) proper(length(X)) = X + 1 >= X + 1 = length(proper(X)) proper(nil()) = 1 >= 1 = ok(nil()) proper(s(X)) = X + 2 >= X + 2 = s(proper(X)) proper(take(X1,X2)) = X1 + 1 >= X1 + 1 = take(proper(X1),proper(X2)) cons(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(cons(X1,X2)) and(ok(X1),ok(X2)) = X1 + 2 >= X1 + 2 = ok(and(X1,X2)) length(ok(X)) = X + 1 >= X + 1 = ok(length(X)) s(ok(X)) = X + 2 >= X + 2 = ok(s(X)) take(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(take(X1,X2)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) problem: DPs: TRS: active(zeros()) -> mark(cons(0(),zeros())) active(and(tt(),X)) -> mark(X) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(s(length(L))) active(take(0(),IL)) -> mark(nil()) active(take(s(M),cons(N,IL))) -> mark(cons(N,take(M,IL))) active(cons(X1,X2)) -> cons(active(X1),X2) active(and(X1,X2)) -> and(active(X1),X2) active(length(X)) -> length(active(X)) active(s(X)) -> s(active(X)) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) cons(mark(X1),X2) -> mark(cons(X1,X2)) and(mark(X1),X2) -> mark(and(X1,X2)) length(mark(X)) -> mark(length(X)) s(mark(X)) -> mark(s(X)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(length(X)) -> length(proper(X)) proper(nil()) -> ok(nil()) proper(s(X)) -> s(proper(X)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) length(ok(X)) -> ok(length(X)) s(ok(X)) -> ok(s(X)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed DPs: length#(ok(X)) -> length#(X) length#(mark(X)) -> length#(X) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(and(tt(),X)) -> mark(X) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(s(length(L))) active(take(0(),IL)) -> mark(nil()) active(take(s(M),cons(N,IL))) -> mark(cons(N,take(M,IL))) active(cons(X1,X2)) -> cons(active(X1),X2) active(and(X1,X2)) -> and(active(X1),X2) active(length(X)) -> length(active(X)) active(s(X)) -> s(active(X)) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) cons(mark(X1),X2) -> mark(cons(X1,X2)) and(mark(X1),X2) -> mark(and(X1,X2)) length(mark(X)) -> mark(length(X)) s(mark(X)) -> mark(s(X)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(length(X)) -> length(proper(X)) proper(nil()) -> ok(nil()) proper(s(X)) -> s(proper(X)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) length(ok(X)) -> ok(length(X)) s(ok(X)) -> ok(s(X)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dimension: 1 interpretation: [length#](x0) = x0 + 1, [top](x0) = 0, [ok](x0) = x0 + 1, [proper](x0) = 1, [take](x0, x1) = x0, [s](x0) = x0, [length](x0) = x0, [nil] = 0, [and](x0, x1) = x1, [tt] = 0, [mark](x0) = x0, [cons](x0, x1) = x1, [0] = 0, [active](x0) = x0, [zeros] = 0 orientation: length#(ok(X)) = X + 2 >= X + 1 = length#(X) length#(mark(X)) = X + 1 >= X + 1 = length#(X) active(zeros()) = 0 >= 0 = mark(cons(0(),zeros())) active(and(tt(),X)) = X >= X = mark(X) active(length(nil())) = 0 >= 0 = mark(0()) active(length(cons(N,L))) = L >= L = mark(s(length(L))) active(take(0(),IL)) = 0 >= 0 = mark(nil()) active(take(s(M),cons(N,IL))) = M >= M = mark(cons(N,take(M,IL))) active(cons(X1,X2)) = X2 >= X2 = cons(active(X1),X2) active(and(X1,X2)) = X2 >= X2 = and(active(X1),X2) active(length(X)) = X >= X = length(active(X)) active(s(X)) = X >= X = s(active(X)) active(take(X1,X2)) = X1 >= X1 = take(active(X1),X2) active(take(X1,X2)) = X1 >= X1 = take(X1,active(X2)) cons(mark(X1),X2) = X2 >= X2 = mark(cons(X1,X2)) and(mark(X1),X2) = X2 >= X2 = mark(and(X1,X2)) length(mark(X)) = X >= X = mark(length(X)) s(mark(X)) = X >= X = mark(s(X)) take(mark(X1),X2) = X1 >= X1 = mark(take(X1,X2)) take(X1,mark(X2)) = X1 >= X1 = mark(take(X1,X2)) proper(zeros()) = 1 >= 1 = ok(zeros()) proper(cons(X1,X2)) = 1 >= 1 = cons(proper(X1),proper(X2)) proper(0()) = 1 >= 1 = ok(0()) proper(and(X1,X2)) = 1 >= 1 = and(proper(X1),proper(X2)) proper(tt()) = 1 >= 1 = ok(tt()) proper(length(X)) = 1 >= 1 = length(proper(X)) proper(nil()) = 1 >= 1 = ok(nil()) proper(s(X)) = 1 >= 1 = s(proper(X)) proper(take(X1,X2)) = 1 >= 1 = take(proper(X1),proper(X2)) cons(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(cons(X1,X2)) and(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(and(X1,X2)) length(ok(X)) = X + 1 >= X + 1 = ok(length(X)) s(ok(X)) = X + 1 >= X + 1 = ok(s(X)) take(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(take(X1,X2)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) problem: DPs: length#(mark(X)) -> length#(X) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(and(tt(),X)) -> mark(X) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(s(length(L))) active(take(0(),IL)) -> mark(nil()) active(take(s(M),cons(N,IL))) -> mark(cons(N,take(M,IL))) active(cons(X1,X2)) -> cons(active(X1),X2) active(and(X1,X2)) -> and(active(X1),X2) active(length(X)) -> length(active(X)) active(s(X)) -> s(active(X)) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) cons(mark(X1),X2) -> mark(cons(X1,X2)) and(mark(X1),X2) -> mark(and(X1,X2)) length(mark(X)) -> mark(length(X)) s(mark(X)) -> mark(s(X)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(length(X)) -> length(proper(X)) proper(nil()) -> ok(nil()) proper(s(X)) -> s(proper(X)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) length(ok(X)) -> ok(length(X)) s(ok(X)) -> ok(s(X)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dimension: 1 interpretation: [length#](x0) = x0 + 1, [top](x0) = 1, [ok](x0) = 0, [proper](x0) = 0, [take](x0, x1) = x0 + x1, [s](x0) = x0, [length](x0) = x0, [nil] = 0, [and](x0, x1) = x0 + x1, [tt] = 1, [mark](x0) = x0 + 1, [cons](x0, x1) = x0 + x1, [0] = 0, [active](x0) = x0 + 1, [zeros] = 1 orientation: length#(mark(X)) = X + 2 >= X + 1 = length#(X) active(zeros()) = 2 >= 2 = mark(cons(0(),zeros())) active(and(tt(),X)) = X + 2 >= X + 1 = mark(X) active(length(nil())) = 1 >= 1 = mark(0()) active(length(cons(N,L))) = L + N + 1 >= L + 1 = mark(s(length(L))) active(take(0(),IL)) = IL + 1 >= 1 = mark(nil()) active(take(s(M),cons(N,IL))) = IL + M + N + 1 >= IL + M + N + 1 = mark(cons(N,take(M,IL))) active(cons(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = cons(active(X1),X2) active(and(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = and(active(X1),X2) active(length(X)) = X + 1 >= X + 1 = length(active(X)) active(s(X)) = X + 1 >= X + 1 = s(active(X)) active(take(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = take(active(X1),X2) active(take(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = take(X1,active(X2)) cons(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(cons(X1,X2)) and(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(and(X1,X2)) length(mark(X)) = X + 1 >= X + 1 = mark(length(X)) s(mark(X)) = X + 1 >= X + 1 = mark(s(X)) take(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(take(X1,X2)) take(X1,mark(X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(take(X1,X2)) proper(zeros()) = 0 >= 0 = ok(zeros()) proper(cons(X1,X2)) = 0 >= 0 = cons(proper(X1),proper(X2)) proper(0()) = 0 >= 0 = ok(0()) proper(and(X1,X2)) = 0 >= 0 = and(proper(X1),proper(X2)) proper(tt()) = 0 >= 0 = ok(tt()) proper(length(X)) = 0 >= 0 = length(proper(X)) proper(nil()) = 0 >= 0 = ok(nil()) proper(s(X)) = 0 >= 0 = s(proper(X)) proper(take(X1,X2)) = 0 >= 0 = take(proper(X1),proper(X2)) cons(ok(X1),ok(X2)) = 0 >= 0 = ok(cons(X1,X2)) and(ok(X1),ok(X2)) = 0 >= 0 = ok(and(X1,X2)) length(ok(X)) = 0 >= 0 = ok(length(X)) s(ok(X)) = 0 >= 0 = ok(s(X)) take(ok(X1),ok(X2)) = 0 >= 0 = ok(take(X1,X2)) top(mark(X)) = 1 >= 1 = top(proper(X)) top(ok(X)) = 1 >= 1 = top(active(X)) problem: DPs: TRS: active(zeros()) -> mark(cons(0(),zeros())) active(and(tt(),X)) -> mark(X) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(s(length(L))) active(take(0(),IL)) -> mark(nil()) active(take(s(M),cons(N,IL))) -> mark(cons(N,take(M,IL))) active(cons(X1,X2)) -> cons(active(X1),X2) active(and(X1,X2)) -> and(active(X1),X2) active(length(X)) -> length(active(X)) active(s(X)) -> s(active(X)) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) cons(mark(X1),X2) -> mark(cons(X1,X2)) and(mark(X1),X2) -> mark(and(X1,X2)) length(mark(X)) -> mark(length(X)) s(mark(X)) -> mark(s(X)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(length(X)) -> length(proper(X)) proper(nil()) -> ok(nil()) proper(s(X)) -> s(proper(X)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) length(ok(X)) -> ok(length(X)) s(ok(X)) -> ok(s(X)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed DPs: s#(ok(X)) -> s#(X) s#(mark(X)) -> s#(X) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(and(tt(),X)) -> mark(X) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(s(length(L))) active(take(0(),IL)) -> mark(nil()) active(take(s(M),cons(N,IL))) -> mark(cons(N,take(M,IL))) active(cons(X1,X2)) -> cons(active(X1),X2) active(and(X1,X2)) -> and(active(X1),X2) active(length(X)) -> length(active(X)) active(s(X)) -> s(active(X)) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) cons(mark(X1),X2) -> mark(cons(X1,X2)) and(mark(X1),X2) -> mark(and(X1,X2)) length(mark(X)) -> mark(length(X)) s(mark(X)) -> mark(s(X)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(length(X)) -> length(proper(X)) proper(nil()) -> ok(nil()) proper(s(X)) -> s(proper(X)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) length(ok(X)) -> ok(length(X)) s(ok(X)) -> ok(s(X)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dimension: 1 interpretation: [s#](x0) = x0 + 1, [top](x0) = 0, [ok](x0) = x0 + 1, [proper](x0) = 1, [take](x0, x1) = x0, [s](x0) = x0, [length](x0) = x0, [nil] = 0, [and](x0, x1) = x1, [tt] = 0, [mark](x0) = x0, [cons](x0, x1) = x1, [0] = 0, [active](x0) = x0, [zeros] = 0 orientation: s#(ok(X)) = X + 2 >= X + 1 = s#(X) s#(mark(X)) = X + 1 >= X + 1 = s#(X) active(zeros()) = 0 >= 0 = mark(cons(0(),zeros())) active(and(tt(),X)) = X >= X = mark(X) active(length(nil())) = 0 >= 0 = mark(0()) active(length(cons(N,L))) = L >= L = mark(s(length(L))) active(take(0(),IL)) = 0 >= 0 = mark(nil()) active(take(s(M),cons(N,IL))) = M >= M = mark(cons(N,take(M,IL))) active(cons(X1,X2)) = X2 >= X2 = cons(active(X1),X2) active(and(X1,X2)) = X2 >= X2 = and(active(X1),X2) active(length(X)) = X >= X = length(active(X)) active(s(X)) = X >= X = s(active(X)) active(take(X1,X2)) = X1 >= X1 = take(active(X1),X2) active(take(X1,X2)) = X1 >= X1 = take(X1,active(X2)) cons(mark(X1),X2) = X2 >= X2 = mark(cons(X1,X2)) and(mark(X1),X2) = X2 >= X2 = mark(and(X1,X2)) length(mark(X)) = X >= X = mark(length(X)) s(mark(X)) = X >= X = mark(s(X)) take(mark(X1),X2) = X1 >= X1 = mark(take(X1,X2)) take(X1,mark(X2)) = X1 >= X1 = mark(take(X1,X2)) proper(zeros()) = 1 >= 1 = ok(zeros()) proper(cons(X1,X2)) = 1 >= 1 = cons(proper(X1),proper(X2)) proper(0()) = 1 >= 1 = ok(0()) proper(and(X1,X2)) = 1 >= 1 = and(proper(X1),proper(X2)) proper(tt()) = 1 >= 1 = ok(tt()) proper(length(X)) = 1 >= 1 = length(proper(X)) proper(nil()) = 1 >= 1 = ok(nil()) proper(s(X)) = 1 >= 1 = s(proper(X)) proper(take(X1,X2)) = 1 >= 1 = take(proper(X1),proper(X2)) cons(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(cons(X1,X2)) and(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(and(X1,X2)) length(ok(X)) = X + 1 >= X + 1 = ok(length(X)) s(ok(X)) = X + 1 >= X + 1 = ok(s(X)) take(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(take(X1,X2)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) problem: DPs: s#(mark(X)) -> s#(X) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(and(tt(),X)) -> mark(X) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(s(length(L))) active(take(0(),IL)) -> mark(nil()) active(take(s(M),cons(N,IL))) -> mark(cons(N,take(M,IL))) active(cons(X1,X2)) -> cons(active(X1),X2) active(and(X1,X2)) -> and(active(X1),X2) active(length(X)) -> length(active(X)) active(s(X)) -> s(active(X)) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) cons(mark(X1),X2) -> mark(cons(X1,X2)) and(mark(X1),X2) -> mark(and(X1,X2)) length(mark(X)) -> mark(length(X)) s(mark(X)) -> mark(s(X)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(length(X)) -> length(proper(X)) proper(nil()) -> ok(nil()) proper(s(X)) -> s(proper(X)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) length(ok(X)) -> ok(length(X)) s(ok(X)) -> ok(s(X)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dimension: 1 interpretation: [s#](x0) = x0 + 1, [top](x0) = 1, [ok](x0) = 0, [proper](x0) = 0, [take](x0, x1) = x0 + x1, [s](x0) = x0, [length](x0) = x0, [nil] = 0, [and](x0, x1) = x0 + x1, [tt] = 1, [mark](x0) = x0 + 1, [cons](x0, x1) = x0 + x1, [0] = 0, [active](x0) = x0 + 1, [zeros] = 1 orientation: s#(mark(X)) = X + 2 >= X + 1 = s#(X) active(zeros()) = 2 >= 2 = mark(cons(0(),zeros())) active(and(tt(),X)) = X + 2 >= X + 1 = mark(X) active(length(nil())) = 1 >= 1 = mark(0()) active(length(cons(N,L))) = L + N + 1 >= L + 1 = mark(s(length(L))) active(take(0(),IL)) = IL + 1 >= 1 = mark(nil()) active(take(s(M),cons(N,IL))) = IL + M + N + 1 >= IL + M + N + 1 = mark(cons(N,take(M,IL))) active(cons(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = cons(active(X1),X2) active(and(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = and(active(X1),X2) active(length(X)) = X + 1 >= X + 1 = length(active(X)) active(s(X)) = X + 1 >= X + 1 = s(active(X)) active(take(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = take(active(X1),X2) active(take(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = take(X1,active(X2)) cons(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(cons(X1,X2)) and(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(and(X1,X2)) length(mark(X)) = X + 1 >= X + 1 = mark(length(X)) s(mark(X)) = X + 1 >= X + 1 = mark(s(X)) take(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(take(X1,X2)) take(X1,mark(X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(take(X1,X2)) proper(zeros()) = 0 >= 0 = ok(zeros()) proper(cons(X1,X2)) = 0 >= 0 = cons(proper(X1),proper(X2)) proper(0()) = 0 >= 0 = ok(0()) proper(and(X1,X2)) = 0 >= 0 = and(proper(X1),proper(X2)) proper(tt()) = 0 >= 0 = ok(tt()) proper(length(X)) = 0 >= 0 = length(proper(X)) proper(nil()) = 0 >= 0 = ok(nil()) proper(s(X)) = 0 >= 0 = s(proper(X)) proper(take(X1,X2)) = 0 >= 0 = take(proper(X1),proper(X2)) cons(ok(X1),ok(X2)) = 0 >= 0 = ok(cons(X1,X2)) and(ok(X1),ok(X2)) = 0 >= 0 = ok(and(X1,X2)) length(ok(X)) = 0 >= 0 = ok(length(X)) s(ok(X)) = 0 >= 0 = ok(s(X)) take(ok(X1),ok(X2)) = 0 >= 0 = ok(take(X1,X2)) top(mark(X)) = 1 >= 1 = top(proper(X)) top(ok(X)) = 1 >= 1 = top(active(X)) problem: DPs: TRS: active(zeros()) -> mark(cons(0(),zeros())) active(and(tt(),X)) -> mark(X) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(s(length(L))) active(take(0(),IL)) -> mark(nil()) active(take(s(M),cons(N,IL))) -> mark(cons(N,take(M,IL))) active(cons(X1,X2)) -> cons(active(X1),X2) active(and(X1,X2)) -> and(active(X1),X2) active(length(X)) -> length(active(X)) active(s(X)) -> s(active(X)) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) cons(mark(X1),X2) -> mark(cons(X1,X2)) and(mark(X1),X2) -> mark(and(X1,X2)) length(mark(X)) -> mark(length(X)) s(mark(X)) -> mark(s(X)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(length(X)) -> length(proper(X)) proper(nil()) -> ok(nil()) proper(s(X)) -> s(proper(X)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) length(ok(X)) -> ok(length(X)) s(ok(X)) -> ok(s(X)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed DPs: take#(ok(X1),ok(X2)) -> take#(X1,X2) take#(X1,mark(X2)) -> take#(X1,X2) take#(mark(X1),X2) -> take#(X1,X2) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(and(tt(),X)) -> mark(X) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(s(length(L))) active(take(0(),IL)) -> mark(nil()) active(take(s(M),cons(N,IL))) -> mark(cons(N,take(M,IL))) active(cons(X1,X2)) -> cons(active(X1),X2) active(and(X1,X2)) -> and(active(X1),X2) active(length(X)) -> length(active(X)) active(s(X)) -> s(active(X)) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) cons(mark(X1),X2) -> mark(cons(X1,X2)) and(mark(X1),X2) -> mark(and(X1,X2)) length(mark(X)) -> mark(length(X)) s(mark(X)) -> mark(s(X)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(length(X)) -> length(proper(X)) proper(nil()) -> ok(nil()) proper(s(X)) -> s(proper(X)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) length(ok(X)) -> ok(length(X)) s(ok(X)) -> ok(s(X)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dimension: 1 interpretation: [take#](x0, x1) = x0, [top](x0) = 0, [ok](x0) = x0, [proper](x0) = 0, [take](x0, x1) = x0 + x1, [s](x0) = x0, [length](x0) = x0, [nil] = 0, [and](x0, x1) = x0 + x1, [tt] = 0, [mark](x0) = x0 + 1, [cons](x0, x1) = x0 + x1, [0] = 0, [active](x0) = x0 + 1, [zeros] = 0 orientation: take#(ok(X1),ok(X2)) = X1 >= X1 = take#(X1,X2) take#(X1,mark(X2)) = X1 >= X1 = take#(X1,X2) take#(mark(X1),X2) = X1 + 1 >= X1 = take#(X1,X2) active(zeros()) = 1 >= 1 = mark(cons(0(),zeros())) active(and(tt(),X)) = X + 1 >= X + 1 = mark(X) active(length(nil())) = 1 >= 1 = mark(0()) active(length(cons(N,L))) = L + N + 1 >= L + 1 = mark(s(length(L))) active(take(0(),IL)) = IL + 1 >= 1 = mark(nil()) active(take(s(M),cons(N,IL))) = IL + M + N + 1 >= IL + M + N + 1 = mark(cons(N,take(M,IL))) active(cons(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = cons(active(X1),X2) active(and(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = and(active(X1),X2) active(length(X)) = X + 1 >= X + 1 = length(active(X)) active(s(X)) = X + 1 >= X + 1 = s(active(X)) active(take(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = take(active(X1),X2) active(take(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = take(X1,active(X2)) cons(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(cons(X1,X2)) and(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(and(X1,X2)) length(mark(X)) = X + 1 >= X + 1 = mark(length(X)) s(mark(X)) = X + 1 >= X + 1 = mark(s(X)) take(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(take(X1,X2)) take(X1,mark(X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(take(X1,X2)) proper(zeros()) = 0 >= 0 = ok(zeros()) proper(cons(X1,X2)) = 0 >= 0 = cons(proper(X1),proper(X2)) proper(0()) = 0 >= 0 = ok(0()) proper(and(X1,X2)) = 0 >= 0 = and(proper(X1),proper(X2)) proper(tt()) = 0 >= 0 = ok(tt()) proper(length(X)) = 0 >= 0 = length(proper(X)) proper(nil()) = 0 >= 0 = ok(nil()) proper(s(X)) = 0 >= 0 = s(proper(X)) proper(take(X1,X2)) = 0 >= 0 = take(proper(X1),proper(X2)) cons(ok(X1),ok(X2)) = X1 + X2 >= X1 + X2 = ok(cons(X1,X2)) and(ok(X1),ok(X2)) = X1 + X2 >= X1 + X2 = ok(and(X1,X2)) length(ok(X)) = X >= X = ok(length(X)) s(ok(X)) = X >= X = ok(s(X)) take(ok(X1),ok(X2)) = X1 + X2 >= X1 + X2 = ok(take(X1,X2)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) problem: DPs: take#(ok(X1),ok(X2)) -> take#(X1,X2) take#(X1,mark(X2)) -> take#(X1,X2) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(and(tt(),X)) -> mark(X) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(s(length(L))) active(take(0(),IL)) -> mark(nil()) active(take(s(M),cons(N,IL))) -> mark(cons(N,take(M,IL))) active(cons(X1,X2)) -> cons(active(X1),X2) active(and(X1,X2)) -> and(active(X1),X2) active(length(X)) -> length(active(X)) active(s(X)) -> s(active(X)) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) cons(mark(X1),X2) -> mark(cons(X1,X2)) and(mark(X1),X2) -> mark(and(X1,X2)) length(mark(X)) -> mark(length(X)) s(mark(X)) -> mark(s(X)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(length(X)) -> length(proper(X)) proper(nil()) -> ok(nil()) proper(s(X)) -> s(proper(X)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) length(ok(X)) -> ok(length(X)) s(ok(X)) -> ok(s(X)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dimension: 1 interpretation: [take#](x0, x1) = x0, [top](x0) = 0, [ok](x0) = x0 + 1, [proper](x0) = x0 + 1, [take](x0, x1) = x0, [s](x0) = x0 + 1, [length](x0) = x0, [nil] = 1, [and](x0, x1) = x1, [tt] = 0, [mark](x0) = 0, [cons](x0, x1) = x0, [0] = 0, [active](x0) = x0, [zeros] = 0 orientation: take#(ok(X1),ok(X2)) = X1 + 1 >= X1 = take#(X1,X2) take#(X1,mark(X2)) = X1 >= X1 = take#(X1,X2) active(zeros()) = 0 >= 0 = mark(cons(0(),zeros())) active(and(tt(),X)) = X >= 0 = mark(X) active(length(nil())) = 1 >= 0 = mark(0()) active(length(cons(N,L))) = N >= 0 = mark(s(length(L))) active(take(0(),IL)) = 0 >= 0 = mark(nil()) active(take(s(M),cons(N,IL))) = M + 1 >= 0 = mark(cons(N,take(M,IL))) active(cons(X1,X2)) = X1 >= X1 = cons(active(X1),X2) active(and(X1,X2)) = X2 >= X2 = and(active(X1),X2) active(length(X)) = X >= X = length(active(X)) active(s(X)) = X + 1 >= X + 1 = s(active(X)) active(take(X1,X2)) = X1 >= X1 = take(active(X1),X2) active(take(X1,X2)) = X1 >= X1 = take(X1,active(X2)) cons(mark(X1),X2) = 0 >= 0 = mark(cons(X1,X2)) and(mark(X1),X2) = X2 >= 0 = mark(and(X1,X2)) length(mark(X)) = 0 >= 0 = mark(length(X)) s(mark(X)) = 1 >= 0 = mark(s(X)) take(mark(X1),X2) = 0 >= 0 = mark(take(X1,X2)) take(X1,mark(X2)) = X1 >= 0 = mark(take(X1,X2)) proper(zeros()) = 1 >= 1 = ok(zeros()) proper(cons(X1,X2)) = X1 + 1 >= X1 + 1 = cons(proper(X1),proper(X2)) proper(0()) = 1 >= 1 = ok(0()) proper(and(X1,X2)) = X2 + 1 >= X2 + 1 = and(proper(X1),proper(X2)) proper(tt()) = 1 >= 1 = ok(tt()) proper(length(X)) = X + 1 >= X + 1 = length(proper(X)) proper(nil()) = 2 >= 2 = ok(nil()) proper(s(X)) = X + 2 >= X + 2 = s(proper(X)) proper(take(X1,X2)) = X1 + 1 >= X1 + 1 = take(proper(X1),proper(X2)) cons(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(cons(X1,X2)) and(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(and(X1,X2)) length(ok(X)) = X + 1 >= X + 1 = ok(length(X)) s(ok(X)) = X + 2 >= X + 2 = ok(s(X)) take(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(take(X1,X2)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) problem: DPs: take#(X1,mark(X2)) -> take#(X1,X2) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(and(tt(),X)) -> mark(X) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(s(length(L))) active(take(0(),IL)) -> mark(nil()) active(take(s(M),cons(N,IL))) -> mark(cons(N,take(M,IL))) active(cons(X1,X2)) -> cons(active(X1),X2) active(and(X1,X2)) -> and(active(X1),X2) active(length(X)) -> length(active(X)) active(s(X)) -> s(active(X)) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) cons(mark(X1),X2) -> mark(cons(X1,X2)) and(mark(X1),X2) -> mark(and(X1,X2)) length(mark(X)) -> mark(length(X)) s(mark(X)) -> mark(s(X)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(length(X)) -> length(proper(X)) proper(nil()) -> ok(nil()) proper(s(X)) -> s(proper(X)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) length(ok(X)) -> ok(length(X)) s(ok(X)) -> ok(s(X)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dimension: 1 interpretation: [take#](x0, x1) = x1 + 1, [top](x0) = 0, [ok](x0) = 0, [proper](x0) = 0, [take](x0, x1) = x0 + x1, [s](x0) = x0, [length](x0) = x0, [nil] = 0, [and](x0, x1) = x0 + x1, [tt] = 0, [mark](x0) = x0 + 1, [cons](x0, x1) = x0 + x1, [0] = 0, [active](x0) = x0 + 1, [zeros] = 0 orientation: take#(X1,mark(X2)) = X2 + 2 >= X2 + 1 = take#(X1,X2) active(zeros()) = 1 >= 1 = mark(cons(0(),zeros())) active(and(tt(),X)) = X + 1 >= X + 1 = mark(X) active(length(nil())) = 1 >= 1 = mark(0()) active(length(cons(N,L))) = L + N + 1 >= L + 1 = mark(s(length(L))) active(take(0(),IL)) = IL + 1 >= 1 = mark(nil()) active(take(s(M),cons(N,IL))) = IL + M + N + 1 >= IL + M + N + 1 = mark(cons(N,take(M,IL))) active(cons(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = cons(active(X1),X2) active(and(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = and(active(X1),X2) active(length(X)) = X + 1 >= X + 1 = length(active(X)) active(s(X)) = X + 1 >= X + 1 = s(active(X)) active(take(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = take(active(X1),X2) active(take(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = take(X1,active(X2)) cons(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(cons(X1,X2)) and(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(and(X1,X2)) length(mark(X)) = X + 1 >= X + 1 = mark(length(X)) s(mark(X)) = X + 1 >= X + 1 = mark(s(X)) take(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(take(X1,X2)) take(X1,mark(X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(take(X1,X2)) proper(zeros()) = 0 >= 0 = ok(zeros()) proper(cons(X1,X2)) = 0 >= 0 = cons(proper(X1),proper(X2)) proper(0()) = 0 >= 0 = ok(0()) proper(and(X1,X2)) = 0 >= 0 = and(proper(X1),proper(X2)) proper(tt()) = 0 >= 0 = ok(tt()) proper(length(X)) = 0 >= 0 = length(proper(X)) proper(nil()) = 0 >= 0 = ok(nil()) proper(s(X)) = 0 >= 0 = s(proper(X)) proper(take(X1,X2)) = 0 >= 0 = take(proper(X1),proper(X2)) cons(ok(X1),ok(X2)) = 0 >= 0 = ok(cons(X1,X2)) and(ok(X1),ok(X2)) = 0 >= 0 = ok(and(X1,X2)) length(ok(X)) = 0 >= 0 = ok(length(X)) s(ok(X)) = 0 >= 0 = ok(s(X)) take(ok(X1),ok(X2)) = 0 >= 0 = ok(take(X1,X2)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) problem: DPs: TRS: active(zeros()) -> mark(cons(0(),zeros())) active(and(tt(),X)) -> mark(X) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(s(length(L))) active(take(0(),IL)) -> mark(nil()) active(take(s(M),cons(N,IL))) -> mark(cons(N,take(M,IL))) active(cons(X1,X2)) -> cons(active(X1),X2) active(and(X1,X2)) -> and(active(X1),X2) active(length(X)) -> length(active(X)) active(s(X)) -> s(active(X)) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) cons(mark(X1),X2) -> mark(cons(X1,X2)) and(mark(X1),X2) -> mark(and(X1,X2)) length(mark(X)) -> mark(length(X)) s(mark(X)) -> mark(s(X)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(length(X)) -> length(proper(X)) proper(nil()) -> ok(nil()) proper(s(X)) -> s(proper(X)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) length(ok(X)) -> ok(length(X)) s(ok(X)) -> ok(s(X)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed