MAYBE Problem: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),L)) -> mark(U12(tt(),L)) active(U12(tt(),L)) -> mark(s(length(L))) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U11(tt(),L)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X1,X2)) -> U11(active(X1),X2) active(U12(X1,X2)) -> U12(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X1),X2) -> mark(U11(X1,X2)) U12(mark(X1),X2) -> mark(U12(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(U12(X1,X2)) -> U12(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(nil()) -> ok(nil()) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) U12(ok(X1),ok(X2)) -> ok(U12(X1,X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Proof: DP Processor: DPs: active#(zeros()) -> cons#(0(),zeros()) active#(U11(tt(),L)) -> U12#(tt(),L) active#(U12(tt(),L)) -> length#(L) active#(U12(tt(),L)) -> s#(length(L)) active#(length(cons(N,L))) -> U11#(tt(),L) active#(cons(X1,X2)) -> active#(X1) active#(cons(X1,X2)) -> cons#(active(X1),X2) active#(U11(X1,X2)) -> active#(X1) active#(U11(X1,X2)) -> U11#(active(X1),X2) active#(U12(X1,X2)) -> active#(X1) active#(U12(X1,X2)) -> U12#(active(X1),X2) active#(s(X)) -> active#(X) active#(s(X)) -> s#(active(X)) active#(length(X)) -> active#(X) active#(length(X)) -> length#(active(X)) cons#(mark(X1),X2) -> cons#(X1,X2) U11#(mark(X1),X2) -> U11#(X1,X2) U12#(mark(X1),X2) -> U12#(X1,X2) s#(mark(X)) -> s#(X) length#(mark(X)) -> length#(X) proper#(cons(X1,X2)) -> proper#(X2) proper#(cons(X1,X2)) -> proper#(X1) proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2)) proper#(U11(X1,X2)) -> proper#(X2) proper#(U11(X1,X2)) -> proper#(X1) proper#(U11(X1,X2)) -> U11#(proper(X1),proper(X2)) proper#(U12(X1,X2)) -> proper#(X2) proper#(U12(X1,X2)) -> proper#(X1) proper#(U12(X1,X2)) -> U12#(proper(X1),proper(X2)) proper#(s(X)) -> proper#(X) proper#(s(X)) -> s#(proper(X)) proper#(length(X)) -> proper#(X) proper#(length(X)) -> length#(proper(X)) cons#(ok(X1),ok(X2)) -> cons#(X1,X2) U11#(ok(X1),ok(X2)) -> U11#(X1,X2) U12#(ok(X1),ok(X2)) -> U12#(X1,X2) s#(ok(X)) -> s#(X) length#(ok(X)) -> length#(X) top#(mark(X)) -> proper#(X) top#(mark(X)) -> top#(proper(X)) top#(ok(X)) -> active#(X) top#(ok(X)) -> top#(active(X)) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),L)) -> mark(U12(tt(),L)) active(U12(tt(),L)) -> mark(s(length(L))) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U11(tt(),L)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X1,X2)) -> U11(active(X1),X2) active(U12(X1,X2)) -> U12(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X1),X2) -> mark(U11(X1,X2)) U12(mark(X1),X2) -> mark(U12(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(U12(X1,X2)) -> U12(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(nil()) -> ok(nil()) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) U12(ok(X1),ok(X2)) -> ok(U12(X1,X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) EDG Processor: DPs: active#(zeros()) -> cons#(0(),zeros()) active#(U11(tt(),L)) -> U12#(tt(),L) active#(U12(tt(),L)) -> length#(L) active#(U12(tt(),L)) -> s#(length(L)) active#(length(cons(N,L))) -> U11#(tt(),L) active#(cons(X1,X2)) -> active#(X1) active#(cons(X1,X2)) -> cons#(active(X1),X2) active#(U11(X1,X2)) -> active#(X1) active#(U11(X1,X2)) -> U11#(active(X1),X2) active#(U12(X1,X2)) -> active#(X1) active#(U12(X1,X2)) -> U12#(active(X1),X2) active#(s(X)) -> active#(X) active#(s(X)) -> s#(active(X)) active#(length(X)) -> active#(X) active#(length(X)) -> length#(active(X)) cons#(mark(X1),X2) -> cons#(X1,X2) U11#(mark(X1),X2) -> U11#(X1,X2) U12#(mark(X1),X2) -> U12#(X1,X2) s#(mark(X)) -> s#(X) length#(mark(X)) -> length#(X) proper#(cons(X1,X2)) -> proper#(X2) proper#(cons(X1,X2)) -> proper#(X1) proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2)) proper#(U11(X1,X2)) -> proper#(X2) proper#(U11(X1,X2)) -> proper#(X1) proper#(U11(X1,X2)) -> U11#(proper(X1),proper(X2)) proper#(U12(X1,X2)) -> proper#(X2) proper#(U12(X1,X2)) -> proper#(X1) proper#(U12(X1,X2)) -> U12#(proper(X1),proper(X2)) proper#(s(X)) -> proper#(X) proper#(s(X)) -> s#(proper(X)) proper#(length(X)) -> proper#(X) proper#(length(X)) -> length#(proper(X)) cons#(ok(X1),ok(X2)) -> cons#(X1,X2) U11#(ok(X1),ok(X2)) -> U11#(X1,X2) U12#(ok(X1),ok(X2)) -> U12#(X1,X2) s#(ok(X)) -> s#(X) length#(ok(X)) -> length#(X) top#(mark(X)) -> proper#(X) top#(mark(X)) -> top#(proper(X)) top#(ok(X)) -> active#(X) top#(ok(X)) -> top#(active(X)) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),L)) -> mark(U12(tt(),L)) active(U12(tt(),L)) -> mark(s(length(L))) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U11(tt(),L)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X1,X2)) -> U11(active(X1),X2) active(U12(X1,X2)) -> U12(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X1),X2) -> mark(U11(X1,X2)) U12(mark(X1),X2) -> mark(U12(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(U12(X1,X2)) -> U12(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(nil()) -> ok(nil()) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) U12(ok(X1),ok(X2)) -> ok(U12(X1,X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) 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#(U11(tt(),L)) -> U12#(tt(),L) top#(ok(X)) -> active#(X) -> active#(U12(tt(),L)) -> length#(L) top#(ok(X)) -> active#(X) -> active#(U12(tt(),L)) -> s#(length(L)) top#(ok(X)) -> active#(X) -> active#(length(cons(N,L))) -> U11#(tt(),L) 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#(U11(X1,X2)) -> active#(X1) top#(ok(X)) -> active#(X) -> active#(U11(X1,X2)) -> U11#(active(X1),X2) top#(ok(X)) -> active#(X) -> active#(U12(X1,X2)) -> active#(X1) top#(ok(X)) -> active#(X) -> active#(U12(X1,X2)) -> U12#(active(X1),X2) 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#(length(X)) -> active#(X) top#(ok(X)) -> active#(X) -> active#(length(X)) -> length#(active(X)) 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#(U11(X1,X2)) -> proper#(X2) top#(mark(X)) -> proper#(X) -> proper#(U11(X1,X2)) -> proper#(X1) top#(mark(X)) -> proper#(X) -> proper#(U11(X1,X2)) -> U11#(proper(X1),proper(X2)) top#(mark(X)) -> proper#(X) -> proper#(U12(X1,X2)) -> proper#(X2) top#(mark(X)) -> proper#(X) -> proper#(U12(X1,X2)) -> proper#(X1) top#(mark(X)) -> proper#(X) -> proper#(U12(X1,X2)) -> U12#(proper(X1),proper(X2)) 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#(length(X)) -> proper#(X) top#(mark(X)) -> proper#(X) -> proper#(length(X)) -> length#(proper(X)) 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#(U11(X1,X2)) -> proper#(X2) proper#(s(X)) -> proper#(X) -> proper#(U11(X1,X2)) -> proper#(X1) proper#(s(X)) -> proper#(X) -> proper#(U11(X1,X2)) -> U11#(proper(X1),proper(X2)) proper#(s(X)) -> proper#(X) -> proper#(U12(X1,X2)) -> proper#(X2) proper#(s(X)) -> proper#(X) -> proper#(U12(X1,X2)) -> proper#(X1) proper#(s(X)) -> proper#(X) -> proper#(U12(X1,X2)) -> U12#(proper(X1),proper(X2)) 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#(length(X)) -> proper#(X) proper#(s(X)) -> proper#(X) -> proper#(length(X)) -> length#(proper(X)) 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#(U11(X1,X2)) -> proper#(X2) proper#(length(X)) -> proper#(X) -> proper#(U11(X1,X2)) -> proper#(X1) proper#(length(X)) -> proper#(X) -> proper#(U11(X1,X2)) -> U11#(proper(X1),proper(X2)) proper#(length(X)) -> proper#(X) -> proper#(U12(X1,X2)) -> proper#(X2) proper#(length(X)) -> proper#(X) -> proper#(U12(X1,X2)) -> proper#(X1) proper#(length(X)) -> proper#(X) -> proper#(U12(X1,X2)) -> U12#(proper(X1),proper(X2)) 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#(length(X)) -> proper#(X) proper#(length(X)) -> proper#(X) -> proper#(length(X)) -> length#(proper(X)) proper#(length(X)) -> length#(proper(X)) -> length#(mark(X)) -> length#(X) proper#(length(X)) -> length#(proper(X)) -> length#(ok(X)) -> length#(X) proper#(U12(X1,X2)) -> proper#(X2) -> proper#(cons(X1,X2)) -> proper#(X2) proper#(U12(X1,X2)) -> proper#(X2) -> proper#(cons(X1,X2)) -> proper#(X1) proper#(U12(X1,X2)) -> proper#(X2) -> proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2)) proper#(U12(X1,X2)) -> proper#(X2) -> proper#(U11(X1,X2)) -> proper#(X2) proper#(U12(X1,X2)) -> proper#(X2) -> proper#(U11(X1,X2)) -> proper#(X1) proper#(U12(X1,X2)) -> proper#(X2) -> proper#(U11(X1,X2)) -> U11#(proper(X1),proper(X2)) proper#(U12(X1,X2)) -> proper#(X2) -> proper#(U12(X1,X2)) -> proper#(X2) proper#(U12(X1,X2)) -> proper#(X2) -> proper#(U12(X1,X2)) -> proper#(X1) proper#(U12(X1,X2)) -> proper#(X2) -> proper#(U12(X1,X2)) -> U12#(proper(X1),proper(X2)) proper#(U12(X1,X2)) -> proper#(X2) -> proper#(s(X)) -> proper#(X) proper#(U12(X1,X2)) -> proper#(X2) -> proper#(s(X)) -> s#(proper(X)) proper#(U12(X1,X2)) -> proper#(X2) -> proper#(length(X)) -> proper#(X) proper#(U12(X1,X2)) -> proper#(X2) -> proper#(length(X)) -> length#(proper(X)) proper#(U12(X1,X2)) -> proper#(X1) -> proper#(cons(X1,X2)) -> proper#(X2) proper#(U12(X1,X2)) -> proper#(X1) -> proper#(cons(X1,X2)) -> proper#(X1) proper#(U12(X1,X2)) -> proper#(X1) -> proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2)) proper#(U12(X1,X2)) -> proper#(X1) -> proper#(U11(X1,X2)) -> proper#(X2) proper#(U12(X1,X2)) -> proper#(X1) -> proper#(U11(X1,X2)) -> proper#(X1) proper#(U12(X1,X2)) -> proper#(X1) -> proper#(U11(X1,X2)) -> U11#(proper(X1),proper(X2)) proper#(U12(X1,X2)) -> proper#(X1) -> proper#(U12(X1,X2)) -> proper#(X2) proper#(U12(X1,X2)) -> proper#(X1) -> proper#(U12(X1,X2)) -> proper#(X1) proper#(U12(X1,X2)) -> proper#(X1) -> proper#(U12(X1,X2)) -> U12#(proper(X1),proper(X2)) proper#(U12(X1,X2)) -> proper#(X1) -> proper#(s(X)) -> proper#(X) proper#(U12(X1,X2)) -> proper#(X1) -> proper#(s(X)) -> s#(proper(X)) proper#(U12(X1,X2)) -> proper#(X1) -> proper#(length(X)) -> proper#(X) proper#(U12(X1,X2)) -> proper#(X1) -> proper#(length(X)) -> length#(proper(X)) proper#(U12(X1,X2)) -> U12#(proper(X1),proper(X2)) -> U12#(mark(X1),X2) -> U12#(X1,X2) proper#(U12(X1,X2)) -> U12#(proper(X1),proper(X2)) -> U12#(ok(X1),ok(X2)) -> U12#(X1,X2) proper#(U11(X1,X2)) -> proper#(X2) -> proper#(cons(X1,X2)) -> proper#(X2) proper#(U11(X1,X2)) -> proper#(X2) -> proper#(cons(X1,X2)) -> proper#(X1) proper#(U11(X1,X2)) -> proper#(X2) -> proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2)) proper#(U11(X1,X2)) -> proper#(X2) -> proper#(U11(X1,X2)) -> proper#(X2) proper#(U11(X1,X2)) -> proper#(X2) -> proper#(U11(X1,X2)) -> proper#(X1) proper#(U11(X1,X2)) -> proper#(X2) -> proper#(U11(X1,X2)) -> U11#(proper(X1),proper(X2)) proper#(U11(X1,X2)) -> proper#(X2) -> proper#(U12(X1,X2)) -> proper#(X2) proper#(U11(X1,X2)) -> proper#(X2) -> proper#(U12(X1,X2)) -> proper#(X1) proper#(U11(X1,X2)) -> proper#(X2) -> proper#(U12(X1,X2)) -> U12#(proper(X1),proper(X2)) proper#(U11(X1,X2)) -> proper#(X2) -> proper#(s(X)) -> proper#(X) proper#(U11(X1,X2)) -> proper#(X2) -> proper#(s(X)) -> s#(proper(X)) proper#(U11(X1,X2)) -> proper#(X2) -> proper#(length(X)) -> proper#(X) proper#(U11(X1,X2)) -> proper#(X2) -> proper#(length(X)) -> length#(proper(X)) proper#(U11(X1,X2)) -> proper#(X1) -> proper#(cons(X1,X2)) -> proper#(X2) proper#(U11(X1,X2)) -> proper#(X1) -> proper#(cons(X1,X2)) -> proper#(X1) proper#(U11(X1,X2)) -> proper#(X1) -> proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2)) proper#(U11(X1,X2)) -> proper#(X1) -> proper#(U11(X1,X2)) -> proper#(X2) proper#(U11(X1,X2)) -> proper#(X1) -> proper#(U11(X1,X2)) -> proper#(X1) proper#(U11(X1,X2)) -> proper#(X1) -> proper#(U11(X1,X2)) -> U11#(proper(X1),proper(X2)) proper#(U11(X1,X2)) -> proper#(X1) -> proper#(U12(X1,X2)) -> proper#(X2) proper#(U11(X1,X2)) -> proper#(X1) -> proper#(U12(X1,X2)) -> proper#(X1) proper#(U11(X1,X2)) -> proper#(X1) -> proper#(U12(X1,X2)) -> U12#(proper(X1),proper(X2)) proper#(U11(X1,X2)) -> proper#(X1) -> proper#(s(X)) -> proper#(X) proper#(U11(X1,X2)) -> proper#(X1) -> proper#(s(X)) -> s#(proper(X)) proper#(U11(X1,X2)) -> proper#(X1) -> proper#(length(X)) -> proper#(X) proper#(U11(X1,X2)) -> proper#(X1) -> proper#(length(X)) -> length#(proper(X)) proper#(U11(X1,X2)) -> U11#(proper(X1),proper(X2)) -> U11#(mark(X1),X2) -> U11#(X1,X2) proper#(U11(X1,X2)) -> U11#(proper(X1),proper(X2)) -> U11#(ok(X1),ok(X2)) -> U11#(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#(U11(X1,X2)) -> proper#(X2) proper#(cons(X1,X2)) -> proper#(X2) -> proper#(U11(X1,X2)) -> proper#(X1) proper#(cons(X1,X2)) -> proper#(X2) -> proper#(U11(X1,X2)) -> U11#(proper(X1),proper(X2)) proper#(cons(X1,X2)) -> proper#(X2) -> proper#(U12(X1,X2)) -> proper#(X2) proper#(cons(X1,X2)) -> proper#(X2) -> proper#(U12(X1,X2)) -> proper#(X1) proper#(cons(X1,X2)) -> proper#(X2) -> proper#(U12(X1,X2)) -> U12#(proper(X1),proper(X2)) 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#(length(X)) -> proper#(X) proper#(cons(X1,X2)) -> proper#(X2) -> proper#(length(X)) -> length#(proper(X)) 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#(U11(X1,X2)) -> proper#(X2) proper#(cons(X1,X2)) -> proper#(X1) -> proper#(U11(X1,X2)) -> proper#(X1) proper#(cons(X1,X2)) -> proper#(X1) -> proper#(U11(X1,X2)) -> U11#(proper(X1),proper(X2)) proper#(cons(X1,X2)) -> proper#(X1) -> proper#(U12(X1,X2)) -> proper#(X2) proper#(cons(X1,X2)) -> proper#(X1) -> proper#(U12(X1,X2)) -> proper#(X1) proper#(cons(X1,X2)) -> proper#(X1) -> proper#(U12(X1,X2)) -> U12#(proper(X1),proper(X2)) 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#(length(X)) -> proper#(X) proper#(cons(X1,X2)) -> proper#(X1) -> proper#(length(X)) -> length#(proper(X)) 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) U11#(ok(X1),ok(X2)) -> U11#(X1,X2) -> U11#(mark(X1),X2) -> U11#(X1,X2) U11#(ok(X1),ok(X2)) -> U11#(X1,X2) -> U11#(ok(X1),ok(X2)) -> U11#(X1,X2) U11#(mark(X1),X2) -> U11#(X1,X2) -> U11#(mark(X1),X2) -> U11#(X1,X2) U11#(mark(X1),X2) -> U11#(X1,X2) -> U11#(ok(X1),ok(X2)) -> U11#(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) U12#(ok(X1),ok(X2)) -> U12#(X1,X2) -> U12#(mark(X1),X2) -> U12#(X1,X2) U12#(ok(X1),ok(X2)) -> U12#(X1,X2) -> U12#(ok(X1),ok(X2)) -> U12#(X1,X2) U12#(mark(X1),X2) -> U12#(X1,X2) -> U12#(mark(X1),X2) -> U12#(X1,X2) U12#(mark(X1),X2) -> U12#(X1,X2) -> U12#(ok(X1),ok(X2)) -> U12#(X1,X2) 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#(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#(U11(tt(),L)) -> U12#(tt(),L) active#(s(X)) -> active#(X) -> active#(U12(tt(),L)) -> length#(L) active#(s(X)) -> active#(X) -> active#(U12(tt(),L)) -> s#(length(L)) active#(s(X)) -> active#(X) -> active#(length(cons(N,L))) -> U11#(tt(),L) 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#(U11(X1,X2)) -> active#(X1) active#(s(X)) -> active#(X) -> active#(U11(X1,X2)) -> U11#(active(X1),X2) active#(s(X)) -> active#(X) -> active#(U12(X1,X2)) -> active#(X1) active#(s(X)) -> active#(X) -> active#(U12(X1,X2)) -> U12#(active(X1),X2) 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#(length(X)) -> active#(X) active#(s(X)) -> active#(X) -> active#(length(X)) -> length#(active(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#(U11(tt(),L)) -> U12#(tt(),L) active#(length(X)) -> active#(X) -> active#(U12(tt(),L)) -> length#(L) active#(length(X)) -> active#(X) -> active#(U12(tt(),L)) -> s#(length(L)) active#(length(X)) -> active#(X) -> active#(length(cons(N,L))) -> U11#(tt(),L) 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#(U11(X1,X2)) -> active#(X1) active#(length(X)) -> active#(X) -> active#(U11(X1,X2)) -> U11#(active(X1),X2) active#(length(X)) -> active#(X) -> active#(U12(X1,X2)) -> active#(X1) active#(length(X)) -> active#(X) -> active#(U12(X1,X2)) -> U12#(active(X1),X2) 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#(length(X)) -> active#(X) active#(length(X)) -> active#(X) -> active#(length(X)) -> length#(active(X)) active#(U12(tt(),L)) -> s#(length(L)) -> s#(mark(X)) -> s#(X) active#(U12(tt(),L)) -> s#(length(L)) -> s#(ok(X)) -> s#(X) active#(U12(tt(),L)) -> length#(L) -> length#(mark(X)) -> length#(X) active#(U12(tt(),L)) -> length#(L) -> length#(ok(X)) -> length#(X) active#(U12(X1,X2)) -> U12#(active(X1),X2) -> U12#(mark(X1),X2) -> U12#(X1,X2) active#(U12(X1,X2)) -> U12#(active(X1),X2) -> U12#(ok(X1),ok(X2)) -> U12#(X1,X2) active#(U12(X1,X2)) -> active#(X1) -> active#(zeros()) -> cons#(0(),zeros()) active#(U12(X1,X2)) -> active#(X1) -> active#(U11(tt(),L)) -> U12#(tt(),L) active#(U12(X1,X2)) -> active#(X1) -> active#(U12(tt(),L)) -> length#(L) active#(U12(X1,X2)) -> active#(X1) -> active#(U12(tt(),L)) -> s#(length(L)) active#(U12(X1,X2)) -> active#(X1) -> active#(length(cons(N,L))) -> U11#(tt(),L) active#(U12(X1,X2)) -> active#(X1) -> active#(cons(X1,X2)) -> active#(X1) active#(U12(X1,X2)) -> active#(X1) -> active#(cons(X1,X2)) -> cons#(active(X1),X2) active#(U12(X1,X2)) -> active#(X1) -> active#(U11(X1,X2)) -> active#(X1) active#(U12(X1,X2)) -> active#(X1) -> active#(U11(X1,X2)) -> U11#(active(X1),X2) active#(U12(X1,X2)) -> active#(X1) -> active#(U12(X1,X2)) -> active#(X1) active#(U12(X1,X2)) -> active#(X1) -> active#(U12(X1,X2)) -> U12#(active(X1),X2) active#(U12(X1,X2)) -> active#(X1) -> active#(s(X)) -> active#(X) active#(U12(X1,X2)) -> active#(X1) -> active#(s(X)) -> s#(active(X)) active#(U12(X1,X2)) -> active#(X1) -> active#(length(X)) -> active#(X) active#(U12(X1,X2)) -> active#(X1) -> active#(length(X)) -> length#(active(X)) active#(U11(X1,X2)) -> U11#(active(X1),X2) -> U11#(mark(X1),X2) -> U11#(X1,X2) active#(U11(X1,X2)) -> U11#(active(X1),X2) -> U11#(ok(X1),ok(X2)) -> U11#(X1,X2) active#(U11(X1,X2)) -> active#(X1) -> active#(zeros()) -> cons#(0(),zeros()) active#(U11(X1,X2)) -> active#(X1) -> active#(U11(tt(),L)) -> U12#(tt(),L) active#(U11(X1,X2)) -> active#(X1) -> active#(U12(tt(),L)) -> length#(L) active#(U11(X1,X2)) -> active#(X1) -> active#(U12(tt(),L)) -> s#(length(L)) active#(U11(X1,X2)) -> active#(X1) -> active#(length(cons(N,L))) -> U11#(tt(),L) active#(U11(X1,X2)) -> active#(X1) -> active#(cons(X1,X2)) -> active#(X1) active#(U11(X1,X2)) -> active#(X1) -> active#(cons(X1,X2)) -> cons#(active(X1),X2) active#(U11(X1,X2)) -> active#(X1) -> active#(U11(X1,X2)) -> active#(X1) active#(U11(X1,X2)) -> active#(X1) -> active#(U11(X1,X2)) -> U11#(active(X1),X2) active#(U11(X1,X2)) -> active#(X1) -> active#(U12(X1,X2)) -> active#(X1) active#(U11(X1,X2)) -> active#(X1) -> active#(U12(X1,X2)) -> U12#(active(X1),X2) active#(U11(X1,X2)) -> active#(X1) -> active#(s(X)) -> active#(X) active#(U11(X1,X2)) -> active#(X1) -> active#(s(X)) -> s#(active(X)) active#(U11(X1,X2)) -> active#(X1) -> active#(length(X)) -> active#(X) active#(U11(X1,X2)) -> active#(X1) -> active#(length(X)) -> length#(active(X)) 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#(U11(tt(),L)) -> U12#(tt(),L) active#(cons(X1,X2)) -> active#(X1) -> active#(U12(tt(),L)) -> length#(L) active#(cons(X1,X2)) -> active#(X1) -> active#(U12(tt(),L)) -> s#(length(L)) active#(cons(X1,X2)) -> active#(X1) -> active#(length(cons(N,L))) -> U11#(tt(),L) 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#(U11(X1,X2)) -> active#(X1) active#(cons(X1,X2)) -> active#(X1) -> active#(U11(X1,X2)) -> U11#(active(X1),X2) active#(cons(X1,X2)) -> active#(X1) -> active#(U12(X1,X2)) -> active#(X1) active#(cons(X1,X2)) -> active#(X1) -> active#(U12(X1,X2)) -> U12#(active(X1),X2) 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#(length(X)) -> active#(X) active#(cons(X1,X2)) -> active#(X1) -> active#(length(X)) -> length#(active(X)) SCC Processor: #sccs: 8 #rules: 25 #arcs: 259/1764 DPs: top#(ok(X)) -> top#(active(X)) top#(mark(X)) -> top#(proper(X)) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),L)) -> mark(U12(tt(),L)) active(U12(tt(),L)) -> mark(s(length(L))) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U11(tt(),L)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X1,X2)) -> U11(active(X1),X2) active(U12(X1,X2)) -> U12(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X1),X2) -> mark(U11(X1,X2)) U12(mark(X1),X2) -> mark(U12(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(U12(X1,X2)) -> U12(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(nil()) -> ok(nil()) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) U12(ok(X1),ok(X2)) -> ok(U12(X1,X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Open DPs: proper#(length(X)) -> proper#(X) proper#(s(X)) -> proper#(X) proper#(U12(X1,X2)) -> proper#(X1) proper#(U12(X1,X2)) -> proper#(X2) proper#(U11(X1,X2)) -> proper#(X1) proper#(U11(X1,X2)) -> proper#(X2) proper#(cons(X1,X2)) -> proper#(X1) proper#(cons(X1,X2)) -> proper#(X2) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),L)) -> mark(U12(tt(),L)) active(U12(tt(),L)) -> mark(s(length(L))) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U11(tt(),L)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X1,X2)) -> U11(active(X1),X2) active(U12(X1,X2)) -> U12(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X1),X2) -> mark(U11(X1,X2)) U12(mark(X1),X2) -> mark(U12(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(U12(X1,X2)) -> U12(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(nil()) -> ok(nil()) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) U12(ok(X1),ok(X2)) -> ok(U12(X1,X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dimension: 1 interpretation: [proper#](x0) = x0 + 1, [top](x0) = 0, [ok](x0) = x0, [proper](x0) = x0, [nil] = 1, [s](x0) = x0 + 1, [length](x0) = x0 + 1, [U12](x0, x1) = x0 + x1, [U11](x0, x1) = x0 + x1, [tt] = 1, [mark](x0) = 1, [cons](x0, x1) = x0 + x1, [0] = 1, [active](x0) = x0, [zeros] = 1 orientation: proper#(length(X)) = X + 2 >= X + 1 = proper#(X) proper#(s(X)) = X + 2 >= X + 1 = proper#(X) proper#(U12(X1,X2)) = X1 + X2 + 1 >= X1 + 1 = proper#(X1) proper#(U12(X1,X2)) = X1 + X2 + 1 >= X2 + 1 = proper#(X2) proper#(U11(X1,X2)) = X1 + X2 + 1 >= X1 + 1 = proper#(X1) proper#(U11(X1,X2)) = X1 + X2 + 1 >= 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(U11(tt(),L)) = L + 1 >= 1 = mark(U12(tt(),L)) active(U12(tt(),L)) = L + 1 >= 1 = mark(s(length(L))) active(length(nil())) = 2 >= 1 = mark(0()) active(length(cons(N,L))) = L + N + 1 >= 1 = mark(U11(tt(),L)) active(cons(X1,X2)) = X1 + X2 >= X1 + X2 = cons(active(X1),X2) active(U11(X1,X2)) = X1 + X2 >= X1 + X2 = U11(active(X1),X2) active(U12(X1,X2)) = X1 + X2 >= X1 + X2 = U12(active(X1),X2) active(s(X)) = X + 1 >= X + 1 = s(active(X)) active(length(X)) = X + 1 >= X + 1 = length(active(X)) cons(mark(X1),X2) = X2 + 1 >= 1 = mark(cons(X1,X2)) U11(mark(X1),X2) = X2 + 1 >= 1 = mark(U11(X1,X2)) U12(mark(X1),X2) = X2 + 1 >= 1 = mark(U12(X1,X2)) s(mark(X)) = 2 >= 1 = mark(s(X)) length(mark(X)) = 2 >= 1 = mark(length(X)) proper(zeros()) = 1 >= 1 = ok(zeros()) proper(cons(X1,X2)) = X1 + X2 >= X1 + X2 = cons(proper(X1),proper(X2)) proper(0()) = 1 >= 1 = ok(0()) proper(U11(X1,X2)) = X1 + X2 >= X1 + X2 = U11(proper(X1),proper(X2)) proper(tt()) = 1 >= 1 = ok(tt()) proper(U12(X1,X2)) = X1 + X2 >= X1 + X2 = U12(proper(X1),proper(X2)) proper(s(X)) = X + 1 >= X + 1 = s(proper(X)) proper(length(X)) = X + 1 >= X + 1 = length(proper(X)) proper(nil()) = 1 >= 1 = ok(nil()) cons(ok(X1),ok(X2)) = X1 + X2 >= X1 + X2 = ok(cons(X1,X2)) U11(ok(X1),ok(X2)) = X1 + X2 >= X1 + X2 = ok(U11(X1,X2)) U12(ok(X1),ok(X2)) = X1 + X2 >= X1 + X2 = ok(U12(X1,X2)) s(ok(X)) = X + 1 >= X + 1 = ok(s(X)) length(ok(X)) = X + 1 >= X + 1 = ok(length(X)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) problem: DPs: proper#(U12(X1,X2)) -> proper#(X1) proper#(U12(X1,X2)) -> proper#(X2) proper#(U11(X1,X2)) -> proper#(X1) proper#(U11(X1,X2)) -> proper#(X2) proper#(cons(X1,X2)) -> proper#(X1) proper#(cons(X1,X2)) -> proper#(X2) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),L)) -> mark(U12(tt(),L)) active(U12(tt(),L)) -> mark(s(length(L))) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U11(tt(),L)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X1,X2)) -> U11(active(X1),X2) active(U12(X1,X2)) -> U12(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X1),X2) -> mark(U11(X1,X2)) U12(mark(X1),X2) -> mark(U12(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(U12(X1,X2)) -> U12(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(nil()) -> ok(nil()) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) U12(ok(X1),ok(X2)) -> ok(U12(X1,X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dimension: 1 interpretation: [proper#](x0) = x0 + 1, [top](x0) = 0, [ok](x0) = x0, [proper](x0) = x0, [nil] = 0, [s](x0) = 0, [length](x0) = 0, [U12](x0, x1) = x0 + x1 + 1, [U11](x0, x1) = x0 + x1, [tt] = 0, [mark](x0) = 0, [cons](x0, x1) = x0 + x1, [0] = 1, [active](x0) = x0, [zeros] = 0 orientation: proper#(U12(X1,X2)) = X1 + X2 + 2 >= X1 + 1 = proper#(X1) proper#(U12(X1,X2)) = X1 + X2 + 2 >= X2 + 1 = proper#(X2) proper#(U11(X1,X2)) = X1 + X2 + 1 >= X1 + 1 = proper#(X1) proper#(U11(X1,X2)) = X1 + X2 + 1 >= X2 + 1 = proper#(X2) proper#(cons(X1,X2)) = X1 + X2 + 1 >= X1 + 1 = proper#(X1) proper#(cons(X1,X2)) = X1 + X2 + 1 >= X2 + 1 = proper#(X2) active(zeros()) = 0 >= 0 = mark(cons(0(),zeros())) active(U11(tt(),L)) = L >= 0 = mark(U12(tt(),L)) active(U12(tt(),L)) = L + 1 >= 0 = mark(s(length(L))) active(length(nil())) = 0 >= 0 = mark(0()) active(length(cons(N,L))) = 0 >= 0 = mark(U11(tt(),L)) active(cons(X1,X2)) = X1 + X2 >= X1 + X2 = cons(active(X1),X2) active(U11(X1,X2)) = X1 + X2 >= X1 + X2 = U11(active(X1),X2) active(U12(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = U12(active(X1),X2) active(s(X)) = 0 >= 0 = s(active(X)) active(length(X)) = 0 >= 0 = length(active(X)) cons(mark(X1),X2) = X2 >= 0 = mark(cons(X1,X2)) U11(mark(X1),X2) = X2 >= 0 = mark(U11(X1,X2)) U12(mark(X1),X2) = X2 + 1 >= 0 = mark(U12(X1,X2)) s(mark(X)) = 0 >= 0 = mark(s(X)) length(mark(X)) = 0 >= 0 = mark(length(X)) proper(zeros()) = 0 >= 0 = ok(zeros()) proper(cons(X1,X2)) = X1 + X2 >= X1 + X2 = cons(proper(X1),proper(X2)) proper(0()) = 1 >= 1 = ok(0()) proper(U11(X1,X2)) = X1 + X2 >= X1 + X2 = U11(proper(X1),proper(X2)) proper(tt()) = 0 >= 0 = ok(tt()) proper(U12(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = U12(proper(X1),proper(X2)) proper(s(X)) = 0 >= 0 = s(proper(X)) proper(length(X)) = 0 >= 0 = length(proper(X)) proper(nil()) = 0 >= 0 = ok(nil()) cons(ok(X1),ok(X2)) = X1 + X2 >= X1 + X2 = ok(cons(X1,X2)) U11(ok(X1),ok(X2)) = X1 + X2 >= X1 + X2 = ok(U11(X1,X2)) U12(ok(X1),ok(X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = ok(U12(X1,X2)) s(ok(X)) = 0 >= 0 = ok(s(X)) length(ok(X)) = 0 >= 0 = ok(length(X)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) problem: DPs: proper#(U11(X1,X2)) -> proper#(X1) proper#(U11(X1,X2)) -> proper#(X2) proper#(cons(X1,X2)) -> proper#(X1) proper#(cons(X1,X2)) -> proper#(X2) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),L)) -> mark(U12(tt(),L)) active(U12(tt(),L)) -> mark(s(length(L))) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U11(tt(),L)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X1,X2)) -> U11(active(X1),X2) active(U12(X1,X2)) -> U12(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X1),X2) -> mark(U11(X1,X2)) U12(mark(X1),X2) -> mark(U12(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(U12(X1,X2)) -> U12(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(nil()) -> ok(nil()) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) U12(ok(X1),ok(X2)) -> ok(U12(X1,X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dimension: 1 interpretation: [proper#](x0) = x0 + 1, [top](x0) = 1, [ok](x0) = 0, [proper](x0) = x0, [nil] = 0, [s](x0) = 0, [length](x0) = x0, [U12](x0, x1) = 0, [U11](x0, x1) = x0 + x1 + 1, [tt] = 0, [mark](x0) = 0, [cons](x0, x1) = x0 + x1 + 1, [0] = 0, [active](x0) = x0, [zeros] = 0 orientation: proper#(U11(X1,X2)) = X1 + X2 + 2 >= X1 + 1 = proper#(X1) proper#(U11(X1,X2)) = X1 + X2 + 2 >= X2 + 1 = proper#(X2) proper#(cons(X1,X2)) = X1 + X2 + 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(U11(tt(),L)) = L + 1 >= 0 = mark(U12(tt(),L)) active(U12(tt(),L)) = 0 >= 0 = mark(s(length(L))) active(length(nil())) = 0 >= 0 = mark(0()) active(length(cons(N,L))) = L + N + 1 >= 0 = mark(U11(tt(),L)) active(cons(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = cons(active(X1),X2) active(U11(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = U11(active(X1),X2) active(U12(X1,X2)) = 0 >= 0 = U12(active(X1),X2) active(s(X)) = 0 >= 0 = s(active(X)) active(length(X)) = X >= X = length(active(X)) cons(mark(X1),X2) = X2 + 1 >= 0 = mark(cons(X1,X2)) U11(mark(X1),X2) = X2 + 1 >= 0 = mark(U11(X1,X2)) U12(mark(X1),X2) = 0 >= 0 = mark(U12(X1,X2)) s(mark(X)) = 0 >= 0 = mark(s(X)) length(mark(X)) = 0 >= 0 = mark(length(X)) proper(zeros()) = 0 >= 0 = ok(zeros()) proper(cons(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = cons(proper(X1),proper(X2)) proper(0()) = 0 >= 0 = ok(0()) proper(U11(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = U11(proper(X1),proper(X2)) proper(tt()) = 0 >= 0 = ok(tt()) proper(U12(X1,X2)) = 0 >= 0 = U12(proper(X1),proper(X2)) proper(s(X)) = 0 >= 0 = s(proper(X)) proper(length(X)) = X >= X = length(proper(X)) proper(nil()) = 0 >= 0 = ok(nil()) cons(ok(X1),ok(X2)) = 1 >= 0 = ok(cons(X1,X2)) U11(ok(X1),ok(X2)) = 1 >= 0 = ok(U11(X1,X2)) U12(ok(X1),ok(X2)) = 0 >= 0 = ok(U12(X1,X2)) s(ok(X)) = 0 >= 0 = ok(s(X)) length(ok(X)) = 0 >= 0 = ok(length(X)) top(mark(X)) = 1 >= 1 = top(proper(X)) top(ok(X)) = 1 >= 1 = top(active(X)) problem: DPs: TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),L)) -> mark(U12(tt(),L)) active(U12(tt(),L)) -> mark(s(length(L))) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U11(tt(),L)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X1,X2)) -> U11(active(X1),X2) active(U12(X1,X2)) -> U12(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X1),X2) -> mark(U11(X1,X2)) U12(mark(X1),X2) -> mark(U12(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(U12(X1,X2)) -> U12(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(nil()) -> ok(nil()) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) U12(ok(X1),ok(X2)) -> ok(U12(X1,X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed DPs: active#(length(X)) -> active#(X) active#(s(X)) -> active#(X) active#(U12(X1,X2)) -> active#(X1) active#(U11(X1,X2)) -> active#(X1) active#(cons(X1,X2)) -> active#(X1) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),L)) -> mark(U12(tt(),L)) active(U12(tt(),L)) -> mark(s(length(L))) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U11(tt(),L)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X1,X2)) -> U11(active(X1),X2) active(U12(X1,X2)) -> U12(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X1),X2) -> mark(U11(X1,X2)) U12(mark(X1),X2) -> mark(U12(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(U12(X1,X2)) -> U12(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(nil()) -> ok(nil()) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) U12(ok(X1),ok(X2)) -> ok(U12(X1,X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dimension: 1 interpretation: [active#](x0) = x0 + 1, [top](x0) = 0, [ok](x0) = x0, [proper](x0) = x0, [nil] = 0, [s](x0) = x0 + 1, [length](x0) = x0 + 1, [U12](x0, x1) = x0 + 1, [U11](x0, x1) = x0 + 1, [tt] = 0, [mark](x0) = 1, [cons](x0, x1) = x0 + 1, [0] = 1, [active](x0) = x0, [zeros] = 1 orientation: active#(length(X)) = X + 2 >= X + 1 = active#(X) active#(s(X)) = X + 2 >= X + 1 = active#(X) active#(U12(X1,X2)) = X1 + 2 >= X1 + 1 = active#(X1) active#(U11(X1,X2)) = X1 + 2 >= X1 + 1 = active#(X1) active#(cons(X1,X2)) = X1 + 2 >= X1 + 1 = active#(X1) active(zeros()) = 1 >= 1 = mark(cons(0(),zeros())) active(U11(tt(),L)) = 1 >= 1 = mark(U12(tt(),L)) active(U12(tt(),L)) = 1 >= 1 = mark(s(length(L))) active(length(nil())) = 1 >= 1 = mark(0()) active(length(cons(N,L))) = N + 2 >= 1 = mark(U11(tt(),L)) active(cons(X1,X2)) = X1 + 1 >= X1 + 1 = cons(active(X1),X2) active(U11(X1,X2)) = X1 + 1 >= X1 + 1 = U11(active(X1),X2) active(U12(X1,X2)) = X1 + 1 >= X1 + 1 = U12(active(X1),X2) active(s(X)) = X + 1 >= X + 1 = s(active(X)) active(length(X)) = X + 1 >= X + 1 = length(active(X)) cons(mark(X1),X2) = 2 >= 1 = mark(cons(X1,X2)) U11(mark(X1),X2) = 2 >= 1 = mark(U11(X1,X2)) U12(mark(X1),X2) = 2 >= 1 = mark(U12(X1,X2)) s(mark(X)) = 2 >= 1 = mark(s(X)) length(mark(X)) = 2 >= 1 = mark(length(X)) 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(U11(X1,X2)) = X1 + 1 >= X1 + 1 = U11(proper(X1),proper(X2)) proper(tt()) = 0 >= 0 = ok(tt()) proper(U12(X1,X2)) = X1 + 1 >= X1 + 1 = U12(proper(X1),proper(X2)) proper(s(X)) = X + 1 >= X + 1 = s(proper(X)) proper(length(X)) = X + 1 >= X + 1 = length(proper(X)) proper(nil()) = 0 >= 0 = ok(nil()) cons(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(cons(X1,X2)) U11(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(U11(X1,X2)) U12(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(U12(X1,X2)) s(ok(X)) = X + 1 >= X + 1 = ok(s(X)) length(ok(X)) = X + 1 >= X + 1 = ok(length(X)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) problem: DPs: TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),L)) -> mark(U12(tt(),L)) active(U12(tt(),L)) -> mark(s(length(L))) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U11(tt(),L)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X1,X2)) -> U11(active(X1),X2) active(U12(X1,X2)) -> U12(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X1),X2) -> mark(U11(X1,X2)) U12(mark(X1),X2) -> mark(U12(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(U12(X1,X2)) -> U12(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(nil()) -> ok(nil()) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) U12(ok(X1),ok(X2)) -> ok(U12(X1,X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed DPs: cons#(ok(X1),ok(X2)) -> cons#(X1,X2) cons#(mark(X1),X2) -> cons#(X1,X2) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),L)) -> mark(U12(tt(),L)) active(U12(tt(),L)) -> mark(s(length(L))) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U11(tt(),L)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X1,X2)) -> U11(active(X1),X2) active(U12(X1,X2)) -> U12(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X1),X2) -> mark(U11(X1,X2)) U12(mark(X1),X2) -> mark(U12(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(U12(X1,X2)) -> U12(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(nil()) -> ok(nil()) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) U12(ok(X1),ok(X2)) -> ok(U12(X1,X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dimension: 1 interpretation: [cons#](x0, x1) = x1 + 1, [top](x0) = 0, [ok](x0) = x0 + 1, [proper](x0) = x0 + 1, [nil] = 0, [s](x0) = x0, [length](x0) = x0, [U12](x0, x1) = x1, [U11](x0, x1) = x1, [tt] = 0, [mark](x0) = x0, [cons](x0, x1) = x1, [0] = 0, [active](x0) = x0, [zeros] = 1 orientation: cons#(ok(X1),ok(X2)) = X2 + 2 >= X2 + 1 = cons#(X1,X2) cons#(mark(X1),X2) = X2 + 1 >= X2 + 1 = cons#(X1,X2) active(zeros()) = 1 >= 1 = mark(cons(0(),zeros())) active(U11(tt(),L)) = L >= L = mark(U12(tt(),L)) active(U12(tt(),L)) = L >= L = mark(s(length(L))) active(length(nil())) = 0 >= 0 = mark(0()) active(length(cons(N,L))) = L >= L = mark(U11(tt(),L)) active(cons(X1,X2)) = X2 >= X2 = cons(active(X1),X2) active(U11(X1,X2)) = X2 >= X2 = U11(active(X1),X2) active(U12(X1,X2)) = X2 >= X2 = U12(active(X1),X2) active(s(X)) = X >= X = s(active(X)) active(length(X)) = X >= X = length(active(X)) cons(mark(X1),X2) = X2 >= X2 = mark(cons(X1,X2)) U11(mark(X1),X2) = X2 >= X2 = mark(U11(X1,X2)) U12(mark(X1),X2) = X2 >= X2 = mark(U12(X1,X2)) s(mark(X)) = X >= X = mark(s(X)) length(mark(X)) = X >= X = mark(length(X)) proper(zeros()) = 2 >= 2 = ok(zeros()) proper(cons(X1,X2)) = X2 + 1 >= X2 + 1 = cons(proper(X1),proper(X2)) proper(0()) = 1 >= 1 = ok(0()) proper(U11(X1,X2)) = X2 + 1 >= X2 + 1 = U11(proper(X1),proper(X2)) proper(tt()) = 1 >= 1 = ok(tt()) proper(U12(X1,X2)) = X2 + 1 >= X2 + 1 = U12(proper(X1),proper(X2)) proper(s(X)) = X + 1 >= X + 1 = s(proper(X)) proper(length(X)) = X + 1 >= X + 1 = length(proper(X)) proper(nil()) = 1 >= 1 = ok(nil()) cons(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(cons(X1,X2)) U11(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(U11(X1,X2)) U12(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(U12(X1,X2)) s(ok(X)) = X + 1 >= X + 1 = ok(s(X)) length(ok(X)) = X + 1 >= X + 1 = ok(length(X)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) problem: DPs: cons#(mark(X1),X2) -> cons#(X1,X2) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),L)) -> mark(U12(tt(),L)) active(U12(tt(),L)) -> mark(s(length(L))) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U11(tt(),L)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X1,X2)) -> U11(active(X1),X2) active(U12(X1,X2)) -> U12(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X1),X2) -> mark(U11(X1,X2)) U12(mark(X1),X2) -> mark(U12(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(U12(X1,X2)) -> U12(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(nil()) -> ok(nil()) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) U12(ok(X1),ok(X2)) -> ok(U12(X1,X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dimension: 1 interpretation: [cons#](x0, x1) = x0 + 1, [top](x0) = 0, [ok](x0) = 0, [proper](x0) = 0, [nil] = 0, [s](x0) = x0, [length](x0) = x0, [U12](x0, x1) = x0 + x1, [U11](x0, x1) = x0 + x1, [tt] = 0, [mark](x0) = x0 + 1, [cons](x0, x1) = x0 + x1, [0] = 0, [active](x0) = x0 + 1, [zeros] = 0 orientation: cons#(mark(X1),X2) = X1 + 2 >= X1 + 1 = cons#(X1,X2) active(zeros()) = 1 >= 1 = mark(cons(0(),zeros())) active(U11(tt(),L)) = L + 1 >= L + 1 = mark(U12(tt(),L)) active(U12(tt(),L)) = L + 1 >= L + 1 = mark(s(length(L))) active(length(nil())) = 1 >= 1 = mark(0()) active(length(cons(N,L))) = L + N + 1 >= L + 1 = mark(U11(tt(),L)) active(cons(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = cons(active(X1),X2) active(U11(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = U11(active(X1),X2) active(U12(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = U12(active(X1),X2) active(s(X)) = X + 1 >= X + 1 = s(active(X)) active(length(X)) = X + 1 >= X + 1 = length(active(X)) cons(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(cons(X1,X2)) U11(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(U11(X1,X2)) U12(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(U12(X1,X2)) s(mark(X)) = X + 1 >= X + 1 = mark(s(X)) length(mark(X)) = X + 1 >= X + 1 = mark(length(X)) proper(zeros()) = 0 >= 0 = ok(zeros()) proper(cons(X1,X2)) = 0 >= 0 = cons(proper(X1),proper(X2)) proper(0()) = 0 >= 0 = ok(0()) proper(U11(X1,X2)) = 0 >= 0 = U11(proper(X1),proper(X2)) proper(tt()) = 0 >= 0 = ok(tt()) proper(U12(X1,X2)) = 0 >= 0 = U12(proper(X1),proper(X2)) proper(s(X)) = 0 >= 0 = s(proper(X)) proper(length(X)) = 0 >= 0 = length(proper(X)) proper(nil()) = 0 >= 0 = ok(nil()) cons(ok(X1),ok(X2)) = 0 >= 0 = ok(cons(X1,X2)) U11(ok(X1),ok(X2)) = 0 >= 0 = ok(U11(X1,X2)) U12(ok(X1),ok(X2)) = 0 >= 0 = ok(U12(X1,X2)) s(ok(X)) = 0 >= 0 = ok(s(X)) length(ok(X)) = 0 >= 0 = ok(length(X)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) problem: DPs: TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),L)) -> mark(U12(tt(),L)) active(U12(tt(),L)) -> mark(s(length(L))) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U11(tt(),L)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X1,X2)) -> U11(active(X1),X2) active(U12(X1,X2)) -> U12(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X1),X2) -> mark(U11(X1,X2)) U12(mark(X1),X2) -> mark(U12(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(U12(X1,X2)) -> U12(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(nil()) -> ok(nil()) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) U12(ok(X1),ok(X2)) -> ok(U12(X1,X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed DPs: U11#(ok(X1),ok(X2)) -> U11#(X1,X2) U11#(mark(X1),X2) -> U11#(X1,X2) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),L)) -> mark(U12(tt(),L)) active(U12(tt(),L)) -> mark(s(length(L))) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U11(tt(),L)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X1,X2)) -> U11(active(X1),X2) active(U12(X1,X2)) -> U12(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X1),X2) -> mark(U11(X1,X2)) U12(mark(X1),X2) -> mark(U12(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(U12(X1,X2)) -> U12(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(nil()) -> ok(nil()) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) U12(ok(X1),ok(X2)) -> ok(U12(X1,X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dimension: 1 interpretation: [U11#](x0, x1) = x1 + 1, [top](x0) = 0, [ok](x0) = x0 + 1, [proper](x0) = x0 + 1, [nil] = 0, [s](x0) = x0, [length](x0) = x0, [U12](x0, x1) = x1, [U11](x0, x1) = x1, [tt] = 0, [mark](x0) = x0, [cons](x0, x1) = x1, [0] = 0, [active](x0) = x0, [zeros] = 1 orientation: U11#(ok(X1),ok(X2)) = X2 + 2 >= X2 + 1 = U11#(X1,X2) U11#(mark(X1),X2) = X2 + 1 >= X2 + 1 = U11#(X1,X2) active(zeros()) = 1 >= 1 = mark(cons(0(),zeros())) active(U11(tt(),L)) = L >= L = mark(U12(tt(),L)) active(U12(tt(),L)) = L >= L = mark(s(length(L))) active(length(nil())) = 0 >= 0 = mark(0()) active(length(cons(N,L))) = L >= L = mark(U11(tt(),L)) active(cons(X1,X2)) = X2 >= X2 = cons(active(X1),X2) active(U11(X1,X2)) = X2 >= X2 = U11(active(X1),X2) active(U12(X1,X2)) = X2 >= X2 = U12(active(X1),X2) active(s(X)) = X >= X = s(active(X)) active(length(X)) = X >= X = length(active(X)) cons(mark(X1),X2) = X2 >= X2 = mark(cons(X1,X2)) U11(mark(X1),X2) = X2 >= X2 = mark(U11(X1,X2)) U12(mark(X1),X2) = X2 >= X2 = mark(U12(X1,X2)) s(mark(X)) = X >= X = mark(s(X)) length(mark(X)) = X >= X = mark(length(X)) proper(zeros()) = 2 >= 2 = ok(zeros()) proper(cons(X1,X2)) = X2 + 1 >= X2 + 1 = cons(proper(X1),proper(X2)) proper(0()) = 1 >= 1 = ok(0()) proper(U11(X1,X2)) = X2 + 1 >= X2 + 1 = U11(proper(X1),proper(X2)) proper(tt()) = 1 >= 1 = ok(tt()) proper(U12(X1,X2)) = X2 + 1 >= X2 + 1 = U12(proper(X1),proper(X2)) proper(s(X)) = X + 1 >= X + 1 = s(proper(X)) proper(length(X)) = X + 1 >= X + 1 = length(proper(X)) proper(nil()) = 1 >= 1 = ok(nil()) cons(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(cons(X1,X2)) U11(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(U11(X1,X2)) U12(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(U12(X1,X2)) s(ok(X)) = X + 1 >= X + 1 = ok(s(X)) length(ok(X)) = X + 1 >= X + 1 = ok(length(X)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) problem: DPs: U11#(mark(X1),X2) -> U11#(X1,X2) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),L)) -> mark(U12(tt(),L)) active(U12(tt(),L)) -> mark(s(length(L))) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U11(tt(),L)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X1,X2)) -> U11(active(X1),X2) active(U12(X1,X2)) -> U12(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X1),X2) -> mark(U11(X1,X2)) U12(mark(X1),X2) -> mark(U12(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(U12(X1,X2)) -> U12(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(nil()) -> ok(nil()) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) U12(ok(X1),ok(X2)) -> ok(U12(X1,X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dimension: 1 interpretation: [U11#](x0, x1) = x0 + 1, [top](x0) = 0, [ok](x0) = 0, [proper](x0) = 0, [nil] = 0, [s](x0) = x0, [length](x0) = x0, [U12](x0, x1) = x0 + x1, [U11](x0, x1) = x0 + x1, [tt] = 0, [mark](x0) = x0 + 1, [cons](x0, x1) = x0 + x1, [0] = 0, [active](x0) = x0 + 1, [zeros] = 0 orientation: U11#(mark(X1),X2) = X1 + 2 >= X1 + 1 = U11#(X1,X2) active(zeros()) = 1 >= 1 = mark(cons(0(),zeros())) active(U11(tt(),L)) = L + 1 >= L + 1 = mark(U12(tt(),L)) active(U12(tt(),L)) = L + 1 >= L + 1 = mark(s(length(L))) active(length(nil())) = 1 >= 1 = mark(0()) active(length(cons(N,L))) = L + N + 1 >= L + 1 = mark(U11(tt(),L)) active(cons(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = cons(active(X1),X2) active(U11(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = U11(active(X1),X2) active(U12(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = U12(active(X1),X2) active(s(X)) = X + 1 >= X + 1 = s(active(X)) active(length(X)) = X + 1 >= X + 1 = length(active(X)) cons(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(cons(X1,X2)) U11(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(U11(X1,X2)) U12(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(U12(X1,X2)) s(mark(X)) = X + 1 >= X + 1 = mark(s(X)) length(mark(X)) = X + 1 >= X + 1 = mark(length(X)) proper(zeros()) = 0 >= 0 = ok(zeros()) proper(cons(X1,X2)) = 0 >= 0 = cons(proper(X1),proper(X2)) proper(0()) = 0 >= 0 = ok(0()) proper(U11(X1,X2)) = 0 >= 0 = U11(proper(X1),proper(X2)) proper(tt()) = 0 >= 0 = ok(tt()) proper(U12(X1,X2)) = 0 >= 0 = U12(proper(X1),proper(X2)) proper(s(X)) = 0 >= 0 = s(proper(X)) proper(length(X)) = 0 >= 0 = length(proper(X)) proper(nil()) = 0 >= 0 = ok(nil()) cons(ok(X1),ok(X2)) = 0 >= 0 = ok(cons(X1,X2)) U11(ok(X1),ok(X2)) = 0 >= 0 = ok(U11(X1,X2)) U12(ok(X1),ok(X2)) = 0 >= 0 = ok(U12(X1,X2)) s(ok(X)) = 0 >= 0 = ok(s(X)) length(ok(X)) = 0 >= 0 = ok(length(X)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) problem: DPs: TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),L)) -> mark(U12(tt(),L)) active(U12(tt(),L)) -> mark(s(length(L))) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U11(tt(),L)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X1,X2)) -> U11(active(X1),X2) active(U12(X1,X2)) -> U12(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X1),X2) -> mark(U11(X1,X2)) U12(mark(X1),X2) -> mark(U12(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(U12(X1,X2)) -> U12(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(nil()) -> ok(nil()) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) U12(ok(X1),ok(X2)) -> ok(U12(X1,X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed DPs: U12#(ok(X1),ok(X2)) -> U12#(X1,X2) U12#(mark(X1),X2) -> U12#(X1,X2) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),L)) -> mark(U12(tt(),L)) active(U12(tt(),L)) -> mark(s(length(L))) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U11(tt(),L)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X1,X2)) -> U11(active(X1),X2) active(U12(X1,X2)) -> U12(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X1),X2) -> mark(U11(X1,X2)) U12(mark(X1),X2) -> mark(U12(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(U12(X1,X2)) -> U12(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(nil()) -> ok(nil()) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) U12(ok(X1),ok(X2)) -> ok(U12(X1,X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dimension: 1 interpretation: [U12#](x0, x1) = x1 + 1, [top](x0) = 0, [ok](x0) = x0 + 1, [proper](x0) = x0 + 1, [nil] = 0, [s](x0) = x0, [length](x0) = x0, [U12](x0, x1) = x1, [U11](x0, x1) = x1, [tt] = 0, [mark](x0) = x0, [cons](x0, x1) = x1, [0] = 0, [active](x0) = x0, [zeros] = 1 orientation: U12#(ok(X1),ok(X2)) = X2 + 2 >= X2 + 1 = U12#(X1,X2) U12#(mark(X1),X2) = X2 + 1 >= X2 + 1 = U12#(X1,X2) active(zeros()) = 1 >= 1 = mark(cons(0(),zeros())) active(U11(tt(),L)) = L >= L = mark(U12(tt(),L)) active(U12(tt(),L)) = L >= L = mark(s(length(L))) active(length(nil())) = 0 >= 0 = mark(0()) active(length(cons(N,L))) = L >= L = mark(U11(tt(),L)) active(cons(X1,X2)) = X2 >= X2 = cons(active(X1),X2) active(U11(X1,X2)) = X2 >= X2 = U11(active(X1),X2) active(U12(X1,X2)) = X2 >= X2 = U12(active(X1),X2) active(s(X)) = X >= X = s(active(X)) active(length(X)) = X >= X = length(active(X)) cons(mark(X1),X2) = X2 >= X2 = mark(cons(X1,X2)) U11(mark(X1),X2) = X2 >= X2 = mark(U11(X1,X2)) U12(mark(X1),X2) = X2 >= X2 = mark(U12(X1,X2)) s(mark(X)) = X >= X = mark(s(X)) length(mark(X)) = X >= X = mark(length(X)) proper(zeros()) = 2 >= 2 = ok(zeros()) proper(cons(X1,X2)) = X2 + 1 >= X2 + 1 = cons(proper(X1),proper(X2)) proper(0()) = 1 >= 1 = ok(0()) proper(U11(X1,X2)) = X2 + 1 >= X2 + 1 = U11(proper(X1),proper(X2)) proper(tt()) = 1 >= 1 = ok(tt()) proper(U12(X1,X2)) = X2 + 1 >= X2 + 1 = U12(proper(X1),proper(X2)) proper(s(X)) = X + 1 >= X + 1 = s(proper(X)) proper(length(X)) = X + 1 >= X + 1 = length(proper(X)) proper(nil()) = 1 >= 1 = ok(nil()) cons(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(cons(X1,X2)) U11(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(U11(X1,X2)) U12(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(U12(X1,X2)) s(ok(X)) = X + 1 >= X + 1 = ok(s(X)) length(ok(X)) = X + 1 >= X + 1 = ok(length(X)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) problem: DPs: U12#(mark(X1),X2) -> U12#(X1,X2) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),L)) -> mark(U12(tt(),L)) active(U12(tt(),L)) -> mark(s(length(L))) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U11(tt(),L)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X1,X2)) -> U11(active(X1),X2) active(U12(X1,X2)) -> U12(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X1),X2) -> mark(U11(X1,X2)) U12(mark(X1),X2) -> mark(U12(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(U12(X1,X2)) -> U12(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(nil()) -> ok(nil()) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) U12(ok(X1),ok(X2)) -> ok(U12(X1,X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dimension: 1 interpretation: [U12#](x0, x1) = x0 + 1, [top](x0) = 0, [ok](x0) = 0, [proper](x0) = 0, [nil] = 0, [s](x0) = x0, [length](x0) = x0, [U12](x0, x1) = x0 + x1, [U11](x0, x1) = x0 + x1, [tt] = 0, [mark](x0) = x0 + 1, [cons](x0, x1) = x0 + x1, [0] = 0, [active](x0) = x0 + 1, [zeros] = 0 orientation: U12#(mark(X1),X2) = X1 + 2 >= X1 + 1 = U12#(X1,X2) active(zeros()) = 1 >= 1 = mark(cons(0(),zeros())) active(U11(tt(),L)) = L + 1 >= L + 1 = mark(U12(tt(),L)) active(U12(tt(),L)) = L + 1 >= L + 1 = mark(s(length(L))) active(length(nil())) = 1 >= 1 = mark(0()) active(length(cons(N,L))) = L + N + 1 >= L + 1 = mark(U11(tt(),L)) active(cons(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = cons(active(X1),X2) active(U11(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = U11(active(X1),X2) active(U12(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = U12(active(X1),X2) active(s(X)) = X + 1 >= X + 1 = s(active(X)) active(length(X)) = X + 1 >= X + 1 = length(active(X)) cons(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(cons(X1,X2)) U11(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(U11(X1,X2)) U12(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(U12(X1,X2)) s(mark(X)) = X + 1 >= X + 1 = mark(s(X)) length(mark(X)) = X + 1 >= X + 1 = mark(length(X)) proper(zeros()) = 0 >= 0 = ok(zeros()) proper(cons(X1,X2)) = 0 >= 0 = cons(proper(X1),proper(X2)) proper(0()) = 0 >= 0 = ok(0()) proper(U11(X1,X2)) = 0 >= 0 = U11(proper(X1),proper(X2)) proper(tt()) = 0 >= 0 = ok(tt()) proper(U12(X1,X2)) = 0 >= 0 = U12(proper(X1),proper(X2)) proper(s(X)) = 0 >= 0 = s(proper(X)) proper(length(X)) = 0 >= 0 = length(proper(X)) proper(nil()) = 0 >= 0 = ok(nil()) cons(ok(X1),ok(X2)) = 0 >= 0 = ok(cons(X1,X2)) U11(ok(X1),ok(X2)) = 0 >= 0 = ok(U11(X1,X2)) U12(ok(X1),ok(X2)) = 0 >= 0 = ok(U12(X1,X2)) s(ok(X)) = 0 >= 0 = ok(s(X)) length(ok(X)) = 0 >= 0 = ok(length(X)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) problem: DPs: TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),L)) -> mark(U12(tt(),L)) active(U12(tt(),L)) -> mark(s(length(L))) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U11(tt(),L)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X1,X2)) -> U11(active(X1),X2) active(U12(X1,X2)) -> U12(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X1),X2) -> mark(U11(X1,X2)) U12(mark(X1),X2) -> mark(U12(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(U12(X1,X2)) -> U12(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(nil()) -> ok(nil()) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) U12(ok(X1),ok(X2)) -> ok(U12(X1,X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed DPs: s#(ok(X)) -> s#(X) s#(mark(X)) -> s#(X) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),L)) -> mark(U12(tt(),L)) active(U12(tt(),L)) -> mark(s(length(L))) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U11(tt(),L)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X1,X2)) -> U11(active(X1),X2) active(U12(X1,X2)) -> U12(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X1),X2) -> mark(U11(X1,X2)) U12(mark(X1),X2) -> mark(U12(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(U12(X1,X2)) -> U12(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(nil()) -> ok(nil()) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) U12(ok(X1),ok(X2)) -> ok(U12(X1,X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dimension: 1 interpretation: [s#](x0) = x0, [top](x0) = 0, [ok](x0) = x0 + 1, [proper](x0) = 1, [nil] = 0, [s](x0) = x0, [length](x0) = x0, [U12](x0, x1) = x1, [U11](x0, x1) = x1, [tt] = 0, [mark](x0) = x0, [cons](x0, x1) = x1, [0] = 0, [active](x0) = x0, [zeros] = 0 orientation: s#(ok(X)) = X + 1 >= X = s#(X) s#(mark(X)) = X >= X = s#(X) active(zeros()) = 0 >= 0 = mark(cons(0(),zeros())) active(U11(tt(),L)) = L >= L = mark(U12(tt(),L)) active(U12(tt(),L)) = L >= L = mark(s(length(L))) active(length(nil())) = 0 >= 0 = mark(0()) active(length(cons(N,L))) = L >= L = mark(U11(tt(),L)) active(cons(X1,X2)) = X2 >= X2 = cons(active(X1),X2) active(U11(X1,X2)) = X2 >= X2 = U11(active(X1),X2) active(U12(X1,X2)) = X2 >= X2 = U12(active(X1),X2) active(s(X)) = X >= X = s(active(X)) active(length(X)) = X >= X = length(active(X)) cons(mark(X1),X2) = X2 >= X2 = mark(cons(X1,X2)) U11(mark(X1),X2) = X2 >= X2 = mark(U11(X1,X2)) U12(mark(X1),X2) = X2 >= X2 = mark(U12(X1,X2)) s(mark(X)) = X >= X = mark(s(X)) length(mark(X)) = X >= X = mark(length(X)) proper(zeros()) = 1 >= 1 = ok(zeros()) proper(cons(X1,X2)) = 1 >= 1 = cons(proper(X1),proper(X2)) proper(0()) = 1 >= 1 = ok(0()) proper(U11(X1,X2)) = 1 >= 1 = U11(proper(X1),proper(X2)) proper(tt()) = 1 >= 1 = ok(tt()) proper(U12(X1,X2)) = 1 >= 1 = U12(proper(X1),proper(X2)) proper(s(X)) = 1 >= 1 = s(proper(X)) proper(length(X)) = 1 >= 1 = length(proper(X)) proper(nil()) = 1 >= 1 = ok(nil()) cons(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(cons(X1,X2)) U11(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(U11(X1,X2)) U12(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(U12(X1,X2)) s(ok(X)) = X + 1 >= X + 1 = ok(s(X)) length(ok(X)) = X + 1 >= X + 1 = ok(length(X)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) problem: DPs: s#(mark(X)) -> s#(X) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),L)) -> mark(U12(tt(),L)) active(U12(tt(),L)) -> mark(s(length(L))) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U11(tt(),L)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X1,X2)) -> U11(active(X1),X2) active(U12(X1,X2)) -> U12(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X1),X2) -> mark(U11(X1,X2)) U12(mark(X1),X2) -> mark(U12(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(U12(X1,X2)) -> U12(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(nil()) -> ok(nil()) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) U12(ok(X1),ok(X2)) -> ok(U12(X1,X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dimension: 1 interpretation: [s#](x0) = x0 + 1, [top](x0) = 1, [ok](x0) = 0, [proper](x0) = 0, [nil] = 0, [s](x0) = x0, [length](x0) = x0, [U12](x0, x1) = x0 + x1, [U11](x0, x1) = x0 + x1, [tt] = 0, [mark](x0) = x0 + 1, [cons](x0, x1) = x0 + x1, [0] = 0, [active](x0) = x0 + 1, [zeros] = 0 orientation: s#(mark(X)) = X + 2 >= X + 1 = s#(X) active(zeros()) = 1 >= 1 = mark(cons(0(),zeros())) active(U11(tt(),L)) = L + 1 >= L + 1 = mark(U12(tt(),L)) active(U12(tt(),L)) = L + 1 >= L + 1 = mark(s(length(L))) active(length(nil())) = 1 >= 1 = mark(0()) active(length(cons(N,L))) = L + N + 1 >= L + 1 = mark(U11(tt(),L)) active(cons(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = cons(active(X1),X2) active(U11(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = U11(active(X1),X2) active(U12(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = U12(active(X1),X2) active(s(X)) = X + 1 >= X + 1 = s(active(X)) active(length(X)) = X + 1 >= X + 1 = length(active(X)) cons(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(cons(X1,X2)) U11(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(U11(X1,X2)) U12(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(U12(X1,X2)) s(mark(X)) = X + 1 >= X + 1 = mark(s(X)) length(mark(X)) = X + 1 >= X + 1 = mark(length(X)) proper(zeros()) = 0 >= 0 = ok(zeros()) proper(cons(X1,X2)) = 0 >= 0 = cons(proper(X1),proper(X2)) proper(0()) = 0 >= 0 = ok(0()) proper(U11(X1,X2)) = 0 >= 0 = U11(proper(X1),proper(X2)) proper(tt()) = 0 >= 0 = ok(tt()) proper(U12(X1,X2)) = 0 >= 0 = U12(proper(X1),proper(X2)) proper(s(X)) = 0 >= 0 = s(proper(X)) proper(length(X)) = 0 >= 0 = length(proper(X)) proper(nil()) = 0 >= 0 = ok(nil()) cons(ok(X1),ok(X2)) = 0 >= 0 = ok(cons(X1,X2)) U11(ok(X1),ok(X2)) = 0 >= 0 = ok(U11(X1,X2)) U12(ok(X1),ok(X2)) = 0 >= 0 = ok(U12(X1,X2)) s(ok(X)) = 0 >= 0 = ok(s(X)) length(ok(X)) = 0 >= 0 = ok(length(X)) top(mark(X)) = 1 >= 1 = top(proper(X)) top(ok(X)) = 1 >= 1 = top(active(X)) problem: DPs: TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),L)) -> mark(U12(tt(),L)) active(U12(tt(),L)) -> mark(s(length(L))) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U11(tt(),L)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X1,X2)) -> U11(active(X1),X2) active(U12(X1,X2)) -> U12(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X1),X2) -> mark(U11(X1,X2)) U12(mark(X1),X2) -> mark(U12(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(U12(X1,X2)) -> U12(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(nil()) -> ok(nil()) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) U12(ok(X1),ok(X2)) -> ok(U12(X1,X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed DPs: length#(ok(X)) -> length#(X) length#(mark(X)) -> length#(X) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),L)) -> mark(U12(tt(),L)) active(U12(tt(),L)) -> mark(s(length(L))) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U11(tt(),L)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X1,X2)) -> U11(active(X1),X2) active(U12(X1,X2)) -> U12(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X1),X2) -> mark(U11(X1,X2)) U12(mark(X1),X2) -> mark(U12(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(U12(X1,X2)) -> U12(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(nil()) -> ok(nil()) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) U12(ok(X1),ok(X2)) -> ok(U12(X1,X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dimension: 1 interpretation: [length#](x0) = x0, [top](x0) = 0, [ok](x0) = x0 + 1, [proper](x0) = 1, [nil] = 0, [s](x0) = x0, [length](x0) = x0, [U12](x0, x1) = x1, [U11](x0, x1) = x1, [tt] = 0, [mark](x0) = x0, [cons](x0, x1) = x1, [0] = 0, [active](x0) = x0, [zeros] = 0 orientation: length#(ok(X)) = X + 1 >= X = length#(X) length#(mark(X)) = X >= X = length#(X) active(zeros()) = 0 >= 0 = mark(cons(0(),zeros())) active(U11(tt(),L)) = L >= L = mark(U12(tt(),L)) active(U12(tt(),L)) = L >= L = mark(s(length(L))) active(length(nil())) = 0 >= 0 = mark(0()) active(length(cons(N,L))) = L >= L = mark(U11(tt(),L)) active(cons(X1,X2)) = X2 >= X2 = cons(active(X1),X2) active(U11(X1,X2)) = X2 >= X2 = U11(active(X1),X2) active(U12(X1,X2)) = X2 >= X2 = U12(active(X1),X2) active(s(X)) = X >= X = s(active(X)) active(length(X)) = X >= X = length(active(X)) cons(mark(X1),X2) = X2 >= X2 = mark(cons(X1,X2)) U11(mark(X1),X2) = X2 >= X2 = mark(U11(X1,X2)) U12(mark(X1),X2) = X2 >= X2 = mark(U12(X1,X2)) s(mark(X)) = X >= X = mark(s(X)) length(mark(X)) = X >= X = mark(length(X)) proper(zeros()) = 1 >= 1 = ok(zeros()) proper(cons(X1,X2)) = 1 >= 1 = cons(proper(X1),proper(X2)) proper(0()) = 1 >= 1 = ok(0()) proper(U11(X1,X2)) = 1 >= 1 = U11(proper(X1),proper(X2)) proper(tt()) = 1 >= 1 = ok(tt()) proper(U12(X1,X2)) = 1 >= 1 = U12(proper(X1),proper(X2)) proper(s(X)) = 1 >= 1 = s(proper(X)) proper(length(X)) = 1 >= 1 = length(proper(X)) proper(nil()) = 1 >= 1 = ok(nil()) cons(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(cons(X1,X2)) U11(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(U11(X1,X2)) U12(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(U12(X1,X2)) s(ok(X)) = X + 1 >= X + 1 = ok(s(X)) length(ok(X)) = X + 1 >= X + 1 = ok(length(X)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) problem: DPs: length#(mark(X)) -> length#(X) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),L)) -> mark(U12(tt(),L)) active(U12(tt(),L)) -> mark(s(length(L))) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U11(tt(),L)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X1,X2)) -> U11(active(X1),X2) active(U12(X1,X2)) -> U12(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X1),X2) -> mark(U11(X1,X2)) U12(mark(X1),X2) -> mark(U12(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(U12(X1,X2)) -> U12(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(nil()) -> ok(nil()) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) U12(ok(X1),ok(X2)) -> ok(U12(X1,X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dimension: 1 interpretation: [length#](x0) = x0 + 1, [top](x0) = 1, [ok](x0) = 0, [proper](x0) = 0, [nil] = 0, [s](x0) = x0, [length](x0) = x0, [U12](x0, x1) = x0 + x1, [U11](x0, x1) = x0 + x1, [tt] = 0, [mark](x0) = x0 + 1, [cons](x0, x1) = x0 + x1, [0] = 0, [active](x0) = x0 + 1, [zeros] = 0 orientation: length#(mark(X)) = X + 2 >= X + 1 = length#(X) active(zeros()) = 1 >= 1 = mark(cons(0(),zeros())) active(U11(tt(),L)) = L + 1 >= L + 1 = mark(U12(tt(),L)) active(U12(tt(),L)) = L + 1 >= L + 1 = mark(s(length(L))) active(length(nil())) = 1 >= 1 = mark(0()) active(length(cons(N,L))) = L + N + 1 >= L + 1 = mark(U11(tt(),L)) active(cons(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = cons(active(X1),X2) active(U11(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = U11(active(X1),X2) active(U12(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = U12(active(X1),X2) active(s(X)) = X + 1 >= X + 1 = s(active(X)) active(length(X)) = X + 1 >= X + 1 = length(active(X)) cons(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(cons(X1,X2)) U11(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(U11(X1,X2)) U12(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(U12(X1,X2)) s(mark(X)) = X + 1 >= X + 1 = mark(s(X)) length(mark(X)) = X + 1 >= X + 1 = mark(length(X)) proper(zeros()) = 0 >= 0 = ok(zeros()) proper(cons(X1,X2)) = 0 >= 0 = cons(proper(X1),proper(X2)) proper(0()) = 0 >= 0 = ok(0()) proper(U11(X1,X2)) = 0 >= 0 = U11(proper(X1),proper(X2)) proper(tt()) = 0 >= 0 = ok(tt()) proper(U12(X1,X2)) = 0 >= 0 = U12(proper(X1),proper(X2)) proper(s(X)) = 0 >= 0 = s(proper(X)) proper(length(X)) = 0 >= 0 = length(proper(X)) proper(nil()) = 0 >= 0 = ok(nil()) cons(ok(X1),ok(X2)) = 0 >= 0 = ok(cons(X1,X2)) U11(ok(X1),ok(X2)) = 0 >= 0 = ok(U11(X1,X2)) U12(ok(X1),ok(X2)) = 0 >= 0 = ok(U12(X1,X2)) s(ok(X)) = 0 >= 0 = ok(s(X)) length(ok(X)) = 0 >= 0 = ok(length(X)) top(mark(X)) = 1 >= 1 = top(proper(X)) top(ok(X)) = 1 >= 1 = top(active(X)) problem: DPs: TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),L)) -> mark(U12(tt(),L)) active(U12(tt(),L)) -> mark(s(length(L))) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U11(tt(),L)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X1,X2)) -> U11(active(X1),X2) active(U12(X1,X2)) -> U12(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X1),X2) -> mark(U11(X1,X2)) U12(mark(X1),X2) -> mark(U12(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(U12(X1,X2)) -> U12(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(nil()) -> ok(nil()) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) U12(ok(X1),ok(X2)) -> ok(U12(X1,X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed