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: Matrix Interpretation Processor: dim=3 interpretation: [1 1 0] [top](x0) = [0 0 0]x0 [0 0 0] , [1 1 0] [ok](x0) = [0 0 0]x0 [0 0 1] , [1 1 0] [proper](x0) = [0 0 0]x0 [0 0 0] , [1] [nil] = [0] [0], [1 1 0] [s](x0) = [0 0 0]x0 [0 0 0] , [1 1 0] [length](x0) = [0 0 0]x0 [0 0 0] , [1 0 0] [1 1 0] [U12](x0, x1) = [0 1 0]x0 + [0 0 0]x1 [0 0 0] [0 0 0] , [1 0 0] [1 1 0] [U11](x0, x1) = [0 1 0]x0 + [0 0 0]x1 [0 0 0] [0 0 0] , [0] [tt] = [0] [0], [1 0 0] [mark](x0) = [0 1 0]x0 [0 0 0] , [1 0 0] [1 1 1] [cons](x0, x1) = [0 1 0]x0 + [0 0 0]x1 [0 0 0] [0 0 0] , [0] [0] = [0] [0], [1 1 0] [active](x0) = [0 0 0]x0 [1 0 0] , [0] [zeros] = [1] [0] orientation: [1] [1] active(zeros()) = [0] >= [0] = mark(cons(0(),zeros())) [0] [0] [1 1 0] [1 1 0] active(U11(tt(),L)) = [0 0 0]L >= [0 0 0]L = mark(U12(tt(),L)) [1 1 0] [0 0 0] [1 1 0] [1 1 0] active(U12(tt(),L)) = [0 0 0]L >= [0 0 0]L = mark(s(length(L))) [1 1 0] [0 0 0] [1] [0] active(length(nil())) = [0] >= [0] = mark(0()) [1] [0] [1 1 1] [1 1 0] [1 1 0] active(length(cons(N,L))) = [0 0 0]L + [0 0 0]N >= [0 0 0]L = mark(U11(tt(),L)) [1 1 1] [1 1 0] [0 0 0] [1 1 0] [1 1 1] [1 1 0] [1 1 1] active(cons(X1,X2)) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = cons(active(X1),X2) [1 0 0] [1 1 1] [0 0 0] [0 0 0] [1 1 0] [1 1 0] [1 1 0] [1 1 0] active(U11(X1,X2)) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = U11(active(X1),X2) [1 0 0] [1 1 0] [0 0 0] [0 0 0] [1 1 0] [1 1 0] [1 1 0] [1 1 0] active(U12(X1,X2)) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = U12(active(X1),X2) [1 0 0] [1 1 0] [0 0 0] [0 0 0] [1 1 0] [1 1 0] active(s(X)) = [0 0 0]X >= [0 0 0]X = s(active(X)) [1 1 0] [0 0 0] [1 1 0] [1 1 0] active(length(X)) = [0 0 0]X >= [0 0 0]X = length(active(X)) [1 1 0] [0 0 0] [1 0 0] [1 1 1] [1 0 0] [1 1 1] cons(mark(X1),X2) = [0 1 0]X1 + [0 0 0]X2 >= [0 1 0]X1 + [0 0 0]X2 = mark(cons(X1,X2)) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 0 0] [1 1 0] [1 0 0] [1 1 0] U11(mark(X1),X2) = [0 1 0]X1 + [0 0 0]X2 >= [0 1 0]X1 + [0 0 0]X2 = mark(U11(X1,X2)) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 0 0] [1 1 0] [1 0 0] [1 1 0] U12(mark(X1),X2) = [0 1 0]X1 + [0 0 0]X2 >= [0 1 0]X1 + [0 0 0]X2 = mark(U12(X1,X2)) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 1 0] [1 1 0] s(mark(X)) = [0 0 0]X >= [0 0 0]X = mark(s(X)) [0 0 0] [0 0 0] [1 1 0] [1 1 0] length(mark(X)) = [0 0 0]X >= [0 0 0]X = mark(length(X)) [0 0 0] [0 0 0] [1] [1] proper(zeros()) = [0] >= [0] = ok(zeros()) [0] [0] [1 1 0] [1 1 1] [1 1 0] [1 1 0] proper(cons(X1,X2)) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = cons(proper(X1),proper(X2)) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0] [0] proper(0()) = [0] >= [0] = ok(0()) [0] [0] [1 1 0] [1 1 0] [1 1 0] [1 1 0] proper(U11(X1,X2)) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = U11(proper(X1),proper(X2)) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0] [0] proper(tt()) = [0] >= [0] = ok(tt()) [0] [0] [1 1 0] [1 1 0] [1 1 0] [1 1 0] proper(U12(X1,X2)) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = U12(proper(X1),proper(X2)) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 1 0] [1 1 0] proper(s(X)) = [0 0 0]X >= [0 0 0]X = s(proper(X)) [0 0 0] [0 0 0] [1 1 0] [1 1 0] proper(length(X)) = [0 0 0]X >= [0 0 0]X = length(proper(X)) [0 0 0] [0 0 0] [1] [1] proper(nil()) = [0] >= [0] = ok(nil()) [0] [0] [1 1 0] [1 1 1] [1 1 0] [1 1 1] cons(ok(X1),ok(X2)) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = ok(cons(X1,X2)) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 1 0] [1 1 0] [1 1 0] [1 1 0] U11(ok(X1),ok(X2)) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = ok(U11(X1,X2)) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 1 0] [1 1 0] [1 1 0] [1 1 0] U12(ok(X1),ok(X2)) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = ok(U12(X1,X2)) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 1 0] [1 1 0] s(ok(X)) = [0 0 0]X >= [0 0 0]X = ok(s(X)) [0 0 0] [0 0 0] [1 1 0] [1 1 0] length(ok(X)) = [0 0 0]X >= [0 0 0]X = ok(length(X)) [0 0 0] [0 0 0] [1 1 0] [1 1 0] top(mark(X)) = [0 0 0]X >= [0 0 0]X = top(proper(X)) [0 0 0] [0 0 0] [1 1 0] [1 1 0] top(ok(X)) = [0 0 0]X >= [0 0 0]X = top(active(X)) [0 0 0] [0 0 0] 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(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)) 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(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)) TDG 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(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#(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#(length(X)) -> length#(active(X)) top#(ok(X)) -> active#(X) -> active#(length(X)) -> active#(X) 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#(U12(X1,X2)) -> U12#(active(X1),X2) top#(ok(X)) -> active#(X) -> active#(U12(X1,X2)) -> active#(X1) top#(ok(X)) -> active#(X) -> active#(U11(X1,X2)) -> U11#(active(X1),X2) top#(ok(X)) -> active#(X) -> active#(U11(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#(length(cons(N,L))) -> U11#(tt(),L) top#(ok(X)) -> active#(X) -> active#(U12(tt(),L)) -> s#(length(L)) top#(ok(X)) -> active#(X) -> active#(U12(tt(),L)) -> length#(L) top#(ok(X)) -> active#(X) -> active#(U11(tt(),L)) -> U12#(tt(),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#(length(X)) -> length#(proper(X)) top#(mark(X)) -> proper#(X) -> proper#(length(X)) -> proper#(X) 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#(U12(X1,X2)) -> U12#(proper(X1),proper(X2)) top#(mark(X)) -> proper#(X) -> proper#(U12(X1,X2)) -> proper#(X1) top#(mark(X)) -> proper#(X) -> proper#(U12(X1,X2)) -> proper#(X2) top#(mark(X)) -> proper#(X) -> proper#(U11(X1,X2)) -> U11#(proper(X1),proper(X2)) top#(mark(X)) -> proper#(X) -> proper#(U11(X1,X2)) -> proper#(X1) top#(mark(X)) -> proper#(X) -> proper#(U11(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#(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#(s(X)) -> s#(proper(X)) proper#(s(X)) -> proper#(X) -> proper#(s(X)) -> proper#(X) proper#(s(X)) -> proper#(X) -> proper#(U12(X1,X2)) -> U12#(proper(X1),proper(X2)) proper#(s(X)) -> proper#(X) -> proper#(U12(X1,X2)) -> proper#(X1) proper#(s(X)) -> proper#(X) -> proper#(U12(X1,X2)) -> proper#(X2) proper#(s(X)) -> proper#(X) -> proper#(U11(X1,X2)) -> U11#(proper(X1),proper(X2)) proper#(s(X)) -> proper#(X) -> proper#(U11(X1,X2)) -> proper#(X1) proper#(s(X)) -> proper#(X) -> proper#(U11(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#(length(X)) -> length#(proper(X)) proper#(length(X)) -> proper#(X) -> proper#(length(X)) -> proper#(X) 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#(U12(X1,X2)) -> U12#(proper(X1),proper(X2)) proper#(length(X)) -> proper#(X) -> proper#(U12(X1,X2)) -> proper#(X1) proper#(length(X)) -> proper#(X) -> proper#(U12(X1,X2)) -> proper#(X2) proper#(length(X)) -> proper#(X) -> proper#(U11(X1,X2)) -> U11#(proper(X1),proper(X2)) proper#(length(X)) -> proper#(X) -> proper#(U11(X1,X2)) -> proper#(X1) proper#(length(X)) -> proper#(X) -> proper#(U11(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#(U12(X1,X2)) -> proper#(X2) -> proper#(length(X)) -> length#(proper(X)) proper#(U12(X1,X2)) -> proper#(X2) -> proper#(length(X)) -> proper#(X) proper#(U12(X1,X2)) -> proper#(X2) -> proper#(s(X)) -> s#(proper(X)) proper#(U12(X1,X2)) -> proper#(X2) -> proper#(s(X)) -> proper#(X) proper#(U12(X1,X2)) -> proper#(X2) -> proper#(U12(X1,X2)) -> U12#(proper(X1),proper(X2)) proper#(U12(X1,X2)) -> proper#(X2) -> proper#(U12(X1,X2)) -> proper#(X1) proper#(U12(X1,X2)) -> proper#(X2) -> proper#(U12(X1,X2)) -> proper#(X2) proper#(U12(X1,X2)) -> proper#(X2) -> proper#(U11(X1,X2)) -> U11#(proper(X1),proper(X2)) proper#(U12(X1,X2)) -> proper#(X2) -> proper#(U11(X1,X2)) -> proper#(X1) proper#(U12(X1,X2)) -> proper#(X2) -> proper#(U11(X1,X2)) -> proper#(X2) proper#(U12(X1,X2)) -> proper#(X2) -> proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2)) proper#(U12(X1,X2)) -> proper#(X2) -> proper#(cons(X1,X2)) -> proper#(X1) proper#(U12(X1,X2)) -> proper#(X2) -> proper#(cons(X1,X2)) -> proper#(X2) proper#(U12(X1,X2)) -> proper#(X1) -> proper#(length(X)) -> length#(proper(X)) proper#(U12(X1,X2)) -> proper#(X1) -> proper#(length(X)) -> proper#(X) proper#(U12(X1,X2)) -> proper#(X1) -> proper#(s(X)) -> s#(proper(X)) proper#(U12(X1,X2)) -> proper#(X1) -> proper#(s(X)) -> proper#(X) proper#(U12(X1,X2)) -> proper#(X1) -> proper#(U12(X1,X2)) -> U12#(proper(X1),proper(X2)) proper#(U12(X1,X2)) -> proper#(X1) -> proper#(U12(X1,X2)) -> proper#(X1) proper#(U12(X1,X2)) -> proper#(X1) -> proper#(U12(X1,X2)) -> proper#(X2) proper#(U12(X1,X2)) -> proper#(X1) -> proper#(U11(X1,X2)) -> U11#(proper(X1),proper(X2)) proper#(U12(X1,X2)) -> proper#(X1) -> proper#(U11(X1,X2)) -> proper#(X1) proper#(U12(X1,X2)) -> proper#(X1) -> proper#(U11(X1,X2)) -> proper#(X2) proper#(U12(X1,X2)) -> proper#(X1) -> proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2)) proper#(U12(X1,X2)) -> proper#(X1) -> proper#(cons(X1,X2)) -> proper#(X1) proper#(U12(X1,X2)) -> proper#(X1) -> proper#(cons(X1,X2)) -> proper#(X2) proper#(U12(X1,X2)) -> U12#(proper(X1),proper(X2)) -> U12#(ok(X1),ok(X2)) -> U12#(X1,X2) proper#(U12(X1,X2)) -> U12#(proper(X1),proper(X2)) -> U12#(mark(X1),X2) -> U12#(X1,X2) proper#(U11(X1,X2)) -> proper#(X2) -> proper#(length(X)) -> length#(proper(X)) proper#(U11(X1,X2)) -> proper#(X2) -> proper#(length(X)) -> proper#(X) proper#(U11(X1,X2)) -> proper#(X2) -> proper#(s(X)) -> s#(proper(X)) proper#(U11(X1,X2)) -> proper#(X2) -> proper#(s(X)) -> proper#(X) proper#(U11(X1,X2)) -> proper#(X2) -> proper#(U12(X1,X2)) -> U12#(proper(X1),proper(X2)) proper#(U11(X1,X2)) -> proper#(X2) -> proper#(U12(X1,X2)) -> proper#(X1) proper#(U11(X1,X2)) -> proper#(X2) -> proper#(U12(X1,X2)) -> proper#(X2) proper#(U11(X1,X2)) -> proper#(X2) -> proper#(U11(X1,X2)) -> U11#(proper(X1),proper(X2)) proper#(U11(X1,X2)) -> proper#(X2) -> proper#(U11(X1,X2)) -> proper#(X1) proper#(U11(X1,X2)) -> proper#(X2) -> proper#(U11(X1,X2)) -> proper#(X2) proper#(U11(X1,X2)) -> proper#(X2) -> proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2)) proper#(U11(X1,X2)) -> proper#(X2) -> proper#(cons(X1,X2)) -> proper#(X1) proper#(U11(X1,X2)) -> proper#(X2) -> proper#(cons(X1,X2)) -> proper#(X2) proper#(U11(X1,X2)) -> proper#(X1) -> proper#(length(X)) -> length#(proper(X)) proper#(U11(X1,X2)) -> proper#(X1) -> proper#(length(X)) -> proper#(X) proper#(U11(X1,X2)) -> proper#(X1) -> proper#(s(X)) -> s#(proper(X)) proper#(U11(X1,X2)) -> proper#(X1) -> proper#(s(X)) -> proper#(X) proper#(U11(X1,X2)) -> proper#(X1) -> proper#(U12(X1,X2)) -> U12#(proper(X1),proper(X2)) proper#(U11(X1,X2)) -> proper#(X1) -> proper#(U12(X1,X2)) -> proper#(X1) proper#(U11(X1,X2)) -> proper#(X1) -> proper#(U12(X1,X2)) -> proper#(X2) proper#(U11(X1,X2)) -> proper#(X1) -> proper#(U11(X1,X2)) -> U11#(proper(X1),proper(X2)) proper#(U11(X1,X2)) -> proper#(X1) -> proper#(U11(X1,X2)) -> proper#(X1) proper#(U11(X1,X2)) -> proper#(X1) -> proper#(U11(X1,X2)) -> proper#(X2) proper#(U11(X1,X2)) -> proper#(X1) -> proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2)) proper#(U11(X1,X2)) -> proper#(X1) -> proper#(cons(X1,X2)) -> proper#(X1) proper#(U11(X1,X2)) -> proper#(X1) -> proper#(cons(X1,X2)) -> proper#(X2) proper#(U11(X1,X2)) -> U11#(proper(X1),proper(X2)) -> U11#(ok(X1),ok(X2)) -> U11#(X1,X2) proper#(U11(X1,X2)) -> U11#(proper(X1),proper(X2)) -> U11#(mark(X1),X2) -> U11#(X1,X2) 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#(s(X)) -> s#(proper(X)) proper#(cons(X1,X2)) -> proper#(X2) -> proper#(s(X)) -> proper#(X) proper#(cons(X1,X2)) -> proper#(X2) -> proper#(U12(X1,X2)) -> U12#(proper(X1),proper(X2)) proper#(cons(X1,X2)) -> proper#(X2) -> proper#(U12(X1,X2)) -> proper#(X1) proper#(cons(X1,X2)) -> proper#(X2) -> proper#(U12(X1,X2)) -> proper#(X2) proper#(cons(X1,X2)) -> proper#(X2) -> proper#(U11(X1,X2)) -> U11#(proper(X1),proper(X2)) proper#(cons(X1,X2)) -> proper#(X2) -> proper#(U11(X1,X2)) -> proper#(X1) proper#(cons(X1,X2)) -> proper#(X2) -> proper#(U11(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#(length(X)) -> length#(proper(X)) proper#(cons(X1,X2)) -> proper#(X1) -> proper#(length(X)) -> proper#(X) 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#(U12(X1,X2)) -> U12#(proper(X1),proper(X2)) proper#(cons(X1,X2)) -> proper#(X1) -> proper#(U12(X1,X2)) -> proper#(X1) proper#(cons(X1,X2)) -> proper#(X1) -> proper#(U12(X1,X2)) -> proper#(X2) proper#(cons(X1,X2)) -> proper#(X1) -> proper#(U11(X1,X2)) -> U11#(proper(X1),proper(X2)) proper#(cons(X1,X2)) -> proper#(X1) -> proper#(U11(X1,X2)) -> proper#(X1) proper#(cons(X1,X2)) -> proper#(X1) -> proper#(U11(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) U11#(ok(X1),ok(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#(ok(X1),ok(X2)) -> U11#(X1,X2) U11#(mark(X1),X2) -> U11#(X1,X2) -> U11#(mark(X1),X2) -> U11#(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) U12#(ok(X1),ok(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#(ok(X1),ok(X2)) -> U12#(X1,X2) U12#(mark(X1),X2) -> U12#(X1,X2) -> U12#(mark(X1),X2) -> U12#(X1,X2) 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#(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#(length(X)) -> length#(active(X)) active#(s(X)) -> active#(X) -> active#(length(X)) -> active#(X) 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#(U12(X1,X2)) -> U12#(active(X1),X2) active#(s(X)) -> active#(X) -> active#(U12(X1,X2)) -> active#(X1) active#(s(X)) -> active#(X) -> active#(U11(X1,X2)) -> U11#(active(X1),X2) active#(s(X)) -> active#(X) -> active#(U11(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#(length(cons(N,L))) -> U11#(tt(),L) active#(s(X)) -> active#(X) -> active#(U12(tt(),L)) -> s#(length(L)) active#(s(X)) -> active#(X) -> active#(U12(tt(),L)) -> length#(L) active#(s(X)) -> active#(X) -> active#(U11(tt(),L)) -> U12#(tt(),L) active#(s(X)) -> active#(X) -> active#(zeros()) -> cons#(0(),zeros()) active#(length(cons(N,L))) -> U11#(tt(),L) -> U11#(ok(X1),ok(X2)) -> U11#(X1,X2) active#(length(cons(N,L))) -> U11#(tt(),L) -> U11#(mark(X1),X2) -> U11#(X1,X2) 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#(length(X)) -> length#(active(X)) active#(length(X)) -> active#(X) -> active#(length(X)) -> active#(X) 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#(U12(X1,X2)) -> U12#(active(X1),X2) active#(length(X)) -> active#(X) -> active#(U12(X1,X2)) -> active#(X1) active#(length(X)) -> active#(X) -> active#(U11(X1,X2)) -> U11#(active(X1),X2) active#(length(X)) -> active#(X) -> active#(U11(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#(length(cons(N,L))) -> U11#(tt(),L) active#(length(X)) -> active#(X) -> active#(U12(tt(),L)) -> s#(length(L)) active#(length(X)) -> active#(X) -> active#(U12(tt(),L)) -> length#(L) active#(length(X)) -> active#(X) -> active#(U11(tt(),L)) -> U12#(tt(),L) active#(length(X)) -> active#(X) -> active#(zeros()) -> cons#(0(),zeros()) active#(U12(tt(),L)) -> s#(length(L)) -> s#(ok(X)) -> s#(X) active#(U12(tt(),L)) -> s#(length(L)) -> s#(mark(X)) -> s#(X) active#(U12(tt(),L)) -> length#(L) -> length#(ok(X)) -> length#(X) active#(U12(tt(),L)) -> length#(L) -> length#(mark(X)) -> length#(X) active#(U12(X1,X2)) -> U12#(active(X1),X2) -> U12#(ok(X1),ok(X2)) -> U12#(X1,X2) active#(U12(X1,X2)) -> U12#(active(X1),X2) -> U12#(mark(X1),X2) -> U12#(X1,X2) active#(U12(X1,X2)) -> active#(X1) -> active#(length(X)) -> length#(active(X)) active#(U12(X1,X2)) -> active#(X1) -> active#(length(X)) -> active#(X) active#(U12(X1,X2)) -> active#(X1) -> active#(s(X)) -> s#(active(X)) active#(U12(X1,X2)) -> active#(X1) -> active#(s(X)) -> active#(X) active#(U12(X1,X2)) -> active#(X1) -> active#(U12(X1,X2)) -> U12#(active(X1),X2) active#(U12(X1,X2)) -> active#(X1) -> active#(U12(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#(U11(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#(cons(X1,X2)) -> active#(X1) active#(U12(X1,X2)) -> active#(X1) -> active#(length(cons(N,L))) -> U11#(tt(),L) active#(U12(X1,X2)) -> active#(X1) -> active#(U12(tt(),L)) -> s#(length(L)) active#(U12(X1,X2)) -> active#(X1) -> active#(U12(tt(),L)) -> length#(L) active#(U12(X1,X2)) -> active#(X1) -> active#(U11(tt(),L)) -> U12#(tt(),L) active#(U12(X1,X2)) -> active#(X1) -> active#(zeros()) -> cons#(0(),zeros()) active#(U11(tt(),L)) -> U12#(tt(),L) -> U12#(ok(X1),ok(X2)) -> U12#(X1,X2) active#(U11(tt(),L)) -> U12#(tt(),L) -> U12#(mark(X1),X2) -> U12#(X1,X2) active#(U11(X1,X2)) -> U11#(active(X1),X2) -> U11#(ok(X1),ok(X2)) -> U11#(X1,X2) active#(U11(X1,X2)) -> U11#(active(X1),X2) -> U11#(mark(X1),X2) -> U11#(X1,X2) active#(U11(X1,X2)) -> active#(X1) -> active#(length(X)) -> length#(active(X)) active#(U11(X1,X2)) -> active#(X1) -> active#(length(X)) -> active#(X) active#(U11(X1,X2)) -> active#(X1) -> active#(s(X)) -> s#(active(X)) active#(U11(X1,X2)) -> active#(X1) -> active#(s(X)) -> active#(X) active#(U11(X1,X2)) -> active#(X1) -> active#(U12(X1,X2)) -> U12#(active(X1),X2) active#(U11(X1,X2)) -> active#(X1) -> active#(U12(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#(U11(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#(cons(X1,X2)) -> active#(X1) active#(U11(X1,X2)) -> active#(X1) -> active#(length(cons(N,L))) -> U11#(tt(),L) active#(U11(X1,X2)) -> active#(X1) -> active#(U12(tt(),L)) -> s#(length(L)) active#(U11(X1,X2)) -> active#(X1) -> active#(U12(tt(),L)) -> length#(L) active#(U11(X1,X2)) -> active#(X1) -> active#(U11(tt(),L)) -> U12#(tt(),L) active#(U11(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#(length(X)) -> length#(active(X)) active#(cons(X1,X2)) -> active#(X1) -> active#(length(X)) -> active#(X) 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#(U12(X1,X2)) -> U12#(active(X1),X2) active#(cons(X1,X2)) -> active#(X1) -> active#(U12(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#(U11(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#(length(cons(N,L))) -> U11#(tt(),L) active#(cons(X1,X2)) -> active#(X1) -> active#(U12(tt(),L)) -> s#(length(L)) active#(cons(X1,X2)) -> active#(X1) -> active#(U12(tt(),L)) -> length#(L) active#(cons(X1,X2)) -> active#(X1) -> active#(U11(tt(),L)) -> U12#(tt(),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) SCC Processor: #sccs: 8 #rules: 25 #arcs: 265/1764 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(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: active#(cons(X1,X2)) -> active#(X1) active#(U11(X1,X2)) -> active#(X1) active#(U12(X1,X2)) -> active#(X1) active#(s(X)) -> active#(X) active#(length(X)) -> 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(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)) Subterm Criterion Processor: simple projection: pi(active#) = 0 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(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: proper#(cons(X1,X2)) -> proper#(X2) proper#(cons(X1,X2)) -> proper#(X1) proper#(U11(X1,X2)) -> proper#(X2) proper#(U11(X1,X2)) -> proper#(X1) proper#(U12(X1,X2)) -> proper#(X2) proper#(U12(X1,X2)) -> proper#(X1) proper#(s(X)) -> proper#(X) proper#(length(X)) -> 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(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)) Subterm Criterion Processor: simple projection: pi(proper#) = 0 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(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#(mark(X)) -> length#(X) length#(ok(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(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)) Subterm Criterion Processor: simple projection: pi(length#) = 0 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(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#(mark(X)) -> s#(X) s#(ok(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(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)) Subterm Criterion Processor: simple projection: pi(s#) = 0 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(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#(mark(X1),X2) -> U12#(X1,X2) U12#(ok(X1),ok(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(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)) Subterm Criterion Processor: simple projection: pi(U12#) = 1 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(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)) Subterm Criterion Processor: simple projection: pi(U12#) = 0 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(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#(mark(X1),X2) -> U11#(X1,X2) U11#(ok(X1),ok(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(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)) Subterm Criterion Processor: simple projection: pi(U11#) = 1 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(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)) Subterm Criterion Processor: simple projection: pi(U11#) = 0 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(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#(mark(X1),X2) -> cons#(X1,X2) cons#(ok(X1),ok(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(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)) Subterm Criterion Processor: simple projection: pi(cons#) = 1 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(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)) Subterm Criterion Processor: simple projection: pi(cons#) = 0 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(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