YES Problem: active(eq(0(),0())) -> mark(true()) active(eq(s(X),s(Y))) -> mark(eq(X,Y)) active(eq(X,Y)) -> mark(false()) active(inf(X)) -> mark(cons(X,inf(s(X)))) active(take(0(),X)) -> mark(nil()) active(take(s(X),cons(Y,L))) -> mark(cons(Y,take(X,L))) active(length(nil())) -> mark(0()) active(length(cons(X,L))) -> mark(s(length(L))) active(inf(X)) -> inf(active(X)) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) active(length(X)) -> length(active(X)) inf(mark(X)) -> mark(inf(X)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) length(mark(X)) -> mark(length(X)) proper(eq(X1,X2)) -> eq(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(true()) -> ok(true()) proper(s(X)) -> s(proper(X)) proper(false()) -> ok(false()) proper(inf(X)) -> inf(proper(X)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) proper(nil()) -> ok(nil()) proper(length(X)) -> length(proper(X)) eq(ok(X1),ok(X2)) -> ok(eq(X1,X2)) s(ok(X)) -> ok(s(X)) inf(ok(X)) -> ok(inf(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) length(ok(X)) -> ok(length(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Proof: DP Processor: DPs: active#(eq(s(X),s(Y))) -> eq#(X,Y) active#(inf(X)) -> s#(X) active#(inf(X)) -> inf#(s(X)) active#(inf(X)) -> cons#(X,inf(s(X))) active#(take(s(X),cons(Y,L))) -> take#(X,L) active#(take(s(X),cons(Y,L))) -> cons#(Y,take(X,L)) active#(length(cons(X,L))) -> length#(L) active#(length(cons(X,L))) -> s#(length(L)) active#(inf(X)) -> active#(X) active#(inf(X)) -> inf#(active(X)) active#(take(X1,X2)) -> active#(X1) active#(take(X1,X2)) -> take#(active(X1),X2) active#(take(X1,X2)) -> active#(X2) active#(take(X1,X2)) -> take#(X1,active(X2)) active#(length(X)) -> active#(X) active#(length(X)) -> length#(active(X)) inf#(mark(X)) -> inf#(X) take#(mark(X1),X2) -> take#(X1,X2) take#(X1,mark(X2)) -> take#(X1,X2) length#(mark(X)) -> length#(X) proper#(eq(X1,X2)) -> proper#(X2) proper#(eq(X1,X2)) -> proper#(X1) proper#(eq(X1,X2)) -> eq#(proper(X1),proper(X2)) proper#(s(X)) -> proper#(X) proper#(s(X)) -> s#(proper(X)) proper#(inf(X)) -> proper#(X) proper#(inf(X)) -> inf#(proper(X)) proper#(cons(X1,X2)) -> proper#(X2) proper#(cons(X1,X2)) -> proper#(X1) proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2)) proper#(take(X1,X2)) -> proper#(X2) proper#(take(X1,X2)) -> proper#(X1) proper#(take(X1,X2)) -> take#(proper(X1),proper(X2)) proper#(length(X)) -> proper#(X) proper#(length(X)) -> length#(proper(X)) eq#(ok(X1),ok(X2)) -> eq#(X1,X2) s#(ok(X)) -> s#(X) inf#(ok(X)) -> inf#(X) cons#(ok(X1),ok(X2)) -> cons#(X1,X2) take#(ok(X1),ok(X2)) -> take#(X1,X2) 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(eq(0(),0())) -> mark(true()) active(eq(s(X),s(Y))) -> mark(eq(X,Y)) active(eq(X,Y)) -> mark(false()) active(inf(X)) -> mark(cons(X,inf(s(X)))) active(take(0(),X)) -> mark(nil()) active(take(s(X),cons(Y,L))) -> mark(cons(Y,take(X,L))) active(length(nil())) -> mark(0()) active(length(cons(X,L))) -> mark(s(length(L))) active(inf(X)) -> inf(active(X)) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) active(length(X)) -> length(active(X)) inf(mark(X)) -> mark(inf(X)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) length(mark(X)) -> mark(length(X)) proper(eq(X1,X2)) -> eq(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(true()) -> ok(true()) proper(s(X)) -> s(proper(X)) proper(false()) -> ok(false()) proper(inf(X)) -> inf(proper(X)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) proper(nil()) -> ok(nil()) proper(length(X)) -> length(proper(X)) eq(ok(X1),ok(X2)) -> ok(eq(X1,X2)) s(ok(X)) -> ok(s(X)) inf(ok(X)) -> ok(inf(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) length(ok(X)) -> ok(length(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) TDG Processor: DPs: active#(eq(s(X),s(Y))) -> eq#(X,Y) active#(inf(X)) -> s#(X) active#(inf(X)) -> inf#(s(X)) active#(inf(X)) -> cons#(X,inf(s(X))) active#(take(s(X),cons(Y,L))) -> take#(X,L) active#(take(s(X),cons(Y,L))) -> cons#(Y,take(X,L)) active#(length(cons(X,L))) -> length#(L) active#(length(cons(X,L))) -> s#(length(L)) active#(inf(X)) -> active#(X) active#(inf(X)) -> inf#(active(X)) active#(take(X1,X2)) -> active#(X1) active#(take(X1,X2)) -> take#(active(X1),X2) active#(take(X1,X2)) -> active#(X2) active#(take(X1,X2)) -> take#(X1,active(X2)) active#(length(X)) -> active#(X) active#(length(X)) -> length#(active(X)) inf#(mark(X)) -> inf#(X) take#(mark(X1),X2) -> take#(X1,X2) take#(X1,mark(X2)) -> take#(X1,X2) length#(mark(X)) -> length#(X) proper#(eq(X1,X2)) -> proper#(X2) proper#(eq(X1,X2)) -> proper#(X1) proper#(eq(X1,X2)) -> eq#(proper(X1),proper(X2)) proper#(s(X)) -> proper#(X) proper#(s(X)) -> s#(proper(X)) proper#(inf(X)) -> proper#(X) proper#(inf(X)) -> inf#(proper(X)) proper#(cons(X1,X2)) -> proper#(X2) proper#(cons(X1,X2)) -> proper#(X1) proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2)) proper#(take(X1,X2)) -> proper#(X2) proper#(take(X1,X2)) -> proper#(X1) proper#(take(X1,X2)) -> take#(proper(X1),proper(X2)) proper#(length(X)) -> proper#(X) proper#(length(X)) -> length#(proper(X)) eq#(ok(X1),ok(X2)) -> eq#(X1,X2) s#(ok(X)) -> s#(X) inf#(ok(X)) -> inf#(X) cons#(ok(X1),ok(X2)) -> cons#(X1,X2) take#(ok(X1),ok(X2)) -> take#(X1,X2) 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(eq(0(),0())) -> mark(true()) active(eq(s(X),s(Y))) -> mark(eq(X,Y)) active(eq(X,Y)) -> mark(false()) active(inf(X)) -> mark(cons(X,inf(s(X)))) active(take(0(),X)) -> mark(nil()) active(take(s(X),cons(Y,L))) -> mark(cons(Y,take(X,L))) active(length(nil())) -> mark(0()) active(length(cons(X,L))) -> mark(s(length(L))) active(inf(X)) -> inf(active(X)) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) active(length(X)) -> length(active(X)) inf(mark(X)) -> mark(inf(X)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) length(mark(X)) -> mark(length(X)) proper(eq(X1,X2)) -> eq(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(true()) -> ok(true()) proper(s(X)) -> s(proper(X)) proper(false()) -> ok(false()) proper(inf(X)) -> inf(proper(X)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) proper(nil()) -> ok(nil()) proper(length(X)) -> length(proper(X)) eq(ok(X1),ok(X2)) -> ok(eq(X1,X2)) s(ok(X)) -> ok(s(X)) inf(ok(X)) -> ok(inf(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) 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#(take(X1,X2)) -> take#(X1,active(X2)) top#(ok(X)) -> active#(X) -> active#(take(X1,X2)) -> active#(X2) top#(ok(X)) -> active#(X) -> active#(take(X1,X2)) -> take#(active(X1),X2) top#(ok(X)) -> active#(X) -> active#(take(X1,X2)) -> active#(X1) top#(ok(X)) -> active#(X) -> active#(inf(X)) -> inf#(active(X)) top#(ok(X)) -> active#(X) -> active#(inf(X)) -> active#(X) top#(ok(X)) -> active#(X) -> active#(length(cons(X,L))) -> s#(length(L)) top#(ok(X)) -> active#(X) -> active#(length(cons(X,L))) -> length#(L) top#(ok(X)) -> active#(X) -> active#(take(s(X),cons(Y,L))) -> cons#(Y,take(X,L)) top#(ok(X)) -> active#(X) -> active#(take(s(X),cons(Y,L))) -> take#(X,L) top#(ok(X)) -> active#(X) -> active#(inf(X)) -> cons#(X,inf(s(X))) top#(ok(X)) -> active#(X) -> active#(inf(X)) -> inf#(s(X)) top#(ok(X)) -> active#(X) -> active#(inf(X)) -> s#(X) top#(ok(X)) -> active#(X) -> active#(eq(s(X),s(Y))) -> eq#(X,Y) 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#(take(X1,X2)) -> take#(proper(X1),proper(X2)) top#(mark(X)) -> proper#(X) -> proper#(take(X1,X2)) -> proper#(X1) top#(mark(X)) -> proper#(X) -> proper#(take(X1,X2)) -> proper#(X2) top#(mark(X)) -> proper#(X) -> proper#(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) top#(mark(X)) -> proper#(X) -> proper#(inf(X)) -> inf#(proper(X)) top#(mark(X)) -> proper#(X) -> proper#(inf(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#(eq(X1,X2)) -> eq#(proper(X1),proper(X2)) top#(mark(X)) -> proper#(X) -> proper#(eq(X1,X2)) -> proper#(X1) top#(mark(X)) -> proper#(X) -> proper#(eq(X1,X2)) -> proper#(X2) 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#(take(X1,X2)) -> take#(proper(X1),proper(X2)) proper#(length(X)) -> proper#(X) -> proper#(take(X1,X2)) -> proper#(X1) proper#(length(X)) -> proper#(X) -> proper#(take(X1,X2)) -> proper#(X2) proper#(length(X)) -> proper#(X) -> proper#(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)) -> proper#(X) -> proper#(inf(X)) -> inf#(proper(X)) proper#(length(X)) -> proper#(X) -> proper#(inf(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#(eq(X1,X2)) -> eq#(proper(X1),proper(X2)) proper#(length(X)) -> proper#(X) -> proper#(eq(X1,X2)) -> proper#(X1) proper#(length(X)) -> proper#(X) -> proper#(eq(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#(take(X1,X2)) -> proper#(X2) -> proper#(length(X)) -> length#(proper(X)) proper#(take(X1,X2)) -> proper#(X2) -> proper#(length(X)) -> proper#(X) proper#(take(X1,X2)) -> proper#(X2) -> proper#(take(X1,X2)) -> take#(proper(X1),proper(X2)) proper#(take(X1,X2)) -> proper#(X2) -> proper#(take(X1,X2)) -> proper#(X1) proper#(take(X1,X2)) -> proper#(X2) -> proper#(take(X1,X2)) -> proper#(X2) proper#(take(X1,X2)) -> proper#(X2) -> proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2)) proper#(take(X1,X2)) -> proper#(X2) -> proper#(cons(X1,X2)) -> proper#(X1) proper#(take(X1,X2)) -> proper#(X2) -> proper#(cons(X1,X2)) -> proper#(X2) proper#(take(X1,X2)) -> proper#(X2) -> proper#(inf(X)) -> inf#(proper(X)) proper#(take(X1,X2)) -> proper#(X2) -> proper#(inf(X)) -> proper#(X) proper#(take(X1,X2)) -> proper#(X2) -> proper#(s(X)) -> s#(proper(X)) proper#(take(X1,X2)) -> proper#(X2) -> proper#(s(X)) -> proper#(X) proper#(take(X1,X2)) -> proper#(X2) -> proper#(eq(X1,X2)) -> eq#(proper(X1),proper(X2)) proper#(take(X1,X2)) -> proper#(X2) -> proper#(eq(X1,X2)) -> proper#(X1) proper#(take(X1,X2)) -> proper#(X2) -> proper#(eq(X1,X2)) -> proper#(X2) proper#(take(X1,X2)) -> proper#(X1) -> proper#(length(X)) -> length#(proper(X)) proper#(take(X1,X2)) -> proper#(X1) -> proper#(length(X)) -> proper#(X) proper#(take(X1,X2)) -> proper#(X1) -> proper#(take(X1,X2)) -> take#(proper(X1),proper(X2)) proper#(take(X1,X2)) -> proper#(X1) -> proper#(take(X1,X2)) -> proper#(X1) proper#(take(X1,X2)) -> proper#(X1) -> proper#(take(X1,X2)) -> proper#(X2) proper#(take(X1,X2)) -> proper#(X1) -> proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2)) proper#(take(X1,X2)) -> proper#(X1) -> proper#(cons(X1,X2)) -> proper#(X1) proper#(take(X1,X2)) -> proper#(X1) -> proper#(cons(X1,X2)) -> proper#(X2) proper#(take(X1,X2)) -> proper#(X1) -> proper#(inf(X)) -> inf#(proper(X)) proper#(take(X1,X2)) -> proper#(X1) -> proper#(inf(X)) -> proper#(X) proper#(take(X1,X2)) -> proper#(X1) -> proper#(s(X)) -> s#(proper(X)) proper#(take(X1,X2)) -> proper#(X1) -> proper#(s(X)) -> proper#(X) proper#(take(X1,X2)) -> proper#(X1) -> proper#(eq(X1,X2)) -> eq#(proper(X1),proper(X2)) proper#(take(X1,X2)) -> proper#(X1) -> proper#(eq(X1,X2)) -> proper#(X1) proper#(take(X1,X2)) -> proper#(X1) -> proper#(eq(X1,X2)) -> proper#(X2) proper#(take(X1,X2)) -> take#(proper(X1),proper(X2)) -> take#(ok(X1),ok(X2)) -> take#(X1,X2) proper#(take(X1,X2)) -> take#(proper(X1),proper(X2)) -> take#(X1,mark(X2)) -> take#(X1,X2) proper#(take(X1,X2)) -> take#(proper(X1),proper(X2)) -> take#(mark(X1),X2) -> take#(X1,X2) proper#(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#(take(X1,X2)) -> take#(proper(X1),proper(X2)) proper#(cons(X1,X2)) -> proper#(X2) -> proper#(take(X1,X2)) -> proper#(X1) proper#(cons(X1,X2)) -> proper#(X2) -> proper#(take(X1,X2)) -> proper#(X2) proper#(cons(X1,X2)) -> proper#(X2) -> proper#(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#(X2) -> proper#(inf(X)) -> inf#(proper(X)) proper#(cons(X1,X2)) -> proper#(X2) -> proper#(inf(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#(eq(X1,X2)) -> eq#(proper(X1),proper(X2)) proper#(cons(X1,X2)) -> proper#(X2) -> proper#(eq(X1,X2)) -> proper#(X1) proper#(cons(X1,X2)) -> proper#(X2) -> proper#(eq(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#(take(X1,X2)) -> take#(proper(X1),proper(X2)) proper#(cons(X1,X2)) -> proper#(X1) -> proper#(take(X1,X2)) -> proper#(X1) proper#(cons(X1,X2)) -> proper#(X1) -> proper#(take(X1,X2)) -> proper#(X2) proper#(cons(X1,X2)) -> proper#(X1) -> proper#(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)) -> proper#(X1) -> proper#(inf(X)) -> inf#(proper(X)) proper#(cons(X1,X2)) -> proper#(X1) -> proper#(inf(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#(eq(X1,X2)) -> eq#(proper(X1),proper(X2)) proper#(cons(X1,X2)) -> proper#(X1) -> proper#(eq(X1,X2)) -> proper#(X1) proper#(cons(X1,X2)) -> proper#(X1) -> proper#(eq(X1,X2)) -> proper#(X2) proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2)) -> cons#(ok(X1),ok(X2)) -> cons#(X1,X2) proper#(inf(X)) -> proper#(X) -> proper#(length(X)) -> length#(proper(X)) proper#(inf(X)) -> proper#(X) -> proper#(length(X)) -> proper#(X) proper#(inf(X)) -> proper#(X) -> proper#(take(X1,X2)) -> take#(proper(X1),proper(X2)) proper#(inf(X)) -> proper#(X) -> proper#(take(X1,X2)) -> proper#(X1) proper#(inf(X)) -> proper#(X) -> proper#(take(X1,X2)) -> proper#(X2) proper#(inf(X)) -> proper#(X) -> proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2)) proper#(inf(X)) -> proper#(X) -> proper#(cons(X1,X2)) -> proper#(X1) proper#(inf(X)) -> proper#(X) -> proper#(cons(X1,X2)) -> proper#(X2) proper#(inf(X)) -> proper#(X) -> proper#(inf(X)) -> inf#(proper(X)) proper#(inf(X)) -> proper#(X) -> proper#(inf(X)) -> proper#(X) proper#(inf(X)) -> proper#(X) -> proper#(s(X)) -> s#(proper(X)) proper#(inf(X)) -> proper#(X) -> proper#(s(X)) -> proper#(X) proper#(inf(X)) -> proper#(X) -> proper#(eq(X1,X2)) -> eq#(proper(X1),proper(X2)) proper#(inf(X)) -> proper#(X) -> proper#(eq(X1,X2)) -> proper#(X1) proper#(inf(X)) -> proper#(X) -> proper#(eq(X1,X2)) -> proper#(X2) proper#(inf(X)) -> inf#(proper(X)) -> inf#(ok(X)) -> inf#(X) proper#(inf(X)) -> inf#(proper(X)) -> inf#(mark(X)) -> inf#(X) proper#(s(X)) -> proper#(X) -> proper#(length(X)) -> length#(proper(X)) proper#(s(X)) -> proper#(X) -> proper#(length(X)) -> proper#(X) proper#(s(X)) -> proper#(X) -> proper#(take(X1,X2)) -> take#(proper(X1),proper(X2)) proper#(s(X)) -> proper#(X) -> proper#(take(X1,X2)) -> proper#(X1) proper#(s(X)) -> proper#(X) -> proper#(take(X1,X2)) -> proper#(X2) proper#(s(X)) -> proper#(X) -> proper#(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)) -> proper#(X) -> proper#(inf(X)) -> inf#(proper(X)) proper#(s(X)) -> proper#(X) -> proper#(inf(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#(eq(X1,X2)) -> eq#(proper(X1),proper(X2)) proper#(s(X)) -> proper#(X) -> proper#(eq(X1,X2)) -> proper#(X1) proper#(s(X)) -> proper#(X) -> proper#(eq(X1,X2)) -> proper#(X2) proper#(s(X)) -> s#(proper(X)) -> s#(ok(X)) -> s#(X) proper#(eq(X1,X2)) -> proper#(X2) -> proper#(length(X)) -> length#(proper(X)) proper#(eq(X1,X2)) -> proper#(X2) -> proper#(length(X)) -> proper#(X) proper#(eq(X1,X2)) -> proper#(X2) -> proper#(take(X1,X2)) -> take#(proper(X1),proper(X2)) proper#(eq(X1,X2)) -> proper#(X2) -> proper#(take(X1,X2)) -> proper#(X1) proper#(eq(X1,X2)) -> proper#(X2) -> proper#(take(X1,X2)) -> proper#(X2) proper#(eq(X1,X2)) -> proper#(X2) -> proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2)) proper#(eq(X1,X2)) -> proper#(X2) -> proper#(cons(X1,X2)) -> proper#(X1) proper#(eq(X1,X2)) -> proper#(X2) -> proper#(cons(X1,X2)) -> proper#(X2) proper#(eq(X1,X2)) -> proper#(X2) -> proper#(inf(X)) -> inf#(proper(X)) proper#(eq(X1,X2)) -> proper#(X2) -> proper#(inf(X)) -> proper#(X) proper#(eq(X1,X2)) -> proper#(X2) -> proper#(s(X)) -> s#(proper(X)) proper#(eq(X1,X2)) -> proper#(X2) -> proper#(s(X)) -> proper#(X) proper#(eq(X1,X2)) -> proper#(X2) -> proper#(eq(X1,X2)) -> eq#(proper(X1),proper(X2)) proper#(eq(X1,X2)) -> proper#(X2) -> proper#(eq(X1,X2)) -> proper#(X1) proper#(eq(X1,X2)) -> proper#(X2) -> proper#(eq(X1,X2)) -> proper#(X2) proper#(eq(X1,X2)) -> proper#(X1) -> proper#(length(X)) -> length#(proper(X)) proper#(eq(X1,X2)) -> proper#(X1) -> proper#(length(X)) -> proper#(X) proper#(eq(X1,X2)) -> proper#(X1) -> proper#(take(X1,X2)) -> take#(proper(X1),proper(X2)) proper#(eq(X1,X2)) -> proper#(X1) -> proper#(take(X1,X2)) -> proper#(X1) proper#(eq(X1,X2)) -> proper#(X1) -> proper#(take(X1,X2)) -> proper#(X2) proper#(eq(X1,X2)) -> proper#(X1) -> proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2)) proper#(eq(X1,X2)) -> proper#(X1) -> proper#(cons(X1,X2)) -> proper#(X1) proper#(eq(X1,X2)) -> proper#(X1) -> proper#(cons(X1,X2)) -> proper#(X2) proper#(eq(X1,X2)) -> proper#(X1) -> proper#(inf(X)) -> inf#(proper(X)) proper#(eq(X1,X2)) -> proper#(X1) -> proper#(inf(X)) -> proper#(X) proper#(eq(X1,X2)) -> proper#(X1) -> proper#(s(X)) -> s#(proper(X)) proper#(eq(X1,X2)) -> proper#(X1) -> proper#(s(X)) -> proper#(X) proper#(eq(X1,X2)) -> proper#(X1) -> proper#(eq(X1,X2)) -> eq#(proper(X1),proper(X2)) proper#(eq(X1,X2)) -> proper#(X1) -> proper#(eq(X1,X2)) -> proper#(X1) proper#(eq(X1,X2)) -> proper#(X1) -> proper#(eq(X1,X2)) -> proper#(X2) proper#(eq(X1,X2)) -> eq#(proper(X1),proper(X2)) -> eq#(ok(X1),ok(X2)) -> eq#(X1,X2) 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) take#(ok(X1),ok(X2)) -> take#(X1,X2) -> take#(ok(X1),ok(X2)) -> take#(X1,X2) take#(ok(X1),ok(X2)) -> take#(X1,X2) -> take#(X1,mark(X2)) -> take#(X1,X2) take#(ok(X1),ok(X2)) -> take#(X1,X2) -> take#(mark(X1),X2) -> take#(X1,X2) take#(mark(X1),X2) -> take#(X1,X2) -> take#(ok(X1),ok(X2)) -> take#(X1,X2) take#(mark(X1),X2) -> take#(X1,X2) -> take#(X1,mark(X2)) -> take#(X1,X2) take#(mark(X1),X2) -> take#(X1,X2) -> take#(mark(X1),X2) -> take#(X1,X2) take#(X1,mark(X2)) -> take#(X1,X2) -> take#(ok(X1),ok(X2)) -> take#(X1,X2) take#(X1,mark(X2)) -> take#(X1,X2) -> take#(X1,mark(X2)) -> take#(X1,X2) take#(X1,mark(X2)) -> take#(X1,X2) -> take#(mark(X1),X2) -> take#(X1,X2) cons#(ok(X1),ok(X2)) -> cons#(X1,X2) -> cons#(ok(X1),ok(X2)) -> cons#(X1,X2) inf#(ok(X)) -> inf#(X) -> inf#(ok(X)) -> inf#(X) inf#(ok(X)) -> inf#(X) -> inf#(mark(X)) -> inf#(X) inf#(mark(X)) -> inf#(X) -> inf#(ok(X)) -> inf#(X) inf#(mark(X)) -> inf#(X) -> inf#(mark(X)) -> inf#(X) s#(ok(X)) -> s#(X) -> s#(ok(X)) -> s#(X) eq#(ok(X1),ok(X2)) -> eq#(X1,X2) -> eq#(ok(X1),ok(X2)) -> eq#(X1,X2) active#(length(cons(X,L))) -> length#(L) -> length#(ok(X)) -> length#(X) active#(length(cons(X,L))) -> length#(L) -> length#(mark(X)) -> length#(X) active#(length(cons(X,L))) -> s#(length(L)) -> s#(ok(X)) -> s#(X) active#(length(X)) -> length#(active(X)) -> length#(ok(X)) -> length#(X) active#(length(X)) -> length#(active(X)) -> length#(mark(X)) -> length#(X) active#(length(X)) -> active#(X) -> active#(length(X)) -> length#(active(X)) active#(length(X)) -> active#(X) -> active#(length(X)) -> active#(X) active#(length(X)) -> active#(X) -> active#(take(X1,X2)) -> take#(X1,active(X2)) active#(length(X)) -> active#(X) -> active#(take(X1,X2)) -> active#(X2) active#(length(X)) -> active#(X) -> active#(take(X1,X2)) -> take#(active(X1),X2) active#(length(X)) -> active#(X) -> active#(take(X1,X2)) -> active#(X1) active#(length(X)) -> active#(X) -> active#(inf(X)) -> inf#(active(X)) active#(length(X)) -> active#(X) -> active#(inf(X)) -> active#(X) active#(length(X)) -> active#(X) -> active#(length(cons(X,L))) -> s#(length(L)) active#(length(X)) -> active#(X) -> active#(length(cons(X,L))) -> length#(L) active#(length(X)) -> active#(X) -> active#(take(s(X),cons(Y,L))) -> cons#(Y,take(X,L)) active#(length(X)) -> active#(X) -> active#(take(s(X),cons(Y,L))) -> take#(X,L) active#(length(X)) -> active#(X) -> active#(inf(X)) -> cons#(X,inf(s(X))) active#(length(X)) -> active#(X) -> active#(inf(X)) -> inf#(s(X)) active#(length(X)) -> active#(X) -> active#(inf(X)) -> s#(X) active#(length(X)) -> active#(X) -> active#(eq(s(X),s(Y))) -> eq#(X,Y) active#(take(s(X),cons(Y,L))) -> take#(X,L) -> take#(ok(X1),ok(X2)) -> take#(X1,X2) active#(take(s(X),cons(Y,L))) -> take#(X,L) -> take#(X1,mark(X2)) -> take#(X1,X2) active#(take(s(X),cons(Y,L))) -> take#(X,L) -> take#(mark(X1),X2) -> take#(X1,X2) active#(take(s(X),cons(Y,L))) -> cons#(Y,take(X,L)) -> cons#(ok(X1),ok(X2)) -> cons#(X1,X2) active#(take(X1,X2)) -> take#(active(X1),X2) -> take#(ok(X1),ok(X2)) -> take#(X1,X2) active#(take(X1,X2)) -> take#(active(X1),X2) -> take#(X1,mark(X2)) -> take#(X1,X2) active#(take(X1,X2)) -> take#(active(X1),X2) -> take#(mark(X1),X2) -> take#(X1,X2) active#(take(X1,X2)) -> take#(X1,active(X2)) -> take#(ok(X1),ok(X2)) -> take#(X1,X2) active#(take(X1,X2)) -> take#(X1,active(X2)) -> take#(X1,mark(X2)) -> take#(X1,X2) active#(take(X1,X2)) -> take#(X1,active(X2)) -> take#(mark(X1),X2) -> take#(X1,X2) active#(take(X1,X2)) -> active#(X2) -> active#(length(X)) -> length#(active(X)) active#(take(X1,X2)) -> active#(X2) -> active#(length(X)) -> active#(X) active#(take(X1,X2)) -> active#(X2) -> active#(take(X1,X2)) -> take#(X1,active(X2)) active#(take(X1,X2)) -> active#(X2) -> active#(take(X1,X2)) -> active#(X2) active#(take(X1,X2)) -> active#(X2) -> active#(take(X1,X2)) -> take#(active(X1),X2) active#(take(X1,X2)) -> active#(X2) -> active#(take(X1,X2)) -> active#(X1) active#(take(X1,X2)) -> active#(X2) -> active#(inf(X)) -> inf#(active(X)) active#(take(X1,X2)) -> active#(X2) -> active#(inf(X)) -> active#(X) active#(take(X1,X2)) -> active#(X2) -> active#(length(cons(X,L))) -> s#(length(L)) active#(take(X1,X2)) -> active#(X2) -> active#(length(cons(X,L))) -> length#(L) active#(take(X1,X2)) -> active#(X2) -> active#(take(s(X),cons(Y,L))) -> cons#(Y,take(X,L)) active#(take(X1,X2)) -> active#(X2) -> active#(take(s(X),cons(Y,L))) -> take#(X,L) active#(take(X1,X2)) -> active#(X2) -> active#(inf(X)) -> cons#(X,inf(s(X))) active#(take(X1,X2)) -> active#(X2) -> active#(inf(X)) -> inf#(s(X)) active#(take(X1,X2)) -> active#(X2) -> active#(inf(X)) -> s#(X) active#(take(X1,X2)) -> active#(X2) -> active#(eq(s(X),s(Y))) -> eq#(X,Y) active#(take(X1,X2)) -> active#(X1) -> active#(length(X)) -> length#(active(X)) active#(take(X1,X2)) -> active#(X1) -> active#(length(X)) -> active#(X) active#(take(X1,X2)) -> active#(X1) -> active#(take(X1,X2)) -> take#(X1,active(X2)) active#(take(X1,X2)) -> active#(X1) -> active#(take(X1,X2)) -> active#(X2) active#(take(X1,X2)) -> active#(X1) -> active#(take(X1,X2)) -> take#(active(X1),X2) active#(take(X1,X2)) -> active#(X1) -> active#(take(X1,X2)) -> active#(X1) active#(take(X1,X2)) -> active#(X1) -> active#(inf(X)) -> inf#(active(X)) active#(take(X1,X2)) -> active#(X1) -> active#(inf(X)) -> active#(X) active#(take(X1,X2)) -> active#(X1) -> active#(length(cons(X,L))) -> s#(length(L)) active#(take(X1,X2)) -> active#(X1) -> active#(length(cons(X,L))) -> length#(L) active#(take(X1,X2)) -> active#(X1) -> active#(take(s(X),cons(Y,L))) -> cons#(Y,take(X,L)) active#(take(X1,X2)) -> active#(X1) -> active#(take(s(X),cons(Y,L))) -> take#(X,L) active#(take(X1,X2)) -> active#(X1) -> active#(inf(X)) -> cons#(X,inf(s(X))) active#(take(X1,X2)) -> active#(X1) -> active#(inf(X)) -> inf#(s(X)) active#(take(X1,X2)) -> active#(X1) -> active#(inf(X)) -> s#(X) active#(take(X1,X2)) -> active#(X1) -> active#(eq(s(X),s(Y))) -> eq#(X,Y) active#(inf(X)) -> cons#(X,inf(s(X))) -> cons#(ok(X1),ok(X2)) -> cons#(X1,X2) active#(inf(X)) -> inf#(s(X)) -> inf#(ok(X)) -> inf#(X) active#(inf(X)) -> inf#(s(X)) -> inf#(mark(X)) -> inf#(X) active#(inf(X)) -> inf#(active(X)) -> inf#(ok(X)) -> inf#(X) active#(inf(X)) -> inf#(active(X)) -> inf#(mark(X)) -> inf#(X) active#(inf(X)) -> s#(X) -> s#(ok(X)) -> s#(X) active#(inf(X)) -> active#(X) -> active#(length(X)) -> length#(active(X)) active#(inf(X)) -> active#(X) -> active#(length(X)) -> active#(X) active#(inf(X)) -> active#(X) -> active#(take(X1,X2)) -> take#(X1,active(X2)) active#(inf(X)) -> active#(X) -> active#(take(X1,X2)) -> active#(X2) active#(inf(X)) -> active#(X) -> active#(take(X1,X2)) -> take#(active(X1),X2) active#(inf(X)) -> active#(X) -> active#(take(X1,X2)) -> active#(X1) active#(inf(X)) -> active#(X) -> active#(inf(X)) -> inf#(active(X)) active#(inf(X)) -> active#(X) -> active#(inf(X)) -> active#(X) active#(inf(X)) -> active#(X) -> active#(length(cons(X,L))) -> s#(length(L)) active#(inf(X)) -> active#(X) -> active#(length(cons(X,L))) -> length#(L) active#(inf(X)) -> active#(X) -> active#(take(s(X),cons(Y,L))) -> cons#(Y,take(X,L)) active#(inf(X)) -> active#(X) -> active#(take(s(X),cons(Y,L))) -> take#(X,L) active#(inf(X)) -> active#(X) -> active#(inf(X)) -> cons#(X,inf(s(X))) active#(inf(X)) -> active#(X) -> active#(inf(X)) -> inf#(s(X)) active#(inf(X)) -> active#(X) -> active#(inf(X)) -> s#(X) active#(inf(X)) -> active#(X) -> active#(eq(s(X),s(Y))) -> eq#(X,Y) active#(eq(s(X),s(Y))) -> eq#(X,Y) -> eq#(ok(X1),ok(X2)) -> eq#(X1,X2) SCC Processor: #sccs: 9 #rules: 25 #arcs: 290/2025 DPs: top#(ok(X)) -> top#(active(X)) top#(mark(X)) -> top#(proper(X)) TRS: active(eq(0(),0())) -> mark(true()) active(eq(s(X),s(Y))) -> mark(eq(X,Y)) active(eq(X,Y)) -> mark(false()) active(inf(X)) -> mark(cons(X,inf(s(X)))) active(take(0(),X)) -> mark(nil()) active(take(s(X),cons(Y,L))) -> mark(cons(Y,take(X,L))) active(length(nil())) -> mark(0()) active(length(cons(X,L))) -> mark(s(length(L))) active(inf(X)) -> inf(active(X)) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) active(length(X)) -> length(active(X)) inf(mark(X)) -> mark(inf(X)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) length(mark(X)) -> mark(length(X)) proper(eq(X1,X2)) -> eq(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(true()) -> ok(true()) proper(s(X)) -> s(proper(X)) proper(false()) -> ok(false()) proper(inf(X)) -> inf(proper(X)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) proper(nil()) -> ok(nil()) proper(length(X)) -> length(proper(X)) eq(ok(X1),ok(X2)) -> ok(eq(X1,X2)) s(ok(X)) -> ok(s(X)) inf(ok(X)) -> ok(inf(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) length(ok(X)) -> ok(length(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Usable Rule Processor: DPs: top#(ok(X)) -> top#(active(X)) top#(mark(X)) -> top#(proper(X)) TRS: active(eq(0(),0())) -> mark(true()) active(eq(s(X),s(Y))) -> mark(eq(X,Y)) active(eq(X,Y)) -> mark(false()) active(inf(X)) -> mark(cons(X,inf(s(X)))) active(take(0(),X)) -> mark(nil()) active(take(s(X),cons(Y,L))) -> mark(cons(Y,take(X,L))) active(length(nil())) -> mark(0()) active(length(cons(X,L))) -> mark(s(length(L))) active(inf(X)) -> inf(active(X)) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) active(length(X)) -> length(active(X)) eq(ok(X1),ok(X2)) -> ok(eq(X1,X2)) s(ok(X)) -> ok(s(X)) inf(mark(X)) -> mark(inf(X)) inf(ok(X)) -> ok(inf(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) length(mark(X)) -> mark(length(X)) length(ok(X)) -> ok(length(X)) proper(eq(X1,X2)) -> eq(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(true()) -> ok(true()) proper(s(X)) -> s(proper(X)) proper(false()) -> ok(false()) proper(inf(X)) -> inf(proper(X)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) proper(nil()) -> ok(nil()) proper(length(X)) -> length(proper(X)) Matrix Interpretation Processor: dim=3 usable rules: active(eq(0(),0())) -> mark(true()) active(eq(s(X),s(Y))) -> mark(eq(X,Y)) active(eq(X,Y)) -> mark(false()) active(inf(X)) -> mark(cons(X,inf(s(X)))) active(take(0(),X)) -> mark(nil()) active(take(s(X),cons(Y,L))) -> mark(cons(Y,take(X,L))) active(length(nil())) -> mark(0()) active(length(cons(X,L))) -> mark(s(length(L))) active(inf(X)) -> inf(active(X)) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) active(length(X)) -> length(active(X)) eq(ok(X1),ok(X2)) -> ok(eq(X1,X2)) s(ok(X)) -> ok(s(X)) inf(mark(X)) -> mark(inf(X)) inf(ok(X)) -> ok(inf(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) length(mark(X)) -> mark(length(X)) length(ok(X)) -> ok(length(X)) proper(eq(X1,X2)) -> eq(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(true()) -> ok(true()) proper(s(X)) -> s(proper(X)) proper(false()) -> ok(false()) proper(inf(X)) -> inf(proper(X)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) proper(nil()) -> ok(nil()) proper(length(X)) -> length(proper(X)) interpretation: [top#](x0) = [0 0 1]x0, [0 0 0] [ok](x0) = [0 1 0]x0 [0 0 1] , [0 0 0] [proper](x0) = [0 1 0]x0 [0 0 1] , [0 0 0] [0] [length](x0) = [0 0 0]x0 + [0] [0 0 1] [1], [0] [nil] = [0] [0], [0 0 0] [0 0 0] [0] [take](x0, x1) = [0 0 0]x0 + [0 0 0]x1 + [0] [0 0 1] [0 0 1] [1], [0] [cons](x0, x1) = [0] [0], [0 0 0] [0] [inf](x0) = [0 0 0]x0 + [0] [0 0 1] [1], [0] [false] = [1] [0], [0 0 0] [0] [s](x0) = [0 1 0]x0 + [1] [0 0 0] [0], [0 0 0] [0] [mark](x0) = [0 0 0]x0 + [0] [0 0 1] [1], [0] [true] = [0] [0], [0 0 0] [active](x0) = [0 0 0]x0 [0 0 1] , [0 0 0] [0] [eq](x0, x1) = [0 0 0]x1 + [0] [0 1 0] [1], [0] [0] = [0] [0] orientation: top#(ok(X)) = [0 0 1]X >= [0 0 1]X = top#(active(X)) top#(mark(X)) = [0 0 1]X + [1] >= [0 0 1]X = top#(proper(X)) [0] [0] active(eq(0(),0())) = [0] >= [0] = mark(true()) [1] [1] [0 0 0] [0] [0 0 0] [0] active(eq(s(X),s(Y))) = [0 0 0]Y + [0] >= [0 0 0]Y + [0] = mark(eq(X,Y)) [0 1 0] [2] [0 1 0] [2] [0 0 0] [0] [0] active(eq(X,Y)) = [0 0 0]Y + [0] >= [0] = mark(false()) [0 1 0] [1] [1] [0 0 0] [0] [0] active(inf(X)) = [0 0 0]X + [0] >= [0] = mark(cons(X,inf(s(X)))) [0 0 1] [1] [1] [0 0 0] [0] [0] active(take(0(),X)) = [0 0 0]X + [0] >= [0] = mark(nil()) [0 0 1] [1] [1] [0] [0] active(take(s(X),cons(Y,L))) = [0] >= [0] = mark(cons(Y,take(X,L))) [1] [1] [0] [0] active(length(nil())) = [0] >= [0] = mark(0()) [1] [1] [0] [0] active(length(cons(X,L))) = [0] >= [0] = mark(s(length(L))) [1] [1] [0 0 0] [0] [0 0 0] [0] active(inf(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = inf(active(X)) [0 0 1] [1] [0 0 1] [1] [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [0] active(take(X1,X2)) = [0 0 0]X1 + [0 0 0]X2 + [0] >= [0 0 0]X1 + [0 0 0]X2 + [0] = take(active(X1),X2) [0 0 1] [0 0 1] [1] [0 0 1] [0 0 1] [1] [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [0] active(take(X1,X2)) = [0 0 0]X1 + [0 0 0]X2 + [0] >= [0 0 0]X1 + [0 0 0]X2 + [0] = take(X1,active(X2)) [0 0 1] [0 0 1] [1] [0 0 1] [0 0 1] [1] [0 0 0] [0] [0 0 0] [0] active(length(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = length(active(X)) [0 0 1] [1] [0 0 1] [1] [0 0 0] [0] [0 0 0] [0] eq(ok(X1),ok(X2)) = [0 0 0]X2 + [0] >= [0 0 0]X2 + [0] = ok(eq(X1,X2)) [0 1 0] [1] [0 1 0] [1] [0 0 0] [0] [0 0 0] [0] s(ok(X)) = [0 1 0]X + [1] >= [0 1 0]X + [1] = ok(s(X)) [0 0 0] [0] [0 0 0] [0] [0 0 0] [0] [0 0 0] [0] inf(mark(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = mark(inf(X)) [0 0 1] [2] [0 0 1] [2] [0 0 0] [0] [0 0 0] [0] inf(ok(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = ok(inf(X)) [0 0 1] [1] [0 0 1] [1] [0] [0] cons(ok(X1),ok(X2)) = [0] >= [0] = ok(cons(X1,X2)) [0] [0] [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [0] take(mark(X1),X2) = [0 0 0]X1 + [0 0 0]X2 + [0] >= [0 0 0]X1 + [0 0 0]X2 + [0] = mark(take(X1,X2)) [0 0 1] [0 0 1] [2] [0 0 1] [0 0 1] [2] [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [0] take(X1,mark(X2)) = [0 0 0]X1 + [0 0 0]X2 + [0] >= [0 0 0]X1 + [0 0 0]X2 + [0] = mark(take(X1,X2)) [0 0 1] [0 0 1] [2] [0 0 1] [0 0 1] [2] [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [0] take(ok(X1),ok(X2)) = [0 0 0]X1 + [0 0 0]X2 + [0] >= [0 0 0]X1 + [0 0 0]X2 + [0] = ok(take(X1,X2)) [0 0 1] [0 0 1] [1] [0 0 1] [0 0 1] [1] [0 0 0] [0] [0 0 0] [0] length(mark(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = mark(length(X)) [0 0 1] [2] [0 0 1] [2] [0 0 0] [0] [0 0 0] [0] length(ok(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = ok(length(X)) [0 0 1] [1] [0 0 1] [1] [0 0 0] [0] [0 0 0] [0] proper(eq(X1,X2)) = [0 0 0]X2 + [0] >= [0 0 0]X2 + [0] = eq(proper(X1),proper(X2)) [0 1 0] [1] [0 1 0] [1] [0] [0] proper(0()) = [0] >= [0] = ok(0()) [0] [0] [0] [0] proper(true()) = [0] >= [0] = ok(true()) [0] [0] [0 0 0] [0] [0 0 0] [0] proper(s(X)) = [0 1 0]X + [1] >= [0 1 0]X + [1] = s(proper(X)) [0 0 0] [0] [0 0 0] [0] [0] [0] proper(false()) = [1] >= [1] = ok(false()) [0] [0] [0 0 0] [0] [0 0 0] [0] proper(inf(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = inf(proper(X)) [0 0 1] [1] [0 0 1] [1] [0] [0] proper(cons(X1,X2)) = [0] >= [0] = cons(proper(X1),proper(X2)) [0] [0] [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [0] proper(take(X1,X2)) = [0 0 0]X1 + [0 0 0]X2 + [0] >= [0 0 0]X1 + [0 0 0]X2 + [0] = take(proper(X1),proper(X2)) [0 0 1] [0 0 1] [1] [0 0 1] [0 0 1] [1] [0] [0] proper(nil()) = [0] >= [0] = ok(nil()) [0] [0] [0 0 0] [0] [0 0 0] [0] proper(length(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = length(proper(X)) [0 0 1] [1] [0 0 1] [1] problem: DPs: top#(ok(X)) -> top#(active(X)) TRS: active(eq(0(),0())) -> mark(true()) active(eq(s(X),s(Y))) -> mark(eq(X,Y)) active(eq(X,Y)) -> mark(false()) active(inf(X)) -> mark(cons(X,inf(s(X)))) active(take(0(),X)) -> mark(nil()) active(take(s(X),cons(Y,L))) -> mark(cons(Y,take(X,L))) active(length(nil())) -> mark(0()) active(length(cons(X,L))) -> mark(s(length(L))) active(inf(X)) -> inf(active(X)) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) active(length(X)) -> length(active(X)) eq(ok(X1),ok(X2)) -> ok(eq(X1,X2)) s(ok(X)) -> ok(s(X)) inf(mark(X)) -> mark(inf(X)) inf(ok(X)) -> ok(inf(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) length(mark(X)) -> mark(length(X)) length(ok(X)) -> ok(length(X)) proper(eq(X1,X2)) -> eq(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(true()) -> ok(true()) proper(s(X)) -> s(proper(X)) proper(false()) -> ok(false()) proper(inf(X)) -> inf(proper(X)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) proper(nil()) -> ok(nil()) proper(length(X)) -> length(proper(X)) Restore Modifier: DPs: top#(ok(X)) -> top#(active(X)) TRS: active(eq(0(),0())) -> mark(true()) active(eq(s(X),s(Y))) -> mark(eq(X,Y)) active(eq(X,Y)) -> mark(false()) active(inf(X)) -> mark(cons(X,inf(s(X)))) active(take(0(),X)) -> mark(nil()) active(take(s(X),cons(Y,L))) -> mark(cons(Y,take(X,L))) active(length(nil())) -> mark(0()) active(length(cons(X,L))) -> mark(s(length(L))) active(inf(X)) -> inf(active(X)) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) active(length(X)) -> length(active(X)) inf(mark(X)) -> mark(inf(X)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) length(mark(X)) -> mark(length(X)) proper(eq(X1,X2)) -> eq(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(true()) -> ok(true()) proper(s(X)) -> s(proper(X)) proper(false()) -> ok(false()) proper(inf(X)) -> inf(proper(X)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) proper(nil()) -> ok(nil()) proper(length(X)) -> length(proper(X)) eq(ok(X1),ok(X2)) -> ok(eq(X1,X2)) s(ok(X)) -> ok(s(X)) inf(ok(X)) -> ok(inf(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) length(ok(X)) -> ok(length(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Usable Rule Processor: DPs: top#(ok(X)) -> top#(active(X)) TRS: active(eq(0(),0())) -> mark(true()) active(eq(s(X),s(Y))) -> mark(eq(X,Y)) active(eq(X,Y)) -> mark(false()) active(inf(X)) -> mark(cons(X,inf(s(X)))) active(take(0(),X)) -> mark(nil()) active(take(s(X),cons(Y,L))) -> mark(cons(Y,take(X,L))) active(length(nil())) -> mark(0()) active(length(cons(X,L))) -> mark(s(length(L))) active(inf(X)) -> inf(active(X)) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) active(length(X)) -> length(active(X)) eq(ok(X1),ok(X2)) -> ok(eq(X1,X2)) s(ok(X)) -> ok(s(X)) inf(mark(X)) -> mark(inf(X)) inf(ok(X)) -> ok(inf(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) length(mark(X)) -> mark(length(X)) length(ok(X)) -> ok(length(X)) Arctic Interpretation Processor: dimension: 1 usable rules: active(eq(0(),0())) -> mark(true()) active(eq(s(X),s(Y))) -> mark(eq(X,Y)) active(eq(X,Y)) -> mark(false()) active(inf(X)) -> mark(cons(X,inf(s(X)))) active(take(0(),X)) -> mark(nil()) active(take(s(X),cons(Y,L))) -> mark(cons(Y,take(X,L))) active(length(nil())) -> mark(0()) active(length(cons(X,L))) -> mark(s(length(L))) active(inf(X)) -> inf(active(X)) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) active(length(X)) -> length(active(X)) eq(ok(X1),ok(X2)) -> ok(eq(X1,X2)) s(ok(X)) -> ok(s(X)) inf(mark(X)) -> mark(inf(X)) inf(ok(X)) -> ok(inf(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) length(mark(X)) -> mark(length(X)) length(ok(X)) -> ok(length(X)) interpretation: [top#](x0) = x0, [ok](x0) = 1x0 + 0, [length](x0) = 2x0 + 0, [nil] = 0, [take](x0, x1) = x0 + x1, [cons](x0, x1) = x0 + x1, [inf](x0) = x0, [false] = 0, [s](x0) = x0, [mark](x0) = x0, [true] = 5, [active](x0) = x0, [eq](x0, x1) = x0 + 4x1 + 0, [0] = 2 orientation: top#(ok(X)) = 1X + 0 >= X = top#(active(X)) active(eq(0(),0())) = 6 >= 5 = mark(true()) active(eq(s(X),s(Y))) = X + 4Y + 0 >= X + 4Y + 0 = mark(eq(X,Y)) active(eq(X,Y)) = X + 4Y + 0 >= 0 = mark(false()) active(inf(X)) = X >= X = mark(cons(X,inf(s(X)))) active(take(0(),X)) = X + 2 >= 0 = mark(nil()) active(take(s(X),cons(Y,L))) = L + X + Y >= L + X + Y = mark(cons(Y,take(X,L))) active(length(nil())) = 2 >= 2 = mark(0()) active(length(cons(X,L))) = 2L + 2X + 0 >= 2L + 0 = mark(s(length(L))) active(inf(X)) = X >= X = inf(active(X)) active(take(X1,X2)) = X1 + X2 >= X1 + X2 = take(active(X1),X2) active(take(X1,X2)) = X1 + X2 >= X1 + X2 = take(X1,active(X2)) active(length(X)) = 2X + 0 >= 2X + 0 = length(active(X)) eq(ok(X1),ok(X2)) = 1X1 + 5X2 + 4 >= 1X1 + 5X2 + 1 = ok(eq(X1,X2)) s(ok(X)) = 1X + 0 >= 1X + 0 = ok(s(X)) inf(mark(X)) = X >= X = mark(inf(X)) inf(ok(X)) = 1X + 0 >= 1X + 0 = ok(inf(X)) cons(ok(X1),ok(X2)) = 1X1 + 1X2 + 0 >= 1X1 + 1X2 + 0 = ok(cons(X1,X2)) take(mark(X1),X2) = X1 + X2 >= X1 + X2 = mark(take(X1,X2)) take(X1,mark(X2)) = X1 + X2 >= X1 + X2 = mark(take(X1,X2)) take(ok(X1),ok(X2)) = 1X1 + 1X2 + 0 >= 1X1 + 1X2 + 0 = ok(take(X1,X2)) length(mark(X)) = 2X + 0 >= 2X + 0 = mark(length(X)) length(ok(X)) = 3X + 2 >= 3X + 1 = ok(length(X)) problem: DPs: TRS: active(eq(0(),0())) -> mark(true()) active(eq(s(X),s(Y))) -> mark(eq(X,Y)) active(eq(X,Y)) -> mark(false()) active(inf(X)) -> mark(cons(X,inf(s(X)))) active(take(0(),X)) -> mark(nil()) active(take(s(X),cons(Y,L))) -> mark(cons(Y,take(X,L))) active(length(nil())) -> mark(0()) active(length(cons(X,L))) -> mark(s(length(L))) active(inf(X)) -> inf(active(X)) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) active(length(X)) -> length(active(X)) eq(ok(X1),ok(X2)) -> ok(eq(X1,X2)) s(ok(X)) -> ok(s(X)) inf(mark(X)) -> mark(inf(X)) inf(ok(X)) -> ok(inf(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) length(mark(X)) -> mark(length(X)) length(ok(X)) -> ok(length(X)) Qed DPs: active#(inf(X)) -> active#(X) active#(take(X1,X2)) -> active#(X1) active#(take(X1,X2)) -> active#(X2) active#(length(X)) -> active#(X) TRS: active(eq(0(),0())) -> mark(true()) active(eq(s(X),s(Y))) -> mark(eq(X,Y)) active(eq(X,Y)) -> mark(false()) active(inf(X)) -> mark(cons(X,inf(s(X)))) active(take(0(),X)) -> mark(nil()) active(take(s(X),cons(Y,L))) -> mark(cons(Y,take(X,L))) active(length(nil())) -> mark(0()) active(length(cons(X,L))) -> mark(s(length(L))) active(inf(X)) -> inf(active(X)) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) active(length(X)) -> length(active(X)) inf(mark(X)) -> mark(inf(X)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) length(mark(X)) -> mark(length(X)) proper(eq(X1,X2)) -> eq(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(true()) -> ok(true()) proper(s(X)) -> s(proper(X)) proper(false()) -> ok(false()) proper(inf(X)) -> inf(proper(X)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) proper(nil()) -> ok(nil()) proper(length(X)) -> length(proper(X)) eq(ok(X1),ok(X2)) -> ok(eq(X1,X2)) s(ok(X)) -> ok(s(X)) inf(ok(X)) -> ok(inf(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) 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(eq(0(),0())) -> mark(true()) active(eq(s(X),s(Y))) -> mark(eq(X,Y)) active(eq(X,Y)) -> mark(false()) active(inf(X)) -> mark(cons(X,inf(s(X)))) active(take(0(),X)) -> mark(nil()) active(take(s(X),cons(Y,L))) -> mark(cons(Y,take(X,L))) active(length(nil())) -> mark(0()) active(length(cons(X,L))) -> mark(s(length(L))) active(inf(X)) -> inf(active(X)) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) active(length(X)) -> length(active(X)) inf(mark(X)) -> mark(inf(X)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) length(mark(X)) -> mark(length(X)) proper(eq(X1,X2)) -> eq(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(true()) -> ok(true()) proper(s(X)) -> s(proper(X)) proper(false()) -> ok(false()) proper(inf(X)) -> inf(proper(X)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) proper(nil()) -> ok(nil()) proper(length(X)) -> length(proper(X)) eq(ok(X1),ok(X2)) -> ok(eq(X1,X2)) s(ok(X)) -> ok(s(X)) inf(ok(X)) -> ok(inf(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) length(ok(X)) -> ok(length(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed DPs: proper#(eq(X1,X2)) -> proper#(X2) proper#(eq(X1,X2)) -> proper#(X1) proper#(s(X)) -> proper#(X) proper#(inf(X)) -> proper#(X) proper#(cons(X1,X2)) -> proper#(X2) proper#(cons(X1,X2)) -> proper#(X1) proper#(take(X1,X2)) -> proper#(X2) proper#(take(X1,X2)) -> proper#(X1) proper#(length(X)) -> proper#(X) TRS: active(eq(0(),0())) -> mark(true()) active(eq(s(X),s(Y))) -> mark(eq(X,Y)) active(eq(X,Y)) -> mark(false()) active(inf(X)) -> mark(cons(X,inf(s(X)))) active(take(0(),X)) -> mark(nil()) active(take(s(X),cons(Y,L))) -> mark(cons(Y,take(X,L))) active(length(nil())) -> mark(0()) active(length(cons(X,L))) -> mark(s(length(L))) active(inf(X)) -> inf(active(X)) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) active(length(X)) -> length(active(X)) inf(mark(X)) -> mark(inf(X)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) length(mark(X)) -> mark(length(X)) proper(eq(X1,X2)) -> eq(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(true()) -> ok(true()) proper(s(X)) -> s(proper(X)) proper(false()) -> ok(false()) proper(inf(X)) -> inf(proper(X)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) proper(nil()) -> ok(nil()) proper(length(X)) -> length(proper(X)) eq(ok(X1),ok(X2)) -> ok(eq(X1,X2)) s(ok(X)) -> ok(s(X)) inf(ok(X)) -> ok(inf(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) 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(eq(0(),0())) -> mark(true()) active(eq(s(X),s(Y))) -> mark(eq(X,Y)) active(eq(X,Y)) -> mark(false()) active(inf(X)) -> mark(cons(X,inf(s(X)))) active(take(0(),X)) -> mark(nil()) active(take(s(X),cons(Y,L))) -> mark(cons(Y,take(X,L))) active(length(nil())) -> mark(0()) active(length(cons(X,L))) -> mark(s(length(L))) active(inf(X)) -> inf(active(X)) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) active(length(X)) -> length(active(X)) inf(mark(X)) -> mark(inf(X)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) length(mark(X)) -> mark(length(X)) proper(eq(X1,X2)) -> eq(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(true()) -> ok(true()) proper(s(X)) -> s(proper(X)) proper(false()) -> ok(false()) proper(inf(X)) -> inf(proper(X)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) proper(nil()) -> ok(nil()) proper(length(X)) -> length(proper(X)) eq(ok(X1),ok(X2)) -> ok(eq(X1,X2)) s(ok(X)) -> ok(s(X)) inf(ok(X)) -> ok(inf(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) 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(eq(0(),0())) -> mark(true()) active(eq(s(X),s(Y))) -> mark(eq(X,Y)) active(eq(X,Y)) -> mark(false()) active(inf(X)) -> mark(cons(X,inf(s(X)))) active(take(0(),X)) -> mark(nil()) active(take(s(X),cons(Y,L))) -> mark(cons(Y,take(X,L))) active(length(nil())) -> mark(0()) active(length(cons(X,L))) -> mark(s(length(L))) active(inf(X)) -> inf(active(X)) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) active(length(X)) -> length(active(X)) inf(mark(X)) -> mark(inf(X)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) length(mark(X)) -> mark(length(X)) proper(eq(X1,X2)) -> eq(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(true()) -> ok(true()) proper(s(X)) -> s(proper(X)) proper(false()) -> ok(false()) proper(inf(X)) -> inf(proper(X)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) proper(nil()) -> ok(nil()) proper(length(X)) -> length(proper(X)) eq(ok(X1),ok(X2)) -> ok(eq(X1,X2)) s(ok(X)) -> ok(s(X)) inf(ok(X)) -> ok(inf(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) 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(eq(0(),0())) -> mark(true()) active(eq(s(X),s(Y))) -> mark(eq(X,Y)) active(eq(X,Y)) -> mark(false()) active(inf(X)) -> mark(cons(X,inf(s(X)))) active(take(0(),X)) -> mark(nil()) active(take(s(X),cons(Y,L))) -> mark(cons(Y,take(X,L))) active(length(nil())) -> mark(0()) active(length(cons(X,L))) -> mark(s(length(L))) active(inf(X)) -> inf(active(X)) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) active(length(X)) -> length(active(X)) inf(mark(X)) -> mark(inf(X)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) length(mark(X)) -> mark(length(X)) proper(eq(X1,X2)) -> eq(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(true()) -> ok(true()) proper(s(X)) -> s(proper(X)) proper(false()) -> ok(false()) proper(inf(X)) -> inf(proper(X)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) proper(nil()) -> ok(nil()) proper(length(X)) -> length(proper(X)) eq(ok(X1),ok(X2)) -> ok(eq(X1,X2)) s(ok(X)) -> ok(s(X)) inf(ok(X)) -> ok(inf(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) length(ok(X)) -> ok(length(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed DPs: take#(mark(X1),X2) -> take#(X1,X2) take#(X1,mark(X2)) -> take#(X1,X2) take#(ok(X1),ok(X2)) -> take#(X1,X2) TRS: active(eq(0(),0())) -> mark(true()) active(eq(s(X),s(Y))) -> mark(eq(X,Y)) active(eq(X,Y)) -> mark(false()) active(inf(X)) -> mark(cons(X,inf(s(X)))) active(take(0(),X)) -> mark(nil()) active(take(s(X),cons(Y,L))) -> mark(cons(Y,take(X,L))) active(length(nil())) -> mark(0()) active(length(cons(X,L))) -> mark(s(length(L))) active(inf(X)) -> inf(active(X)) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) active(length(X)) -> length(active(X)) inf(mark(X)) -> mark(inf(X)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) length(mark(X)) -> mark(length(X)) proper(eq(X1,X2)) -> eq(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(true()) -> ok(true()) proper(s(X)) -> s(proper(X)) proper(false()) -> ok(false()) proper(inf(X)) -> inf(proper(X)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) proper(nil()) -> ok(nil()) proper(length(X)) -> length(proper(X)) eq(ok(X1),ok(X2)) -> ok(eq(X1,X2)) s(ok(X)) -> ok(s(X)) inf(ok(X)) -> ok(inf(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) length(ok(X)) -> ok(length(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Subterm Criterion Processor: simple projection: pi(take#) = 0 problem: DPs: take#(X1,mark(X2)) -> take#(X1,X2) TRS: active(eq(0(),0())) -> mark(true()) active(eq(s(X),s(Y))) -> mark(eq(X,Y)) active(eq(X,Y)) -> mark(false()) active(inf(X)) -> mark(cons(X,inf(s(X)))) active(take(0(),X)) -> mark(nil()) active(take(s(X),cons(Y,L))) -> mark(cons(Y,take(X,L))) active(length(nil())) -> mark(0()) active(length(cons(X,L))) -> mark(s(length(L))) active(inf(X)) -> inf(active(X)) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) active(length(X)) -> length(active(X)) inf(mark(X)) -> mark(inf(X)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) length(mark(X)) -> mark(length(X)) proper(eq(X1,X2)) -> eq(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(true()) -> ok(true()) proper(s(X)) -> s(proper(X)) proper(false()) -> ok(false()) proper(inf(X)) -> inf(proper(X)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) proper(nil()) -> ok(nil()) proper(length(X)) -> length(proper(X)) eq(ok(X1),ok(X2)) -> ok(eq(X1,X2)) s(ok(X)) -> ok(s(X)) inf(ok(X)) -> ok(inf(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) length(ok(X)) -> ok(length(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Subterm Criterion Processor: simple projection: pi(take#) = 1 problem: DPs: TRS: active(eq(0(),0())) -> mark(true()) active(eq(s(X),s(Y))) -> mark(eq(X,Y)) active(eq(X,Y)) -> mark(false()) active(inf(X)) -> mark(cons(X,inf(s(X)))) active(take(0(),X)) -> mark(nil()) active(take(s(X),cons(Y,L))) -> mark(cons(Y,take(X,L))) active(length(nil())) -> mark(0()) active(length(cons(X,L))) -> mark(s(length(L))) active(inf(X)) -> inf(active(X)) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) active(length(X)) -> length(active(X)) inf(mark(X)) -> mark(inf(X)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) length(mark(X)) -> mark(length(X)) proper(eq(X1,X2)) -> eq(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(true()) -> ok(true()) proper(s(X)) -> s(proper(X)) proper(false()) -> ok(false()) proper(inf(X)) -> inf(proper(X)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) proper(nil()) -> ok(nil()) proper(length(X)) -> length(proper(X)) eq(ok(X1),ok(X2)) -> ok(eq(X1,X2)) s(ok(X)) -> ok(s(X)) inf(ok(X)) -> ok(inf(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) 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) TRS: active(eq(0(),0())) -> mark(true()) active(eq(s(X),s(Y))) -> mark(eq(X,Y)) active(eq(X,Y)) -> mark(false()) active(inf(X)) -> mark(cons(X,inf(s(X)))) active(take(0(),X)) -> mark(nil()) active(take(s(X),cons(Y,L))) -> mark(cons(Y,take(X,L))) active(length(nil())) -> mark(0()) active(length(cons(X,L))) -> mark(s(length(L))) active(inf(X)) -> inf(active(X)) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) active(length(X)) -> length(active(X)) inf(mark(X)) -> mark(inf(X)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) length(mark(X)) -> mark(length(X)) proper(eq(X1,X2)) -> eq(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(true()) -> ok(true()) proper(s(X)) -> s(proper(X)) proper(false()) -> ok(false()) proper(inf(X)) -> inf(proper(X)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) proper(nil()) -> ok(nil()) proper(length(X)) -> length(proper(X)) eq(ok(X1),ok(X2)) -> ok(eq(X1,X2)) s(ok(X)) -> ok(s(X)) inf(ok(X)) -> ok(inf(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) 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(eq(0(),0())) -> mark(true()) active(eq(s(X),s(Y))) -> mark(eq(X,Y)) active(eq(X,Y)) -> mark(false()) active(inf(X)) -> mark(cons(X,inf(s(X)))) active(take(0(),X)) -> mark(nil()) active(take(s(X),cons(Y,L))) -> mark(cons(Y,take(X,L))) active(length(nil())) -> mark(0()) active(length(cons(X,L))) -> mark(s(length(L))) active(inf(X)) -> inf(active(X)) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) active(length(X)) -> length(active(X)) inf(mark(X)) -> mark(inf(X)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) length(mark(X)) -> mark(length(X)) proper(eq(X1,X2)) -> eq(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(true()) -> ok(true()) proper(s(X)) -> s(proper(X)) proper(false()) -> ok(false()) proper(inf(X)) -> inf(proper(X)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) proper(nil()) -> ok(nil()) proper(length(X)) -> length(proper(X)) eq(ok(X1),ok(X2)) -> ok(eq(X1,X2)) s(ok(X)) -> ok(s(X)) inf(ok(X)) -> ok(inf(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) length(ok(X)) -> ok(length(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed DPs: inf#(mark(X)) -> inf#(X) inf#(ok(X)) -> inf#(X) TRS: active(eq(0(),0())) -> mark(true()) active(eq(s(X),s(Y))) -> mark(eq(X,Y)) active(eq(X,Y)) -> mark(false()) active(inf(X)) -> mark(cons(X,inf(s(X)))) active(take(0(),X)) -> mark(nil()) active(take(s(X),cons(Y,L))) -> mark(cons(Y,take(X,L))) active(length(nil())) -> mark(0()) active(length(cons(X,L))) -> mark(s(length(L))) active(inf(X)) -> inf(active(X)) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) active(length(X)) -> length(active(X)) inf(mark(X)) -> mark(inf(X)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) length(mark(X)) -> mark(length(X)) proper(eq(X1,X2)) -> eq(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(true()) -> ok(true()) proper(s(X)) -> s(proper(X)) proper(false()) -> ok(false()) proper(inf(X)) -> inf(proper(X)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) proper(nil()) -> ok(nil()) proper(length(X)) -> length(proper(X)) eq(ok(X1),ok(X2)) -> ok(eq(X1,X2)) s(ok(X)) -> ok(s(X)) inf(ok(X)) -> ok(inf(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) length(ok(X)) -> ok(length(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Subterm Criterion Processor: simple projection: pi(inf#) = 0 problem: DPs: TRS: active(eq(0(),0())) -> mark(true()) active(eq(s(X),s(Y))) -> mark(eq(X,Y)) active(eq(X,Y)) -> mark(false()) active(inf(X)) -> mark(cons(X,inf(s(X)))) active(take(0(),X)) -> mark(nil()) active(take(s(X),cons(Y,L))) -> mark(cons(Y,take(X,L))) active(length(nil())) -> mark(0()) active(length(cons(X,L))) -> mark(s(length(L))) active(inf(X)) -> inf(active(X)) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) active(length(X)) -> length(active(X)) inf(mark(X)) -> mark(inf(X)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) length(mark(X)) -> mark(length(X)) proper(eq(X1,X2)) -> eq(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(true()) -> ok(true()) proper(s(X)) -> s(proper(X)) proper(false()) -> ok(false()) proper(inf(X)) -> inf(proper(X)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) proper(nil()) -> ok(nil()) proper(length(X)) -> length(proper(X)) eq(ok(X1),ok(X2)) -> ok(eq(X1,X2)) s(ok(X)) -> ok(s(X)) inf(ok(X)) -> ok(inf(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) 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) TRS: active(eq(0(),0())) -> mark(true()) active(eq(s(X),s(Y))) -> mark(eq(X,Y)) active(eq(X,Y)) -> mark(false()) active(inf(X)) -> mark(cons(X,inf(s(X)))) active(take(0(),X)) -> mark(nil()) active(take(s(X),cons(Y,L))) -> mark(cons(Y,take(X,L))) active(length(nil())) -> mark(0()) active(length(cons(X,L))) -> mark(s(length(L))) active(inf(X)) -> inf(active(X)) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) active(length(X)) -> length(active(X)) inf(mark(X)) -> mark(inf(X)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) length(mark(X)) -> mark(length(X)) proper(eq(X1,X2)) -> eq(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(true()) -> ok(true()) proper(s(X)) -> s(proper(X)) proper(false()) -> ok(false()) proper(inf(X)) -> inf(proper(X)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) proper(nil()) -> ok(nil()) proper(length(X)) -> length(proper(X)) eq(ok(X1),ok(X2)) -> ok(eq(X1,X2)) s(ok(X)) -> ok(s(X)) inf(ok(X)) -> ok(inf(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) 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(eq(0(),0())) -> mark(true()) active(eq(s(X),s(Y))) -> mark(eq(X,Y)) active(eq(X,Y)) -> mark(false()) active(inf(X)) -> mark(cons(X,inf(s(X)))) active(take(0(),X)) -> mark(nil()) active(take(s(X),cons(Y,L))) -> mark(cons(Y,take(X,L))) active(length(nil())) -> mark(0()) active(length(cons(X,L))) -> mark(s(length(L))) active(inf(X)) -> inf(active(X)) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) active(length(X)) -> length(active(X)) inf(mark(X)) -> mark(inf(X)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) length(mark(X)) -> mark(length(X)) proper(eq(X1,X2)) -> eq(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(true()) -> ok(true()) proper(s(X)) -> s(proper(X)) proper(false()) -> ok(false()) proper(inf(X)) -> inf(proper(X)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) proper(nil()) -> ok(nil()) proper(length(X)) -> length(proper(X)) eq(ok(X1),ok(X2)) -> ok(eq(X1,X2)) s(ok(X)) -> ok(s(X)) inf(ok(X)) -> ok(inf(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) length(ok(X)) -> ok(length(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed DPs: eq#(ok(X1),ok(X2)) -> eq#(X1,X2) TRS: active(eq(0(),0())) -> mark(true()) active(eq(s(X),s(Y))) -> mark(eq(X,Y)) active(eq(X,Y)) -> mark(false()) active(inf(X)) -> mark(cons(X,inf(s(X)))) active(take(0(),X)) -> mark(nil()) active(take(s(X),cons(Y,L))) -> mark(cons(Y,take(X,L))) active(length(nil())) -> mark(0()) active(length(cons(X,L))) -> mark(s(length(L))) active(inf(X)) -> inf(active(X)) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) active(length(X)) -> length(active(X)) inf(mark(X)) -> mark(inf(X)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) length(mark(X)) -> mark(length(X)) proper(eq(X1,X2)) -> eq(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(true()) -> ok(true()) proper(s(X)) -> s(proper(X)) proper(false()) -> ok(false()) proper(inf(X)) -> inf(proper(X)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) proper(nil()) -> ok(nil()) proper(length(X)) -> length(proper(X)) eq(ok(X1),ok(X2)) -> ok(eq(X1,X2)) s(ok(X)) -> ok(s(X)) inf(ok(X)) -> ok(inf(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) length(ok(X)) -> ok(length(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Subterm Criterion Processor: simple projection: pi(eq#) = 0 problem: DPs: TRS: active(eq(0(),0())) -> mark(true()) active(eq(s(X),s(Y))) -> mark(eq(X,Y)) active(eq(X,Y)) -> mark(false()) active(inf(X)) -> mark(cons(X,inf(s(X)))) active(take(0(),X)) -> mark(nil()) active(take(s(X),cons(Y,L))) -> mark(cons(Y,take(X,L))) active(length(nil())) -> mark(0()) active(length(cons(X,L))) -> mark(s(length(L))) active(inf(X)) -> inf(active(X)) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) active(length(X)) -> length(active(X)) inf(mark(X)) -> mark(inf(X)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) length(mark(X)) -> mark(length(X)) proper(eq(X1,X2)) -> eq(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(true()) -> ok(true()) proper(s(X)) -> s(proper(X)) proper(false()) -> ok(false()) proper(inf(X)) -> inf(proper(X)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) proper(nil()) -> ok(nil()) proper(length(X)) -> length(proper(X)) eq(ok(X1),ok(X2)) -> ok(eq(X1,X2)) s(ok(X)) -> ok(s(X)) inf(ok(X)) -> ok(inf(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) length(ok(X)) -> ok(length(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed