MAYBE 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)) EDG 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#(mark(X)) -> proper#(X) top#(ok(X)) -> top#(active(X)) -> top#(mark(X)) -> top#(proper(X)) top#(ok(X)) -> top#(active(X)) -> top#(ok(X)) -> active#(X) top#(ok(X)) -> top#(active(X)) -> top#(ok(X)) -> top#(active(X)) top#(ok(X)) -> active#(X) -> active#(eq(s(X),s(Y))) -> eq#(X,Y) top#(ok(X)) -> active#(X) -> active#(inf(X)) -> s#(X) top#(ok(X)) -> active#(X) -> active#(inf(X)) -> inf#(s(X)) top#(ok(X)) -> active#(X) -> active#(inf(X)) -> cons#(X,inf(s(X))) top#(ok(X)) -> active#(X) -> active#(take(s(X),cons(Y,L))) -> take#(X,L) top#(ok(X)) -> active#(X) -> active#(take(s(X),cons(Y,L))) -> cons#(Y,take(X,L)) top#(ok(X)) -> active#(X) -> active#(length(cons(X,L))) -> length#(L) top#(ok(X)) -> active#(X) -> active#(length(cons(X,L))) -> s#(length(L)) top#(ok(X)) -> active#(X) -> active#(inf(X)) -> active#(X) top#(ok(X)) -> active#(X) -> active#(inf(X)) -> inf#(active(X)) top#(ok(X)) -> active#(X) -> active#(take(X1,X2)) -> active#(X1) top#(ok(X)) -> active#(X) -> active#(take(X1,X2)) -> take#(active(X1),X2) top#(ok(X)) -> active#(X) -> active#(take(X1,X2)) -> active#(X2) top#(ok(X)) -> active#(X) -> active#(take(X1,X2)) -> take#(X1,active(X2)) top#(ok(X)) -> active#(X) -> active#(length(X)) -> active#(X) top#(ok(X)) -> active#(X) -> active#(length(X)) -> length#(active(X)) top#(mark(X)) -> top#(proper(X)) -> top#(mark(X)) -> proper#(X) top#(mark(X)) -> top#(proper(X)) -> top#(mark(X)) -> top#(proper(X)) top#(mark(X)) -> top#(proper(X)) -> top#(ok(X)) -> active#(X) top#(mark(X)) -> top#(proper(X)) -> top#(ok(X)) -> top#(active(X)) top#(mark(X)) -> proper#(X) -> proper#(eq(X1,X2)) -> proper#(X2) top#(mark(X)) -> proper#(X) -> proper#(eq(X1,X2)) -> proper#(X1) top#(mark(X)) -> proper#(X) -> proper#(eq(X1,X2)) -> eq#(proper(X1),proper(X2)) top#(mark(X)) -> proper#(X) -> proper#(s(X)) -> proper#(X) top#(mark(X)) -> proper#(X) -> proper#(s(X)) -> s#(proper(X)) top#(mark(X)) -> proper#(X) -> proper#(inf(X)) -> proper#(X) top#(mark(X)) -> proper#(X) -> proper#(inf(X)) -> inf#(proper(X)) top#(mark(X)) -> proper#(X) -> proper#(cons(X1,X2)) -> proper#(X2) top#(mark(X)) -> proper#(X) -> proper#(cons(X1,X2)) -> proper#(X1) top#(mark(X)) -> proper#(X) -> proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2)) top#(mark(X)) -> proper#(X) -> proper#(take(X1,X2)) -> proper#(X2) top#(mark(X)) -> proper#(X) -> proper#(take(X1,X2)) -> proper#(X1) top#(mark(X)) -> proper#(X) -> proper#(take(X1,X2)) -> take#(proper(X1),proper(X2)) top#(mark(X)) -> proper#(X) -> proper#(length(X)) -> proper#(X) top#(mark(X)) -> proper#(X) -> proper#(length(X)) -> length#(proper(X)) proper#(length(X)) -> proper#(X) -> proper#(eq(X1,X2)) -> proper#(X2) proper#(length(X)) -> proper#(X) -> proper#(eq(X1,X2)) -> proper#(X1) proper#(length(X)) -> proper#(X) -> proper#(eq(X1,X2)) -> eq#(proper(X1),proper(X2)) proper#(length(X)) -> proper#(X) -> proper#(s(X)) -> proper#(X) proper#(length(X)) -> proper#(X) -> proper#(s(X)) -> s#(proper(X)) proper#(length(X)) -> proper#(X) -> proper#(inf(X)) -> proper#(X) proper#(length(X)) -> proper#(X) -> proper#(inf(X)) -> inf#(proper(X)) proper#(length(X)) -> proper#(X) -> proper#(cons(X1,X2)) -> proper#(X2) proper#(length(X)) -> proper#(X) -> proper#(cons(X1,X2)) -> proper#(X1) proper#(length(X)) -> proper#(X) -> proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2)) proper#(length(X)) -> proper#(X) -> proper#(take(X1,X2)) -> proper#(X2) proper#(length(X)) -> proper#(X) -> proper#(take(X1,X2)) -> proper#(X1) proper#(length(X)) -> proper#(X) -> proper#(take(X1,X2)) -> take#(proper(X1),proper(X2)) proper#(length(X)) -> proper#(X) -> proper#(length(X)) -> proper#(X) proper#(length(X)) -> proper#(X) -> proper#(length(X)) -> length#(proper(X)) proper#(length(X)) -> length#(proper(X)) -> length#(mark(X)) -> length#(X) proper#(length(X)) -> length#(proper(X)) -> length#(ok(X)) -> length#(X) proper#(take(X1,X2)) -> proper#(X2) -> proper#(eq(X1,X2)) -> proper#(X2) proper#(take(X1,X2)) -> proper#(X2) -> proper#(eq(X1,X2)) -> proper#(X1) proper#(take(X1,X2)) -> proper#(X2) -> proper#(eq(X1,X2)) -> eq#(proper(X1),proper(X2)) proper#(take(X1,X2)) -> proper#(X2) -> proper#(s(X)) -> proper#(X) proper#(take(X1,X2)) -> proper#(X2) -> proper#(s(X)) -> s#(proper(X)) proper#(take(X1,X2)) -> proper#(X2) -> proper#(inf(X)) -> proper#(X) proper#(take(X1,X2)) -> proper#(X2) -> proper#(inf(X)) -> inf#(proper(X)) proper#(take(X1,X2)) -> proper#(X2) -> proper#(cons(X1,X2)) -> proper#(X2) proper#(take(X1,X2)) -> proper#(X2) -> proper#(cons(X1,X2)) -> proper#(X1) proper#(take(X1,X2)) -> proper#(X2) -> proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2)) proper#(take(X1,X2)) -> proper#(X2) -> proper#(take(X1,X2)) -> proper#(X2) proper#(take(X1,X2)) -> proper#(X2) -> proper#(take(X1,X2)) -> proper#(X1) proper#(take(X1,X2)) -> proper#(X2) -> proper#(take(X1,X2)) -> take#(proper(X1),proper(X2)) proper#(take(X1,X2)) -> proper#(X2) -> proper#(length(X)) -> proper#(X) proper#(take(X1,X2)) -> proper#(X2) -> proper#(length(X)) -> length#(proper(X)) proper#(take(X1,X2)) -> proper#(X1) -> proper#(eq(X1,X2)) -> proper#(X2) proper#(take(X1,X2)) -> proper#(X1) -> proper#(eq(X1,X2)) -> proper#(X1) proper#(take(X1,X2)) -> proper#(X1) -> proper#(eq(X1,X2)) -> eq#(proper(X1),proper(X2)) proper#(take(X1,X2)) -> proper#(X1) -> proper#(s(X)) -> proper#(X) proper#(take(X1,X2)) -> proper#(X1) -> proper#(s(X)) -> s#(proper(X)) proper#(take(X1,X2)) -> proper#(X1) -> proper#(inf(X)) -> proper#(X) proper#(take(X1,X2)) -> proper#(X1) -> proper#(inf(X)) -> inf#(proper(X)) proper#(take(X1,X2)) -> proper#(X1) -> proper#(cons(X1,X2)) -> proper#(X2) proper#(take(X1,X2)) -> proper#(X1) -> proper#(cons(X1,X2)) -> proper#(X1) proper#(take(X1,X2)) -> proper#(X1) -> proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2)) proper#(take(X1,X2)) -> proper#(X1) -> proper#(take(X1,X2)) -> proper#(X2) proper#(take(X1,X2)) -> proper#(X1) -> proper#(take(X1,X2)) -> proper#(X1) proper#(take(X1,X2)) -> proper#(X1) -> proper#(take(X1,X2)) -> take#(proper(X1),proper(X2)) proper#(take(X1,X2)) -> proper#(X1) -> proper#(length(X)) -> proper#(X) proper#(take(X1,X2)) -> proper#(X1) -> proper#(length(X)) -> length#(proper(X)) proper#(take(X1,X2)) -> take#(proper(X1),proper(X2)) -> take#(mark(X1),X2) -> take#(X1,X2) proper#(take(X1,X2)) -> take#(proper(X1),proper(X2)) -> take#(X1,mark(X2)) -> take#(X1,X2) proper#(take(X1,X2)) -> take#(proper(X1),proper(X2)) -> take#(ok(X1),ok(X2)) -> take#(X1,X2) proper#(cons(X1,X2)) -> proper#(X2) -> proper#(eq(X1,X2)) -> proper#(X2) proper#(cons(X1,X2)) -> proper#(X2) -> proper#(eq(X1,X2)) -> proper#(X1) proper#(cons(X1,X2)) -> proper#(X2) -> proper#(eq(X1,X2)) -> eq#(proper(X1),proper(X2)) proper#(cons(X1,X2)) -> proper#(X2) -> proper#(s(X)) -> proper#(X) proper#(cons(X1,X2)) -> proper#(X2) -> proper#(s(X)) -> s#(proper(X)) proper#(cons(X1,X2)) -> proper#(X2) -> proper#(inf(X)) -> proper#(X) proper#(cons(X1,X2)) -> proper#(X2) -> proper#(inf(X)) -> inf#(proper(X)) proper#(cons(X1,X2)) -> proper#(X2) -> proper#(cons(X1,X2)) -> proper#(X2) proper#(cons(X1,X2)) -> proper#(X2) -> proper#(cons(X1,X2)) -> proper#(X1) proper#(cons(X1,X2)) -> proper#(X2) -> proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2)) proper#(cons(X1,X2)) -> proper#(X2) -> proper#(take(X1,X2)) -> proper#(X2) proper#(cons(X1,X2)) -> proper#(X2) -> proper#(take(X1,X2)) -> proper#(X1) proper#(cons(X1,X2)) -> proper#(X2) -> proper#(take(X1,X2)) -> take#(proper(X1),proper(X2)) proper#(cons(X1,X2)) -> proper#(X2) -> proper#(length(X)) -> proper#(X) proper#(cons(X1,X2)) -> proper#(X2) -> proper#(length(X)) -> length#(proper(X)) proper#(cons(X1,X2)) -> proper#(X1) -> proper#(eq(X1,X2)) -> proper#(X2) proper#(cons(X1,X2)) -> proper#(X1) -> proper#(eq(X1,X2)) -> proper#(X1) proper#(cons(X1,X2)) -> proper#(X1) -> proper#(eq(X1,X2)) -> eq#(proper(X1),proper(X2)) proper#(cons(X1,X2)) -> proper#(X1) -> proper#(s(X)) -> proper#(X) proper#(cons(X1,X2)) -> proper#(X1) -> proper#(s(X)) -> s#(proper(X)) proper#(cons(X1,X2)) -> proper#(X1) -> proper#(inf(X)) -> proper#(X) proper#(cons(X1,X2)) -> proper#(X1) -> proper#(inf(X)) -> inf#(proper(X)) proper#(cons(X1,X2)) -> proper#(X1) -> proper#(cons(X1,X2)) -> proper#(X2) proper#(cons(X1,X2)) -> proper#(X1) -> proper#(cons(X1,X2)) -> proper#(X1) proper#(cons(X1,X2)) -> proper#(X1) -> proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2)) proper#(cons(X1,X2)) -> proper#(X1) -> proper#(take(X1,X2)) -> proper#(X2) proper#(cons(X1,X2)) -> proper#(X1) -> proper#(take(X1,X2)) -> proper#(X1) proper#(cons(X1,X2)) -> proper#(X1) -> proper#(take(X1,X2)) -> take#(proper(X1),proper(X2)) proper#(cons(X1,X2)) -> proper#(X1) -> proper#(length(X)) -> proper#(X) proper#(cons(X1,X2)) -> proper#(X1) -> proper#(length(X)) -> length#(proper(X)) proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2)) -> cons#(ok(X1),ok(X2)) -> cons#(X1,X2) proper#(inf(X)) -> proper#(X) -> proper#(eq(X1,X2)) -> proper#(X2) proper#(inf(X)) -> proper#(X) -> proper#(eq(X1,X2)) -> proper#(X1) proper#(inf(X)) -> proper#(X) -> proper#(eq(X1,X2)) -> eq#(proper(X1),proper(X2)) proper#(inf(X)) -> proper#(X) -> proper#(s(X)) -> proper#(X) proper#(inf(X)) -> proper#(X) -> proper#(s(X)) -> s#(proper(X)) proper#(inf(X)) -> proper#(X) -> proper#(inf(X)) -> proper#(X) proper#(inf(X)) -> proper#(X) -> proper#(inf(X)) -> inf#(proper(X)) proper#(inf(X)) -> proper#(X) -> proper#(cons(X1,X2)) -> proper#(X2) proper#(inf(X)) -> proper#(X) -> proper#(cons(X1,X2)) -> proper#(X1) proper#(inf(X)) -> proper#(X) -> proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2)) proper#(inf(X)) -> proper#(X) -> proper#(take(X1,X2)) -> proper#(X2) proper#(inf(X)) -> proper#(X) -> proper#(take(X1,X2)) -> proper#(X1) proper#(inf(X)) -> proper#(X) -> proper#(take(X1,X2)) -> take#(proper(X1),proper(X2)) proper#(inf(X)) -> proper#(X) -> proper#(length(X)) -> proper#(X) proper#(inf(X)) -> proper#(X) -> proper#(length(X)) -> length#(proper(X)) proper#(inf(X)) -> inf#(proper(X)) -> inf#(mark(X)) -> inf#(X) proper#(inf(X)) -> inf#(proper(X)) -> inf#(ok(X)) -> inf#(X) proper#(s(X)) -> proper#(X) -> proper#(eq(X1,X2)) -> proper#(X2) proper#(s(X)) -> proper#(X) -> proper#(eq(X1,X2)) -> proper#(X1) proper#(s(X)) -> proper#(X) -> proper#(eq(X1,X2)) -> eq#(proper(X1),proper(X2)) proper#(s(X)) -> proper#(X) -> proper#(s(X)) -> proper#(X) proper#(s(X)) -> proper#(X) -> proper#(s(X)) -> s#(proper(X)) proper#(s(X)) -> proper#(X) -> proper#(inf(X)) -> proper#(X) proper#(s(X)) -> proper#(X) -> proper#(inf(X)) -> inf#(proper(X)) proper#(s(X)) -> proper#(X) -> proper#(cons(X1,X2)) -> proper#(X2) proper#(s(X)) -> proper#(X) -> proper#(cons(X1,X2)) -> proper#(X1) proper#(s(X)) -> proper#(X) -> proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2)) proper#(s(X)) -> proper#(X) -> proper#(take(X1,X2)) -> proper#(X2) proper#(s(X)) -> proper#(X) -> proper#(take(X1,X2)) -> proper#(X1) proper#(s(X)) -> proper#(X) -> proper#(take(X1,X2)) -> take#(proper(X1),proper(X2)) proper#(s(X)) -> proper#(X) -> proper#(length(X)) -> proper#(X) proper#(s(X)) -> proper#(X) -> proper#(length(X)) -> length#(proper(X)) proper#(s(X)) -> s#(proper(X)) -> s#(ok(X)) -> s#(X) proper#(eq(X1,X2)) -> proper#(X2) -> proper#(eq(X1,X2)) -> proper#(X2) proper#(eq(X1,X2)) -> proper#(X2) -> proper#(eq(X1,X2)) -> proper#(X1) proper#(eq(X1,X2)) -> proper#(X2) -> proper#(eq(X1,X2)) -> eq#(proper(X1),proper(X2)) proper#(eq(X1,X2)) -> proper#(X2) -> proper#(s(X)) -> proper#(X) proper#(eq(X1,X2)) -> proper#(X2) -> proper#(s(X)) -> s#(proper(X)) proper#(eq(X1,X2)) -> proper#(X2) -> proper#(inf(X)) -> proper#(X) proper#(eq(X1,X2)) -> proper#(X2) -> proper#(inf(X)) -> inf#(proper(X)) proper#(eq(X1,X2)) -> proper#(X2) -> proper#(cons(X1,X2)) -> proper#(X2) proper#(eq(X1,X2)) -> proper#(X2) -> proper#(cons(X1,X2)) -> proper#(X1) proper#(eq(X1,X2)) -> proper#(X2) -> proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2)) proper#(eq(X1,X2)) -> proper#(X2) -> proper#(take(X1,X2)) -> proper#(X2) proper#(eq(X1,X2)) -> proper#(X2) -> proper#(take(X1,X2)) -> proper#(X1) proper#(eq(X1,X2)) -> proper#(X2) -> proper#(take(X1,X2)) -> take#(proper(X1),proper(X2)) proper#(eq(X1,X2)) -> proper#(X2) -> proper#(length(X)) -> proper#(X) proper#(eq(X1,X2)) -> proper#(X2) -> proper#(length(X)) -> length#(proper(X)) proper#(eq(X1,X2)) -> proper#(X1) -> proper#(eq(X1,X2)) -> proper#(X2) proper#(eq(X1,X2)) -> proper#(X1) -> proper#(eq(X1,X2)) -> proper#(X1) proper#(eq(X1,X2)) -> proper#(X1) -> proper#(eq(X1,X2)) -> eq#(proper(X1),proper(X2)) proper#(eq(X1,X2)) -> proper#(X1) -> proper#(s(X)) -> proper#(X) proper#(eq(X1,X2)) -> proper#(X1) -> proper#(s(X)) -> s#(proper(X)) proper#(eq(X1,X2)) -> proper#(X1) -> proper#(inf(X)) -> proper#(X) proper#(eq(X1,X2)) -> proper#(X1) -> proper#(inf(X)) -> inf#(proper(X)) proper#(eq(X1,X2)) -> proper#(X1) -> proper#(cons(X1,X2)) -> proper#(X2) proper#(eq(X1,X2)) -> proper#(X1) -> proper#(cons(X1,X2)) -> proper#(X1) proper#(eq(X1,X2)) -> proper#(X1) -> proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2)) proper#(eq(X1,X2)) -> proper#(X1) -> proper#(take(X1,X2)) -> proper#(X2) proper#(eq(X1,X2)) -> proper#(X1) -> proper#(take(X1,X2)) -> proper#(X1) proper#(eq(X1,X2)) -> proper#(X1) -> proper#(take(X1,X2)) -> take#(proper(X1),proper(X2)) proper#(eq(X1,X2)) -> proper#(X1) -> proper#(length(X)) -> proper#(X) proper#(eq(X1,X2)) -> proper#(X1) -> proper#(length(X)) -> length#(proper(X)) proper#(eq(X1,X2)) -> eq#(proper(X1),proper(X2)) -> eq#(ok(X1),ok(X2)) -> eq#(X1,X2) length#(ok(X)) -> length#(X) -> length#(mark(X)) -> length#(X) length#(ok(X)) -> length#(X) -> length#(ok(X)) -> length#(X) length#(mark(X)) -> length#(X) -> length#(mark(X)) -> length#(X) length#(mark(X)) -> length#(X) -> length#(ok(X)) -> length#(X) take#(ok(X1),ok(X2)) -> take#(X1,X2) -> take#(mark(X1),X2) -> take#(X1,X2) take#(ok(X1),ok(X2)) -> take#(X1,X2) -> take#(X1,mark(X2)) -> take#(X1,X2) take#(ok(X1),ok(X2)) -> take#(X1,X2) -> take#(ok(X1),ok(X2)) -> take#(X1,X2) take#(mark(X1),X2) -> take#(X1,X2) -> take#(mark(X1),X2) -> take#(X1,X2) take#(mark(X1),X2) -> take#(X1,X2) -> take#(X1,mark(X2)) -> take#(X1,X2) take#(mark(X1),X2) -> take#(X1,X2) -> take#(ok(X1),ok(X2)) -> take#(X1,X2) take#(X1,mark(X2)) -> take#(X1,X2) -> take#(mark(X1),X2) -> take#(X1,X2) take#(X1,mark(X2)) -> take#(X1,X2) -> take#(X1,mark(X2)) -> take#(X1,X2) take#(X1,mark(X2)) -> take#(X1,X2) -> take#(ok(X1),ok(X2)) -> take#(X1,X2) cons#(ok(X1),ok(X2)) -> cons#(X1,X2) -> cons#(ok(X1),ok(X2)) -> cons#(X1,X2) inf#(ok(X)) -> inf#(X) -> inf#(mark(X)) -> inf#(X) inf#(ok(X)) -> inf#(X) -> inf#(ok(X)) -> inf#(X) inf#(mark(X)) -> inf#(X) -> inf#(mark(X)) -> inf#(X) inf#(mark(X)) -> inf#(X) -> inf#(ok(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#(mark(X)) -> length#(X) active#(length(cons(X,L))) -> length#(L) -> length#(ok(X)) -> length#(X) active#(length(cons(X,L))) -> s#(length(L)) -> s#(ok(X)) -> s#(X) active#(length(X)) -> length#(active(X)) -> length#(mark(X)) -> length#(X) active#(length(X)) -> length#(active(X)) -> length#(ok(X)) -> length#(X) active#(length(X)) -> active#(X) -> active#(eq(s(X),s(Y))) -> eq#(X,Y) active#(length(X)) -> active#(X) -> active#(inf(X)) -> s#(X) active#(length(X)) -> active#(X) -> active#(inf(X)) -> inf#(s(X)) active#(length(X)) -> active#(X) -> active#(inf(X)) -> cons#(X,inf(s(X))) active#(length(X)) -> active#(X) -> active#(take(s(X),cons(Y,L))) -> take#(X,L) active#(length(X)) -> active#(X) -> active#(take(s(X),cons(Y,L))) -> cons#(Y,take(X,L)) active#(length(X)) -> active#(X) -> active#(length(cons(X,L))) -> length#(L) active#(length(X)) -> active#(X) -> active#(length(cons(X,L))) -> s#(length(L)) active#(length(X)) -> active#(X) -> active#(inf(X)) -> active#(X) active#(length(X)) -> active#(X) -> active#(inf(X)) -> inf#(active(X)) active#(length(X)) -> active#(X) -> active#(take(X1,X2)) -> active#(X1) active#(length(X)) -> active#(X) -> active#(take(X1,X2)) -> take#(active(X1),X2) active#(length(X)) -> active#(X) -> active#(take(X1,X2)) -> active#(X2) active#(length(X)) -> active#(X) -> active#(take(X1,X2)) -> take#(X1,active(X2)) active#(length(X)) -> active#(X) -> active#(length(X)) -> active#(X) active#(length(X)) -> active#(X) -> active#(length(X)) -> length#(active(X)) active#(take(s(X),cons(Y,L))) -> take#(X,L) -> take#(mark(X1),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))) -> cons#(Y,take(X,L)) -> cons#(ok(X1),ok(X2)) -> cons#(X1,X2) active#(take(X1,X2)) -> take#(active(X1),X2) -> take#(mark(X1),X2) -> take#(X1,X2) active#(take(X1,X2)) -> take#(active(X1),X2) -> take#(ok(X1),ok(X2)) -> take#(X1,X2) active#(take(X1,X2)) -> take#(X1,active(X2)) -> take#(X1,mark(X2)) -> take#(X1,X2) active#(take(X1,X2)) -> take#(X1,active(X2)) -> take#(ok(X1),ok(X2)) -> take#(X1,X2) active#(take(X1,X2)) -> active#(X2) -> active#(eq(s(X),s(Y))) -> eq#(X,Y) active#(take(X1,X2)) -> active#(X2) -> active#(inf(X)) -> s#(X) active#(take(X1,X2)) -> active#(X2) -> active#(inf(X)) -> inf#(s(X)) active#(take(X1,X2)) -> active#(X2) -> active#(inf(X)) -> cons#(X,inf(s(X))) active#(take(X1,X2)) -> active#(X2) -> active#(take(s(X),cons(Y,L))) -> take#(X,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#(length(cons(X,L))) -> length#(L) active#(take(X1,X2)) -> active#(X2) -> active#(length(cons(X,L))) -> s#(length(L)) active#(take(X1,X2)) -> active#(X2) -> active#(inf(X)) -> active#(X) active#(take(X1,X2)) -> active#(X2) -> active#(inf(X)) -> inf#(active(X)) active#(take(X1,X2)) -> active#(X2) -> active#(take(X1,X2)) -> active#(X1) active#(take(X1,X2)) -> active#(X2) -> active#(take(X1,X2)) -> take#(active(X1),X2) active#(take(X1,X2)) -> active#(X2) -> active#(take(X1,X2)) -> active#(X2) active#(take(X1,X2)) -> active#(X2) -> active#(take(X1,X2)) -> take#(X1,active(X2)) active#(take(X1,X2)) -> active#(X2) -> active#(length(X)) -> active#(X) active#(take(X1,X2)) -> active#(X2) -> active#(length(X)) -> length#(active(X)) active#(take(X1,X2)) -> active#(X1) -> active#(eq(s(X),s(Y))) -> eq#(X,Y) active#(take(X1,X2)) -> active#(X1) -> active#(inf(X)) -> s#(X) active#(take(X1,X2)) -> active#(X1) -> active#(inf(X)) -> inf#(s(X)) active#(take(X1,X2)) -> active#(X1) -> active#(inf(X)) -> cons#(X,inf(s(X))) active#(take(X1,X2)) -> active#(X1) -> active#(take(s(X),cons(Y,L))) -> take#(X,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#(length(cons(X,L))) -> length#(L) active#(take(X1,X2)) -> active#(X1) -> active#(length(cons(X,L))) -> s#(length(L)) active#(take(X1,X2)) -> active#(X1) -> active#(inf(X)) -> active#(X) active#(take(X1,X2)) -> active#(X1) -> active#(inf(X)) -> inf#(active(X)) active#(take(X1,X2)) -> active#(X1) -> active#(take(X1,X2)) -> active#(X1) active#(take(X1,X2)) -> active#(X1) -> active#(take(X1,X2)) -> take#(active(X1),X2) active#(take(X1,X2)) -> active#(X1) -> active#(take(X1,X2)) -> active#(X2) active#(take(X1,X2)) -> active#(X1) -> active#(take(X1,X2)) -> take#(X1,active(X2)) active#(take(X1,X2)) -> active#(X1) -> active#(length(X)) -> active#(X) active#(take(X1,X2)) -> active#(X1) -> active#(length(X)) -> length#(active(X)) active#(inf(X)) -> inf#(active(X)) -> inf#(mark(X)) -> inf#(X) active#(inf(X)) -> inf#(active(X)) -> inf#(ok(X)) -> inf#(X) active#(inf(X)) -> active#(X) -> active#(eq(s(X),s(Y))) -> eq#(X,Y) active#(inf(X)) -> active#(X) -> active#(inf(X)) -> s#(X) active#(inf(X)) -> active#(X) -> active#(inf(X)) -> inf#(s(X)) active#(inf(X)) -> active#(X) -> active#(inf(X)) -> cons#(X,inf(s(X))) active#(inf(X)) -> active#(X) -> active#(take(s(X),cons(Y,L))) -> take#(X,L) active#(inf(X)) -> active#(X) -> active#(take(s(X),cons(Y,L))) -> cons#(Y,take(X,L)) active#(inf(X)) -> active#(X) -> active#(length(cons(X,L))) -> length#(L) active#(inf(X)) -> active#(X) -> active#(length(cons(X,L))) -> s#(length(L)) active#(inf(X)) -> active#(X) -> active#(inf(X)) -> active#(X) active#(inf(X)) -> active#(X) -> active#(inf(X)) -> inf#(active(X)) active#(inf(X)) -> active#(X) -> active#(take(X1,X2)) -> active#(X1) active#(inf(X)) -> active#(X) -> active#(take(X1,X2)) -> take#(active(X1),X2) active#(inf(X)) -> active#(X) -> active#(take(X1,X2)) -> active#(X2) active#(inf(X)) -> active#(X) -> active#(take(X1,X2)) -> take#(X1,active(X2)) active#(inf(X)) -> active#(X) -> active#(length(X)) -> active#(X) active#(inf(X)) -> active#(X) -> active#(length(X)) -> length#(active(X)) SCC Processor: #sccs: 9 #rules: 25 #arcs: 282/2025 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)) Open DPs: proper#(length(X)) -> proper#(X) proper#(take(X1,X2)) -> proper#(X1) proper#(take(X1,X2)) -> proper#(X2) proper#(cons(X1,X2)) -> proper#(X1) proper#(cons(X1,X2)) -> proper#(X2) proper#(inf(X)) -> proper#(X) proper#(s(X)) -> proper#(X) proper#(eq(X1,X2)) -> proper#(X1) proper#(eq(X1,X2)) -> proper#(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)) Matrix Interpretation Processor: dimension: 1 interpretation: [proper#](x0) = x0 + 1, [top](x0) = 0, [ok](x0) = 0, [proper](x0) = x0, [length](x0) = x0 + 1, [nil] = 0, [take](x0, x1) = x0 + x1, [cons](x0, x1) = x0 + x1 + 1, [inf](x0) = x0 + 1, [false] = 0, [s](x0) = x0 + 1, [mark](x0) = 0, [true] = 0, [active](x0) = x0 + 1, [eq](x0, x1) = x0 + x1, [0] = 0 orientation: proper#(length(X)) = X + 2 >= X + 1 = proper#(X) proper#(take(X1,X2)) = X1 + X2 + 1 >= X1 + 1 = proper#(X1) proper#(take(X1,X2)) = X1 + X2 + 1 >= X2 + 1 = proper#(X2) proper#(cons(X1,X2)) = X1 + X2 + 2 >= X1 + 1 = proper#(X1) proper#(cons(X1,X2)) = X1 + X2 + 2 >= X2 + 1 = proper#(X2) proper#(inf(X)) = X + 2 >= X + 1 = proper#(X) proper#(s(X)) = X + 2 >= X + 1 = proper#(X) proper#(eq(X1,X2)) = X1 + X2 + 1 >= X1 + 1 = proper#(X1) proper#(eq(X1,X2)) = X1 + X2 + 1 >= X2 + 1 = proper#(X2) active(eq(0(),0())) = 1 >= 0 = mark(true()) active(eq(s(X),s(Y))) = X + Y + 3 >= 0 = mark(eq(X,Y)) active(eq(X,Y)) = X + Y + 1 >= 0 = mark(false()) active(inf(X)) = X + 2 >= 0 = mark(cons(X,inf(s(X)))) active(take(0(),X)) = X + 1 >= 0 = mark(nil()) active(take(s(X),cons(Y,L))) = L + X + Y + 3 >= 0 = mark(cons(Y,take(X,L))) active(length(nil())) = 2 >= 0 = mark(0()) active(length(cons(X,L))) = L + X + 3 >= 0 = mark(s(length(L))) active(inf(X)) = X + 2 >= X + 2 = inf(active(X)) active(take(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = take(active(X1),X2) active(take(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = take(X1,active(X2)) active(length(X)) = X + 2 >= X + 2 = length(active(X)) inf(mark(X)) = 1 >= 0 = mark(inf(X)) take(mark(X1),X2) = X2 >= 0 = mark(take(X1,X2)) take(X1,mark(X2)) = X1 >= 0 = mark(take(X1,X2)) length(mark(X)) = 1 >= 0 = mark(length(X)) proper(eq(X1,X2)) = X1 + X2 >= X1 + X2 = eq(proper(X1),proper(X2)) proper(0()) = 0 >= 0 = ok(0()) proper(true()) = 0 >= 0 = ok(true()) proper(s(X)) = X + 1 >= X + 1 = s(proper(X)) proper(false()) = 0 >= 0 = ok(false()) proper(inf(X)) = X + 1 >= X + 1 = inf(proper(X)) proper(cons(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = cons(proper(X1),proper(X2)) proper(take(X1,X2)) = X1 + X2 >= X1 + X2 = take(proper(X1),proper(X2)) proper(nil()) = 0 >= 0 = ok(nil()) proper(length(X)) = X + 1 >= X + 1 = length(proper(X)) eq(ok(X1),ok(X2)) = 0 >= 0 = ok(eq(X1,X2)) s(ok(X)) = 1 >= 0 = ok(s(X)) inf(ok(X)) = 1 >= 0 = ok(inf(X)) cons(ok(X1),ok(X2)) = 1 >= 0 = ok(cons(X1,X2)) take(ok(X1),ok(X2)) = 0 >= 0 = ok(take(X1,X2)) length(ok(X)) = 1 >= 0 = ok(length(X)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) problem: DPs: proper#(take(X1,X2)) -> proper#(X1) proper#(take(X1,X2)) -> proper#(X2) proper#(eq(X1,X2)) -> proper#(X1) proper#(eq(X1,X2)) -> proper#(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)) Matrix Interpretation Processor: dimension: 1 interpretation: [proper#](x0) = x0 + 1, [top](x0) = 0, [ok](x0) = x0, [proper](x0) = x0, [length](x0) = 1, [nil] = 0, [take](x0, x1) = x0 + x1 + 1, [cons](x0, x1) = 0, [inf](x0) = 1, [false] = 0, [s](x0) = 0, [mark](x0) = 1, [true] = 0, [active](x0) = x0, [eq](x0, x1) = x0 + x1 + 1, [0] = 0 orientation: proper#(take(X1,X2)) = X1 + X2 + 2 >= X1 + 1 = proper#(X1) proper#(take(X1,X2)) = X1 + X2 + 2 >= X2 + 1 = proper#(X2) proper#(eq(X1,X2)) = X1 + X2 + 2 >= X1 + 1 = proper#(X1) proper#(eq(X1,X2)) = X1 + X2 + 2 >= X2 + 1 = proper#(X2) active(eq(0(),0())) = 1 >= 1 = mark(true()) active(eq(s(X),s(Y))) = 1 >= 1 = mark(eq(X,Y)) active(eq(X,Y)) = X + Y + 1 >= 1 = mark(false()) active(inf(X)) = 1 >= 1 = mark(cons(X,inf(s(X)))) active(take(0(),X)) = X + 1 >= 1 = mark(nil()) active(take(s(X),cons(Y,L))) = 1 >= 1 = mark(cons(Y,take(X,L))) active(length(nil())) = 1 >= 1 = mark(0()) active(length(cons(X,L))) = 1 >= 1 = mark(s(length(L))) active(inf(X)) = 1 >= 1 = inf(active(X)) active(take(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = take(active(X1),X2) active(take(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = take(X1,active(X2)) active(length(X)) = 1 >= 1 = length(active(X)) inf(mark(X)) = 1 >= 1 = mark(inf(X)) take(mark(X1),X2) = X2 + 2 >= 1 = mark(take(X1,X2)) take(X1,mark(X2)) = X1 + 2 >= 1 = mark(take(X1,X2)) length(mark(X)) = 1 >= 1 = mark(length(X)) proper(eq(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = eq(proper(X1),proper(X2)) proper(0()) = 0 >= 0 = ok(0()) proper(true()) = 0 >= 0 = ok(true()) proper(s(X)) = 0 >= 0 = s(proper(X)) proper(false()) = 0 >= 0 = ok(false()) proper(inf(X)) = 1 >= 1 = inf(proper(X)) proper(cons(X1,X2)) = 0 >= 0 = cons(proper(X1),proper(X2)) proper(take(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = take(proper(X1),proper(X2)) proper(nil()) = 0 >= 0 = ok(nil()) proper(length(X)) = 1 >= 1 = length(proper(X)) eq(ok(X1),ok(X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = ok(eq(X1,X2)) s(ok(X)) = 0 >= 0 = ok(s(X)) inf(ok(X)) = 1 >= 1 = ok(inf(X)) cons(ok(X1),ok(X2)) = 0 >= 0 = ok(cons(X1,X2)) take(ok(X1),ok(X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = ok(take(X1,X2)) length(ok(X)) = 1 >= 1 = ok(length(X)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(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)) 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)) Matrix Interpretation Processor: dimension: 1 interpretation: [eq#](x0, x1) = x1 + 1, [top](x0) = 0, [ok](x0) = x0 + 1, [proper](x0) = 1, [length](x0) = x0, [nil] = 0, [take](x0, x1) = x1, [cons](x0, x1) = x0, [inf](x0) = x0, [false] = 0, [s](x0) = x0, [mark](x0) = 0, [true] = 0, [active](x0) = x0, [eq](x0, x1) = x1, [0] = 0 orientation: eq#(ok(X1),ok(X2)) = X2 + 2 >= X2 + 1 = eq#(X1,X2) active(eq(0(),0())) = 0 >= 0 = mark(true()) active(eq(s(X),s(Y))) = Y >= 0 = mark(eq(X,Y)) active(eq(X,Y)) = Y >= 0 = mark(false()) active(inf(X)) = X >= 0 = mark(cons(X,inf(s(X)))) active(take(0(),X)) = X >= 0 = mark(nil()) active(take(s(X),cons(Y,L))) = Y >= 0 = mark(cons(Y,take(X,L))) active(length(nil())) = 0 >= 0 = mark(0()) active(length(cons(X,L))) = X >= 0 = mark(s(length(L))) active(inf(X)) = X >= X = inf(active(X)) active(take(X1,X2)) = X2 >= X2 = take(active(X1),X2) active(take(X1,X2)) = X2 >= X2 = take(X1,active(X2)) active(length(X)) = X >= X = length(active(X)) inf(mark(X)) = 0 >= 0 = mark(inf(X)) take(mark(X1),X2) = X2 >= 0 = mark(take(X1,X2)) take(X1,mark(X2)) = 0 >= 0 = mark(take(X1,X2)) length(mark(X)) = 0 >= 0 = mark(length(X)) proper(eq(X1,X2)) = 1 >= 1 = eq(proper(X1),proper(X2)) proper(0()) = 1 >= 1 = ok(0()) proper(true()) = 1 >= 1 = ok(true()) proper(s(X)) = 1 >= 1 = s(proper(X)) proper(false()) = 1 >= 1 = ok(false()) proper(inf(X)) = 1 >= 1 = inf(proper(X)) proper(cons(X1,X2)) = 1 >= 1 = cons(proper(X1),proper(X2)) proper(take(X1,X2)) = 1 >= 1 = take(proper(X1),proper(X2)) proper(nil()) = 1 >= 1 = ok(nil()) proper(length(X)) = 1 >= 1 = length(proper(X)) eq(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(eq(X1,X2)) s(ok(X)) = X + 1 >= X + 1 = ok(s(X)) inf(ok(X)) = X + 1 >= X + 1 = ok(inf(X)) cons(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(cons(X1,X2)) take(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(take(X1,X2)) length(ok(X)) = X + 1 >= X + 1 = ok(length(X)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) problem: DPs: TRS: active(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: active#(length(X)) -> active#(X) active#(take(X1,X2)) -> active#(X2) active#(take(X1,X2)) -> active#(X1) active#(inf(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)) Matrix Interpretation Processor: dimension: 1 interpretation: [active#](x0) = x0 + 1, [top](x0) = 0, [ok](x0) = x0, [proper](x0) = x0, [length](x0) = x0 + 1, [nil] = 1, [take](x0, x1) = x0 + x1 + 1, [cons](x0, x1) = x1, [inf](x0) = x0 + 1, [false] = 0, [s](x0) = x0, [mark](x0) = x0, [true] = 0, [active](x0) = x0, [eq](x0, x1) = 0, [0] = 0 orientation: active#(length(X)) = X + 2 >= X + 1 = active#(X) active#(take(X1,X2)) = X1 + X2 + 2 >= X2 + 1 = active#(X2) active#(take(X1,X2)) = X1 + X2 + 2 >= X1 + 1 = active#(X1) active#(inf(X)) = X + 2 >= X + 1 = active#(X) active(eq(0(),0())) = 0 >= 0 = mark(true()) active(eq(s(X),s(Y))) = 0 >= 0 = mark(eq(X,Y)) active(eq(X,Y)) = 0 >= 0 = mark(false()) active(inf(X)) = X + 1 >= X + 1 = mark(cons(X,inf(s(X)))) active(take(0(),X)) = X + 1 >= 1 = mark(nil()) active(take(s(X),cons(Y,L))) = L + X + 1 >= L + X + 1 = mark(cons(Y,take(X,L))) active(length(nil())) = 2 >= 0 = mark(0()) active(length(cons(X,L))) = L + 1 >= L + 1 = mark(s(length(L))) active(inf(X)) = X + 1 >= X + 1 = inf(active(X)) active(take(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = take(active(X1),X2) active(take(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = take(X1,active(X2)) active(length(X)) = X + 1 >= X + 1 = length(active(X)) inf(mark(X)) = X + 1 >= X + 1 = mark(inf(X)) take(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(take(X1,X2)) take(X1,mark(X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(take(X1,X2)) length(mark(X)) = X + 1 >= X + 1 = mark(length(X)) proper(eq(X1,X2)) = 0 >= 0 = eq(proper(X1),proper(X2)) proper(0()) = 0 >= 0 = ok(0()) proper(true()) = 0 >= 0 = ok(true()) proper(s(X)) = X >= X = s(proper(X)) proper(false()) = 0 >= 0 = ok(false()) proper(inf(X)) = X + 1 >= X + 1 = inf(proper(X)) proper(cons(X1,X2)) = X2 >= X2 = cons(proper(X1),proper(X2)) proper(take(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = take(proper(X1),proper(X2)) proper(nil()) = 1 >= 1 = ok(nil()) proper(length(X)) = X + 1 >= X + 1 = length(proper(X)) eq(ok(X1),ok(X2)) = 0 >= 0 = ok(eq(X1,X2)) s(ok(X)) = X >= X = ok(s(X)) inf(ok(X)) = X + 1 >= X + 1 = ok(inf(X)) cons(ok(X1),ok(X2)) = X2 >= X2 = ok(cons(X1,X2)) take(ok(X1),ok(X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = ok(take(X1,X2)) length(ok(X)) = X + 1 >= X + 1 = ok(length(X)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) problem: DPs: TRS: active(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)) Matrix Interpretation Processor: dimension: 1 interpretation: [cons#](x0, x1) = x1 + 1, [top](x0) = 0, [ok](x0) = x0 + 1, [proper](x0) = 1, [length](x0) = x0, [nil] = 0, [take](x0, x1) = x1, [cons](x0, x1) = x0, [inf](x0) = x0, [false] = 0, [s](x0) = x0, [mark](x0) = 0, [true] = 0, [active](x0) = x0, [eq](x0, x1) = x1, [0] = 0 orientation: cons#(ok(X1),ok(X2)) = X2 + 2 >= X2 + 1 = cons#(X1,X2) active(eq(0(),0())) = 0 >= 0 = mark(true()) active(eq(s(X),s(Y))) = Y >= 0 = mark(eq(X,Y)) active(eq(X,Y)) = Y >= 0 = mark(false()) active(inf(X)) = X >= 0 = mark(cons(X,inf(s(X)))) active(take(0(),X)) = X >= 0 = mark(nil()) active(take(s(X),cons(Y,L))) = Y >= 0 = mark(cons(Y,take(X,L))) active(length(nil())) = 0 >= 0 = mark(0()) active(length(cons(X,L))) = X >= 0 = mark(s(length(L))) active(inf(X)) = X >= X = inf(active(X)) active(take(X1,X2)) = X2 >= X2 = take(active(X1),X2) active(take(X1,X2)) = X2 >= X2 = take(X1,active(X2)) active(length(X)) = X >= X = length(active(X)) inf(mark(X)) = 0 >= 0 = mark(inf(X)) take(mark(X1),X2) = X2 >= 0 = mark(take(X1,X2)) take(X1,mark(X2)) = 0 >= 0 = mark(take(X1,X2)) length(mark(X)) = 0 >= 0 = mark(length(X)) proper(eq(X1,X2)) = 1 >= 1 = eq(proper(X1),proper(X2)) proper(0()) = 1 >= 1 = ok(0()) proper(true()) = 1 >= 1 = ok(true()) proper(s(X)) = 1 >= 1 = s(proper(X)) proper(false()) = 1 >= 1 = ok(false()) proper(inf(X)) = 1 >= 1 = inf(proper(X)) proper(cons(X1,X2)) = 1 >= 1 = cons(proper(X1),proper(X2)) proper(take(X1,X2)) = 1 >= 1 = take(proper(X1),proper(X2)) proper(nil()) = 1 >= 1 = ok(nil()) proper(length(X)) = 1 >= 1 = length(proper(X)) eq(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(eq(X1,X2)) s(ok(X)) = X + 1 >= X + 1 = ok(s(X)) inf(ok(X)) = X + 1 >= X + 1 = ok(inf(X)) cons(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(cons(X1,X2)) take(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(take(X1,X2)) length(ok(X)) = X + 1 >= X + 1 = ok(length(X)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) problem: DPs: TRS: active(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)) Matrix Interpretation Processor: dimension: 1 interpretation: [s#](x0) = x0 + 1, [top](x0) = 0, [ok](x0) = x0 + 1, [proper](x0) = 1, [length](x0) = x0, [nil] = 0, [take](x0, x1) = x1, [cons](x0, x1) = x1, [inf](x0) = x0, [false] = 0, [s](x0) = x0, [mark](x0) = 0, [true] = 0, [active](x0) = x0, [eq](x0, x1) = x1, [0] = 0 orientation: s#(ok(X)) = X + 2 >= X + 1 = s#(X) active(eq(0(),0())) = 0 >= 0 = mark(true()) active(eq(s(X),s(Y))) = Y >= 0 = mark(eq(X,Y)) active(eq(X,Y)) = Y >= 0 = mark(false()) active(inf(X)) = X >= 0 = mark(cons(X,inf(s(X)))) active(take(0(),X)) = X >= 0 = mark(nil()) active(take(s(X),cons(Y,L))) = L >= 0 = mark(cons(Y,take(X,L))) active(length(nil())) = 0 >= 0 = mark(0()) active(length(cons(X,L))) = L >= 0 = mark(s(length(L))) active(inf(X)) = X >= X = inf(active(X)) active(take(X1,X2)) = X2 >= X2 = take(active(X1),X2) active(take(X1,X2)) = X2 >= X2 = take(X1,active(X2)) active(length(X)) = X >= X = length(active(X)) inf(mark(X)) = 0 >= 0 = mark(inf(X)) take(mark(X1),X2) = X2 >= 0 = mark(take(X1,X2)) take(X1,mark(X2)) = 0 >= 0 = mark(take(X1,X2)) length(mark(X)) = 0 >= 0 = mark(length(X)) proper(eq(X1,X2)) = 1 >= 1 = eq(proper(X1),proper(X2)) proper(0()) = 1 >= 1 = ok(0()) proper(true()) = 1 >= 1 = ok(true()) proper(s(X)) = 1 >= 1 = s(proper(X)) proper(false()) = 1 >= 1 = ok(false()) proper(inf(X)) = 1 >= 1 = inf(proper(X)) proper(cons(X1,X2)) = 1 >= 1 = cons(proper(X1),proper(X2)) proper(take(X1,X2)) = 1 >= 1 = take(proper(X1),proper(X2)) proper(nil()) = 1 >= 1 = ok(nil()) proper(length(X)) = 1 >= 1 = length(proper(X)) eq(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(eq(X1,X2)) s(ok(X)) = X + 1 >= X + 1 = ok(s(X)) inf(ok(X)) = X + 1 >= X + 1 = ok(inf(X)) cons(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(cons(X1,X2)) take(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(take(X1,X2)) length(ok(X)) = X + 1 >= X + 1 = ok(length(X)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) problem: DPs: TRS: active(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#(ok(X)) -> inf#(X) inf#(mark(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)) Matrix Interpretation Processor: dimension: 1 interpretation: [inf#](x0) = x0 + 1, [top](x0) = 0, [ok](x0) = x0 + 1, [proper](x0) = x0 + 1, [length](x0) = x0, [nil] = 1, [take](x0, x1) = x0 + 1, [cons](x0, x1) = x1, [inf](x0) = x0, [false] = 0, [s](x0) = x0, [mark](x0) = x0, [true] = 0, [active](x0) = x0, [eq](x0, x1) = x0, [0] = 0 orientation: inf#(ok(X)) = X + 2 >= X + 1 = inf#(X) inf#(mark(X)) = X + 1 >= X + 1 = inf#(X) active(eq(0(),0())) = 0 >= 0 = mark(true()) active(eq(s(X),s(Y))) = X >= X = mark(eq(X,Y)) active(eq(X,Y)) = X >= 0 = mark(false()) active(inf(X)) = X >= X = mark(cons(X,inf(s(X)))) active(take(0(),X)) = 1 >= 1 = mark(nil()) active(take(s(X),cons(Y,L))) = X + 1 >= X + 1 = mark(cons(Y,take(X,L))) active(length(nil())) = 1 >= 0 = mark(0()) active(length(cons(X,L))) = L >= L = mark(s(length(L))) active(inf(X)) = X >= X = inf(active(X)) active(take(X1,X2)) = X1 + 1 >= X1 + 1 = take(active(X1),X2) active(take(X1,X2)) = X1 + 1 >= X1 + 1 = take(X1,active(X2)) active(length(X)) = X >= X = length(active(X)) inf(mark(X)) = X >= X = mark(inf(X)) take(mark(X1),X2) = X1 + 1 >= X1 + 1 = mark(take(X1,X2)) take(X1,mark(X2)) = X1 + 1 >= X1 + 1 = mark(take(X1,X2)) length(mark(X)) = X >= X = mark(length(X)) proper(eq(X1,X2)) = X1 + 1 >= X1 + 1 = eq(proper(X1),proper(X2)) proper(0()) = 1 >= 1 = ok(0()) proper(true()) = 1 >= 1 = ok(true()) proper(s(X)) = X + 1 >= X + 1 = s(proper(X)) proper(false()) = 1 >= 1 = ok(false()) proper(inf(X)) = X + 1 >= X + 1 = inf(proper(X)) proper(cons(X1,X2)) = X2 + 1 >= X2 + 1 = cons(proper(X1),proper(X2)) proper(take(X1,X2)) = X1 + 2 >= X1 + 2 = take(proper(X1),proper(X2)) proper(nil()) = 2 >= 2 = ok(nil()) proper(length(X)) = X + 1 >= X + 1 = length(proper(X)) eq(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(eq(X1,X2)) s(ok(X)) = X + 1 >= X + 1 = ok(s(X)) inf(ok(X)) = X + 1 >= X + 1 = ok(inf(X)) cons(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(cons(X1,X2)) take(ok(X1),ok(X2)) = X1 + 2 >= X1 + 2 = ok(take(X1,X2)) length(ok(X)) = X + 1 >= X + 1 = ok(length(X)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) problem: DPs: inf#(mark(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)) Matrix Interpretation Processor: dimension: 1 interpretation: [inf#](x0) = x0 + 1, [top](x0) = 0, [ok](x0) = 0, [proper](x0) = 0, [length](x0) = x0, [nil] = 0, [take](x0, x1) = x0 + x1, [cons](x0, x1) = 0, [inf](x0) = x0, [false] = 0, [s](x0) = 0, [mark](x0) = x0 + 1, [true] = 0, [active](x0) = x0 + 1, [eq](x0, x1) = 0, [0] = 0 orientation: inf#(mark(X)) = X + 2 >= X + 1 = inf#(X) active(eq(0(),0())) = 1 >= 1 = mark(true()) active(eq(s(X),s(Y))) = 1 >= 1 = mark(eq(X,Y)) active(eq(X,Y)) = 1 >= 1 = mark(false()) active(inf(X)) = X + 1 >= 1 = mark(cons(X,inf(s(X)))) active(take(0(),X)) = X + 1 >= 1 = mark(nil()) active(take(s(X),cons(Y,L))) = 1 >= 1 = mark(cons(Y,take(X,L))) active(length(nil())) = 1 >= 1 = mark(0()) active(length(cons(X,L))) = 1 >= 1 = mark(s(length(L))) active(inf(X)) = X + 1 >= X + 1 = inf(active(X)) active(take(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = take(active(X1),X2) active(take(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = take(X1,active(X2)) active(length(X)) = X + 1 >= X + 1 = length(active(X)) inf(mark(X)) = X + 1 >= X + 1 = mark(inf(X)) take(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(take(X1,X2)) take(X1,mark(X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(take(X1,X2)) length(mark(X)) = X + 1 >= X + 1 = mark(length(X)) proper(eq(X1,X2)) = 0 >= 0 = eq(proper(X1),proper(X2)) proper(0()) = 0 >= 0 = ok(0()) proper(true()) = 0 >= 0 = ok(true()) proper(s(X)) = 0 >= 0 = s(proper(X)) proper(false()) = 0 >= 0 = ok(false()) proper(inf(X)) = 0 >= 0 = inf(proper(X)) proper(cons(X1,X2)) = 0 >= 0 = cons(proper(X1),proper(X2)) proper(take(X1,X2)) = 0 >= 0 = take(proper(X1),proper(X2)) proper(nil()) = 0 >= 0 = ok(nil()) proper(length(X)) = 0 >= 0 = length(proper(X)) eq(ok(X1),ok(X2)) = 0 >= 0 = ok(eq(X1,X2)) s(ok(X)) = 0 >= 0 = ok(s(X)) inf(ok(X)) = 0 >= 0 = ok(inf(X)) cons(ok(X1),ok(X2)) = 0 >= 0 = ok(cons(X1,X2)) take(ok(X1),ok(X2)) = 0 >= 0 = ok(take(X1,X2)) length(ok(X)) = 0 >= 0 = ok(length(X)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) problem: DPs: TRS: active(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#(ok(X1),ok(X2)) -> take#(X1,X2) take#(X1,mark(X2)) -> take#(X1,X2) take#(mark(X1),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)) Matrix Interpretation Processor: dimension: 1 interpretation: [take#](x0, x1) = x0, [top](x0) = 0, [ok](x0) = x0, [proper](x0) = 0, [length](x0) = x0, [nil] = 0, [take](x0, x1) = x0 + x1, [cons](x0, x1) = 0, [inf](x0) = x0, [false] = 0, [s](x0) = 0, [mark](x0) = x0 + 1, [true] = 0, [active](x0) = x0 + 1, [eq](x0, x1) = 0, [0] = 0 orientation: take#(ok(X1),ok(X2)) = X1 >= X1 = take#(X1,X2) take#(X1,mark(X2)) = X1 >= X1 = take#(X1,X2) take#(mark(X1),X2) = X1 + 1 >= X1 = take#(X1,X2) active(eq(0(),0())) = 1 >= 1 = mark(true()) active(eq(s(X),s(Y))) = 1 >= 1 = mark(eq(X,Y)) active(eq(X,Y)) = 1 >= 1 = mark(false()) active(inf(X)) = X + 1 >= 1 = mark(cons(X,inf(s(X)))) active(take(0(),X)) = X + 1 >= 1 = mark(nil()) active(take(s(X),cons(Y,L))) = 1 >= 1 = mark(cons(Y,take(X,L))) active(length(nil())) = 1 >= 1 = mark(0()) active(length(cons(X,L))) = 1 >= 1 = mark(s(length(L))) active(inf(X)) = X + 1 >= X + 1 = inf(active(X)) active(take(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = take(active(X1),X2) active(take(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = take(X1,active(X2)) active(length(X)) = X + 1 >= X + 1 = length(active(X)) inf(mark(X)) = X + 1 >= X + 1 = mark(inf(X)) take(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(take(X1,X2)) take(X1,mark(X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(take(X1,X2)) length(mark(X)) = X + 1 >= X + 1 = mark(length(X)) proper(eq(X1,X2)) = 0 >= 0 = eq(proper(X1),proper(X2)) proper(0()) = 0 >= 0 = ok(0()) proper(true()) = 0 >= 0 = ok(true()) proper(s(X)) = 0 >= 0 = s(proper(X)) proper(false()) = 0 >= 0 = ok(false()) proper(inf(X)) = 0 >= 0 = inf(proper(X)) proper(cons(X1,X2)) = 0 >= 0 = cons(proper(X1),proper(X2)) proper(take(X1,X2)) = 0 >= 0 = take(proper(X1),proper(X2)) proper(nil()) = 0 >= 0 = ok(nil()) proper(length(X)) = 0 >= 0 = length(proper(X)) eq(ok(X1),ok(X2)) = 0 >= 0 = ok(eq(X1,X2)) s(ok(X)) = 0 >= 0 = ok(s(X)) inf(ok(X)) = X >= X = ok(inf(X)) cons(ok(X1),ok(X2)) = 0 >= 0 = ok(cons(X1,X2)) take(ok(X1),ok(X2)) = X1 + X2 >= X1 + X2 = ok(take(X1,X2)) length(ok(X)) = X >= X = ok(length(X)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) problem: DPs: take#(ok(X1),ok(X2)) -> take#(X1,X2) take#(X1,mark(X2)) -> take#(X1,X2) TRS: active(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)) Matrix Interpretation Processor: dimension: 1 interpretation: [take#](x0, x1) = x1 + 1, [top](x0) = 0, [ok](x0) = x0 + 1, [proper](x0) = 1, [length](x0) = x0, [nil] = 0, [take](x0, x1) = x1, [cons](x0, x1) = x1, [inf](x0) = x0, [false] = 0, [s](x0) = x0, [mark](x0) = x0, [true] = 0, [active](x0) = x0 + 1, [eq](x0, x1) = x0, [0] = 0 orientation: take#(ok(X1),ok(X2)) = X2 + 2 >= X2 + 1 = take#(X1,X2) take#(X1,mark(X2)) = X2 + 1 >= X2 + 1 = take#(X1,X2) active(eq(0(),0())) = 1 >= 0 = mark(true()) active(eq(s(X),s(Y))) = X + 1 >= X = mark(eq(X,Y)) active(eq(X,Y)) = X + 1 >= 0 = mark(false()) active(inf(X)) = X + 1 >= X = mark(cons(X,inf(s(X)))) active(take(0(),X)) = X + 1 >= 0 = mark(nil()) active(take(s(X),cons(Y,L))) = L + 1 >= L = mark(cons(Y,take(X,L))) active(length(nil())) = 1 >= 0 = mark(0()) active(length(cons(X,L))) = L + 1 >= L = mark(s(length(L))) active(inf(X)) = X + 1 >= X + 1 = inf(active(X)) active(take(X1,X2)) = X2 + 1 >= X2 = take(active(X1),X2) active(take(X1,X2)) = X2 + 1 >= X2 + 1 = take(X1,active(X2)) active(length(X)) = X + 1 >= X + 1 = length(active(X)) inf(mark(X)) = X >= X = mark(inf(X)) take(mark(X1),X2) = X2 >= X2 = mark(take(X1,X2)) take(X1,mark(X2)) = X2 >= X2 = mark(take(X1,X2)) length(mark(X)) = X >= X = mark(length(X)) proper(eq(X1,X2)) = 1 >= 1 = eq(proper(X1),proper(X2)) proper(0()) = 1 >= 1 = ok(0()) proper(true()) = 1 >= 1 = ok(true()) proper(s(X)) = 1 >= 1 = s(proper(X)) proper(false()) = 1 >= 1 = ok(false()) proper(inf(X)) = 1 >= 1 = inf(proper(X)) proper(cons(X1,X2)) = 1 >= 1 = cons(proper(X1),proper(X2)) proper(take(X1,X2)) = 1 >= 1 = take(proper(X1),proper(X2)) proper(nil()) = 1 >= 1 = ok(nil()) proper(length(X)) = 1 >= 1 = length(proper(X)) eq(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(eq(X1,X2)) s(ok(X)) = X + 1 >= X + 1 = ok(s(X)) inf(ok(X)) = X + 1 >= X + 1 = ok(inf(X)) cons(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(cons(X1,X2)) take(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(take(X1,X2)) length(ok(X)) = X + 1 >= X + 1 = ok(length(X)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) problem: DPs: 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)) Matrix Interpretation Processor: dimension: 1 interpretation: [take#](x0, x1) = x1 + 1, [top](x0) = 0, [ok](x0) = x0, [proper](x0) = x0, [length](x0) = x0, [nil] = 0, [take](x0, x1) = x0 + x1, [cons](x0, x1) = x0, [inf](x0) = x0, [false] = 1, [s](x0) = 0, [mark](x0) = x0 + 1, [true] = 0, [active](x0) = x0 + 1, [eq](x0, x1) = 1, [0] = 0 orientation: take#(X1,mark(X2)) = X2 + 2 >= X2 + 1 = take#(X1,X2) active(eq(0(),0())) = 2 >= 1 = mark(true()) active(eq(s(X),s(Y))) = 2 >= 2 = mark(eq(X,Y)) active(eq(X,Y)) = 2 >= 2 = mark(false()) active(inf(X)) = X + 1 >= X + 1 = mark(cons(X,inf(s(X)))) active(take(0(),X)) = X + 1 >= 1 = mark(nil()) active(take(s(X),cons(Y,L))) = Y + 1 >= Y + 1 = mark(cons(Y,take(X,L))) active(length(nil())) = 1 >= 1 = mark(0()) active(length(cons(X,L))) = X + 1 >= 1 = mark(s(length(L))) active(inf(X)) = X + 1 >= X + 1 = inf(active(X)) active(take(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = take(active(X1),X2) active(take(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = take(X1,active(X2)) active(length(X)) = X + 1 >= X + 1 = length(active(X)) inf(mark(X)) = X + 1 >= X + 1 = mark(inf(X)) take(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(take(X1,X2)) take(X1,mark(X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(take(X1,X2)) length(mark(X)) = X + 1 >= X + 1 = mark(length(X)) proper(eq(X1,X2)) = 1 >= 1 = eq(proper(X1),proper(X2)) proper(0()) = 0 >= 0 = ok(0()) proper(true()) = 0 >= 0 = ok(true()) proper(s(X)) = 0 >= 0 = s(proper(X)) proper(false()) = 1 >= 1 = ok(false()) proper(inf(X)) = X >= X = inf(proper(X)) proper(cons(X1,X2)) = X1 >= X1 = cons(proper(X1),proper(X2)) proper(take(X1,X2)) = X1 + X2 >= X1 + X2 = take(proper(X1),proper(X2)) proper(nil()) = 0 >= 0 = ok(nil()) proper(length(X)) = X >= X = length(proper(X)) eq(ok(X1),ok(X2)) = 1 >= 1 = ok(eq(X1,X2)) s(ok(X)) = 0 >= 0 = ok(s(X)) inf(ok(X)) = X >= X = ok(inf(X)) cons(ok(X1),ok(X2)) = X1 >= X1 = ok(cons(X1,X2)) take(ok(X1),ok(X2)) = X1 + X2 >= X1 + X2 = ok(take(X1,X2)) length(ok(X)) = X >= X = ok(length(X)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(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)) 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#(ok(X)) -> length#(X) length#(mark(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)) Matrix Interpretation Processor: dimension: 1 interpretation: [length#](x0) = x0 + 1, [top](x0) = 0, [ok](x0) = x0 + 1, [proper](x0) = x0 + 1, [length](x0) = x0, [nil] = 1, [take](x0, x1) = x0 + 1, [cons](x0, x1) = x1, [inf](x0) = x0, [false] = 0, [s](x0) = x0, [mark](x0) = x0, [true] = 0, [active](x0) = x0, [eq](x0, x1) = x0, [0] = 0 orientation: length#(ok(X)) = X + 2 >= X + 1 = length#(X) length#(mark(X)) = X + 1 >= X + 1 = length#(X) active(eq(0(),0())) = 0 >= 0 = mark(true()) active(eq(s(X),s(Y))) = X >= X = mark(eq(X,Y)) active(eq(X,Y)) = X >= 0 = mark(false()) active(inf(X)) = X >= X = mark(cons(X,inf(s(X)))) active(take(0(),X)) = 1 >= 1 = mark(nil()) active(take(s(X),cons(Y,L))) = X + 1 >= X + 1 = mark(cons(Y,take(X,L))) active(length(nil())) = 1 >= 0 = mark(0()) active(length(cons(X,L))) = L >= L = mark(s(length(L))) active(inf(X)) = X >= X = inf(active(X)) active(take(X1,X2)) = X1 + 1 >= X1 + 1 = take(active(X1),X2) active(take(X1,X2)) = X1 + 1 >= X1 + 1 = take(X1,active(X2)) active(length(X)) = X >= X = length(active(X)) inf(mark(X)) = X >= X = mark(inf(X)) take(mark(X1),X2) = X1 + 1 >= X1 + 1 = mark(take(X1,X2)) take(X1,mark(X2)) = X1 + 1 >= X1 + 1 = mark(take(X1,X2)) length(mark(X)) = X >= X = mark(length(X)) proper(eq(X1,X2)) = X1 + 1 >= X1 + 1 = eq(proper(X1),proper(X2)) proper(0()) = 1 >= 1 = ok(0()) proper(true()) = 1 >= 1 = ok(true()) proper(s(X)) = X + 1 >= X + 1 = s(proper(X)) proper(false()) = 1 >= 1 = ok(false()) proper(inf(X)) = X + 1 >= X + 1 = inf(proper(X)) proper(cons(X1,X2)) = X2 + 1 >= X2 + 1 = cons(proper(X1),proper(X2)) proper(take(X1,X2)) = X1 + 2 >= X1 + 2 = take(proper(X1),proper(X2)) proper(nil()) = 2 >= 2 = ok(nil()) proper(length(X)) = X + 1 >= X + 1 = length(proper(X)) eq(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(eq(X1,X2)) s(ok(X)) = X + 1 >= X + 1 = ok(s(X)) inf(ok(X)) = X + 1 >= X + 1 = ok(inf(X)) cons(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(cons(X1,X2)) take(ok(X1),ok(X2)) = X1 + 2 >= X1 + 2 = ok(take(X1,X2)) length(ok(X)) = X + 1 >= X + 1 = ok(length(X)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) problem: DPs: length#(mark(X)) -> length#(X) TRS: active(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)) Matrix Interpretation Processor: dimension: 1 interpretation: [length#](x0) = x0 + 1, [top](x0) = 0, [ok](x0) = 0, [proper](x0) = 0, [length](x0) = x0, [nil] = 0, [take](x0, x1) = x0 + x1, [cons](x0, x1) = 0, [inf](x0) = x0, [false] = 0, [s](x0) = 0, [mark](x0) = x0 + 1, [true] = 0, [active](x0) = x0 + 1, [eq](x0, x1) = 0, [0] = 0 orientation: length#(mark(X)) = X + 2 >= X + 1 = length#(X) active(eq(0(),0())) = 1 >= 1 = mark(true()) active(eq(s(X),s(Y))) = 1 >= 1 = mark(eq(X,Y)) active(eq(X,Y)) = 1 >= 1 = mark(false()) active(inf(X)) = X + 1 >= 1 = mark(cons(X,inf(s(X)))) active(take(0(),X)) = X + 1 >= 1 = mark(nil()) active(take(s(X),cons(Y,L))) = 1 >= 1 = mark(cons(Y,take(X,L))) active(length(nil())) = 1 >= 1 = mark(0()) active(length(cons(X,L))) = 1 >= 1 = mark(s(length(L))) active(inf(X)) = X + 1 >= X + 1 = inf(active(X)) active(take(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = take(active(X1),X2) active(take(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = take(X1,active(X2)) active(length(X)) = X + 1 >= X + 1 = length(active(X)) inf(mark(X)) = X + 1 >= X + 1 = mark(inf(X)) take(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(take(X1,X2)) take(X1,mark(X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(take(X1,X2)) length(mark(X)) = X + 1 >= X + 1 = mark(length(X)) proper(eq(X1,X2)) = 0 >= 0 = eq(proper(X1),proper(X2)) proper(0()) = 0 >= 0 = ok(0()) proper(true()) = 0 >= 0 = ok(true()) proper(s(X)) = 0 >= 0 = s(proper(X)) proper(false()) = 0 >= 0 = ok(false()) proper(inf(X)) = 0 >= 0 = inf(proper(X)) proper(cons(X1,X2)) = 0 >= 0 = cons(proper(X1),proper(X2)) proper(take(X1,X2)) = 0 >= 0 = take(proper(X1),proper(X2)) proper(nil()) = 0 >= 0 = ok(nil()) proper(length(X)) = 0 >= 0 = length(proper(X)) eq(ok(X1),ok(X2)) = 0 >= 0 = ok(eq(X1,X2)) s(ok(X)) = 0 >= 0 = ok(s(X)) inf(ok(X)) = 0 >= 0 = ok(inf(X)) cons(ok(X1),ok(X2)) = 0 >= 0 = ok(cons(X1,X2)) take(ok(X1),ok(X2)) = 0 >= 0 = ok(take(X1,X2)) length(ok(X)) = 0 >= 0 = ok(length(X)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) problem: DPs: TRS: active(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