/export/starexec/sandbox/solver/bin/starexec_run_ttt2-1.17+nonreach /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files -------------------------------------------------------------------------------- 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(any(X1),X2)) -> cons(any(any(proper(X1))),any(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)) any(X) -> s(X) any(proper(X)) -> any(any(any(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(any(X1),X2)) -> proper#(X2) proper#(cons(any(X1),X2)) -> any#(proper(X2)) proper#(cons(any(X1),X2)) -> proper#(X1) proper#(cons(any(X1),X2)) -> any#(proper(X1)) proper#(cons(any(X1),X2)) -> any#(any(proper(X1))) proper#(cons(any(X1),X2)) -> cons#(any(any(proper(X1))),any(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)) any#(X) -> s#(X) any#(proper(X)) -> any#(X) any#(proper(X)) -> any#(any(X)) any#(proper(X)) -> any#(any(any(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(any(X1),X2)) -> cons(any(any(proper(X1))),any(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)) any(X) -> s(X) any(proper(X)) -> any(any(any(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(any(X1),X2)) -> proper#(X2) proper#(cons(any(X1),X2)) -> any#(proper(X2)) proper#(cons(any(X1),X2)) -> proper#(X1) proper#(cons(any(X1),X2)) -> any#(proper(X1)) proper#(cons(any(X1),X2)) -> any#(any(proper(X1))) proper#(cons(any(X1),X2)) -> cons#(any(any(proper(X1))),any(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)) any#(X) -> s#(X) any#(proper(X)) -> any#(X) any#(proper(X)) -> any#(any(X)) any#(proper(X)) -> any#(any(any(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(any(X1),X2)) -> cons(any(any(proper(X1))),any(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)) any(X) -> s(X) any(proper(X)) -> any(any(any(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(any(X1),X2)) -> cons#(any(any(proper(X1))),any(proper(X2))) top#(mark(X)) -> proper#(X) -> proper#(cons(any(X1),X2)) -> any#(any(proper(X1))) top#(mark(X)) -> proper#(X) -> proper#(cons(any(X1),X2)) -> any#(proper(X1)) top#(mark(X)) -> proper#(X) -> proper#(cons(any(X1),X2)) -> proper#(X1) top#(mark(X)) -> proper#(X) -> proper#(cons(any(X1),X2)) -> any#(proper(X2)) top#(mark(X)) -> proper#(X) -> proper#(cons(any(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) any#(proper(X)) -> any#(any(any(X))) -> any#(proper(X)) -> any#(any(any(X))) any#(proper(X)) -> any#(any(any(X))) -> any#(proper(X)) -> any#(any(X)) any#(proper(X)) -> any#(any(any(X))) -> any#(proper(X)) -> any#(X) any#(proper(X)) -> any#(any(any(X))) -> any#(X) -> s#(X) any#(proper(X)) -> any#(any(X)) -> any#(proper(X)) -> any#(any(any(X))) any#(proper(X)) -> any#(any(X)) -> any#(proper(X)) -> any#(any(X)) any#(proper(X)) -> any#(any(X)) -> any#(proper(X)) -> any#(X) any#(proper(X)) -> any#(any(X)) -> any#(X) -> s#(X) any#(proper(X)) -> any#(X) -> any#(proper(X)) -> any#(any(any(X))) any#(proper(X)) -> any#(X) -> any#(proper(X)) -> any#(any(X)) any#(proper(X)) -> any#(X) -> any#(proper(X)) -> any#(X) any#(proper(X)) -> any#(X) -> any#(X) -> s#(X) any#(X) -> s#(X) -> s#(ok(X)) -> s#(X) proper#(length(X)) -> proper#(X) -> proper#(length(X)) -> length#(proper(X)) proper#(length(X)) -> proper#(X) -> proper#(length(X)) -> proper#(X) proper#(length(X)) -> proper#(X) -> proper#(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(any(X1),X2)) -> cons#(any(any(proper(X1))),any(proper(X2))) proper#(length(X)) -> proper#(X) -> proper#(cons(any(X1),X2)) -> any#(any(proper(X1))) proper#(length(X)) -> proper#(X) -> proper#(cons(any(X1),X2)) -> any#(proper(X1)) proper#(length(X)) -> proper#(X) -> proper#(cons(any(X1),X2)) -> proper#(X1) proper#(length(X)) -> proper#(X) -> proper#(cons(any(X1),X2)) -> any#(proper(X2)) proper#(length(X)) -> proper#(X) -> proper#(cons(any(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(any(X1),X2)) -> cons#(any(any(proper(X1))),any(proper(X2))) proper#(take(X1,X2)) -> proper#(X2) -> proper#(cons(any(X1),X2)) -> any#(any(proper(X1))) proper#(take(X1,X2)) -> proper#(X2) -> proper#(cons(any(X1),X2)) -> any#(proper(X1)) proper#(take(X1,X2)) -> proper#(X2) -> proper#(cons(any(X1),X2)) -> proper#(X1) proper#(take(X1,X2)) -> proper#(X2) -> proper#(cons(any(X1),X2)) -> any#(proper(X2)) proper#(take(X1,X2)) -> proper#(X2) -> proper#(cons(any(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(any(X1),X2)) -> cons#(any(any(proper(X1))),any(proper(X2))) proper#(take(X1,X2)) -> proper#(X1) -> proper#(cons(any(X1),X2)) -> any#(any(proper(X1))) proper#(take(X1,X2)) -> proper#(X1) -> proper#(cons(any(X1),X2)) -> any#(proper(X1)) proper#(take(X1,X2)) -> proper#(X1) -> proper#(cons(any(X1),X2)) -> proper#(X1) proper#(take(X1,X2)) -> proper#(X1) -> proper#(cons(any(X1),X2)) -> any#(proper(X2)) proper#(take(X1,X2)) -> proper#(X1) -> proper#(cons(any(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(any(X1),X2)) -> any#(any(proper(X1))) -> any#(proper(X)) -> any#(any(any(X))) proper#(cons(any(X1),X2)) -> any#(any(proper(X1))) -> any#(proper(X)) -> any#(any(X)) proper#(cons(any(X1),X2)) -> any#(any(proper(X1))) -> any#(proper(X)) -> any#(X) proper#(cons(any(X1),X2)) -> any#(any(proper(X1))) -> any#(X) -> s#(X) proper#(cons(any(X1),X2)) -> any#(proper(X2)) -> any#(proper(X)) -> any#(any(any(X))) proper#(cons(any(X1),X2)) -> any#(proper(X2)) -> any#(proper(X)) -> any#(any(X)) proper#(cons(any(X1),X2)) -> any#(proper(X2)) -> any#(proper(X)) -> any#(X) proper#(cons(any(X1),X2)) -> any#(proper(X2)) -> any#(X) -> s#(X) proper#(cons(any(X1),X2)) -> any#(proper(X1)) -> any#(proper(X)) -> any#(any(any(X))) proper#(cons(any(X1),X2)) -> any#(proper(X1)) -> any#(proper(X)) -> any#(any(X)) proper#(cons(any(X1),X2)) -> any#(proper(X1)) -> any#(proper(X)) -> any#(X) proper#(cons(any(X1),X2)) -> any#(proper(X1)) -> any#(X) -> s#(X) proper#(cons(any(X1),X2)) -> proper#(X2) -> proper#(length(X)) -> length#(proper(X)) proper#(cons(any(X1),X2)) -> proper#(X2) -> proper#(length(X)) -> proper#(X) proper#(cons(any(X1),X2)) -> proper#(X2) -> proper#(take(X1,X2)) -> take#(proper(X1),proper(X2)) proper#(cons(any(X1),X2)) -> proper#(X2) -> proper#(take(X1,X2)) -> proper#(X1) proper#(cons(any(X1),X2)) -> proper#(X2) -> proper#(take(X1,X2)) -> proper#(X2) proper#(cons(any(X1),X2)) -> proper#(X2) -> proper#(cons(any(X1),X2)) -> cons#(any(any(proper(X1))),any(proper(X2))) proper#(cons(any(X1),X2)) -> proper#(X2) -> proper#(cons(any(X1),X2)) -> any#(any(proper(X1))) proper#(cons(any(X1),X2)) -> proper#(X2) -> proper#(cons(any(X1),X2)) -> any#(proper(X1)) proper#(cons(any(X1),X2)) -> proper#(X2) -> proper#(cons(any(X1),X2)) -> proper#(X1) proper#(cons(any(X1),X2)) -> proper#(X2) -> proper#(cons(any(X1),X2)) -> any#(proper(X2)) proper#(cons(any(X1),X2)) -> proper#(X2) -> proper#(cons(any(X1),X2)) -> proper#(X2) proper#(cons(any(X1),X2)) -> proper#(X2) -> proper#(inf(X)) -> inf#(proper(X)) proper#(cons(any(X1),X2)) -> proper#(X2) -> proper#(inf(X)) -> proper#(X) proper#(cons(any(X1),X2)) -> proper#(X2) -> proper#(s(X)) -> s#(proper(X)) proper#(cons(any(X1),X2)) -> proper#(X2) -> proper#(s(X)) -> proper#(X) proper#(cons(any(X1),X2)) -> proper#(X2) -> proper#(eq(X1,X2)) -> eq#(proper(X1),proper(X2)) proper#(cons(any(X1),X2)) -> proper#(X2) -> proper#(eq(X1,X2)) -> proper#(X1) proper#(cons(any(X1),X2)) -> proper#(X2) -> proper#(eq(X1,X2)) -> proper#(X2) proper#(cons(any(X1),X2)) -> proper#(X1) -> proper#(length(X)) -> length#(proper(X)) proper#(cons(any(X1),X2)) -> proper#(X1) -> proper#(length(X)) -> proper#(X) proper#(cons(any(X1),X2)) -> proper#(X1) -> proper#(take(X1,X2)) -> take#(proper(X1),proper(X2)) proper#(cons(any(X1),X2)) -> proper#(X1) -> proper#(take(X1,X2)) -> proper#(X1) proper#(cons(any(X1),X2)) -> proper#(X1) -> proper#(take(X1,X2)) -> proper#(X2) proper#(cons(any(X1),X2)) -> proper#(X1) -> proper#(cons(any(X1),X2)) -> cons#(any(any(proper(X1))),any(proper(X2))) proper#(cons(any(X1),X2)) -> proper#(X1) -> proper#(cons(any(X1),X2)) -> any#(any(proper(X1))) proper#(cons(any(X1),X2)) -> proper#(X1) -> proper#(cons(any(X1),X2)) -> any#(proper(X1)) proper#(cons(any(X1),X2)) -> proper#(X1) -> proper#(cons(any(X1),X2)) -> proper#(X1) proper#(cons(any(X1),X2)) -> proper#(X1) -> proper#(cons(any(X1),X2)) -> any#(proper(X2)) proper#(cons(any(X1),X2)) -> proper#(X1) -> proper#(cons(any(X1),X2)) -> proper#(X2) proper#(cons(any(X1),X2)) -> proper#(X1) -> proper#(inf(X)) -> inf#(proper(X)) proper#(cons(any(X1),X2)) -> proper#(X1) -> proper#(inf(X)) -> proper#(X) proper#(cons(any(X1),X2)) -> proper#(X1) -> proper#(s(X)) -> s#(proper(X)) proper#(cons(any(X1),X2)) -> proper#(X1) -> proper#(s(X)) -> proper#(X) proper#(cons(any(X1),X2)) -> proper#(X1) -> proper#(eq(X1,X2)) -> eq#(proper(X1),proper(X2)) proper#(cons(any(X1),X2)) -> proper#(X1) -> proper#(eq(X1,X2)) -> proper#(X1) proper#(cons(any(X1),X2)) -> proper#(X1) -> proper#(eq(X1,X2)) -> proper#(X2) proper#(cons(any(X1),X2)) -> cons#(any(any(proper(X1))),any(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(any(X1),X2)) -> cons#(any(any(proper(X1))),any(proper(X2))) proper#(inf(X)) -> proper#(X) -> proper#(cons(any(X1),X2)) -> any#(any(proper(X1))) proper#(inf(X)) -> proper#(X) -> proper#(cons(any(X1),X2)) -> any#(proper(X1)) proper#(inf(X)) -> proper#(X) -> proper#(cons(any(X1),X2)) -> proper#(X1) proper#(inf(X)) -> proper#(X) -> proper#(cons(any(X1),X2)) -> any#(proper(X2)) proper#(inf(X)) -> proper#(X) -> proper#(cons(any(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(any(X1),X2)) -> cons#(any(any(proper(X1))),any(proper(X2))) proper#(s(X)) -> proper#(X) -> proper#(cons(any(X1),X2)) -> any#(any(proper(X1))) proper#(s(X)) -> proper#(X) -> proper#(cons(any(X1),X2)) -> any#(proper(X1)) proper#(s(X)) -> proper#(X) -> proper#(cons(any(X1),X2)) -> proper#(X1) proper#(s(X)) -> proper#(X) -> proper#(cons(any(X1),X2)) -> any#(proper(X2)) proper#(s(X)) -> proper#(X) -> proper#(cons(any(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(any(X1),X2)) -> cons#(any(any(proper(X1))),any(proper(X2))) proper#(eq(X1,X2)) -> proper#(X2) -> proper#(cons(any(X1),X2)) -> any#(any(proper(X1))) proper#(eq(X1,X2)) -> proper#(X2) -> proper#(cons(any(X1),X2)) -> any#(proper(X1)) proper#(eq(X1,X2)) -> proper#(X2) -> proper#(cons(any(X1),X2)) -> proper#(X1) proper#(eq(X1,X2)) -> proper#(X2) -> proper#(cons(any(X1),X2)) -> any#(proper(X2)) proper#(eq(X1,X2)) -> proper#(X2) -> proper#(cons(any(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(any(X1),X2)) -> cons#(any(any(proper(X1))),any(proper(X2))) proper#(eq(X1,X2)) -> proper#(X1) -> proper#(cons(any(X1),X2)) -> any#(any(proper(X1))) proper#(eq(X1,X2)) -> proper#(X1) -> proper#(cons(any(X1),X2)) -> any#(proper(X1)) proper#(eq(X1,X2)) -> proper#(X1) -> proper#(cons(any(X1),X2)) -> proper#(X1) proper#(eq(X1,X2)) -> proper#(X1) -> proper#(cons(any(X1),X2)) -> any#(proper(X2)) proper#(eq(X1,X2)) -> proper#(X1) -> proper#(cons(any(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: 10 #rules: 28 #arcs: 345/2704 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(any(X1),X2)) -> cons(any(any(proper(X1))),any(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)) any(X) -> s(X) any(proper(X)) -> any(any(any(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(any(X1),X2)) -> cons(any(any(proper(X1))),any(proper(X2))) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) proper(nil()) -> ok(nil()) proper(length(X)) -> length(proper(X)) any(X) -> s(X) any(proper(X)) -> any(any(any(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(any(X1),X2)) -> cons(any(any(proper(X1))),any(proper(X2))) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) proper(nil()) -> ok(nil()) proper(length(X)) -> length(proper(X)) interpretation: [top#](x0) = [1 0 0]x0, [1 1 1] [any](x0) = [1 0 0]x0 [0 0 0] , [1 0 0] [ok](x0) = [0 0 0]x0 [0 1 1] , [1 0 0] [proper](x0) = [0 0 0]x0 [0 0 1] , [1 0 0] [1] [length](x0) = [0 0 0]x0 + [0] [0 0 0] [0], [0] [nil] = [0] [0], [1 0 0] [1 0 0] [1] [take](x0, x1) = [0 0 0]x0 + [0 0 0]x1 + [0] [0 0 1] [0 0 0] [0], [0] [cons](x0, x1) = [0] [0], [1 0 0] [1] [inf](x0) = [0 0 0]x0 + [0] [0 0 0] [0], [0] [false] = [0] [0], [0 0 1] [0] [s](x0) = [0 0 0]x0 + [0] [0 0 1] [1], [1 0 0] [1] [mark](x0) = [0 0 0]x0 + [0] [0 0 0] [0], [0] [true] = [0] [0], [1 0 0] [0] [active](x0) = [0 0 1]x0 + [0] [1 0 1] [1], [0 0 1] [1] [eq](x0, x1) = [0 0 0]x1 + [0] [0 0 0] [1], [0] [0] = [0] [1] orientation: top#(ok(X)) = [1 0 0]X >= [1 0 0]X = top#(active(X)) top#(mark(X)) = [1 0 0]X + [1] >= [1 0 0]X = top#(proper(X)) [2] [1] active(eq(0(),0())) = [1] >= [0] = mark(true()) [4] [0] [0 0 1] [2] [0 0 1] [2] active(eq(s(X),s(Y))) = [0 0 0]Y + [1] >= [0 0 0]Y + [0] = mark(eq(X,Y)) [0 0 1] [4] [0 0 0] [0] [0 0 1] [1] [1] active(eq(X,Y)) = [0 0 0]Y + [1] >= [0] = mark(false()) [0 0 1] [3] [0] [1 0 0] [1] [1] active(inf(X)) = [0 0 0]X + [0] >= [0] = mark(cons(X,inf(s(X)))) [1 0 0] [2] [0] [1 0 0] [1] [1] active(take(0(),X)) = [0 0 0]X + [1] >= [0] = mark(nil()) [1 0 0] [3] [0] [0 0 1] [1] [1] active(take(s(X),cons(Y,L))) = [0 0 1]X + [1] >= [0] = mark(cons(Y,take(X,L))) [0 0 2] [3] [0] [1] [1] active(length(nil())) = [0] >= [0] = mark(0()) [2] [0] [1] [1] active(length(cons(X,L))) = [0] >= [0] = mark(s(length(L))) [2] [0] [1 0 0] [1] [1 0 0] [1] active(inf(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = inf(active(X)) [1 0 0] [2] [0 0 0] [0] [1 0 0] [1 0 0] [1] [1 0 0] [1 0 0] [1] active(take(X1,X2)) = [0 0 1]X1 + [0 0 0]X2 + [0] >= [0 0 0]X1 + [0 0 0]X2 + [0] = take(active(X1),X2) [1 0 1] [1 0 0] [2] [1 0 1] [0 0 0] [1] [1 0 0] [1 0 0] [1] [1 0 0] [1 0 0] [1] active(take(X1,X2)) = [0 0 1]X1 + [0 0 0]X2 + [0] >= [0 0 0]X1 + [0 0 0]X2 + [0] = take(X1,active(X2)) [1 0 1] [1 0 0] [2] [0 0 1] [0 0 0] [0] [1 0 0] [1] [1 0 0] [1] active(length(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = length(active(X)) [1 0 0] [2] [0 0 0] [0] [0 1 1] [1] [0 0 1] [1] eq(ok(X1),ok(X2)) = [0 0 0]X2 + [0] >= [0 0 0]X2 + [0] = ok(eq(X1,X2)) [0 0 0] [1] [0 0 0] [1] [0 1 1] [0] [0 0 1] [0] s(ok(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = ok(s(X)) [0 1 1] [1] [0 0 1] [1] [1 0 0] [2] [1 0 0] [2] inf(mark(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = mark(inf(X)) [0 0 0] [0] [0 0 0] [0] [1 0 0] [1] [1 0 0] [1] inf(ok(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = ok(inf(X)) [0 0 0] [0] [0 0 0] [0] [0] [0] cons(ok(X1),ok(X2)) = [0] >= [0] = ok(cons(X1,X2)) [0] [0] [1 0 0] [1 0 0] [2] [1 0 0] [1 0 0] [2] 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 0] [0 0 0] [0] [0 0 0] [0 0 0] [0] [1 0 0] [1 0 0] [2] [1 0 0] [1 0 0] [2] 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 0] [0] [0 0 0] [0 0 0] [0] [1 0 0] [1 0 0] [1] [1 0 0] [1 0 0] [1] 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 1 1] [0 0 0] [0] [0 0 1] [0 0 0] [0] [1 0 0] [2] [1 0 0] [2] length(mark(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = mark(length(X)) [0 0 0] [0] [0 0 0] [0] [1 0 0] [1] [1 0 0] [1] length(ok(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = ok(length(X)) [0 0 0] [0] [0 0 0] [0] [0 0 1] [1] [0 0 1] [1] proper(eq(X1,X2)) = [0 0 0]X2 + [0] >= [0 0 0]X2 + [0] = eq(proper(X1),proper(X2)) [0 0 0] [1] [0 0 0] [1] [0] [0] proper(0()) = [0] >= [0] = ok(0()) [1] [1] [0] [0] proper(true()) = [0] >= [0] = ok(true()) [0] [0] [0 0 1] [0] [0 0 1] [0] proper(s(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = s(proper(X)) [0 0 1] [1] [0 0 1] [1] [0] [0] proper(false()) = [0] >= [0] = ok(false()) [0] [0] [1 0 0] [1] [1 0 0] [1] proper(inf(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = inf(proper(X)) [0 0 0] [0] [0 0 0] [0] [0] [0] proper(cons(any(X1),X2)) = [0] >= [0] = cons(any(any(proper(X1))),any(proper(X2))) [0] [0] [1 0 0] [1 0 0] [1] [1 0 0] [1 0 0] [1] 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 0] [0] [0 0 1] [0 0 0] [0] [0] [0] proper(nil()) = [0] >= [0] = ok(nil()) [0] [0] [1 0 0] [1] [1 0 0] [1] proper(length(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = length(proper(X)) [0 0 0] [0] [0 0 0] [0] [1 1 1] [0 0 1] [0] any(X) = [1 0 0]X >= [0 0 0]X + [0] = s(X) [0 0 0] [0 0 1] [1] [1 0 1] [3 2 2] any(proper(X)) = [1 0 0]X >= [2 1 1]X = any(any(any(X))) [0 0 0] [0 0 0] 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(any(X1),X2)) -> cons(any(any(proper(X1))),any(proper(X2))) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) proper(nil()) -> ok(nil()) proper(length(X)) -> length(proper(X)) any(X) -> s(X) any(proper(X)) -> any(any(any(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(any(X1),X2)) -> cons(any(any(proper(X1))),any(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)) any(X) -> s(X) any(proper(X)) -> any(any(any(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)) KBO Processor: argument filtering: pi(0) = [] pi(eq) = [] pi(active) = 0 pi(true) = [] pi(mark) = [] pi(s) = [] pi(false) = [] pi(inf) = [0] pi(cons) = [] pi(take) = [0,1] pi(nil) = [] pi(length) = 0 pi(ok) = [0] pi(top#) = 0 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)) inf(mark(X)) -> mark(inf(X)) inf(ok(X)) -> ok(inf(X)) 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)) weight function: w0 = 1 w(top#) = w(ok) = w(nil) = w(cons) = w(inf) = w(false) = w( s) = w(mark) = w(true) = w(eq) = w(0) = 1 w(length) = w(take) = w(active) = 0 precedence: nil ~ cons ~ inf ~ eq > top# ~ ok ~ length ~ take ~ false ~ s ~ mark ~ true ~ 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)) 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(any(X1),X2)) -> cons(any(any(proper(X1))),any(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)) any(X) -> s(X) any(proper(X)) -> any(any(any(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(any(X1),X2)) -> cons(any(any(proper(X1))),any(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)) any(X) -> s(X) any(proper(X)) -> any(any(any(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(any(X1),X2)) -> proper#(X2) proper#(cons(any(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(any(X1),X2)) -> cons(any(any(proper(X1))),any(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)) any(X) -> s(X) any(proper(X)) -> any(any(any(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(any(X1),X2)) -> cons(any(any(proper(X1))),any(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)) any(X) -> s(X) any(proper(X)) -> any(any(any(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(any(X1),X2)) -> cons(any(any(proper(X1))),any(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)) any(X) -> s(X) any(proper(X)) -> any(any(any(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(any(X1),X2)) -> cons(any(any(proper(X1))),any(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)) any(X) -> s(X) any(proper(X)) -> any(any(any(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(any(X1),X2)) -> cons(any(any(proper(X1))),any(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)) any(X) -> s(X) any(proper(X)) -> any(any(any(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(any(X1),X2)) -> cons(any(any(proper(X1))),any(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)) any(X) -> s(X) any(proper(X)) -> any(any(any(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(any(X1),X2)) -> cons(any(any(proper(X1))),any(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)) any(X) -> s(X) any(proper(X)) -> any(any(any(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(any(X1),X2)) -> cons(any(any(proper(X1))),any(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)) any(X) -> s(X) any(proper(X)) -> any(any(any(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(any(X1),X2)) -> cons(any(any(proper(X1))),any(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)) any(X) -> s(X) any(proper(X)) -> any(any(any(X))) Qed DPs: any#(proper(X)) -> any#(X) any#(proper(X)) -> any#(any(X)) any#(proper(X)) -> any#(any(any(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(any(X1),X2)) -> cons(any(any(proper(X1))),any(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)) any(X) -> s(X) any(proper(X)) -> any(any(any(X))) EDG Processor: DPs: any#(proper(X)) -> any#(X) any#(proper(X)) -> any#(any(X)) any#(proper(X)) -> any#(any(any(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(any(X1),X2)) -> cons(any(any(proper(X1))),any(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)) any(X) -> s(X) any(proper(X)) -> any(any(any(X))) graph: any#(proper(X)) -> any#(X) -> any#(proper(X)) -> any#(X) any#(proper(X)) -> any#(X) -> any#(proper(X)) -> any#(any(X)) any#(proper(X)) -> any#(X) -> any#(proper(X)) -> any#(any(any(X))) SCC Processor: #sccs: 1 #rules: 1 #arcs: 3/9 DPs: any#(proper(X)) -> any#(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(any(X1),X2)) -> cons(any(any(proper(X1))),any(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)) any(X) -> s(X) any(proper(X)) -> any(any(any(X))) Size-Change Termination Processor: 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(any(X1),X2)) -> cons(any(any(proper(X1))),any(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)) any(X) -> s(X) any(proper(X)) -> any(any(any(X))) The DP: any#(proper(X)) -> any#(X) has the edges: 0 > 0 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(any(X1),X2)) -> cons(any(any(proper(X1))),any(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)) any(X) -> s(X) any(proper(X)) -> any(any(any(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(any(X1),X2)) -> cons(any(any(proper(X1))),any(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)) any(X) -> s(X) any(proper(X)) -> any(any(any(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(any(X1),X2)) -> cons(any(any(proper(X1))),any(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)) any(X) -> s(X) any(proper(X)) -> any(any(any(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(any(X1),X2)) -> cons(any(any(proper(X1))),any(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)) any(X) -> s(X) any(proper(X)) -> any(any(any(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(any(X1),X2)) -> cons(any(any(proper(X1))),any(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)) any(X) -> s(X) any(proper(X)) -> any(any(any(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(any(X1),X2)) -> cons(any(any(proper(X1))),any(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)) any(X) -> s(X) any(proper(X)) -> any(any(any(X))) Qed