/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(from(X)) -> mark(cons(X,from(s(X)))) active(length(nil())) -> mark(0()) active(length(cons(X,Y))) -> mark(s(length1(Y))) active(length1(X)) -> mark(length(X)) active(from(X)) -> from(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(s(X)) -> s(active(X)) from(mark(X)) -> mark(from(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) s(mark(X)) -> mark(s(X)) proper(from(X)) -> from(proper(X)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(nil()) -> ok(nil()) proper(0()) -> ok(0()) proper(length1(X)) -> length1(proper(X)) from(ok(X)) -> ok(from(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) length1(ok(X)) -> ok(length1(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Proof: DP Processor: DPs: active#(from(X)) -> s#(X) active#(from(X)) -> from#(s(X)) active#(from(X)) -> cons#(X,from(s(X))) active#(length(cons(X,Y))) -> length1#(Y) active#(length(cons(X,Y))) -> s#(length1(Y)) active#(length1(X)) -> length#(X) active#(from(X)) -> active#(X) active#(from(X)) -> from#(active(X)) active#(cons(X1,X2)) -> active#(X1) active#(cons(X1,X2)) -> cons#(active(X1),X2) active#(s(X)) -> active#(X) active#(s(X)) -> s#(active(X)) from#(mark(X)) -> from#(X) cons#(mark(X1),X2) -> cons#(X1,X2) s#(mark(X)) -> s#(X) proper#(from(X)) -> proper#(X) proper#(from(X)) -> from#(proper(X)) proper#(cons(X1,X2)) -> proper#(X2) proper#(cons(X1,X2)) -> proper#(X1) proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2)) proper#(s(X)) -> proper#(X) proper#(s(X)) -> s#(proper(X)) proper#(length(X)) -> proper#(X) proper#(length(X)) -> length#(proper(X)) proper#(length1(X)) -> proper#(X) proper#(length1(X)) -> length1#(proper(X)) from#(ok(X)) -> from#(X) cons#(ok(X1),ok(X2)) -> cons#(X1,X2) s#(ok(X)) -> s#(X) length#(ok(X)) -> length#(X) length1#(ok(X)) -> length1#(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(from(X)) -> mark(cons(X,from(s(X)))) active(length(nil())) -> mark(0()) active(length(cons(X,Y))) -> mark(s(length1(Y))) active(length1(X)) -> mark(length(X)) active(from(X)) -> from(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(s(X)) -> s(active(X)) from(mark(X)) -> mark(from(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) s(mark(X)) -> mark(s(X)) proper(from(X)) -> from(proper(X)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(nil()) -> ok(nil()) proper(0()) -> ok(0()) proper(length1(X)) -> length1(proper(X)) from(ok(X)) -> ok(from(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) length1(ok(X)) -> ok(length1(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) TDG Processor: DPs: active#(from(X)) -> s#(X) active#(from(X)) -> from#(s(X)) active#(from(X)) -> cons#(X,from(s(X))) active#(length(cons(X,Y))) -> length1#(Y) active#(length(cons(X,Y))) -> s#(length1(Y)) active#(length1(X)) -> length#(X) active#(from(X)) -> active#(X) active#(from(X)) -> from#(active(X)) active#(cons(X1,X2)) -> active#(X1) active#(cons(X1,X2)) -> cons#(active(X1),X2) active#(s(X)) -> active#(X) active#(s(X)) -> s#(active(X)) from#(mark(X)) -> from#(X) cons#(mark(X1),X2) -> cons#(X1,X2) s#(mark(X)) -> s#(X) proper#(from(X)) -> proper#(X) proper#(from(X)) -> from#(proper(X)) proper#(cons(X1,X2)) -> proper#(X2) proper#(cons(X1,X2)) -> proper#(X1) proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2)) proper#(s(X)) -> proper#(X) proper#(s(X)) -> s#(proper(X)) proper#(length(X)) -> proper#(X) proper#(length(X)) -> length#(proper(X)) proper#(length1(X)) -> proper#(X) proper#(length1(X)) -> length1#(proper(X)) from#(ok(X)) -> from#(X) cons#(ok(X1),ok(X2)) -> cons#(X1,X2) s#(ok(X)) -> s#(X) length#(ok(X)) -> length#(X) length1#(ok(X)) -> length1#(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(from(X)) -> mark(cons(X,from(s(X)))) active(length(nil())) -> mark(0()) active(length(cons(X,Y))) -> mark(s(length1(Y))) active(length1(X)) -> mark(length(X)) active(from(X)) -> from(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(s(X)) -> s(active(X)) from(mark(X)) -> mark(from(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) s(mark(X)) -> mark(s(X)) proper(from(X)) -> from(proper(X)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(nil()) -> ok(nil()) proper(0()) -> ok(0()) proper(length1(X)) -> length1(proper(X)) from(ok(X)) -> ok(from(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) length1(ok(X)) -> ok(length1(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) graph: top#(ok(X)) -> top#(active(X)) -> top#(ok(X)) -> top#(active(X)) top#(ok(X)) -> top#(active(X)) -> top#(ok(X)) -> active#(X) top#(ok(X)) -> top#(active(X)) -> top#(mark(X)) -> top#(proper(X)) top#(ok(X)) -> top#(active(X)) -> top#(mark(X)) -> proper#(X) top#(ok(X)) -> active#(X) -> active#(s(X)) -> s#(active(X)) top#(ok(X)) -> active#(X) -> active#(s(X)) -> active#(X) top#(ok(X)) -> active#(X) -> active#(cons(X1,X2)) -> cons#(active(X1),X2) top#(ok(X)) -> active#(X) -> active#(cons(X1,X2)) -> active#(X1) top#(ok(X)) -> active#(X) -> active#(from(X)) -> from#(active(X)) top#(ok(X)) -> active#(X) -> active#(from(X)) -> active#(X) top#(ok(X)) -> active#(X) -> active#(length1(X)) -> length#(X) top#(ok(X)) -> active#(X) -> active#(length(cons(X,Y))) -> s#(length1(Y)) top#(ok(X)) -> active#(X) -> active#(length(cons(X,Y))) -> length1#(Y) top#(ok(X)) -> active#(X) -> active#(from(X)) -> cons#(X,from(s(X))) top#(ok(X)) -> active#(X) -> active#(from(X)) -> from#(s(X)) top#(ok(X)) -> active#(X) -> active#(from(X)) -> s#(X) 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#(length1(X)) -> length1#(proper(X)) top#(mark(X)) -> proper#(X) -> proper#(length1(X)) -> proper#(X) top#(mark(X)) -> proper#(X) -> proper#(length(X)) -> length#(proper(X)) top#(mark(X)) -> proper#(X) -> proper#(length(X)) -> proper#(X) top#(mark(X)) -> proper#(X) -> proper#(s(X)) -> s#(proper(X)) top#(mark(X)) -> proper#(X) -> proper#(s(X)) -> proper#(X) top#(mark(X)) -> proper#(X) -> proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2)) top#(mark(X)) -> proper#(X) -> proper#(cons(X1,X2)) -> proper#(X1) top#(mark(X)) -> proper#(X) -> proper#(cons(X1,X2)) -> proper#(X2) top#(mark(X)) -> proper#(X) -> proper#(from(X)) -> from#(proper(X)) top#(mark(X)) -> proper#(X) -> proper#(from(X)) -> proper#(X) proper#(length1(X)) -> proper#(X) -> proper#(length1(X)) -> length1#(proper(X)) proper#(length1(X)) -> proper#(X) -> proper#(length1(X)) -> proper#(X) proper#(length1(X)) -> proper#(X) -> proper#(length(X)) -> length#(proper(X)) proper#(length1(X)) -> proper#(X) -> proper#(length(X)) -> proper#(X) proper#(length1(X)) -> proper#(X) -> proper#(s(X)) -> s#(proper(X)) proper#(length1(X)) -> proper#(X) -> proper#(s(X)) -> proper#(X) proper#(length1(X)) -> proper#(X) -> proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2)) proper#(length1(X)) -> proper#(X) -> proper#(cons(X1,X2)) -> proper#(X1) proper#(length1(X)) -> proper#(X) -> proper#(cons(X1,X2)) -> proper#(X2) proper#(length1(X)) -> proper#(X) -> proper#(from(X)) -> from#(proper(X)) proper#(length1(X)) -> proper#(X) -> proper#(from(X)) -> proper#(X) proper#(length1(X)) -> length1#(proper(X)) -> length1#(ok(X)) -> length1#(X) proper#(length(X)) -> proper#(X) -> proper#(length1(X)) -> length1#(proper(X)) proper#(length(X)) -> proper#(X) -> proper#(length1(X)) -> proper#(X) proper#(length(X)) -> proper#(X) -> proper#(length(X)) -> length#(proper(X)) proper#(length(X)) -> proper#(X) -> proper#(length(X)) -> proper#(X) proper#(length(X)) -> proper#(X) -> proper#(s(X)) -> s#(proper(X)) proper#(length(X)) -> proper#(X) -> proper#(s(X)) -> proper#(X) proper#(length(X)) -> proper#(X) -> proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2)) proper#(length(X)) -> proper#(X) -> proper#(cons(X1,X2)) -> proper#(X1) proper#(length(X)) -> proper#(X) -> proper#(cons(X1,X2)) -> proper#(X2) proper#(length(X)) -> proper#(X) -> proper#(from(X)) -> from#(proper(X)) proper#(length(X)) -> proper#(X) -> proper#(from(X)) -> proper#(X) proper#(length(X)) -> length#(proper(X)) -> length#(ok(X)) -> length#(X) proper#(cons(X1,X2)) -> proper#(X2) -> proper#(length1(X)) -> length1#(proper(X)) proper#(cons(X1,X2)) -> proper#(X2) -> proper#(length1(X)) -> proper#(X) proper#(cons(X1,X2)) -> proper#(X2) -> proper#(length(X)) -> length#(proper(X)) proper#(cons(X1,X2)) -> proper#(X2) -> proper#(length(X)) -> proper#(X) proper#(cons(X1,X2)) -> proper#(X2) -> proper#(s(X)) -> s#(proper(X)) proper#(cons(X1,X2)) -> proper#(X2) -> proper#(s(X)) -> proper#(X) proper#(cons(X1,X2)) -> proper#(X2) -> proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2)) proper#(cons(X1,X2)) -> proper#(X2) -> proper#(cons(X1,X2)) -> proper#(X1) proper#(cons(X1,X2)) -> proper#(X2) -> proper#(cons(X1,X2)) -> proper#(X2) proper#(cons(X1,X2)) -> proper#(X2) -> proper#(from(X)) -> from#(proper(X)) proper#(cons(X1,X2)) -> proper#(X2) -> proper#(from(X)) -> proper#(X) proper#(cons(X1,X2)) -> proper#(X1) -> proper#(length1(X)) -> length1#(proper(X)) proper#(cons(X1,X2)) -> proper#(X1) -> proper#(length1(X)) -> proper#(X) proper#(cons(X1,X2)) -> proper#(X1) -> proper#(length(X)) -> length#(proper(X)) proper#(cons(X1,X2)) -> proper#(X1) -> proper#(length(X)) -> proper#(X) proper#(cons(X1,X2)) -> proper#(X1) -> proper#(s(X)) -> s#(proper(X)) proper#(cons(X1,X2)) -> proper#(X1) -> proper#(s(X)) -> proper#(X) proper#(cons(X1,X2)) -> proper#(X1) -> proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2)) proper#(cons(X1,X2)) -> proper#(X1) -> proper#(cons(X1,X2)) -> proper#(X1) proper#(cons(X1,X2)) -> proper#(X1) -> proper#(cons(X1,X2)) -> proper#(X2) proper#(cons(X1,X2)) -> proper#(X1) -> proper#(from(X)) -> from#(proper(X)) proper#(cons(X1,X2)) -> proper#(X1) -> proper#(from(X)) -> proper#(X) proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2)) -> cons#(ok(X1),ok(X2)) -> cons#(X1,X2) proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2)) -> cons#(mark(X1),X2) -> cons#(X1,X2) proper#(s(X)) -> proper#(X) -> proper#(length1(X)) -> length1#(proper(X)) proper#(s(X)) -> proper#(X) -> proper#(length1(X)) -> proper#(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#(s(X)) -> s#(proper(X)) proper#(s(X)) -> proper#(X) -> proper#(s(X)) -> proper#(X) proper#(s(X)) -> proper#(X) -> proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2)) proper#(s(X)) -> proper#(X) -> proper#(cons(X1,X2)) -> proper#(X1) proper#(s(X)) -> proper#(X) -> proper#(cons(X1,X2)) -> proper#(X2) proper#(s(X)) -> proper#(X) -> proper#(from(X)) -> from#(proper(X)) proper#(s(X)) -> proper#(X) -> proper#(from(X)) -> proper#(X) proper#(s(X)) -> s#(proper(X)) -> s#(ok(X)) -> s#(X) proper#(s(X)) -> s#(proper(X)) -> s#(mark(X)) -> s#(X) proper#(from(X)) -> proper#(X) -> proper#(length1(X)) -> length1#(proper(X)) proper#(from(X)) -> proper#(X) -> proper#(length1(X)) -> proper#(X) proper#(from(X)) -> proper#(X) -> proper#(length(X)) -> length#(proper(X)) proper#(from(X)) -> proper#(X) -> proper#(length(X)) -> proper#(X) proper#(from(X)) -> proper#(X) -> proper#(s(X)) -> s#(proper(X)) proper#(from(X)) -> proper#(X) -> proper#(s(X)) -> proper#(X) proper#(from(X)) -> proper#(X) -> proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2)) proper#(from(X)) -> proper#(X) -> proper#(cons(X1,X2)) -> proper#(X1) proper#(from(X)) -> proper#(X) -> proper#(cons(X1,X2)) -> proper#(X2) proper#(from(X)) -> proper#(X) -> proper#(from(X)) -> from#(proper(X)) proper#(from(X)) -> proper#(X) -> proper#(from(X)) -> proper#(X) proper#(from(X)) -> from#(proper(X)) -> from#(ok(X)) -> from#(X) proper#(from(X)) -> from#(proper(X)) -> from#(mark(X)) -> from#(X) length#(ok(X)) -> length#(X) -> length#(ok(X)) -> length#(X) length1#(ok(X)) -> length1#(X) -> length1#(ok(X)) -> length1#(X) cons#(ok(X1),ok(X2)) -> cons#(X1,X2) -> cons#(ok(X1),ok(X2)) -> cons#(X1,X2) cons#(ok(X1),ok(X2)) -> cons#(X1,X2) -> cons#(mark(X1),X2) -> cons#(X1,X2) cons#(mark(X1),X2) -> cons#(X1,X2) -> cons#(ok(X1),ok(X2)) -> cons#(X1,X2) cons#(mark(X1),X2) -> cons#(X1,X2) -> cons#(mark(X1),X2) -> cons#(X1,X2) from#(ok(X)) -> from#(X) -> from#(ok(X)) -> from#(X) from#(ok(X)) -> from#(X) -> from#(mark(X)) -> from#(X) from#(mark(X)) -> from#(X) -> from#(ok(X)) -> from#(X) from#(mark(X)) -> from#(X) -> from#(mark(X)) -> from#(X) s#(ok(X)) -> s#(X) -> s#(ok(X)) -> s#(X) s#(ok(X)) -> s#(X) -> s#(mark(X)) -> s#(X) s#(mark(X)) -> s#(X) -> s#(ok(X)) -> s#(X) s#(mark(X)) -> s#(X) -> s#(mark(X)) -> s#(X) active#(length1(X)) -> length#(X) -> length#(ok(X)) -> length#(X) active#(length(cons(X,Y))) -> length1#(Y) -> length1#(ok(X)) -> length1#(X) active#(length(cons(X,Y))) -> s#(length1(Y)) -> s#(ok(X)) -> s#(X) active#(length(cons(X,Y))) -> s#(length1(Y)) -> s#(mark(X)) -> s#(X) active#(cons(X1,X2)) -> cons#(active(X1),X2) -> cons#(ok(X1),ok(X2)) -> cons#(X1,X2) active#(cons(X1,X2)) -> cons#(active(X1),X2) -> cons#(mark(X1),X2) -> cons#(X1,X2) active#(cons(X1,X2)) -> active#(X1) -> active#(s(X)) -> s#(active(X)) active#(cons(X1,X2)) -> active#(X1) -> active#(s(X)) -> active#(X) active#(cons(X1,X2)) -> active#(X1) -> active#(cons(X1,X2)) -> cons#(active(X1),X2) active#(cons(X1,X2)) -> active#(X1) -> active#(cons(X1,X2)) -> active#(X1) active#(cons(X1,X2)) -> active#(X1) -> active#(from(X)) -> from#(active(X)) active#(cons(X1,X2)) -> active#(X1) -> active#(from(X)) -> active#(X) active#(cons(X1,X2)) -> active#(X1) -> active#(length1(X)) -> length#(X) active#(cons(X1,X2)) -> active#(X1) -> active#(length(cons(X,Y))) -> s#(length1(Y)) active#(cons(X1,X2)) -> active#(X1) -> active#(length(cons(X,Y))) -> length1#(Y) active#(cons(X1,X2)) -> active#(X1) -> active#(from(X)) -> cons#(X,from(s(X))) active#(cons(X1,X2)) -> active#(X1) -> active#(from(X)) -> from#(s(X)) active#(cons(X1,X2)) -> active#(X1) -> active#(from(X)) -> s#(X) active#(s(X)) -> s#(active(X)) -> s#(ok(X)) -> s#(X) active#(s(X)) -> s#(active(X)) -> s#(mark(X)) -> s#(X) active#(s(X)) -> active#(X) -> active#(s(X)) -> s#(active(X)) active#(s(X)) -> active#(X) -> active#(s(X)) -> active#(X) active#(s(X)) -> active#(X) -> active#(cons(X1,X2)) -> cons#(active(X1),X2) active#(s(X)) -> active#(X) -> active#(cons(X1,X2)) -> active#(X1) active#(s(X)) -> active#(X) -> active#(from(X)) -> from#(active(X)) active#(s(X)) -> active#(X) -> active#(from(X)) -> active#(X) active#(s(X)) -> active#(X) -> active#(length1(X)) -> length#(X) active#(s(X)) -> active#(X) -> active#(length(cons(X,Y))) -> s#(length1(Y)) active#(s(X)) -> active#(X) -> active#(length(cons(X,Y))) -> length1#(Y) active#(s(X)) -> active#(X) -> active#(from(X)) -> cons#(X,from(s(X))) active#(s(X)) -> active#(X) -> active#(from(X)) -> from#(s(X)) active#(s(X)) -> active#(X) -> active#(from(X)) -> s#(X) active#(from(X)) -> cons#(X,from(s(X))) -> cons#(ok(X1),ok(X2)) -> cons#(X1,X2) active#(from(X)) -> cons#(X,from(s(X))) -> cons#(mark(X1),X2) -> cons#(X1,X2) active#(from(X)) -> from#(s(X)) -> from#(ok(X)) -> from#(X) active#(from(X)) -> from#(s(X)) -> from#(mark(X)) -> from#(X) active#(from(X)) -> from#(active(X)) -> from#(ok(X)) -> from#(X) active#(from(X)) -> from#(active(X)) -> from#(mark(X)) -> from#(X) active#(from(X)) -> s#(X) -> s#(ok(X)) -> s#(X) active#(from(X)) -> s#(X) -> s#(mark(X)) -> s#(X) active#(from(X)) -> active#(X) -> active#(s(X)) -> s#(active(X)) active#(from(X)) -> active#(X) -> active#(s(X)) -> active#(X) active#(from(X)) -> active#(X) -> active#(cons(X1,X2)) -> cons#(active(X1),X2) active#(from(X)) -> active#(X) -> active#(cons(X1,X2)) -> active#(X1) active#(from(X)) -> active#(X) -> active#(from(X)) -> from#(active(X)) active#(from(X)) -> active#(X) -> active#(from(X)) -> active#(X) active#(from(X)) -> active#(X) -> active#(length1(X)) -> length#(X) active#(from(X)) -> active#(X) -> active#(length(cons(X,Y))) -> s#(length1(Y)) active#(from(X)) -> active#(X) -> active#(length(cons(X,Y))) -> length1#(Y) active#(from(X)) -> active#(X) -> active#(from(X)) -> cons#(X,from(s(X))) active#(from(X)) -> active#(X) -> active#(from(X)) -> from#(s(X)) active#(from(X)) -> active#(X) -> active#(from(X)) -> s#(X) SCC Processor: #sccs: 8 #rules: 19 #arcs: 171/1225 DPs: top#(ok(X)) -> top#(active(X)) top#(mark(X)) -> top#(proper(X)) TRS: active(from(X)) -> mark(cons(X,from(s(X)))) active(length(nil())) -> mark(0()) active(length(cons(X,Y))) -> mark(s(length1(Y))) active(length1(X)) -> mark(length(X)) active(from(X)) -> from(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(s(X)) -> s(active(X)) from(mark(X)) -> mark(from(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) s(mark(X)) -> mark(s(X)) proper(from(X)) -> from(proper(X)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(nil()) -> ok(nil()) proper(0()) -> ok(0()) proper(length1(X)) -> length1(proper(X)) from(ok(X)) -> ok(from(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) length1(ok(X)) -> ok(length1(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Usable Rule Processor: DPs: top#(ok(X)) -> top#(active(X)) top#(mark(X)) -> top#(proper(X)) TRS: active(from(X)) -> mark(cons(X,from(s(X)))) active(length(nil())) -> mark(0()) active(length(cons(X,Y))) -> mark(s(length1(Y))) active(length1(X)) -> mark(length(X)) active(from(X)) -> from(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(s(X)) -> s(active(X)) s(mark(X)) -> mark(s(X)) s(ok(X)) -> ok(s(X)) from(mark(X)) -> mark(from(X)) from(ok(X)) -> ok(from(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) length1(ok(X)) -> ok(length1(X)) length(ok(X)) -> ok(length(X)) proper(from(X)) -> from(proper(X)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(nil()) -> ok(nil()) proper(0()) -> ok(0()) proper(length1(X)) -> length1(proper(X)) Matrix Interpretation Processor: dim=3 usable rules: active(from(X)) -> mark(cons(X,from(s(X)))) active(length(nil())) -> mark(0()) active(length(cons(X,Y))) -> mark(s(length1(Y))) active(length1(X)) -> mark(length(X)) active(from(X)) -> from(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(s(X)) -> s(active(X)) s(mark(X)) -> mark(s(X)) s(ok(X)) -> ok(s(X)) from(mark(X)) -> mark(from(X)) from(ok(X)) -> ok(from(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) length1(ok(X)) -> ok(length1(X)) length(ok(X)) -> ok(length(X)) proper(from(X)) -> from(proper(X)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(nil()) -> ok(nil()) proper(0()) -> ok(0()) proper(length1(X)) -> length1(proper(X)) interpretation: [top#](x0) = [0 0 1]x0, [0 1 0] [ok](x0) = [1 0 0]x0 [0 0 1] , [0 1 0] [proper](x0) = [1 0 0]x0 [0 0 1] , [0 0 0] [0] [length1](x0) = [0 0 0]x0 + [0] [1 1 0] [1], [0] [0] = [0] [0], [0 0 0] [length](x0) = [0 0 0]x0 [1 1 0] , [0] [nil] = [1] [0], [0 0 0] [0] [mark](x0) = [0 0 0]x0 + [0] [0 0 1] [1], [0 0 0] [1 1 0] [1] [cons](x0, x1) = [0 0 0]x0 + [1 1 0]x1 + [1] [0 0 1] [0 0 0] [0], [0 0 0] [s](x0) = [0 0 0]x0 [0 0 1] , [active](x0) = x0 , [0] [from](x0) = x0 + [0] [1] orientation: top#(ok(X)) = [0 0 1]X >= [0 0 1]X = top#(active(X)) top#(mark(X)) = [0 0 1]X + [1] >= [0 0 1]X = top#(proper(X)) [0] [0 0 0] [0] active(from(X)) = X + [0] >= [0 0 0]X + [0] = mark(cons(X,from(s(X)))) [1] [0 0 1] [1] [0] [0] active(length(nil())) = [0] >= [0] = mark(0()) [1] [1] [0 0 0] [0] [0 0 0] [0] active(length(cons(X,Y))) = [0 0 0]Y + [0] >= [0 0 0]Y + [0] = mark(s(length1(Y))) [2 2 0] [2] [1 1 0] [2] [0 0 0] [0] [0 0 0] [0] active(length1(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = mark(length(X)) [1 1 0] [1] [1 1 0] [1] [0] [0] active(from(X)) = X + [0] >= X + [0] = from(active(X)) [1] [1] [0 0 0] [1 1 0] [1] [0 0 0] [1 1 0] [1] active(cons(X1,X2)) = [0 0 0]X1 + [1 1 0]X2 + [1] >= [0 0 0]X1 + [1 1 0]X2 + [1] = cons(active(X1),X2) [0 0 1] [0 0 0] [0] [0 0 1] [0 0 0] [0] [0 0 0] [0 0 0] active(s(X)) = [0 0 0]X >= [0 0 0]X = s(active(X)) [0 0 1] [0 0 1] [0 0 0] [0] [0 0 0] [0] s(mark(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = mark(s(X)) [0 0 1] [1] [0 0 1] [1] [0 0 0] [0 0 0] s(ok(X)) = [0 0 0]X >= [0 0 0]X = ok(s(X)) [0 0 1] [0 0 1] [0 0 0] [0] [0 0 0] [0] from(mark(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = mark(from(X)) [0 0 1] [2] [0 0 1] [2] [0 1 0] [0] [0 1 0] [0] from(ok(X)) = [1 0 0]X + [0] >= [1 0 0]X + [0] = ok(from(X)) [0 0 1] [1] [0 0 1] [1] [0 0 0] [1 1 0] [1] [0 0 0] [0] cons(mark(X1),X2) = [0 0 0]X1 + [1 1 0]X2 + [1] >= [0 0 0]X1 + [0] = mark(cons(X1,X2)) [0 0 1] [0 0 0] [1] [0 0 1] [1] [0 0 0] [1 1 0] [1] [0 0 0] [1 1 0] [1] cons(ok(X1),ok(X2)) = [0 0 0]X1 + [1 1 0]X2 + [1] >= [0 0 0]X1 + [1 1 0]X2 + [1] = ok(cons(X1,X2)) [0 0 1] [0 0 0] [0] [0 0 1] [0 0 0] [0] [0 0 0] [0] [0 0 0] [0] length1(ok(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = ok(length1(X)) [1 1 0] [1] [1 1 0] [1] [0 0 0] [0 0 0] length(ok(X)) = [0 0 0]X >= [0 0 0]X = ok(length(X)) [1 1 0] [1 1 0] [0 1 0] [0] [0 1 0] [0] proper(from(X)) = [1 0 0]X + [0] >= [1 0 0]X + [0] = from(proper(X)) [0 0 1] [1] [0 0 1] [1] [0 0 0] [1 1 0] [1] [0 0 0] [1 1 0] [1] proper(cons(X1,X2)) = [0 0 0]X1 + [1 1 0]X2 + [1] >= [0 0 0]X1 + [1 1 0]X2 + [1] = cons(proper(X1),proper(X2)) [0 0 1] [0 0 0] [0] [0 0 1] [0 0 0] [0] [0 0 0] [0 0 0] proper(s(X)) = [0 0 0]X >= [0 0 0]X = s(proper(X)) [0 0 1] [0 0 1] [0 0 0] [0 0 0] proper(length(X)) = [0 0 0]X >= [0 0 0]X = length(proper(X)) [1 1 0] [1 1 0] [1] [1] proper(nil()) = [0] >= [0] = ok(nil()) [0] [0] [0] [0] proper(0()) = [0] >= [0] = ok(0()) [0] [0] [0 0 0] [0] [0 0 0] [0] proper(length1(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = length1(proper(X)) [1 1 0] [1] [1 1 0] [1] problem: DPs: top#(ok(X)) -> top#(active(X)) TRS: active(from(X)) -> mark(cons(X,from(s(X)))) active(length(nil())) -> mark(0()) active(length(cons(X,Y))) -> mark(s(length1(Y))) active(length1(X)) -> mark(length(X)) active(from(X)) -> from(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(s(X)) -> s(active(X)) s(mark(X)) -> mark(s(X)) s(ok(X)) -> ok(s(X)) from(mark(X)) -> mark(from(X)) from(ok(X)) -> ok(from(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) length1(ok(X)) -> ok(length1(X)) length(ok(X)) -> ok(length(X)) proper(from(X)) -> from(proper(X)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(nil()) -> ok(nil()) proper(0()) -> ok(0()) proper(length1(X)) -> length1(proper(X)) Restore Modifier: DPs: top#(ok(X)) -> top#(active(X)) TRS: active(from(X)) -> mark(cons(X,from(s(X)))) active(length(nil())) -> mark(0()) active(length(cons(X,Y))) -> mark(s(length1(Y))) active(length1(X)) -> mark(length(X)) active(from(X)) -> from(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(s(X)) -> s(active(X)) from(mark(X)) -> mark(from(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) s(mark(X)) -> mark(s(X)) proper(from(X)) -> from(proper(X)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(nil()) -> ok(nil()) proper(0()) -> ok(0()) proper(length1(X)) -> length1(proper(X)) from(ok(X)) -> ok(from(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) length1(ok(X)) -> ok(length1(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Usable Rule Processor: DPs: top#(ok(X)) -> top#(active(X)) TRS: active(from(X)) -> mark(cons(X,from(s(X)))) active(length(nil())) -> mark(0()) active(length(cons(X,Y))) -> mark(s(length1(Y))) active(length1(X)) -> mark(length(X)) active(from(X)) -> from(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(s(X)) -> s(active(X)) s(mark(X)) -> mark(s(X)) s(ok(X)) -> ok(s(X)) from(mark(X)) -> mark(from(X)) from(ok(X)) -> ok(from(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) length1(ok(X)) -> ok(length1(X)) length(ok(X)) -> ok(length(X)) KBO Processor: argument filtering: pi(from) = 0 pi(active) = [] pi(s) = 0 pi(cons) = 0 pi(mark) = [] pi(nil) = [] pi(length) = [] pi(0) = [] pi(length1) = [] pi(ok) = [0] pi(top#) = 0 usable rules: active(from(X)) -> mark(cons(X,from(s(X)))) active(length(nil())) -> mark(0()) active(length(cons(X,Y))) -> mark(s(length1(Y))) active(length1(X)) -> mark(length(X)) active(from(X)) -> from(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(s(X)) -> s(active(X)) s(mark(X)) -> mark(s(X)) s(ok(X)) -> ok(s(X)) from(mark(X)) -> mark(from(X)) from(ok(X)) -> ok(from(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) weight function: w0 = 1 w(top#) = w(ok) = w(length1) = w(0) = w(length) = w(nil) = w( mark) = w(cons) = w(s) = w(active) = w(from) = 1 precedence: active > top# ~ ok ~ length1 ~ 0 ~ length ~ nil ~ mark ~ cons ~ s ~ from problem: DPs: TRS: active(from(X)) -> mark(cons(X,from(s(X)))) active(length(nil())) -> mark(0()) active(length(cons(X,Y))) -> mark(s(length1(Y))) active(length1(X)) -> mark(length(X)) active(from(X)) -> from(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(s(X)) -> s(active(X)) s(mark(X)) -> mark(s(X)) s(ok(X)) -> ok(s(X)) from(mark(X)) -> mark(from(X)) from(ok(X)) -> ok(from(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) length1(ok(X)) -> ok(length1(X)) length(ok(X)) -> ok(length(X)) Qed DPs: active#(from(X)) -> active#(X) active#(cons(X1,X2)) -> active#(X1) active#(s(X)) -> active#(X) TRS: active(from(X)) -> mark(cons(X,from(s(X)))) active(length(nil())) -> mark(0()) active(length(cons(X,Y))) -> mark(s(length1(Y))) active(length1(X)) -> mark(length(X)) active(from(X)) -> from(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(s(X)) -> s(active(X)) from(mark(X)) -> mark(from(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) s(mark(X)) -> mark(s(X)) proper(from(X)) -> from(proper(X)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(nil()) -> ok(nil()) proper(0()) -> ok(0()) proper(length1(X)) -> length1(proper(X)) from(ok(X)) -> ok(from(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) length1(ok(X)) -> ok(length1(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Subterm Criterion Processor: simple projection: pi(active#) = 0 problem: DPs: TRS: active(from(X)) -> mark(cons(X,from(s(X)))) active(length(nil())) -> mark(0()) active(length(cons(X,Y))) -> mark(s(length1(Y))) active(length1(X)) -> mark(length(X)) active(from(X)) -> from(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(s(X)) -> s(active(X)) from(mark(X)) -> mark(from(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) s(mark(X)) -> mark(s(X)) proper(from(X)) -> from(proper(X)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(nil()) -> ok(nil()) proper(0()) -> ok(0()) proper(length1(X)) -> length1(proper(X)) from(ok(X)) -> ok(from(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) length1(ok(X)) -> ok(length1(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed DPs: proper#(from(X)) -> proper#(X) proper#(cons(X1,X2)) -> proper#(X2) proper#(cons(X1,X2)) -> proper#(X1) proper#(s(X)) -> proper#(X) proper#(length(X)) -> proper#(X) proper#(length1(X)) -> proper#(X) TRS: active(from(X)) -> mark(cons(X,from(s(X)))) active(length(nil())) -> mark(0()) active(length(cons(X,Y))) -> mark(s(length1(Y))) active(length1(X)) -> mark(length(X)) active(from(X)) -> from(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(s(X)) -> s(active(X)) from(mark(X)) -> mark(from(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) s(mark(X)) -> mark(s(X)) proper(from(X)) -> from(proper(X)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(nil()) -> ok(nil()) proper(0()) -> ok(0()) proper(length1(X)) -> length1(proper(X)) from(ok(X)) -> ok(from(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) length1(ok(X)) -> ok(length1(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Subterm Criterion Processor: simple projection: pi(proper#) = 0 problem: DPs: TRS: active(from(X)) -> mark(cons(X,from(s(X)))) active(length(nil())) -> mark(0()) active(length(cons(X,Y))) -> mark(s(length1(Y))) active(length1(X)) -> mark(length(X)) active(from(X)) -> from(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(s(X)) -> s(active(X)) from(mark(X)) -> mark(from(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) s(mark(X)) -> mark(s(X)) proper(from(X)) -> from(proper(X)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(nil()) -> ok(nil()) proper(0()) -> ok(0()) proper(length1(X)) -> length1(proper(X)) from(ok(X)) -> ok(from(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) length1(ok(X)) -> ok(length1(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed DPs: length1#(ok(X)) -> length1#(X) TRS: active(from(X)) -> mark(cons(X,from(s(X)))) active(length(nil())) -> mark(0()) active(length(cons(X,Y))) -> mark(s(length1(Y))) active(length1(X)) -> mark(length(X)) active(from(X)) -> from(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(s(X)) -> s(active(X)) from(mark(X)) -> mark(from(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) s(mark(X)) -> mark(s(X)) proper(from(X)) -> from(proper(X)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(nil()) -> ok(nil()) proper(0()) -> ok(0()) proper(length1(X)) -> length1(proper(X)) from(ok(X)) -> ok(from(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) length1(ok(X)) -> ok(length1(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Subterm Criterion Processor: simple projection: pi(length1#) = 0 problem: DPs: TRS: active(from(X)) -> mark(cons(X,from(s(X)))) active(length(nil())) -> mark(0()) active(length(cons(X,Y))) -> mark(s(length1(Y))) active(length1(X)) -> mark(length(X)) active(from(X)) -> from(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(s(X)) -> s(active(X)) from(mark(X)) -> mark(from(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) s(mark(X)) -> mark(s(X)) proper(from(X)) -> from(proper(X)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(nil()) -> ok(nil()) proper(0()) -> ok(0()) proper(length1(X)) -> length1(proper(X)) from(ok(X)) -> ok(from(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) length1(ok(X)) -> ok(length1(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed DPs: length#(ok(X)) -> length#(X) TRS: active(from(X)) -> mark(cons(X,from(s(X)))) active(length(nil())) -> mark(0()) active(length(cons(X,Y))) -> mark(s(length1(Y))) active(length1(X)) -> mark(length(X)) active(from(X)) -> from(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(s(X)) -> s(active(X)) from(mark(X)) -> mark(from(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) s(mark(X)) -> mark(s(X)) proper(from(X)) -> from(proper(X)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(nil()) -> ok(nil()) proper(0()) -> ok(0()) proper(length1(X)) -> length1(proper(X)) from(ok(X)) -> ok(from(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) length1(ok(X)) -> ok(length1(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Subterm Criterion Processor: simple projection: pi(length#) = 0 problem: DPs: TRS: active(from(X)) -> mark(cons(X,from(s(X)))) active(length(nil())) -> mark(0()) active(length(cons(X,Y))) -> mark(s(length1(Y))) active(length1(X)) -> mark(length(X)) active(from(X)) -> from(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(s(X)) -> s(active(X)) from(mark(X)) -> mark(from(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) s(mark(X)) -> mark(s(X)) proper(from(X)) -> from(proper(X)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(nil()) -> ok(nil()) proper(0()) -> ok(0()) proper(length1(X)) -> length1(proper(X)) from(ok(X)) -> ok(from(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) length1(ok(X)) -> ok(length1(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed DPs: s#(mark(X)) -> s#(X) s#(ok(X)) -> s#(X) TRS: active(from(X)) -> mark(cons(X,from(s(X)))) active(length(nil())) -> mark(0()) active(length(cons(X,Y))) -> mark(s(length1(Y))) active(length1(X)) -> mark(length(X)) active(from(X)) -> from(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(s(X)) -> s(active(X)) from(mark(X)) -> mark(from(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) s(mark(X)) -> mark(s(X)) proper(from(X)) -> from(proper(X)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(nil()) -> ok(nil()) proper(0()) -> ok(0()) proper(length1(X)) -> length1(proper(X)) from(ok(X)) -> ok(from(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) length1(ok(X)) -> ok(length1(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Subterm Criterion Processor: simple projection: pi(s#) = 0 problem: DPs: TRS: active(from(X)) -> mark(cons(X,from(s(X)))) active(length(nil())) -> mark(0()) active(length(cons(X,Y))) -> mark(s(length1(Y))) active(length1(X)) -> mark(length(X)) active(from(X)) -> from(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(s(X)) -> s(active(X)) from(mark(X)) -> mark(from(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) s(mark(X)) -> mark(s(X)) proper(from(X)) -> from(proper(X)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(nil()) -> ok(nil()) proper(0()) -> ok(0()) proper(length1(X)) -> length1(proper(X)) from(ok(X)) -> ok(from(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) length1(ok(X)) -> ok(length1(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed DPs: cons#(mark(X1),X2) -> cons#(X1,X2) cons#(ok(X1),ok(X2)) -> cons#(X1,X2) TRS: active(from(X)) -> mark(cons(X,from(s(X)))) active(length(nil())) -> mark(0()) active(length(cons(X,Y))) -> mark(s(length1(Y))) active(length1(X)) -> mark(length(X)) active(from(X)) -> from(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(s(X)) -> s(active(X)) from(mark(X)) -> mark(from(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) s(mark(X)) -> mark(s(X)) proper(from(X)) -> from(proper(X)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(nil()) -> ok(nil()) proper(0()) -> ok(0()) proper(length1(X)) -> length1(proper(X)) from(ok(X)) -> ok(from(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) length1(ok(X)) -> ok(length1(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Subterm Criterion Processor: simple projection: pi(cons#) = 0 problem: DPs: TRS: active(from(X)) -> mark(cons(X,from(s(X)))) active(length(nil())) -> mark(0()) active(length(cons(X,Y))) -> mark(s(length1(Y))) active(length1(X)) -> mark(length(X)) active(from(X)) -> from(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(s(X)) -> s(active(X)) from(mark(X)) -> mark(from(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) s(mark(X)) -> mark(s(X)) proper(from(X)) -> from(proper(X)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(nil()) -> ok(nil()) proper(0()) -> ok(0()) proper(length1(X)) -> length1(proper(X)) from(ok(X)) -> ok(from(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) length1(ok(X)) -> ok(length1(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed DPs: from#(mark(X)) -> from#(X) from#(ok(X)) -> from#(X) TRS: active(from(X)) -> mark(cons(X,from(s(X)))) active(length(nil())) -> mark(0()) active(length(cons(X,Y))) -> mark(s(length1(Y))) active(length1(X)) -> mark(length(X)) active(from(X)) -> from(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(s(X)) -> s(active(X)) from(mark(X)) -> mark(from(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) s(mark(X)) -> mark(s(X)) proper(from(X)) -> from(proper(X)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(nil()) -> ok(nil()) proper(0()) -> ok(0()) proper(length1(X)) -> length1(proper(X)) from(ok(X)) -> ok(from(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) length1(ok(X)) -> ok(length1(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Subterm Criterion Processor: simple projection: pi(from#) = 0 problem: DPs: TRS: active(from(X)) -> mark(cons(X,from(s(X)))) active(length(nil())) -> mark(0()) active(length(cons(X,Y))) -> mark(s(length1(Y))) active(length1(X)) -> mark(length(X)) active(from(X)) -> from(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(s(X)) -> s(active(X)) from(mark(X)) -> mark(from(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) s(mark(X)) -> mark(s(X)) proper(from(X)) -> from(proper(X)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(nil()) -> ok(nil()) proper(0()) -> ok(0()) proper(length1(X)) -> length1(proper(X)) from(ok(X)) -> ok(from(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) length1(ok(X)) -> ok(length1(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed