YES Problem: active(first(0(),X)) -> mark(nil()) active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z))) active(from(X)) -> mark(cons(X,from(s(X)))) active(first(X1,X2)) -> first(active(X1),X2) active(first(X1,X2)) -> first(X1,active(X2)) active(s(X)) -> s(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(from(X)) -> from(active(X)) first(mark(X1),X2) -> mark(first(X1,X2)) first(X1,mark(X2)) -> mark(first(X1,X2)) s(mark(X)) -> mark(s(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) from(mark(X)) -> mark(from(X)) proper(first(X1,X2)) -> first(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(nil()) -> ok(nil()) proper(s(X)) -> s(proper(X)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(from(X)) -> from(proper(X)) first(ok(X1),ok(X2)) -> ok(first(X1,X2)) s(ok(X)) -> ok(s(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) from(ok(X)) -> ok(from(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Proof: DP Processor: DPs: active#(first(s(X),cons(Y,Z))) -> first#(X,Z) active#(first(s(X),cons(Y,Z))) -> cons#(Y,first(X,Z)) active#(from(X)) -> s#(X) active#(from(X)) -> from#(s(X)) active#(from(X)) -> cons#(X,from(s(X))) active#(first(X1,X2)) -> active#(X1) active#(first(X1,X2)) -> first#(active(X1),X2) active#(first(X1,X2)) -> active#(X2) active#(first(X1,X2)) -> first#(X1,active(X2)) active#(s(X)) -> active#(X) active#(s(X)) -> s#(active(X)) active#(cons(X1,X2)) -> active#(X1) active#(cons(X1,X2)) -> cons#(active(X1),X2) active#(from(X)) -> active#(X) active#(from(X)) -> from#(active(X)) first#(mark(X1),X2) -> first#(X1,X2) first#(X1,mark(X2)) -> first#(X1,X2) s#(mark(X)) -> s#(X) cons#(mark(X1),X2) -> cons#(X1,X2) from#(mark(X)) -> from#(X) proper#(first(X1,X2)) -> proper#(X2) proper#(first(X1,X2)) -> proper#(X1) proper#(first(X1,X2)) -> first#(proper(X1),proper(X2)) proper#(s(X)) -> proper#(X) proper#(s(X)) -> s#(proper(X)) proper#(cons(X1,X2)) -> proper#(X2) proper#(cons(X1,X2)) -> proper#(X1) proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2)) proper#(from(X)) -> proper#(X) proper#(from(X)) -> from#(proper(X)) first#(ok(X1),ok(X2)) -> first#(X1,X2) s#(ok(X)) -> s#(X) cons#(ok(X1),ok(X2)) -> cons#(X1,X2) from#(ok(X)) -> from#(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(first(0(),X)) -> mark(nil()) active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z))) active(from(X)) -> mark(cons(X,from(s(X)))) active(first(X1,X2)) -> first(active(X1),X2) active(first(X1,X2)) -> first(X1,active(X2)) active(s(X)) -> s(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(from(X)) -> from(active(X)) first(mark(X1),X2) -> mark(first(X1,X2)) first(X1,mark(X2)) -> mark(first(X1,X2)) s(mark(X)) -> mark(s(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) from(mark(X)) -> mark(from(X)) proper(first(X1,X2)) -> first(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(nil()) -> ok(nil()) proper(s(X)) -> s(proper(X)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(from(X)) -> from(proper(X)) first(ok(X1),ok(X2)) -> ok(first(X1,X2)) s(ok(X)) -> ok(s(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) from(ok(X)) -> ok(from(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) TDG Processor: DPs: active#(first(s(X),cons(Y,Z))) -> first#(X,Z) active#(first(s(X),cons(Y,Z))) -> cons#(Y,first(X,Z)) active#(from(X)) -> s#(X) active#(from(X)) -> from#(s(X)) active#(from(X)) -> cons#(X,from(s(X))) active#(first(X1,X2)) -> active#(X1) active#(first(X1,X2)) -> first#(active(X1),X2) active#(first(X1,X2)) -> active#(X2) active#(first(X1,X2)) -> first#(X1,active(X2)) active#(s(X)) -> active#(X) active#(s(X)) -> s#(active(X)) active#(cons(X1,X2)) -> active#(X1) active#(cons(X1,X2)) -> cons#(active(X1),X2) active#(from(X)) -> active#(X) active#(from(X)) -> from#(active(X)) first#(mark(X1),X2) -> first#(X1,X2) first#(X1,mark(X2)) -> first#(X1,X2) s#(mark(X)) -> s#(X) cons#(mark(X1),X2) -> cons#(X1,X2) from#(mark(X)) -> from#(X) proper#(first(X1,X2)) -> proper#(X2) proper#(first(X1,X2)) -> proper#(X1) proper#(first(X1,X2)) -> first#(proper(X1),proper(X2)) proper#(s(X)) -> proper#(X) proper#(s(X)) -> s#(proper(X)) proper#(cons(X1,X2)) -> proper#(X2) proper#(cons(X1,X2)) -> proper#(X1) proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2)) proper#(from(X)) -> proper#(X) proper#(from(X)) -> from#(proper(X)) first#(ok(X1),ok(X2)) -> first#(X1,X2) s#(ok(X)) -> s#(X) cons#(ok(X1),ok(X2)) -> cons#(X1,X2) from#(ok(X)) -> from#(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(first(0(),X)) -> mark(nil()) active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z))) active(from(X)) -> mark(cons(X,from(s(X)))) active(first(X1,X2)) -> first(active(X1),X2) active(first(X1,X2)) -> first(X1,active(X2)) active(s(X)) -> s(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(from(X)) -> from(active(X)) first(mark(X1),X2) -> mark(first(X1,X2)) first(X1,mark(X2)) -> mark(first(X1,X2)) s(mark(X)) -> mark(s(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) from(mark(X)) -> mark(from(X)) proper(first(X1,X2)) -> first(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(nil()) -> ok(nil()) proper(s(X)) -> s(proper(X)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(from(X)) -> from(proper(X)) first(ok(X1),ok(X2)) -> ok(first(X1,X2)) s(ok(X)) -> ok(s(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) from(ok(X)) -> ok(from(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#(from(X)) -> from#(active(X)) top#(ok(X)) -> active#(X) -> active#(from(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#(s(X)) -> s#(active(X)) top#(ok(X)) -> active#(X) -> active#(s(X)) -> active#(X) top#(ok(X)) -> active#(X) -> active#(first(X1,X2)) -> first#(X1,active(X2)) top#(ok(X)) -> active#(X) -> active#(first(X1,X2)) -> active#(X2) top#(ok(X)) -> active#(X) -> active#(first(X1,X2)) -> first#(active(X1),X2) top#(ok(X)) -> active#(X) -> active#(first(X1,X2)) -> active#(X1) 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#(ok(X)) -> active#(X) -> active#(first(s(X),cons(Y,Z))) -> cons#(Y,first(X,Z)) top#(ok(X)) -> active#(X) -> active#(first(s(X),cons(Y,Z))) -> first#(X,Z) 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#(from(X)) -> from#(proper(X)) top#(mark(X)) -> proper#(X) -> proper#(from(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#(s(X)) -> s#(proper(X)) top#(mark(X)) -> proper#(X) -> proper#(s(X)) -> proper#(X) top#(mark(X)) -> proper#(X) -> proper#(first(X1,X2)) -> first#(proper(X1),proper(X2)) top#(mark(X)) -> proper#(X) -> proper#(first(X1,X2)) -> proper#(X1) top#(mark(X)) -> proper#(X) -> proper#(first(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)) -> 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#(s(X)) -> s#(proper(X)) proper#(from(X)) -> proper#(X) -> proper#(s(X)) -> proper#(X) proper#(from(X)) -> proper#(X) -> proper#(first(X1,X2)) -> first#(proper(X1),proper(X2)) proper#(from(X)) -> proper#(X) -> proper#(first(X1,X2)) -> proper#(X1) proper#(from(X)) -> proper#(X) -> proper#(first(X1,X2)) -> proper#(X2) proper#(from(X)) -> from#(proper(X)) -> from#(ok(X)) -> from#(X) proper#(from(X)) -> from#(proper(X)) -> from#(mark(X)) -> from#(X) 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#(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#(s(X)) -> s#(proper(X)) proper#(cons(X1,X2)) -> proper#(X2) -> proper#(s(X)) -> proper#(X) proper#(cons(X1,X2)) -> proper#(X2) -> proper#(first(X1,X2)) -> first#(proper(X1),proper(X2)) proper#(cons(X1,X2)) -> proper#(X2) -> proper#(first(X1,X2)) -> proper#(X1) proper#(cons(X1,X2)) -> proper#(X2) -> proper#(first(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)) -> 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#(s(X)) -> s#(proper(X)) proper#(cons(X1,X2)) -> proper#(X1) -> proper#(s(X)) -> proper#(X) proper#(cons(X1,X2)) -> proper#(X1) -> proper#(first(X1,X2)) -> first#(proper(X1),proper(X2)) proper#(cons(X1,X2)) -> proper#(X1) -> proper#(first(X1,X2)) -> proper#(X1) proper#(cons(X1,X2)) -> proper#(X1) -> proper#(first(X1,X2)) -> proper#(X2) proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2)) -> cons#(ok(X1),ok(X2)) -> cons#(X1,X2) proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2)) -> cons#(mark(X1),X2) -> cons#(X1,X2) proper#(s(X)) -> proper#(X) -> proper#(from(X)) -> from#(proper(X)) proper#(s(X)) -> proper#(X) -> proper#(from(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#(s(X)) -> s#(proper(X)) proper#(s(X)) -> proper#(X) -> proper#(s(X)) -> proper#(X) proper#(s(X)) -> proper#(X) -> proper#(first(X1,X2)) -> first#(proper(X1),proper(X2)) proper#(s(X)) -> proper#(X) -> proper#(first(X1,X2)) -> proper#(X1) proper#(s(X)) -> proper#(X) -> proper#(first(X1,X2)) -> proper#(X2) proper#(s(X)) -> s#(proper(X)) -> s#(ok(X)) -> s#(X) proper#(s(X)) -> s#(proper(X)) -> s#(mark(X)) -> s#(X) proper#(first(X1,X2)) -> proper#(X2) -> proper#(from(X)) -> from#(proper(X)) proper#(first(X1,X2)) -> proper#(X2) -> proper#(from(X)) -> proper#(X) proper#(first(X1,X2)) -> proper#(X2) -> proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2)) proper#(first(X1,X2)) -> proper#(X2) -> proper#(cons(X1,X2)) -> proper#(X1) proper#(first(X1,X2)) -> proper#(X2) -> proper#(cons(X1,X2)) -> proper#(X2) proper#(first(X1,X2)) -> proper#(X2) -> proper#(s(X)) -> s#(proper(X)) proper#(first(X1,X2)) -> proper#(X2) -> proper#(s(X)) -> proper#(X) proper#(first(X1,X2)) -> proper#(X2) -> proper#(first(X1,X2)) -> first#(proper(X1),proper(X2)) proper#(first(X1,X2)) -> proper#(X2) -> proper#(first(X1,X2)) -> proper#(X1) proper#(first(X1,X2)) -> proper#(X2) -> proper#(first(X1,X2)) -> proper#(X2) proper#(first(X1,X2)) -> proper#(X1) -> proper#(from(X)) -> from#(proper(X)) proper#(first(X1,X2)) -> proper#(X1) -> proper#(from(X)) -> proper#(X) proper#(first(X1,X2)) -> proper#(X1) -> proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2)) proper#(first(X1,X2)) -> proper#(X1) -> proper#(cons(X1,X2)) -> proper#(X1) proper#(first(X1,X2)) -> proper#(X1) -> proper#(cons(X1,X2)) -> proper#(X2) proper#(first(X1,X2)) -> proper#(X1) -> proper#(s(X)) -> s#(proper(X)) proper#(first(X1,X2)) -> proper#(X1) -> proper#(s(X)) -> proper#(X) proper#(first(X1,X2)) -> proper#(X1) -> proper#(first(X1,X2)) -> first#(proper(X1),proper(X2)) proper#(first(X1,X2)) -> proper#(X1) -> proper#(first(X1,X2)) -> proper#(X1) proper#(first(X1,X2)) -> proper#(X1) -> proper#(first(X1,X2)) -> proper#(X2) proper#(first(X1,X2)) -> first#(proper(X1),proper(X2)) -> first#(ok(X1),ok(X2)) -> first#(X1,X2) proper#(first(X1,X2)) -> first#(proper(X1),proper(X2)) -> first#(X1,mark(X2)) -> first#(X1,X2) proper#(first(X1,X2)) -> first#(proper(X1),proper(X2)) -> first#(mark(X1),X2) -> first#(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) 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) first#(ok(X1),ok(X2)) -> first#(X1,X2) -> first#(ok(X1),ok(X2)) -> first#(X1,X2) first#(ok(X1),ok(X2)) -> first#(X1,X2) -> first#(X1,mark(X2)) -> first#(X1,X2) first#(ok(X1),ok(X2)) -> first#(X1,X2) -> first#(mark(X1),X2) -> first#(X1,X2) first#(mark(X1),X2) -> first#(X1,X2) -> first#(ok(X1),ok(X2)) -> first#(X1,X2) first#(mark(X1),X2) -> first#(X1,X2) -> first#(X1,mark(X2)) -> first#(X1,X2) first#(mark(X1),X2) -> first#(X1,X2) -> first#(mark(X1),X2) -> first#(X1,X2) first#(X1,mark(X2)) -> first#(X1,X2) -> first#(ok(X1),ok(X2)) -> first#(X1,X2) first#(X1,mark(X2)) -> first#(X1,X2) -> first#(X1,mark(X2)) -> first#(X1,X2) first#(X1,mark(X2)) -> first#(X1,X2) -> first#(mark(X1),X2) -> first#(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)) -> 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)) -> active#(X) -> active#(from(X)) -> from#(active(X)) active#(from(X)) -> active#(X) -> active#(from(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#(s(X)) -> s#(active(X)) active#(from(X)) -> active#(X) -> active#(s(X)) -> active#(X) active#(from(X)) -> active#(X) -> active#(first(X1,X2)) -> first#(X1,active(X2)) active#(from(X)) -> active#(X) -> active#(first(X1,X2)) -> active#(X2) active#(from(X)) -> active#(X) -> active#(first(X1,X2)) -> first#(active(X1),X2) active#(from(X)) -> active#(X) -> active#(first(X1,X2)) -> active#(X1) 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) active#(from(X)) -> active#(X) -> active#(first(s(X),cons(Y,Z))) -> cons#(Y,first(X,Z)) active#(from(X)) -> active#(X) -> active#(first(s(X),cons(Y,Z))) -> first#(X,Z) 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#(from(X)) -> from#(active(X)) active#(cons(X1,X2)) -> active#(X1) -> active#(from(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#(s(X)) -> s#(active(X)) active#(cons(X1,X2)) -> active#(X1) -> active#(s(X)) -> active#(X) active#(cons(X1,X2)) -> active#(X1) -> active#(first(X1,X2)) -> first#(X1,active(X2)) active#(cons(X1,X2)) -> active#(X1) -> active#(first(X1,X2)) -> active#(X2) active#(cons(X1,X2)) -> active#(X1) -> active#(first(X1,X2)) -> first#(active(X1),X2) active#(cons(X1,X2)) -> active#(X1) -> active#(first(X1,X2)) -> active#(X1) 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#(cons(X1,X2)) -> active#(X1) -> active#(first(s(X),cons(Y,Z))) -> cons#(Y,first(X,Z)) active#(cons(X1,X2)) -> active#(X1) -> active#(first(s(X),cons(Y,Z))) -> first#(X,Z) 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#(from(X)) -> from#(active(X)) active#(s(X)) -> active#(X) -> active#(from(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#(s(X)) -> s#(active(X)) active#(s(X)) -> active#(X) -> active#(s(X)) -> active#(X) active#(s(X)) -> active#(X) -> active#(first(X1,X2)) -> first#(X1,active(X2)) active#(s(X)) -> active#(X) -> active#(first(X1,X2)) -> active#(X2) active#(s(X)) -> active#(X) -> active#(first(X1,X2)) -> first#(active(X1),X2) active#(s(X)) -> active#(X) -> active#(first(X1,X2)) -> active#(X1) 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#(s(X)) -> active#(X) -> active#(first(s(X),cons(Y,Z))) -> cons#(Y,first(X,Z)) active#(s(X)) -> active#(X) -> active#(first(s(X),cons(Y,Z))) -> first#(X,Z) active#(first(s(X),cons(Y,Z))) -> cons#(Y,first(X,Z)) -> cons#(ok(X1),ok(X2)) -> cons#(X1,X2) active#(first(s(X),cons(Y,Z))) -> cons#(Y,first(X,Z)) -> cons#(mark(X1),X2) -> cons#(X1,X2) active#(first(s(X),cons(Y,Z))) -> first#(X,Z) -> first#(ok(X1),ok(X2)) -> first#(X1,X2) active#(first(s(X),cons(Y,Z))) -> first#(X,Z) -> first#(X1,mark(X2)) -> first#(X1,X2) active#(first(s(X),cons(Y,Z))) -> first#(X,Z) -> first#(mark(X1),X2) -> first#(X1,X2) active#(first(X1,X2)) -> first#(active(X1),X2) -> first#(ok(X1),ok(X2)) -> first#(X1,X2) active#(first(X1,X2)) -> first#(active(X1),X2) -> first#(X1,mark(X2)) -> first#(X1,X2) active#(first(X1,X2)) -> first#(active(X1),X2) -> first#(mark(X1),X2) -> first#(X1,X2) active#(first(X1,X2)) -> first#(X1,active(X2)) -> first#(ok(X1),ok(X2)) -> first#(X1,X2) active#(first(X1,X2)) -> first#(X1,active(X2)) -> first#(X1,mark(X2)) -> first#(X1,X2) active#(first(X1,X2)) -> first#(X1,active(X2)) -> first#(mark(X1),X2) -> first#(X1,X2) active#(first(X1,X2)) -> active#(X2) -> active#(from(X)) -> from#(active(X)) active#(first(X1,X2)) -> active#(X2) -> active#(from(X)) -> active#(X) active#(first(X1,X2)) -> active#(X2) -> active#(cons(X1,X2)) -> cons#(active(X1),X2) active#(first(X1,X2)) -> active#(X2) -> active#(cons(X1,X2)) -> active#(X1) active#(first(X1,X2)) -> active#(X2) -> active#(s(X)) -> s#(active(X)) active#(first(X1,X2)) -> active#(X2) -> active#(s(X)) -> active#(X) active#(first(X1,X2)) -> active#(X2) -> active#(first(X1,X2)) -> first#(X1,active(X2)) active#(first(X1,X2)) -> active#(X2) -> active#(first(X1,X2)) -> active#(X2) active#(first(X1,X2)) -> active#(X2) -> active#(first(X1,X2)) -> first#(active(X1),X2) active#(first(X1,X2)) -> active#(X2) -> active#(first(X1,X2)) -> active#(X1) active#(first(X1,X2)) -> active#(X2) -> active#(from(X)) -> cons#(X,from(s(X))) active#(first(X1,X2)) -> active#(X2) -> active#(from(X)) -> from#(s(X)) active#(first(X1,X2)) -> active#(X2) -> active#(from(X)) -> s#(X) active#(first(X1,X2)) -> active#(X2) -> active#(first(s(X),cons(Y,Z))) -> cons#(Y,first(X,Z)) active#(first(X1,X2)) -> active#(X2) -> active#(first(s(X),cons(Y,Z))) -> first#(X,Z) active#(first(X1,X2)) -> active#(X1) -> active#(from(X)) -> from#(active(X)) active#(first(X1,X2)) -> active#(X1) -> active#(from(X)) -> active#(X) active#(first(X1,X2)) -> active#(X1) -> active#(cons(X1,X2)) -> cons#(active(X1),X2) active#(first(X1,X2)) -> active#(X1) -> active#(cons(X1,X2)) -> active#(X1) active#(first(X1,X2)) -> active#(X1) -> active#(s(X)) -> s#(active(X)) active#(first(X1,X2)) -> active#(X1) -> active#(s(X)) -> active#(X) active#(first(X1,X2)) -> active#(X1) -> active#(first(X1,X2)) -> first#(X1,active(X2)) active#(first(X1,X2)) -> active#(X1) -> active#(first(X1,X2)) -> active#(X2) active#(first(X1,X2)) -> active#(X1) -> active#(first(X1,X2)) -> first#(active(X1),X2) active#(first(X1,X2)) -> active#(X1) -> active#(first(X1,X2)) -> active#(X1) active#(first(X1,X2)) -> active#(X1) -> active#(from(X)) -> cons#(X,from(s(X))) active#(first(X1,X2)) -> active#(X1) -> active#(from(X)) -> from#(s(X)) active#(first(X1,X2)) -> active#(X1) -> active#(from(X)) -> s#(X) active#(first(X1,X2)) -> active#(X1) -> active#(first(s(X),cons(Y,Z))) -> cons#(Y,first(X,Z)) active#(first(X1,X2)) -> active#(X1) -> active#(first(s(X),cons(Y,Z))) -> first#(X,Z) SCC Processor: #sccs: 7 #rules: 22 #arcs: 221/1444 DPs: top#(ok(X)) -> top#(active(X)) top#(mark(X)) -> top#(proper(X)) TRS: active(first(0(),X)) -> mark(nil()) active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z))) active(from(X)) -> mark(cons(X,from(s(X)))) active(first(X1,X2)) -> first(active(X1),X2) active(first(X1,X2)) -> first(X1,active(X2)) active(s(X)) -> s(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(from(X)) -> from(active(X)) first(mark(X1),X2) -> mark(first(X1,X2)) first(X1,mark(X2)) -> mark(first(X1,X2)) s(mark(X)) -> mark(s(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) from(mark(X)) -> mark(from(X)) proper(first(X1,X2)) -> first(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(nil()) -> ok(nil()) proper(s(X)) -> s(proper(X)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(from(X)) -> from(proper(X)) first(ok(X1),ok(X2)) -> ok(first(X1,X2)) s(ok(X)) -> ok(s(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) from(ok(X)) -> ok(from(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(first(0(),X)) -> mark(nil()) active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z))) active(from(X)) -> mark(cons(X,from(s(X)))) active(first(X1,X2)) -> first(active(X1),X2) active(first(X1,X2)) -> first(X1,active(X2)) active(s(X)) -> s(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(from(X)) -> from(active(X)) first(mark(X1),X2) -> mark(first(X1,X2)) first(X1,mark(X2)) -> mark(first(X1,X2)) first(ok(X1),ok(X2)) -> ok(first(X1,X2)) cons(mark(X1),X2) -> mark(cons(X1,X2)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) s(mark(X)) -> mark(s(X)) s(ok(X)) -> ok(s(X)) from(mark(X)) -> mark(from(X)) from(ok(X)) -> ok(from(X)) proper(first(X1,X2)) -> first(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(nil()) -> ok(nil()) proper(s(X)) -> s(proper(X)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(from(X)) -> from(proper(X)) KBO Processor: argument filtering: pi(0) = [] pi(first) = [0,1] pi(active) = 0 pi(nil) = [] pi(mark) = [0] pi(s) = [0] pi(cons) = 0 pi(from) = [0] pi(proper) = 0 pi(ok) = 0 pi(top#) = 0 usable rules: active(first(0(),X)) -> mark(nil()) active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z))) active(from(X)) -> mark(cons(X,from(s(X)))) active(first(X1,X2)) -> first(active(X1),X2) active(first(X1,X2)) -> first(X1,active(X2)) active(s(X)) -> s(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(from(X)) -> from(active(X)) first(mark(X1),X2) -> mark(first(X1,X2)) first(X1,mark(X2)) -> mark(first(X1,X2)) first(ok(X1),ok(X2)) -> ok(first(X1,X2)) cons(mark(X1),X2) -> mark(cons(X1,X2)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) s(mark(X)) -> mark(s(X)) s(ok(X)) -> ok(s(X)) from(mark(X)) -> mark(from(X)) from(ok(X)) -> ok(from(X)) proper(first(X1,X2)) -> first(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(nil()) -> ok(nil()) proper(s(X)) -> s(proper(X)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(from(X)) -> from(proper(X)) weight function: w0 = 1 w(top#) = w(from) = w(s) = w(mark) = w(nil) = w(0) = 1 w(ok) = w(proper) = w(cons) = w(active) = w(first) = 0 precedence: from ~ s > first > top# ~ ok ~ proper ~ cons ~ mark ~ nil ~ active ~ 0 problem: DPs: top#(ok(X)) -> top#(active(X)) TRS: active(first(0(),X)) -> mark(nil()) active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z))) active(from(X)) -> mark(cons(X,from(s(X)))) active(first(X1,X2)) -> first(active(X1),X2) active(first(X1,X2)) -> first(X1,active(X2)) active(s(X)) -> s(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(from(X)) -> from(active(X)) first(mark(X1),X2) -> mark(first(X1,X2)) first(X1,mark(X2)) -> mark(first(X1,X2)) first(ok(X1),ok(X2)) -> ok(first(X1,X2)) cons(mark(X1),X2) -> mark(cons(X1,X2)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) s(mark(X)) -> mark(s(X)) s(ok(X)) -> ok(s(X)) from(mark(X)) -> mark(from(X)) from(ok(X)) -> ok(from(X)) proper(first(X1,X2)) -> first(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(nil()) -> ok(nil()) proper(s(X)) -> s(proper(X)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(from(X)) -> from(proper(X)) Restore Modifier: DPs: top#(ok(X)) -> top#(active(X)) TRS: active(first(0(),X)) -> mark(nil()) active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z))) active(from(X)) -> mark(cons(X,from(s(X)))) active(first(X1,X2)) -> first(active(X1),X2) active(first(X1,X2)) -> first(X1,active(X2)) active(s(X)) -> s(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(from(X)) -> from(active(X)) first(mark(X1),X2) -> mark(first(X1,X2)) first(X1,mark(X2)) -> mark(first(X1,X2)) s(mark(X)) -> mark(s(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) from(mark(X)) -> mark(from(X)) proper(first(X1,X2)) -> first(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(nil()) -> ok(nil()) proper(s(X)) -> s(proper(X)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(from(X)) -> from(proper(X)) first(ok(X1),ok(X2)) -> ok(first(X1,X2)) s(ok(X)) -> ok(s(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) from(ok(X)) -> ok(from(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(first(0(),X)) -> mark(nil()) active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z))) active(from(X)) -> mark(cons(X,from(s(X)))) active(first(X1,X2)) -> first(active(X1),X2) active(first(X1,X2)) -> first(X1,active(X2)) active(s(X)) -> s(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(from(X)) -> from(active(X)) first(mark(X1),X2) -> mark(first(X1,X2)) first(X1,mark(X2)) -> mark(first(X1,X2)) first(ok(X1),ok(X2)) -> ok(first(X1,X2)) cons(mark(X1),X2) -> mark(cons(X1,X2)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) s(mark(X)) -> mark(s(X)) s(ok(X)) -> ok(s(X)) from(mark(X)) -> mark(from(X)) from(ok(X)) -> ok(from(X)) Arctic Interpretation Processor: dimension: 1 usable rules: active(first(0(),X)) -> mark(nil()) active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z))) active(from(X)) -> mark(cons(X,from(s(X)))) active(first(X1,X2)) -> first(active(X1),X2) active(first(X1,X2)) -> first(X1,active(X2)) active(s(X)) -> s(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(from(X)) -> from(active(X)) first(mark(X1),X2) -> mark(first(X1,X2)) first(X1,mark(X2)) -> mark(first(X1,X2)) first(ok(X1),ok(X2)) -> ok(first(X1,X2)) cons(mark(X1),X2) -> mark(cons(X1,X2)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) s(mark(X)) -> mark(s(X)) s(ok(X)) -> ok(s(X)) from(mark(X)) -> mark(from(X)) from(ok(X)) -> ok(from(X)) interpretation: [top#](x0) = x0, [ok](x0) = 1x0 + 2, [from](x0) = x0, [cons](x0, x1) = x0, [s](x0) = x0 + 1, [mark](x0) = x0, [nil] = 3, [active](x0) = x0, [first](x0, x1) = 2x0 + 1x1 + 3, [0] = 1 orientation: top#(ok(X)) = 1X + 2 >= X = top#(active(X)) active(first(0(),X)) = 1X + 3 >= 3 = mark(nil()) active(first(s(X),cons(Y,Z))) = 2X + 1Y + 3 >= Y = mark(cons(Y,first(X,Z))) active(from(X)) = X >= X = mark(cons(X,from(s(X)))) active(first(X1,X2)) = 2X1 + 1X2 + 3 >= 2X1 + 1X2 + 3 = first(active(X1),X2) active(first(X1,X2)) = 2X1 + 1X2 + 3 >= 2X1 + 1X2 + 3 = first(X1,active(X2)) active(s(X)) = X + 1 >= X + 1 = s(active(X)) active(cons(X1,X2)) = X1 >= X1 = cons(active(X1),X2) active(from(X)) = X >= X = from(active(X)) first(mark(X1),X2) = 2X1 + 1X2 + 3 >= 2X1 + 1X2 + 3 = mark(first(X1,X2)) first(X1,mark(X2)) = 2X1 + 1X2 + 3 >= 2X1 + 1X2 + 3 = mark(first(X1,X2)) first(ok(X1),ok(X2)) = 3X1 + 2X2 + 4 >= 3X1 + 2X2 + 4 = ok(first(X1,X2)) cons(mark(X1),X2) = X1 >= X1 = mark(cons(X1,X2)) cons(ok(X1),ok(X2)) = 1X1 + 2 >= 1X1 + 2 = ok(cons(X1,X2)) s(mark(X)) = X + 1 >= X + 1 = mark(s(X)) s(ok(X)) = 1X + 2 >= 1X + 2 = ok(s(X)) from(mark(X)) = X >= X = mark(from(X)) from(ok(X)) = 1X + 2 >= 1X + 2 = ok(from(X)) problem: DPs: TRS: active(first(0(),X)) -> mark(nil()) active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z))) active(from(X)) -> mark(cons(X,from(s(X)))) active(first(X1,X2)) -> first(active(X1),X2) active(first(X1,X2)) -> first(X1,active(X2)) active(s(X)) -> s(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(from(X)) -> from(active(X)) first(mark(X1),X2) -> mark(first(X1,X2)) first(X1,mark(X2)) -> mark(first(X1,X2)) first(ok(X1),ok(X2)) -> ok(first(X1,X2)) cons(mark(X1),X2) -> mark(cons(X1,X2)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) s(mark(X)) -> mark(s(X)) s(ok(X)) -> ok(s(X)) from(mark(X)) -> mark(from(X)) from(ok(X)) -> ok(from(X)) Qed DPs: active#(first(X1,X2)) -> active#(X1) active#(first(X1,X2)) -> active#(X2) active#(s(X)) -> active#(X) active#(cons(X1,X2)) -> active#(X1) active#(from(X)) -> active#(X) TRS: active(first(0(),X)) -> mark(nil()) active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z))) active(from(X)) -> mark(cons(X,from(s(X)))) active(first(X1,X2)) -> first(active(X1),X2) active(first(X1,X2)) -> first(X1,active(X2)) active(s(X)) -> s(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(from(X)) -> from(active(X)) first(mark(X1),X2) -> mark(first(X1,X2)) first(X1,mark(X2)) -> mark(first(X1,X2)) s(mark(X)) -> mark(s(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) from(mark(X)) -> mark(from(X)) proper(first(X1,X2)) -> first(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(nil()) -> ok(nil()) proper(s(X)) -> s(proper(X)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(from(X)) -> from(proper(X)) first(ok(X1),ok(X2)) -> ok(first(X1,X2)) s(ok(X)) -> ok(s(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) from(ok(X)) -> ok(from(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(first(0(),X)) -> mark(nil()) active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z))) active(from(X)) -> mark(cons(X,from(s(X)))) active(first(X1,X2)) -> first(active(X1),X2) active(first(X1,X2)) -> first(X1,active(X2)) active(s(X)) -> s(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(from(X)) -> from(active(X)) first(mark(X1),X2) -> mark(first(X1,X2)) first(X1,mark(X2)) -> mark(first(X1,X2)) s(mark(X)) -> mark(s(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) from(mark(X)) -> mark(from(X)) proper(first(X1,X2)) -> first(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(nil()) -> ok(nil()) proper(s(X)) -> s(proper(X)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(from(X)) -> from(proper(X)) first(ok(X1),ok(X2)) -> ok(first(X1,X2)) s(ok(X)) -> ok(s(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) from(ok(X)) -> ok(from(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed DPs: proper#(first(X1,X2)) -> proper#(X2) proper#(first(X1,X2)) -> proper#(X1) proper#(s(X)) -> proper#(X) proper#(cons(X1,X2)) -> proper#(X2) proper#(cons(X1,X2)) -> proper#(X1) proper#(from(X)) -> proper#(X) TRS: active(first(0(),X)) -> mark(nil()) active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z))) active(from(X)) -> mark(cons(X,from(s(X)))) active(first(X1,X2)) -> first(active(X1),X2) active(first(X1,X2)) -> first(X1,active(X2)) active(s(X)) -> s(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(from(X)) -> from(active(X)) first(mark(X1),X2) -> mark(first(X1,X2)) first(X1,mark(X2)) -> mark(first(X1,X2)) s(mark(X)) -> mark(s(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) from(mark(X)) -> mark(from(X)) proper(first(X1,X2)) -> first(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(nil()) -> ok(nil()) proper(s(X)) -> s(proper(X)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(from(X)) -> from(proper(X)) first(ok(X1),ok(X2)) -> ok(first(X1,X2)) s(ok(X)) -> ok(s(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) from(ok(X)) -> ok(from(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(first(0(),X)) -> mark(nil()) active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z))) active(from(X)) -> mark(cons(X,from(s(X)))) active(first(X1,X2)) -> first(active(X1),X2) active(first(X1,X2)) -> first(X1,active(X2)) active(s(X)) -> s(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(from(X)) -> from(active(X)) first(mark(X1),X2) -> mark(first(X1,X2)) first(X1,mark(X2)) -> mark(first(X1,X2)) s(mark(X)) -> mark(s(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) from(mark(X)) -> mark(from(X)) proper(first(X1,X2)) -> first(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(nil()) -> ok(nil()) proper(s(X)) -> s(proper(X)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(from(X)) -> from(proper(X)) first(ok(X1),ok(X2)) -> ok(first(X1,X2)) s(ok(X)) -> ok(s(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) from(ok(X)) -> ok(from(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(first(0(),X)) -> mark(nil()) active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z))) active(from(X)) -> mark(cons(X,from(s(X)))) active(first(X1,X2)) -> first(active(X1),X2) active(first(X1,X2)) -> first(X1,active(X2)) active(s(X)) -> s(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(from(X)) -> from(active(X)) first(mark(X1),X2) -> mark(first(X1,X2)) first(X1,mark(X2)) -> mark(first(X1,X2)) s(mark(X)) -> mark(s(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) from(mark(X)) -> mark(from(X)) proper(first(X1,X2)) -> first(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(nil()) -> ok(nil()) proper(s(X)) -> s(proper(X)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(from(X)) -> from(proper(X)) first(ok(X1),ok(X2)) -> ok(first(X1,X2)) s(ok(X)) -> ok(s(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) from(ok(X)) -> ok(from(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(first(0(),X)) -> mark(nil()) active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z))) active(from(X)) -> mark(cons(X,from(s(X)))) active(first(X1,X2)) -> first(active(X1),X2) active(first(X1,X2)) -> first(X1,active(X2)) active(s(X)) -> s(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(from(X)) -> from(active(X)) first(mark(X1),X2) -> mark(first(X1,X2)) first(X1,mark(X2)) -> mark(first(X1,X2)) s(mark(X)) -> mark(s(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) from(mark(X)) -> mark(from(X)) proper(first(X1,X2)) -> first(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(nil()) -> ok(nil()) proper(s(X)) -> s(proper(X)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(from(X)) -> from(proper(X)) first(ok(X1),ok(X2)) -> ok(first(X1,X2)) s(ok(X)) -> ok(s(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) from(ok(X)) -> ok(from(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(first(0(),X)) -> mark(nil()) active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z))) active(from(X)) -> mark(cons(X,from(s(X)))) active(first(X1,X2)) -> first(active(X1),X2) active(first(X1,X2)) -> first(X1,active(X2)) active(s(X)) -> s(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(from(X)) -> from(active(X)) first(mark(X1),X2) -> mark(first(X1,X2)) first(X1,mark(X2)) -> mark(first(X1,X2)) s(mark(X)) -> mark(s(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) from(mark(X)) -> mark(from(X)) proper(first(X1,X2)) -> first(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(nil()) -> ok(nil()) proper(s(X)) -> s(proper(X)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(from(X)) -> from(proper(X)) first(ok(X1),ok(X2)) -> ok(first(X1,X2)) s(ok(X)) -> ok(s(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) from(ok(X)) -> ok(from(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Subterm Criterion Processor: simple projection: pi(cons#) = 1 problem: DPs: cons#(mark(X1),X2) -> cons#(X1,X2) TRS: active(first(0(),X)) -> mark(nil()) active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z))) active(from(X)) -> mark(cons(X,from(s(X)))) active(first(X1,X2)) -> first(active(X1),X2) active(first(X1,X2)) -> first(X1,active(X2)) active(s(X)) -> s(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(from(X)) -> from(active(X)) first(mark(X1),X2) -> mark(first(X1,X2)) first(X1,mark(X2)) -> mark(first(X1,X2)) s(mark(X)) -> mark(s(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) from(mark(X)) -> mark(from(X)) proper(first(X1,X2)) -> first(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(nil()) -> ok(nil()) proper(s(X)) -> s(proper(X)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(from(X)) -> from(proper(X)) first(ok(X1),ok(X2)) -> ok(first(X1,X2)) s(ok(X)) -> ok(s(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) from(ok(X)) -> ok(from(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(first(0(),X)) -> mark(nil()) active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z))) active(from(X)) -> mark(cons(X,from(s(X)))) active(first(X1,X2)) -> first(active(X1),X2) active(first(X1,X2)) -> first(X1,active(X2)) active(s(X)) -> s(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(from(X)) -> from(active(X)) first(mark(X1),X2) -> mark(first(X1,X2)) first(X1,mark(X2)) -> mark(first(X1,X2)) s(mark(X)) -> mark(s(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) from(mark(X)) -> mark(from(X)) proper(first(X1,X2)) -> first(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(nil()) -> ok(nil()) proper(s(X)) -> s(proper(X)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(from(X)) -> from(proper(X)) first(ok(X1),ok(X2)) -> ok(first(X1,X2)) s(ok(X)) -> ok(s(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) from(ok(X)) -> ok(from(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(first(0(),X)) -> mark(nil()) active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z))) active(from(X)) -> mark(cons(X,from(s(X)))) active(first(X1,X2)) -> first(active(X1),X2) active(first(X1,X2)) -> first(X1,active(X2)) active(s(X)) -> s(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(from(X)) -> from(active(X)) first(mark(X1),X2) -> mark(first(X1,X2)) first(X1,mark(X2)) -> mark(first(X1,X2)) s(mark(X)) -> mark(s(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) from(mark(X)) -> mark(from(X)) proper(first(X1,X2)) -> first(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(nil()) -> ok(nil()) proper(s(X)) -> s(proper(X)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(from(X)) -> from(proper(X)) first(ok(X1),ok(X2)) -> ok(first(X1,X2)) s(ok(X)) -> ok(s(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) from(ok(X)) -> ok(from(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(first(0(),X)) -> mark(nil()) active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z))) active(from(X)) -> mark(cons(X,from(s(X)))) active(first(X1,X2)) -> first(active(X1),X2) active(first(X1,X2)) -> first(X1,active(X2)) active(s(X)) -> s(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(from(X)) -> from(active(X)) first(mark(X1),X2) -> mark(first(X1,X2)) first(X1,mark(X2)) -> mark(first(X1,X2)) s(mark(X)) -> mark(s(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) from(mark(X)) -> mark(from(X)) proper(first(X1,X2)) -> first(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(nil()) -> ok(nil()) proper(s(X)) -> s(proper(X)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(from(X)) -> from(proper(X)) first(ok(X1),ok(X2)) -> ok(first(X1,X2)) s(ok(X)) -> ok(s(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) from(ok(X)) -> ok(from(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed DPs: first#(mark(X1),X2) -> first#(X1,X2) first#(X1,mark(X2)) -> first#(X1,X2) first#(ok(X1),ok(X2)) -> first#(X1,X2) TRS: active(first(0(),X)) -> mark(nil()) active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z))) active(from(X)) -> mark(cons(X,from(s(X)))) active(first(X1,X2)) -> first(active(X1),X2) active(first(X1,X2)) -> first(X1,active(X2)) active(s(X)) -> s(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(from(X)) -> from(active(X)) first(mark(X1),X2) -> mark(first(X1,X2)) first(X1,mark(X2)) -> mark(first(X1,X2)) s(mark(X)) -> mark(s(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) from(mark(X)) -> mark(from(X)) proper(first(X1,X2)) -> first(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(nil()) -> ok(nil()) proper(s(X)) -> s(proper(X)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(from(X)) -> from(proper(X)) first(ok(X1),ok(X2)) -> ok(first(X1,X2)) s(ok(X)) -> ok(s(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) from(ok(X)) -> ok(from(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Subterm Criterion Processor: simple projection: pi(first#) = 1 problem: DPs: first#(mark(X1),X2) -> first#(X1,X2) TRS: active(first(0(),X)) -> mark(nil()) active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z))) active(from(X)) -> mark(cons(X,from(s(X)))) active(first(X1,X2)) -> first(active(X1),X2) active(first(X1,X2)) -> first(X1,active(X2)) active(s(X)) -> s(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(from(X)) -> from(active(X)) first(mark(X1),X2) -> mark(first(X1,X2)) first(X1,mark(X2)) -> mark(first(X1,X2)) s(mark(X)) -> mark(s(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) from(mark(X)) -> mark(from(X)) proper(first(X1,X2)) -> first(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(nil()) -> ok(nil()) proper(s(X)) -> s(proper(X)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(from(X)) -> from(proper(X)) first(ok(X1),ok(X2)) -> ok(first(X1,X2)) s(ok(X)) -> ok(s(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) from(ok(X)) -> ok(from(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Subterm Criterion Processor: simple projection: pi(first#) = 0 problem: DPs: TRS: active(first(0(),X)) -> mark(nil()) active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z))) active(from(X)) -> mark(cons(X,from(s(X)))) active(first(X1,X2)) -> first(active(X1),X2) active(first(X1,X2)) -> first(X1,active(X2)) active(s(X)) -> s(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(from(X)) -> from(active(X)) first(mark(X1),X2) -> mark(first(X1,X2)) first(X1,mark(X2)) -> mark(first(X1,X2)) s(mark(X)) -> mark(s(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) from(mark(X)) -> mark(from(X)) proper(first(X1,X2)) -> first(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(nil()) -> ok(nil()) proper(s(X)) -> s(proper(X)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(from(X)) -> from(proper(X)) first(ok(X1),ok(X2)) -> ok(first(X1,X2)) s(ok(X)) -> ok(s(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) from(ok(X)) -> ok(from(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed