MAYBE Problem: active(fib(N)) -> mark(sel(N,fib1(s(0()),s(0())))) active(fib1(X,Y)) -> mark(cons(X,fib1(Y,add(X,Y)))) active(add(0(),X)) -> mark(X) active(add(s(X),Y)) -> mark(s(add(X,Y))) active(sel(0(),cons(X,XS))) -> mark(X) active(sel(s(N),cons(X,XS))) -> mark(sel(N,XS)) active(fib(X)) -> fib(active(X)) active(sel(X1,X2)) -> sel(active(X1),X2) active(sel(X1,X2)) -> sel(X1,active(X2)) active(fib1(X1,X2)) -> fib1(active(X1),X2) active(fib1(X1,X2)) -> fib1(X1,active(X2)) active(s(X)) -> s(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(add(X1,X2)) -> add(active(X1),X2) active(add(X1,X2)) -> add(X1,active(X2)) fib(mark(X)) -> mark(fib(X)) sel(mark(X1),X2) -> mark(sel(X1,X2)) sel(X1,mark(X2)) -> mark(sel(X1,X2)) fib1(mark(X1),X2) -> mark(fib1(X1,X2)) fib1(X1,mark(X2)) -> mark(fib1(X1,X2)) s(mark(X)) -> mark(s(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) add(mark(X1),X2) -> mark(add(X1,X2)) add(X1,mark(X2)) -> mark(add(X1,X2)) proper(fib(X)) -> fib(proper(X)) proper(sel(X1,X2)) -> sel(proper(X1),proper(X2)) proper(fib1(X1,X2)) -> fib1(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(0()) -> ok(0()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(add(X1,X2)) -> add(proper(X1),proper(X2)) fib(ok(X)) -> ok(fib(X)) sel(ok(X1),ok(X2)) -> ok(sel(X1,X2)) fib1(ok(X1),ok(X2)) -> ok(fib1(X1,X2)) s(ok(X)) -> ok(s(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) add(ok(X1),ok(X2)) -> ok(add(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Proof: DP Processor: DPs: active#(fib(N)) -> s#(0()) active#(fib(N)) -> fib1#(s(0()),s(0())) active#(fib(N)) -> sel#(N,fib1(s(0()),s(0()))) active#(fib1(X,Y)) -> add#(X,Y) active#(fib1(X,Y)) -> fib1#(Y,add(X,Y)) active#(fib1(X,Y)) -> cons#(X,fib1(Y,add(X,Y))) active#(add(s(X),Y)) -> add#(X,Y) active#(add(s(X),Y)) -> s#(add(X,Y)) active#(sel(s(N),cons(X,XS))) -> sel#(N,XS) active#(fib(X)) -> active#(X) active#(fib(X)) -> fib#(active(X)) active#(sel(X1,X2)) -> active#(X1) active#(sel(X1,X2)) -> sel#(active(X1),X2) active#(sel(X1,X2)) -> active#(X2) active#(sel(X1,X2)) -> sel#(X1,active(X2)) active#(fib1(X1,X2)) -> active#(X1) active#(fib1(X1,X2)) -> fib1#(active(X1),X2) active#(fib1(X1,X2)) -> active#(X2) active#(fib1(X1,X2)) -> fib1#(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#(add(X1,X2)) -> active#(X1) active#(add(X1,X2)) -> add#(active(X1),X2) active#(add(X1,X2)) -> active#(X2) active#(add(X1,X2)) -> add#(X1,active(X2)) fib#(mark(X)) -> fib#(X) sel#(mark(X1),X2) -> sel#(X1,X2) sel#(X1,mark(X2)) -> sel#(X1,X2) fib1#(mark(X1),X2) -> fib1#(X1,X2) fib1#(X1,mark(X2)) -> fib1#(X1,X2) s#(mark(X)) -> s#(X) cons#(mark(X1),X2) -> cons#(X1,X2) add#(mark(X1),X2) -> add#(X1,X2) add#(X1,mark(X2)) -> add#(X1,X2) proper#(fib(X)) -> proper#(X) proper#(fib(X)) -> fib#(proper(X)) proper#(sel(X1,X2)) -> proper#(X2) proper#(sel(X1,X2)) -> proper#(X1) proper#(sel(X1,X2)) -> sel#(proper(X1),proper(X2)) proper#(fib1(X1,X2)) -> proper#(X2) proper#(fib1(X1,X2)) -> proper#(X1) proper#(fib1(X1,X2)) -> fib1#(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#(add(X1,X2)) -> proper#(X2) proper#(add(X1,X2)) -> proper#(X1) proper#(add(X1,X2)) -> add#(proper(X1),proper(X2)) fib#(ok(X)) -> fib#(X) sel#(ok(X1),ok(X2)) -> sel#(X1,X2) fib1#(ok(X1),ok(X2)) -> fib1#(X1,X2) s#(ok(X)) -> s#(X) cons#(ok(X1),ok(X2)) -> cons#(X1,X2) add#(ok(X1),ok(X2)) -> add#(X1,X2) top#(mark(X)) -> proper#(X) top#(mark(X)) -> top#(proper(X)) top#(ok(X)) -> active#(X) top#(ok(X)) -> top#(active(X)) TRS: active(fib(N)) -> mark(sel(N,fib1(s(0()),s(0())))) active(fib1(X,Y)) -> mark(cons(X,fib1(Y,add(X,Y)))) active(add(0(),X)) -> mark(X) active(add(s(X),Y)) -> mark(s(add(X,Y))) active(sel(0(),cons(X,XS))) -> mark(X) active(sel(s(N),cons(X,XS))) -> mark(sel(N,XS)) active(fib(X)) -> fib(active(X)) active(sel(X1,X2)) -> sel(active(X1),X2) active(sel(X1,X2)) -> sel(X1,active(X2)) active(fib1(X1,X2)) -> fib1(active(X1),X2) active(fib1(X1,X2)) -> fib1(X1,active(X2)) active(s(X)) -> s(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(add(X1,X2)) -> add(active(X1),X2) active(add(X1,X2)) -> add(X1,active(X2)) fib(mark(X)) -> mark(fib(X)) sel(mark(X1),X2) -> mark(sel(X1,X2)) sel(X1,mark(X2)) -> mark(sel(X1,X2)) fib1(mark(X1),X2) -> mark(fib1(X1,X2)) fib1(X1,mark(X2)) -> mark(fib1(X1,X2)) s(mark(X)) -> mark(s(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) add(mark(X1),X2) -> mark(add(X1,X2)) add(X1,mark(X2)) -> mark(add(X1,X2)) proper(fib(X)) -> fib(proper(X)) proper(sel(X1,X2)) -> sel(proper(X1),proper(X2)) proper(fib1(X1,X2)) -> fib1(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(0()) -> ok(0()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(add(X1,X2)) -> add(proper(X1),proper(X2)) fib(ok(X)) -> ok(fib(X)) sel(ok(X1),ok(X2)) -> ok(sel(X1,X2)) fib1(ok(X1),ok(X2)) -> ok(fib1(X1,X2)) s(ok(X)) -> ok(s(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) add(ok(X1),ok(X2)) -> ok(add(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) TDG Processor: DPs: active#(fib(N)) -> s#(0()) active#(fib(N)) -> fib1#(s(0()),s(0())) active#(fib(N)) -> sel#(N,fib1(s(0()),s(0()))) active#(fib1(X,Y)) -> add#(X,Y) active#(fib1(X,Y)) -> fib1#(Y,add(X,Y)) active#(fib1(X,Y)) -> cons#(X,fib1(Y,add(X,Y))) active#(add(s(X),Y)) -> add#(X,Y) active#(add(s(X),Y)) -> s#(add(X,Y)) active#(sel(s(N),cons(X,XS))) -> sel#(N,XS) active#(fib(X)) -> active#(X) active#(fib(X)) -> fib#(active(X)) active#(sel(X1,X2)) -> active#(X1) active#(sel(X1,X2)) -> sel#(active(X1),X2) active#(sel(X1,X2)) -> active#(X2) active#(sel(X1,X2)) -> sel#(X1,active(X2)) active#(fib1(X1,X2)) -> active#(X1) active#(fib1(X1,X2)) -> fib1#(active(X1),X2) active#(fib1(X1,X2)) -> active#(X2) active#(fib1(X1,X2)) -> fib1#(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#(add(X1,X2)) -> active#(X1) active#(add(X1,X2)) -> add#(active(X1),X2) active#(add(X1,X2)) -> active#(X2) active#(add(X1,X2)) -> add#(X1,active(X2)) fib#(mark(X)) -> fib#(X) sel#(mark(X1),X2) -> sel#(X1,X2) sel#(X1,mark(X2)) -> sel#(X1,X2) fib1#(mark(X1),X2) -> fib1#(X1,X2) fib1#(X1,mark(X2)) -> fib1#(X1,X2) s#(mark(X)) -> s#(X) cons#(mark(X1),X2) -> cons#(X1,X2) add#(mark(X1),X2) -> add#(X1,X2) add#(X1,mark(X2)) -> add#(X1,X2) proper#(fib(X)) -> proper#(X) proper#(fib(X)) -> fib#(proper(X)) proper#(sel(X1,X2)) -> proper#(X2) proper#(sel(X1,X2)) -> proper#(X1) proper#(sel(X1,X2)) -> sel#(proper(X1),proper(X2)) proper#(fib1(X1,X2)) -> proper#(X2) proper#(fib1(X1,X2)) -> proper#(X1) proper#(fib1(X1,X2)) -> fib1#(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#(add(X1,X2)) -> proper#(X2) proper#(add(X1,X2)) -> proper#(X1) proper#(add(X1,X2)) -> add#(proper(X1),proper(X2)) fib#(ok(X)) -> fib#(X) sel#(ok(X1),ok(X2)) -> sel#(X1,X2) fib1#(ok(X1),ok(X2)) -> fib1#(X1,X2) s#(ok(X)) -> s#(X) cons#(ok(X1),ok(X2)) -> cons#(X1,X2) add#(ok(X1),ok(X2)) -> add#(X1,X2) top#(mark(X)) -> proper#(X) top#(mark(X)) -> top#(proper(X)) top#(ok(X)) -> active#(X) top#(ok(X)) -> top#(active(X)) TRS: active(fib(N)) -> mark(sel(N,fib1(s(0()),s(0())))) active(fib1(X,Y)) -> mark(cons(X,fib1(Y,add(X,Y)))) active(add(0(),X)) -> mark(X) active(add(s(X),Y)) -> mark(s(add(X,Y))) active(sel(0(),cons(X,XS))) -> mark(X) active(sel(s(N),cons(X,XS))) -> mark(sel(N,XS)) active(fib(X)) -> fib(active(X)) active(sel(X1,X2)) -> sel(active(X1),X2) active(sel(X1,X2)) -> sel(X1,active(X2)) active(fib1(X1,X2)) -> fib1(active(X1),X2) active(fib1(X1,X2)) -> fib1(X1,active(X2)) active(s(X)) -> s(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(add(X1,X2)) -> add(active(X1),X2) active(add(X1,X2)) -> add(X1,active(X2)) fib(mark(X)) -> mark(fib(X)) sel(mark(X1),X2) -> mark(sel(X1,X2)) sel(X1,mark(X2)) -> mark(sel(X1,X2)) fib1(mark(X1),X2) -> mark(fib1(X1,X2)) fib1(X1,mark(X2)) -> mark(fib1(X1,X2)) s(mark(X)) -> mark(s(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) add(mark(X1),X2) -> mark(add(X1,X2)) add(X1,mark(X2)) -> mark(add(X1,X2)) proper(fib(X)) -> fib(proper(X)) proper(sel(X1,X2)) -> sel(proper(X1),proper(X2)) proper(fib1(X1,X2)) -> fib1(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(0()) -> ok(0()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(add(X1,X2)) -> add(proper(X1),proper(X2)) fib(ok(X)) -> ok(fib(X)) sel(ok(X1),ok(X2)) -> ok(sel(X1,X2)) fib1(ok(X1),ok(X2)) -> ok(fib1(X1,X2)) s(ok(X)) -> ok(s(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) add(ok(X1),ok(X2)) -> ok(add(X1,X2)) 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#(add(X1,X2)) -> add#(X1,active(X2)) top#(ok(X)) -> active#(X) -> active#(add(X1,X2)) -> active#(X2) top#(ok(X)) -> active#(X) -> active#(add(X1,X2)) -> add#(active(X1),X2) top#(ok(X)) -> active#(X) -> active#(add(X1,X2)) -> active#(X1) 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#(fib1(X1,X2)) -> fib1#(X1,active(X2)) top#(ok(X)) -> active#(X) -> active#(fib1(X1,X2)) -> active#(X2) top#(ok(X)) -> active#(X) -> active#(fib1(X1,X2)) -> fib1#(active(X1),X2) top#(ok(X)) -> active#(X) -> active#(fib1(X1,X2)) -> active#(X1) top#(ok(X)) -> active#(X) -> active#(sel(X1,X2)) -> sel#(X1,active(X2)) top#(ok(X)) -> active#(X) -> active#(sel(X1,X2)) -> active#(X2) top#(ok(X)) -> active#(X) -> active#(sel(X1,X2)) -> sel#(active(X1),X2) top#(ok(X)) -> active#(X) -> active#(sel(X1,X2)) -> active#(X1) top#(ok(X)) -> active#(X) -> active#(fib(X)) -> fib#(active(X)) top#(ok(X)) -> active#(X) -> active#(fib(X)) -> active#(X) top#(ok(X)) -> active#(X) -> active#(sel(s(N),cons(X,XS))) -> sel#(N,XS) top#(ok(X)) -> active#(X) -> active#(add(s(X),Y)) -> s#(add(X,Y)) top#(ok(X)) -> active#(X) -> active#(add(s(X),Y)) -> add#(X,Y) top#(ok(X)) -> active#(X) -> active#(fib1(X,Y)) -> cons#(X,fib1(Y,add(X,Y))) top#(ok(X)) -> active#(X) -> active#(fib1(X,Y)) -> fib1#(Y,add(X,Y)) top#(ok(X)) -> active#(X) -> active#(fib1(X,Y)) -> add#(X,Y) top#(ok(X)) -> active#(X) -> active#(fib(N)) -> sel#(N,fib1(s(0()),s(0()))) top#(ok(X)) -> active#(X) -> active#(fib(N)) -> fib1#(s(0()),s(0())) top#(ok(X)) -> active#(X) -> active#(fib(N)) -> s#(0()) 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#(add(X1,X2)) -> add#(proper(X1),proper(X2)) top#(mark(X)) -> proper#(X) -> proper#(add(X1,X2)) -> proper#(X1) top#(mark(X)) -> proper#(X) -> proper#(add(X1,X2)) -> proper#(X2) top#(mark(X)) -> proper#(X) -> proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2)) top#(mark(X)) -> proper#(X) -> proper#(cons(X1,X2)) -> proper#(X1) top#(mark(X)) -> proper#(X) -> proper#(cons(X1,X2)) -> proper#(X2) top#(mark(X)) -> proper#(X) -> proper#(s(X)) -> s#(proper(X)) top#(mark(X)) -> proper#(X) -> proper#(s(X)) -> proper#(X) top#(mark(X)) -> proper#(X) -> proper#(fib1(X1,X2)) -> fib1#(proper(X1),proper(X2)) top#(mark(X)) -> proper#(X) -> proper#(fib1(X1,X2)) -> proper#(X1) top#(mark(X)) -> proper#(X) -> proper#(fib1(X1,X2)) -> proper#(X2) top#(mark(X)) -> proper#(X) -> proper#(sel(X1,X2)) -> sel#(proper(X1),proper(X2)) top#(mark(X)) -> proper#(X) -> proper#(sel(X1,X2)) -> proper#(X1) top#(mark(X)) -> proper#(X) -> proper#(sel(X1,X2)) -> proper#(X2) top#(mark(X)) -> proper#(X) -> proper#(fib(X)) -> fib#(proper(X)) top#(mark(X)) -> proper#(X) -> proper#(fib(X)) -> proper#(X) proper#(cons(X1,X2)) -> proper#(X2) -> proper#(add(X1,X2)) -> add#(proper(X1),proper(X2)) proper#(cons(X1,X2)) -> proper#(X2) -> proper#(add(X1,X2)) -> proper#(X1) proper#(cons(X1,X2)) -> proper#(X2) -> proper#(add(X1,X2)) -> proper#(X2) proper#(cons(X1,X2)) -> proper#(X2) -> proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2)) proper#(cons(X1,X2)) -> proper#(X2) -> proper#(cons(X1,X2)) -> proper#(X1) proper#(cons(X1,X2)) -> proper#(X2) -> proper#(cons(X1,X2)) -> proper#(X2) proper#(cons(X1,X2)) -> proper#(X2) -> proper#(s(X)) -> s#(proper(X)) proper#(cons(X1,X2)) -> proper#(X2) -> proper#(s(X)) -> proper#(X) proper#(cons(X1,X2)) -> proper#(X2) -> proper#(fib1(X1,X2)) -> fib1#(proper(X1),proper(X2)) proper#(cons(X1,X2)) -> proper#(X2) -> proper#(fib1(X1,X2)) -> proper#(X1) proper#(cons(X1,X2)) -> proper#(X2) -> proper#(fib1(X1,X2)) -> proper#(X2) proper#(cons(X1,X2)) -> proper#(X2) -> proper#(sel(X1,X2)) -> sel#(proper(X1),proper(X2)) proper#(cons(X1,X2)) -> proper#(X2) -> proper#(sel(X1,X2)) -> proper#(X1) proper#(cons(X1,X2)) -> proper#(X2) -> proper#(sel(X1,X2)) -> proper#(X2) proper#(cons(X1,X2)) -> proper#(X2) -> proper#(fib(X)) -> fib#(proper(X)) proper#(cons(X1,X2)) -> proper#(X2) -> proper#(fib(X)) -> proper#(X) proper#(cons(X1,X2)) -> proper#(X1) -> proper#(add(X1,X2)) -> add#(proper(X1),proper(X2)) proper#(cons(X1,X2)) -> proper#(X1) -> proper#(add(X1,X2)) -> proper#(X1) proper#(cons(X1,X2)) -> proper#(X1) -> proper#(add(X1,X2)) -> proper#(X2) proper#(cons(X1,X2)) -> proper#(X1) -> proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2)) proper#(cons(X1,X2)) -> proper#(X1) -> proper#(cons(X1,X2)) -> proper#(X1) proper#(cons(X1,X2)) -> proper#(X1) -> proper#(cons(X1,X2)) -> proper#(X2) proper#(cons(X1,X2)) -> proper#(X1) -> proper#(s(X)) -> s#(proper(X)) proper#(cons(X1,X2)) -> proper#(X1) -> proper#(s(X)) -> proper#(X) proper#(cons(X1,X2)) -> proper#(X1) -> proper#(fib1(X1,X2)) -> fib1#(proper(X1),proper(X2)) proper#(cons(X1,X2)) -> proper#(X1) -> proper#(fib1(X1,X2)) -> proper#(X1) proper#(cons(X1,X2)) -> proper#(X1) -> proper#(fib1(X1,X2)) -> proper#(X2) proper#(cons(X1,X2)) -> proper#(X1) -> proper#(sel(X1,X2)) -> sel#(proper(X1),proper(X2)) proper#(cons(X1,X2)) -> proper#(X1) -> proper#(sel(X1,X2)) -> proper#(X1) proper#(cons(X1,X2)) -> proper#(X1) -> proper#(sel(X1,X2)) -> proper#(X2) proper#(cons(X1,X2)) -> proper#(X1) -> proper#(fib(X)) -> fib#(proper(X)) proper#(cons(X1,X2)) -> proper#(X1) -> proper#(fib(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#(add(X1,X2)) -> proper#(X2) -> proper#(add(X1,X2)) -> add#(proper(X1),proper(X2)) proper#(add(X1,X2)) -> proper#(X2) -> proper#(add(X1,X2)) -> proper#(X1) proper#(add(X1,X2)) -> proper#(X2) -> proper#(add(X1,X2)) -> proper#(X2) proper#(add(X1,X2)) -> proper#(X2) -> proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2)) proper#(add(X1,X2)) -> proper#(X2) -> proper#(cons(X1,X2)) -> proper#(X1) proper#(add(X1,X2)) -> proper#(X2) -> proper#(cons(X1,X2)) -> proper#(X2) proper#(add(X1,X2)) -> proper#(X2) -> proper#(s(X)) -> s#(proper(X)) proper#(add(X1,X2)) -> proper#(X2) -> proper#(s(X)) -> proper#(X) proper#(add(X1,X2)) -> proper#(X2) -> proper#(fib1(X1,X2)) -> fib1#(proper(X1),proper(X2)) proper#(add(X1,X2)) -> proper#(X2) -> proper#(fib1(X1,X2)) -> proper#(X1) proper#(add(X1,X2)) -> proper#(X2) -> proper#(fib1(X1,X2)) -> proper#(X2) proper#(add(X1,X2)) -> proper#(X2) -> proper#(sel(X1,X2)) -> sel#(proper(X1),proper(X2)) proper#(add(X1,X2)) -> proper#(X2) -> proper#(sel(X1,X2)) -> proper#(X1) proper#(add(X1,X2)) -> proper#(X2) -> proper#(sel(X1,X2)) -> proper#(X2) proper#(add(X1,X2)) -> proper#(X2) -> proper#(fib(X)) -> fib#(proper(X)) proper#(add(X1,X2)) -> proper#(X2) -> proper#(fib(X)) -> proper#(X) proper#(add(X1,X2)) -> proper#(X1) -> proper#(add(X1,X2)) -> add#(proper(X1),proper(X2)) proper#(add(X1,X2)) -> proper#(X1) -> proper#(add(X1,X2)) -> proper#(X1) proper#(add(X1,X2)) -> proper#(X1) -> proper#(add(X1,X2)) -> proper#(X2) proper#(add(X1,X2)) -> proper#(X1) -> proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2)) proper#(add(X1,X2)) -> proper#(X1) -> proper#(cons(X1,X2)) -> proper#(X1) proper#(add(X1,X2)) -> proper#(X1) -> proper#(cons(X1,X2)) -> proper#(X2) proper#(add(X1,X2)) -> proper#(X1) -> proper#(s(X)) -> s#(proper(X)) proper#(add(X1,X2)) -> proper#(X1) -> proper#(s(X)) -> proper#(X) proper#(add(X1,X2)) -> proper#(X1) -> proper#(fib1(X1,X2)) -> fib1#(proper(X1),proper(X2)) proper#(add(X1,X2)) -> proper#(X1) -> proper#(fib1(X1,X2)) -> proper#(X1) proper#(add(X1,X2)) -> proper#(X1) -> proper#(fib1(X1,X2)) -> proper#(X2) proper#(add(X1,X2)) -> proper#(X1) -> proper#(sel(X1,X2)) -> sel#(proper(X1),proper(X2)) proper#(add(X1,X2)) -> proper#(X1) -> proper#(sel(X1,X2)) -> proper#(X1) proper#(add(X1,X2)) -> proper#(X1) -> proper#(sel(X1,X2)) -> proper#(X2) proper#(add(X1,X2)) -> proper#(X1) -> proper#(fib(X)) -> fib#(proper(X)) proper#(add(X1,X2)) -> proper#(X1) -> proper#(fib(X)) -> proper#(X) proper#(add(X1,X2)) -> add#(proper(X1),proper(X2)) -> add#(ok(X1),ok(X2)) -> add#(X1,X2) proper#(add(X1,X2)) -> add#(proper(X1),proper(X2)) -> add#(X1,mark(X2)) -> add#(X1,X2) proper#(add(X1,X2)) -> add#(proper(X1),proper(X2)) -> add#(mark(X1),X2) -> add#(X1,X2) proper#(sel(X1,X2)) -> proper#(X2) -> proper#(add(X1,X2)) -> add#(proper(X1),proper(X2)) proper#(sel(X1,X2)) -> proper#(X2) -> proper#(add(X1,X2)) -> proper#(X1) proper#(sel(X1,X2)) -> proper#(X2) -> proper#(add(X1,X2)) -> proper#(X2) proper#(sel(X1,X2)) -> proper#(X2) -> proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2)) proper#(sel(X1,X2)) -> proper#(X2) -> proper#(cons(X1,X2)) -> proper#(X1) proper#(sel(X1,X2)) -> proper#(X2) -> proper#(cons(X1,X2)) -> proper#(X2) proper#(sel(X1,X2)) -> proper#(X2) -> proper#(s(X)) -> s#(proper(X)) proper#(sel(X1,X2)) -> proper#(X2) -> proper#(s(X)) -> proper#(X) proper#(sel(X1,X2)) -> proper#(X2) -> proper#(fib1(X1,X2)) -> fib1#(proper(X1),proper(X2)) proper#(sel(X1,X2)) -> proper#(X2) -> proper#(fib1(X1,X2)) -> proper#(X1) proper#(sel(X1,X2)) -> proper#(X2) -> proper#(fib1(X1,X2)) -> proper#(X2) proper#(sel(X1,X2)) -> proper#(X2) -> proper#(sel(X1,X2)) -> sel#(proper(X1),proper(X2)) proper#(sel(X1,X2)) -> proper#(X2) -> proper#(sel(X1,X2)) -> proper#(X1) proper#(sel(X1,X2)) -> proper#(X2) -> proper#(sel(X1,X2)) -> proper#(X2) proper#(sel(X1,X2)) -> proper#(X2) -> proper#(fib(X)) -> fib#(proper(X)) proper#(sel(X1,X2)) -> proper#(X2) -> proper#(fib(X)) -> proper#(X) proper#(sel(X1,X2)) -> proper#(X1) -> proper#(add(X1,X2)) -> add#(proper(X1),proper(X2)) proper#(sel(X1,X2)) -> proper#(X1) -> proper#(add(X1,X2)) -> proper#(X1) proper#(sel(X1,X2)) -> proper#(X1) -> proper#(add(X1,X2)) -> proper#(X2) proper#(sel(X1,X2)) -> proper#(X1) -> proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2)) proper#(sel(X1,X2)) -> proper#(X1) -> proper#(cons(X1,X2)) -> proper#(X1) proper#(sel(X1,X2)) -> proper#(X1) -> proper#(cons(X1,X2)) -> proper#(X2) proper#(sel(X1,X2)) -> proper#(X1) -> proper#(s(X)) -> s#(proper(X)) proper#(sel(X1,X2)) -> proper#(X1) -> proper#(s(X)) -> proper#(X) proper#(sel(X1,X2)) -> proper#(X1) -> proper#(fib1(X1,X2)) -> fib1#(proper(X1),proper(X2)) proper#(sel(X1,X2)) -> proper#(X1) -> proper#(fib1(X1,X2)) -> proper#(X1) proper#(sel(X1,X2)) -> proper#(X1) -> proper#(fib1(X1,X2)) -> proper#(X2) proper#(sel(X1,X2)) -> proper#(X1) -> proper#(sel(X1,X2)) -> sel#(proper(X1),proper(X2)) proper#(sel(X1,X2)) -> proper#(X1) -> proper#(sel(X1,X2)) -> proper#(X1) proper#(sel(X1,X2)) -> proper#(X1) -> proper#(sel(X1,X2)) -> proper#(X2) proper#(sel(X1,X2)) -> proper#(X1) -> proper#(fib(X)) -> fib#(proper(X)) proper#(sel(X1,X2)) -> proper#(X1) -> proper#(fib(X)) -> proper#(X) proper#(sel(X1,X2)) -> sel#(proper(X1),proper(X2)) -> sel#(ok(X1),ok(X2)) -> sel#(X1,X2) proper#(sel(X1,X2)) -> sel#(proper(X1),proper(X2)) -> sel#(X1,mark(X2)) -> sel#(X1,X2) proper#(sel(X1,X2)) -> sel#(proper(X1),proper(X2)) -> sel#(mark(X1),X2) -> sel#(X1,X2) proper#(fib1(X1,X2)) -> proper#(X2) -> proper#(add(X1,X2)) -> add#(proper(X1),proper(X2)) proper#(fib1(X1,X2)) -> proper#(X2) -> proper#(add(X1,X2)) -> proper#(X1) proper#(fib1(X1,X2)) -> proper#(X2) -> proper#(add(X1,X2)) -> proper#(X2) proper#(fib1(X1,X2)) -> proper#(X2) -> proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2)) proper#(fib1(X1,X2)) -> proper#(X2) -> proper#(cons(X1,X2)) -> proper#(X1) proper#(fib1(X1,X2)) -> proper#(X2) -> proper#(cons(X1,X2)) -> proper#(X2) proper#(fib1(X1,X2)) -> proper#(X2) -> proper#(s(X)) -> s#(proper(X)) proper#(fib1(X1,X2)) -> proper#(X2) -> proper#(s(X)) -> proper#(X) proper#(fib1(X1,X2)) -> proper#(X2) -> proper#(fib1(X1,X2)) -> fib1#(proper(X1),proper(X2)) proper#(fib1(X1,X2)) -> proper#(X2) -> proper#(fib1(X1,X2)) -> proper#(X1) proper#(fib1(X1,X2)) -> proper#(X2) -> proper#(fib1(X1,X2)) -> proper#(X2) proper#(fib1(X1,X2)) -> proper#(X2) -> proper#(sel(X1,X2)) -> sel#(proper(X1),proper(X2)) proper#(fib1(X1,X2)) -> proper#(X2) -> proper#(sel(X1,X2)) -> proper#(X1) proper#(fib1(X1,X2)) -> proper#(X2) -> proper#(sel(X1,X2)) -> proper#(X2) proper#(fib1(X1,X2)) -> proper#(X2) -> proper#(fib(X)) -> fib#(proper(X)) proper#(fib1(X1,X2)) -> proper#(X2) -> proper#(fib(X)) -> proper#(X) proper#(fib1(X1,X2)) -> proper#(X1) -> proper#(add(X1,X2)) -> add#(proper(X1),proper(X2)) proper#(fib1(X1,X2)) -> proper#(X1) -> proper#(add(X1,X2)) -> proper#(X1) proper#(fib1(X1,X2)) -> proper#(X1) -> proper#(add(X1,X2)) -> proper#(X2) proper#(fib1(X1,X2)) -> proper#(X1) -> proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2)) proper#(fib1(X1,X2)) -> proper#(X1) -> proper#(cons(X1,X2)) -> proper#(X1) proper#(fib1(X1,X2)) -> proper#(X1) -> proper#(cons(X1,X2)) -> proper#(X2) proper#(fib1(X1,X2)) -> proper#(X1) -> proper#(s(X)) -> s#(proper(X)) proper#(fib1(X1,X2)) -> proper#(X1) -> proper#(s(X)) -> proper#(X) proper#(fib1(X1,X2)) -> proper#(X1) -> proper#(fib1(X1,X2)) -> fib1#(proper(X1),proper(X2)) proper#(fib1(X1,X2)) -> proper#(X1) -> proper#(fib1(X1,X2)) -> proper#(X1) proper#(fib1(X1,X2)) -> proper#(X1) -> proper#(fib1(X1,X2)) -> proper#(X2) proper#(fib1(X1,X2)) -> proper#(X1) -> proper#(sel(X1,X2)) -> sel#(proper(X1),proper(X2)) proper#(fib1(X1,X2)) -> proper#(X1) -> proper#(sel(X1,X2)) -> proper#(X1) proper#(fib1(X1,X2)) -> proper#(X1) -> proper#(sel(X1,X2)) -> proper#(X2) proper#(fib1(X1,X2)) -> proper#(X1) -> proper#(fib(X)) -> fib#(proper(X)) proper#(fib1(X1,X2)) -> proper#(X1) -> proper#(fib(X)) -> proper#(X) proper#(fib1(X1,X2)) -> fib1#(proper(X1),proper(X2)) -> fib1#(ok(X1),ok(X2)) -> fib1#(X1,X2) proper#(fib1(X1,X2)) -> fib1#(proper(X1),proper(X2)) -> fib1#(X1,mark(X2)) -> fib1#(X1,X2) proper#(fib1(X1,X2)) -> fib1#(proper(X1),proper(X2)) -> fib1#(mark(X1),X2) -> fib1#(X1,X2) proper#(s(X)) -> proper#(X) -> proper#(add(X1,X2)) -> add#(proper(X1),proper(X2)) proper#(s(X)) -> proper#(X) -> proper#(add(X1,X2)) -> proper#(X1) proper#(s(X)) -> proper#(X) -> proper#(add(X1,X2)) -> proper#(X2) proper#(s(X)) -> proper#(X) -> proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2)) proper#(s(X)) -> proper#(X) -> proper#(cons(X1,X2)) -> proper#(X1) proper#(s(X)) -> proper#(X) -> proper#(cons(X1,X2)) -> proper#(X2) proper#(s(X)) -> proper#(X) -> proper#(s(X)) -> s#(proper(X)) proper#(s(X)) -> proper#(X) -> proper#(s(X)) -> proper#(X) proper#(s(X)) -> proper#(X) -> proper#(fib1(X1,X2)) -> fib1#(proper(X1),proper(X2)) proper#(s(X)) -> proper#(X) -> proper#(fib1(X1,X2)) -> proper#(X1) proper#(s(X)) -> proper#(X) -> proper#(fib1(X1,X2)) -> proper#(X2) proper#(s(X)) -> proper#(X) -> proper#(sel(X1,X2)) -> sel#(proper(X1),proper(X2)) proper#(s(X)) -> proper#(X) -> proper#(sel(X1,X2)) -> proper#(X1) proper#(s(X)) -> proper#(X) -> proper#(sel(X1,X2)) -> proper#(X2) proper#(s(X)) -> proper#(X) -> proper#(fib(X)) -> fib#(proper(X)) proper#(s(X)) -> proper#(X) -> proper#(fib(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#(fib(X)) -> proper#(X) -> proper#(add(X1,X2)) -> add#(proper(X1),proper(X2)) proper#(fib(X)) -> proper#(X) -> proper#(add(X1,X2)) -> proper#(X1) proper#(fib(X)) -> proper#(X) -> proper#(add(X1,X2)) -> proper#(X2) proper#(fib(X)) -> proper#(X) -> proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2)) proper#(fib(X)) -> proper#(X) -> proper#(cons(X1,X2)) -> proper#(X1) proper#(fib(X)) -> proper#(X) -> proper#(cons(X1,X2)) -> proper#(X2) proper#(fib(X)) -> proper#(X) -> proper#(s(X)) -> s#(proper(X)) proper#(fib(X)) -> proper#(X) -> proper#(s(X)) -> proper#(X) proper#(fib(X)) -> proper#(X) -> proper#(fib1(X1,X2)) -> fib1#(proper(X1),proper(X2)) proper#(fib(X)) -> proper#(X) -> proper#(fib1(X1,X2)) -> proper#(X1) proper#(fib(X)) -> proper#(X) -> proper#(fib1(X1,X2)) -> proper#(X2) proper#(fib(X)) -> proper#(X) -> proper#(sel(X1,X2)) -> sel#(proper(X1),proper(X2)) proper#(fib(X)) -> proper#(X) -> proper#(sel(X1,X2)) -> proper#(X1) proper#(fib(X)) -> proper#(X) -> proper#(sel(X1,X2)) -> proper#(X2) proper#(fib(X)) -> proper#(X) -> proper#(fib(X)) -> fib#(proper(X)) proper#(fib(X)) -> proper#(X) -> proper#(fib(X)) -> proper#(X) proper#(fib(X)) -> fib#(proper(X)) -> fib#(ok(X)) -> fib#(X) proper#(fib(X)) -> fib#(proper(X)) -> fib#(mark(X)) -> fib#(X) fib#(ok(X)) -> fib#(X) -> fib#(ok(X)) -> fib#(X) fib#(ok(X)) -> fib#(X) -> fib#(mark(X)) -> fib#(X) fib#(mark(X)) -> fib#(X) -> fib#(ok(X)) -> fib#(X) fib#(mark(X)) -> fib#(X) -> fib#(mark(X)) -> fib#(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) add#(ok(X1),ok(X2)) -> add#(X1,X2) -> add#(ok(X1),ok(X2)) -> add#(X1,X2) add#(ok(X1),ok(X2)) -> add#(X1,X2) -> add#(X1,mark(X2)) -> add#(X1,X2) add#(ok(X1),ok(X2)) -> add#(X1,X2) -> add#(mark(X1),X2) -> add#(X1,X2) add#(mark(X1),X2) -> add#(X1,X2) -> add#(ok(X1),ok(X2)) -> add#(X1,X2) add#(mark(X1),X2) -> add#(X1,X2) -> add#(X1,mark(X2)) -> add#(X1,X2) add#(mark(X1),X2) -> add#(X1,X2) -> add#(mark(X1),X2) -> add#(X1,X2) add#(X1,mark(X2)) -> add#(X1,X2) -> add#(ok(X1),ok(X2)) -> add#(X1,X2) add#(X1,mark(X2)) -> add#(X1,X2) -> add#(X1,mark(X2)) -> add#(X1,X2) add#(X1,mark(X2)) -> add#(X1,X2) -> add#(mark(X1),X2) -> add#(X1,X2) sel#(ok(X1),ok(X2)) -> sel#(X1,X2) -> sel#(ok(X1),ok(X2)) -> sel#(X1,X2) sel#(ok(X1),ok(X2)) -> sel#(X1,X2) -> sel#(X1,mark(X2)) -> sel#(X1,X2) sel#(ok(X1),ok(X2)) -> sel#(X1,X2) -> sel#(mark(X1),X2) -> sel#(X1,X2) sel#(mark(X1),X2) -> sel#(X1,X2) -> sel#(ok(X1),ok(X2)) -> sel#(X1,X2) sel#(mark(X1),X2) -> sel#(X1,X2) -> sel#(X1,mark(X2)) -> sel#(X1,X2) sel#(mark(X1),X2) -> sel#(X1,X2) -> sel#(mark(X1),X2) -> sel#(X1,X2) sel#(X1,mark(X2)) -> sel#(X1,X2) -> sel#(ok(X1),ok(X2)) -> sel#(X1,X2) sel#(X1,mark(X2)) -> sel#(X1,X2) -> sel#(X1,mark(X2)) -> sel#(X1,X2) sel#(X1,mark(X2)) -> sel#(X1,X2) -> sel#(mark(X1),X2) -> sel#(X1,X2) fib1#(ok(X1),ok(X2)) -> fib1#(X1,X2) -> fib1#(ok(X1),ok(X2)) -> fib1#(X1,X2) fib1#(ok(X1),ok(X2)) -> fib1#(X1,X2) -> fib1#(X1,mark(X2)) -> fib1#(X1,X2) fib1#(ok(X1),ok(X2)) -> fib1#(X1,X2) -> fib1#(mark(X1),X2) -> fib1#(X1,X2) fib1#(mark(X1),X2) -> fib1#(X1,X2) -> fib1#(ok(X1),ok(X2)) -> fib1#(X1,X2) fib1#(mark(X1),X2) -> fib1#(X1,X2) -> fib1#(X1,mark(X2)) -> fib1#(X1,X2) fib1#(mark(X1),X2) -> fib1#(X1,X2) -> fib1#(mark(X1),X2) -> fib1#(X1,X2) fib1#(X1,mark(X2)) -> fib1#(X1,X2) -> fib1#(ok(X1),ok(X2)) -> fib1#(X1,X2) fib1#(X1,mark(X2)) -> fib1#(X1,X2) -> fib1#(X1,mark(X2)) -> fib1#(X1,X2) fib1#(X1,mark(X2)) -> fib1#(X1,X2) -> fib1#(mark(X1),X2) -> fib1#(X1,X2) 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#(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#(add(X1,X2)) -> add#(X1,active(X2)) active#(cons(X1,X2)) -> active#(X1) -> active#(add(X1,X2)) -> active#(X2) active#(cons(X1,X2)) -> active#(X1) -> active#(add(X1,X2)) -> add#(active(X1),X2) active#(cons(X1,X2)) -> active#(X1) -> active#(add(X1,X2)) -> active#(X1) 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#(fib1(X1,X2)) -> fib1#(X1,active(X2)) active#(cons(X1,X2)) -> active#(X1) -> active#(fib1(X1,X2)) -> active#(X2) active#(cons(X1,X2)) -> active#(X1) -> active#(fib1(X1,X2)) -> fib1#(active(X1),X2) active#(cons(X1,X2)) -> active#(X1) -> active#(fib1(X1,X2)) -> active#(X1) active#(cons(X1,X2)) -> active#(X1) -> active#(sel(X1,X2)) -> sel#(X1,active(X2)) active#(cons(X1,X2)) -> active#(X1) -> active#(sel(X1,X2)) -> active#(X2) active#(cons(X1,X2)) -> active#(X1) -> active#(sel(X1,X2)) -> sel#(active(X1),X2) active#(cons(X1,X2)) -> active#(X1) -> active#(sel(X1,X2)) -> active#(X1) active#(cons(X1,X2)) -> active#(X1) -> active#(fib(X)) -> fib#(active(X)) active#(cons(X1,X2)) -> active#(X1) -> active#(fib(X)) -> active#(X) active#(cons(X1,X2)) -> active#(X1) -> active#(sel(s(N),cons(X,XS))) -> sel#(N,XS) active#(cons(X1,X2)) -> active#(X1) -> active#(add(s(X),Y)) -> s#(add(X,Y)) active#(cons(X1,X2)) -> active#(X1) -> active#(add(s(X),Y)) -> add#(X,Y) active#(cons(X1,X2)) -> active#(X1) -> active#(fib1(X,Y)) -> cons#(X,fib1(Y,add(X,Y))) active#(cons(X1,X2)) -> active#(X1) -> active#(fib1(X,Y)) -> fib1#(Y,add(X,Y)) active#(cons(X1,X2)) -> active#(X1) -> active#(fib1(X,Y)) -> add#(X,Y) active#(cons(X1,X2)) -> active#(X1) -> active#(fib(N)) -> sel#(N,fib1(s(0()),s(0()))) active#(cons(X1,X2)) -> active#(X1) -> active#(fib(N)) -> fib1#(s(0()),s(0())) active#(cons(X1,X2)) -> active#(X1) -> active#(fib(N)) -> s#(0()) active#(add(s(X),Y)) -> add#(X,Y) -> add#(ok(X1),ok(X2)) -> add#(X1,X2) active#(add(s(X),Y)) -> add#(X,Y) -> add#(X1,mark(X2)) -> add#(X1,X2) active#(add(s(X),Y)) -> add#(X,Y) -> add#(mark(X1),X2) -> add#(X1,X2) active#(add(s(X),Y)) -> s#(add(X,Y)) -> s#(ok(X)) -> s#(X) active#(add(s(X),Y)) -> s#(add(X,Y)) -> s#(mark(X)) -> s#(X) active#(add(X1,X2)) -> add#(active(X1),X2) -> add#(ok(X1),ok(X2)) -> add#(X1,X2) active#(add(X1,X2)) -> add#(active(X1),X2) -> add#(X1,mark(X2)) -> add#(X1,X2) active#(add(X1,X2)) -> add#(active(X1),X2) -> add#(mark(X1),X2) -> add#(X1,X2) active#(add(X1,X2)) -> add#(X1,active(X2)) -> add#(ok(X1),ok(X2)) -> add#(X1,X2) active#(add(X1,X2)) -> add#(X1,active(X2)) -> add#(X1,mark(X2)) -> add#(X1,X2) active#(add(X1,X2)) -> add#(X1,active(X2)) -> add#(mark(X1),X2) -> add#(X1,X2) active#(add(X1,X2)) -> active#(X2) -> active#(add(X1,X2)) -> add#(X1,active(X2)) active#(add(X1,X2)) -> active#(X2) -> active#(add(X1,X2)) -> active#(X2) active#(add(X1,X2)) -> active#(X2) -> active#(add(X1,X2)) -> add#(active(X1),X2) active#(add(X1,X2)) -> active#(X2) -> active#(add(X1,X2)) -> active#(X1) active#(add(X1,X2)) -> active#(X2) -> active#(cons(X1,X2)) -> cons#(active(X1),X2) active#(add(X1,X2)) -> active#(X2) -> active#(cons(X1,X2)) -> active#(X1) active#(add(X1,X2)) -> active#(X2) -> active#(s(X)) -> s#(active(X)) active#(add(X1,X2)) -> active#(X2) -> active#(s(X)) -> active#(X) active#(add(X1,X2)) -> active#(X2) -> active#(fib1(X1,X2)) -> fib1#(X1,active(X2)) active#(add(X1,X2)) -> active#(X2) -> active#(fib1(X1,X2)) -> active#(X2) active#(add(X1,X2)) -> active#(X2) -> active#(fib1(X1,X2)) -> fib1#(active(X1),X2) active#(add(X1,X2)) -> active#(X2) -> active#(fib1(X1,X2)) -> active#(X1) active#(add(X1,X2)) -> active#(X2) -> active#(sel(X1,X2)) -> sel#(X1,active(X2)) active#(add(X1,X2)) -> active#(X2) -> active#(sel(X1,X2)) -> active#(X2) active#(add(X1,X2)) -> active#(X2) -> active#(sel(X1,X2)) -> sel#(active(X1),X2) active#(add(X1,X2)) -> active#(X2) -> active#(sel(X1,X2)) -> active#(X1) active#(add(X1,X2)) -> active#(X2) -> active#(fib(X)) -> fib#(active(X)) active#(add(X1,X2)) -> active#(X2) -> active#(fib(X)) -> active#(X) active#(add(X1,X2)) -> active#(X2) -> active#(sel(s(N),cons(X,XS))) -> sel#(N,XS) active#(add(X1,X2)) -> active#(X2) -> active#(add(s(X),Y)) -> s#(add(X,Y)) active#(add(X1,X2)) -> active#(X2) -> active#(add(s(X),Y)) -> add#(X,Y) active#(add(X1,X2)) -> active#(X2) -> active#(fib1(X,Y)) -> cons#(X,fib1(Y,add(X,Y))) active#(add(X1,X2)) -> active#(X2) -> active#(fib1(X,Y)) -> fib1#(Y,add(X,Y)) active#(add(X1,X2)) -> active#(X2) -> active#(fib1(X,Y)) -> add#(X,Y) active#(add(X1,X2)) -> active#(X2) -> active#(fib(N)) -> sel#(N,fib1(s(0()),s(0()))) active#(add(X1,X2)) -> active#(X2) -> active#(fib(N)) -> fib1#(s(0()),s(0())) active#(add(X1,X2)) -> active#(X2) -> active#(fib(N)) -> s#(0()) active#(add(X1,X2)) -> active#(X1) -> active#(add(X1,X2)) -> add#(X1,active(X2)) active#(add(X1,X2)) -> active#(X1) -> active#(add(X1,X2)) -> active#(X2) active#(add(X1,X2)) -> active#(X1) -> active#(add(X1,X2)) -> add#(active(X1),X2) active#(add(X1,X2)) -> active#(X1) -> active#(add(X1,X2)) -> active#(X1) active#(add(X1,X2)) -> active#(X1) -> active#(cons(X1,X2)) -> cons#(active(X1),X2) active#(add(X1,X2)) -> active#(X1) -> active#(cons(X1,X2)) -> active#(X1) active#(add(X1,X2)) -> active#(X1) -> active#(s(X)) -> s#(active(X)) active#(add(X1,X2)) -> active#(X1) -> active#(s(X)) -> active#(X) active#(add(X1,X2)) -> active#(X1) -> active#(fib1(X1,X2)) -> fib1#(X1,active(X2)) active#(add(X1,X2)) -> active#(X1) -> active#(fib1(X1,X2)) -> active#(X2) active#(add(X1,X2)) -> active#(X1) -> active#(fib1(X1,X2)) -> fib1#(active(X1),X2) active#(add(X1,X2)) -> active#(X1) -> active#(fib1(X1,X2)) -> active#(X1) active#(add(X1,X2)) -> active#(X1) -> active#(sel(X1,X2)) -> sel#(X1,active(X2)) active#(add(X1,X2)) -> active#(X1) -> active#(sel(X1,X2)) -> active#(X2) active#(add(X1,X2)) -> active#(X1) -> active#(sel(X1,X2)) -> sel#(active(X1),X2) active#(add(X1,X2)) -> active#(X1) -> active#(sel(X1,X2)) -> active#(X1) active#(add(X1,X2)) -> active#(X1) -> active#(fib(X)) -> fib#(active(X)) active#(add(X1,X2)) -> active#(X1) -> active#(fib(X)) -> active#(X) active#(add(X1,X2)) -> active#(X1) -> active#(sel(s(N),cons(X,XS))) -> sel#(N,XS) active#(add(X1,X2)) -> active#(X1) -> active#(add(s(X),Y)) -> s#(add(X,Y)) active#(add(X1,X2)) -> active#(X1) -> active#(add(s(X),Y)) -> add#(X,Y) active#(add(X1,X2)) -> active#(X1) -> active#(fib1(X,Y)) -> cons#(X,fib1(Y,add(X,Y))) active#(add(X1,X2)) -> active#(X1) -> active#(fib1(X,Y)) -> fib1#(Y,add(X,Y)) active#(add(X1,X2)) -> active#(X1) -> active#(fib1(X,Y)) -> add#(X,Y) active#(add(X1,X2)) -> active#(X1) -> active#(fib(N)) -> sel#(N,fib1(s(0()),s(0()))) active#(add(X1,X2)) -> active#(X1) -> active#(fib(N)) -> fib1#(s(0()),s(0())) active#(add(X1,X2)) -> active#(X1) -> active#(fib(N)) -> s#(0()) active#(sel(s(N),cons(X,XS))) -> sel#(N,XS) -> sel#(ok(X1),ok(X2)) -> sel#(X1,X2) active#(sel(s(N),cons(X,XS))) -> sel#(N,XS) -> sel#(X1,mark(X2)) -> sel#(X1,X2) active#(sel(s(N),cons(X,XS))) -> sel#(N,XS) -> sel#(mark(X1),X2) -> sel#(X1,X2) active#(sel(X1,X2)) -> sel#(active(X1),X2) -> sel#(ok(X1),ok(X2)) -> sel#(X1,X2) active#(sel(X1,X2)) -> sel#(active(X1),X2) -> sel#(X1,mark(X2)) -> sel#(X1,X2) active#(sel(X1,X2)) -> sel#(active(X1),X2) -> sel#(mark(X1),X2) -> sel#(X1,X2) active#(sel(X1,X2)) -> sel#(X1,active(X2)) -> sel#(ok(X1),ok(X2)) -> sel#(X1,X2) active#(sel(X1,X2)) -> sel#(X1,active(X2)) -> sel#(X1,mark(X2)) -> sel#(X1,X2) active#(sel(X1,X2)) -> sel#(X1,active(X2)) -> sel#(mark(X1),X2) -> sel#(X1,X2) active#(sel(X1,X2)) -> active#(X2) -> active#(add(X1,X2)) -> add#(X1,active(X2)) active#(sel(X1,X2)) -> active#(X2) -> active#(add(X1,X2)) -> active#(X2) active#(sel(X1,X2)) -> active#(X2) -> active#(add(X1,X2)) -> add#(active(X1),X2) active#(sel(X1,X2)) -> active#(X2) -> active#(add(X1,X2)) -> active#(X1) active#(sel(X1,X2)) -> active#(X2) -> active#(cons(X1,X2)) -> cons#(active(X1),X2) active#(sel(X1,X2)) -> active#(X2) -> active#(cons(X1,X2)) -> active#(X1) active#(sel(X1,X2)) -> active#(X2) -> active#(s(X)) -> s#(active(X)) active#(sel(X1,X2)) -> active#(X2) -> active#(s(X)) -> active#(X) active#(sel(X1,X2)) -> active#(X2) -> active#(fib1(X1,X2)) -> fib1#(X1,active(X2)) active#(sel(X1,X2)) -> active#(X2) -> active#(fib1(X1,X2)) -> active#(X2) active#(sel(X1,X2)) -> active#(X2) -> active#(fib1(X1,X2)) -> fib1#(active(X1),X2) active#(sel(X1,X2)) -> active#(X2) -> active#(fib1(X1,X2)) -> active#(X1) active#(sel(X1,X2)) -> active#(X2) -> active#(sel(X1,X2)) -> sel#(X1,active(X2)) active#(sel(X1,X2)) -> active#(X2) -> active#(sel(X1,X2)) -> active#(X2) active#(sel(X1,X2)) -> active#(X2) -> active#(sel(X1,X2)) -> sel#(active(X1),X2) active#(sel(X1,X2)) -> active#(X2) -> active#(sel(X1,X2)) -> active#(X1) active#(sel(X1,X2)) -> active#(X2) -> active#(fib(X)) -> fib#(active(X)) active#(sel(X1,X2)) -> active#(X2) -> active#(fib(X)) -> active#(X) active#(sel(X1,X2)) -> active#(X2) -> active#(sel(s(N),cons(X,XS))) -> sel#(N,XS) active#(sel(X1,X2)) -> active#(X2) -> active#(add(s(X),Y)) -> s#(add(X,Y)) active#(sel(X1,X2)) -> active#(X2) -> active#(add(s(X),Y)) -> add#(X,Y) active#(sel(X1,X2)) -> active#(X2) -> active#(fib1(X,Y)) -> cons#(X,fib1(Y,add(X,Y))) active#(sel(X1,X2)) -> active#(X2) -> active#(fib1(X,Y)) -> fib1#(Y,add(X,Y)) active#(sel(X1,X2)) -> active#(X2) -> active#(fib1(X,Y)) -> add#(X,Y) active#(sel(X1,X2)) -> active#(X2) -> active#(fib(N)) -> sel#(N,fib1(s(0()),s(0()))) active#(sel(X1,X2)) -> active#(X2) -> active#(fib(N)) -> fib1#(s(0()),s(0())) active#(sel(X1,X2)) -> active#(X2) -> active#(fib(N)) -> s#(0()) active#(sel(X1,X2)) -> active#(X1) -> active#(add(X1,X2)) -> add#(X1,active(X2)) active#(sel(X1,X2)) -> active#(X1) -> active#(add(X1,X2)) -> active#(X2) active#(sel(X1,X2)) -> active#(X1) -> active#(add(X1,X2)) -> add#(active(X1),X2) active#(sel(X1,X2)) -> active#(X1) -> active#(add(X1,X2)) -> active#(X1) active#(sel(X1,X2)) -> active#(X1) -> active#(cons(X1,X2)) -> cons#(active(X1),X2) active#(sel(X1,X2)) -> active#(X1) -> active#(cons(X1,X2)) -> active#(X1) active#(sel(X1,X2)) -> active#(X1) -> active#(s(X)) -> s#(active(X)) active#(sel(X1,X2)) -> active#(X1) -> active#(s(X)) -> active#(X) active#(sel(X1,X2)) -> active#(X1) -> active#(fib1(X1,X2)) -> fib1#(X1,active(X2)) active#(sel(X1,X2)) -> active#(X1) -> active#(fib1(X1,X2)) -> active#(X2) active#(sel(X1,X2)) -> active#(X1) -> active#(fib1(X1,X2)) -> fib1#(active(X1),X2) active#(sel(X1,X2)) -> active#(X1) -> active#(fib1(X1,X2)) -> active#(X1) active#(sel(X1,X2)) -> active#(X1) -> active#(sel(X1,X2)) -> sel#(X1,active(X2)) active#(sel(X1,X2)) -> active#(X1) -> active#(sel(X1,X2)) -> active#(X2) active#(sel(X1,X2)) -> active#(X1) -> active#(sel(X1,X2)) -> sel#(active(X1),X2) active#(sel(X1,X2)) -> active#(X1) -> active#(sel(X1,X2)) -> active#(X1) active#(sel(X1,X2)) -> active#(X1) -> active#(fib(X)) -> fib#(active(X)) active#(sel(X1,X2)) -> active#(X1) -> active#(fib(X)) -> active#(X) active#(sel(X1,X2)) -> active#(X1) -> active#(sel(s(N),cons(X,XS))) -> sel#(N,XS) active#(sel(X1,X2)) -> active#(X1) -> active#(add(s(X),Y)) -> s#(add(X,Y)) active#(sel(X1,X2)) -> active#(X1) -> active#(add(s(X),Y)) -> add#(X,Y) active#(sel(X1,X2)) -> active#(X1) -> active#(fib1(X,Y)) -> cons#(X,fib1(Y,add(X,Y))) active#(sel(X1,X2)) -> active#(X1) -> active#(fib1(X,Y)) -> fib1#(Y,add(X,Y)) active#(sel(X1,X2)) -> active#(X1) -> active#(fib1(X,Y)) -> add#(X,Y) active#(sel(X1,X2)) -> active#(X1) -> active#(fib(N)) -> sel#(N,fib1(s(0()),s(0()))) active#(sel(X1,X2)) -> active#(X1) -> active#(fib(N)) -> fib1#(s(0()),s(0())) active#(sel(X1,X2)) -> active#(X1) -> active#(fib(N)) -> s#(0()) active#(fib1(X1,X2)) -> fib1#(active(X1),X2) -> fib1#(ok(X1),ok(X2)) -> fib1#(X1,X2) active#(fib1(X1,X2)) -> fib1#(active(X1),X2) -> fib1#(X1,mark(X2)) -> fib1#(X1,X2) active#(fib1(X1,X2)) -> fib1#(active(X1),X2) -> fib1#(mark(X1),X2) -> fib1#(X1,X2) active#(fib1(X1,X2)) -> fib1#(X1,active(X2)) -> fib1#(ok(X1),ok(X2)) -> fib1#(X1,X2) active#(fib1(X1,X2)) -> fib1#(X1,active(X2)) -> fib1#(X1,mark(X2)) -> fib1#(X1,X2) active#(fib1(X1,X2)) -> fib1#(X1,active(X2)) -> fib1#(mark(X1),X2) -> fib1#(X1,X2) active#(fib1(X1,X2)) -> active#(X2) -> active#(add(X1,X2)) -> add#(X1,active(X2)) active#(fib1(X1,X2)) -> active#(X2) -> active#(add(X1,X2)) -> active#(X2) active#(fib1(X1,X2)) -> active#(X2) -> active#(add(X1,X2)) -> add#(active(X1),X2) active#(fib1(X1,X2)) -> active#(X2) -> active#(add(X1,X2)) -> active#(X1) active#(fib1(X1,X2)) -> active#(X2) -> active#(cons(X1,X2)) -> cons#(active(X1),X2) active#(fib1(X1,X2)) -> active#(X2) -> active#(cons(X1,X2)) -> active#(X1) active#(fib1(X1,X2)) -> active#(X2) -> active#(s(X)) -> s#(active(X)) active#(fib1(X1,X2)) -> active#(X2) -> active#(s(X)) -> active#(X) active#(fib1(X1,X2)) -> active#(X2) -> active#(fib1(X1,X2)) -> fib1#(X1,active(X2)) active#(fib1(X1,X2)) -> active#(X2) -> active#(fib1(X1,X2)) -> active#(X2) active#(fib1(X1,X2)) -> active#(X2) -> active#(fib1(X1,X2)) -> fib1#(active(X1),X2) active#(fib1(X1,X2)) -> active#(X2) -> active#(fib1(X1,X2)) -> active#(X1) active#(fib1(X1,X2)) -> active#(X2) -> active#(sel(X1,X2)) -> sel#(X1,active(X2)) active#(fib1(X1,X2)) -> active#(X2) -> active#(sel(X1,X2)) -> active#(X2) active#(fib1(X1,X2)) -> active#(X2) -> active#(sel(X1,X2)) -> sel#(active(X1),X2) active#(fib1(X1,X2)) -> active#(X2) -> active#(sel(X1,X2)) -> active#(X1) active#(fib1(X1,X2)) -> active#(X2) -> active#(fib(X)) -> fib#(active(X)) active#(fib1(X1,X2)) -> active#(X2) -> active#(fib(X)) -> active#(X) active#(fib1(X1,X2)) -> active#(X2) -> active#(sel(s(N),cons(X,XS))) -> sel#(N,XS) active#(fib1(X1,X2)) -> active#(X2) -> active#(add(s(X),Y)) -> s#(add(X,Y)) active#(fib1(X1,X2)) -> active#(X2) -> active#(add(s(X),Y)) -> add#(X,Y) active#(fib1(X1,X2)) -> active#(X2) -> active#(fib1(X,Y)) -> cons#(X,fib1(Y,add(X,Y))) active#(fib1(X1,X2)) -> active#(X2) -> active#(fib1(X,Y)) -> fib1#(Y,add(X,Y)) active#(fib1(X1,X2)) -> active#(X2) -> active#(fib1(X,Y)) -> add#(X,Y) active#(fib1(X1,X2)) -> active#(X2) -> active#(fib(N)) -> sel#(N,fib1(s(0()),s(0()))) active#(fib1(X1,X2)) -> active#(X2) -> active#(fib(N)) -> fib1#(s(0()),s(0())) active#(fib1(X1,X2)) -> active#(X2) -> active#(fib(N)) -> s#(0()) active#(fib1(X1,X2)) -> active#(X1) -> active#(add(X1,X2)) -> add#(X1,active(X2)) active#(fib1(X1,X2)) -> active#(X1) -> active#(add(X1,X2)) -> active#(X2) active#(fib1(X1,X2)) -> active#(X1) -> active#(add(X1,X2)) -> add#(active(X1),X2) active#(fib1(X1,X2)) -> active#(X1) -> active#(add(X1,X2)) -> active#(X1) active#(fib1(X1,X2)) -> active#(X1) -> active#(cons(X1,X2)) -> cons#(active(X1),X2) active#(fib1(X1,X2)) -> active#(X1) -> active#(cons(X1,X2)) -> active#(X1) active#(fib1(X1,X2)) -> active#(X1) -> active#(s(X)) -> s#(active(X)) active#(fib1(X1,X2)) -> active#(X1) -> active#(s(X)) -> active#(X) active#(fib1(X1,X2)) -> active#(X1) -> active#(fib1(X1,X2)) -> fib1#(X1,active(X2)) active#(fib1(X1,X2)) -> active#(X1) -> active#(fib1(X1,X2)) -> active#(X2) active#(fib1(X1,X2)) -> active#(X1) -> active#(fib1(X1,X2)) -> fib1#(active(X1),X2) active#(fib1(X1,X2)) -> active#(X1) -> active#(fib1(X1,X2)) -> active#(X1) active#(fib1(X1,X2)) -> active#(X1) -> active#(sel(X1,X2)) -> sel#(X1,active(X2)) active#(fib1(X1,X2)) -> active#(X1) -> active#(sel(X1,X2)) -> active#(X2) active#(fib1(X1,X2)) -> active#(X1) -> active#(sel(X1,X2)) -> sel#(active(X1),X2) active#(fib1(X1,X2)) -> active#(X1) -> active#(sel(X1,X2)) -> active#(X1) active#(fib1(X1,X2)) -> active#(X1) -> active#(fib(X)) -> fib#(active(X)) active#(fib1(X1,X2)) -> active#(X1) -> active#(fib(X)) -> active#(X) active#(fib1(X1,X2)) -> active#(X1) -> active#(sel(s(N),cons(X,XS))) -> sel#(N,XS) active#(fib1(X1,X2)) -> active#(X1) -> active#(add(s(X),Y)) -> s#(add(X,Y)) active#(fib1(X1,X2)) -> active#(X1) -> active#(add(s(X),Y)) -> add#(X,Y) active#(fib1(X1,X2)) -> active#(X1) -> active#(fib1(X,Y)) -> cons#(X,fib1(Y,add(X,Y))) active#(fib1(X1,X2)) -> active#(X1) -> active#(fib1(X,Y)) -> fib1#(Y,add(X,Y)) active#(fib1(X1,X2)) -> active#(X1) -> active#(fib1(X,Y)) -> add#(X,Y) active#(fib1(X1,X2)) -> active#(X1) -> active#(fib(N)) -> sel#(N,fib1(s(0()),s(0()))) active#(fib1(X1,X2)) -> active#(X1) -> active#(fib(N)) -> fib1#(s(0()),s(0())) active#(fib1(X1,X2)) -> active#(X1) -> active#(fib(N)) -> s#(0()) active#(fib1(X,Y)) -> cons#(X,fib1(Y,add(X,Y))) -> cons#(ok(X1),ok(X2)) -> cons#(X1,X2) active#(fib1(X,Y)) -> cons#(X,fib1(Y,add(X,Y))) -> cons#(mark(X1),X2) -> cons#(X1,X2) active#(fib1(X,Y)) -> add#(X,Y) -> add#(ok(X1),ok(X2)) -> add#(X1,X2) active#(fib1(X,Y)) -> add#(X,Y) -> add#(X1,mark(X2)) -> add#(X1,X2) active#(fib1(X,Y)) -> add#(X,Y) -> add#(mark(X1),X2) -> add#(X1,X2) active#(fib1(X,Y)) -> fib1#(Y,add(X,Y)) -> fib1#(ok(X1),ok(X2)) -> fib1#(X1,X2) active#(fib1(X,Y)) -> fib1#(Y,add(X,Y)) -> fib1#(X1,mark(X2)) -> fib1#(X1,X2) active#(fib1(X,Y)) -> fib1#(Y,add(X,Y)) -> fib1#(mark(X1),X2) -> fib1#(X1,X2) 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#(add(X1,X2)) -> add#(X1,active(X2)) active#(s(X)) -> active#(X) -> active#(add(X1,X2)) -> active#(X2) active#(s(X)) -> active#(X) -> active#(add(X1,X2)) -> add#(active(X1),X2) active#(s(X)) -> active#(X) -> active#(add(X1,X2)) -> active#(X1) 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#(fib1(X1,X2)) -> fib1#(X1,active(X2)) active#(s(X)) -> active#(X) -> active#(fib1(X1,X2)) -> active#(X2) active#(s(X)) -> active#(X) -> active#(fib1(X1,X2)) -> fib1#(active(X1),X2) active#(s(X)) -> active#(X) -> active#(fib1(X1,X2)) -> active#(X1) active#(s(X)) -> active#(X) -> active#(sel(X1,X2)) -> sel#(X1,active(X2)) active#(s(X)) -> active#(X) -> active#(sel(X1,X2)) -> active#(X2) active#(s(X)) -> active#(X) -> active#(sel(X1,X2)) -> sel#(active(X1),X2) active#(s(X)) -> active#(X) -> active#(sel(X1,X2)) -> active#(X1) active#(s(X)) -> active#(X) -> active#(fib(X)) -> fib#(active(X)) active#(s(X)) -> active#(X) -> active#(fib(X)) -> active#(X) active#(s(X)) -> active#(X) -> active#(sel(s(N),cons(X,XS))) -> sel#(N,XS) active#(s(X)) -> active#(X) -> active#(add(s(X),Y)) -> s#(add(X,Y)) active#(s(X)) -> active#(X) -> active#(add(s(X),Y)) -> add#(X,Y) active#(s(X)) -> active#(X) -> active#(fib1(X,Y)) -> cons#(X,fib1(Y,add(X,Y))) active#(s(X)) -> active#(X) -> active#(fib1(X,Y)) -> fib1#(Y,add(X,Y)) active#(s(X)) -> active#(X) -> active#(fib1(X,Y)) -> add#(X,Y) active#(s(X)) -> active#(X) -> active#(fib(N)) -> sel#(N,fib1(s(0()),s(0()))) active#(s(X)) -> active#(X) -> active#(fib(N)) -> fib1#(s(0()),s(0())) active#(s(X)) -> active#(X) -> active#(fib(N)) -> s#(0()) active#(fib(X)) -> fib#(active(X)) -> fib#(ok(X)) -> fib#(X) active#(fib(X)) -> fib#(active(X)) -> fib#(mark(X)) -> fib#(X) active#(fib(X)) -> active#(X) -> active#(add(X1,X2)) -> add#(X1,active(X2)) active#(fib(X)) -> active#(X) -> active#(add(X1,X2)) -> active#(X2) active#(fib(X)) -> active#(X) -> active#(add(X1,X2)) -> add#(active(X1),X2) active#(fib(X)) -> active#(X) -> active#(add(X1,X2)) -> active#(X1) active#(fib(X)) -> active#(X) -> active#(cons(X1,X2)) -> cons#(active(X1),X2) active#(fib(X)) -> active#(X) -> active#(cons(X1,X2)) -> active#(X1) active#(fib(X)) -> active#(X) -> active#(s(X)) -> s#(active(X)) active#(fib(X)) -> active#(X) -> active#(s(X)) -> active#(X) active#(fib(X)) -> active#(X) -> active#(fib1(X1,X2)) -> fib1#(X1,active(X2)) active#(fib(X)) -> active#(X) -> active#(fib1(X1,X2)) -> active#(X2) active#(fib(X)) -> active#(X) -> active#(fib1(X1,X2)) -> fib1#(active(X1),X2) active#(fib(X)) -> active#(X) -> active#(fib1(X1,X2)) -> active#(X1) active#(fib(X)) -> active#(X) -> active#(sel(X1,X2)) -> sel#(X1,active(X2)) active#(fib(X)) -> active#(X) -> active#(sel(X1,X2)) -> active#(X2) active#(fib(X)) -> active#(X) -> active#(sel(X1,X2)) -> sel#(active(X1),X2) active#(fib(X)) -> active#(X) -> active#(sel(X1,X2)) -> active#(X1) active#(fib(X)) -> active#(X) -> active#(fib(X)) -> fib#(active(X)) active#(fib(X)) -> active#(X) -> active#(fib(X)) -> active#(X) active#(fib(X)) -> active#(X) -> active#(sel(s(N),cons(X,XS))) -> sel#(N,XS) active#(fib(X)) -> active#(X) -> active#(add(s(X),Y)) -> s#(add(X,Y)) active#(fib(X)) -> active#(X) -> active#(add(s(X),Y)) -> add#(X,Y) active#(fib(X)) -> active#(X) -> active#(fib1(X,Y)) -> cons#(X,fib1(Y,add(X,Y))) active#(fib(X)) -> active#(X) -> active#(fib1(X,Y)) -> fib1#(Y,add(X,Y)) active#(fib(X)) -> active#(X) -> active#(fib1(X,Y)) -> add#(X,Y) active#(fib(X)) -> active#(X) -> active#(fib(N)) -> sel#(N,fib1(s(0()),s(0()))) active#(fib(X)) -> active#(X) -> active#(fib(N)) -> fib1#(s(0()),s(0())) active#(fib(X)) -> active#(X) -> active#(fib(N)) -> s#(0()) active#(fib(N)) -> sel#(N,fib1(s(0()),s(0()))) -> sel#(ok(X1),ok(X2)) -> sel#(X1,X2) active#(fib(N)) -> sel#(N,fib1(s(0()),s(0()))) -> sel#(X1,mark(X2)) -> sel#(X1,X2) active#(fib(N)) -> sel#(N,fib1(s(0()),s(0()))) -> sel#(mark(X1),X2) -> sel#(X1,X2) active#(fib(N)) -> fib1#(s(0()),s(0())) -> fib1#(ok(X1),ok(X2)) -> fib1#(X1,X2) active#(fib(N)) -> fib1#(s(0()),s(0())) -> fib1#(X1,mark(X2)) -> fib1#(X1,X2) active#(fib(N)) -> fib1#(s(0()),s(0())) -> fib1#(mark(X1),X2) -> fib1#(X1,X2) active#(fib(N)) -> s#(0()) -> s#(ok(X)) -> s#(X) active#(fib(N)) -> s#(0()) -> s#(mark(X)) -> s#(X) SCC Processor: #sccs: 9 #rules: 36 #arcs: 556/3844 DPs: top#(ok(X)) -> top#(active(X)) top#(mark(X)) -> top#(proper(X)) TRS: active(fib(N)) -> mark(sel(N,fib1(s(0()),s(0())))) active(fib1(X,Y)) -> mark(cons(X,fib1(Y,add(X,Y)))) active(add(0(),X)) -> mark(X) active(add(s(X),Y)) -> mark(s(add(X,Y))) active(sel(0(),cons(X,XS))) -> mark(X) active(sel(s(N),cons(X,XS))) -> mark(sel(N,XS)) active(fib(X)) -> fib(active(X)) active(sel(X1,X2)) -> sel(active(X1),X2) active(sel(X1,X2)) -> sel(X1,active(X2)) active(fib1(X1,X2)) -> fib1(active(X1),X2) active(fib1(X1,X2)) -> fib1(X1,active(X2)) active(s(X)) -> s(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(add(X1,X2)) -> add(active(X1),X2) active(add(X1,X2)) -> add(X1,active(X2)) fib(mark(X)) -> mark(fib(X)) sel(mark(X1),X2) -> mark(sel(X1,X2)) sel(X1,mark(X2)) -> mark(sel(X1,X2)) fib1(mark(X1),X2) -> mark(fib1(X1,X2)) fib1(X1,mark(X2)) -> mark(fib1(X1,X2)) s(mark(X)) -> mark(s(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) add(mark(X1),X2) -> mark(add(X1,X2)) add(X1,mark(X2)) -> mark(add(X1,X2)) proper(fib(X)) -> fib(proper(X)) proper(sel(X1,X2)) -> sel(proper(X1),proper(X2)) proper(fib1(X1,X2)) -> fib1(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(0()) -> ok(0()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(add(X1,X2)) -> add(proper(X1),proper(X2)) fib(ok(X)) -> ok(fib(X)) sel(ok(X1),ok(X2)) -> ok(sel(X1,X2)) fib1(ok(X1),ok(X2)) -> ok(fib1(X1,X2)) s(ok(X)) -> ok(s(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) add(ok(X1),ok(X2)) -> ok(add(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Open DPs: active#(fib(X)) -> active#(X) active#(sel(X1,X2)) -> active#(X1) active#(sel(X1,X2)) -> active#(X2) active#(fib1(X1,X2)) -> active#(X1) active#(fib1(X1,X2)) -> active#(X2) active#(s(X)) -> active#(X) active#(cons(X1,X2)) -> active#(X1) active#(add(X1,X2)) -> active#(X1) active#(add(X1,X2)) -> active#(X2) TRS: active(fib(N)) -> mark(sel(N,fib1(s(0()),s(0())))) active(fib1(X,Y)) -> mark(cons(X,fib1(Y,add(X,Y)))) active(add(0(),X)) -> mark(X) active(add(s(X),Y)) -> mark(s(add(X,Y))) active(sel(0(),cons(X,XS))) -> mark(X) active(sel(s(N),cons(X,XS))) -> mark(sel(N,XS)) active(fib(X)) -> fib(active(X)) active(sel(X1,X2)) -> sel(active(X1),X2) active(sel(X1,X2)) -> sel(X1,active(X2)) active(fib1(X1,X2)) -> fib1(active(X1),X2) active(fib1(X1,X2)) -> fib1(X1,active(X2)) active(s(X)) -> s(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(add(X1,X2)) -> add(active(X1),X2) active(add(X1,X2)) -> add(X1,active(X2)) fib(mark(X)) -> mark(fib(X)) sel(mark(X1),X2) -> mark(sel(X1,X2)) sel(X1,mark(X2)) -> mark(sel(X1,X2)) fib1(mark(X1),X2) -> mark(fib1(X1,X2)) fib1(X1,mark(X2)) -> mark(fib1(X1,X2)) s(mark(X)) -> mark(s(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) add(mark(X1),X2) -> mark(add(X1,X2)) add(X1,mark(X2)) -> mark(add(X1,X2)) proper(fib(X)) -> fib(proper(X)) proper(sel(X1,X2)) -> sel(proper(X1),proper(X2)) proper(fib1(X1,X2)) -> fib1(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(0()) -> ok(0()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(add(X1,X2)) -> add(proper(X1),proper(X2)) fib(ok(X)) -> ok(fib(X)) sel(ok(X1),ok(X2)) -> ok(sel(X1,X2)) fib1(ok(X1),ok(X2)) -> ok(fib1(X1,X2)) s(ok(X)) -> ok(s(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) add(ok(X1),ok(X2)) -> ok(add(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Subterm Criterion Processor: simple projection: pi(active#) = 0 problem: DPs: TRS: active(fib(N)) -> mark(sel(N,fib1(s(0()),s(0())))) active(fib1(X,Y)) -> mark(cons(X,fib1(Y,add(X,Y)))) active(add(0(),X)) -> mark(X) active(add(s(X),Y)) -> mark(s(add(X,Y))) active(sel(0(),cons(X,XS))) -> mark(X) active(sel(s(N),cons(X,XS))) -> mark(sel(N,XS)) active(fib(X)) -> fib(active(X)) active(sel(X1,X2)) -> sel(active(X1),X2) active(sel(X1,X2)) -> sel(X1,active(X2)) active(fib1(X1,X2)) -> fib1(active(X1),X2) active(fib1(X1,X2)) -> fib1(X1,active(X2)) active(s(X)) -> s(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(add(X1,X2)) -> add(active(X1),X2) active(add(X1,X2)) -> add(X1,active(X2)) fib(mark(X)) -> mark(fib(X)) sel(mark(X1),X2) -> mark(sel(X1,X2)) sel(X1,mark(X2)) -> mark(sel(X1,X2)) fib1(mark(X1),X2) -> mark(fib1(X1,X2)) fib1(X1,mark(X2)) -> mark(fib1(X1,X2)) s(mark(X)) -> mark(s(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) add(mark(X1),X2) -> mark(add(X1,X2)) add(X1,mark(X2)) -> mark(add(X1,X2)) proper(fib(X)) -> fib(proper(X)) proper(sel(X1,X2)) -> sel(proper(X1),proper(X2)) proper(fib1(X1,X2)) -> fib1(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(0()) -> ok(0()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(add(X1,X2)) -> add(proper(X1),proper(X2)) fib(ok(X)) -> ok(fib(X)) sel(ok(X1),ok(X2)) -> ok(sel(X1,X2)) fib1(ok(X1),ok(X2)) -> ok(fib1(X1,X2)) s(ok(X)) -> ok(s(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) add(ok(X1),ok(X2)) -> ok(add(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed DPs: proper#(fib(X)) -> proper#(X) proper#(sel(X1,X2)) -> proper#(X2) proper#(sel(X1,X2)) -> proper#(X1) proper#(fib1(X1,X2)) -> proper#(X2) proper#(fib1(X1,X2)) -> proper#(X1) proper#(s(X)) -> proper#(X) proper#(cons(X1,X2)) -> proper#(X2) proper#(cons(X1,X2)) -> proper#(X1) proper#(add(X1,X2)) -> proper#(X2) proper#(add(X1,X2)) -> proper#(X1) TRS: active(fib(N)) -> mark(sel(N,fib1(s(0()),s(0())))) active(fib1(X,Y)) -> mark(cons(X,fib1(Y,add(X,Y)))) active(add(0(),X)) -> mark(X) active(add(s(X),Y)) -> mark(s(add(X,Y))) active(sel(0(),cons(X,XS))) -> mark(X) active(sel(s(N),cons(X,XS))) -> mark(sel(N,XS)) active(fib(X)) -> fib(active(X)) active(sel(X1,X2)) -> sel(active(X1),X2) active(sel(X1,X2)) -> sel(X1,active(X2)) active(fib1(X1,X2)) -> fib1(active(X1),X2) active(fib1(X1,X2)) -> fib1(X1,active(X2)) active(s(X)) -> s(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(add(X1,X2)) -> add(active(X1),X2) active(add(X1,X2)) -> add(X1,active(X2)) fib(mark(X)) -> mark(fib(X)) sel(mark(X1),X2) -> mark(sel(X1,X2)) sel(X1,mark(X2)) -> mark(sel(X1,X2)) fib1(mark(X1),X2) -> mark(fib1(X1,X2)) fib1(X1,mark(X2)) -> mark(fib1(X1,X2)) s(mark(X)) -> mark(s(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) add(mark(X1),X2) -> mark(add(X1,X2)) add(X1,mark(X2)) -> mark(add(X1,X2)) proper(fib(X)) -> fib(proper(X)) proper(sel(X1,X2)) -> sel(proper(X1),proper(X2)) proper(fib1(X1,X2)) -> fib1(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(0()) -> ok(0()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(add(X1,X2)) -> add(proper(X1),proper(X2)) fib(ok(X)) -> ok(fib(X)) sel(ok(X1),ok(X2)) -> ok(sel(X1,X2)) fib1(ok(X1),ok(X2)) -> ok(fib1(X1,X2)) s(ok(X)) -> ok(s(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) add(ok(X1),ok(X2)) -> ok(add(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Subterm Criterion Processor: simple projection: pi(proper#) = 0 problem: DPs: TRS: active(fib(N)) -> mark(sel(N,fib1(s(0()),s(0())))) active(fib1(X,Y)) -> mark(cons(X,fib1(Y,add(X,Y)))) active(add(0(),X)) -> mark(X) active(add(s(X),Y)) -> mark(s(add(X,Y))) active(sel(0(),cons(X,XS))) -> mark(X) active(sel(s(N),cons(X,XS))) -> mark(sel(N,XS)) active(fib(X)) -> fib(active(X)) active(sel(X1,X2)) -> sel(active(X1),X2) active(sel(X1,X2)) -> sel(X1,active(X2)) active(fib1(X1,X2)) -> fib1(active(X1),X2) active(fib1(X1,X2)) -> fib1(X1,active(X2)) active(s(X)) -> s(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(add(X1,X2)) -> add(active(X1),X2) active(add(X1,X2)) -> add(X1,active(X2)) fib(mark(X)) -> mark(fib(X)) sel(mark(X1),X2) -> mark(sel(X1,X2)) sel(X1,mark(X2)) -> mark(sel(X1,X2)) fib1(mark(X1),X2) -> mark(fib1(X1,X2)) fib1(X1,mark(X2)) -> mark(fib1(X1,X2)) s(mark(X)) -> mark(s(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) add(mark(X1),X2) -> mark(add(X1,X2)) add(X1,mark(X2)) -> mark(add(X1,X2)) proper(fib(X)) -> fib(proper(X)) proper(sel(X1,X2)) -> sel(proper(X1),proper(X2)) proper(fib1(X1,X2)) -> fib1(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(0()) -> ok(0()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(add(X1,X2)) -> add(proper(X1),proper(X2)) fib(ok(X)) -> ok(fib(X)) sel(ok(X1),ok(X2)) -> ok(sel(X1,X2)) fib1(ok(X1),ok(X2)) -> ok(fib1(X1,X2)) s(ok(X)) -> ok(s(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) add(ok(X1),ok(X2)) -> ok(add(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed DPs: add#(mark(X1),X2) -> add#(X1,X2) add#(X1,mark(X2)) -> add#(X1,X2) add#(ok(X1),ok(X2)) -> add#(X1,X2) TRS: active(fib(N)) -> mark(sel(N,fib1(s(0()),s(0())))) active(fib1(X,Y)) -> mark(cons(X,fib1(Y,add(X,Y)))) active(add(0(),X)) -> mark(X) active(add(s(X),Y)) -> mark(s(add(X,Y))) active(sel(0(),cons(X,XS))) -> mark(X) active(sel(s(N),cons(X,XS))) -> mark(sel(N,XS)) active(fib(X)) -> fib(active(X)) active(sel(X1,X2)) -> sel(active(X1),X2) active(sel(X1,X2)) -> sel(X1,active(X2)) active(fib1(X1,X2)) -> fib1(active(X1),X2) active(fib1(X1,X2)) -> fib1(X1,active(X2)) active(s(X)) -> s(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(add(X1,X2)) -> add(active(X1),X2) active(add(X1,X2)) -> add(X1,active(X2)) fib(mark(X)) -> mark(fib(X)) sel(mark(X1),X2) -> mark(sel(X1,X2)) sel(X1,mark(X2)) -> mark(sel(X1,X2)) fib1(mark(X1),X2) -> mark(fib1(X1,X2)) fib1(X1,mark(X2)) -> mark(fib1(X1,X2)) s(mark(X)) -> mark(s(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) add(mark(X1),X2) -> mark(add(X1,X2)) add(X1,mark(X2)) -> mark(add(X1,X2)) proper(fib(X)) -> fib(proper(X)) proper(sel(X1,X2)) -> sel(proper(X1),proper(X2)) proper(fib1(X1,X2)) -> fib1(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(0()) -> ok(0()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(add(X1,X2)) -> add(proper(X1),proper(X2)) fib(ok(X)) -> ok(fib(X)) sel(ok(X1),ok(X2)) -> ok(sel(X1,X2)) fib1(ok(X1),ok(X2)) -> ok(fib1(X1,X2)) s(ok(X)) -> ok(s(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) add(ok(X1),ok(X2)) -> ok(add(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Subterm Criterion Processor: simple projection: pi(add#) = 1 problem: DPs: add#(mark(X1),X2) -> add#(X1,X2) TRS: active(fib(N)) -> mark(sel(N,fib1(s(0()),s(0())))) active(fib1(X,Y)) -> mark(cons(X,fib1(Y,add(X,Y)))) active(add(0(),X)) -> mark(X) active(add(s(X),Y)) -> mark(s(add(X,Y))) active(sel(0(),cons(X,XS))) -> mark(X) active(sel(s(N),cons(X,XS))) -> mark(sel(N,XS)) active(fib(X)) -> fib(active(X)) active(sel(X1,X2)) -> sel(active(X1),X2) active(sel(X1,X2)) -> sel(X1,active(X2)) active(fib1(X1,X2)) -> fib1(active(X1),X2) active(fib1(X1,X2)) -> fib1(X1,active(X2)) active(s(X)) -> s(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(add(X1,X2)) -> add(active(X1),X2) active(add(X1,X2)) -> add(X1,active(X2)) fib(mark(X)) -> mark(fib(X)) sel(mark(X1),X2) -> mark(sel(X1,X2)) sel(X1,mark(X2)) -> mark(sel(X1,X2)) fib1(mark(X1),X2) -> mark(fib1(X1,X2)) fib1(X1,mark(X2)) -> mark(fib1(X1,X2)) s(mark(X)) -> mark(s(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) add(mark(X1),X2) -> mark(add(X1,X2)) add(X1,mark(X2)) -> mark(add(X1,X2)) proper(fib(X)) -> fib(proper(X)) proper(sel(X1,X2)) -> sel(proper(X1),proper(X2)) proper(fib1(X1,X2)) -> fib1(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(0()) -> ok(0()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(add(X1,X2)) -> add(proper(X1),proper(X2)) fib(ok(X)) -> ok(fib(X)) sel(ok(X1),ok(X2)) -> ok(sel(X1,X2)) fib1(ok(X1),ok(X2)) -> ok(fib1(X1,X2)) s(ok(X)) -> ok(s(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) add(ok(X1),ok(X2)) -> ok(add(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Subterm Criterion Processor: simple projection: pi(add#) = 0 problem: DPs: TRS: active(fib(N)) -> mark(sel(N,fib1(s(0()),s(0())))) active(fib1(X,Y)) -> mark(cons(X,fib1(Y,add(X,Y)))) active(add(0(),X)) -> mark(X) active(add(s(X),Y)) -> mark(s(add(X,Y))) active(sel(0(),cons(X,XS))) -> mark(X) active(sel(s(N),cons(X,XS))) -> mark(sel(N,XS)) active(fib(X)) -> fib(active(X)) active(sel(X1,X2)) -> sel(active(X1),X2) active(sel(X1,X2)) -> sel(X1,active(X2)) active(fib1(X1,X2)) -> fib1(active(X1),X2) active(fib1(X1,X2)) -> fib1(X1,active(X2)) active(s(X)) -> s(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(add(X1,X2)) -> add(active(X1),X2) active(add(X1,X2)) -> add(X1,active(X2)) fib(mark(X)) -> mark(fib(X)) sel(mark(X1),X2) -> mark(sel(X1,X2)) sel(X1,mark(X2)) -> mark(sel(X1,X2)) fib1(mark(X1),X2) -> mark(fib1(X1,X2)) fib1(X1,mark(X2)) -> mark(fib1(X1,X2)) s(mark(X)) -> mark(s(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) add(mark(X1),X2) -> mark(add(X1,X2)) add(X1,mark(X2)) -> mark(add(X1,X2)) proper(fib(X)) -> fib(proper(X)) proper(sel(X1,X2)) -> sel(proper(X1),proper(X2)) proper(fib1(X1,X2)) -> fib1(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(0()) -> ok(0()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(add(X1,X2)) -> add(proper(X1),proper(X2)) fib(ok(X)) -> ok(fib(X)) sel(ok(X1),ok(X2)) -> ok(sel(X1,X2)) fib1(ok(X1),ok(X2)) -> ok(fib1(X1,X2)) s(ok(X)) -> ok(s(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) add(ok(X1),ok(X2)) -> ok(add(X1,X2)) 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(fib(N)) -> mark(sel(N,fib1(s(0()),s(0())))) active(fib1(X,Y)) -> mark(cons(X,fib1(Y,add(X,Y)))) active(add(0(),X)) -> mark(X) active(add(s(X),Y)) -> mark(s(add(X,Y))) active(sel(0(),cons(X,XS))) -> mark(X) active(sel(s(N),cons(X,XS))) -> mark(sel(N,XS)) active(fib(X)) -> fib(active(X)) active(sel(X1,X2)) -> sel(active(X1),X2) active(sel(X1,X2)) -> sel(X1,active(X2)) active(fib1(X1,X2)) -> fib1(active(X1),X2) active(fib1(X1,X2)) -> fib1(X1,active(X2)) active(s(X)) -> s(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(add(X1,X2)) -> add(active(X1),X2) active(add(X1,X2)) -> add(X1,active(X2)) fib(mark(X)) -> mark(fib(X)) sel(mark(X1),X2) -> mark(sel(X1,X2)) sel(X1,mark(X2)) -> mark(sel(X1,X2)) fib1(mark(X1),X2) -> mark(fib1(X1,X2)) fib1(X1,mark(X2)) -> mark(fib1(X1,X2)) s(mark(X)) -> mark(s(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) add(mark(X1),X2) -> mark(add(X1,X2)) add(X1,mark(X2)) -> mark(add(X1,X2)) proper(fib(X)) -> fib(proper(X)) proper(sel(X1,X2)) -> sel(proper(X1),proper(X2)) proper(fib1(X1,X2)) -> fib1(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(0()) -> ok(0()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(add(X1,X2)) -> add(proper(X1),proper(X2)) fib(ok(X)) -> ok(fib(X)) sel(ok(X1),ok(X2)) -> ok(sel(X1,X2)) fib1(ok(X1),ok(X2)) -> ok(fib1(X1,X2)) s(ok(X)) -> ok(s(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) add(ok(X1),ok(X2)) -> ok(add(X1,X2)) 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(fib(N)) -> mark(sel(N,fib1(s(0()),s(0())))) active(fib1(X,Y)) -> mark(cons(X,fib1(Y,add(X,Y)))) active(add(0(),X)) -> mark(X) active(add(s(X),Y)) -> mark(s(add(X,Y))) active(sel(0(),cons(X,XS))) -> mark(X) active(sel(s(N),cons(X,XS))) -> mark(sel(N,XS)) active(fib(X)) -> fib(active(X)) active(sel(X1,X2)) -> sel(active(X1),X2) active(sel(X1,X2)) -> sel(X1,active(X2)) active(fib1(X1,X2)) -> fib1(active(X1),X2) active(fib1(X1,X2)) -> fib1(X1,active(X2)) active(s(X)) -> s(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(add(X1,X2)) -> add(active(X1),X2) active(add(X1,X2)) -> add(X1,active(X2)) fib(mark(X)) -> mark(fib(X)) sel(mark(X1),X2) -> mark(sel(X1,X2)) sel(X1,mark(X2)) -> mark(sel(X1,X2)) fib1(mark(X1),X2) -> mark(fib1(X1,X2)) fib1(X1,mark(X2)) -> mark(fib1(X1,X2)) s(mark(X)) -> mark(s(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) add(mark(X1),X2) -> mark(add(X1,X2)) add(X1,mark(X2)) -> mark(add(X1,X2)) proper(fib(X)) -> fib(proper(X)) proper(sel(X1,X2)) -> sel(proper(X1),proper(X2)) proper(fib1(X1,X2)) -> fib1(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(0()) -> ok(0()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(add(X1,X2)) -> add(proper(X1),proper(X2)) fib(ok(X)) -> ok(fib(X)) sel(ok(X1),ok(X2)) -> ok(sel(X1,X2)) fib1(ok(X1),ok(X2)) -> ok(fib1(X1,X2)) s(ok(X)) -> ok(s(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) add(ok(X1),ok(X2)) -> ok(add(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Subterm Criterion Processor: simple projection: pi(cons#) = 0 problem: DPs: TRS: active(fib(N)) -> mark(sel(N,fib1(s(0()),s(0())))) active(fib1(X,Y)) -> mark(cons(X,fib1(Y,add(X,Y)))) active(add(0(),X)) -> mark(X) active(add(s(X),Y)) -> mark(s(add(X,Y))) active(sel(0(),cons(X,XS))) -> mark(X) active(sel(s(N),cons(X,XS))) -> mark(sel(N,XS)) active(fib(X)) -> fib(active(X)) active(sel(X1,X2)) -> sel(active(X1),X2) active(sel(X1,X2)) -> sel(X1,active(X2)) active(fib1(X1,X2)) -> fib1(active(X1),X2) active(fib1(X1,X2)) -> fib1(X1,active(X2)) active(s(X)) -> s(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(add(X1,X2)) -> add(active(X1),X2) active(add(X1,X2)) -> add(X1,active(X2)) fib(mark(X)) -> mark(fib(X)) sel(mark(X1),X2) -> mark(sel(X1,X2)) sel(X1,mark(X2)) -> mark(sel(X1,X2)) fib1(mark(X1),X2) -> mark(fib1(X1,X2)) fib1(X1,mark(X2)) -> mark(fib1(X1,X2)) s(mark(X)) -> mark(s(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) add(mark(X1),X2) -> mark(add(X1,X2)) add(X1,mark(X2)) -> mark(add(X1,X2)) proper(fib(X)) -> fib(proper(X)) proper(sel(X1,X2)) -> sel(proper(X1),proper(X2)) proper(fib1(X1,X2)) -> fib1(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(0()) -> ok(0()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(add(X1,X2)) -> add(proper(X1),proper(X2)) fib(ok(X)) -> ok(fib(X)) sel(ok(X1),ok(X2)) -> ok(sel(X1,X2)) fib1(ok(X1),ok(X2)) -> ok(fib1(X1,X2)) s(ok(X)) -> ok(s(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) add(ok(X1),ok(X2)) -> ok(add(X1,X2)) 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(fib(N)) -> mark(sel(N,fib1(s(0()),s(0())))) active(fib1(X,Y)) -> mark(cons(X,fib1(Y,add(X,Y)))) active(add(0(),X)) -> mark(X) active(add(s(X),Y)) -> mark(s(add(X,Y))) active(sel(0(),cons(X,XS))) -> mark(X) active(sel(s(N),cons(X,XS))) -> mark(sel(N,XS)) active(fib(X)) -> fib(active(X)) active(sel(X1,X2)) -> sel(active(X1),X2) active(sel(X1,X2)) -> sel(X1,active(X2)) active(fib1(X1,X2)) -> fib1(active(X1),X2) active(fib1(X1,X2)) -> fib1(X1,active(X2)) active(s(X)) -> s(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(add(X1,X2)) -> add(active(X1),X2) active(add(X1,X2)) -> add(X1,active(X2)) fib(mark(X)) -> mark(fib(X)) sel(mark(X1),X2) -> mark(sel(X1,X2)) sel(X1,mark(X2)) -> mark(sel(X1,X2)) fib1(mark(X1),X2) -> mark(fib1(X1,X2)) fib1(X1,mark(X2)) -> mark(fib1(X1,X2)) s(mark(X)) -> mark(s(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) add(mark(X1),X2) -> mark(add(X1,X2)) add(X1,mark(X2)) -> mark(add(X1,X2)) proper(fib(X)) -> fib(proper(X)) proper(sel(X1,X2)) -> sel(proper(X1),proper(X2)) proper(fib1(X1,X2)) -> fib1(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(0()) -> ok(0()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(add(X1,X2)) -> add(proper(X1),proper(X2)) fib(ok(X)) -> ok(fib(X)) sel(ok(X1),ok(X2)) -> ok(sel(X1,X2)) fib1(ok(X1),ok(X2)) -> ok(fib1(X1,X2)) s(ok(X)) -> ok(s(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) add(ok(X1),ok(X2)) -> ok(add(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Subterm Criterion Processor: simple projection: pi(s#) = 0 problem: DPs: TRS: active(fib(N)) -> mark(sel(N,fib1(s(0()),s(0())))) active(fib1(X,Y)) -> mark(cons(X,fib1(Y,add(X,Y)))) active(add(0(),X)) -> mark(X) active(add(s(X),Y)) -> mark(s(add(X,Y))) active(sel(0(),cons(X,XS))) -> mark(X) active(sel(s(N),cons(X,XS))) -> mark(sel(N,XS)) active(fib(X)) -> fib(active(X)) active(sel(X1,X2)) -> sel(active(X1),X2) active(sel(X1,X2)) -> sel(X1,active(X2)) active(fib1(X1,X2)) -> fib1(active(X1),X2) active(fib1(X1,X2)) -> fib1(X1,active(X2)) active(s(X)) -> s(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(add(X1,X2)) -> add(active(X1),X2) active(add(X1,X2)) -> add(X1,active(X2)) fib(mark(X)) -> mark(fib(X)) sel(mark(X1),X2) -> mark(sel(X1,X2)) sel(X1,mark(X2)) -> mark(sel(X1,X2)) fib1(mark(X1),X2) -> mark(fib1(X1,X2)) fib1(X1,mark(X2)) -> mark(fib1(X1,X2)) s(mark(X)) -> mark(s(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) add(mark(X1),X2) -> mark(add(X1,X2)) add(X1,mark(X2)) -> mark(add(X1,X2)) proper(fib(X)) -> fib(proper(X)) proper(sel(X1,X2)) -> sel(proper(X1),proper(X2)) proper(fib1(X1,X2)) -> fib1(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(0()) -> ok(0()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(add(X1,X2)) -> add(proper(X1),proper(X2)) fib(ok(X)) -> ok(fib(X)) sel(ok(X1),ok(X2)) -> ok(sel(X1,X2)) fib1(ok(X1),ok(X2)) -> ok(fib1(X1,X2)) s(ok(X)) -> ok(s(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) add(ok(X1),ok(X2)) -> ok(add(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed DPs: fib1#(mark(X1),X2) -> fib1#(X1,X2) fib1#(X1,mark(X2)) -> fib1#(X1,X2) fib1#(ok(X1),ok(X2)) -> fib1#(X1,X2) TRS: active(fib(N)) -> mark(sel(N,fib1(s(0()),s(0())))) active(fib1(X,Y)) -> mark(cons(X,fib1(Y,add(X,Y)))) active(add(0(),X)) -> mark(X) active(add(s(X),Y)) -> mark(s(add(X,Y))) active(sel(0(),cons(X,XS))) -> mark(X) active(sel(s(N),cons(X,XS))) -> mark(sel(N,XS)) active(fib(X)) -> fib(active(X)) active(sel(X1,X2)) -> sel(active(X1),X2) active(sel(X1,X2)) -> sel(X1,active(X2)) active(fib1(X1,X2)) -> fib1(active(X1),X2) active(fib1(X1,X2)) -> fib1(X1,active(X2)) active(s(X)) -> s(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(add(X1,X2)) -> add(active(X1),X2) active(add(X1,X2)) -> add(X1,active(X2)) fib(mark(X)) -> mark(fib(X)) sel(mark(X1),X2) -> mark(sel(X1,X2)) sel(X1,mark(X2)) -> mark(sel(X1,X2)) fib1(mark(X1),X2) -> mark(fib1(X1,X2)) fib1(X1,mark(X2)) -> mark(fib1(X1,X2)) s(mark(X)) -> mark(s(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) add(mark(X1),X2) -> mark(add(X1,X2)) add(X1,mark(X2)) -> mark(add(X1,X2)) proper(fib(X)) -> fib(proper(X)) proper(sel(X1,X2)) -> sel(proper(X1),proper(X2)) proper(fib1(X1,X2)) -> fib1(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(0()) -> ok(0()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(add(X1,X2)) -> add(proper(X1),proper(X2)) fib(ok(X)) -> ok(fib(X)) sel(ok(X1),ok(X2)) -> ok(sel(X1,X2)) fib1(ok(X1),ok(X2)) -> ok(fib1(X1,X2)) s(ok(X)) -> ok(s(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) add(ok(X1),ok(X2)) -> ok(add(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Subterm Criterion Processor: simple projection: pi(fib1#) = 1 problem: DPs: fib1#(mark(X1),X2) -> fib1#(X1,X2) TRS: active(fib(N)) -> mark(sel(N,fib1(s(0()),s(0())))) active(fib1(X,Y)) -> mark(cons(X,fib1(Y,add(X,Y)))) active(add(0(),X)) -> mark(X) active(add(s(X),Y)) -> mark(s(add(X,Y))) active(sel(0(),cons(X,XS))) -> mark(X) active(sel(s(N),cons(X,XS))) -> mark(sel(N,XS)) active(fib(X)) -> fib(active(X)) active(sel(X1,X2)) -> sel(active(X1),X2) active(sel(X1,X2)) -> sel(X1,active(X2)) active(fib1(X1,X2)) -> fib1(active(X1),X2) active(fib1(X1,X2)) -> fib1(X1,active(X2)) active(s(X)) -> s(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(add(X1,X2)) -> add(active(X1),X2) active(add(X1,X2)) -> add(X1,active(X2)) fib(mark(X)) -> mark(fib(X)) sel(mark(X1),X2) -> mark(sel(X1,X2)) sel(X1,mark(X2)) -> mark(sel(X1,X2)) fib1(mark(X1),X2) -> mark(fib1(X1,X2)) fib1(X1,mark(X2)) -> mark(fib1(X1,X2)) s(mark(X)) -> mark(s(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) add(mark(X1),X2) -> mark(add(X1,X2)) add(X1,mark(X2)) -> mark(add(X1,X2)) proper(fib(X)) -> fib(proper(X)) proper(sel(X1,X2)) -> sel(proper(X1),proper(X2)) proper(fib1(X1,X2)) -> fib1(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(0()) -> ok(0()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(add(X1,X2)) -> add(proper(X1),proper(X2)) fib(ok(X)) -> ok(fib(X)) sel(ok(X1),ok(X2)) -> ok(sel(X1,X2)) fib1(ok(X1),ok(X2)) -> ok(fib1(X1,X2)) s(ok(X)) -> ok(s(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) add(ok(X1),ok(X2)) -> ok(add(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Subterm Criterion Processor: simple projection: pi(fib1#) = 0 problem: DPs: TRS: active(fib(N)) -> mark(sel(N,fib1(s(0()),s(0())))) active(fib1(X,Y)) -> mark(cons(X,fib1(Y,add(X,Y)))) active(add(0(),X)) -> mark(X) active(add(s(X),Y)) -> mark(s(add(X,Y))) active(sel(0(),cons(X,XS))) -> mark(X) active(sel(s(N),cons(X,XS))) -> mark(sel(N,XS)) active(fib(X)) -> fib(active(X)) active(sel(X1,X2)) -> sel(active(X1),X2) active(sel(X1,X2)) -> sel(X1,active(X2)) active(fib1(X1,X2)) -> fib1(active(X1),X2) active(fib1(X1,X2)) -> fib1(X1,active(X2)) active(s(X)) -> s(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(add(X1,X2)) -> add(active(X1),X2) active(add(X1,X2)) -> add(X1,active(X2)) fib(mark(X)) -> mark(fib(X)) sel(mark(X1),X2) -> mark(sel(X1,X2)) sel(X1,mark(X2)) -> mark(sel(X1,X2)) fib1(mark(X1),X2) -> mark(fib1(X1,X2)) fib1(X1,mark(X2)) -> mark(fib1(X1,X2)) s(mark(X)) -> mark(s(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) add(mark(X1),X2) -> mark(add(X1,X2)) add(X1,mark(X2)) -> mark(add(X1,X2)) proper(fib(X)) -> fib(proper(X)) proper(sel(X1,X2)) -> sel(proper(X1),proper(X2)) proper(fib1(X1,X2)) -> fib1(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(0()) -> ok(0()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(add(X1,X2)) -> add(proper(X1),proper(X2)) fib(ok(X)) -> ok(fib(X)) sel(ok(X1),ok(X2)) -> ok(sel(X1,X2)) fib1(ok(X1),ok(X2)) -> ok(fib1(X1,X2)) s(ok(X)) -> ok(s(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) add(ok(X1),ok(X2)) -> ok(add(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed DPs: sel#(mark(X1),X2) -> sel#(X1,X2) sel#(X1,mark(X2)) -> sel#(X1,X2) sel#(ok(X1),ok(X2)) -> sel#(X1,X2) TRS: active(fib(N)) -> mark(sel(N,fib1(s(0()),s(0())))) active(fib1(X,Y)) -> mark(cons(X,fib1(Y,add(X,Y)))) active(add(0(),X)) -> mark(X) active(add(s(X),Y)) -> mark(s(add(X,Y))) active(sel(0(),cons(X,XS))) -> mark(X) active(sel(s(N),cons(X,XS))) -> mark(sel(N,XS)) active(fib(X)) -> fib(active(X)) active(sel(X1,X2)) -> sel(active(X1),X2) active(sel(X1,X2)) -> sel(X1,active(X2)) active(fib1(X1,X2)) -> fib1(active(X1),X2) active(fib1(X1,X2)) -> fib1(X1,active(X2)) active(s(X)) -> s(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(add(X1,X2)) -> add(active(X1),X2) active(add(X1,X2)) -> add(X1,active(X2)) fib(mark(X)) -> mark(fib(X)) sel(mark(X1),X2) -> mark(sel(X1,X2)) sel(X1,mark(X2)) -> mark(sel(X1,X2)) fib1(mark(X1),X2) -> mark(fib1(X1,X2)) fib1(X1,mark(X2)) -> mark(fib1(X1,X2)) s(mark(X)) -> mark(s(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) add(mark(X1),X2) -> mark(add(X1,X2)) add(X1,mark(X2)) -> mark(add(X1,X2)) proper(fib(X)) -> fib(proper(X)) proper(sel(X1,X2)) -> sel(proper(X1),proper(X2)) proper(fib1(X1,X2)) -> fib1(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(0()) -> ok(0()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(add(X1,X2)) -> add(proper(X1),proper(X2)) fib(ok(X)) -> ok(fib(X)) sel(ok(X1),ok(X2)) -> ok(sel(X1,X2)) fib1(ok(X1),ok(X2)) -> ok(fib1(X1,X2)) s(ok(X)) -> ok(s(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) add(ok(X1),ok(X2)) -> ok(add(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Subterm Criterion Processor: simple projection: pi(sel#) = 1 problem: DPs: sel#(mark(X1),X2) -> sel#(X1,X2) TRS: active(fib(N)) -> mark(sel(N,fib1(s(0()),s(0())))) active(fib1(X,Y)) -> mark(cons(X,fib1(Y,add(X,Y)))) active(add(0(),X)) -> mark(X) active(add(s(X),Y)) -> mark(s(add(X,Y))) active(sel(0(),cons(X,XS))) -> mark(X) active(sel(s(N),cons(X,XS))) -> mark(sel(N,XS)) active(fib(X)) -> fib(active(X)) active(sel(X1,X2)) -> sel(active(X1),X2) active(sel(X1,X2)) -> sel(X1,active(X2)) active(fib1(X1,X2)) -> fib1(active(X1),X2) active(fib1(X1,X2)) -> fib1(X1,active(X2)) active(s(X)) -> s(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(add(X1,X2)) -> add(active(X1),X2) active(add(X1,X2)) -> add(X1,active(X2)) fib(mark(X)) -> mark(fib(X)) sel(mark(X1),X2) -> mark(sel(X1,X2)) sel(X1,mark(X2)) -> mark(sel(X1,X2)) fib1(mark(X1),X2) -> mark(fib1(X1,X2)) fib1(X1,mark(X2)) -> mark(fib1(X1,X2)) s(mark(X)) -> mark(s(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) add(mark(X1),X2) -> mark(add(X1,X2)) add(X1,mark(X2)) -> mark(add(X1,X2)) proper(fib(X)) -> fib(proper(X)) proper(sel(X1,X2)) -> sel(proper(X1),proper(X2)) proper(fib1(X1,X2)) -> fib1(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(0()) -> ok(0()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(add(X1,X2)) -> add(proper(X1),proper(X2)) fib(ok(X)) -> ok(fib(X)) sel(ok(X1),ok(X2)) -> ok(sel(X1,X2)) fib1(ok(X1),ok(X2)) -> ok(fib1(X1,X2)) s(ok(X)) -> ok(s(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) add(ok(X1),ok(X2)) -> ok(add(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Subterm Criterion Processor: simple projection: pi(sel#) = 0 problem: DPs: TRS: active(fib(N)) -> mark(sel(N,fib1(s(0()),s(0())))) active(fib1(X,Y)) -> mark(cons(X,fib1(Y,add(X,Y)))) active(add(0(),X)) -> mark(X) active(add(s(X),Y)) -> mark(s(add(X,Y))) active(sel(0(),cons(X,XS))) -> mark(X) active(sel(s(N),cons(X,XS))) -> mark(sel(N,XS)) active(fib(X)) -> fib(active(X)) active(sel(X1,X2)) -> sel(active(X1),X2) active(sel(X1,X2)) -> sel(X1,active(X2)) active(fib1(X1,X2)) -> fib1(active(X1),X2) active(fib1(X1,X2)) -> fib1(X1,active(X2)) active(s(X)) -> s(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(add(X1,X2)) -> add(active(X1),X2) active(add(X1,X2)) -> add(X1,active(X2)) fib(mark(X)) -> mark(fib(X)) sel(mark(X1),X2) -> mark(sel(X1,X2)) sel(X1,mark(X2)) -> mark(sel(X1,X2)) fib1(mark(X1),X2) -> mark(fib1(X1,X2)) fib1(X1,mark(X2)) -> mark(fib1(X1,X2)) s(mark(X)) -> mark(s(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) add(mark(X1),X2) -> mark(add(X1,X2)) add(X1,mark(X2)) -> mark(add(X1,X2)) proper(fib(X)) -> fib(proper(X)) proper(sel(X1,X2)) -> sel(proper(X1),proper(X2)) proper(fib1(X1,X2)) -> fib1(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(0()) -> ok(0()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(add(X1,X2)) -> add(proper(X1),proper(X2)) fib(ok(X)) -> ok(fib(X)) sel(ok(X1),ok(X2)) -> ok(sel(X1,X2)) fib1(ok(X1),ok(X2)) -> ok(fib1(X1,X2)) s(ok(X)) -> ok(s(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) add(ok(X1),ok(X2)) -> ok(add(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed DPs: fib#(mark(X)) -> fib#(X) fib#(ok(X)) -> fib#(X) TRS: active(fib(N)) -> mark(sel(N,fib1(s(0()),s(0())))) active(fib1(X,Y)) -> mark(cons(X,fib1(Y,add(X,Y)))) active(add(0(),X)) -> mark(X) active(add(s(X),Y)) -> mark(s(add(X,Y))) active(sel(0(),cons(X,XS))) -> mark(X) active(sel(s(N),cons(X,XS))) -> mark(sel(N,XS)) active(fib(X)) -> fib(active(X)) active(sel(X1,X2)) -> sel(active(X1),X2) active(sel(X1,X2)) -> sel(X1,active(X2)) active(fib1(X1,X2)) -> fib1(active(X1),X2) active(fib1(X1,X2)) -> fib1(X1,active(X2)) active(s(X)) -> s(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(add(X1,X2)) -> add(active(X1),X2) active(add(X1,X2)) -> add(X1,active(X2)) fib(mark(X)) -> mark(fib(X)) sel(mark(X1),X2) -> mark(sel(X1,X2)) sel(X1,mark(X2)) -> mark(sel(X1,X2)) fib1(mark(X1),X2) -> mark(fib1(X1,X2)) fib1(X1,mark(X2)) -> mark(fib1(X1,X2)) s(mark(X)) -> mark(s(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) add(mark(X1),X2) -> mark(add(X1,X2)) add(X1,mark(X2)) -> mark(add(X1,X2)) proper(fib(X)) -> fib(proper(X)) proper(sel(X1,X2)) -> sel(proper(X1),proper(X2)) proper(fib1(X1,X2)) -> fib1(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(0()) -> ok(0()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(add(X1,X2)) -> add(proper(X1),proper(X2)) fib(ok(X)) -> ok(fib(X)) sel(ok(X1),ok(X2)) -> ok(sel(X1,X2)) fib1(ok(X1),ok(X2)) -> ok(fib1(X1,X2)) s(ok(X)) -> ok(s(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) add(ok(X1),ok(X2)) -> ok(add(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Subterm Criterion Processor: simple projection: pi(fib#) = 0 problem: DPs: TRS: active(fib(N)) -> mark(sel(N,fib1(s(0()),s(0())))) active(fib1(X,Y)) -> mark(cons(X,fib1(Y,add(X,Y)))) active(add(0(),X)) -> mark(X) active(add(s(X),Y)) -> mark(s(add(X,Y))) active(sel(0(),cons(X,XS))) -> mark(X) active(sel(s(N),cons(X,XS))) -> mark(sel(N,XS)) active(fib(X)) -> fib(active(X)) active(sel(X1,X2)) -> sel(active(X1),X2) active(sel(X1,X2)) -> sel(X1,active(X2)) active(fib1(X1,X2)) -> fib1(active(X1),X2) active(fib1(X1,X2)) -> fib1(X1,active(X2)) active(s(X)) -> s(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(add(X1,X2)) -> add(active(X1),X2) active(add(X1,X2)) -> add(X1,active(X2)) fib(mark(X)) -> mark(fib(X)) sel(mark(X1),X2) -> mark(sel(X1,X2)) sel(X1,mark(X2)) -> mark(sel(X1,X2)) fib1(mark(X1),X2) -> mark(fib1(X1,X2)) fib1(X1,mark(X2)) -> mark(fib1(X1,X2)) s(mark(X)) -> mark(s(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) add(mark(X1),X2) -> mark(add(X1,X2)) add(X1,mark(X2)) -> mark(add(X1,X2)) proper(fib(X)) -> fib(proper(X)) proper(sel(X1,X2)) -> sel(proper(X1),proper(X2)) proper(fib1(X1,X2)) -> fib1(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(0()) -> ok(0()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(add(X1,X2)) -> add(proper(X1),proper(X2)) fib(ok(X)) -> ok(fib(X)) sel(ok(X1),ok(X2)) -> ok(sel(X1,X2)) fib1(ok(X1),ok(X2)) -> ok(fib1(X1,X2)) s(ok(X)) -> ok(s(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) add(ok(X1),ok(X2)) -> ok(add(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed