MAYBE
MAYBE
TRS:
 {
              active(and(X1, X2)) -> and(active(X1), X2),
           active(and(true(), X)) -> mark(X),
          active(and(false(), Y)) -> mark(false()),
           active(if(X1, X2, X3)) -> if(active(X1), X2, X3),
         active(if(true(), X, Y)) -> mark(X),
        active(if(false(), X, Y)) -> mark(Y),
              active(add(X1, X2)) -> add(active(X1), X2),
              active(add(0(), X)) -> mark(X),
             active(add(s(X), Y)) -> mark(s(add(X, Y))),
            active(first(X1, X2)) -> first(X1, active(X2)),
            active(first(X1, X2)) -> first(active(X1), X2),
            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)))),
                and(mark(X1), X2) -> mark(and(X1, X2)),
              and(ok(X1), ok(X2)) -> ok(and(X1, X2)),
             if(mark(X1), X2, X3) -> mark(if(X1, X2, X3)),
       if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3)),
                add(mark(X1), X2) -> mark(add(X1, X2)),
              add(ok(X1), ok(X2)) -> ok(add(X1, X2)),
                         s(ok(X)) -> ok(s(X)),
              first(X1, mark(X2)) -> mark(first(X1, X2)),
              first(mark(X1), X2) -> mark(first(X1, X2)),
            first(ok(X1), ok(X2)) -> ok(first(X1, X2)),
             cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)),
                      from(ok(X)) -> ok(from(X)),
              proper(and(X1, X2)) -> and(proper(X1), proper(X2)),
                   proper(true()) -> ok(true()),
                  proper(false()) -> ok(false()),
           proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3)),
              proper(add(X1, X2)) -> add(proper(X1), proper(X2)),
                      proper(0()) -> ok(0()),
                     proper(s(X)) -> s(proper(X)),
                    proper(nil()) -> ok(nil()),
            proper(first(X1, X2)) -> first(proper(X1), proper(X2)),
             proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)),
                  proper(from(X)) -> from(proper(X)),
                     top(mark(X)) -> top(proper(X)),
                       top(ok(X)) -> top(active(X))
 }
 DUP: We consider a duplicating system.
  Trs:
   {
                active(and(X1, X2)) -> and(active(X1), X2),
             active(and(true(), X)) -> mark(X),
            active(and(false(), Y)) -> mark(false()),
             active(if(X1, X2, X3)) -> if(active(X1), X2, X3),
           active(if(true(), X, Y)) -> mark(X),
          active(if(false(), X, Y)) -> mark(Y),
                active(add(X1, X2)) -> add(active(X1), X2),
                active(add(0(), X)) -> mark(X),
               active(add(s(X), Y)) -> mark(s(add(X, Y))),
              active(first(X1, X2)) -> first(X1, active(X2)),
              active(first(X1, X2)) -> first(active(X1), X2),
              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)))),
                  and(mark(X1), X2) -> mark(and(X1, X2)),
                and(ok(X1), ok(X2)) -> ok(and(X1, X2)),
               if(mark(X1), X2, X3) -> mark(if(X1, X2, X3)),
         if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3)),
                  add(mark(X1), X2) -> mark(add(X1, X2)),
                add(ok(X1), ok(X2)) -> ok(add(X1, X2)),
                           s(ok(X)) -> ok(s(X)),
                first(X1, mark(X2)) -> mark(first(X1, X2)),
                first(mark(X1), X2) -> mark(first(X1, X2)),
              first(ok(X1), ok(X2)) -> ok(first(X1, X2)),
               cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)),
                        from(ok(X)) -> ok(from(X)),
                proper(and(X1, X2)) -> and(proper(X1), proper(X2)),
                     proper(true()) -> ok(true()),
                    proper(false()) -> ok(false()),
             proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3)),
                proper(add(X1, X2)) -> add(proper(X1), proper(X2)),
                        proper(0()) -> ok(0()),
                       proper(s(X)) -> s(proper(X)),
                      proper(nil()) -> ok(nil()),
              proper(first(X1, X2)) -> first(proper(X1), proper(X2)),
               proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)),
                    proper(from(X)) -> from(proper(X)),
                       top(mark(X)) -> top(proper(X)),
                         top(ok(X)) -> top(active(X))
   }
  Fail