MAYBE
MAYBE
TRS:
 {
                  active(dbl(X)) -> dbl(active(X)),
                active(dbl(0())) -> mark(0()),
               active(dbl(s(X))) -> mark(s(s(dbl(X)))),
                 active(dbls(X)) -> dbls(active(X)),
             active(dbls(nil())) -> mark(nil()),
        active(dbls(cons(X, Y))) -> mark(cons(dbl(X), dbls(Y))),
             active(sel(X1, X2)) -> sel(X1, active(X2)),
             active(sel(X1, X2)) -> sel(active(X1), X2),
    active(sel(0(), cons(X, Y))) -> mark(X),
   active(sel(s(X), cons(Y, Z))) -> mark(sel(X, Z)),
            active(indx(X1, X2)) -> indx(active(X1), X2),
          active(indx(nil(), X)) -> mark(nil()),
     active(indx(cons(X, Y), Z)) -> mark(cons(sel(X, Z), indx(Y, Z))),
                 active(from(X)) -> mark(cons(X, from(s(X)))),
                 active(dbl1(X)) -> dbl1(active(X)),
               active(dbl1(0())) -> mark(01()),
              active(dbl1(s(X))) -> mark(s1(s1(dbl1(X)))),
                   active(s1(X)) -> s1(active(X)),
            active(sel1(X1, X2)) -> sel1(X1, active(X2)),
            active(sel1(X1, X2)) -> sel1(active(X1), X2),
   active(sel1(0(), cons(X, Y))) -> mark(X),
  active(sel1(s(X), cons(Y, Z))) -> mark(sel1(X, Z)),
                active(quote(X)) -> quote(active(X)),
              active(quote(0())) -> mark(01()),
           active(quote(dbl(X))) -> mark(dbl1(X)),
             active(quote(s(X))) -> mark(s1(quote(X))),
        active(quote(sel(X, Y))) -> mark(sel1(X, Y)),
                    dbl(mark(X)) -> mark(dbl(X)),
                      dbl(ok(X)) -> ok(dbl(X)),
                        s(ok(X)) -> ok(s(X)),
                   dbls(mark(X)) -> mark(dbls(X)),
                     dbls(ok(X)) -> ok(dbls(X)),
            cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)),
               sel(X1, mark(X2)) -> mark(sel(X1, X2)),
               sel(mark(X1), X2) -> mark(sel(X1, X2)),
             sel(ok(X1), ok(X2)) -> ok(sel(X1, X2)),
              indx(mark(X1), X2) -> mark(indx(X1, X2)),
            indx(ok(X1), ok(X2)) -> ok(indx(X1, X2)),
                     from(ok(X)) -> ok(from(X)),
                   dbl1(mark(X)) -> mark(dbl1(X)),
                     dbl1(ok(X)) -> ok(dbl1(X)),
                     s1(mark(X)) -> mark(s1(X)),
                       s1(ok(X)) -> ok(s1(X)),
              sel1(X1, mark(X2)) -> mark(sel1(X1, X2)),
              sel1(mark(X1), X2) -> mark(sel1(X1, X2)),
            sel1(ok(X1), ok(X2)) -> ok(sel1(X1, X2)),
                  quote(mark(X)) -> mark(quote(X)),
                    quote(ok(X)) -> ok(quote(X)),
                     proper(0()) -> ok(0()),
                  proper(dbl(X)) -> dbl(proper(X)),
                    proper(s(X)) -> s(proper(X)),
                   proper(nil()) -> ok(nil()),
                 proper(dbls(X)) -> dbls(proper(X)),
            proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)),
             proper(sel(X1, X2)) -> sel(proper(X1), proper(X2)),
            proper(indx(X1, X2)) -> indx(proper(X1), proper(X2)),
                 proper(from(X)) -> from(proper(X)),
                    proper(01()) -> ok(01()),
                 proper(dbl1(X)) -> dbl1(proper(X)),
                   proper(s1(X)) -> s1(proper(X)),
            proper(sel1(X1, X2)) -> sel1(proper(X1), proper(X2)),
                proper(quote(X)) -> quote(proper(X)),
                    top(mark(X)) -> top(proper(X)),
                      top(ok(X)) -> top(active(X))
 }
 DUP: We consider a duplicating system.
  Trs:
   {
                    active(dbl(X)) -> dbl(active(X)),
                  active(dbl(0())) -> mark(0()),
                 active(dbl(s(X))) -> mark(s(s(dbl(X)))),
                   active(dbls(X)) -> dbls(active(X)),
               active(dbls(nil())) -> mark(nil()),
          active(dbls(cons(X, Y))) -> mark(cons(dbl(X), dbls(Y))),
               active(sel(X1, X2)) -> sel(X1, active(X2)),
               active(sel(X1, X2)) -> sel(active(X1), X2),
      active(sel(0(), cons(X, Y))) -> mark(X),
     active(sel(s(X), cons(Y, Z))) -> mark(sel(X, Z)),
              active(indx(X1, X2)) -> indx(active(X1), X2),
            active(indx(nil(), X)) -> mark(nil()),
       active(indx(cons(X, Y), Z)) -> mark(cons(sel(X, Z), indx(Y, Z))),
                   active(from(X)) -> mark(cons(X, from(s(X)))),
                   active(dbl1(X)) -> dbl1(active(X)),
                 active(dbl1(0())) -> mark(01()),
                active(dbl1(s(X))) -> mark(s1(s1(dbl1(X)))),
                     active(s1(X)) -> s1(active(X)),
              active(sel1(X1, X2)) -> sel1(X1, active(X2)),
              active(sel1(X1, X2)) -> sel1(active(X1), X2),
     active(sel1(0(), cons(X, Y))) -> mark(X),
    active(sel1(s(X), cons(Y, Z))) -> mark(sel1(X, Z)),
                  active(quote(X)) -> quote(active(X)),
                active(quote(0())) -> mark(01()),
             active(quote(dbl(X))) -> mark(dbl1(X)),
               active(quote(s(X))) -> mark(s1(quote(X))),
          active(quote(sel(X, Y))) -> mark(sel1(X, Y)),
                      dbl(mark(X)) -> mark(dbl(X)),
                        dbl(ok(X)) -> ok(dbl(X)),
                          s(ok(X)) -> ok(s(X)),
                     dbls(mark(X)) -> mark(dbls(X)),
                       dbls(ok(X)) -> ok(dbls(X)),
              cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)),
                 sel(X1, mark(X2)) -> mark(sel(X1, X2)),
                 sel(mark(X1), X2) -> mark(sel(X1, X2)),
               sel(ok(X1), ok(X2)) -> ok(sel(X1, X2)),
                indx(mark(X1), X2) -> mark(indx(X1, X2)),
              indx(ok(X1), ok(X2)) -> ok(indx(X1, X2)),
                       from(ok(X)) -> ok(from(X)),
                     dbl1(mark(X)) -> mark(dbl1(X)),
                       dbl1(ok(X)) -> ok(dbl1(X)),
                       s1(mark(X)) -> mark(s1(X)),
                         s1(ok(X)) -> ok(s1(X)),
                sel1(X1, mark(X2)) -> mark(sel1(X1, X2)),
                sel1(mark(X1), X2) -> mark(sel1(X1, X2)),
              sel1(ok(X1), ok(X2)) -> ok(sel1(X1, X2)),
                    quote(mark(X)) -> mark(quote(X)),
                      quote(ok(X)) -> ok(quote(X)),
                       proper(0()) -> ok(0()),
                    proper(dbl(X)) -> dbl(proper(X)),
                      proper(s(X)) -> s(proper(X)),
                     proper(nil()) -> ok(nil()),
                   proper(dbls(X)) -> dbls(proper(X)),
              proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)),
               proper(sel(X1, X2)) -> sel(proper(X1), proper(X2)),
              proper(indx(X1, X2)) -> indx(proper(X1), proper(X2)),
                   proper(from(X)) -> from(proper(X)),
                      proper(01()) -> ok(01()),
                   proper(dbl1(X)) -> dbl1(proper(X)),
                     proper(s1(X)) -> s1(proper(X)),
              proper(sel1(X1, X2)) -> sel1(proper(X1), proper(X2)),
                  proper(quote(X)) -> quote(proper(X)),
                      top(mark(X)) -> top(proper(X)),
                        top(ok(X)) -> top(active(X))
   }
  Fail