MAYBE
MAYBE
TRS:
 {
                 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)),
               active(sel(X1, X2)) -> sel(X1, active(X2)),
               active(sel(X1, X2)) -> sel(active(X1), X2),
     active(sel(s(X), cons(Y, Z))) -> mark(sel(X, Z)),
      active(sel(0(), cons(X, Z))) -> mark(X),
                      active(s(X)) -> s(active(X)),
              active(cons(X1, X2)) -> cons(active(X1), X2),
             active(first(X1, X2)) -> first(X1, active(X2)),
             active(first(X1, X2)) -> first(active(X1), X2),
   active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z))),
             active(first(0(), Z)) -> mark(nil()),
                   active(from(X)) -> mark(cons(X, from(s(X)))),
                   active(from(X)) -> from(active(X)),
              active(sel1(X1, X2)) -> sel1(X1, active(X2)),
              active(sel1(X1, X2)) -> sel1(active(X1), X2),
    active(sel1(s(X), cons(Y, Z))) -> mark(sel1(X, Z)),
     active(sel1(0(), cons(X, Z))) -> mark(quote(X)),
          active(quote(sel(X, Z))) -> mark(sel1(X, Z)),
               active(quote(s(X))) -> mark(s1(quote(X))),
                active(quote(0())) -> mark(01()),
            active(first1(X1, X2)) -> first1(X1, active(X2)),
            active(first1(X1, X2)) -> first1(active(X1), X2),
  active(first1(s(X), cons(Y, Z))) -> mark(cons1(quote(Y), first1(X, Z))),
            active(first1(0(), Z)) -> mark(nil1()),
             active(cons1(X1, X2)) -> cons1(X1, active(X2)),
             active(cons1(X1, X2)) -> cons1(active(X1), X2),
        active(quote1(cons(X, Z))) -> mark(cons1(quote(X), quote1(Z))),
             active(quote1(nil())) -> mark(nil1()),
       active(quote1(first(X, Z))) -> mark(first1(X, Z)),
                     active(s1(X)) -> s1(active(X)),
                active(unquote(X)) -> unquote(active(X)),
             active(unquote(01())) -> mark(0()),
            active(unquote(s1(X))) -> mark(s(unquote(X))),
               active(unquote1(X)) -> unquote1(active(X)),
          active(unquote1(nil1())) -> mark(nil()),
     active(unquote1(cons1(X, Z))) -> mark(fcons(unquote(X), unquote1(Z))),
               active(fcons(X, Z)) -> mark(cons(X, Z)),
             active(fcons(X1, X2)) -> fcons(X1, active(X2)),
             active(fcons(X1, X2)) -> fcons(active(X1), X2),
                        s(mark(X)) -> mark(s(X)),
                          s(ok(X)) -> ok(s(X)),
                cons(mark(X1), X2) -> mark(cons(X1, X2)),
              cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)),
               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)),
                     from(mark(X)) -> mark(from(X)),
                       from(ok(X)) -> ok(from(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(ok(X)) -> ok(quote(X)),
              first1(X1, mark(X2)) -> mark(first1(X1, X2)),
              first1(mark(X1), X2) -> mark(first1(X1, X2)),
            first1(ok(X1), ok(X2)) -> ok(first1(X1, X2)),
               cons1(X1, mark(X2)) -> mark(cons1(X1, X2)),
               cons1(mark(X1), X2) -> mark(cons1(X1, X2)),
             cons1(ok(X1), ok(X2)) -> ok(cons1(X1, X2)),
                     quote1(ok(X)) -> ok(quote1(X)),
                       s1(mark(X)) -> mark(s1(X)),
                         s1(ok(X)) -> ok(s1(X)),
                  unquote(mark(X)) -> mark(unquote(X)),
                    unquote(ok(X)) -> ok(unquote(X)),
                 unquote1(mark(X)) -> mark(unquote1(X)),
                   unquote1(ok(X)) -> ok(unquote1(X)),
               fcons(X1, mark(X2)) -> mark(fcons(X1, X2)),
               fcons(mark(X1), X2) -> mark(fcons(X1, X2)),
             fcons(ok(X1), ok(X2)) -> ok(fcons(X1, X2)),
               proper(sel(X1, X2)) -> sel(proper(X1), proper(X2)),
                      proper(s(X)) -> s(proper(X)),
              proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)),
                       proper(0()) -> ok(0()),
                     proper(nil()) -> ok(nil()),
             proper(first(X1, X2)) -> first(proper(X1), proper(X2)),
                   proper(from(X)) -> from(proper(X)),
              proper(sel1(X1, X2)) -> sel1(proper(X1), proper(X2)),
                  proper(quote(X)) -> quote(proper(X)),
                    proper(nil1()) -> ok(nil1()),
            proper(first1(X1, X2)) -> first1(proper(X1), proper(X2)),
             proper(cons1(X1, X2)) -> cons1(proper(X1), proper(X2)),
                      proper(01()) -> ok(01()),
                 proper(quote1(X)) -> quote1(proper(X)),
                     proper(s1(X)) -> s1(proper(X)),
                proper(unquote(X)) -> unquote(proper(X)),
               proper(unquote1(X)) -> unquote1(proper(X)),
             proper(fcons(X1, X2)) -> fcons(proper(X1), proper(X2)),
                      top(mark(X)) -> top(proper(X)),
                        top(ok(X)) -> top(active(X))
 }
 DUP: We consider a duplicating system.
  Trs:
   {
                   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)),
                 active(sel(X1, X2)) -> sel(X1, active(X2)),
                 active(sel(X1, X2)) -> sel(active(X1), X2),
       active(sel(s(X), cons(Y, Z))) -> mark(sel(X, Z)),
        active(sel(0(), cons(X, Z))) -> mark(X),
                        active(s(X)) -> s(active(X)),
                active(cons(X1, X2)) -> cons(active(X1), X2),
               active(first(X1, X2)) -> first(X1, active(X2)),
               active(first(X1, X2)) -> first(active(X1), X2),
     active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z))),
               active(first(0(), Z)) -> mark(nil()),
                     active(from(X)) -> mark(cons(X, from(s(X)))),
                     active(from(X)) -> from(active(X)),
                active(sel1(X1, X2)) -> sel1(X1, active(X2)),
                active(sel1(X1, X2)) -> sel1(active(X1), X2),
      active(sel1(s(X), cons(Y, Z))) -> mark(sel1(X, Z)),
       active(sel1(0(), cons(X, Z))) -> mark(quote(X)),
            active(quote(sel(X, Z))) -> mark(sel1(X, Z)),
                 active(quote(s(X))) -> mark(s1(quote(X))),
                  active(quote(0())) -> mark(01()),
              active(first1(X1, X2)) -> first1(X1, active(X2)),
              active(first1(X1, X2)) -> first1(active(X1), X2),
    active(first1(s(X), cons(Y, Z))) -> mark(cons1(quote(Y), first1(X, Z))),
              active(first1(0(), Z)) -> mark(nil1()),
               active(cons1(X1, X2)) -> cons1(X1, active(X2)),
               active(cons1(X1, X2)) -> cons1(active(X1), X2),
          active(quote1(cons(X, Z))) -> mark(cons1(quote(X), quote1(Z))),
               active(quote1(nil())) -> mark(nil1()),
         active(quote1(first(X, Z))) -> mark(first1(X, Z)),
                       active(s1(X)) -> s1(active(X)),
                  active(unquote(X)) -> unquote(active(X)),
               active(unquote(01())) -> mark(0()),
              active(unquote(s1(X))) -> mark(s(unquote(X))),
                 active(unquote1(X)) -> unquote1(active(X)),
            active(unquote1(nil1())) -> mark(nil()),
       active(unquote1(cons1(X, Z))) -> mark(fcons(unquote(X), unquote1(Z))),
                 active(fcons(X, Z)) -> mark(cons(X, Z)),
               active(fcons(X1, X2)) -> fcons(X1, active(X2)),
               active(fcons(X1, X2)) -> fcons(active(X1), X2),
                          s(mark(X)) -> mark(s(X)),
                            s(ok(X)) -> ok(s(X)),
                  cons(mark(X1), X2) -> mark(cons(X1, X2)),
                cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)),
                 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)),
                       from(mark(X)) -> mark(from(X)),
                         from(ok(X)) -> ok(from(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(ok(X)) -> ok(quote(X)),
                first1(X1, mark(X2)) -> mark(first1(X1, X2)),
                first1(mark(X1), X2) -> mark(first1(X1, X2)),
              first1(ok(X1), ok(X2)) -> ok(first1(X1, X2)),
                 cons1(X1, mark(X2)) -> mark(cons1(X1, X2)),
                 cons1(mark(X1), X2) -> mark(cons1(X1, X2)),
               cons1(ok(X1), ok(X2)) -> ok(cons1(X1, X2)),
                       quote1(ok(X)) -> ok(quote1(X)),
                         s1(mark(X)) -> mark(s1(X)),
                           s1(ok(X)) -> ok(s1(X)),
                    unquote(mark(X)) -> mark(unquote(X)),
                      unquote(ok(X)) -> ok(unquote(X)),
                   unquote1(mark(X)) -> mark(unquote1(X)),
                     unquote1(ok(X)) -> ok(unquote1(X)),
                 fcons(X1, mark(X2)) -> mark(fcons(X1, X2)),
                 fcons(mark(X1), X2) -> mark(fcons(X1, X2)),
               fcons(ok(X1), ok(X2)) -> ok(fcons(X1, X2)),
                 proper(sel(X1, X2)) -> sel(proper(X1), proper(X2)),
                        proper(s(X)) -> s(proper(X)),
                proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)),
                         proper(0()) -> ok(0()),
                       proper(nil()) -> ok(nil()),
               proper(first(X1, X2)) -> first(proper(X1), proper(X2)),
                     proper(from(X)) -> from(proper(X)),
                proper(sel1(X1, X2)) -> sel1(proper(X1), proper(X2)),
                    proper(quote(X)) -> quote(proper(X)),
                      proper(nil1()) -> ok(nil1()),
              proper(first1(X1, X2)) -> first1(proper(X1), proper(X2)),
               proper(cons1(X1, X2)) -> cons1(proper(X1), proper(X2)),
                        proper(01()) -> ok(01()),
                   proper(quote1(X)) -> quote1(proper(X)),
                       proper(s1(X)) -> s1(proper(X)),
                  proper(unquote(X)) -> unquote(proper(X)),
                 proper(unquote1(X)) -> unquote1(proper(X)),
               proper(fcons(X1, X2)) -> fcons(proper(X1), proper(X2)),
                        top(mark(X)) -> top(proper(X)),
                          top(ok(X)) -> top(active(X))
   }
  Fail