MAYBE
MAYBE
TRS:
 {
                       sieve(mark(X)) -> mark(sieve(X)),
                         sieve(ok(X)) -> ok(sieve(X)),
                        from(mark(X)) -> mark(from(X)),
                          from(ok(X)) -> ok(from(X)),
                           s(mark(X)) -> mark(s(X)),
                             s(ok(X)) -> ok(s(X)),
                     active(sieve(X)) -> sieve(active(X)),
            active(sieve(cons(X, Y))) -> mark(cons(X, filter(X, sieve(Y)))),
                      active(from(X)) -> mark(cons(X, from(s(X)))),
                      active(from(X)) -> from(active(X)),
                         active(s(X)) -> s(active(X)),
                     active(primes()) -> mark(sieve(from(s(s(0()))))),
                 active(cons(X1, X2)) -> cons(active(X1), X2),
                      active(head(X)) -> head(active(X)),
             active(head(cons(X, Y))) -> mark(X),
                      active(tail(X)) -> tail(active(X)),
             active(tail(cons(X, Y))) -> mark(Y),
               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(divides(X1, X2)) -> divides(X1, active(X2)),
              active(divides(X1, X2)) -> divides(active(X1), X2),
               active(filter(X1, X2)) -> filter(X1, active(X2)),
               active(filter(X1, X2)) -> filter(active(X1), X2),
  active(filter(s(s(X)), cons(Y, Z))) ->
  mark(
      if(
        divides(s(s(X)), Y), filter(s(s(X)), Z), cons(Y, filter(X, sieve(Y)))
      )
  ),
                   cons(mark(X1), X2) -> mark(cons(X1, X2)),
                 cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)),
                        head(mark(X)) -> mark(head(X)),
                          head(ok(X)) -> ok(head(X)),
                        tail(mark(X)) -> mark(tail(X)),
                          tail(ok(X)) -> ok(tail(X)),
                 if(mark(X1), X2, X3) -> mark(if(X1, X2, X3)),
           if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3)),
                divides(X1, mark(X2)) -> mark(divides(X1, X2)),
                divides(mark(X1), X2) -> mark(divides(X1, X2)),
              divides(ok(X1), ok(X2)) -> ok(divides(X1, X2)),
                 filter(X1, mark(X2)) -> mark(filter(X1, X2)),
                 filter(mark(X1), X2) -> mark(filter(X1, X2)),
               filter(ok(X1), ok(X2)) -> ok(filter(X1, X2)),
                     proper(sieve(X)) -> sieve(proper(X)),
                      proper(from(X)) -> from(proper(X)),
                         proper(s(X)) -> s(proper(X)),
                          proper(0()) -> ok(0()),
                     proper(primes()) -> ok(primes()),
                 proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)),
                      proper(head(X)) -> head(proper(X)),
                      proper(tail(X)) -> tail(proper(X)),
               proper(if(X1, X2, X3)) ->
  if(proper(X1), proper(X2), proper(X3)),
                       proper(true()) -> ok(true()),
                      proper(false()) -> ok(false()),
              proper(divides(X1, X2)) -> divides(proper(X1), proper(X2)),
               proper(filter(X1, X2)) -> filter(proper(X1), proper(X2)),
                         top(mark(X)) -> top(proper(X)),
                           top(ok(X)) -> top(active(X))
 }
 DUP: We consider a duplicating system.
  Trs:
   {
                         sieve(mark(X)) -> mark(sieve(X)),
                           sieve(ok(X)) -> ok(sieve(X)),
                          from(mark(X)) -> mark(from(X)),
                            from(ok(X)) -> ok(from(X)),
                             s(mark(X)) -> mark(s(X)),
                               s(ok(X)) -> ok(s(X)),
                       active(sieve(X)) -> sieve(active(X)),
              active(sieve(cons(X, Y))) -> mark(cons(X, filter(X, sieve(Y)))),
                        active(from(X)) -> mark(cons(X, from(s(X)))),
                        active(from(X)) -> from(active(X)),
                           active(s(X)) -> s(active(X)),
                       active(primes()) -> mark(sieve(from(s(s(0()))))),
                   active(cons(X1, X2)) -> cons(active(X1), X2),
                        active(head(X)) -> head(active(X)),
               active(head(cons(X, Y))) -> mark(X),
                        active(tail(X)) -> tail(active(X)),
               active(tail(cons(X, Y))) -> mark(Y),
                 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(divides(X1, X2)) -> divides(X1, active(X2)),
                active(divides(X1, X2)) -> divides(active(X1), X2),
                 active(filter(X1, X2)) -> filter(X1, active(X2)),
                 active(filter(X1, X2)) -> filter(active(X1), X2),
    active(filter(s(s(X)), cons(Y, Z))) ->
    mark(
        if(
          divides(s(s(X)), Y), filter(s(s(X)), Z),
          cons(Y, filter(X, sieve(Y)))
        )
    ),
                     cons(mark(X1), X2) -> mark(cons(X1, X2)),
                   cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)),
                          head(mark(X)) -> mark(head(X)),
                            head(ok(X)) -> ok(head(X)),
                          tail(mark(X)) -> mark(tail(X)),
                            tail(ok(X)) -> ok(tail(X)),
                   if(mark(X1), X2, X3) -> mark(if(X1, X2, X3)),
             if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3)),
                  divides(X1, mark(X2)) -> mark(divides(X1, X2)),
                  divides(mark(X1), X2) -> mark(divides(X1, X2)),
                divides(ok(X1), ok(X2)) -> ok(divides(X1, X2)),
                   filter(X1, mark(X2)) -> mark(filter(X1, X2)),
                   filter(mark(X1), X2) -> mark(filter(X1, X2)),
                 filter(ok(X1), ok(X2)) -> ok(filter(X1, X2)),
                       proper(sieve(X)) -> sieve(proper(X)),
                        proper(from(X)) -> from(proper(X)),
                           proper(s(X)) -> s(proper(X)),
                            proper(0()) -> ok(0()),
                       proper(primes()) -> ok(primes()),
                   proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)),
                        proper(head(X)) -> head(proper(X)),
                        proper(tail(X)) -> tail(proper(X)),
                 proper(if(X1, X2, X3)) ->
    if(proper(X1), proper(X2), proper(X3)),
                         proper(true()) -> ok(true()),
                        proper(false()) -> ok(false()),
                proper(divides(X1, X2)) -> divides(proper(X1), proper(X2)),
                 proper(filter(X1, X2)) -> filter(proper(X1), proper(X2)),
                           top(mark(X)) -> top(proper(X)),
                             top(ok(X)) -> top(active(X))
   }
  Fail