MAYBE
MAYBE
TRS:
 {
          active(U11(X1, X2)) -> U11(active(X1), X2),
         active(U11(tt(), N)) -> mark(N),
                 active(s(X)) -> s(active(X)),
        active(plus(N, s(M))) -> mark(U21(and(isNat(M), isNat(N)), M, N)),
         active(plus(N, 0())) -> mark(U11(isNat(N), N)),
         active(plus(X1, X2)) -> plus(X1, active(X2)),
         active(plus(X1, X2)) -> plus(active(X1), X2),
      active(U21(X1, X2, X3)) -> U21(active(X1), X2, X3),
      active(U21(tt(), M, N)) -> mark(s(plus(N, M))),
               active(U31(X)) -> U31(active(X)),
            active(U31(tt())) -> mark(0()),
           active(x(N, s(M))) -> mark(U41(and(isNat(M), isNat(N)), M, N)),
            active(x(N, 0())) -> mark(U31(isNat(N))),
            active(x(X1, X2)) -> x(X1, active(X2)),
            active(x(X1, X2)) -> x(active(X1), X2),
      active(U41(X1, X2, X3)) -> U41(active(X1), X2, X3),
      active(U41(tt(), M, N)) -> mark(plus(x(N, M), N)),
          active(and(X1, X2)) -> and(active(X1), X2),
         active(and(tt(), X)) -> mark(X),
         active(isNat(s(V1))) -> mark(isNat(V1)),
  active(isNat(plus(V1, V2))) -> mark(and(isNat(V1), isNat(V2))),
           active(isNat(0())) -> mark(tt()),
     active(isNat(x(V1, V2))) -> mark(and(isNat(V1), isNat(V2))),
            U11(mark(X1), X2) -> mark(U11(X1, X2)),
          U11(ok(X1), ok(X2)) -> ok(U11(X1, X2)),
                   s(mark(X)) -> mark(s(X)),
                     s(ok(X)) -> ok(s(X)),
           plus(X1, mark(X2)) -> mark(plus(X1, X2)),
           plus(mark(X1), X2) -> mark(plus(X1, X2)),
         plus(ok(X1), ok(X2)) -> ok(plus(X1, X2)),
        U21(mark(X1), X2, X3) -> mark(U21(X1, X2, X3)),
  U21(ok(X1), ok(X2), ok(X3)) -> ok(U21(X1, X2, X3)),
                 U31(mark(X)) -> mark(U31(X)),
                   U31(ok(X)) -> ok(U31(X)),
              x(X1, mark(X2)) -> mark(x(X1, X2)),
              x(mark(X1), X2) -> mark(x(X1, X2)),
            x(ok(X1), ok(X2)) -> ok(x(X1, X2)),
        U41(mark(X1), X2, X3) -> mark(U41(X1, X2, X3)),
  U41(ok(X1), ok(X2), ok(X3)) -> ok(U41(X1, X2, X3)),
            and(mark(X1), X2) -> mark(and(X1, X2)),
          and(ok(X1), ok(X2)) -> ok(and(X1, X2)),
                 isNat(ok(X)) -> ok(isNat(X)),
          proper(U11(X1, X2)) -> U11(proper(X1), proper(X2)),
                 proper(tt()) -> ok(tt()),
                 proper(s(X)) -> s(proper(X)),
         proper(plus(X1, X2)) -> plus(proper(X1), proper(X2)),
      proper(U21(X1, X2, X3)) -> U21(proper(X1), proper(X2), proper(X3)),
                  proper(0()) -> ok(0()),
               proper(U31(X)) -> U31(proper(X)),
            proper(x(X1, X2)) -> x(proper(X1), proper(X2)),
      proper(U41(X1, X2, X3)) -> U41(proper(X1), proper(X2), proper(X3)),
          proper(and(X1, X2)) -> and(proper(X1), proper(X2)),
             proper(isNat(X)) -> isNat(proper(X)),
                 top(mark(X)) -> top(proper(X)),
                   top(ok(X)) -> top(active(X))
 }
 DUP: We consider a duplicating system.
  Trs:
   {
            active(U11(X1, X2)) -> U11(active(X1), X2),
           active(U11(tt(), N)) -> mark(N),
                   active(s(X)) -> s(active(X)),
          active(plus(N, s(M))) -> mark(U21(and(isNat(M), isNat(N)), M, N)),
           active(plus(N, 0())) -> mark(U11(isNat(N), N)),
           active(plus(X1, X2)) -> plus(X1, active(X2)),
           active(plus(X1, X2)) -> plus(active(X1), X2),
        active(U21(X1, X2, X3)) -> U21(active(X1), X2, X3),
        active(U21(tt(), M, N)) -> mark(s(plus(N, M))),
                 active(U31(X)) -> U31(active(X)),
              active(U31(tt())) -> mark(0()),
             active(x(N, s(M))) -> mark(U41(and(isNat(M), isNat(N)), M, N)),
              active(x(N, 0())) -> mark(U31(isNat(N))),
              active(x(X1, X2)) -> x(X1, active(X2)),
              active(x(X1, X2)) -> x(active(X1), X2),
        active(U41(X1, X2, X3)) -> U41(active(X1), X2, X3),
        active(U41(tt(), M, N)) -> mark(plus(x(N, M), N)),
            active(and(X1, X2)) -> and(active(X1), X2),
           active(and(tt(), X)) -> mark(X),
           active(isNat(s(V1))) -> mark(isNat(V1)),
    active(isNat(plus(V1, V2))) -> mark(and(isNat(V1), isNat(V2))),
             active(isNat(0())) -> mark(tt()),
       active(isNat(x(V1, V2))) -> mark(and(isNat(V1), isNat(V2))),
              U11(mark(X1), X2) -> mark(U11(X1, X2)),
            U11(ok(X1), ok(X2)) -> ok(U11(X1, X2)),
                     s(mark(X)) -> mark(s(X)),
                       s(ok(X)) -> ok(s(X)),
             plus(X1, mark(X2)) -> mark(plus(X1, X2)),
             plus(mark(X1), X2) -> mark(plus(X1, X2)),
           plus(ok(X1), ok(X2)) -> ok(plus(X1, X2)),
          U21(mark(X1), X2, X3) -> mark(U21(X1, X2, X3)),
    U21(ok(X1), ok(X2), ok(X3)) -> ok(U21(X1, X2, X3)),
                   U31(mark(X)) -> mark(U31(X)),
                     U31(ok(X)) -> ok(U31(X)),
                x(X1, mark(X2)) -> mark(x(X1, X2)),
                x(mark(X1), X2) -> mark(x(X1, X2)),
              x(ok(X1), ok(X2)) -> ok(x(X1, X2)),
          U41(mark(X1), X2, X3) -> mark(U41(X1, X2, X3)),
    U41(ok(X1), ok(X2), ok(X3)) -> ok(U41(X1, X2, X3)),
              and(mark(X1), X2) -> mark(and(X1, X2)),
            and(ok(X1), ok(X2)) -> ok(and(X1, X2)),
                   isNat(ok(X)) -> ok(isNat(X)),
            proper(U11(X1, X2)) -> U11(proper(X1), proper(X2)),
                   proper(tt()) -> ok(tt()),
                   proper(s(X)) -> s(proper(X)),
           proper(plus(X1, X2)) -> plus(proper(X1), proper(X2)),
        proper(U21(X1, X2, X3)) -> U21(proper(X1), proper(X2), proper(X3)),
                    proper(0()) -> ok(0()),
                 proper(U31(X)) -> U31(proper(X)),
              proper(x(X1, X2)) -> x(proper(X1), proper(X2)),
        proper(U41(X1, X2, X3)) -> U41(proper(X1), proper(X2), proper(X3)),
            proper(and(X1, X2)) -> and(proper(X1), proper(X2)),
               proper(isNat(X)) -> isNat(proper(X)),
                   top(mark(X)) -> top(proper(X)),
                     top(ok(X)) -> top(active(X))
   }
  Fail