YES
TRS:
 {ack_in(s(m), s(n)) -> u21(ack_in(s(m), n), m),
   ack_in(s(m), 0()) -> u11(ack_in(m, s(0()))),
      ack_in(0(), n) -> ack_out(s(n)),
     u11(ack_out(n)) -> ack_out(n),
  u21(ack_out(n), m) -> u22(ack_in(m, n)),
     u22(ack_out(n)) -> ack_out(n)}
 DP:
  Strict:
   {ack_in#(s(m), s(n)) -> ack_in#(s(m), n),
    ack_in#(s(m), s(n)) -> u21#(ack_in(s(m), n), m),
     ack_in#(s(m), 0()) -> ack_in#(m, s(0())),
     ack_in#(s(m), 0()) -> u11#(ack_in(m, s(0()))),
    u21#(ack_out(n), m) -> ack_in#(m, n),
    u21#(ack_out(n), m) -> u22#(ack_in(m, n))}
  Weak:
  {ack_in(s(m), s(n)) -> u21(ack_in(s(m), n), m),
    ack_in(s(m), 0()) -> u11(ack_in(m, s(0()))),
       ack_in(0(), n) -> ack_out(s(n)),
      u11(ack_out(n)) -> ack_out(n),
   u21(ack_out(n), m) -> u22(ack_in(m, n)),
      u22(ack_out(n)) -> ack_out(n)}
  EDG:
   {(u21#(ack_out(n), m) -> ack_in#(m, n), ack_in#(s(m), 0()) -> u11#(ack_in(m, s(0()))))
    (u21#(ack_out(n), m) -> ack_in#(m, n), ack_in#(s(m), 0()) -> ack_in#(m, s(0())))
    (u21#(ack_out(n), m) -> ack_in#(m, n), ack_in#(s(m), s(n)) -> u21#(ack_in(s(m), n), m))
    (u21#(ack_out(n), m) -> ack_in#(m, n), ack_in#(s(m), s(n)) -> ack_in#(s(m), n))
    (ack_in#(s(m), s(n)) -> ack_in#(s(m), n), ack_in#(s(m), 0()) -> u11#(ack_in(m, s(0()))))
    (ack_in#(s(m), s(n)) -> ack_in#(s(m), n), ack_in#(s(m), 0()) -> ack_in#(m, s(0())))
    (ack_in#(s(m), s(n)) -> ack_in#(s(m), n), ack_in#(s(m), s(n)) -> u21#(ack_in(s(m), n), m))
    (ack_in#(s(m), s(n)) -> ack_in#(s(m), n), ack_in#(s(m), s(n)) -> ack_in#(s(m), n))
    (ack_in#(s(m), s(n)) -> u21#(ack_in(s(m), n), m), u21#(ack_out(n), m) -> ack_in#(m, n))
    (ack_in#(s(m), s(n)) -> u21#(ack_in(s(m), n), m), u21#(ack_out(n), m) -> u22#(ack_in(m, n)))
    (ack_in#(s(m), 0()) -> ack_in#(m, s(0())), ack_in#(s(m), s(n)) -> ack_in#(s(m), n))
    (ack_in#(s(m), 0()) -> ack_in#(m, s(0())), ack_in#(s(m), s(n)) -> u21#(ack_in(s(m), n), m))}
   SCCS:
    Scc:
     {ack_in#(s(m), s(n)) -> ack_in#(s(m), n),
      ack_in#(s(m), s(n)) -> u21#(ack_in(s(m), n), m),
       ack_in#(s(m), 0()) -> ack_in#(m, s(0())),
      u21#(ack_out(n), m) -> ack_in#(m, n)}
    SCC:
     Strict:
      {ack_in#(s(m), s(n)) -> ack_in#(s(m), n),
       ack_in#(s(m), s(n)) -> u21#(ack_in(s(m), n), m),
        ack_in#(s(m), 0()) -> ack_in#(m, s(0())),
       u21#(ack_out(n), m) -> ack_in#(m, n)}
     Weak:
     {ack_in(s(m), s(n)) -> u21(ack_in(s(m), n), m),
       ack_in(s(m), 0()) -> u11(ack_in(m, s(0()))),
          ack_in(0(), n) -> ack_out(s(n)),
         u11(ack_out(n)) -> ack_out(n),
      u21(ack_out(n), m) -> u22(ack_in(m, n)),
         u22(ack_out(n)) -> ack_out(n)}
     SPSC:
      Simple Projection:
       pi(u21#) = 1, pi(ack_in#) = 0
      Strict:
       {ack_in#(s(m), s(n)) -> ack_in#(s(m), n),
         ack_in#(s(m), 0()) -> ack_in#(m, s(0())),
        u21#(ack_out(n), m) -> ack_in#(m, n)}
      EDG:
       {(ack_in#(s(m), 0()) -> ack_in#(m, s(0())), ack_in#(s(m), s(n)) -> ack_in#(s(m), n))
        (u21#(ack_out(n), m) -> ack_in#(m, n), ack_in#(s(m), s(n)) -> ack_in#(s(m), n))
        (u21#(ack_out(n), m) -> ack_in#(m, n), ack_in#(s(m), 0()) -> ack_in#(m, s(0())))
        (ack_in#(s(m), s(n)) -> ack_in#(s(m), n), ack_in#(s(m), s(n)) -> ack_in#(s(m), n))
        (ack_in#(s(m), s(n)) -> ack_in#(s(m), n), ack_in#(s(m), 0()) -> ack_in#(m, s(0())))}
       SCCS:
        Scc:
         {ack_in#(s(m), s(n)) -> ack_in#(s(m), n),
           ack_in#(s(m), 0()) -> ack_in#(m, s(0()))}
        SCC:
         Strict:
          {ack_in#(s(m), s(n)) -> ack_in#(s(m), n),
            ack_in#(s(m), 0()) -> ack_in#(m, s(0()))}
         Weak:
         {ack_in(s(m), s(n)) -> u21(ack_in(s(m), n), m),
           ack_in(s(m), 0()) -> u11(ack_in(m, s(0()))),
              ack_in(0(), n) -> ack_out(s(n)),
             u11(ack_out(n)) -> ack_out(n),
          u21(ack_out(n), m) -> u22(ack_in(m, n)),
             u22(ack_out(n)) -> ack_out(n)}
         SPSC:
          Simple Projection:
           pi(ack_in#) = 0
          Strict:
           {ack_in#(s(m), s(n)) -> ack_in#(s(m), n)}
          EDG:
           {(ack_in#(s(m), s(n)) -> ack_in#(s(m), n), ack_in#(s(m), s(n)) -> ack_in#(s(m), n))}
           SCCS:
            Scc:
             {ack_in#(s(m), s(n)) -> ack_in#(s(m), n)}
            SCC:
             Strict:
              {ack_in#(s(m), s(n)) -> ack_in#(s(m), n)}
             Weak:
             {ack_in(s(m), s(n)) -> u21(ack_in(s(m), n), m),
               ack_in(s(m), 0()) -> u11(ack_in(m, s(0()))),
                  ack_in(0(), n) -> ack_out(s(n)),
                 u11(ack_out(n)) -> ack_out(n),
              u21(ack_out(n), m) -> u22(ack_in(m, n)),
                 u22(ack_out(n)) -> ack_out(n)}
             SPSC:
              Simple Projection:
               pi(ack_in#) = 1
              Strict:
               {}
              Qed