YES

Problem:
 a__U11(tt(),M,N) -> a__U12(tt(),M,N)
 a__U12(tt(),M,N) -> s(a__plus(mark(N),mark(M)))
 a__plus(N,0()) -> mark(N)
 a__plus(N,s(M)) -> a__U11(tt(),M,N)
 mark(U11(X1,X2,X3)) -> a__U11(mark(X1),X2,X3)
 mark(U12(X1,X2,X3)) -> a__U12(mark(X1),X2,X3)
 mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2))
 mark(tt()) -> tt()
 mark(s(X)) -> s(mark(X))
 mark(0()) -> 0()
 a__U11(X1,X2,X3) -> U11(X1,X2,X3)
 a__U12(X1,X2,X3) -> U12(X1,X2,X3)
 a__plus(X1,X2) -> plus(X1,X2)

Proof:
 DP Processor:
  DPs:
   a__U11#(tt(),M,N) -> a__U12#(tt(),M,N)
   a__U12#(tt(),M,N) -> mark#(M)
   a__U12#(tt(),M,N) -> mark#(N)
   a__U12#(tt(),M,N) -> a__plus#(mark(N),mark(M))
   a__plus#(N,0()) -> mark#(N)
   a__plus#(N,s(M)) -> a__U11#(tt(),M,N)
   mark#(U11(X1,X2,X3)) -> mark#(X1)
   mark#(U11(X1,X2,X3)) -> a__U11#(mark(X1),X2,X3)
   mark#(U12(X1,X2,X3)) -> mark#(X1)
   mark#(U12(X1,X2,X3)) -> a__U12#(mark(X1),X2,X3)
   mark#(plus(X1,X2)) -> mark#(X2)
   mark#(plus(X1,X2)) -> mark#(X1)
   mark#(plus(X1,X2)) -> a__plus#(mark(X1),mark(X2))
   mark#(s(X)) -> mark#(X)
  TRS:
   a__U11(tt(),M,N) -> a__U12(tt(),M,N)
   a__U12(tt(),M,N) -> s(a__plus(mark(N),mark(M)))
   a__plus(N,0()) -> mark(N)
   a__plus(N,s(M)) -> a__U11(tt(),M,N)
   mark(U11(X1,X2,X3)) -> a__U11(mark(X1),X2,X3)
   mark(U12(X1,X2,X3)) -> a__U12(mark(X1),X2,X3)
   mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2))
   mark(tt()) -> tt()
   mark(s(X)) -> s(mark(X))
   mark(0()) -> 0()
   a__U11(X1,X2,X3) -> U11(X1,X2,X3)
   a__U12(X1,X2,X3) -> U12(X1,X2,X3)
   a__plus(X1,X2) -> plus(X1,X2)
  TDG Processor:
   DPs:
    a__U11#(tt(),M,N) -> a__U12#(tt(),M,N)
    a__U12#(tt(),M,N) -> mark#(M)
    a__U12#(tt(),M,N) -> mark#(N)
    a__U12#(tt(),M,N) -> a__plus#(mark(N),mark(M))
    a__plus#(N,0()) -> mark#(N)
    a__plus#(N,s(M)) -> a__U11#(tt(),M,N)
    mark#(U11(X1,X2,X3)) -> mark#(X1)
    mark#(U11(X1,X2,X3)) -> a__U11#(mark(X1),X2,X3)
    mark#(U12(X1,X2,X3)) -> mark#(X1)
    mark#(U12(X1,X2,X3)) -> a__U12#(mark(X1),X2,X3)
    mark#(plus(X1,X2)) -> mark#(X2)
    mark#(plus(X1,X2)) -> mark#(X1)
    mark#(plus(X1,X2)) -> a__plus#(mark(X1),mark(X2))
    mark#(s(X)) -> mark#(X)
   TRS:
    a__U11(tt(),M,N) -> a__U12(tt(),M,N)
    a__U12(tt(),M,N) -> s(a__plus(mark(N),mark(M)))
    a__plus(N,0()) -> mark(N)
    a__plus(N,s(M)) -> a__U11(tt(),M,N)
    mark(U11(X1,X2,X3)) -> a__U11(mark(X1),X2,X3)
    mark(U12(X1,X2,X3)) -> a__U12(mark(X1),X2,X3)
    mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2))
    mark(tt()) -> tt()
    mark(s(X)) -> s(mark(X))
    mark(0()) -> 0()
    a__U11(X1,X2,X3) -> U11(X1,X2,X3)
    a__U12(X1,X2,X3) -> U12(X1,X2,X3)
    a__plus(X1,X2) -> plus(X1,X2)
   graph:
    a__plus#(N,0()) -> mark#(N) -> mark#(s(X)) -> mark#(X)
    a__plus#(N,0()) -> mark#(N) ->
    mark#(plus(X1,X2)) -> a__plus#(mark(X1),mark(X2))
    a__plus#(N,0()) -> mark#(N) -> mark#(plus(X1,X2)) -> mark#(X1)
    a__plus#(N,0()) -> mark#(N) -> mark#(plus(X1,X2)) -> mark#(X2)
    a__plus#(N,0()) -> mark#(N) ->
    mark#(U12(X1,X2,X3)) -> a__U12#(mark(X1),X2,X3)
    a__plus#(N,0()) -> mark#(N) -> mark#(U12(X1,X2,X3)) -> mark#(X1)
    a__plus#(N,0()) -> mark#(N) ->
    mark#(U11(X1,X2,X3)) -> a__U11#(mark(X1),X2,X3)
    a__plus#(N,0()) -> mark#(N) ->
    mark#(U11(X1,X2,X3)) -> mark#(X1)
    a__plus#(N,s(M)) -> a__U11#(tt(),M,N) ->
    a__U11#(tt(),M,N) -> a__U12#(tt(),M,N)
    mark#(plus(X1,X2)) -> a__plus#(mark(X1),mark(X2)) ->
    a__plus#(N,s(M)) -> a__U11#(tt(),M,N)
    mark#(plus(X1,X2)) -> a__plus#(mark(X1),mark(X2)) ->
    a__plus#(N,0()) -> mark#(N)
    mark#(plus(X1,X2)) -> mark#(X2) -> mark#(s(X)) -> mark#(X)
    mark#(plus(X1,X2)) -> mark#(X2) ->
    mark#(plus(X1,X2)) -> a__plus#(mark(X1),mark(X2))
    mark#(plus(X1,X2)) -> mark#(X2) ->
    mark#(plus(X1,X2)) -> mark#(X1)
    mark#(plus(X1,X2)) -> mark#(X2) ->
    mark#(plus(X1,X2)) -> mark#(X2)
    mark#(plus(X1,X2)) -> mark#(X2) ->
    mark#(U12(X1,X2,X3)) -> a__U12#(mark(X1),X2,X3)
    mark#(plus(X1,X2)) -> mark#(X2) ->
    mark#(U12(X1,X2,X3)) -> mark#(X1)
    mark#(plus(X1,X2)) -> mark#(X2) ->
    mark#(U11(X1,X2,X3)) -> a__U11#(mark(X1),X2,X3)
    mark#(plus(X1,X2)) -> mark#(X2) ->
    mark#(U11(X1,X2,X3)) -> mark#(X1)
    mark#(plus(X1,X2)) -> mark#(X1) -> mark#(s(X)) -> mark#(X)
    mark#(plus(X1,X2)) -> mark#(X1) ->
    mark#(plus(X1,X2)) -> a__plus#(mark(X1),mark(X2))
    mark#(plus(X1,X2)) -> mark#(X1) ->
    mark#(plus(X1,X2)) -> mark#(X1)
    mark#(plus(X1,X2)) -> mark#(X1) ->
    mark#(plus(X1,X2)) -> mark#(X2)
    mark#(plus(X1,X2)) -> mark#(X1) ->
    mark#(U12(X1,X2,X3)) -> a__U12#(mark(X1),X2,X3)
    mark#(plus(X1,X2)) -> mark#(X1) ->
    mark#(U12(X1,X2,X3)) -> mark#(X1)
    mark#(plus(X1,X2)) -> mark#(X1) ->
    mark#(U11(X1,X2,X3)) -> a__U11#(mark(X1),X2,X3)
    mark#(plus(X1,X2)) -> mark#(X1) ->
    mark#(U11(X1,X2,X3)) -> mark#(X1)
    mark#(U12(X1,X2,X3)) -> mark#(X1) -> mark#(s(X)) -> mark#(X)
    mark#(U12(X1,X2,X3)) -> mark#(X1) ->
    mark#(plus(X1,X2)) -> a__plus#(mark(X1),mark(X2))
    mark#(U12(X1,X2,X3)) -> mark#(X1) ->
    mark#(plus(X1,X2)) -> mark#(X1)
    mark#(U12(X1,X2,X3)) -> mark#(X1) ->
    mark#(plus(X1,X2)) -> mark#(X2)
    mark#(U12(X1,X2,X3)) -> mark#(X1) ->
    mark#(U12(X1,X2,X3)) -> a__U12#(mark(X1),X2,X3)
    mark#(U12(X1,X2,X3)) -> mark#(X1) ->
    mark#(U12(X1,X2,X3)) -> mark#(X1)
    mark#(U12(X1,X2,X3)) -> mark#(X1) ->
    mark#(U11(X1,X2,X3)) -> a__U11#(mark(X1),X2,X3)
    mark#(U12(X1,X2,X3)) -> mark#(X1) ->
    mark#(U11(X1,X2,X3)) -> mark#(X1)
    mark#(U12(X1,X2,X3)) -> a__U12#(mark(X1),X2,X3) ->
    a__U12#(tt(),M,N) -> a__plus#(mark(N),mark(M))
    mark#(U12(X1,X2,X3)) -> a__U12#(mark(X1),X2,X3) ->
    a__U12#(tt(),M,N) -> mark#(N)
    mark#(U12(X1,X2,X3)) -> a__U12#(mark(X1),X2,X3) ->
    a__U12#(tt(),M,N) -> mark#(M)
    mark#(U11(X1,X2,X3)) -> mark#(X1) -> mark#(s(X)) -> mark#(X)
    mark#(U11(X1,X2,X3)) -> mark#(X1) ->
    mark#(plus(X1,X2)) -> a__plus#(mark(X1),mark(X2))
    mark#(U11(X1,X2,X3)) -> mark#(X1) ->
    mark#(plus(X1,X2)) -> mark#(X1)
    mark#(U11(X1,X2,X3)) -> mark#(X1) ->
    mark#(plus(X1,X2)) -> mark#(X2)
    mark#(U11(X1,X2,X3)) -> mark#(X1) ->
    mark#(U12(X1,X2,X3)) -> a__U12#(mark(X1),X2,X3)
    mark#(U11(X1,X2,X3)) -> mark#(X1) ->
    mark#(U12(X1,X2,X3)) -> mark#(X1)
    mark#(U11(X1,X2,X3)) -> mark#(X1) ->
    mark#(U11(X1,X2,X3)) -> a__U11#(mark(X1),X2,X3)
    mark#(U11(X1,X2,X3)) -> mark#(X1) ->
    mark#(U11(X1,X2,X3)) -> mark#(X1)
    mark#(U11(X1,X2,X3)) -> a__U11#(mark(X1),X2,X3) ->
    a__U11#(tt(),M,N) -> a__U12#(tt(),M,N)
    mark#(s(X)) -> mark#(X) -> mark#(s(X)) -> mark#(X)
    mark#(s(X)) -> mark#(X) ->
    mark#(plus(X1,X2)) -> a__plus#(mark(X1),mark(X2))
    mark#(s(X)) -> mark#(X) -> mark#(plus(X1,X2)) -> mark#(X1)
    mark#(s(X)) -> mark#(X) -> mark#(plus(X1,X2)) -> mark#(X2)
    mark#(s(X)) -> mark#(X) ->
    mark#(U12(X1,X2,X3)) -> a__U12#(mark(X1),X2,X3)
    mark#(s(X)) -> mark#(X) -> mark#(U12(X1,X2,X3)) -> mark#(X1)
    mark#(s(X)) -> mark#(X) ->
    mark#(U11(X1,X2,X3)) -> a__U11#(mark(X1),X2,X3)
    mark#(s(X)) -> mark#(X) ->
    mark#(U11(X1,X2,X3)) -> mark#(X1)
    a__U12#(tt(),M,N) -> a__plus#(mark(N),mark(M)) ->
    a__plus#(N,s(M)) -> a__U11#(tt(),M,N)
    a__U12#(tt(),M,N) -> a__plus#(mark(N),mark(M)) ->
    a__plus#(N,0()) -> mark#(N)
    a__U12#(tt(),M,N) -> mark#(N) -> mark#(s(X)) -> mark#(X)
    a__U12#(tt(),M,N) -> mark#(N) ->
    mark#(plus(X1,X2)) -> a__plus#(mark(X1),mark(X2))
    a__U12#(tt(),M,N) -> mark#(N) -> mark#(plus(X1,X2)) -> mark#(X1)
    a__U12#(tt(),M,N) -> mark#(N) -> mark#(plus(X1,X2)) -> mark#(X2)
    a__U12#(tt(),M,N) -> mark#(N) ->
    mark#(U12(X1,X2,X3)) -> a__U12#(mark(X1),X2,X3)
    a__U12#(tt(),M,N) -> mark#(N) -> mark#(U12(X1,X2,X3)) -> mark#(X1)
    a__U12#(tt(),M,N) -> mark#(N) ->
    mark#(U11(X1,X2,X3)) -> a__U11#(mark(X1),X2,X3)
    a__U12#(tt(),M,N) -> mark#(N) -> mark#(U11(X1,X2,X3)) -> mark#(X1)
    a__U12#(tt(),M,N) -> mark#(M) -> mark#(s(X)) -> mark#(X)
    a__U12#(tt(),M,N) -> mark#(M) ->
    mark#(plus(X1,X2)) -> a__plus#(mark(X1),mark(X2))
    a__U12#(tt(),M,N) -> mark#(M) -> mark#(plus(X1,X2)) -> mark#(X1)
    a__U12#(tt(),M,N) -> mark#(M) -> mark#(plus(X1,X2)) -> mark#(X2)
    a__U12#(tt(),M,N) -> mark#(M) ->
    mark#(U12(X1,X2,X3)) -> a__U12#(mark(X1),X2,X3)
    a__U12#(tt(),M,N) -> mark#(M) -> mark#(U12(X1,X2,X3)) -> mark#(X1)
    a__U12#(tt(),M,N) -> mark#(M) ->
    mark#(U11(X1,X2,X3)) -> a__U11#(mark(X1),X2,X3)
    a__U12#(tt(),M,N) -> mark#(M) ->
    mark#(U11(X1,X2,X3)) -> mark#(X1)
    a__U11#(tt(),M,N) -> a__U12#(tt(),M,N) ->
    a__U12#(tt(),M,N) -> a__plus#(mark(N),mark(M))
    a__U11#(tt(),M,N) -> a__U12#(tt(),M,N) ->
    a__U12#(tt(),M,N) -> mark#(N)
    a__U11#(tt(),M,N) -> a__U12#(tt(),M,N) -> a__U12#(tt(),M,N) -> mark#(M)
   Arctic Interpretation Processor:
    dimension: 1
    interpretation:
     [a__plus#](x0, x1) = x0 + 1x1,
     
     [mark#](x0) = x0,
     
     [a__U12#](x0, x1, x2) = x0 + 1x1 + x2 + 0,
     
     [a__U11#](x0, x1, x2) = 1x1 + x2 + 0,
     
     [plus](x0, x1) = x0 + 1x1,
     
     [U12](x0, x1, x2) = x0 + 1x1 + x2 + 0,
     
     [U11](x0, x1, x2) = x0 + 1x1 + x2 + 0,
     
     [0] = 0,
     
     [s](x0) = x0 + 0,
     
     [a__plus](x0, x1) = x0 + 1x1,
     
     [mark](x0) = x0,
     
     [a__U12](x0, x1, x2) = x0 + 1x1 + x2 + 0,
     
     [a__U11](x0, x1, x2) = x0 + 1x1 + x2 + 0,
     
     [tt] = 0
    orientation:
     a__U11#(tt(),M,N) = 1M + N + 0 >= 1M + N + 0 = a__U12#(tt(),M,N)
     
     a__U12#(tt(),M,N) = 1M + N + 0 >= M = mark#(M)
     
     a__U12#(tt(),M,N) = 1M + N + 0 >= N = mark#(N)
     
     a__U12#(tt(),M,N) = 1M + N + 0 >= 1M + N = a__plus#(mark(N),mark(M))
     
     a__plus#(N,0()) = N + 1 >= N = mark#(N)
     
     a__plus#(N,s(M)) = 1M + N + 1 >= 1M + N + 0 = a__U11#(tt(),M,N)
     
     mark#(U11(X1,X2,X3)) = X1 + 1X2 + X3 + 0 >= X1 = mark#(X1)
     
     mark#(U11(X1,X2,X3)) = X1 + 1X2 + X3 + 0 >= 1X2 + X3 + 0 = a__U11#(mark(X1),X2,X3)
     
     mark#(U12(X1,X2,X3)) = X1 + 1X2 + X3 + 0 >= X1 = mark#(X1)
     
     mark#(U12(X1,X2,X3)) = X1 + 1X2 + X3 + 0 >= X1 + 1X2 + X3 + 0 = a__U12#(mark(X1),X2,X3)
     
     mark#(plus(X1,X2)) = X1 + 1X2 >= X2 = mark#(X2)
     
     mark#(plus(X1,X2)) = X1 + 1X2 >= X1 = mark#(X1)
     
     mark#(plus(X1,X2)) = X1 + 1X2 >= X1 + 1X2 = a__plus#(mark(X1),mark(X2))
     
     mark#(s(X)) = X + 0 >= X = mark#(X)
     
     a__U11(tt(),M,N) = 1M + N + 0 >= 1M + N + 0 = a__U12(tt(),M,N)
     
     a__U12(tt(),M,N) = 1M + N + 0 >= 1M + N + 0 = s(a__plus(mark(N),mark(M)))
     
     a__plus(N,0()) = N + 1 >= N = mark(N)
     
     a__plus(N,s(M)) = 1M + N + 1 >= 1M + N + 0 = a__U11(tt(),M,N)
     
     mark(U11(X1,X2,X3)) = X1 + 1X2 + X3 + 0 >= X1 + 1X2 + X3 + 0 = a__U11(mark(X1),X2,X3)
     
     mark(U12(X1,X2,X3)) = X1 + 1X2 + X3 + 0 >= X1 + 1X2 + X3 + 0 = a__U12(mark(X1),X2,X3)
     
     mark(plus(X1,X2)) = X1 + 1X2 >= X1 + 1X2 = a__plus(mark(X1),mark(X2))
     
     mark(tt()) = 0 >= 0 = tt()
     
     mark(s(X)) = X + 0 >= X + 0 = s(mark(X))
     
     mark(0()) = 0 >= 0 = 0()
     
     a__U11(X1,X2,X3) = X1 + 1X2 + X3 + 0 >= X1 + 1X2 + X3 + 0 = U11(X1,X2,X3)
     
     a__U12(X1,X2,X3) = X1 + 1X2 + X3 + 0 >= X1 + 1X2 + X3 + 0 = U12(X1,X2,X3)
     
     a__plus(X1,X2) = X1 + 1X2 >= X1 + 1X2 = plus(X1,X2)
    problem:
     DPs:
      a__U11#(tt(),M,N) -> a__U12#(tt(),M,N)
      a__U12#(tt(),M,N) -> mark#(N)
      a__U12#(tt(),M,N) -> a__plus#(mark(N),mark(M))
      a__plus#(N,0()) -> mark#(N)
      a__plus#(N,s(M)) -> a__U11#(tt(),M,N)
      mark#(U11(X1,X2,X3)) -> mark#(X1)
      mark#(U11(X1,X2,X3)) -> a__U11#(mark(X1),X2,X3)
      mark#(U12(X1,X2,X3)) -> mark#(X1)
      mark#(U12(X1,X2,X3)) -> a__U12#(mark(X1),X2,X3)
      mark#(plus(X1,X2)) -> mark#(X1)
      mark#(plus(X1,X2)) -> a__plus#(mark(X1),mark(X2))
      mark#(s(X)) -> mark#(X)
     TRS:
      a__U11(tt(),M,N) -> a__U12(tt(),M,N)
      a__U12(tt(),M,N) -> s(a__plus(mark(N),mark(M)))
      a__plus(N,0()) -> mark(N)
      a__plus(N,s(M)) -> a__U11(tt(),M,N)
      mark(U11(X1,X2,X3)) -> a__U11(mark(X1),X2,X3)
      mark(U12(X1,X2,X3)) -> a__U12(mark(X1),X2,X3)
      mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2))
      mark(tt()) -> tt()
      mark(s(X)) -> s(mark(X))
      mark(0()) -> 0()
      a__U11(X1,X2,X3) -> U11(X1,X2,X3)
      a__U12(X1,X2,X3) -> U12(X1,X2,X3)
      a__plus(X1,X2) -> plus(X1,X2)
    KBO Processor:
     argument filtering:
      pi(tt) = []
      pi(a__U11) = [0,1,2]
      pi(a__U12) = [0,1,2]
      pi(mark) = [0]
      pi(a__plus) = [0,1]
      pi(s) = [0]
      pi(0) = []
      pi(U11) = [0,1,2]
      pi(U12) = [0,1,2]
      pi(plus) = [0,1]
      pi(a__U11#) = [0,1,2]
      pi(a__U12#) = [1,2]
      pi(mark#) = [0]
      pi(a__plus#) = [0,1]
     weight function:
      w0 = 1
      w(0) = w(s) = w(tt) = 1
      w(a__plus#) = w(mark#) = w(a__U12#) = w(a__U11#) = w(plus) = w(
      U12) = w(U11) = w(a__plus) = w(mark) = w(a__U12) = w(a__U11) = 0
     precedence:
      mark# ~ mark > a__U12# ~ a__plus > a__plus# ~ a__U11 > a__U12 > 
      a__U11# ~ plus ~ U12 ~ U11 ~ 0 ~ s ~ tt
     problem:
      DPs:
       
      TRS:
       a__U11(tt(),M,N) -> a__U12(tt(),M,N)
       a__U12(tt(),M,N) -> s(a__plus(mark(N),mark(M)))
       a__plus(N,0()) -> mark(N)
       a__plus(N,s(M)) -> a__U11(tt(),M,N)
       mark(U11(X1,X2,X3)) -> a__U11(mark(X1),X2,X3)
       mark(U12(X1,X2,X3)) -> a__U12(mark(X1),X2,X3)
       mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2))
       mark(tt()) -> tt()
       mark(s(X)) -> s(mark(X))
       mark(0()) -> 0()
       a__U11(X1,X2,X3) -> U11(X1,X2,X3)
       a__U12(X1,X2,X3) -> U12(X1,X2,X3)
       a__plus(X1,X2) -> plus(X1,X2)
     Qed