YES

Problem:
 times(x,plus(y,1())) -> plus(times(x,plus(y,times(1(),0()))),x)
 times(x,1()) -> x
 plus(x,0()) -> x
 times(x,0()) -> 0()

Proof:
 DP Processor:
  DPs:
   times#(x,plus(y,1())) -> times#(1(),0())
   times#(x,plus(y,1())) -> plus#(y,times(1(),0()))
   times#(x,plus(y,1())) -> times#(x,plus(y,times(1(),0())))
   times#(x,plus(y,1())) -> plus#(times(x,plus(y,times(1(),0()))),x)
  TRS:
   times(x,plus(y,1())) -> plus(times(x,plus(y,times(1(),0()))),x)
   times(x,1()) -> x
   plus(x,0()) -> x
   times(x,0()) -> 0()
  TDG Processor:
   DPs:
    times#(x,plus(y,1())) -> times#(1(),0())
    times#(x,plus(y,1())) -> plus#(y,times(1(),0()))
    times#(x,plus(y,1())) -> times#(x,plus(y,times(1(),0())))
    times#(x,plus(y,1())) -> plus#(times(x,plus(y,times(1(),0()))),x)
   TRS:
    times(x,plus(y,1())) -> plus(times(x,plus(y,times(1(),0()))),x)
    times(x,1()) -> x
    plus(x,0()) -> x
    times(x,0()) -> 0()
   graph:
    times#(x,plus(y,1())) -> times#(1(),0()) ->
    times#(x,plus(y,1())) -> plus#(times(x,plus(y,times(1(),0()))),x)
    times#(x,plus(y,1())) -> times#(1(),0()) ->
    times#(x,plus(y,1())) -> times#(x,plus(y,times(1(),0())))
    times#(x,plus(y,1())) -> times#(1(),0()) ->
    times#(x,plus(y,1())) -> plus#(y,times(1(),0()))
    times#(x,plus(y,1())) -> times#(1(),0()) ->
    times#(x,plus(y,1())) -> times#(1(),0())
    times#(x,plus(y,1())) -> times#(x,plus(y,times(1(),0()))) ->
    times#(x,plus(y,1())) -> plus#(times(x,plus(y,times(1(),0()))),x)
    times#(x,plus(y,1())) -> times#(x,plus(y,times(1(),0()))) ->
    times#(x,plus(y,1())) -> times#(x,plus(y,times(1(),0())))
    times#(x,plus(y,1())) -> times#(x,plus(y,times(1(),0()))) ->
    times#(x,plus(y,1())) -> plus#(y,times(1(),0()))
    times#(x,plus(y,1())) -> times#(x,plus(y,times(1(),0()))) ->
    times#(x,plus(y,1())) -> times#(1(),0())
   SCC Processor:
    #sccs: 1
    #rules: 2
    #arcs: 8/16
    DPs:
     times#(x,plus(y,1())) -> times#(1(),0())
     times#(x,plus(y,1())) -> times#(x,plus(y,times(1(),0())))
    TRS:
     times(x,plus(y,1())) -> plus(times(x,plus(y,times(1(),0()))),x)
     times(x,1()) -> x
     plus(x,0()) -> x
     times(x,0()) -> 0()
    EDG Processor:
     DPs:
      times#(x,plus(y,1())) -> times#(1(),0())
      times#(x,plus(y,1())) -> times#(x,plus(y,times(1(),0())))
     TRS:
      times(x,plus(y,1())) -> plus(times(x,plus(y,times(1(),0()))),x)
      times(x,1()) -> x
      plus(x,0()) -> x
      times(x,0()) -> 0()
     graph:
      times#(x,plus(y,1())) -> times#(x,plus(y,times(1(),0()))) ->
      times#(x,plus(y,1())) -> times#(1(),0())
      times#(x,plus(y,1())) -> times#(x,plus(y,times(1(),0()))) ->
      times#(x,plus(y,1())) -> times#(x,plus(y,times(1(),0())))
     SCC Processor:
      #sccs: 1
      #rules: 1
      #arcs: 2/4
      DPs:
       times#(x,plus(y,1())) -> times#(x,plus(y,times(1(),0())))
      TRS:
       times(x,plus(y,1())) -> plus(times(x,plus(y,times(1(),0()))),x)
       times(x,1()) -> x
       plus(x,0()) -> x
       times(x,0()) -> 0()
      Usable Rule Processor:
       DPs:
        times#(x,plus(y,1())) -> times#(x,plus(y,times(1(),0())))
       TRS:
        times(x,0()) -> 0()
        plus(x,0()) -> x
       Semantic Labeling Processor:
        dimension: 1
        usable rules:
         times(x,0()) -> 0()
         plus(x,0()) -> x
        interpretation:
         [plus](x0, x1) = x0 + x1,
         
         [0] = 1,
         
         [1] = 0,
         
         [times](x0, x1) = 1
         labeled:
          times#
          times
          0
         usable (for model):
          times#
          plus
          1
          times
          0
         argument filtering:
          pi(1) = []
          pi(plus) = [0,1]
          pi(times) = []
          pi(0) = []
          pi(times#) = 1
        precedence:
         1 > plus > times > times# ~ 0
        problem:
         DPs:
          
         TRS:
          times(x,0()) -> 0()
          plus(x,0()) -> x
        Qed