YES
Time: 0.040

Problem:
 Equations:
  plusAC(plusAC(x2,x3),x4) -> plusAC(x2,plusAC(x3,x4))
  plusAC(x2,x3) -> plusAC(x3,x2)
  plusAC(x2,plusAC(x3,x4)) -> plusAC(plusAC(x2,x3),x4)
  plusAC(x3,x2) -> plusAC(x2,x3)
 TRS:
  f(plusAC(x,y)) -> plusAC(f(x),f(y))
  plusAC(x,0()) -> 0()
  f(0()) -> 0()

Proof:
 DP Processor:
  Equations#:
   plus{AC,#}(plusAC(x2,x3),x4) -> plus{AC,#}(x2,plusAC(x3,x4))
   plus{AC,#}(x2,x3) -> plus{AC,#}(x3,x2)
   plus{AC,#}(x2,plusAC(x3,x4)) -> plus{AC,#}(plusAC(x2,x3),x4)
   plus{AC,#}(x3,x2) -> plus{AC,#}(x2,x3)
  DPs:
   f#(plusAC(x,y)) -> f#(y)
   f#(plusAC(x,y)) -> f#(x)
   f#(plusAC(x,y)) -> plus{AC,#}(f(x),f(y))
   plus{AC,#}(x5,plusAC(x,0())) -> plus{AC,#}(x5,0())
  Equations:
   plusAC(plusAC(x2,x3),x4) -> plusAC(x2,plusAC(x3,x4))
   plusAC(x2,x3) -> plusAC(x3,x2)
   plusAC(x2,plusAC(x3,x4)) -> plusAC(plusAC(x2,x3),x4)
   plusAC(x3,x2) -> plusAC(x2,x3)
  TRS:
   f(plusAC(x,y)) -> plusAC(f(x),f(y))
   plusAC(x,0()) -> 0()
   f(0()) -> 0()
  S:
   plus{AC,#}(plusAC(x6,x7),x8) -> plus{AC,#}(x6,x7)
   plus{AC,#}(x6,plusAC(x7,x8)) -> plus{AC,#}(x7,x8)
  AC-EDG Processor:
   Equations#:
    plus{AC,#}(plusAC(x2,x3),x4) -> plus{AC,#}(x2,plusAC(x3,x4))
    plus{AC,#}(x2,x3) -> plus{AC,#}(x3,x2)
    plus{AC,#}(x2,plusAC(x3,x4)) -> plus{AC,#}(plusAC(x2,x3),x4)
    plus{AC,#}(x3,x2) -> plus{AC,#}(x2,x3)
   DPs:
    f#(plusAC(x,y)) -> f#(y)
    f#(plusAC(x,y)) -> f#(x)
    f#(plusAC(x,y)) -> plus{AC,#}(f(x),f(y))
    plus{AC,#}(x5,plusAC(x,0())) -> plus{AC,#}(x5,0())
   Equations:
    plusAC(plusAC(x2,x3),x4) -> plusAC(x2,plusAC(x3,x4))
    plusAC(x2,x3) -> plusAC(x3,x2)
    plusAC(x2,plusAC(x3,x4)) -> plusAC(plusAC(x2,x3),x4)
    plusAC(x3,x2) -> plusAC(x2,x3)
   TRS:
    f(plusAC(x,y)) -> plusAC(f(x),f(y))
    plusAC(x,0()) -> 0()
    f(0()) -> 0()
   S:
    plus{AC,#}(plusAC(x6,x7),x8) -> plus{AC,#}(x6,x7)
    plus{AC,#}(x6,plusAC(x7,x8)) -> plus{AC,#}(x7,x8)
   SCC Processor:
    #sccs: 2
    #rules: 3
    #arcs: 8/16
    Equations#:
     plus{AC,#}(plusAC(x2,x3),x4) -> plus{AC,#}(x2,plusAC(x3,x4))
     plus{AC,#}(x2,x3) -> plus{AC,#}(x3,x2)
     plus{AC,#}(x2,plusAC(x3,x4)) -> plus{AC,#}(plusAC(x2,x3),x4)
     plus{AC,#}(x3,x2) -> plus{AC,#}(x2,x3)
    DPs:
     f#(plusAC(x,y)) -> f#(y)
     f#(plusAC(x,y)) -> f#(x)
    Equations:
     plusAC(plusAC(x2,x3),x4) -> plusAC(x2,plusAC(x3,x4))
     plusAC(x2,x3) -> plusAC(x3,x2)
     plusAC(x2,plusAC(x3,x4)) -> plusAC(plusAC(x2,x3),x4)
     plusAC(x3,x2) -> plusAC(x2,x3)
    TRS:
     f(plusAC(x,y)) -> plusAC(f(x),f(y))
     plusAC(x,0()) -> 0()
     f(0()) -> 0()
    S:
     plus{AC,#}(plusAC(x6,x7),x8) -> plus{AC,#}(x6,x7)
     plus{AC,#}(x6,plusAC(x7,x8)) -> plus{AC,#}(x7,x8)
    AC-DP unlabeling:
     Equations#:
      plusAC(plusAC(x2,x3),x4) -> plusAC(x2,plusAC(x3,x4))
      plusAC(x2,x3) -> plusAC(x3,x2)
      plusAC(x2,plusAC(x3,x4)) -> plusAC(plusAC(x2,x3),x4)
      plusAC(x3,x2) -> plusAC(x2,x3)
     DPs:
      f#(plusAC(x,y)) -> f#(y)
      f#(plusAC(x,y)) -> f#(x)
     Equations:
      plusAC(plusAC(x2,x3),x4) -> plusAC(x2,plusAC(x3,x4))
      plusAC(x2,x3) -> plusAC(x3,x2)
      plusAC(x2,plusAC(x3,x4)) -> plusAC(plusAC(x2,x3),x4)
      plusAC(x3,x2) -> plusAC(x2,x3)
     TRS:
      f(plusAC(x,y)) -> plusAC(f(x),f(y))
      plusAC(x,0()) -> 0()
      f(0()) -> 0()
     S:
      plusAC(plusAC(x6,x7),x8) -> plusAC(x6,x7)
      plusAC(x6,plusAC(x7,x8)) -> plusAC(x7,x8)
     Usable Rule Processor:
      Equations#:
       plusAC(plusAC(x2,x3),x4) -> plusAC(x2,plusAC(x3,x4))
       plusAC(x2,x3) -> plusAC(x3,x2)
       plusAC(x2,plusAC(x3,x4)) -> plusAC(plusAC(x2,x3),x4)
       plusAC(x3,x2) -> plusAC(x2,x3)
      DPs:
       f#(plusAC(x,y)) -> f#(y)
       f#(plusAC(x,y)) -> f#(x)
      Equations:
       plusAC(plusAC(x2,x3),x4) -> plusAC(x2,plusAC(x3,x4))
       plusAC(x2,x3) -> plusAC(x3,x2)
       plusAC(x2,plusAC(x3,x4)) -> plusAC(plusAC(x2,x3),x4)
       plusAC(x3,x2) -> plusAC(x2,x3)
      TRS:
       
      S:
       plusAC(plusAC(x6,x7),x8) -> plusAC(x6,x7)
       plusAC(x6,plusAC(x7,x8)) -> plusAC(x7,x8)
      AC-KBO Processor:
       argument filtering:
        pi(plusAC) = [0,1]
        pi(f#) = 0
       precedence:
        f# ~ plusAC
        weight function:
         w0 = 6
         w(f#) = 7
         w(plusAC) = 4
         usable rules:
          
         problem:
          Equations#:
           plusAC(plusAC(x2,x3),x4) -> plusAC(x2,plusAC(x3,x4))
           plusAC(x2,x3) -> plusAC(x3,x2)
           plusAC(x2,plusAC(x3,x4)) -> plusAC(plusAC(x2,x3),x4)
           plusAC(x3,x2) -> plusAC(x2,x3)
          DPs:
           
          Equations:
           plusAC(plusAC(x2,x3),x4) -> plusAC(x2,plusAC(x3,x4))
           plusAC(x2,x3) -> plusAC(x3,x2)
           plusAC(x2,plusAC(x3,x4)) -> plusAC(plusAC(x2,x3),x4)
           plusAC(x3,x2) -> plusAC(x2,x3)
          TRS:
           
          S:
           plusAC(plusAC(x6,x7),x8) -> plusAC(x6,x7)
           plusAC(x6,plusAC(x7,x8)) -> plusAC(x7,x8)
         Qed
     
     Equations#:
      plus{AC,#}(plusAC(x2,x3),x4) -> plus{AC,#}(x2,plusAC(x3,x4))
      plus{AC,#}(x2,x3) -> plus{AC,#}(x3,x2)
      plus{AC,#}(x2,plusAC(x3,x4)) -> plus{AC,#}(plusAC(x2,x3),x4)
      plus{AC,#}(x3,x2) -> plus{AC,#}(x2,x3)
     DPs:
      plus{AC,#}(x5,plusAC(x,0())) -> plus{AC,#}(x5,0())
     Equations:
      plusAC(plusAC(x2,x3),x4) -> plusAC(x2,plusAC(x3,x4))
      plusAC(x2,x3) -> plusAC(x3,x2)
      plusAC(x2,plusAC(x3,x4)) -> plusAC(plusAC(x2,x3),x4)
      plusAC(x3,x2) -> plusAC(x2,x3)
     TRS:
      f(plusAC(x,y)) -> plusAC(f(x),f(y))
      plusAC(x,0()) -> 0()
      f(0()) -> 0()
     S:
      plus{AC,#}(plusAC(x6,x7),x8) -> plus{AC,#}(x6,x7)
      plus{AC,#}(x6,plusAC(x7,x8)) -> plus{AC,#}(x7,x8)
     AC-DP unlabeling:
      Equations#:
       plusAC(plusAC(x2,x3),x4) -> plusAC(x2,plusAC(x3,x4))
       plusAC(x2,x3) -> plusAC(x3,x2)
       plusAC(x2,plusAC(x3,x4)) -> plusAC(plusAC(x2,x3),x4)
       plusAC(x3,x2) -> plusAC(x2,x3)
      DPs:
       plusAC(x5,plusAC(x,0())) -> plusAC(x5,0())
      Equations:
       plusAC(plusAC(x2,x3),x4) -> plusAC(x2,plusAC(x3,x4))
       plusAC(x2,x3) -> plusAC(x3,x2)
       plusAC(x2,plusAC(x3,x4)) -> plusAC(plusAC(x2,x3),x4)
       plusAC(x3,x2) -> plusAC(x2,x3)
      TRS:
       f(plusAC(x,y)) -> plusAC(f(x),f(y))
       plusAC(x,0()) -> 0()
       f(0()) -> 0()
      S:
       plusAC(plusAC(x6,x7),x8) -> plusAC(x6,x7)
       plusAC(x6,plusAC(x7,x8)) -> plusAC(x7,x8)
      Usable Rule Processor:
       Equations#:
        plusAC(plusAC(x2,x3),x4) -> plusAC(x2,plusAC(x3,x4))
        plusAC(x2,x3) -> plusAC(x3,x2)
        plusAC(x2,plusAC(x3,x4)) -> plusAC(plusAC(x2,x3),x4)
        plusAC(x3,x2) -> plusAC(x2,x3)
       DPs:
        plusAC(x5,plusAC(x,0())) -> plusAC(x5,0())
       Equations:
        plusAC(plusAC(x2,x3),x4) -> plusAC(x2,plusAC(x3,x4))
        plusAC(x2,x3) -> plusAC(x3,x2)
        plusAC(x2,plusAC(x3,x4)) -> plusAC(plusAC(x2,x3),x4)
        plusAC(x3,x2) -> plusAC(x2,x3)
       TRS:
        plusAC(x,0()) -> 0()
       S:
        plusAC(plusAC(x6,x7),x8) -> plusAC(x6,x7)
        plusAC(x6,plusAC(x7,x8)) -> plusAC(x7,x8)
       AC-KBO Processor:
        argument filtering:
         pi(plusAC) = [0,1]
         pi(0) = []
        precedence:
         0 ~ plusAC
         weight function:
          w0 = 2
          w(0) = 4
          w(plusAC) = 1
          usable rules:
           plusAC(x,0()) -> 0()
          problem:
           Equations#:
            plusAC(plusAC(x2,x3),x4) -> plusAC(x2,plusAC(x3,x4))
            plusAC(x2,x3) -> plusAC(x3,x2)
            plusAC(x2,plusAC(x3,x4)) -> plusAC(plusAC(x2,x3),x4)
            plusAC(x3,x2) -> plusAC(x2,x3)
           DPs:
            
           Equations:
            plusAC(plusAC(x2,x3),x4) -> plusAC(x2,plusAC(x3,x4))
            plusAC(x2,x3) -> plusAC(x3,x2)
            plusAC(x2,plusAC(x3,x4)) -> plusAC(plusAC(x2,x3),x4)
            plusAC(x3,x2) -> plusAC(x2,x3)
           TRS:
            plusAC(x,0()) -> 0()
           S:
            plusAC(plusAC(x6,x7),x8) -> plusAC(x6,x7)
            plusAC(x6,plusAC(x7,x8)) -> plusAC(x7,x8)
          Qed