YES
Time: 0.082

Problem:
 Equations:
  fAC(fAC(x3,x4),x5) -> fAC(x3,fAC(x4,x5))
  fAC(x3,x4) -> fAC(x4,x3)
  fAC(x3,fAC(x4,x5)) -> fAC(fAC(x3,x4),x5)
  fAC(x4,x3) -> fAC(x3,x4)
 TRS:
  fAC(x,one()) -> x
  fAC(i(x),x) -> one()
  phi(one(),y1) -> y1
  phi(y1,phi(y2,x)) -> phi(fAC(y1,y2),x)

Proof:
 DP Processor:
  Equations#:
   f{AC,#}(fAC(x3,x4),x5) -> f{AC,#}(x3,fAC(x4,x5))
   f{AC,#}(x3,x4) -> f{AC,#}(x4,x3)
   f{AC,#}(x3,fAC(x4,x5)) -> f{AC,#}(fAC(x3,x4),x5)
   f{AC,#}(x4,x3) -> f{AC,#}(x3,x4)
  DPs:
   phi#(y1,phi(y2,x)) -> f{AC,#}(y1,y2)
   phi#(y1,phi(y2,x)) -> phi#(fAC(y1,y2),x)
   f{AC,#}(x6,fAC(x,one())) -> f{AC,#}(x6,x)
   f{AC,#}(x7,fAC(i(x),x)) -> f{AC,#}(x7,one())
  Equations:
   fAC(fAC(x3,x4),x5) -> fAC(x3,fAC(x4,x5))
   fAC(x3,x4) -> fAC(x4,x3)
   fAC(x3,fAC(x4,x5)) -> fAC(fAC(x3,x4),x5)
   fAC(x4,x3) -> fAC(x3,x4)
  TRS:
   fAC(x,one()) -> x
   fAC(i(x),x) -> one()
   phi(one(),y1) -> y1
   phi(y1,phi(y2,x)) -> phi(fAC(y1,y2),x)
  S:
   f{AC,#}(fAC(x8,x9),x10) -> f{AC,#}(x8,x9)
   f{AC,#}(x8,fAC(x9,x10)) -> f{AC,#}(x9,x10)
  AC-EDG Processor:
   Equations#:
    f{AC,#}(fAC(x3,x4),x5) -> f{AC,#}(x3,fAC(x4,x5))
    f{AC,#}(x3,x4) -> f{AC,#}(x4,x3)
    f{AC,#}(x3,fAC(x4,x5)) -> f{AC,#}(fAC(x3,x4),x5)
    f{AC,#}(x4,x3) -> f{AC,#}(x3,x4)
   DPs:
    phi#(y1,phi(y2,x)) -> f{AC,#}(y1,y2)
    phi#(y1,phi(y2,x)) -> phi#(fAC(y1,y2),x)
    f{AC,#}(x6,fAC(x,one())) -> f{AC,#}(x6,x)
    f{AC,#}(x7,fAC(i(x),x)) -> f{AC,#}(x7,one())
   Equations:
    fAC(fAC(x3,x4),x5) -> fAC(x3,fAC(x4,x5))
    fAC(x3,x4) -> fAC(x4,x3)
    fAC(x3,fAC(x4,x5)) -> fAC(fAC(x3,x4),x5)
    fAC(x4,x3) -> fAC(x3,x4)
   TRS:
    fAC(x,one()) -> x
    fAC(i(x),x) -> one()
    phi(one(),y1) -> y1
    phi(y1,phi(y2,x)) -> phi(fAC(y1,y2),x)
   S:
    f{AC,#}(fAC(x8,x9),x10) -> f{AC,#}(x8,x9)
    f{AC,#}(x8,fAC(x9,x10)) -> f{AC,#}(x9,x10)
   SCC Processor:
    #sccs: 2
    #rules: 3
    #arcs: 8/16
    Equations#:
     f{AC,#}(fAC(x3,x4),x5) -> f{AC,#}(x3,fAC(x4,x5))
     f{AC,#}(x3,x4) -> f{AC,#}(x4,x3)
     f{AC,#}(x3,fAC(x4,x5)) -> f{AC,#}(fAC(x3,x4),x5)
     f{AC,#}(x4,x3) -> f{AC,#}(x3,x4)
    DPs:
     phi#(y1,phi(y2,x)) -> phi#(fAC(y1,y2),x)
    Equations:
     fAC(fAC(x3,x4),x5) -> fAC(x3,fAC(x4,x5))
     fAC(x3,x4) -> fAC(x4,x3)
     fAC(x3,fAC(x4,x5)) -> fAC(fAC(x3,x4),x5)
     fAC(x4,x3) -> fAC(x3,x4)
    TRS:
     fAC(x,one()) -> x
     fAC(i(x),x) -> one()
     phi(one(),y1) -> y1
     phi(y1,phi(y2,x)) -> phi(fAC(y1,y2),x)
    S:
     f{AC,#}(fAC(x8,x9),x10) -> f{AC,#}(x8,x9)
     f{AC,#}(x8,fAC(x9,x10)) -> f{AC,#}(x9,x10)
    AC-DP unlabeling:
     Equations#:
      fAC(fAC(x3,x4),x5) -> fAC(x3,fAC(x4,x5))
      fAC(x3,x4) -> fAC(x4,x3)
      fAC(x3,fAC(x4,x5)) -> fAC(fAC(x3,x4),x5)
      fAC(x4,x3) -> fAC(x3,x4)
     DPs:
      phi#(y1,phi(y2,x)) -> phi#(fAC(y1,y2),x)
     Equations:
      fAC(fAC(x3,x4),x5) -> fAC(x3,fAC(x4,x5))
      fAC(x3,x4) -> fAC(x4,x3)
      fAC(x3,fAC(x4,x5)) -> fAC(fAC(x3,x4),x5)
      fAC(x4,x3) -> fAC(x3,x4)
     TRS:
      fAC(x,one()) -> x
      fAC(i(x),x) -> one()
      phi(one(),y1) -> y1
      phi(y1,phi(y2,x)) -> phi(fAC(y1,y2),x)
     S:
      fAC(fAC(x8,x9),x10) -> fAC(x8,x9)
      fAC(x8,fAC(x9,x10)) -> fAC(x9,x10)
     Usable Rule Processor:
      Equations#:
       fAC(fAC(x3,x4),x5) -> fAC(x3,fAC(x4,x5))
       fAC(x3,x4) -> fAC(x4,x3)
       fAC(x3,fAC(x4,x5)) -> fAC(fAC(x3,x4),x5)
       fAC(x4,x3) -> fAC(x3,x4)
      DPs:
       phi#(y1,phi(y2,x)) -> phi#(fAC(y1,y2),x)
      Equations:
       fAC(fAC(x3,x4),x5) -> fAC(x3,fAC(x4,x5))
       fAC(x3,x4) -> fAC(x4,x3)
       fAC(x3,fAC(x4,x5)) -> fAC(fAC(x3,x4),x5)
       fAC(x4,x3) -> fAC(x3,x4)
      TRS:
       fAC(x,one()) -> x
       fAC(i(x),x) -> one()
      S:
       fAC(fAC(x8,x9),x10) -> fAC(x8,x9)
       fAC(x8,fAC(x9,x10)) -> fAC(x9,x10)
      AC-KBO Processor:
       argument filtering:
        pi(fAC) = [0,1]
        pi(one) = []
        pi(i) = 0
        pi(phi) = [1]
        pi(phi#) = 1
       precedence:
        phi# ~ phi ~ i ~ one ~ fAC
        weight function:
         w0 = 4
         w(phi#) = w(i) = w(one) = 4
         w(phi) = w(fAC) = 2
         usable rules:
          
         problem:
          Equations#:
           fAC(fAC(x3,x4),x5) -> fAC(x3,fAC(x4,x5))
           fAC(x3,x4) -> fAC(x4,x3)
           fAC(x3,fAC(x4,x5)) -> fAC(fAC(x3,x4),x5)
           fAC(x4,x3) -> fAC(x3,x4)
          DPs:
           
          Equations:
           fAC(fAC(x3,x4),x5) -> fAC(x3,fAC(x4,x5))
           fAC(x3,x4) -> fAC(x4,x3)
           fAC(x3,fAC(x4,x5)) -> fAC(fAC(x3,x4),x5)
           fAC(x4,x3) -> fAC(x3,x4)
          TRS:
           fAC(x,one()) -> x
           fAC(i(x),x) -> one()
          S:
           fAC(fAC(x8,x9),x10) -> fAC(x8,x9)
           fAC(x8,fAC(x9,x10)) -> fAC(x9,x10)
         Qed
     
     Equations#:
      f{AC,#}(fAC(x3,x4),x5) -> f{AC,#}(x3,fAC(x4,x5))
      f{AC,#}(x3,x4) -> f{AC,#}(x4,x3)
      f{AC,#}(x3,fAC(x4,x5)) -> f{AC,#}(fAC(x3,x4),x5)
      f{AC,#}(x4,x3) -> f{AC,#}(x3,x4)
     DPs:
      f{AC,#}(x7,fAC(i(x),x)) -> f{AC,#}(x7,one())
      f{AC,#}(x6,fAC(x,one())) -> f{AC,#}(x6,x)
     Equations:
      fAC(fAC(x3,x4),x5) -> fAC(x3,fAC(x4,x5))
      fAC(x3,x4) -> fAC(x4,x3)
      fAC(x3,fAC(x4,x5)) -> fAC(fAC(x3,x4),x5)
      fAC(x4,x3) -> fAC(x3,x4)
     TRS:
      fAC(x,one()) -> x
      fAC(i(x),x) -> one()
      phi(one(),y1) -> y1
      phi(y1,phi(y2,x)) -> phi(fAC(y1,y2),x)
     S:
      f{AC,#}(fAC(x8,x9),x10) -> f{AC,#}(x8,x9)
      f{AC,#}(x8,fAC(x9,x10)) -> f{AC,#}(x9,x10)
     AC-DP unlabeling:
      Equations#:
       fAC(fAC(x3,x4),x5) -> fAC(x3,fAC(x4,x5))
       fAC(x3,x4) -> fAC(x4,x3)
       fAC(x3,fAC(x4,x5)) -> fAC(fAC(x3,x4),x5)
       fAC(x4,x3) -> fAC(x3,x4)
      DPs:
       fAC(x7,fAC(i(x),x)) -> fAC(x7,one())
       fAC(x6,fAC(x,one())) -> fAC(x6,x)
      Equations:
       fAC(fAC(x3,x4),x5) -> fAC(x3,fAC(x4,x5))
       fAC(x3,x4) -> fAC(x4,x3)
       fAC(x3,fAC(x4,x5)) -> fAC(fAC(x3,x4),x5)
       fAC(x4,x3) -> fAC(x3,x4)
      TRS:
       fAC(x,one()) -> x
       fAC(i(x),x) -> one()
       phi(one(),y1) -> y1
       phi(y1,phi(y2,x)) -> phi(fAC(y1,y2),x)
      S:
       fAC(fAC(x8,x9),x10) -> fAC(x8,x9)
       fAC(x8,fAC(x9,x10)) -> fAC(x9,x10)
      Usable Rule Processor:
       Equations#:
        fAC(fAC(x3,x4),x5) -> fAC(x3,fAC(x4,x5))
        fAC(x3,x4) -> fAC(x4,x3)
        fAC(x3,fAC(x4,x5)) -> fAC(fAC(x3,x4),x5)
        fAC(x4,x3) -> fAC(x3,x4)
       DPs:
        fAC(x7,fAC(i(x),x)) -> fAC(x7,one())
        fAC(x6,fAC(x,one())) -> fAC(x6,x)
       Equations:
        fAC(fAC(x3,x4),x5) -> fAC(x3,fAC(x4,x5))
        fAC(x3,x4) -> fAC(x4,x3)
        fAC(x3,fAC(x4,x5)) -> fAC(fAC(x3,x4),x5)
        fAC(x4,x3) -> fAC(x3,x4)
       TRS:
        fAC(x,one()) -> x
        fAC(i(x),x) -> one()
       S:
        fAC(fAC(x8,x9),x10) -> fAC(x8,x9)
        fAC(x8,fAC(x9,x10)) -> fAC(x9,x10)
       AC-KBO Processor:
        argument filtering:
         pi(fAC) = [0,1]
         pi(one) = []
         pi(i) = []
        precedence:
         i ~ one ~ fAC
         weight function:
          w0 = 2
          w(i) = w(one) = 4
          w(fAC) = 1
          usable rules:
           fAC(x,one()) -> x
           fAC(i(x),x) -> one()
          problem:
           Equations#:
            fAC(fAC(x3,x4),x5) -> fAC(x3,fAC(x4,x5))
            fAC(x3,x4) -> fAC(x4,x3)
            fAC(x3,fAC(x4,x5)) -> fAC(fAC(x3,x4),x5)
            fAC(x4,x3) -> fAC(x3,x4)
           DPs:
            
           Equations:
            fAC(fAC(x3,x4),x5) -> fAC(x3,fAC(x4,x5))
            fAC(x3,x4) -> fAC(x4,x3)
            fAC(x3,fAC(x4,x5)) -> fAC(fAC(x3,x4),x5)
            fAC(x4,x3) -> fAC(x3,x4)
           TRS:
            fAC(x,one()) -> x
            fAC(i(x),x) -> one()
           S:
            fAC(fAC(x8,x9),x10) -> fAC(x8,x9)
            fAC(x8,fAC(x9,x10)) -> fAC(x9,x10)
          Qed