YES
Time: 0.362

Problem:
 Equations:
  pAC(pAC(x3,x4),x5) -> pAC(x3,pAC(x4,x5))
  pAC(x3,x4) -> pAC(x4,x3)
  pAC(x3,pAC(x4,x5)) -> pAC(pAC(x3,x4),x5)
  pAC(x4,x3) -> pAC(x3,x4)
 TRS:
  pAC(x,zero()) -> x
  pAC(i(x),x) -> zero()
  f(zero()) -> zero()
  pAC(f(x),f(y)) -> f(pAC(x,y))
  g(zero()) -> zero()
  pAC(g(x),g(y)) -> g(pAC(x,y))
  m(one(),x) -> x
  m(x,one()) -> x
  m(z,pAC(x,y)) -> pAC(m(z,x),m(z,y))
  m(pAC(x,y),z) -> pAC(m(x,z),m(y,z))

Proof:
 DP Processor:
  Equations#:
   p{AC,#}(pAC(x3,x4),x5) -> p{AC,#}(x3,pAC(x4,x5))
   p{AC,#}(x3,x4) -> p{AC,#}(x4,x3)
   p{AC,#}(x3,pAC(x4,x5)) -> p{AC,#}(pAC(x3,x4),x5)
   p{AC,#}(x4,x3) -> p{AC,#}(x3,x4)
  DPs:
   p{AC,#}(f(x),f(y)) -> p{AC,#}(x,y)
   p{AC,#}(f(x),f(y)) -> f#(pAC(x,y))
   p{AC,#}(g(x),g(y)) -> p{AC,#}(x,y)
   p{AC,#}(g(x),g(y)) -> g#(pAC(x,y))
   m#(z,pAC(x,y)) -> m#(z,y)
   m#(z,pAC(x,y)) -> m#(z,x)
   m#(z,pAC(x,y)) -> p{AC,#}(m(z,x),m(z,y))
   m#(pAC(x,y),z) -> m#(y,z)
   m#(pAC(x,y),z) -> m#(x,z)
   m#(pAC(x,y),z) -> p{AC,#}(m(x,z),m(y,z))
   p{AC,#}(x6,pAC(x,zero())) -> p{AC,#}(x6,x)
   p{AC,#}(x7,pAC(i(x),x)) -> p{AC,#}(x7,zero())
   p{AC,#}(x8,pAC(f(x),f(y))) -> p{AC,#}(x,y)
   p{AC,#}(x8,pAC(f(x),f(y))) -> f#(pAC(x,y))
   p{AC,#}(x8,pAC(f(x),f(y))) -> p{AC,#}(x8,f(pAC(x,y)))
   p{AC,#}(x9,pAC(g(x),g(y))) -> p{AC,#}(x,y)
   p{AC,#}(x9,pAC(g(x),g(y))) -> g#(pAC(x,y))
   p{AC,#}(x9,pAC(g(x),g(y))) -> p{AC,#}(x9,g(pAC(x,y)))
  Equations:
   pAC(pAC(x3,x4),x5) -> pAC(x3,pAC(x4,x5))
   pAC(x3,x4) -> pAC(x4,x3)
   pAC(x3,pAC(x4,x5)) -> pAC(pAC(x3,x4),x5)
   pAC(x4,x3) -> pAC(x3,x4)
  TRS:
   pAC(x,zero()) -> x
   pAC(i(x),x) -> zero()
   f(zero()) -> zero()
   pAC(f(x),f(y)) -> f(pAC(x,y))
   g(zero()) -> zero()
   pAC(g(x),g(y)) -> g(pAC(x,y))
   m(one(),x) -> x
   m(x,one()) -> x
   m(z,pAC(x,y)) -> pAC(m(z,x),m(z,y))
   m(pAC(x,y),z) -> pAC(m(x,z),m(y,z))
  S:
   p{AC,#}(pAC(x10,x11),x12) -> p{AC,#}(x10,x11)
   p{AC,#}(x10,pAC(x11,x12)) -> p{AC,#}(x11,x12)
  AC-EDG Processor:
   Equations#:
    p{AC,#}(pAC(x3,x4),x5) -> p{AC,#}(x3,pAC(x4,x5))
    p{AC,#}(x3,x4) -> p{AC,#}(x4,x3)
    p{AC,#}(x3,pAC(x4,x5)) -> p{AC,#}(pAC(x3,x4),x5)
    p{AC,#}(x4,x3) -> p{AC,#}(x3,x4)
   DPs:
    p{AC,#}(f(x),f(y)) -> p{AC,#}(x,y)
    p{AC,#}(f(x),f(y)) -> f#(pAC(x,y))
    p{AC,#}(g(x),g(y)) -> p{AC,#}(x,y)
    p{AC,#}(g(x),g(y)) -> g#(pAC(x,y))
    m#(z,pAC(x,y)) -> m#(z,y)
    m#(z,pAC(x,y)) -> m#(z,x)
    m#(z,pAC(x,y)) -> p{AC,#}(m(z,x),m(z,y))
    m#(pAC(x,y),z) -> m#(y,z)
    m#(pAC(x,y),z) -> m#(x,z)
    m#(pAC(x,y),z) -> p{AC,#}(m(x,z),m(y,z))
    p{AC,#}(x6,pAC(x,zero())) -> p{AC,#}(x6,x)
    p{AC,#}(x7,pAC(i(x),x)) -> p{AC,#}(x7,zero())
    p{AC,#}(x8,pAC(f(x),f(y))) -> p{AC,#}(x,y)
    p{AC,#}(x8,pAC(f(x),f(y))) -> f#(pAC(x,y))
    p{AC,#}(x8,pAC(f(x),f(y))) -> p{AC,#}(x8,f(pAC(x,y)))
    p{AC,#}(x9,pAC(g(x),g(y))) -> p{AC,#}(x,y)
    p{AC,#}(x9,pAC(g(x),g(y))) -> g#(pAC(x,y))
    p{AC,#}(x9,pAC(g(x),g(y))) -> p{AC,#}(x9,g(pAC(x,y)))
   Equations:
    pAC(pAC(x3,x4),x5) -> pAC(x3,pAC(x4,x5))
    pAC(x3,x4) -> pAC(x4,x3)
    pAC(x3,pAC(x4,x5)) -> pAC(pAC(x3,x4),x5)
    pAC(x4,x3) -> pAC(x3,x4)
   TRS:
    pAC(x,zero()) -> x
    pAC(i(x),x) -> zero()
    f(zero()) -> zero()
    pAC(f(x),f(y)) -> f(pAC(x,y))
    g(zero()) -> zero()
    pAC(g(x),g(y)) -> g(pAC(x,y))
    m(one(),x) -> x
    m(x,one()) -> x
    m(z,pAC(x,y)) -> pAC(m(z,x),m(z,y))
    m(pAC(x,y),z) -> pAC(m(x,z),m(y,z))
   S:
    p{AC,#}(pAC(x10,x11),x12) -> p{AC,#}(x10,x11)
    p{AC,#}(x10,pAC(x11,x12)) -> p{AC,#}(x11,x12)
   SCC Processor:
    #sccs: 2
    #rules: 12
    #arcs: 144/324
    Equations#:
     p{AC,#}(pAC(x3,x4),x5) -> p{AC,#}(x3,pAC(x4,x5))
     p{AC,#}(x3,x4) -> p{AC,#}(x4,x3)
     p{AC,#}(x3,pAC(x4,x5)) -> p{AC,#}(pAC(x3,x4),x5)
     p{AC,#}(x4,x3) -> p{AC,#}(x3,x4)
    DPs:
     m#(pAC(x,y),z) -> m#(y,z)
     m#(pAC(x,y),z) -> m#(x,z)
     m#(z,pAC(x,y)) -> m#(z,x)
     m#(z,pAC(x,y)) -> m#(z,y)
    Equations:
     pAC(pAC(x3,x4),x5) -> pAC(x3,pAC(x4,x5))
     pAC(x3,x4) -> pAC(x4,x3)
     pAC(x3,pAC(x4,x5)) -> pAC(pAC(x3,x4),x5)
     pAC(x4,x3) -> pAC(x3,x4)
    TRS:
     pAC(x,zero()) -> x
     pAC(i(x),x) -> zero()
     f(zero()) -> zero()
     pAC(f(x),f(y)) -> f(pAC(x,y))
     g(zero()) -> zero()
     pAC(g(x),g(y)) -> g(pAC(x,y))
     m(one(),x) -> x
     m(x,one()) -> x
     m(z,pAC(x,y)) -> pAC(m(z,x),m(z,y))
     m(pAC(x,y),z) -> pAC(m(x,z),m(y,z))
    S:
     p{AC,#}(pAC(x10,x11),x12) -> p{AC,#}(x10,x11)
     p{AC,#}(x10,pAC(x11,x12)) -> p{AC,#}(x11,x12)
    AC-DP unlabeling:
     Equations#:
      pAC(pAC(x3,x4),x5) -> pAC(x3,pAC(x4,x5))
      pAC(x3,x4) -> pAC(x4,x3)
      pAC(x3,pAC(x4,x5)) -> pAC(pAC(x3,x4),x5)
      pAC(x4,x3) -> pAC(x3,x4)
     DPs:
      m#(pAC(x,y),z) -> m#(y,z)
      m#(pAC(x,y),z) -> m#(x,z)
      m#(z,pAC(x,y)) -> m#(z,x)
      m#(z,pAC(x,y)) -> m#(z,y)
     Equations:
      pAC(pAC(x3,x4),x5) -> pAC(x3,pAC(x4,x5))
      pAC(x3,x4) -> pAC(x4,x3)
      pAC(x3,pAC(x4,x5)) -> pAC(pAC(x3,x4),x5)
      pAC(x4,x3) -> pAC(x3,x4)
     TRS:
      pAC(x,zero()) -> x
      pAC(i(x),x) -> zero()
      f(zero()) -> zero()
      pAC(f(x),f(y)) -> f(pAC(x,y))
      g(zero()) -> zero()
      pAC(g(x),g(y)) -> g(pAC(x,y))
      m(one(),x) -> x
      m(x,one()) -> x
      m(z,pAC(x,y)) -> pAC(m(z,x),m(z,y))
      m(pAC(x,y),z) -> pAC(m(x,z),m(y,z))
     S:
      pAC(pAC(x10,x11),x12) -> pAC(x10,x11)
      pAC(x10,pAC(x11,x12)) -> pAC(x11,x12)
     Usable Rule Processor:
      Equations#:
       pAC(pAC(x3,x4),x5) -> pAC(x3,pAC(x4,x5))
       pAC(x3,x4) -> pAC(x4,x3)
       pAC(x3,pAC(x4,x5)) -> pAC(pAC(x3,x4),x5)
       pAC(x4,x3) -> pAC(x3,x4)
      DPs:
       m#(pAC(x,y),z) -> m#(y,z)
       m#(pAC(x,y),z) -> m#(x,z)
       m#(z,pAC(x,y)) -> m#(z,x)
       m#(z,pAC(x,y)) -> m#(z,y)
      Equations:
       pAC(pAC(x3,x4),x5) -> pAC(x3,pAC(x4,x5))
       pAC(x3,x4) -> pAC(x4,x3)
       pAC(x3,pAC(x4,x5)) -> pAC(pAC(x3,x4),x5)
       pAC(x4,x3) -> pAC(x3,x4)
      TRS:
       
      S:
       pAC(pAC(x10,x11),x12) -> pAC(x10,x11)
       pAC(x10,pAC(x11,x12)) -> pAC(x11,x12)
      AC-RPO Processor:
       argument filtering:
        pi(pAC) = [0,1]
        pi(m#) = [1]
       precedence:
        pAC > m#
       status:
        m#:mul  
       problem:
        Equations#:
         pAC(pAC(x3,x4),x5) -> pAC(x3,pAC(x4,x5))
         pAC(x3,x4) -> pAC(x4,x3)
         pAC(x3,pAC(x4,x5)) -> pAC(pAC(x3,x4),x5)
         pAC(x4,x3) -> pAC(x3,x4)
        DPs:
         m#(pAC(x,y),z) -> m#(y,z)
         m#(pAC(x,y),z) -> m#(x,z)
        Equations:
         pAC(pAC(x3,x4),x5) -> pAC(x3,pAC(x4,x5))
         pAC(x3,x4) -> pAC(x4,x3)
         pAC(x3,pAC(x4,x5)) -> pAC(pAC(x3,x4),x5)
         pAC(x4,x3) -> pAC(x3,x4)
        TRS:
         
        S:
         pAC(pAC(x10,x11),x12) -> pAC(x10,x11)
         pAC(x10,pAC(x11,x12)) -> pAC(x11,x12)
       Restore Modifier:
        Equations#:
         pAC(pAC(x3,x4),x5) -> pAC(x3,pAC(x4,x5))
         pAC(x3,x4) -> pAC(x4,x3)
         pAC(x3,pAC(x4,x5)) -> pAC(pAC(x3,x4),x5)
         pAC(x4,x3) -> pAC(x3,x4)
        DPs:
         m#(pAC(x,y),z) -> m#(y,z)
         m#(pAC(x,y),z) -> m#(x,z)
        Equations:
         pAC(pAC(x3,x4),x5) -> pAC(x3,pAC(x4,x5))
         pAC(x3,x4) -> pAC(x4,x3)
         pAC(x3,pAC(x4,x5)) -> pAC(pAC(x3,x4),x5)
         pAC(x4,x3) -> pAC(x3,x4)
        TRS:
         
        S:
         pAC(pAC(x10,x11),x12) -> pAC(x10,x11)
         pAC(x10,pAC(x11,x12)) -> pAC(x11,x12)
        AC-DP unlabeling:
         Equations#:
          pAC(pAC(x3,x4),x5) -> pAC(x3,pAC(x4,x5))
          pAC(x3,x4) -> pAC(x4,x3)
          pAC(x3,pAC(x4,x5)) -> pAC(pAC(x3,x4),x5)
          pAC(x4,x3) -> pAC(x3,x4)
         DPs:
          m#(pAC(x,y),z) -> m#(y,z)
          m#(pAC(x,y),z) -> m#(x,z)
         Equations:
          pAC(pAC(x3,x4),x5) -> pAC(x3,pAC(x4,x5))
          pAC(x3,x4) -> pAC(x4,x3)
          pAC(x3,pAC(x4,x5)) -> pAC(pAC(x3,x4),x5)
          pAC(x4,x3) -> pAC(x3,x4)
         TRS:
          
         S:
          pAC(pAC(x10,x11),x12) -> pAC(x10,x11)
          pAC(x10,pAC(x11,x12)) -> pAC(x11,x12)
         AC-RPO Processor:
          argument filtering:
           pi(pAC) = [0,1]
           pi(m#) = [0]
          precedence:
           pAC > m#
          status:
           m#:lex  
          problem:
           Equations#:
            pAC(pAC(x3,x4),x5) -> pAC(x3,pAC(x4,x5))
            pAC(x3,x4) -> pAC(x4,x3)
            pAC(x3,pAC(x4,x5)) -> pAC(pAC(x3,x4),x5)
            pAC(x4,x3) -> pAC(x3,x4)
           DPs:
            
           Equations:
            pAC(pAC(x3,x4),x5) -> pAC(x3,pAC(x4,x5))
            pAC(x3,x4) -> pAC(x4,x3)
            pAC(x3,pAC(x4,x5)) -> pAC(pAC(x3,x4),x5)
            pAC(x4,x3) -> pAC(x3,x4)
           TRS:
            
           S:
            pAC(pAC(x10,x11),x12) -> pAC(x10,x11)
            pAC(x10,pAC(x11,x12)) -> pAC(x11,x12)
          Qed
    
    Equations#:
     p{AC,#}(pAC(x3,x4),x5) -> p{AC,#}(x3,pAC(x4,x5))
     p{AC,#}(x3,x4) -> p{AC,#}(x4,x3)
     p{AC,#}(x3,pAC(x4,x5)) -> p{AC,#}(pAC(x3,x4),x5)
     p{AC,#}(x4,x3) -> p{AC,#}(x3,x4)
    DPs:
     p{AC,#}(x9,pAC(g(x),g(y))) -> p{AC,#}(x9,g(pAC(x,y)))
     p{AC,#}(x9,pAC(g(x),g(y))) -> p{AC,#}(x,y)
     p{AC,#}(x8,pAC(f(x),f(y))) -> p{AC,#}(x8,f(pAC(x,y)))
     p{AC,#}(x8,pAC(f(x),f(y))) -> p{AC,#}(x,y)
     p{AC,#}(x7,pAC(i(x),x)) -> p{AC,#}(x7,zero())
     p{AC,#}(x6,pAC(x,zero())) -> p{AC,#}(x6,x)
     p{AC,#}(g(x),g(y)) -> p{AC,#}(x,y)
     p{AC,#}(f(x),f(y)) -> p{AC,#}(x,y)
    Equations:
     pAC(pAC(x3,x4),x5) -> pAC(x3,pAC(x4,x5))
     pAC(x3,x4) -> pAC(x4,x3)
     pAC(x3,pAC(x4,x5)) -> pAC(pAC(x3,x4),x5)
     pAC(x4,x3) -> pAC(x3,x4)
    TRS:
     pAC(x,zero()) -> x
     pAC(i(x),x) -> zero()
     f(zero()) -> zero()
     pAC(f(x),f(y)) -> f(pAC(x,y))
     g(zero()) -> zero()
     pAC(g(x),g(y)) -> g(pAC(x,y))
     m(one(),x) -> x
     m(x,one()) -> x
     m(z,pAC(x,y)) -> pAC(m(z,x),m(z,y))
     m(pAC(x,y),z) -> pAC(m(x,z),m(y,z))
    S:
     p{AC,#}(pAC(x10,x11),x12) -> p{AC,#}(x10,x11)
     p{AC,#}(x10,pAC(x11,x12)) -> p{AC,#}(x11,x12)
    AC-DP unlabeling:
     Equations#:
      pAC(pAC(x3,x4),x5) -> pAC(x3,pAC(x4,x5))
      pAC(x3,x4) -> pAC(x4,x3)
      pAC(x3,pAC(x4,x5)) -> pAC(pAC(x3,x4),x5)
      pAC(x4,x3) -> pAC(x3,x4)
     DPs:
      pAC(x9,pAC(g(x),g(y))) -> pAC(x9,g(pAC(x,y)))
      pAC(x9,pAC(g(x),g(y))) -> pAC(x,y)
      pAC(x8,pAC(f(x),f(y))) -> pAC(x8,f(pAC(x,y)))
      pAC(x8,pAC(f(x),f(y))) -> pAC(x,y)
      pAC(x7,pAC(i(x),x)) -> pAC(x7,zero())
      pAC(x6,pAC(x,zero())) -> pAC(x6,x)
      pAC(g(x),g(y)) -> pAC(x,y)
      pAC(f(x),f(y)) -> pAC(x,y)
     Equations:
      pAC(pAC(x3,x4),x5) -> pAC(x3,pAC(x4,x5))
      pAC(x3,x4) -> pAC(x4,x3)
      pAC(x3,pAC(x4,x5)) -> pAC(pAC(x3,x4),x5)
      pAC(x4,x3) -> pAC(x3,x4)
     TRS:
      pAC(x,zero()) -> x
      pAC(i(x),x) -> zero()
      f(zero()) -> zero()
      pAC(f(x),f(y)) -> f(pAC(x,y))
      g(zero()) -> zero()
      pAC(g(x),g(y)) -> g(pAC(x,y))
      m(one(),x) -> x
      m(x,one()) -> x
      m(z,pAC(x,y)) -> pAC(m(z,x),m(z,y))
      m(pAC(x,y),z) -> pAC(m(x,z),m(y,z))
     S:
      pAC(pAC(x10,x11),x12) -> pAC(x10,x11)
      pAC(x10,pAC(x11,x12)) -> pAC(x11,x12)
     Usable Rule Processor:
      Equations#:
       pAC(pAC(x3,x4),x5) -> pAC(x3,pAC(x4,x5))
       pAC(x3,x4) -> pAC(x4,x3)
       pAC(x3,pAC(x4,x5)) -> pAC(pAC(x3,x4),x5)
       pAC(x4,x3) -> pAC(x3,x4)
      DPs:
       pAC(x9,pAC(g(x),g(y))) -> pAC(x9,g(pAC(x,y)))
       pAC(x9,pAC(g(x),g(y))) -> pAC(x,y)
       pAC(x8,pAC(f(x),f(y))) -> pAC(x8,f(pAC(x,y)))
       pAC(x8,pAC(f(x),f(y))) -> pAC(x,y)
       pAC(x7,pAC(i(x),x)) -> pAC(x7,zero())
       pAC(x6,pAC(x,zero())) -> pAC(x6,x)
       pAC(g(x),g(y)) -> pAC(x,y)
       pAC(f(x),f(y)) -> pAC(x,y)
      Equations:
       pAC(pAC(x3,x4),x5) -> pAC(x3,pAC(x4,x5))
       pAC(x3,x4) -> pAC(x4,x3)
       pAC(x3,pAC(x4,x5)) -> pAC(pAC(x3,x4),x5)
       pAC(x4,x3) -> pAC(x3,x4)
      TRS:
       pAC(x,zero()) -> x
       pAC(i(x),x) -> zero()
       pAC(f(x),f(y)) -> f(pAC(x,y))
       pAC(g(x),g(y)) -> g(pAC(x,y))
       f(zero()) -> zero()
       g(zero()) -> zero()
      S:
       pAC(pAC(x10,x11),x12) -> pAC(x10,x11)
       pAC(x10,pAC(x11,x12)) -> pAC(x11,x12)
      AC-RPO Processor:
       argument filtering:
        pi(pAC) = [0,1]
        pi(zero) = []
        pi(i) = []
        pi(f) = [0]
        pi(g) = [0]
       precedence:
        i > pAC > f > zero > g
       status:
        
       problem:
        Equations#:
         pAC(pAC(x3,x4),x5) -> pAC(x3,pAC(x4,x5))
         pAC(x3,x4) -> pAC(x4,x3)
         pAC(x3,pAC(x4,x5)) -> pAC(pAC(x3,x4),x5)
         pAC(x4,x3) -> pAC(x3,x4)
        DPs:
         
        Equations:
         pAC(pAC(x3,x4),x5) -> pAC(x3,pAC(x4,x5))
         pAC(x3,x4) -> pAC(x4,x3)
         pAC(x3,pAC(x4,x5)) -> pAC(pAC(x3,x4),x5)
         pAC(x4,x3) -> pAC(x3,x4)
        TRS:
         pAC(x,zero()) -> x
         pAC(i(x),x) -> zero()
         pAC(f(x),f(y)) -> f(pAC(x,y))
         pAC(g(x),g(y)) -> g(pAC(x,y))
         f(zero()) -> zero()
         g(zero()) -> zero()
        S:
         pAC(pAC(x10,x11),x12) -> pAC(x10,x11)
         pAC(x10,pAC(x11,x12)) -> pAC(x11,x12)
       Qed