MAYBE

Problem:
 plus(0(),x) -> x
 plus(s(x),y) -> s(plus(p(s(x)),y))
 times(0(),y) -> 0()
 times(s(x),y) -> plus(y,times(p(s(x)),y))
 p(s(0())) -> 0()
 p(s(s(x))) -> s(p(s(x)))
 fac(0(),x) -> x
 fac(s(x),y) -> fac(p(s(x)),times(s(x),y))
 factorial(x) -> fac(x,s(0()))

Proof:
 DP Processor:
  DPs:
   plus#(s(x),y) -> p#(s(x))
   plus#(s(x),y) -> plus#(p(s(x)),y)
   times#(s(x),y) -> p#(s(x))
   times#(s(x),y) -> times#(p(s(x)),y)
   times#(s(x),y) -> plus#(y,times(p(s(x)),y))
   p#(s(s(x))) -> p#(s(x))
   fac#(s(x),y) -> times#(s(x),y)
   fac#(s(x),y) -> p#(s(x))
   fac#(s(x),y) -> fac#(p(s(x)),times(s(x),y))
   factorial#(x) -> fac#(x,s(0()))
  TRS:
   plus(0(),x) -> x
   plus(s(x),y) -> s(plus(p(s(x)),y))
   times(0(),y) -> 0()
   times(s(x),y) -> plus(y,times(p(s(x)),y))
   p(s(0())) -> 0()
   p(s(s(x))) -> s(p(s(x)))
   fac(0(),x) -> x
   fac(s(x),y) -> fac(p(s(x)),times(s(x),y))
   factorial(x) -> fac(x,s(0()))
  Usable Rule Processor:
   DPs:
    plus#(s(x),y) -> p#(s(x))
    plus#(s(x),y) -> plus#(p(s(x)),y)
    times#(s(x),y) -> p#(s(x))
    times#(s(x),y) -> times#(p(s(x)),y)
    times#(s(x),y) -> plus#(y,times(p(s(x)),y))
    p#(s(s(x))) -> p#(s(x))
    fac#(s(x),y) -> times#(s(x),y)
    fac#(s(x),y) -> p#(s(x))
    fac#(s(x),y) -> fac#(p(s(x)),times(s(x),y))
    factorial#(x) -> fac#(x,s(0()))
   TRS:
    f12(x,y) -> x
    f12(x,y) -> y
    p(s(0())) -> 0()
    p(s(s(x))) -> s(p(s(x)))
    times(0(),y) -> 0()
    times(s(x),y) -> plus(y,times(p(s(x)),y))
    plus(0(),x) -> x
    plus(s(x),y) -> s(plus(p(s(x)),y))
   CDG Processor:
    DPs:
     plus#(s(x),y) -> p#(s(x))
     plus#(s(x),y) -> plus#(p(s(x)),y)
     times#(s(x),y) -> p#(s(x))
     times#(s(x),y) -> times#(p(s(x)),y)
     times#(s(x),y) -> plus#(y,times(p(s(x)),y))
     p#(s(s(x))) -> p#(s(x))
     fac#(s(x),y) -> times#(s(x),y)
     fac#(s(x),y) -> p#(s(x))
     fac#(s(x),y) -> fac#(p(s(x)),times(s(x),y))
     factorial#(x) -> fac#(x,s(0()))
    TRS:
     f12(x,y) -> x
     f12(x,y) -> y
     p(s(0())) -> 0()
     p(s(s(x))) -> s(p(s(x)))
     times(0(),y) -> 0()
     times(s(x),y) -> plus(y,times(p(s(x)),y))
     plus(0(),x) -> x
     plus(s(x),y) -> s(plus(p(s(x)),y))
    graph:
     factorial#(x) -> fac#(x,s(0())) ->
     fac#(s(x),y) -> times#(s(x),y)
     factorial#(x) -> fac#(x,s(0())) -> fac#(s(x),y) -> p#(s(x))
     factorial#(x) -> fac#(x,s(0())) ->
     fac#(s(x),y) -> fac#(p(s(x)),times(s(x),y))
     fac#(s(x),y) -> fac#(p(s(x)),times(s(x),y)) ->
     fac#(s(x),y) -> times#(s(x),y)
     fac#(s(x),y) -> fac#(p(s(x)),times(s(x),y)) ->
     fac#(s(x),y) -> p#(s(x))
     fac#(s(x),y) -> fac#(p(s(x)),times(s(x),y)) ->
     fac#(s(x),y) -> fac#(p(s(x)),times(s(x),y))
     fac#(s(x),y) -> times#(s(x),y) -> times#(s(x),y) -> p#(s(x))
     fac#(s(x),y) -> times#(s(x),y) ->
     times#(s(x),y) -> times#(p(s(x)),y)
     fac#(s(x),y) -> times#(s(x),y) ->
     times#(s(x),y) -> plus#(y,times(p(s(x)),y))
     fac#(s(x),y) -> p#(s(x)) -> p#(s(s(x))) -> p#(s(x))
     times#(s(x),y) -> times#(p(s(x)),y) ->
     times#(s(x),y) -> p#(s(x))
     times#(s(x),y) -> times#(p(s(x)),y) ->
     times#(s(x),y) -> times#(p(s(x)),y)
     times#(s(x),y) -> times#(p(s(x)),y) ->
     times#(s(x),y) -> plus#(y,times(p(s(x)),y))
     times#(s(x),y) -> p#(s(x)) ->
     p#(s(s(x))) -> p#(s(x))
     times#(s(x),y) -> plus#(y,times(p(s(x)),y)) ->
     plus#(s(x),y) -> p#(s(x))
     times#(s(x),y) -> plus#(y,times(p(s(x)),y)) ->
     plus#(s(x),y) -> plus#(p(s(x)),y)
     p#(s(s(x))) -> p#(s(x)) -> p#(s(s(x))) -> p#(s(x))
     plus#(s(x),y) -> p#(s(x)) -> p#(s(s(x))) -> p#(s(x))
     plus#(s(x),y) -> plus#(p(s(x)),y) ->
     plus#(s(x),y) -> p#(s(x))
     plus#(s(x),y) -> plus#(p(s(x)),y) -> plus#(s(x),y) -> plus#(p(s(x)),y)
    Restore Modifier:
     DPs:
      plus#(s(x),y) -> p#(s(x))
      plus#(s(x),y) -> plus#(p(s(x)),y)
      times#(s(x),y) -> p#(s(x))
      times#(s(x),y) -> times#(p(s(x)),y)
      times#(s(x),y) -> plus#(y,times(p(s(x)),y))
      p#(s(s(x))) -> p#(s(x))
      fac#(s(x),y) -> times#(s(x),y)
      fac#(s(x),y) -> p#(s(x))
      fac#(s(x),y) -> fac#(p(s(x)),times(s(x),y))
      factorial#(x) -> fac#(x,s(0()))
     TRS:
      plus(0(),x) -> x
      plus(s(x),y) -> s(plus(p(s(x)),y))
      times(0(),y) -> 0()
      times(s(x),y) -> plus(y,times(p(s(x)),y))
      p(s(0())) -> 0()
      p(s(s(x))) -> s(p(s(x)))
      fac(0(),x) -> x
      fac(s(x),y) -> fac(p(s(x)),times(s(x),y))
      factorial(x) -> fac(x,s(0()))
     SCC Processor:
      #sccs: 4
      #rules: 4
      #arcs: 20/100
      DPs:
       fac#(s(x),y) -> fac#(p(s(x)),times(s(x),y))
      TRS:
       plus(0(),x) -> x
       plus(s(x),y) -> s(plus(p(s(x)),y))
       times(0(),y) -> 0()
       times(s(x),y) -> plus(y,times(p(s(x)),y))
       p(s(0())) -> 0()
       p(s(s(x))) -> s(p(s(x)))
       fac(0(),x) -> x
       fac(s(x),y) -> fac(p(s(x)),times(s(x),y))
       factorial(x) -> fac(x,s(0()))
      Open
      
      DPs:
       times#(s(x),y) -> times#(p(s(x)),y)
      TRS:
       plus(0(),x) -> x
       plus(s(x),y) -> s(plus(p(s(x)),y))
       times(0(),y) -> 0()
       times(s(x),y) -> plus(y,times(p(s(x)),y))
       p(s(0())) -> 0()
       p(s(s(x))) -> s(p(s(x)))
       fac(0(),x) -> x
       fac(s(x),y) -> fac(p(s(x)),times(s(x),y))
       factorial(x) -> fac(x,s(0()))
      Open
      
      DPs:
       plus#(s(x),y) -> plus#(p(s(x)),y)
      TRS:
       plus(0(),x) -> x
       plus(s(x),y) -> s(plus(p(s(x)),y))
       times(0(),y) -> 0()
       times(s(x),y) -> plus(y,times(p(s(x)),y))
       p(s(0())) -> 0()
       p(s(s(x))) -> s(p(s(x)))
       fac(0(),x) -> x
       fac(s(x),y) -> fac(p(s(x)),times(s(x),y))
       factorial(x) -> fac(x,s(0()))
      Open
      
      DPs:
       p#(s(s(x))) -> p#(s(x))
      TRS:
       plus(0(),x) -> x
       plus(s(x),y) -> s(plus(p(s(x)),y))
       times(0(),y) -> 0()
       times(s(x),y) -> plus(y,times(p(s(x)),y))
       p(s(0())) -> 0()
       p(s(s(x))) -> s(p(s(x)))
       fac(0(),x) -> x
       fac(s(x),y) -> fac(p(s(x)),times(s(x),y))
       factorial(x) -> fac(x,s(0()))
      Open