MAYBE

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

Proof:
 DP Processor:
  DPs:
   plus#(x,s(y)) -> plus#(x,y)
   times#(s(x),y) -> times#(x,y)
   times#(s(x),y) -> plus#(times(x,y),y)
   p#(s(s(x))) -> p#(s(x))
   fac#(s(x)) -> p#(s(x))
   fac#(s(x)) -> fac#(p(s(x)))
   fac#(s(x)) -> times#(fac(p(s(x))),s(x))
  TRS:
   plus(x,0()) -> x
   plus(x,s(y)) -> s(plus(x,y))
   times(0(),y) -> 0()
   times(x,0()) -> 0()
   times(s(x),y) -> plus(times(x,y),y)
   p(s(s(x))) -> s(p(s(x)))
   p(s(0())) -> 0()
   fac(s(x)) -> times(fac(p(s(x))),s(x))
  TDG Processor:
   DPs:
    plus#(x,s(y)) -> plus#(x,y)
    times#(s(x),y) -> times#(x,y)
    times#(s(x),y) -> plus#(times(x,y),y)
    p#(s(s(x))) -> p#(s(x))
    fac#(s(x)) -> p#(s(x))
    fac#(s(x)) -> fac#(p(s(x)))
    fac#(s(x)) -> times#(fac(p(s(x))),s(x))
   TRS:
    plus(x,0()) -> x
    plus(x,s(y)) -> s(plus(x,y))
    times(0(),y) -> 0()
    times(x,0()) -> 0()
    times(s(x),y) -> plus(times(x,y),y)
    p(s(s(x))) -> s(p(s(x)))
    p(s(0())) -> 0()
    fac(s(x)) -> times(fac(p(s(x))),s(x))
   graph:
    fac#(s(x)) -> fac#(p(s(x))) ->
    fac#(s(x)) -> times#(fac(p(s(x))),s(x))
    fac#(s(x)) -> fac#(p(s(x))) -> fac#(s(x)) -> fac#(p(s(x)))
    fac#(s(x)) -> fac#(p(s(x))) -> fac#(s(x)) -> p#(s(x))
    fac#(s(x)) -> p#(s(x)) -> p#(s(s(x))) -> p#(s(x))
    fac#(s(x)) -> times#(fac(p(s(x))),s(x)) ->
    times#(s(x),y) -> plus#(times(x,y),y)
    fac#(s(x)) -> times#(fac(p(s(x))),s(x)) -> times#(s(x),y) -> times#(x,y)
    p#(s(s(x))) -> p#(s(x)) -> p#(s(s(x))) -> p#(s(x))
    times#(s(x),y) -> times#(x,y) ->
    times#(s(x),y) -> plus#(times(x,y),y)
    times#(s(x),y) -> times#(x,y) ->
    times#(s(x),y) -> times#(x,y)
    times#(s(x),y) -> plus#(times(x,y),y) -> plus#(x,s(y)) -> plus#(x,y)
    plus#(x,s(y)) -> plus#(x,y) -> plus#(x,s(y)) -> plus#(x,y)
   SCC Processor:
    #sccs: 4
    #rules: 4
    #arcs: 11/49
    DPs:
     fac#(s(x)) -> fac#(p(s(x)))
    TRS:
     plus(x,0()) -> x
     plus(x,s(y)) -> s(plus(x,y))
     times(0(),y) -> 0()
     times(x,0()) -> 0()
     times(s(x),y) -> plus(times(x,y),y)
     p(s(s(x))) -> s(p(s(x)))
     p(s(0())) -> 0()
     fac(s(x)) -> times(fac(p(s(x))),s(x))
    Open
    
    DPs:
     times#(s(x),y) -> times#(x,y)
    TRS:
     plus(x,0()) -> x
     plus(x,s(y)) -> s(plus(x,y))
     times(0(),y) -> 0()
     times(x,0()) -> 0()
     times(s(x),y) -> plus(times(x,y),y)
     p(s(s(x))) -> s(p(s(x)))
     p(s(0())) -> 0()
     fac(s(x)) -> times(fac(p(s(x))),s(x))
    Subterm Criterion Processor:
     simple projection:
      pi(times#) = 0
     problem:
      DPs:
       
      TRS:
       plus(x,0()) -> x
       plus(x,s(y)) -> s(plus(x,y))
       times(0(),y) -> 0()
       times(x,0()) -> 0()
       times(s(x),y) -> plus(times(x,y),y)
       p(s(s(x))) -> s(p(s(x)))
       p(s(0())) -> 0()
       fac(s(x)) -> times(fac(p(s(x))),s(x))
     Qed
    
    DPs:
     plus#(x,s(y)) -> plus#(x,y)
    TRS:
     plus(x,0()) -> x
     plus(x,s(y)) -> s(plus(x,y))
     times(0(),y) -> 0()
     times(x,0()) -> 0()
     times(s(x),y) -> plus(times(x,y),y)
     p(s(s(x))) -> s(p(s(x)))
     p(s(0())) -> 0()
     fac(s(x)) -> times(fac(p(s(x))),s(x))
    Subterm Criterion Processor:
     simple projection:
      pi(plus#) = 1
     problem:
      DPs:
       
      TRS:
       plus(x,0()) -> x
       plus(x,s(y)) -> s(plus(x,y))
       times(0(),y) -> 0()
       times(x,0()) -> 0()
       times(s(x),y) -> plus(times(x,y),y)
       p(s(s(x))) -> s(p(s(x)))
       p(s(0())) -> 0()
       fac(s(x)) -> times(fac(p(s(x))),s(x))
     Qed
    
    DPs:
     p#(s(s(x))) -> p#(s(x))
    TRS:
     plus(x,0()) -> x
     plus(x,s(y)) -> s(plus(x,y))
     times(0(),y) -> 0()
     times(x,0()) -> 0()
     times(s(x),y) -> plus(times(x,y),y)
     p(s(s(x))) -> s(p(s(x)))
     p(s(0())) -> 0()
     fac(s(x)) -> times(fac(p(s(x))),s(x))
    Subterm Criterion Processor:
     simple projection:
      pi(p#) = 0
     problem:
      DPs:
       
      TRS:
       plus(x,0()) -> x
       plus(x,s(y)) -> s(plus(x,y))
       times(0(),y) -> 0()
       times(x,0()) -> 0()
       times(s(x),y) -> plus(times(x,y),y)
       p(s(s(x))) -> s(p(s(x)))
       p(s(0())) -> 0()
       fac(s(x)) -> times(fac(p(s(x))),s(x))
     Qed