YES

Problem:
 fac(s(x)) -> *(fac(p(s(x))),s(x))
 p(s(0())) -> 0()
 p(s(s(x))) -> s(p(s(x)))

Proof:
 DP Processor:
  DPs:
   fac#(s(x)) -> p#(s(x))
   fac#(s(x)) -> fac#(p(s(x)))
   p#(s(s(x))) -> p#(s(x))
  TRS:
   fac(s(x)) -> *(fac(p(s(x))),s(x))
   p(s(0())) -> 0()
   p(s(s(x))) -> s(p(s(x)))
  Matrix Interpretation Processor: dim=2
   
   interpretation:
    [p#](x0) = [0 1]x0,
    
    [fac#](x0) = [2 1]x0,
    
          [0]
    [0] = [0],
    
                  [0 0]     [2 0]     [2]
    [*](x0, x1) = [0 1]x0 + [0 0]x1 + [0],
    
              [0 1]  
    [p](x0) = [0 3]x0,
    
                [2 0]     [3]
    [fac](x0) = [2 1]x0 + [2],
    
              [2 0]     [1]
    [s](x0) = [1 0]x0 + [0]
   orientation:
    fac#(s(x)) = [5 0]x + [2] >= [1 0]x = p#(s(x))
    
    fac#(s(x)) = [5 0]x + [2] >= [5 0]x = fac#(p(s(x)))
    
    p#(s(s(x))) = [2 0]x + [1] >= [1 0]x = p#(s(x))
    
                [4 0]    [5]    [4 0]    [4]                       
    fac(s(x)) = [5 0]x + [4] >= [5 0]x + [2] = *(fac(p(s(x))),s(x))
    
                [0]    [0]      
    p(s(0())) = [0] >= [0] = 0()
    
                 [2 0]    [1]    [2 0]    [1]             
    p(s(s(x))) = [6 0]x + [3] >= [1 0]x + [0] = s(p(s(x)))
   problem:
    DPs:
     
    TRS:
     fac(s(x)) -> *(fac(p(s(x))),s(x))
     p(s(0())) -> 0()
     p(s(s(x))) -> s(p(s(x)))
   Qed