YES

Problem:
 a__f(0()) -> cons(0(),f(s(0())))
 a__f(s(0())) -> a__f(a__p(s(0())))
 a__p(s(X)) -> mark(X)
 mark(f(X)) -> a__f(mark(X))
 mark(p(X)) -> a__p(mark(X))
 mark(0()) -> 0()
 mark(cons(X1,X2)) -> cons(mark(X1),X2)
 mark(s(X)) -> s(mark(X))
 a__f(X) -> f(X)
 a__p(X) -> p(X)

Proof:
 DP Processor:
  DPs:
   a__f#(s(0())) -> a__p#(s(0()))
   a__f#(s(0())) -> a__f#(a__p(s(0())))
   a__p#(s(X)) -> mark#(X)
   mark#(f(X)) -> mark#(X)
   mark#(f(X)) -> a__f#(mark(X))
   mark#(p(X)) -> mark#(X)
   mark#(p(X)) -> a__p#(mark(X))
   mark#(cons(X1,X2)) -> mark#(X1)
   mark#(s(X)) -> mark#(X)
  TRS:
   a__f(0()) -> cons(0(),f(s(0())))
   a__f(s(0())) -> a__f(a__p(s(0())))
   a__p(s(X)) -> mark(X)
   mark(f(X)) -> a__f(mark(X))
   mark(p(X)) -> a__p(mark(X))
   mark(0()) -> 0()
   mark(cons(X1,X2)) -> cons(mark(X1),X2)
   mark(s(X)) -> s(mark(X))
   a__f(X) -> f(X)
   a__p(X) -> p(X)
  Matrix Interpretation Processor: dim=3
   
   interpretation:
    [mark#](x0) = [1 0 1]x0,
    
    [a__p#](x0) = [1 0 0]x0,
    
    [a__f#](x0) = [1 0 0]x0 + [1],
    
              [0 0 1]     [0]
    [p](x0) = [0 0 1]x0 + [0]
              [1 0 0]     [1],
    
                   
    [mark](x0) = x0
                   ,
    
                 [0 0 1]     [0]
    [a__p](x0) = [0 0 1]x0 + [0]
                 [1 0 0]     [1],
    
                     [0 1 0]     [0 1 0]     [1]
    [cons](x0, x1) = [0 0 0]x0 + [0 0 0]x1 + [0]
                     [1 0 1]     [0 0 0]     [0],
    
              [1 1 1]     [1]
    [f](x0) = [0 1 0]x0 + [0]
              [0 0 0]     [1],
    
              [1 0 1]     [1]
    [s](x0) = [0 0 0]x0 + [1]
              [1 1 0]     [0],
    
                 [1 1 1]     [1]
    [a__f](x0) = [0 1 0]x0 + [0]
                 [0 0 0]     [1],
    
          [0]
    [0] = [0]
          [1]
   orientation:
    a__f#(s(0())) = 3 >= 2 = a__p#(s(0()))
    
    a__f#(s(0())) = 3 >= 1 = a__f#(a__p(s(0())))
    
    a__p#(s(X)) = [1 0 1]X + [1] >= [1 0 1]X = mark#(X)
    
    mark#(f(X)) = [1 1 1]X + [2] >= [1 0 1]X = mark#(X)
    
    mark#(f(X)) = [1 1 1]X + [2] >= [1 0 0]X + [1] = a__f#(mark(X))
    
    mark#(p(X)) = [1 0 1]X + [1] >= [1 0 1]X = mark#(X)
    
    mark#(p(X)) = [1 0 1]X + [1] >= [1 0 0]X = a__p#(mark(X))
    
    mark#(cons(X1,X2)) = [1 1 1]X1 + [0 1 0]X2 + [1] >= [1 0 1]X1 = mark#(X1)
    
    mark#(s(X)) = [2 1 1]X + [1] >= [1 0 1]X = mark#(X)
    
                [2]    [2]                      
    a__f(0()) = [0] >= [0] = cons(0(),f(s(0())))
                [1]    [1]                      
    
                   [4]    [4]                     
    a__f(s(0())) = [1] >= [0] = a__f(a__p(s(0())))
                   [1]    [1]                     
    
                 [1 1 0]    [0]               
    a__p(s(X)) = [1 1 0]X + [0] >= X = mark(X)
                 [1 0 1]    [2]               
    
                 [1 1 1]    [1]    [1 1 1]    [1]                
    mark(f(X)) = [0 1 0]X + [0] >= [0 1 0]X + [0] = a__f(mark(X))
                 [0 0 0]    [1]    [0 0 0]    [1]                
    
                 [0 0 1]    [0]    [0 0 1]    [0]                
    mark(p(X)) = [0 0 1]X + [0] >= [0 0 1]X + [0] = a__p(mark(X))
                 [1 0 0]    [1]    [1 0 0]    [1]                
    
                [0]    [0]      
    mark(0()) = [0] >= [0] = 0()
                [1]    [1]      
    
                        [0 1 0]     [0 1 0]     [1]    [0 1 0]     [0 1 0]     [1]                    
    mark(cons(X1,X2)) = [0 0 0]X1 + [0 0 0]X2 + [0] >= [0 0 0]X1 + [0 0 0]X2 + [0] = cons(mark(X1),X2)
                        [1 0 1]     [0 0 0]     [0]    [1 0 1]     [0 0 0]     [0]                    
    
                 [1 0 1]    [1]    [1 0 1]    [1]             
    mark(s(X)) = [0 0 0]X + [1] >= [0 0 0]X + [1] = s(mark(X))
                 [1 1 0]    [0]    [1 1 0]    [0]             
    
              [1 1 1]    [1]    [1 1 1]    [1]       
    a__f(X) = [0 1 0]X + [0] >= [0 1 0]X + [0] = f(X)
              [0 0 0]    [1]    [0 0 0]    [1]       
    
              [0 0 1]    [0]    [0 0 1]    [0]       
    a__p(X) = [0 0 1]X + [0] >= [0 0 1]X + [0] = p(X)
              [1 0 0]    [1]    [1 0 0]    [1]       
   problem:
    DPs:
     
    TRS:
     a__f(0()) -> cons(0(),f(s(0())))
     a__f(s(0())) -> a__f(a__p(s(0())))
     a__p(s(X)) -> mark(X)
     mark(f(X)) -> a__f(mark(X))
     mark(p(X)) -> a__p(mark(X))
     mark(0()) -> 0()
     mark(cons(X1,X2)) -> cons(mark(X1),X2)
     mark(s(X)) -> s(mark(X))
     a__f(X) -> f(X)
     a__p(X) -> p(X)
   Qed