YES

Problem:
 f(c(s(x),y)) -> f(c(x,s(y)))
 f(c(s(x),s(y))) -> g(c(x,y))
 g(c(x,s(y))) -> g(c(s(x),y))
 g(c(s(x),s(y))) -> f(c(x,y))

Proof:
 DP Processor:
  DPs:
   f#(c(s(x),y)) -> f#(c(x,s(y)))
   f#(c(s(x),s(y))) -> g#(c(x,y))
   g#(c(x,s(y))) -> g#(c(s(x),y))
   g#(c(s(x),s(y))) -> f#(c(x,y))
  TRS:
   f(c(s(x),y)) -> f(c(x,s(y)))
   f(c(s(x),s(y))) -> g(c(x,y))
   g(c(x,s(y))) -> g(c(s(x),y))
   g(c(s(x),s(y))) -> f(c(x,y))
  Usable Rule Processor:
   DPs:
    f#(c(s(x),y)) -> f#(c(x,s(y)))
    f#(c(s(x),s(y))) -> g#(c(x,y))
    g#(c(x,s(y))) -> g#(c(s(x),y))
    g#(c(s(x),s(y))) -> f#(c(x,y))
   TRS:
    
   CDG Processor:
    DPs:
     f#(c(s(x),y)) -> f#(c(x,s(y)))
     f#(c(s(x),s(y))) -> g#(c(x,y))
     g#(c(x,s(y))) -> g#(c(s(x),y))
     g#(c(s(x),s(y))) -> f#(c(x,y))
    TRS:
     
    graph:
     g#(c(s(x),s(y))) -> f#(c(x,y)) -> f#(c(s(x),y)) -> f#(c(x,s(y)))
     g#(c(s(x),s(y))) -> f#(c(x,y)) -> f#(c(s(x),s(y))) -> g#(c(x,y))
     g#(c(x,s(y))) -> g#(c(s(x),y)) -> g#(c(x,s(y))) -> g#(c(s(x),y))
     g#(c(x,s(y))) -> g#(c(s(x),y)) -> g#(c(s(x),s(y))) -> f#(c(x,y))
     f#(c(s(x),s(y))) -> g#(c(x,y)) -> g#(c(x,s(y))) -> g#(c(s(x),y))
     f#(c(s(x),s(y))) -> g#(c(x,y)) -> g#(c(s(x),s(y))) -> f#(c(x,y))
     f#(c(s(x),y)) -> f#(c(x,s(y))) -> f#(c(s(x),y)) -> f#(c(x,s(y)))
     f#(c(s(x),y)) -> f#(c(x,s(y))) -> f#(c(s(x),s(y))) -> g#(c(x,y))
    Restore Modifier:
     DPs:
      f#(c(s(x),y)) -> f#(c(x,s(y)))
      f#(c(s(x),s(y))) -> g#(c(x,y))
      g#(c(x,s(y))) -> g#(c(s(x),y))
      g#(c(s(x),s(y))) -> f#(c(x,y))
     TRS:
      f(c(s(x),y)) -> f(c(x,s(y)))
      f(c(s(x),s(y))) -> g(c(x,y))
      g(c(x,s(y))) -> g(c(s(x),y))
      g(c(s(x),s(y))) -> f(c(x,y))
     SCC Processor:
      #sccs: 1
      #rules: 4
      #arcs: 8/16
      DPs:
       g#(c(s(x),s(y))) -> f#(c(x,y))
       f#(c(s(x),s(y))) -> g#(c(x,y))
       g#(c(x,s(y))) -> g#(c(s(x),y))
       f#(c(s(x),y)) -> f#(c(x,s(y)))
      TRS:
       f(c(s(x),y)) -> f(c(x,s(y)))
       f(c(s(x),s(y))) -> g(c(x,y))
       g(c(x,s(y))) -> g(c(s(x),y))
       g(c(s(x),s(y))) -> f(c(x,y))
      Matrix Interpretation Processor:
       dimension: 1
       interpretation:
        [g#](x0) = x0,
        
        [f#](x0) = x0,
        
        [g](x0) = 0,
        
        [f](x0) = 0,
        
        [c](x0, x1) = x0 + x1,
        
        [s](x0) = x0 + 1
       orientation:
        g#(c(s(x),s(y))) = x + y + 2 >= x + y = f#(c(x,y))
        
        f#(c(s(x),s(y))) = x + y + 2 >= x + y = g#(c(x,y))
        
        g#(c(x,s(y))) = x + y + 1 >= x + y + 1 = g#(c(s(x),y))
        
        f#(c(s(x),y)) = x + y + 1 >= x + y + 1 = f#(c(x,s(y)))
        
        f(c(s(x),y)) = 0 >= 0 = f(c(x,s(y)))
        
        f(c(s(x),s(y))) = 0 >= 0 = g(c(x,y))
        
        g(c(x,s(y))) = 0 >= 0 = g(c(s(x),y))
        
        g(c(s(x),s(y))) = 0 >= 0 = f(c(x,y))
       problem:
        DPs:
         g#(c(x,s(y))) -> g#(c(s(x),y))
         f#(c(s(x),y)) -> f#(c(x,s(y)))
        TRS:
         f(c(s(x),y)) -> f(c(x,s(y)))
         f(c(s(x),s(y))) -> g(c(x,y))
         g(c(x,s(y))) -> g(c(s(x),y))
         g(c(s(x),s(y))) -> f(c(x,y))
       Matrix Interpretation Processor:
        dimension: 1
        interpretation:
         [g#](x0) = 0,
         
         [f#](x0) = x0 + 1,
         
         [g](x0) = 0,
         
         [f](x0) = 0,
         
         [c](x0, x1) = x0 + 1,
         
         [s](x0) = x0 + 1
        orientation:
         g#(c(x,s(y))) = 0 >= 0 = g#(c(s(x),y))
         
         f#(c(s(x),y)) = x + 3 >= x + 2 = f#(c(x,s(y)))
         
         f(c(s(x),y)) = 0 >= 0 = f(c(x,s(y)))
         
         f(c(s(x),s(y))) = 0 >= 0 = g(c(x,y))
         
         g(c(x,s(y))) = 0 >= 0 = g(c(s(x),y))
         
         g(c(s(x),s(y))) = 0 >= 0 = f(c(x,y))
        problem:
         DPs:
          g#(c(x,s(y))) -> g#(c(s(x),y))
         TRS:
          f(c(s(x),y)) -> f(c(x,s(y)))
          f(c(s(x),s(y))) -> g(c(x,y))
          g(c(x,s(y))) -> g(c(s(x),y))
          g(c(s(x),s(y))) -> f(c(x,y))
        Matrix Interpretation Processor:
         dimension: 1
         interpretation:
          [g#](x0) = x0,
          
          [g](x0) = 0,
          
          [f](x0) = 0,
          
          [c](x0, x1) = x1,
          
          [s](x0) = x0 + 1
         orientation:
          g#(c(x,s(y))) = y + 1 >= y = g#(c(s(x),y))
          
          f(c(s(x),y)) = 0 >= 0 = f(c(x,s(y)))
          
          f(c(s(x),s(y))) = 0 >= 0 = g(c(x,y))
          
          g(c(x,s(y))) = 0 >= 0 = g(c(s(x),y))
          
          g(c(s(x),s(y))) = 0 >= 0 = f(c(x,y))
         problem:
          DPs:
           
          TRS:
           f(c(s(x),y)) -> f(c(x,s(y)))
           f(c(s(x),s(y))) -> g(c(x,y))
           g(c(x,s(y))) -> g(c(s(x),y))
           g(c(s(x),s(y))) -> f(c(x,y))
         Qed