YES

Problem:
 f(f(x)) -> f(c(f(x)))
 f(f(x)) -> f(d(f(x)))
 g(c(x)) -> x
 g(d(x)) -> x
 g(c(0())) -> g(d(1()))
 g(c(1())) -> g(d(0()))

Proof:
 Matrix Interpretation Processor: dim=3
  
  interpretation:
         [0]
   [0] = [0]
         [0],
   
             [1 1 0]  
   [c](x0) = [0 0 1]x0
             [0 0 0]  ,
   
             [1 0 1]     [1]
   [g](x0) = [1 0 0]x0 + [1]
             [0 1 1]     [1],
   
         [0]
   [1] = [0]
         [0],
   
             [1 0 1]     [0]
   [f](x0) = [0 0 0]x0 + [0]
             [0 0 0]     [1],
   
             [1 1 0]  
   [d](x0) = [0 0 0]x0
             [0 0 1]  
  orientation:
             [1 0 1]    [1]    [1 0 1]    [0]             
   f(f(x)) = [0 0 0]x + [0] >= [0 0 0]x + [0] = f(c(f(x)))
             [0 0 0]    [1]    [0 0 0]    [1]             
   
             [1 0 1]    [1]    [1 0 1]    [1]             
   f(f(x)) = [0 0 0]x + [0] >= [0 0 0]x + [0] = f(d(f(x)))
             [0 0 0]    [1]    [0 0 0]    [1]             
   
             [1 1 0]    [1]         
   g(c(x)) = [1 1 0]x + [1] >= x = x
             [0 0 1]    [1]         
   
             [1 1 1]    [1]         
   g(d(x)) = [1 1 0]x + [1] >= x = x
             [0 0 1]    [1]         
   
               [1]    [1]            
   g(c(0())) = [1] >= [1] = g(d(1()))
               [1]    [1]            
   
               [1]    [1]            
   g(c(1())) = [1] >= [1] = g(d(0()))
               [1]    [1]            
  problem:
   f(f(x)) -> f(d(f(x)))
   g(c(0())) -> g(d(1()))
   g(c(1())) -> g(d(0()))
  Matrix Interpretation Processor: dim=3
   
   interpretation:
          [0]
    [0] = [0]
          [0],
    
              [1 0 0]     [0]
    [c](x0) = [0 0 0]x0 + [0]
              [0 0 0]     [1],
    
              [1 0 1]  
    [g](x0) = [0 0 0]x0
              [0 0 0]  ,
    
          [0]
    [1] = [0]
          [0],
    
              [1 0 1]     [0]
    [f](x0) = [0 0 0]x0 + [0]
              [0 0 0]     [1],
    
              [1 0 0]  
    [d](x0) = [0 0 0]x0
              [0 0 0]  
   orientation:
              [1 0 1]    [1]    [1 0 1]    [0]             
    f(f(x)) = [0 0 0]x + [0] >= [0 0 0]x + [0] = f(d(f(x)))
              [0 0 0]    [1]    [0 0 0]    [1]             
    
                [1]    [0]            
    g(c(0())) = [0] >= [0] = g(d(1()))
                [0]    [0]            
    
                [1]    [0]            
    g(c(1())) = [0] >= [0] = g(d(0()))
                [0]    [0]            
   problem:
    
   Qed