YES

Problem:
 strict:
  g(c(x,s(y))) -> g(c(s(x),y))
  f(c(s(x),y)) -> f(c(x,s(y)))
  f(f(x)) -> f(d(f(x)))
  f(x) -> x
 weak:
  rand(x) -> x
  rand(x) -> rand(s(x))

Proof:
 Matrix Interpretation Processor: dim=2
  
  interpretation:
             [1 0]  
   [d](x0) = [0 0]x0,
   
                 [1 1]     [1 0]     [1]
   [c](x0, x1) = [2 2]x0 + [1 0]x1 + [2],
   
               
   [f](x0) = x0,
   
                [2 0]     [2]
   [rand](x0) = [0 1]x0 + [0],
   
               
   [s](x0) = x0,
   
             [2 0]     [1]
   [g](x0) = [1 1]x0 + [0]
  orientation:
                  [2 2]    [2 0]    [3]    [2 2]    [2 0]    [3]               
   g(c(x,s(y))) = [3 3]x + [2 0]y + [3] >= [3 3]x + [2 0]y + [3] = g(c(s(x),y))
   
                  [1 1]    [1 0]    [1]    [1 1]    [1 0]    [1]               
   f(c(s(x),y)) = [2 2]x + [1 0]y + [2] >= [2 2]x + [1 0]y + [2] = f(c(x,s(y)))
   
                  [1 0]              
   f(f(x)) = x >= [0 0]x = f(d(f(x)))
   
                    
   f(x) = x >= x = x
   
             [2 0]    [2]         
   rand(x) = [0 1]x + [0] >= x = x
   
             [2 0]    [2]    [2 0]    [2]             
   rand(x) = [0 1]x + [0] >= [0 1]x + [0] = rand(s(x))
  problem:
   strict:
    g(c(x,s(y))) -> g(c(s(x),y))
    f(c(s(x),y)) -> f(c(x,s(y)))
    f(f(x)) -> f(d(f(x)))
    f(x) -> x
   weak:
    rand(x) -> rand(s(x))
  Matrix Interpretation Processor: dim=2
   
   interpretation:
              [1 0]  
    [d](x0) = [0 0]x0,
    
                  [1 1]       
    [c](x0, x1) = [0 0]x0 + x1,
    
              [1 0]  
    [f](x0) = [1 1]x0,
    
                 [1 0]  
    [rand](x0) = [0 0]x0,
    
                   [0]
    [s](x0) = x0 + [1],
    
              [2 2]  
    [g](x0) = [0 2]x0
   orientation:
                   [2 2]    [2 2]    [2]    [2 2]    [2 2]    [2]               
    g(c(x,s(y))) = [0 0]x + [0 2]y + [2] >= [0 0]x + [0 2]y + [0] = g(c(s(x),y))
    
                   [1 1]    [1 0]    [1]    [1 1]    [1 0]    [0]               
    f(c(s(x),y)) = [1 1]x + [1 1]y + [1] >= [1 1]x + [1 1]y + [1] = f(c(x,s(y)))
    
              [1 0]     [1 0]              
    f(f(x)) = [2 1]x >= [1 0]x = f(d(f(x)))
    
           [1 0]          
    f(x) = [1 1]x >= x = x
    
              [1 0]     [1 0]              
    rand(x) = [0 0]x >= [0 0]x = rand(s(x))
   problem:
    strict:
     g(c(x,s(y))) -> g(c(s(x),y))
     f(f(x)) -> f(d(f(x)))
     f(x) -> x
    weak:
     rand(x) -> rand(s(x))
   Matrix Interpretation Processor: dim=2
    
    interpretation:
               [1 0]  
     [d](x0) = [0 0]x0,
     
                   [2 1]     [1 2]     [0]
     [c](x0, x1) = [0 0]x0 + [0 0]x1 + [1],
     
                 
     [f](x0) = x0,
     
                  [1 0]     [0]
     [rand](x0) = [2 0]x0 + [1],
     
                    [0]
     [s](x0) = x0 + [1],
     
               [1 0]  
     [g](x0) = [1 1]x0
    orientation:
                    [2 1]    [1 2]    [2]    [2 1]    [1 2]    [1]               
     g(c(x,s(y))) = [2 1]x + [1 2]y + [3] >= [2 1]x + [1 2]y + [2] = g(c(s(x),y))
     
                    [1 0]              
     f(f(x)) = x >= [0 0]x = f(d(f(x)))
     
                      
     f(x) = x >= x = x
     
               [1 0]    [0]    [1 0]    [0]             
     rand(x) = [2 0]x + [1] >= [2 0]x + [1] = rand(s(x))
    problem:
     strict:
      f(f(x)) -> f(d(f(x)))
      f(x) -> x
     weak:
      rand(x) -> rand(s(x))
    String Reversal Processor:
     strict:
      f(f(x)) -> f(d(f(x)))
      f(x) -> x
     weak:
      rand(x) -> s(rand(x))
     Arctic Interpretation Processor:
      dimension: 2
      interpretation:
                 [0  -&]  
       [d](x0) = [0  -&]x0,
       
                 [0 0]  
       [f](x0) = [1 4]x0,
       
                    [2 6]  
       [rand](x0) = [0 6]x0,
       
                 [0  0 ]  
       [s](x0) = [-& 0 ]x0
      orientation:
                 [1 4]     [0 0]              
       f(f(x)) = [5 8]x >= [4 4]x = f(d(f(x)))
       
              [0 0]          
       f(x) = [1 4]x >= x = x
       
                 [2 6]     [2 6]              
       rand(x) = [0 6]x >= [0 6]x = s(rand(x))
      problem:
       strict:
        f(x) -> x
       weak:
        rand(x) -> s(rand(x))
      Arctic Interpretation Processor:
       dimension: 4
       interpretation:
                  [1 0 0 0]  
                  [0 2 0 0]  
        [f](x0) = [0 0 1 0]x0
                  [0 0 0 1]  ,
        
                     [0  -& 0  -&]  
                     [0  0  0  0 ]  
        [rand](x0) = [0  1  0  2 ]x0
                     [0  0  0  -&]  ,
        
                  [0  -& -& -&]  
                  [-& -& -& 0 ]  
        [s](x0) = [0  -& 0  0 ]x0
                  [-& -& -& -&]  
       orientation:
               [1 0 0 0]          
               [0 2 0 0]          
        f(x) = [0 0 1 0]x >= x = x
               [0 0 0 1]          
        
                  [0  -& 0  -&]     [0  -& 0  -&]              
                  [0  0  0  0 ]     [0  0  0  -&]              
        rand(x) = [0  1  0  2 ]x >= [0  1  0  2 ]x = s(rand(x))
                  [0  0  0  -&]     [-& -& -& -&]              
       problem:
        strict:
         
        weak:
         rand(x) -> s(rand(x))
       Qed