YES

Problem:
 strict:
  f(x) -> s(x)
  f(s(s(x))) -> s(f(f(x)))
 weak:
  rand(x) -> x
  rand(x) -> rand(s(x))

Proof:
 Arctic Interpretation Processor:
  dimension: 1
  interpretation:
   [s](x0) = x0,
   
   [f](x0) = x0,
   
   [rand](x0) = 8x0
  orientation:
   f(x) = x >= x = s(x)
   
   f(s(s(x))) = x >= x = s(f(f(x)))
   
   rand(x) = 8x >= x = x
   
   rand(x) = 8x >= 8x = rand(s(x))
  problem:
   strict:
    f(x) -> s(x)
    f(s(s(x))) -> s(f(f(x)))
   weak:
    rand(x) -> rand(s(x))
  String Reversal Processor:
   strict:
    f(x) -> s(x)
    s(s(f(x))) -> f(f(s(x)))
   weak:
    rand(x) -> s(rand(x))
   Matrix Interpretation Processor: dim=3
    
    interpretation:
               [1 0 2]  
     [s](x0) = [0 1 1]x0
               [0 1 0]  ,
     
               [1 0 2]     [0]
     [f](x0) = [0 1 1]x0 + [1]
               [0 1 0]     [0],
     
                  [2 3 0]     [1]
     [rand](x0) = [0 0 0]x0 + [0]
                  [0 0 0]     [0]
    orientation:
            [1 0 2]    [0]    [1 0 2]        
     f(x) = [0 1 1]x + [1] >= [0 1 1]x = s(x)
            [0 1 0]    [0]    [0 1 0]        
     
                  [1 4 4]    [2]    [1 4 4]    [0]             
     s(s(f(x))) = [0 3 2]x + [2] >= [0 3 2]x + [2] = f(f(s(x)))
                  [0 2 1]    [1]    [0 2 1]    [1]             
     
               [2 3 0]    [1]    [2 3 0]    [1]             
     rand(x) = [0 0 0]x + [0] >= [0 0 0]x + [0] = s(rand(x))
               [0 0 0]    [0]    [0 0 0]    [0]             
    problem:
     strict:
      f(x) -> s(x)
     weak:
      rand(x) -> s(rand(x))
    String Reversal Processor:
     strict:
      f(x) -> s(x)
     weak:
      rand(x) -> rand(s(x))
     Arctic Interpretation Processor:
      dimension: 2
      interpretation:
                 [0  -&]  
       [s](x0) = [0  0 ]x0,
       
                 [7 0]  
       [f](x0) = [7 7]x0,
       
                    [4 4]  
       [rand](x0) = [0 0]x0
      orientation:
              [7 0]     [0  -&]        
       f(x) = [7 7]x >= [0  0 ]x = s(x)
       
                 [4 4]     [4 4]              
       rand(x) = [0 0]x >= [0 0]x = rand(s(x))
      problem:
       strict:
        
       weak:
        rand(x) -> rand(s(x))
      Qed