YES

Problem:
 strict:
  f(0()) -> s(0())
  f(s(0())) -> s(0())
  f(s(s(x))) -> f(f(s(x)))
 weak:
  rand(x) -> x
  rand(x) -> rand(s(x))

Proof:
 Arctic Interpretation Processor:
  dimension: 2
  interpretation:
             [0  5 ]  
   [f](x0) = [-& 0 ]x0,
   
                [0 0]  
   [rand](x0) = [3 3]x0,
   
         [2]
   [0] = [6],
   
             [0  -&]  
   [s](x0) = [0  -&]x0
  orientation:
            [11]    [2]         
   f(0()) = [6 ] >= [2] = s(0())
   
               [7]    [2]         
   f(s(0())) = [2] >= [2] = s(0())
   
                [5  -&]     [5  -&]              
   f(s(s(x))) = [0  -&]x >= [0  -&]x = f(f(s(x)))
   
             [0 0]          
   rand(x) = [3 3]x >= x = x
   
             [0 0]     [0  -&]              
   rand(x) = [3 3]x >= [3  -&]x = rand(s(x))
  problem:
   strict:
    f(s(0())) -> s(0())
    f(s(s(x))) -> f(f(s(x)))
   weak:
    rand(x) -> x
    rand(x) -> rand(s(x))
  Arctic Interpretation Processor:
   dimension: 2
   interpretation:
              [0  2 ]  
    [f](x0) = [-& 0 ]x0,
    
                 [1  6 ]  
    [rand](x0) = [-& 1 ]x0,
    
          [0 ]
    [0] = [-&],
    
              [0  5 ]  
    [s](x0) = [-& -&]x0
   orientation:
                [0 ]    [0 ]         
    f(s(0())) = [-&] >= [-&] = s(0())
    
                 [0  5 ]     [0  5 ]              
    f(s(s(x))) = [-& -&]x >= [-& -&]x = f(f(s(x)))
    
              [1  6 ]          
    rand(x) = [-& 1 ]x >= x = x
    
              [1  6 ]     [1  6 ]              
    rand(x) = [-& 1 ]x >= [-& -&]x = rand(s(x))
   problem:
    strict:
     f(s(0())) -> s(0())
     f(s(s(x))) -> f(f(s(x)))
    weak:
     rand(x) -> rand(s(x))
   Arctic Interpretation Processor:
    dimension: 2
    interpretation:
               [0 0]  
     [f](x0) = [0 0]x0,
     
                  [0  -&]  
     [rand](x0) = [-& -&]x0,
     
           [0 ]
     [0] = [-&],
     
               [0  -&]  
     [s](x0) = [0  4 ]x0
    orientation:
                 [0]    [0]         
     f(s(0())) = [0] >= [0] = s(0())
     
                  [4 8]     [0 4]              
     f(s(s(x))) = [4 8]x >= [0 4]x = f(f(s(x)))
     
               [0  -&]     [0  -&]              
     rand(x) = [-& -&]x >= [-& -&]x = rand(s(x))
    problem:
     strict:
      f(s(0())) -> s(0())
     weak:
      rand(x) -> rand(s(x))
    Arctic Interpretation Processor:
     dimension: 4
     interpretation:
                [0  0  2  0 ]  
                [2  2  3  -&]  
      [f](x0) = [1  3  1  -&]x0
                [3  0  1  0 ]  ,
      
                   [0  1  0  0 ]  
                   [0  1  0  0 ]  
      [rand](x0) = [-& 2  1  1 ]x0
                   [-& 0  -& -&]  ,
      
            [0 ]
            [-&]
      [0] = [2 ]
            [0 ],
      
                [0  0  -& -&]  
                [-& -& -& -&]  
      [s](x0) = [-& 1  -& -&]x0
                [-& 1  0  0 ]  
     orientation:
                  [2]    [0 ]         
                  [2]    [-&]         
      f(s(0())) = [1] >= [-&] = s(0())
                  [3]    [2 ]         
      
                [0  1  0  0 ]     [0  1  0  0 ]              
                [0  1  0  0 ]     [0  1  0  0 ]              
      rand(x) = [-& 2  1  1 ]x >= [-& 2  1  1 ]x = rand(s(x))
                [-& 0  -& -&]     [-& -& -& -&]              
     problem:
      strict:
       
      weak:
       rand(x) -> rand(s(x))
     Qed