YES

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

Proof:
 Arctic Interpretation Processor:
  dimension: 2
  interpretation:
                [0 0]  
   [rand](x0) = [0 0]x0,
   
                [0  -&]  
   [half](x0) = [0  0 ]x0,
   
               [0 4]  
   [log](x0) = [4 6]x0,
   
         [4 ]
   [0] = [-&],
   
             [0  -&]  
   [s](x0) = [0  -&]x0
  orientation:
               [4]    [4 ]      
   half(0()) = [4] >= [-&] = 0()
   
                   [0  -&]     [0  -&]              
   half(s(s(x))) = [0  -&]x >= [0  -&]x = s(half(x))
   
                 [8 ]    [4 ]      
   log(s(0())) = [10] >= [-&] = 0()
   
                  [4  -&]     [4  -&]                      
   log(s(s(x))) = [6  -&]x >= [4  -&]x = s(log(s(half(x))))
   
             [0 0]          
   rand(x) = [0 0]x >= x = x
   
             [0 0]     [0  -&]              
   rand(x) = [0 0]x >= [0  -&]x = rand(s(x))
  problem:
   strict:
    half(0()) -> 0()
    half(s(s(x))) -> s(half(x))
    log(s(s(x))) -> s(log(s(half(x))))
   weak:
    rand(x) -> x
    rand(x) -> rand(s(x))
  Arctic Interpretation Processor:
   dimension: 1
   interpretation:
    [rand](x0) = 1x0,
    
    [half](x0) = x0,
    
    [log](x0) = 2x0,
    
    [0] = 15,
    
    [s](x0) = x0
   orientation:
    half(0()) = 15 >= 15 = 0()
    
    half(s(s(x))) = x >= x = s(half(x))
    
    log(s(s(x))) = 2x >= 2x = s(log(s(half(x))))
    
    rand(x) = 1x >= x = x
    
    rand(x) = 1x >= 1x = rand(s(x))
   problem:
    strict:
     half(0()) -> 0()
     half(s(s(x))) -> s(half(x))
     log(s(s(x))) -> s(log(s(half(x))))
    weak:
     rand(x) -> rand(s(x))
   Arctic Interpretation Processor:
    dimension: 2
    interpretation:
                  [0  -&]  
     [rand](x0) = [-& -&]x0,
     
                  [0  3 ]  
     [half](x0) = [-& 0 ]x0,
     
                 [6  4 ]  
     [log](x0) = [-& 2 ]x0,
     
           [4]
     [0] = [0],
     
               [0  -&]  
     [s](x0) = [0  4 ]x0
    orientation:
                 [4]    [4]      
     half(0()) = [0] >= [0] = 0()
     
                     [7  11]     [0 3]              
     half(s(s(x))) = [4  8 ]x >= [0 4]x = s(half(x))
     
                    [8  12]     [6  9 ]                      
     log(s(s(x))) = [6  10]x >= [6  10]x = s(log(s(half(x))))
     
               [0  -&]     [0  -&]              
     rand(x) = [-& -&]x >= [-& -&]x = rand(s(x))
    problem:
     strict:
      half(0()) -> 0()
      log(s(s(x))) -> s(log(s(half(x))))
     weak:
      rand(x) -> rand(s(x))
    Arctic Interpretation Processor:
     dimension: 2
     interpretation:
                   [0  0 ]  
      [rand](x0) = [-& -&]x0,
      
                   [0 0]  
      [half](x0) = [1 1]x0,
      
                  [0 1]  
      [log](x0) = [0 1]x0,
      
            [0]
      [0] = [1],
      
                [0  0 ]  
      [s](x0) = [0  -&]x0
     orientation:
                  [1]    [0]      
      half(0()) = [2] >= [1] = 0()
      
                     [1 1]     [1 1]                      
      log(s(s(x))) = [1 1]x >= [1 1]x = s(log(s(half(x))))
      
                [0  0 ]     [0  0 ]              
      rand(x) = [-& -&]x >= [-& -&]x = rand(s(x))
     problem:
      strict:
       log(s(s(x))) -> s(log(s(half(x))))
      weak:
       rand(x) -> rand(s(x))
     Matrix Interpretation Processor: dim=2
      
      interpretation:
                    [2 0]  
       [rand](x0) = [0 0]x0,
       
                    [1 0]  
       [half](x0) = [0 0]x0,
       
                   [2 1]     [0]
       [log](x0) = [0 1]x0 + [1],
       
                      [0]
       [s](x0) = x0 + [1]
      orientation:
                      [2 1]    [2]    [2 0]    [1]                     
       log(s(s(x))) = [0 1]x + [3] >= [0 0]x + [3] = s(log(s(half(x))))
       
                 [2 0]     [2 0]              
       rand(x) = [0 0]x >= [0 0]x = rand(s(x))
      problem:
       strict:
        
       weak:
        rand(x) -> rand(s(x))
      Qed