YES(?,O(n^1))

Problem:
 half(0()) -> 0()
 half(s(0())) -> 0()
 half(s(s(x))) -> s(half(x))
 s(log(0())) -> s(0())
 log(s(x)) -> s(log(half(s(x))))

Proof:
 RT Transformation Processor:
  strict:
   half(0()) -> 0()
   half(s(0())) -> 0()
   half(s(s(x))) -> s(half(x))
   s(log(0())) -> s(0())
   log(s(x)) -> s(log(half(s(x))))
  weak:
   
  Matrix Interpretation Processor:
   dimension: 1
   interpretation:
    [log](x0) = x0,
    
    [s](x0) = x0,
    
    [half](x0) = x0 + 16,
    
    [0] = 1
   orientation:
    half(0()) = 17 >= 1 = 0()
    
    half(s(0())) = 17 >= 1 = 0()
    
    half(s(s(x))) = x + 16 >= x + 16 = s(half(x))
    
    s(log(0())) = 1 >= 1 = s(0())
    
    log(s(x)) = x >= x + 16 = s(log(half(s(x))))
   problem:
    strict:
     half(s(s(x))) -> s(half(x))
     s(log(0())) -> s(0())
     log(s(x)) -> s(log(half(s(x))))
    weak:
     half(0()) -> 0()
     half(s(0())) -> 0()
   Matrix Interpretation Processor:
    dimension: 1
    interpretation:
     [log](x0) = x0,
     
     [s](x0) = x0 + 8,
     
     [half](x0) = x0,
     
     [0] = 2
    orientation:
     half(s(s(x))) = x + 16 >= x + 8 = s(half(x))
     
     s(log(0())) = 10 >= 10 = s(0())
     
     log(s(x)) = x + 8 >= x + 16 = s(log(half(s(x))))
     
     half(0()) = 2 >= 2 = 0()
     
     half(s(0())) = 10 >= 2 = 0()
    problem:
     strict:
      s(log(0())) -> s(0())
      log(s(x)) -> s(log(half(s(x))))
     weak:
      half(s(s(x))) -> s(half(x))
      half(0()) -> 0()
      half(s(0())) -> 0()
    Matrix Interpretation Processor:
     dimension: 1
     interpretation:
      [log](x0) = x0 + 4,
      
      [s](x0) = x0,
      
      [half](x0) = x0,
      
      [0] = 6
     orientation:
      s(log(0())) = 10 >= 6 = s(0())
      
      log(s(x)) = x + 4 >= x + 4 = s(log(half(s(x))))
      
      half(s(s(x))) = x >= x = s(half(x))
      
      half(0()) = 6 >= 6 = 0()
      
      half(s(0())) = 6 >= 6 = 0()
     problem:
      strict:
       log(s(x)) -> s(log(half(s(x))))
      weak:
       s(log(0())) -> s(0())
       half(s(s(x))) -> s(half(x))
       half(0()) -> 0()
       half(s(0())) -> 0()
     Arctic Interpretation Processor:
      dimension: 3
      interpretation:
                   [0 0 0]  
       [log](x0) = [0 2 2]x0
                   [0 1 1]  ,
       
                 [0  -& 0 ]  
       [s](x0) = [0  0  2 ]x0
                 [2  0  0 ]  ,
       
                    [0  -& -&]  
       [half](x0) = [0  -& -&]x0
                    [0  -& -&]  ,
       
             [0 ]
       [0] = [-&]
             [-&]
      orientation:
                   [2 0 2]     [1  -& 1 ]                      
       log(s(x)) = [4 2 4]x >= [3  -& 3 ]x = s(log(half(s(x))))
                   [3 1 3]     [2  -& 2 ]                      
       
                     [0]    [0]         
       s(log(0())) = [2] >= [0] = s(0())
                     [2]    [2]         
       
                       [2 0 0]     [0  -& -&]              
       half(s(s(x))) = [2 0 0]x >= [2  -& -&]x = s(half(x))
                       [2 0 0]     [2  -& -&]              
       
                   [0]    [0 ]      
       half(0()) = [0] >= [-&] = 0()
                   [0]    [-&]      
       
                      [0]    [0 ]      
       half(s(0())) = [0] >= [-&] = 0()
                      [0]    [-&]      
      problem:
       strict:
        
       weak:
        log(s(x)) -> s(log(half(s(x))))
        s(log(0())) -> s(0())
        half(s(s(x))) -> s(half(x))
        half(0()) -> 0()
        half(s(0())) -> 0()
      Qed