YES(?,O(n^1))

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

Proof:
 RT Transformation Processor:
  strict:
   half(0()) -> 0()
   half(s(0())) -> 0()
   half(s(s(x))) -> s(half(x))
   bits(0()) -> 0()
   bits(s(x)) -> s(bits(half(s(x))))
  weak:
   
  Matrix Interpretation Processor:
   dimension: 1
   interpretation:
    [bits](x0) = x0 + 1,
    
    [s](x0) = x0,
    
    [half](x0) = x0 + 2,
    
    [0] = 4
   orientation:
    half(0()) = 6 >= 4 = 0()
    
    half(s(0())) = 6 >= 4 = 0()
    
    half(s(s(x))) = x + 2 >= x + 2 = s(half(x))
    
    bits(0()) = 5 >= 4 = 0()
    
    bits(s(x)) = x + 1 >= x + 3 = s(bits(half(s(x))))
   problem:
    strict:
     half(s(s(x))) -> s(half(x))
     bits(s(x)) -> s(bits(half(s(x))))
    weak:
     half(0()) -> 0()
     half(s(0())) -> 0()
     bits(0()) -> 0()
   Matrix Interpretation Processor:
    dimension: 1
    interpretation:
     [bits](x0) = x0 + 2,
     
     [s](x0) = x0 + 16,
     
     [half](x0) = x0,
     
     [0] = 0
    orientation:
     half(s(s(x))) = x + 32 >= x + 16 = s(half(x))
     
     bits(s(x)) = x + 18 >= x + 34 = s(bits(half(s(x))))
     
     half(0()) = 0 >= 0 = 0()
     
     half(s(0())) = 16 >= 0 = 0()
     
     bits(0()) = 2 >= 0 = 0()
    problem:
     strict:
      bits(s(x)) -> s(bits(half(s(x))))
     weak:
      half(s(s(x))) -> s(half(x))
      half(0()) -> 0()
      half(s(0())) -> 0()
      bits(0()) -> 0()
    Arctic Interpretation Processor:
     dimension: 3
     interpretation:
                   [0 0 0]  
      [bits](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  -& -&]  ,
      
            [1]
      [0] = [0]
            [0]
     orientation:
                   [2 0 2]     [1  -& 1 ]                       
      bits(s(x)) = [4 2 4]x >= [3  -& 3 ]x = s(bits(half(s(x))))
                   [3 1 3]     [2  -& 2 ]                       
      
                      [2 0 0]     [0  -& -&]              
      half(s(s(x))) = [2 0 0]x >= [2  -& -&]x = s(half(x))
                      [2 0 0]     [2  -& -&]              
      
                  [1]    [1]      
      half(0()) = [1] >= [0] = 0()
                  [1]    [0]      
      
                     [1]    [1]      
      half(s(0())) = [1] >= [0] = 0()
                     [1]    [0]      
      
                  [1]    [1]      
      bits(0()) = [2] >= [0] = 0()
                  [1]    [0]      
     problem:
      strict:
       
      weak:
       bits(s(x)) -> s(bits(half(s(x))))
       half(s(s(x))) -> s(half(x))
       half(0()) -> 0()
       half(s(0())) -> 0()
       bits(0()) -> 0()
     Qed