YES(?,O(n^2))

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

Proof:
 Complexity Transformation Processor:
  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:
   
  Matrix Interpretation Processor:
   dimension: 1
   max_matrix:
    1
    interpretation:
     [log](x0) = x0 + 1,
     
     [s](x0) = x0,
     
     [half](x0) = x0,
     
     [0] = 1
    orientation:
     half(0()) = 1 >= 1 = 0()
     
     half(s(s(x))) = x >= x = s(half(x))
     
     log(s(0())) = 2 >= 1 = 0()
     
     log(s(s(x))) = x + 1 >= x + 1 = s(log(s(half(x))))
    problem:
     strict:
      half(0()) -> 0()
      half(s(s(x))) -> s(half(x))
      log(s(s(x))) -> s(log(s(half(x))))
     weak:
      log(s(0())) -> 0()
    Matrix Interpretation Processor:
     dimension: 1
     max_matrix:
      1
      interpretation:
       [log](x0) = x0,
       
       [s](x0) = x0 + 1,
       
       [half](x0) = x0 + 1,
       
       [0] = 1
      orientation:
       half(0()) = 2 >= 1 = 0()
       
       half(s(s(x))) = x + 3 >= x + 2 = s(half(x))
       
       log(s(s(x))) = x + 2 >= x + 3 = s(log(s(half(x))))
       
       log(s(0())) = 2 >= 1 = 0()
      problem:
       strict:
        log(s(s(x))) -> s(log(s(half(x))))
       weak:
        half(0()) -> 0()
        half(s(s(x))) -> s(half(x))
        log(s(0())) -> 0()
      Matrix Interpretation Processor:
       dimension: 2
       max_matrix:
        [1 1]
        [0 1]
        interpretation:
                     [1 1]     [1]
         [log](x0) = [0 1]x0 + [0],
         
                        [0]
         [s](x0) = x0 + [1],
         
                        
         [half](x0) = x0,
         
               [0]
         [0] = [0]
        orientation:
                        [1 1]    [3]    [1 1]    [2]                     
         log(s(s(x))) = [0 1]x + [2] >= [0 1]x + [2] = s(log(s(half(x))))
         
                     [0]    [0]      
         half(0()) = [0] >= [0] = 0()
         
                             [0]        [0]             
         half(s(s(x))) = x + [2] >= x + [1] = s(half(x))
         
                       [2]    [0]      
         log(s(0())) = [1] >= [0] = 0()
        problem:
         strict:
          
         weak:
          log(s(s(x))) -> s(log(s(half(x))))
          half(0()) -> 0()
          half(s(s(x))) -> s(half(x))
          log(s(0())) -> 0()
        Qed