We consider the following Problem:

  Strict Trs:
    {  half(0()) -> 0()
     , half(s(s(x))) -> s(half(x))
     , log(s(0())) -> 0()
     , log(s(s(x))) -> s(log(s(half(x))))}
  StartTerms: basic terms
  Strategy: innermost

Certificate: YES(?,O(n^2))

Proof:
  We consider the following Problem:
  
    Strict Trs:
      {  half(0()) -> 0()
       , half(s(s(x))) -> s(half(x))
       , log(s(0())) -> 0()
       , log(s(s(x))) -> s(log(s(half(x))))}
    StartTerms: basic terms
    Strategy: innermost
  
  Certificate: YES(?,O(n^2))
  
  Proof:
    The weightgap principle applies, where following rules are oriented strictly:
    
    TRS Component:
      {  half(0()) -> 0()
       , log(s(0())) -> 0()}
    
    Interpretation of nonconstant growth:
    -------------------------------------
      The following argument positions are usable:
        Uargs(half) = {}, Uargs(s) = {1}, Uargs(log) = {1}
      We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
      Interpretation Functions:
       half(x1) = [0 0] x1 + [1]
                  [0 0]      [1]
       0() = [0]
             [0]
       s(x1) = [1 0] x1 + [0]
               [0 0]      [1]
       log(x1) = [1 0] x1 + [1]
                 [0 0]      [1]
    
    The strictly oriented rules are moved into the weak component.
    
    We consider the following Problem:
    
      Strict Trs:
        {  half(s(s(x))) -> s(half(x))
         , log(s(s(x))) -> s(log(s(half(x))))}
      Weak Trs:
        {  half(0()) -> 0()
         , log(s(0())) -> 0()}
      StartTerms: basic terms
      Strategy: innermost
    
    Certificate: YES(?,O(n^2))
    
    Proof:
      The weightgap principle applies, where following rules are oriented strictly:
      
      TRS Component: {half(s(s(x))) -> s(half(x))}
      
      Interpretation of nonconstant growth:
      -------------------------------------
        The following argument positions are usable:
          Uargs(half) = {}, Uargs(s) = {1}, Uargs(log) = {1}
        We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
        Interpretation Functions:
         half(x1) = [1 0] x1 + [0]
                    [0 1]      [0]
         0() = [0]
               [1]
         s(x1) = [1 1] x1 + [0]
                 [0 0]      [2]
         log(x1) = [1 0] x1 + [0]
                   [0 0]      [1]
      
      The strictly oriented rules are moved into the weak component.
      
      We consider the following Problem:
      
        Strict Trs: {log(s(s(x))) -> s(log(s(half(x))))}
        Weak Trs:
          {  half(s(s(x))) -> s(half(x))
           , half(0()) -> 0()
           , log(s(0())) -> 0()}
        StartTerms: basic terms
        Strategy: innermost
      
      Certificate: YES(?,O(n^2))
      
      Proof:
        We consider the following Problem:
        
          Strict Trs: {log(s(s(x))) -> s(log(s(half(x))))}
          Weak Trs:
            {  half(s(s(x))) -> s(half(x))
             , half(0()) -> 0()
             , log(s(0())) -> 0()}
          StartTerms: basic terms
          Strategy: innermost
        
        Certificate: YES(?,O(n^2))
        
        Proof:
          The following argument positions are usable:
            Uargs(half) = {}, Uargs(s) = {1}, Uargs(log) = {1}
          We have the following restricted  polynomial interpretation:
          Interpretation Functions:
           [half](x1) = x1
           [0]() = 1
           [s](x1) = 2 + x1
           [log](x1) = x1 + 2*x1^2

Hurray, we answered YES(?,O(n^2))