We consider the following Problem:

  Strict Trs:
    {  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))))}
  StartTerms: basic terms
  Strategy: innermost

Certificate: YES(?,O(n^1))

Proof:
  We consider the following Problem:
  
    Strict Trs:
      {  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))))}
    StartTerms: basic terms
    Strategy: innermost
  
  Certificate: YES(?,O(n^1))
  
  Proof:
    The weightgap principle applies, where following rules are oriented strictly:
    
    TRS Component:
      {  half(0()) -> 0()
       , half(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 + [0]
                 [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))
         , s(log(0())) -> s(0())
         , log(s(x)) -> s(log(half(s(x))))}
      Weak Trs:
        {  half(0()) -> 0()
         , half(s(0())) -> 0()}
      StartTerms: basic terms
      Strategy: innermost
    
    Certificate: YES(?,O(n^1))
    
    Proof:
      The weightgap principle applies, where following rules are oriented strictly:
      
      TRS Component: {s(log(0())) -> s(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() = [1]
               [0]
         s(x1) = [1 0] x1 + [0]
                 [0 0]      [1]
         log(x1) = [1 0] x1 + [2]
                   [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(x)) -> s(log(half(s(x))))}
        Weak Trs:
          {  s(log(0())) -> s(0())
           , half(0()) -> 0()
           , half(s(0())) -> 0()}
        StartTerms: basic terms
        Strategy: innermost
      
      Certificate: YES(?,O(n^1))
      
      Proof:
        The weightgap principle applies, where following rules are oriented strictly:
        
        TRS Component: {log(s(x)) -> s(log(half(s(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) = [0 0] x1 + [0]
                      [0 0]      [0]
           0() = [0]
                 [0]
           s(x1) = [1 0] x1 + [0]
                   [0 0]      [3]
           log(x1) = [1 2] x1 + [2]
                     [0 0]      [3]
        
        The strictly oriented rules are moved into the weak component.
        
        We consider the following Problem:
        
          Strict Trs: {half(s(s(x))) -> s(half(x))}
          Weak Trs:
            {  log(s(x)) -> s(log(half(s(x))))
             , s(log(0())) -> s(0())
             , half(0()) -> 0()
             , half(s(0())) -> 0()}
          StartTerms: basic terms
          Strategy: innermost
        
        Certificate: YES(?,O(n^1))
        
        Proof:
          We consider the following Problem:
          
            Strict Trs: {half(s(s(x))) -> s(half(x))}
            Weak Trs:
              {  log(s(x)) -> s(log(half(s(x))))
               , s(log(0())) -> s(0())
               , half(0()) -> 0()
               , half(s(0())) -> 0()}
            StartTerms: basic terms
            Strategy: innermost
          
          Certificate: YES(?,O(n^1))
          
          Proof:
            The problem is match-bounded by 0.
            The enriched problem is compatible with the following automaton:
            {  half_0(2) -> 1
             , 0_0() -> 1
             , 0_0() -> 2
             , s_0(2) -> 1
             , log_0(2) -> 1}

Hurray, we answered YES(?,O(n^1))