We consider the following Problem:

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

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

Proof:
  We consider the following Problem:
  
    Strict Trs:
      {  average(s(x), y) -> average(x, s(y))
       , average(x, s(s(s(y)))) -> s(average(s(x), y))
       , average(0(), 0()) -> 0()
       , average(0(), s(0())) -> 0()
       , average(0(), s(s(0()))) -> s(0())}
    StartTerms: basic terms
    Strategy: innermost
  
  Certificate: YES(?,O(n^1))
  
  Proof:
    The weightgap principle applies, where following rules are oriented strictly:
    
    TRS Component:
      {  average(0(), 0()) -> 0()
       , average(0(), s(0())) -> 0()
       , average(0(), s(s(0()))) -> s(0())}
    
    Interpretation of nonconstant growth:
    -------------------------------------
      The following argument positions are usable:
        Uargs(average) = {}, Uargs(s) = {1}
      We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
      Interpretation Functions:
       average(x1, x2) = [0 0] x1 + [0 0] x2 + [1]
                         [0 0]      [0 0]      [1]
       s(x1) = [1 2] x1 + [0]
               [0 0]      [1]
       0() = [0]
             [0]
    
    The strictly oriented rules are moved into the weak component.
    
    We consider the following Problem:
    
      Strict Trs:
        {  average(s(x), y) -> average(x, s(y))
         , average(x, s(s(s(y)))) -> s(average(s(x), y))}
      Weak Trs:
        {  average(0(), 0()) -> 0()
         , average(0(), s(0())) -> 0()
         , average(0(), s(s(0()))) -> s(0())}
      StartTerms: basic terms
      Strategy: innermost
    
    Certificate: YES(?,O(n^1))
    
    Proof:
      The weightgap principle applies, where following rules are oriented strictly:
      
      TRS Component: {average(x, s(s(s(y)))) -> s(average(s(x), y))}
      
      Interpretation of nonconstant growth:
      -------------------------------------
        The following argument positions are usable:
          Uargs(average) = {}, Uargs(s) = {1}
        We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
        Interpretation Functions:
         average(x1, x2) = [0 0] x1 + [0 1] x2 + [0]
                           [0 0]      [0 1]      [0]
         s(x1) = [1 0] x1 + [0]
                 [0 1]      [1]
         0() = [0]
               [0]
      
      The strictly oriented rules are moved into the weak component.
      
      We consider the following Problem:
      
        Strict Trs: {average(s(x), y) -> average(x, s(y))}
        Weak Trs:
          {  average(x, s(s(s(y)))) -> s(average(s(x), y))
           , average(0(), 0()) -> 0()
           , average(0(), s(0())) -> 0()
           , average(0(), s(s(0()))) -> s(0())}
        StartTerms: basic terms
        Strategy: innermost
      
      Certificate: YES(?,O(n^1))
      
      Proof:
        The weightgap principle applies, where following rules are oriented strictly:
        
        TRS Component: {average(s(x), y) -> average(x, s(y))}
        
        Interpretation of nonconstant growth:
        -------------------------------------
          The following argument positions are usable:
            Uargs(average) = {}, Uargs(s) = {1}
          We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
          Interpretation Functions:
           average(x1, x2) = [0 1] x1 + [1 0] x2 + [0]
                             [0 1]      [0 1]      [0]
           s(x1) = [1 0] x1 + [1]
                   [0 1]      [2]
           0() = [0]
                 [0]
        
        The strictly oriented rules are moved into the weak component.
        
        We consider the following Problem:
        
          Weak Trs:
            {  average(s(x), y) -> average(x, s(y))
             , average(x, s(s(s(y)))) -> s(average(s(x), y))
             , average(0(), 0()) -> 0()
             , average(0(), s(0())) -> 0()
             , average(0(), s(s(0()))) -> s(0())}
          StartTerms: basic terms
          Strategy: innermost
        
        Certificate: YES(O(1),O(1))
        
        Proof:
          We consider the following Problem:
          
            Weak Trs:
              {  average(s(x), y) -> average(x, s(y))
               , average(x, s(s(s(y)))) -> s(average(s(x), y))
               , average(0(), 0()) -> 0()
               , average(0(), s(0())) -> 0()
               , average(0(), s(s(0()))) -> s(0())}
            StartTerms: basic terms
            Strategy: innermost
          
          Certificate: YES(O(1),O(1))
          
          Proof:
            Empty rules are trivially bounded

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