We consider the following Problem:

  Strict Trs:
    {  a(a(f(b(), a(x)))) -> f(a(a(a(x))), b())
     , a(a(x)) -> f(b(), a(f(a(x), b())))
     , f(a(x), b()) -> f(b(), a(x))}
  StartTerms: basic terms
  Strategy: innermost

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

Proof:
  We consider the following Problem:
  
    Strict Trs:
      {  a(a(f(b(), a(x)))) -> f(a(a(a(x))), b())
       , a(a(x)) -> f(b(), a(f(a(x), b())))
       , f(a(x), b()) -> f(b(), a(x))}
    StartTerms: basic terms
    Strategy: innermost
  
  Certificate: YES(?,O(n^1))
  
  Proof:
    The weightgap principle applies, where following rules are oriented strictly:
    
    TRS Component: {a(a(f(b(), a(x)))) -> f(a(a(a(x))), b())}
    
    Interpretation of nonconstant growth:
    -------------------------------------
      We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
      Interpretation Functions:
       a(x1) = [1 0] x1 + [2]
               [0 0]      [0]
       f(x1, x2) = [1 1] x1 + [1 0] x2 + [1]
                   [0 0]      [0 0]      [0]
       b() = [0]
             [1]
    
    The strictly oriented rules are moved into the weak component.
    
    We consider the following Problem:
    
      Strict Trs:
        {  a(a(x)) -> f(b(), a(f(a(x), b())))
         , f(a(x), b()) -> f(b(), a(x))}
      Weak Trs: {a(a(f(b(), a(x)))) -> f(a(a(a(x))), b())}
      StartTerms: basic terms
      Strategy: innermost
    
    Certificate: YES(?,O(n^1))
    
    Proof:
      The weightgap principle applies, where following rules are oriented strictly:
      
      TRS Component: {a(a(x)) -> f(b(), a(f(a(x), b())))}
      
      Interpretation of nonconstant growth:
      -------------------------------------
        We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
        Interpretation Functions:
         a(x1) = [1 2] x1 + [0]
                 [0 0]      [2]
         f(x1, x2) = [1 0] x1 + [1 0] x2 + [0]
                     [0 0]      [0 1]      [0]
         b() = [0]
               [0]
      
      The strictly oriented rules are moved into the weak component.
      
      We consider the following Problem:
      
        Strict Trs: {f(a(x), b()) -> f(b(), a(x))}
        Weak Trs:
          {  a(a(x)) -> f(b(), a(f(a(x), b())))
           , a(a(f(b(), a(x)))) -> f(a(a(a(x))), b())}
        StartTerms: basic terms
        Strategy: innermost
      
      Certificate: YES(?,O(n^1))
      
      Proof:
        We consider the following Problem:
        
          Strict Trs: {f(a(x), b()) -> f(b(), a(x))}
          Weak Trs:
            {  a(a(x)) -> f(b(), a(f(a(x), b())))
             , a(a(f(b(), a(x)))) -> f(a(a(a(x))), b())}
          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:
          {  a_0(2) -> 1
           , f_0(2, 2) -> 1
           , b_0() -> 2}

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