We consider the following Problem:

  Strict Trs:
    {  a__f(f(a())) -> a__f(g(f(a())))
     , mark(f(X)) -> a__f(mark(X))
     , mark(a()) -> a()
     , mark(g(X)) -> g(X)
     , a__f(X) -> f(X)}
  StartTerms: basic terms
  Strategy: innermost

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

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

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