We consider the following Problem:

  Strict Trs:
    {  g(f(x), y) -> f(h(x, y))
     , h(x, y) -> g(x, f(y))}
  StartTerms: basic terms
  Strategy: innermost

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

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

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