We consider the following Problem:

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

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

Proof:
  We consider the following Problem:
  
    Strict Trs:
      {  f(h(x)) -> f(i(x))
       , g(i(x)) -> g(h(x))
       , h(a()) -> b()
       , i(a()) -> b()}
    StartTerms: basic terms
    Strategy: innermost
  
  Certificate: YES(?,O(n^1))
  
  Proof:
    The weightgap principle applies, where following rules are oriented strictly:
    
    TRS Component: {g(i(x)) -> g(h(x))}
    
    Interpretation of nonconstant growth:
    -------------------------------------
      The following argument positions are usable:
        Uargs(f) = {1}, Uargs(h) = {}, Uargs(i) = {}, Uargs(g) = {1}
      We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
      Interpretation Functions:
       f(x1) = [1 0] x1 + [1]
               [0 0]      [1]
       h(x1) = [0 0] x1 + [0]
               [0 0]      [0]
       i(x1) = [0 0] x1 + [0]
               [0 0]      [2]
       g(x1) = [1 1] x1 + [1]
               [0 0]      [1]
       a() = [0]
             [0]
       b() = [0]
             [0]
    
    The strictly oriented rules are moved into the weak component.
    
    We consider the following Problem:
    
      Strict Trs:
        {  f(h(x)) -> f(i(x))
         , h(a()) -> b()
         , i(a()) -> b()}
      Weak Trs: {g(i(x)) -> g(h(x))}
      StartTerms: basic terms
      Strategy: innermost
    
    Certificate: YES(?,O(n^1))
    
    Proof:
      The weightgap principle applies, where following rules are oriented strictly:
      
      TRS Component: {i(a()) -> b()}
      
      Interpretation of nonconstant growth:
      -------------------------------------
        The following argument positions are usable:
          Uargs(f) = {1}, Uargs(h) = {}, Uargs(i) = {}, Uargs(g) = {1}
        We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
        Interpretation Functions:
         f(x1) = [1 0] x1 + [1]
                 [0 0]      [1]
         h(x1) = [0 0] x1 + [0]
                 [0 0]      [0]
         i(x1) = [0 0] x1 + [1]
                 [0 0]      [1]
         g(x1) = [1 0] x1 + [0]
                 [0 0]      [1]
         a() = [0]
               [0]
         b() = [0]
               [0]
      
      The strictly oriented rules are moved into the weak component.
      
      We consider the following Problem:
      
        Strict Trs:
          {  f(h(x)) -> f(i(x))
           , h(a()) -> b()}
        Weak Trs:
          {  i(a()) -> b()
           , g(i(x)) -> g(h(x))}
        StartTerms: basic terms
        Strategy: innermost
      
      Certificate: YES(?,O(n^1))
      
      Proof:
        The weightgap principle applies, where following rules are oriented strictly:
        
        TRS Component: {f(h(x)) -> f(i(x))}
        
        Interpretation of nonconstant growth:
        -------------------------------------
          The following argument positions are usable:
            Uargs(f) = {1}, Uargs(h) = {}, Uargs(i) = {}, Uargs(g) = {1}
          We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
          Interpretation Functions:
           f(x1) = [1 1] x1 + [1]
                   [0 0]      [1]
           h(x1) = [0 0] x1 + [0]
                   [0 0]      [2]
           i(x1) = [0 0] x1 + [0]
                   [0 0]      [0]
           g(x1) = [1 0] x1 + [1]
                   [0 0]      [1]
           a() = [0]
                 [0]
           b() = [0]
                 [0]
        
        The strictly oriented rules are moved into the weak component.
        
        We consider the following Problem:
        
          Strict Trs: {h(a()) -> b()}
          Weak Trs:
            {  f(h(x)) -> f(i(x))
             , i(a()) -> b()
             , g(i(x)) -> g(h(x))}
          StartTerms: basic terms
          Strategy: innermost
        
        Certificate: YES(?,O(n^1))
        
        Proof:
          The weightgap principle applies, where following rules are oriented strictly:
          
          TRS Component: {h(a()) -> b()}
          
          Interpretation of nonconstant growth:
          -------------------------------------
            The following argument positions are usable:
              Uargs(f) = {1}, Uargs(h) = {}, Uargs(i) = {}, Uargs(g) = {1}
            We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
            Interpretation Functions:
             f(x1) = [1 0] x1 + [0]
                     [0 0]      [1]
             h(x1) = [0 0] x1 + [1]
                     [0 0]      [1]
             i(x1) = [0 0] x1 + [0]
                     [0 0]      [3]
             g(x1) = [1 3] x1 + [0]
                     [0 0]      [1]
             a() = [0]
                   [0]
             b() = [0]
                   [0]
          
          The strictly oriented rules are moved into the weak component.
          
          We consider the following Problem:
          
            Weak Trs:
              {  h(a()) -> b()
               , f(h(x)) -> f(i(x))
               , i(a()) -> b()
               , g(i(x)) -> g(h(x))}
            StartTerms: basic terms
            Strategy: innermost
          
          Certificate: YES(O(1),O(1))
          
          Proof:
            We consider the following Problem:
            
              Weak Trs:
                {  h(a()) -> b()
                 , f(h(x)) -> f(i(x))
                 , i(a()) -> b()
                 , g(i(x)) -> g(h(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))