We consider the following Problem:

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

Certificate: YES(?,O(n^2))

Proof:
  We consider the following Problem:
  
    Strict Trs:
      {  f(a()) -> b()
       , f(c()) -> d()
       , f(g(x, y)) -> g(f(x), f(y))
       , f(h(x, y)) -> g(h(y, f(x)), h(x, f(y)))
       , g(x, x) -> h(e(), x)}
    StartTerms: basic terms
    Strategy: innermost
  
  Certificate: YES(?,O(n^2))
  
  Proof:
    The weightgap principle applies, where following rules are oriented strictly:
    
    TRS Component:
      {  f(a()) -> b()
       , f(c()) -> d()}
    
    Interpretation of nonconstant growth:
    -------------------------------------
      The following argument positions are usable:
        Uargs(f) = {}, Uargs(g) = {1, 2}, Uargs(h) = {2}
      We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
      Interpretation Functions:
       f(x1) = [0 0] x1 + [1]
               [0 0]      [1]
       a() = [0]
             [0]
       b() = [0]
             [0]
       c() = [0]
             [0]
       d() = [0]
             [0]
       g(x1, x2) = [1 1] x1 + [1 0] x2 + [0]
                   [0 0]      [0 0]      [1]
       h(x1, x2) = [0 0] x1 + [1 0] x2 + [0]
                   [0 0]      [0 0]      [1]
       e() = [0]
             [0]
    
    The strictly oriented rules are moved into the weak component.
    
    We consider the following Problem:
    
      Strict Trs:
        {  f(g(x, y)) -> g(f(x), f(y))
         , f(h(x, y)) -> g(h(y, f(x)), h(x, f(y)))
         , g(x, x) -> h(e(), x)}
      Weak Trs:
        {  f(a()) -> b()
         , f(c()) -> d()}
      StartTerms: basic terms
      Strategy: innermost
    
    Certificate: YES(?,O(n^2))
    
    Proof:
      The weightgap principle applies, where following rules are oriented strictly:
      
      TRS Component: {g(x, x) -> h(e(), x)}
      
      Interpretation of nonconstant growth:
      -------------------------------------
        The following argument positions are usable:
          Uargs(f) = {}, Uargs(g) = {1, 2}, Uargs(h) = {2}
        We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
        Interpretation Functions:
         f(x1) = [0 0] x1 + [1]
                 [0 0]      [0]
         a() = [0]
               [0]
         b() = [0]
               [0]
         c() = [0]
               [0]
         d() = [0]
               [0]
         g(x1, x2) = [1 1] x1 + [1 0] x2 + [1]
                     [0 0]      [0 1]      [1]
         h(x1, x2) = [0 0] x1 + [1 0] x2 + [0]
                     [0 0]      [0 0]      [0]
         e() = [0]
               [0]
      
      The strictly oriented rules are moved into the weak component.
      
      We consider the following Problem:
      
        Strict Trs:
          {  f(g(x, y)) -> g(f(x), f(y))
           , f(h(x, y)) -> g(h(y, f(x)), h(x, f(y)))}
        Weak Trs:
          {  g(x, x) -> h(e(), x)
           , f(a()) -> b()
           , f(c()) -> d()}
        StartTerms: basic terms
        Strategy: innermost
      
      Certificate: YES(?,O(n^2))
      
      Proof:
        The weightgap principle applies, where following rules are oriented strictly:
        
        TRS Component: {f(g(x, y)) -> g(f(x), f(y))}
        
        Interpretation of nonconstant growth:
        -------------------------------------
          The following argument positions are usable:
            Uargs(f) = {}, Uargs(g) = {1, 2}, Uargs(h) = {2}
          We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
          Interpretation Functions:
           f(x1) = [0 1] x1 + [0]
                   [0 1]      [0]
           a() = [0]
                 [0]
           b() = [0]
                 [0]
           c() = [0]
                 [0]
           d() = [0]
                 [0]
           g(x1, x2) = [1 0] x1 + [1 0] x2 + [0]
                       [0 1]      [0 1]      [1]
           h(x1, x2) = [0 0] x1 + [1 0] x2 + [0]
                       [0 1]      [0 1]      [0]
           e() = [0]
                 [1]
        
        The strictly oriented rules are moved into the weak component.
        
        We consider the following Problem:
        
          Strict Trs: {f(h(x, y)) -> g(h(y, f(x)), h(x, f(y)))}
          Weak Trs:
            {  f(g(x, y)) -> g(f(x), f(y))
             , g(x, x) -> h(e(), x)
             , f(a()) -> b()
             , f(c()) -> d()}
          StartTerms: basic terms
          Strategy: innermost
        
        Certificate: YES(?,O(n^2))
        
        Proof:
          We consider the following Problem:
          
            Strict Trs: {f(h(x, y)) -> g(h(y, f(x)), h(x, f(y)))}
            Weak Trs:
              {  f(g(x, y)) -> g(f(x), f(y))
               , g(x, x) -> h(e(), x)
               , f(a()) -> b()
               , f(c()) -> d()}
            StartTerms: basic terms
            Strategy: innermost
          
          Certificate: YES(?,O(n^2))
          
          Proof:
            The following argument positions are usable:
              Uargs(f) = {}, Uargs(g) = {1, 2}, Uargs(h) = {2}
            We have the following restricted  polynomial interpretation:
            Interpretation Functions:
             [f](x1) = 3*x1 + 2*x1^2
             [a]() = 0
             [b]() = 0
             [c]() = 0
             [d]() = 0
             [g](x1, x2) = 1 + x1 + x2
             [h](x1, x2) = 1 + x1 + x2
             [e]() = 0

Hurray, we answered YES(?,O(n^2))