YES(?,O(n^2))

Problem:
 f(h(x)) -> f(i(x))
 g(i(x)) -> g(h(x))
 h(a()) -> b()
 i(a()) -> b()

Proof:
 Complexity Transformation Processor:
  strict:
   f(h(x)) -> f(i(x))
   g(i(x)) -> g(h(x))
   h(a()) -> b()
   i(a()) -> b()
  weak:
   
  Matrix Interpretation Processor:
   dimension: 1
   max_matrix:
    1
    interpretation:
     [b] = 0,
     
     [a] = 1,
     
     [g](x0) = x0,
     
     [i](x0) = x0,
     
     [f](x0) = x0,
     
     [h](x0) = x0 + 1
    orientation:
     f(h(x)) = x + 1 >= x = f(i(x))
     
     g(i(x)) = x >= x + 1 = g(h(x))
     
     h(a()) = 2 >= 0 = b()
     
     i(a()) = 1 >= 0 = b()
    problem:
     strict:
      g(i(x)) -> g(h(x))
     weak:
      f(h(x)) -> f(i(x))
      h(a()) -> b()
      i(a()) -> b()
    Matrix Interpretation Processor:
     dimension: 2
     max_matrix:
      [1 1]
      [0 0]
      interpretation:
             [0]
       [b] = [0],
       
             [1]
       [a] = [0],
       
                 [1 1]     [0]
       [g](x0) = [0 0]x0 + [1],
       
                 [1 0]     [0]
       [i](x0) = [0 0]x0 + [1],
       
                 [1 0]  
       [f](x0) = [0 0]x0,
       
                 [1 0]  
       [h](x0) = [0 0]x0
      orientation:
                 [1 0]    [1]    [1 0]    [0]          
       g(i(x)) = [0 0]x + [1] >= [0 0]x + [1] = g(h(x))
       
                 [1 0]     [1 0]           
       f(h(x)) = [0 0]x >= [0 0]x = f(i(x))
       
                [1]    [0]      
       h(a()) = [0] >= [0] = b()
       
                [1]    [0]      
       i(a()) = [1] >= [0] = b()
      problem:
       strict:
        
       weak:
        g(i(x)) -> g(h(x))
        f(h(x)) -> f(i(x))
        h(a()) -> b()
        i(a()) -> b()
      Qed