YES(?,O(n^2))

Problem:
 p(f(f(x))) -> q(f(g(x)))
 p(g(g(x))) -> q(g(f(x)))
 q(f(f(x))) -> p(f(g(x)))
 q(g(g(x))) -> p(g(f(x)))

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