YES
O(n^2)
TRS:
 {
  q(f(f(x))) -> p(f(g(x))),
  q(g(g(x))) -> p(g(f(x))),
  p(f(f(x))) -> q(f(g(x))),
  p(g(g(x))) -> q(g(f(x)))
 }
 DUP: We consider a non-duplicating system.
  Trs:
   {
    q(f(f(x))) -> p(f(g(x))),
    q(g(g(x))) -> p(g(f(x))),
    p(f(f(x))) -> q(f(g(x))),
    p(g(g(x))) -> q(g(f(x)))
   }
  Matrix Interpretation:
   Interpretation class: triangular
       [X1]    [1 3][X1]   [0]
   [p]([X0]) = [0 0][X0] + [0]
   
       [X1]    [1 0][X1]   [1]
   [g]([X0]) = [0 0][X0] + [0]
   
       [X1]    [1 0][X1]   [0]
   [f]([X0]) = [0 1][X0] + [2]
   
       [X1]    [1 2][X1]   [0]
   [q]([X0]) = [0 0][X0] + [0]
   
   
   Qed