YES

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:
 DP Processor:
  DPs:
   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)))
  TRS:
   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)))
  Arctic Interpretation Processor:
   dimension: 2
   interpretation:
    [q#](x0) = [-& 0 ]x0 + [0],
    
    [p#](x0) = [-& 0 ]x0 + [0],
    
              [-& 0 ]     [0]
    [q](x0) = [-& 0 ]x0 + [0],
    
              [0  2 ]     [3]
    [g](x0) = [0  -&]x0 + [2],
    
              [-& 0 ]     [0]
    [p](x0) = [-& 0 ]x0 + [0],
    
              [-& 0 ]     [0]
    [f](x0) = [1  2 ]x0 + [3]
   orientation:
    p#(f(f(x))) = [3 4]x + [5] >= [2 3]x + [4] = q#(f(g(x)))
    
    p#(g(g(x))) = [0 2]x + [3] >= [-& 0 ]x + [2] = q#(g(f(x)))
    
    q#(f(f(x))) = [3 4]x + [5] >= [2 3]x + [4] = p#(f(g(x)))
    
    q#(g(g(x))) = [0 2]x + [3] >= [-& 0 ]x + [2] = p#(g(f(x)))
    
                 [3 4]    [5]    [2 3]    [4]             
    p(f(f(x))) = [3 4]x + [5] >= [2 3]x + [4] = q(f(g(x)))
    
                 [0 2]    [3]    [-& 0 ]    [2]             
    p(g(g(x))) = [0 2]x + [3] >= [-& 0 ]x + [2] = q(g(f(x)))
    
                 [3 4]    [5]    [2 3]    [4]             
    q(f(f(x))) = [3 4]x + [5] >= [2 3]x + [4] = p(f(g(x)))
    
                 [0 2]    [3]    [-& 0 ]    [2]             
    q(g(g(x))) = [0 2]x + [3] >= [-& 0 ]x + [2] = p(g(f(x)))
   problem:
    DPs:
     
    TRS:
     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)))
   Qed