Problem:
 f(g(x),g(y)) -> f(g(x),h(y))
 f(h(x),g(y)) -> f(g(x),g(y))
 f(g(x),h(y)) -> f(x,y)
 f(h(x),h(y)) -> f(y,x)
 f(x,y) -> f(y,x)
 g(x) -> h(x)
 h(x) -> g(x)

Proof:
 AT confluence processor
  Complete TRS T' of input TRS:
  f(g(x),h(y)) -> f(x,y)
  f(h(x),h(y)) -> f(y,x)
  f(g(x),g(y)) -> f(y,x)
  f(h(x),g(y)) -> f(x,y)
  f(g(x),g(y)) -> f(g(x),h(y))
  f(h(x),g(y)) -> f(g(x),g(y))
  f(x,y) -> f(y,x)
  g(x) -> h(x)
  h(x) -> g(x)
  
   T' = (P union S) with
  
   TRS P:f(g(x),g(y)) -> f(g(x),h(y))
         f(h(x),g(y)) -> f(g(x),g(y))
         f(x,y) -> f(y,x)
         g(x) -> h(x)
         h(x) -> g(x)
  
   TRS S:f(g(x),h(y)) -> f(x,y)
         f(h(x),h(y)) -> f(y,x)
         f(g(x),g(y)) -> f(y,x)
         f(h(x),g(y)) -> f(x,y)
  
  S is linear and P is reversible.
  
   CP(S,S) = 
  
   CP(S,P union P^-1) = 
  f(x837,x838) = f(h(x838),g(x837)), f(x,y) = f(g(x),g(y)), f(x841,x842) = 
  f(h(x842),g(x841)), f(x844,x843) = f(h(x844),h(x843)), f(x846,x845) = 
  f(h(x846),h(x845)), f(y,x) = f(g(x),h(y)), f(x850,x849) = f(g(x850),g(x849)), 
  f(y,x) = f(h(x),g(y)), f(x854,x853) = f(g(x854),g(x853)), f(x857,x858) = 
  f(g(x858),h(x857)), f(x859,x860) = f(g(x860),h(x859))
  
   CP(P union P^-1,S) = 
  f(g(x),h(y)) = f(y,x), f(g(x),g(y)) = f(x,y), f(h(y),g(x)) = f(x,y), 
  f(h(y),h(x)) = f(y,x), f(g(y),g(x)) = f(y,x), f(g(y),h(x)) = f(x,y), 
  f(h(x),h(y)) = f(x,y), f(h(x),g(y)) = f(y,x)
  
  
  We have to check termination of S:
  
  Matrix Interpretation Processor: dim=1
   
   interpretation:
    [f](x0, x1) = x0 + 4x1,
    
    [g](x0) = 6x0 + 4,
    
    [h](x0) = 7x0
   orientation:
    f(g(x),h(y)) = 6x + 28y + 4 >= x + 4y = f(x,y)
    
    f(h(x),h(y)) = 7x + 28y >= 4x + y = f(y,x)
    
    f(g(x),g(y)) = 6x + 24y + 20 >= 4x + y = f(y,x)
    
    f(h(x),g(y)) = 7x + 24y + 16 >= x + 4y = f(x,y)
   problem:
    f(h(x),h(y)) -> f(y,x)
   Matrix Interpretation Processor: dim=1
    
    interpretation:
     [f](x0, x1) = 2x0 + x1,
     
     [h](x0) = 3x0 + 1
    orientation:
     f(h(x),h(y)) = 6x + 3y + 3 >= x + 2y = f(y,x)
    problem:
     
    Qed