YES

Problem:
 f(f(x,y,z),u,f(x,y,v)) -> f(x,y,f(z,u,v))
 f(x,y,y) -> y
 f(x,y,g(y)) -> x
 f(x,x,y) -> x
 f(g(x),x,y) -> y

Proof:
 Matrix Interpretation Processor: dim=1
  
  interpretation:
   [g](x0) = 4x0 + 4,
   
   [f](x0, x1, x2) = 2x0 + 2x1 + x2
  orientation:
   f(f(x,y,z),u,f(x,y,v)) = 2u + v + 6x + 6y + 2z >= 2u + v + 2x + 2y + 2z = f(x,y,f(z,u,v))
   
   f(x,y,y) = 2x + 3y >= y = y
   
   f(x,y,g(y)) = 2x + 6y + 4 >= x = x
   
   f(x,x,y) = 4x + y >= x = x
   
   f(g(x),x,y) = 10x + y + 8 >= y = y
  problem:
   f(f(x,y,z),u,f(x,y,v)) -> f(x,y,f(z,u,v))
   f(x,y,y) -> y
   f(x,x,y) -> x
  Matrix Interpretation Processor: dim=1
   
   interpretation:
    [f](x0, x1, x2) = x0 + 2x1 + x2 + 7
   orientation:
    f(f(x,y,z),u,f(x,y,v)) = 2u + v + 2x + 4y + z + 21 >= 2u + v + x + 2y + z + 14 = f(x,y,f(z,u,v))
    
    f(x,y,y) = x + 3y + 7 >= y = y
    
    f(x,x,y) = 3x + y + 7 >= x = x
   problem:
    
   Qed