YES

Problem:
 if(true(),x,y) -> x
 if(false(),x,y) -> y
 if(x,y,y) -> y
 if(if(x,y,z),u(),v()) -> if(x,if(y,u(),v()),if(z,u(),v()))

Proof:
 Polynomial Interpretation Processor:
  dimension: 1
  interpretation:
   [v] = 0,
   
   [u] = 0,
   
   [false] = 7,
   
   [if](x0, x1, x2) = 7x0 + x1 + 4x2 + 1,
   
   [true] = 0
  orientation:
   if(true(),x,y) = x + 4y + 1 >= x = x
   
   if(false(),x,y) = x + 4y + 50 >= y = y
   
   if(x,y,y) = 7x + 5y + 1 >= y = y
   
   if(if(x,y,z),u(),v()) = 49x + 7y + 28z + 8 >= 7x + 7y + 28z + 6 = if(x,if(y,u(),v()),if(z,u(),v()))
  problem:
   
  Qed