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:
 Matrix Interpretation Processor: dim=1
  
  interpretation:
   [v] = 0,
   
   [u] = 0,
   
   [false] = 1,
   
   [if](x0, x1, x2) = x0 + x1 + x2,
   
   [true] = 1
  orientation:
   if(true(),x,y) = x + y + 1 >= x = x
   
   if(false(),x,y) = x + y + 1 >= y = y
   
   if(x,y,y) = x + 2y >= y = y
   
   if(if(x,y,z),u(),v()) = x + y + z >= x + y + z = if(x,if(y,u(),v()),if(z,u(),v()))
  problem:
   if(x,y,y) -> y
   if(if(x,y,z),u(),v()) -> if(x,if(y,u(),v()),if(z,u(),v()))
  Matrix Interpretation Processor: dim=1
   
   interpretation:
    [v] = 0,
    
    [u] = 1,
    
    [if](x0, x1, x2) = 5x0 + 2x1 + x2 + 2
   orientation:
    if(x,y,y) = 5x + 3y + 2 >= y = y
    
    if(if(x,y,z),u(),v()) = 25x + 10y + 5z + 14 >= 5x + 10y + 5z + 14 = if(x,if(y,u(),v()),if(z,u(),v()))
   problem:
    if(if(x,y,z),u(),v()) -> if(x,if(y,u(),v()),if(z,u(),v()))
   Matrix Interpretation Processor: dim=1
    
    interpretation:
     [v] = 5,
     
     [u] = 0,
     
     [if](x0, x1, x2) = 3x0 + x1 + x2 + 6
    orientation:
     if(if(x,y,z),u(),v()) = 9x + 3y + 3z + 29 >= 3x + 3y + 3z + 28 = if(x,if(y,u(),v()),if(z,u(),v()))
    problem:
     
    Qed