2. (a) You are given the following lambda expressions:

          pair   = Lx.Ly.Lf.((f x) y)
          first  = Lx.Ly.x
          second = Lx.Ly.y

       Evaluate the following function applications, showing
       each reduction step (=>) in your evaluations.

        (i)    P = ((pair a ) b)
        (ii)  (P first) 
        (iii) (P second)
                                                               (10 marks)

   (b) There are two approaches to evaluating function applications,
       namely "applicative order" (->) and "normal order" (=>). Explain 
       what these approaches mean and discuss the advantages and 
       disadvantages of each approach. 
                                                               (8 marks)

   (c) In ML we might define a replacement for the "if then else"
       construct as follows:

              fun cond c e1 e2 = if c then e1 else e2;

       We might now attempt to define the factorial function in terms 
       of the function cond as follows:

              fun fact n = cond (n = 0) 1 (n * fact(n-1));
  
       Show that the above function does not work, and explain what 
       this tells us about ML in general, and about the "if then else"
       construct in particular.
                                                               (7 marks)