409 

   410 The other interesting case with simplification is the 

   411 sequence case. Here the main simplification function 

   412 is as follows

   413 

   414 \begin{center}

   415 \begin{tabular}{l}

   416 $simp(r)$:\qquad\qquad (continued)\\

   417 \quad case $r = r_1 \cdot r_2$\\

   418 \qquad let $(r_{1s}, f_{1s}) = simp(r_1)$\\

   419 \qquad \phantom{let} $(r_{2s}, f_{2s}) = simp(r_2)$\smallskip\\

   420 \qquad case $r_{1s} = \varnothing$: 

   421        return $(\varnothing, f_{error})$\\

   422 \qquad case $r_{2s} = \varnothing$: 

   423        return $(\varnothing, f_{error})$\\

   424 \qquad case $r_{1s} = \epsilon$: 

   425 return $(r_{2s}, \lambda v. \,Seq(f_{1s}(Empty), f_{2s}(v)))$\\

   426 \qquad case $r_{2s} = \epsilon$: 

   427 return $(r_{1s}, \lambda v. \,Seq(f_{1s}(v), f_{2s}(Empty)))$\\

   428 \qquad otherwise: 

   429        return $(r_{1s} \cdot r_{2s}, f_{seq}(f_{1s}, f_{2s}))$\\

   430 \end{tabular}

   431 \end{center}

   432 

   433 \noindent whereby $f_{seq}$ is again pushing the two 

   434 rectification functions into the two components of

   435 the Seq-value:

   436 

   437 \begin{center}

   438 \begin{tabular}{l@{\hspace{1mm}}l}

   439 $f_{seq}(f_1, f_2) \dn$\\

   440 \qquad $\lambda v.\,$ case $v = Seq(v_1, v_2)$: 

   441       & return $Seq(f_1(v_1), f_2(v_2))$\\

   442 \end{tabular}

   443 \end{center}

   444 

   445 \noindent Note that in the case of $r_{1s}$ (similarly $r_{2s}$)

   446 we use the function $f_{error}$ for rectification. If you 

   447 think carefully, then you will see that this function will

   448 actually never been called. Because a sequence with 

   449 $\varnothing$ will never recognise any string and therefore

   450 the second phase of the algorithm would never been called.

   451 The simplification function still expects us to give a 

   452 function. So in my own implementation I just returned 

   453 a function which raises an error. 

   454 

   455