 
 
 
 
20.2 The variables map
It is frequent to have an expression built with + and
 ×, but rarely on variables only.
Let us associate a number to each subterm of a ring
expression in the Gallina language. For example in the ring
nat, consider the expression:
(plus (mult (plus (f (5)) x) x)
      (mult (if b then (4) else (f (3))) (2)))
As a ring expression, is has 3 subterms. Give each subterm a
number in an arbitrary order:
| 0 | |® | if b then (4) else (f (3)) | 
| 1 | |® | (f (5)) | 
| 2 | |® | x | 
Then normalize the ``abstract'' polynomial 
((V1 Ä V2) Å V2) Å (V0 Ä 2) 
In our example the normal form is:
(2 Ä V0) Å (V1 Ä V2) Å (V2 Ä V2)
Then substitute the variables by their values in the variables map to
get the concrete normal polynomial:
(plus (mult (2) (if b then (4) else (f (3)))) 
      (plus (mult (f (5)) x) (mult x x))) 
 
 
 
 
