19.2 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 Gallinalangage. 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'' polynom
((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 normalized polynom:
(plus (mult (2) (if b then (4) else (f (3))))
(plus (mult (f (5)) x) (mult x x)))