Duality and De Morgan Principles for Lists

Abstract

Lists are the most important data type in functional programming. This is because they replace the use of iteration in imperative programming. For example, if we are asked to sum the values of f(i), for i ranging from 1 to 1000, a problem that the imperative programmer would be likely to solve by writing, say

$$ \begin{gathered} x\,: \equiv 0; \hfill \\ for\,i\,: \equiv \,1\,to\,1000 \hfill \\ do\,x\,: \equiv \,x\, + \,f\left( i \right); \hfill \\ \end{gathered} $$

the functional programmer would probably write something like

$$sum\left( {map\,f[1..1000]} \right)$$

and we see that the sequence of values assumed over time by i and x, respectively, in the imperative program are each now represented by a list.