4.3.5 boolean expressions

A boolean expression is really an int expression used in a logical context:

An int expression (<> 0 evaluates to TRUE (represented by 1), 0 represents FALSE).

The following is the list of available comparisons of objects of the same type.

Note: There are no comparisons for ideals and modules, resolution and maps.

1. an integer comparison:

   i == j i != j // or i <> j i <= j i >= j i > j i < j

2. a number comparison:

   m == n m != n // or m <> n m < n m > n m <= n m >= n

   For numbers from Z/p or from field extensions not all operations are useful:
   - 0 is always the smallest element,
   - in Z/p the representatives in the range -(p-1)/2..(p-1)/2 when p>2 resp. 0 and 1 for p=2 are used for comparisons,
   - in field extensions the last two operations (>=,<=) yield always TRUE (1) and the < and > are equivalent to !=.

3. a polynomial or vector comparison:

   f == g f != g // or f <> g f <= g // comparing the leading term w.r.t. the monomial order f < g f >= g f > g

4. an intmat or matrix comparison:

   v == w v != w // or v <> w

5. an intvec or string comparison:

   f == g f != g // or f <> g f <= g // comparing lexicographically f >= g // w.r.t. the order specified by ASCII f > g f < g

6. boolean expressions combined by boolean operations (and, or, not)

Note: All arguments of a logical expression are first evaluated and then the value of the logical expression is determined. For example, the logical expression (a || b) is evaluated by first evaluating a and b, even though the value of b has no influence on the value of (a || b), if a evaluates to true.

Note that this evaluation is different from the left-to-right, conditional evaluation of logical expressions (as found in most programming languages). For example, in these other languages, the value of (1 || b) is determined without ever evaluating b.

See Major differences to the C programming language.