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4.5.5 boolean expressions

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

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

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

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

  1. integer comparisons:
      i == j
      i != j    // or     i <> j
      i <= j
      i >= j
      i > j
      i < j
  2. number comparisons:
      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. polynomial or vector comparisons:
      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. intmat or matrix comparisons:
      v == w
      v != w    // or     v <> w
  5. intvec or string comparisons:
      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.