20 Booleans
  
  The  two  main boolean values are true and false. They stand for the logical
  values  of  the  same  name.  They  appear  as  values  of the conditions in
  if-statements  and while-loops. Booleans are also important as return values
  of  filters  (see 13.2)  such as IsFinite (30.4-2) and IsBool (20.1-1). Note
  that  it  is  a  convention that the name of a function that returns true or
  false according to the outcome, starts with Is.
  
  For  technical  reasons,  also  the  value  fail (see 20.2) is regarded as a
  boolean.
  
  
  20.1 IsBool (Filter)
  
  20.1-1 IsBool
  
  IsBool( obj )  Category
  
  tests whether obj is true, false or fail.
  
    Example  
    gap> IsBool( true );  IsBool( false );  IsBool( 17 );
    true
    true
    false
  
  
  
  20.2 Fail (Variable)
  
  20.2-1 fail
  
  fail  global variable
  
  The value fail is used to indicate situations when an operation could not be
  performed  for  the  given  arguments, either because of shortcomings of the
  arguments or because of restrictions in the implementation or computability.
  So for example Position (21.16-1) will return fail if the point searched for
  is not in the list.
  
  fail  is  simply  an  object  that is different from every other object than
  itself.
  
  For technical reasons, fail is a boolean value. But note that fail cannot be
  used to form boolean expressions with and, or, and not (see 20.4 below), and
  fail cannot appear in boolean lists (see Chapter 22).
  
  
  20.3 Comparisons of Booleans
  
  
  20.3-1 Equality and inequality of Booleans
  
  bool1 = bool2
  
  bool1 <> bool2
  
  The  equality  operator  = evaluates to true if the two boolean values bool1
  and bool2 are equal, i.e., both are true or both are false or both fail, and
  false  otherwise.  The  inequality  operator <> evaluates to true if the two
  boolean  values  bool1,  bool2  are  different,  and  false  otherwise. This
  operation  is  also  called  the  exclusive or, because its value is true if
  exactly one of bool1 or bool2 is true.
  
  You  can  compare boolean values with objects of other types. Of course they
  are never equal.
  
    Example  
    gap> true = false;
    false
    gap> false = (true = fail);
    true
    gap> true <> 17;
    true
  
  
  
  20.3-2 Ordering of Booleans
  
  bool1 < bool2
  
  The  ordering  of  boolean values is defined by true < false < fail. For the
  comparison of booleans with other GAP objects, see Section 4.13.
  
    Example  
    gap> true < false;  fail >= false;
    true
    true
  
  
  
  20.4 Operations for Booleans
  
  The following boolean operations are only applicable to true and false.
  
  
  20.4-1 Logical disjunction
  
  bool1 or bool2
  
  The logical operator or evaluates to true if at least one of the two boolean
  operands bool1 and bool2 is true, and to false otherwise.
  
  or first evaluates bool1. If the value is neither true nor false an error is
  signalled.  If  the  value  is true, then or returns true without evaluating
  bool2.  If  the value is false, then or evaluates bool2. Again, if the value
  is  neither  true  nor false an error is signalled. Otherwise or returns the
  value of bool2. This short-circuited evaluation is important if the value of
  bool1  is  true  and  evaluation  of  bool2 would take much time or cause an
  error.
  
  or  is  associative,  i.e.,  it is allowed to write b1 or b2 or b3, which is
  interpreted as (b1 or b2) or b3. or has the lowest precedence of the logical
  operators.  All  logical operators have lower precedence than the comparison
  operators =, <, in, etc.
  
    Example  
    gap> true or false;
    true
    gap> false or false;
    false
    gap> i := -1;; l := [1,2,3];;
    gap> if i <= 0 or l[i] = false then     # this does not cause an error,
    >    Print("aha\n"); fi;                # because `l[i]' is not evaluated
    aha
  
  
  
  20.4-2 Logical conjunction
  
  bool1 and bool2
  
  fil1 and fil2
  
  The  logical  operator and evaluates to true if both boolean operands bool1,
  bool2 are true, and to false otherwise.
  
  and  first  evaluates bool1. If the value is neither true nor false an error
  is  signalled.  If  the  value  is  false,  then  and  returns false without
  evaluating  bool2. If the value is true, then and evaluates bool2. Again, if
  the  value  is  neither  true nor false an error is signalled. Otherwise and
  returns  the value of bool2. This short-circuited evaluation is important if
  the  value of bool1 is false and evaluation of bool2 would take much time or
  cause an error.
  
  and  is associative, i.e., it is allowed to write b1 and b2 and b3, which is
  interpreted  as  (b1  and  b2)  and  b3.  and has higher precedence than the
  logical  or  operator,  but  lower  than the unary logical not operator. All
  logical  operators have lower precedence than the comparison operators =, <,
  in, etc.
  
    Example  
    gap> true and false;
    false
    gap> true and true;
    true
    gap> false and 17;  # does not cause error, because 17 is never looked at
    false
  
  
  and can also be applied to filters. It returns a filter that when applied to
  some argument x, tests fil1(x) and fil2(x).
  
    Example  
    gap> andfilt:= IsPosRat and IsInt;;
    gap> andfilt( 17 );  andfilt( 1/2 );
    true
    false
  
  
  
  20.4-3 Logical negation
  
  not bool
  
  The  logical  operator  not returns true if the boolean value bool is false,
  and  true otherwise. An error is signalled if bool does not evaluate to true
  or false.
  
  not  has higher precedence than the other logical operators, or and and. All
  logical  operators have lower precedence than the comparison operators =, <,
  in, etc.
  
    Example  
    gap> true and false;
    false
    gap> not true;
    false
    gap> not false;
    true