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{xrst_begin base_ordered}
{xrst_spell
    geq
}

Base Type Requirements for Ordered Comparisons
##############################################

Purpose
*******
The following operations
(in the CppAD namespace) are required to use the type
``AD`` < *Base* > :

.. list-table::
    :widths: auto

    * - **Syntax**
      - **Result**
    * - *b* = ``GreaterThanZero`` ( *x* )
      - :math:`x > 0`
    * - *b* = ``GreaterThanOrZero`` ( *x* )
      - :math:`x \geq 0`
    * - *b* = ``LessThanZero`` ( *x* )
      - :math:`x < 0`
    * - *b* = ``LessThanOrZero`` ( *x* )
      - :math:`x \leq 0`
    * - *b* = ``abs_geq`` ( *x* , *y* )
      - :math:`|x| \geq |y|`.

where the arguments and return value have the prototypes

| |tab| ``const`` *Base* & *x*
| |tab| ``const`` *Base* & *y*
| |tab| ``bool`` *b*

Ordered Type
************
If the type *Base* supports ordered operations,
these functions should have their corresponding definitions.
For example,

| ``namespace CppAD`` {
| |tab| ``inline bool GreaterThanZero`` ( ``const`` *Base* & ``x`` )
| |tab| { ``return`` ( ``x > 0`` );
| |tab| }
| }

The other functions would replace ``>`` by the corresponding operator.
For example, see
:ref:`base_alloc<base_alloc.hpp@Ordered>` .

Not Ordered
***********
If the type *Base* does not support ordering,
one might (but need not) define ``GreaterThanZero`` as follows:

| ``namespace CppAD`` {
| |tab| ``inline bool GreaterThanZero`` ( ``const`` *Base* & ``x`` )
| |tab| {  // ``attempt to use GreaterThanZero with a`` *Base* ``argument``
| |tab| |tab| ``assert`` (0);
| |tab| |tab| ``return x`` ;
| |tab| }
| }

The other functions would have the corresponding definition.
For example, see
:ref:`complex Ordered<base_complex.hpp@Ordered>` .

Special Requirements
********************
The following are special requirements when
there is no ordered comparison for the base type:

Independent
===========
:ref:`Independent@record_compare` must be false
in the call to ``Independent`` .

Optimize
========
The :ref:`optimize@options@no_conditional_skip`
option must be present when ``optimize`` is used.

PrintFor
========
The :ref:`PrintFor-name` operator cannot be used.

{xrst_end base_ordered}