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================
Logical Criteria
================

In order to process arbitrary expression-based rules, PEAK-Rules needs to
"understand" the way that conditions logically relate to each other.  This
document describes the design (and tests the implementation) of its logical
criteria management.  You do not need to read this unless you are extending or
interfacing with this subsystem directly, or just want to understand how this
stuff actually works!

The most important ideas here are implication, intersection, and disjunctive
normal form.  But don't panic if you don't know what those terms mean!  They're
really quite simple.

Implication means that if one thing is true, then so is another.  A implies B
if B is always true whenever A is true.  It doesn't matter what B is when A is
not true, however.  It could be true or false, we don't care.  Implication is
important for prioritizing which rules are "more specific" than others.

Intersection just means that both things have to be true for a condition to
be true - it's like the "and" of two conditions.  But rather than performing
an actual "and", we're creating a *new condition* that will only be true when
the two original conditions would be true.

And finally, disjunctive normal form (DNF) means "an OR of ANDs".  For example,
this expression is in DNF::

    (A and C) or (B and C) or (A and D) or (B and D)

But this equivalent expression is **not** in DNF::

    (A or B) and (C or D)

The criteria used to define generic function methods are likely to look more
like this, than they are to be in disjunctive normal form.  Therefore, we must
convert them in order to implement the Chambers & Chen dispatch algorithm
correctly (see `Indexing.txt`_).

.. _Indexing.txt: http://peak.telecommunity.com/DevCenter/PEAK-Rules/Indexing

We do this using the ``DisjunctionSet`` and ``OrElse`` classes to represent
overall expressions (sets or sequences of "ors"), and the ``Signature`` and
``Conjunction`` classes to represent sequences or sets of "and"-ed conditions.

Within a ``Signature``, the things that are "and"-ed together are a sequence
of ``Test`` instances.  A ``Test`` pairs a "dispatch expression" with a
"criterion".  For example, this expression::

    isinstance(x, Y)

would be represented internally as a ``Test`` instance like this::

    Test(IsInstance(Local('x')), Class(Y))

``Conjunction`` instances, on the other hand, are used to "and" together
criteria that apply to the same dispatch expression.  For example, this
expression::

    isinstance(x, Y) and isinstance(x, Z)

would be represented internally like this::

    Test(IsInstance(Local('x')), Conjunction([Class(Y), Class(Z)]))

The rest of this document describes how predicates, signatures, tests, dispatch
expressions, and criteria work together to create expressions in disjunctive
normal form, and whose implication of other expressions can be determined.

The basic logical functions we will use are ``implies()``, ``intersect()``,
``disjuncts()``, and ``negate()``, all of which are defined in
``peak.rules.core``::

    >>> from peak.rules.core import implies, intersect, disjuncts, negate, long



----------------------------------------
Boolean Conditions and Logical Operators
----------------------------------------

The most fundamental conditions are simply ``True`` and ``False``.  ``True``
represents a rule that *always* applies, while ``False`` represents a rule that
*never* applies.  Therefore, the result of intersecting ``True`` and any other
object, always returns that object, while intersecting ``False`` with any other
object returns ``False``::

    >>> intersect(False, False)
    False
    >>> intersect(False, True)
    False
    >>> intersect(True, False)
    False
    >>> intersect(True, True)
    True

    >>> intersect(object(), True)
    <object object at ...>

    >>> intersect(True, object())
    <object object at ...>

    >>> intersect(object(), False)
    False

    >>> intersect(False, object())
    False

Because ``True`` means "condition that always applies", *everything* implies
``True``, but ``True`` only implies itself::

    >>> implies(object(), True)
    True

    >>> implies(True, object())
    False

    >>> implies(True, True)
    True

On the other hand, because ``False`` means "condition that never applies",
``False`` implies *everything*.  (Because if you start from a false premise,
you can arrive at any conclusion!)::

    >>> implies(False, True)
    True

    >>> implies(False, object())
    True

However, no condition other than ``False`` can ever imply ``False``, because
all other conditions can *sometimes* apply::

    >>> implies(object(), False)
    False

    >>> implies(True, False)
    False

    >>> implies(False, False)
    True

Notice, by the way, a few important differences between ``implies()`` and
``intersect()``.  ``implies()`` *always* returns a boolean value, ``True`` or
``False``, because it's an immediate answer to the question of, "does the
second condition always apply if the first condition applies?"

``intersect()``, on the other hand, returns a *condition* that will always be
true when the original conditions apply.  So, if it returns a boolean value,
that's just an indication that the intersection of the two input conditions
would always apply or never apply.  Also, ``intersect()`` is logically
symmetrical, in that it doesn't matter what order the arguments are in, whereas
the order is critically important for ``implies()``.

However, ``intersect()`` methods must be order *preserving*, because the order
in which logical "and" operations occur is important.  Consider, for example,
the condition ``"y!=0 and z>x/y"``, in which it would be a bad thing to skip
the zero check before the division!

So, as we will see later on, when working with more complex conditions,
``intersect()`` methods must ensure that the subparts of the output condition
are in the same relative order as they were in the input.

(Also, note that in general, when you intersect two conditions, if one condition
implies the other, the result of the intersection is the implying condition.
This general rule greatly simplifies the implementation of most intersect
operations, since as long as there is an implication relationship defined
between conditions, many common cases of intersection can be handled
automatically.)

In contrast to both ``implies()`` and ``intersects()``, the ``disjuncts()``
function takes only a single argument, and returns a list of the "disjuncts"
(or-ed-together conditions) of its argument.  More precisely, it returns a list
of conditions that each imply the original condition.  That is, if any of the
disjuncts were true, then the original condition would also be true.

Thus, the ``disjuncts()`` of an arbitrary object will normally be a list
containing just that object::

    >>> disjuncts(object())
    [<object object at ...>]

    >>> disjuncts(True)
    [True]

But ``False`` is a special case; ``False`` has *no* disjuncts, since no other
condition can ever imply ``False``::

    >>> disjuncts(False)
    []

As a result, "or"-ing ``False`` with other conditions will simply remove the
``False`` from the resulting predicate, and conditions that can never be true
are not used for indexing or dispatching.

Another special case is tuples containing nested tuples::

    >>> disjuncts( (float, (int, str)) )
    [(<... 'float'>, <... 'int'>),
     (<... 'float'>, <... 'str'>)] 

    >>> disjuncts( ((int, str), object) )
    [(<... 'int'>, <... 'object'>),
     (<... 'str'>, <... 'object'>)] 

    >>> disjuncts( (object, (int, str), float) )
    [(<... 'object'>, <... 'int'>, <... 'float'>),
     (<... 'object'>, <... 'str'>, <... 'float'>)] 

    >>> disjuncts( ((int, str), (int, str)) )
    [(<... 'int'>, <... 'int'>),
     (<... 'str'>, <... 'int'>),
     (<... 'int'>, <... 'str'>),
     (<... 'str'>, <... 'str'>)] 

This lets you avoid writing lots of decorators for the cases where you want
more than one type (or ``istype()`` instance) to match in a given argument
position.  (As you can see, it's equivalent to specifying all the individual
combinations of specified types.)

Finally, the ``negate()`` function inverts the truth of a condition, e.g.::

    >>> negate(True)
    False

    >>> negate(False)
    True

Of course, it also applies to criteria other than pure boolean values, as we'll
see in the upcoming sections.


-------------------
"Criterion" Objects
-------------------

A criterion object describes a set of possible values for a dispatch
expression. There are several criterion types supplied with PEAK-Rules, but you
can also add your own, as long as they can be tested for implication with
``implies()``, and intersected with ``intersect()``.  (And if they represent an
"or" of sub-criteria, they should be able to provide their list of
``disjuncts()``.  They'll also need to be indexable, but more on that later in
other documents!)


"And"-ed Criteria
=================

Sometimes, more than one criterion is applied to the same dispatch expression.
For example in the expression ``x is not y and x is not z``, two criteria are
being applied to the identity of ``x``.  To represent this, we need a way to
represent a set of "and-ed" criteria.  ``peak.rules.criteria`` provides a base
class for this, called ``Conjunction``::

    >>> from peak.rules.criteria import Conjunction

This class is a subclass of ``frozenset``, but has a few additional features.
First, a ``Conjunction`` never contains redundant (implied) items.
For example, the conjunction of the classes ``object`` and ``int`` is ``int``,
because ``int`` already implies ``object``::

    >>> Conjunction([int, object])
    <... 'int'>

    >>> Conjunction([object, int])
    <... 'int'>

Notice also that instead of getting back a set with one member, we got back the
item that would have been in the set.  This helps to simplify the expression
structure.  As a further simplification, creating an empty conjunction returns
``True``, because "no conditions required" is the same as "always true"::

    >>> Conjunction([])
    True

A conjunction implies a condition, if any condition in the conjunction
implies the other condition::

    >>> implies(Conjunction([str, int]), str)
    True
    >>> implies(Conjunction([str, int]), int)
    True
    >>> implies(Conjunction([str, int]), object)
    True
    >>> implies(Conjunction([str, int]), float)
    False

A condition implies a conjunction, however, only if the condition implies
every part of the conjunction::

    >>> class a: pass
    >>> class b: pass
    >>> class c(a,b): pass
    >>> class d(a, int): pass

    >>> implies(c, Conjunction([a, b]))
    True
    >>> implies(a, Conjunction([a, b]))
    False

    >>> implies(Conjunction([c,d]), Conjunction([a, int]))
    True
    >>> implies(Conjunction([c, int]), Conjunction([a, int]))
    True
    >>> implies(Conjunction([a, int]), Conjunction([c, int]))
    False

(By the way, on a more sophisticated level of reasoning, you could say that
``Conjunction([str, int])`` should have equalled ``False`` above, since
there's no way for an object to be both an ``int`` and a ``str`` at the same
time.  But that would be an excursion into semantics and outside the bounds of
what PEAK-Rules can "reason" about using only logical implication as defined by
the ``implies()`` generic function.)

``Conjunction`` objects can be intersected with one another, or with
additional conditions, and the result is another ``Conjunction`` of the
same type as the leftmost set.  So, if we use subclasses of our own, the result
of intersecting them will be a conjunction of the correct subclass::

    >>> class MySet(Conjunction): pass

    >>> type(intersect(MySet([int, str]), float))
    <class 'MySet'>
    
    >>> intersect(MySet([int, str]), float) == MySet([int, str, float])
    True

    >>> intersect(float, MySet([int, str])) == MySet([float, int, str])
    True

    >>> intersect(MySet([d, c]), MySet([int, str])) == MySet([d,c,str])
    True

If you want to ensure that all items in a set are of appropriate type or value,
you can override ``__init__`` to do the checking, and raise an appropriate
error.  PEAK-Rules does this for its specialized conjunction classes, but uses
``if __debug__:`` and ``assert`` statements to avoid the extra overhead when
run with ``python -O``.  You may wish to do the same for your subclasses.


"Or"-ed Criteria
================

The ``DisjunctionSet`` and ``OrElse`` classes are used to represent sets and
sequences of "or"-ed criteria::

    >>> from peak.rules.criteria import DisjunctionSet, OrElse

Both types automatically exclude redundant (i.e. more-specific) criteria, and
can never contain less than 2 entries.  For example, "or"-ing ``object`` and
``int`` always returns ``object``, because ``object`` is implied by ``int``::

    >>> DisjunctionSet([int, object])
    <... 'object'>

    >>> DisjunctionSet([object, int])
    <... 'object'>

    >>> OrElse([int, object])
    <... 'object'>

    >>> OrElse([object, int])
    <... 'object'>

Notice that instead of getting back a set or sequence with one member, we got
back the item that would have been in the set.  This helps to simplify the
expression structure.  As a further simplification, creating an empty
disjunction returns ``False``, because "no conditions are sufficient" is the
same as "always false"::

    >>> DisjunctionSet([])
    False

    >>> OrElse([])
    False

In addition to eliminating redundancy, disjunction *sets* also flatten any
nested disjunctions::

    >>> DisjunctionSet([DisjunctionSet([1, 2]), DisjunctionSet([3, 4])])
    DisjunctionSet([1, 2, 3, 4])

This is because it uses the ``disjuncts()`` generic function to determine
whether any of the items it was given are "or"-ed conditions of some kind.  And
the ``disjuncts()`` of a ``DisjunctionSet`` are a list of its contents::

    >>> disjuncts(DisjunctionSet([1, 2, 3, 4]))
    [1, 2, 3, 4]

But ``OrElse`` sequences do not do this flattening, in order to avoid imposing
an arbitrary sequence on their contents::

    >>> OrElse([DisjunctionSet([1, 2]), DisjunctionSet([3, 4])])
    OrElse([DisjunctionSet([1, 2]), DisjunctionSet([3, 4])])

(The ``disjuncts()`` of an ``OrElse`` are much more complicated, as the
disjuncts of a Python expression like ``"a or b or c"`` reduce to ``"a"``,
``"(not a) and b"``, and ``"(not a and not b) and c"``!  We'll talk more about
this later, in the section on `Predicates`_ below.)

A disjunction only implies a condition if *all* conditions in the disjunction
imply the other condition::

    >>> implies(DisjunctionSet([str, int]), str)
    False
    >>> implies(DisjunctionSet([str, int]), int)
    False
    >>> implies(DisjunctionSet([str, int]), float)
    False
    >>> implies(DisjunctionSet([str, int]), object)
    True

    >>> implies(OrElse([str, int]), str)
    False
    >>> implies(OrElse([str, int]), int)
    False
    >>> implies(OrElse([str, int]), float)
    False
    >>> implies(OrElse([str, int]), object)
    True

A condition implies a disjunction, however, if the condition implies any part
of the disjunction::

    >>> class a: pass
    >>> class b: pass
    >>> class c(a,b): pass
    >>> class d(a, int): pass

    >>> implies(c, DisjunctionSet([a, b]))
    True
    >>> implies(a, DisjunctionSet([a, b]))
    True
    >>> implies(a, DisjunctionSet([int, str]))
    False
    >>> implies(DisjunctionSet([c,d]), DisjunctionSet([a, int]))
    True
    >>> implies(DisjunctionSet([c,int]), DisjunctionSet([a, int]))
    True

    >>> implies(DisjunctionSet([c, int]), True)
    True
    >>> implies(False, DisjunctionSet([c, int]))
    True

    >>> implies(c, OrElse([a, b]))
    True
    >>> implies(a, OrElse([a, b]))
    True
    >>> implies(a, OrElse([int, str]))
    False
    >>> implies(OrElse([c,d]), OrElse([a, int]))
    True
    >>> implies(OrElse([c,int]), OrElse([a, int]))
    True

    >>> implies(OrElse([c, int]), True)
    True
    >>> implies(False, OrElse([c, int]))
    True

The intersection of a disjunction and any other object is a disjunction
containing the intersection of that object with the original disjunctions'
contents.  In other words::

    >>> int_or_str = DisjunctionSet([int, str])
    >>> long_or_float = DisjunctionSet([long, float])

    >>> intersect(int_or_str, float) == DisjunctionSet([
    ...     Conjunction([int, float]), Conjunction([str, float])
    ... ])
    True

    >>> intersect(long, int_or_str) == DisjunctionSet([
    ...     Conjunction([long, int]), Conjunction([long, str])
    ... ])
    True

    >>> intersect(int_or_str, long_or_float) == DisjunctionSet([
    ...     Conjunction([int,long]), Conjunction([int, float]),
    ...     Conjunction([str,long]), Conjunction([str, float]),
    ... ])
    True

    >>> intersect(int_or_str, Conjunction([long, float])) == \
    ...     DisjunctionSet(
    ...         [Conjunction([int, long, float]),
    ...          Conjunction([str, long, float])]
    ... )
    True

    >>> intersect(Conjunction([int, str]), long_or_float) == \
    ...     DisjunctionSet(
    ...         [Conjunction([int, str, long]), Conjunction([int, str, float])]
    ...     )
    True

As you can see, this is the heart of the process that allows expressions like
``(A or B) and (C or D)`` to be transformed into their disjunctive normal
form (i.e. ``(A and C) or (A and D) or (B and C) or (B and D)``).

(In other words, by using ``Disjunction()`` as an "or" operator and
``intersect()`` as the "and" operator, we always end up with a DNF result!)


Object Identity
===============

The ``IsObject`` criterion type represents the set of objects which either
are -- or are *not* -- one specific object instance.  ``IsObject(x)`` (or
``IsObject(x, True)``) represents the set of objects ``y`` for which the
``y is x`` condition would be true.  Conversely, ``IsObject(x, False)``
represents the set of objects ``y`` for whom ``y is not x``::

    >>> from peak.rules.criteria import IsObject, Conjunction

    >>> o = object()
    >>> is_o = IsObject(o)
    >>> is_not_o = IsObject(o, False)

    >>> is_o
    IsObject(<object object at ...>, True)

    >>> is_not_o
    IsObject(<object object at ...>, False)

    >>> is_not_o == negate(is_o)
    True

    >>> is_o == negate(is_not_o)
    True

The intersection of two different ``is`` identities is ``False``, since an
object cannot be both itself and another object::

    >>> intersect(is_o, IsObject("foo"))
    False

    >>> implies(is_o, IsObject("foo"))
    False

Similarly, an object can't be both itself, and not itself::

    >>> intersect(is_o, is_not_o)
    False

    >>> intersect(is_not_o, is_o)
    False

    >>> implies(is_o, is_not_o)
    False

But it *can* be itself and itself::

    >>> intersect(is_o, is_o) == is_o
    True

    >>> implies(is_o, is_o)
    True

Or not itself and not itself::

    >>> intersect(is_not_o, is_not_o) == is_not_o
    True

    >>> implies(is_not_o, is_not_o)
    True

And an object can be itself, while not being something else::

    >>> intersect(is_o, IsObject("foo", False)) == is_o
    True

    >>> intersect(IsObject("foo", False), is_o) == is_o
    True

    >>> implies(is_o, IsObject("foo", False))
    True

But just because an object is not something, doesn't mean it's something else::

    >>> implies(is_not_o, IsObject("foo"))
    False

And the intersection of multiple ``is not`` conditions produces a
``Conjunction``::

    >>> not_foo = IsObject("foo", False)
    >>> not_bar = IsObject("bar", False)
    >>> not_foobar = intersect(not_foo, not_bar)
    >>> not_foobar
    Conjunction([IsObject('foo', False), IsObject('bar', False)])

Which of course then implies each of the individual "not" conditions::

    >>> implies(not_foobar, not_bar)
    True
    >>> implies(not_foobar, not_foo)
    True

But not their opposites::

    >>> implies(not_foobar, IsObject("bar"))
    False

Oh, and an ``is`` condition implies any ``Conjunction`` that don't contain its
opposite::

    >>> implies(is_o, not_foobar)
    True

But not the other way around::

    >>> implies(not_foobar, is_o)
    False

Finally, negating a ``Conjunction`` of is-nots returns a disjunction of true
``IsObject`` tests, and vice versa::

    >>> negate(not_foobar)
    DisjunctionSet([IsObject('foo', True), IsObject('bar', True)])

    >>> negate(DisjunctionSet([IsObject('foo'), IsObject('bar')]))
    Conjunction([IsObject('foo', False), IsObject('bar', False)])


Values and Ranges
=================

``Value`` objects are used to represent ``==`` and ``!=`` comparisons.
``Value(x)`` represents ``==x`` and ``Value(x, False)`` represents ``!=x``.

A ``Value`` implies another ``Value`` if the two are identical::

    >>> from peak.rules.criteria import Value, Range, Min, Max

    >>> implies(Value(27), Value(42))
    False
    >>> implies(Value(27, False), Value(42))
    False
    >>> implies(Value(27), Value(27))
    True
    >>> implies(Value(99), Value(99, False))
    False
    >>> implies(Value(99, False), Value(99, False))
    True    

Or, if they have different target values, but the first is an ``==``
comparison, and the second is a ``!=`` comparison::

    >>> implies(Value(27), Value(99, False))
    True

    >>> intersect(Value(27), Value(99, False))
    Value(27, True)

The negation of a ``Value`` is of course another ``Value`` of the same
target but the reverse operator::

    >>> negate(Value(27))
    Value(27, False)

    >>> negate(Value(99, False))
    Value(99, True)

The intersection of two different ``==`` values, or a ``!=`` and ``==`` of the
same value, is ``False`` (i.e., no possible match::

    >>> intersect(Value(27), Value(42))
    False
    >>> intersect(Value(27), Value(27, False))
    False

But the intersection of two different ``!=`` values produces a disjunction of
three ``Range()`` objects::

    >>> one_two = intersect(Value(1, False), Value(2, False))

    >>> one_two == DisjunctionSet([
    ...     Range((Min, -1), (1, -1)),
    ...     Range((1, 1), (2, -1)),
    ...     Range((2, 1), (Max, 1))
    ... ])
    True

    >>> intersect(one_two, Value(3,False)) == DisjunctionSet([
    ...     Range((Min, -1), (1, -1)),
    ...     Range((1, 1), (2, -1)),
    ...     Range((2, 1), (3, -1)),
    ...     Range((3, 1), (Max, 1))
    ... ])
    True
    

The ``Range()`` criterion type represents an inequality such as ``lo < x < hi``
or ``x >= lo``.  The lows and highs given have to be a 2-tuple, consisting of
a value and a "direction".  The direction is an integer (either -1 or 1) that
indicates whether the edge is on the low or high side of the target value.
Thus, a tuple ``(27, -1)`` means "the low edge of 27", while ``(99, 1)``
means "the high edge of 99".  In this way, any simple inequality or range
can be represented by a pair of edges.

Thus, the intersection of two different ``!=`` values produces a disjunction of
three ``Range()`` objects, representing the intervals that "surround" the
original ``!=`` values::

    >>> from peak.rules.criteria import Range

    >>> intersect(Value(27, False), Value(42, False)) == DisjunctionSet([
    ...     Range(hi=(27, -1)),     # below Min ... below 27
    ...     Range((27,1), (42,-1)), # above  27 ... below 42
    ...     Range(lo=(42, 1)),      # above  42 ... above Max
    ... ])
    True

Notice that if we omit the ``hi`` or ``lo``, end of the range, it's replaced
with "below ``Min``" or "above ``Max``", as appropriate.  (The ``Min`` and
``Max`` values are special objects that compare below or above any other
object.)

When creating range and value objects, it can be useful to use the
``Inequality`` constructor, which takes a comparison operator and a value::

    >>> from peak.rules.criteria import Inequality

    >>> Inequality('>=', 27)    # >=27 : below 27 ... above Max
    Range((27, -1), (Max, 1))

    >>> negate(Inequality('<', 27))
    Range((27, -1), (Max, 1))
    
    >>> Inequality('>', 27)     # > 27 : above 27 ... above Max
    Range((27, 1), (Max, 1))

    >>> Inequality('<', 99)     # < 99 : below Min ... below 99
    Range((Min, -1), (99, -1))

    >>> Inequality('<=',99)     # <=99 : below Min ... above 99
    Range((Min, -1), (99, 1))

    >>> negate(Inequality('>', 99))
    Range((Min, -1), (99, 1))

    >>> Inequality('==', 66)
    Value(66, True)

    >>> Inequality('!=', 77)
    Value(77, False)

Intersecting two ranges (or a range and an ``==`` value) produces a smaller
range or value, or ``False`` if there is no overlap::

    >>> intersect(Inequality('<', 27), Inequality(">",19))
    Range((19, 1), (27, -1))

    >>> intersect(Inequality('>=', 27), Inequality("<=",19))
    False

    >>> intersect(Value(27), Inequality('>=', 27))
    Value(27, True)    
    >>> intersect(Inequality('<=', 27), Value(27))
    Value(27, True)

    >>> intersect(Value(27), Inequality('<',27))
    False
    >>> intersect(Inequality('>',27), Value(27))
    False    
    
Last, but not least, a range (or value) implies another range or value if it
lies entirely within it::

    >>> implies(Range((42,-1), (42,1)), Value(42))
    True

    >>> implies(Range((27,-1), (42,1)), Range((15,1),(99,-1)))
    True

    >>> implies(Range((27,-1), (42,1)), Value(99, False))
    True

But not if it overlaps or lies outside of it::

    >>> implies(Range((15,-1),(42,1)), Range((15,1),(99,-1)))
    False

    >>> implies(Range((27,-1), (42,1)), Value(99))
    False


Class/Type Tests
================

``Class`` objects represent ``issubclass()`` or ``isinstance()`` sets.
``Class(x)`` is a instance/subclass match, while ``Class(x, False)`` is a
non-match.  Implication, negation, and intersection are defined accordingly::

    >>> from peak.rules.criteria import Class

    >>> implies(Class(int), Class(object))
    True
    >>> implies(Class(object, False), Class(int, False))
    True

    >>> negate(Class(int))
    Class(<... 'int'>, False)

    >>> negate(Class(object, False))
    Class(<... 'object'>, True)
    
    >>> implies(Class(int), Class(str))
    False
    >>> implies(Class(object), Class(int, False))
    False
    >>> implies(Class(object), Class(int))
    False
    >>> implies(Class(int), Class(int))
    True

    >>> intersect(Class(int), Class(object))
    Class(<... 'int'>, True)

    >>> intersect(Class(object), Class(int))
    Class(<... 'int'>, True)

The intersection of two or more unrelated ``Class`` criteria is represented by
a ``Conjunction``::

    >>> from peak.rules.criteria import Conjunction

    >>> intersect(Class(int, False), Class(str, False)) == Conjunction(
    ...     [Class(int, False), Class(str, False)]
    ... )
    True


Exact-Type and Type-Exclusion Tests
===================================

Exact type tests are expressed using ``istype(x)``, and type exclusion tests
are represented as ``istype(x, False)``::

    >>> from peak.rules import istype

    >>> negate(istype(int))
    istype(<... 'int'>, False)

    >>> negate(istype(object, False))
    istype(<... 'object'>, True)

One ``istype()`` test implies another only if they're equal::

    >>> implies(istype(int), istype(int))
    True

    >>> implies(istype(int, False), istype(int, False))
    True

    >>> implies(istype(int, False), istype(int))
    False

Or if the first is an exact match, and the second is an exclusion test for
a different type::

    >>> implies(istype(int), istype(str, False))
    True

Thus, the intersection of two ``istype()`` tests will be either one of the
input tests, or ``False`` (meaning no overlap)::

    >>> intersect(istype(int), istype(int))
    istype(<... 'int'>, True)

    >>> intersect(istype(int), istype(str, False))
    istype(<... 'int'>, True)

    >>> intersect(istype(int, False), istype(int, False))
    istype(<... 'int'>, False)

    >>> intersect(istype(int), istype(str))
    False

Unless both are exclusion tests on different types, in which case their
intersection is a ``Conjunction`` of the two::

    >>> intersect(istype(str, False), istype(int, False)) == Conjunction([
    ...     istype(int, False), istype(str, False)
    ... ])
    True

An ``istype(x)`` implies ``Class(y)`` only if x is y or a subtype thereof::

    >>> implies(istype(int), Class(str))
    False

    >>> implies(istype(int), Class(object))
    True

And it implies ``Class(y, False)`` only if x is *not* y or a subtype thereof::

    >>> implies(istype(int), Class(str, False))
    True

    >>> implies(istype(int), Class(object, False))
    False

But ``istype(x, False)`` implies nothing about any ``Class`` test, since it
refers to exactly one type, while the ``Class`` may refer to infinitely many
types::

    >>> implies(istype(int, False), Class(int, False))
    False

    >>> implies(istype(int, False), Class(object))
    False

Meanwhile, ``Class(x)`` tests can only imply ``istype(y, False)``, where y
is a *superclass* of x::

    >>> implies(Class(int), istype(int))
    False

    >>> implies(Class(int), istype(object))
    False    

    >>> implies(Class(int), istype(object, False))
    True

And ``Class(x, False)`` cannot imply anything about any ``istype()`` test,
whether true or false::

    >>> implies(Class(int, False), istype(int))
    False

    >>> implies(Class(int, False), istype(int, False))
    False

When ``Class()`` is intersected with an exact type test, the result is either
the exact type test, or ``False``::

    >>> intersect(Class(int), istype(int))
    istype(<... 'int'>, True)

    >>> intersect(istype(int), Class(int))
    istype(<... 'int'>, True)

    >>> intersect(Class(int), istype(object))
    False

    >>> intersect(istype(object), Class(int))
    False

    >>> intersect(Class(int, False), istype(object))
    istype(<... 'object'>, True)

    >>> intersect(istype(object), Class(int, False))
    istype(<... 'object'>, True)

But when it's intersected with a type exclusion test, the result is a
``Conjunction``::

    >>> intersect(istype(int, False), Class(str)) == Conjunction([
    ...     istype(int, False), Class(str, True)
    ... ])
    True

    >>> s = intersect(Class(str), istype(int, False))
    >>> s == Conjunction([istype(int, False), Class(str, True)])
    True

    >>> intersect(s, istype(int))
    False

    >>> intersect(s, istype(int, False)) == s
    True

    >>> intersect(s, istype(str))
    istype(<... 'str'>, True)


--------------------
Tests and Signatures
--------------------


``Test`` Objects
================

A ``Test`` is the combination of a "dispatch expression" and a criterion to
be applied to it::

    >>> from peak.rules.criteria import Test
    >>> x_isa_int = Test("x", Class(int))

(Note that although these examples use strings, actual dispatch expressions
will be AST-like structures.)

Creating a test with disjunct criteria actually returns a set of tests::

    >>> Test("x", DisjunctionSet([int, str])) == \
    ...           DisjunctionSet([Test('x', int), Test('x', str)])
    True

So the ``disjuncts()`` of a test will always just be the test itself::

    >>> disjuncts(x_isa_int)
    [Test('x', Class(<... 'int'>, True))]

Negating a test usually just negates its criterion, leaving the expression
intact::

    >>> negate(x_isa_int)
    Test('x', Class(<... 'int'>, False))

But if the test criterion is a conjunction or range, negating it can produce
a disjunction of tests::

    >>> negate(
    ...     Test('x',
    ...         Conjunction([IsObject('foo',False), IsObject('bar',False)])
    ...     )
    ... ) == DisjunctionSet(
    ...     [Test('x', IsObject('foo', True)),
    ...      Test('x', IsObject('bar', True))])
    True

Intersecting two tests for the same dispatch expression returns a test whose
criterion is the intersection of the original tests' criteria::

    >>> intersect(x_isa_int, Test("x", Class(str))) == Test(
    ...     'x', Conjunction([Class(int), Class(str)])
    ... )
    True

And similarly, a test only implies another test if they have equal dispatch
expressions, and the second test's criterion is implied by the first's::

    >>> implies(x_isa_int, Test("x", Class(str)))
    False
    >>> implies(x_isa_int, Test("x", Class(object)))
    True
    >>> implies(x_isa_int, Test("y", Class(int)))
    False

But the intersection of two tests with *different* dispatch expressions
produces a ``Signature`` object::

    >>> y_isa_str = Test("y", Class(str))
    >>> x_int_y_str = intersect(x_isa_int, y_isa_str)
    >>> x_int_y_str
    Signature([Test('x', Class(...int...)), Test('y', Class(...str...))])


``Signature`` Objects
=====================

``Signature`` objects are similar to ``Conjunction`` objects, except for three
important differences.

First, signatures are sequences, not sets.  They preserve the ordering they
were created with::

    >>> intersect(x_isa_int, y_isa_str)
    Signature([Test('x', Class(...int...)), Test('y', Class(...str...))])

    >>> intersect(y_isa_str, x_isa_int)
    Signature([Test('y', Class(...str...)), Test('x', Class(...int...))])

and their negations preserve the order as well (using ``OrElse`` instances)::

    >>> negate(intersect(x_isa_int, y_isa_str))
    OrElse([Test('x', Class(...int..., False)),
            Test('y', Class(...str..., False))])

    >>> negate(intersect(y_isa_str, x_isa_int))
    OrElse([Test('y', Class(...str..., False)),
            Test('x', Class(...int..., False))])

Second, signatures can only contain ``Test`` instances, and they automatically
``intersect()`` any tests that apply to the same dispatch signatures::

    >>> from peak.rules.criteria import Signature

    >>> intersect(x_int_y_str, Test("y", Class(float))) == Signature([
    ...     Test('x', Class(int)),
    ...     Test('y', Conjunction([Class(str), Class(float)]))
    ... ])
    True

    >>> intersect(x_int_y_str, Test("x", Class(float))) == Signature([
    ...     Test('x', Conjunction([Class(int), Class(float)])),
    ...     Test('y', Class(str))
    ... ])
    True

    >>> intersect(Test("x", Class(float)), x_int_y_str) == Signature([
    ...     Test('x', Conjunction([Class(int), Class(float)])),
    ...     Test('y', Class(str))
    ... ])
    True

But, as with conjunctions, you can't create a signature with less than two
items in it::

    >>> Signature([Test("x",1)])
    Test('x', 1)

    >>> Signature([True])
    True

    >>> Signature([False])
    False   

    >>> Signature([])
    True


----------
Predicates
----------

Now that we've got all the basic pieces in place, we can now operationally
define predicates for the Chambers & Chen dispatch algorithm.

Specifically, a predicate can be any of the following:

* ``True`` (meaning a condition that always applies)

* ``False`` (meaning a condition that *never* applies)

* A ``Test`` or ``Signature`` instance

* A ``DisjunctionSet`` or ``OrElse`` containing two or more ``Test`` or
  ``Signature`` instances

In each case, invoking ``disjuncts()`` on the object in question will return
a list of objects suitable for constructing dispatch "cases" -- i.e., sets of
simple "and-ed" criteria that can easily be indexed.

The ``tests_for()`` function can then be used to yield the component tests of
each case signature.  When called on a ``Test``, it yields the given test::

    >>> from peak.rules.criteria import tests_for

    >>> list(tests_for(Test('y',42)))
    [Test('y', 42)]
    
But called on a ``Signature``, it yields the tests contained within::
    
    >>> list(tests_for(x_int_y_str))
    [Test('x', Class(...int...)), Test('y', Class(...str...))]

And called on ``True``, it yields nothing::

    >>> list(tests_for(True))
    []

``tests_for(False)``, however, is undefined, because ``False`` cannot be
represented as a conjunction of tests.  ``False`` is still a valid predicate,
of course, because it represents an empty disjunction.

In normal predicate processing, one loops over the ``disjuncts()`` of a
predicate, and only then uses ``tests_for()`` to inspect the individual items.
But since ``disjuncts(False)`` is an empty list, it should never be necessary
to invoke ``tests_for(False)``.

There is an important distinction, however, in how ``disjuncts()`` works on
``OrElse`` objects, compared to all other kinds of predicates.  ``disjuncts()``
is used to obtain the *unordered* disjunctions of a logical condition, but
``OrElse`` is *ordered*, because it represents a series of applications of the
Python "or" operator.

In Python, a condition on the right-hand side of an "or" operator is not tested
*unless the condition on the left is false*.  PEAK-Rules, however, tests the
``disjuncts()`` of a predicate *independently*.  Thus, in order to properly
translate "or" conditions in a predicate, the ``disjuncts()`` of an ``OrElse``
must include additional and-ed conditions to force them to be tested in order.

Specifically, the ``disjuncts()`` of ``OrElse([a, b, c])`` will be:

  * ``a``,
  * ``intersect(negate(a), b)``, and
  * ``intersect(intersect(negate(a), negate(b)), c)``!

This expansion ensures that ``b`` will never be tested unless ``a`` is false,
and ``c`` will never be tested unless ``a`` and ``b`` are both false, just like
in a regular Python expression.  Observe::

    >>> DisjunctionSet([OrElse([Class(a), Class(b)])]) == DisjunctionSet([
    ...     Class(a, True),
    ...     Conjunction([Class(a, False), Class(b, True)])
    ... ])
    True
    
Also, because ``OrElse`` objects don't expand their contents' disjuncts at
creation time, they must be expanded as part of the ``disjuncts()`` operation::

    >>> a_or_b = DisjunctionSet([Class(a), Class(b)])

    >>> try:
    ...     set = set
    ... except NameError:
    ...     from sets import Set as set     # 2.3, ugh

    >>> set(disjuncts(OrElse([istype(int), a_or_b]))) == set([
    ...     istype(int),
    ...     Conjunction([istype(int, False), Class(b)]),
    ...     Conjunction([istype(int, False), Class(a)])
    ... ])
    True

This delayed expansion "preserves the unorderedness" of the contents, by not
forcing them to be evaluated in any specific sequence, apart from the
requirements imposed by their position within the ``OrElse``.

We'll do one more test, to show that the disjuncts of the negated portions of
the ``OrElse`` are also expanded::

    >>> a_and_b = Conjunction([Class(a), Class(b)])
    >>> not_a = Class(a, False)
    >>> not_b = Class(b, False)
    >>> int_or_str = DisjunctionSet([Class(int), Class(str)])

    >>> set(disjuncts(OrElse([a_and_b, int_or_str]))) == set([
    ...     a_and_b, Conjunction([not_a, Class(int)]), Conjunction([not_a, Class(str)]),
    ...              Conjunction([not_b, Class(int)]), Conjunction([not_b, Class(str)])
    ... ])
    True

per this expansion logic (using ``|`` for "symmetric or")::

    (a and b) or (int|str) => (a and b) | not (a and b) and (int|str)

    not (a and b) and (int|str) => (not a | not b) and (int|str)

    (not a | not b) and (int|str) => (
        (not a and int) | (not a and str) | (not b and int) | (not b and str)
    )