Coercion
========

Preliminaries
--------------

What is coercion all about?
~~~~~~~~~~~~~~~~~~~~~~~~~~~

*The primary goal of coercion is to be able to transparently do arithmetic, comparisons, etc. between elements of distinct sets.*

As a concrete example, when one writes `1 + 1/2` one wants to perform
arithmetic on the operands as rational numbers, despite the left being
an integer. This makes sense given the obvious and natural inclusion
of the integers into the rational numbers. The goal of the coercion
system is to facilitate this (and more complicated arithmetic) without
having to explicitly map everything over into the same domain, and at
the same time being strict enough to not resolve ambiguity or accept
nonsense. Here are some examples::

    sage: 1 + 1/2
    3/2
    sage: R.<x,y> = ZZ[]
    sage: R
    Multivariate Polynomial Ring in x, y over Integer Ring
    sage: parent(x)
    Multivariate Polynomial Ring in x, y over Integer Ring
    sage: parent(1/3)
    Rational Field
    sage: x+1/3
    x + 1/3
    sage: parent(x+1/3)
    Multivariate Polynomial Ring in x, y over Rational Field

    sage: GF(5)(1) + CC(I)
    Traceback (most recent call last):
    ...
    TypeError: unsupported operand parent(s) for +: 'Finite Field of size 5' and 'Complex Field with 53 bits of precision'

Parents and Elements
~~~~~~~~~~~~~~~~~~~~

Parents are objects in concrete categories, and Elements are their
members. Parents are first-class objects.  Most things in Sage are
either parents or have a parent. Typically whenever one sees the word
*Parent* one can think *Set*. Here are some examples::

    sage: parent(1)
    Integer Ring
    sage: parent(1) is ZZ
    True
    sage: ZZ
    Integer Ring
    sage: parent(1.50000000000000000000000000000000000)
    Real Field with 120 bits of precision
    sage: parent(x)
    Symbolic Ring
    sage: x^sin(x)
    x^sin(x)
    sage: R.<t> = Qp(5)[]
    sage: f = t^3-5; f
    (1 + O(5^20))*t^3 + (4*5 + 4*5^2 + 4*5^3 + 4*5^4 + 4*5^5 + 4*5^6 + 4*5^7 + 4*5^8 + 4*5^9 + 4*5^10 + 4*5^11 + 4*5^12 + 4*5^13 + 4*5^14 + 4*5^15 + 4*5^16 + 4*5^17 + 4*5^18 + 4*5^19 + 4*5^20 + O(5^21))
    sage: parent(f)
    Univariate Polynomial Ring in t over 5-adic Field with capped relative precision 20
    sage: f = EllipticCurve('37a').lseries().taylor_series(10); f
    0.990010459847588 + 0.0191338632530789*z - 0.0197489006172923*z^2 + 0.0137240085327618*z^3 - 0.00703880791607153*z^4 + 0.00280906165766519*z^5 + O(z^6)           # 32-bit
    0.997997869801216 + 0.00140712894524925*z - 0.000498127610960097*z^2 + 0.000118835596665956*z^3 - 0.0000215906522442707*z^4 + (3.20363155418419e-6)*z^5 + O(z^6)  # 64-bit
    sage: parent(f)
    Power Series Ring in z over Complex Field with 53 bits of precision

There is an important distinction between Parents and types::

    sage: a = GF(5).random_element()
    sage: b = GF(7).random_element()
    sage: type(a)
    <type 'sage.rings.finite_rings.integer_mod.IntegerMod_int'>
    sage: type(b)
    <type 'sage.rings.finite_rings.integer_mod.IntegerMod_int'>
    sage: type(a) == type(b)
    True
    sage: parent(a)
    Finite Field of size 5
    sage: parent(a) == parent(b)
    False

However, non-Sage objects don't really have parents, but we still want
to be able to reason with them, so their type is used instead::

    sage: a = int(10)
    sage: parent(a)
    <... 'int'>

In fact, under the hood, a special kind of parent "The set of all
Python objects of class T" is used in these cases.

Note that parents are **not** always as tight as possible.

::

    sage: parent(1/2)
    Rational Field
    sage: parent(2/1)
    Rational Field

Maps between Parents
~~~~~~~~~~~~~~~~~~~~

Many parents come with maps to and from other parents.

Sage makes a distinction between being able to **convert** between
various parents, and **coerce** between them. Conversion is explicit
and tries to make sense of an object in the target domain if at all
possible. It is invoked by calling::

    sage: ZZ(5)
    5
    sage: ZZ(10/5)
    2
    sage: QQ(10)
    10
    sage: parent(QQ(10))
    Rational Field
    sage: a = GF(5)(2); a
    2
    sage: parent(a)
    Finite Field of size 5
    sage: parent(ZZ(a))
    Integer Ring
    sage: GF(71)(1/5)
    57
    sage: ZZ(1/2)
    Traceback (most recent call last):
    ...
    TypeError: no conversion of this rational to integer

Conversions need not be canonical (they may for example involve a
choice of lift) or even make sense mathematically (e.g. constructions
of some kind).

::

    sage: ZZ("123")
    123
    sage: ZZ(GF(5)(14))
    4
    sage: ZZ['x']([4,3,2,1])
    x^3 + 2*x^2 + 3*x + 4
    sage: a = Qp(5, 10)(1/3); a
    2 + 3*5 + 5^2 + 3*5^3 + 5^4 + 3*5^5 + 5^6 + 3*5^7 + 5^8 + 3*5^9 + O(5^10)
    sage: ZZ(a)
    6510417

On the other hand, Sage has the notion of a **coercion**, which is a
canonical morphism (occasionally up to a conventional choice made by
developers) between parents. A coercion from one parent to another
**must** be defined on the whole domain, and always succeeds. As it
may be invoked implicitly, it should be obvious and natural (in both
the mathematically rigorous and colloquial sense of the word). Up to
inescapable rounding issues that arise with inexact representations,
these coercion morphisms should all commute.  In particular, if there
are coercion maps `A \to B` and `B \to A`, then their composites
must be the identity maps.

Coercions can be discovered via the :meth:`Parent.has_coerce_map_from`
method, and if needed explicitly invoked with the
:meth:`Parent.coerce` method::

    sage: QQ.has_coerce_map_from(ZZ)
    True
    sage: QQ.has_coerce_map_from(RR)
    False
    sage: ZZ['x'].has_coerce_map_from(QQ)
    False
    sage: ZZ['x'].has_coerce_map_from(ZZ)
    True
    sage: ZZ['x'].coerce(5)
    5
    sage: ZZ['x'].coerce(5).parent()
    Univariate Polynomial Ring in x over Integer Ring
    sage: ZZ['x'].coerce(5/1)
    Traceback (most recent call last):
    ...
    TypeError: no canonical coercion from Rational Field to Univariate Polynomial Ring in x over Integer Ring

Basic Arithmetic Rules
----------------------

Suppose we want to add two element, a and b, whose parents are A and B
respectively. When we type ``a+b`` then

1. If A ``is`` B, call a._add_(b)

2. If there is a coercion `\phi: B \rightarrow A`, call a._add_( `\phi` (b))

3. If there is a coercion `\phi: A \rightarrow B`, call `\phi` (a)._add_(b)

4. Look for `Z` such that there is a coercion `\phi_A: A \rightarrow Z` and
   `\phi_B: B \rightarrow Z`, call `\phi_A` (a)._add_( `\phi_B` (b))

These rules are evaluated in order; therefore if there are coercions
in both directions, then the parent of a._add_b is A -- the parent
of the left-hand operand is used in such cases.

The same rules are used for subtraction, multiplication, and
division. This logic is embedded in a coercion model object, which can
be obtained and queried.

::

    sage: parent(1 + 1/2)
    Rational Field
    sage: cm = sage.structure.element.get_coercion_model(); cm
    <sage.structure.coerce.CoercionModel_cache_maps object at ...>
    sage: cm.explain(ZZ, QQ)
    Coercion on left operand via
       Natural morphism:
         From: Integer Ring
         To:   Rational Field
    Arithmetic performed after coercions.
    Result lives in Rational Field
    Rational Field

    sage: cm.explain(ZZ['x','y'], QQ['x'])
    Coercion on left operand via
       Coercion map:
         From: Multivariate Polynomial Ring in x, y over Integer Ring
         To:   Multivariate Polynomial Ring in x, y over Rational Field
    Coercion on right operand via
       Coercion map:
         From: Univariate Polynomial Ring in x over Rational Field
         To:   Multivariate Polynomial Ring in x, y over Rational Field
    Arithmetic performed after coercions.
    Result lives in Multivariate Polynomial Ring in x, y over Rational Field
    Multivariate Polynomial Ring in x, y over Rational Field

The coercion model can be used directly for any binary operation
(callable taking two arguments).

.. link

::

    sage: cm.bin_op(77, 9, gcd)
    1

There are also **actions** in the sense that a field `K` acts on a
module over `K`, or a permutation group acts on a set. These are
discovered between steps 1 and 2 above.

.. link

::

    sage: cm.explain(ZZ['x'], ZZ, operator.mul)
    Action discovered.
       Right scalar multiplication by Integer Ring on Univariate Polynomial Ring in x over Integer Ring
    Result lives in Univariate Polynomial Ring in x over Integer Ring
    Univariate Polynomial Ring in x over Integer Ring

    sage: cm.explain(ZZ['x'], ZZ, operator.truediv)
    Action discovered.
       Right inverse action by Rational Field on Univariate Polynomial Ring in x over Integer Ring
       with precomposition on right by Natural morphism:
         From: Integer Ring
         To:   Rational Field
    Result lives in Univariate Polynomial Ring in x over Rational Field
    Univariate Polynomial Ring in x over Rational Field

    sage: f = QQ.coerce_map_from(ZZ)
    sage: f(3).parent()
    Rational Field

Note that by :trac:`14711` Sage's coercion system uses maps with weak
references to the domain. Such maps should only be used internally, and so a
copy should be used instead (unless one knows what one is doing)::

    sage: QQ._internal_coerce_map_from(int)
    (map internal to coercion system -- copy before use)
    Native morphism:
      From: Set of Python objects of class 'int'
      To:   Rational Field
    sage: copy(QQ._internal_coerce_map_from(int))
    Native morphism:
      From: Set of Python objects of class 'int'
      To:   Rational Field

Note that the user-visible method (without underscore) automates this copy::

    sage: copy(QQ.coerce_map_from(int))
    Native morphism:
      From: Set of Python objects of class 'int'
      To:   Rational Field

::

    sage: QQ.has_coerce_map_from(RR)
    False
    sage: QQ['x'].get_action(QQ)
    Right scalar multiplication by Rational Field on Univariate Polynomial Ring in x over Rational Field
    sage: QQ2 = QQ^2
    sage: (QQ2).get_action(QQ)
    Right scalar multiplication by Rational Field on Vector space of dimension 2 over Rational Field
    sage: QQ['x'].get_action(RR)
    Right scalar multiplication by Real Field with 53 bits of precision on Univariate Polynomial Ring in x over Rational Field

How to Implement
----------------

Methods to implement
~~~~~~~~~~~~~~~~~~~~

* Arithmetic on Elements: ``_add_``, ``_sub_``, ``_mul_``, ``_div_``

  This is where the binary arithmetic operators should be
  implemented. Unlike Python's ``__add__``, both operands are
  *guaranteed* to have the same Parent at this point.

* Coercion for Parents: ``_coerce_map_from_``

  Given two parents R and S, ``R._coerce_map_from_(S)`` is called to
  determine if there is a coercion `\phi: S \rightarrow R`.  Note that
  the function is called on the potential codomain.  To indicate that
  there is no coercion from S to R (self), return ``False`` or
  ``None``. This is the default behavior.  If there is a coercion,
  return ``True`` (in which case an morphism using
  ``R._element_constructor_`` will be created) or an actual
  :class:`Morphism` object with S as the domain and R as the codomain.

* Actions for Parents: ``_get_action_`` or ``_rmul_``, ``_lmul_``, ``_r_action_``, ``_l_action_``

  Suppose one wants R to act on S. Some examples of this could be
  `R = \QQ`, `S = \QQ[x]` or `R = {\rm Gal}(S/\QQ)`
  where `S`  is a number field. There are several ways to implement this:

  * If `R` is the base of `S` (as in the first example), simply
    implement ``_rmul_`` and/or ``_lmul_`` on the Elements of `S`.
    In this case ``r * s`` gets handled as ``s._rmul_(r)`` and
    ``s * r`` as ``s._lmul_(r)``.  The argument to ``_rmul_``
    and ``_lmul_`` are *guaranteed* to be Elements of the base of
    `S` (with coercion happening beforehand if necessary).

  * If `R` acts on `S`, one can alternatively define the methods
    ``_r_action_`` and/or ``_l_action_`` on the Elements of `R`.
    There is no constraint on the type or parents of objects passed to
    these methods; raise a ``TypeError`` or ``ValueError`` if the
    wrong kind of object is passed in to indicate the action is not
    appropriate here.

  * If either `R` acts on `S` *or* `S` acts on `R`, one may implement
    ``R._get_action_`` to return an actual
    :class:`~sage.categories.action.Action` object to be used.  This
    is how non-multiplicative actions must be implemented, and is the
    most powerful (and completed) way to do things.

* Element conversion/construction for Parents: use
  ``_element_constructor_`` **not** ``__call__``

  The :meth:`Parent.__call__` method dispatches to
  ``_element_constructor_``. When someone writes ``R(x, ...)``, this is
  the method that eventually gets called in most cases.  See the
  documentation on the ``__call__`` method below.

Parents may also call the ``self._populate_coercion_lists_`` method in
their ``__init__`` functions to pass any callable for use instead of
``_element_constructor_``, provide a list of Parents with coercions to
self (as an alternative to implementing ``_coerce_map_from_``),
provide special construction methods (like ``_integer_`` for ZZ),
etc. This also allows one to specify a single coercion embedding *out*
of self (whereas the rest of the coercion functions all specify maps
*into* self). There is extensive documentation in the docstring of the
``_populate_coercion_lists_`` method.

Example
~~~~~~~

Sometimes a simple example is worth a thousand words. Here is a
minimal example of setting up a simple Ring that handles coercion. (It
is easy to imagine much more sophisticated and powerful localizations,
but that would obscure the main points being made here.)

::

    class Localization(Ring):
       def __init__(self, primes):
           """
           Localization of `\ZZ` away from primes.
           """
           Ring.__init__(self, base=ZZ)
           self._primes = primes
           self._populate_coercion_lists_()

       def _repr_(self):
           """
           How to print self.
           """
           return "%s localized at %s" % (self.base(), self._primes)

       def _element_constructor_(self, x):
           """
           Make sure x is a valid member of self, and return the constructed element.
           """
           if isinstance(x, LocalizationElement):
               x = x._value
           else:
               x = QQ(x)
           for p, e in x.denominator().factor():
               if p not in self._primes:
                   raise ValueError("Not integral at %s" % p)
           return LocalizationElement(self, x)

       def _coerce_map_from_(self, S):
           """
           The only things that coerce into this ring are:

           - the integer ring

           - other localizations away from fewer primes
           """
           if S is ZZ:
               return True
           elif isinstance(S, Localization):
               return all(p in self._primes for p in S._primes)


    class LocalizationElement(RingElement):

       def __init__(self, parent, x):
           RingElement.__init__(self, parent)
           self._value = x


       # We're just printing out this way to make it easy to see what's going on in the examples.

       def _repr_(self):
           return "LocalElt(%s)" % self._value

       # Now define addition, subtraction, and multiplication of elements.
       # Note that left and right always have the same parent.

       def _add_(left, right):
           return LocalizationElement(left.parent(), left._value + right._value)

       def _sub_(left, right):
           return LocalizationElement(left.parent(), left._value - right._value)

       def _mul_(left, right):
           return LocalizationElement(left.parent(), left._value * right._value)

       # The basering was set to ZZ, so c is guaranteed to be in ZZ

       def _rmul_(self, c):
           return LocalizationElement(self.parent(), c * self._value)

       def _lmul_(self, c):
           return LocalizationElement(self.parent(), self._value * c)

That's all there is to it. Now we can test it out:

.. skip

::

    sage: R = Localization([2]); R
    Integer Ring localized at [2]
    sage: R(1)
    LocalElt(1)
    sage: R(1/2)
    LocalElt(1/2)
    sage: R(1/3)
    Traceback (most recent call last):
    ...
    ValueError: Not integral at 3

    sage: R.coerce(1)
    LocalElt(1)
    sage: R.coerce(1/4)
    Traceback (click to the left for traceback)
    ...
    TypeError: no canonical coercion from Rational Field to Integer Ring localized at [2]

    sage: R(1/2) + R(3/4)
    LocalElt(5/4)
    sage: R(1/2) + 5
    LocalElt(11/2)
    sage: 5 + R(1/2)
    LocalElt(11/2)
    sage: R(1/2) + 1/7
    Traceback (most recent call last):
    ...
    TypeError: unsupported operand parent(s) for +: 'Integer Ring localized at [2]' and 'Rational Field'
    sage: R(3/4) * 7
    LocalElt(21/4)

    sage: R.get_action(ZZ)
    Right scalar multiplication by Integer Ring on Integer Ring localized at [2]
    sage: cm = sage.structure.element.get_coercion_model()
    sage: cm.explain(R, ZZ, operator.add)
    Coercion on right operand via
       Conversion map:
         From: Integer Ring
         To:   Integer Ring localized at [2]
    Arithmetic performed after coercions.
    Result lives in Integer Ring localized at [2]
    Integer Ring localized at [2]

    sage: cm.explain(R, ZZ, operator.mul)
    Action discovered.
       Right scalar multiplication by Integer Ring on Integer Ring localized at [2]
    Result lives in Integer Ring localized at [2]
    Integer Ring localized at [2]

    sage: R6 = Localization([2,3]); R6
    Integer Ring localized at [2, 3]
    sage: R6(1/3) - R(1/2)
    LocalElt(-1/6)
    sage: parent(R6(1/3) - R(1/2))
    Integer Ring localized at [2, 3]

    sage: R.has_coerce_map_from(ZZ)
    True
    sage: R.coerce_map_from(ZZ)
    Conversion map:
     From: Integer Ring
     To:   Integer Ring localized at [2]

    sage: R6.coerce_map_from(R)
    Conversion map:
     From: Integer Ring localized at [2]
     To:   Integer Ring localized at [2, 3]

    sage: R6.coerce(R(1/2))
    LocalElt(1/2)

    sage: cm.explain(R, R6, operator.mul)
    Coercion on left operand via
       Conversion map:
         From: Integer Ring localized at [2]
         To:   Integer Ring localized at [2, 3]
    Arithmetic performed after coercions.
    Result lives in Integer Ring localized at [2, 3]
    Integer Ring localized at [2, 3]

Provided Methods
~~~~~~~~~~~~~~~~

* ``__call__``

  This provides a consistent interface for element construction. In
  particular, it makes sure that conversion always gives the same
  result as coercion, if a coercion exists. (This used to be violated
  for some Rings in Sage as the code for conversion and coercion got
  edited separately.) Let R be a Parent and assume the user types
  R(x), where x has parent X.  Roughly speaking, the following occurs:

  1. If X ``is`` R, return x (*)

  2. If there is a coercion `f: X \rightarrow R`, return `f(x)`

  3. If there is a coercion `f: R \rightarrow X`, try to return `{f^{-1}}(x)`

  4. Return ``R._element_constructor_(x)`` (**)

  Keywords and extra arguments are passed on. The result of all this logic is cached.

  (*) Unless there is a "copy" keyword like R(x, copy=False)

  (**) Technically, a generic morphism is created from X to R, which
  may use magic methods like ``_integer_`` or other data provided by
  ``_populate_coercion_lists_``.

* ``coerce``

  Coerces elements into self, raising a type error if there is no
  coercion map.

* ``coerce_map_from, convert_map_from``

  Returns an actual ``Morphism`` object to coerce/convert from
  another Parent to self. Barring direct construction of elements of
  R, ``R.convert_map_from(S)`` will provide a callable Python object
  which is the fastest way to convert elements of S to elements of
  R.  From Cython, it can be invoked via the cdef ``_call_`` method.

* ``has_coerce_map_from``

  Returns ``True`` or ``False`` depending on whether or not there is
  a coercion.  ``R.has_coerce_map_from(S)`` is shorthand for
  ``R.coerce_map_from(S) is not None``

* ``get_action``

  This will unwind all the
  ``_rmul_, _lmul_, _r_action_, _l_action_, ...`` methods to provide
  an actual ``Action`` object, if one exists.


Discovering new parents
-----------------------

New parents are discovered using an algorithm in
sage/category/pushout.py.  The fundamental idea is that most Parents
in Sage are constructed from simpler objects via various functors.
These are accessed via the :meth:`construction` method, which returns a
(simpler) Parent along with a functor with which one can create self.

::

    sage: CC.construction()
    (AlgebraicClosureFunctor, Real Field with 53 bits of precision)
    sage: RR.construction()
    (Completion[+Infinity], Rational Field)
    sage: QQ.construction()
    (FractionField, Integer Ring)
    sage: ZZ.construction()  # None

    sage: Qp(5).construction()
    (Completion[5], Rational Field)
    sage: QQ.completion(5, 100, {})
    5-adic Field with capped relative precision 100
    sage: c, R = RR.construction()
    sage: a = CC.construction()[0]
    sage: a.commutes(c)
    False
    sage: RR == c(QQ)
    True

    sage: sage.categories.pushout.construction_tower(Frac(CDF['x']))
    [(None,
     Fraction Field of Univariate Polynomial Ring in x over Complex Double Field),
    (FractionField, Univariate Polynomial Ring in x over Complex Double Field),
    (Poly[x], Complex Double Field),
    (AlgebraicClosureFunctor, Real Double Field),
    (Completion[+Infinity], Rational Field),
    (FractionField, Integer Ring)]

Given Parents R and S, such that there is no coercion either from R to
S or from S to R, one can find a common Z with coercions
`R \rightarrow Z` and `S \rightarrow Z` by considering the sequence of
construction functors to get from a common ancestor to both R and S.
We then use a *heuristic* algorithm to interleave these constructors
in an attempt to arrive at a suitable Z (if one exists). For example::

    sage: ZZ['x'].construction()
    (Poly[x], Integer Ring)
    sage: QQ.construction()
    (FractionField, Integer Ring)
    sage: sage.categories.pushout.pushout(ZZ['x'], QQ)
    Univariate Polynomial Ring in x over Rational Field
    sage: sage.categories.pushout.pushout(ZZ['x'], QQ).construction()
    (Poly[x], Rational Field)

The common ancestor is `Z` and our options for Z are
`\mathrm{Frac}(\ZZ[x])` or `\mathrm{Frac}(\ZZ)[x]`.
In Sage we choose the later, treating the fraction
field functor as binding "more tightly" than the polynomial functor,
as most people agree that `\QQ[x]` is the more natural choice. The same
procedure is applied to more complicated Parents, returning a new
Parent if one can be unambiguously determined.

::

    sage: sage.categories.pushout.pushout(Frac(ZZ['x,y,z']), QQ['z, t'])
    Univariate Polynomial Ring in t over Fraction Field of Multivariate Polynomial Ring in x, y, z over Rational Field

Modules
-------

.. toctree::
    :maxdepth: 2

    sage/structure/coerce
    sage/structure/coerce_actions
    sage/structure/coerce_maps

    sage/categories/pushout
    sage/categories/action

    sage/structure/coerce_dict
    sage/structure/coerce_exceptions

.. include:: ../footer.txt
