.. _coercions:

Implicit Coercions
====================

:Author: Amokrane Saïbi

General Presentation
---------------------

This section describes the inheritance mechanism of Coq. In Coq with
inheritance, we are not interested in adding any expressive power to
our theory, but only convenience. Given a term, possibly not typable,
we are interested in the problem of determining if it can be well
typed modulo insertion of appropriate coercions. We allow to write:

 * :g:`f a` where :g:`f:(forall x:A,B)` and :g:`a:A'` when ``A'`` can
   be seen in some sense as a subtype of ``A``.
 * :g:`x:A` when ``A`` is not a type, but can be seen in
   a certain sense as a type: set, group, category etc.
 * :g:`f a` when ``f`` is not a function, but can be seen in a certain sense
   as a function: bijection, functor, any structure morphism etc.


Classes
-------

A class with :math:`n` parameters is any defined name with a type
:n:`forall (@ident__1 : @type__1)..(@ident__n:@type__n), @sort`.  Thus a class with
parameters is considered as a single class and not as a family of
classes.  An object of a class is any term of type :n:`@class @term__1 .. @term__n`.
In addition to these user-defined classes, we have two built-in classes:


  * ``Sortclass``, the class of sorts; its objects are the terms whose type is a
    sort (e.g. :g:`Prop` or :g:`Type`).
  * ``Funclass``, the class of functions; its objects are all the terms with a functional
    type, i.e. of form :g:`forall x:A,B`.

Formally, the syntax of classes is defined as:

   .. insertprodn class class

   .. prodn::
      class ::= Funclass
      | Sortclass
      | @reference



Coercions
---------

A name ``f`` can be declared as a coercion between a source user-defined class
``C`` with :math:`n` parameters and a target class ``D`` if one of these
conditions holds:

 * ``D`` is a user-defined class, then the type of ``f`` must have the form
   :g:`forall (x₁:A₁)..(xₙ:Aₙ)(y:C x₁..xₙ), D u₁..uₘ` where :math:`m`
   is the number of parameters of ``D``.
 * ``D`` is ``Funclass``, then the type of ``f`` must have the form
   :g:`forall (x₁:A₁)..(xₙ:Aₙ)(y:C x₁..xₙ)(x:A), B`.
 * ``D`` is ``Sortclass``, then the type of ``f`` must have the form
   :g:`forall (x₁:A₁)..(xₙ:Aₙ)(y:C x₁..xₙ), s` with ``s`` a sort.

We then write :g:`f : C >-> D`. The restriction on the type
of coercions is called *the uniform inheritance condition*.

.. note:: The built-in class ``Sortclass`` can be used as a source class, but
          the built-in class ``Funclass`` cannot.

To coerce an object :g:`t:C t₁..tₙ` of ``C`` towards ``D``, we have to
apply the coercion ``f`` to it; the obtained term :g:`f t₁..tₙ t` is
then an object of ``D``.


Identity Coercions
-------------------

Identity coercions are special cases of coercions used to go around
the uniform inheritance condition. Let ``C`` and ``D`` be two classes
with respectively `n` and `m` parameters and
:g:`f:forall (x₁:T₁)..(xₖ:Tₖ)(y:C u₁..uₙ), D v₁..vₘ` a function which
does not verify the uniform inheritance condition. To declare ``f`` as
coercion, one has first to declare a subclass ``C'`` of ``C``:

  :g:`C' := fun (x₁:T₁)..(xₖ:Tₖ) => C u₁..uₙ`

We then define an *identity coercion* between ``C'`` and ``C``:

  :g:`Id_C'_C  := fun (x₁:T₁)..(xₖ:Tₖ)(y:C' x₁..xₖ) => (y:C u₁..uₙ)`

We can now declare ``f`` as coercion from ``C'`` to ``D``, since we can
"cast" its type as
:g:`forall (x₁:T₁)..(xₖ:Tₖ)(y:C' x₁..xₖ),D v₁..vₘ`.

The identity coercions have a special status: to coerce an object
:g:`t:C' t₁..tₖ`
of ``C'`` towards ``C``, we do not have to insert explicitly ``Id_C'_C``
since :g:`Id_C'_C t₁..tₖ t` is convertible with ``t``.  However we
"rewrite" the type of ``t`` to become an object of ``C``; in this case,
it becomes :g:`C uₙ'..uₖ'` where each ``uᵢ'`` is the result of the
substitution in ``uᵢ`` of the variables ``xⱼ`` by ``tⱼ``.

Inheritance Graph
------------------

Coercions form an inheritance graph with classes as nodes.  We call
*coercion path* an ordered list of coercions between two nodes of
the graph.  A class ``C`` is said to be a subclass of ``D`` if there is a
coercion path in the graph from ``C`` to ``D``; we also say that ``C``
inherits from ``D``. Our mechanism supports multiple inheritance since a
class may inherit from several classes, contrary to simple inheritance
where a class inherits from at most one class.  However there must be
at most one path between two classes. If this is not the case, only
the *oldest* one is valid and the others are ignored. So the order
of declaration of coercions is important.

We extend notations for coercions to coercion paths. For instance
:g:`[f₁;..;fₖ] : C >-> D` is the coercion path composed
by the coercions ``f₁..fₖ``.  The application of a coercion path to a
term consists of the successive application of its coercions.


Declaring Coercions
-------------------------

.. cmd:: Coercion @reference : @class >-> @class
         Coercion @ident {? @univ_decl } @def_body

   :name: Coercion; _

  The first form declares the construction denoted by :token:`reference` as a coercion between
  the two given classes.  The second form defines :token:`ident`
  just like :cmd:`Definition` :n:`@ident {? @univ_decl } @def_body`
  and then declares :token:`ident` as a coercion between it source and its target.
  Both forms support the :attr:`local` attribute, which makes the coercion local to the current section.

  .. exn:: @qualid not declared.
     :undocumented:

  .. exn:: @qualid is already a coercion.
     :undocumented:

  .. exn:: Funclass cannot be a source class.
     :undocumented:

  .. exn:: @qualid is not a function.
     :undocumented:

  .. exn:: Cannot find the source class of @qualid.
     :undocumented:

  .. exn:: Cannot recognize @class as a source class of @qualid.
     :undocumented:

  .. warn:: @qualid does not respect the uniform inheritance condition.
     :undocumented:

  .. exn:: Found target class ... instead of ...
     :undocumented:

  .. warn:: New coercion path ... is ambiguous with existing ...

     When the coercion :token:`qualid` is added to the inheritance graph, new
     coercion paths which have the same classes as existing ones are ignored.
     The :cmd:`Coercion` command tries to check the convertibility of new ones and
     existing ones. The paths for which this check fails are displayed by a warning
     in the form :g:`[f₁;..;fₙ] : C >-> D`.

     The convertibility checking procedure for coercion paths is complete for
     paths consisting of coercions satisfying the uniform inheritance condition,
     but some coercion paths could be reported as ambiguous even if they are
     convertible with existing ones when they have coercions that don't satisfy
     the uniform inheritance condition.

  .. warn:: ... is not definitionally an identity function.

     If a coercion path has the same source and target class, that is said to be
     circular. When a new circular coercion path is not convertible with the
     identity function, it will be reported as ambiguous.

Some objects can be declared as coercions when they are defined.
This applies to :ref:`assumptions<gallina-assumptions>` and
constructors of :ref:`inductive types and record fields<gallina-inductive-definitions>`.
Use :n:`:>` instead of :n:`:` before the
:n:`@type` of the assumption to do so.  See :n:`@of_type`.


.. cmd:: Identity Coercion @ident : @class >-> @class

   If ``C`` is the source `class` and ``D`` the destination, we check
   that ``C`` is a :term:`constant` with a :term:`body` of the form
   :g:`fun (x₁:T₁)..(xₙ:Tₙ) => D t₁..tₘ` where `m` is the
   number of parameters of ``D``.  Then we define an identity
   function with type :g:`forall (x₁:T₁)..(xₙ:Tₙ)(y:C x₁..xₙ),D t₁..tₘ`,
   and we declare it as an identity coercion between ``C`` and ``D``.

   This command supports the :attr:`local` attribute, which makes the coercion local to the current section.

   .. exn:: @class must be a transparent constant.
      :undocumented:

   .. cmd:: SubClass @ident_decl @def_body

      If :n:`@type` is a class :n:`@ident'` applied to some arguments then
      :n:`@ident` is defined and an identity coercion of name
      :n:`Id_@ident_@ident'` is
      declared. Otherwise said, this is an abbreviation for

      :n:`Definition @ident := @type.`
      :n:`Identity Coercion Id_@ident_@ident' : @ident >-> @ident'`.

      This command supports the :attr:`local` attribute, which makes the coercion local to the current section.


Displaying Available Coercions
-------------------------------

.. cmd:: Print Classes

   Print the list of declared classes in the current context.

.. cmd:: Print Coercions

   Print the list of declared coercions in the current context.

.. cmd:: Print Graph

   Print the list of valid coercion paths in the current context.

.. cmd:: Print Coercion Paths @class @class

   Print the list of valid coercion paths between the two given classes.

Activating the Printing of Coercions
-------------------------------------

.. flag:: Printing Coercions

   When on, this :term:`flag` forces all the coercions to be printed.
   By default, coercions are not printed.

.. table:: Printing Coercion @qualid

   This :term:`table` specifies a set of qualids for which coercions are always displayed.  Use the
   :cmd:`Add` and :cmd:`Remove` commands to update the set of qualids.

.. _coercions-classes-as-records:

Classes as Records
------------------

.. index:: :> (coercion)

*Structures with Inheritance* may be defined using the :cmd:`Record` command.

Use `>` before the record name to declare the constructor name as
a coercion from the class of the last field type to the record name
(this may fail if the uniform inheritance condition is not
satisfied).  See :token:`record_definition`.

Use `:>` in the field type to declare the field as a coercion from the record name
to the class of the field type.  See :token:`of_type`.

Coercions and Sections
----------------------

The inheritance mechanism is compatible with the section
mechanism. The global classes and coercions defined inside a section
are redefined after its closing, using their new value and new
type. The classes and coercions which are local to the section are
simply forgotten.
Coercions with a local source class or a local target class, and
coercions which do not verify the uniform inheritance condition any longer
are also forgotten.

Coercions and Modules
---------------------

The coercions present in a module are activated only when the module is
explicitly imported.

Examples
--------

There are three situations:

Coercion at function application
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

:g:`f a` is ill-typed where :g:`f:forall x:A,B` and :g:`a:A'`. If there is a
coercion path between ``A'`` and ``A``, then :g:`f a` is transformed into
:g:`f a'` where ``a'`` is the result of the application of this
coercion path to ``a``.

We first give an example of coercion between atomic inductive types

.. coqtop:: all

  Definition bool_in_nat (b:bool) := if b then 0 else 1.
  Coercion bool_in_nat : bool >-> nat.
  Check (0 = true).
  Set Printing Coercions.
  Check (0 = true).
  Unset Printing Coercions.


.. warning::

  Note that ``Check (true = O)`` would fail. This is "normal" behavior of
  coercions. To validate ``true=O``, the coercion is searched from
  ``nat`` to ``bool``. There is none.

We give an example of coercion between classes with parameters.

.. coqtop:: all

  Parameters (C : nat -> Set) (D : nat -> bool -> Set) (E : bool -> Set).
  Parameter f : forall n:nat, C n -> D (S n) true.
  Coercion f : C >-> D.
  Parameter g : forall (n:nat) (b:bool), D n b -> E b.
  Coercion g : D >-> E.
  Parameter c : C 0.
  Parameter T : E true -> nat.
  Check (T c).
  Set Printing Coercions.
  Check (T c).
  Unset Printing Coercions.

We give now an example using identity coercions.

.. coqtop:: all

  Definition D' (b:bool) := D 1 b.
  Identity Coercion IdD'D : D' >-> D.
  Print IdD'D.
  Parameter d' : D' true.
  Check (T d').
  Set Printing Coercions.
  Check (T d').
  Unset Printing Coercions.


In the case of functional arguments, we use the monotonic rule of
sub-typing. To coerce :g:`t : forall x : A, B` towards
:g:`forall x : A', B'`, we have to coerce ``A'`` towards ``A`` and ``B``
towards ``B'``. An example is given below:

.. coqtop:: all

  Parameters (A B : Set) (h : A -> B).
  Coercion h : A >-> B.
  Parameter U : (A -> E true) -> nat.
  Parameter t : B -> C 0.
  Check (U t).
  Set Printing Coercions.
  Check (U t).
  Unset Printing Coercions.

Remark the changes in the result following the modification of the
previous example.

.. coqtop:: all

  Parameter U' : (C 0 -> B) -> nat.
  Parameter t' : E true -> A.
  Check (U' t').
  Set Printing Coercions.
  Check (U' t').
  Unset Printing Coercions.


Coercion to a type
~~~~~~~~~~~~~~~~~~

An assumption ``x:A`` when ``A`` is not a type, is ill-typed.  It is
replaced by ``x:A'`` where ``A'`` is the result of the application to
``A`` of the coercion path between the class of ``A`` and
``Sortclass`` if it exists.  This case occurs in the abstraction
:g:`fun x:A => t`, universal quantification :g:`forall x:A,B`, global
variables and parameters of (co)inductive definitions and
functions. In :g:`forall x:A,B`, such a coercion path may also be applied
to ``B`` if necessary.

.. coqtop:: all

  Parameter Graph : Type.
  Parameter Node : Graph -> Type.
  Coercion Node : Graph >-> Sortclass.
  Parameter G : Graph.
  Parameter Arrows : G -> G -> Type.
  Check Arrows.
  Parameter fg : G -> G.
  Check fg.
  Set Printing Coercions.
  Check fg.
  Unset Printing Coercions.


Coercion to a function
~~~~~~~~~~~~~~~~~~~~~~

``f a`` is ill-typed because ``f:A`` is not a function. The term
``f`` is replaced by the term obtained by applying to ``f`` the
coercion path between ``A`` and ``Funclass`` if it exists.

.. coqtop:: all

  Parameter bij : Set -> Set -> Set.
  Parameter ap : forall A B:Set, bij A B -> A -> B.
  Coercion ap : bij >-> Funclass.
  Parameter b : bij nat nat.
  Check (b 0).
  Set Printing Coercions.
  Check (b 0).
  Unset Printing Coercions.

Let us see the resulting graph after all these examples.

.. coqtop:: all

  Print Graph.
