From 8a35af641d92d7a65e35a3838f9cbbef4e2b2c8b Mon Sep 17 00:00:00 2001
From: Théo Zimmermann
Date: Wed, 13 May 2020 23:56:27 +0200
Subject: New file on existential variables.

---
 doc/sphinx/language/extensions/evars.rst | 97 ++++++++++++++++++++++++++++++++
 1 file changed, 97 insertions(+)
 create mode 100644 doc/sphinx/language/extensions/evars.rst

(limited to 'doc/sphinx/language/extensions')

diff --git a/doc/sphinx/language/extensions/evars.rst b/doc/sphinx/language/extensions/evars.rst
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+.. _existential-variables:
+
+Existential variables
+---------------------
+
+.. insertprodn term_evar term_evar
+
+.. prodn::
+   term_evar ::= _
+   | ?[ @ident ]
+   | ?[ ?@ident ]
+   | ?@ident {? @%{ {+; @ident := @term } %} }
+
+|Coq| terms can include existential variables which represents unknown
+subterms to eventually be replaced by actual subterms.
+
+Existential variables are generated in place of unsolvable implicit
+arguments or “_” placeholders when using commands such as ``Check`` (see
+Section :ref:`requests-to-the-environment`) or when using tactics such as
+:tacn:`refine`, as well as in place of unsolvable instances when using
+tactics such that :tacn:`eapply`. An existential
+variable is defined in a context, which is the context of variables of
+the placeholder which generated the existential variable, and a type,
+which is the expected type of the placeholder.
+
+As a consequence of typing constraints, existential variables can be
+duplicated in such a way that they possibly appear in different
+contexts than their defining context. Thus, any occurrence of a given
+existential variable comes with an instance of its original context.
+In the simple case, when an existential variable denotes the
+placeholder which generated it, or is used in the same context as the
+one in which it was generated, the context is not displayed and the
+existential variable is represented by “?” followed by an identifier.
+
+.. coqtop:: all
+
+   Parameter identity : forall (X:Set), X -> X.
+
+   Check identity _ _.
+
+   Check identity _ (fun x => _).
+
+In the general case, when an existential variable :n:`?@ident` appears
+outside of its context of definition, its instance, written under the
+form :n:`{ {*; @ident := @term} }` is appending to its name, indicating
+how the variables of its defining context are instantiated.
+The variables of the context of the existential variables which are
+instantiated by themselves are not written, unless the :flag:`Printing Existential Instances` flag
+is on (see Section :ref:`explicit-display-existentials`), and this is why an
+existential variable used in the same context as its context of definition is written with no instance.
+
+.. coqtop:: all
+
+   Check (fun x y => _) 0 1.
+
+   Set Printing Existential Instances.
+
+   Check (fun x y => _) 0 1.
+
+Existential variables can be named by the user upon creation using
+the syntax :n:`?[@ident]`. This is useful when the existential
+variable needs to be explicitly handled later in the script (e.g.
+with a named-goal selector, see :ref:`goal-selectors`).
+
+.. _explicit-display-existentials:
+
+Explicit displaying of existential instances for pretty-printing
+~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
+
+.. flag:: Printing Existential Instances
+
+   This flag (off by default) activates the full display of how the
+   context of an existential variable is instantiated at each of the
+   occurrences of the existential variable.
+
+.. _tactics-in-terms:
+
+Solving existential variables using tactics
+~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
+
+Instead of letting the unification engine try to solve an existential
+variable by itself, one can also provide an explicit hole together
+with a tactic to solve it. Using the syntax ``ltac:(``\ `tacexpr`\ ``)``, the user
+can put a tactic anywhere a term is expected. The order of resolution
+is not specified and is implementation-dependent. The inner tactic may
+use any variable defined in its scope, including repeated alternations
+between variables introduced by term binding as well as those
+introduced by tactic binding. The expression `tacexpr` can be any tactic
+expression as described in :ref:`ltac`.
+
+.. coqtop:: all
+
+   Definition foo (x : nat) : nat := ltac:(exact x).
+
+This construction is useful when one wants to define complicated terms
+using highly automated tactics without resorting to writing the proof-term
+by means of the interactive proof engine.
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