README.rst

Documenting Coq with Sphinx

Coq's reference manual is written in reStructuredText (“reST”), and compiled with Sphinx.

In addition to standard reST directives (a directive is similar to a LaTeX environment) and roles (a role is similar to a LaTeX command), the coqrst plugin loaded by the documentation uses a custom Coq domain — a set of Coq-specific directives that define objects like tactics, commands (vernacs), warnings, etc. —, some custom directives, and a few custom roles. Finally, this manual uses a small DSL to describe tactic invocations and commands.

Coq objects

Our Coq domain define multiple objects. Each object has a signature (think type signature), followed by an optional body (a description of that object). The following example defines two objects: a variant of the simpl tactic, and an error that it may raise:

.. tacv:: simpl @pattern at {+ @natural}
   :name: simpl_at

   This applies ``simpl`` only to the :n:`{+ @natural}` occurrences of the subterms
   matching :n:`@pattern` in the current goal.

   .. exn:: Too few occurrences
      :undocumented:

Objects are automatically collected into indices, and can be linked to using the role version of the object's directive. For example, you could link to the tactic variant above using :tacv:`simpl_at`, and to its exception using :exn:`Too few occurrences`.

Names (link targets) are auto-generated for most simple objects, though they can always be overwritten using a :name: option, as shown above.

• Options, errors, warnings have their name set to their signature, with ... replacing all notation bits. For example, the auto-generated name of .. exn:: @qualid is not a module is ... is not a module, and a link to it would take the form :exn:`... is not a module`.
• Vernacs (commands) have their name set to the first word of their signature. For example, the auto-generated name of Axiom @ident : @term is Axiom, and a link to it would take the form :cmd:`Axiom`.
• Vernac variants, tactic notations, and tactic variants do not have a default name.

Most objects should have a body (i.e. a block of indented text following the signature, called “contents” in Sphinx terms). Undocumented objects should have the :undocumented: flag instead, as shown above. When multiple objects have a single description, they can be grouped into a single object, like this (semicolons can be used to separate the names of the objects; names starting with _ will be omitted from the indexes):

.. cmdv:: Lemma @ident {* @binder } : @type
          Remark @ident {* @binder } : @type
          Fact @ident {* @binder } : @type
          Corollary @ident {* @binder } : @type
          Proposition @ident {* @binder } : @type
   :name: Lemma; Remark; Fact; Corollary; Proposition

   These commands are all synonyms of :n:`Theorem @ident {* @binder } : type`.

Notations

The signatures of most objects can be written using a succinct DSL for Coq notations (think regular expressions written with a Lispy syntax). A typical signature might look like Hint Extern @natural {? @pattern} => @tactic, which means that the Hint Extern command takes a number (natural), followed by an optional pattern, and a mandatory tactic. The language has the following constructs (the full grammar is in TacticNotations.g):

@…

A placeholder (@ident, @natural, @tactic…)

{? …}

an optional block

{* …}, {+ …}

an optional (*) or mandatory (+) block that can be repeated, with repetitions separated by spaces

{*, …}, {+, …}

an optional or mandatory repeatable block, with repetitions separated by commas

{| … | … | … }

an alternative, indicating than one of multiple constructs can be used

%{, %}, %|

an escaped character (rendered without the leading %). In most cases, escaping is not necessary. In particular, the following expressions are all parsed as plain text, and do not need escaping: { xyz }, x |- y. But the following escapes are needed: {| a b %| c | d }, all: %{. (We use % instead of the usual \ because you'd have to type \ twice in your reStructuredText file.)

For more details and corner cases, see Advanced uses of notations below.

As an exercise, what do the following patterns mean?

pattern {+, @term {? at {+ @natural}}}
generalize {+, @term at {+ @natural} as @ident}
fix @ident @natural with {+ (@ident {+ @binder} {? {struct @ident'}} : @type)}

Objects

Here is the list of all objects of the Coq domain (The symbol ✒️ indicates an object whose signature can be written using the notations DSL):

.. attr:: ✒️ An attribute.

Example:

.. attr:: local

.. cmd:: ✒️ A Coq command.

Example:

.. cmd:: Infix @string := @one_term {? ( {+, @syntax_modifier } ) } {? : @ident }

   This command is equivalent to :n:`…`.

.. cmdv:: ✒️ A variant of a Coq command.

Example:

.. cmd:: Axiom @ident : @term.

   This command links :token:`term` to the name :token:`term` as its specification in
   the global context. The fact asserted by :token:`term` is thus assumed as a
   postulate.

   .. cmdv:: Parameter @ident : @term.

      This is equivalent to :n:`Axiom @ident : @term`.

.. exn:: ✒️ An error raised by a Coq command or tactic.

This commonly appears nested in the .. tacn:: that raises the exception.

Example:

.. tacv:: assert @form by @tactic

   This tactic applies :n:`@tactic` to solve the subgoals generated by
   ``assert``.

   .. exn:: Proof is not complete

      Raised if :n:`@tactic` does not fully solve the goal.

.. flag:: ✒️ A Coq flag (i.e. a boolean setting).

Example:

.. flag:: Nonrecursive Elimination Schemes

   Controls whether types declared with the keywords
   :cmd:`Variant` and :cmd:`Record` get an automatic declaration of
   induction principles.

.. opt:: ✒️ A Coq option (a setting with non-boolean value, e.g. a string or numeric value).

Example:

.. opt:: Hyps Limit @natural
   :name Hyps Limit

   Controls the maximum number of hypotheses displayed in goals after
   application of a tactic.

.. prodn:: A grammar production.

Use .. prodn to document grammar productions instead of Sphinx production lists.

prodn displays multiple productions together with alignment similar to .. productionlist, however unlike .. productionlists, this directive accepts notation syntax.

Example:

 .. prodn:: occ_switch ::= { {? {| + | - } } {* @natural } }
 term += let: @pattern := @term in @term
 | second_production

The first line defines "occ_switch", which must be unique in the document.  The second
references and expands the definition of "term", whose main definition is elsewhere
in the document.  The third form is for continuing the
definition of a nonterminal when it has multiple productions.  It leaves the first
column in the output blank.

.. table:: ✒️ A Coq table, i.e. a setting that is a set of values.

Example:

.. table:: Search Blacklist @string
   :name: Search Blacklist

   Controls ...

.. tacn:: ✒️ A tactic, or a tactic notation.

Example:

.. tacn:: do @natural @expr

   :token:`expr` is evaluated to ``v`` which must be a tactic value. …

.. tacv:: ✒️ A variant of a tactic.

Example:

.. tacn:: fail

   This is the always-failing tactic: it does not solve any goal. It is
   useful for defining other tacticals since it can be caught by
   :tacn:`try`, :tacn:`repeat`, :tacn:`match goal`, or the branching
   tacticals. …

   .. tacv:: fail @natural

      The number is the failure level. If no level is specified, it
      defaults to 0. …

.. thm:: A theorem.

Example:

.. thm:: Bound on the ceiling function

   Let :math:`p` be an integer and :math:`c` a rational constant. Then
   :math:`p \ge c \rightarrow p \ge \lceil{c}\rceil`.

.. warn:: ✒️ An warning raised by a Coq command or tactic..

Do not mistake this for .. warning::; this directive is for warning messages produced by Coq.

Example:

.. warn:: Ambiguous path

   When the coercion :token:`qualid` is added to the inheritance graph, non
   valid coercion paths are ignored.

Coq directives

In addition to the objects above, the coqrst Sphinx plugin defines the following directives:

.. coqtop:: A reST directive to describe interactions with Coqtop.

Usage:

.. coqtop:: options…

   Coq code to send to coqtop

Example:

.. coqtop:: in reset

   Print nat.
   Definition a := 1.

The blank line after the directive is required. If you begin a proof, use the abort option to reset coqtop for the next example.

Here is a list of permissible options:

• Display options (choose exactly one)
  • all: Display input and output
  • in: Display only input
  • out: Display only output
  • none: Display neither (useful for setup commands)
• Behavior options
  • reset: Send a Reset Initial command before running this block
  • fail: Don't die if a command fails, implies warn (so no need to put both)
  • warn: Don't die if a command emits a warning
  • restart: Send a Restart command before running this block (only works in proof mode)
  • abort: Send an Abort All command after running this block (leaves all pending proofs if any)

coqtop's state is preserved across consecutive .. coqtop:: blocks of the same document (coqrst creates a single coqtop process per reST source file). Use the reset option to reset Coq's state.

.. coqdoc:: A reST directive to display Coqtop-formatted source code.

Usage:

.. coqdoc::

   Coq code to highlight

Example:

.. coqdoc::

   Definition test := 1.

.. example:: A reST directive for examples.

This behaves like a generic admonition; see http://docutils.sourceforge.net/docs/ref/rst/directives.html#generic-admonition for more details.

Optionally, any text immediately following the .. example:: header is used as the example's title.

Example:

.. example:: Adding a hint to a database

   The following adds ``plus_comm`` to the ``plu`` database:

   .. coqdoc::

      Hint Resolve plus_comm : plu.

.. inference:: A reST directive to format inference rules.

This also serves as a small illustration of the way to create new Sphinx directives.

Usage:

.. inference:: name

   newline-separated premises
   --------------------------
   conclusion

Example:

.. inference:: Prod-Pro

   \WTEG{T}{s}
   s \in \Sort
   \WTE{\Gamma::(x:T)}{U}{\Prop}
   -----------------------------
   \WTEG{\forall~x:T,U}{\Prop}

.. preamble:: A reST directive to include a TeX file.

Mostly useful to let MathJax know about defs and newcommands. The contents of the TeX file are wrapped in a math environment, as MathJax doesn't process LaTeX definitions otherwise.

Usage:

.. preamble:: preamble.tex

Coq roles

In addition to the objects and directives above, the coqrst Sphinx plugin defines the following roles:

:g: Coq code.

Use this for Gallina and Ltac snippets:

:g:`apply plus_comm; reflexivity`
:g:`Set Printing All.`
:g:`forall (x: t), P(x)`

:n: Any text using the notation syntax (@id, {+, …}, etc.).

Use this to explain tactic equivalences. For example, you might write this:

:n:`generalize @term as @ident` is just like :n:`generalize @term`, but
it names the introduced hypothesis :token:`ident`.

Note that this example also uses :token:. That's because ident is defined in the Coq manual as a grammar production, and :token: creates a link to that. When referring to a placeholder that happens to be a grammar production, :token:`…` is typically preferable to :n:`@…`.

:production: A grammar production not included in a prodn directive.

Useful to informally introduce a production, as part of running text.

Example:

:production:`string` indicates a quoted string.

You're not likely to use this role very commonly; instead, use a prodn directive and reference its tokens using :token:`…`.

:gdef: Marks the definition of a glossary term inline in the text. Matching :term:`XXX`

constructs will link to it. Use the form :gdef:`text <term>` to display "text" for the definition of "term", such as when "term" must be capitalized or plural for grammatical reasons. The term will also appear in the Glossary Index.

Examples:

A :gdef:`prime` number is divisible only by itself and 1.
:gdef:`Composite <composite>` numbers are the non-prime numbers.

Common mistakes

Improper nesting

DO

.. cmd:: Foo @bar

   Foo the first instance of :token:`bar`\ s.

   .. cmdv:: Foo All

      Foo all the :token:`bar`\ s in
      the current context

DON'T

.. cmd:: Foo @bar

Foo the first instance of :token:`bar`\ s.

.. cmdv:: Foo All

Foo all the :token:`bar`\ s in
the current context

You can set the report_undocumented_coq_objects setting in conf.py to "info" or "warning" to get a list of all Coq objects without a description.

Overusing :token:

DO

This is equivalent to :n:`Axiom @ident : @term`.

DON'T

This is equivalent to ``Axiom`` :token:`ident` : :token:`term`.

DO

:n:`power_tac @term [@ltac]`
  allows :tacn:`ring` and :tacn:`ring_simplify` to recognize …

DON'T

power_tac :n:`@term` [:n:`@ltac`]
  allows :tacn:`ring` and :tacn:`ring_simplify` to recognize …

DO

:n:`name={*; attr}`

DON'T

``name=``:n:`{*; attr}`

Omitting annotations

DO

.. tacv:: assert @form as @simple_intropattern

DON'T

.. tacv:: assert form as simple_intropattern

Using the .. coqtop:: directive for syntax highlighting

DO

A tactic of the form:

.. coqdoc::

   do [ t1 | … | tn ].

is equivalent to the standard Ltac expression:

.. coqdoc::

   first [ t1 | … | tn ].

DON'T

A tactic of the form:

.. coqtop:: in

   do [ t1 | … | tn ].

is equivalent to the standard Ltac expression:

.. coqtop:: in

   first [ t1 | … | tn ].

Overusing plain quotes

DO

The :tacn:`refine` tactic can raise the :exn:`Invalid argument` exception.
The term :g:`let a = 1 in a a` is ill-typed.

DON'T

The ``refine`` tactic can raise the ``Invalid argument`` exception.
The term ``let a = 1 in a a`` is ill-typed.

Plain quotes produce plain text, without highlighting or cross-references.

Overusing the example directive

DO

Here is a useful axiom:

.. coqdoc::

   Axiom proof_irrelevance : forall (P : Prop) (x y : P), x=y.

DO

.. example:: Using proof-irrelevance

   If you assume the axiom above, …

DON'T

Here is a useful axiom:

.. example::

   .. coqdoc::

      Axiom proof_irrelevance : forall (P : Prop) (x y : P), x=y.

Tips and tricks

Nested lemmas

The .. coqtop:: directive does not reset Coq after running its contents. That is, the following will create two nested lemmas (which by default results in a failure):

.. coqtop:: all

   Lemma l1: 1 + 1 = 2.

.. coqtop:: all

   Lemma l2: 2 + 2 <> 1.

Add either abort to the first block or reset to the second block to avoid nesting lemmas.

Abbreviations and macros

Substitutions for specially-formatted names (like |Cic|, |Ltac| and |Latex|), along with some useful LaTeX macros, are defined in a separate file. This file is automatically included in all manual pages.

Emacs

The dev/tools/coqdev.el folder contains a convenient Emacs function to quickly insert Sphinx roles and quotes. It takes a single character (one of gntm:`), and inserts one of :g:, :n:, :t:, or an arbitrary role, or double quotes. You can also select a region of text, and wrap it in single or double backticks using that function.

Use the following snippet to bind it to F12 in rst-mode:

(with-eval-after-load 'rst
  (define-key rst-mode-map (kbd "<f12>") #'coqdev-sphinx-rst-coq-action))

Advanced uses of notations

• Use % to escape grammar literal strings that are the same as metasyntax, such as {, |, } and {|. (While this is optional for | and { ... } outside of {| ... }, always using the escape requires less thought.)
• Literals such as |- and || don't need to be escaped.
• The literal % shouldn't be escaped.
• Don't use the escape for a | separator in {* and {+. These should appear as {*| and {+|.