# Documentation style #

All pull requests must meet the following documentation standards. See the doc-gen repo for information about the automatically generated doc pages.

You can preview the markdown processing of a GitHub page or pull request using the Lean doc preview page.

## Header comment #

Each mathlib file should start with:

(See the example below.)

Headers use atx-style headers (with hash signs, no underlying dash). The open and close delimiters /-! and -/ should appear on their own lines.

The mandatory title of the file is a first level header. It is followed by a summary of the content of the file.

The other sections, with second level headers are (in this order):

• Main definitions (optional, can be covered in the summary)
• Main statements (optional, can be covered in the summary)
• Notations (omitted only if no notation is introduced in this file)
• Implementation notes (description of important design decisions or interface features, including use of type classes and simp canonical form for new definitions)
• References (references to textbooks or papers, or Wikipedia pages)
• Tags (a list of keywords that could be useful when doing text search in mathlib to find where something is covered)

References should refer to bibtex entries in the mathlib citations file. See the Citing other works section below.

The following code block is an example of a file header.

/-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis
-/
import data.rat
import algebra.gcd_domain
import algebra.field_power
import ring_theory.multiplicity
import tactic.ring
import data.real.cau_seq
import tactic.norm_cast

/-!
# p-adic norm

This file defines the p-adic valuation and the p-adic norm on ℚ.

The p-adic valuation on ℚ is the difference of the multiplicities of p in the numerator and
denominator of q. This function obeys the standard properties of a valuation, with the appropriate
assumptions on p.

The valuation induces a norm on ℚ. This norm is a nonarchimedean absolute value.
It takes values in {0} ∪ {1/p^k | k ∈ ℤ}.

## Notations

This file uses the local notation /. for rat.mk.

## Implementation notes

Much, but not all, of this file assumes that p is prime. This assumption is inferred automatically
by taking (prime p) as a type class argument.

## References

* [F. Q. Gouvêa, *p-adic numbers*][gouvea1997]
* <https://en.wikipedia.org/wiki/P-adic_number>

## Tags

p-adic, p adic, padic, norm, valuation
-/


## Doc strings #

Every definition and major theorem is required to have a doc string. (Doc strings on lemmas are also encouraged, particularly if the lemma has any mathematical content or might be useful in another file.) These are introduced using /-- and closed by -/ above the definition, with either newlines or single spaces between the markers and the text. They can contain Markdown and LaTeX as well: see the next section. If a doc string is a complete sentence, then it should end in a period. Named theorems, such as the mean value theorem should be bold faced (i.e., with two asterisks before and after).

Doc strings should convey the mathematical meaning of the definition. They are allowed to lie slightly about the actual implementation. The following is a doc string example:

/--
If q ≠ 0, the p-adic norm of a rational q is p ^ (-(padic_val_rat p q)).
If q = 0, the p-adic norm of q is 0.
-/
def padic_norm (p : ℕ) (q : ℚ) : ℚ :=
if q = 0 then 0 else (p : ℚ) ^ (-(padic_val_rat p q))


An example that is slightly lying but still describes the mathematical content would be:

/--
For p ≠ 1, the p-adic valuation of an integer z ≠ 0 is the largest natural number n such that
p^n divides z.
padic_val_rat defines the valuation of a rational q to be the valuation of q.num minus the
valuation of q.denom.
If q = 0 or p = 1, then padic_val_rat p q defaults to 0.
-/
def padic_val_rat (p : ℕ) (q : ℚ) : ℤ :=
if h : q ≠ 0 ∧ p ≠ 1
then (multiplicity (p : ℤ) q.num).get
(multiplicity.finite_int_iff.2 ⟨h.2, rat.num_ne_zero_of_ne_zero h.1⟩) -
(multiplicity (p : ℤ) q.denom).get
(multiplicity.finite_int_iff.2 ⟨h.2, by exact_mod_cast rat.denom_ne_zero _⟩)
else 0


The doc_blame linter lists all definitions that do not have doc strings. The doc_blame_thm linter will lists theorems and lemmas that do not have doc strings.

To run only the doc_blame linter, add the following to the end of your lean file:

#lint only doc_blame


To run only the doc_blame and doc_blame_thm linters, add the following to the end of your lean file:

#lint only doc_blame doc_blame_thm


To run the all default linters, including doc_blame, add the following to the end of your lean file:

#lint


To run the all default linters, including doc_blame and run doc_blame_thm, add the following to the end of your lean file:

#lint doc_blame_thm


## LaTeX and Markdown #

We generally put references to Lean declarations or variables in between backticks. Writing the fully-qualified name (e.g. finset.card_pos instead of just card_pos) will turn the name into a link on our online docs.

Raw URLs should be enclosed in angle brackets <...> to ensure that they will be clickable online. (Some URLs, especially those with parentheses or other special symbols, may not be parsed correctly by the markdown renderer.)

When talking about mathematical symbols instead, it may be preferable to use LaTeX. LaTeX can be included in doc strings in three ways:

• using single dollar signs $...$ to render math inline,
• using double dollar signs $$...$$ to render math in "display mode", or
• using environments \begin{*} ... \end{*} (without dollar signs).

These correspond to the MathJax settings of our online docs. The interaction between the Markdown and LaTeX there is similar to that on https://math.stackexchange.com and https://mathoverflow.net, so you can paste a doc string into an editing sandbox there to preview the final result. See also the math.stackexchange MathJax tutorial.

## Tactic doc entries #

Interactive tactics, user commands, hole commands and user attributes should have doc strings explaining how they can be used. The add_tactic_doc command should be invoked afterwards to add a doc entry to the appropriate page in the online docs.

Example:

/--
describe what the command does here
-/
add_tactic_doc
{ name := "display name of the tactic",
category := cat,
decl_names := [dcl_1, dcl_2],
tags := ["tag_1", "tag_2"]
}


The argument to add_tactic_doc is a structure of type tactic_doc_entry.

• name refers to the display name of the tactic; it is used as the header of the doc entry.
• cat refers to the category of doc entry. Options: doc_category.tactic, doc_category.cmd, doc_category.hole_cmd, doc_category.attr
• decl_names is a list of the declarations associated with this doc. For instance, the entry for linarith would set decl_names := [tactic.interactive.linarith]. Some entries may cover multiple declarations. It is only necessary to list the interactive versions of tactics.
• tags is an optional list of strings used to categorize entries.
• The doc string is the body of the entry. It can be formatted with markdown. What you are reading now is taken from the description of add_tactic_doc.

If only one related declaration is listed in decl_names and if this invocation of add_tactic_doc does not have a doc string, the doc string of that declaration will become the body of the tactic doc entry. If there are multiple declarations, you can select the one to be used by passing a name to the inherit_description_from field.

If you prefer a tactic to have a doc string that is different then the doc entry, then between the /-- -/ markers, write the desired doc string first, then --- surrounded by new lines, and then the doc entry.

Note that providing a badly formed tactic_doc_entry to the command can result in strange error messages.

## Sectioning comments #

It is common to structure a file in sections, where each section contains related declarations. By describing the sections with module documentation /-! ... -/ at the beginning, these sections can be seen in the documentation.

While these sectioning comments will often correspond to section or namespace commands, this is not required. You can use sectioning comments inside of a section or namespace, and you can have multiple sections or namespaces following one sectioning comment.

Sectioning comments are for display and readability only. They have no semantic meaning.

Third-level headers ### should be used for titles inside sectioning comments.

If the comment is more than one line long, the delimiters /-! and -/ should appear on their own lines.

See meta/expr.lean for an example in practice.

namespace binder_info

/-!
### Declarations about binder_info

This section develops an extended API for the type binder_info.
-/

instance : inhabited binder_info := ⟨ binder_info.default ⟩

/-- The brackets corresponding to a given binder_info. -/
def brackets : binder_info → string × string
| binder_info.implicit        := ("{", "}")
| binder_info.strict_implicit := ("{{", "}}")
| binder_info.inst_implicit   := ("[", "]")
| _                           := ("(", ")")

end binder_info

namespace name

/-! ### Declarations about name -/

/-- Find the largest prefix n of a name such that f n ≠ none, then replace this prefix
with the value of f n. -/
def map_prefix (f : name → option name) : name → name
| anonymous := anonymous
| (mk_string s n') := (f (mk_string s n')).get_or_else (mk_string s $map_prefix n') | (mk_numeral d n') := (f (mk_numeral d n')).get_or_else (mk_numeral d$ map_prefix n')


## Theories documentation #

In addition to documentation living in Lean files, we have theories documentation where we give overviews spanning several Lean files, and more mathematical explanations in cases where formalization requires slightly exotic points of view, see for instance the topology documentation.

## Citing other works #

To cite papers and books in doc strings, the references should first be added to the BibTeX file: docs/references.bib. To normalize the file with bibtool, you can run:

bibtool --preserve.key.case=on --preserve.keys=on --print.use.tab=off --pass.comments=on -s -i docs/references.bib -o docs/references.bib


To ensure that your citations become links in the online docs, you can use either of the following two styles:

First, you may enclose the citation key used in docs/references.bib in square brackets:

The proof can be found in [Boole1854].


In the online docs, this will become something like:

The proof can be found in [Boo54]

(The key will change into an alpha style label and become a link to the References page of the docs.)

Alternatively, you can use custom text for the citation by putting text in square brackets ahead of the citation key:

See [Grundlagen der Geometrie][hilbert1999] for an alternative axiomatization.


See Grundlagen der Geometrie for an alternative axiomatization.

Note that you currently cannot use the closing square bracket ] symbol in the link text. So the following will not result in a working link:

We follow [Euclid's *Elements* [Prop. 1]][heath1956a].
`

We follow [Euclid's Elements [Prop. 1]][heath1956a].

## Examples #

The following files are maintained as examples of good documentation style: