CONTRIBUTING.md

Contribution Guide for the Mathematical Components library

We describe here best practices for contributing to the library. In particular we explain what conventions are used in the library. When contributing, you should comply to these conventions to get your code integrated to the library.

This file is not comprehensive yet and might still contain mistakes or unclear indications, please consider contributing to its improvement.

Proof style

General guidelines

• One important guideline is to structure proofs in blocks, i.e., forward steps, to limit the scope of errors.
  • See G. Gonthier, A. Mahboubi, "An introduction to small scale reflection in Coq", p.103 for an illustration
• A line should have no more than 80 characters. If a line is longer than that, it should be cut semantically. If there is no way to make a semantic cut (e.g. the user provides an explicit term that is too long to fit on one line), then just cut it over several lines to make it readable.
• Lines end with a point . and only have ; inside them.
• Lines that close a goal must start with a terminator (by or exact). You should consider using an editor that highlights those terminators in a specific color (e.g. red).
• Chaining too many optional rewrites makes error detection hard. The idiom is

  rewrite conditional_rule ?simplify_side_condition // next_rule.
  

• Do not use Focus or {}, use the relevant indentation, along with terminators like by or exact.

Spaces

We write

• move=> and move: (no space between move and => or :)
• apply/ and apply: (no space between apply and / or :)
• rewrite /definition (there is a space between rewrite and an unfold)

Indentation in proof scripts

• When two subgoals are created, the first subgoal is indented by 2 spaces, the second is not. Use last first to bring the smallest/less meaningful goal first, and keep the main flow of the proof unindented.

• When more than two subgoals are created, bullets are used - for the first level, + for the second and * for the third as in:

  tactic.
  - tactic.
    + tactic.
      * tactic.
      * tactic.
      * tactic.
    + tactic.
    + tactic.
  - tactic
  - tactic
  

  If all the subgoals have the same importance, use bullets for all of them, however, if one goal is more important than the others (i.e. is main flow of the proof). Then you might remove the bullet for this last one and unindent it as in:

  tactic.
  - tactic. (* secondary subgoal 1 *)
  - tactic. (* secondary subgoal 2 *)
  tactic. (* third subgoal is the main one *)
  

Statements of lemmas, theorems and definitions

• Universal quantifications with dependent variable should appear on the left hand side of the colon, until we reach the first non dependent variables. e.g. Lemma example x y : x < y -> x >= y = false

Term style

• Operators are surrounded by space, for example n*m should be written n * m. This particular example can be problematic if matrix.v is imported because then, M *m N is matrix product.

Statement-macros

• There is a number of "macros" that are available to state lemmas, like commutative, associative,... (see ssrfun.v)

• There are also macros that are available to localize a statement, like {in A, P},... (see ssrbool.v)

Naming of variables

• Variable/hypothesis names follow the following conventions.
  • Hypothesis should not be named H, H',... (these collide with subgroup variable conventions) but have meaningful names. For example, an hypothesis n > 0 should be named n_gt0.
  • Induction Hypotheses are prefixed by IH or ih (e.g. induction hypothesis on n is called IHn).
  • Natural numbers and integers should be named m, n, p, d, ...
  • Elements of another ring should be named x, y, z, u, v, w, ...
  • Polynomials should be named by lower case letter p, q, r ... (to avoid collision with properties named P, Q, ...)
  • Matrices should be named A, B, ..., M, N, ...
  • Polymorphic variables should be named x, ...
• Variables/hypotheses that do not survive the line can be introduced using ?.
• Variables/hypotheses with a very short scope (~ 1-5 lines) can have a short name.
• Variables/hypotheses with a longer scope (> 5 lines) must have a meaningful name.

Naming conventions for lemmas (non exhaustive)

Names in the library usually obey one of the following conventions:

• (condition_)?mainSymbol_suffixes
• mainSymbol_suffixes(_condition)?

Or in the presence of a property denoted by an n-ary or unary predicate:

• naryPredicate_mainSymbol+
• mainSymbol_unaryPredicate

Where:

• mainSymbol is the most meaningful part of the lemma. It generally is the head symbol of the right-hand side of an equation or the head symbol of a theorem. It can also simply be the main object of study, head symbol or not. It is usually either:
  • one of the main symbols of the theory at hand (e.g., it will be opp, add, mul, etc.) or
  • a special "canonical" operation, such as a ring morphism or a subtype predicate (e.g. linear, raddf, rmorph, rpred, etc.)
• condition is used when the lemma applies under some hypothesis.
• suffixes are there to refine what shape and/or what other symbols the lemma has. It can either be the name of a symbol (add, mul, etc.), or the (short) name of a predicate (inj for injective, id for "identity", etc.) or an abbreviation.

There is an underscore before suffixes when suffixes starts with a one-letter small identifier (i.e., not a capital letter or a number or a longer identifier such as if). Since the intent is to make the suffixes readable enough, there are exceptions for short names (e.g., lern1).

Abbreviations are in the header of the file which introduces them. We list here the main abbreviations.

• A -- associativity, as in andbA : associative andb
• AC -- right commutativity
• ACA -- self-interchange (inner commutativity), e.g., orbACA : (a || b) || (c || d) = (a || c) || (b || d)
• b -- a boolean argument, as in andbb : idempotent andb
• C -- commutativity, as in andbC : commutative andb -- alternatively, predicate or set complement, as in predC -- alternatively, constant
• CA -- left commutativity
• D -- predicate or set difference, as in predD
• E -- elimination lemma, as in negbFE : ~~ b = false -> b
• F or f -- boolean false, as in andbF : b && false = false
• F -- alternatively, about a finite type
• g -- a group argument.
• I -- left/right injectivity, as in addbI : right_injective addb -- alternatively predicate or set intersection, as in predI
• l -- the left-hand of an operation, as in
  • andb_orl : left_distributive andb orb
  • ltr_norml x y : (`|x| < y) = (- y < x < y)
• L -- the left-hand of a relation, as in ltn_subrL : n - m < n = (0 < m) && (0 < n)
• LR -- moving an operator from the left-hand to the right-hand of an relation, as in leq_subLR : (m - n <= p) = (m <= n + p)
• N or n -- boolean negation, as in andbN : a && (~~ a) = false
• n -- alternatively, it is a natural number argument
• N -- alternatively ring negation, as in mulNr : (- x) * y = - (x * y)
• P -- a characteristic property, often a reflection lemma, as in andP : reflect (a /\ b) (a && b)
• r -- a right-hand operation, as in
  • orb_andr : right_distributive orb andb
  • ler_normr x y : (x <= `|y|) = (x <= y) || (x <= - y)
  • alternatively, it is a ring argument
• R -- the right-hand of a relation, as in ltn_subrR : n < n - m = false
• RL -- moving an operator from the right-hand to the left-hand of an relation, as in ltn_subRL : (n < p - m) = (m + n < p)
• T or t -- boolean truth, as in andbT: right_id true andb
• T -- alternatively, total set
• U -- predicate or set union, as in predU
• W -- weakening, as in in1W : {in D, forall x, P} -> forall x, P
• 0 -- ring or nat 0, or empty set, as in addr0 : x + 0 = x
• 1 -- ring; nat or group 1, as in mulr1 : x * 1 = x
• D -- addition, as in linearD : f (u + v) = f u + f v
• B -- subtraction, as in opprB : - (x - y) = y - x
• M -- multiplication, as in invfM : (x * y)^-1 = x^-1 * y^-1
• Mn -- ring nat multiplication, as in raddfMn : f (x *+ n) = f x *+ n
• V -- multiplicative inverse, as in mulVr : x^-1 * x = 1
• X -- exponentiation, as in rmorphXn : f (x ^+ n) = f x ^+ n
  • Xn -- nat exponentiation
  • Xz -- int exponentiation
• Z -- (left) module scaling, as in linearZ : f (a *: v) = s *: f v
• z -- an int argument
• p -- positive number, as in ltr_pM2l x : 0 < x -> {mono *%R x : x y / x < y}
• n -- negative number
• w -- non strict (weak) monotony, as in ler_wpM2r x : 0 <= x -> {homo *%R^~ x : y z / y <= z}
• wp -- non-negative number
• wn -- non-positive number

Special naming conventions (non exhaustive)

• For the infix membership predicate _ \in _, the prefix in_ is used for lemmas that unfold specific predicates, possibly propagating the infix membership (e.g, in_cons or in_set0). These lemmas are generally part of the inE multirule. Other lemmas involving the infix membership predicated use the generic prefix mem_ (e.g., mem_head or mem_map).

Typical search pattern

• Search _ "prefix" "suffix"* (symbol|pattern)* in library. (for coq < 8.12)
• Search "prefix" "suffix"* (symbol|pattern)* inside library. (for coq >= 8.12)

Naming conventions for definitions (non exhaustive)

• types of mathematical structures
  • Mixed case, the first letter lowercase and the first letter of each internal word capitalized, end with Type
  • e.g., unitRingType
• HB structures
  • Mixed case, the first letter of each internal word capitalized
  • e.g., UnitRing
• interfaces (mixins, factories)
  • when the interface sits at the bottom of a hierarchy: mixed case, starts with is or has, the first letter of each internal word capitalized
    • e.g., hasChoice, isZsemimodule
  • when the interface extends a structure A into a structure B using C: A_C_isB or A_C_hasB where B and C are mixed case, the first letter of each internal word capitalized
    • e.g., Zsemimodule_isZmodule, SemiRing_hasCommutativeMul, Lattice_Meet_isDistrLattice
    • exceptions: Choice_, Equality_ can be omitted
• Coq modules
  • Mixed case, the first letter of each internal word capitalized
  • e.g., NumDomain in ssrnum.v

Abbreviations

• The following are considered as single words and are abbreviated when used as prefixes
  • Z-module becomes zmod/Zmod, e.g., ZmodType in ssralg.v, normedZmodType in ssrnum.v
  • L-module becomes lmod/Lmod
  • L-algebra becomes lalg/Lalg
• Partial order is abbreviated to porder or POrder, e.g., porderType, CanPOrderMixin in order.v

Deprecations

Definitions and lemmas should never be removed or renamed without warning users: they should be deprecated first, during at least one release. We use the pragma deprecated to implement deprecation of definitions and lemmas; see the many examples in the source code. We try to keep deprecation warnings for at least two years unless we need to remove them (for example because of a name clash, and in any case after at least one release); after this period, deprecation warnings might disappear at any moment, making the deletion or the renaming definitive.

Doc style

See this wiki entry

Instantiating structures with Hierarchy Builder

First

From HB Require Import structures.

The structure names can be found in the header comments, for instance, the eqType structure is defined in eqtype.v. Basic information about structures can be obtained via HB.about, for instance

HB.about eqType.

Regular factories

Factories enabling to build a structure can be discovered with HB.howto, for instance

HB.howto eqType.

tells us that eqType instances can be built with hasDecEq.Build. (Note that by default HB.howto may not return all the available factories; it might be necessary to increase the depth search using a natural number as in HB.howto xyzType 5.) One can then

HB.about hasDecEq.Build.

to learn that hasDecEq.Build is expecting a type T, a predicate eq_op : rel T (implicit argument, as indicated by the square brackets) and proof of Equality.axiom eq_op. One can thus instantiate an eqType on some type T with

HB.instance Definition _ := hasDecEq.Build T proof_of_Equality_axiom.

or

HB.instance Definition _ := @hasDecEq.Build T eq_op proof_of_Equality_axiom.

which should output a few lines among which:

module_T__canonical__eqtype_Equality is defined

(beware that the output may not be visible by default with VSCoq). Absence of such a line indicates failure of the command.

Aliases / feather factories

In addition to the regular factories, listed by HB.howto, the library defines some aliases (aka feather factories). Those aliases are documented in the header comments. For instance, an eqType instance on some type T can be derived from some T' already equipped with an eqType structure, given a function f : T -> T' and a proof injf : injective f

HB.instance Definition _ := Equality.copy T (inj_type injf).

Listing instances on a given type

Instances a type is already equipped with can be listed with HB.about, for instance

HB.about bool.

lists all the structures bool is already equipped with.

Graph of the hierarchy

A graph of the hierarchy can be obtained with

HB.graph "hierarchy.dot".

then

tred hierarchy.dot | xdot

or

tred hierarchy.dot | dot -Tpng > hierarchy.png

Adding new structures with Hierarchy Builder

See the documentation of Hierarchy Builder.