Conventions And Style Guide

1. Organization

1.1. The Core library

We do not use the Coq standard library, instead working from scratch. We do not use the Coq universes Prop and SProp.

The Coq files of the HoTT library live in the theories/ directory. Many are in subdirectories, and a subdirectory Foo/ often has a corresponding file Foo.v that imports everything in the subdirectory.

Basics/*: These files contain basic definitions that underlie everything else. Nothing in the Basics directory should depend on anything outside the Basics directory. The file Basics in the root imports everything from the directory Basics/, so most other files in the library start with Require Import HoTT.Basics. (See remarks below on qualified imports.)
Types/*: This subdirectory contains a file corresponding to each basic type former (e.g. sigma-types, pi-types, etc.), which proves the "computational" rules for the path-types, transport, functorial action, etc. of that type former. It also Types/Equiv, which proves that being an equivalence is an hprop. The univalence axiom is introduced, as a typeclass (see below) in Types/Universe. Function extensionality is introduced in Basics/Overture for dependency reasons, but developed further in Types/Forall and Types/Arrow. Some type formers are defined in their corresponding Types/ file, while others are defined in Basics/Overture for dependency reasons but studied further in their Types/ file. Files in Types/ should not depend on anything except Basics and other Types/ files.
Universes/*: Files related to universe levels, classifying maps, or to particular subuniverses, including UniverseLevel, Smallness, ObjectClassifier, BAut, HProp, HSet, DProp, and TruncType. The files here depend on files in Types/, and occasionally on some files mentioned below.
There are files in the root theories/ directory, including EquivGroupoids, ExcludedMiddle, Factorization, HFiber, Extensions, NullHomotopy, Projective, Idempotents, Constant, BoundedSearch, etc. These contain more advanced results which may depend on files in the whole library. We try to limit the number of files in the top-level folder, and would like to reduce the number.
WildCat/*: Files related to wild categories. They are used extensively in the library, so we try to minimize the files they depend on.
Modalities/*: Files involving modalities. The most important files here are Modalities/ReflectiveSubuniverse and Modalities/Modality.
Basics/Trunc, TruncType, Truncations/*: Files involving truncations. There are interdependencies between the files in Truncations/ and some of the Modalities/* files.
Equiv/*: Files showing that various definitions of equivalence agree.
Pointed/*: Files related to pointed types.
HIT/*: Files involving higher inductive types. Each higher inductive type is defined in a corresponding file (see conventions on defining HITs, below). These are only lightly used in the rest of the library. See Colimits/* for the HIT that is most commonly used.
Diagrams/*: Files involving graphs and diagrams, used for colimits and limits.
Colimits/*: Files involving colimits. Colimits/GraphQuotient defines graph quotients as a HIT, and other constructions are built from this.
Limits/*: Files involving limits.
Cubical/*: Files involving cubical methods in HoTT.
Algebra/*: Files related to algebra.
Analysis/*: Files related to analysis.
Sets/*: Files related to set theory.
Spaces/*: Files involving various spaces, including Spaces/Nat/*, Spaces/Pos/*, Spaces/Int/*, Spaces/Finite/*, Spaces/List/*, Spaces/Circle, Spaces/Torus/*, Spaces/Universe, Spaces/BAut/* and Spaces/No/* (the surreal numbers),
Homotopy/*: Files related to synthetic homotopy theory. Also contains Homotopy/IdentitySystems and Homotopy/EncodeDecode, which give results for computing path types.
Spectra/*: Files related to spectra in the sense of stable homotopy theory.
Tactics, Tactics/*: some more advanced tactics.
HoTT: This file imports and exports everything in the core (that is, everything mentioned above). Thus, in a development based on the HoTT library, you can say simply Require Import HoTT to pull in everything (but don't do this for files in the core itself).
theories/Classes/*: The math classes library. While we don't regard this as part of the core library, and don't explicitly export the contents in HoTT, some files in the classes library are used by files in the core library.

1.2. Non-core files

theories/Axioms/: Contains FunextAxiom and UnivalenceAxiom: You can import these files to assume the axioms globally (in the core, we track them with typeclasses).
theories/Metatheory/*: Contains UnivalenceImpliesFunext, IntervalImpliesFunext, ImpredicativeTruncation, PropTrunc, Nat (a definition of the natural numbers using univalence and propositional resizing), and other meta-theoretic results.
theories/Utf8 and theories/Utf8Minimal: optional Unicode notations for the basic definitions (we avoid Unicode in the core library).
theories/Categories/*: The categories library, which is not considered part of the core (e.g. it uses unicode), but nevertheless lives in theories/.
contrib/HoTTBook: This file lists all the definitions and theorems from the HoTT Book and gives pointers to where the corresponding fact is defined in the library. It is not intended to be a formalization of the book, but rather a guide to the library for a reader familiar with the book.
contrib/HoTTBookExercises: This file contains both formalizations of the exercises from the HoTT Book, and pointers to the corresponding facts in the library. The latter pointers serve the same purpose as contrib/HoTTBook, but the former may contain alternative solutions, such as ones that depend only on the concepts that are introduced prior to the exercise in the book.
contrib/*: Other work in progress, or files not judged appropriate for the core.

1.3. Tests

test/*: Tests of the library. See the file test/README.md for more information.

1.4. Dependencies

A dependency graph of all the files in the library can be found on the wiki; this may be helpful in avoiding circular dependencies. It is updated automatically by Travis (see below) on every push to the master branch.

2. Naming Conventions

2.1. General principles

In general, the name of a theorem (or definition, or instance, etc.) should begin with the property (or structure, or class, or record, etc.) being proven, and then state the object or construction it is being proven about. For instance, isequiv_idmap proves IsEquiv idmap, and equiv_compose constructs an "Equiv" record by composing two given equivalences.

In particular, a prefix of path_ means a theorem that constructs a path in some type. For instance, path_sigma is a theorem constructing paths in a sigma-type.

More generally, where applicable the order of parts in a lemma name should roughly respect their placement in (the syntax tree of) the goal, from outermost to deepest. For instance, path_equiv constructs a path in the type Equiv f g, while isequiv_path_equiv shows that path_equiv is an equivalence.

2.2. Capitalization and spacing

Names of types, such as Unit and Equiv and IsHProp, should generally be capitalized. Names of functions and definitions should be lowercase, including the names of types when they appear therein. Thus, for instance, the theorem that Unit is contractible is contr_unit : Contr Unit.

Multiple-word names, especially in lowercase, should generally be separated with underscores. We make an exception for names of types beginning with is, such as IsEquiv and IsTrunc.

2.3. Suffixes

A suffix of D indicates a dependent version of something ordinarily non-dependent. For instance, ap applies to non-dependent functions while apD applies to dependent ones. When possible, the non-dependent version should be an instantiation of the dependent one using constant type families, but sometimes they are more different than this, usually due to the fact that transport_const is not the identity (e.g. ap and apD).

When there is more than one theorem that seems to merit the same name, and no obvious concise way to distinguish them, one of them can be given a prime suffix, e.g. we have path_sigma and path_sigma'. Do this with caution.

2.4. Induction and recursion principles

In conformity with the HoTT Book, the induction principle of a (perhaps higher) inductive type thing (that is, its dependent eliminator) should be named thing_ind, while its recursion principle (non-dependent eliminator) should be named thing_rec.

However, by default, when you declare a (non-higher) inductive type, Coq automatically defines induction principles named thing_rect, thing_rec, and thing_ind that vary only in the sort of their target (Type, Set, or Prop). In order to turn this off, you must say Local Unset Elimination Schemes before defining an inductive type. You can then have Coq automatically generate the correctly named induction principles with

Scheme thing_ind := Induction for thing Sort Type.
Scheme thing_rec := Minimality for thing Sort Type.

Unfortunately, Coq's built-in tactics induction and elim assume that the induction principles are named in Coq's default manner. We are hoping that this will be fixed eventually, but in the meantime, to make those tactics work, you need to also say

Definition thing_rect := thing_ind.

(We have not turned on Global Unset Elimination Schemes because this would cause induction and elim to fail for all newly defined inductive types unless these Scheme commands are also given, which might be an unpleasant and confusing surprise to people who haven't read (or don't remember) these instructions.)

Note that elimination schemes are always off for Private Inductive types such as are used to hack HITs. For HITs, you must always define both the induction and recursion principles by hand, as described in the appropriate section below.

Some types have a "coinduction" or "corecursion" principle; these should have instead the suffix _coind or _corec.

Finally, a type will often have a universal property expressed by saying that its induction or recursion (or coinduction or corecursion) principle is an equivalence. These should be named according to the naming conventions for equivalences below, e.g. isequiv_thing_rec and equiv_thing_rec.

2.5. Path algebra functions

The path algebra functions defined mainly in Basics/PathGroupoids follow a particular set of naming conventions. Generally they are named according to the head constant of their primary input and the pattern of paths appearing therein. For more details, see the comments in Basics/PathGroupoids.

2.6. Equivalences

When defining an equivalence, the standard naming procedure is to

Define the function in one direction with a name, say foo.
Define an IsEquiv instance for this function, called isequiv_foo.
Define an Equiv record putting them together, called equiv_foo.

The inverse function can then be referred to as foo ^-1. Avoid using equiv_foo unless you really do need an Equiv object, rather than a function which happens to be an equivalence.

On the other hand, sometimes it is easier to define an equivalence by composing together existing equivalences, in which case one goes immediately to the last step, defining equiv_foo. If the equivalence is used frequently as an ordinary function, one might also define foo as the underlying function of equiv_foo; see path_equiv, as an example.

3. Records, Structures, Typeclasses

We use Coq Records when appropriate for important definitions. For instance, being an equivalence (IsEquiv) and equivalences (Equiv) are both Record types (in fact, the former is a typeclass). The file Basics/Tactics contains a tactic issig for proving semiautomatically that record types are equivalent to the corresponding sigma-types, so that the relevant general theorems can be applied to them.

3.1. Two-component records

Sometimes a two-component record is better defined as a sigma-type, especially if it is a "subset type" whose second component is an hprop. For instance, this has the advantage that we do not need new names for its constructor and its fields, and we can apply theorems in Types/Sigma to it directly rather than via issig.

TODO: Decide about hProp and hSet and TruncType (issue #514).

3.2. Typeclasses

We are using typeclasses in preference to canonical structures. Typeclasses are particularly convenient for h-properties of objects. Here are some of the typeclasses we are using:

equivalences: IsEquiv
truncation levels: IsTrunc (a notation for IsTrunc_internal)
axioms (see below): Funext, Univalence
subuniverses: In, Replete, MapIn, IsConnected, IsConnMap

IsHSet, IsHProp, and Contr are notations for IsTrunc 0, IsTrunc -1, and IsTrunc -2 respectively. See the comments in Overture.v for more details.

3.3. When to declare instances

When constructing terms in a typeclass record such as IsEquiv, Contr, or IsTrunc, one has the choice to declare it as an Instance, in which case it is added to the hint database that is searched when Coq tries to do typeclass instance resolution. Care must be taken with this, as indiscriminately adding theorems to this database can easily result in infinite loops (or at least very long loops).

In general, it seems to be better not to add instances which suggest an open-ended search. E.g. the theorem that truncation levels are closed under equivalence is a bad candidate for an Instance, because when Coq is searching for a proof of Contr B this theorem could cause it to look through all possible types A for an equivalence A <~> B and a proof of Contr A. Results of this sort should be proven as Definitions or Theorems, not as Instances. If you need to use a result of this sort in the middle of a proof, use a tactic like pose or assert to add a particular instance of its conclusion to the context; then it will be found by subsequent typeclass resolution.

If you have determined through trial and error that a particular result should not be an Instance (e.g. when making it an Instance, a tactic in some other proof loops, but changing it to a Definition prevents this), please add a comment to that effect where it is defined. This way no one else will come along and helpfully change it back to an Instance.

If a particular fact should not be made an ordinary instance, it can still be made an "immediate instance", meaning that Coq will use it automatically to solve a goal if its hypotheses are already present in the context, but will not initiate an instance search for those hypotheses otherwise. This avoids infinite instance-search loops. To declare a fact as an immediate instance, make it a Definition rather than an Instance and then say

Hint Immediate foo : typeclass_instances.

3.4. Local and Global Instances

When declaring an Instance you should always use either the Local or the Global keyword. The former makes the instance local to the current section, module, or file (although its definition will still be visible globally, it won't be in the instance database for typeclass resolution outside its containing section, module or file), while the latter puts it in the instance database globally. If you write Instance without Local or Global, Coq will sometimes make it local and sometimes global, so to avoid confusion it is better to always specify explicitly which you intend.

3.5. Using Typeclasses

Try to avoid ever giving a name to variables inhabiting typeclasses. When introducing such a variable, you can write intros ? to put it in the hypotheses without specifying a name for it. When using such a variable, typeclass resolution means you shouldn't even need to refer to it by name: you can write _ in tactics such as refine and Coq will find typeclass instances from the context. Even exact _ works. (You can usually also use typeclasses eauto or eauto with typeclass_instances, but exact _ is preferable when it works, as it is shorter and uses a tactic name the reader is presumably already familiar with.)

Unfortunately, it is not currently possible to write _ in a refined term for an inhabitant of a typeclass and have Coq generate a subgoal if it can't find an instance; Coq will fail if it can't resolve a typeclass variable from the context. You have to assert or pose such an inhabitant first, or give an explicit term for it.

Note that when you don't give a name to a variable, Coq often names it H or some modification thereof. For that reason, it's often better avoid using H for your own explicitly named variables, since if you do and later on someone introduces a new unnamed hypothesis that Coq names H, your name will result in a conflict. Conversely, we sometimes give a hypothesis a name that won't be used, to preempt such conflicts, such as {ua : Univalence} or {fs : Funext}.

One gotcha about typeclass arguments is that they cannot be inferred automatically when preceded by non-implicit arguments. So for instance if we write

Definition foo (A : Type) `{Funext}

then the Funext argument will not generally be inferable. Thus, typeclass arguments should generally come first if possible. In addition, note that when section variables are generalized at the close of a section, they appear first. Thus, if anything in a section requires Funext or Univalence, those hypotheses should go in the Context at the top of the section in order that they'll come first in the eventual argument lists.

3.6. Truncation

The conventions for the typeclass IsTrunc are:

We prefer to do inference with IsTrunc n A rather than IsTrunc n (a = b).
We prefer to expand IsTrunc n (forall a, P a) into forall a, IsTrunc n (P a), and similarly for other data types. For instance, IsTrunc n (A * B) gets transformed to IsTrunc n A and IsTrunc n B, as a goal.
Due to the desire to use IsTrunc rather than Contr, we have Contr as a notation for IsTrunc minus_two.

3.7. Coercions and Existing Instances

A "coercion" from A to B is a function that Coq will insert silently if given an A when it expects a B, and which it doesn't display. For example, we have declared equiv_fun as a coercion from A <~> B to A -> B, so that we can use an equivalence as a function without needing to manually apply the projection equiv_fun. Coercions can make code easier to read and write, but when used carelessly they can have the opposite effect.

When defining a Record, Coq allows you to declare a field as a coercion by writing its type with :> instead of :. When defining a Class, the :> notation has a different meaning: it declares a field as an Existing Instance.

(In addition, the :> notation is used when needed for path types to indicate the type in which the paths are taken: x = y :> A for x, y : A.)

4. Axioms

4.1. Univalence, function extensionality and propositional resizing

The "axioms" of Univalence, Funext (function extensionality) and PropResizing (propositional resizing) are typeclasses rather than Coq Axioms. (But see the technical note below on universe polymorphism.) In the core, we use these typeclasses to keep track of which theorems depend on the axioms and which don't.

This means that any theorem which depends on one or the other must take an argument of the appropriate type. It is simple to write this using typeclass magic as follows:

Theorem uses_univalence `{Univalence} (A : Type) ...

The axiom-term witnessing univalence does not have to be named, nor does it have to be passed explicitly to any other lemma which uses univalence; once it is in the typeclass context, it should be found automatically.

For longer developments using Univalence or Funext, it is probably preferable to assume it as part of the context.

Section UsesUnivalence.
  Context `{Univalence}.

Now everything defined and proven in this section can use univalence without saying so explicitly, and at the end of the section it will be implicitly generalized if necessary. (Results that don't use univalence won't get an univalence argument.) The backquote syntax ``{Univalence}` allows us to avoid giving a name to the hypothesis. (Backquote syntax is also used for implicit generalization of variables, but that is not needed for univalence and funext.)

4.2. Higher inductive types

Every higher inductive type technically assumes some Axioms. These axioms are asserted globally by the corresponding HIT/ file or Colimits/GraphQuotient, since there's not much point to assuming a HIT without the axioms that make it work.

4.3. Relationships between axioms

The file UnivalenceImpliesFunext shows, as its name implies, that univalence implies funext. However, this file is not needed in practice, since in Types/Universe we assert axiomatically that univalence implies funext.

Similarly, the file HIT/IntervalImpliesFunext proves funext from the interval type assumed in HIT/Interval, so if you import this file then funext is always true automatically (just as if you'd imported FunextAxiom). Of course, once you've imported HIT/Interval it is always possible to prove funext by hand, but by importing HIT/Interval without HIT/IntervalImpliesFunext you can still use the interval in some places and track moral uses of funext elsewhere.

4.4. Assuming axioms

When working in a derived development using the HoTT library, you may import the files FunextAxiom and/or UnivalenceAxiom to assume these axioms globally. You should not assume these axioms yourself by writing something such as Axiom fs : Funext. The problem with this is that if two different files do this, and then a third file imports them both, it ends up with two different witnesses for Funext, not definitionally equal. Thus, derived objects that should be judgmentally equal might fail to be so because they use different witnesses.

4.5. Technical note: Universe-polymorphic axioms

In order for the "axioms" univalence and funext to be usable at different universe levels, the types Univalence and Funext do not technically assert the axioms themselves. Rather they assert inhabitants of dummy types, while the axioms are actually declared globally but depending on elements of those dummy types. This is not something you generally need to worry about; see the comments in Basics/Overture for more information.

However, one situation in which this matters is when proving an implication between axioms. Because Univalence and Funext are dummy types, we cannot actually prove that Univalence -> Funext. Instead we define placeholders with names like Funext_type and Univalence_type that have the actual type that the axiom would have except for the polymorphism trick, and prove that Univalence_type -> Funext_type. Then we feel justified in asserting as a further Axiom that Univalence -> Funext.

When introducing further axioms, please use this same naming convention. For another example, see ExcludedMiddle.v.

5. Higher Inductive Types

Most higher inductive types are defined in the HIT/ directory, but Colimits/GraphQuotient also uses a HIT. All are defined using Dan Licata's "private inductive types" hack which was implemented in Coq by Yves Bertot. This means the procedure for defining a HIT is:

Wrap the entire definition in a module, which you will usually want to export to the rest of the file containing the definition.
Define a Private Inductive type whose constructors are the desired point-constructors of your HIT.
Assert the desired path-constructors as Axioms.
Define the induction principle, with all the correct hypotheses, by matching against the point-constructors. There is an important additional hack here. If the path-hypotheses are not "used" anywhere in the match, then Coq will notice and will consider two invocations of the induction principle to be judgmentally equal if they have the same point-hypotheses, even if their path-hypotheses differ. Thus, it is important to "use" the path-hypotheses trivially by making the match return a function which is then applied to all the path-hypotheses. For example, with the circle we write fun x => match x with base => fun _ => b end l instead of fun x => match x with base => b end.
Assert the "computation rules" for the path-constructors, in the form of propositional equalities, as Axioms.
Close the module. It is important to do this immediately after defining the induction principle, so that the private inductive type can't be matched against anywhere else; any other uses of it have to call the correct induction principle.
Usually, you will want to also define a non-dependent recursor and its computation rules as well.

Look at some of the existing files in HIT/* for examples.

Going forward, we try to use Colimits/GraphQuotient to define further higher inductive types.

5.1. Case analysis on private inductive

You may get this error at Qed/Defined if unification unfolded the induction principle and used its value to produce the proof term.

To fix this, you need to identify which tactic produced the problematic term, then either avoid unification by annotating more (e.g. apply (@foo bla) instead of apply foo), or guide unification by manipulating the goal (e.g. using rewrite with the lemma witnessing the computation rule of the inductive principle) or making related definitions opaque.

6. Universe Polymorphism

We have Coq's "universe polymorphism" feature turned on throughout the library. Thus, all definitions are universe polymorphic by default, i.e. they can be applied to types that live in any universe level.

Usually, this is not something you have to worry about, as Coq tries to automatically make everything maximally polymorphic, but sometimes a bit of attention is required. If Coq is claiming that an instance is not found which is "obviously" present, or a term doesn't have a type that it "clearly" does (or, of course, if it complains about a universe inconsistency), then a universe problem may be the culprit.

6.1. Displaying universes

If you suspect a universe problem, usually the first thing to do is to turn on the display of universes with the command Set Printing Universes. This causes Coq to print the universe parameters of every occurrence of a definition when displaying the current proof state or when giving an error message, and also to print the universe parameters and the constraints imposed on them when displaying a definition with Print or About or a typechecking a term with Check. (Nowadays Coq is sometimes smart enough to display universes automatically when giving an error message that would otherwise look like "unable to unify A with A".) To display the current universe constraints during a proof, use Show Universes (this is not to be confused with Print Universes, which displays the current list of global universes; the latter is usually quite small with universe polymorphism enabled).

The universe parameters of an occurrence of a definition are displayed as foo@{Top.1 Top.2}. Here foo is a definition which takes two universe parameters, and this occurrence of foo has those two parameters instantiated to the universes Top.1 and Top.2. When displaying a definition with Print or About, its universe parameters are shown in a comment below the definition, followed by |- and a list of the constraints on those parameters.

In general, the universe parameters of a definition are automatically computed from the parameters of its constituents, and the order of the parameters is likewise induced by the order in which they occur in the definition. This means you must generally pay close attention to the output of Print or About to learn which universe parameter is which, and insignificant-seeming changes in a definition can sometimes cause changes in the number or order of its universe parameters.

Note that Check foo will often give a different list of universes than Print foo and About foo. This is because the latter two display information about foo as a definition, while Check treats its argument as a term to be typechecked, and Coq is willing to collapse some universes during typechecking.

6.2. Universe annotations

You can exert a certain degree of control over universe polymorphism by using explicit universe annotations, which use the same syntax as the display of universes: if you write foo@{i j}, that means foo with its two universe parameters instantiated to the universes i and j. You are required to supply exactly the right number of universes, and be careful about the order in which they occur.

It is very important to note that universe names such as i and j are definition local and implicitly declared. This means that whenever you write i inside a universe annotation, Coq implicitly declares a universe named i, and all occurrences of the universe i in the same definition refer to the same universe. When the definition is complete, this universe will become one of its universe parameters. An annotation named i in a different definition will instead become one of that definition's parameters. Thus, if you define foo using a universe i, and then define bar which uses foo, in order to force a particular universe parameter of bar to coincide with i in foo, you must annotate the occurrences of foo in bar appropriately.

It is also possible to explicitly declare and name universes globally with the Universe command, usually within a section. Universes declared with Universe will be discharged on each section definition independently.

The universe variables of a definition can be declared and their constraints can be chosen by hand. This is good practice for delicate situations, as it serves to both document the expected constraints and to cause an error if a future change causes the universe variables or constraints to change. We also use Check commands, mostly in test/*, to verify that universe variables before as expected.

There are several uses for universe annotations. One is to force a definition to have fewer universe parameters than it would otherwise. This can sometimes improve performance: if you know that in practice, several of the universes occurring in a definition will always be the same, then saving Coq the burden of carrying them all around separately can sometimes make it run faster. Additionally, reducing the number of universe parameters in a definition can make it significantly easier to universe-annotate uses of that definition later on.

Another reason for universe annotations is to make a definition more universe polymorphic. In some situations, in the absence of annotations Coq will automatically collapse one or more universe parameters which could be kept separate if annotated. It is not clear under exactly what situations this occurs, but one culprit appears to be section variables: if you declare a section variable which you need to be universe polymorphic, you may need to annotate it.

(Another occasional culprit of less-polymorphic-than-expected definitions seems to be giving type parameters without a type. At least in some situations, writing Definition foo {A B} ... rather than Definition foo {A B : Type} ... can cause A and B to live in the same universe.)

Finally, universe annotations can also be necessary to instruct Coq how to instantiate the universes when using a definition. In some situations, Coq seems to make a default guess that doesn't work (perhaps collapsing some universes that need to remain distinct) and then complains without trying anything else; an annotation can point it in the right direction.

In Coq, the bottom universe is denoted Set, but this does not mean that its elements 0-truncated. In Basics/Overture, we define a notation Type0 for this universe.

Coq also has universes Prop and SProp which we do not use in the library.

6.3. Unexpected universes

If you ever need to pay close attention to universes, it is useful to know that there are several ways in which extra universe parameters can creep into a definition unexpectedly. Here are a few.

The induction tactic invokes the appropriate induction principle, which is a function generally named *_ind or *_rect (see notes on naming conventions above). This function, in turn, requires a universe parameter describing the size of its output. Therefore, if you prove something by induction that is generalized over a "large" argument (e.g. a type or a type family), the resulting definition will pick up an extra universe parameter that's strictly larger than the argument in question. One way to avoid this is to instead use a Fixpoint definition, or the tactic fix, along with destruct. There is a tactic simple_induction defined in Basics/Tactics whose interface is almost the same as induction but does this internally, although it only works for induction over nat and trunc_index.

If you apply the symmetry tactic when constructing an equivalence to reverse the direction of the goal, then rather than applying equiv_inverse directly it goes through the Symmetric typeclass. This involves a universe for the size of the type on which the symmetric relation lives, which in the case of Equiv is the universe. Thus, applying symmetry to an Equiv introduces a strictly larger universe. A solution is to apply equiv_inverse instead. Similarly, use equiv_compose' instead of transitivity.

Typeclass inference doesn't always find the simplest solution, and may insert unnecessary calls to instances that introduce additional universes. One solution is to alter the proofs of those instances as described above; another is to call the instance(s) you need explicitly, rather than relying on typeclass inference to find them.

Sometimes binders without type annotations, like forall n, foo n where foo : nat -> Type0, will produce a fresh universe for the variable's type, eg forall n : (? : Type@{fresh}), foo n, which will remain in the definition as a phantom type: fresh |= forall n : nat, foo n. Annotating the binder will get rid of it. See also bug #4868.

6.4. Lifting and lowering

The file Universes/UniverseLevel contains an operation Lift which lifts a type from one universe to a larger one, with maps lift and lower relating the two types and forming an equivalence. There are primed versions Lift', lift', and lower' which allow the two universe levels to possibly be the same.

In the past, Lift was used to force universe levels to be distinct, but now that Coq supports constraints between universe variables, this is no longer needed in practice.

The file Universes/Smallness contains results allowing us to show that a type lives in a certain universe.

6.5. Universes and HITs

Another use for universe annotations is to force HITs to live in the correct universe. Coq assigns a universe level to an inductive type based on the levels of its indices and constructors, which is correct for ordinary inductive types. However, the universe level of a HIT should depend also on the levels of its path-constructors, but since these are not actually constructors of the Private Inductive, Coq doesn't take them into account.

We have not yet formulated a general method for resolving this. In the few cases when it arises, it should be solvable with universe annotations, but we have not yet implemented such a fix; see bug #565.

7. Transparency and Opacity

If the value of something being defined matters, then you must either give an explicit term defining it, or construct it with tactics and end the proof with Defined. But if all that matters is that you have defined something with a given type, you can construct it with tactics and end the proof with Qed. The latter makes the term "opaque" so that it doesn't "compute".

If something can be made opaque, it is generally preferable to do so, for performance reasons. However, many things which a traditional type theorist would make opaque cannot be opaque in homotopy type theory. For instance, none of the higher-groupoid structure in PathGroupoids can be made opaque, not even the "coherence laws". If you doubt this, try making some of it opaque and you will find that the "higher coherences" such as pentagon and eckmann_hilton will fail to typecheck.

In general, it is okay to construct something transparent using tactics; it's often a matter of aesthetics whether an explicit proof term or a tactic proof is more readable or elegant, and personal aesthetics may differ. Consider, for example, the explicit proof term given for eckmann_hilton: some may consider it optimally elegant, while others would prefer to be able to step through a tactic proof to understand what is happening step-by-step.

One thing to beware of is explicit match terms that require in or return annotations, as these are particularly difficult for newcomers to understand. Avoiding them is not a hard and fast rule, but when there is a short proof using tactics that produces an acceptable proof term, it should probably be preferred.

The important thing is that when defining a transparent term with tactics, you should restrict yourself to tactics which maintain a high degree of control over the resulting term; "blast" tactics like autorewrite should be eschewed. Even plain rewrite is usually to be avoided in this context: although the terms it produces are at least predictable, they are one big transport (under a different name) whereas a term we would want to reason about ought to be constructed using smaller pieces like ap and concat which we can understand.

Here are some acceptable tactics to use in transparent definitions (this is not an exhaustive list):

intros, revert, generalize, generalize dependent
pose, assert, set, cut
transparent assert (see below)
fold, unfold, simpl, cbn, hnf, change
case, elim, destruct, induction
apply, eapply, assumption, eassumption, exact
refine, nrefine, srefine, snrefine (see below for the last three)
rapply, nrapply, srapply, snrapply (see below)
lhs, lhs_V, rhs, rhs_V
reflexivity, symmetry, transitivity, etransitivity
by, done

Conversely, if you want to use rewrite, that is fine, but you might then make the thing you are defining opaque. If it turns out later that you need it to be transparent, then you should go back and prove it without using rewrite.

Currently, there are some basic facts in the library, such as the "adjointify" lemma, which are proven using rewrite and hence are at least partially opaque. It might be desirable one day to prove these more explicitly and make them transparent, but so far it has not been necessary.

Note that it is acceptable for the definition of a transparent theorem to invoke other theorems which are opaque. For instance, the isequiv_adjointify lemma itself is actually transparent, but it invokes an opaque sublemma that computes the triangle identity (using rewrite). Making the main lemma transparent is necessary so that the other parts of an equivalence -- the inverse function and homotopies -- will compute. Thus, a transparent definition will not generally be "completely transparent".

It is possible to make subterms of a term opaque by using the abstract tactic, although that requires grouping the entire proof script to be abstracted into a single command with semicolons, e.g. abstract (apply lem1; apply lem2; reflexivity). Note that the assert tactic produces subterms that cannot be inspected by the rest of the proof script, but they remain transparent in the resulting proof term (at least if the proof is ended with Defined.).

For a transparent subterm, use refine or transparent assert (the latter defined in Basics/Overture; see "Available tactics", below).

8. Imports/exports

Most Require commands should be just Require Import rather than Require Export: imports should not be re-exported, by default.

However, if you can't imagine making practical use of file Foo without file Bar, then Bar may export Foo via Require Export Foo. For instance, Modality exports ReflectiveSubuniverse because so many of the theorems about modalities are actually theorems about reflective subuniverses.

9. Formatting

9.1. Location of commands

All Require commands should be placed at the top of a file. Ideally, they should be grouped onto lines according to the rough grouping of files listed under "Organization". Requires should generally be followed by all [Local] Open Scope commands, and then by Generalizable Variables commands.

The latter two might also occur in Sections later on in the file, but in that case they should usually come at the beginning of the Section. The assumptions of a section, such as Variable and Context, should also generally come at the beginning of that section.

9.2. Indentation

In general, the bodies of sections and modules should be indented, two spaces per nested section or module. This is the default behavior of ProofGeneral.

However, when enclosing existing code in a new section or module, it is acceptable to avoid re-indenting it at the same time, to avoid excessive churn in the diff. If you wish, you can submit a separate pull request or commit for the re-indentation, labeled as "only whitespace changes" so that no one bothers reading the diff carefully.

9.3. Line lengths and comments

Lines of code should be of limited width; try to restrict yourself to not much more than 70 characters. Remember that when Coq code is often edited in split-screen so that the screen width is cut in half, and that not everyone's screen is as wide as yours.

coqdoc is used to produce a browsable view of the library. coqdoc treats comments specially, so comments should follow the conventions described on the coqdoc page. The most important ones are that Coq expressions within comments are surrounded by square brackets, and that headings are indicated with comments of the form

(** * This is a top-level section *)
(** ** This is a subsection *)
(** *** This is a sub-subsection *)

Section titles should be less than 80 characters, on one line, and not end in a period. Other comments are generally written using the style (** This is a comment. *) or (* This is a comment. *), with the latter generally used for inline comments during a proof.

Text in comments should not contain hard newlines. Putting hard newlines in text makes it extremely ugly when viewed in a window that is narrower than the width to which you filled it. If editing in Emacs, turn off auto-fill-mode and turn on visual-line-mode; then you'll be able to read comment paragraphs without scrolling horizontally, no matter how narrow your window is. The repository contains a file .dir-locals.el in the top-level directory which turns on visual-line-mode when emacs visits any file in the repository.

Documentation can be built locally using make doc or dune build @theories/doc. The HTML files generated by Dune are placed in _build/default/HoTT.html/.

Unfortunately, when viewing source code on Github, these long comment lines are not wrapped, making them hard to read. If you use the Stylish plugin, you can make them wrap by adding the following style:

@-moz-document domain(github.com) {
    div.line {
        white-space: pre-wrap;
    }
}

This messes up the line-numbering, though, you'll have to turn it off in order to link to or comment on a particular line.

9.4. Tactic scripts

When writing tactic scripts, Proof. and Defined. should be given as individual lines, and the tactic code should be indented. Within the tactic script, use newlines as a "logical grouping" construct. Important tactic invocations, such as a top-level induction which create a branch point in the proof, should generally be on lines by themselves. Other lines can contain several short tactic commands (separated by either periods or semicolons), if they together implement a single idea or finish off a subgoal.

For long proofs with multiple significant subgoals, use branching constructs such as bullets and braces to clarify the structure. See the section of the Coq Reference Manual entitled "Navigation in the proof tree".

9.5. Placement of Arguments and types

If the entire type of a theorem or definition does not fit on one line, then it is better to put the result type (the part after the colon) on an indented line by itself, together with the colon to make it clear that this is the result type.

Definition triangulator {A : Type} {x y z : A} (p : x = y) (q : y = z)
  : concat_p_pp p 1 q @ whiskerR (concat_p1 p) q.

Of course, if the list of input types does not fit on a line by itself, it should be broken across lines as well, with later lines indented, and similarly for the result type.

Definition pentagon {A : Type} {v w x y z : A}
  (p : v = w) (q : w = x) (r : x = y) (s : y = z)
  : whiskerL p (concat_p_pp q r s)
      @ concat_p_pp p (q@r) s
      @ whiskerR (concat_p_pp p q r) s.

For definitions given with an explicit term, that term should usually also be on an indented line by itself, together with the := to make it clear that this is the definition.

Definition concat_p1 {A : Type} {x y : A} (p : x = y) : p @ 1 = p
  := match p with idpath => 1 end.

Of course, if the term is longer than one line, it should be broken across lines, with later lines indented further.

10. Implicit Arguments

Do not use Set Implicit Arguments in the core. It makes it difficult for a newcomer to read the code; use braces {...} to explicitly mark which arguments are implicit. If necessary, you can use the Arguments command to adjust implicitness of arguments after a function is defined, but braces are preferable when possible.

If you want to use Arguments to make all the arguments of a function explicit, the obvious-looking syntax Arguments foo a b does not work: you have to write Arguments foo : clear implicits instead.

11. Coding Hints

11.1. Notations

The operation compose, notation g o f, is simply a notation for fun x => g (f x) rather than a defined constant. We define compose := (fun g f x => g (f x)) so that typeclass inference can pick up isequiv_compose instances. This has the unfortunate side-effect that simpl/cbn is enough to "unfold" compose, and there's no way to prevent this. We could additionally define g o f := (fun x => g (f x)) to change this, but this would result in identically looking goals which are really different. We consider it poor style to use compose as a partially applied constant, such as compose g; we take the point of view that fun f => g o f is more readable anyway.

11.2. Unfolding definitions

When a definition has to be unfolded repeatedly in the middle of proofs, you can say Local Arguments name / ., which tells simpl and related tactics to automatically unfold name. In particular, this allows the tactic simpl rewrite (defined in Tactics) to apply theorems containing name to goals in which it has been unfolded. It seems better not to make these declarations globally, however.

It may not always be obvious which definition this needs to be applied to; sometimes the unification failure happens in an implicit argument that is not directly visible in the output. One way to discover where the problem lies is to turn on printing of all implicit arguments with Set Printing All; another is to use Set Debug Tactic Unification and inspect the output to see where rewrite is failing to unify.

11.3. Finding theorems

The naming conventions mentioned above often help to guess the name of a theorem. However, it still may happen that you expect that a theorem should exist but don't know what it is called. One approach to finding it is to guess what file it should live in and look there; for instance, theorems about sigma-types are often in Types/Sigma.v, and so on.

Another approach is to use Coq's command Search to display all the theorems that relate to a particular definition. For example, Search (IsHProp ?A) will show all results in which the expression IsHProp A appears for some A, and Search "ishprop" will show all results having "ishprop" in the name. However, some results about HProp are true for any truncation level, so you may want to expand your search to include IsTrunc. In general, if you can't find something at first using Search, think about ways that your desired theorem might be generalized and search for those instead.

Generalizing from a particular truncation level (like IsHProp) to all truncation levels is a good example. Another one that it's important to be aware of is a generalization from truncation (IsTrunc and Trunc) to all reflective subuniverses or modalities; many many theorems about truncation are actually proven more generally in the latter situations.

11.4. Simpl nomatch

If a theorem or definition is defined by destruct or match (as many operations on paths are), and if its value needs to be reasoned about in tactic proofs, then it is helpful to declare its arguments as Arguments foo ... : simpl nomatch. This instructs simpl and related tactics never to simplify it if doing so would result in a match that doesn't reduce, which is usually what you want.

11.5. Available tactics

Here are some tactics defined in the core that you may find useful. They are described more fully, usually with examples, in the files where they are defined.

nrefine, srefine, snrefine: Defined in Basics/Overture, these are shorthands for notypeclasses refine, simple refine, and simple notypeclasses refine. It's good to avoid typeclass search if it isn't needed.
rapply, nrapply, srapply, snrapply: Defined in Basics/Overture, these tactics use refine, nrefine, srefine and snrefine, except that additional holes are added to the function so they behave like apply does. The unification algorithm used by apply is different and often less powerful than the one used by refine, though it is occasionally better at pattern matching.

Here are some tips:
- If apply fails with a unification error you think it shouldn't have, try nrapply.
- If you want to use type class resolution as well, try rapply. But it's better to use nrapply if it works.
- You could add a prime to the tactic, to try with many arguments first, decreasing the number on each try.
- If you don't want Coq to create evars for certain subgoals, add an s to the tactic name to make it use simple refine.
lhs, lhs_V, rhs, rhs_V: Defined in Basics/Tactics. These are tacticals that apply a specified tactic to one side of an equality. E.g. lhs nrapply concat_1p.
transparent assert: Defined in Basics/Overture, this tactic is like assert but produces a transparent subterm rather than an opaque one. Due to restrictions of tactic notations, you have to write transparent assert (foo : (bar baz)) rather than transparent assert (foo : bar baz).
issig: Defined in Basics/Tactics, this tactic proves automatically that a record type is equivalent to a nested sigma-type.
make_equiv: Defined in Basics/Equivalences, this tactic can prove two types are equivalent if the proof involves juggling components. See also make_equiv_contr_basedpaths.
pointed_reduce: Defined in Pointed/Core, this tactic lets you assume that functions are strictly pointed. See related tactics there.
strip_truncations, strip_modalities and strip_reflections: These let you lift an element from O X to X when the goal is O-local, for various O.
simpl rewrite: Defined in Tactics, this tactic applies simpl to the type of a lemma, and to the goal, before rewriting the goal with the lemma. In particular, this is useful for rewriting with lemmas containing definitions like compose that appear unfolded in the goal: if the operation has Arguments ... / . as above then simpl will unfold it.
binder apply: Defined in Tactics/BinderApply, this tactic applies a lemma inside of a lambda abstraction, in the goal or in a hypothesis.

12. Contributing to the library

12.1. Fork & Pull

We mainly work by the "Fork & Pull" model. Briefly: to contribute, create your own fork of the repository, do your main work there, and issue pull requests when work is ready to be brought into the main library. Direct pushes to the library should never be made.

There are various reasons for preferring the fork/pull workflow. Firstly, it helps maintain code consistency. Secondly, it makes it easier for all to keep track of what is being added --- it’s easier to survey changes grouped into pull requests than in individual commits. Thirdly, it means we can make our work in progress as messy and uncertain as we want, while keeping the main library clean and tidy.

Submit your pull request not from the master branch of your fork, but from another branch created specially for that purpose. Among other things, this allows you to continue developing on your fork without changing the pull request, since a pull request is automatically updated to contain all commits pushed to the branch that it was made from. It also allows you to submit multiple unrelated pull requests at the same time that do not depend on each other.

12.2. Approval of pull requests

Before being merged, pull requests must be approved by one of the core developers, not counting whoever submitted it. An approval can be an official "Approving review" through the GitHub UI, or just a comment such as LGTM ("Looks Good To Me"). Currently the rules are:

Any objections or requested changes must be addressed somehow before merging (which doesn't always mean making the changes, but a discussion must be had and resolved).
In general, a pull request should not be merged unless the CI tests confirm that it builds successfully. Exceptions to this rule sometimes have to be made if the CI configuration is broken for some unrelated reason, but in that case it is better if the person(s) approving the pull request confirms locally that it builds successfully.

Note also that the CI doesn't automatically restart itself on a pull request when the master branch changes. Thus, if other pull requests have been merged in the interval since a given pull request was first submitted, it may be necessary to rebase that pull request against the new master, to make sure before merging it that it won't break the master branch.
In the absence of objections, one approval suffices for a pull request to be merged.
In the absence of objections, a minor pull request may be merged if at least 48 hours have passed after its submission. In some cases, this can be done immediately.

If a pull request is lacking approval and hasn't received any discussion, feel free to bump it back to attention with a comment.

12.3. Commit messages

Please try to keep commit messages clear and informative. We don’t currently have a specific preferred convention, but the answers here and here (not just the top answers!) give excellent, if varied, advice. That said, this is a minor point. Good code with bad commit messages is much better than the reverse!

Some good examples, showing what kind of change was made (additions, updates, fixes), and what material it was on:

"adapt to the new version of coqtop by disabling the native compiler"
"resolved universe inconsistency in Freudenthal"
"epis are surjective"

Some bad examples:

"further progress" Progress in what files?
"Bug in [Equivalences.v]." Was the bug fixed, or just noticed and flagged in comments, or what?
"asdfjkl"

12.4. Creating new files

If you create a new file, make will only compile it if it is being tracked by git, so you will need to git add it.

You will probably also want to add your new file to HoTT.v, unless it is outside the core (e.g. in contrib/) or should not be exported for some other reason.

12.5. Travis

We use the Travis Continuous Integration Platform to check that pull requests do not break anything, and also to automatically update various things (such as the documentation and dependency graph linked on the project wiki). Normally you shouldn't need to know anything about this; Travis automatically checks every pull request made to the central repository.

12.6. Git rebase

If the master branch has diverged in some significant way since a pull request was made, then merging it may result in non-compiling code. Or the changes may conflict so that github becomes unable to merge it automatically. In either case, the situation should be resolved manually before the pull request can be merged, and the resolution should generally be done by the submitter of the pull request.

One way to do the resolution is to merge the current master branch into the branch from which the pull request was made, resolving conflicts manually, and then make and commit whatever other changes may be necessary. This has the disadvantage of creating new merge commits, so another option is to git rebase against the master branch. We encourage the use of rebase if you are comfortable with it; but for newcomers to git, rebasing can be intimidating, so merges are also perfectly acceptable.

12.7. Timing scripts

There are scripts in etc/timing to track (compile-time) performance changes in the library. When you make large changes, you may want to include performance information in your commit message (recommended, but certainly not required!).

Note: Make sure you have gnu time installed. Many terminals have their own time command but this will not work. To install time run sudo apt install time. You will also need to rerun the ./configure script so that it detects the newly installed time.

To run the timing scripts, use the following work-flow from the root of the repository after you have made your edits. To make use of these scripts, you must first run git submodule update --init --recursive.

$ git status
$ git add <all files mentioned above which you care to have in the repo>
$ git status

It's good practice at this point to ensure that there are no .v files mentioned.

$ ./etc/coq-scripts/timing/make-pretty-timed-diff.sh
$ make

Ensure that make succeeds, since make-pretty-timed-diff.sh will succeed even if some files fail to build.

$ git commit -at ./time-of-build-both.log

This pops open an editor. You should add lines at the beginning of the commit message, leaving at least one blank line before the performance table.

See the comments at the top of make-pretty-timed-diff.sh for more detailed instructions and caveats.

13. Bugs in Coq

More often than we would like, we run across bugs in Coq. A sure sign of a bug in Coq is when you get a message about an "Anomaly", but a bug can also be unjustifiable behavior. If you aren't sure whether something is a bug in Coq, feel free to open an issue about it on the Coq-HoTT GitHub project.

13.1. Reporting bugs

Bugs in Coq should be reported on the Coq bug tracker. You should search the tracker first to see whether your bug has already been reported.

After reporting a bug, you may need to add a temporary workaround to the HoTT library until the bug is fixed. In this case, please add a comment labeling this as a workaround and citing the bug report. That way when the bug is fixed, we can remove the workaround.

13.2. Minimizing bugs

When submitting a bug report, it is appreciated to submit a minimal test example. Since the HoTT library is quite large, it can be quite some work to track down the actual trigger of a bug. Fortunately, Jason Gross has written a convenient "bug minimizing" script, which is available in his coq-tools repository. To use it:

Clone the coq-tools repository somewhere (usually somewhere outside the HoTT library directory).
Attempt to compile the file where the bug occurs, e.g. by running make theories/Path/To/Buggy.vo from the root of the HoTT library directory. This creates a .glob file which the bug-finder needs.
cd theories and then run the bug-finder script find-bug.py. It will combine the file with all the rest of the library that it needs, ask you to confirm the error, and then proceed to minimize it as much as possible.

You will need to pass the bug-finder several arguments to tell it to pass the right flags and where to find the rest of the library; a common invocation would be something like

$ /path/to/find-bug.py --arg -noinit --arg -indices-matter -R . HoTT Path/To/Buggy.v bug_minimized.v

When it exits, the minimized code producing the bug will be in bug_minimized.v.

Note that sometimes coqc and coqtop can exhibit different behavior, and one may produce a bug while the other doesn't. (One reason for this is that they give different names to universe parameters, Top.1 versus Filename.1, and this can result in different results from sorting, which can affect the output of the universe minimization algorithm, yielding different numbers or different ordering of universe parameters for the same definitions. This is itself a bug, but as of June 2015 it has not yet been fixed.) The bug-finder normally uses both coqc and coqtop, but you can tell it to "fake" coqc using coqtop by passing the argument --coqc-as-coqtop instead of --coqc.

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