README.md

Univalent Mathematics Coq files

Each subdirectory of this directory consists of a separate package, with various authors, as recorded in the README (or README.md) file in it.

Contributing code to UniMath

Volunteers may look at unassigned issues at GitHub and volunteer to be assigned one of them. New proposals and ideas may be submitted as issues at GitHub for discussion and feedback.

Contributions are submitted in the form of pull requests at GitHub and are subject to approval by the UniMath Development Team.

Changes to the package "Foundations" are normally not accepted, for we are trying to keep it in a state close to what Vladimir Voevodsky originally intended. A warning is issued if you run make or make all and have changed a file in the Foundations package.

Adding a file to a package

Each package contains a subdirectory called ".package". The file ".packages/files" consists of a list of the paths to the *.v files of the package, in order, i.e., a file is listed after files it depends on. (That's just so the TAGS file will be correctly sequenced.) To add a file to a package, add its path to that file.

Adding a new package

Create a subdirectory of this directory, populate it with your files, add a README (or README.md) file, and add a file .package/files, listing the *.v files of your package, as above. Then add the name of your package to the head of the list assigned to "PACKAGES" in the file "./Makefile", or, alternatively, if you'd like to test your package without modifying "./Makefile", which you might accidentally commit and push, add its name to the head of the list in "../build/Makefile-configuration", which is created from "../build/Makefile-configuration-template".

The UniMath formal language

The formal language used in the UniMath project is based on Martin-Löf type theory, as present in MLTT79 below. We are currently on version 2.

UniMath-1 is MLTT79 except:

• the bound variable in a λ-expression is annotated with its type
• we omit W-types
• just the finite types of cardinality 0, 1, and 2 are used, although there would be no problem with introducing further ones
• we omit reflection from identities to judgmental equalities
• we add the resizing rules from the slides of Voevodsky's 2011 talk in Bergen

UniMath-2 is UniMath-1 except:

• we add η for pairs

The axioms accepted are: the univalence axiom, the law of excluded middle, the axiom of choice, and a few new variants of the axiom of choice, validated by the semantic model.

MLTT79 is this paper:

@incollection {MLTT79,
    AUTHOR = {Martin-L\"of, Per},
     TITLE = {Constructive mathematics and computer programming},
 BOOKTITLE = {Logic, methodology and philosophy of science, {VI} ({H}annover, 1979)},
    SERIES = {Stud. Logic Found. Math.},
    VOLUME = {104},
     PAGES = {153--175},
 PUBLISHER = {North-Holland, Amsterdam},
      YEAR = {1982},
   MRCLASS = {03F50 (03B70 03F55 68Q45)},
  MRNUMBER = {682410},
MRREVIEWER = {B. H. Mayoh},
       DOI = {10.1016/S0049-237X(09)70189-2},
       URL = {http://dx.doi.org/10.1016/S0049-237X(09)70189-2},
}

UniMath coding style

In the following rules, we purposely restrict our use of Coq to a subset whose semantics is more likely to be rigorously verifiable and portable to new proof checking systems, and we follow a style of coding designed to render proofs less fragile and to make the files have a more uniform and pleasing appearance.

• Identifiers and function names
  • Form identifiers by concatenating English words or existing identifiers in lower case, separating them by underscores.
  • Unless it impedes clarity or goes against common practice avoid using abbreviations.
  • In some parts of the library uppercase is used for bundled mathematical objects (e.g. Pullback, Topos). It is sometimes justified to introduce new identifiers using this naming scheme. The following guidelines should then be applied:
    • Identifiers with capital letters must not use underscores to separate words, they must use CamelCase.
    • Only use CamelCase when it is already used in the parts of the library you are working in or there is some compelling reason for it to be introduced.
    • Do not use CamelCase for intermediary structures. Example: if CamelCasecontains a data part and a property part then name these camel_case_data and is_camel_case, do not call them CamelCaseData and IsCamelCase.
    • Upper-case letters should not be used in function names unless there is specific good reason to do so. In general name your functions make_camel_case and camel_case_property, not make_CamelCase and CamelCase_property, even if the object is called CamelCase.
• Do not use Admitted or introduce new axioms.
• Do not use apply with a term that needs no additional arguments filled in, because using exact would be clearer.
• Do not use Prop or Set, and ensure definitions don't produce elements of them.
• Do not use Inductive or Record. Their use is limited to just a few basic types, which are defined in Foundations/Preamble.v.
• Do not use Structure.
• Use Module only naively, to create blocks of code that can be imported. Do not use Module Type.
• Do not use Fixpoint.
• Do not use destruct, match, case, square brackets with intros, or nested square brackets with induction. (The goal is to prevent generation of proof terms using match.)
• Use do with a specific numerical count, rather than repeat, to make proofs easier to repair.
• Use as to name all new variables introduced by induction or destruct, if the corresponding type is defined in a remote location, because different names might be used by Coq when the definition of the type is changed. Name all variables introduced by assert, if they are used by name later, with as or to the left of a colon.
• Avoid ending proofs with Qed, because that may prevent future computation. If you decide to make a proof opaque, then make sure that its type is a proposition. It is undesirable to write multiple opaque proofs of properties, for then proofs of equality of objects containing them cannot be accomplished by reflexivity.
• Start all proofs with Proof. on a separate line and end it with Defined. on a separate line, as this makes it possible for us to generate HTML with expansible/collapsible proofs.
• Use Lemma, Proposition, or Theorem for proofs of propositions; for defining elements of types that are not propositions, use Definition.
• Use Unicode notation freely, but make the parsing conventions uniform across files. All notations, except for certain notations in the Foundations package used everywhere, should be local or in a scope. All scopes, if opened, should be opened only locally. Consider also putting them into a submodule, for then they won't be activated even for printing.
• When introducing a notation using Unicode characters, document in a comment how to input that character using the Agda input method.
• Each line should be limited to at most 100 (Unicode) characters. The makefile target enforce-max-line-length can be used to detect nonconforming files, and the target show-long-lines can be used to display the nonconforming lines.
• Always use Coq's proof structuring syntax ( { } + - * ) to focus on a single goal immediately after a tactic creates additional goals.
• Indentation should normally be that produced automatically by emacs' coq-mode.
• When using abstract in a proof, it is unsound to refer later by name to the abstracted lemma (whose name typically ends with _subproof), because its type may vary from one version of Coq to another. Coq's current behavior is also unlikely to be duplicated precisely by a future proof assistant.
• Define and use accessor functions for structures instead of chains of pr1 and pr2. This makes the code easier to maintain in the long run (if the structure is rearranged the proofs will still work if the accessor functions are changed accordingly).
• Define constructor functions for structures taking all of the required data in the right order. This way one can write use constructor instead of having a nested chain of use tpair leading to flatter proof scripts for instantiating structures.

Our files don't adhere yet to all of these conventions, but it's a goal we strive for.

Another advantage of coding in this style is that the proofs should be easier to transport to another proof assistant.