Cubical Type Theory

Experimental implementation of a cubical type theory in which the user can directly manipulate n-dimensional cubes. The language extends type theory with:

• Path abstraction and application
• Composition and transport
• Equivalences can be transformed into equalities (and univalence can be proved, see "examples/univalence.ctt")
• Some higher inductive types (see "examples/circle.ctt" and "examples/integer.ctt")

Because of this it is not necessary to have a special file of primitives (like in cubical), for instance function extensionality is provable in the system by:

funExt (A : U) (B : A -> U) (f g : (x : A) -> B x)
       (p : (x : A) -> Id (B x) (f x) (g x)) :
       Id ((y : A) -> B y) f g = <i> \(a : A) -> (p a) @ i

For more examples, see "examples/demo.ctt" and "examples/aim.ctt".

The following keywords are reserved:

module, where, let, in, split, mutual, import, data, hdata, undefined,
IdP, comp, transport, fill, glue, glueElem, unglueElem, U

Install

To compile the project using cabal, first install the build-time dependencies (either globally or in a cabal sandbox):

cabal install alex happy bnfc

Then the project can be built (and installed):

cabal install

Alternatively, a Makefile is provided:

    make

This assumes that the following Haskell packages are installed using cabal:

mtl, haskeline, directory, BNFC, alex, happy, QuickCheck

To build the TAGS file, run:

    make TAGS

This assumes that hasktags has been installed.

To clean up, run:

    make clean

Usage

To run the system type

cubical <filename>

To enable the debugging mode add the -d flag. In the interaction loop type :h to get a list of available commands. Note that the current directory will be taken as the search path for the imports.

When using cabal sandboxes, cubical can be invoked using

cabal exec cubical <filename>

To enable emacs to edit *.ctt files in ctt-mode, add the following line to your .emacs file:

(autoload 'ctt-mode "cubicaltt" "cubical editing mode" t)
(setq auto-mode-alist (append auto-mode-alist '(("\\.ctt$" . ctt-mode))))

and ensure that the file cubicaltt.el is visible in one of the diretories on emacs' load-path, or else load it in advance, either manually with M-x load-file, or with something like the following line in .emacs:

(load-file "cubicaltt.el")

References and notes

• Voevodsky's lectures and texts on univalent foundations

• HoTT book and webpage: http://homotopytypetheory.org/

• Cubical Type Theory: a constructive interpretation of the univalence axiom, Cyril Cohen, Thierry Coquand, Simon Huber, and Anders Mörtberg

• Cubical Type Theory - The typing rules of the system. See this for a variation using isomorphisms instead of equivalences.

• Internal version of the uniform Kan filling condition

• A category of cubical sets - main definitions towards a formalization

• hoq - A language based on homotopy type theory with an interval (documentation available here).

• A Cubical Approach to Synthetic Homotopy Theory, Dan Licata, Guillaume Brunerie.

• Type Theory in Color, Jean-Philippe Bernardy, Guilhem Moulin.

• A simple type-theoretic language: Mini-TT, Thierry Coquand, Yoshiki Kinoshita, Bengt Nordström and Makoto Takeyama - This presents the type theory that the system is based on.

• A cubical set model of type theory, Marc Bezem, Thierry Coquand and Simon Huber.

• An equivalent presentation of the Bezem-Coquand-Huber category of cubical sets, Andrew Pitts - This gives a presentation of the cubical set model in nominal sets.

• Remark on singleton types, Thierry Coquand.

• Note on Kripke model, Marc Bezem and Thierry Coquand.

Authors

Cyril Cohen, Thierry Coquand, Simon Huber, Anders Mörtberg