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".
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 bnfc && make
This assumes that the following Haskell packages are installed:
mtl, haskeline, directory, BNFC, alex, happy
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>
-
Voevodsky's lectures and texts on univalent foundations
-
HoTT book and webpage: http://homotopytypetheory.org/
-
Cubical Type Theory - The typing rules of the system. See this for a variation using isomorphisms instead of equivalences.
-
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.
Cyril Cohen, Thierry Coquand, Simon Huber, Anders Mörtberg