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-# cubical type theory
+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
+* Isomorphisms can be transformed into equalities
+* Some higher inductive types
+
+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`
+
+
+Install
+-------
+
+To compile the program type:
+
+  `make bnfc && make`
+
+This assumes that the following Haskell packages are installed:
+
+  mtl, haskeline, directory, BNFC, alex, happy
+
+
+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.
+
+
+References and notes
+--------------------
+
+ * Voevodsky's home page on univalent foundation
+
+ * HoTT book and webpage:
+   [http://homotopytypetheory.org/](http://homotopytypetheory.org/)
+
+ * Type Theory in Color, J.P. Bernardy, G. Moulin
+
+ * A simple type-theoretic language: Mini-TT, Th. Coquand,
+   Y. Kinoshita, B. Nordström and M. Takeyama
+
+ * [A cubical set model of type
+   theory](http://www.cse.chalmers.se/~coquand/model1.pdf), M. Bezem,
+   Th. Coquand and S. Huber.
+
+ * [A remark on contractible family of
+   type](http://www.cse.chalmers.se/~coquand/contr.pdf), Th. Coquand.
+
+   This note explains how to derive univalence.
+
+ * [An equivalent presentation of the Bezem-Coquand-Huber category of
+   cubical sets](http://arxiv.org/abs/1401.7807), A. Pitts.
+
+   This gives a presentation of the cubical set model in nominal sets.
+
+ * [Remark on singleton
+   types](http://www.cse.chalmers.se/~coquand/singl.pdf), Th. Coquand.
+
+ * [Note on Kripke
+   model](http://www.cse.chalmers.se/~coquand/countermodel.pdf), M. Bezem
+   and Th. Coquand.
+
+ * [Some connections between cubical sets and
+   parametricity](http://www.cse.chalmers.se/~coquand/param.pdf),
+   Th. Coquand.
+
+
+Authors
+-------
+
+Cyril Cohen, Thierry Coquand, Simon Huber, Anders Mörtberg