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Implementation of Univalence in Cubical Sets
Haskell Emacs Lisp

README.md

CUBICAL

Cubical implements an experimental simple type-checker for type theory with univalence with an evaluator for closed terms.

INSTALL

To install cubical, a working Haskell and cabal installation are required. To build cubical go to the main directory and do

cabal install

To only build (not install) cubical do

cabal configure

cabal build

Alternatively one can also use the Makefile to build the system by typing:

make bnfc && make

However this requires that the following Haskell packages are installed:

mtl, haskeline, directory, BNFC, alex, happy

Note: In order to make the mutual keyword work a patched version of BNFC is needed. To install this download the patched version from

https://github.com/simhu/bnfc

and then cabal install it.

Emacs mode:

To install syntax highlighting for cubical files load the cubical.el file into emacs. In order to load it automatically add

(load-file "/path/to/cubical.el")

(add-to-list 'auto-mode-alist '("\\.cub\\'" . cub-mode))

to your .emacs file.

USAGE

To run cubical 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.

OVERVIEW

The program is organized as follows:

  • the files are parsed and produce a list of definitions; the syntax is described in the file Exp/Doc.txt or Exp/Doc.tex (auto generated by bnfc);

  • this list of definitions is type-checked;

  • if successful, we can then write an expression which is type-checked w.r.t. these definitions;

  • if the expression is well-typed it is translated to the cubical syntax and evaluated by a "cubical abstract machine", which computes its semantics in cubical sets; the result is shown after "EVAL:" (to enable the trace of the evaluation run cubical with the -d flag);

During type-checking, we consider the primitives listed in examples/primitive.cub as non interpreted constants. The type-checker is in the file TypeChecker.hs and is rudimentary (200 lines), without good error messages.

These primitives however have a meaning in cubical sets, and the evaluation function computes this meaning. This semantics/evaluation is described in the file Eval.hs, which is the main file. The most complex part corresponds to the computations witnessing that the universe has Kan filling operations.

For writing this semantics, it was convenient to use the alternative presentation of cubical sets as nominal sets with 01-substitutions (see A. Pitts' note, references listed below).

The primitives needed to get univalence are.

DESCRIPTION OF THE LANGUAGE

We have

  • dependent function types (x:A) -> B; non-dependent function types can be written as A -> B

  • abstraction \x -> e

  • named/labelled sum c1 (x1:A1)...(xn:An) | c2 ... | ... a data type is a recursively defined named sum

  • function defined by case f = split c1 x1 ... xn -> e1 | c2 ... -> ... | ...

  • sigma types (x:A) * B, with the pair constructor (e1,e2) and eliminators e.1 and e.2

  • a universe U and assume U:U for simplicity

  • let/where: let D in e where D is a list of definitions an alternative syntax is e where D

  • undefined like in Haskell

  • mutual definitions (this requires a patched version of BNFC, see the install instructions above).

The syntax allows Landin's offside rule similar to Haskell.

The basic (untyped) language has a direct simple denotational semantics. Type theory works with the total part of this language (it is possible to define totality at the denotational semantics level). Our evaluator works in a nominal version of this semantics. The type-checker assumes that we work in this total part, however, there is no termination check.

DESCRIPTION OF THE SEMANTICS/EVALUATION

The values depend on a new class of names, also called directions, which can be thought of as varying over the unit interval [0,1]. A path connecting a0 and a1 in the direction x is a value p(x) such that p(0) = a0 and p(1) = a1. An element in the identity type a0 = a1 is then of the form <x>p(x) where the name x is bound. An identity proof in an identity type will then be interpreted as a "square" of the form <x><y>p(x,y). See examples/hedberg.cub and the example test3 (in the current implementation directions/names are represented by numbers).

Operationally, a type is explained by giving what are its Kan filling operation. For instance, we have to explain what are the Kan filling for the dependent product.

The main step for interpreting univalence is to transform an equivalence A -> B to a path in any direction x connecting A and B. This is a new basic element of the universe, called VEquivEq in the file Eval.hs which takes a name and arguments A,B,f and the proof that f is an equivalence. The main part of the work is then to explain the Kan filling operation for this new type.

The Kan filling for the universe can be seen as a generalization of the operation of composition of relation.

DESCRIPTION OF THE EXAMPLES

The directory examples contains some examples of proofs. The file examples/primitive.cub list the new primitives that have cubical set semantics. These primitive notions imply the axiom of univalence. The file examples/primitive.cub should be the basis of any development using univalence.

Most of the example files contain simple test examples of computations:

  • the file hedberg.cub contains a test computation of a square in Nat; the example is test. In the type Nat or Bool, any square (proof of identity of two identity proofs) is constant.

  • The file nIso.cub contains a non trivial example of a transport of a section of a dependent type along the isomorphism between N and N+1; the examples are testSN, testSN1, testSN2, testSN3.

  • The file testInh.cub contains examples of computation for the propositional reflection. It gives an example test which produces a (surprisingly complex) composition of squares in the universe.

  • The file quotient.cub contains an example of a computation from an equivalence class. The relation R over Nat is to have the same parity, and the map is Nat/R -> Bool which returns true if the equivalence class contains even number. The examples are test5 and test8 which computes the value of this map on the equivalence class of five and eight respectively. This uses the file description.cub which justifies the axiom of description.

  • The file Kraus.cub contains the example of Nicolai Kraus of the myst function, which also shows that we can extract computational information from propositions; the example is testMyst zero; the computation does not create higher dimensional objects.

  • The file swap.cub contains examples of transport along the isomorphism between A x B and B x A; the examples are test14, test15.

NEWS (to be detailed)

  • Some constants have a direct cubical semantics having better behavior w.r.t. equality. For instance the constant

    mapOnPath : (A B : U) (f : A -> B) (a b : A) (p : Id A a b) -> Id B (f a) (f b)

    has a semantics which satisfies the definitional equalities:

    mapOnPath (id A) = id A

    mapOnPath (g o f) = (mapOnPath g) o (mapOnPath f)

    mapOnPath f (refl A a) = refl B (f a)

    The evaluation is now used for conversion during type-checking, and then we get these equalities definitionally.

    Some proofs are now much simpler than before, e.g. the proof of the Graduate Lemma.

  • Similarly we also have eta conversion and surjective pairing.

  • As a test, the particular case of the circle (S1) and the interval (I) has been added.

FURTHER WORK (non-exhaustive)

  • The Kan filling operations should be formally proved correct and tested on higher inductive types.

  • For higher inductive types, like the circle or the sphere, it would be appropriate to extend the syntax of type theory, in order to get natural elimination rules (see the paper on cubical sets).

  • To explore the termination of the resizing rule. Computationally the resizing rule does not do anything, but the termination seems to be an interesting proof-theoretical problem.

REFERENCES AND NOTES

AUTHORS

Cyril Cohen, Thierry Coquand, Simon Huber, Anders Mörtberg

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