Section: New Results

Type theory and the foundations of Coq

Participants : Pierre Boutillier, Pierre-Louis Curien, Hugo Herbelin, Pierre-Marie Pédrot, Yann Régis-Gianas, Alexis Saurin, Matthieu Sozeau.

Calculus of inductive constructions and typed equality

The work of Hugo Herbelin and Vincent Siles on the equivalence of Pure Type Systems with typed or untyped equality has been published [17] .

Substitutions and isomorphisms

Pierre-Louis Curien completed his joint work with Martin Hofmann (Univ. of Munich) and Richard Garner (MacQuarie University, Sydney) on comparing two categorical interpretations of (extensional) type theories. More precisely, we wanted to compare two ways of giving a categorical interpretation of Martin-Löf type theory, both overcoming the following mismatch: syntax has exact substitutions, while their categorical interpretation, in terms of pullbacks or fibrations, “implements” substitutions only up to isomorphism. One can then either change the model (strictification) [55] , or modify the syntax (by introducing explicit substitutions and more importantly explicit coercions between types that are now only isomorphic) [2] . In the latter case, one has to prove a coherence theorem to show that the interpretation is in the end independent from these coercion decorations. Such a proof was given in [2] , using rewriting methods. These approaches turn out to be related through a general machinery that relates three kinds of categories, with strict or non strict objects and morphisms. As a bonus we get a new, more conceptual proof of coherence. These results are now being written up for a special issue in honour of Glynn Winskel. In further work, we wish to address intensional, and homotopy type-theoretic versions of these coherence problems.

Homotopy type theory

The univalence axiom proposed by Voevodsky states that for any two types to be equal exactly means being of same cardinality. This new axiom for type theory turns to have very interesting consequences for the practical foundations of formal mathematical reasoning: it smoothly implies other axioms such as functional extensionality or propositional existential but before all it says that any property proved about some mathematical structure immediately applies to any other other type (“sets” informally) which it is isomorphic to.

This axiom however contradicts the current logical foundations of Coq (in the presence of Streicher's axiom K). Investigations have then been started to understand how to weaken the Calculus of Inductive Constructions implemented in Coq so as to make it compatible with univalence. In a first step, this resulted in the design of a new rule for singleton elimination that has been implemented by Hugo Herbelin as an optional feature of Coq (singleton elimination is the ability to build objects in datatypes from canonically-proved propositional properties such as equality).

Models of type theory

The existing models of homotopy type theory are based on simplicial sets or on their extensions as Kan complexes. Hugo Herbelin developed a concrete type-theoretic formalisation of semi-simplicial sets following ideas from Steve Awodey, Peter LeFanu Lumsdaine and other researchers both at Carnegie-Mellon University and at the Institute of Advanced Study. The technique he used seems to straightforwardly generalise to provide type-theoretic constructions for arbitrary presheaves on inductively generated categories.

Forcing in type theory

Together with Nicolas Tabareau and Guilhem Jaber (Inria Ascola team, École des mines de Nantes), Matthieu Sozeau investigated an internalisation of the presheaf model of the Calculus of Inductive Constructions (CIC). They published their work at LICS'12 [23] . This work corresponds to adapting the idea of Forcing due to Cohen in Type Theory. An internal model construction allows to enrich the logical type theory with new modalities and define their semantics by translation to CIC. The usual Cohen forcing can be realised using this framework to show the independence of the continuum hypothesis in CIC, but more practical applications are possible as well. Notably, the step-indexed technique for building models of imperative languages with rich type structure can be phrased as a forcing/presheaf construction. Sozeau, Tabareau and Jaber developed a plugin that can handle this example [32] which relies on a modified Coq version implementing proof-irrelevance and eta-rules for records.

Proof irrelevance, eta-rules

Matthieu Sozeau continued his work on proof-irrelevance by implementing a variant of Werner's proof-irrelevant CIC in Coq [72] . An article describing this work is in preparation. The new system also handles the extensional eta-rules for records, extending the technique implemented by Hugo Herbelin to handle eta-expansion of functions in Coq.

Unification

The unification algorithm of Coq now essentially dwells in the $\lambda$-calculus part of the language. Pierre Boutillier started a refactoring of the code in order to deal with algebraic datatypes. Hugo Herbelin and Pierre Boutillier investigated how to reformulate unification on top of an abstract machine (i.e. on top of sequent calculus). Hugo Herbelin added various heuristics to the unification algorithm of Coq, making them both more powerful and customisable.

Matthieu Sozeau is continuing work in collaboration with Beta Ziliani (PhD student of Derek Dreyer at MPI Saarbrücken, two one week visits in 2012), and Aleksandar Nanevski (Researcher at IMDEA Madrid) on giving a clear formalisation for the unification algorithm of Coq. This will help understand better the working of advanced features like Canonical Structures and Type Classes that are heavily used in big developments, as the spectacular recently completed formalisation of the proof of Feit-Thompson's Odd theorem by the Mathematical Components team.

Matthieu Sozeau adapted the existing unification algorithm to be universe-aware, resulting in more predictability and earlier error-reporting in both the type inference and tactic unification algorithms of Coq.