We want to mention the creation of a workshop on homotopy type theory in 2015 at the start of the project.
The workshop happens every year since then (
https://hott-uf.github.io/(si apre in una nuova finestra)).
We now proceed to an overview of the main results of the project:
- Univalent parametricity.
Reasoning modulo equivalences is natural for everyone, including mathematicians. Unfortunately, in proof assistants based on type theory, which are frequently used to
mechanize mathematical results and carry out program verification efforts, equality is appallingly syntactic and, as a result, exploiting equivalences is cumbersome at best.
Parametricity and univalence are two major concepts that have been explored in the literature to transport programs and proofs across type equivalences, but they fall short
of achieving seamless, automatic transport. Our work devises a fruitful marriage between both.
The resulting concept, called univalent parametricity, is an extension of parametricity strengthened with univalence that fully realizes programming and proving modulo equivalences.
In addition to the study of the theory of univalent parametricity, we have developed a lightweight framework implemented in the Coq proof assistant
that allows the user to transparently transfer definitions and theorems for a type to an equivalent one, as if they were equal.
Published at ICFP (rank A* conference) and Journal of the ACM
Implemented using the Coq proof assistant
- SProp.
Definitional equality---or conversion---for a type theory with a decidable type checking is the simplest tool to prove that two objects are the same, letting the system decide just using computation.
Therefore, the more things are equal by conversion, the simpler it is to use a language based on type theory. Proof-irrelevance, stating that any two proofs of the same proposition are equal, i
s a possible way to extend conversion to make a type theory more powerful.
Taking inspiration from homotopy type theory, we have proposed a general way to extend a type theory with definitional
proof irrelevance, in a way that keeps type checking decidable and is compatible with univalence.
Published at POPL (the best conference of our field).
Implementation in both Coq and Agda
- Rewrite Rules.
Dependently typed programming languages and proof assistants such as Agda and Coq rely on computation to automatically simplify
expressions during type checking. To overcome the lack of certain programming primitives or logical principles in those systems, it is common to appeal to axioms
to postulate their existence. However, one can only postulate the bare existence of an axiom, not its computational behaviour.
To solve this tension, we have developed a type theory where it is possible to add computational assumptions in the
form of rewrite rules. Rewrite rules go beyond the computational capabilities of intensional type theory, but in contrast to extensional type theory,
they are applied automatically so type checking does not require input from the user.
Published at POPL (the best conference of our field).
Implementation in Agda
- Certification of Coq in MetaCoq.
Coq is built around a well-delimited kernel that perfoms typechecking for definitions in a variant of the Calculus of Inductive
Constructions (CIC). Although the metatheory of CIC is very stable and reliable, the correctness of its implementation in
Coq is less clear. Indeed, implementing an efficient type checker for CIC is a rather complex task, and many parts of the code rely
on implicit invariants which can easily be broken by further evolution of the code. Therefore, on average, one critical bug has
been found every year in Coq.
We have been able to develop the first implementation of a type checker for the kernel of Coq which is proven correct in Coq with respect to its
formal specification and axiomatisation of part of its metatheory.
Published at POPL (the best conference of our field).
Formalization in Coq
