Fibred Fibration Categories

We introduce fibred type-theoretic fibration categories which are fibred categories between categorical models of Martin-L\"{o}f type theory. Fibred type-theoretic fibration categories give a categorical description of logical predicates for identity types. As an application, we show a relational parametricity result for homotopy type theory. As a corollary, it follows that every closed term of type of polymorphic endofunctions on a loop space is homotopic to some iterated concatenation of a loop

A Relationally Parametric Model of Dependent Type Theory

Reynolds’ theory of relational parametricity captures the invariance of polymorphically typed programs under change of data representation. Reynolds’ original work exploited the typing discipline of the polymorphically typed -calculus System F, but there is now considerable interest in extending relational parametricity to type systems that are richer and more expressive than that of System F.This paper constructs parametric models of predicative and impredicative dependent type theory. The significance of our models is twofold. Firstly, in the impredicative variant we are able to deduce the existence of initial algebras for all indexed functors. To our knowledge, ours is the first account of parametricity for dependent types that is able to lift the useful deduction of the existence of initial algebras in parametric models of System F to the dependently typed setting. Secondly, our models offer conceptual clarity by uniformly expressing relational parametricity for dependent types in terms of reflexive graphs, which allows us to unify the interpretations of types and kinds, instead of taking the relational interpretation of types as a primitive notion. Expressing our model in terms of reflexive graphs ensures that it has canonical choices for the interpretations of the standard type constructors of dependent type theory, except for the interpretation of the universe of small types, where we formulate a refined interpretation tailored for relational parametricity. Moreover, our reflexive graph model opens the door to generalizations of relational parametricity, for example to higher-dimensional relational parametricity

Comprehensive parametric polymorphism : categorical models and type theory

This paper combines reflexive-graph-category structure for relational parametricity with fibrational models of impredicative polymorphism. To achieve this, we modify the definition of fibrational model of impredicative polymorphism by adding one further ingredient to the structure: comprehension in the sense of Lawvere. Our main result is that such comprehensive models, once further endowed with reflexive-graph-category structure, enjoy the expected consequences of parametricity. This is proved using a type-theoretic presentation of the category-theoretic structure, within which the desired consequences of parametricity are derived. The formalisation requires new techniques because equality relations are not available, and standard arguments that exploit equality need to be reworked

Trocq: Proof Transfer for Free, With or Without Univalence

Libraries of formalized mathematics use a possibly broad range of different representations for a same mathematical concept. Yet light to major manual input from users remains most often required for obtaining the corresponding variants of theorems, when such obvious replacements are typically left implicit on paper. This article presents Trocq, a new proof transfer framework for dependent type theory. Trocq is based on a novel formulation of type equivalence, used to generalize the univalent parametricity translation. This framework takes care of avoiding dependency on the axiom of univalence when possible, and may be used with more relations than just equivalences. We have implemented a corresponding plugin for the Coq proof assistant, in the CoqElpi meta-language. We use this plugin on a gallery of representative examples of proof transfer issues in interactive theorem proving, and illustrate how Trocq covers the spectrum of several existing tools, used in program verification as well as in formalized mathematics in the broad sense

Type‐Preserving CPS Translation of Σ and Π Types is Not Not Possible

International audienceDependently typed languages like Coq are used to specify and prove functional correctness of source programs,but what we ultimately need are guarantees about correctness of compiled code. By preserving dependenttypes through each compiler pass, we could preserve source-level specifications and correctness proofs intothe generated target-language programs. Unfortunately, type-preserving compilation of dependent types isnontrivial. In 2002, Barthe and Uustalu showed that type-preserving CPS is not possible for languages likeCoq. Specifically, they showed that for strong dependent pairs (Σ types), the standard typed call-by-name CPSis not type preserving. They further proved that for dependent case analysis on sums, a class of typed CPStranslations—including the standard translation—is not possible. In 2016, Morrisett noticed a similar problemwith the standard call-by-value CPS translation for dependent functions (Π types). In essence, the problem isthat the standard typed CPS translation by double-negation, in which computations are assigned types of theform (A → ⊥) → ⊥, disrupts the term/type equivalence that is used during type checking in a dependentlytyped language.In this paper, we prove that type-preserving CPS translation for dependently typed languages is not notpossible. We develop both call-by-name and call-by-value CPS translations from the Calculus of Constructionswith both Π and Σ types (CC) to a dependently typed target language, and prove type preservation andcompiler correctness of each translation. Our target language is CC extended with an additional equivalencerule and an additional typing rule, which we prove consistent by giving a model in the extensional Calculus ofConstructions. Our key observation is that we can use a CPS translation that employs answer-type polymorphism,where CPS-translated computations have type ∀α.(A → α) → α. This type justifies, by a free theorem,the new equality rule in our target language and allows us to recover the term/type equivalences that CPStranslation disrupts. Finally, we conjecture that our translation extends to dependent case analysis on sums,despite the impossibility result, and provide a proof sketch

A Presheaf Model of Parametric Type Theory

Abstract We extend Martin-Löf's Logical Framework with special constructions and typing rules providing internalized parametricity. Compared to previous similar proposals, this version comes with a denotational semantics which is a refinement of the standard presheaf semantics of dependent type theory. Further, this presheaf semantics is a refinement of the one used to interpret nominal sets with restrictions. The present calculus is a candidate for the core of a proof assistant with internalized parametricity

The Marriage of Univalence and Parametricity

International audienceReasoning 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. This work first clarifies the limitations of these two concepts when considered in isolation, and then 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. Our approach handles both type and term dependency, as well as type-level computation. In addition to the theory of univalent parametricity, we present 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. For instance, this makes it possible to conveniently switch between an easy-to-reason-about representation and a computationally-efficient representation, as soon as they are proven equivalent. The combination of parametricity and univalence supports transport à la carte: basic univalent transport, which stems from a type equivalence, can be complemented with additional proofs of equivalences between functions over these types, in order to be able to transport more programs and proofs, as well as to yield more efficient terms. We illustrate the use of univalent parametricity on several examples, including a recent integration of native integers in Coq. This work paves the way to easier-to-use proof assistants by supporting seamless programming and proving modulo equivalences