Sharing a Library between Proof Assistants: Reaching out to the HOL Family

We observe today a large diversity of proof systems. This diversity has the negative consequence that a lot of theorems are proved many times. Unlike programming languages, it is difficult for these systems to co-operate because they do not implement the same logic. Logical frameworks are a class of theorem provers that overcome this issue by their capacity of implementing various logics. In this work, we study the STTforall logic, an extension of Simple Type Theory that has been encoded in the logical framework Dedukti. We present a translation from this logic to OpenTheory, a proof system and interoperability tool between provers of the HOL family. We have used this translation to export an arithmetic library containing Fermat's little theorem to OpenTheory and to two other proof systems that are Coq and Matita.Comment: In Proceedings LFMTP 2018, arXiv:1807.0135

Impredicative Encodings of (Higher) Inductive Types

Postulating an impredicative universe in dependent type theory allows System F style encodings of finitary inductive types, but these fail to satisfy the relevant {\eta}-equalities and consequently do not admit dependent eliminators. To recover {\eta} and dependent elimination, we present a method to construct refinements of these impredicative encodings, using ideas from homotopy type theory. We then extend our method to construct impredicative encodings of some higher inductive types, such as 1-truncation and the unit circle S1

Goal Translation for a Hammer for Coq (Extended Abstract)

Hammers are tools that provide general purpose automation for formal proof assistants. Despite the gaining popularity of the more advanced versions of type theory, there are no hammers for such systems. We present an extension of the various hammer components to type theory: (i) a translation of a significant part of the Coq logic into the format of automated proof systems; (ii) a proof reconstruction mechanism based on a Ben-Yelles-type algorithm combined with limited rewriting, congruence closure and a first-order generalization of the left rules of Dyckhoff's system LJT.Comment: In Proceedings HaTT 2016, arXiv:1606.0542

Reasoning about Knowledge in Linear Logic: Modalities and Complexity

In a recent paper, Jean-Yves Girard commented that ”it has been a long time since philosophy has stopped intereacting with logic”[17]. Actually, it has no

Rewriting Modulo \beta in the \lambda\Pi-Calculus Modulo

The lambda-Pi-calculus Modulo is a variant of the lambda-calculus with dependent types where beta-conversion is extended with user-defined rewrite rules. It is an expressive logical framework and has been used to encode logics and type systems in a shallow way. Basic properties such as subject reduction or uniqueness of types do not hold in general in the lambda-Pi-calculus Modulo. However, they hold if the rewrite system generated by the rewrite rules together with beta-reduction is confluent. But this is too restrictive. To handle the case where non confluence comes from the interference between the beta-reduction and rewrite rules with lambda-abstraction on their left-hand side, we introduce a notion of rewriting modulo beta for the lambda-Pi-calculus Modulo. We prove that confluence of rewriting modulo beta is enough to ensure subject reduction and uniqueness of types. We achieve our goal by encoding the lambda-Pi-calculus Modulo into Higher-Order Rewrite System (HRS). As a consequence, we also make the confluence results for HRSs available for the lambda-Pi-calculus Modulo.Comment: In Proceedings LFMTP 2015, arXiv:1507.0759

Translating HOL to Dedukti

Dedukti is a logical framework based on the lambda-Pi-calculus modulo rewriting, which extends the lambda-Pi-calculus with rewrite rules. In this paper, we show how to translate the proofs of a family of HOL proof assistants to Dedukti. The translation preserves binding, typing, and reduction. We implemented this translation in an automated tool and used it to successfully translate the OpenTheory standard library.Comment: In Proceedings PxTP 2015, arXiv:1507.0837

Relational parametricity for higher kinds

Reynolds’ notion of relational parametricity has been extremely influential and well studied for polymorphic programming languages and type theories based on System F. The extension of relational parametricity to higher kinded polymorphism, which allows quantification over type operators as well as types, has not received as much attention. We present a model of relational parametricity for System Fω, within the impredicative Calculus of Inductive Constructions, and show how it forms an instance of a general class of models defined by Hasegawa. We investigate some of the consequences of our model and show that it supports the definition of inductive types, indexed by an arbitrary kind, and with reasoning principles provided by initiality

Investigations on a Pedagogical Calculus of Constructions

In the last few years appeared pedagogical propositional natural deduction systems. In these systems, one must satisfy the pedagogical constraint: the user must give an example of any introduced notion. First we expose the reasons of such a constraint and properties of these "pedagogical" calculi: the absence of negation at logical side, and the "usefulness" feature of terms at computational side (through the Curry-Howard correspondence). Then we construct a simple pedagogical restriction of the calculus of constructions (CC) called CCr. We establish logical limitations of this system, and compare its computational expressiveness to Godel system T. Finally, guided by the logical limitations of CCr, we propose a formal and general definition of what a pedagogical calculus of constructions should be.Comment: 18 page

The Arity Hierarchy in the Polyadic μ\mu-Calculus

The polyadic mu-calculus is a modal fixpoint logic whose formulas define relations of nodes rather than just sets in labelled transition systems. It can express exactly the polynomial-time computable and bisimulation-invariant queries on finite graphs. In this paper we show a hierarchy result with respect to expressive power inside the polyadic mu-calculus: for every level of fixpoint alternation, greater arity of relations gives rise to higher expressive power. The proof uses a diagonalisation argument.Comment: In Proceedings FICS 2015, arXiv:1509.0282