A Case Study on Logical Relations using Contextual Types

Proofs by logical relations play a key role to establish rich properties such as normalization or contextual equivalence. They are also challenging to mechanize. In this paper, we describe the completeness proof of algorithmic equality for simply typed lambda-terms by Crary where we reason about logically equivalent terms in the proof environment Beluga. There are three key aspects we rely upon: 1) we encode lambda-terms together with their operational semantics and algorithmic equality using higher-order abstract syntax 2) we directly encode the corresponding logical equivalence of well-typed lambda-terms using recursive types and higher-order functions 3) we exploit Beluga's support for contexts and the equational theory of simultaneous substitutions. This leads to a direct and compact mechanization, demonstrating Beluga's strength at formalizing logical relations proofs.Comment: In Proceedings LFMTP 2015, arXiv:1507.0759

Lincx: A Linear Logical Framework with First-class Contexts

Linear logic provides an elegant framework for modelling stateful, imperative and con- current systems by viewing a context of assumptions as a set of resources. However, mech- anizing the meta-theory of such systems remains a challenge, as we need to manage and reason about mixed contexts of linear and intuitionistic assumptions. We present Lincx, a contextual linear logical framework with first-class mixed contexts. Lincx allows us to model (linear) abstract syntax trees as syntactic structures that may depend on intuitionistic and linear assumptions. It can also serve as a foundation for reasoning about such structures. Lincx extends the linear logical framework LLF with first-class (linear) contexts and an equational theory of context joins that can otherwise be very tedious and intricate to develop. This work may be also viewed as a generalization of contextual LF that supports both intuitionistic and linear variables, functions, and assumptions. We describe a decidable type-theoretic foundation for Lincx that only characterizes canonical forms and show that our equational theory of context joins is associative and commu- tative. Finally, we outline how Lincx may serve as a practical foundation for mechanizing the meta-theory of stateful systems.La logique linéaire represente une structure élégante pour modeler des systèmes im- pératifs, concurrents et avec des systèmes a états, en représentant un contexte d'hypothèses comme une collection de ressources. Cependant, la mécanisation de la métathéorie de ces systèmes demeure un défi, puisque nous devons gérer et raisonner à propos de contextes d'hypothèses mixtes linéaires et intuitionistiques. Nous présentons Lincx, une structure logique linéaire et contextuelle avec des contextes mixtes de première classe. Lincx nous permet d'établir des modèles (linéaires) d'arbres de syntaxe abstraits en tant que structures syntactiques qui peuvent dependre d'hypothèses intuitionistiques et linéaires. Lincx peut également servir de fondation pour raisonner à propos de telles structures. Lincx étend la structure logique linéaire LLF avec des contextes (linéaires) de premier ordre et une théorie d'equations d'assemblage de contextes qui peut autrement être très fastidieux et complexe à développer. Cet oeuvre peut également être perçu comme une généralisation du LF contextuel qui supporte les fonctions, les hypothéses et les variables intuitionistiques et linéaires. Nous décrivons une fondation de la théorie des types décidable pour Lincx qui ne décrit que les formes canoniques et montrons que notre theorie d'équations d'assemblage de contextes est associative et commutative. Finalement, nous donnons un aperçu de comment Lincx peut servir de fondation pratique pour la mécanisation de la métathéorie de systèmes à états

POPLMark reloaded: Mechanizing proofs by logical relations

We propose a new collection of benchmark problems in mechanizing the metatheory of programming languages, in order to compare and push the state of the art of proof assistants. In particular, we focus on proofs using logical relations (LRs) and propose establishing strong normalization of a simply typed calculus with a proof by Kripke-style LRs as a benchmark. We give a modern view of this well-understood problem by formulating our LR on well-typed terms. Using this case study, we share some of the lessons learned tackling this problem in different dependently typed proof environments. In particular, we consider the mechanization in Beluga, a proof environment that supports higher-order abstract syntax encodings and contrast it to the development and strategies used in general-purpose proof assistants such as Coq and Agda. The goal of this paper is to engage the community in discussions on what support in proof environments is needed to truly bring mechanized metatheory to the masses and engage said community in the crafting of future benchmarks