Equations for Hereditary Substitution in Leivant's Predicative System F: A Case Study

This paper presents a case study of formalizing a normalization proof for Leivant's Predicative System F using the Equations package. Leivant's Predicative System F is a stratified version of System F, where type quantification is annotated with kinds representing universe levels. A weaker variant of this system was studied by Stump & Eades, employing the hereditary substitution method to show normalization. We improve on this result by showing normalization for Leivant's original system using hereditary substitutions and a novel multiset ordering on types. Our development is done in the Coq proof assistant using the Equations package, which provides an interface to define dependently-typed programs with well-founded recursion and full dependent pattern- matching. Equations allows us to define explicitly the hereditary substitution function, clarifying its algorithmic behavior in presence of term and type substitutions. From this definition, consistency can easily be derived. The algorithmic nature of our development is crucial to reflect languages with type quantification, enlarging the class of languages on which reflection methods can be used in the proof assistant.Comment: In Proceedings LFMTP 2015, arXiv:1507.07597. www: http://equations-fpred.gforge.inria.fr

Eliminating Dependent Pattern-Matching in Coq

International audienceCoq [1] is a proof assistant which relies on the Curry-Howard isomorphism to construct certified proofs. Proving a theorem is the same thing as providing a term which inhabits a type corresponding to this theorem. In order to trust Coq, it is enough to trust its kernel, which is intentionally kept small enough for a motivated reader to understand and, hopefully, trust it. This approach can have some drawbacks, as high-level constructs have to be translated down to simpler constructs, by the user or by some part of code external to the kernel. For example, writing dependent pattern-matching in Coq can be complicated. Simpifying this task is one of the purposes of the Equations [2] plugin. Given a high-level specification of a function, which can use dependent pattern-matching and complex recursion schemes, it will compile it to pure Coq terms. Equations is the result of the work of Matthieu Sozeau[10], largely based on the research of Goguen et al[7]. This internship revolves around Equations as a tool to benchmark, improve and adapt to new settings. This in turn involves the study of dependent pattern-matching. Research problem The initial goal of this internship was to rewrite a part of Equations, in order to better control the use of the axiom K during the compilation phase. This axiom states that to prove a property depending on a proof of equality, it is enough to consider the case where this proof is the reflexivity. Equivalently, it says that any proof of equality is propositionally equal to the reflexivity. While it is useful in some cases, and even provable for a lot of types, it can be harmful when working in some contexts, like Homotopy Type Theory[12] – abbreviated HoTT. Another problem is that, as an axiom, it will block any computation in Coq that involves it

Equations reloaded

International audienceEquations is a plugin for the Coq proof assistant which provides a notation for defining programs by dependent pattern-matching and structural or well-founded recursion. It additionally derives useful proof principles for demonstrating properties about them. We present a general design and implementation that provides a robust and expressive function definition package as a definitional extension to the Coq kernel. At the core of the system is a new simplifier for dependent equalities that can be reused to define enhanced versions of dependent elimination tactics. We introduce verified optimizations of the simplifier that allow generating smaller and simpler Equations definitions and proof terms for these tactics in general

mattam82/Coq-Equations: Equations 1.0 for Coq 8.7

Equations is a function definition plugin for Coq (supporting Coq 8.6 and 8.7), that allows the definition of functions by dependent pattern-matching and well-founded, mutual or nested structural recursion and compiles them into core terms. It automatically derives the clauses equations, the graph of the function and its associated elimination principle. See the release post for more information about this version

mattam82/Coq-Equations: Equations v1.2-beta for Coq 8.9

Release 1.2beta of Coq-Equations, see announcement here: http://mattam82.github.io/Coq-Equations/equations/2019/01/28/1.2beta.htm