Equations Reloaded Accompanying Material

Sozeau, Matthieu; Mangin, Cyprien

doi:10.1145/3342526

Cited by 2 publications

(4 citation statements)

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“…They are proven to exist and enjoy a strong definitional property in the metatheoretical proofs only. While we were able to reproduce this result [Sozeau and Mangin 2019a, file theories/telescopes.v], any change to the core calculus implies a requirement of trust towards its implementation, whose burden we avoid in the case of Eqations by providing a definitional translation.…”

Section: A Short History Of Dependent Pattern-matchingmentioning

confidence: 88%

“…In a system with sized types, a direct call to regular map would be allowed here, but requires the term datatype and substitution function to be indexed by sizes (Sozeau and Mangin [2019a, file agda/nested sized.agda] formalizes this). With size annotations the l argument of App has type list (term j k) while the initial term t has type term i k, with size relation j < i.…”

Section: Well-founded Recursion On Nested Typesmentioning

confidence: 99%

“…Lean handles nested and mutual inductive types by rewriting inductive definitions using an isomorphism with regular inductive definitions, resulting in back and forth translations. The translation appears to be partial: it cannot handle the definition of term from section 2.4 and requires to write a mutual type of term lists with its own map function [Sozeau and Mangin 2019a, file lean/nested.lean] (tested with Lean 3.4.2). We have not been able to handle the case of the abstraction constructor either in that case, Lean fails to check termination of the lifting and substitution functions.…”

Section: Elimination Principlementioning

confidence: 99%

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Equations reloaded: high-level dependently-typed functional programming and proving in Coq

Sozeau

Mangin

2019

Proc. ACM Program. Lang.

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Eqations 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 high-level proof principles for demonstrating properties about them, abstracting away from the implementation details of the function and its compiled form. 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 based on an original handling of the no-confusion property of constructors.

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Section: A Short History Of Dependent Pattern-matchingmentioning

confidence: 88%

Section: Well-founded Recursion On Nested Typesmentioning

confidence: 99%

Section: Elimination Principlementioning

confidence: 99%

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Equations reloaded: high-level dependently-typed functional programming and proving in Coq

Sozeau

Mangin

2019

Proc. ACM Program. Lang.

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“…We can now write a function qperm that computes permutations nondeterministically. For this purpose, we use the C Equations command (Sozeau, 2009;Sozeau and Mangin, 2019) that provides support to prove the termination of functions whose recursion is not structural. The annotation by wf (size s) lt indicates that the relation between the sizes of lists is well-founded.…”

Section: Nondeterministic Computation Of Permutationsmentioning

confidence: 99%

A Library for Formalization of Linear Error-Correcting Codes

2020

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One can perform equational reasoning about computational effects with a purely functional programming language thanks to monads. Even though equational reasoning for effectful programs is desirable, it is not yet mainstream. This is partly because it is difficult to maintain pencil-and-paper proofs of large examples. We propose a formalization of a hierarchy of effects using monads in the Coq proof assistant that makes monadic equational reasoning practical. Our main idea is to formalize the hierarchy of effects and algebraic laws as interfaces like it is done when formalizing hierarchy of algebras in dependent type theory. Thanks to this approach, we clearly separate equational laws from models. We can then take advantage of the sophisticated rewriting capabilities of Coq and build libraries of lemmas to achieve concise proofs of programs. We can also use the resulting framework to leverage on Coq's mathematical theories and formalize models of monads. In this article, we explain how we formalize a rich hierarchy of effects (nondeterminism, state, probability, etc.), how we mechanize examples of monadic equational reasoning from the literature, and how we apply our framework to the design of equational laws for a subset of ML with references.

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Equations Reloaded Accompanying Material

Cited by 2 publications

References 0 publications

Equations reloaded: high-level dependently-typed functional programming and proving in Coq

Equations reloaded: high-level dependently-typed functional programming and proving in Coq

A Library for Formalization of Linear Error-Correcting Codes

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