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Section: New Results

Program Certifications and Formalisation of Mathematics

Participants : Danil Annenkov, Assia Mahboubi, Étienne Miquey.

Certified Compilation of Financial Contracts.

In [11], we present an extension to a certified financial contract management system that allows for templated declarative financial contracts and for integration with financial stochastic models through verified compilation into so-called payoff-expressions. Such expressions readily allow for determining the value of a contract in a given evaluation context, such as contexts created for stochastic simulations. The templating mechanism is useful both at the contract specification level, for writing generic reusable contracts, and for reuse of code that, without the templating mechanism, needs to be recompiled for different evaluation contexts. We report on the effect of using the certified system in the context of a GPGPU-based Monte Carlo simulation engine for pricing various over-the-counter (OTC) financial contracts. The full contract-management system, including the payoff-language compilation, is verified in the Coq proof assistant and certified Haskell code is extracted from our Coq development along with Futhark code for use in a data-parallel pricing engine.

Static interpretation of higher-order modules in Futhark: functional GPU programming in the large.

In [12], we present a higher-order module system for the purely functional data-parallel array language Futhark. The module language has the property that it is completely eliminated at compile time, yet it serves as a powerful tool for organising libraries and complete programs. The presentation includes a static and a dynamic semantics for the language in terms of, respectively, a static type system and a provably terminating elaboration of terms into terms of an underlying target language. The development is formalised in Coq using a novel encoding of semantic objects based on products, sets, and finite maps. The module language features a unified treatment of module type abstraction and core language polymorphism and is rich enough for expressing practical forms of module composition.

Formalising Implicative Algebras in Coq.

In [15], we present a Coq formalisation of Alexandre Miquel’s implicative algebras, which aim at providing a general algebraic framework for the study of classical realisability models. We first give a self-contained presentation of the underlying implicative structures, which roughly consists of a complete lattice equipped with a binary law representing the implication. We then explain how these structures can beturned into models by adding separators, giving rise to the so-called implicative algebras. Additionally, we show how they generalise Boolean and Heyting algebras as well as the usual algebraic structures used in the analysis of classical realisability.

Formally Verified Approximations of Definite Integrals.

Finding an elementary form for an antiderivative is often a difficult task, so numerical integration has become a common tool when it comes to making sense of a definite integral. Some of the numerical integration methods can even be made rigorous: not only do they compute an approximation of the integral value but they also bound its inaccuracy. Yet numerical integration is still missing from the toolbox when performing formal proofs in analysis. In [5], we present an efficient method for automatically computing and proving bounds on some definite integrals inside the Coq formal system. Our approach is not based on traditional quadrature methods such as Newton-Cotes formulas. Instead, it relies on computing and evaluating antiderivatives of rigorous polynomial approximations, combined with an adaptive domain splitting. Our approach also handles improper integrals, provided that a factor of the integrand belongs to a catalog of identified integrable functions. This work has been integrated to the CoqInterval library.