Extensive games are tools largely used in economics to describe decision processes of a community of agents. In this paper we propose a formal presentation based on the proof assistant COQ which focuses mostly on infinite extensive games and their characteristics. COQ proposes a feature called types'', which means that the type of an object may depend on the type of its components. For instance, the set of choices or the set of utilities of an agent may depend on the agent herself. Using dependent types, we describe formally a very general class of games and strategy profiles, which corresponds somewhat to what game theorists are used to. We also discuss the notions of infiniteness in game theory and how this can be precisely described.

This paper presents a formally verified decision procedure for determinining the satisfiability of a system of univariate polynomial relations over the real line. The procedure combines a root counting function, based on Sturm's theorem, with an interval subdivision algorithm. Given a system of polynomial relations over the same variable, the decision procedure progressively subdivides the real interval into smaller intervals. The subdivision continues until the satisfiability of the system can be determined on each subinterval using Sturm's theorem on a subset of the system's polynomials. The decision procedure has been formally verified in the Prototype Verification System (PVS). In PVS, the decision procedure is specified as a computable boolean function on a deep embedding of polynomial relations. This function is used to define a proof producing strategy for automatically proving existential and universal statements on polynomial systems. The soundness of the strategy solely depends on the internal logic of PVS.

The main goal of this thesis is to define an extension of Godel not-not translation to all truncated types, in the setting of homotopy type theory. This goal will use some existing theories, like Lawvere-Tierney sheaves theory in toposes, we will adapt in the setting of homotopy type theory. In particular, we will define a Lawvere-Tierney sheafification functor, which is the main theorem presented in this thesis.To define it, we will need some concepts, either already defined in type theory, either not existing yet. In particular, we will define a theory of colimits over graphs as well as their truncated version, and the notion of truncated modalities, based on the existing definition of modalities.Almost all the result presented in this thesis are formalized with the proof assistant Coq together with the library [HoTT/Coq]

We discuss the formalization, in the Matita Interactive Theorem Prover, of some results by Chebyshev concerning the distribution of prime numbers, subsuming, as a corollary, Bertrand's postulate. Even if Chebyshev's result has been later superseded by the stronger prime number theorem, his machinery, and in particular the two functions psi and theta still play a central role in the modern development of number theory. The proof makes use of most part of the machinery of elementary arithmetics, and in particular of properties of prime numbers, gcd, products and summations, providing a natural benchmark for assessing the actual development of the arithmetical knowledge base.

In this thesis we give an algebraic characterization of the syntax and semantics of simply–typed languages. More precisely, we characterize simply–typed binding syntax equipped with reduction rules via a universal property, namely as the initial object of some category. We specify a language by a 2–signature (Σ, A), that is, a signature on two levels: the syntactic level Σ specifies the sorts and terms of the language, and associates a sort to each term. The semantic level A specifies, through inequations, reduction rules on the terms of the language. To any given 2–signature (Σ, A) we associate a category of “models” of (Σ, A). We prove that this category has an initial object, which integrates the terms freely generated by Σ and the reduction relation — on those terms — generated by A. We call this object the programming language generated by (Σ, A). Initiality provides an iteration principle which allows to specify translations on the syntax, possibly to a language over different sorts. Furthermore, translations specified via the iteration principle are by construction type–safe and faithful with respect to reduction. To illustrate our results, we consider two examples extensively: firstly, we specify a double negation translation from classical to intuitionistic propositional logic via the category–theoretic iteration principle. Secondly, we specify a translation from PCF to the untyped lambda calculus which is faithful with respect to reduction in the source and target languages. In a second part, we formalize some of our initiality theorems in the proof assistant Coq. The implementation yields a machinery which, when given a 2–signature, returns an implementation of its associated abstract syntax together with certified substitution operation, iteration operator and a reduction relation generated by the specified reduction rules.

Relative monads are a generalisation of ordinary monads where the underlying functor need not be an endofunctor. In this paper, we describe a formalisation of the basic theory of relative monads in the interactive theorem prover and dependently typed programming language Agda. The formalisation comprises the requisite basic category theory, the central concepts of the theory of relative monads and adjunctions, which are compared to their ordinary counterparts, and two running examples from programming theory.

This tutorial presents the Ssreflect extension to the Coq system. This extension consists of an extension to the Coq language of script, and of a set of libraries, originating from the formal proof of the Four Color theorem. This tutorial proposes a guided tour in some of the basic libraries distributed in the Ssreflect package. It focuses on the application of the small scale reflection methodology to the formalization of finite objects in intuitionistic type theory.

Rewriting is an essential tool for computer-based reasoning, both automated and assisted. This is because rewriting is a general notion that permits modeling a wide range of problems and provides a means to effectively solve them. In a proof assistant, rewriting can be used to replace terms in arbitrary contexts, generalizing the usual equational reasoning to reasoning modulo arbitrary relations. This can be done provided the necessary proofs that functions appearing in goals are congruent with respect to specific relations. We present a new implementation of generalized rewriting in the Coq proof assistant, making essential use of the expressive power of dependent types and the recently implemented type class mechanism. The new rewrite tactic improves on and generalizes previous versions by natively supporting higher-order functions, polymorphism and subrelations. The type class system inspired by Haskell provides a perfect interface between the user and the tactic, making it easily extensible.

We describe the formalization using the HOL Light theorem prover of Dirichlet's theorem on primes in arithmetic progression. The proof turned out to be more straightforward than expected, but this depended on a careful choice of an informal proof to use as a starting-point. The goal of this paper iis twofold. First we describe a simple and efficient proof of the theorem informally, which iis otherwise difficult to find in one self-contained place at an elementary level. We also describe its, largely routine, HOL Light formalization, a task that took only a few days.

In this article we present the implementation and formal verification, using the Coq system, of a generalized version of the crossover operator applied to genetic algorithms (GA). The first part of this work defines the multiple crossover of two lists in any finite number of points, generalizing a previous definition by Uchibori. In the second part, we show that the definition does not depend on the order of the list of points. Then, a more efficient definition of the crossover operation is provided, and formally proved to be equivalent to the previous one, exploiting the notion of difference list.