A formal quantifier elimination for algebraically closed fields

The final publication is available at www.springerlink.comInternational audienceWe prove formally that the first order theory of algebraically closed fields enjoy quantifier elimination, and hence is decidable. This proof is organized in two modular parts. We first reify the first order theory of rings and prove that quantifier elimination leads to decidability. Then we implement an algorithm which constructs a quantifier free formula from any first order formula in the theory of ring. If the underlying ring is in fact an algebraically closed field, we prove that the two formulas have the same semantic. The algorithm producing the quantifier free formula is programmed in continuation passing style, which leads to both a concise program and an elegant proof of semantic correctness

An introduction to small scale reflection in Coq

International audienceThis 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

A Generic Formalised Framework for Reasoning About Weak Memory Models

This paper describes Coq libraries devoted to the semantic of relaxed memory models. These libraries formalise a framework which covers a large class of industrial models. Implementing this framework inside a proof assistant has significantly helped improving its design and crafting the most concise and relevant specifications. Similarly the use of a proof assistant has been instrumental in the study of the semantic of synchronisation primitives, which we illustrate by the formal proof of a barrier placement theorem. We explain the choices we made to re-design our Coq libraries, and in particular what we gained from adopting a small-scale reflection methodology

Machine-checked mathematics

International audienceIn this article she gives an overview about machine-checked mathematics

An Induction Principle over Real Numbers

International audienceWe give a constructive proof of the open induction principle on real numbers, using bar induction and enumerative open sets. We comment the algorithmic content of this result

Simulating the DPLL(T ) procedure in a sequent calculus with focusing

This paper gives an abstract description of decision procedures for Satisfiability Modulo Theory (SMT) as proof search procedures in a sequent calculus with polarities and focusing. In particular, we show how to simulate the execution of standard techniques based on the Davis-Putnam- Logemann-Loveland (DPLL) procedure modulo theory as the gradual construction of a proof tree in sequent calculus. The construction mimicking a run of DPLL-modulo-Theory can be obtained by a meta-logical control on the proof-search in sequent calculus. This control is provided by polarities and focusing features, which there- fore narrow the corresponding search space in a sense we discuss. This simulation can also account for backjumping and learning steps, which correspond to the use of general cuts in sequent calculus

Trocq: Proof Transfer for Free, With or Without Univalence

Libraries of formalized mathematics use a possibly broad range of different representations for a same mathematical concept. Yet light to major manual input from users remains most often required for obtaining the corresponding variants of theorems, when such obvious replacements are typically left implicit on paper. This article presents Trocq, a new proof transfer framework for dependent type theory. Trocq is based on a novel formulation of type equivalence, used to generalize the univalent parametricity translation. This framework takes care of avoiding dependency on the axiom of univalence when possible, and may be used with more relations than just equivalences. We have implemented a corresponding plugin for the Coq proof assistant, in the CoqElpi meta-language. We use this plugin on a gallery of representative examples of proof transfer issues in interactive theorem proving, and illustrate how Trocq covers the spectrum of several existing tools, used in program verification as well as in formalized mathematics in the broad sense

A bisimulation between DPLL(T) and a proof-search strategy for the focused sequent calculus

International audienceWe describe how the Davis-Putnam-Logemann-Loveland proced- ure DPLL is bisimilar to the goal-directed proof-search mechanism described by a standard but carefully chosen sequent calculus. We thus relate a procedure described as a transition system on states to the gradual completion of incomplete proof-trees. For this we use a focused sequent calculus for polarised clas- sical logic, for which we allow analytic cuts. The focusing mech- anisms, together with an appropriate management of polarities, then allows the bisimulation to hold: The class of sequent calculus proofs that are the images of the DPLL runs finishing on UNSAT, is identified with a simple criterion involving polarities. We actually provide those results for a version DPLL(T ) of the procedure that is parameterised by a background theory T for which we can decide whether conjunctions of literals are con- sistent. This procedure is used for Satisfiability Modulo Theor- ies (SMT) generalising propositional SAT. For this, we extend the standard focused sequent calculus for propositional logic in the same way DPLL(T ) extends DPLL: with the ability to call the de- cision procedure for T . DPLL(T ) is implemented as a plugin for P SYCHE, a proof- search engine for this sequent calculus, to provide a sequent- calculus based SMT-solver