Formalizing real analysis for polynomials

When reasoning formally with polynomials over real numbers, or more generally real closed fields, we need to be able to manipulate easily statements featuring an order relation, either in their conditions or in their conclusion. For instance, we need to state the intermediate value theorem and the mean value theorem and we need tools to ease both their proof and their further use. For that purpose we propose a Coq library for ordered integral domains and ordered fields with decidable comparison. In this paper we present the design choices of this libraries, and show how it has been used as a basis for developing a fare amount of basic real algebraic geometry

Construction des nombres algébriques réels en Coq

National audienceCet article présente une construction en Coq de l'ensemble des nombres algébriques réels, ainsi qu'une preuve formelle que cet ensemble est muni d'une structure de corps réel clos discret archimédien. Cette construction vient ainsi implémenter une interface de corps réel clos réalisée dans un travail antérieur et bénéficie alors de la propriété d'élimination des quantificateurs, formellement prouvée pour toute instance de l'interface. Ce travail est destiné à servir de fondement à une construction de l'ensemble des nombres algébriques complexes, ainsi que d'implémentation de référence pour la certification des nombreux algorithmes de calcul formel qui utilisent des nombres algébriques

Formalized linear algebra over Elementary Divisor Rings in Coq

This paper presents a Coq formalization of linear algebra over elementary divisor rings, that is, rings where every matrix is equivalent to a matrix in Smith normal form. The main results are the formalization that these rings support essential operations of linear algebra, the classification theorem of finitely presented modules over such rings and the uniqueness of the Smith normal form up to multiplication by units. We present formally verified algorithms computing this normal form on a variety of coefficient structures including Euclidean domains and constructive principal ideal domains. We also study different ways to extend B\'ezout domains in order to be able to compute the Smith normal form of matrices. The extensions we consider are: adequacy (i.e. the existence of a gdco operation), Krull dimension $\leq 1$ and well-founded strict divisibility

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

Cubical Type Theory: A Constructive Interpretation of the Univalence Axiom

This paper presents a type theory in which it is possible to directly manipulate $n$-dimensional cubes (points, lines, squares, cubes, etc.) based on an interpretation of dependent type theory in a cubical set model. This enables new ways to reason about identity types, for instance, function extensionality is directly provable in the system. Further, Voevodsky\u27s univalence axiom is provable in this system. We also explain an extension with some higher inductive types like the circle and propositional truncation. Finally we provide semantics for this cubical type theory in a constructive meta-theory

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

Formal Proofs of Tarjan\u27s Strongly Connected Components Algorithm in Why3, Coq and Isabelle

Comparing provers on a formalization of the same problem is always a valuable exercise. In this paper, we present the formal proof of correctness of a non-trivial algorithm from graph theory that was carried out in three proof assistants: Why3, Coq, and Isabelle

Pragmatic Quotient Types in Coq

International audienceIn intensional type theory, it is not always possible to form the quotient of a type by an equivalence relation. However, quotients are extremely useful when formalizing mathematics, especially in algebra. We provide a Coq library with a pragmatic approach in two complementary components. First, we provide a framework to work with quotient types in an axiomatic manner. Second, we program construction mechanisms for some specific cases where it is possible to build a quotient type. This library was helpful in implementing the types of rational fractions, multivariate polynomials, field extensions and real algebraic numbers