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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,(More)
This paper reports on a six-year collaborative effort that culminated in a complete formalization of a proof of the Feit-Thompson Odd Order Theorem in the Coq proof assistant. The formalized proof is constructive , and relies on nothing but the axioms and rules of the founda-tional framework implemented by Coq. To support the formalization, we developed a(More)
This paper describes a formalization of discrete real closed fields in the Coq proof assistant. This abstract structure captures for instance the theory of real algebraic numbers, a decidable subset of real numbers with good algorithmic properties. The theory of real algebraic numbers and more generally of semi-algebraic varieties is at the core of a number(More)
Merci beaucoup à Lydie puis à Valérie ainsi qu'à Martine pour avoir toujours su m'accueillir avec le sourire, malgré tout ce que j'ai eu à vous demander, et pour avoir réglé tous mes problèmes avec une efficacité remarquable. Je n'en serais pas là sans ceux dont les enseignements ont marqué ma vie. Merci à Nicolas Tosel, pour m'avoir initié à l'algèbre.(More)
Human T lymphocyte virus type I (HTLV-I)-associated chronic inflammatory neurological disease (HTLV-I-associated myelopathy/tropical spastic paraparesis [HAM/TSP]) is suggested to be an immunopathologically mediated disorder characterized by large numbers of HTLV-I Tax-specific CD8+ T cells. The frequency of these cells in the peripheral blood and(More)
This paper shows a construction in Coq of the set of real algebraic numbers, together with a formal proof that this set has a structure of discrete Archimedean real closed field. This construction hence implements an interface of real closed field. Instances of such an interface immediately enjoy quantifier elimination thanks to a previous work. This work(More)
We prove formally that the first order theory of algebraically closed fields enjoys 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(More)
In 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(More)