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- Georges Gonthier, Andrea Asperti, Jeremy Avigad, Yves Bertot, Cyril Cohen, François Garillot +9 others
- ITP
- 2013

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)

A constructive proof of the Gödel-Rosser incompleteness theorem [9] has been completed using the Coq proof assistant. Some theory of classical first-order logic over an arbitrary language is formalized. A development of primitive recursive functions is given, and all primitive recursive functions are proved to be representable in a weak axiom system.… (More)

Reasoning about real number expressions in a proof assistant is challenging. Several problems in theorem proving can be solved by using exact real number computation. I have implemented a library for reasoning and computing with complete metric spaces in the Coq proof assistant and used this library to build a constructive real number implementation… (More)

Large scale real number computation is an essential ingredient in several modern mathematical proofs. Because such lengthy computations cannot be verified by hand, some mathematicians want to use software proof assistants to verify the correctness of these proofs. This paper develops a new implementation of the constructive real numbers and elementary… (More)

The following full text is a preprint version which may differ from the publisher's version. Abstract We provide a computer verified exact monadic functional implementation of the Riemann integral in type theory. Together with previous work by O'Connor, this may be seen as the beginning of the realization of Bishop's vision to use constructive mathematics… (More)

This paper gives two new categorical characterisations of lenses: one as a coalgebra of the store comonad, and the other as a monoidal natural transformation on a category of a certain class of coalgebras. The store comonad of the first characterisation can be generalized to a Cartesian store comonad, and the coalgebras of this Cartesian store comonad turn… (More)

There are two incompatible Coq libraries that have a theory of the real numbers; the Coq standard library gives an axiomatic treatment of classical real numbers, while the CoRN library from Nijmegen defines constructively valid real numbers. Unfortunately, this means results about one structure cannot easily be used in the other structure. We present a way… (More)

We present some of the experiments we have performed to best test our design for a library for MathScheme, the mechanized mathematics software system we are building. We wish for our library design to use and reflect, as much as possible, the mathematical structure present in the objects which populate the library.

The mission of mechanized mathematics is to develop software systems that support the process people use to create, explore, connect, and apply mathematics. Working mathematicians routinely leverage a powerful synergy between deduction and computation. The artificial division between (axiomatic) theorem proving systems and (algorithmic) computer algebra… (More)

Compact sets in constructive mathematics capture our intuition of what computable subsets of the plane (or any other complete metric space) ought to be. A good representation of compact sets provides an efficient means of creating and displaying images with a computer. In this paper, I build upon existing work about complete metric spaces to define compact… (More)