We present C-CoRN, the Constructive Coq Repository at Nijmegen. It consists of a library of constructive algebra and analysis, formalized in the theorem prover Coq. In this paper we explain the structure, the contents and the use of the library. Moreover we discuss the motivation and the (possible) applications of such a library.

It is a well-known fact that algorithms are often hidden inside mathematical proofs. If these proofs are formalized inside a proof assistant, then a mechanism called extraction can generate the corresponding programs automatically. Previous work has focused on the difficulties in obtaining a program from a formalization of the Fundamental Theorem of Algebra inside the Coq proof assistant. In theory, this program allows one to compute approximations of roots of polynomials. However, as we show in this work, there is currently a big gap between theory and practice. We study the complexity of the extracted program and analyze the reasons of its inefficiency, showing that this is a direct consequence of the approach used throughout the formalization.

Abstract. We provide a monadic view on implicit and noncomputational arguments. This allows us to treat Berger’s non-computational quantifiers in the Coq-system. We use Tait’s normalization proof and the concatenation of vectors as case studies for the extraction of programs. With little effort one can eliminate noncomputational arguments from extracted programs. One thus obtains extracted code that is not only closer to the intended one, but also decreases both the running time and the memory usage dramatically. We also study the connection between Harrop formulas, lax modal logic and the Coq type theory.

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 as a programming language for exact analysis. 1 1.