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.

Contents 1. Real Numbers 3 2 3 1.2. Reals, Equality of Reals 5 1.3. The Archimedian Axiom 6 1.4. Nonnegative and Positive Reals 6 1.5. Arithmetical Functions 7 1.6. Comparison of Reals 8 1.7. Non-Countability 10 1.8. Cleaning of Reals 11 2. Sequences and Series of Real Numbers 11 2.1. Completeness 11 2.2. Limits and Inequalities 13 2.3. Series 13 2.4. Redundant Dyadic Representation of Reals 14 2.5. Convergence Tests 15 2.6. Reordering Theorem 17 2.7. The Exponential Series 18 3. The Exponential Function for Complex Numbers 21 4. Continuous Functions 23 4.1. Suprema and In ma 24 4.2. Continuous Functions 25 4.3. Application of a Continuous Function to a Real 27 4.4. Continuous Functions and Limits 28 4.5. Composition of Continuous Functions 28 4.6. Properties of Continuous Functions 29 4.7. Intermediate Value Theorem 30 4.8. Continuity of Functions with More Than One Variable 32 5. Dierentiation 33 5.1. Derivatives 33 5.2. Bounds on the Slope 33 5.3. Properties of Derivatives 34 5

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. 1