We formalize the deenition and the metatheory of the Calculus of Constructions (CC) using the proof assistant Coq. In particular, we prove strong nor-malization and decidability of type inference. From the latter proof, we extract a certiied Objective Caml program which performs type inference in CC and use this code to build a small-scale certiied proof-checker.

Proof assistants based on dependent type theory are progressively used as a tool to develop certified programs. A successful example is the Coq proof assistant, an implementation of a dependent type theory called the Calculus of Inductive Constructions (CIC). Coq is a functional programming language with an expressive type system that allows to specify and prove properties of programs in a high...

In this article, we present the development of a library of formal proofs for theorem proving in plane geometry in a pedagogical context. We use the Coq proof assistant [4]. This library includes the basic geometric notions to state theorems and provides a database of theorems to construct interactive proofs more easily. It is an extension of the library of F. Guilhot for interactive theorem pr...

We address here the problem of automatically translating the Natural Semantics of programming languages to Coq, in order to prove formally general properties of languages. Natural Semantics 18] is a formalism for specifying semantics of programming languages inspired by Plotkin's Structural Operational Semantics 22]. The Coq proof development system 12], based on the Calculus of Constructions e...

We describe a method for building composable and extensible verification procedures within the Coq proof assistant. Unlike traditional methods that rely on run-time generation and checking of proofs, we use verified-correct procedures with Coq soundness proofs. Though they are internalized in Coq’s logic, our provers support sound extension by users with hints over new domains, enabling automat...

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 interfacing these two libraries by sho...

The focal language (formerly Foc) allows a programmer to incrementally build mathematical structures and to formally prove their correctness. focal encourages a development process by refinement, deriving step-by-step implementations from specifications. This refinement process is realized using an inheritance mechanism on structures which can mix primitive operations, axioms, algorithms and pr...

Data representation is an important aspect of software composition. It is often the case that different software components are programmed to represent data in the ways which are the most appropriate for their problem domains. Sometimes, converting data from one representation to another is a non-trivial task. This is the case with computer algebra systems and type-theory based interactive theo...

Eqations is a plugin for the Coq proof assistant which provides a notation for defining programs by dependent pattern-matching and well-founded recursion and derives useful proof principles for demonstrating properties about them. We present here an improved design and rewriting of its implementation that avoids the use of axioms and reliance on Ltac programming, providing a feature-rich functi...