Formalisation en Coq et visualisation d'un cours de géométrie pour le lycée

Abstract

Enseigner les mathématiques aux élèves de lycée en utilisant la démonstration assistée par ordinateur devient un objectif accessible dans un futur proche. Nous présentons dans cet article une formalisation en Coq de la géométrie enseignée au lycée ayant la particularité de présenter des définitions, théorèmes et preuves très proches de ceux donnés dans les cours de mathématiques. Pour rendre les textes formels accessibles aux élèves, nous utilisons une interface graphique et un outil de construction automatique de figure dynamique. Nous présentons dans cet article plusieurs exemples d'énoncés, démonstrations et figures correspondantes puis abordons les difficultés rencontrées dans ce travail.

A Decision Procedure for Geometry in Coq

“…Gilles Kahn has formalized Jan von Plato's constructive geometry in the Coq system [12,17]. Frédérique Guilhot has done a large development in Coq dealing with French high school geometry [7]. We believe that automated theorem proving and interactive proof development are complementary to formal proof generation.…”

Formalizing Projective Plane Geometry in Coq

“…The proofs can be made more rigorous by machine assistance. Frédérique Guilhot realized a large Coq development about Euclidean geometry following a presentation suitable for use in french high-school [13]. In [24,25], Julien Narboux presented the formalization and implementation in the Coq proof assistant of the area decision procedure of Chou, Gao and Zhang [5] and a formalization of foundations of Euclidean geometry based on Tarski's axiom system [33,30].…”

Mechanical Theorem Proving in Tarski’s Geometry

“…The formal proof of the theorems in the degenerated cases is often tedious and even sometimes difficult. These cases often do not even appear in the informal proof 10 . In order to limit the size of the proofs, we tried to automate some tasks.…”

“…Frédérique Guilhot has realized a large Coq development about euclidean geometry following a presentation suitable for use in french high-school [10]. Concerning the proof of programs in the field of computational geometry we can cite the formalization of convex hulls algorithms by David Pichardie and Yves Bertot in Coq [11] and by Laura Meikle and Jacques Fleuriot in Isabelle [12] and the formalization of an image segmentation algorithm by Jean-François Dufourd [13].…”

“…9 The theorem 4.17 states that A = B ∧ ColABC ∧ AP ≡ AQ ∧ BP ≡ BQ ⇒ CP ≡ CQ. 10 It seems that degenerated cases play the same role in geometry as α-conversion in lambda calculus: they are a great source of difficulties in the context of a mechanization.…”