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Formalizing Projective Plane Geometry in Coq
, 2008
"... We investigate how projective plane geometry can be formalized in a proof assistant such as Coq. Such a formalization increases the reliability of textbook proofs whose details and particular cases are often overlooked and left to the reader as exercises. Projective plane geometry is described thro ..."
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We investigate how projective plane geometry can be formalized in a proof assistant such as Coq. Such a formalization increases the reliability of textbook proofs whose details and particular cases are often overlooked and left to the reader as exercises. Projective plane geometry is described through two different axiom systems which are formally proved equivalent. Usual properties such as decidability of equality of points (and lines) are then proved in a constructive way. The duality principle as well as formal models of projective plane geometry are then studied and implemented in Coq. Finally, we formally prove in Coq that Desargues’ property is independent of the axioms of projective plane geometry.
Formalization and Implementation of Algebraic Methods in Geometry
, 2012
"... We describe our ongoing project of formalization of algebraic methods for geometry theorem proving (Wu’s method and the Gröbner bases method), their implementation and integration in educational tools. The project includes formal verification of the algebraic methods within Isabelle/HOL proof assis ..."
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We describe our ongoing project of formalization of algebraic methods for geometry theorem proving (Wu’s method and the Gröbner bases method), their implementation and integration in educational tools. The project includes formal verification of the algebraic methods within Isabelle/HOL proof assistant and development of a new, opensource Java implementation of the algebraic methods. The project should fillin some gaps still existing in this area (e.g., the lack of formal links between algebraic methods and synthetic geometry and the lack of selfcontained implementations of algebraic methods suitable for integration with dynamic geometry tools) and should enable new applications of theorem proving in education.
THedu'11 EPTCS ??
, 2012
"... We describe our ongoing project of formalization of algebraic methods for geometry theorem proving (Wu's method and the Gröbner bases method), their implementation and integration in educational tools. The project includes formal verification of the algebraic methods within Isabelle/HOL proof ..."
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We describe our ongoing project of formalization of algebraic methods for geometry theorem proving (Wu's method and the Gröbner bases method), their implementation and integration in educational tools. The project includes formal verification of the algebraic methods within Isabelle/HOL proof assistant and development of a new, opensource Java implementation of the algebraic methods. The project should fillin some gaps still existing in this area (e.g., the lack of formal links between algebraic methods and synthetic geometry and the lack of selfcontained implementations of algebraic methods suitable for integration with dynamic geometry tools) and should enable new applications of theorem proving in education.
Proof and Computation in Geometry
"... We consider the relationships between algebra, geometry, computation, and proof. Computers have been used to verify geometrical facts by reducing them to algebraic computations. But this does not produce computercheckable firstorder proofs in geometry. We might try to produce such proofs directl ..."
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We consider the relationships between algebra, geometry, computation, and proof. Computers have been used to verify geometrical facts by reducing them to algebraic computations. But this does not produce computercheckable firstorder proofs in geometry. We might try to produce such proofs directly, or we might try to develop a “backtranslation” from algebra to geometry, following Descartes but with computer in hand. This paper discusses the relations between the two approaches, the attempts that have been made, and the obstacles remaining. On the theoretical side we give a new firstorder theory of “vector geometry”, suitable for formalizing geometry and algebra and the relations between them. On the practical side we report on some experiments in automated deduction in these areas.
A Case Study in Formalizing Projective Geometry in Coq: Desargues Theorem
, 2012
"... Formalizing geometry theorems in a proof assistant like Coq is challenging. As emphasized in the literature, the nondegeneracy conditions lead to long technical proofs. In addition, when considering higherdimensions, the amount of incidence relations (e.g. pointline, pointplane, lineplane) indu ..."
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Formalizing geometry theorems in a proof assistant like Coq is challenging. As emphasized in the literature, the nondegeneracy conditions lead to long technical proofs. In addition, when considering higherdimensions, the amount of incidence relations (e.g. pointline, pointplane, lineplane) induce numerous technical lemmas. In this article, we investigate formalizing projective plane geometry as well as projective space geometry. We mainly focus on one of the fundamental properties of the projective space, namely Desargues property. We formally prove that it is independent of projective plane geometry axioms but can be derived from Pappus property in a twodimensional setting. Regarding at least three dimensional projective geometry, we present an original approach based on the notion of rank which allows to describe incidence and nonincidence relations such as equality, collinearity and coplanarity homogeneously. This approach allows to carry out proofs in a more systematic way and was successfully used to fairly easily formalize Desargues theorem in Coq. This illustrates the power and efficiency of our approach (using only ranks) to prove properties of the projective space.
SCIENCES ET TECHNOLOGIES DE L’INFORMATION ET DE LA COMMUNICATION
, 2015
"... HAL is a multidisciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte p ..."
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HAL is a multidisciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et a ̀ la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.
From Tarski to Hilbert
"... Abstract. In this paper, we report on the formal proof that Hilbert’s axiom system can be derived from Tarski’s system. For this purpose we mechanized the proofs of the first twelve chapters of Schwabäuser, Szmielew and Tarski’s book: Metamathematische Methoden in der Geometrie. The proofs are che ..."
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Abstract. In this paper, we report on the formal proof that Hilbert’s axiom system can be derived from Tarski’s system. For this purpose we mechanized the proofs of the first twelve chapters of Schwabäuser, Szmielew and Tarski’s book: Metamathematische Methoden in der Geometrie. The proofs are checked formally within classical logic using the Coq proof assistant. The goal of this development is to provide clear foundations for other formalizations of geometry and implementations of decision procedures. 1