A combination of a dynamic geometry software with a proof assistant for interactive formal proofs

International audienceThis paper presents an interface for geometry proving. It is a combination of a dynamic geometry software - Geogebra[11] with a proof assistant - Coq[8]. Thanks to the features of Geogebra, users can create and manipulate geometric constructions, they discover conjectures and interactively build formal proofs with the support of Coq. Our system allows users to construct fully traditional proofs in the same style as the ones in high school. For each step of proving, we provide a set of applicable rules veri ed in Coq for users, we also provide tactics in Coq by which minor steps of reasoning are solved automatically

A Coq-based Library for Interactive and Automated Theorem Proving in Plane Geometry

International audienceIn 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. 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 proving at the level of high-school geometry, where we eliminate redundant axioms and give formalizations for the geometric concepts using a vector approach. We also enrich this library by offering an automated deduction method which can be used as a complement to interactive proof. For that purpose, we integrate the formalization of the area method which was developed by J. Narboux in Coq

Formal systems for proving incidence results

U ovoj tezi razvijen je formalni sistem za dokazivanje teorema incidencije u projektivnoj geometiji. Osnova sistema je Čeva/Menelaj metod za dokazivanje teorema incidencije. Formalizacija o kojoj je ovdje riječ izvedena je korišćenjem Δ-kompleksa, pa su tako u disertaciji spojene oblasti logike, geometrije i algebarske topologije. Aksiomatski sekventi proizilaze iz 2-ciklova Δ-kompleksa. Definisana je Euklidska i projektivna interpretacija sekvenata i dokazana je saglasnost i odlučivost sistema. Dati su primjeri iščitavanja teorema incidencije iz dokazivih sekvenata sistema. U tezi je data i procedura za provjeru da li je skup od n šestorki tačaka aksiomatski sekvent.In this thesis, a formal sequent system for proving incidence theorems in projective geometry is introduced. This system is based on the Ceva/Menelaus method for proving theorems. This formalization is performed using Δ-complexes, so the areas of logic, geometry and algebraic topology are combined in the dissertation. The axiomatic sequents of the system stem from 2-cycles of Δ-complexes. The Euclidean and projective interpretations of the sequents are defined and the decidability and soundness of the system are proved. Patterns for extracting formulation and proof of the incidence result from derivable sequents of system are exemplified. The procedure for deciding if set of n sextuples represent an axiomatic sequent is presented within the thesis

Exploring grade 11 learners’ functional understanding of proof in relation to argumentation in selected high schools.

Doctoral Degrees. University of KwaZulu-Natal, Durban.Research has established that understanding the functions of proof in mathematics and argumentation ability provide learners with a firm foundation for constructing proofs. Yet, little is known about the extent to which learners appreciate the functions of proof and whether an association between functional understanding of proof and argumentation ability exists. Guided by van Hiele’s and Toulmin’s theories, this study utilised a sequential explanatory design to randomly select three schools from a cluster grouping of ten Dinaledi high schools in the Pinetown district. Three survey questionnaires, Learners’ Functional Understanding of Proof (LFUP), self-efficacy scale, and Argumentation Framework for Euclidean Geometry (AFEG), were administered to a sample of 135 Grade 11 learners to measure their understanding of the functions of proof and argumentation ability, and to explore the relationship between argumentation ability and functional understanding of proof. Then, Presh N (pseudonym)—a female learner who obtained the highest LFUP score despite attending a historically under-resourced township school—was purposively selected from the larger sample. In addition to her responses on the questionnaires, a semistructured interview, and a standard proof-related task served as data sources to explain the origins of her functional understanding of proof. Statistical analyses were conducted on data obtained from questionnaires while pattern matching method was used to analyse the interview data. The analyses revealed that learners held hybrid functional understanding of proof, the quality of their argumentation was poor, the relationship between functional understanding of proof and argumentation ability was weak and statistically significant, and the collectivist culture and the teacher were the two factors which largely accounted for Presh N’s informed beliefs about the functions of proof. In addition, although she constructed a deductive proof, she did not perform the inductive segment prior to formally proving the proposition. The recommendation that Euclidean geometry curriculum needs to be revamped for the purpose of making functional understanding of proof and argumentation explicit and assessable content has implications for two constituencies. Instructional practices in high schools and methods modules at higher education institutions need to include these exploratory activities (functional understanding of proof and argumentation) prior to engaging in the final step of formal proof construction. Future research initiatives need to blend close-ended items with open-ended questions to enhance insights into learners’ functional understanding of proof. This study not only provides high school teachers and researchers with a single, reliable tool to assess functional understanding of proof but also proposes a model for studying factors affecting functional understanding of proof. Overall, the results of this study are offered as a contribution to the field’s growing understanding of learners’ activities prior to constructing proofs

Ordered geometry in Hilbert’s Grundlagen der Geometrie

The Grundlagen der Geometrie brought Euclid’s ancient axioms up to the standards of modern logic, anticipating a completely mechanical verification of their theorems. There are five groups of axioms, each focused on a logical feature of Euclidean geometry. The first two groups give us ordered geometry, a highly limited setting where there is no talk of measure or angle. From these, we mechanically verify the Polygonal Jordan Curve Theorem, a result of much generality given the setting, and subtle enough to warrant a full verification. Along the way, we describe and implement a general-purpose algebraic language for proof search, which we use to automate arguments from the first axiom group. We then follow Hilbert through the preliminary definitions and theorems that lead up to his statement of the Polygonal Jordan Curve Theorem. These, once formalised and verified, give us a final piece of automation. Suitably armed, we can then tackle the main theorem

Intuition in formal proof : a novel framework for combining mathematical tools

This doctoral thesis addresses one major difficulty in formal proof: removing obstructions to intuition which hamper the proof endeavour. We investigate this in the context of formally verifying geometric algorithms using the theorem prover Isabelle, by first proving the Graham’s Scan algorithm for finding convex hulls, then using the challenges we encountered as motivations for the design of a general, modular framework for combining mathematical tools. We introduce our integration framework — the Prover’s Palette, describing in detail the guiding principles from software engineering and the key differentiator of our approach — emphasising the role of the user. Two integrations are described, using the framework to extend Eclipse Proof General so that the computer algebra systems QEPCAD and Maple are directly available in an Isabelle proof context, capable of running either fully automated or with user customisation. The versatility of the approach is illustrated by showing a variety of ways that these tools can be used to streamline the theorem proving process, enriching the user’s intuition rather than disrupting it. The usefulness of our approach is then demonstrated through the formal verification of an algorithm for computing Delaunay triangulations in the Prover’s Palette

A decision procedure for geometry in coq

We present in this paper the development of a decision procedure for affine plane geometry in the Coq proof assistant. Among the existing decision methods, we have chosen to implement one based on the area method developed by Chou, Gao and Zhang, which provides short and “readable ” proofs for geometry theorems. The idea of the method is to express the goal to be proved using three geometric quantities and eliminate points in the reverse order of their construction thanks to some elimination lemmas