6 research outputs found
Pascalâs Theorem in Real Projective Plane
SummaryIn this article we check, with the Mizar system [2], Pascalâs theorem in the real projective plane (in projective geometry Pascalâs theorem is also known as the Hexagrammum Mysticum Theorem)1. Pappusâ theorem is a special case of a degenerate conic of two lines. For proving Pascalâs theorem, we use the techniques developed in the section âProjective Proofs of Pappusâ Theoremâ in the chapter âPappusâ Theorem: Nine proofs and three variationsâ [11]. We also follow some ideas from Harrisonâs work. With HOL Light, he has the proof of Pascalâs theorem2. For a lemma, we use PROVER93 and OTT2MIZ by Josef Urban4 [12, 6, 7]. We note, that we donât use Skolem/Herbrand functions (see âSkolemizationâ in [1]).Rue de la Brasserie 5, 7100 La LouviĂšre, BelgiumJesse Alama. Escape to Mizar for ATPs. arXiv preprint arXiv:1204.6615, 2012.Grzegorz Bancerek, CzesĆaw ByliĆski, Adam Grabowski, Artur KorniĆowicz, Roman Matuszewski, Adam Naumowicz, Karol PÄ
k, and Josef Urban. Mizar: State-of-the-art and beyond. In Manfred Kerber, Jacques Carette, Cezary Kaliszyk, Florian Rabe, and Volker Sorge, editors, Intelligent Computer Mathematics, volume 9150 of Lecture Notes in Computer Science, pages 261â279. Springer International Publishing, 2015. ISBN 978-3-319-20614-1. doi: 10.1007/978-3-319-20615-817.Roland Coghetto. Homography in â â2. Formalized Mathematics, 24(4):239â251, 2016. doi: 10.1515/forma-2016-0020.Roland Coghetto. Group of homography in real projective plane. Formalized Mathematics, 25(1):55â62, 2017. doi: 10.1515/forma-2017-0005.Agata DarmochwaĆ. The Euclidean space. Formalized Mathematics, 2(4):599â603, 1991.Adam Grabowski. Solving two problems in general topology via types. In Types for Proofs and Programs, International Workshop, TYPES 2004, Jouy-en-Josas, France, December 15-18, 2004, Revised Selected Papers, pages 138â153, 2004. doi: 10.1007/116179909.Adam Grabowski. Mechanizing complemented lattices within Mizar system. Journal of Automated Reasoning, 55:211â221, 2015. doi: 10.1007/s10817-015-9333-5.Kanchun, Hiroshi Yamazaki, and Yatsuka Nakamura. Cross products and tripple vector products in 3-dimensional Euclidean space. Formalized Mathematics, 11(4):381â383, 2003.Wojciech LeoĆczuk and Krzysztof PraĆŒmowski. A construction of analytical projective space. Formalized Mathematics, 1(4):761â766, 1990.Wojciech LeoĆczuk and Krzysztof PraĆŒmowski. Projective spaces â part I. Formalized Mathematics, 1(4):767â776, 1990.JĂŒrgen Richter-Gebert. Papposâs Theorem: Nine Proofs and Three Variations, pages 3â31. Springer Berlin Heidelberg, 2011. ISBN 978-3-642-17286-1. doi: 10.1007/978-3-642-17286-11.Piotr Rudnicki and Josef Urban. Escape to ATP for Mizar. In First International Workshop on Proof eXchange for Theorem Proving-PxTP 2011, 2011.Wojciech Skaba. The collinearity structure. Formalized Mathematics, 1(4):657â659, 1990.Nobuyuki Tamura and Yatsuka Nakamura. Determinant and inverse of matrices of real elements. Formalized Mathematics, 15(3):127â136, 2007. doi: 10.2478/v10037-007-0014-7.25210711
Automated Unbounded Analysis of Cryptographic Constructions in the Generic Group Model
We develop a new method to automatically prove security statements in the Generic Group Model as they occur in actual papers. We start by defining (i) a general language to describe security definitions, (ii) a class of logical formulas that characterize how an adversary can win, and (iii) a translation from security definitions to such formulas. We prove a Master Theorem that relates the security of the construction to the existence of a solution for the associated logical formulas. Moreover, we define a constraint solving algorithm that proves the security of a construction by proving the absence of solutions.
We implement our approach in a fully automated tool, the tool, and use it to verify different examples from the literature. The results improve on the tool by Barthe et al. (CRYPTO\u2714, PKC\u2715): for many constructions, succeeds in proving standard (unbounded) security, whereas Barthe\u27s tool is only able to prove security for a small number of oracle queries
Boundary-Border Extensions of the Kuratowski Monoid
The Kuratowski monoid is generated under operator composition by
closure and complement in a nonempty topological space. It satisfies
. The Gaida-Eremenko (or GE) monoid
extends by adding the boundary operator. It satisfies
. We show that when the GE monoid
is determined by . When if the interior of the
boundary of every subset is clopen, then . This defines a new
type of topological space we call . Otherwise
. When applied to an arbitrary subset the GE monoid collapses
in one of possible ways. We investigate how these collapses and
interdepend, settling two questions raised by Gardner and
Jackson. Computer experimentation played a key role in our research.Comment: 48 pages, 9 figure
Verified programming with explicit coercions
Type systems have proved to be a powerful means of specifying and proving
important program invariants. In dependently typed programming languages
types can depend on values and hence express arbitrarily complicated
propositions and their machine checkable proofs. The type-based approach
to program specification allows for the programmer to not only transcribe
their intentions, but arranges for their direct involvement in the proving
process, thus aiding the machine in its attempt to satisfy difficult obligations.
In this thesis we develop a series of patterns for programming in a correct-by-construction style making use of constraints and coercions to prove
properties within a dependently typed host. This allows for the development
of a verified, kernel which can be built upon using the host system features.
In particular this should allow for the development of âtacticsâ or semiautomated
solvers invoked when coercing types all within a single language.
The efficacy of this approach is given by the development of a system of
expressions indexed by their, exposing a case analysis feature serving to
generate value constraints. These constraints are directly reflected into
the host allowing for their involvement in the type-checking process. A
motivating use case of this design shows how a termâs semantic index
information admits an exact, formalized cost analysis amenable to reasoning
within the host. Finally we show how such a system is used to identify
unreachable dead-code, trivially admitting the design and verification of
an SSA style compiler with this optimization. We think such a design
of explicitly proving the local correctness of type-transformations in the
presence of accumulated constraints can form the basis of a flexible language
in concert with a variety of trusted solver
Formal verification of the equivalence of system F and the pure type system L2
We develop a formal proof of the equivalence of two different variants of System F. The first is close to the original presentation where expressions are separated into distinct syntactic classes of types and terms. The second, L2 (also written as λ2), is a particular pure type system (PTS) where the notions of types and terms, and the associated expressions are unified in a single syntactic class. The employed notion of equivalence is a bidirectional reduction of the respective typing relations. A machine-verified proof of this result turns out to be surprisingly intricate, since the two variants noticeably differ in their expression languages, their type systems and the binding of local variables. Most of this work is executed in the Coq theorem prover and encompasses a general development of the PTS metatheory, an equivalence result for a stratified and a PTS variant of the simply typed λ-calculus as well as the subsequent extension to the full equivalence result for System F. We utilise nameless de Bruijn syntax with parallel substitutions for the representation of variable binding and develop an extended notion of context morphism lemmas as a structured proof method for this setting. We also provide two developments of the equivalence result in the proof systems Abella and Beluga, where we rely on higher-order abstract syntax (HOAS). This allows us to compare the three proof systems, as well as HOAS and de Bruijn for the purpose of developing formal metatheory.Wir prĂ€sentieren einen maschinell verifizierten Beweis der Ăquivalenz zweier Darstellungen des Lambda-KalkĂŒls System F. Die erste unterscheidet syntaktisch zwischen Termen und Typen und entspricht somit der gelĂ€ufigen Form. Die zweite, L2 bzw. λ2, ist ein sog. Pure Type System (PTS), bei welchem alle AusdrĂŒcke in einer syntaktischen Klasse zusammen fallen. Unser Ăquivalenzbegriff ist eine bidirektionale Reduktion der jeweiligen Typrelationen. Ein formaler Beweis dieser Eigenschaft ist aufgrund der Unterschiede der Ausdruckssprachen, der Typrelationen und der Bindung lokaler Variablen ĂŒberraschend anspruchsvoll. Der Hauptteil dieser Arbeit wurde in dem Beweisassistenten Coq entwickelt und umfasst eine Abhandlung der PTS Metatheorie, sowie einen Ăquivalenzbeweis fĂŒr das einfach getypte Lambda-KalkĂŒl, welcher dann zu dem vollen Ergebnis fĂŒr System F skaliert wird. FĂŒr die Darstellung lokaler Variablenbindung verwenden wir de Bruijn Syntax, gepaart mit parallelen Substitutionen. AuĂerdem entwickeln wir eine generalisierte Form von Kontext-Morphismen Lemmas, welche eine strukturierte Beweismethodik in diesem Umfeld liefern. DarĂŒber hinaus betrachten wir zwei weitere Formalisierungen des Ăquivalenzresultats in den Beweissystemen Abella und Beluga, welche beide höherstufige abstrakte Syntax (HOAS) zur Darstellung lokaler Bindung verwenden. Dies ermöglicht es uns, sowohl die drei Beweissysteme, als auch den HOAS und den de Bruijn Ansatz mit Hinblick auf die Entwicklung formaler Metatheorie zu vergleichen