Formalising Mathematics in Simple Type Theory

Despite the considerable interest in new dependent type theories, simple type theory (which dates from 1940) is sufficient to formalise serious topics in mathematics. This point is seen by examining formal proofs of a theorem about stereographic projections. A formalisation using the HOL Light proof assistant is contrasted with one using Isabelle/HOL. Harrison's technique for formalising Euclidean spaces is contrasted with an approach using Isabelle/HOL's axiomatic type classes. However, every formal system can be outgrown, and mathematics should be formalised with a view that it will eventually migrate to a new formalism

The Court of Justice of the Economic Community of West African States as a Constitutional Court

Eine der wichtigsten Innovationen der Wirtschaftsgemeinschaft Westafrikanischer Staaten (ECOWAS) ist die eindeutige Gewährung einer supranationalen Rolle an den Gerichtshof der Organisation. Sein Menschenrechtsmandat hat jedoch zu realen und potenziellen Spannungen innerhalb der ECOWAS-Rechtsordnung geführt. Die Spannungen ergeben sich aus der Rechtskraft von Urteilen von Verfassungsgerichten der Mitgliedstaaten und der Zulässigkeit von Individualbeschwerden vor dem Gerichtshof. Diese Arbeit zeigt einige Mängel in der derzeitigen Regelung des Menschenrechtsmandats des Gerichtshofs auf. Lücken bestehen sowohl auf der Ebene der verfassungsmäßigen Ordnung der Mitgliedstaaten als auch auf der Ebene der Gemeinschaft. Die supranationale Menschenrechtsgerichtsbarkeit muss durch die Möglichkeit umgesetzt werden, konkrete Abhilfemaßnahmen zur Behebung von Menschenrechtsverletzungen an die betroffenen Mitgliedsstaaten anzuordnen. In dieser Arbeit werden innovative Lösungen vorgeschlagen, um prozessuale und substantielle Lücken im etablierten Menschenrechtsschutzsystem in Westafrika zu schließen

ISABELLE - THE NEXT 700 THEOREM PROVERS

Isabelle is a generic theorem prover, designed for interactive reasoning in a variety of formal theories. At present it provides useful proof procedures for Constructive Type Theory, various first-order logics, Zermelo-Fraenkel set theory, and higher-order logic. This survey of Isabelle serves as an introduction to the literature. It explains why generic theorem proving is beneficial. It gives a thorough history of Isabelle, beginning with its origins in the LCF system. It presents an account of how logics are represented, illustrated using classical logic. The approach is compared with the Edinburgh Logical Framework. Several of the Isabelle object-logics are presented

Higher-Order Tarski Grothendieck as a Foundation for Formal Proof

We formally introduce a foundation for computer verified proofs based on higher-order Tarski-Grothendieck set theory. We show that this theory has a model if a 2-inaccessible cardinal exists. This assumption is the same as the one needed for a model of plain Tarski-Grothendieck set theory. The foundation allows the co-existence of proofs based on two major competing foundations for formal proofs: higher-order logic and TG set theory. We align two co-existing Isabelle libraries, Isabelle/HOL and Isabelle/Mizar, in a single foundation in the Isabelle logical framework. We do this by defining isomorphisms between the basic concepts, including integers, functions, lists, and algebraic structures that preserve the important operations. With this we can transfer theorems proved in higher-order logic to TG set theory and vice versa. We practically show this by formally transferring Lagrange\u27s four-square theorem, Fermat 3-4, and other theorems between the foundations in the Isabelle framework