Isabelle/PIDE as Platform for Educational Tools

The Isabelle/PIDE platform addresses the question whether proof assistants of the LCF family are suitable as technological basis for educational tools. The traditionally strong logical foundations of systems like HOL, Coq, or Isabelle have so far been counter-balanced by somewhat inaccessible interaction via the TTY (or minor variations like the well-known Proof General / Emacs interface). Thus the fundamental question of math education tools with fully-formal background theories has often been answered negatively due to accidental weaknesses of existing proof engines. The idea of "PIDE" (which means "Prover IDE") is to integrate existing provers like Isabelle into a larger environment, that facilitates access by end-users and other tools. We use Scala to expose the proof engine in ML to the JVM world, where many user-interfaces, editor frameworks, and educational tools already exist. This shall ultimately lead to combined mathematical assistants, where the logical engine is in the background, without obstructing the view on applications of formal methods, formalized mathematics, and math education in particular.Comment: In Proceedings THedu'11, arXiv:1202.453

Verifying mixed real-integer quantifier elimination

We present a formally verified quantifier elimination procedure for the first order theory over linear mixed real-integer arithmetics in higher-order logic based on a work by Weispfenning. To this end we provide two verified quantifier elimination procedures: for Presburger arithmitics and for linear real arithmetics

Automatische Methoden für formale Beweise in einfachen Arithmetiken und Algebren

In an LCF-like theorem prover, any proof must be produced from a small set of inference rules. The development of automated proof methods in such systems is extremely important. In this thesis we study the following question: How should we integrate a proof procedure in an LCF-like theorem prover, both in general and in the special case of arithmetics? We investigate three integration paradigms and present several proof procedures. These include universal and weak existential problems over rings, universal polynomial problems over the reals, quantifier elimination for parametric linear problems over ordered fields, Presburger arithmetic, mixed real-integer linear arithmetic, algebraically and real closed fields. Our work has been carried out in the Isabelle framework.In einem LCF-ähnlichen Theorembeweiser, stammt jeder Beweis aus einer minimalen Menge von Inferenzregeln ab. Somit sind Verfahren zur Generierung solcher Beweise von enormer Wichtigkeit. Das Ziel dieser Abhandlung ist folgende Frage zu studieren: Wie soll, allgemein und im Spezialfall der Arithmetik, ein LCF-ähnlicher Theorembeweiser um eine Entscheidungsprozedur erweitert werden? Wir betrachten drei verschiedene Ansätze für eine solche Integration und präsentieren mehrere Beweisverfahren im Detail. Die wichtigsten präsentierten Verfahren sind: a) Entscheidungsprozeduren für universelle und schwach existentielle Probleme in Ringen, b) Univerelle Probleme reeller Polynome, c) Quantoren-elimination für parametrische lineare Formeln über geordnete Körper, Presburger Arithmetik, die gemischte lineare Theorie der reelen und ganzen Zahlen, Algebraisch- und Reel-abgeschlossene Körper. Alle unsere Arbeiten basieren auf dem Isabelle Theorembeweiser

Mechanized quantifier elimination for linear real-arithmetic in Isabelle/HOL

We integrate Ferrante and Rackoff’s quantifier elimination procedure for linear real arithmetic in Isabelle/HOL in two manners: (a) tactic-style, i.e. for every problem instance a proof is generated by invoking a series of inference rules, and (b) reflection, where the whole algorithm is implemented and verified within Isabelle/HOL. We discuss the performance obtained for both integrations

Generic proof synthesis for Presburger arithmetic

We develop in complete detail an extension of Cooper’s decision procedure for Presburger arithmetic that returns a proof of the equivalence of the input formula to a quantifier-free formula. For closed input formulae this is a proof of their validity or unsatisfiability. The algorithm is formulated as a functional program that makes only very minimal assumptions w.r.t. the underlying logical system and is therefore easily adaptable to specific theorem provers