Daniel Winterstein

A Framework for Interactive Proof

This paper introduces Proof General Kit, a framework for software components tailored to interactive proof development. The goal of the framework is to enable flexible environments for managing formal proofs across their life-cycle: creation, maintenance and exploitation. The framework connects together different kinds of component, exchanging messages using a common communication infrastructure and protocol called PGIP. The main channel connects proversto displays. Provers are the back-end interactive proof engines and displays are components for interacting with the user, allowing browsing or editing of proofs. At the core of the framework is a brokermiddleware component which manages proof-in-progress and mediates between components.

Using Animation in Diagrammatic Theorem Proving

Diagrams have many uses in mathematics, one of the most ambitious of which is as a form of proof. The domain we consider is real analysis, where quantification issues are subtle but crucial. Computers offer new possibilities in diagrammatic reasoning, one of which is animation. Here we develop animated rules as a solution to problems of quantification. We show a simple application of this to constraint diagrams, and also how it can deal with the more complex questions of quantification and generalisation in diagrams that use more specific representations. This allows us to tackle difficult theorems that previously could only be proved algebraically.

Dr.Doodle: A Diagrammatic Theorem Prover

This paper presents the Dr.Doodle system, an interactive theorem prover that uses diagrammatic representations. The assumption underlying this project is that, for some domains (principally geometry), diagrammatic reasoning is easier to understand than conventional algebraic approaches – at least for a significant number of people. The Dr.Doodle system was developed for the domain of metric-space analysis (a geometric domain, but traditionally taught using a dry algebraic formalism). Pilot experiments were conducted to evaluate its potential as the basis of an educational tool, with encouraging results.

Proof general in Eclipse: system and architecture overview

Interactive theorem proving is the art of constructing electronic proofs. Proof development, based around a proof script, has much in common with program development, based around a program text. Proof developers use rather primitive tools for developing and manipulating proof scripts at present. The Proof General project aims at to change this, by providing powerful generic tools and interfaces. The flagship tool is our Eclipse plugin, which brings the features of a industrial-strength IDE to theorem proving for the first time. In this paper we give an overview of the Eclipse plugin and its underlying architecture.

On Differences between the Real and Physical Plane

When formalising diagrammatic systems, it is quite common to situate diagrams in the real plane, \({\mathbb R}^{\rm 2}\). However this is not necessarily sound unless the link between formal and physical diagrams is examined. We explore some issues relating to this, and potential mistakes that can arise. This demonstrates that the effects of drawing resolution and the limits of perception can change the meaning of a diagram in surprising ways. These effects should therefore be taken into account when giving formalisations based on \({\mathbb R}^{\rm 2}\).

An Experimental Comparison of Diagrammatic and Algebraic Logics

We have developed a diagrammatic logic for theorem proving, focusing on the domain of metric-space analysis (a geometric domain, but traditionally taught using a dry algebraic formalism). To evaluate its pragmatic value, pilot experiments were conducted using this logic – implemented in an interactive theorem prover – to teach undergraduate students (and comparing performance against an equivalent algebraic logic). Our results show significantly better performance for students using diagrammatic reasoning. We conclude that diagrams are a useful tool for reasoning in such domains.