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**1 - 3**of**3**### Web Based GUI for Natural Deduction Proofs In Isabelle

, 2007

"... It is fair to say that the use of interactive theorem provers is mostly limited to experts in the field. This project attributed this mainly to the high barrier of entry associated with using interactive theorem provers, and that most current systems do not aid the user in visualizing proofs. A web- ..."

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It is fair to say that the use of interactive theorem provers is mostly limited to experts in the field. This project attributed this mainly to the high barrier of entry associated with using interactive theorem provers, and that most current systems do not aid the user in visualizing proofs. A web-based client/server system with a graphical user interface was designed and implemented that users could use to perform point-and-click natural deduction theorem proving. The system did not require client users to install software in order to perform proofs, as the system was accessible through the use of a web browser. Proofs were visualized in box-style notation, and proof construction done by performing point-andclick actions on this. The sound and widely used interactive theorem prover Isabelle was used for verifying the proofs created. The system was deemed as successful, based on the analysis of a user test perfomed.

### Combining diagrammatic and symbolic reasoning

"... We introduce a domain-independent framework for heterogeneous natural deduction that combines diagrammatic and sentential reasoning. The framework is presented in the form of a family of denotational proof languages (DPLs). Diagrams are represented as possibly partial descriptions of finite system s ..."

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We introduce a domain-independent framework for heterogeneous natural deduction that combines diagrammatic and sentential reasoning. The framework is presented in the form of a family of denotational proof languages (DPLs). Diagrams are represented as possibly partial descriptions of finite system states. This allows us to deal with incomplete information, which we formalize by admitting sets as attribute values. We introduce a notion of attribute interpretations that enables us to interpret first-order signatures into such system states, and develop a formal semantic framework based on Kleeneās strong three-valued logic. We extend the assumption-base semantics of DPLs to accodomodate diagrammatic reasoning by introducing general inference mechanisms for the valid extraction of information from diagrams and for the incorporation of sentential information into diagrams. A rigorous big-step operational semantics is given, on the basis of which we prove that our framework is sound. In addition, we specify detailed algorithms for implementing proof checkers for the resulting languages, and discuss associated efficiency issues. 1.1

### Mathematical knowledge

, 2007

"... Abstract The original proof of the four-color theorem by Appel and Haken sparked a controversy when Tymoczko used it to argue that the justification provided by unsurveyable proofs carried out by computers cannot be a priori. It also created a lingering impression to the effect that such proofs depe ..."

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Abstract The original proof of the four-color theorem by Appel and Haken sparked a controversy when Tymoczko used it to argue that the justification provided by unsurveyable proofs carried out by computers cannot be a priori. It also created a lingering impression to the effect that such proofs depend heavily for their soundness on large amounts of computation-intensive custom-built software. Contra Tymoczko, we argue that the justification provided by certain computerized mathematical proofs is not fundamentally different from that provided by surveyable proofs, and can be sensibly regarded as a priori. We also show that the aforementioned impression is mistaken because it fails to distinguish between proof search (the context of discovery) and proof checking (the context of justification). By using mechanized proof assistants capable of producing certificates that can be independently checked, it is possible to carry out complex proofs without the need to trust arbitrary custom-written code. We only need to trust one fixed, small, and simple piece of software: the proof checker. This is not only possible in principle, but is in fact becoming a viable methodology for performing complicated mathematical reasoning. This is evinced by a new proof of the four-color theorem that appeared in 2005, and which was developed and checked in its entirety by a mechanical proof system.