2,377 research outputs found
Natural Deduction and the Isabelle Proof Assistant
We describe our Natural Deduction Assistant (NaDeA) and the interfaces between the Isabelle proof assistant and NaDeA. In particular, we explain how NaDeA, using a generated prover that has been verified in Isabelle, provides feedback to the student, and also how NaDeA, for each formula proved by the student, provides a generated theorem that can be verified in Isabelle.<br/
Formalization, Mechanization and Automation of G\"odel's Proof of God's Existence
G\"odel's ontological proof has been analysed for the first-time with an
unprecedent degree of detail and formality with the help of higher-order
theorem provers. The following has been done (and in this order): A detailed
natural deduction proof. A formalization of the axioms, definitions and
theorems in the TPTP THF syntax. Automatic verification of the consistency of
the axioms and definitions with Nitpick. Automatic demonstration of the
theorems with the provers LEO-II and Satallax. A step-by-step formalization
using the Coq proof assistant. A formalization using the Isabelle proof
assistant, where the theorems (and some additional lemmata) have been automated
with Sledgehammer and Metis.Comment: 2 page
NaDeA: A Natural Deduction Assistant with a Formalization in Isabelle
We present a new software tool for teaching logic based on natural deduction.
Its proof system is formalized in the proof assistant Isabelle such that its
definition is very precise. Soundness of the formalization has been proved in
Isabelle. The tool is open source software developed in TypeScript / JavaScript
and can thus be used directly in a browser without any further installation.
Although developed for undergraduate computer science students who are used to
study and program concrete computer code in a programming language we consider
the approach relevant for a broader audience and for other proof systems as
well.Comment: Proceedings of the Fourth International Conference on Tools for
Teaching Logic (TTL2015), Rennes, France, June 9-12, 2015. Editors: M.
Antonia Huertas, Jo\~ao Marcos, Mar\'ia Manzano, Sophie Pinchinat,
Fran\c{c}ois Schwarzentrube
Natural Deduction Assistant (NaDeA)
We present the Natural Deduction Assistant (NaDeA) and discuss its advantages
and disadvantages as a tool for teaching logic. NaDeA is available online and
is based on a formalization of natural deduction in the Isabelle proof
assistant. We first provide concise formulations of the main formalization
results. We then elaborate on the prerequisites for NaDeA, in particular we
describe a formalization in Isabelle of "Hilbert's Axioms" that we use as a
starting point in our bachelor course on mathematical logic. We discuss a
recent evaluation of NaDeA and also give an overview of the exercises in NaDeA.Comment: In Proceedings ThEdu'18, arXiv:1903.1240
Isabelle/HOL as a Meta-Language for Teaching Logic
Proof assistants are important tools for teaching logic. We support this
claim by discussing three formalizations in Isabelle/HOL used in a recent
course on automated reasoning. The first is a formalization of System W (a
system of classical propositional logic with only two primitive symbols), the
second is the Natural Deduction Assistant (NaDeA), and the third is a one-sided
sequent calculus that uses our Sequent Calculus Verifier (SeCaV). We describe
each formalization in turn, concentrating on how we used them in our teaching,
and commenting on features that are interesting or useful from a logic
education perspective. In the conclusion, we reflect on the lessons learned and
where they might lead us next.Comment: In Proceedings ThEdu'20, arXiv:2010.1583
A Vernacular for Coherent Logic
We propose a simple, yet expressive proof representation from which proofs
for different proof assistants can easily be generated. The representation uses
only a few inference rules and is based on a frag- ment of first-order logic
called coherent logic. Coherent logic has been recognized by a number of
researchers as a suitable logic for many ev- eryday mathematical developments.
The proposed proof representation is accompanied by a corresponding XML format
and by a suite of XSL transformations for generating formal proofs for
Isabelle/Isar and Coq, as well as proofs expressed in a natural language form
(formatted in LATEX or in HTML). Also, our automated theorem prover for
coherent logic exports proofs in the proposed XML format. All tools are
publicly available, along with a set of sample theorems.Comment: CICM 2014 - Conferences on Intelligent Computer Mathematics (2014
Automated verification of termination certificates
In order to increase user confidence, many automated theorem provers provide
certificates that can be independently verified. In this paper, we report on
our progress in developing a standalone tool for checking the correctness of
certificates for the termination of term rewrite systems, and formally proving
its correctness in the proof assistant Coq. To this end, we use the extraction
mechanism of Coq and the library on rewriting theory and termination called
CoLoR
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