scholarly journals CoLoR: a Coq library on well-founded rewrite relations and its application to the automated verification of termination certificates

2011 ◽  
Vol 21 (4) ◽  
pp. 827-859 ◽  
Author(s):  
FRÉDÉRIC BLANQUI ◽  
ADAM KOPROWSKI
Keyword(s):  
Term Rewriting ◽  
Automated Verification ◽  
Proof Assistant ◽  
Proof Assistants ◽  
Active Subject ◽  
Turing Complete

Termination is an important property of programs, and is notably required for programs formulated in proof assistants. It is a very active subject of research in the Turing-complete formalism of term rewriting. Over the years, many methods and tools have been developed to address the problem of deciding termination for specific problems (since it is undecidable in general). Ensuring the reliability of those tools is therefore an important issue.In this paper we present a library formalising important results of the theory of well-founded (rewrite) relations in the proof assistant Coq. We also present its application to the automated verification of termination certificates, as produced by termination tools.The sources are freely available athttp://color.inria.fr/.

scholarly journals A Theory for Abstract Reduction Systems in PVS

2008 ◽  
Vol 11 (2) ◽  
Author(s):  
Andre Luiz Galdino ◽  
Mauricio Ayala Rincon
Keyword(s):  
Normal Form ◽  
Term Rewriting ◽  
General Treatment ◽  
Reduction Theory ◽  
Proof Assistant ◽  
Proof Assistants ◽  
Specification Languages ◽  
Term Rewriting Systems ◽  
Rewriting Systems ◽  
Prototype Verification

A theory for Abstract Reduction Systems (ARS) in the proof assistant PVS (Prototype Verification System) called ars is described. Adequate specifications of basic definitions and notions of the theory of ARSs such as reduction, confluence and normal form are given and well-known results formalized. The formalizations include non trivial results of the theory of ARSs such as the correctness of the principle of Noetherian Induction, Newman’s Lemma and its generalizations, and Commutation Lemmas, among others. Although term rewriting proving technologies have been provided in several specification languages and proof assistants, to our knowledge, before the development presented in this paper there was no complete formalization of an abstract reduction theory in PVS. This makes relevant the presented ars specification as the basis of a PVStheory called trs for the general treatment of Term Rewriting Systems.

scholarly journals Competitive Proving for Fun

10.29007/ktx8 ◽  
2019 ◽  
Author(s):  
Maximilian Paul Louis Haslbeck ◽  
Simon Wimmer
Keyword(s):  
Theorem Proving ◽  
System Architecture ◽  
Large Scale ◽  
Research Field ◽  
Proof Assistant ◽  
Proof Assistants ◽  
Prototype Implementation ◽  
New Generation

We propose a system for large-scale theorem proving contests. We hope that such contests could spark interest in the research field, attract a new generation of theorem proving savants, and foster competition among proof assistants. For the proof assistant Isabelle, we construct and evaluate two iterations of a prototype implementation of our proposed system architecture.

scholarly journals TacticToe: Learning to Reason with HOL4 Tactics

10.29007/ntlb ◽  
2018 ◽  
Author(s):  
Thibault Gauthier ◽  
Cezary Kaliszyk ◽  
Josef Urban
Keyword(s):  
Machine Learning ◽  
Automated Reasoning ◽  
Formal Proof ◽  
Small Scale ◽  
Proof Assistant ◽  
Proof Assistants ◽  
A Algorithm ◽  
Learning To Reason ◽  
Traditional Proof ◽  
Selection Of

Techniques combining machine learning with translation to automated reasoning have recently become an important component of formal proof assistants. Such “hammer” techniques complement traditional proof assistant automation as implemented by tactics and decision procedures. In this paper we present a unified proof assistant automation approach which attempts to automate the selection of appropriate tactics and tactic-sequences combined with an optimized small-scale hammering approach. We implement the technique as a tactic-level automation for HOL4: TacticToe. It implements a modified A*-algorithm directly in HOL4 that explores different tactic-level proof paths, guiding their selection by learning from a large number of previous tactic-level proofs. Unlike the existing hammer methods, TacticToe avoids translation to FOL, working directly on the HOL level. By combining tactic prediction and premise selection, TacticToe is able to re-prove 39% of 7902 HOL4 theorems in 5 seconds whereas the best single HOL(y)Hammer strategy solves 32% in the same amount of time.

scholarly journals Formalising Mathematics – in Praxis; A Mathematician’s First Experiences with Isabelle/HOL and the Why and How of Getting Started

2020 ◽  
Author(s):  
Angeliki Koutsoukou-Argyraki
Keyword(s):  
Theorem Prover ◽  
Proof Assistant ◽  
Proof Assistants

Abstract This is an account of a mathematician’s first experiences with the proof assistant (interactive theorem prover) Isabelle/HOL, including a discussion on the rationale behind formalising mathematics and the choice of Isabelle/HOL in particular, some instructions for new users, some technical and conceptual observations focussing on some of the first difficulties encountered, and some thoughts on the use and potential of proof assistants for mathematics.

scholarly journals EMTST: Engineering the Meta-theory of Session Types

2020 ◽  
pp. 278-285
Author(s):  
David Castro ◽  
Francisco Ferreira ◽  
Nobuko Yoshida
Keyword(s):  
Wide Spectrum ◽  
Type Systems ◽  
Small Scale ◽  
Proof Assistant ◽  
Proof Assistants ◽  
Type Safety ◽  
Session Types ◽  
The Coq Proof Assistant ◽  
Coq Proof Assistant ◽  
Type Preservation

Abstract Session types provide a principled programming discipline for structured interactions. They represent a wide spectrum of type-systems for concurrency. Their type safety is thus extremely important. EMTST is a tool to aid in representing and validating theorems about session types in the Coq proof assistant. On paper, these proofs are often tricky, and error prone. In proof assistants, they are typically long and difficult to prove. In this work, we propose a library that helps validate the theory of session types calculi in proof assistants. As a case study, we study two of the most used binary session types systems: we show the impossibility of representing the first system in $$\alpha $$-equivalent representations, and we prove type preservation for the revisited system. We develop our tool in the Coq proof assistant, using locally nameless for binders and small scale reflection to simplify the handling of linear typing environments.

scholarly journals Proof Assistants and the Dynamic Nature of Formal Theories

10.29007/d2df ◽  
2018 ◽  
Author(s):  
Robert L. Constable
Keyword(s):  
Proof Assistant ◽  
Proof Assistants ◽  
Dynamic Nature ◽  
Problem Space ◽  
Domain Specific ◽  
Critical Components ◽  
Deployed Systems ◽  
The Impact ◽  
Over Time ◽  
Formal Theories

This article shows that theory exploration arises naturally from the need to progressively modify applied formal theories, especially those underpinning deployed systems that change over time or need to be attack-tolerant. Such formal theories require us to explore a problem space with a proof assistant and are naturally dynamic.The examples in this article are from our on-going decade-long effort to formally synthesize critical components of modern distributed systems. Using the Nuprl proof assistant we created event logic and its protocol theories. I also mention the impact over this period of extensions to the constructive type theory implemented by Nuprl. One of them led to our solution of a long standing open problem in constructive logic.Proof exchange among theorem provers is promising for improving the "super tactics" that provide domain specific reasoners for our protocol theories. Both theory exploration and proof exchange illustrate the dynamic nature of applied formal theories built using modern proof assistants. These activities dispel the false impression that formal theories are rigid and brittle artifacts that become less relevant over time in a fast moving field like computer science.

scholarly journals Initial Experiments on Deriving a Complete HOL Simplification Set

10.29007/95qb ◽  
2018 ◽  
Author(s):  
Cezary Kaliszyk ◽  
Thomas Sternagel
Keyword(s):  
State Of The Art ◽  
Desirable Property ◽  
Proof Assistant ◽  
Proof Assistants ◽  
First Order

Rewriting is a common functionality in proof assistants, that allows to simplify theorems and goals. The set of equations to use in a rewrite step has to be manually specified, and therefore often includes rules which may lead to non-termination. Even in the case of termination another desirable property of a simplification set would be confluence. A well-known technique from rewriting to transform a terminating system into a terminating and confluent one is completion. But the sets of equations we find in the context of proof assistants are typically huge and most state-of-the-art completion tools only work on relatively small problems. In this paper we describe our initial experiments with the aim to close the gap and use rewriting to compute a complete first-order simplification set for a HOL-based proof assistant fully automatically.

scholarly journals Experiences from Exporting Major Proof Assistant Libraries

2021 ◽  
Author(s):  
Michael Kohlhase ◽  
Florian Rabe
Keyword(s):  
Theorem Proving ◽  
Proof Assistant ◽  
Proof Assistants ◽  
Social Challenges ◽  
Future System ◽  
Hol Light ◽  
System Interoperability

AbstractThe interoperability of proof assistants and the integration of their libraries is a highly valued but elusive goal in the field of theorem proving. As a preparatory step, in previous work, we translated the libraries of multiple proof assistants, specifically the ones of Coq, HOL Light, IMPS, Isabelle, Mizar, and PVS into a universal format: OMDoc/MMT. Each translation presented great theoretical, technical, and social challenges, some universal and some system-specific, some solvable and some still open. In this paper, we survey these challenges and compare and evaluate the solutions we chose. We believe similar library translations will be an essential part of any future system interoperability solution, and our experiences will prove valuable to others undertaking such efforts.

INEVITABILITY OF REVIVAL: PRO AND COUNTER ANTINATALISM

2019 ◽  
Vol 19 (1) ◽  
pp. 80-88
Author(s):  
Nikolay S. Savkin
Keyword(s):  
Social Relations ◽  
Systematic Approach ◽  
Human Life ◽  
Laws Of Nature ◽  
Arthur Schopenhauer ◽  
Expected Effect ◽  
Philosophy Of Life ◽  
Active Subject ◽  
Over Time ◽  
Natural Way

Introduction. Radical pessimism and militant anti-natalism of Arthur Schopenhauer and David Benathar create an optimistic philosophy of life, according to which life is not meaningless. It is given by nature in a natural way, and a person lives, studies, works, makes a career, achieves results, grows, develops. Being an active subject of his own social relations, a person does not refuse to continue the race, no matter what difficulties, misfortunes and sufferings would be experienced. Benathar convinces that all life is continuous suffering, and existence is constant dying. Therefore, it is better not to be born. Materials and Methods. As the main theoretical and methodological direction of research, the dialectical materialist and integrative approaches are used, the realization of which, in conjunction with the synergetic technique, provides a certain result: is convinced that the idea of anti-natalism is inadequate, the idea of giving up life. A systematic approach and a comprehensive assessment of the studied processes provide for the disclosure of the contradictory nature of anti-natalism. Results of the study are presented in the form of conclusions that human life is naturally given by nature itself. Instincts, needs, interests embodied in a person, stimulate to active actions, and he lives. But even if we finish off with all of humanity by agreement, then over time, according to the laws of nature and according to evolutionary theory, man will inevitably, objectively, and naturally reappear. Discussion and Conclusion. The expected effect of the idea of inevitability of rebirth can be the formation of an optimistic orientation of a significant part of the youth, the idea of continuing life and building happiness, development. As a social being, man is universal, and the awareness of this universality allows one to understand one’s purpose – continuous versatile development.

Verification of Content-Centric Networking Using Proof Assistant

2016 ◽  
Vol E99.B (11) ◽  
pp. 2297-2304
Author(s):  
Sosuke MORIGUCHI ◽  
Takashi MORISHIMA ◽  
Mizuki GOTO ◽  
Kazuko TAKAHASHI
Keyword(s):  
Proof Assistant ◽  
Content Centric Networking
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