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Experiments with ZF set theory in HOL and Isabelle [chapter]

Sten Agerholm, Mike Gordon
1995 Lecture Notes in Computer Science  
This paper compares the representation and use of ZF set theory within both HOL and Isabelle. The main case study is the construction of D∞.  ...  The advantages and disadvantages of higher-order set theory versus firstorder set theory are explored experimentally.  ...  Francisco Corella, whose PhD thesis [6] contains many insights into mechanizing set theory (including the use of simple type theory as the underlying logic) has had a significant influence on the second  ... 
doi:10.1007/3-540-60275-5_55 fatcat:3ormlf77dvdp7ld77ltd7kulhm

Experiments with ZF Set Theory in HOL and Isabelle

Sten Agerholm, Mike Gordon
1995 BRICS Report Series  
<br /><br />This paper compares the representation and use of ZF set theory within<br />both HOL and Isabelle. The main case study is the construction of D_inf.  ...  <br />The advantages and disadvantages of higher-order set theory versus first-order<br />set theory are explored experimentally.  ...  Francisco Corella, whose PhD thesis [6] contains many insights into mechanizing set theory (including the use of simple type theory as the underlying logic) has had a significant influence on the second  ... 
doi:10.7146/brics.v2i37.19940 fatcat:4ptzes5o2vbmhh3m3wmwjn6ake

Higher-Order Logic and Theorem Proving for Structured Specifications [chapter]

Tomasz Borzyszkowski
2000 Lecture Notes in Computer Science  
Then we show that for maps of institutions into HOL that satisfy certain technical conditions we can reuse the proof system of the higher-order logic to reason about structured specifications built over  ...  We also show some maps of institutions underlying the CASL specification formalism (see [CASL 99]) into HOL that satisfy conditions needed for reusing proof systems.  ...  A task for the future is to extend presented work to the CASL language and to design a tool for automatic translation of the CASL specifications into one of the theorem-provers for HOL logic together with  ... 
doi:10.1007/978-3-540-44616-3_23 fatcat:3dfhrzw3wvgpxjbob3masrh54y

Unique Solutions of Contractions, CCS, and their HOL Formalisation

Chun Tian, Davide Sangiorgi
2018 Electronic Proceedings in Theoretical Computer Science  
The paper presents an overview of a rather comprehensive formalisation of the core of the theory of CCS in the HOL theorem prover (HOL4), with a focus towards the theory of unique solutions of contractions  ...  (The formalisation consists of about 20,000 lines of proof scripts in Standard ML.) Some refinements of the theory itself are obtained.  ...  The second half of this paper was written in memory of Michael J. C. Gordon, the creator of HOL theorem prover.  ... 
doi:10.4204/eptcs.276.10 fatcat:5p4xhummsvd5fh3hamg6pito5e

Mechanizing coinduction and corecursion in higher-order logic

L. Paulson
1997 Journal of Logic and Computation  
A theory of recursive and corecursive definitions has been developed in higher-order logic (HOL) and mechanized using Isabelle.  ...  The theory rests on a traditional formalization of infinite trees. The theory is intended for use in specification and verification.  ...  Martin Coen helped to set up the environment for coinduction proofs. Andrew Pitts gave much advice, for example on proving that equality is a gfp.  ... 
doi:10.1093/logcom/7.2.175 fatcat:lmmrync4qnhxzbmggldyyvppje

Page 8785 of Mathematical Reviews Vol. , Issue 99m [page]

1999 Mathematical Reviews  
Chapter 6 contrasts computer algebra systems and theorem provers, pointing out the strengths and weaknesses of each.  ...  Chapter 5 describes an HOL implementation of a quantifier elimination procedure for the first order theory of reals.  ... 

Formal verification of a partial-order reduction technique for model checking [chapter]

Ching -Tsun Chou, Doron Peled
1996 Lecture Notes in Computer Science  
As a case study, we use the mechanical theorem prover HOL to verify the correctness of a partial-order reduction technique for cutting down the amount of state search performed by model checkers.  ...  In this paper we explore another possibility: we use mechanical theorem proving to formally verify a meta-theory of model checking.  ...  Acknowledgements We would like to thank Elsa Gunter for her penetrating remarks (especially regarding the formalization of deterministic transition systems), and Rajeev Alur, Gerard Holzmann, Doug Howe  ... 
doi:10.1007/3-540-61042-1_48 fatcat:mzcv3l3bgbbbvpllnzgynp6zg4

Isomorphisms — A Link Between the Shallow and the Deep [chapter]

Thomas Santen
1999 Lecture Notes in Computer Science  
We present a theory of isomorphisms between typed sets in Isabelle/HOL. Those isomorphisms can serve to link a shallow embedding with a theory that defines certain concepts directly in HOL.  ...  Thus, it becomes possible to use the advantage of a shallow embedding that it allows for efficient proofs about concrete terms of the embedded formalism with the advantage of a deeper theory that establishes  ...  Thanks to the referees for their constructive comments and for their patience with the first version of the paper.  ... 
doi:10.1007/3-540-48256-3_4 fatcat:pzwrcinecrc5xafccf6lj7wemq

Heterogeneous Development Graphs and Heterogeneous Borrowing [chapter]

Till Mossakowski
2002 Lecture Notes in Computer Science  
Often, different aspects of a software system have to be specified in different logics, since the construction of a huge logic covering all needed features would be too complex to be feasible.  ...  Therefore, we introduce heterogeneous development graphs as a means to cope with heterogeneous specifications. We cover both the semantics and the proof theory of heterogeneous development graphs.  ...  Acknowledgments This work came out of both a cooperation with Andrzej Tarlecki on the semantic aspects and with Serge Autexier and Dieter Hutter on the proof theoretic aspects of specifications.  ... 
doi:10.1007/3-540-45931-6_23 fatcat:i6gfn65dfzacrbiuezalt36epu

A Formalization of the Process Algebra CCS in HOL4 [article]

Chun Tian
2017 arXiv   pre-print
We have used HOL4's new co-inductive relation support to re-define strong and weak bisimulation equivalances, and shows that the new definitions are equivalent with old ones.  ...  The aim is to provide an up-to-date sound and effective tool to support verification and reasoning about CCS, and to provide a formal logic basis for further theoretical developments in Concurrency Theory  ...  And the use of CURRY and UNCURRY is to transform the relation from types between CCS -> CCS -> bool and CCS × CCS -> bool, since relations in HOL cannot be treated directly as mathematical sets.  ... 
arXiv:1705.07313v2 fatcat:atvk2h3jfbghtpqcldme3ahkiy

Foundational, Compositional (Co)datatypes for Higher-Order Logic: Category Theory Applied to Theorem Proving

Dmitry Traytel, Andrei Popescu, Jasmin C. Blanchette
2012 2012 27th Annual IEEE Symposium on Logic in Computer Science  
Our framework draws heavily from category theory.  ...  Interactive theorem provers based on higher-order logic (HOL) traditionally follow the definitional approach, reducing high-level specifications to logical primitives.  ...  HOL supports a restrictive, simply typed flavor of set theory. We write α set for the powertype of α, consisting of sets of α elements; it is isomorphic to α → bool.  ... 
doi:10.1109/lics.2012.75 dblp:conf/lics/TraytelPB12 fatcat:gxwn5ptan5b6niq3qf4ahwbcua

Traces of I/O-automata in Isabelle/HOLCF [chapter]

Olaf Müller, Tobias Nipkow
1997 Lecture Notes in Computer Science  
We make use of the logic HOLCF, an extension of HOL with domain theory and show how to move between HOL and HOLCF.  ...  This allows us to restrict the use of HOLCF to metatheoretic arguments while actual re nement proofs between I/O automata are carried out within the simpler logic HOL.  ...  (Unless noted otherwise, HOL will refer to Isabelle/HOL rather than Gordon's HOL system 7] .)  ... 
doi:10.1007/bfb0030627 fatcat:dyxboezwgrf5xgcowqdg543igy

A new interface for HOL — Ideas, issues and implementation [chapter]

Donald Syme
1995 Lecture Notes in Computer Science  
TkHolWorkbench is a new set of interface tools for HOL implemented using the Tk toolkit. It aims to be robust, extensible, lightweight and user-friendly.  ...  The tools are designed to augment the existing HOL interface. The project applies rapid prototyping and the use of an interpreted toolkit to the eld of theorem proving interfaces.  ...  It draws from several sources 11, 4 , 9 , 7 ] . { The current i n terface to HOL is very weak, particularly in the area of visualisation.  ... 
doi:10.1007/3-540-60275-5_74 fatcat:te7fkrxfcjagnbb7madijixvbq

Further Formalization of the Process Algebra CCS in HOL4 [article]

Chun Tian
2017 arXiv   pre-print
We have added full supports on weak bisimulation equivalence and observation congruence (rooted weak equivalence), with related definitions, theorems and algebraic laws.  ...  Some deep lemmas were also formally proved in this project, including Deng Lemma, Hennessy Lemma and several versions of the "Coarsest congruence contained in weak equivalence".  ...  It's know that, the typed logic implemented in the various HOL systems (including Isabelle/HOL) is not strong enough to define a type for all possible ordinal values (a proper class in a set theory like  ... 
arXiv:1707.04894v2 fatcat:wfejjkivvvafrdlmxeqelolkay

A Formalization of Unique Solutions of Equations in Process Algebra [article]

Chun Tian
2017 arXiv   pre-print
This includes all classical properties of strong/weak bisimulation equivalences and observation congruence, a theory of congruence for CCS, various versions of "bisimulation up to" techniques, and several  ...  This work is further extended to support recent developments in Concurrency Theory, namely the "contraction" relation and the related "unique solutions of contractions" theorem found by Prof.  ...  The HOL deductive system The deductive system of the HOL logic is specified by the following eight rules of inference.  ... 
arXiv:1712.09402v1 fatcat:hgzwipj3qfekbioxaus2jzjtzq
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