Dear Members of the Research Community,
As part of my efforts to try to understand the type system of Isabelle/HOL, I
have gathered the references from the mails written by Corey Richardson and
Michael Norrish and from The Isabelle/Isar Reference Manual, especially from
a) Chapter 10: Higher-Order Logic (pp. 236 f.)
b) § 11.7 Semantic subtype definitions (pp. 260-263)
c) § 11.8 Functorial structure of types (pp. 263 f.).
In all cases, with one exception, I was able to obtain either the link to the
online version or the ISBN number.
The exception is Mike Gordon's text "HOL: A machine oriented formulation of
higher order logic" (Technical Report 68, University of Cambridge Computer
Laboratory), registered at
http://www.cl.cam.ac.uk/techreports/UCAM-CL-TR-68.html
As Isabelle is provided by the University of Cambridge Computer Laboratory,
among others, would it be possible for the Isabelle developers to make this
technical report available online as a PDF file? This was done for the first
five reports listed at
http://www.cl.cam.ac.uk/techreports/UCAM-CL-TR-table.html
According to the references supplied with the traditional (Mike Gordon's) HOL
variant, a revised version exists: "Technical Report No.\ 68 (revised
version)", quoted from
https://github.com/HOL-Theorem-Prover/HOL/blob/master/Manual/Description/references.tex
Either both versions or the revised version should be made available.
I would like to take this opportunity to thank Michael Norrish for his advice.
The LOGIC part of HOL4 system's documentation at
http://sourceforge.net/projects/hol/files/hol/kananaskis-10/kananaskis-10-logic.pdf
is the kind of short and precise formulation of the logical core I recently
proposed for Isabelle/HOL, too:
https://lists.cam.ac.uk/pipermail/cl-isabelle-users/2016-July/msg00105.html
In this HOL4 document, it is stated: "From a logical point of view it would be
better to have a simpler substitution primitive, such as 'Rule R' of Andrews'
logic Q0, and then to derive more complex rules from it." (p. 27)
This is exactly the same idea that I presented recently: "The main criterion is
expressiveness [cf. Andrews, 2002, p. 205], or, vice versa, the reducibility of
mathematics to a minimal set of primitive rules and symbols. [...] For the
philosopher it is not proof automation but expressiveness that is the main
criterion, such that the method with the least amount of rules of inferences,
i.e., a Hilbert-style system, is preferred, and all other rules become derived
rules."
https://lists.cam.ac.uk/pipermail/cl-isabelle-users/2016-July/msg00028.html
Similarly, the HOL4 document argues in favor of definitions instead of
introducing new axioms: "The advantages of postulation over definition have
been likened by Bertrand Russell to the advantages of theft over honest toil.
As it is all too easy to introduce inconsistent axiomatizations, users of the
HOL system are strongly advised to resist the temptation to add axioms, but to
toil through definitional theories honestly." [p. 33]
In the same manner, I wrote: "One consequence [...] is to avoid the use of
non-logical axioms. [...] The appropriate way to deal with objects of certain
properties is to create a set of objects with these properties." [Kubota, 2015,
p. 29]
For this reason, in the R0 implementation, the group axioms are expressed not
as axioms, but as properties in the definition of "Grp" (wff 266):
http://www.kenkubota.de/files/R0_draft_excerpt_with_definitions_and_groups.pdf
(p. 359)
Kind regards,
Ken Kubota
Sources
[S1] The Isabelle/Isar Reference Manual (February 17, 2016)
http://www.cl.cam.ac.uk/research/hvg/Isabelle/dist/Isabelle2016/doc/isar-ref.pdf
[S2] Reference in the e-mail written by Corey Richardson
https://lists.cam.ac.uk/pipermail/cl-isabelle-users/2016-July/msg00070.html
Dmitry Traytel / Andrei Popescu / Jasmin C. Blanchette, Foundational,
Compositional (Co)datatypes for Higher-Order Logic: Category Theory Applied to
Theorem Proving
http://dx.doi.org/10.1109/LICS.2012.75
https://www21.in.tum.de/~traytel/papers/lics12-codatatypes/codat.pdf
[S3] Reference in the e-mail written by Michael Norrish
"The chapters from the HOL book mentioned above are available as part of the
HOL4 system's documentation at:
http://sourceforge.net/projects/hol/files/hol/kananaskis-10/kananaskis-10-logic.pdf/download
See §2.5.4 in particular for a discussion of how the system can be extended
with new types."
The HOL System: LOGIC (November 6, 2014)
http://sourceforge.net/projects/hol/files/hol/kananaskis-10/kananaskis-10-logic.pdf
Derived References (from [S1], pp. 260-263)
[15] W. M. Farmer. The seven virtues of simple type theory. J. Applied Logic,
6(3):267-286, 2008.
http://imps.mcmaster.ca/doc/seven-virtues.pdf
[18] M. J. C. Gordon. HOL: A machine oriented formulation of higher order
logic. Technical Report 68, University of Cambridge Computer Laboratory, 1985.
[50] A. Pitts. The HOL logic. In M. J. C. Gordon and T. F. Melham, editors,
Introduction to HOL: A Theorem Proving Environment for Higher Order Logic,
pages 191-232. Cambridge University Press, 1993.
ISBN 0-521-44189-7
[57] M. Wenzel. Type classes and overloading in higher-order logic. In E. L.
Gunter and A. Felty, editors, Theorem Proving in Higher Order Logics: TPHOLs
'97, volume 1275 of Lecture Notes in Computer Science. Springer-Verlag, 1997.
http://www4.in.tum.de/~wenzelm/papers/axclass-TPHOLs97.pdf
[22] F. Haftmann and M. Wenzel. Constructive type classes in Isabelle. In T.
Altenkirch and C. McBride, editors, Types for Proofs and Programs, TYPES 2006,
volume 4502 of LNCS. Springer, 2007.
http://www4.in.tum.de/~wenzelm/papers/constructive_classes.pdf
[27] O. Kuncar and A. Popescu. A consistent foundation for Isabelle/HOL. In C.
Urban and X. Zhang, editors, Interactive Theorem Proving - 6th International
Conference, ITP 2015, Nanjing, China, August 24-27, 2015, Proceedings, volume
9236 of Lecture Notes in Computer Science. Springer, 2015.
http://www21.in.tum.de/~kuncar/kuncar-popescu-jar2016.pdf
http://andreipopescu.uk/pdf/ITP2015.pdf
Standard References
Andrews, Peter B. (1986), An Introduction to Mathematical Logic and Type
Theory: To Truth Through Proof. Orlando: Academic Press. ISBN 0-12-058535-9
(Hardcover). ISBN 0-12-058536-7 (Paperback).
Andrews, Peter B. (2002), An Introduction to Mathematical Logic and Type
Theory: To Truth Through Proof. Second edition. Dordrecht / Boston / London:
Kluwer Academic Publishers. ISBN 1-4020-0763-9. DOI: 10.1007/978-94-015-9934-4.
Andrews, Peter B. (2014), "Church's Type Theory". In: Stanford Encyclopedia of
Philosophy (Spring 2014 Edition). Ed. by Edward N. Zalta. Available online at
http://plato.stanford.edu/archives/spr2014/entries/type-theory-church/ (July 2,
2016).
Church, Alonzo (1940), "A Formulation of the Simple Theory of Types". In:
Journal of Symbolic Logic 5, pp. 56-68.
Further References
Kubota, Ken (2015), On the Theory of Mathematical Forms (draft from May 18,
2015). Unpublished manuscript. SHA-512: a0dfe205eb1a2cb29efaa579d68fa2e5
45af74d8cd6c270cf4c95ed1ba6f7944 fdcffaef2e761c8215945a9dcd535a50
011d8303fd59f2c8a4e6f64125867dc4. DOI: 10.4444/100.10. See:
http://dx.doi.org/10.4444/100.10
Kubota, Ken (2015a), Excerpt from [Kubota, 2015] (pp. 356-364, pp. 411-420, and
pp. 754-755). Available online at
http://www.kenkubota.de/files/R0_draft_excerpt_with_definitions_and_groups.pdf
(January 24, 2016). SHA-512: 67a48ee12a61583bd82c286136459c0c
9fd77f5fd7bd820e8b6fe6384d977eda 4d92705dac5b3f97e7981d7b709fc783
3d1047d1831bc357eb57b263b44c885b.
____________________
Ken Kubota
doi: 10.4444/100
http://dx.doi.org/10.4444/100
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