[isabelle] Literature on the type system of Isabelle/HOL; text by Mike Gordon

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) Â 11.7 Semantic subtype definitions (pp. 260-263)
	c) Â 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

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

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
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
is the kind of short and precise formulation of the logical core I recently 
proposed for Isabelle/HOL, too:

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 

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


[S1] The Isabelle/Isar Reference Manual (February 17, 2016)

[S2] Reference in the e-mail written by Corey Richardson

    Dmitry Traytel / Andrei Popescu / Jasmin C. Blanchette, Foundational, 
Compositional (Co)datatypes for Higher-Order Logic: Category Theory Applied to 
Theorem Proving

[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:
See Â2.5.4 in particular for a discussion of how the system can be extended 
with new types."

    The HOL System: LOGIC (November 6, 2014)

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.

[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.

[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.

[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.

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, 

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: 

Kubota, Ken (2015a), Excerpt from [Kubota, 2015] (pp. 356-364, pp. 411-420, and 
pp. 754-755). Available online at 
(January 24, 2016). SHA-512: 67a48ee12a61583bd82c286136459c0c 
9fd77f5fd7bd820e8b6fe6384d977eda 4d92705dac5b3f97e7981d7b709fc783 


Ken Kubota
doi: 10.4444/100

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