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Abstract Automath MATHEMATICAL CENTRE TRACTS 160 ABSTRACT AUTOMATH A. REZUS MATH EMATISCH CENTRUM AMST EW DAM 1983 1980 Mathematics subject classification: Primary: 03B40 Secondary: 03-04, 03F99, 68-04, 68C20 ISBN 90 6196 256 0 copyrightO 1983, Mathematisch Centrum, Amsterdam "De objectiveering der wereld in wiskun- dige systemen bij verschillende individu- en wordt in onderling verband gehouden door de passielooze taa1,die bi den hoor- der het identieke wiskundige sssteem als bij den spreker doet oprijzeh,terwijl de gevoelsinhoud van dat systeem bij beiden totaal verschillend kan zijn ..." (L.E.J.BROUWER: "Wiskunde en Ervaring".) PREFACE The idea of a computer-assisted proof-checking (applied to concrete proofs,as they appear in the mathematician's everyday life) has probably occurred to many mindsteven before the von Neumann compu- ter was conceived. In particular,an attempt to increase the reliability of mathematical texts (proofs) by having them processed and checked on computer has also been the main motivation behind the specific AUTOmated MATHema- tics Project,developed since 1967 at the Eindhoven University of Technology. Abstracting from the pragmatic motivations (cf. 02. below),a suffi- ciently complex AUTOMATH-system (the "classical" version: AUT-68, say) can beviewed as an (applied) typed lambda-calculus with a "poly- morphic" type-structure (to coin a word from R.~ilner). Although the underlying 17type-polymorphism"appears to be - at a first look - a very specific one,we realize soon that an AUTOMATH- system is powerful enough such as to allow interpreting (in it) fami- liar typed lambda-calculi as,e.g.,the Curry Theory of Functionality (= "First-order Typed Lambda-calculus") or the Girard-Reynolds Second-order ("parametric") Typed Lambda-calculus.In this respect, an AUTOMATH-system is a "generalized typed lambda-calculus". At a closer examination,the "pure" (i.e.,constantless) part of the- se systems turns out to be much similar if not equivalent to other typed lambda-calculi occurring in the recent literature on the foun- dations of logic and mathematics.(Specifically,"Pure AUT-68" is,in fact,a fully formalized version of J.P.Seldinls Theory of Generali- zed Functionality (cf.Annals Math.Log. -12,1979,29-59),and - it becomes - under an appropriate translation - equivalent to the "!-fragmentn of Martin-Lof's Theori(es) of Types (cf. 8eferences,under MABTIX-LgF'.) On the other hand,the specific manipulation of the AUT(OMATH)-cons- tants - a .E& generis feature of these systems - is,up to a certain point,independent from the underlying lambda-calculus part and can be used,with similar effects,in connection with any other typed lamb- da-calculus. The present work is intended to discuss in detail a theoretical as- pect of the main AUTOMATH-systems,viz. the possibility of fvsepara- ting" the "lambda-calculus-freevfpart (usually identified as "Primi- tive AUTOMATH") from a "full AUT-systemn.Despite the fact that the so-called "Primitive AUTOMATH" can be defined independently,it is not immediately clear,from the language-definition of "biggervfsystems, that the corresponding "correctness categoriesffof the latter are actually "conservativeu over those of the former one. The affirmative answer (given in Z.,below) insures the fact that the "definitional mechanismu of an AUT-system is actually an indepen- dent,nsuper-imposed" structure on a "Pure AUTOMATHIT-system and shows that a (theoretical) study of the latter (which is,properly speaking, a Chapter of ~ambda-calculus) might be also profitable for the speci- fic purposes of the AUTOMATH Project. Other theoretical aspects of these systems,as,e.g.,a "global proof- theoryvvand a umathematicalfvsemantics (model theory) will be dis- cussed elsewhere. ACKNOWLEDGEMENTS. This book is the outcome of a not too remote and particularly fruit- ful period of time when I was employed by the Eindhoven.University of Technology (TITHE"),~~the Department of Mathematics and Computing Science. It is probably premature,for me,to be aware of the various influen- ces I have undergone during my stay in Eindhoven.Still,I have a num- ber of specific debts to record. First among these is to Professor N.G.de Bruijn who has suggested the subject of this work and has been the main source of counse1,encou- ragement and criticism during the months I was writing the book. His views on AUTOMATH and (Typed) Lambda-calculus,in genera1,have exerced a decisive influence on the present work. Next I am indebted to my colleagues in the AUTOMATH research group, who have helped me improve my understanding of the subject-matter. iii I would like to thank,in particular,L.S.van Benthem Jutting,who has spent - altogether - long weeks on several "preliminary" drafts of the main text,suggesting numerous improvements and providing detailed criticism. Many ideas on a rigorous language description of the main AUTOMATH systems - which have been used in the book - should be actually credited to D.T.van Daalen,who advised me on a couple of critical aspects of the proof-theory of AUTOMATH. Of course,I still remain the only responsible one for the possible shortcomings of the text. My debts to the literature should be obvious from the long list of Reference~~placedat the end of the book.(Quotations are given by author and year of publication/pre-print - the last two digits - for all items that are not official AUT-reports,while the AUT-reports are referred to by author,year of publication/pre-print report number.So DE BRUIJN 80-72 is Number 72 in the official THE-list of the AUT-reports.) At this point I should also acknowledge,again,my debt to Henk Baren- dregt who taught me Lambda-calculus. Hartfelt thanks to a philosopher: David Meredith,for reading and commenting on an early version of the Introduction.(I should perhaps ultimately agree with his "Proof is talk." by humbly noticing that it is so ---once we have talked one and we are able to do it again and again,just to make everybody see what we see.) Thanks are due to Professor A.S.Troelstra for listening to and commenting on my (hasty) views on solipsism in mathematics (pp. 7 -et sqq. below).In a more ~philo~ophical~~setting I would have been more careful in finding the due nuances when speaking about the ll(in)dependence of mathematical results on a communication-process", taking into account the genuine (intuitionistic) counter-examples he has produced.The final form of the discussion was kept un- changed only because I still think it reflects accurately the ll~hilo-- sophyll behind the AUTOMATH Project. I am grateful to the Mathematical Centre which accepted this work for publica- tion as a "Mathematical Centre Tract1'. Final I y, I am indebted to Mrs. Jos6e Brands (~in2hovenUniversity of Technologyland Mr. P.A.J. Hoeben (University of Utrecht) for clerical help in the final preparation of the text. CONTENTS 0. INTRODUCTION 01.AUTOMATH: historical landmarks 02 .AUTOMATH: tracing back motivations 03.AUTOMATH versus semantics 04.The contents of this work 05.Abstract AUTOMATH: definition scheme 1.LANGUAGE DEFINITION: WELL-FoRMEDNESS I0 .Alphabets to use 10 1 .Further structure on (floating) constants 11.Variable-strings 12.Term structures 121 .Terms 122.Canonical terms and definitions 123. Combinatory reduction sys tems 13. Term-strings 14 .E-sentences- 15. Contexts 16. Constructions and sites 2. LANGUAGE DEFINITION: CORRECTNESS 65 20.Correctness categories 65 2 1. Correctness for Primitive and Classical (Abstract) AUTOMATH 7 1 2 1 1. Correctness rules for PA and CA 72 212.Basic epi-theory for PA,CA: "structural lemmas" 80 2 13. Basic epi-theory : invariance under site- and/or context- expansion 9 1 214.Substitution and category correctness 95 22.Correctness for Q-A and QA 9 8 22 1. Correctness rules for Q-A and QA 103 222.Basic epi-theory for Q-A and QA 106 3.GLOBAL STRUCTURE OF CORRECTNESS: PA-SEPARATION 30.What is going on: heuristics 3 1. Tree-analysis 311.Well-formedness part 312.Analytic trees 313.Genetic trees: a digression 314.0n isolating wfe's "on" PA 32.Global structure of (LA-compatible) sites 321.Reduced sites.Reference order in LA-sites 322 .Proof theory: correctness proofs and their indices 323.Analytic sites -qua reducts and some topology 33.Conserving PA-correctness 33 1. "Conservation" over PA 332 .PA-separation and conservativity over PA REFERENCES INDEX OF NOTIONS This work is concerned with the abstract structure of some representative formal languages in the family of mathematical languages AUTOMATH (cf. DE BRUIJN 70-02, 73-3,73-34,80-72,JUTTING 79-46,VAN DAALEN 73-35,80-73).The basic analysis was mainly motivated by the need for a theoretical approach to a couple of open pro- blems concerning conservativity situations in AUTOMATH and closely related formal systems.Soon it turned out the approach which was taken in analysis is more significant for the understanding of the nature of an AUTOMATH-language than was initially thought of. Though essentially self-contained,the expository parts of the present notes do - in general - presurpose sore ~ini~albackground of mathematical logic (as,e.g., first-order logic,set theory and lambda-calculus) and of what is usually called 'If inite mathematics'' (viz. rather elementary fragments of combinatorics) .For ins- tance,some familiarity with the basics of (both "type-free" and "typed") lambda- calculus should certainly enable the reader to follow or to reconstruct,on his own, remote details of the "language theory" of AUTOMATH on which we will not generally insist (and which are copiously documented in recent monographs as BARENDREGT 81,KLOP 80,REZUS 81 and - essentially -VAN DAALEN 80-73).Otherwise,the main exposition is primarily intended for logicians and this is largely reflected in our terminological habits (diverging in several respects from standard termino- logy in AUTOMATH but very close to generally accepted ways of speaking in mathema- tical logic).Of course,mathematicians,computer scientists and philosophers having some basic acquaintance with the methods of symbolic logic are thereby expected to find the present text easily readable.
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