A LANGUAG

Par Nicolaas Govert de BRUTJN

Technological University, Eindhoven, Pays-Bas

1973

LES PRESSES DE L ' U N I V E R S I ~ ' ~ D E M O N T R ~ A L

C.P. 6128, XION IX$,AL 101. C.AN.\I).Z

CONTENTS

page ...................................... In t roduc t i on 6

. . . . . . . . . . . . . . . . . . . . . . Informal i n t roduc t ion t o LSP 14

..................... PAL - pre l iminary o r i e n t a t i o n 24

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D e f i n i t i o n of PAL 26

......... How t o use PAL f o r mathematical reasoning 29

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Metalanguage 35

............................... The Lambda Calculus 39

Descr ip t ion of AUTOMATM . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1

Extens ions of AUT ................................. 5 2

...................................... Bibl iography 56

No te s taken by Barrg Fawcett,

ISotes du cows donne par le professeur Nicohas G. de Bruijn a la d i x i h e session du Sh ina i r e de mathkmatiques supkrieures de La , le 1'Universitt de MontrCal, tenue I'erk 1971. Le SCrninaire es: placc :I. sous les auspices de la SociktC Mathkmatique tlu Canada et d u

Wc have t o s t ~ r t with , in apology f o r trcbknt lrrg AU'I'OMAl'll I n $ 1

semin,lr devoted t o combinatorics . First, AUl'CMA'fll and r e l a t e d languages

were devised a s a s e r i o u s at tempt t o br idge t h e c r e d l b l l ~ t y gap en-

gendred hy long and excess ive ly d e t a i l e d proofs i n r ,~athematics ; t h i s

need is f e l t s t rong ly in p a r t i c u l a r i n combinatorics . Seconi ly, t h e

s tudy of t h e s e languages i t s e l f has many combinatorial a spec t s but i s

a l s o r e l a t e d t o var ious o the r f i e l d s ; l og i c ( i n p a r t i c u l a r combinato-

r i a l l o g i c ) , foundat ions, t h e philosophy of mathematics, t h2 h i s t o r y

of mathematics, mathematics educat ion, t h e u n i f i c a t i o n of matnematicai

d i s c i p l i n e s and computer sc ience .

The speake r ' s i n t e r e s t i n t hese mat te rs began during an a t -

tempt t o c lean h i s desk. Among a sheaf of r e p r i n t s was discovered a

paper which cons i s t ed of only one page, and which could n o t , because

of i t s b r e v i t y , be ignored i n good conscience. The t o p i c of t h e paper

was multiprogramming i n computer systems, with an expos i t ion of ce r -

t a i n procedures which were claimed t o make a system reach c e r t a i n goa ls .

Following a somewhat i n t u i t i v e argument it was remarked, with unusual

and poss ib ly unintended candaur, t h a t " t h i s , t h e au thor be l i eves , com-

p l e t e s t h e proof". The speaker , i n at tempting t o prove a c o r r e c t "Leo-

rem, encountered horrendous d e t a i l s . Here was a p i ece of mathematics

which was d i f f i c u l t t o w r i t e up c o r r e c t l y and a l s o very d i f f i c u l t t o

v e r i f y with confidence. Thus t h e r e i s a need f o r a language which w i l l

al low machines t o check whether such complicated theorems a r e s t a t e d

and proved c o r r e c t l y . The at ta inment of such a language t h a t w i l l em-

I)r.~cc> t i l t b i ) o ~ l y 01 ' ~ . u ~ ~ t - c n t ~ ~ ~ i t t h t ~ l ~ ~ ~ ! i c \ s 8 ~ \ < * , i \ v i * ~ ~ - , l ~ ~ o ~ ~ t \ st[ 1 i ' I

J istant goa 1 . Somc 1)r;lnchcs ot' n~;~tllorn:rt lcs, c.. g . < I 1gc>J~r-;a, \c.: t ! l r x o v i ,

can be formalized e a s i l y . However, problems which a r e expressed i n

i n t u i t i v e language, e . g . t h e ce l eb ra t ed problem of covering a t run -

ca t ed chessboard with dominoes, a r e more d i f f i c u l t t o fo rma l i ze , a t

l e a s t i f we s t a r t from t h e o r i g i n a l geometr ical formula t ion . The s tudy

of t h e s e languages is a branch of cornbinatorial l og i c . As such, it i s

a p i e c e of combinator ics , d e s p i t e t h e f a c t t n a t t h e methods appro-

p r i a t e f o r d i s cus s ion of t h i s t o p i c a r e n o t common combinator ics . We

s h a l l be concerned wi th t h e fo l lowing group of languages.

LS P

SEMIPAL

PAL *

AUTOMATH ( 1968)

AUT-QE (1969)

The languages, arranged i n ascending order of complexity, a r e

not programming languages. That i s , they do not express s e t s of machine

i n s t r u c t i o n s . They a r e simply schemes which ensure t h a t every th ing t h a t

i s s t a t e d c o r r e c t l y i s mathematically c o r r e c t .

To begin in formal ly , l e t us cons ider a book t h a t has been

w r i t t e n l i n e by l i n e . The usual approach is t o p o s i t a pool of assump-

t i o n s about foundat ions , and then t o t e s t t h e s ta tements of t h e book

* P A L must not be confused with t h e programming language which sha re s t h e

same acronym.

a g a i n s t t h e e t e r n a l va lues of t h e pool . Here, however, we s h a l l adopt

t h e novel approach t h a t no foundat iona l system i s given i n advance,

but t h a t one is w r i t t e n i n t o t h e t e x t l i ne -by - l i ne . The problem of i n -

t e r p r e t a t i o n t h a t a r i s e s w i l l be d e a l t with l a t e r .

A s f o r t h e usua l confusion between language and metalanguage,

l e t us say, i n t h e absence of a formal d e f i n i t i o n , t h a t a language i s ,

roughly, a system f o r w r i t i n g books. When we s t e p back and t a l k about

books t h a t have been, o r may be, w r i t t e n , a metalanguage must be used.

Often a language i s extended by inco rpo ra t i on of m e t a l i n p a l f e a t u r e s .

The jump from PAL t o AUTOMATH descr ibed below i s such an ex tens ion .

Matnematical languages or books may have va r ious i n t e r p r e t a -

t i o n s . For example, A . A . Morsefs Theory of S e t s ( C 2 2 3 ) has theorems

which can be i n t e r p r e t a t e d both a s l o g i c a l and a s s e t - t h e o r e t i c a l

theorems. Our aim i s t o d e f i n e a language formally i n such a way t h a t

a computer can check whether a t e x t i s w r i t t e n according t o t h e r u l e s

of t h a t language. The computer cannot t ake r e s p o n s i b i l i t y f o r t h e va-

r i o u s i n t e r p r e t a t i o n s of t h e t e x t , but we ourse lves w i l l be concerned

with i n t e r p r e t a t i o n s f o r t h e sake of t h e a p p l i c a t i o n .

Some a s p e c t s of mathematical p roo f s , l i k e hand-waving and

r h e t o r i c a l dev i ce s , a r e impossible t o formal ize . These techniques can-

not convince a machine of t h e v a l i d i t y of a proof un l e s s they can be

expressed i n a formal language. Although we should l i k e t o be l i eve

t h a t a l l mathematics can be presen ted formal ly , we ought t o r e a l i z e

t h a t mathematics i s a s o c i a l a f f a i r , and r e l a t e d t o t h e o u t s i d e world.

Vague communication i s sometimes t o be p r e f e r r e d t o fo rma l i za t i on . For-

ma l i za t i on i s not every th ing , ye t it i s c e r t a i n l y something.

H i s t o r i c a l l y , t h e idea of a f i x e d formal language d a t e s from

t h e Vienna ( u n t i l 1933) C i r c l e of P o s i t i v i s t s . Witkgenstein s a i d "Don't

ask f o r t h e meaning, ask f o r t h e use", o r t o paraphrase, "ignore i n t e r -

p r e t a t i o n s " . I n t e r e s t i n formal languages increased with t h e advent of

computer languages. We were suddenly made aware of t h e r e a l r equ i r e -

ments. Computers re fused our hand-waving and e t e r n a l language-chan

Formalizat ion beyond t h e s t a g e of foundat ions became ind ispensable .

What a r e t h e impl ica t ions f o r pedagogy ? Some say our teach-

ing i s s u i t e d f o r those who a l r eady know t h e ropes . Some confused s t u -

d e n t s j u s t copy t h e mysterious s ta tements ( they are o f t en c a l l e d " g i f t -

ed"). Is it p o s s i b l e t h a t we j u s t l ack a s u i t a b l e language ? Let u s go

a s f a r a s p o s s i b l e and make something l i k e AUTOMATH a c c e s s i b l e t o

four teen-year o ld s . For a t p r e s e n t , we teach by i n t imida t ion and l e a r n

by i m i t a t i o n . The mys te r ies a r e repea ted i n a r a i s e d vo ice . A t last

t h e s tudent g ives i n . But computers a r e immune t o such t r ea tmen t .

The s tudent might ask why mathematics i s c a s t i n t o t h e DEFINI-

TION-THEOREM-PROOF r e c i p e . I s t h i s t r a d i t i o n ? O r i s it e s s e n t i a l ?

Clea r ly a language is needed t o t a c k l e t h i s ques t i on . One p o s i t i v e a s -

p e c t of PAL and AUTOMATH i s t h e expos i t ion of t h e s t r u c t u r e under ly ing

t h a t r e c i p e .

Our endeavour i s u s e f u l f o r proof-checking and f o r avoiding

t h e danger of i n c o r r e c t usage of otherwise c o r r e c t t h e o r i e s . The pro-

blems of teamwork, man-machine cooperat ion, and p ro ram checking can

be brought up. Today, computer s c i e n t i s t s need e f f i c i e n t techniques

t o check t h e v a l i d i t y of programs ( i . e . , t o check whether t h e program

achieves what t h e programmer c l a ims ) , Hopefully, a la rge p a r t of t h i s

may be done by machines. As it is now, it may happen t h a t i n f e r i o r pro-

grams w r i t t e n by not -so-c lever programmers a r e checked by c l e v e r ex-

p e r t s . To avoid t h e waste of brainpower, it might be b e t t e r t h a t t h e s e

expe r t s w r i t e both t h e program and a s e t of h i n t s by which a computer

can check t h e v a l i d i t y of t h e program.

The study of AUTOMATH has led t o new v i s i o n s of t h e b a s i s

of mathematics, inc luding an a n a l y s i s of i t s s t r u c t u r e and sugges t ions

a s t o how it might be changed. Sometimes it may be revealed whether a

process has i n t r i n s i c o r only h i s t o r i c a l j u s t i f i c a t i o n . There e x i s t s

a popular b e l i e f , not shared by t h e speaker , t h a t a s a t i s f a c t o r y ba-

s i s f o r mathematics i s d i f f i c u l t o r impossible t o a t t a i n . Whitehead and

R u s s e l l s l P r i n c i p i a had a nega t ive e f f e c t on t h i s endeavour (because

it i s t o o complex) . The i s s u e was complicated by Can to r ' s Paradise

(which i s e s s e n t i a l l y a mixing of language and metalanguage). Cantor

invented t h e a lephs ; l a t e r mathematicians considered and adopted them

a s f a m i l i a r cons t ruc t ions . In t h e twent ie th century it is widely be-

l i eved t h a t Cantor ' s Paradise e x i s t s and t h a t i t i s important . In t h e

t w e n t y - f i r s t cen tury , mathematicians may th ink it b e t t e r t o leave Can-

t o r ' s Paradise . They may ge t t h e idea t h a t t h e pa rad i se is a poor b a s i s

f o r mathematics, no ma t t e r how b e a u t i f u l i t may be.

The u s e of t h e s e languages may lead t o new d i s c o v e r i e s ,

pe rhaps by analogy and i n s p e c t i o n . H i s t o r i c a l l y , new l a n g ~ a g e s p r e -

ceded major developments i n mathematics. Although AIJTOMATH h s s not

y e t g iven r i s e t o i n t e r e s t i n g new theorems, t h e f u t u r e p o r t e n d s such

developments a s a g e n e r a l mathemat ical l i b r a r y (DIAL-A-THEOREM ?) ,

improved p u b l i c a t i o n s t a n d a r d s and h i g h e r l e v e l s of man-machine i n -

t e r a c t i o n .

C e r t a i n l y t h e e n t i r e language of mathematics w i l l change

i n t h e f u t u r e . The Greeks i n a n t i q u i t y s t u d i e d geomet r ica l f i g u r e s .

Geometr ical f i g u r e s , however, c o n s t i t u t e on ly a rough approximation

t o a mathemat ical language. Van d e r Waerden ( [ 2 6 ] ) remarked t h a t des -

p i t e t h e obvious i n g e n u i t y of t h e Greeks, t h e i r n o t a t i o n f o r t h e i n -

t e g e r s p rec luded f u r t h e r developments ( they r a n o u t o f B e t t e r s ) . The

Arabs ' tremendous achievement was t h e i n t r o d u c t i o n of B e t t e r v a r i a -

b l e s . When modern a l g e b r a i c n o t a t i o n came t o Europe, Descar tes and

Fermat were a b l e t o "formal ize" geometry i n t h i s new language. Even

t h e development of decimal n o t a t i o n a l r e a d y f a c i l i t a t e d t h e communi-

c a t i o n o f a fragment of mathematics. The s t o r y of t h e Greeks ' f a i l u -

r e caused by l a c k of language might c o n t a i n a moral . Leibniz ' s dreamof

a u n i v e r s a l s c i e n t i f i c language I n which t h i n k i n g is r e p l a c e d by c a l -

c u l a t i o n i s an e x t e n s i o n of D e s c a r t e s ' i d e a .

Boole was i n f l u e n c e d by a s i m i l a r i d e a . We d e v i s e d a langua-

ge f o r a p a r t of l o g i c , but h i s e f f o r t s had no r e l a t i o n t o t h e s t a n -

d a r d mathematics of h i s con temporar ies . The very e x t e n s i v e language o f

Whitehead and R u s s e l l which combined Logic and mathematics, and t h e

formidable na tu re of i t s p r e s e n t a t i o n , has been rnentioriecr p r ev ious ly .

P c , ~ n o wrotc a d c t a I lccl form;^ 1 cncyclol~ctl ~ , i o f n l ~ t h c m a t l c s wh rch rc-

a ; i ~ r l \ toclay . I S , I I)c.r.~oJ p ~ c c c ; w ~ t h a l l i t 5 c lC ' l ; ~~ ls . i t w;i \ f;ir f r t m

bclng accessible t o something l i k c mcchanicai c h c c k ~ n g .

In our p r e s e n t a t i o n , comments may be d e s i r a b l e f o r f l a v o u r ,

r e f e r ences and phys i ca l imp l i ca t i ons , but a r e unnecessary f o r checking

t h e t e x t . This r e f l e c t s t h e s t imulus of computer technology. compute;^

a r e extremely s t u p i d , and don ' t understand what t hey do not hea r . But

they a r e dependable and f a s t .

L .E . . J . Brouwer, Kronecker and H , WeyP r e j e c t e d formalism.

Many paradoxes emerged from Cantor ' s Parad ise , some involv ing mixing

of language and metalanguage, and some, l i k e R u s s e l l ' s paradox, neces-

s i t a t i n g new suppor t s f o r a f a l l i n g bu i ld ing . There i s no guarantee

a g a i n s t t h e appearance of new ant inomies. Brouwer opted f o r cons t ruc-

t i v i t y and fought h i s f o r m a l i s t i c contemporar ies , but h i s wr i t i ngs

g ive us t h e impression t h a t he lacked a s u i t a b l e language f o r h i s c r i -

t i c i s m . Brouwer's i n t u i t i o n i s m seemed t o cause many t e c h n i c a l i t i e s and

t h e r e f o r e d i d no t get much suppor t . Recently, E . Bishop ( [S]) has r e -

vived Brouwer's i deas . Rather than concen t r a t i ng on complicated i n t u i -

t i o n i s t i c counter-examples, he t h inks h i s t a s k i s simply t o prove or-

d ina ry t h i n g s .

S ince 1960, John McCarthy, J .A. Robinson ( f 2S] ) , J . B . Rosser ,

Hao Wang and o t h e r s have s t u d i e d automatic theorem proving. The i r a l -

gorithms lead t o long proofs which do not gene ra l l y correspond t o proofs

:I inathc?n~;it i c i i ~ n wou l k l g i v c . 'I'll i:, i 5 d i l'l'c'rcnt w i t il I;iij!:ii;lgcs i i kc A i l -

'i'OMA'l'11, which a r c , oil t h c other. Ii;rnd, not cf't'cctivc. {'or ;rutonlatic thco-

ry proving. In 141 appears a mathematical language based on t h e pro-

gramming LISP. A proof i s a LISP procedure and some forms of s u b s t i t u -

t i o n a r e permi t ted . The languages VAT168 and VAT'70 a r e d i scussed i n

C11, [21 and C31. The language AUTOMATH (or AUT) was developped by

mathematicians a t Eindhoven, ( see : [ 6 3 , [71 , Ell], [12], Cl41, Cl51,

C201, C211 and C231). AUT may seem t o be t h e worst of a l l a s f a r a s ac-

t u a l w r i t i n g i s concerned, but it seems t o be t h e j e s t i n terms of

f l e x i b i l i t y and genera l a p p l i c a b i l i t y .

We expect t h a t a u se fu l language w i l l dependably t ransmi t o r -

d ina ry mathematics by means of formulae which a r e no t excess ive ly long.

Re l i ab l e methods of extending languages, and of incorpora t ing a u x i l i a -

ry languages a r e r equ i r ed . We expect t h a t a machine can check a book

w r i t t e n i n t h e formal language and dec ide whether it i s c o r r e c t . Compa-

r e t h i s s i t u a t i o n with t h a t of a machine which v e r i f i e s whether a chess

game has been conducted according t o t h e r u l e s . There i s a formal lan-

guage f o r chess which employs symbols l i k e 1. e2 - e4, e7 - e§ 2.

r e s i g n s , e t c . , and t h e r e i s a ma te r i a l i n t e r p r e t a t i o n : p ieces being

moved on a board. Other i tems must be considered, such a s whether c a s t -

l i n g i s s t i l l p o s s i b l e o r whether a pawn may be captured en pas san t . A

more complicated f e a t u r e r s t h c a p p l i c a t i o n of t h e r u l e s about a drawn

game.

What d id we achieve ? An AUTOMATH checker I s a t p r e sen t i n

opera t ion a t Eindhoven. An AUT t e x t can be typed i n l i ne -by - l i ne a t a

te rmina l : t h e computer s i g n a l s e i t h e r 'WRONGv, t oge the r with a d i a -

gnos i s , o r "CORRECT". L.S. J u t t i n g has developped cons iderab le expe-

r i e n c e with t h e proofchecker . A t p r e s e n t , Landau's booklet on t h e foun-

d a t i o n s of a n a l y s i s i s being t r a n s l a t e d i n t o AUT. Although t h e booklet

i s very d e t a i l e d , t h e t r a n s l a t o r must f i l l s e v e r a l gaps. Eventua l ly ,

t h e machine w i l l c e r t a i n l y be a b l e t o read Landau's book i n something

l i k e an hour .

A novel aspec t of AUT is t h e p r e s e n t a t i o n of p roofs as cons-

t r u c t i o n s . Cons t ruc t ion of o b j e c t s and cons t ruc t ion of p roofs a r e

t r e a t e d i n l i k e manner. This s o r t of mixture occurs n a t u r a l l y i n t h e

w r i t i n g of o rd inary mathematical t e x t s . For example, r u l e r and compass

cons t ruc t ions i n s y n t h e t i c geometry u sua l ly involve t h e fo l lowing types

of cons t ruc t ion : 1) d e f i n i t i o n s o f geometr ical o b j e c t s i n terms

of o t h e r s , 2 ) d e s c r i p t i o n s of geometr ical cons t ruc t ions with t h e r u l e r

and compass, 3) cons t ruc t ion of p ropos i t i ons and pred ica tes ,and 4) t h e

cons t ruc t ion of p roo f s . I f i n a geometr ical cons t ruc t ion we connect two

p o i n t s by a s t r a i g h t l i n e , we must f i r s t c o n s t r u c t a proof t h a t they a r e

d i s t i n c t . I f we a s s e r t t h a t a cons t ruc ted po in t i s t h e barycent re of

a t r i a n g l e , then a d e f i n i t i o n cf t h a t no t ion ought t o precede.

Informal In t roduc t ion t o LSP

LSP has analphabet c o n s i s t i n g of two kinds of symbols,

(1) cons t an t s a , b , c ,d , . . . ( 2 ) v a r i a b l e s s , t , u , v , ...

An e x t r a symbol, PN ("pr imir ive no t ionu ) and t h e s i g n s ( 1 , : =

Lxample of an LSP book (Pnterpre t , i t ion : t h e definrtlon of' functmonj

Observe t h a t every Xlne h a s i t s own identifier on t h e l e f t , and

3 (poss ib l e empty) sequence of d i s t i n c t variables a t t ached t o i t . On t h e

r i g h t occurs e i t h e r t h e symbol PN or an expression. AT1 v a r i a b l e s occur-

ing i n an expression on t h e r i g h t a l s o occur on t h e l e f t , but not necessarily

conversely. Every cons tan t on t h e r i g h t has occurred i n a previous Pine. Eve-

r y cons tan t on t h e r i g h t is t h e head of an expression with a s many subex-

p re s s ions a s t h e number of v a r i a b l e s a s s o c i a t e d with t h a t cons tan t a t t h e

l i n e where it was introduced. This nunber i s c a l l e d t h e length of t h e

cons t an t . In t h e i n t e r p r e t a t i o n , we say t h a t t h e l e f t hand s i d e of a l i n e i s

def ined by t h e r ight-hand s i d e (unless t h e r igh t -hand s i d e i s PN).

The not ion ":orrect LSP" book i s def ined r e c u r s i v e l y . F i r s t t h e

empty book is c o r r e c t . Assuming char we have a c o r r e c t book B , we d e f i n e

an admiss ib le expression w i t h r e spec t t o t h a t book as fol lows :

1) each v a r i a b l e i s & d n i s s i b l e ;

2 ) cons t an t s of length ze ro are admiss ib le ;

3) a cons tan t of p o s i t i v e length k i s admiss ib le when it i s

followed by a s t r i n g of k admissible expressions separaeed by commas,

and surrounded by parentheses . In both cases , "constant" means "constant

t h a t has appeared on t h e le f t -hand s i d e i n t h e bookv.

Assuming t h a t t h e book B i s c o r r e c t , a c o r r e c t one- l ine ex-

t ens ion of B i s w r i t t e n by choosing any - new cons tan t t oge the r with any

sequence of v a r i a b l e s followed by := . On t h e r i g h t , we w r i t e any expression

admiss ib le with respec t t o t h e old book, provided t h a t only v a r i a b l e s

which appear on t h e l e f t a r e used. O r we may w r i t e PN with no condi t ion .

Let us cons ider an example with an i n t e r p r e t a t i o n from Boolean log ic .

non (x)

imp1 (X,Y)

notboth (x ,y)

and (x, Y)

or (x ,y )

a t l e a s t o n e ( x , y , z )

a l l t h r e e ( x , y , z )

. - - PN

. - . - PN

. .- - impl(x,non(y)j

. . - - non(notboth (x ,y j )

. - . - not both (non (x) , non (y) )

:= o r ( x , o r ( y , z ) ]

. - . - and (x ,and (y , z) )

The not ion of d e f i n i t i o n a l equivalence (DEFEQUAL), i s def ined by

means of t h e opera tor DEF which a p p l i e s t o admiss ib le express ions . The head - of an express ion i s t h e f i r s t symbol which appears . A cons tan t i s primi-

t i v e i f it is def ined by PN . D E F i s undefined i f t h e head i s p r i m i t i v e - o r a v a r i a b l e . The a p p l i c a t i o n qf t h e opera tor DEF w i l l be i l l u s t r a t e d by

t h e f i r s t example of an LSP t e x t , i i n e 6 . To c a l c u l a t e

f i r s t t ake t h e head (= d ) , look u~ i t s d e f i n i t i o n ,

substitute

applied to the - x and - t that are underlined and not to those appearing

on the right in the substitution. The result is

B Now the notion DEFEQUAL (notation =), w i t h respect to a book,

D is defined recursively. We shall write C = C (the C's are metalingual 1 2

symbols) . We require

D (1) if C1 = DEF C2, then C1 - C2 t

(2) if C and C' have the same head, with subexpressions

D C1, C2, ..., Cr and Ci, Xi, ..., C:, respectively, then if L i = Ci then

(3) DEFEQUAL is ref kxive , symmetric and transitive.

primitive.

THEOREM 1

then there

An expression C is called normal if a11 its constants are

(For example, a(t,a(t,t)),) Now we state without proof:

If Z is admissible with respect to a correct LSP book,

is exactly one normal e-;pression (called the normal form of -- U C and written NF(C)) such that C -- NF(C).

Roughly speaking, we obtain the normal form i f we contirius

to apply the operator DEF until all non-primitive constants are gone.

In our example,

Note that this example was taken from the very beginning of our book; it

can be expected that it gets much worse later on, so that normal forms

have theoretical interest only.

THEOREM 2

D C, = C iff NF(C1) = N F ( C 2 ) .

2

Let us remark that the notions of DEF, NF and DEFEQUAL play

no role in the definition of language, nor in testing the language. A

computer programme which checks LSP books is easy to write. Given a correct

book B and two expressions, Cl and C2, THEOREM 2 produces an algorithm to

determine whether they are admissible and definitionally equivalent. The

head of N F ( C ) is easy to find by applying DEF. The structure of an LSP

book is that of a directed graph without loops. The exact linear order of

D the lines is immaterial. To decide whether C = C take the expression 1 2 '

with the younger head, apply DEF and look again. If the heads are equal,

check whether the subexpressions are definitionally equivalent. CAUTION:

One must check wherher an expression does really depend on all its variables.

Possibly some of them mighr be Ir.active (i.e., they do not occur in the --- normal form of the expression). It is not difficult to keep a list of

active variables from line to linz. It would be better t9 write texts

without inactive variables; in practice they do not often occur.

SEMIPAL arises from LSP by declaring variables, coneex:

indication and abbreviations. With the new symbol: x:= - called a block opener, we declare the variable x . No longer are lists of

variables written on the left. Declared variables form together a

limited context. A set of nested blocks indicates the sequence of

variables to be employed. For clear exposition, the block structure

of a SEMIPAL book is indicated by vertical bars. Machines (and typists)

abhor vertical lines, preferring horizontal indicator strings. The

context indicator of a line is the last previously declared variable or

0 (an extra symbol of SEMIPAL) in case there are none.

Consider the example:

Indicator String

Empty

Context Indicator

The t r a n s l a t i o n t o LSP i s s t r a i g h t forward.

In t r a n s l a t i n g an LSP t e x t i n t o SEMIPAL i t might become

necessary t o a l t e r t h e v a r i a b l e s f i r s t . Note t h a t t h e block s t r u c t u r e

and i n d i c a t o r s t r i n g s convey i d e n t i c a l information; one may be r e t r i e v e d

from knowledge of t h e o t h e r . Let us remark t h a t i e i s p e r a i t t e d t o re -

open an o ld block.

An example of an LSP hook t h a t cannot be t r a n s l a t e d immediately

i n t o SEMIPAL i s e a s i l y cons t ruc ted , v i z . :

The names of t h e v a r i a b l e s must be a l t e r e d s o t h a t , i f two

s t r i n g s of v a r i a b l e s occurr ing t o t h e l e f t o f : = have v a r i a b l e s i n common,

then t h e s e common v a r i a b l e s should form an i d e n t i c a l i n i t i a l segment of

both s t r i n g s . (For example: x,y,:,w and x ,y ,p ,q , a r e compatible, bu t

not x,y,z,w and x,y,p,q,w). Tke zondi t ion t h a t a v a r i a b l e may not be

dec l a red twice seems inipractica! f o r wr i t i ng mathematics (althoogh computers

d i s a g r e e ) . With a c e r t a i n amourit of ca re , conventions may be s t r u c k t o govern

t h e re -use of old v a r i a b l e s . Yet l r i s b e t t e r f o r t h e o r e t i c a i purposes t o

avoid t h i s .

In SEMIPAL an e x ~ r e s s i o n l i k e bj) : , , . . . , C i can be 1 k

abbrevia ted : i f x l Y . . - , x k i s t h e i n d i c a t o r s t r i n g o f b ; and i f

L1 = x E = x ..,I. = x . , then we may w r i t e 3(C. . . . > I $ ' o r 1 ' 2 2 ' ' 7 3 1'1"

j u s t b i f j = k. That i s , i f i n t he abbreviated form t h e r e a r e not

enough subexpressions, they a r e completed a t t h e f r o n t by adding t h e

beginning of t h e i n d i c a t o r s t r i n g of b . A d i r e c t d e f i n i t i o n of a

B-acceptable abbrevia ted expression i s poss i l e b u t t h i s would complicate

somewhat t h e add i t i on of a l i n e t o t h e book.

D The not ions = and NF were def ined by means of t h e unabbrevinted

express ions . There i s an i n t e r p r e t a t i o n i n terms of t h e abbreviated forms

but t h a t is hard ly necessary . An exain2le of an abbrevia ted book fo l lows .

I t i s t h e same example from Boolean

t h i s form i t i s not very readable .

X

non

Y

Y imp 1

Y notboth

and

o r

z

a t l e a s t o n e

a i l t h r e e

log ic given previous ly . Note t h a t i n

PN

PN

imp1 (non (y) )

non (notboth)

notboth (non'non (y) )

o r ( o r ( r , z j >

and(and{y,z)) .

I t seems overdone t o devote a s p e c i a l l i n e t o each "block opener".

I t does s t r e s s t h e f a c t t h a t a l l i d e n t i f i e r s a r e d i s t i n c t . The e x t r a l i n e s -

will have a greater signiflcance in the language PAL. One advantage of

the outlined way of context indication and abbreviation r s tkaL quite

often a large number of lines in a book will depend upon a fixed nunber

of variables. These need not be repeated in every line. The block

structure allows us to introduce "localt1 constants which appear only in

a limited context. This feature parallels the similar practice in every-

day mathematical writing.

A block is not necessarily a connected piece of information.

Lines which are written in an old context may be added at a later stage;

old variables may be revived, and old blocks re-opened. In our examples,

the blocks consisted of sets of consecutive lines. This occurred merely

to sharpen the exposition.

In speaking about a SEMIPAL book we use metalingual terminology

like, "something is written in context y", or, "the indicator string in

context y (the empty string if y = O ) " , or, "in context y the only live

variables are those of the indicator string". This terminology, and its

metalingual character are conspicuous. Thus the indicator string in context

x is a string xyx2, . . . , xk where x = x and x = indicator of x k i itli'

Substitution

Let xl,x Z , . . . ,xk be disrinct variables and A1,A 2'...,Ak,

expressions. The symbol S I' denotes the result x1 + Al' x2 + A2, . . . ,xk + Ak

of substituting simultaneously

in 7 (i.e. replace each x. in r by the metalingual symbol A . , then 1 I

replace each A by the expression for which it stands). i

Formally speaking, a (metalingual) notation to distinguish

objects from their names is required. The customary practice of placing

the name of an object in quotation marks is ill-advised, since the name

ought to determine the object and not conversely. Our convention shall

be to underline the object to distinguish it from its name. For example,

we do not write

Montreal is very clean

"Montreal" has eight letters

but

Montreal is very clean

Montreal has eight letters.

Accordingly, our substitution recipe should read:

Ilowever, we shall not apply this notation in these notes.

THEOREM 3

If X~,X~,..~.X. are distinct variables and if A i? k 1, 2 , . . , A k >

are acceptable expressions with r-esvect to a correct LSP book B , then

The proof is omitted.

PAL - Pre l iminary Or i en t a t i on

To cap tu re t h e essence of s ta tements l i k e

i x i s a po in t

y i s a f i n e

1 t h e d i s t a n c e between x and y i s a r e a l iiumber

we should be a b l e t o w r i t e l i n e s of t h e form

i x . - . - p o i n t

. - . - l i ne 1 :ist := . . . . . . r e a l ,

That i s : we should be a b l e t o a t t a c h c a t e g o r i e s t o t h e o b j e c t s

we a r e d i s cus s ing .

PAL i s an ex tens ion of SEMIPAL which al lows f o r a t t a t c h i n g ca-

t e g o r i e s . When s u b s t i t u t i n g express ions , f o r example w r i t i n g d i s t (Cl ,CZ) ,

we s h a l l r e q u i r e t h a t t h e express ions C l and C 2 be of t h e proper ca-

t ego ry . A ca tegory i s a t t ached t o every express ion . This i s a r e s t r i c t i o n

on t h e express ions we can admit. Moreover,we should Pike t o be a b l e t o

in t roduce new c a t e g o r i e s , pos s ib ly a s a func t ion of some v a r i a b l e s . To

expedi te t h i s , we in t roduce t h e symbol : type. PAL i s adequate f o r ex-

p r e s s i n g elementary geometry and f i r s t o rde r i o g i c (without f u n c t i o n a l

a b s t r a c t i o n ) . Consider two examples of a p i ece of a PAL book. The f i r s t ,

a t r a n s l a t i o n of H i l b e r t ' s axioms f o r geometry does not ge t very f a r . The

second has an i n t e r p r e t a t i o n &s t h e d e f i n i t i o n of a @ a r t e s i a n product .

Example

po in t : = P~ type

l i n e : = PN 9E

1 X := - p o i n t . .- - ------ l i n e

Example

O

n

prod

I p r o j

I p r 0 j 2

X

Y

1 oc

number

c a r t p l a n e : : prod (number, nuinber)

p rod

number

number

Here a g a i n , C1 and C 2 a r e m e t a l i n g u a l symbols d e n o t i n g c e r -

t a i n e x p r e s s i o n s .

Among t h e e x p r e s s i o n s i n PAL we make t h e f o l l o w i n g (meta l in -

g u a l ) d i s t i n c t i o n s :

a 1 - e x p r e s s i o n i s t h e symbol t y p e ;

a 2-express ion is t h e name of a c a t e g o r y j e . ~ . p o i n t , l i n e ) ;

a 3 -express ion i s t h e name of a n o b j e c t (e .g . sum(prod(x ,y )z ) ) .

Next we d e f i n e a mapping CAT which a s s o c i a t e s w i t h e v e r y 3-ex-

p r e s s i o n i t s c a t e g o r y , and which a s s o c i a t e s wi th e v e r y 2 - e x p r e s s i o n t h e

symbol t y p e ; CAT(type) i s u d e f i n e d . Thus CAT maps 3 - e x p r e s s i o n s i n t o

2 - e x p r e s s i o n s , and 2 - e x p r e s s i o n s i n t o 1 - e x p r e s s i o n s .

A PAL book s h a l l have t h e form of a SEMIPAL book with an ex t r a

column. I f B i s a c o r r e c t PAL book, then B* , t h e book obtained by

omi t t ing t h e l a s t column, i s a c o r r e c t SEMIPAL book. In t h e l a s t column

t h e en t ry is e i t h e r an expression o r t h e symbol type . I f B i s a c o r r e c t

PAL book, then it remains co r r ec t i f t h e l a s t l i n e i s de l e t ed . The not ions

1) = and NF w i l l be i n t e r p r e t e d with respec t t o R* .

The d e f i n i t i o n of t h e not ion "cor rec t PAL book" s h a l l be arranged

along s i m i l a r l i n e s a s t h e d e f i n i t i o n of t h e not ion ' k c o r e c t LSP book1'. Na-

mely :

1) The empty book i s c o r r e c t ;

2 ) If 3 i s a c o r r e c t PAL book,then it i s ind ica ted how admissi-

b l e 0-expressions a r e b u i l t . Once a new expression has been

cons t ruc t ed , i t s category i s def ined ;

3) How B may be extended by one l i n e i s descr ibed .

DEFINITION OF PAL -- --

1 ) The empty book i s co r r ec t ;

2 ) Lrt 5 be a c o r r e c t book,and 0 one of it s v a r i a b l e s or O . We def ine a B-a6sissjbLe expression X and i t s category CAT(C)

r e c u r s i v e l y :

i ) type i s admissible

i i ) if x occurs i n t h e i n d i c a t o r s t r i n g a t 8 , then

x i s admiss ib le and CAT(xj is the entry i n t h e ca tegory column a t

7-

x ( i .e , t h e expression A when x i s def ined i n t h e l i n e

x: = - r iii) l e t h t e a cons tan t of 0 with i ~ d i c a t o r s t r i n g

x!,.. and ca t egor i e s I' ..., r. r e spec t ive ly ; t h a t i s , 1 ' K

I p k . . - - rk i . . . b := Q c

are (not necessarily adjacent) lines of B ,

Let A . . A be B-admissible expressions and all different

from type, so that CAT(Al), . . . ,CAT(Ak) are defined. Then b(Al,. . . ,Ak) is admissible provided that

D rmyh 1-r 1 1

In that case CATIb ( A l y . , . , Ak)) is defined as

or, if is PN then S C . xl -+ A1>...,xk -+ hk

3) If B is a correct book and 0 either 0 or a variable of B,

then each of thc fo:Iowing lines in context O provides a correct extension

of R .

q , o and r are new i d e n t i f l e r r , C1 is type or a B-admissible 2-ex-

pression, C is a B-admissibl2 2-expression or 3-expression and 2

I n consequence, i f B is a c o r r e c t PAL. hook :tnd A i s a 1 3 - a c l -

m i s s ib l c cxprcss ion J i f ' fc rcn t from t y p e ) then

a ) A i s X3*-:~tlmissible ;

b) CAT ( A ) i s R - admiss ib le (and R*- admissible) ;

C ) NF (CAT(A)) = NF (CAT(NF (A))) .

Now we may check t h e t e x t below by means of t h e d e f i n i t i o n of

Let us remark t h a t a PAC book becomes a SEMIPAL book by cance l -

l i n g :he l a s t column. A SEMIPAL book becomes a PAL book by adding a f i r s t

l i n e 8 : = PN , 2nd t h e n adding a l a s t column with a s t h e only

e n t r y , o r s i w l y by a d d i n g a list column with type a s t h e only entry..41so,

Let us n o t i c e t h a t we maintail1 our abbrevia t ion f a c i l i t i e s . That i s :

Suppose a cons tan t 6 i s introduced i n context x 1'. ' .,XL . Supnosc IP some coqtext 8 we m e C (C4, . . . , F j , i. e . we use C with l e s s a.

-l' C ( x l , . . . , x ~ - ~ , , , ~ , . . . \ ' ) ; we of c o u r s e requil .c t l la t x ' ' , l? 1 ' * * ' ,Xk- 1

is a s u b s t r i n g o f t h e i n d i c a t o r - s t r i n g 8 .

How t o u s e PAL f o r ma themat ica l r eason ing , ( s e e [ 6 ] , p . 15)

So f a r o u r concern h a s been merely t o e x p r e s s t h i n g s by means

of LSP, SEMIPAL and PAL. However, mathemat ic ians a r e u s u a l l y more i n t e r e s -

t e d i n how t o p rove theorems, r a t h e r t h a n e x p r e s s i n g t h i n g s . Mathematics

has t h e same b lock s t r u c t u r e a s PAL, b u t t h e r e a r e two ways t o open a

b l o c k . One i s by i n t r o d u c i n g a v a r i a b l e t h a t w i l l have meaning throughout

t h e b lock ; t h e o t h e r i s by making a n assumpt ion t h a t i s v a l i d th roughout

t h e b l o c k . The second c a s e s h a l l be d e a l t w i t h by r e p r e s e n t i n g s t a t e m e n t s

by c a t e g o r i e s . C o n s t r u c t i n e a n obj e c t wi th t h a t c a t e g o r y means a s s e r t -

i n g t h e s t a t e m e n t . T h i s may be done by means.of , PN o r an expres -

s f o n , c o r r e s p o n d i n g t o a s s e r t i o n by assumpt ion, a n axiom, o r p r o o f , res-

p e c t i v e l y . Thus a n a s s e r t i o n may be a l i n e of t h e form

A i s c a l l e d a p roof f o r C , and t h e c a t e g o r y C may be though t of a s

t h e c l a s s of a l l i t s p r o o f s . Recall t h e t e x t on page 2 3 . A s an example of

a theorem and i t s p r o o f , l e t u s w r i t e from t h e t e x t , i n t h e n o t a t i o n of

Lindenbaum, t h e s t a t e m e n t s :

I f we wish t o d e r i v e t h e s t a t e m e n t s ,

and

we may do s o by t h e f o l l o w i n g argument. To p rove 111, f i r s t d e r i v e

y % y from I . Next u s e 11 w i t h z + y t o d e r i v e y % x . To prove TV

r e p l a c e x -, z , z -+ x i n 11 t o o b t a i n % Y ' " x ' b z . Then o b t a i n

z % y from y 'L z by 111 (wi th x -+ y, y -t z).

I n a fo rmat s u g g e s t i v e o f PAL we a r r a n g e t h e p r o o f as:

I (x) i s a ( p r i m i t i v e ) p roof f o r x % x .

I1 (x ,y ,p roof x % y , z , p roof z pb y) i s a ( p r i m i t i v e ) p roof

f o r x s x .

I I I ( x , y , p r o o f x % y) : = I I ( x , y , p roof x 'L y , y , I ( y ) ) i s a

p roof f o r y % x .

IV(x,y , p roof x % y , z, p r o o f y % z) := I I ( z , y , I I I ( y , z , p roof

x 'L y ) , x , p roof x % y ) is a p roof f o r x % z .

Now t h i s t r a n s l a t e s i n t o PAL a u t o m a t i c a l l y . We add t h e t y p e 6

e v e r y t h i n g i s about ; but s i n c e t h a t i s n o t s u b s t i t u t e d i n t o t h e t e x t , no

d i f f i c u l t i e s a r e p r e s e n t e d .

An a s s e r t i o n seems t o be a more n a t u r a l n o t i o n t h a n t h a t o f a

p r o p o s i t i o n (which may o r may not be t r u e ) . However, p r o p o s i t i o n s ( o r

boo leans ) may be i n t r o d u c e d i n t o "AL by a d m i t t i n g t h e c a t e g o r y "bool",

c o n s i s t i n g o f a l l p r o p o s i t i o n s . The book may begin a s f o l l o w s :

/ TRUE : = PN t y p e

I f i n a c e r t a i n c o n t e x t a p p e a r s a l i n e l l h e :

. - . . . . - . . . . . . TRUE (a )

where a is a boolean ( i n t h a t c o n t e x t ) , t h e n t h e i n t e r p r e t a t i o n i n us -

u a l mathemat ica l t e r m s i s t h a t a i s a s s e r t e d .

Modus Ponens p r e s e n t e d i n t h i s format ( s e e [ 6 ] , p . 1 8 ) becomes

0

0

b

0

X

v

Y

a s p 4

a s p 5

boo1 : = PN

/ b : = ---------

/ TRUE : = PN

Y . . . _ --

i m p 1 : = PN

a s p 4 : =

a s p 5 :=:

modpon : - PN

t YP e

boo1

type

bool

boo 1

boo1

'TRUE (x j

TRUE (imp 1 (x , y ) )

TRUE 0.1

To u s e such a p i e c e of t e x t , suppose t h a t we have l i n e s l i k e :

t h e n we nay add -

' fhc I l c . r i v ; ~ t io11 r u l ~ s , c v iJc11t l y , :i rc. i I!.;; t heo rc~~~ l s . N c w rli! c , \

can be devised and used. Note t h a t t h e t e x t i s not subdivided i n t o p; l r is

a long t h e usua l def ini t ion-theorem-proof model. Every l i n e is a r e s u i t

t h a t may be used whenever we wish. A theorem i s never announced p r i o r

t o i t s proof ; a r e s u l t cannot be s t a t e d u n t i l it i s der ived .

In t h e next example we d e f i n e something of t h e form : " i f ...,

then ..., e l s e ...". Some assumptions a r e needed ; t h e f i r s t requirement

i s t h a t axioms f o r e q u a l i t y a r e given before ( t h i s i s an awkward f e a t u r e ) .

Also, negat ion should be defined beforehand.

[axioms f o r IS , f o r example, Lindenbaumts]

condi t ion : = type

value 1 . - . - 8

value 2 . - . - e

i f t h e n e l s e := PN 8

i f . - . - condi t ion

thenl := PN IS(€), va lue I , i f t h e n e l s e )

ifhowever := -- NON (condit ion)

then2 := PN IS(0, va lue 2 , i f t h e n e l s e ) .

Now suppose t h a t , i n t h e presence of expressions "real" , "O",

1 1 1 1 1 I ~ ~ ~ ~ I I , and "grea te r (a , b) l f e t c . , we wish t o d e f i n e t h e func t ion

given by t h e r u l e :

'l'hc dcr iv; i t io r~ r i ~ l d s , c v ~ d c r l t l y , arc' j i i q t t h c ~ o r ' c ~ ~ i ~ s . N c ~ v I - i i i

can be devised and used. Note t h a t t h e t e x t i s not subdivided i n t o p r t s

along t h e usua l def ini t ion-theorem-proof model. Every l i n e is a r e s u l t

t h a t may be used whenever we wish. A theorem is never announced p r i o r

t o i t s proof ; a r e s u l t cannot be s t a t e d u n t i l it i s der ived .

In t h e next example we d e f i n e something of t h e form : "if ..., then . . . , e l s e ...". Some assumptions a r e needed ; t h e f i r s t requirement

is t h a t axioms f o r e q u a l i t y a r e given before ( t h i s is an awkward f e a t u r e ) .

Also, negat ion should be defined beforehand.

type

[axioms f o r IS , f o r example, Lindenbaumls]

condi t ion : = type

value 1 . - . - 8

va lue 2 . - . - e

i f t h e n e l s e := PN 8

i f , - . - condi t ion

thenl := PN IS(8, va lue 1, i f t h e n e l s e )

ifhowever := -- NON (condit ion)

then2 := PN IS(8, va lue 2 , i f t h e n e l s e )

Now suppose t h a t , i n t n e presence of express ions " r ea l t1 , " O w ,

, 1111 , w ~ ~ ~ I I , and "grea te r (a ,b)" e t c . , we wish t o d e f i n e t h e func t ion

given by t h e r u l e :

f (x) - Then we may proceed :

x := r e a i

, f : = i f t h e n e l s e ( r e a l , g r e a t e r [ x , O ) , x , sum(x,i)) r e 3 i

And we might app ly t h e t e x t a s f o l l o w s :

a : = . . . . real

weknow := .... g r e a t e r (a, 0)

hence : = t h e n ( r e a l , g r e a t e r (a,O) ,a, sum(a, 1) J ~ e k ~ ~ ~ ) IS(real,a, Et'a:i> 1

One f e e l s t h a t such l i n e s ought t o be inven ted by machines. An a u x i l i a r y

language, o r a smal l handbook, might inform us how t o c o n s t r u c t such

t h i n g s wi thou t t h i n k i n g .

Thc l i m i t a t i o n s of PAL

I n a book wi th a b lock l i k e :

l l i I x : = number

f := C (x) number

where C(x) deno tes an e x p r e s s i o n c o n t a i n i n g x , a f u n c t i o n i s a v a i l a b l e ,

a c t u a l l y g iven by an e x p l i c i t c o n s t r u c t i o n . Rut we cannot say s%ssume we

have a block l i k e t h i s " , o r , " l e t f be a f u n c t i o n mapping something t o

something e l s e " , o r , " f o r every f u n c t i o n i t i s t r u e t ha t " .

Then we cannot say t h a t "a --> b i s t rue" . That would amounr t o

saying t h a t , i n t h e language, t h e r e - i s a block l i k e t h a t .

In a d d i t i o n , t h e induc t ion axiom f o r t h e n a t u r a l numbers can-

no t be descr ibed p rope r ly . Tn t h e 18 th cen tury impl ica t ion and induc t ion

were c e r t a i n l y somewhat meta l ingua l , and t h e r e f o r e , myster ious. In cur -

r e n t mathematics we have incorpora ted them i n t o our formalisms. Likewise

f o r P A L , some metal ingual ex tens ion i s indicated.

Change of V a r i a b l e s

The p o s s i b i l i t y of in te rchang in? : v a r i a b l e s r e q u i r e s a meta lan-

guage. For example, In e lementa ry geometry, a proof might be given i n -

v o i v l n g a t r l a n g l e whose v e r t l c e s a r e l e t t e r e d A , B and C . Then it

might be s t a t e d t h a t t h e p roof caii be ;eyeaxell w i t h K and A i n t e r -

changed. T h i s is economical , but n o r i e s s e n t r a i , u s e of meta language. [One

c o u l d , a f t e r a l l , a v o i d metalanguage by r eyea t l ag :ce p rdo f ) . T t shou ld be

remarked t h a t i n PAL t h i s u s e of metalanguage i s p o s s l b i e .

Models

Motalsngual f e a t u r e s p r e s e n t themselves I n t h e c a s e of models

Assume t h e r e is an a x l o n , i n con tex t o ,

fo i lowed by a chapter k o i t h e book, of " c o n c i u s i o n " . And l a t e r , let

ds suppose something i n c a t e g o r y 1s o b t a i n e d I n some c o n t e x t .

We shou ld l i k e t o s a y , ' T v e r y t h i n g t3a t was cieriveb f o r k can be d e r i -

ved w i t h a ". We have d mode; f o r t h e axiom. Indeed, t h i s can be done

by r e w r i t i n g c h a p t e r K . However, we shou ld p r e f e r t o a b b r e v i a t e t h i s .

An n b b r e v l a t l o n is p o s s l o l e ;f t h e Ph can h e r e p l a c e d by a b l o c k opener .

I f however, t h e axiom i s "covered" t h u s :

then t h e abbrcvio t inn i s more d i f f i c u l t . I n some ianguagcs of AU'T t y p e

(depending on whether r and a r e 1- o r 2-exprcssions) t h i s can be

accomplished. In t h e case of a s e t of axioms, complicat ions abound. ( I t - can happen t h a t t h e second axiom can only be formulated a f t e r t h e f i r s t

i s assumed, e t c . ) . Axioms can be dispensed with i n an ex tens ion of

AUTOMATH c a l l e d AUT-SL ( s i n g l e - l i n e AUTOMATH). AUT-SL i s however mainly

of t h e o r e t i c a l importance.

Local Axioms

Let u s s a y t h a t , In a mathematical encyclopaedia, i t i s d e s i r e d

t o p l acc a s e t of axioms, f o r p r o j e c t r v c geometry. Af te r t h e s e axioms

a r e w r i t t e n down, they a r c a v a i l a b l e f o r use everywhere i n t h e book.

T h i s very unaes the t i c idiosyncrasy can be removed by a lock-and-key tech-

n ique . We s t a r t with one harmless axiom :

projgeom := PN

T h e r e a f t e r , we cont inue :

i f := projgeom ( the only appearance of t h i s type)

followed by whatever axioms arc needcd.

Outside t h e block t h e axioms cannot be used a s long a s t h e r e i s nothing

i n ;.rojgeom. f h_.rd cjf c h n s r i t y b e l t !

Composite Not ions

.'+. composite noZion l i k e " l e t G be a grccp" cannot be i n t r o - . duced i n t h e same manner z s .

x := number

One must be a b l e t o e x p r e s s f o r [ G , o, * ) " l e t G he n s e t , l e t a

be a b i n a r y o p e r a t i o n on G , l e t * be a proof f o r t h c group axioms".

One needs a scheme approximately l i k e :

G . - . - -- s e t

p rod . . - - map G x G + G

assump 1 := . . .

j assump 2 := ...

The c h a s t i t y b e l t t e c h n i q u e p e r m i t s a q u i c k r e f e r e n c e t o a composi te d e c l a -

r a t i o n . Thus :

GROUP : = PN type

key . - . - GROUP

G := PN s e t

prod : = PN . . . . . assump : = PN . . . . .

e t c . . .

The e x t r a axioms i n t r o d u c e d by t h e t e c h n i q u e a r e harmless ; n o t h i n g can

be d e r i v e d from them a s long as we do n o t say t h a r wc assume we have a

key. But on t h e o t h e r hand, i f we do have an o b j e c t whlch does s a t i s f y

t h e group axioms, we a r e n o t y e t i n t h e p o s i t i o n where we can produce n

key . Without a f u r t h e r e x t e n s i o n o f t h e language i t a p p e a r s t h a t , i n

t h o s e c i r c u m s t a n c e s , a r e bound t o do t h e fo l lowing

G* . - s e t

1 prod* . . - - map

axiom* := PN GROUP

S u b s e q u e n t l y , it would be n e c e s s a r y t o w r i t e , by means of a x i o m a t i c

e q u a l i t y t h a t G* i s e q u a l t o G(axiom*) e t c ... A l l t h i s a p p e a r s t o

be v e r y clumsy. N e v e r t h e l e s s t h e lock-and-key t e c h n i q u e i s u s e f u l i n

many s i t u a t i o n s . AUT h a s no f a c i l i t i e s f o r condensing composi te de-

c l a r a t i o n s i n t o s i n g l e l i n e s . We need a u x i l i a r y languages f o r t h i s .

A thorough i n v e s t i g a t i o n i n t o t h e m e t a i i n g u a l r equ i rements

of mathemat ica l languages h a s y e t t o be under taken .

The Lambda C a l c u l u s (A . Church

I t i s s t r a n g e t h a t t h e e f f i c i e n t n o t a t i o n of Church f o r r e -

p r e s e n t i n g f u n c t i o n s i s n o t i n g e n e r a l u s e . Perhaps Bourbaki is r e s -

2 p o n s i b l e . I n Church 's n o t a t i o n , t h e f u n c t i o n which sends x t o x + x

i s r e p r e s e n t e d by

L The f u n c t i o n , which f o r a c e r t a i n pa ramete r a , sends x t o x + ax " i s denoted by Ax(x t ax) . For a more involved example, c o n s i d e r t h e

mapping i n H i l b e r t space which sends g t o t h e map L , where I, j f i = < f , p g 6

( i n n e r p r o d u c t ) . The map L sending f -t <f ,g> i s h <f ,g> . The f u n c t i o n R f

d e s c r i b e d above i s simply X X-<f ,g> . g r

I n a n a l y s i s F r e u d e n t h a l ' s Y-nota t ion i s conven ien t , mos t ly because

i t a v o i d s t h e use of a l e t t e r which i s n i c e t o have a v a i l a b l e f o r o t h e r p u r -

poses . For example, t h e F o u r i e r t r a n s f o r m i n , L2 can be denoted :

J u s t as w i t h t h e q u a l i f i e r s , 3 , t h e r e i s an obvious need f o r i n d i -

c a t i n g s e t s o r t y p e s . So, i n analogy wi th \/ x e R . . . , we should r e -

w r j t e t h e p r e c e d i n g d e f i n i t i o n :

2 In a formula l i k e Xx(x + xy + z ) , two k inds o f s u b s t i t u t i o n

a r e p o s s i b l e , v i z . f o r t h e "parameters" y , z and f o r t h e " v a r i a b l e " x .

p l a c e d t h e v a r i a b l e x i s i l l - a d a p t e d t o the language PAL arid ~ t ; ex-

t e n s i o n s . I n t h e f i r s t p l a c e , t h e p a r e n t h e s e s ( , ) a r e used t o i n d l -

c a t e a change o f c o n t e x t i n PAL (1.e. a s u b s t i t u t i o n f o r p a r a m e t e r s ) .

Fur thermore, w i t h t h e " q u a n t i f i e r 1 ' w r i t t e n on t h e l e f t , it would be

unwieldy t o p l a c e t h e " i n v e r s e o p e r a t i o n t 1 ( s u b s t i t u t i o n f o r a v a r i a b l e )

on t h e r i g h t . T h e r e f o r e we i n t r o d u c e new symbols, ( b r a c e s ) , and w r i t e

{b) f i n p l a c e of f (b) as used i n o r d i n a r y mathematics.

F o l d i n g Rule (B-reduct i o n ) :

2 {b) XX(x + x + ex) reduces t o b2 t b T e b

( S u ~ s e q u e n t l y "reduces t o " w i l l be r e p r e s e n t e d by t h e symboi /. j . There -

i s a l s o a - r e d u c t i o n :

and q - r e d u c t i o n :

X {x) A > A i f x does n o t occur i n A . X -

The Normal Form Problem

I n an e x p r e s s i o n l i k e :

we can a t t e m p t t o s i m p l i f y by f o l d i n g , hoping t o o b t a i n an e x p ~ . e s s i o n of

t h e t y p e : h A x {rl){T2}f . i n t h i s e x p r e s s i o n f o l d i n g i s n o t p o s s i - X1 X2 3

b l e any more, suppos ing t h a t i t i s no t p o s s i b l e i n I'l and r 2 . We say

t h a t t h e e x p r e s s i o n i s i n normal form. But f o l d i n g o f t e n makes t h i n g s

worse. Church gave t h e c l e v e r example :

I f we agree t o denote i x { x l x by r . then fo ld ing jus r glvc-: t ! ' r \ '

a g a i n .

Expressions l i k e {x)x do not occur i n mathematrcs. i f x i s a

f u n c t i o n , then i t s v a r i a b l e s a r e of a d i f f e r e n t t ype . In AUTOMATH we t h i n k

i n terms of c a t e g o r i e s , and we extend t h e A-calculus with t h e s e . I n f a c t ,

AUTOMATH and r e l a t e d languages a r e j u s t PAL augmented by such a A-calculus .

For t h e o r i g i n a l v e r s i o n of AUTOMATH, t h e Normal Form Theorem, i . e . t h e

s ta tement t h a t every express ion reduces t o an express ion i n normal form,

has n o t been proved. I t has been proved however f o r c e r t a i n c l o s e l y r e l a -

t e d languages.

Desc r ip t i on of AUTOMAT11

In p l a c e of X x E r e a l 9 we s h a l l w r i t e [x , rea l ! . And i f , f o r

x r e a l , C has ca tegory "point", t hen t h e mapping x -+ C is de-

noted by [ x , r e a l l C (where C may depend upon x ) and is s a i d t o have

ca tegory [ x , r e a l l p o i n t . ( In desc r ib ing AUTOMATH we s h a l l o f t e n p l a c e

numerals above meta l ingua l symbols t o denote what s o r t of express ion they 0

a r e : t hus A r e p r e s e n t s a 2-express ion , e t c . ) . I f we have a block :

0 I x : = A (A may not con ta in x but C and I -

i 0 0 I' may) q : = C I'

we a l low ourse lves t o w r i t e

0 0 That i s , i n contex t 0 t h e "expression1' [x,A] 1 is admiss ib le , and i t s

O Q CAT i s Lx,A] r ; t h e express ion Cx,l\l r i s admiss ib le and has catego-

ry t ype . Th i s innovat ion i s c a l l e d f u n c t i o n a l a b s t r a c t i o n . For formal r c a -

s o n s ~t may be p r e f e r a b l e t o r e p l a c e x i n t h e P a t t e r l i n e by a s p e c i a l

symbol t (a bound v a r i a b l e ) ; one can imagine t h a t it i s used wherever

c o n f u s i o n i s p o s s i b l e , and t h a t , rough ly , two e x p r e s s i o n s a r e c o n s i d e r e d

t o be " the samet' p rov ided t h e y can be t r ans formed i n t o one a n o t h e r by

" l e g i t i m a t e " a - r e d u c t ion .

Suppose we have a l i n e :

The b lock i t s e l f may be r e c o n s t r u c t e d i f we s t i p u l a t e t h a t

D ( 2 ) i f r[ is a c c e p t a b l e , and CAT(II) = A , t h e n (li) C t , A l C

i s a c c e p t a b l e and CAT{ll}[t,Al!: = Stin r .

We a l s o admit t h i s i n a more r e s t r i c t i v e c o n t e x t (as observed i.n P A L ) ,

The o r i g i n a l b lock appeared a s :

0 The l a t t e r e x p r e s s i o n s a r e d e f i n i t i o n a i l y e q u i v a l e n t t o C

6 and T , r e s p e c t i v e l y . Let us s u p p r e s s t h e p r e c i s e d e t a i l s of t h e l an-

guage d e f i n i t i o n ( s e e C71 f o r q u i t e an e x t e n s i v e t r e a t m e n t ) ; f i r s t ,

some examples.

EXAMPLE 1 ( d i s t r i b u t i v i t y of i n n e r p r o d u c t )

n a t

r e a 1

a

b

s um

prod

E Q

/ d i s t r

. . - . . . t y p e

. - .- . . . t y p e

.- - . - r e a l

- - - . - r e a l . - .- ... r e a l . - . - . . . r e a 1 . - .- . . . t y p e . - . -- - r e a l . - . - . . . EQ(prod(sum(a,b) , c ) ,sum(prod(a,b) , p r o d ( b , c ) l )

m

f i n s e t

Y

v e c t o r

f

rowsum

g

sumvec

provdec

EQV

i n p r o d

h

assum

1 emma

d i s t v

theorem

. - -

. - . - . . . . . . := f i n s e t (m)

:= [ t , y ] r e a l

. - -

. - . - . . . . . .

. - . -

:= Lt , y l s u m ( i t ) f , { t ) g )

:= Ct , y I p r o d ( { t ) f , { t l g )

:= C ~ , Y I E Q ( I t ) f , { t l g )

: = rowsum (prodvec)

. - . -

. - . -

. - .- . . . . . . := [ t , y l d i s t r ( { t ] f , { t ) g , { t ] h )

:= lemma ( f h , g h , ( f t g ) h , d i s t v )

n a t

( f i n i t e s e t of m e l ements )

v e c t o r

r e a l ( t h e p r o p e r d e s c r i p t i o n of t h e sum of t h e compo-

v e c t o r n e n t s of v e c t o r f can- not be g iven h e r e )

v e c t o r

v e c t o r

r e a l

v e c t o r

EQV(h, sumvec)

EQV ( ( f + g ) h , f h t g h ) ( a b b r e v i a t e d )

44

EXAMPLE 2 : t h e i n d u c t i o n axiom

n a t

bool

b

TRUE

s u c c

1 j assume 1

assume 2

i n d u c t ax

t E t y p e

bool

t y p e

n a t

[ x , n a t ] n a t ( t h e s u c c e s s o r f u n c t i o n )

[ x , n a t l b o o l (a p r e d i c a t e )

TRUE ((1)P)

'To i l l u s t r a t e i t s u s c , suppose t h a t i n some c o n t e x t we have CA1'(Q) =

L x , n a t J b o o l and t h a t we have CA'l'(dumo) = 'TRUE({ 1;Q) and CATfk) = n a t

We shou ld l i k e t o a s s e r t TRUE({kjQ) , assuming t h a t we a l r e a d y have :

I rn . - . - n a t

I F . - . - TRUE ((m)Q)

THEN . - . - . . . . TRUE (((m) succ}Q)

I n t h i s s i t u a t i o n we w r i t e s imply :

r e s u l t := ( k j i n d u c t a x (Q,demo, [ n , n a t l [ t ,TRUE ((n]Q) 1 ?'nEN(n, t ) ) TRuE((k}Q).

The machine f i n d s t h e middle e x p r e s s i o n a c c e p t a b l e and g i v e s f o r i t s CAT ,

Ck) [ n , n a t l TRUE ({n}Q) , which i s f o l d e d i n t o TRUE ( i k ) Q ) .

Which f u n c t i o n a l a b s t r a c t i o n t o choose ? I n AUT we beg in w i t h map- 3 0 0

p i n g s from 3 - e x p r e s s i o n s t o 3 - e x p r e s s i o n s . Along wi th [x,).] L' we must ac- O Q O

c e p t Cx,Al I' . A t t h i s s t a g e , l i t t l e h a s been a c h i e v e d . A n o t a t i o n i s a-

v a i l a b l e f o r "cons ider f a s a f u n c t i o n of x" , b u t n o t y e t f o r ' ! let

f be n f u n c t i o n " . T h e r e f o r e , we open t h e p o s s i b i l i t y t o w r i t e :

prov ided t h a t t h e 2 - e x p r e s s i o n s A and r a r e a c c e p t a b l e a t 0 . I n

t h i s b lock {n}f is a c c e p t a b l e w i t h CAT{II}I = St+$ whenever I' i s accep-

t a b l e . And what shou ld we w r i t e f o r CAT([t,AJ,r) , which shou ld be a 1-

e x p r e s s i o n ? Must we adhere t o t h e former conven t ion t h a t t y p e i s t h e 0 0

only 1 - e x p r e s s i o n ? O r s h a l l we s a y t h a t i t is [ t , A ] CAT(r] ? I n t h e

l a t t e r c a s e , i f CAT(T) = t y p e , we have CAT([t,A]r) = [ t , A l t y p e . The

f i r s t c h o i c e is t a k e n i n AUT. I t i s n a t u r a l t o want " r e a l f u n c t i o n 1 ' and

" r e a l number" both a s t y p e s

The i n t e r m e d i a t e p o i n t o f view o f f e r e d by AUT-QE admi t s both

C t , A l t y p e and t y p e , s o t h a t i f we know

1 ) : = C C t , A 1 t y p e

w e a l l o w 2 ) : = C t y p e

Note t h a t l i n e (1) b e a r s more i n f o r m a t i o n t h a n l i n e ( 2 ) . AUT-QE is a ve-

r y handy language, b u t q u i t e s o p h i s t i c a t e d ; a new c o n ~ p l r c a t i o n i s t h e

nonuniqueness of CAT. That un iqueness may be p r e s e r v e d i f we make t h e con-

v e n t i o n t h a t t h e r e d u c t i o n r u i e [ t , A l t y p e -* type a p p l i e s on ly t o sub.;-

t i t u t i o n r i g h t s . So t h a t , g iven a block l i k e :

0 ' - . - type

b := ...... r e a l , when we have

... := C [ t , A l type , we s h a l l a l s o accept

. . . := b(C) r e a l .

By abuse of metalanguage, C t , A l [s,C] type c C t , A l type c type . Of

course , we might have formulated t h e s e cond i t i ons by another word, su- -

p e r t y p e , and then agreed t h a t i n

0 := type . - .- . . . . . . s we may only s u b s t i t u t e I: f o r 8

i f CAT(C) = type . However, i f we have

r l : = supertype

. - .- . . . . . . , we may a l s o s u b s t i t u t e

r f o r q i f CAT(r) = [ x , ... I Cy, ... I type , and s o f o r t h .

This would open up a v a s t new a r e a ; we approach such an undertaking with

t r e p i d a t i o n . Present experience with AUT-QE sugges ts t h a t s h o r t e r w r i t -

ing i s p o s s i b l e than with AUT . Note t h a t some obvious r u l e s about DE-

FEQUAL must be observed, f o r example :

D D D i f h l = A and il = I'2 , then Lt,iilIT1 = [ t ,h21T2 .

2

Before in spec t ing t h e next p i ece of t e x t , l e t us t ake note of

t h e fo l lowing p o s s i b i l i t y . Wen we have a block l i k e :

a . .- - . . . . . . boo1

b . . - - . . . . . . boo1 - . - TRUE (a) 1 t i e n : = x TRUE (b)

Funct ional a b s t r a c t i o n permi ts us t o w r i t e

.... := [t,TRIJE(a)lC [t,TRUE(a)] TRUE(bj

This observa t ion exp la in s t h e d e r i v a t i o n of t h e s i x t h l i n e i n t h e p i e c e

of t e x t below, a d e r i v a t i o n of modus ponens i n AUT.

boo1 : = PN

b . - . - TRUE := PN

a , - . - b . - . - boo1

IMP L : = Cx,TRUE(a)lTRUE(bj type

a s s 1 . - . - TRUE (a) 1

. - . - modpon := { a s s l ) a s s2

IMP L

TRUE (b)

Now we i n v e s t i g a t e q u a n t i f i c a t i o n . F i r s t we in t roduce t h e A L L

symbol (over a type) ; it i s s l i g h t l y ha rde r t o handle over a subse t of

a type .

I I A L L := Cu,Ol TRUE(lu}P) type

I I observe t h a t :

a . - . - 0

i Roughly, if P(x)

a l l t r u e := AEL(O,Pj i s t r u e f o r a l l x , / then := ( a ) a l l t r u e TRUE ( (a lp) then P (a) i s t r u e

and if t h e r e i s a block :

i b . - - 0

1 then : = Z TRUE ({u)F)

f u n c t i o n a l a b s t r a c t i o n a l lows

. . . . := Cu,OlC Cu,07TRUE (CulP)

. . , . := lu,OlC A L L

Note t h e analogy between t h e texts f o r XMPL and ALL.

Exis tence is an example of a not ion t h a t has been taught by kn-

t imida t ion . Actua l ly t h e r e a r e s e v e r a l forms of ex i s t ence ; we begin with

a s t rong form (Note t h e s l i g h t dev ia t ion from i61, p . 7 3 ) .

EXISTS := PN 922

I . - - 0 i an axiom

a s s 1 . - - TRUE ({ V) P)

then 1 := PN EXISTS

1 f o r any kind

,/ of ex i s t ence

a s s2 . - - EXISTS 1 1 H i l b e r t l s

e p s i l o n := PN 0 I axiom i t s a t i s f i e s := PN TRUE((epsi1on)P) j

I n a s impler form (with P i d e n t i c a l l y t r u e ) we have t h e no t ion NONEMPTY,

0 . - -

NONEMPTY := PN

. - . -

:= PN

The axiom of choice i s implemented by a t e x t l i k e :

type

92.E

NONEMPTY

a s s 4 . - . - i t ,ol NONEMPTY (F ( t ) )

then2 : = C t , O l i o t a i F ( t > , I t ) a s s4 ) C t , O l F ( t )

There is a weaker form of ex i s t ence ( e s s e n t i a l l y 3 = T V I-) which is

s t i l l workable. Let us r e s t r i c t it t o nonempty ; we c a l l t h e not ion

NEPTY. F i r s t l e t us remark how nonemptiness i s used i n mathematics. We - have a p ropos i t i on p , and a s e t S ; we know t h a t S is nonempty.

Then, i f f o r a l l a c S t h e p ropos i t i on p i s t r u e , t hen p i s t r u e .

So we d e f i n e nonemptiness by 'd ( 'd p => p ) . pcbool aeS

Negation may be approached i n t h e fo l lowing manner :

0 . - .- - t ype

NEPTY := [c,booll[u,[x,OlTRUE(c)l TRUE(c) type

F i r s t come t h e l i n e s

. - . - 0

then3 1 a := [c,booll[u,Cx,Ol TRUE(c)l ( a )u NEPTY

A s f o r t h e use of NEPTY, suppose we have :

a s s 5 . - . - NEPTY

P . - . - boo1

Y . - - 0

then3 := c TRUE (P)

/ then we may w r i t e

conclusion := ICy,03Ll i P I a s s 5 TRUE (P)

CON (TRAD ICT ION)

0

nonempty

1 a s s 6

( non

:= [x ,bool] TRUE(x)

. - . - := PN

. - . - := PN

. - . - := PN . - . - : = [ x,TRUE (c) 1 CON

: = nonempty (NON (c) )

type

w bool

TRUE (nonempty (0) )

NEPTY (@)

0

TRUE (nonempty (O ) )

boo1

type

bool

We leave it a s an e x e r c i s e t o prove t h a t :

a s s 8

then6 I ass9

TRUE (c)

TRUE (non (c) )

CON

A s a consequence, TRUE (non (c) ) impl ies NEPTY (NON (c) ) , and t o -

ge the r w i th TRUE(c) , t h i s l e ads t o CON . The awkward jumps from 0 t o

TRUE(nonempty(0)) involve many e x t r a l i n e s , a very annoying f e a t u r e .

S e t theory may be formulated i n two ways. I n t h e f i r s t of t h e s e ,

an expos i t i on of ax iomat ic s e t t heo ry , it is permi t ted t h a t s e t s a r e e l e -

ments of o t h e r s e t s , p o s s i b l y even themselves. Thus :

set := PN type

a . - . - s e t

b . - . - s e t

E := PN boo P

c := PN boo 1

u := PN s e t

n := PI4 s e t

Now some axioms a r e added l i k e :

if . - . - TRUE (E ( a , b)

I C . - - s e t

and :=--- TRUE(c ( b , ~ ) )

t hen := PN TRUE (E ( a , c ) )

An e n t i r e l y d i f f e r e n t approach is t o t r e a t s e t s a s s e t s of

t h i n g s of a c e r t a i n k ind . We do no t need t o form t h e union of a s e t of

t r i a n g l e s w i t h a s e t of r e a l numbers. We t h i n k of t y p e s as s e t s ; " s e t f t

becomes a 1 -express ion depending on a 2 -express ion .

s e t := PN t p c

Here E may be i n t r o d u c e d a s f o l l o w s

I s . - . - s -.t (0) I

In t h i s f o r m a t i o n x E x does not occur . The requ i rements o f t h e axiom

of e x t e n s i o n a l i t y makes us r e a l i z e t h a t it i s e a s i e r o t d e f i n e t h e no-

t i o n " s e t f 1 by means o f a p r e d i c a t e ; v i d e :

Here a s e t i s a s u b s e t o f a 2 -express ion . Accord ing ly , e v e r y t y p e i s it-

self a s e t . I f , f u r t h e r m o r e , t h e n a t u r a l numbers N a r e a v a i l a b l e , t h e

language f a c i l i t i e s of AUTOM.4T'tl pe rmi t LIS t o c o n s t r u c t t h e s e t s

N~ N , N ~ , N , . . . but n o t t h e i r union. For i f we have n a t := FN t y p e , - N we can form [n,N]N o r N , which i s , essentially, t h e s e t of real num-

b e r s . N~ i s aga in a t y p e and we may q u a n t i f y o v e r i f . Cantor c o n s t r u c t e d

a l l t h e s e s e t s and took t h e i r union which l e d him i n t o C m t o r g s Parsdnsa. -*

An i n f i n i t e book would be r equ i r ed t o enable us t o do t h a t h e r e . 'icjwtver,

we a r e a l r eady i n t h e Analys t s ' Pa rad i se . No more t han t h i s i s ever en-

countered by t h e a n a l y s t .

Extensions of AUT

Let h and l' be 2-express ions . In AUT t h e express ion i x , A i r

has CAT type . The ex tens ion of AUT c a l l e d AUT-QE permits us t o w r i t e CAT

of t h e exp re s s ion [x,A]r , a s C x , A J t y p e ; t h e l a t t e r i s a new l-ex-

p r e s s i o n - a mapping type . We may reduce [x,Al type t o t ype but i t i s

no t o b l i g a t o r y . An express ion C , with CATCC) = [x,A] t ype nay be subs-

t i t u t e d f o r a v a r i a b l e wi th CAT(q) = type ; but it i s forb idden t o

s u b s t i t u t e an exp re s s ion C with ca tegory type f o r a v a r i a b l e wi th ca-

t ego ry Lx,hl t ype . Eor can we s u b s t i t u t e Z with C A T ( C ) = [x,A! t ype

i! f o r a v a r i a b l e with ca tegory Cx,Rl w e , if A # B . Tn ca se CAT(C) = O -

and CAT(IZ) = lx,OI L I ... t I = , we must add t h e r i g h t t o use { C ] A

and t h e corresponding f o l d i n g rule . We do not admit q u a n t i f i c a t i o n over -- anyth ing o t h e r t han 2-express ions .

In t h e language AUT-SL , q u a n t i f i c a t i o n over l - exp re s s ions is

admi t ted . Then eve ry th ing can be w r i t t e n a t l e v e l 0 ; P N 1 s a r e e l imina t ed .

A s i m i l a r l eve l -0 language employing AUT r u l e s has been desc r ibed by Ne-

d e r p e l t (X-AUTOMATH) . Actua l ly , w r i t i n g a t l e v e l 0 with un l imi t ed quan t i -

f i c a t i o n does no t appear t o b e h a r m f u l ; even f o r AUT-SL t h e normal form

theorem can be proved. The tough quest ion is , %ow f a r can we go with type

reduction and/or s u b s t i t u t i o n r i g h t s ? The el imination of P N s s has the

g r e a t advantage t h a t models can be used : i f we have a model f o r a PN ,

then a l l t h a t was derived f o r the PN i s a t once ava i l ab le f o r the model.

The language AUT-SL i s defined by a computer programme. Pn f a c t , t he same

computer programme is used t o check the language. Since the lan,guage i s

defined by the programme, the necess i ty of proving t h a t the programme des-

c r ibes the language i s sidestepped.

Let us conclude by examining a p iece of t e x t i n AUT-QE. The same

t e x t i n AUT appears i n [I17 ; t h e i n t e r p r e t a t i o n is t h e in t roduct ion of

limits. Since type p red ica tes can be used exclus ively , t h e ALL-quantifier

i s unnecessary.

0 . - - w??. . - I .- Cx,Ol type (Q i s a t n e p red ica te i n 0)

D Note t h a t i f CA'l?(a) = O , then q(a) = {a) Q

ALL : = Cu,Ol (u) Q type (see : in t roduct ion of ALL)

D By r)-reduction, Cx,O1 {x) Q = Q , so t h a t t h e ~ e i s something i n Q iff

t h e r e i s something i n ALL. The type p red ica te i s " the same" as t h e A L L

q u a n t i f i e r . For t h e present t e x t it i s i r r e l e v a n t what notion of exis tence

i s chosen.

. - . - Cx,Ol type

EXISTS := PN Q!E? (not s t r i c t l y necessary -see p re - vious d i scuss ion o f ex i s t ence )

i i a . - . - 0

a s s . - . - {a1 Q

axiom := PN EXISTS

The a u x i l i a r y no t ions a r e i d e n t i c a l i n appearance t o t h e i r AU-

TOMATH ve r s ions ( see Clll p. 9 ) .

r e a l : = PN type

. - . - real

'2 . - . - r e a l

d i s t : = PN r e a l

LESS : = PN type

nu1 l : = PN r e a l

n a t : = PN trpe

1 : = PN n a t

if . . - - LESS (nu1 l , r 2 )

1 emma : = PN LESS ( d i s t (ri , r l ) , r e )

1 . - . - n a t

k2 . - . - n a t

LESSNAT : =' PN type

Sequence := Cx,na t l r ea l type

I f a i s a sequence and 1 is r e a l , t h e L I M w i l l a s s e r t t h a t l i m an = 1 . n-m

CONV a s s e r t s t h a t such a l i m i t e x i s t s .

1 . - . - re;i

6 . - . - r e a l

t . - . - tESS(nul: ,d j

I I) := CnO,nOICn,natlCu,LESSNAT(no,n)lLESS(dist(in)a,l) , 6 ) Cno,natl=

LIM := C6,reallCt,LESS(nul1,6)1EXISTS(nat,p(b,t]) t p e

1 CONV : = EXISTS ( r e a l , C 1, r e a l ] LIM(1) ) LYE

The theorem t h a t t h e sequence with constant value c converges t o

t h e l i m i t c follows. Compare the proof wr i t t en i n AUT (Clll p.10) with the

proof i n AUT-QE. The log ica l s t r u c t u r e i s the same as it IS f o r any l i m i t

theorem ; t echn ica l ly i t is q u i t e s i

r e a l

sequence

The f%echnical" p a r t of t h e proof i s simply t h e segment

lemma (Cnlp,F,ass)

appearing i n l i n e 6. The r e s t i s t h e " l o g i ~ a l ' ~ p a r t of the proof.

AUT-QE i s somewhat e a s i e r t o w r i t e than AUT, judging by t h i s example;

yet AUT-QE i s a s t ronger language. Hopefully, t h i s one example may serve

f o r many. Those i n t e r e s t e d i n pursuing t h e matter may consult t h e bibl iography.

I - 6 . - . - r e a l

I

ass := LESS (nu1 l , 8 )

Q : = P (p , c,X, ass) Cno,natl type

now : = Cn,natl Cu,LESSNAT(l ,n) 1 lemma((n)p,%,ass) I13 Q

thus : = axiom(nat ,Q , 1 ,now) EXIST(nat ,Q)

T H M ~ : = ~ 6 , r e a l ] i t , LESS (nu1 1 ,X) 1 thus (T, t ) LIM(P ,c)

THM2 : = axiom(rea1, Cs , r e a l l LIM(p,s) , c,THMl) CONV (p, c)

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(1970).

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du Labora to i re de Calcul de l a Facul t6 des Sciences de

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L i l l e (1968).

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t u r e Notes i n Mathematics N R 125 (1970).

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t e r . Colloquiumvoordracht, THE.

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