Some abbreviations in the input language for Automath

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Some abbreviations in the input language for Automath

Citation for published version (APA):

de Bruijn, N. G. (1972). Some abbreviations in the input language for Automath. Eindhoven University of Technology.

Document status and date: Published: 01/01/1972

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T e c h n o l o g i c a l U n i v e r s i t y , Eindhoven Department of Mathematics N o t i t i e 15,

17

A p r i l 1 9 7 2 . Some a b b r e v i a t i o n s i n t h e i n p u t l a n g u a g e f o r AUTOMATH by N . G . d e B r u i j n . Q u i t e o f t e n w r i t i n g t e x t s i n AUTONXTH c a n b e t e d i o u s b e c a u s e o f t h e n e c e s s i t y t o w r i t e and r e - w r i t e t h i n g s t h a t had b e e n w r i t t e n b e f o r e . Ad- m i t t e d l y , t h i s c a n , a t l e a s t t o some e x t e n t , b e c i r c u m v e n t e d i f we w r i t e e x t r a l i n e s : t h e s e e n a b l e us t o a b b r e v i a t e e x p r e s s i o n s by s i n g l e i d e n t i - f i e r s . U n f o r t u n a t e l y we o f t e n do n o t know b e f o r e h a n d w h e t h e r a c e r t a i n e x p r e s s i o n w i l l b e u s e d a g a i n . I f i t does t u r n up l a t e r , we s t i l l h a v e t o w r i t e i t a s e c o n d t i m e i f we w a n t t o i n t r o d u c e t h e a b b r e v i a t i o n , and we w i l l do t h i s u n l e s s we d e c i d e t h a t i t i s n o t l i k e l y t o t u r n up a t h i r d t i m e ,

A l l t h i s may mean t h a t i t m i g h t come i n handy t o h a v e a s y s t e m f o r r e f e r r i n g t o e x p r e s s i o n s by i n d i c a t i n g t h e i r p o s i t i o n i n t h e p r e v i o u s t e x t s . A t p r e s e n t we o n l y e x p l a i n a s i n g l e d e v i c e o f t h i s k i n d , t h a t can b e a p p l i e d

i f t h e e x p r e s s i o n i s t h e f u l l e n t r y of t h e c a t e g o r y of one o f t h e p r e v i o u s l i n e s .

Assume we h a v e i n t h e p r e v i o u s t e x t a l i n e l i k e

where

X

i s t h e c o n t e x t i n d i c a t o r and where - A i s a 2 - e x p r e s s i o n ( l . e . a n e x p r e s s i o n whose c a t e g o r y i s t y p e ) . Then we s h a l l t a k e t h e l i b e r t y t o r e f e r t o

-

A by means of c a t b . To b e p r e c i s e , we a c t a s i f c a t b

_-

i s a g a i n a n i d e n t i - f i e r , and a s i f t h e r e had b e e n two l i n e s i n t h e book, v i z .

_-

c a t b

-

T h i s c a t b c a n b e used f r e e l y a s i f

_-

i t had b e e n d e f i n e d i n t h e book, s o w e a l s o admit t h i n g s l i k e c a t b (

,

) and o t h e r AUTOMATH c o n s t r u c t i o n s .

We do t h e same i f

-

C

i s n o t a n e x p r e s s i o n , b u t PN. I f we h a v e a b l o c k o p e n e r ( w i t h _-

A

$

_-

t y p e )

w e admit t h e u s e of c a t x a s i f t h e r e had b e e n two l i n e s

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(Note t h a t c a t x i s n o t a v a r i a b l e ; x can b e u s e 3 o n l y i n i t s b l o c k , f o r c a t x t h e r e i s no s u c h r e s t r i c t i o n ) . We emphasize t h a t

-

c a t a l w a y s b e l o n g s t o t h e i d e n t i f i e r f o l l o w i n g i t ; t o g e t h e r t h e y form a s i n g l e i d e n t i f i e r . We a l s o p r o p o s e a n a b b r e v i a t i o n of a d i f f e r e n t n a t u r e . F l a t h a p p e n s q u i t e o f t e n i n AUTOMATH i s t h i s . We h a v e Assuming t h a t C L 2 , i g ,

2

a r e n o t t y p e , we may w r i t e h e r e

- - -

Wewant t o a b b r e v i a t e t h e s e e x p r e s s i o n s a s I f

n

= cype we c a n s t i l l u s e @ p b u t we do n o t u s e

The e n c i r c l e d i n t e g e r ( h e r e 3) i n d i c a t e s how many b l o c k s we h a v e l e f t . Note t h a t

O p

i s t o b e c r e a t e d a s a new i d e n t i f i e r , and t h a t we a c t a s i f i t had b e e n d e f i n e d i n t h e c o n t e x t 1, i m n e d i a t e l y a f t e r t h e l i n e of p, by NOW t h a t @ p i s a n i d e n t i f i e r , we c a n a g r e e t h a t c a t o p i s a g a i n a n i d e n t - n

-

i f i e r ; o b v i o u s l y

-

c a t Q) p and

@

-

c a t p a r e d e f i n e d by t h e same e x p r e s s i o n . As examples we s h a l l r e w r i t e a p i e c e of t e x t w r i t t e n p r e v i o u s l y ( D e f i n i t i o n of l i m i t s i o AUTOMATH, I n t e r n a l R e p o r t , T e c h n o l o g i c a l U n i v e r s i t y , Eindhoven, t h e N e t h e r l a n d s , 1968).

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1. Introduction to mathematics.

1.10

P

1.11 v

1.12 ass

I

1.13 P

1.14 P

1.15 0

1.16 a

1.17 b

1.18 b

boo1

.

= 3

i

: =

m e m p

ty

.

=

a

.

-

then 2

.

=

nxis

ts

: =

ass 1

: =

*

t h e n 1 3

:= ALL

.

-

ai

1 .

= PN

Cu,ksilTRUE

({u)P)

nonemp

ty

(ALL)

CU,TI~UE

(a) ITRUE

(b)

nonempty (IMPL)

type

bool

type

tY

Pe

bool

ks

i

TRUE

(nonempty)

[u,ksilbool

bool

ks

i

TRUE

({VIP)

TRUE(exists)

type

bool

type

bool

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2 . ~ e f i n i t i o n of l i m i t . r e a l

lrl

r 2 d i s t a n c e l e s s n u l l n a t s e q u e n c e 1 k i? l e s s n a t PN

PN

Cn,

n a t l r e a l PN

l

t y p e r e a l e p s m 0 m W Z Y iq r e a l r e a l b o o l

I

) ~ . i m conv r e a l t Y Pe t y p e n a t n a t n a t b o o l s e q u e n c e r e a l r e a l n a t n a t i m p l ( l e s s n a t ( r n o , m ) , boo1 l e s s ( d i s t a n c e ( { m } a , a )

,

e p s ) ) a l l ( n a t ,

O w )

boo1 e x i s t r ( n a t ,

@

z ) boo1 i m p l ( l e s s ( n u l l , e p s ) , y ) b o o l a l l ( r e a l ,

($

q ) boo1 e x i s t s ( r e a l ,

@

l i m ) boo1

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(Section 3 of the 196C version is superfluous in the present version.)

4. How to show convergence of a sequence.

4.1 o

i

P

:=

----

4.2 P

4.3 C

4.4 Mo

4.5 DAN

4.6

E

4.7 assu

4.8 assu

4.9 m

4.10 als

4.11 m

4.12 assu

4.13 assu

4.14

E

4.15 DAN

4.16 DAN

.=

-

SSU :=

-

sequence

real

[G,realllh,TRUE(less(nu11,6))~

nat

[G,reaP][~,TR~E(less(null,Sj)l

[M,nat][w,TRUE(iessnat

( i A l { d } N O , E I ) ) !

TRUE(less(distance({M}P,C),6))

real

TRUE

(less

(null,

E))

nat

TRUE

(les

snat

(mo

,m)

)

Idan

:=

ialsj

{mliassu:

{EIDAN

TRUE(1ess (dis

tance({m}P,C)

, E ) )

b :=

then 2

(cat

-

@

dan,

@

dan)

TRUE

(w (P

,

C,

E

,mo

,m) )

L :=

then 2(cat@b,@b)

TRUE(Z(P,C,E,K))

0 *

! :=

then 13 (nar,

@

z(P,C,~),m~,h)

TRUE(y(P,C,&))

:=

hen ?(cat

@

d,

@

d)

TRUE(q(P,C,&))

?orem1

:=

then

2(cat

-

@

e,

3 e )

TRUE(lim(P,C))

*

?oren2

:=

ther. 13 (real,

@

lim(P) ,C,lheorem

I )

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5 . An a p p l i c a t i o n .

Assume we h a v e t h e f o l l o w i n g p i e c e of t e x t i n some c o n t e x t A ( i t i s t h e s t a n d a r d p a t t e r n by which we show t h a t a s e q u e n c e c o n v e r g e s t o a num- b e r ) .

X

A E 1 assume a s s ume n 2 a l s 1 C i =

.

r e a l P 1

.

=

...

s e q u e n c e E 1

.=

-

r e a l assume :=

-

T R U E ( l e s s ( n u l 1 , ~ l ) ) n l

.

=

_...

_{n a t} n2 .=

----

n a t a l s l :=

-

T R U E ( l e s s n a t ( n l , n 2 ) ) d a n l :=

. .

.

. .

TRUE(less(di~tance({n2}~l

, C l ) , E l ) ) Now we c a n w r i t e i n a s i n g l e l i n e 5 . 9 A c o n c l := Theorem I ( p ~ , ~ ~ , @ n l , O d a n l ) TRUE(lim(p1

,CI))

6 . Comments on u s e of q u a n t i f i e r s .

As may b e s e e n from t h e p r e v i o u s t e x t s , t h e r e i s a s t r o n g a n a l o g y between " a l l " and "impl" ( l i n e s 1 . 1 4 and 1 . 1 8 ) . Yet t h e t r e a t m e n t was d i f f e r - e n t (compare l i n e 2.21 w i t h 2 . 2 2 ) . The r e a s o n i s t h a t IMPL ( l i n e 1.17) i s s i m p l e r t h a n ALL ( l i n e 1.13) s i n c e u i s c o n t a i n e d i n TRUE({u}P) b u t n o t i n TRUE ( b )

.

However, i f we w l s h , we c a n e n f o r c e t h e p a r a l l e l i s m . L e t u s t a k e a f o r m u l a l i k e

(where A may depend on x ; B and C may depend on x and Y ) .

We

can

p u t

this

as f o l l o w s

i n

AUTOMATH

(assuming

t h a t

5

and

n

h a v e b e e n i n t r o d u c e d p r e v i o u s l y a s t y p e , al,u A, B and C a s b o o l ) :

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.=

TRUE (A) ? :=

-

rl :=

-

TRUE (B) v := C boo1 a := a l l ( c a t $ ,

@

v) boo1 := a l l ( c a t y ,

-

a

a ) boo1

-

:= a l l ( c a t c p ,

_-

@

b ) boo1 := a l l ( c a t x ,

@

c ) boo1 i n s t e a d of t h e s h o r t e r form X :=

---

F

I

Y

:=

---

n

e

.

= imp1 (B, C) boo1 f := a l l ( n , @ e ) boo1 g : = imp1 ( A , f ) b oo 1

Now assume we h a v e a p i e c e of t e x t l i k e t h i s ( i n some c o n t e x t A ) :

Then w e c a n d e s c e n d t o t h e t r u t h of d a s f o l l o w s : 6 . 1 6

X

1 .

- x**

.-

-

5

k

*

6 . 2 1 y* \ I j i * := t h e n 2 ( c a t @ k * , @ k * ) TRUE(a(x , p ; y * ) )

*

: _

*

6 . 2 2 q* t h e n 2 ( c a t

@

L*,

@

L*) TRUE(b ( x

,

cp*) )

*

6 . 2 3 x

_I

_'_I _:₌ _{t h e n}₂_{( c a t}

_@

_m*,

_@

_m*) _TRUE_{( c (x*)}₎ I * 6 . 2 h

X

cC, : = tiler. 2 ( c a t L

G)

n*

,

@

r,*) TRUE ( G )

*

6 . 1 7 x 6 . 1 8

cp*

6. 19 y* CP :=

-

TRUE (A)

*

_:=

_-

n

1

*

I +

:=

-

TRUE

(B) 6.20 i* I l k * : =

...

TRUE (C)

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The descent to the truth of h (which has the same interpretation

as

the

truth of d) is not shorter at all; actually it is entirely equivalent:

The difference between the two treatments lies mainly in the block

openers 6.2 and 6.4; for the rest, the text 6.1-6.9

+

6.21-6.24 is the

simpler one.

6.25 y*

6.26 cp*

The combination

( A 3

)

occurs very often, and therefore there is

X

use for a shortcut. We shall introduce "allimpl" for this purpose. We ex-

/ q *

:=

then 2

(cat

@

k*,

@

k*)

TRUE(~(X, y))

*

r

_:=

then 2

(cat

@

q*,

@

q*)

TRUE(^

(x*)

)

plain the machinery for deriving the truth of allimpl, but we omit the text

6.27

r*

is*

: =

then 2

(cat

@

r*,

@

r*) TRUE

( g

(x*)

)

6.28 X

t*

:=

then 2

(cat

@

s*,

@

s*)

TRUE(h)

.

needed for the use of that truth.

I

IQ

.=

-

_Cx,

_Slbool

allimpl

:=

all

(5,

@

a)

boo1

4

.

-

--

[ x , 5 1 [ ~ , T R U E i { x ~ P ) l T R U E ( C x ) Q )

5 6.37 y

:=

Iy}a

C~,TRUE(CY

~P>ITRUE({YIQ)

: =

then 2

(catb,

-

b)

TRUE(a(y))

*

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NOW assume w e have, in a certain context

A ,

a piece of text like this:

Then w e can conclude in a single line