Some auxiliary operators in AUT-PI

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Some auxiliary operators in AUT-PI

Citation for published version (APA):

Bruijn, de, N. G. (1977). Some auxiliary operators in AUT-PI. (Eindhoven University of Technology : Dept of Mathematics : memorandum; Vol. 7715). Technische Hogeschool Eindhoven.

Document status and date: Published: 01/01/1977 Document Version:

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•

EINDHOVEN UNIVERSITY OF TECHNOLOGY

Department of Mathematics

Memorandum 1977-15. Issued November 1977 •

SOME AUXILIARY OPERATORS IN AUT-TI.

by

N.G. de Bruijn.

University of Technology Department of Mathematics P.O. Box 513 Eindhoven. The Netherlands.

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SOME AUXILIARY OPERATORS IN AUT-IT.

by

N.G. de Bruijn.

For AUT-IT we refer to Zucker [2J. If we omit all those features that the languages of the AUTOMATH family have in common (cf. the description of AUT-QE in D. van Daalen [IJ), the basic rules are the following (i)

-(vii). Two simplifications are made here. First, we use a symbol T which • may be either type or prop. Secondly, we omit all IT's in expressions of

degree 1, which does not make any essential difference. And we use the notation (x : a) ~ in order to indicate that something is valid in the context extended by x (of type (1) • .\8 in

rll, b/AJ

X means thAt in Z we have to replace x by A.

The rules are

(i)

fl-

T (ii) ~ a : T (x : a)·~ P j.l [x : aJ P (iii) _{(x : a)}

f2.

Q : P

f1.

[x aJQ : [x : aJP (iv)

0

A : a T

t?-

_{Q :} [x : ciJp

r£

{A}Q [x/AD p (v) [x aJT .~ ITQ T (vi)

J1.

a : T (x : a) ~ _R _Q

.

T 3 [x : ciJR [x : ciJQ r

(vii) ~A : a : T

?

_{R : ITQ}

F-g

[x _aJT

~ {A}R {A} Q We shall now define operators 8

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•

2

-The symbols are metalinguistic: 8.Q is used in the metalanguage to indicate

J

a certain expression in the language, viz.

8₂Q = [x) : atJ .•• [x' 2 : a _m- _m-2

In [

x _I : a _m- _m-I] IT {x _m-

)l. ..

{x)}Q . . . " . . . <II . . . ..

Note that 8. is built by starting from the expression just given for 8 Q and then

J m

omitting the first m-j IT's. If m=1 we just have 6]Q = ITQ. If m=2 then 6] Q =

=

[xI: atJ IT {xl }Q and 8

2 Q

=

IT[xt : 0,1] IT {x]}Q. If m=O none of the 6j's are

defined.

We can now prove the validity of a new rule, viz:

(viii) ~ I a : T

!1-

6. Q

J

a ]T

m

for I s j ~ m. If j=~1 it is just the old rule (v).

For shortness we shall write [x.] and (x.) instead of [x.

1 1 1

(x. : a.).

1 1

Let us start from

T

Applying (iv) we get

and m-2 more applications of the same rule leads to

Next we apply (v):

a.] and

1

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•

3

-arid by (iii) this gives

Now m-2 further applications of (iii) g~ves

On the other hand, if we apply (v) to (2) followed by a single application of (iii) we get

Now me3 more applications of (iii) lead to

On the other hand, if we apply (v) followed by (iii) to (3) we get

This way we get, indeed

jZ-

e.Q : [xIJ ••• [x .J T J meJ for all j (l ::; j ::; m) • (3) [x ~JT. m-j (4)

We shall also show that

e. e.

~

e ...

More precisely, i f

~

Q : [xl : a IJ •••

~ J 1+J

••• [ X : ex ] T, and i f i ~ I, j ~ 1, i + j ::; m,

m m then

e.8.

]. J Q reduces to

e

~ '+' J Q by

means of repeated S-reduction. First we have (4), i.e.

f-[xtJ···[x .J rr [x ]

rr •••

rr[x...-1J rr{ x_m_1 ••• {x1}Q

m-] m-j+l U4 r

: [x I J ••• [x • J1" •

me] Applying

e.

to this we get

1

~

e. e.

Q [xlJ ••• [ x . .J 1"

1 J m-J-].

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4

-e. -e.

Q = [ y 1 J ••. [y . 1 J II [ y . . + 1 J II ... II [y . I J II {y . J}'"

~ J m - J - . m-J-~ m-J-

m-]-.•• {YI} [x)J ••• [x .J II [x . IJII m-]-.•• n[x )J II {x l}· .• {xt}Q.

m-J m-]+ m-

m-The sequence {y . t1,"{Yt} [x1J ... [x . IJ is annihilated by m applications

m-J-

m-J-of s-reduction. After that, we change the names Yl""y . 1 into x1""'x . l'

m-J-

m-J-thus arriving at e . . Q.

~+J

Above we extended rule (v) to rule (viii). Similarly we shall extend rule (vii) to the following rule (ix) for m ~ I:

(ix) ~ {A} R

e

_mQ {A} ffu-l Q a Jr m I f m=1 we have 8

mQ = II Q, and 8m-1 Q has to be explained as Q itself

(8

0 was not defined before).

Rule (ix) is not hard to derive. Noting that

e

Q =II8 1 Q,and~

e

t Q

m m-

m-[xI : a

l JT by (viii), we can apply (vii) with Q replaced by e m-l Q, which

leads to ~ {A} R : {Ale

m_t Q.

We note that in all rules formulas of the type ~ R : Q lead to ~ Q :T . Indeed, in (vi) we have!1 11 [x : aJQ : T by (v), and in (ix) we have

jl

{A}em_tQ : T

by (iv), according to the typing of e 1 Q just derived.

m-Instead of the lower kind of (ix) we may as well get

~ {A} R :

e

m-l {A} Q

since {A}8

m_1 Q reduces to 8m- 1 {A} Q by a single beta reduction.

More generally we observe that

{A}

e. o

reduces to e. {A} Q

J ' J

by a single beta reduction if j < m.

The symbols e. also commute with abstraction if

J if j :0; m. a }r then m [y : BJ 8. Q reduces to e. [y : BJQ J ]

These observations mean that in composite expressions like

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•

5

-References:

I. D. van Daalen: A description of AUTOMATH and some aspects of its language theory. Proceedings of the Symposium APLASM. Vol. I, ed. P. Braffort, (Orsay) France (December 1973).

2. J.Zllcker:

(This paper is reproduced in L.S. van Benthem Jutting, Checking Landau's "Grundlagen" in the AUTOMATH system. Thesis, Technolo-gical University Eindhoven, ]977).

Formalization of classical mathematics in AUTOMATH. Colloque international de logique, Clermont-Ferrand ]8-25 July 1975. Colloques internationaux du Centre National de la Recherche Scientifique Nr. 249.