STRONG NORMALIZATION IN

A TYPED LAMBDA-CALCULUS WITH

LAMBDA-STRUCTURED TYPES

STRONG NORMALIZATION IN A TYPED LAMBDA CALCULUS WITH

LAMBDA STRUCTURED TYPES

PROEFSCHRI F T

TER V E R K R I J G I N G VAN DE GRAAD VAN DOCTOR I N DE

TECHNISCHE WETENSCHAPPEN AAN DE TECHNISCHE

HOGESCHOOL EINDHOVENt OP GEZAG VAN DE RECTOR

M A G N I F I C U S t PROF, D R I IRm G I VOSSERSt VOOR EEN

COMMISSIE AANGEWEZEN DOOR HET COLLEGE VAN

DEKANEN I N HET OPENBAAR T E VERDEDIGEN OP

DINSDAG 12 JUNI 1973 TE 1 L 0 0 UUR

DOOR

ROBERT P I ETER NEDEPPELT LAZAROR

GEBOREN T E IS-GRAVENHAGE

C 1973 by R.P. Nederpelt, Eindhoven, The Netherlands -

D I T PROEFSCHRIFT I S GOEDGEKEURD

DOOR DE PROMOTOREN

PROF.DR. N.G. DE BRUIJN

en

PROF. DR. W. PEREMANS

v

CONTENTS

Conventions as regards references

CHAPTER I. Introduction and summary

§ 1. Lambda-calculus

§ 2 , Normalization and strong normalization

§ 3. Normalization in systems of typed lambda-calculus

§ 4 , The relation to the mathematical language Automath

5 5. Change of notational conventions

§ 6 . Summary of the contents of this thesis

CHAPTER 11. The formal system A

§ 1. Alphabet and syntactical variables

§ 2 . Expressions

§ 3. Bound expressions

§ 4 . Replacement, renovation and a-reduction

§ 5. Substitution and 6-reduction

§ 6 . Other 6-reductions

§ 7. q-reduction, reduction and lambda-equivalence

§ 8. Type and degree

CHAPTER 111. The formal system A

§ 1 , Legitimate expressions

§ 2. The normalization theorem

§ 3. Strong normalization

VII

References

Samenvatting

Curriculum Vitae

CONVENTIONS AS REGARDS REFERENCES

Refe rences t o l i t e r a t u r e a r e denoted by t h e name(s) of t h e

a u t h o r ( s ) and a number i n s q u a r e b r a c k e t s which i s sometimes f o l -

lowed by a f u r t h e r i n d i c a t i o n ( e . g . Church C2, p. 363) . These r e f -

e r e n c e s a r e l i s t e d a t t h e end of t h i s t h e s i s .

T h i s t h e s i s c o n t a i n s t h r e e c h a p t e r s , which a r e denoted by cap-

i t a l Roman numerals . Each c h a p t e r c o n s i s t s of a number of s e c t i o n s .

I n r e f e r r i n g , f o r example, t o § 3 of Chapter 11, we w r i t e S e c t i o n

11.3 . I n each s e c t i o n we number d e f i n i t i o n s , theorems and n o t a t i o n

r u l e s c o n s e c u t i v e l y , i n d i c a t i n g t h e s e c t i o n number i n f r o n t of t h e

s e r i a l number. Thus, i n S e c t i o n 1 1 . 3 we may p ropose Theorem 3 . 5 . A

r e f e r e n c e t o t h i s theorem i n t h e p e r t a i n i n g c h a p t e r i s w r i t t e n a s

Th. 3.5, i n o t h e r c h a p t e r s a s Th. 11 .3 .5 .

CHAPTER I, INTRODUCTION AND SUMMARY

§ 1. LAMBDA-CALCULUS

The lambda-notation was originally introduced as a useful

notation by Church in two papers developing a system of formal

logic 121. He extended this notation in his calculus of lambda-

conversion (Zarnbda-caZcuZus). This calculus was meant to describe a

general class of functions which have the feature that they can be

applied to functions of this same class.

For historical comment see Curry and Feys [3, Ch. 0, § D and

Ch. 3, § S11 and Barendregt [ I , Ch. 1, § 1.11. In the latter refer-

ence the importance of lambda-calculus for the development of re-

cursive functions is mentioned. The calculus has also been brought

into relation with the theory of ordinal numbers, predicate calcu-

lus and other theories. From the very beginning, lambda-calculus

was strongly linked to the theory of combinatory logic.

We shall later mention some major results achieved concerning

lambda-calculus. Right here we stress the contribution of lambda-

calculus to ordinary mathematics at a purely notational level. The

mathematical custom to use the notation f(x), both for the function

itself and for the value of this function at an undetermined argu- 11 ment x, obscures the mathematical notion function". According to

Curry and Feys "this defect is especially striking in theories

which employ functional operations (functions which admit other

functions as arguments)". For an example showing that the usual

mathematical function notation is defective not only for under-

standing, but also in use, see Curry and Feys C3, Ch. 3, § A21.

We shall give an example of the lambda-notation. Consider the

function which assigns to x the value x + 2. This function is denot-

ed in lambda-notation as Ax-x+2. We can apply the function to an

argument, say 3. The application of this function to the argument 3

is denoted as (Ax.x+2)3. The result of this application must clear-

This suggests t h a t the re e x i s t s an order between the terms

(Ax*x+2)3 and 3 + 2 ( the l a t t e r term i s "closer t o the outcome").

The t r a n s i t i v e and r e f l e x i v e r e l a t i o n corresponding t o such an

order i s c a l l e d a reduct ion. I n the above case i t i s c a l l e d a B-

reduct ion , o f t e n denoted by 2 Thus we have the r e l a t i o n B (Ax*x+2)3 2 3+2.

B The reduct ion r e l a t i o n i s a l s o monotonous, i . e . : i f term S re-

duces t o term T , then Ax*S reduces t o Ax*T, (U)S t o (U)T and (S)U

t o (T)U. So from t h e r e l a t i o n (Ax*x+2)3 2 3+2 follows, f o r example, B

t h a t Aye ((Ax*x+2)3) 2 Ay*(3+2). P The r e l a t i o n compares two terms (v iz . (Ax*x+2)3 and 3+2); the

f a c t t h a t these terms have the common value 5 i n t h e usual i n t e r -

p r e t a t i o n , plays no rCle here.

I f we do no t take 3, but x as argument f o r the above funct ion ,

then we ob ta in (Ax-x+2)x 2 x+2. So lambda-calculus makes a c l e a r 13

d i s t i n c t i o n between the funct ion: Ax-x+2 and the value of t h i s

funct ion f o r an undetermined argument: x+2.

We a r e used t o the f a c t t h a t the terms Ax*x+2 and Ay-y+2 de- 1

note the same funct ion. The two terms a r e c a l l e d a-equivalent , and

the passage of the one i n t o the o ther i s c a l l e d a-reduction, o f t en

denoted by I n t h i s way we a l s o have the r e l a t i o n Ax-x+22 hy*y+2. a

It i s q u i t e a nuisance t h a t t h i s a-reduction, which i s simply a re-

naming of v a r i a b l e s , plays a rCle i n the lambda-notation. One can

avoid t h i s by considering a-equivalence c l a s s e s ins t ead of sepa ra te

terms. Another n i c e and p r a c t i c a l way out i s given by De Brui jn

C81, who completely suppresses the use of names of v a r i a b l e s by

means of a n o t a t i o n a l system r e f e r r i n g t o the p o s i t i o n s of a var i -

a b l e i n a term.

We wish t o s t a t e t h a t the d e s i r e t o e l imina te v a r i a b l e s i s one

of t h e th ings g iv ing r i s e t o combinatory logic . The method used i n

combinatory l o g i c t o obta in t h i s e l iminat ion i s , however, d i f f e r e n t

from De Bru i jn ' s .

A t h i r d reduct ion , which i s commonly used and s t rong ly r e l a t e d

t o e x t e n s i o n a l i t y (see Barendregt [ I , Th. 1.1.17 and Th. 1.1.181),

is called q-reduction. This relation, commonly denoted by 2 is rl'

based on the following rule: If x is not free in the term M, then

Ax. (M)x 2 M e An intuitive justification is that, for any argument rl

X, the sides of the relation have comparable values: this value is

(Ax. (M)x)X for the left-hand side and (M)X for the right-hand side,

and (AX-(M)x)X 2 (M)X. 6

A sequence of reductions obtained by successive application of

reductions is called a reduction sequence.

For each of the reduction relations explained above, the corre-

sponding symmetric and transitive closure is called a conversion

relation. One of the first important rezults in lambda-calculus

concerns the dependence between conversion and reduction. This is

called the Church-Rossep theorem, which states: If X converts to Y,

then there is a Z such that X reduces to Z and Y reduces to Z (see

Curry and Feys E3, Ch. 41). For interesting historical comments see

Barendregt [ I , Th. 1.2.9 and the remarks in 1.2.18 plus footnote].

In Appendix I1 of the latter reference the latest and nicest proof

of the Church-Rosser theorem is given (1971 by W.W. Tait and P.

arti in-~gf). For a precise description see Schulte ~gnting C241.

In this thesis we shall use the name "Church-Rosser propertyt'

for the following statement: If A reduces to B and to C, then there

is a D such that B and C reduce to D. This property is equivalent

to the Church-Rosser theorem.

§ 2. NORMALIZATION AND STRONG NORMALIZATION

An important issue in lambda-calculus is the question of the

normaZization of terms. This is a termination problem. For example,

a 6-reduction such as (Xx.x)y 2 y cannot be continued in a non- B

trivial manner: there is no reduction for y, except those trivial

on account of the reflexivity of a-, 8- and q-reduction. In this

case (Ax*x)y is said to normalize into a normal form y.

In lambda-calculus, which allows all functions as arguments of

functions, such a termination of the reduction is not guaranteed,

See Church's nice example: w2 = (Ax=xx)(Xx.xx). There is a non-

trivial f3-reduction, by applying the rule (hx=xx)A 2B (A)A with

A = hx.xx. This produces w2 tg w2. It is clear that the reduction

of u2 by repeated use of the above non-trivial 8-reduction will

never come to an end.

There are more and stranger examples of such terms, the reduc-

tion of which never terminates. For example: put w = Xx=xxx. Then 3 W W W W3W3 > B 3 3 3 >B

... . Barendregt even constructed a universal

generator with the property that it has a reduction sequence in

which all terms of lambda-calculus occur as subterms.

A term in lambda-calculus is called normaZizabZe if there is

some reduction sequence which terminates. A term is strongly nor-

maZizabZe if each of its reduction sequences terminates. The last

term of a terminating sequence is called a normal j"orm.

It is obvious that strong normalization implies normalization.

The reverse implication does not hold. For example: put again

u2 = Ax*xx, then (hx=(Ay-y))(w2w2) reduces to Xy=y if the function

Xx=(Ay=y) is applied to the argument w w but it reduces to itself 2 2' if the function w2 is applied to the argument w2. Since hy-y is in

normal form, (Ax* (hymy)) (w202) is normalizable, but not strongly

normalizable.

In this example we see a term that normalizes if one applica-

tion of a function to an argument is assigned priority over anoth-

er. There is a general theorem in lambda-calculus (the standardiza-

t i o n theorem, cf. Curry and Feys [3, Ch. 4 E 1 1 ) , which states that

any normalizable term can be normalized by assigning priority to

the "leftmost" application in the term.

The fact that some term in lambda-calculus have non-terminat-

ing reduction sequences is related to the feature that one can use

all functions as arguments for functions. (Even the function itself

can be used as an argument, see the above-mentioned example by

Church. This is called self-appZication, )

The same things can happen in programming languages and in the

theory of partial recursive functions, where normalization- (or

termination-) problems arise too.

In lambda-calculus the question of the normalizability of

terms has been shown to be undecidable.

There are systems in which normalization implies strong nor-

malization. For example, in a restricted lambda-calculus (AI-calcu-

lus) this implication holds (the so-called second Church-Rosser

theorem, see Curry and Feys [ 3 , Ch. 4, § El), but the proof is not

trivial.

Prawitz [ I 7 1 proved normalization for derivations in natural

deduction. He also proved strong normalization for these deriva-

tions in C181. Note that in the latter proof he does not use his

results from C171, but quite a different proof technique developed

by Tait C211.

An interesting problem concerning normalization is the ques-

tion of uniqueness of normal forms. If a term A has the property

that every terminating reduction sequence leads to the same normal

form (but for a-reduction), then A is said to have a unique normal

form. We note that the Church-Rosser theorem implies the uniqueness

of the normal form if this exists.

In this thesis we shall show that, if in a system all terms

are normalizable into a unique normal form, then each term is

strongly normalizable. This will be proved for a certain lambda-

calculus called A, the method can, however, be applied to more

systems, and we suggest this as a field of further investigation.

§ 3. NORMALIZATION IN SYSTEMS OF TYPE3 LAMBDA-CALCULUS

In ordinary mathematics one, sometimes tacitly, assumes that

each object has a certain type (in our example of a term in lambda-

notation: Ax-x+2, we assumed that x has a type (e.g. that of the

natural numbers) in which addition is possible). In systems of

typed lambda-calculus one attaches a type to each term. In so doing

and in restricting the formation of terms in accordance with the

types (see the "applicability condition1' explained in Section I. 4)

one brings lambda-calculus nearer to usual mathematical systems.

We note here that there is a strong correspondence between de-

rivations in systems of natural deduction and terms in systems of

typed lambda-calculus, as well as between formulae in the one and

types in the other: a derivation D proving a formula F corresponds It to a term D' with type F', This is called the formulae-as-type

notion".

The latter notion has recently been investigated by various

authors in developing a theory of construction and in studying

functional interpretations. The first indication in this direction

was given in Curry and Feys C3, p. 312-3151. We further mention

~iuchli C11 1 , De Bruijn, who developed and applied this notion with a large variety of types in his mathematical language Automath

(C41), Howard C101, Prawitz C181 and Girard C91,

Normalization problems also arise in systems of typed lambda-

calculus. Sanchis C191 investigated a lambda-calculus with types

(essentially ~odel's theory of functionals of finite type) and

found all terms in this calculus to be strongly normalizable.

art in- of El21 admitted more general types and obtained nor- malization for his terms. His system is close to the requirements

of common mathematics in the sense that usual mathematical notions

such as the logical connectives and the recursion operator are in-

corporated.

In this thesis we shall regard a typed lambda-calculus, in

which the types themselves have lambda-structure. Our typed lambda-

calculus, which we call A, has a large overlap with the mathematic-

al language Automath C41. (See the following section for the rela-

tion between Automath and our system A,)

In particular, a single-line version of Automath (AUT-SL, see

C71) introduced by De Bruijn has led us to the investigations in

this thesis. Preliminary work in the direction of AUT-SL can be

found in our notes on Lambda-Automath (El31 and C141), in which

some syntactical notions of Automath were unified. In AUT-SL this

unification was extended considerably.

De Bruijn defined AUT-SL by means of a recursive programme.

Our definition of system A (given in Chapter 111) follows more

orthodox recursive lines. Nevertheless, the resulting systems are

the same.

In these systems there is no syntactical distinction between

terms and types. We therefore use the word expression rather than

term or type. There is one basic constant in the system, called T.

To each expression which does not end in T we shall assign a type

in a natural manner.

We say that expressions ending in T have degree 1. Each other

expression has some degree n > 1, while the degree of such an ex-

pression A is defined to be one more than the degree of the type of

A. In this manner we have expressions of any finite degree at our

disposal.

In Automath and in arti in- sf's system there is a restriction

to the degrees permitted. Both systems have only terms and types of

degree 1 , 2 or 3.

Our system has in connnon with Automath that logical connec-

tives, the recursion operator and a basic set of numbers (e.g. nat-

ural numbers) are not incorporated. The proofs of normalization

results concerning these systems can be formalized in first order

arithmetic.

Yet it is possible to interpret into these systems mathematic-

al theories containing, for instance, logical connectives and the

recursion operator by introducing new primitive equality .relations

which extend the existing equality relations which correspond to

conversion.

We shall prove normalization and strong normalization for our

system in Chapter 111. As mentioned above, we shall introduce a

method for deriving strong normalization from normalization togeth-

er with the uniqueness of normal forms (see Section 1.6).

5 4. THE RELATION TO THE MATHEMATICAL LANGUAGE AUTOMATH

Automath (see C41 and C51) was designed by De Bruijn as a

language for mathematics. It has the property that the interpreta-

tion of a text written in Automath is correct mathematics if the

text is syntactically correct.

Many such systems have been developed for logic. For mathemat-

ics, Russell and Whitehead's Principia Mathematica was the first

successful attempt in the direction of formalization. There have

since been many other attempts.

However, in the majority of these systems important parts of

the mathematical argumentation were not incorporated in the formal

system, but were dealt with at a meta-level. For example, in sys-

tems based on axioms and inference rules a theorem is true if it

can be inferred by successive application of a number of axioms and

rules. But one hardly ever says exactly (in terms of the formal

language) which axioms and rules were used, and in which order.

Moreover, the use of an axiom scheme was usually not substantiated

by a formalized indication of the substitution instance employed.

Admittedly, there is a gap in the completeness of the formal-

ization in Automath, too. The gap is that, in the case of "defini-

tionally equal1' expressions, there is no indication of how this

equality can be established on the basis of the language definition.

It is left to algorithms to justify these definitional equalities.

The existence of terminating algorithms for this purpose can

be proved by means of normalization properties. The question of

practical efficiency of such algorithms is, of course, a different

one, and is not considered in this thesis.

Two expressions in Automath are called definitionaZZy equal if

one expression can be transferred into the other by (1) conversions

and (2) the elimination of abbreviations.

A major problem forautomaticchecking in Automath is whether

definitional equality of two expressions is decidable. The latter

is clearly the case if each expression is effectively normalizable

into a unique normal form. In this respect, see Kreisel C231.

The main aim of this thesis is to prove the existence and

uniqueness of normal forms for A. Since A does not use an abbrevia-

tion system as a syntactical element like Automath does, we may re-

strict ourselves to conversions. We note that the omission of ab-

breviations is no severe restriction, since abbreviations are re-

latively simple operations usually? considered to be only notational

devices without mathematical content.

The mere typing of lambda-calculus expressions does not guar-

antee the property of normalization. We need more.

Automath permits only a restricted class of expressions. In

this class only those expressions E are included which obey the so-

called appZicabiZity condition: for each part of E which has the

form of a function F applied to an argument A it is required that

(1 ) F has a domain D, and (2) the type of A is definitionally equal

to D.

These requirements are natural for a system which is so close-

ly linked to ordinary mathematics. The following examples in lambda-

notation will make this clear. In the first place, it would be un-

natural to supply an expression which is not a function with an

argument: one can attach an argument to Ax.x+2, but it looks strange

to provide the number 7 with an argument.

Secondly, let us assume that x in Ax.x+2 is required to have

the natural numbers as type. This defines the domain of the func-

tion. Then one may write the application (Ax.x+2)3, since 3 has the

same type as x. But it would be quite unnatural to write the appli-

cation (Ax*x+2)a, - where - a represents a vector in R3. In AUT-SL and in A, expressions have to obey the applicability

condition, like in Automath, This condition is sufficiently strong

to guarantee normalizability (even a weaker condition suffices, see

Section 1.6).

We note that Automath has the property that assignment of a

type to an expression of degree 3 is different to that for expres-

sions of degree 2. Expressions of degree 3 have lambda-structured

types, whereas expressions of degree 2 all have the same type, viz,

t h e e x p r e s s i o n denoted by t h e u n d e r l i n e d symbol - type . (This symbol

t ype i s t h e Automath v e r s i o n of t h e symbol T used i n ou r sys tem A.) - As an i l l u s t r a t i o n we g i v e a n example i n lambda-notat icn.

Suppose t h a t t h e term hx*x+2 has Nat a s t ype f o r x, and t ype a s

t ype f o r Nat. Then hx.x+2 has deg ree 3. I n t h e manner of Automath

i t has Ax~Nat a s type. The l a t t e r e x p r e s s i o n , hav ing deg ree 2, ha s

a s type t h e e x p r e s s i o n - type .

I n AUT-SL and i n A , however, t h e ass ignment of t ypes t o ex-

p r e s s i o n s of any degree 2 2 i s t r e a t e d i n a uni form manner, com-

p a r a b l e t o t h e ass ignment of types t o e x p r e s s i o n s of deg ree 3 i n

Automath.

I f t h e term i n t h e above example (Ax*x+2) were t r e a t e d i n t h e

A-way, i t s t ype would a g a i n b e AxeNat, b u t t h e type of hx.Nat would

b e X X ~ T .

We n o t e t h a t an e x t e n s i o n of Automath, c a l l e d AUT-QE ("Automath

w i t h quas i - exp re s s ions " , s e e [ 6 ] ) , has more e x p r e s s i o n s of deg ree 1

t h an on ly -- t ype ; i t admits a s e x p r e s s i o n s of deg ree 1 some of t h o s e

admi t t ed i n AUT-SL and A. However, AUT-QE a l lows a cho i ce t o b e

made f o r some e x p r e s s i o n s of deg ree 2 , between e s s e n t i a l l y d i f f e r -

e n t types .

Again u s i n g t h e above example as an i l l u s t r a t i o n : i n t h e man-

n e r of AUT-QE one may choose e i t h e r Axetype o r t ype a s t ype of - - X x ~ N a t .

I t i s t o b e no ted t h a t t h e above-mentioned d i f f e r e n c e between

Automath ( o r AUT-QE) and AUT-SL ( o r A) ha s t h e impor t an t conse-

quence t h a t n e i t h e r Automath no r AUT-QE i s a subsystem of AUT-SL

( o r A). The r e s u l t s f o r A o b t a i n e d i n t h i s t h e s i s a r e t h e r e f o r e n o t

immediately t r a n s f e r a b l e e i t h e r t o Automath o r t o AUT-QE.

Normal iza t ion f o r a s imp le r form of AUT-QE, which does form a

subsys tem of AUT-SL, was proved by Van Benthem J u t t i n g [221, u s i n g

t h e norm i n t r o d u c e d i n t h i s t h e s i s (we s h a l l c a l l t h i s norm p ; c f .

S e c t i o n 1 . 6 ) . The n o r m a l i z a t i o n theorem of t h i s t h e s i s i s a gener-

alization of t h a t of C221,

Strong normalization for a system resembling Automath was re-

cently studied by R.C. de Vrijer on the basis of Tait's ideas ex-

posed in C211, and for Automath and AUT-QE by D.T. van Daalen (pri-

vate communications).

The uniqueness of normal form has only been proved with re-

spect to @-reduction. Uniqueness'of normal form with respect to

B-q-reduction is as yet an open question (see also Section 1.6).

§ 5. CHANGE OF NOTATIONAL CONVENTIONS

In the lambda-notation as usually employed the quantifiers

(such as Ax) are written to the left of the expressions they oper-

ate upon, whereas applications are written to the right. This cor-

responds to mathematical notational traditions to write quantifiers m

(such as Vx, En=], ., .) to the left, and to write the argument of a function f to the right (as in f(a)).

In ordinary mathematics these two kinds of operations have

nothing in common, but in lambda-calculus they are closely related

by @- and q-reduction. In a sense, quantification (also called ab-

straction) and application are inverse operations. Sequences of

such operations can be applied in various orders, and it is most

convenient to write them all on the same side of an expression,

thus showing clearly in which order the expression has been formed

from its constituents.

In Automath applications and abstractions are all written to

the left. Instead of writing abstractions in the form Ax, Automath

writes Cx,A], in which A stands for the type of the variable x. Ap-

plications are indicated by writing the expression in braces; in-

stead of the usual mathematical notation £(a) we write {a)f.

For example: the term given in lambda-notation as (Axax+2)3

reads in Automath as : {3)[x,Nat]plus (x, 2). (Here we assume that x

has as type the natural numbers, abbreviated Nat; a minor differ-

ence is that Automath uses only prefix notation for operators.)

Note t h a t t h e p a i r i n d i c a t e s t h e p o s s i b i l i t y c f B-reduction.

Sometimes, b u t n o t always, t h e p a i r I{ i n d i c a t e s t h e p o s s i b i l i t y of

q- reduct ion.

Th i s n o t a t i o n f o r a b s t r a c t i o n and a p p l i c a t i o n r e n d e r s t h e u se

of p a r e n t h e s e s ( ) e n t i r e l y s u p e r f l u o u s , s i n c e t h e r e can b e no

doubt a s t o t h e o r d e r i n which a b s t r a c t i o n s and a p p l i c a t i o n s ap-

pea r . The s e p a r a t i o n d o t a s used i n Xx*x+2 d i s a p p e a r s a s w e l l .

Automath u se s t h e pa r en the se s ( ) , b u t f o r a d i f f e r e n t purpose.

I n AUT-QE, AUT-SL and i n t h e sys tem A which we s h a l l develop

i n t h i s t h e s i s , t h e s e s l i g h t l y d i f f e r e n t n o t a t i o n a l conven t ions a r e

a l s o adopted.

§ 6 . SUMMARY OF THE CONTENTS OF THIS THESIS

Th i s t h e s i s c o n t a i n s a chap t e r on t h e fo rmal sys tem A (Chapter

11) and a c h a p t e r on t h e fo rmal sys tem A (Chapter 111) . I n t h e

l a t t e r c h a p t e r we develop t h e main r e s u l t s of t h e t h e s i s .

System A forms p a r t of sys tem A , c o n t a i n i n g t h o s e exp re s s ions

of A which obey t h e a p p l i c a b i l i t y c o n d i t i o n ( exp l a ined i n S e c t i o n

1 . 4 ) .

We s h a l l now d i s c u s s t h e c o n t e n t s of Chapter 11. There we de-

f i n e e x p r e s s i o n s i n d u c t i v e l y by: x and T a r e e x p r e s s i o n s ; CX,AIB and I A ) B a r e e x p r e s s i o n s i f A and B a r e s o ( x i s a v a r i a b l e ) .

11 I n sys tem A we on ly i n c l u d e t h o s e e x p r e s s i o n s which a r e d i s -

t i n c t l y bound", i . e . ( 1 ) which do n o t c o n t a i n f r e e v a r i a b l e s and

(2) which have d i s t i n c t b ind ing v a r i a b l e s .

Our p r e f e r e n c e f o r bound ( a l s o c a l l e d c l o s e d ) e x p r e s s i o n s (ex-

p r e s s i o n s w i t h o u t f r e e v a r i a b l e s ) is n o t i c e a b l e throughout t h i s

t h e s i s . We g i v e t h e f o l l o w i n g j u s t i f i c a t i o n f o r t h i s p r e f e r ence . We

b e l i e v e t h a t i n a typed lambda-calculus t h e f e a t u r e of t y p i n g can

only be meaningful i f every t y p a b l e e x p r e s s i o n h a s an e f f e c t i v e l y

computable type . S i n c e f r e e v a r i a b l e s have no t r a c e a b l e type i n our

sys tem, t h i s i m p l i e s t h a t on ly bound e x p r e s s i o n s a r e admi s s ib l e . I f

i n t h i s t h e s i s w e d e v i a t e from t h i s agreement by c o n s i d e r i n g ex-

pressions with free variables, this will be in cases in which it is

clear from the context which types belong to these free variables.

The consequence of the above agreement is that many expres-

sions under discussion begin with an abstractor chain Q. (An ab-

stractor chain is a string of abstractors; an abstractor has the

form [x,A], A being an expression,)

The fact that we require all binding variables in an expres-

sion in A to be distinct has only practical reasons (cf. Section

11.5).

We stress that system A is not a typed lambda-calculus in the

usual sense, since the types have no influence whatsoever on the

formation of expressions. The types, which themselves have a lambda-

structure, will only be treated as formal expressions. It is not

until Chapter 111, dealing with the restricted system A , that the

types will play the usual r81e in the formation of expressions.

This is due to the applicability condition imposed upon expressions

in A.

We shall formulate the relations a-, 6- and q-reduction inside

A and we shall prove a number of properties of these reductions in

the system A (in Sections 11.4, 11.5 and 11.7, respectively).

In Section 11.6 we shall consider somereductions related to B-

reduction. Our proof of strong normalization in A (Section 111.3)

is based on these reductions. The more important one of these re-

ductions will be called B1-reduction.

We shall explain its characteristic property by reducing the

term which we previously used as an example: (Ax*x+2)3, or, in

Automath-notation: {3)[x,Nat]plus(x,2) (cf. the previous section).

As for @-reduction, we have the relation {3)[x,Natlplus(x,2) 2 B

'B plus(3,Z). But with B1-reduction, which we denote by 2 , we

1 have: {3)[x,Nat]plus(x,2) > {3)[x,Nat]plus(3,2). Here the part

1 {3}Cx,Nat] is left intact on the right-hand side. (Actually B1-re-

duction is more complicated; see Section 11.6.)

The following feature of B1-reduction is worth noting: Appli-

cation of 8-reduction sometimes enables one to eliminate a non-

no rma l i zab l e sub te rm ( i n t h i s r e s p e c t w e r e c a l l t h e example

(Ax* ( X y ~ y ) ) (w2w2) 2B Xy*y of S e c t i o n I. 2 ) , bu t w i t h BI - reduc t ion

t h i s i s imposs ib le .

I n S e c t i o n 11.6 we prove t h e Church-Rosser p r o p e r t y f o r 1 - r e d u c t i o n s , u s i n g a proof t echn ique of T a i t and arti in-~zf. This

p r o p e r t y imp l i e s t h e uniqueness of normal form f o r B1-reduct ions .

From t h e Church-Rosser p r o p e r t y f o r @ l - r e d u c t i o n s fo l lows t h e

Church-Rosser p r o p e r t y f o r @-reduc t ions (a l though t h e l a t t e r cou ld

a l s o be proved d i r e c t l y ) .

Unfo r tuna t e ly t h e Church-Rosser p r o p e r t y f o r 6-q-reductions

does n o t ho ld i n our sys tem A . The t r o u b l e h e r e a r i s e s from t h e

typed c h a r a c t e r of our lambda-calculus. We e x p l a i n t h i s i n g r e a t e r

d e t a i l i n S e c t i o n 11.7 . (However, we c o n j e c t u r e t h e Church-Rosser

p r o p e r t y f o r 6-q-reductions i n A ; s e e t h e end of t h e p r e s e n t sec-

t i o n . ) I 1 I n S e c t i o n 11.7 we a l s o prove a theorem concern ing t h e pos t -

ponement of n- reduct ions" i n a sequence of 6- and n - reduc t ions , by

means of a method sugges ted by Barendregt .

A t t h e end of S e c t i o n 11.7 we d e f i n e lambda-equivalence f o r A :

A and B a r e lambda-equivalent i f t h e r e i s a C such t h a t A and B re-

duce t o C. Th i s lambda-equivalence i s n o t n e c e s s a r i l y t r a n s i t i v e

s i n c e t h e Church-Rosser p r o p e r t y f o r 6-n-reductions does n o t ho ld

i n A .

I n S e c t i o n 11.8 we d e f i n e a formal type-opera to r c a l l e d Typ,

which a s s i g n s a type t o an exp re s s ion n o t end ing i n T. The a c t i o n

of t h i s type-opera tor i s s y n t a c t i c a l l y s imple and i s i n agreement

w i t h what we mentioned about t h e ass ignment of t ypes i n S e c t i o n I . 3 .

I n S e c t i o n 11.8 we a l s o d e f i n e t h e degree - func t ion Deg, which

i s i n agreement w i t h ou r d e s c r i p t i o n of deg ree a s g iven i n S e c t i o n

1.3. I n ou r sys tem A we can app ly t h e type-opera to r Typ a f i n i t e

number o f t imes .

For each e x p r e s s i o n A i n A t h e r e i s an n 2 0 such t h a t T~~~ A

ends i n r , which i m p l i e s t h a t T~~~ A h a s no type . (Here T~~~ A i s

ob t a ined by n a p p l i c a t i o n s of t h e type-opera to r . ) Th i s n i s t h e

* degree of A minus one. We define Typ A to be T~~~ A for that par-

ticular n.

We begin Chapter 111 with the definition of the formal system

A (Section 111.1). Among the theorems in Section 111.1 there is one

which states that the type of an expression in f~ again belongstoA,

In Section 111.2 we prove the normalization theorem for A. We

use a norm p, which is a partial function on A. The norm p(A) for a

certain A in A is itself an expression in A. The norm of A is de-

fined if A obeys a weak form of the applicability condition, which

amounts to the following: for each part of an expression E which

has the form of a function F applied to an argument A: (1) F has a

domain D, and (2) the norms of A and D are defined and equal (apart

from a-reduction).

Applied to expressions for which the norm is defined (so-

called p-normable expressions), the norm p has two powerful proper-

ties: ( 1 ) If A reduces to B, the norms of A and 5 are (essentially)

equal, and (2) the norm of an expression is (essentially) the same

as the norm of its type.

The norm p (A) of a p-normable expression A can be obtained by

(1) replacing non-binding variables by their types, repeating this

process until no non-binding variable remains, and (2) cancelling

adjacent pairs {C)CX,DI. We show in Section 111.2 that all expressions in A are p-

normable. We subsequently show that each p-normable expression has

a normal form for @-reductions. It follows in particular that A is

normalizable for @-reductions. It now easily follows that A is also

normalizable for 8-n-reductions. Our proofs show that the normal

form of A in A is effectively (viz. primitively recursively) com-

putab le.

In Section 111.3 we prove strong normalization for A. We use

the B1-reduction introduced in Section 11.6. We show that expres-

sions in A are normalizable for 8,-reductions, using the same meth-

ods as in the corresponding proof for 6-reductions in Section 111.2.

By using the Church-Rosser property for BI-reductions as proved in

S e c t i o n 11.6 we o b t a i n t h e uniqueness of normal form f o r 6 I -reduc-

t i o n s .

The s p e c i a l f e a t u r e s of Bl- reduct ion enab l e us t o conclude

s t r o n g n o r m a l i z a t i o n i n A f o r Bl- reduct ions from t h e no rma l i za t i on

and t h e uniqueness of normal form. S t rong n o r m a l i z a t i o n i n A f o r 6-

r e d u c t i o n s i s a consequence, a s w e l l a s s t r o n g n o r m a l i z a t i o n f o r 6-

q - r educ t i ons .

The uniqueness of normal form i n A i s proved f o r B-reductions

b u t n o t f o r 6-q-reductions. Neve r the l e s s , t h e l a t t e r would be a

consequence i f we could prove t h e fo l l owing c o n j e c t u r e :

Con j ec tu r e I. I n A t h e Church-Rosser p r o p e r t y h o l d s f o r B-q-reduc-

t i o n s .

(The d i f f i c u l t i e s i n p rov ing t h i s a r i s e i n t h e same p l a c e where t h e

co r r e spond ing s t a t emen t f o r A t u r n s o u t t o be f a l s e ; s e e S e c t i o n

As t o A , t h e r e i s an impor tan t c o n j e c t u r e on c l o s u r e :

Conjec tu re 11. I f A i s an exp re s s ion i n A and i f A reduces t o B,

t hen B i s an e x p r e s s i o n i n A.

I n [ I 5 1 we s t a t e d t h i s a s a theorem, b u t the proof t u rned o u t t o be

i n c o r r e c t . The l a t t e r c o n j e c t u r e has no i n f l u e n c e upon t h e r e s u l t s

i n t h i s t h e s i s ; i t i s , however, of impor tance f o r t h e c o n s t r u c t i o n

of an e f f i c i e n t checking-a lgor i thm f o r e x p r e s s i o n s i n A.

- 17 -

CHAPTER 1 1 , THE FORMAL SYSTEM A

§ 1. ALPHABET AND SYNTACTICAL VARIABLES

We use the following symbols as our alphabet:

(i) an infinite set of (individual) variables:

a, B, Y, a], BI, Y1,*-• ; (ii) a single constant, called the base: T ;

(iii) theimproper symbols: C , 1 , { , , , . As syntactical variables denoting certain well-structured

symbol strings (possibly empty) we use small Latin letters a, b,

c,... and Latin capitals A, B, C,.. . (primed or subscripted if required). In special definitions, called Notation Rules, we re-

strict the use of some syntactical variables (and its primed or

subscripted variants),

For example, we agree upon:

Notation Rule 1.1. As a syntactical variable for arbitrary strings

of symbols from the alphabet we use the Latin capital S. 0

Such a string can be empty. The empty string itself is denoted by 4 .

Notation Rule 1.2. As syntactical variables for individual variables

we use the small Latin letters x, y and z. 0

(Instead of "individual variable" we often say "variable".)

Hence from now on each use of a syntactical variable S (or S 1' S f , etc.) denotes a string of symbols from the alphabet, and each

use of a syntactical variable x (or y, x l , etc.) denotes an indi-

vidual variable.

It is usual to build strings of symbols from the alphabet and

syntactical variables, concatenated. For example, Cx,al[y,a]x is

such a string. We shall call this kind of string a mixed string.

Equality of mixed strings will be expressed in the discussion

language by the symbol E. For example, if we wish to express that

the strings S and Cx,a]B are the same, we write S E [x,a]B. The

symbol $ is the negation of r . 1 I There are said to be two occurrences" of a in the string

Ca,Bla. We shall formalize this notion occurrence. We define that -11 1 Sf' occurs i n S a f t e r S' if there is an b such that S E S' S1'S1" .

Hence, in the above example: cl occurs in [a,Bla after [, and a

occurs in [a,B]a after [a,@]. In this manner we can distinguish

between occurrences.

The following statement is clear: If S occurs in S after S' 1 and S occurs in S after S", then S occurs in S after S'S".

2 1 2 Consider the mixed string [x,ylx, in which there are two oc-

currences of x. If x denotes a, then Cx,ylx denotes [a,yla: both

occurrences of x are replaced by a. If, moreover, y denotes B, then

[x,~]x denotes [a,@]@. It is, however, also possible that both x

and y denote a. Then [x,y]x denotes [a,ala (see also Schoenfield

C20, p. 71).

Syntactical variables are used in two hardly distinguishable

rGles: as abbreviations ("we abbreviate Cx,alB as Sf') and as vari-

ables ("Let S be a string of the form ..."). It is also good usage to state something in the nature of: " ~ e t A I [x,B]c", meaning:

"Assume that A has the form [x,BIC for certain x,B and C" (in this

manner one economizes in the use of the existential quantifier).

We shall define many specific sets and relations in an induc-

t i v e manner (see Schoenfield C20, p. 41). The proof technique linked

with this kind of definition, which amounts to induction on the

construction, is often called (somewhat confusingly) induction on

the length o f proof (or induction on theorems, see Schoenfield C20,

p. 51).

We shall call an application of one rule of the inductive def-

inition a derivation-step. If a relation is defined inductively by

a number of r u l e s , then the r e l a t i o n i s also s a i d t o be generated

by these rules. When speaking of a transitive (or reflexive, etc.)

relation generated by a number of rules, one wishes to express that

the r u l e of t r a n s i t i v i t y (or r e f l e x i v i t y , e t c , ) i s t o be added t o

t h a t number of r u l e s .

If S denotes a c e r t a i n symbol s t r i n g , then the length of S i s

the number of symbols i n t h a t s t r i n g , We denote the l eng th ~ f S by

I s I . For example, i f S [a ,Bla, then I s I = 6.

§ 2. EXPRESSIONS

The expressions of our systems a r e induct ive ly def ined as

follows (we use the word expression r a t h e r than the words term o r

type) :

- ~

( 1 ) A v a r i a b l e i s an expression.

( 2 ) T i s an expression.

( 3 ) I f x i s a v a r i a b l e and i f A and B a r e expressions, then CX,AIB

i s an expression.

(4) If A and B a r e expressions, then (A)B i s an expression. 0

Note t h a t t h i s d e f i n i t i o n g ives a unique cons t ruc t ion of an

expression.

Notation Rule 2.2. As s y n t a c t i c a l . v a r i a b l e s f o r expressions we use

the La t in c a p i t a l s A, B , C , , . . , N, 0

Def in i t ion 2.3. A symbol s t r i n g of the form [ x , ~ ] i s c a l l e d an

a b s t r a c t o r , a symbol s t r i n g of the form ( D ) an a p p l i c a t o r . A

lambda-phrase i s e i t h e r an a b s t r a c t o r o r an app l i ca to r . A (possibly

empty) s t r i n g of a b s t r a c t o r s ( app l i ca to r s , A-phrases) i s c a l l e d an

a b s t r a c t o r chain (an (appl ica tor chain, a lambda-phrase chain) . 0

Notation Rule 2.4. As a s y n t a c t i c a l v a r i a b l e f o r a b s t r a c t o r chains

we use the La t in c a p i t a l Q , f o r app l i ca to r chains the L a t i n cap-

i t a l R and f o r A-phrase chains the Lat in c a p i t a l P. 0

The number of entries in a string forming an abstractor chain

Q (an applicator chain P , a lambda-phrase chain R) is denoted by

l l Q l l (I1 P I I , l l R l l respectively). Hence l l Q l l = 0 if Q Z 4 , and

I I Q C x , C l l l = IiQll + 1.

An expression B can be a subexpression of an expression A, de-

noted B c A. This relation is inductively defined as follows:

Definition 2.5.

(1) A c A .

(2) If C c A or C c B, then C c [x,A]B and C c {A)B.

Note: if B c A, then A E S BS i.e.: a subexpression of an 1 2' expression A is an expression which forms a connected part of A.

Instead of B c A we sometimes say: A contains B, If B c A and

B 9 A, we call B a proper subexpression of A.

Theorem 2.6. If F c E and E c D, then F c D.

Proof. Induction on ID I . If B c A, then B occurs in A , but there may evidently be more

occurrences of B in A. In the following we wish to be able to dis-

tinguish between such occurrences of B in A. We shall indicate the

occurrence meant by saying "B c A after S" if B c A and B occurs

in A after S.

Definition 2.7. Let B occur in A after S and let C occur in A 1

after S2. We call these occurrences d i s j o i n t if either S2 E SIBS'

Theorem 2.8. Let B occur in A after SI, let C occur in A after S 2' let B c A and C c A . Then ( 1 ) B and C occur disjointly in A or (2)

B c C or (3) C c B.

Proof. Induction on the length of proof of B c A.

Let B c A after S. We shall inductively define the factor of A

with respect to S and B (denoted A I < S;B > ) in definition 2.9.

I n t h i s d e f i n i t i o n t h e o c c u r r e n c e of B meant i s p r e c i s e l y d e s c r i b e d .

However, i t w i l l o f t e n be c l e a r from t h e c o n t e x t which occur-

r e n c e of B i s meant i n c a s e B c A, I n t h a t c a s e t h e p r e c i s e i n d i c a -

t i o n of t h i s o c c u r r e n c e i s s u p e r f l u o u s , and i n s t e a d of A 1 < S;B >

we s h a l l w r i t e A / B .

I n f o r m a l l y we can c o n c e n t r a t e t h e i n d u c t i v e d e f i n i t i o n of A I B , under t h e c o n d i t i o n t h a t i n each of t h e f o l l o w i n g r u l e s t h e occur-

r e n c e s of B under d i s c u s s i o n a r e " i n c o r r e s p o n d i n g p laces" :

(1 ) I f A E B , t hen A I B E A.

(2 ) I f A - C X , C ] D , t hen A I B E C I B i n c a s e B c C and A I B ~ [ X , C ] ( D ~ B )

i n c a s e B c D.

( 3 ) I f A Z K I D , t h e n A I B 5 C I B i n c a s e B c C and A I B Z D I B i n c a s e

B c D .

For a d e s c r i p t i o n of a c h a r a c t e r i s t i c p r o p e r t y of A I B , which

j u s t i f i e s i t s i n t r o d u c t i o n , s e e t h e f o l l o w i n g s e c t i o n ( a f t e r Th.

3 .6) .

The fo rmal i n d u c t i v e d e f i n i t i o n of A I < S;B > i s t h e f o l l o w i n g :

D e f i n i t i o n 2.9. L e t B c A a f t e r S.

(I ) I f A B, t h e n A I < S;B > E A .

( 2 ) L e t A E [x,CID. I f B c C a f t e r S I and [x ,SI - S , then

A I < S;B > Z C I < S ; B > . I f B c D a f t e r S2 and [x,ClS2 - S, t h e n 1

A I < S ; B > E CX,CI(DI < S ~ ; B > ).

( 3 ) L e t A {c ID. I f B c C a f t e r S1 and {S1 Z S , t h e n

A \ < S ; B > Z C I < SI ; B > . I f B c D a f t e r S2 and {CIS2 5 S , then

Note: t h e p a r e n t h e s e s ( ) i n [ X , C I ( D I < S2;B > ) belong t o t h e

d i s c u s s i o n l anguage and a r e meant t o f i x t h e scope of I .

L e t B c A a f t e r S. It w i l l b e c l e a r t h a t B c A I B , o r , a f o r -

t i o r i : A I B ends i n B ( h e r e , of c o u r s e , A I B i s meant t o b e

A1 < S;B > ). It i s a l s o e v i d e n t t h a t A I B 5 QB and (QA) I B 5 Q ( A ~ B ) .

We s t a t e t h e f o l l o w i n g theorems:

Theorem 2.10. I f C c B c A , then (AIB) I C A I C . P roof . I n d u c t i o n on / A I .

Theorem 2.11 . I f B c A and A / B E Q [x,C]Q B , t h e n C c A. 1 2

P roof . I n d u c t i o n on I A ~ , u s i n g Th. 2.6. 0

Theorem 2.12. I f E Q [x,CID and B c C a f t e r S, t h e n E I B E Q ( c IB) 1 1 ( h e r e E I B i s E l <QICx,S ;B> and C I B i s C I < S ; B > ).

Proof . I n d u c t i o n on I I Q1 I I . 0

Theorem 2.13.

(1 ) I f [x,C]D c A a f t e r SI and B c D a f t e r S2, t h e n A I B Z Q I C X , C ] Q ~ B

( h e r e A I B i s A1 < S C X , C ] S ~ ; B > ). 1

(2 ) I f B c A a f t e r S and A I B E Q [x,C]Q B, t h e n t h e r e i s a D such 1 2

t h a t [x,C]D c A a f t e r S l , B c D a f t e r S2 and S : S [x,C]S 1 2

( h e r e A I B i s A I < S;B > ).

Proof . I n b o t h p a r t s of t h e theorem: i n d u c t i o n on I A 1 .

We conc lude w i t h a n i n d u c t i v e d e f i n i t i o n of t h e f u n c t i o n T a i l ,

which maps e x p r e s s i o n s t o e x p r e s s i o n s :

D e f i n i t i o n 2.14.

(1 ) T a i l ( x ) E x.

(2) tail(^) : r .

( 3 ) T a i l (Cx ,AIB) E T a i l (B) . ( 4 ) Tai l({A)B) E T a i l ( B ) .

Note t h a t Ta i l (A) can o n l y b e a v a r i a b l e o r r. An e x p r e s s i o n A

c a n always b e w r i t t e n ( u n i q u e l y ) a s A E P T a i l ( A ) ( i n which P de-

n o t e s a A-phrase c h a i n , s e e N o t a t i o n Rule 2.4) .

§ 3. BOUND EXPRESSIONS

An occurrence of a variable in an expression can be a free, a

bound or a binding occurrence. We shall introduce these well-known

notions in our system too. An occurrence of a variable in an ex-

pression is binding if and only if that occurrence immediately

follows an opening bracket [. If D contains an occurrence of x (i.e.: x c D), then that

occurrence of x is either bound (and there is a unique binding

occurrence of x which binds that bound occurrence) or free. A

formal description is given in the following inductive definition. 11 I n this definition we often encounter corresponding" occurrences

of x. For easy understanding we shall not use our formalism con- 11 cerning occurrences (see Section II.I), but we shall introduce a

certain x" and refer to it as "that x".

~efinition 3 . 1 .

( 1 ) x is free in x.

(2) Let a certain x be free in A or B. Then that x is free in {A)B.

(3) Let a certain x be free in A, Then that x is free in [y,AIB

(both if y r x and if y f x).

(4) Let a certain x be free in B, Then that x is free in CY,AIB if

y f x, but that x is bound in [x,A]B (by the binding x occur-

ring in [x,A]B after [).

(5) Let a certain x in A be bound by a binding x in A, or let a

certain x in B be bound by a binding x in B, Then that x is

bound by the corresponding binding x in both {A)B and [y,AIB

(also if y : x). 0

The binding x occurring in Cx,AIB after C binds precisely the

free x's in B (if any).

We shall mainly be interested in expressions in which no vari-

able is free, called bound expressions (in the literature also

called closed expressions), In bound expressions the same binding

variable can occur in different instances, This cannot, however,

g i v e r i s e t o confusion a s t o t h e connect ion between a bound v a r i -

a b l e and t h e binding v a r i a b l e by which i t i s bound.

Yet, f o r p r a c t i c a l reasons , we wish t o avoid such express ions .

We c a l l a bound express ion i n which a l l b inding v a r i a b l e s a r e d i f -

f e r e n t , a d i s t i n c t z y bound express ion , and we r e s t r i c t ou r se lves t o

t he s e t of a l l d i s t i n c t l y bound express ions , which we c a l l A .

This i s no e s s e n t i a l r e s t r i c t i o n . Every i n t e r e s t i n g theory

concerning bound expres s ions can be r e s t r i c t e d t o d i s t i n c t l y bound

express ions .

Le t x c D a f t e r S and l e t t h i s x be bound i n D. I t fo l lows

from Def. 3.1 t h a t we have D - S C X , E I F S ~ such t h a t CX,EIF c D , 1

x c F a f t e r S S Cx,EIS r S and t h e x occur r ing i n D a f t e r S [ 3' 1 3 1

b inds t h e x occu r r ing i n D a f t e r S. We s h a l l c a l l CX,EI t h e binding

abstractor of t h e bound x .

From t h i s and Th. 2.1 3 (1 ) i t fo l lows:

Theorem 3.2. I f x c D E A , then D ~ X E QICx,EIQ x and [x ,El i s t h e 2

b ind ing a b s t r a c t o r of t h e bound x i n D. 0

It fo l lows from Th. 2.13 (2 ) :

Theorem 3.3. I f f o r each x c D t h e r e a r e Q l , A and Q such t h a t 2 D / X E QI[x,A]Q2x, then D i s a bound express ion . 0

The fol lowing theorem expresses i n an i n t r i c a t e manner t h e

obvious o b s e r v a t i o n t h a t , i n ca se x c K c D E

a b ind ing a b s t r a c t o r e i t h e r o u t s i d e o r i n s i d e

Theorem 3.4. I f x c K c D E A , then e i t h e r

( i ) D I K E Q 1 Cx,AlQ2K, D ~ X : Q l Cx,AIQ2Qfx and

( i i ) K ( X a Ql Cx,A]Q2x and D ~ X E Q Q I Cx,A]Q x. 2

I n bo th c a s e s Cx,Al i s t h e binding a b s t r a c t o r

Proof. Let D I K QK, then D lx (by Th. 2.10)

A , t he x i s bound by

K.

Q(K(X) - QQ'x. Hence, by Th. 3 . 2 , e i t h e r Q E Q , Cx,AIQ2, o r

Q' Q, Cx ,AlQ,. 0

Theorem 3.5. I f QCAIB E A , then QA and QB E A ; i f QCx,A]B E L , then

QA E A .

P roof . Apply Th. 3.3.

Theorem 3.6. I f A E A and B c A a f t e r S, then A I B E A .

Proof . F i r s t assume t h a t x c B . Thenby Th.2. IO: (A!B) IX:A!X:Q 3 C X , C ] Q 4 x.

Next assume t h a t A / B QB and x c Q. Then Q 5 Q1[y,D]Q2 and x c D.

BY ~ h . 2.11: D c A. S O ( Q B ) ~ X E Q ~ ( D ~ x ) " Q ~ D ) [ x ( A / D ) [ x = A ~ X E

Q3[x,C]Q4x. Apply Th. 3 .3 , and n o t e t h a t t h e b i n d i n g v a r i a b l e s i n

A I B must b e d i s t i n c t . 0

The above theorem s t a t e s an e s s e n t i a l f e a t u r e of t h e f a c t o r

A I B . I f A E A and B c A, t hen no t n e c e s s a r i l y B E A . But A I B , which 11 i s QB f o r a c e r t a i n a b s t r a c t o r cha in Q , c l o s e s " B i n A by p l a c i n g

i n f r o n t of B t h o s e a b s t r a c t o r s which n e c e s s a r i l y b ind a l l f r e e

v a r i a b l e s of B. T h i s , t o g e t h e r w i t h our p r e v i o u s l y u t t e r e d wish t o

r e s t r i c t ou r e x p r e s s i o n s a s much a s p o s s i b l e t o A, j u s t i f i e s our

i n t r o d u c t i o n i n S e c t i o n 11.2 of t h e f a c t o r A / B .

We c o n t i n u e w i t h t h r e e theorems, r e l a t e d t o one. ano the r .

Theorem 3.7. I f Q[x,A]B E A and x # B , t h en QB E A .

P roof . Observe t h e v a r i o u s p l a c e s where a v a r i a b l e y ($ x ) can

occur i n QB. 0

Theorem 3.8. I f QB E A and Q [ x , A I B E A , t h e n x 6 B.

Proof . The assumpt ion x c B l e a d s t o a c o n t r a d i c t i o n .

Theorem 3.9. I f QA and QB E A , i f t h e b ind ing v a r i a b l e s i n A and B

a r e d i s t i n c t and i f x does n o t occur a s a b ind ing v a r i a b l e i n QA o r

QB, t h en Q[x,AlB E A .

P roof . Again obse rve t h e v a r i o u s p l a c e s where a v a r i a b l e y ( $ x )

can occur i n Q[x,A]B,

In general: If QB and QPB E A and x occurs as a binding vari-

able in P, then not x c B, We say: P has no binding influence on B,

The description of A, which we gave so far, began with (gen-

eral) expressions and selected the distinctly bound expressions

among these. This method is not so practical for theoretical in-

vestigations. In the following two theorems we shall indicate how

we can compose ex3ressions in A from expressions in A, Or, rather,

we shall show how expressions in A can be decomposed into smaller

expressions, which also belocg to A.

Theorem 3.10.

(I) r E A.

(2) If QA E A and if x does not occur in QA, then QCx,Alx E A and

QCx,Al-r E A.

(3) If QA and Qy E A, if x does not occur in QA and if x $ y, then

QCx,Aly E A *

(4) If QA and QB E A and if the binding variables in A and B are

distinct, then Q{A)B E A.

Proof. It is trivial that Q[x,A]x, Q[x,A]T, QCx,Aly and Q{A)B

respectively are again expressions, These are also clearly bound

expressions, moreover distinctly bound by the conditions given in

the theorem. 0

We may consider the four parts of the previous theorem as

derivation rules.

Definition 3.11. We call K A-constructible if we can establish

that K E A by a (finite) number of applications of the rules in

Th. 3.10. 0

The proof of the following theorem is technical. Yet it is

interesting to see how we can establish A-constructibility. For

better understanding, we shall express the main lines at the end

of the present section.

Theorem 3.12. If K E A, then K is A-constructible.

Proof. Induction on I K / . If I K I = I , then K z T and K is A-con-

structible by rule (1).

Assume that / K / > 1 , and let all distinctly bound expressions

K' with IK' / < ( K I be A-constructible (first induction hypothesis).

Then K PIP2 ... P Tail(K) for some n r 1 , where each of the P. n 1

is a lambda-phrase (i.e. either an abstractor or an applicator). I t We can now prove the lemma: For all i for which 1 6 i B n+l

it holds that: K I (Pi . . .P n Tail(K)) is A-constructible", by in-

duction on n+l-i. - (1) Let i = n+l (i.e. P. 1 ... P n = 4 ) . If K T ~ ( K ) K then the

first induction hypothesis leaves nothing to prove. If IK(~ai1 (K) ( =

I K I , then Kl~ail(~) K E [zl ,El] ... [Z n' E n ]Tail(K) for n 2 1. - (la) Assume that Tail(K) x. Then for exactly one s : zs = x.

Abbreviate [z, ,El] . . . [zt ,Et] as Qt for 0 L t n. We distinguish - the cases z 2 x and z $ x. If z = x, then K is A-constructible n n n

by rule (2) since Q E E A by Th. 3.5. If z $ x, then K is A- n-l n n

constructible by rule ( 3 ) , since Q n-1 E n E A and Qn - lzn E A (the

latter by Th. 3.7).

E E A by Th. 3.5 and A- (lb) Assume that Tail(K) r. Then Qn-l n

constructible by induction. By rule (2) we then find that K is A-

constructible.

(2) Let I B i B n and assume that K((P. 1 ... P n Tail(K)) is A-con- J

structible if i < j L n+l (second induction hypothesis). If

I K( (Pi . . . P Tail (K)) I < I K I , then again the first induction hy- n pothesis leaves nothing to prove. So let I K I (Pi . . . P n Tail(K)) / =

I K ( . Then K . K/(pi ... P n Tail(K)) [ z 1 ,EI]...[zt,EtIPi ... P n Tail (K) . (2a) Let P. 1. Then KI(P. ... P Tail(K)) 5

1 1 n K( (Pi+l . . . P Tail(K)), which is A-constructible by the second n induction hypothesis.

(2b) Let P. IF}. Then QtF E A by Th. 3.5 and A-constructible by 1

the second induction hypothesis, and the same holds for

QtPi+l . . . P Tail (K) . Hence, by rule (4) : n

Q t w P i + l ... P Ta i l (K) 5 K i s A - c o n s t r u c t i b l e . n From t h i s lemma i t fo l l ows t h a t K I (PI . . . P T a i l ( K ) ) E K I K 5 K i s n A-cons t r u c t i b l e .

( I n t h i s proof we d i d n o t check t h e c o n d i t i o n s concern ing v a r i a b l e s

i n r u l e s ( 1 ) t o ( 4 ) . I t i s e a sy t o s e e t h a t t h e s e a r e f u l f i l l e d i n

t h e a p p r o p r i a t e p l a c e s . ) 0

From Th. 3.10 and Th. 3.12 fo l l ows t h a t r u l e s (1 ) t o ( 4 ) gen-

e r a t e t h e r e l a t i o n K E A . Hence, w e can c o n s i d e r t h e s e f o u r r u l e s

a s a second d e f i n i t i o n of A . The advantage of t h e l a t t e r i s t h a t we

have an i n d u c t i v e d e f i n i t i o n of A , whereas t h e o r i g i n a l d e f i n i t i o n

was r e s t r i c t i v e w i t h r e s p e c t t o t h e s e t of a l l e x p r e s s i o n s .

I n a proof by i n d u c t i o n t h e f o u r r u l e s of Th, 3.10 a r e much

e a s i e r t o use . The n o t i o n " i nduc t i on on t h e l e n g t h of proof" usu-

a l l y r e f e r s t o a proof ( i n f a c t a c o n s t r u c t i o n ) produced by an in -

d u c t i v e d e f i n i t i o n , a s i s t h e second d e f i n i t i o n of A. From now on

we s h a l l r e f e r t o t h i s l a t t e r d e f i n i t i o n when g i v i n g a proof by I1 i n d u c t i o n on t h e l e n g t h of proof of K E A".

Note t h a t , g i ven a K E A , o n l y one of t h e d e r i v a t i o n s t e p s i n

Th. 3.10 can g i v e K E A a s a conc lu s ion ( i f K G T, t h e n t h i s must

be r u l e ( 1 ) . I f K : Q{A)B, t hen t h i s must be r u l e ( 4 ) . I f K E QT

and Q $ $I , t h en t h i s must be p a r t 2 of r u l e ( 2 ) . I f K 5 Qx and

Q r Q1Cx,A1, t hen t h i s must be p a r t 1 of r u l e ( 2 ) . I f K : Qx and

Q G Q1[y,Al f o r some y $ x , t h e n t h i s must be r u l e ( 3 ) ) .

The proof of t h e p r ev ious theorem s u g g e s t s how w e can e s t a b l i s h

t h a t K E A by u s i n g t h e f o u r r u l e s of Th. 3 -10 ( i - e . t h a t K i s A-

c o n s t r u c t i b l e ) . We can exp re s s t h i s i n words a s f o l l o w s :

Let K be a d i s t i n c t l y bound exp re s s ion .

( 1 ) we f i r s t e s t a b l i s h t h a t ~ l ~ a i l ( ~ ) i s A-cons t ruc t i b l e :

( l a ) i f Ta i l (K) : x , t hen f i n d t h e b ind ing a b s t r a c t o r Ex,Al i n K

which b inds K , e s t a b l i s h t h a t K I A QA i s A-cons t ruc t i b l e , and

app ly r u l e ( 2 ) t o o b t a i n QCx,AIx E A , Le t Q ' Q;,...,Qt be t h e

a b s t r a c t o r s o c c u r r i n g in i K I ail (K) "between" CX,AI and x. I n s e r t

t h e s e a b s t r a c t o r s , s t a r t i n g w i t h Q' (from l e f t t o r i g h t ) , by i n - 1

s e r t i n g Q! i n QCx,AIQ; . . . Q;-]x between Q! and x (by r u l e ( 3 ) ) . 1 1- 1

In this manner we establish that K I Tail (K) is A-cons truc tib! c .

(Ib) If Tail(K) E T, then K/Tail(K) 5 r and we may use rule (I)

immediately, or KITail (K) QCx,A]r.

In the latter case: establish that QA is A-constructible, and

apply rule (2) to establish that K/Tail(~) is also A-constructible.

(2) We establishedthat ail(^) [xl ,AI I . . . [x n ,A n ]Tail(K) is A-

constructible. In K we find applicators B , . . . B "between"

the abstractors Cx.,A.]. Insert these B. starting with B and 1 1 1 ' R

ending with B ( from right to left) in the appropriate places, 1 using rule (4). In this manner we establish that K is A-construct-

ible.

(Note the following. If we establish that K is A-constructible,

then we use the A-constructibility of K(E for all E c K. We can

prove this by induction on the length of proof of K E A.)

§ 4. REPLACEMENT, RENOVATION AND a-REDUCTION

If we replace a certain variable x in all its occurrences

(free, bound or binding) in an expression A by a variable y, then

we denote the result of this replacement by ((x:=y))A. An inductive

definition of simple replacement is the following (induction here

is on the length of the expression):

Definition 4.1. For each pair x and y, ((x:=y)) is a function from

expressions to expressions.

(1) ((x:=y))x r y; ((x:=y))z % z if z f x; ((x:=y))r 5 T.

(2) ((x:=y))[z,A]~ % [((x:=y))z, ((x:=y))Al((x:=y))B.

(3) ((x:=y)){A)B { ((x:=y))A)((x:=y))B. 0

The simple replacement of certain variables by others will be

used for making the binding variables in an expression distinct.

This we shall call the renovation of the expression.

Renovation is in fact nothing but a repeated renaming of vari-

ables. Renaming does not affect relevant properties of expressions

(under a r e a s o n a b l e i n t e r p r e t a t i o n of "va r i ab l e " ; s e e a l s o what we

s a i d concern ing a - equ iva l en t terms i n S e c t i o n 1 .1 ) .

We have main ta ined names f o r v a r i a b l e s f o r r e a sons of t r a d i -

t i o n and l e g i b i l i t y . Th i s i s a t t h e expense of t h e r enova t i on se-

l e c t o r ( t o be i n t roduced i n t h i s s e c t i o n ) and t h e so -ca l l ed a-

r e d u c t i o n (our wishes concern ing bound e x p r e s s i o n s i n t h e p r ev ious

s e c t i o n had no th ing t o do w i t h names f o r v a r i a b l e s , b u t w i t h our

d i s l i k e of t h e occur rence of f r e e v a r i a b l e s ; t h e a d d i t i o n a l wish

t o have d i s t i n c t l y bound e x p r e s s i o n s , however, does concern names

f o r v a r i a b l e s , and l e a d s u s t o i n t r o d u c e r e n o v a t i o n ) .

We s h a l l d i s c u s s t h e p roce s s of r enova t ion , The mathemat ica l

meaning of r e n o v a t i o n i s n o t profound. I f one d i s l i k e s a formal-

i z a t i o n of a n i n t u i t i v e l y c l e a r concep t , one can c o n t i n u e r e a d i n g

a t Def. 4 .4 (concerning a - r educ t i on ) .

There i s , of cou r se , one p r e c a u t i o n which one must t a k e i n

t h e r e n o v a t i o n p roce s s : t h e r e l a t i o n between bound and b ind ing

v a r i a b l e s should remain u n a f f e c t e d i n a n a t u r a l way, For example,

i n [ x , y lCx ,x lx t h e b ind ing x o c c u r r i n g a f t e r C b inds t h e bound x

occu r r i ng a f t e r Cx,yl[x, ; t h e b ind ing x o c c u r r i n g a f t e r [x,ylC

b inds t h e bound x o c c u r r i n g a t t h e end of t h e express ion . I n

changing t h i s e x p r e s s i o n i n t o a n e x p r e s s i o n w i t h d i s t i n c t b ind ing

v a r i a b l e s , we might o b t a i n : [ x , y ] [ z , x l z f o r a c e r t a i n z f x.

Such a v a r i a b l e z a s i n t roduced i n t h e l a t t e r example by t h e

r e n o v a t i o n p roce s s p l ays a s p e c i a l r81e. It ha s t o be chosen w i t h

c a r e . A t any r a t e i t should be d i f f e r e n t t o a l l b ind ing v a r i a b l e s

i n t h e e x p r e s s i o n under d i s c u s s i o n , o r , a s we s h a l l say: i t ha s t o

be f r e s h w i t h r e s p e c t t o t h a t exp re s s ion .

We i n t r o d u c e t h e r e n o v a t i o n s e l e c t o r F r o p e r a t i n g on ex- v' p r e s s i o n s . I n u s i n g t h e r enova t i on s e l e c t o r F r w i t h a n e x p r e s s i o n v A we have i t preceded by a lambda p h r a s e c h a i n P, g iv ing PFr A. v The s u b s c r i p t V deno t e s a f i n i t e s e t of v a r i a b l e s , which can b e

empty. We s h a l l n o t s p e c i f y t h e v a r i a b l e s be long ing t o V u n t i l t h e

f o l l owing s e c t i o n , where w e u s e F r i n t h e formal d e f i n i t i o n of v subs t i t u t i o n .

The e x p r e s s i o n PFr A can i n f o r m a l l y be d e s c r i b e d a s be ing PA', v where ( 1 ) A' i s ob t a ined from A by r e n o v a t i o n of A and ( 2 ) t h e

f r e s h v a r i a b l e s chosen du r ing t h i s r e n o v a t i o n do n o t occur i n P o r

V and a r e mu tua l l y d i s t i n c t .

The fo l l owing i n d u c t i v e d e f i n i t i o n g i v e s a f o r m a l i z a t i o n of

t h i s concep t :

D e f i n i t i o n 4.2. Le t V be a f i n i t e s e t of v a r i a b l e s . -

( 1 ) PFr$ E Px; PFr r 2 PT. v (2 ) PFrV([y,B]C) ~ [ z , ~ ' ] ~ r ~ ( ( ( y : = z ) ) C , w i t h PB' PFr V B , wh i l e

z does n o t occur i n PB' and z 4 V .

( 3 ) pFrV({B}C) - P{Bf}FrVC, w i t h PB' : PFrVB. 0

From t h e above d e f i n i t i o n we s e e t h a t t h e r e n o v a t i o n of an

e x p r e s s i o n t a k e s p l a c e from l e f t t o r i g h t . For i n s t a n c e , t h e re -

nova t i on r e s u l t i n g i n PFr ( IB )c ) f i r s t r e q u i r e s t h e r e n o v a t i o n r e - v s u l t i n g i n PFr B, and subsequen t ly t h e one r e s u l t i n g i n P{Bf}FrVC. v This i m p l i e s t h a t t h e f r e s h v a r i a b l e s chosen i n t h e r enova t i on

p roce s s a r e mu tua l l y d i s t i n c t ,

Of c o u r s e u n c e r t a i n t i e s remain a s t o t h e c h o i c e of f r e s h

v a r i a b l e s . An o r d e r i n t h e s e t of a l l v a r i a b l e s could t u r n t h e

s e l e c t o r F r i n t o an o p e r a t o r , We s h a l l , however, n o t push t h e v f o r m a l i z a t i o n t h i s f a r .

I n t h e fo l l owing s e c t i o n we s h a l l d e s c r i b e s u b s t i t u t i o n by

t h e a i d of t h e r e n o v a t i o n s e l e c t o r , and we s h a l l i n t u r n u s e sub-

s t i t u t i o n i n d e s c r i b i n g t h e 6-reduct ion. Our u s e of t h e r enova t i on

s e l e c t o r i s meant t o keep a n e x p r e s s i o n d i s t i n c t l y bound a f t e r B-

r e d u c t i o n . We s h a l l . u s e t h e r e n o v a t i o n s e l e c t o r i n t yp ing an ex-

p r e s s i o n , a s i s d e s c r i b e d i n S e c t i o n 11.8, w i t h t h e same purpose.

We u s u a l l y beg in r enova t i on w i t h V = 4 ( h e r e , of cou r se , $

deno t e s t h e empty s e t and n o t t h e empty s t r i n g ) .

~ e f i n i t i o n 4.3. PFrA r PFr A. @

I n f a c t PFrA i s t h e c o n c a t e n a t i o n of P and F r 8, where V i s

t h e s e t of v a r i a b l e s o c c u r r i n g i n P.

L e t PFr A PA' be t h e r e s u l t of a r e n o v a t i o n . I f we a g a i n v w r i t e F r A, i n t h e same c o n t e x t , we do n o t r e q u i r e a new r e n o v a t i o n , v b u t mean A ' . I f we wish a n o t h e r r e n o v a t i o n i n t h e same c o n t e x t ,

t h e n w e s u p p l y F r w i t h primes: P F ~ ' A c a n b e such a new renova t ion . v v We s h a l l now d e f i n e t h e a - r e d u c t i o n r e l a t i o n , For a n in fo rmal

d i s c u s s i o n of a - r e d u c t i o n s e e S e c t i o n 1-1, We r e s t r i c t a - r e d u c t i o n

t o e x p r e s s i o n s i n A :

D e f i n i t i o n 4.4 . a - r e d u c t i o n , denoted by 2 i s t h e t r a n s i t i v e r e - a ' l a t i o n g e n e r a t e d by:

( 1 ) I f A E A and i f y does n o t occur i n A, t h e n A 2 ((x:=y))A. 0 a

The a - r e d u c t i o n i s c l e a r l y an e q u i v a l e n c e r e l a t i o n ( r e f l e x -

i v i t y : t a k e x t o be a v a r i a b l e which does n o t o c c u r i n A; symmetry:

n o t e t h a t x does no l o n g e r occur i n ( (x :=y))A ).

I f two e x p r e s s i o n s a r e r e l a t e d by a - r e d u c t i o n (A 2 B), w e a I t speak of t h e a - r e d u c t i o n A 2 B". T h i s i s c l e a r l y abuse of lan- a

guage, a l t h o u g h i t canno t g i v e r i s e t o confus ion .

A renaming of a s i n g l e v a r i a b l e i n a d i s t i n c t l y bound ex-

p r e s s i o n i s c a l l e d a single-step a - r e d u c t i o n , denoted a s A 2' B a

( s o A 2' B i f and o n l y i f A E A and B ( (x :=y))A where x o c c u r s a

a s a b i n d i n g v a r i a b l e i n A and y does n o t o c c u r i n A).

The f o l l o w i n g theorems a r e t r i v i a l :

Theorem 4.5. I f PA E A , t h e n P F r 8 E A ; i f A E A and A 2 B, t h e n a

B E A and I A I = I B I . 0

Theorem 4.6. L e t PA, PB, PP'A and PP'B E A , Then PA 2 PB i f and a

o n l y i f PP'A 2 PP'B. a 0

§ 5. SUBSTITUTION AND 6-REDUCTION

Substitution is an operation acting on expressions. We denote

"the result of the substitution of A for x in B" by (x:=A)B. One

can use several definitions of substitution which are equivalent.

We shall use the definitions given in Def. 5.1 and Def. 5.2.

These definitions of substitution can informally be described as

follows: P(x:=A) B is the expression which we obtain from PB by v replacing all free x's in B with renovations of A, in which the

fresh variables are chosen in the following manner: they have to

be mutually distinct for all renovations of A replacing the free

x's, and they have to be distinct from the variables occurring in

B, P or V (we do not replace the binding variables in B by fresh

ones). Here V denotes a finite set of variables, which can be

empty . This careful dealing with fresh variables is necessary to

guarantee that an expression with distinct binding variables has

again distinct binding variables after B-reduction (to be defined

in this section.); substitution is an essential part of 6-reduction.

The following part of this section as far as Def, 5.5., will

formalize the above notion of substitution, As with renovation,

our formalization of substitution may be cumbersome to the reader.

One may continue with Def. 5-5, without impairing understanding.

An inductive definition of P(x:=A) B is the following V (induction is here on the length of B):

~efinition 5.1.

(1) P(x:=A) x E PFr A; P(x:=A) y Py if y $ x; P(X:=A)~~ E PT. v v v (2) If y j! x, then P(X:=A)~[~,B]C P[~,B'](X:=A)~C, where

PB' z P(X:=A)~B, W being the union of V and the set of all

variables occurring in C.

If y 5 x, then P(X:=A)~C~,BIC r P[y,B'lC, where B' is obtained

as above,

(3) P(X:=A)~{B)C P{B')(X:=A)~C, where B' is obtained as above. 0

From t h e above d e f i n i t i o n we see t h a t s u b s t i t u t i o n ( a s re-

nova t ion) t akes p l a c e from l e f t t o r i g h t i n an express ion : f o r

i n s t a n c e , t h e s u b s t i t u t i o n r e s u l t i n g i n P (x :=A)~{B}c f i r s t re-

q u i r e s t h e s u b s t i t u t i o n r e s u l t i n g i n P(x:=A) W B, and subsequent ly

t h e s u b s t i t u t i o n r e s u l t i n g i n P{B ' ) (X:=A)~C.

I n p a r t ( 2 ) of t h e d e f i n i t i o n we see t h e importance of t h e

s u b s c r i p t used i n (x:=A): i n execut ing t h e s u b s t i t u t i o n r e s u l t i n g

i n P(x:=A) B we must have a t our d i s p o s a l t h e s e t of v a r i a b l e s W

occur r ing i n C i n o rde r t o be a b l e t o choose f r e s h v a r i a b l e s d i f -

f e r e n t from t h e v a r i a b l e s i n C, The s e t W con ta ins t h e l a t t e r v a r i -

ab l e s .

The s e t V can be t h e empty s e t , When we begin a s u b s t i t u t i o n ,

V i s u s u a l l y empty.

D e f i n i t i o n 5.2. P(x:=A)B 2 P(x:=A) B. 4

I n t he above d e f i n i t i o n s t h e r e a r e two more o r l e s s unusal

p a r t s . Usual ly (x:=A)x i s de f ined a s A; however, w i th a view t o

our wish t o keep d i s t i n c t l y bound express ions d i s t i n c t l y bound

a f t e r some s u b s t i t u t i o n , we d e v i a t e from t h i s . Next, one sometimes

d e f i n e s (x:=A)[y,B]C a s [z,(x:=A)Bl(x:=A)(y:=z)C, i n e i t h e r of t h e

ca ses t h a t x 5 y o r . x f y, z being a f r e s h v a r i a b l e . The l a t t e r

d e f i n i t i o n p reven t s so-cal led "confusion of v a r i a b l e s " ( c f , Curry

and Feys [3 , Ch. 3D21; s e e a l s o t h e example below), b u t g i v e s

r i s e t o an a d d i t i o n a l amount of s imple s u b s t i t u t i o n s of v a r i a b l e s

of t h e form (y :=z) , which we f i n d cumbersome.

I n us ing Def. 5.1. and Def. 5.2. confusion of v a r i a b l e s may

occur i f t h e u s e of t h e s u b s t i t u t i o n o p e r a t o r i s n o t r e s t r i c t e d .

For example: [y ,A] (x :=y) [y ,~ lx r [y,A][y,-rly, where t h e f i n a l y i n

t h e l a t t e r exp res s ion i s in f luenced by [ y , ~ ] , and no t by [y,AI a s

i t should be.

I n gene ra l : confusion of v a r i a b l e s may a r i s e a s a consequence

of t h e s u b s t i t u t i o n r e s u l t i n g i n P(x:=A) B i f a f r e e v a r i a b l e y of v A (wi th y $ x) occurs a s a b ind ing v a r i a b l e i n B, and i f t h e r e i s

11 a f r e e x i n B w i t h i n t h e scopett of t h a t b inding v a r i a b l e y.

A s u f f i c i e n t c o n d i t i o n f o r avo id ing t h i s i s , t h a t t h e f r e e v a r i -

a b l e s of A do n o t occur a s b ind ing v a r i a b l e s i n B.

We u s e s u b s t i t u t i o n o n l y i n t h e r e l a t i o n B-reduction, de f i ned

l a t e r i n t h i s s e c t i o n . The above c o n d i t i o n i s t h e r e f u l f i l l e d .

Hence con fus ion of v a r i a b l e s cannot a r i s e i n ou r system.

Note t h a t (x:=A) o p e r a t e s on f r e e x ' s , ( (x:=A)) on a l l x ' s ,

and F r on a l l b ind ing v a r i a b l e s i n a n exp re s s ion .

We a l s o d e f i n e t h e s u b s t i t u t i o n o p e r a t o r f o r lambda-phrase

cha in s :

D e f i n i t i o n 5.3. I f (x:=A)PT r P ' T , t h e n (x:=A)P - P ' . 0

One may i n t e r c h a n g e t h e s u b s t i t u t i o n o p e r a t o r and t h e reno-

v a t i o n s e l e c t o r under c e r t a i n c o n d i t i o n s :

Theorem 5.4. Le t PA E A and P[x,BID E A . Then

PFr(x:=A)D 2 P(x:=A)FrD i f no b ind ing v a r i a b l e of D o c c u r s i n A. a

Proof . I n d u c t i o n on I D I , u s i n g Th. 4.5 and t h e lemma: - "((y:=z))(x:=A)C (x:=A)((y:=z))C i f y # A (bu t f o r renaming)". U

S u b s t i t u t i o n i s used i n t h e more i n t e r e s t i n g r e d u c t i o n i n

lambda-calculus c a l l e d B- reduc t ion ,whichwedenoteby2 The i n t e r - (3.

, p r e t a t i o n l i n k e d w i t h B-reduction i s t h e a p p l i c a t i o n of a f u n c t i o n

t o a n argument ( s e e a l s o t h e i n fo rma l d e s c r i p t i o n i n S e c t i o n 1.1).

We s h a l l r e s t r i c t B-reduction t o e x p r e s s i o n s be long ing t o A .

This i s unusa l . One u s u a l l y conce ives of a r e d u c t i o n a s a formal

r e l a t i o n between e x p r e s s i o n s i n which f r e e v a r i a b l e s may occur .

It i s o n l y ou r p r e f e r e n c e f o r d i s t i n c t l y bound e x p r e s s i o n s which

makes u s choose t h e d e f i n i t i o n s g iven below.

Note t h a t our @-reduc t ion i s n o t e s s e n t i a l l y d i f f e r e n t from

t h e u s u a l concept . The u s e of t h e Q i n Def, 5 ,5 i s a l i t t l e ob-

s c u r i n g i n t h i s r e s p e c t .

We f i r s t d e f i n e s i n g l e - s t e p B-reduct ion, denoted by 2' : B

D e f i n i t i o n 5.5. S ing l e - s t ep B-reduction i s t h e r e l a t i o n gene ra t ed

by:

(1) If Q{A)Cx,BIC E A, then Q{A)CX,BIC 2' Q(x:=A)C. P

(2) Let Q{A)C and Q{A)D E A. If QC 2' QD, then Q{A)C 2' Q{A)D. B B (3) Let Q[x,A]C and Q[x,B]C E A. If QA 2' QB, then B

(4) Let Q{A)C and Q{B)C E A. If QA 2' QB, then Q{A)C 2' Q{B)C. 0 B B

Note: one rule appears to be missing (viz.: let Q[x,AIC and

Q[x,A]D E A. If QC 2' QD, then Q[x,A]C 2' Q[x,A]D). But this is a 8 B

derived rule, see Cor. 5.14.

Rules (2), (3) and (4) in the above definition are called the

monotony rules of single-step B-reduction; we call rule (1 ) the

rule of elementary 6-reduction.

If K and L are related by a single-step B-reduction, we obtain

L from K b y replacing a certain subexpression {A)Cx,BIC in K by

(x:=A)C. This is, in terms of interpretation, a'single functional

application.

If K 2; L , then we have a construction (a "proof") bhich establishes that relation according to Def. 5.5. Such a construc-

tion consists of one derivation step which is an elementary B-

reduction (rule (1) of Def. 5.5) followed by a number of derivation

steps which are monotony steps (rules (2) to (4)). Note that a

single-step B-reduction is achieved from a nwnber of derivation

steps.

We note that, since Q{A)CX,BIC E A, no free variable of A can

occur as binding variable in C. This is sufficient, as previously

stated, to prevent "confusion of variables".

Definition 5.6. B-reduction is the reflexive and transitive

closure of single-step B-reduction.

(This means that (1) if K 2' L, then K r L; (2) K 2 K; B B B (3) if K 2 L and L 2 M, then K 2 14.)

B B B

If A and B are related by a single-step B-reduction, we speak

of "the single-step B-reduction A 2' B". As with a-reduction, this 6

is abuse of language. Analogously we speak of "the @-reduction

A 2 B". B If A. 2; A,, A 2' A ..., A 2' A we also write

1 B 2 9 n n-1 6 n' ' A >' A 2' . . . 2 A or A. ZB An. We call this a composite single- 0-B 1 B B n step B-reduction, or an n-step B-reduction (a zero-s tep €3-reduction

0 has, of course, the form A 2 A). If A. tg AI ,..., A 2 A we B n-I B n'

also write A A 2 ... 2 A . 0 % I B B n

If follows from the definition of 6-reduction that each 6-

reduction K > L can be presented as an n-step B-reduction B

K E A >' . 2 A E L. This splitting is called a decomposition 0 -B B n

of a B-reduction.

Each of the monotony rules has the form: "If reduction (i)

holds, then reduction (ii) holds". It is usual to call reduction

(ii) the direct consequence of reduction (i), For example:

Q{A}c 2B Q{A}D is a direct consequence of QC 2 QD. We recall that B

"the length of proof of K 2' L" is the total number of derivation B

steps in the proof of K 1' L. B

A proof of K 2' L begins with an elementary B-reduction B

Q{A}Cx,BlC 2B Q(x:=A)C. In this case we say that Q{A}Cx,BIC gen-

erates the single-step B-reduction K 2' L. The following theorem B

holds :

Theorem 5.7. Let K 2 A. Then Q{A}[x,B]C generates a single-step B-

reduction of the form K r ' L if and only if {A}[x ,B]C c K and B Q{A}CX,BIC E K~{A}CX,BIC.

Proof. => : Induction on the length of proof of K 2' L. B

<= : We state the following Lemma: "Let K r Q'M E A, and let

E E {A}CX,BIC c M. Then K I E generates a single-step B-reduction

K E Q'M 2' Q'N 'I. We prove this lemma by induction on I M ~ : B

(1) Let M E. Then K I E 5 K r Q1{A}[x,B]c 2' QF(x:=A)C. B (2a) Let M E [x,F]G and E c F. Call K' r Q'F E A; then E c F, hence,

by induction: K' /E E Q' (F IE) generates a single-step @-reduction

K' Q'F 2' Q'F' . By applying monotony rule (3) it follows that K' I E B also generates Q'M ! Q'CX,F]G 2; QYX,F' IG. Moreover, K I E K' JE.

(2b) Let M Z Cx,FIG and E c G. Take K' 5 K E Q1[x,F]G E Q"G; then

E c G, hence, by induction: K' I E genera tes a s ingle-s tep B-reduc-

t i o n Q"G 2' Q W G ' , which can be r e w r i t t e n as Q ' M 2' Q1[x,F]G'. B B

(3a) Let PI {FIG and E c F. The proof i s analogous t o the one i n

case (2a) , us ing monotony r u l e ( 4 ) ins tead of (3) .

(3b) Let M i {FIG and E c G. Again the proof i s analogous t o the

one i n case (2a) , now using monotony r u l e ( 2 ) i n s t ead of (3) .

This proves the lemma. The "if -par t" of t h e theorem follows

immediately from the lemma.

We s t a t e the "closure theorem f o r A with respec t to s ingle-

s t e p 6-reduction":

Theorem 5.8. I f K E A and K 2' L, then L E A . B

Proof. Induct ion on the length of proof of K 2' L, Note t h a t our B

d e f i n i t i o n of s u b s t i t u t i o n f o r a v a r i a b l e with t h e a i d of the ren-

ovat ion s e l e c t o r i s e s s e n t i a l . In the proof we can use Th, 3 . ~ . 0

Corol lary 5.9. I f K E A and K 2 L, then L E A . B

One may conceive of the B-reduction r e l a t i o n , not a s a r e l a -

t i o n between expressions, but a s a r e l a t i o n between a-equivalence

c l a s s e s ( i n the a-equivalence c l a s s of K we inc lude a l l K' such

t h a t K 2 K'). a

The following theorem gives a j u s t i f i c a t i o n f o r t h i s concep-

t i o n of B-reduc t ion :

Theorem 5.10. Let A E A , A 2' B and A 2 C, Then t h e r e i s a reduc- 8 a

t i o n C 2' D such t h a t B r D. B a

Proof. It i s s u f f i c i e n t t o assume t h a t A 2' C. Apply induct ion on a

t h e l eng th of proof of A 1' B. B 0

I n the sequel we s h a l l sometimes r e f e r t o the above conception 11 of B-reduction, by i n s e r t i n g the words but f o r a-reduction" i n a

statement ( f o r example: "A r B but f o r a-reduction" means t h a t B

t he re a r e A' and B' such t h a t A 2 A' r B' r B ) . However, we a B a I I o f t e n omit the words but f o r a-reduction".

We s h a l l proceed with a number of theorems, i n which e s p e c i a l l y

the r c l e of the a b s t r a c t o r chain Q i n a @-reduction i s considered,

Q occurring i n the beginning of an expression. (The d e f i n i t i o n of

l l P l l was given a f t e r no ta t ion r u l e 2.4.)

Theorem 5.1 1 . I f QE E A , QE 2' Q'F and l l Qll = 11 Q ' 11, then Q - Q' o r B

E z F. In the l a t t e r case Q - Q Cx,KlQ2, Q' Q1Cx,L1Q2 and 1

Proof. Induction on the length of proof of QE 2' Q'F. There a r e B

four poss ib le cases f o r the l a s t d e r i v a t i o n s t e p i n the proof of

QE 2' Q'F. I n t h r e e of these cases the conclusion i s : Q Q ' . The B

f o u r t h case i s t h a t the l a s t d e r i v a t i o n s t e p i n the proof of

QE 2; Q'F i s : " Q ~ K 2; Q I L , SO QE 5 QI[x,K]M 2' Q [x,L]M z Q'F". B 1

Now i f IlQll 5 l lQ1 1 1 , then Q Q ' , and i f IlQll > IIQ 11, then E 5 F, 1 Q 5 Ql[x,KIQ2 and Q' 5 Q,[x,LIQ2. CI

Theorem 5.12. I f QE E A and QE 2' K, then K E Q'F' f o r c e r t a i n Q' B

and F' with 1 1 Q ' 11 = 1 1 Q 1 1 .

Proof. The reduct ion QE 2' K must be the conclusion of one of the B

r u l e s of Def. 5.6. It i s easy t o see t h a t the statement of the

theorem holds good i n a l l t hese cases. 0

If a reduct ion QC ?3

Q ' D i s given, i n which 11 Q I ~ = 11 Q' 11, one 1 I can conceive of an accompanying reduction" of C t o D. The following

theorem shows t h i s .

Theorem 5.13. I f QC, QOC and QOD E A , QC 2 Q'D and IIQII = IlQ' 11, B

then QoC 2 Q D. B 0

Proof. F i r s t assume t h a t QC 2' Q'D. Then by Th. 5.11 Q Q' o r B

C Z D. I n the l a t t e r case i t i s t r i v i a l t h a t Q C 2 Q D. So assume 0 6 0

Q Q ' . Then induct ion on the length of proof of QC 2' Q'D and B

Th. 5.1 2 y i e l d Q C 2' Q D. The general theorem follows. 0 B O 0

The appa ren t ly miss ing monotony r u l e , announced immediately

a f t e r Def. 5.5, i s a consequence:

Coro l l a ry 5.14. L e t QC, QD, Q[X,A]C and Q[x,A]Dc A . I ~ Q c ~ ' 8 QD, then

Q[x,A]C 2' Q[x,A]D. 0 B

We def ined 6-reduction a s being t r a n s i t i v e and r e f l e x i v e .

We s h a l l now show t h a t B-reduction i s a l s o monotonous:

Theorem 5.15. The monotony r u l e s hold f o r B-reduction, i . e . :

(2) I f Q{A}C and Q{I.}D E A , and QC 2 QD, then Q{A}C rB Q{A}D. B

(3) If Q[x,A]C and Q[x,B]C c A , and QA 2 QB, then B QCx,AlC 2 QCx,BlC. B

( 4 ) I f Q{A}C and Q{B}C E A , and QA r QB, then Q{A}C rB Q{B}C. B

Proof. We s h a l l prove r u l e ( 2 ) . - Since QC 2 QD, we know t h a t QC 2' E 2' E . 2' E 2' QD.

B 1 3 1 6 2 B n B From Th. 5.12, Th. 5.13 and induc t ion on n i t fo l lows t h a t t h e r e

i s a l s o a r e d u c t i o n QC 2' QF 2; QF2 .. . >' QF 2' QD. Repeated B 1 -6 n B

a p p l i c a t i o n of t h e corresponding monotony r u l e f o r s ing l e - s t ep 8-

r e d u c t i o n g ives : Q{.A}C 2; Q{A}FI 2; ... 2 ; Q{A}D, hence

Q{AIC rB Q{AID.

The o t h e r two monotony r u l e s f o r @-reduct ion can be proved anal-

ogously . U

One may extend Cor. 5.14 t o 6-reduction:

Theorem 5.16. Le t QC, QD, Q[x,A]C and Q C X , A ] D E A . I f QC 2 B QD, then

Q[x,AlC 2 QCx,AID. B cl

Theorem 5.17. I f QC, PC and PD E A , QC 2 Q'D and 11Q11 = l lQ1 l l , then B

PC 2 PD. B

Proof. Le t P C ~ C 5 QoC, then Q C 2 Q D fo l lows from Th. 5.13. The 0 8 0

theorem i s proved by r epea t ed a p p l i c a t i o n of monotony r u l e (2) f o r

6-reduction ( see Th. 5.15) . 0

Note: t h e converse of t h i s theorem does n o t ho ld ("I£ PC, QC

and QD E A and PC 2 PD, then QC 2 QD "). B B &ample (Q E Cx,r] , P = CX,T]{X)) :

[ X , T ] { X } [ ~ , T ] { X } [ Z , T I ~ >B C x , r l { ~ l C ~ , ~ l ~ , b u t n o t :

[ x , - r l C y , ~ I ~ x ~ C ~ , ~ l y tg C X , T ] [ Z , ~ ] ~ *

Theorem 5.18. I f P{A]CX,B]C E A , t h en P{A}[X,B]C 2 ' 6 P(x:=A)C.

Proof . Th i s i s a consequence of t h e fo l lowing :

(P{A}[x,B]C) I{A}[x,B]c Q{A}[x,B]C 2 ' Q(x:=A)C. Apply Th.5.17. 0 8

Theorem 5.19. I f QK, QM and Q ' M E A , QK 2 Q'L and IlQll = I l Q ' I I , t h en B

QM 1 Q'M. B

Proof . Along t h e same l i n e s a s t h e proof of Th. 5.16.

The fo l l owing theorems a r e t r i v i a l consequences of t h e

preceding.

Theorem 5.20. I f QCy,KIL E A and Q[y,KlL rg QCy,K1 I L ' , t h en

QK 2 QK' . 0 B

Theorem 5.21. I f QK E A , QK 2 Q ' K ' and l lQ l i = 11Q' 1 1 , t hen B

QK t6 QK' >B Q ' K ' and QK 2 Q ' K 2 B Q ' K ' .

We d e f i n e t h e be ta -equ iva lence r e l a t i o n a s f o l l o w s :

D e f i n i t i o n 5.22. L e t A and B E A . We c a l l A beta-equ5vaZent t o B

(denoted: A - B) i f t h e r e i s a n exp re s s ion C such t h a t A 2 C and B B -

It i s c l e a r t h a t be ta -equ iva lence i s r e f l e x i v e and symmetric.

The t r a n s i t i v i t y of be ta -equ iva lence w i l l be proved i n Th. 7.35,

u s i n g Th. 6 .43 ( i n t h e l i t e r a t u r e t h e t r a n s i t i v e c l o s u r e of be ta -

equ iva l ence i s c a l l e d be ta -convers ion) .

Theorem 5.23. L e t QK and QL E A - I f QK - QL, t h e r e i s an N such B

t h a t QK 2 QN and QL 2 QN. B B

Proof. Since QK - QL: (7K 2 A and QL 2 A. From Th. 5.12 it 3 B a

follows that A 5 Q'N with IlQll = 119' 11. Then from Th. 5.21:

QK 2 QN and QL 2 QN. 0 B B

From this theorem, together with Th. 5.15,it easily follows

that the monotony rules hold for beta-equivalence (parts (a), (c)

and (d) of the following theorem); part (b) follows from Th. 5.23

and Th. 5.16:

Theorem 5.24.

(a) If QC, QD, Q{A}C, Q{A}D E A and QC - @ QD, then Q{A}c -@ Q{A}D.

(b) If QC, QD, Q[x,A]C, Q[x,A]D E A and QC - B QD, then QCx,AlC -6 Q[x,AID.

(c) If QA, QB, Q{A}C, Q{B}C E A and QA -B QB, then Q{A}C - @ Q{B}C.

(d) If QA, QB, Q[x,A]C, QCx,B]C E A and QA - QB, then B Q[x,AlC -B Q[x,BlC. 0

§ 6. OTHER B-REDUCTIONS

In a B-reduction we eliminate a pair of the form {A}[x,B] 11 occurring in an expression, obtaining copies" of A (to be precise:

expressions A', A", etc., which are renovations of A) instead of

the non-binding x's in that expression. We sometimes wish to retain I1 information concerning "past't 6-reductions, as a kind of scar" in

an expression.

The easiest way to do this is to maintain the pair {A}[x,B] in

an expression after 8-reduction. We shall formalize this kind of

8-reduction, calling it B1-reduction.. Another B-reduction, called

6,-reduction, will be introduced especially to eliminate the L

I I scars" {A)[x,B] as soon as they are no longer required, We shall

show that a B-reduction can be decomposed into a f3 1 -reduction and

a @2-reduction.

We describe in this section B1- and B2-reduction as a prep- aration for Section 111.3.

The fact that we wish to keep the pair.{A)[x,B] in an ex-

pression after 6 -reduction complicates matters since we wish each 1 sequence of B-reductions to have a corresponding.sequence of B - 1 reductions. For example, a sequence of B-reductions

Q{A){B)[X,T][~,X]~ 2' Q{A}[y,FrB]y 2' QFrA should. have its counter- B B

part in B1-reductions: Q{AHB)[x,~lCy,xly 2' B 1

latter single-s tep B -reduction we have to ignore the scar {B)[x,T] 1

located between { A ) and [y,FrBI (a property of such a scar IBHx,TI

is that the x does not occur in the expression following it).

However, B -reduction permits more. One may ignore the pair 1 {B)[x,r] in Q{A){B)[x,~][y,xly, in spite of the fact that x does

occur in [y,x]y. This gives the single-step B1-reduction:

Q{AHB}CX,TIC~,XI~ 2 ; , Q{A}{B}[x,rlCy,xlFrA, byapplying [y,xly to A.

This is a real extensron of the usual B-reduction concept. We shall

call a string like {B)[x,T] in the above example, which may be lo- 1 I t 1 cated between function" and argument" of a Bl-reduceable ex-

pression, a @-chain.

Moreover, we shall agree that the relation

Q{A)P[y,C]D 2' Q{A}P[y,Cl(y:=A)D (in which P is a B-chain) does B 1 only hold if y occurs in D. If we did not require this, one could

continue the @ -reduction with the latter expression, thus pro- 1

ducing a non-terminating @ -reduction sequence. I

The f3 -reduction relation, on the other hand, eliminates an 2 applicator and an abstractor as in Q{A}P[y,C]E 2' QPE (in which

81 L

P is again a @-chain); the latter relation only holds, however, if

y does not occur in E.

We give an inductive definition of @-chain:

Definition 6.1.

(1) If P 4 , then P is a @-chain.

( 2 ) If P is a @-chain, then {B)P[x,c] is a @-chain.

(3) If PI and P are @-chains, then P P is a B-chain. 2 1 2

N o t a t i o n R u l e 6.2. We w r i t e P t o i n d i c a t e t h a t P i s a B-chain.

We w r i t e [ ~ , B I C t o i n d i c a t e i n [x,B]C t h a t x # C .

A 8-chain h h a s t h e p r o p e r t y t h a t t h e number of a p p l i c a t o r s i n

P i s e q u a l t o t h e number of a b s t r a c t o r s i n P. Moreover, i f ? E P 1 P 2 '

t h e number of a p p l i c a t o r s i n P i s a t l e a s t e q u a l t o t h e number of 1

a b s t r a c t o r s i n P 1 ' We can a l s o e x p r e s s t h i s by means of a vaZuation v of lambda-

p h r a s e c h a i n s , d e f i n e d i n d u c t i v e l y by (1 ) v ( @ ) = 0 ,

( 2 ) v({A)P) = v ( P ) + 1 , ( 3 ) v(Cx,AIP) = v ( P ) - 1 . n

Then f o r a B-chain i t h o l d s t h a t ( i ) v ( P ) = 0 , and ( i i ) i f

P P t h e n v ( P ) 2 0. These c o n d i t i o n s a r e a l s o s u f f i c i e n t . 1 2 ' 1 The f o l l o w i n g theorem concerning B-chains can be proved by t h e

a i d of t h e above-mentioned v a l u a t i o n p r o p e r t i e s :

Theorem 6.3.

CI

( 2 ) P E P P i s a @-chain i f and o n l y i f P' E P P P i s a B-chain. 1 3 1 2 3

Note t h a t f o r each $ @ t h e r e i s a un ique decompos i t ion A CI n

P - P P h w i t h p . $ $ and Fi $ ?!?!I f o r some P! 2 @ and 1 2 " ' n 1 1 1 1

PI! $ @. 1

The f o l l o w i n g theorem shows t h a t a B-chain h a s a compact

s t r u c t u r e :

Theorem 6 . 4 . I f {A}?CX,B]C c P ] F , t h e n e i t h e r {A}PCX,BI i s a p a r t

P roof . The e s s e n t i a l p a r t of t h e theorem i s t h a t t h e f o l l o w i n g

canno t occur : P E P P P1 E P4{A)P and F : P [x,B]C (YA}B[X,BIC 2 3 ' 2 3

o c c u r s p a r t i a l l y i n P p a r t i a l l y i n F"), Th i s can b e proved by 1 '

t h e a i d of t h e v a l u a t i o n p r o p e r t i e s f o r B-chains.

We c o n t i n u e w i t h t h e d e f i n i t i o n s of s i n g l e - s t e p B - and B2- 1

r e d u c t i o n :

D e f i n i t i o n 6.5. S ing l e - s t ep BI - reduc t ion i s t h e r e l a t i o n gene ra t ed

by

( I ) I f Q{A}?[x,B]c E A and x c C , t h en

Q{A}?[x,B]c 2' Q{A)?CX , B ] (x:=A)c, and 1

(2 ) t h e monotony r u l e s ( s e e Def. 5.5 ( 2 ) , ( 3 ) and ( 4 ) , r e a d i n g r' 6 1

i n s t e a d of 2 ' ) . B

'0

D e f i n i t i o n 6.6. B - r educ t i on i s t h e r e f l e x i v e and t r a n s i t i v e c lo - 1 s u r e of s i n g l e - s t e p B - r educ t i on . 1

0

~ e f i n i t i o n 6.7. S ing l e - s t ep B2-reduction i s t h e r e l a t i o n gene ra t ed

by

(I ) I f Q { A ) ~ [ ~ , B ] c E A , t h en Q { A } ~ C X , B ] C 2' Q ~ C , and

(2 ) t h e monotony r u l e s . @2 0

D e f i n i t i o n 6.8. B - r e d u c t i o n i s t h e r e f l e x i v e and t r a n s i t i v e c lo - 2

s u r e of s i n g l e - s t e p B - r educ t i on . 2

Note t h a t i n Def. 6 .7 x may n o t occur i n C .

A s i n t h e c a s e of B-reduction ( s e e t h e p r ev ious s e c t i o n ) , we

speak of e lementa ry B 1 - o r B2-reduction, n-step B 1 - o r a2 - r educ t i on ,

and a s i n g l e - s t e p B 1 - o r B2-reduction g e n e r a t e d by

Q{AI?CX,BIC o r Q { A } ~ c $ , B I C r e s p e c t i v e l y . The " l eng th of proof of

K 2' L I' or ". . . K 2' L " i s d e f i n e d a s f o r s i n g l e - s t e p B-reduction. 1 B2 F i n a l l y we d e f i n e B - equ iva lence (K - L) and B2-equivalence

1 B. 1

(K - L) ana logous ly t o B-equivalence ( s e e t h e p r ev ious s e c t i o n . ) . B,

Theorem 6.9. I f K E A and K 2' L o r K 2 ' L , t hen L E A . 1 B2

Proof . I nduc t i on on t h e l e n g t h of proof of K 2' L o r K 2' L r e s -

p e c t i v e l y . Cf. t h e proof of Th. 5.8. I @2

0

D e f i n i t i o n 6.10. We s h a l l w r i t e P 2 P' e t c . i f PT 2 P ' T e t c . 0 1 1

Theorem 6.11. I f QA E A and QA 2' K ( o r QA 2' K), t h e n K E Q'A'

w i t h I lQl l = 114' 1 1 , and e i t h e r 1 B2

(1 ) Q 2' Q' ( o r Q 2' Q' r e s p e c t i v e l y ) and A z A ' , o r B . B,

( 2 ) Q Q ' and QA 2' QA' ( o r QA 2' QA' r e s p e c t i v e l y ) . B , B,

P roof . Cf. t h e p r o o f s of Th. 5.11 and Th. 5.12.

Theorem 6.1 2. The monotony r u l e s h o l d f o r B 1 - r e d u c t i o n and f o r B 2 - r e d u c t i o n .

P roof . Cf. t h e proof of Th. 5.15.

Theorem 6.13. If QC, QPC and QPDeA, and QC r QD ( o r QC 2 QD) , t h e n B2

QPC 2 QPD ( o r QPC 2 QPD r e s p e c t i v e l y ) . 1 B2

Proof . Cf. t h e proof of Th. 5.17.

The f o l l o w i n g two theorems d e a l w i t h t h e r e l a t i o n between B-

r e d u c t i o n on t h e one hand and B - and Be-reduct ion on t h e o t h e r . 1

Theorem 6.14. I f K 2' L, t h e n K - L. I B

P roof . I n d u c t i o n on t h e l e n g t h of proof of K 2' L. 1

Note t h a t , i f ? 3 +, t h e n E P'{B}[x,c]P '~, and PI - P'P" i s a g a i n

a 6-chain by Th. 6.3. I t f o l l o w s t h a t t h e r e i s a f3-reduction f o r K

and f o r L. C o n t i n u a t i o n of t h i s B-reduct ion p r o c e s s g i v e s :

K z Q{A}[x,B']CT z B 6

Q(x:=A)C1, and L 2 Q{A}[;,BI(X:=A)C' 2 B B

Q(x:=A)C1. I n t h e l a t t e r r e d u c t i o n one shou ld n o t e t h a t t h e sub-

s t i t u t i o n s (y:=D) i n t r o d u c e d i n t h e r e d u c t i o n L> Q{~}[g,B] (x :=A)C1 -8

do n o t i n f l u e n c e A, s i n c e y # A, Together w i t h t h e s t a t e m e n t t h a t

i n t h i s c a s e (y:=D) (x:=A)E Z (x:=A) (y:=D)E ( b u t f o r renaming) t h i s

r e s u l t s i n u s o b t a i n i n g t h e same C ' ( b u t f o r renaming) i n r educ ing

L a s we o b t a i n e d i n r e d u c i n g K. I t f o l l o w s t h a t K 2 L. B

11. K r ' L i s Q{A}c 2' Q{A}D a s a d i r e c t consequence of QC 2' QD. @ 1 1 1

Then, by i n d u c t i o n : QC QD, Hence, by Th. 5.24: Q{A}C -@ Q{A}D.

111, I V . The two o t h e r monotony ca se s a r e proved s i m i l a r l y t o 11. 0

Theorem 6.15, I f K E A and K 2 ' L , t h en K 2' M 2' L o r K r ' L. B 1 B2 B2

Proof . L e t K 2' L b e gene ra t ed by Q{A)[x,B]C 2 ' Q(x:=A)C. B B

I f x c C , t hen Q{A}[x,B]C >i Q{A}[~ ,B] (x :=A)c 2' Q(x:=A)c. t 6, 1 L

I f x # C , then K 2' L i s a B ~ - r e d u c t i o n . The remainder f o l l ows 8

from t h e f a c t t h a t t h e monotony r u l e s f o r B-, e l - and B2-reduction

a r e s i m i l a r .

We s h a l l now prove a number of theorems l e a d i n g t o a theorem

on t h e p o s s i b i l i t y of postponement of B - r educ t i ons a f t e r O - 2 I r e d u c t i o n s (Th. 6 .19) .

n Theorem 6.16. I f K E A , K 2' L 2' El, t h e n K 2' L ' 2 M f o r a

c e r t a i n n 2 1 . @ 2 @1 @ 1 D2

Proof . L e t L >b M be genera ted by Q{A}?[X,BIC t' Q{AI?CX,B](X:=A)C, 1 A 1

and K t' L by Q'{D}:~[;,E]F 2' Q ' ? ~ F . Then {A}P[x,BIC c L and A B2 @2 P I F c L. Now w e can d i s t i n g u i s h t h r e e c a s e s : (1 ) I A I B C X , B I C and

P F occur i n L i n d i s j o i n t p l a c e s , (2 ) {A}?[X,BIC c P l ~ , ( 3 ) 1

I, F c {A}:CX,BIC and P 1 ~ f {A}?[x,BIC.

( 1 ) I n t h i s c a s e i t i s c l e a r t h a t t h e theorem ho ld s f o r n = I .

( 2 ) We may d i s t i n g u i s h ( s e e a l s o Th, 6 . 4 ) :

( a ) P I I P2{G}P3 o r F1 I P Cz,G]P3 and {A}?CX,BIC c G. 2 The theorem h o l d s f o r n = 1 .

( b ) P I P ~ I A I B C X , B I P ~ . Idem.

( c ) { A } ~ [ X , B I C c F. Idem.

These c a s e s ( a ) - ( c ) a r e exhaus t i ve i f {AI?CX,BIC c F I F .

( 3 ) ( a ) Le t ? I ~ c A. Now t h e theorem h o l d s f o r n i s t h e number of

occu r r ences of x i n C p l u s one.

(b) Le t P : P P P The theorem ho ld s f o r n = 1 . 2 1 3' ( c ) L e t $ S P {GIP o r ? : P [z,G]P and p F c G. Idem.

2 3 2 3 1 ( d ) Let B 1 ~ c [x,B]C. Idem.

These c a s e s ( a ) - (d ) a r e exhaus t i ve i f $ F c { A } ~ [ x , B ] c and

B,F $ {A}?[X,BJC. ( I n t h i s proof we s e v e r a l t imes u s e t h e lemma: 1

"If ? P ? P then P { G } B ~ C Z , H I P i s a l s o a (3-chain", which i s a 1 2 3 ' 1 3

consequence of Th. 6 .3 . )

Theorem 6.17. I f K E A , K 2 L 2 ' M , t h en K 2' L ' 2 M. @2 1 (3 1 B2

Proof . I n d u c t i o n on t h e number of s t e p s of K 2 L , u s ing Th. 6 .16. @2 0

P Theorem 6.18. I f K E A , K 2 L 2 M , t hen K 2' L ' 2 M. @2 I I B2

Proof . I n d u c t i o n on p , u s i n g Th. 6 .17 .

Theorem 6.19. I f K r ' K > ' ... 2' K by s i n g l e - s t e p (3 - and (3 - 1 2 - n 1 2

r e d u c t i o n s , t h e t o t a l number of (3,-reductions be ing p , t h e r e i s a

Proof . Combine t h e s u c c e s s i v e s i n g l e - s t e p B - r educ t i ons i n I

K 2' K 2 ' ... 2' Kn, 1 2

and do t h e same w i t h t h e s u c c e s s i v e s i n g l e -

s t e p B2-reductions: we o b t a i n K 5 L 2 M > L > M 2 ... 1 I B 2 1 B 1 2 -B2 2 BI

2 L 2 M r K . I nduc t i on on R y i e l d s t h e p roo f . 13, 2 13, JL B * n

0

We s h a l l now prove what we c a l l t h e Church-Rosser property

(CR) f o r (31-reduction, which we fo rmu la t e a s fo l lows : I f K 2 L 1 and K 2 M , t h e r e i s an N such t h a t L 2 N and M 2 N. We can

1 B1 1 exp re s s t h i s Church-Rosser p r o p e r t y i n a diagram, a s f o l l ows :

C R :

From CR for B1-reduction it easily follows that B1-equivalence

is transitive, hence indeed anequivalence relation (reflexivity and

symmetry of - are trivial). Hence we can also state that - is 8, @ 1

1

the equivalence relation generated which is an alternative

formulation for the Church-Rosser theorem for B 1 -reduction,

In proving CR for B1-reduction we shall use a technique intro-

duced by W.W. Tait and P. arti in-L:£, given in Barendregt [I, appendix 111. We shall discuss the power of this technique in brief.

In order to prove CR for a reduction it is natural to begin I I with single-step reductions K 2' L" and "K 2' MI1. In a usual single-

step (e.g. B-) reduction one can then find an N such that "L t N"

and "M 2 N", but unfortunately only one of these last two reduc-

tions is necessarily a single-step reduction, and one cannot say

in advance which of the two.

If one now begins with multiple-step reductions "K 2 L" and

"K 2 M" and one tries, by the aid of the above, to find an N such

that "L r Nfl and "M 2 N", then the termination of this attempt is

not guaranteed. The following example, drawn in a diagram, suggests

what might happen:

Each rectangle in this diagram represents reductions; three

sides of the rectangle are single-step reductions, one side is two-

step. The diagram can, however, be continued indefinitely in the

place where we draw the dotted lines.

It Now Tait and arti in-L;£ defined a new single1'-step reduction

(which we shall call single-step nested reduction to avoid confu-

sion). The latter reduction has the property that with each pair of 11 * *

single-step nested reductions K r L" and "K 2 M" there can be * *

found an N such that "L 2 N'' and "M 2 N", both last-mentioned

reductions being single-step nested reductions as well. Moreover,

each multiple-step reduction can be decomposed into single-step

nested reductions and each single-step nested reduction is a compo-

sition of (ordinary) single-step reductions. I I If one now begins with multiple-step reductions K 2 L" and

"K 2 MI1, one can decompose these reductions into single-step nested

reductions and apply the above. Then one obtains, for example, a

situation as is expressed in the following diagram:

In each of these restangles the four sides represent single-

step nested reductions. 11 * *

Moreover, the nested reductions L 2 L' 2 N" and It * *

M 2 M' 2 N" can be decomposed into (ordinary) single-step re-

ductions, which combine into "L 2 N" and "14 2 N", Thus we obtain CR.

In the following we shall define a single-step nested B - 1

reduction, which we shall call single-step y-reduction. Our y-

reduction is a little more complex than the nested reduction of

Tait and arti in-LE~, but it yields essentially the same results. The "nested" character of y-reduction can be explained as

follows. Let a B,-reduction be generated by Q{AI?CX,BIC, let

QA 2 ' QA' and Q ~ C X , B I C 2 ' Q?'[x,B1 IC'. Then a l s o Q{A}?CX,BIC 2' Y A Y Y

Q ( A ' ) P ' [ X , B ' ~ ( X : = A ' ) C ' ( i f t h e sugges ted [3 - r e d u c t i o n i s preceded 1

I I by s i n g l e - s t e p n e s t e d r e d u c t i o n s i n s i d e 1 ' A, P, B and C , t h e com-

p o s i t e r e d u c t i o n i s a s i n g l e - s t e p n e s t e d r e d u c t i o n ) . The r e d u c t i o n s

t a k e p l a c e i n a "nes ted" o r d e r .

With t h e a i d of. y - r e d u c t i o n we s h a l l prove CR f o r 8 - r e d u c t i o n . 1

We s h a l l s u b s e q u e n t l y prove CR f o r @-reduc t ion , u s i n g CR f o r @ - 1

r e d u c t i o n (we cou ld a l s o prove CR f o r @ - r e d u c t i o n s d i r e c t l y , a l o n g

t h e same l i n e s p o i n t e d o u t by T a i t and I f a r t i n - L E ~ ) .

D e f i n i t i o n 6.20. S i n g l e - s t e p y - r e d u c t i c z , denoted by 2' i s t h e Y '

r e f l e x i v e r e l a t i o n g e n e r a t e d by

( I ) I f Q{AIBCX,BIC E A , x c C , QA 2' QA' and ~ P C X , B I C 2 ' Y - Y

Q;' C X , B ' IC' , t h e n

Q { A } ~ [ X , BIC 2' Q{A' IF' C X , B ' I (x:=A')c'. Y

( 2 ) I f Q{A}C and Q{A1)C' E A , QA 2; QA' and QC 2' Q C ' , t h e n Y

Q{A)C r' Q{A' )C' . Y

( 3 ) I f Q[X,A]C and Q[x,AIIC' E A , QA 2' QA' and Q[x,A]C 2' Q[x,AIC1, Y Y

t h e n QCX,AIC 2 ' Q C x , A I I C ' . Y

( 4 ) I f A E A , A t ' B and B t B ' , t h e n A 2 ' B ' . a Y

0 Y

We c a l l ( 2 ) and ( 3 ) t h e monotony r u l e s f o r s i n g l e - s t e p y-

r e d u c t i o n . We a l s o d e f i n e y-equivalence (K - L ) a n a l o g o u s l y t o B- Y

e q u i v a l e n c e .

D e f i n i t i o n 6.21. y - reduc t ion , denoted by 2 i s t h e t r a n s i t i v e Y '

c l o s u r e of s i n g l e - s t e p y - reduc t ion . 0

We c o n t i n u e w i t h some theorems concern ing y - r e d u c t i o n ( i t

w i l l b e c l e a r t h a t Q 2 ' Q' i f and o n l y i f Qr 2' Q ' r ) . Y Y

Theorem 6.22. I f QA E A and QA 2' K, t h e n K E Q'A' , where Y

I l Q l l = IlQ' 11, Q 2' Q' and QA 2' QA'. Y Y

P roof . I n d u c t i o n on the l e n g t h of proof of QA 2' K. Y

( 1 ) I f QA 1' K by r e f l e x i v i t y , t h e theorem i s t r i v i a l . Y

( 2 ) I f QA 2 ' K i s Q {B}B[X,C]D 2' Q ~ { B ' } ~ ' [ X , C ' I ( X : = B ' ) D ' a s a Y 0 Y

d i r e c t consequence of Q B 2 ' QoB' and Q B [ X , C I D 2' Q B Y X , C ~ I D ' , 0 Y 0 Y 0

then l l Q l l r l l QO l l , so Qo E QQ", and t h e theorem f 01 lows.

( 3 ) I f QA 2 ' K i s Q {B)C 2 ' Q {B1)C' a s a d i r e c t consequence of Y 0 Y 0

Q B s' Q B ' and Q C 2' QOC' , t h e n a g a i n IIQII 5 I I Qoll and t h e 0 Y O 0 Y

theorem fo l l ows .

( 4 ) Le t QA 2' K be Q [x,B]C 2' Q O [ x , ~ ' ] c ' a s a d i r e c t consequence Y 0 Y

of Q B 2' Q B' and Q [x,BIC r' Q O [ x , ~ l ~ ' . 0 Y O 0 Y

( i) I f 11 Q l l 5 l l Q I I o r Q - Q [x,B] t hen t h e theoreni f o l l ows . 0 0 ( i i ) I f C : Cyl , B 1 I . . . Cyn,BnlCO and Q E QOC~,BICyl , B l I . . .

Cy , B 1, then QA : QOCx,BIC E Q [ x , B l [ y l , B I I 0

. . . C Y ~ , B ~ I A 2' n n Y

B' ]A' (by i n d u c t i o n ) Q ~ C X , B I C ' : Q ~ C X , B I C Z ~ , B ; I . . . Czn,

w i t h QA 2' QA' and Q 2' Q O [ ~ , B I C z l , B ; l ... [zn , B' 1. Y

It fo l l ows t h a t Q 2 ' Q [ x , B ' ] [ ~ , , B ; ] ... [ z , B ' ] . Y 0 n n

- Also C ' : [ z l , B ; ] . . . [ Z ,B;]A', so Q ~ [ X , B ' ] C ' = n

QO[x,B'][zl,B;] ... [ Z B ' I A ' . Consequently t h e theorem n' n

h o l d s i f we t a k e Q ' : QOCx,~ ' ]Cz l , B ; ] . .; [ z B ' ] . n ' n

( 5 ) Le t QA 2' K be a d i r e c t consequence of QA 2 ' K' and K' 2 K. Y Y a

Then by i n d u c t i o n t h e theorem h o l d s f o r QA 2' K ' , a n d * t r i - v i a l l y Y

a l s o f o r QA 2' K. Y

0

Theorem 6.23. The monotony r u l e s ho ld f o r y- reduct ion.

P roof . Cf. t h e proof of Th. 5.15.

Theorem 6.24. I f QC, PC and PD E A , and QC 2' Y QD, ihen PC 2' Y PD.

P roof . Analogous t o t h e proof of Th. 5.17. 0

The fo l l owing two theorems d e a l w i t h t h e r e l a t i o n between I3 - 1

and y-reduct ion.

Theorem 6.25. I f K E A and K 2' L, t h e n K 2' L. I3 1 Y

Proof . I n d u c t i o n on t h e l e n g t h of proof of K 2' L. The r u l e of I3 1

e l ementa ry B - r e d u c t i o n i s covered by Def. 6 a . l l ( l ) ( take QA 2 QA' 1 Y

t o b e QA 2 QA by r e f l e x i v i t y , e t c . ) , t h e monotony r u l e s f o r B - Y 1

r e d u c t i o n a r e covered by t h e monotony r u l e s f o r y - reduc t ion ( aga in

u s ing t h e r e f l e x i v i t y of y - reduc t ion i n a p p r o p r i a t e p l a c e s ) . 0

Theorem 6.26. I f K E A and K 2' L, t h e n K r L b u t f o r a - reduc t ion . Y @ I

Proof . I n d u c t i o n on t h e l e n g t h of proof of K 2' ,, L. For example: I

l e t K 2' L be Q { A } ~ c x , B ] c ~ : Q{A1}? ' [x ,~ ' ] (x :=A')C ' a s a d i r e c t I I

consequence of x c C , QA t ' QA' and QP[x,B]C r ' QPf[x,B']C'. Y Y

By i n d u c t i o n t h e l a s t two r e d u c t i o n s can a l s o b e ob t a ined by B 1 -

r e d u c t i o n s and a - reduc t ions , and Q{A)?[x, BIC t Q{A)?' [ x , ~ ' ]C' B l

Q{A'}?'[X,B']C' r Q{A'}?'[x,B'](x:=A')c' ( b u t f o r a - r educ t i on ) . 1

I n t h e l a s t 3 - r e d u c t i o n we u s e t h e lemma: "If x c C , i f x occu r s 1

I I a s a b ind ing v a r i a b l e i n QI and i f Q C 2 Q C ' , t h en x c C ' . 1 B ] 1

Theorem 6.27. I f K E A and K 2 L, t h e n L E A . Y

P roof . Follows from Th. 6.26 and Th. 6.9.

We i n d u c t i v e l y d e f i n e s i m i l a r i t y of two lambda-phrase cha in s

( n o t n e c e s s a r i l y B-chains):

(I ) I f PI P - $, then P1 and P2 a r e s i m i l a r . 2 =

( 2 ) I f PI and P2 a r e s i m i l a r , then {AlpI and {B}P2 a r e s i m i l a r ,

and [x,A]PI and [x,B]P a r e s i m i l a r . 2

The f o l l o w i n g theorems a r e a p r e p a r a t i o n f o r Th. 6.37, which

e x p r e s s e s CR f o r y- reduct ion. I n o r d e r t o prove Cor. 6.31 and Th.

6.34 i t i s conven ien t t o extend t h e n o t i o n of B-chain, a s i n Def.

6.29: a number of @-chains , connected by a b s t r a c t o r s , w i l l be

c a l l e d a @-chain complex. A 6-chain complex may be empty.

D e f i n i t i o n 6.29. Le t P l , P 2 , ..., Pi be ( p o s s i b l y empty) B-chains.

Then a lambda c h a i n P I [x1 ,A1 lP2 [~2 ,A21 . . .Pi-* [xi-] ,Ai-] ]Pi i s

c a l l e d a 6-chain complex. 0 -

We deno t e a @-chain complex P by P.

The f o l l o w i n g s t a t e m e n t can be proved by t h e a i d of v a l u a t i o n s :

I f P i s a B-chain complex and P 2 P1P29 t h e n P 2 i s a B-chain com-

p l e x .

- Theorem 6.30. I f QFA E A and QFA 2 ' K, t h e n K Z Q'P'A' , where - Y Q = Q Q r ' Q ' , QFA 8' QP'A' and P and P' a r e s i m i l a r .

Y Y

Proof . I f P Q1 ( i n c l u d i n g P Z $) , t h e n Th. 6.22 g i v e s t h e p roof . - L e t P $ Q , . We proceed w i t h i n d u c t i o n on t h e l e n g t h of proof of

Q?A 2' K. Note t h a t t h e r e must b e a t l e a s t one a p p l i c a t o r i n t h e Y

lambda c h a i n P, on accoun t of o u r assumpt ion P 2 Q 1 '

(I ) Assume t h a t QFA 8' K by r e f l e x i v i t y . The proof i s now t r i v i a l . Y

(2 ) Assume t h a t QFA 8' K i s QQl { c } ~ { ~ , D I E 8 QQl { c ' [Y ,D ' ] ( y : = ~ ' ) ~ ' Y

a s a d i r e c t consequence of QQ C 2' QQ C ' and Q Q ~ P ~ C ~ , D I E 8; I Y 1 - = A

QQ F f [ y , ~ ' ] E ' . Now i t must ho ld t h a t E E P 2 A , w h i l e QlPl[y,D]P2 i s 1 1 - -

a f3-chain complex. By i n d u c t i o n : E' I P;A' and P 2 and P; a r e

s i m i l a r . The remainder i s easy.

(3 ) Assume t h a t Q ~ A 8' K i s QQI{C}D 2' QQ1 {C'}D1 a s a d i r e c t con- Y Y

sequence of QQ C 2 ' QQ C ' and Q Q I D 8' Q Q I D 1 . Now F E Q {c}; and - 1- Y 1 Y 1 2 -

D . P A. Here P- i s a 6-chain complex, hence by i n d u c t i o n D ' E % A ' , 2 2

i n which P, and P: a r e s i m i l a r . The remainder f o l l o w s e a s i l y . L L-

(4a ) Assume t h a t QPA 2' K i s QQl[y,CID 2; QQIC.y,C'I~ ' as a d i r e c t Y

consequence of QQ C 2' Q Q I C f and QQ1[y,CID 2' Q Q l [ y , ~ ] ~ ' . Then - - - 1 Y = Y P : QI[y,C]P2 and D z P 2 B. The comple t ion of t h e proof i s s i m i l a r

t o t h a t i n t h e l a s t p a r t of t h e p r e v i o u s c a s e .

(4b) Assume t h a t Q?A 2' K i s Q ~ [ ~ , c I Q ~ ~ A 2; QO[y,C']D' a s a d i r e c t Y -

consequence of Q C 2' QOCt and Q ~ [ ~ , c I Q ~ F A 2; QOCy,CID1. Then, by 0 -Y

i n d u c t i o n , D ' n Q ; P ~ A ' w i t h l lQ1 11 = 11Q; 1 1 , w h i l e F and F' a r e s i m -

i l a r . The remainder f o l l o w s . - (5) Assume t h a t QFA t ' K i s a d i r e c t consequence of QPA t' K' and

Y Y K' 2 K. The proof i s a g a i n by i n d u c t i o n . 0

C1

C o r o l l a r y 6.31. I f QBA E A and QBA 2' K, t h e n K n Q I P ? A ' , where A Y

11Q11 = IIQ'II , Q 2' Q ' , Q?A 2' QP'A', w h i l e P and P' a r e s i m i l a r . 0 Y Y

The fo l l owing f o u r theorems a r e lemmas f o r Th. 6.36. Th. 6.34

might have been c a l l e d " t h e s u b s t i t u t i o n lemma f o r y-reduction".

Theorem 6.32.

( I ) L e t Q B C X , B I C E A and QBLX ,B]C 2' v K. Then K 5 Q ' f ' C X , B ' I C ' such

t h a t Q r ' Q' , Q ~ [ X , B ] C 2' QP? [ x , ~ ' I C ' , l l Q I I = I I Q' I I , w h i l e P and P ' Y Y

a r e s i m i l a r .

( 2 ) Le t Q{A}?[x,B]c E A and Q{A}?CX,B]C 2' K. Then e i t h e r ( i ) Y

K - Q ' { A q } P ? [x ,B ' ]c ' , o r ( i i ) K Q'{A'}P~[x,B'](x:=A')c', where

i n b o t h c a s e s Q 2' Q t , QA 2' QA' , Q ~ C X , B I C 2' QP'CX,B'IC' and Y Y Y

l l Q l l = 11 Q ' 11, w h i l e P and P' a r e s i m i l a r .

(3 ) Le t Q{A)B E A and Q{A}B 2' K. Then e i t h e r ( i ) K r Q'{A'}B ' Y

where Q 2' Q ' , QA 2' Q A ' , QB 2 ' QB' and IlQll = IlQ'11, o r ( i i ) Y Y Y

Q{A}B r Q { A I ? C ~ , C I D , x c D , K Q'{A'IP?[X,C'](X:=A')D', Q 8; Q ' ,

IlQll = IIQ' 11, QA 2' QA' and Q?CX,CID 2' QP?[X,C']D'. Y Y

Theorem 6.33. I f Q?CX,BIC E A , Q ~ [ X , B ] C ~ ' Q ' ~ ' [ x , B ' I c ' and x c C , Y

t h en x c c ' .

P roof . I n a subexp re s s ion we can on ly e l i m i n a t e f r e e v a r i a b l e s by

s u b s t i t u t i o n , and s u b s t i t u t i o n can o n l y o r i g i n a t e from a 6 -reduc- 1

t i o n . Note t h a t a B1-reduct ion y i e l d i n g a s u b s t i t u t i o n ( x : = A ) can-

n o t occur i n t h e above.

Theorem 6.34. L e t QA and QFB E A , l e t no b ind ing v a r i a b l e of h~ - -

occur i n QA and l e t x n o t occu r i n P. Le t Q ~ B 2' QPB' (where Y

P I = P I ) and QA 2' QA'. Then Q?(x:=A)B 2' Q $ ( x : = ~ ' ) ~ ' . Y Y

Proof . F i r s t c o n s i d e r t h e c a s e t h a t P r Q so P' Q ; . 1 '

We prove t h e theorem by i n d u c t i o n on I B I . I f B : y f x o r B 5 r

t h e n t h e theorem i s t r i v i a l . I f B - x , n o t e t h a t QQ A 2' QQ A ' , s o I Y I

also QQIA 2' Q Q i A ' by i n d u c t i o n on t h e l e n g t h of proof of Y

Q Q l x 2; Q Q t x and monotony r u l e ( 3 ) f o r s i n g l e - s t e p y-reduct ion. 1

(1) Let B E C ~ , E I F . Then B ' : CY,E'IF' by Th. 6 .32(1) ,

Q Q I E 2; Q Q ] E ' and QQI [y,ElF 2' QQ1[y,E]F'. So a l s o (by induc t ion ) Y

QQ1 (x:=A)E 2' QQ1 (x:=A')E1 and QQI [y,El(x:=A)F 2; QQl [y,E](x:=A1)F'. Y

It e a s i l y fo l lows from t h e l a t t e r r educ t ion t h a t

QQI Cy, (x:=A)E](x:=A)F 2; QQl [y , (x:=A)El(x :=A1)F'. It fo l lows t h a t

QQl (x:=A)B 2; QQ1 (x:=A')B1, hence a l s o QQ (x:=A)B 2' QQ;(x :=A1)~ ' . 1 Y

(2) Le t B : {E)F. Now by Th. 6.32(3) e i t h e r B ' Z {E'}F' w i t h

Q Q I E 2' Q Q ~ E ' and QQIF 2' QQ;F' , o r B ' { E } ? ~ [y,G]H, and Y Y

QQ 1 B QQ {EIB] [ ~ , G ] H 2' QQ;{E' } P ; ' [ ~ , G ' I ( ~ : = E ' ) H ' Q Q ; B 1 . I Y

I n t he f i r s t c a s e t he proof i s s i m i l a r t o t h e proof i n c a s e (1) .

I n t h e second c a s e we can fo l low analogous l i n e s , u s ing t h e f a c t

t h a t (y:=(x:=A1)E') (x:=A1)H' (x:=A1) (y:=E1)H' b u t f o r re-

naming.

Now cons ider t h e c a s e t h a t P f Q ,whence a t l e a s t one app l i - I c a t o r must occur i n c h a i n P. We proceed wi th i nduc t ion on t h e

l e n g t h of proof of Q ~ B 2' Q$ B' . Y-

( I ) Assume t h a t QFB 2' Q P ~ B ' by r e f l e x i v i t y . This ca se can be V I -

proved s i m i l a r l y t o t h e c a s e t h a t P - Q 1 '

(2) Assume t h a t QFB 2' QP B' i s QQl {c}BI [y,D]E 2' Y Y

QQ {c ' } ~ c ~ , D ' I ( ~ : = c ' ) E ' a s a d i r e c t consequence of QQ C 2 ' QQ C ' 1 1 I Y 1

and QQ 1 [y,D]E 2' QQI [y,D1lE'. Now i t must hold t h a t E E 7 B 1 1 - Y A - 2

and E ' ?B ' , where B ' E (y:=C1)B". Since Q I P I ~ y , ~ ] ~ - 2 i s a l s o a

B-chain complex, i t fo l lows by induc t ion : A - A -

Q Q , P ] C~ ,DIP-~(X:=A)B 2' QQ P ' [ ~ , D ' I ~ ( x : = A ' )B", SO QP(x:=A)B Y 1 1

QQ] {CIS, C ~ , D I < ( X : = A ) B 2; QQ] { c l I?; C ~ , D ' I ( ~ : = C ' ) ( P = ; ( ~ : = A ' ) B I ~ ) = QQ, { c ' } B ; c ~ , D ' I ( ( ~ : = c ' ) ~ ) (x:=Af) ( y : = ~ ' ) ~ " E QP=' (x:=AV)B' (he re we

changed (y:=C')(x:=A1)B" i n t o (x:=A1)(y:=C')B", which i s allowed

by Th. 6.33 and by t h e cond i t i ons imposed upon the v a r i a b l e s ) .

(3 ) Assume t h a t QFB 2' Q ~ B ' i s QQ {C}9 2' QQ {C')D1 a s a d i r e c t Y 1 Y 1

consequence of QQ C 2' QQ C ' and QQ D 2' QQ D ' . Then QI{c}< 1 Y - 1 Y 1 and ? Q1 {C'}?; D B and Dl 5 P;B'. By induc t ion :

2 - QQ 1 2 f (x:=A)B 2' Q Q ~ % ( X : = A ' ) B ' , so a l s o QF(x:=A)B 2' QP='(x:=A')B'.

Y - Y ( 4 ) Assume t h a t QBB 2' QPB' i s QQI[y,CID 2; QQl[y,C'lD1 a s a

Y d i r e c t consequence of Q Q C 2' QQ C ' and QQ C ~ , C I D 2' QQ [y,C]D1.

1 Y 1 1 Y I

- - - - Then p E Q c ~ , c ] F and P" 5 QI [ Y , ~ ' ~ F ; ; D C P-~B and D ' Z P ~ B ' .

1 2 The remainder of t h e proof i s analogous t o t h a t i n c a s e ( 3 ) .

(5) Assume t h a t Q ~ B 2 ' QP'B' i s a d i r e c t consequence of Y -

11 71 11 11 ? I I T 11 71 Q ~ B 2' Q P B and Q P B 5 QP='B' . Then QP(x:=A)B 5' Q f ( ~ : = A ' ) B "

Y C1 Y by i n d u c t i o n ; t h e remainder i s easy . 0

Theorem 6.35. I f QA and Q'A ' E A , ' Q 2 ' Q' and QA 2 ' Q A ' , t h en Y Y

QA 2 ' Q ' A ' . Y

Proof . I n d u c t i o n on 11 411, u s i n g monotony r u l e (3) of s i n g l e - s t e p

y - reduc t ion . 0

Theorem 6.36. I f K E A , K 2' L and K 2' M, t h e r e i s an N such t h a t Y Y

L 2' N and M 2' N. Y Y

P roof . I n d u c t i o n on t h e l e n g t h of proof of K 2' L . Y

We s h a l l u s e Th, 6 . 2 4 and Th, 6.35 s e v e r a l t imes w i t h o u t s ay ing so .

( 1 ) L e t K 2' L by r e f l e x i v i t y . Take N = FI. Y

( 2 ) Le t K 2' L be Q{A}?[X,BIC 2 ' Q { A ' } ~ ' C X , B ' I ( X : = A ' ) C ' a s a d i r e c t Y Y

consequence of QA 2' QA' and Q$[X,BIC 2' Q ? ' C X , B ' I C ' . Now by Th. Y Y

6 . 3 2 ( 2 ) M can have t h e form ( i ) M 5 Q ~ ~ { A ~ ~ } B " C X , B ~ ~ I C ~ ~ o r ( i i )

M - Q ~ ~ { A ~ ~ ) P I " [ x , B"] (x :=A") C ' I , where i n bo th c a s e s Q 2 ' Q", Y

QA 2' QA", QF[X,B]C 2 ' Q P [ X , B ~ ~ I C ~ ~ , IlQII = IIQ1'll and IIPl l = I I P " I I . Y Y

By i n d u c t i o n and by Th. 6 . 2 2 t h e r e i s an A"' such t h a t QA' 5' QA"' Y

and QA" 2' QA"' . Again by i n d u c t i o n and by Th, 6 . 3 2 ( 1 ) t h e r e a r e A Y P" ' , B"' and C"' such t h a t Q ? ' [ x , B 1 ] C ' 2' Q ~ " ' [ x , B " ' I C " ' and

Y Q?~[x,B"]c" 2' QF"' [ x , ~ " ' 1 ~ " ' . By Th. 6 . 3 4 :

Y Q?'[x,B'](x:=A')c ' 2 QF"' [x,B"'](x:=A"')c"' , hence by monotony

Y and Th. 6.35: L E Q{A' } B ' [ x , B ' ] ( x : = A ' ) c ' 2 ' Q"{A"' I B ~ ~ ~ c ~ , B ~ ~ ~ I

Y (X : = A ~ ~ ~ ) c ~ ~ ~ . C a l l t h e l a t t e r exp re s s ion N .

l l A 1 l l t A 1 l 1 It a l s o h o l d s t h a t Q P [x , B ~ I I C ~ ~ 2' Q P [x ,B"' IC" ' , and x c c"' by Y A

- Th. 6.33. So Q"{A"}~"[x , B"]c"~; N ' Q"{AV' }P 'I' [x , B"' ] (X :=A"' )clll by a n e lementa ry y-reduct ion. Th i s completes t h i s p a r t of t h e

proof i n c a s e ( i ) . I n c a s e ( i i ) we f i r s t e s t a b l i s h t h a t A

QP"[x,B"](x:=A")c" 2' QP"'[x,B"'](x:=A"')C"' by Th. 6.34, y i e l d i n g Y

by monotony t h a t M 2' N. Y

(3) Let K 2' L be Q{A)C 2 ' Q{A1)C' a s a d i r e c t consequence of Y Y

QA 2 ' QA' and QC 2 ' Qc ' . Now by Th. 6.32(3) M can have the form Y Y

( i ) M - Q"{A1'}C", with Q 2 Q", I lQll = 11Q"11, QA 2 ' QA" and QC 2' QC", Y Y

o r ( i i ) M z Q"{A"}?"[x,D"](x:=A")E", where K 5 Q{A)?[x,D]E, x c E ,

QA 2' Q A ' , Q 2 ' Q" and QBCX,DIE 2' Q B Y x , D ~ ~ I E ~ ~ . Y Y Y

I n case ( i ) we can f ind by induct ion A"' and C'" such t h a t QAI ? I ~ 1 1 ~ 1 1 1 , Q I I A I ~ 21 ~ l l ~ l t t , QC ' 2' Q"C1" and Q"C" 2' Q"c"' and we

Y Y Y Y can take N 2 Q"{A"')C1". I n case ( i i ) we a r e i n a p o s i t i o n s imi la r

t o case ( i ) of ( 2 ) , with L and M permuted.

( 4 ) Let K 2' L be Q[x,A]C 2 ' Q[x,A']C1 a s a d i r e c t consequence of Y Y

QA 2' QA' and Q[x,A]C 2' Q[x,A]C1. Then PI 2 Q"[x,A"IC" by Th. Y Y a

6.32(1 ) , where Q 2' Q", QA 2' QA" and Q[x,A]C 2' Q[x,A]CU.. Take Y Y Y

N r Q"[X,A"']C''', where A"' and C"' a r e obtained as i n ( 3 ) , case

( i )

(5) Let K 2' L a s a d i r e c t consequence of K 2' L ' and L ' 2 L. Y Y a

Then by induct ion we can f ind an N such t h a t L ' 2' N and M 2' N, Y Y

and a l s o L 2' N. Y

0

Theorem 6.37 (CR f o r y-reduct ion) . I f K E A , K 2 L and K 2 M, Y Y

then L - M. Y

Proof. This i s a consequence of t h e previous theorem. 0

Theorem 6.38 (CR f o r @l-reduuction). I f K E A , K n L and K 2 M,

then L - M but f o r a-reduction. l3 1 @ 1

l31

Proof. Decompose K 2 L and K 2 M, apply Th. @ 1 @ 1

6.25, Th. 6.37 and

Th. 6.26: we ob ta in an N ' such t h a t L 2 N 2 N ' and M 2 N" r N ' . l3 I a l3 1 a

CI

Theorem 6.39. I f K E A , K 2' L and K 2' M, then the re i s an n such

n @ 1 @2 t h a t M 2 ' N and L 2 N with n 2 1 .

1 @2

Proof. Let K 2' L be generated by Q{A}?EX,BIC 2' Q { A I B c ~ , B I ( ~ : = A ) c , 1 1

and K 2' M by Q'{D}?,[;,E]F 2' Q ' B ~ F . I f { D } ? ~ [ ~ , E I F c A, we need n 6, 6,

A 0

B,-reductions f o r t h e n A's i n Q{A}P[x,B](x:=A)c. I f n o t , we need L

o n l y one. The theorem e a s i l y f o l l ows .

Theorem 6.40. I f K E A , K r L and K 2 M, t h e r e i s an N such t h a t

M 2 N a n d L 2 N. 1 @2

1 @2

Proof . Apply Th. 6.39 r e p e a t e d l y . Th i s can be i l l u s t r a t e d by t h e

fo l l owing diagram

Here we assume t h a t K 2 L can be decomposed i n t o K 2' L' 2' L, 1 I f3 1

and K 2 M i n t o K 2 ' M' 2' M. I n t h e diagram a l l edges ( i n t h e 82 B2 B2

s e n s e u s u a l i n graph theory) p a r a l l e l t o t h e edge from K t o L '

r e p r e s e n t s i n g l e - s t e p f31-reductions, t h o s e i n t h e d i r e c t i o n of

K r1 M' r e p r e s e n t s i n g l e - s t e p B2-reductions. B,

Theorem 6.41. I f K E A , K 2' L and K 2' P i , t h e r e i s a n N such t h a t 82 @2

L 2 ' N o r L r N , and M r' N o r M = N. @2 B2

Proof . L e t K 2' L be gene ra t ed by Q{AI?C%,BIC 2' QFC, and K 2' M B2 @2 B2

by Q'{D}?,[;,E]F 2' Q ' ~ , F . I f {D}?] [;,E]F c A o r c B t h e n M 2' L ; B, 6, L L

i f {A}?[~ ,B]c c D o r c E , t h en L 2' M. I f {D}?~[;,E]F E {A}PCR,BIC 6,

t h en L : M. I n a l l o t h e r c a s e s t h e r e i s c l e a r l y an N such t h a t

L r 1 N and M r' N. B2 B2

Theorem 6.42 (CR for B2-reduction). If K E A, K r L and K 2 M,

then L - M. @2 B2

B2

Proof. Apply Th. 6.41 repeatedly.

Theorem 6.43 (CR for 6-reduction). If K E A, K 2 L and K 2 M, B B then L - M. B Proof. Decompose K 2 L and K 2 M, according to Th. 6.19, into B B K 2 L ' 2 L and K 2 M' r M respectively. The remainder of the

B1 B2 1 B2 proof is illustrated by the following diagram:

We find N' from Th. 6.38, N" and N"' from Th. 6.40 and finally N

from Th. 6.42.

§ 7. q-REDUCTION, REDUCTION AND LAMBDA-EQUIVALENCE

A third reduction in lambda-calculus (apart from a- and B- reduction) is called q-reduction and denoted by 2 . We shall in-

rl corporate it in our system.

We first define single-step n-reduction, denoted by 2' : rl

Definition 7.1. Single-step q-reduction is the relation generated

by:

(1) If Q[X,A]{X}B E A and x # B, then Q[X,A]{X}B 2' QB. rl

(2 ) Le t Q{A)C and Q{A)D E A . I f QC 1' Q D , t hen Q{A)C 2 'Q{A]D. n ri

( 3 ) Let Q[x,A]C and Q[x,B]C E A . I f QA 2' Q3, then Q[x,A]C 2' rl n

QCx,BlC.

( 4 ) Le t Q{A}C and Q{B}C E A . I f QA 2 ' QB, t hen Q{A}C 2' Q{B}C. rl rl

Rules ( 2 ) , (3 ) and (4 ) a r e c a l l e d t h e monotony r u l e s of s i n g l e -

s t e p q - reduc t ion ; they a r e s i m i l a r t o t hose of s i n g l e - s t e p B-reduc-

t i o n . Rule ( I ) i s c a l l e d t h e r u l e of e lementary n-reduct ion.

D e f i n i t i o n 7.2. q - reduc t ion i s t h e r e f l e x i v e and t r a n s i t i v e c l o s u r e

of s i n g l e - s t e p q - reduc t ion . 0

I f A and B a r e r e l a t e d by a ( s i n g l e - s t e p ) Q- r educ t i on , w e speak

of " t he ( s i n g l e - s t e p ) q - reduc t ion A 2 ' B". The n o t i o n s n - s tep q- r7

r e d u c t i o n and decomposi t ion of an q - reduc t ion a r e d e f i n e d analogous-

l y t o t h e cor responding n o t i o n s f o r 6- reduct ion. I f t h e f i r s t d e r i -

v a t i o n s t e p of a s i n g l e - s t e p q - reduc t ion has t h e form

Q C x , A l { x l ~ 2' QB, w e s ay t h a t QCx,Al{x}B g e n e r a t e s t h e s i n g l e - s t e p rl

n - reduct ion.

Theorem 7.3. L e t K E A . Then ~ C X , A ~ { X ) B g e n e r a t e s a s i n g l e - s t e p q-

r e d u c t i o n of t h e form K 2' L i f and on ly i f C X , A ~ { X ) B c K and 17

QCX,AI{X}B - K I CX,A~{X}B.

Proof . S i m i l a r t o t h e proof of Th. 5.7.

Theorem 7.4. I f K E A and K 2 ' L, then L E A . rl

Proof . I n d u c t i o n on t h e l e n g t h of proof of K 2' L. The proof i s 17

s i m i l a r t o t h e proof of Th. 5.8. 0

Theorem 7.5. I f QE E A , QE 2' Q'F and I l Q l l = IIQ' 1 1 , t h e n ( i ) Q - . Q ' , rl

( i i ) E F, o r ( i i i ) Q Q [x,A], E E {x}[y,B]E1, Q ' : QO[y,Bl, 0 x # [ y , ~ ] ~ ' a n d F - E ' . I n t h e second c a s e Q Q [x,KIQ 1 2 ' Q' Q l [x,L1Q2 and Q K 2'

1 1 7 Q l L*

Proof . I n d u c t i o n on t h e l e n g t h of proof of QE 2' Q'F. The proof i s rl

comparable t o t h e proof of Th. 5.11, excep t f o r t h e c a s e i n which

QE 2' Q'F is an elementary q-reduction. In this case we have to 7

note the possibility that QE and Q'F are as in (iii). 0

Theorem 7.6. If QE 6 A and QE 2' K, then K - Q'F' for certain Q' n and F ' with 11Q'Il 2 IlQll - 1.

Proof. Similar to the proof of Th. 5.12. - Theorem 7.7. If QE E A and QE 2' K 2 QG, then K Z QF.

r7 n Proof. If Q : QI[x,A] and QE 2' K is QI[x,A]{x}B 2' QIB, then the

rl n binding variable x of K has disappeared, and we cannot regain it

by 17-reduction. Hence by Th. 7.6: K z Q'F and llQ'11 = 11Q11, and the

case expressed in Th. 7.5 (iii) does not hold. In the derivation

steps leading to K 2 QG the final ones of the first IIQII abstractors 17

cannot disappear by an elementary q-reduction for the same reason as

above. Assume that Q $ Q'. Then by Th. 7.5 (ii): Q E QI[x,K]Q2,

Q' E QI[x,L]Q2 and Q K 2' Q I L . It is clear that / L I < I K I . Since the 1 7

length of an expression cannot increase by n-reduction it follows

that we cannot regain Q from Q'. Hence Q E Q'.

Theorem 7.8. The monotony rules hold for q-reduction.

Proof. Similar to the proof of Th. 5.15, using Th. 7.5. 0

Theorem 7.9. If QE, PE and PF E A, and QE 2 QF, t h m P E 2 PF. n n Proof. Analogous to the proof of Th. 5.17; use Th. 7.7. 0

The converse of this theorem holds too.

Given an q-reduction QK r M, it need not follow that rl

M r Q'N' with llQll = 114' 11, since the final abstractors of Q may have

been cancelled in q-reductions. For example: Let Q - QICx,Al and K - {xh, then QK 2' Q'T, where 11Q' 11 = IlQll - 1. This kind of q- n reductions plays an important rEle in the following. We shall call

them 7:-reductions. We shall prove a number of theorems concerning

q!-reductions. In Th. 7.14 we shall show that we can postpone q!-

reductions until after other n-reductions. Cor. 7.17 will result

from our discussions of n!-reductions.

D e f i n i t i o n 7.10.

( I ) K 2: L i s c a l l e d a s i n g l e n : - reduct ion (denoted by K 2':) 17. L) I I

i f K G QCX,AI{X}B and K g e n e r a t e s K 2' 17 L ( i . e . i f K r f 17 L i s an

e lementa ry r l -reduction). Th i s r e d u c t i o n i s c a l l e d of o r d e r p

i f llQCx,AlIl = p.

( 2 ) K r L i s c a l l e d a k-fold n!-reduction (denoted K 2':) L) i f rl 17.

t h e r e a r e K . 1 E QCxI,AI] ... Cx.,A.l{xi} ... {x1}B such t h a t 1 1

(1 ( I ) K : QB G L and K . g e n e r a t e s K : 5 2 n l K , - l . .. 2 , rl . 0 1

( I ) K . . This r e d u c t i o n K 2':) L i s c a l l e d of o r d e r p i f K . 2 , 1 rl. 1 - 1 rl.

( 1 ) 5 5-1 i s of o r d e r p.

Theorem 7.1 1 . I f K i A and K 2': 17. ) L 2' 17 N , t h e n e i t h e r K ' ( f ) n . N o r

r e d u c t i o n .

P roof . K Z Q[x,A]{x}B 2';) rl. QB - L. Consider t h e p o s s i b i l i t i e s f o r

QB G L 2' N. 0 rl

Theorem 7.12. I f K E A and K 2':) rl. L 2 ' rl N , where K 2':) 17. L i s of

o r d e r p , t h e n e i t h e r K r ( k + l ) , N of o r d e r p , o r t h e r e i s a r e d u c t i o n rl.

K r ' M r ( F ) N where M 2':) N i s of o r d e r p and K sf M i s n o t a n q!- rl 17. rl. rl

r e d u c t i o n .

P roof . Compare w i t h t h e p r ev ious theorem.

(k Theorem 7.13. Le t R E A and K 2 rl. , L 2 rl N , where K 2';) rl L i f of

o r d e r p. Then t h e r e i s a r e d u c t i o n K r rl L' 2'5) rl. N , where a decom-

p o s i t i o n of K 2 L c o n t a i n s no n : - reduct ions and L ' 17

('1 N i s of

o r d e r p .

? Proof . Decompose L 2 N i n t o L E E , 2 ... 2' E - N. - n rl r We proceed w i t h i n d u c t i o n on r. I f r = 1 t h e r e i s no th ing t o prove.

L e t r > 1 . Consider t h e r e d u c t i o n K 2 n! (k ) L E E 2' E By t h e pre- 1 rl 2'

(k+ l ) E o r a r e d u c t i o n v i o u s theorem we have e i t h e r K 2 , r 2

K 2' L" 2';) E where K 2' L" i s n o t an n!-reduction. Applying t h e rl rl. 2 rl

) induction hypothesis on K 2 , E2 ?n N or L" 2 n.

(k) E > N , we '0 nl 2 - r l

obtain K 2 L' 2 N, where a decomposition of K 2 L' contains rl n. n

Theorem 7 . 1 4 . If QK E A and QK 2 L , there is a reduction n QK 2 Q ' K ' 2':) L where 11 Qll = I I Q ' 11, Q ' K 2';) L is of order 11 411,

rl n. n. and where a decomposition of QK 2 Q ' K ' contains no q!-reductions n of order 11Q11.

Proof. Decompose QK 2 L into single-step n-reductions n QK = L 2 ' ... 2' L L . Let i be the smallest integer such that

1 n rl n L. 2' L is an n!-reduction of order IIQII. Apply the previous 1 q i + l

( 1 ) theorem on L . r , 2 L L . We obtain a reduction 1 n. L i +~ rl n

QK > L ' 2':) L as desired. The fact that L' Q ' K ' with IIQ'II = I lQll n n . follows from Th. 7 . 5 . 0

Theorem 7 . 1 5 . If QK E A, QK 2 Q ' K ' , I lQl l = 11Q'Il = p and a decom- rl

position of QK 2 Q ' K ' contains no n!-reductions of order p, there n is a reduction QK 2 QK' 2 Q ' K ' and a reduction QK 2 Q ' K r Q ' K ' . n rl rl rl

Proof. See Th. 7 . 5 .

Theorem 7 . 1 6 . If Q I K E A, Q L E A, Q K 2 M, Q L 2 M, 2 I rl 2 n Q ] E [ x ~ , A ~ I . . . [X , A 1 and Q 2 C x 1 , B 1 1 ... C X , B 1, there is an

P P P P N such that Q K r Q I N and Q L 2 Q2N. I rl 2 n

Proof. By the aid of Th. 7 . 1 4 and Th. 7 . 1 5 we can find reductions

Q K r Q K ' Z. Q ' K ' 2':) M and Q L 2 Q L ' r Q 'L ' r ( f ) M, where 1 n 1 n 1 rl . 2 n 2 n 2 rl.

Q ; K ' 2 ' : ) M and Q;L' 2';) M are of order p. Note that ri. n.

Q; : [ x l , A ; l ... Cx , A ' ] and Q; [ x I , B ; l ... Cx B ' l . P P P' P

Now both Q ; K ' and Q;L' E A, so k = k : assume k > L, then

M z Cx , A ; ] ... Cx I

B ' 1 ... A' I M ' [ x I , B ; 1 . . a [ x ~ - ~ , p-k p-k' p-k

[x B ' IN"; it follows that [ x ~ - ~ , B' p-k 1 occurs in M', hence p-2' p-ll

a l s o i n K' ; t h i s c o n t r a d i c t s t h e f a c t t h a t Q ; K 1 E A , s i n c e we

found two b ind ing v a r i a b l e s x i n Q ; K 1 . It fo l l ows t h a t K' L ' ; P-R

we can t ake N 5 K' r L'. 0

C o r o l l a r y 7 . 1 7 . I f QK, QL E A , QK 2 M and QL 2 M, t h e r e i s a n n QN such t h a t QK 2 QN and QL 2 QN.

rl n

Theorem 7.18. I f QD E A , QD 2 Q ' E , Q [ X ~ , A ~ I * - C X ,A 1 and n P P Q' 5 Cx , A ; ] ... [x , A 1 ] , t hen QD 2 QE. 1 P P n

P roof . R e s u l t i n g from Th. 7.14 w e can f i n d a r e d u c t i o n - QD t Q"D' 2':) Q ' E where Q"D' 2':) Q'E i s of o r d e r 11Q11. Now

n n . n . 8 -

Q" s Cx , A f l ... Cx , A n ] , hence k = 0 (because Q"D' E A ; s e e t h e 1 D D

I

proof of Th. 7.16). Then a l s o QD 2 QE by Th. 7.15. n 0

We s h a l l now prove a theorem concerning t h e so - ca l l ed "post-

ponement of rl-reductions" f o r . What we want t o prove i s t h a t

every r e d u c t i o n K 2 M which t a k e s p l a c e by means of s i n g l e - s t e p

6- and q - reduc t ions i n a r b i t r a r y o r d e r , c an be r ep l aced by a reduc-

t i o n K 2 L 2 M , i n which a l l B-reductions p recede a l l n-reduc- B n

t i o n s . It i s ea sy t o show t h a t each r e d u c t i o n A 2' B 2' C can be - n B

r e p l a c e d e i t h e r by a r e d u c t i o n A r ' B' C (where r 2 0) o r by a 8 rl

r e d u c t i o n A 2' B' 2' C. But t h i s does n o t s u f f i c e t o prove the a B theorem. It i s n o t s u r e t h a t t h i s p roce s s of i n t e r c h a n g i n g n's and

B's t e r m i n a t e s f o r a g iven r e d u c t i o n K 2 M.

I n Curry and Feys 13 , Ch. 4 , s D2] a compound B-reduction i s

i n t roduced f o r t h e purpose of p rov ing t h e above mentioned theorem.

I n ou r o p i n i o n t h e r e i s an e r r o r i n t h e i r proof (viz. , t h e c a s e

t h a t R i s %N and L i s some M.y f o r j 5 k i s miss ing) . Neverthe- j - -

l e s s , t h e i r i d e a can be extended i n such a manner t h a t t h e theorem

on t h e postponement of n - reduc t ions can b e proved. We have c a r r i e d 1 I *

t h i s o u t by d e f i n i n g a compound 6-reduction" A r B w i t h t h e * B

p r o p e r t y t h a t each r e d u c t i o n A r B 2 C can be r e p l a c e d by a reduc- * n B

t i o n A 2 B' 2 C. However, t h i s compound B-reduction l ooks r a t h e r B n

compl ica ted .

Barendregt suggested t o us another way of proving t h e theorem 1 I ( p r i v a t e communication). He proposed a nested' ' n-reduction (which

we c a l l K-reduc t ion and denote by 2 ' ) K w i th t h e p rope r ty t h a t a

r e d u c t i o n A 2' B 2 C can be rep laced by a r e d u c t i o n A 2 B' 2' C. K B B K

The nes ted c h a r a c t e r of t h i s K-reduct ion i s comparable t o t h a t of

y-reduct ion d i scussed i n t he prev ious s ec t ion .

We p r e f e r t h e l a t t e r way of proving because i t i s e a s i e r t o

unders tand.

D e f i n i t i o n 7 .19. S ing le -s tep K-reduct ion, denoted by 2' K ' i s t h e

r e f l e x i v e r e l a t i o n genera ted by

(I ) I f QCX,A]{XIB E A , x # B and QB 2' K QC, then Q[X,A]{X}B 2' K QC.

(2) I f Q{A}C E A , QA 2 ' QA' and QC 2' Q C ' , then Q{A}C 2' Q{Af }Cf. K K K

( 3 ) I f Q[x,A]C E A , QA 2' QA' and Q[x,A]C 2: Q[x,A]C', t hen K

Q[x ,A]C 2' Q[x,A1 I C ' . K 0

We c a l l r u l e (1 ) i n t h i s d e f i n i t i o n t h e r u l e of elementary

s ing le - s t ep K-reduct ion, r u l e s (2) and ( 3 ) t h e monotony rules f o r

r r e d u c t i o n .

The fo l lowing two theorems d e a l w i th t h e r e l a t i o n between n-

and K-reduct ion.

Theorem 7.20. I f K E . A and K r ' L, then K 2' L. 17 K

Proof . Induc t ion on t h e l e n g t h of proof of K 2' L. - .- rl

Theorem 7.21. I f K E A and K 2' L, then K 2 L. K n

Proof . Induc t ion on t h e l e n g t h of proof of K 2' L. K

For example, i f K 2' L i s Q[x,A]{xl~ 2: QC, a s a d i r e c t consequence K

of QB 2' QC, then by i n d u c t i o n QB 2 QC, and Q[x,A]{x)B 2' K rl rl

QB 2 QC. 0 rl

We s h a l l now prove a number of theorems which a r e lemmas f o r

t h e theorem on t h e postponement of rl-reductions (Th. 7.28 ).

Theorem 7.22. If K E A and K 2' L, then L E A. K

Proof. Follows from Th. 7.21 and Th. 7.4.

Theorem 7.23. If QE E A and QE 2' QC~,GIH, then K

QE Q C X ~ , A ~ ~ ~ X ~ ~ ~ X ~ , A ~ I ~ X ~ ~ ... [X n n ,A l{x n }Cy,G11H', with

QG' 2' QG, QCY, G'IH' 2: Q[y, Gf1H and xi ,Ai+, K 1 ... {x }[y,G'1H1. n

Proof. Induction on the length of proof of QE 2: Q[~,G]H.

If the latter reduction results from reflexivity, the proof is

completed.

(I ) Let QE 2' Q[y,G]H be ~'[x,Al{x}B 2: Q'C, as a direct consequence K

of Q'B 2' Q'C. If Q' r QQ", induction yields the proof. K

If Q E Q'cx,~], then C begins with [x,A]. This implies that Cx,AI

occurs in B (since K-reduction can only omit abstractors and appli-

cators without influencing the remainder of the expression), which

is impossible since QE E A. SO this latter case cannot apply.

(2) Let QE 2' Q[y,G]H be Qf{A}C 2' Q'{A'}c! Then QE E Q[y,G]F. K K

(3) Let QE 2' Q[y,GIH be QICx,AIC 2' Q'CX,A'IC', as a direct con- K K

sequence of Q'A 2' Q'A' and Qf[x,A]C 2' Q'[x,A]C'. There are the K K

following possibilities: (a) Q : Q', (b) Q Z Q' [x,A]Q, and (c)

Q' - QQ with llQl I1 > 0. In all three cases the proof is easy. 1 0

Theorem 7.24. Let QA and Q[x,BIC E A, QA 2' QA' and QCx,B]C 2' K K

Q[x,B]Cf . Then Q(x:=A)C 2' Q(x:=At)C' . K

Proof. Induction on I c I . If C - r , C E x or C E y $ x, the proof is

easy.

(I) Let C 5 Cy,EIF. There are two possible cases:

(a) QCx,BIC 2' QCx,BICV is QCx,BlCy,EIF 2' QCX,BIC~,E'IF', as a K K

direct consequence of QIx,B]E 2; Q[x,B]E' and Q[x,Bl[y,E]F 2' K

QCx,BICy,EIF1. By induction: Q(x:=A)E 2' Q(x:=A')E', and K

QCy, (x:=A)El(x :=A)F 1' QCy, (x:=A)E](x:=A1)F' (the latter because K

QCy,(x:=A)E]Cx,BlF 2' Q[y,(x:=A)El[x,~lF'), Hence Q(x:=A)C 2' K K

Q(X:=A) )cv.

(b ) QCx,BIC 2' QCx,BIC1 i s QCx,BICy,El{y)G 2' QCx,B]GV, a s a d i r e c t K K

consequence of QCx,B]G 2; Q[x,B]G' . By i n d u c t i o n Q(x:=A)G 2' K

Q(x:=A' )G ' , SO Q(X :=A)CY,EI{Y)G : Q C Y , (x:=A)E]{Y)(x:=A)G 2' K

Q(x:=A' )G ' .

(2) L e t C : { E ~ F . Then QCx,B]C 2 ' Q[x,B]CV i s QCX,B]{E}F 2; K

QCX,BI{E' I F ' , a s a d i r e c t consequence of QCx,BIE 2' Q[x,B]E' and K

Q[x,BIF 2' Q[x,B]F'. The theorem r e s u l t s from t h e i nduc t i on . K

(Note t h a t QCX,BIC 2; QCX,BIC' canno t b e Q[x ,~ l{x}G 2' QG'.) K

0

Theorem 7.25. Le t A E A and A 2 ' B 2' C, Then A 2 B' 2 ' C . K B B K

Proof . I n d u c t i o n on t h e l e n g t h of proof of A 2' B. K

I f t h e l a s t d e r i v a t i o n s t e p r e s u l t s from r e f l e x i v i t y , no th ing

remains t o be proved.

( I ) L e t A 2 ' B be QCx,Dl(x)E 2' QE' a s a d i r e c t consequence of K K

QE 2: Q E ' , and l e t B 2 ' C b e gene ra t ed by Q ' { F } [ ~ , G ] H 2' Q ' ( ~ : = F ) H . B B

The fo l l owing c a s e s may app ly :

( a ) {F)Cy,GIH c Q. There i s c l e a r l y a r e d u c t i o n A 2 B ' 2 ' C. 6 K

( b ) {F)Cy,GIH c E'. Le t B 2 ' C be QE' 2; QE"', t h e n by i n d b c t i o n B

t h e r e i s a r e d u c t i o n

QE 2 QE" 2 ' QE"', hence QCx,D]{x)E 2 Q[x ,D]{x )E~~ 2 ' QE'". B K B K

( 2 ) L e t A 2' B be Q{D)E 2 ' Q{D')E' a s a d i r e c t consequence of K K

QD 2 ' QD' and QE 2' QE', and l e t B 2 ' C be gene ra t ed by K K B

Q ' ( F ) C ~ , G I H 2' Q'(Y:=F)H. B

The fo l l owing c a s e s may app ly :

( a ) {F)C~,G]H c Q, C l e a r l y A 2 B' 2' C . B K

(b ) { F ) C ~ , G I H : {D' )EV. Then QE' r Q[y,G]H, s o

QE E Q[x , A 1 l { ~ I } ... [X , A l{x }Cy,GV]H', w i t h QG' 8' Q G , 1 n n n K

Q[~,G']H' 2' K QCy,G'IH and xi C ,Ai+, I ... {X ICY,G'IH' by ~ h . n 7 . 2 3 . Then Q{D)E 2 Q(y:=D)HV. By Th. 7.24: Q(y:=D)H1 r '

B K

Q(y:=F)H C.

( c ) { F } C ~ , G I H c D ' . Then C 5 Q{D'")E' w i t h QD' 2; QD"', and by

i n d u c t i o n QD 2 QD" 2' QD"', hence Q{D)E 2 Q{D!~)E 8' Q{D"')E' z C. B K B K

( d ) {F}[Y,G]H c E ' . Then C z Q{D')E'" w i t h QE' 2' QE"', and by B

i n d u c t i o n QE r QE" 2' QE"', hence Q(D)E 2 Q{D)E" 2 ' Q{D')E'" E C. 13 K 6 K

( 3 ) L e t A 2' B b e Q[x,D]E 2' Q[x,D']E ' a s a d i r e c t consequence of K K

QD 2 ' QD' and Q[x,D]E 2: Q[x ,D]E ' , and l e t a g a i n B 2' C be g e n e r a t e d K 6

The f o l l o w i n g

( a ) {F}[y,GIH

(b ) {F}Cy,GIH

By i n d u c t i o n :

c a s e s may app ly :

c Q. C l e a r l y A 2 B' 2 ' C. B K

c D ' . Then C Q[x,D"']E1 w i t h QD' t' QD"'. 6

Q [ X , D ~ ~ ] E ' (where we r e q u i r e t h e lemma: Q[x,DIE 2' K Q[x,D]E', t h e n

Q[X,D"IE 2' Q [ X , D ~ ~ ] E ' ) . K

( c ) {F)Cy,G]H c E ' . Then C E Q[X,D']E' ' ' .

Also: Q ~ , D ] E ' 2' QCX,DIE~ ' , s o by i n d u c t i o n Q[x,D]E 2 Q [ x , D ] E ~ ' ~ ~ B B

Q ~ , D ] E " ' . Hence Q[x,DIE 2 QCX,DIE" 2 Q[x,D']E"' E C. B K

n

P Theorem 7.26. L e t A E A and A 2; B 2 C. Then A 2 B 2' C. B B K

Proof . I n d u c t i o n on p , u s i n g t h e p r e v i o u s theorem.

Theorem 7.27. L e t A E A and l e t A 2 C by means of a number of

s i n g l e - s t e p K- and B-reductions i n a r b i t r a r y o r d e r . Then t h e r e i s

a r e d u c t i o n A 2 B 2 C. B K

P r o o f . I n d u c t i o n on t h e number of s i n g l e - s t e p K - r e d u c t i o n s i n A 2 C.

I f t h i s number i s z e r o , t h e proof i s completed. E l s e , l e t A 2 C b e

A 2 A' 2' B 2' C . Apply t h e p r e v i o u s theorem, o b t a i n i n g K B

A 2 A' 2 B' 2' C , and a p p l y t h e i n d u c t i o n on A 2 A ' 6

2B B ' . K

0

Theorem 7.28. L e t A E A and l e t A 2 C by means of a number of

s i n g l e - s t e p 0- and B-reductions i n a r b i t r a r y o r d e r , Then t h e r e i s

a r e d u c t i o n A 2 B 2 C. B 17

P r o o f . S i n c e each q - reduc t ion can b e c o n s i d e r e d a s a K- reduc t ion

(Th. 7 .20) , we can a p p l y t h e p r e v i o u s theorem, o b t a i n i n g

A 2 B 2 C. But B 2 C i m p l i e s B 2 C (Th, 7 - 2 1 ) , s o A 2 B 2 C. B K K rl B " 0

The remainder of t h i s s e c t i o n w i l l concern ( g e n e r a l ) r e d u c t i o n ,

d e f i n e d as a sequence of s i n g l e - s t e p a-, f3- and q - reduc t ions .

Definition 7.29. Single-step reduction (denoted by 2') is the

relation obeying:

A 2' B if and only if A 2' B, A 2' B or A r1 B. C1 B 17

0

Definition 7.30. Reduction (or general reduction,denoted by 2) is

the reflexive and transitive closure of single-step reduction. 0

Theorem 7.31. The monotony rules hold for reduction.

Proof. Use Th. 5.15 and Th. 7.8.

We shall prove a theorem (Th. 7.33) which expresses that the

Q is in a certain sense irrelevant inareductionQC 2QE:itcanbe re-

placed by any P such that PC and PE E A. This corresponds with

general usage in lambda-calculus to define reduction for expressions

which may contain free variables. Our choice to define reductions

inside A is apparently not in disagreement with that general usage.

Theorem 7.32. If QC E A and QC 2 QE by means of 8- and q-reductions,

there is a reduction QC r QD 2 QE. B n

Proof. There is a reduction QC r K r QE by Th, 7.28. B rl Now by Th. 5.12: K r Q'D with IIQll = ilQIII.

I£ Q E [xl,All ... Cx ,A I, then Q' 2 Cxl,A;I ... Cx ,A'], P P P P

so by Th. 7.18: Q'D 2 Q'E. From Th. 5.21 : QC 2 QD, and from Th. rl B

7.9: QD 2 QE. 0 rl

Theorem 7.33. If QC, PC and PE E A, and QC 2 QE, then PC 2 PE.

Proof. See Th. 7.32, Th. 5.17 and Th. 7.9.

Reduction is a non-symmetric relation between expression in A,

which is reflexive and transitive. We shall define lambda-equiv-

alence. The definition of beta-equivalence was given in Def. 5.22.

In Th. 7.35 we shall prove that beta-equivalence is the symmetric

closure of beta-reduction.

Def in i t ion 7.34. Let A and B E A . We c a l l A Lambda-equivaLent to B

(denoted: A - B) i f t h e r e i s an expression C such t h a t A r C and

Theorem 7.35. Beta-equivalence i s r e f l e x i v e , symmetric and t ran-

s i t i v e .

Proof. Ref lex iv i ty and symmetry a r e t r i v i a l . T r a n s i t i v i t y follows

from Th. 6.43 (CR f o r B-reduction): l e t A - B and B - C , then B B

t he re a r e D and E such t h a t A 2 D , B > D , B r E and C 2 E. B B B B

Moreover, t h e r e i s an F such t h a t D 2 F and E 2 F (Th. 6 . 4 3 ) , B B

so A r F and C r F. Hence A - C. B

0 B B

Unfortunately, the re i s no s i m i l a r theorem f o r lambda-equiv-

alence. Of course lambda-equivalence i s symmetric and r e f l e x i v e ,

but not necessa r i ly t r a n s i t i v e , The reason f o r t h i s i s t h a t CR

does not hold f o r (general) reduct ion: f o r example, l e t K 2 L and

K 2 M, l e t K Q[x,A]{x}[y,B]C where x # [Y ,B]C, l e t K 2 L be

Q[x,A]{x)[y,B]C 2 Q[y,B]C and l e t K 2 M be Q[x,A]{x}[~,B]c rB rl

Q[x,A](y:=x)C (>a Q[y,AlC). Now we cannot i n general f i n d an N

such t h a t L 2 N and M 2 N , s ince we know nothing concerning a

r e l a t i o n between A and B.

We note the following. We can embed ordinary lambda-calculus

i n t o A , s ince the re i s a one-to-one correspondence between ex-

press ions from lambda-calculus and those expressions i n A i n which

only a b s t r a c t o r s of t h e form [ x , r ] occur. I f w e r e s t r i c t ourselves

i n A t o the l a t t e r expressions, the example above changes i n t o

K E Q[x,r]{x}[y,r]C, L Q[y,r]C and M r a Q [ y , r l ~ . Now the re i s no

problem as regards CR. Indeed, i n lambda-calculus the Church-

Rosser property holds (see Barendregt [ I , Appendix 111).

The folLowing theorem expresses t h a t lambda-equivalence of QK

and QL implies the exis tence of an N such t h a t QK 2 QN and QL > QN

o r , 0 t h e r w i s e s t a t e d : t h e a b s t r a c t o r chain Q can remain unaffected.

Theorem 7.36.Let QK and QL E A. I f QK - QL, t h e r e e x i s t s a n N such

t h a t QK 2 QN and QL r QN.

P roof . There must b e a n M: QK 2 M and QL 2 M. By postponement of

q - reduc t ions w e o b t a i n r e d u c t i o n s QK > M 2 M and QL 2 M r M. P l r l 6 2 1 1

Th. 5.12 i m p l i e s t h a t MI Q I K' , M2 QZL1, 11 Q 11 = 11 Q, II = 11 Q 2 11.

Then, accord ing t o Th. 5.21, w e a l s o have QK 2 QK' r Q K' 5 M 13 P 1 rl

and QL r QL' r Q L' t M. It i s e a s y t o show t h a t Q and Q2 have P B 2 n 1

t h e form a s r e q u i r e d i n Th. 7.16, hence t h e r e i s a n N such t h a t

Q, K' > Q I N and Q L' > Q N. From Th. 7.9 i t f o l l o w s t h a t QK' 2 QN n 2 -n 2 n and QL' 2 QN. So QK 2 QN and QL 2 QN. 0

rl

The monotony r u l e s a l s o h o l d f o r lambda-equivalence:

Theorem 7.37.

( a ) I f Q C , QD, Q{A}C, Q{A}D E A and QC - QD, t h e n Q{A}C - Q{A}D.

(b ) I f QC, QD, QCx,AIC, QCx,AlB E A and QC - QD, t h e n

QCx,AlC - QCx,AID.

( c ) I f QA, QB, Q{A}C, Q{B}C E A and QA - QB, then Q{A}C - Q{B}C.

(d ) I f QA, QB, Q[x,A]C, Q[x,B]C E A and QA - QB, t h e n

QCx,AlC - Q C x , B l C .

P roof . See Th. 7.36 and Th. 7.33.

Theorem 7.38. I f QC, QD, PC and PD E A and QC - QD, t h e n PC - PD.

Proof . Th. 7.36 and Th. 7.33. 0

§ 8. TYPE AND DEGREE

The n o t i o n s i n t r o d u c e d i n t h e p r e c e d i n g s e c t i o n s a r e from

lambda-calculus ( a s r e d u c t i o n , lambda-equivalence) o r a p p l i c a b l e

t o lambda-calculus ( f a c t o r s , bound e x p r e s s i o n s ) , s i n c e t h e t y p e s

p layed no e s s e n t i a l r81e.

We s h a l l now look i n t o t h e t y p i n g of a n e x p r e s s i o n i n A , With

e v e r y A E A f o r which T a i l A 2 T w e d e f i n e a t y p e , deno ted a s

Typ A, as f o l l o w s :

Definition 8.1. Let A E A and Tail A - x, so A = P 1 [x,B]P 2 x. Then Typ A Z PI [x,B]P2FrB. [7

Informally speaking, we may say that B is the type of x in the

above expression. Note, however, that we allow Typ to operate only

on expressions in A.

Theorem 8.2. If A E A and Tail A $ T, then Typ A E A.

Proof. Let A E P [x,B]P x and let A ~ X - Q [x,B]Q2x. 1 2 1 We prove that Typ A is a bound expression. A11 non-binding variables

in P [x,B]P are clearly also bound in Typ A. Consider a non-binding 1 2

variable z c FrB c Ql[x,B]Q2FrB. There is a corresponding y c B, and

~l~ 5 Q,Q3y. So Typ A ~ Z E Q 1 [x,B]Q 2 Q'z 3 where Q'z 3 is a renovation of

Q3y*

Case I : if y was bound in ~l~ by a binding variable in Q 3 , z is bound in Typ A ~ Z by the corresponding binding variable in Q' 3'

Case 2: if y was bound in ~l~ by a binding variable in Q 1 ' z I y

is still bound by the same binding variable in Q 1 since all binding

variables of [x,B]Q Q' are different from y. 2 3 So Typ A is bound. Clearly Typ A is also distinctly bound by

the renovation of B. 0

We define repeated application of Typ inductively as follows:

0 - n- 1 Definition 8.3. Let A E A, Then Typ A A; if Typ A is defined n- 1 n n- 1 for n r 1 and if Tail Typ A T, then Typ A 5 Typ(Typ A), 0

If A E A and T ~ ~ ~ A is defined, we call n pennissibZe for A

(n = 0 is always permissible for A E A).

n n Theorem 8.4. If A E A and A 2 B, then Typ A 2 Typ B for all n a a permissible for A and B.

Proof. It is sufficient to prove: if A 2' B and Tail A $ T, then a

Typ A >a Typ B. The latter proof is easy. 0

With each e x p r e s s i o n A i n A we d e f i n e a degree , denoted

Deg A:

D e f i n i t i o n 8.5.

( 1 ) I f A E A and T a i l A E T , t h e n Deg(A) = 1 .

( 2 ) I f A E A , T a i l A : x and A 5 PI [x,B]P x , t h e n 2 Deg(A) = Deg(PIB) + I .

I n d u c t i o n on t h e l e n g t h of A shows t h a t Deg(A) i s we l l -

d e f i n e d by Def. 8.5. C l e a r l y Deg A = 1 i f and o n l y i f T a i l A ! T .

We s h a l l now prove a number of theorems, l e a d i n g t o t h e

theorem: i f T a i l A 2 T , t h e n Deg A = Deg Typ A + 1 (Th. 8.12).

We c o u l d have t a k e n t h i s p r o p e r t y a s a d e f i n i t i o n of Deg, I n t h a t

c a s e , however, t h e we l l -de f inedness of Deg would have been h a r d e r

t o p rove .

Theorem 8.6. I f PC E A and P{K)c E A , t h e n Deg PC = Deg P{K)c,

P r o o f . I n d u c t i o n on I PC I . 0

C o r o l l a r y 8.7. I f A E A and A Z PC, t h e n Deg A = D ~ ~ ( A ~ C ) . [3

C o r o l l a r y 8.8. I f A E A , T a i l A E x and A ~ X Q [x,B]Q x, t h e n 1 2 Deg A = Deg Q l B + 1 . 0

Theorem 8.9. I f PC E A and PCX,KIC E A , t h e n Deg PC = Deg PCX,KIC.

P r o o f . By Th. 3.8: x # C. The r e s t of t h e proof f o l l o w s from

i n d u c t i o n on I PC I . 0

Theorem 8.10. I f PC E A and PP'C E A , t h e n Deg PC E Deg PP'C.

P roof . I n d u c t i o n on II P' 11, u s i n g Th. 8.6 and Th. 8.9.

Theorem 8.11. I f A E A and A 2 B, t h e n Deg A = Deg B. 01

Proof . Take A 2' B ; i n d u c t i o n on I A ~ . a 0

Theorem 8.12. I f A E A and T a i l A f T, t h e n Deg Typ A = Deg A-1.

P roof . L e t ail A x and A I P 1 [x,C]P2x, s o Typ A 2 P 1 Cx,c1P2FrC.

Then, P FrC E A and Deg PIFrC = Deg PIC by Th. 4.5 and Th. 8.11 . 1 By Th. 8.10: Deg PIFrC = Deg Typ A.

So Deg A = Deg PIC + 1 = Deg Typ A + 1 . 0

C o r o l l a r y 8.13. I f A 5 A , t h e n ~ a i i ( ~ ~ ~ Deg T e 0

T h i s o p t i m a l exponent of Typ w i t h a c e r t a i n A E A i s of

s p e c i a l impor tance . We s h a l l i n t r o d u c e a n a b b r e v i a t i o n :

* Deg A-IA, D e f i n i t i o n 8.14. I f A E A , t h e n Typ A - Typ

We s t r e s s t h a t t h e a s t e r i s k r e p l a c e s a n exponent n dependent

upon A . Moreover ,note t h a t Typ i s a p a r t i a l f u n c t i o n on A , b u t *

Typ i s a t o t a l f u n c t i o n on A . n *

We proceed w i t h a number of theorems on Typ , Typ and Deg:

Theorem 8.15. I f A E A , Deg A = 1 and A 2 B, t h e n Deg B = 1, 0

n Theorem 8.16. I f PC E A , t h e n f o r p e r m i s s i b l e n Typ PC P C ' ;

* i n p a r t i c u l a r Typ PC r PC". 0

n Theorem 8.17. I f PC E A , PP'C E A , and f o r a p e r m i s s i b l e n Typ PC E

n PC', t h e n n i s p e r m i s s i b l e f o r PP'c, and Typ PP'C 2 a PP'C'.

P roof . It i s s u f f i c i e n t t o assume T a i l PC f T and n = 1 .

and [x,B] a p p e a r s i n e i t h e r P o r C, The remainder f o l l o w s . 0

n Theorem 8.18. I f PC E A , PP'C E A and f o r a p e r m i s s i b l e n Typ PP'C 5

n PP'C', t h e n n i s p e r m i s s i b l e f o r PC and Typ PC 2 a P C ' .

P roof . S i m i l a r t o t h e p r e v i o u s p r o o f .

CHAPTER 1 1 1 , THE FORMAL SYSTEM A

§ 1 . LEGITIMATE EXPRESSIONS

The "meaning" of {A}B is the application of function B to argu-

ment A. So far this application was unrestricted: any expression

could serve as an argument. Besides, it was of no interest whether B

really was a function or not.

In the formal system A, which we shall introduce in this chap-

ter, we only admit the expressions of A which obey the applicability

condition. (For an informal introduction of the applicability condi-

tion: see Section 1.4.) We call this kind of expressions Zegitimate

expressions.

Since A is a part of A , we again provide expressions with ab-

stractor chains Q, as we did with expressions in A (cf. the begin-

ning of Section 1.6). We begin with the definitions of function,

domain and applicability with respect to an abstractor chain Q:

Definition 1.1. Let QB E A. We call QB a Q-function if there are x,

K and L such that T ~ ~ * QB > Q[x,K]L. The expression QK is called a

Q-domain of QB.

Definition 1.2. The expression QB is called Q-appZicabZe to QA if

QB is a Q-function with Q-domain QK, Deg QA > 1 and Typ QA t QK. In

that case .Q{A)B is a Zegitimate Q-appZication of QB to QA. 0

The formal system A is inductively defined by:

Definition 1.3.

( I ) T € A.

(2) If QA E A and if x does not occur in QA, then QCx,AIx E A and

QCx,Alr E A.

(3) If QA and Qy E A, if x does not occur in QA and if x f y, then

QCx,Aly E A .

(4) If QA and QB E A, if the binding variables in A and B are dis-

tinct and if QB is Q-applicable to QA, then Q{A)B E A, 0

The only d i f f e r e n c e t o t h e (second) d e f i n i t i o n of A a s g iven by

Th. 11 .3 . 10 l i e s i n t h e appZicabiZity condit ion i n ( 4 ) : QB must be *

Q-appl icable t o QA, i . e . Typ QB 2 QCY,KIL and Typ QA 2 QK. These

r e d u c t i o n s a r e d e f i n e d f o r e x p r e s s i o n s i n A ( c f . t h e f o l l o w i n g Th.

1.4 andTh. 11 .8 .2) . Note t h a t t h e a p p l i c a b i l i t y c o n d i t i o n does n o t

s t a t e t h a t t h e r e d u c t i o n s mentioned concern e x p r e s s i o n s i n A on ly .

The a p p l i c a b i l i t y c o n d i t i o n ha s t h e powerful consequence t h a t

a l l exp re s s ions i n A normal ize ( c f . S e c t i o n I . 2 ) , which we s h a l l

prove l a t e r i n t h i s c h a p t e r , whereas i n t h e wider sys tem A normal i -

z a t i o n i s n o t guaran teed .

Theorem 1.4. I f A E A , t h e n A E A .

Proof . I n d u c t i o n on t h e l e n g t h of proof of A E A .

R e s t r i c t i n g o u r s e l v e s t o a- and B-reductions, we can weaken t h e

a p p l i c a b i l i t y c o n d i t i o n i n t h e s e n s e t h a t we r e p l a c e 2 by -:

Theorem 1.5. I f QA and QB E A , Q C ~ , K I L E A , T ~ ~ * QB -@ QCy,KIL,

Typ QA -B QK and i f t h e b i n d i n g v a r i a b l e s i n A and B a r e d i s t i n c t ,

t h en Q{A)B E A.

Proof . L ~ ~ T ~ ~ * Q B Q B ' (Th. 11.8.16). Since QB' - Q[y,KIL, t h e r e i s B an M such t h a t Q B ' QM and Q[y,KIL 2 QM (Th. 11.5.12 and Th.

B 11 .5 .16) . From Th. 11.5.16 and Th. 11.5.20: QM 5 Q[y,KfIL' such t h a t

QK 2 Q K ' . B Le t Typ QA E Q A ' . S i n c e QA' - QK, t h e r e i s a K" such t h a t

6 QA' 2 QK" and QK 2 QK". Hence (Church-Rosser theorem f o r B-reduc-

B B t i o n , Th. 11.6.43) QK' - QK", s o t h e r e i s a K " ' : Q K ' 2 QK"' and

B QK" 2 QK"'. Also Q [ ~ , K ' ] L '

B 2B Q C Y , K ~ ~ ' I L ' . *

Resuming: Typ QB sB Q[y,K"'1L1 and Typ QA 2 QK"' . So

Note t h a t t h e above theorem does n o t ho ld i f we u s e lambda-

equ iva l ence (-) i n s t e a d o f B-equivalence (- ) . L e t QA E A and B

* Typ QA E QA1. Le t QB E A f o r some B. Then Typ QB E QB1 - Q C ~ , A ' I { ~ ) B '

f o r some f r e s h y , s i n c e Q[y,A1]{y}B' > n Q B ' . I f t h e above theorem were t o

ho ld w i t h - i n s t e a d of - i t would fo l l ow t h a t Q{A}B E A. Note t h a t B ' A and B a r e a r b i t r a r y . Th i s can c l e a r l y n o t g e n e r a l l y b e t h e ca se .

As a counterexample, t a k e Q E C X , T ] , A B E x. Then Q{A)B E

Cx,r l{x)x , which does n o t be long t o A.

We s h a l l prove a number of theorems concern ing A.

Theorem 1.6. I f A E A and A ra B, t h en B E A.

A s w i t h A , i t ho ld s f o r A t h a t , g iven K E A , on ly one o f t h e

d e r i v a t i o n s t e p s i n Def. 1 . 3 can y i e l d K E A a s a conc lu s ion (unique

A-cons t r u c t i b i l i t y ) .

Theorem 1 . 7 . I f Q{A}B E A , t h en QA and QB E A.

P roof . Follows from t h e unique A - c o n s t r u c t i b i l i t y .

Theorem 1.8. I f QCX,AIB E A , t h e n QA E A.

P roof . I n d u c t i o n on I B I , u s i n g t h e unique A-cons t r u c t i b i l i t y .

Le t B Cyl ,B1l ... [y , B IPS, where P f [ Z , E I P I , and s E T , k k

c a se 1 . P E @, k = 0. Then QA E A from r u l e (2) of Def. 1 . 3 f o r a l l

p o s s i b l e s .

c a s e 2. P 5 fl, k 2 1 . Then Q[x,A]Cyl,B,] ... Cyk-l*Bk-l IB k E A from

r u l e (2) o r ( 3 ) , s o QA E A by i n d u c t i o n .

c a se 3 . P {EIP' . Then QCx,A][yl,B1l ... [yk,BklE E A by Th. 1.7,

hence QA E A by i nduc t i on . 0

Theorem 1.9. I f QA E A , t h en QT E A.

P roof . I n d u c t i o n on I A ~ . I f A E T , t h e r e i s n o t h i n g t o prove. I f

A E x, t hen Q E Q1CyyB1 o r Q QICx,BI. I n bo th c a s e s Q B E A , s o I

a l s o QT E A. I f A E {B)C o r A Z [z,BIC, t h e n QB E A by Th. 1.7 o r

by Th. 1.8, s o by i n d u c t i o n QT E A. 0

Theorem 1.10. If A E A and B c A, then A I B E A.

Proof. Induction on I A ~ . If A 2 r then the proof is trivial. Let

A E Cx1,AlI . . . Cxk,41Ps, where P CZ,E~P'.

(1) If B E [x.,A.] ... [%,$IPS or B P s , t h e n ~ I B = A E A. J J

(2) If B c Ai, then A / B : Cx1 ,Al 1 . . . [xi-] ,Ai-l I(A~~B) (Cxl,All * a I A ~ ) ~ B and Cxl,A11 ... C X ~ - ~ , A ~ - ~ ~ A ~ E A

by Th. 1.8, so by induction A / B E A.

(3) Let B c F s , B $ Ps. If P @ then B E s and A I B 5 A E A. So

assume P S (KIP'. Distinguish the cases B c K and B c P's. In

both cases we may conclude A I B E A by a similar reasoning as in

( 2 ) C

Corollary 1.11. If A E A and x c A, then A ~ X E A.

Theorem 1.12. If QA and QB E A, Q[x,AIB E A, then Q[x,A]B E A.

Proof. Induction on I B I . Let B E Cyl ,BII . . . Cy ,B IPS, where k k P f Cz,EIPV.

case 1. P : @, k = 0. Then QCx,AlB E A by Def. 1.3 (2) or (3).

case 2. P 5 0, k 2 1. Call Cyl ,Bl I . . . [yk-] ,Bk-] 1 9'. (1) Assume s - y ThenQQtBk E A by Th. 1.8, and Q[x,A1QfBk E A k'

(Th. 11.3.8 and Th. II.3.9), so by induction QCx,A1QfBk E A,

hence QCx,AIB E A .

(2) Assume s $ yk. Then QQ'B~ and QQ's E A (by the unique A-con-

structibility), QCX,AIQ'B~ and QCx,A1Qfs E A (Th. 11.3.8 and

Th. II.3.9), so by induction Q[x,A]Q'B~ and Q[x,A]Q's E A. It

follows that QCX,AIB E A.

cae 3. P 5 {E}P1. Call [yI,BII ... [y ,B I ' Qn. Then QQ"E and k k QQ"P's E A by Th. 1.7, T ~ ~ * QQ"P's E QQnF' 2 QQ"[Z,K]L and

Typ QQ"E r QQ"E' 2 QQ"K. It follows from Th. II.7.33,Th. 11.8.9 and

Th. 11.8.17 that T ~ ~ * Q[x,A]Q"P's : QCx,AIQ"F1 2 QCx,AIQ"Cz,KlL and

Typ Q[X,AIQ"E E Q[x,A]Q"E' r Q[x,A]QWK. By Th. 11.3.8 and Th. 11.3.9

QCx,A]Q"P's and Q[x,A]Q"E E A, so by induction they also belong to

A , h e n c e QCX,AIB E A.

Theorem 1 . 1 3 . I f Q[x,AIB E A a n d QB E A , t h e n QB E A.

P r o o f . I n d u c t i o n o n I B I . The p r o o f i s s i m i l a r t o t h e p r o o f o f Th.

1 . 12, w i t h t h e u s e o f Th. 1 1 . 3 . 1 0 i n s t e a d o f Th. 11.3.11. 0

We s h a l l u s e t h e f o l l o w i n g t h e o r e m as a lemma f o r t h e i m p o r t a n t

Th. 1.15.

* * Theorem 1.14. L e t PP'K, PL E A , PP 'L E A a n d Typ PP'K 2 Typ PP 'L. a Then PP 'L E A.

P r o o f . I n d u c t i o n o n IIPP'II. I f PP' fl, t h e p r o o f i s t r i v i a l .

case 1 . Assume P E Q{E}PW. Then QP1'P'K E A , QE E A , T ~ ~ * QP"PIK 2

QCY,MIN a n d Typ QE 2 QM. A l s o : QP1'L E A a n d QP"P1L E A . We now * *

p r o v e t h a t Typ QP1'P'K 2 Typ QPflP'L. L e t T ~ ~ * PP'K E P P ' K ' a n d a Typ* PP 'L P P ' L ' , t h e n by h y p o t h e s i s Q{E}P1'P'K' E PP'K' r a PP'L' E

Q{E}P"P'L', s o a l s o QP1'P'K' 2 QP"P'Lf (Th. 1 1 . 4 . 6 ) . B u t a T ~ ~ * Q P " P ' K 5 QP"P'K' a n d T ~ ~ * QPI1P'L : QP1'P'L' b y Th. 1 1 . 8 . 6 a n d

Th. 11.8. 18. I t f o l l o w s b y i n d u c t i o n t h a t QP1'P'L E A. A l s o

T ~ ~ * Q P ~ P ~ L z Q C Y , M ] N , s o Q { E } P ~ I P ~ L : PP 'L E A .

case 2. Assume P E Q a n d P ' E Q'{E}P". Then QQIP"K E A , QQ'E E A ,

T ~ ~ * QQ'P"K 2 QQICy,M]N a n d Typ QQ'E r QQ'M. A l s o : QQ'P1'L E A a n d * *

Typ QQIP"K Typ QQ'P1'L (which c a n b e p r o v e d as i n case I ) , s o *

b y i n d u c t i o n QQ1P"L E A . S i n c e Typ QQ'P1'L 2 QQ'Cy,M]N i t f o l l o w s

t h a t QQ'{E)P"L Z PP'L E A.

c a s e 3. Assume P E Q a n d P ' E Q ' . I f Q ' E @ t h e r e i s n o t h i n g t o

p r o v e . L e t Q ' E Cxl,M1I ... [X ,M I f o r n 2 1 . S i n c e QL a n d n n QQ'L E A , xi c a n n o t o c c u r i n QL (Th. 11.3.8) o r i n Q[xI,MII ... C X i- 1 sMi- I ]Mi. I t f o l l o w s f r o m QQ'K E A (Th. 1 .8 ) t h a t

Q M I ,Q[xl , M I ]M2, , QCxl 1 Cxn- I ,Mn- I lMn E A. SO a l s o

Q [ x l , M l l L , Q C X ~ ~ M ~ I C X ~ ~ M ~ I L , ..., QQ'L E A b y Th. 11.3.11 a n d Th.

1 . 1 2 . 0

Theorem 1.15. I f A E A, t hen T~~~ A E A f o r a l l p e r m i s s i b l e n.

P roo f . Le t A - Pl[x,B]P2x, then Typ A 5 PI[x,B]P2FrB. S i n c e A E A:

P I B E A (Th. 1 8 ) s o P,FrB E A (Th. 1 .6) . Also Typ A E A (Th. *

11 .8 .2 ) and T ~ ~ * A ra Typ (Typ A). Now, app ly ing Th. 1 . 14, we o b t a i n

Typ A E A. The theorem fo l lows d i r e c t l y . 0

§ 2. THE NORMALIZATION THEOREM

I n t h i s s e c t i o n we s h a l l p rove t h e n o r m a l i z a t i o n theorem: i f

A E A, t h e r e i s a B i n normal form such t h a t A 2 B ( B i s s a i d t o b e

i n normal form i f t h e r e a r e no r e d u c t i o n s B 2 ' B ' o r B 2' B' ) . We do B rl

t h i s by t h e a i d of a norm p , which i s a p a r t i a l f u n c t i o n from ex-

p r e s s i o n s i n A t o exp re s s ions i n A , and which ha s t h e fo l l owing

powerful p r o p e r t i e s w i t h r e l a t i o n t o A:

(1 ) I f A E A, t h e n p(A) i s de f i ned .

( 2 ) I f A E A and A 2 B, t h en p(A) p(B).

( 3 ) I f A E A and Deg A > 1 , t h e n p (A) p (Typ(A) ) . Hence t h i s norm i s i n v a r i a n t ( a p a r t from a - r educ t i on ) w i t h re-

s p e c t t o r e d u c t i o n and typ ing .

We f i r s t d e f i n e p A f o r every A E A . Th i s p A i s a p a r t i a l func-

t i o n from subexpress ions of A t o e x p r e s s i o n s . I t i s r a t h e r i n con-

t r a d i c t i o n t o ou r phi losophy t o d e f i n e t h e norm w i t h r e s p e c t t o sub-

e x p r e s s i o n s , which need n o t be long t o A . We could have avoided t h i s

by g i v i n g a d e f i n i t i o n of t h e norm i n t h e l i n e of our second d e f i n i -

t i o n of A , only c o n s i d e r i n g norms of e x p r e s s i o n s i n A . Th i s , how-

e v e r , would have impaired unde r s t and ing o f t h e fo l l owing and would

have l e d t o l a b o r i o u s d e s c r i p t i o n s . On t h e o t h e r hand, i n t h i s sec-

t i o n t h e c o n t e x t of a subexpress ion w i l l always b e c l e a r , s o t h a t no

con fus ion can a r i s e .

I n t h e fo l l owing i n d u c t i v e d e f i n i t i o n of p A we do n o t e x p l i c i t -

ly indicate which occurrence of a subexpression i n an e x p r e s s i o n i s

meant, s i n c e t h i s w i l l b e c l e a r from t h e c o n t e x t .

D e f i n i t i o n 2. 1 . Le t A E A.

( 1 ) I f r c A then pA(r ) E T.

(2) I f x c A, Alx E Ql[x,B]Q2x and i f pA(B) i s de f ined , t hen

pA(x) ! pA(B)

( 3 ) I f [x,B]C c A, and i f both pA(B) and pA(C) a r e de f ined , then

pA(Cx,BIC) [x,pA(B) IpA(C)

( 4 ) I f {BlC c A, i f bo th pA(B) and pA(C) a r e de f ined and pA(C) E

[y,D]E where D 2 pAB, then pA({~}C) E E. a 0

From t h i s d e f i n i t i o n i t can e a s i l y be seen t h a t , i f pAA i s

de f ined f o r A E A , pAA con ta ins no bound v a r i a b l e s .

The fo l lowing theorem i s obvious :

Theorem 2.2. I f A E A and i f pA(A) i s de f ined , then pA(A) E A ; i f ,

moreover, A 2 B, t hen pA(A) 2 p (B) and I pA(A) I = I P B ( ~ ) I . a a B 0

The b ind ing v a r i a b l e s i n p A w i l l b e i r r e l e v a n t t o our pur- A poses . We might a s w e l l do wi thout them. Our reason f o r r e t a i n i n g

them i s pe r sona l t a s t e : we f i n d t h e proper ty pA(A) A agreeab le .

I n t r y i n g t o c a l c u l a t e pA(A) f o r a c e r t a i n A E A , we apply t h e

fou r r u l e s of Def. 2.1; t he only event i n which t h i s c a l c u l a t i o n

can break down prematurely (before pA(A) i s ob ta ined ) , i s when we

encounter a subexpress ion {B)C c A f o r which t h e cond i t i ons s t a t e d

i n Def. 2.1 (4 ) a r e n o t f u l f i l l e d .

These cond i t i ons may be considered a s a weaker form of t h e ap-

p l i c a b i l i t y cond i t i on ( c f . Sec t ion 1 .6 , where t h i s i s exp la ined i n

an in formal manner): (1) C must have a norm w i t h a f u n c t i o n a l char-

a c t e r : pAC [y,DlE, and (2) B has a norm which behaves a s an ap- I1 p r o p r i a t e argument f o r t h e func t ion" p C: p B 2 D. I f t h e s e condi- A A G.

t i o n s a r e f u l f i l l e d , t h e norm of {B)C i s de f ined as t h e r e s u l t of 1 I I I t h e a p p l i c a t i o n of t he func t ion" pAC t o t h e argument" pAB:

pA({B]C) E E; i f t h e s e cond i t i ons a r e n o t f u l f i l l e d , t h e norm of

{B]C i s n o t de f ined , and n e i t h e r i s the norm of A.

Note t h a t the norm of a bound v a r i a b l e i s defined as the norm

of i t s "typet1: i f [x,B] i s the binding abs t rac to r of x, then

pA(x) pA(B)*

The exis tence of pAA f o r a c e r t a i n A E A i n d i c a t e s t h a t some

weak funct ional condi t ion i s f u l f i l l e d . Surpr is ingly enough, the

exis tence of pAA a l ready guarantees t h a t there i s a normal form f o r

A. We s h a l l prove t h i s i n Th. 2.17. We a r e e s p e c i a l l y i n t e r e s t e d i n

normalization p roper t i e s of expressions i n A , We no te t h a t expres-

s ions i n A have, so t o say, a much s t ronger func t iona l cha rac te r

than i s required f o r the ex i s t ence of the norm of expressions. Th.

2 . 7 , s t a t i n g t h a t p A e x i s t s f o r A E A , i s not hard t o prove. A I f i n the following we speak of the norm of a subexpression B

of a c e r t a i n expression A, i t w i l l be c l e a r which A we mean, even

i f we do not s t a t e t h i s e x p l i c i t l y . I n such cases we s h a l l w r i t e

p (B) ins tead of pA(B).

I f A E A , B c A and pA(B) i s def ined, we c a l l B pA-normab~e.

Here, too, we speak of "p-normable B" i f i t i s c l e a r which A (with

B c A E A) we mean. I f QT E A , i f Qr i s p-normable and ~ ( Q T ) 5 Q ' T ,

we c a l l Q p-normable, and w e abbrevia te p (Q) 5 Q ' .

Theorem 2 . 3 . I f A E A , A i s p-normable and B c A, then B i s p-norm-

able .

Proof. Induct ion on I A ~ , with the use of the d e f i n i t i o n of B c A

(Def. 11.2.5). 0

Theorem 2.4. I f QA E A and QA i s p-normable, then Q and A a r e p-

normable and p(QA) (pQ)pA; i f QA E A , and i f Q and A a r e p-norm-

ab le , then QA i s p-normable and p (QA) (pQ)pA.

Proof. Induction on 11 Q 11.

Theorem 2.5. I f A E A , i f A i s p-normable and A 2 B , then B i s p-

normable and pA 1 pB. 01

Proo f . F i r s t assume t h a t A 2' B. We proceed by i n d u c t i o n on t h e

l e n g t h of p roof t h a t A 2' B. I f A 2' B t hen t h e proof i s t r i v i a l ; a

t h i s c a s e i s exp re s sed i n Th. 2.2.

I a . A 2" B i s Q{C~CX,DIE 2' Q(x := C)E. S ince A i s p-normable B

p([x,DlE) [ x , p D l p E and pC pD 5 px ( s e e Th. 2 .3 and Th.

Moreover, pA 5 p ( Q { c } [ x , ~ l E ) 5 (pQ)pE. We now prove f o r t h i s

and E:

Lemma. I f K c E , t h e n ( x := C)K i s p-normable and p ( x : = C ) K 2 pK. a

Proof o f t h e lemma. I n d u c t i o n on I K I . ( l a ) I f K 5 x, t h e n ( x := C)K 5 FrC and p ( x := C)K z pFrC 2 p C 2

a a px I pK.

( I b ) I f K 5 y f x o r K -r, t h en ( x := C)K 5 K and p ( x := C)K E p K .

( 2 ) I f K 5 Cy,FIG, t hen pK - [y,pFlpG. Note t h a t y f x. By induc-

t i o n : ( x := C)F and ( x := C)G a r e p-normable, p ( x := C)F pF and

p ( x := C)G >a pG. So ( x := C)K i s p-normable and p ( x := C)K Z

[ y , p ( x := c ) F I ~ ( x := C)G C y , p ~ l p ~ : P K .

( 3 ) I f K E {FIG, t h e n pG 5 [y,LIH and L 2 pF, and p K : H . a

duc t i on : (x := C)F and ( x := C)G a r e p-normable, p ( x := C)F

and p ( x := C)G 2 pG. I t fo l l ows t h a t p ( x := C)G [Z ,L ' IH ' a

L' 2 L 2 pF p ( x := C)F, s o (x := C)K i s p-normable and a a p ( x := C)K - H ' 2 H r pK.

a

It fo l l ows t h a t B i s p-normable ( s i n c e E c E ) , and

p B I (pQ)p (x := C)E (pQ)pE 5 pA.

I b . A 2 . B i s Q[x,Cl{x)D 2' QD. S i n c e A i s p-normable: QC i s 11

By in -

2 pF a and

- -

p-norm-

a b l e , Px PC, pD - [ ~ , L ] H and L 2 a px r p c , so r ( p ~ ) [ x , p ~ ] ~ 2 a

(pQ)[y,pC]H (pQ)pD 1 pQD I pB.

11. A 2' B i s a d i r e c t consequence o f a monotony r u l e . I t depends

on t h e monotony r u l e which of t h e f o l l o w i n g t h r e e c a s e s a p p l i e s :

IIa. A 1' B i s Q{C)E 2 ' Q{C)F a s a d i r e c t consequence of QE 2 ' QF.

S i n c e A i s p-normable: QE i s p-normable. By i nduc t i on : QF i s p-

normable, and pQE 5 (pQ)pE 2 (pQ)pF E pQF, s o pE 2 pF. Moreover, a a

pE CY,LIH and pC 2 L, so also pF E [z,L1]H', where.L1 2 L and a a H' 2 H. It follows that (C)F is p-normable, and p{C)F E H' 2 H. a a SO p~ r (pQ)pH r pA. a

IIb. A 2' B is Q[x,C]E 2' Q[x,D]E as a direct consequence of

QC 2' QD.

Then pA E (pQ)[x,pC]pE. By induction: QD is p-normable and pQC 5

(pQ)pC ta (pQ)pD E pQD, so pC pD. Hence B is p-normable and

PA 2 (pQ) [x,pDlpE r pB. a

IIc. A 2' B is Q{C)E 1' Q{D)E as a direct consequence of QC 2' QD.

Then pE [x,LIH and L pC. By induction: QD is p-normable and

pQC (pQ)pC ta (pQ)pD pQD, so pC pD. Hence B is p-normable

and pB E (pQ)H r pA.

Finally, if A 2 B is a multiple-step reduction, decompose the

reduction and apply the above. 0

Theorem 2.6. If A E A, Deg A > 1 and A is p-normable, then Typ A is

p-normable and pA r p Typ A. a

Proof. Let A I Pl~x,B]P2x, then Typ A E Pl[x,~IP2Fr~. It is not hard

to show that pFrB pB 5 px. Let PI[x,~]~2 P'P". Next prove by

induction on 11 P"II that P"F~B is p-normable, and p(P1'FrB) 2 p(P1'x). a 0

Theorem 2.7. If A E A, then A is p-normable (i.e. p is a total func-

tion on A).

Proof. Induction on the length of proof of A E A.

( I ) A E T: trivial.

(2) A E Q[x,B]x or Q[X,B]T E A as a direct consequence of QB E A.

Then by induction QB is p-normable, hence Q is p-normable and

pB E px. Hence A is p-normable.

(3) A E Q[x,~]y E A as a direct consequence of QB E A and Qy E A*

By induction: QB and Qy are p-normable, hence Q, B and y are p-

normable, so A is p-normable.

(4) A Q{B)C E A as a direct consequence of QB E A, QC E A, and

the Q-applicability of QC to QB. Then QB and QC are p-normable

(induction), and so are Q, B and C. The Q-applicability implies

that T ~ ~ * QC > Q[x,K]L and Typ QB 2 QK. From Th. 2.5 and Th. * *

2.6: Typ QC and Q[x,K]L are p-normable, pQC ta p Typ QC ta

(pQ)Cx,pKlpL (so pC 2 a [x,pKlpL) and pQB ra p Typ QB pQK (so

pB pK). Hence {B}C is p-normable and so is A. 0

Instead of: p is total on A , we also say: A is p-norm-

able.

From Th. 2.5 we derive:

Theorem 2.8. If A,B E A and A - By then pA 2 pB. a

We shall now prove the normalization theorem for p-normable

expressions.

Definition 2.9. A E A is normal (or in normal form) if there are no

reductions A 2' B or A 2' B; A is normalizable (or A has a normal B rl form) if there exists a normal C such that A 2 C (C is called a

normal form of A). 0

Definition 2.10. A E A is @-normal if there is no reduction A 2' B; B A is 6-normaZizabZe (or A has a 6-normal form) if there exists a 6-

normal C such that A 2 C (C is called a 6-nomaZ form of A). B 0

Hence A is normal if A admits of neither 6-, nor q-reductions;

A is 6-normal is A admits of no $-reductions (except trivial ones).

Theorem 2.11. If A E A is normal and A La B, then B is normal. If

A E A is B-normal and A 2 B or A 2 B, then B is 8-normal. a rl

Proof. The only non-trivial statement is that B is 6-normal if A is

6-normal and A 2 B. It can, however, easily be seen that a single- n step 0-reduction of a 6-normal expression cannot introduce the pos-

sibility of a single-step B-reduction. 0

We r e s t a t e t h e fo l l owing well-known theorem:

Theorem 2.12. I f A E A i s @-normal izab le , then A i s no rma l i zab l e .

P roo f . A s a r e s u l t of Th. 2.11, q - reduc t ions of A do n o t c a n c e l t h e

6-normal c h a r a c t e r . But t h e p o s s i b l e number of s i n g l e - s t e p q-reduc-

t i o n s a p p l i c a b l e t o A i s f i n i t e , s i n c e t h e exp re s s ion becomes

s h o r t e r w i t h each s t e p . 0

Theorem 2.13. I f A E A i s @-normal izable , t h e @-normal form of A i s

un ique b u t f o r a - reduc t ions .

P roof . Le t C and D be 6-normal, A 2 C and A r D. Then, by t h e B R

Church-Rosser theorem f o r @-reduc t ion (Th. 11.6.43): t h e r e i s an E

such t h a t C r E and D 2 6

E. Hence C 2 D. a, @ a

0

Theorem 2.14. Assume t h a t Q{Ak} . . . {Al }B i s i n A and p-normable. k -.

Then > N PA^^. i= 1

Proof . I n d u c t i o n on k. I f k = 0 t h e proof i s t r i v i a l . L e t k > 0.

Then {Al}B i s p-normable, hence p B [x,M]N and pA1 nu M, s o k

l P ~ l l = I M I . Moreover, p{Al}B E N , hence I N [ > I ~ A . ( 1 by induc- i = 2

t i o n . I t fo l l ows t h a t

D e f i n i t i o n 2.15. Assume t h a t A i s i n A and p-normable, A E Q { C ~ } . . n .. {CI}F f o r some n 2 I and F $- {MIN. Then o(A) = I p c i I . If

i= 1 A E Qs (wi th s E x o r s E T ) , t h e n o(A) = 0.

Theorem 2.16. Assume t h a t A i s i n A and p-normable, A E Q{cn} ... {Cl}F, F j! {M}N, and l e t Q C . ( f o r 1 5 i 5 n ) and QF b e @-normal.

1

Then A i s B-normalizable.

Proof. Induction on u(A)*

(1) If a(A) = 0, then n = 0 and A = QF in @-normal form.

(2) Let a(A) > 0. A is @-normal if n = 0. Let n 2 1. We proceed

by induction on I F I . If F a y, then A is in @-normal form (F T

cannot occur since A is p-normable) . So let I F I > 1. Then F z Cx,DIE.

(i) Assume that E E {Hm} . . . {HI }y where y x and m 2 0. Then

A t Q{Cn} ... {C2}{(x := C )H } ... {(x := C )Hl}y. Now note that 1 m 1 Q{C~}[X,D]H. 1 E A, p-normable, and ~(Q{C~}[X,D]H~) = l p ~ ~ l I o(A).

Moreover, 1 ~ ~ 1 < I E ~ , SO by induction Q{Cl}[x,D]Hi is 8-normaliz-

able. Since QCi, QD and QHi are @-normal, the @-normalization of

Q{Cl }Cx,DIHi must commence with Q{Cl}[x,D]Hi ti Q(x := Cl)Hi, SO

Q(x := C )H. is 6-normalizable; say Q(x := Cl)Hi 2 QKi in @-normal 1 1

form. It follows that A t Q{Cn} ... {C2}{Km} ... {Kl}y in f3-normal form.

(ii) Assume that E r {Hm} ... {H,}T where m t 0. Again, analogous- ly to (i), A is @-normalizable. (Moreover, m must be 0 since A is

p-normable. )

(iii) Assume that E {Hm} . . . {H~}x where m 2 0. Then A 2 A' I

Q{c~} . . . {c~I{K~I . . . { K ~ } F ~ c ~ , where we obtain the @-normal QKi as in (i). If now FrCl is a variable or if FrCl begins with an ap-

plicator, we have obtained a @-normal form. If FrCl [y,MIN, then n m -.

o w ) = 1pcil + 1 I P ~ < lp~rcl 1 (by Th. 2. 14 and Th. 2.5) = i=2 j=1

IPcl 1 5 o(A), so by the induction hypothesis: A' is 6-normalizable,

so A is 6-normalizable too, or n = 1 and m = 0, whence A' is @-normal.

(iv) Assume that E S [yyH 1H Then 1 2' A t Q{Cn} . . . {C2}Cy, (x := Cl)Hll(x := C1)H2 2 Q{C n 1 . . . {C2}CyYKlIK2, where we again obtain the B-normal Q[yyKI]KZ as in (i). since

n

Theorem 2.17 (6-normazization theorem). If A E A is p-normable, then

A is 6-normalizable.

Proof. Induction on the length of proof of A E A.

(1) A E T; trivial.

(2) A E Q[x,B]x or A r Q[x,B]T E A as a direct consequence of

QB E A. Then by induction QB is B-normalizable, so QB 2 Q'B' in B 6-normal form, and A 2 Q'[x,B1]x or A s Q'[x,B']T in 6-normal

B 6 form.

(3) A E Q[x,B]y E A as a direct consequence of QB E A and Qy E A.

Then by induction QB 2 Q'B' in 6-normal form, so A 2 Q'[x,B' Iy 6

in 6-normal form.

(4) A r Q{B}C E A as a direct consequence of QB E A and QC E A.

Then by the induction hypothesis: QB r Q'B' in 6-normal form 6

(with IlQll = 11Q'Il ) and QC r Q"C' in @-normal form (with IlQll = B

11Q" 11). From Th. 2.13 and induction on llQll it follows that

Q"C' r Q'C'. Also Q{B)C 2 Q'{B1)C', which is in 6-normal form a B if C' f [x,D]E. So all that is left to prove is that

Q1{B'}[x,~]E is 6-normalizable. But this follows from Th. 2. I6

and Th. 2.5. 0

Theorem 2.18 (normalization theorem for A ) . If A E A then A is 6-

normalizable and normalizable.

Proof. Follows from Th. 2.17, Th. 2.12 and Th. 2.7.

In fact we proved that A E A is effectively normalizable, since

all our proofs are constructive, which implies that the normal form

of A E A is effectively computable.

5 3, STRONG NORMALIZATION

In the previous section we have proved normalization for A.

This guarantees that for every A E A there is a reduction which

leads to a normal form. However, we do not yet know whether an ar-

bitrary sequence of single-step reductions, beginning with A, ter-

minates (in a normal form). We shall prove this in this section.

The property that an arbitrary sequence of single-step reductions,

beginning with some A, terminates, will be called the property of

strong normaZization.

In the proof we shall use B1-reduction and B2-reduction, in- 1 1 troduced in Section 11.6. A feature of BI-reduction is that scars"

of old Bl-reductions are retained. We shall first prove strong nor-

malization for A as to Bl-reduction, and derive strong normaliza-

tion for A as to B-reduction; finally, we shall incorporate q-re-

ductions .

Definition 3.1. A e A is BI-normal (or in B1-normal form) if there - is no B such that A 2' B; A is Bl-normaZizabZe (or A has a Bl-

1 normal form) if there exists a B -normal C such that A r C (C is 1

1 then called a B1-normaZ form of A). The concepts 2 -normal, 2- normal form and B2-normalizab Ze are defined analogously .

Theorem 3.2. If A e A, A is Bl-normal and A 2 B, then B is B - a 1 normal.

Theorem 3.3. If A A is B1-normalizable, the Bl-normal form is

unique but for a-reduction.

Proof. This follows from CR for B1-reductions (Th. 11.6.38).

Theorem 3.4. If A E A, A is-p-normable and A 2 By then B is p-

normable and pA 2 pB. 1 C1

P r o o f . F i r s t assume t h a t A 2' B. We proceed w i t h i n d u c t i o n on t h e

l e n g t h of p roof of A 2' B. 1

1 A A 0

I. L e t A : Q{C}P[x,D]E 2 Q{C}P[x,D] ( x := C)E E B. Then ?[X,D]E $ 1

i s p-normable. I t i s e a s y t o s e e ( i n d u c t i o n on I1 P 1 1 ) t h a t A

p (P[x,D]E) E p ([x,D]E) 2 [x,pD]pE. So p C 2 a p D . A s i n t h e lemma

o c c u r r i n g i n t h e proof of Th. 2.5 WE! can p rove t h a t ( x := C)E i s p-

normable and t h a t p E 2 p ( ( x := C)E). I t f o l l o w s t h a t B i s p-norm- a a b l e and p A 2 pB. a

11. A 2' B i s a d i r e c t consequence of a monotony. 1

I n a l l t h r e e c a s e s t h e proof i s i d e n t i c a l t o t h a t g i v e n i n p a r t I1

of t h e proof of Th. 2.5.

F i n a l l y , i f A 2 B i s a m u l t i p l e - s t e p r e d u c t i o n , decompose @ 1

t h e r e d u c t i o n i n t o s i n g l e - s t e p 8 1 - r e d u c t i o n s and app ly t h e above. 0

We s h a l l now prove t h e $ l - n o r m a l i z a t i o n theorem. We do t h i s i n

q u i t e t h e same manner i n which we proved t h e B-normal iza t ion theo-

rem. I n f a c t , i f we had begun by p r o v i n g t h e 6 1 - n o r m a l i z a t i o n theo-

rem, t h e $ -normal iza t ion theorem would have been a c o r o l l a r y . We

have n o t chosen t h i s o r d e r because i n t h e f o l l o w i n g proof t h e main A

l i n e s a r e obscured by t h e p r e s e n c e of a number of $-chains P . . 1 I n

c o n t r a s t t o t h i s , t h e l i n e of though t i n t h e proof of t h e $-norma-

l i z a t i o n theorem, g iven i n t h e p r e v i o u s s e c t i o n , i s much more l u c i d .

D e f i n i t i o n 3.5. L e t A E A , A Z P ~ ^ P B and l e t b e such t h a t , f o r each

[x,CI f o r which ? P2[x,CIP3, i t h o l d s t h a t x # P3B. Then we c a l l P a n ineffective @-chain , and w r i t e A E P l f ~ . 0

Theorem 3 . 6 . I f A E A i s $,-normal and B c A, t h e n B h a s t h e form '1

A 39 } ... { B } 9 ' s , w i t h s x . o r B 2 [ x [xn9 0 1 R R 1

P r o o f . I n d u c t i o n on I B I .

A A

Theorem 3 . 7 . Le t QPk+ { A ~ } P ~ { A ~ - }Gk- ... { A I } F 1 ~ be long t o A and

k b e p-normable. Then l p ~ l > I ~ A ~ I .

i= 1

P roof . Analogous t o t h e proof of Th. 2.14. Note a g a i n t h a t f o r p- A n

normable PC i t h o l d s : p(PC) - p C . El

D e f i n i t i o n 3.8. L e t A E A and assume t h a t A i s p-normable, L e t

A Q ? ~ + ~ { C ~ I ~ ~ ... {C1}%,F, where F 5 r , F : x o r F [y,M]N wi th

--

y c N . Then o l ( A ) = I p / c i l i f n > 1 , and o l ( A ) = 0 i n c a s e n = 0 i= 1

and ;n+ j! 0. Moreover, o l ( A ) = 0 i f A Qr o r Qx. 0

Theorem 3.9. L e t A be long t o A and be p-normable, l e t

A : Q H ~ + ~ { C J G ~ . .. { c ~ I ~ ~ F , where F E r , F : x o r F : [x,D]E w i t h

x c E. L e t QCi, QF and Q ? ~ T be 6,-normal. Then A i s f31-normalizable.

P roof . I n d u c t i o n on o l ( A ) . The proof i s analogous t o t h a t of Th.

2.16. However, some m o d i f i c a t i o n s a r e r e q u i r e d due t o t h e G ; . We L

s h a l l b r i e f l y comment on t h i s : As t o ( 2 ) ( i ) : E a % ' { H } . H }%ly , mo 1 0

and A 2 QFn+,{C n }? n ... { c ~ } % ~ { c ~ } % ~ [ ~ , D I % ~ { K ~ } ... P ; { K ~ } ? E ~ ,

where t h e K, a r e ob t a ined a s i n t h e proof of Th. 2.16 and where

Q ~ Y T a r e l.

ed s i n c e

L

t h e 6 -normal

e i t h e r (1) i f

F I ) . Note t h a

forms of Q(x := C l ) % i r . These can b e a

x : f! a f o r (2) i f x c P I : 1 i' 1

c1l 5 o(A) and ( P ~ T I < I F I ( app ly t h e

t f { C }? [ g , ~ ] b " i s an i n e f f e c t i v e 6 2 1 1 - m

ob t a in -

induc-

-chain .

As t o (2) ( i i i ) : FrCl can be P O c x , ~ ] ~ , I f x c N, t h en t h e proof

is s i m i l a r t o t h a t of Th. 2.16. I f x { N , we can t a k e [$,MI a s p a r t

of an i n e f f e c t i v e 6-chain {K I?"? [;,MI, and look a t t h e s t r u c t u r e 1 0 0

of N i n s t e a d o f t h a t of F rCl . 11 This amounts t o l ook ing f o r t h e f i r s t e f f e c t i v e " a b s t r a c t o r

i n N, I f t h e r e i s such a n a b s t r a c t o r and t h e o b t a i n e d e x p r e s s i o n i s

n o t y e t i n 6 -normal form, i n d u c t i o n i s a p p l i c a b l e a s i n Th. 2.16. 1 I f n o t , we have a l r e a d y ob t a ined B1-normal form.

0

As to (2)(iv): E POCy,H1IH2- If y H2 (so y K2), the

proof is obvious. If not, look at K2 instead of E, as in the previ-

ous case.

Theorem 3.10 (f3 l-normaliaation theorem). If A E A is p-normable,

then A is B1-normalizable.

Proof. Induction on the length of proof of A E A, analogously to

the proof of Th. 2.17. As to case (4) of this proof: the only case

worth mentioning is C' 5 ;[X,D]E. If x c E, then Th. 3.9 yields the

desired result, and if x # E we have already obtained fi1-norrnal form. 0

Theorem 3.1 1 (6 l-normalization theorem for A ) . If 4. E A, then A is

Proof. Follows from Th. 2.7 and Th. 3.10.

In fact we have proved that A E A is effectively f3 l-normaliz-

able.

Theorem 3.12. Let A E A and A 2' B. Then I A ~ 5 ( B I . f3 1

Proof. Induction on the length of proof of A 2 ' B. f3 1

Definition 3.13. Let A E A. We write B1-nf A for the B1-normal form

of A which we obtain from the effective computation as suggested by

Th. 3.10 and used in Th. 3.11. 0

Note: this B1-normal form is unique (Th. 3.3).

Definition 3.14. We call K E A strongly f3-normaZizabZe if there is 1 an upper bound for the length L of reduction sequences K : K l >@

K >' . . . 2; Ka. Analogously we define the concepts strong B I - , B2- 2 -6 or n-normalizability of K. 0

Theorem 3.15 (strong B2-normaZization theorem for A). If A E A,

then A is strongly B2-normalizable.

Proof. Induction on I A ~ . 0

Definition 3.16. Let A E A, and let A : A 2 ' A ' . . . 1' A be 1 E2 2 -82 B2

the longest possible sequence of single-step B2-reductions beginning

with A, Then Q2(A) = p .

Note that p 1 ~ 1 .

Theorem 3.17. If A E A and A 2' B, then e2(f1-nfA)= B2(B1 -nfB). @ 1

Proof. Follows from Th. 3.3.

Theorem 3.18. If A 2' B, then B2(A) < 02(B). 1

Proof. Induction on the length of proof of A 2' B. The only inter- I

A - 0

esting case is A E Q{C}P[x,D]E 2' Q{C}PCx,Dl(x := C)E E B, where, 1

indeed, we have at least one single-step B2-reduction more on the

right hand side, The rest of the proof is easy.

Corollary 3.19. If A E A, then B2(A) 5 B2(BI - nf A).

Theorem 3.20 (strong f3 I-normaLization theorem for A ) . If A E A,

then A is strongly B1-normalizable.

Proof. Follows from Th. 3.17, Th. 3.18 and Cor. 3.19. 0

Definition 3.21. Let A E A, and let A E A 2' A > ' . . . 2' A be I B, 2 -B1 61

the longest possible sequence of single-step BI-reductions beginning

with A. Then @,(A) = p . 0

Theorem 3.22. Le t A E A , l e t A be i n B1-normal form and l e t A

Then B i s a l s o i n Bl-normal form.

P roo f . I f B were n o t i n B1-normal form, t hen B 2' C f o r some 1

t h a t c a s e t h e r e would be a r e d u c t i o n A 2' B ' 2 C acco rd ing

Th. 11.6.17. C o n t r a d i c t i o n . 1 52

Theorem 3.23. I f A E A , t hen t h e r e i s an upper bound f o r t h e l e n g t h

k of r e d u c t i o n sequences A E A 2' A2 2' ... 2' 1 A k , where each A . r ' A. i s a s i n g l e - s t e p B 1 - o r B2-reduction. 1 l+ 1

P roof . I n d u c t i o n on B 1 ( A ) I f B1(A) = 0 , t h e n A i s i n 6,-normal .. form. I f w e can app ly B2-reductions on A such t h a t A 2" B (n 2 I ) ,

B2 t h e n B i s a l s o i n 6,-normal form (Th. 3 .22) . The number of p o s s i b l e

s i n g l e - s t e p B2-reductions a p p l i c a b l e i s f i n i t e (5 B2(A); c f . Th.

3. 15) .

So l e t @](A) = p > 0 , and assume t h a t t h e theorem h o l d s f o r

a l l K w i t h B1(K) < p. L e t A 2 D be a r e d u c t i o n sequence c o n s i s t i n g

of s i n g l e - s t e p B l - and B2-reductions. I f no 81 - r educ t i ons occur i n

t h e r e d u c t i o n sequence , t h e l e n g t h of t h e r e d u c t i o n sequence can be n a t most B2(A). E l s e , l e t A 2 D be A 2 C 2 D. Then by Th, B2

11.6.17 t h e r e i s a l s o a r e d u c t i o n sequence A 2 ' B ' 2 C r D. 1 B2

Each B" such t h a t A 2' B" has by i n d u c t i o n ( s i n c e 0 1 ( ~ " ) B 1

1

an upper bound f o r t h e l e n g t h of r e d u c t i o n sequences B" r ' . . . 2' E

i n which each s i n g l e - s t e p r e d u c t i o n i s e i t h e r a B 1 - o r a BE-reduc-

t i o n . Le t m b e t h e maximum of t h e s e upper bounds. Then t h e l eng th

of t h e r e d u c t i o n sequence B ' > C 2 D , hence of C r D, cannot be B2

more t han m. I t fo l l ows t h a t t h e l e n g t h of any r e d u c t i o n sequence

A r D can be a t most 02(A) + m + 1 .

Theorem 3.24 ( s t r o n g B-normaZization theorem for A). I f A E A , t h e n

A i s s t r o n g l y 6-normalizable.

Proof. Each B-reduction sequence of A can be decomposed into single-

step B1- and B2-reductions by Th. 11.6.15. So Th. 3.23 yields the desired result.

Definition 3.25. Let A E A, and let A E A 2' A 2' ... 2' A be 1 B 2 B 6 P the longest possible sequence of single-step 6-reductions beginning

with A. Then @(A) = p, 0

Theorem 3.26 (strong t-rnornalization theorem for A), If A E A, then

A is strongly Q-normalizable.

Proof. Induction on I A ~ . 0

Definition 3.27. We call K E A strongly normalizabZe if there is an

upper bound for the length 1 of reduction sequences K E K > ' K > ' 1 - 2 -

1 ... 2 K where each reduction K. 2' Ki+l is a single-step B- or 11- R 1

reduction. 0

Theorem 3.28 (strong normaZization theorem for A). If A E A, then A

is strongly normalizable.

Proof. Induction on @(A). The proof is similar to that of Th. 3.23.

Use Th. 3.26 instead of Th. 3.22, and instead of Th. 11.6.17, use

the theorem: If K E A, K 2 L 2' M, then K 2' ~ ' 2 M. The latter rl B 8

theorem is easy to prove, since each reduction A 2' B 2' C can be r 11 B

replaced either by a reduction A 2' B' 2 C (where r 2 0) or by a B 11 reduction A 2' B 1 2' C (see p. 65; see also Th. 7.25). B B 0

REFERENCES

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logics and A-calculi, Thesis, Utrecht, 1971.

[2] CHURCH, A., A set of postulates for the foundation of logic,

Ann. of Math. (2) 33, p.346-366 (1932) and 34, p. 839-864

(1 933).

[3] CURRY, H.B. and FEYS, R., Combinatory Logic, Vol. I, North

Holland Publishing Company, Amsterdam, 1958.

[4] DE BRUIJN, N.G., The mathematical language AUTOMATH, its usage,

and some of its extensions, Symposium on Automatic Demon-

stration (Versailles December 1968), Lecture Notes in

Mathematics, Vol. 125, p. 29-61, Springer-Verlag, Berlin,

1970.

[5] DE BRUIJN, N.G., Automath, a language for mathematics, THE-

report 68-WSK-05, Technological University, Eindhoven,

1968.

[6] DE BRUIJN, N.G., Automath, a language for mathematics; notes

(prepared by B. Fawcett) of a series of lectures in the

Sgminaire de Mathgmatiques SupSrieures, Universitg de

Montreal, 1971.

C71 DE BRUIJN, N.G., AUT-SL, a single line version of AUTOMATH,

Technological University Eindhoven, Internal Report,

Notitie 22, 1971.

C81 DE BRUIJN, N.G., Lambda calculus notation with nameless dummies,

a tool for automatic formula manipulation, with applica-

tion to the Church-Rosser theorem, Indag, Math., 34, No. 5,

C91 GIRARD, J.Y., Une extension de l'interpretation de ~;del $

l'analyse, et son application 2 l'slimination des coupures

dans l'analyse et la thgorie des types, in: Proc. 2nd

Scandinavian Logic Symp, (editor Fenstad), North-Holland

Publishing Company, Amsterdam, 1970.

[lo] HOWARD, W.A., The formulae-as-types no t ion .o f . - cons . t ruc t ion ,

mimeographed, 1969.

[Ill ~ C H L I , H., An abstract notion of realizability for which

intuitionistic predicate calculus is complete, in:

Intuitionism and proof theory, Proc. Summer.Conference

at Buffalo 1968 (editors Kino et.al), North-Holland

Pubishing Company, Amsterdam, 1970.

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1972.

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Eindhoven, Internal Report, Notitie 17, 1971.

[I41 NEDERPELT, R.P., Lambda-Automath 11, Technological University

Eindhoven, Internal Report, Notitie 25, 1971.

[I51 NEDERPELT, R.P., The closure theorem in A-typed A-calculus,

Technological University Eindhoven, Internal Report,

Notitie 22, 1972.

[I61 NEDERPELT, R.P., Strong normalisation in a A-calculus with A-

expressions as types, Technological University Eindhoven,

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- 100 -

SAMENVATTING

Aan de lambda-calculus ligt een formele generalisatie van de

wiskundige begrippen functie en functietoepassing ten grondslag.

In een getypeerde lambda-calculus worden de expressies uit de cal-

culus van typen voorzien om de interpreteerbaarheid in gangbare

wiskundige termen te vergemakkelijken. De toepasbaarheid van een

functie op een argument wordt in een getypeerde lambda-calculus

vaak op een natuurlijke manier beperkt in overeenstemming met de

typetoekenning (de toepasbaarheidsvoorwaarde).

In dit proefschrift worden twee systemen van getypeerde lambda-

calculus onderzocht, die A en A worden genoemd. Het systeem A is een

deelsysteem van A. In A wordt de toepasbaarheidsvoorwaarde nog niet

gesteld, in A gebeurt dit wel. De typen in de systemen A en A zijn

algemeen in die zin, dat ze dezelfde structuur hebben als de ex-

pressies. Het systeem A is voortgekomen uit de wiskundige taal Auto-

math.

Hoofdstuk I van dit proefschrift geeft een inleiding in de

lambda-calculus en beschrijft de typetoekenning, Voorts wordt het

verband met Automath besproken, Een uitgebreide samenvatting van

het proefschrift besluit dit hoofdstuk.

In Hoofdstuk I1 wordt het systeem A gerntroduceerd en besproken.

De uit de lambda-calculus bekende relaties worden ook in het systeem

A ingevoerd. Daarnaast worden andere relaties gedefinieerd en onder-

zocht als voorbereiding op Hoofds tuk 111.

In Hoofdstuk I11 wordt het systeem A ingevoerd. Er wordt aan-

getoond dat de toepasbaarheidsvoorwaarde voldoende is om voor alle

expressies in A normaliseerbaarheid te bewerkstelligen: we noemen

een expressie normaliseerbaar als men functietoepassingen binnen de

expressie in eendusdanigevolgorde kan uitvoeren, dat er na een

eindig aantal stappen een expressie ontstaat waarin geen functie-

toepassing meer mogelijk is. In het laatste geval zegt men dat de

normaaZvom is bereikt.

In het system A blijkt zelfs sterke normaliseerbaarheid te

gelden: iedere volgorde van functietoepassingen moet in een eindig

aantal stappen tot de normaalvorm leiden. Dit wordt eveneens bewe-

Zen in Hoofdstuk 111, met gebruikmaking van de hierboven genoemde

(gewone) normaliseerbaarheid.

- 102 -

CURRICULUM V I T A E

De schrijver van dit proefschrift werd op 19 juli 1942 geboren

in Den Haag. Van 1953 tot I959 doorliep hij de gymnasiumafdeling

van het Nederlands Lyceum in deze stad. Daarna begon hij een studie

aan de Rijksuniversiteit te Leiden, Na een studie-onderbreking legde

hij hier in 1965 het candidaatsexarnen sterre- en wiskunde af en in

juni 1969, cum laude, het doctoraalexamen wiskunde met bijvak Rus-

sisch. In het studiejaar 1966-1967 was hij als assistent werkzaam

bij de rekenafdeling van het Mathematisch Centrum te Amsterdam,

Sinds zijn afstuderen is hij als wetenschappelijk medewerker werk-

zaam bij de Onderafdeling der Wiskunde van de Technische Hogeschool

te Eindhoven, met een wetenschappelijke taak in het onderzoek van

de wiskundige taal Automath, onder leiding van Prof.dr. N.G. de

Bruijn, en met een onderwijstaak in het basisonderwijs.

Address of the author:

Technological University, Dept. of Mathematics,

Eindhoven, The Netherlands.