Permutabilityofproofsinintuitionistic sequent

calculi

Roy Dyckho

Scho ol of Mathematical Computational Sciences

St Andrews University St Andrews Scotland

y

Lus Pinto

Departamento de Matematica

Universidade do Minho Braga Portugal

Abstract

Weprove a folklore theorem that two derivations in a cutfree se

quent calculus for intuitioni sti c prop ositional logic based on Kleenes G

are interp ermutable using a set of basic p ermutation reduction rules

derived from Kleenes work in i they determine the same natu

ral deduction The basic rules form a conuentandweakly normalisin g

rewriting system We refer to Schwichtenb ergs pro of elsewhere that a

mo dication of this system is strongly normalising

Key words intuitionistic logic pro of theory natural deduction sequent calcu

lus

Intro duction

There is a folklore theorem that twointuitionistic sequent calculus derivations

are really the same i they are interp ermutable using p ermutations as de

scrib ed by Kleene in Our purp ose here is to make precise and provesuch

a p ermutability theorem

Prawitz showed howintuitionistic sequent calculus derivations determine

LJ to NJ here we consider only natural deductions via a mapping from

the cutfree derivations and normal natural deductions resp ectively and in

eect that this mapping is surjectiveby constructing a rightinverse of from

NJ to LJ Zucker showed that in the negative fragment of the calculus

c

LJ ie LJ including cut two derivations have the same image under i

they are interconvertible using a sequence of p ermutativeconversions eg

p ermutations of logical rules with the cut rule In the presentpaperwe provea

Supp orted by the Europ ean Commission via the ESPRIT BRA GENTZEN

y

Supp orted by the Centro de Matematica da Universidade do Minho Braga Portugal

similar result for a cutfree system making precise the idea referred to ab ove In

fact we showhow certain p ermutation reduction rules can b e used to reduce

an arbitrary derivation to normal form and that the set of such reductions is

conuent With minor changes this system is strongly normalising wepointto

Schwichtenb ergs for a pro of of this

Our interest in these problems arises from the theory of logic programming

regarded as in as based on pro of search in a cutfree system if one asks

not just What problems are solvable but What solutions do these problems

have and Howmany times is each solution obtained one is led to analyse

the manyone relationship b etween sequent calculus derivations suitable for

pro of search and natural deductions suitable for presenting solutions In fact

Herb elins sequent calculus describ ed b elow is a much b etter basis for pro of

search than LJ so the original problem disapp ears nevertheless in view of the

historical imp ortance of Gentzens calculus and Kleenes variantofit

G the p ermutability theorem is of indep endentinterest

Mints pap er on the same topic came to our attention in Octob er

when an early version of this pap er was b eing distributed we discuss the re

lationship b etween his work and our own in x We thank Herb elin Mints

Schwichtenberg and Tro elstra for advance copies of their resp ec

tivelyWe are pleased to acknowledge that Herb elins pap ers lled the

gap b etween the usual denition of normal lamb daterms representing natural

deductions and Prawitz denition of our name for the rightinverse of

Background

Herb elins calculus M

Herb elin gives a nonstandard description with origins in

of terms representing normal natural deductions Consider rst a standard

description of normal terms of the untyp ed lamb da calculus

A apA N j vrV

N V N j anA

where V is a set of variables N is the set of normal terms and A is the set of

application terms We use explicit constructors an and vr to ensure consistency

with our typ echecked implementations The head variable of such a term is

for a large term buried deep inside Herb elins representation brings it to the

wemake the following surface So following Herb elin who calls the calculus

Denition The set M of untypeddeduction terms and the set Ms of lists of

such terms are dened simultaneously as fol lows

M V Ms j V M

Ms j M Ms

Note the use again of the same symbol The notation M M abbreviates

n

the term M M The suggestion that such terms are lists is adequate

n

while we deal with implication alone but not when we add the other connectives

Terms are equal i they are alphaconvertible we use the symbol for this

relation

Adding typ e restrictions gives us a description of the typable deduction

terms We call the asso ciated typ ed system MJasitisintermediate b etween

LJ and NJ rather than use Herb elins name LJT already used in

There is a bijective translation b etween M and N mentioned but not de

tailed in xM M translates into the term apapx N N

n n

usually written as xN N where N is the translation of M and abstraction

n i i

y The bijection extends to the typable terms terms translate in the obvious wa

elsewhere wehave called such sequent calculus permutationfree meaning

that there are no p ermutations ie that the map from MJ to NJ is

Further details of this calculus covering all the connectives and several

pro ofs of admissibility of cut can b e found in We shall im

plicitly use the bijectiveness of the corresp ondences with N and NJ and not

trouble to give pro ofs that eg a result shown for M translates correctly to a

result claimed without pro of for N

The calculus LI

LI is a cutfree sequent calculus for intuitionistic implicational logic First

formulae A are built up from proposition variables p q using just for

implication Second contexts are nite sets of variable formula pairs as

so ciating at most one formula to eachtermvariable in V Third there are

terms dened as in

Denition The set L of terms in cutfree LI derivations is dened as fol lows

L varV j appV L VL j V L

The notions of free and bound variable and of alphaconversion are as usual

there are two binding mechanisms those at the o ccurrences of VL in the ab ove

denition Two terms are said to b e equal i they are alphaconvertible again

we shall use for this relation Note again the overloaded use of We write

x L for x is not free in L similarly x L for x is free in L Fourth there

are judgments L A Fifth there are typing rules inductively dening the

derivations of the calculus

Axiom

x A varxA

L A y B L C

y A L B

L

R

y L A B appx L yL C

with the provisos x A B b elongs to in L and y is new iedoesnot

app ear in the context in b oth L and R

From the term and context parts of the endsequent of a derivation one

can recover the entire derivation the terms mo dulo alpha conversion are

really just a convenient notation for derivations The rules ab out new variables

imply for example that b ound variables are chosen so that the variable y in

appx L yL diers from the variable x and do es not o ccur freely in L

Wemake no distinction b etween the judgment L A and the assertion of its

derivability

Weakening is an admissible rule of LIany derivation can b e transformed to

aweaker derivation by adding an assumption x A to eachantecedent for new x

The two derivations will b e represented by the same term also if a derivation

do es not use an assumption x A then it can b e strengthened byremoving x A

b oth from the endsequents antecedent and inductively with descendants from

the premisses In the following we use b oth the strengthening and the weakening

techniques without comment

The corresp ondence from L to M

Prawitz description see also x of the function from sequent

calculus derivations to natural deductions uses the ordinary notion of subs

titution recursively dened on the structure of the term b eing substituted into

Using Herb elins denition of terms we need a dierentversion of the subs

titution function This should b e based on his cut rules as in x for ease of

exp osition wenow just intro duce it in an ad hoc wayWedoitjustinthe

untyp ed case typing is not necessary for the functions to b e welldened

Denition The functions of substitution of a variable x and a term M for

a variable y in a term resp terms aredened as fol lows

subst V M V M M

substx M y y Ms xM substsx M y M s

def

z substsx M y M s if z y substx M y z Ms

def

substx M y z M z substx M y M

def

substs V M V Ms Ms

substsx M y

def

substsx M y M Ms substx M y M substsx M y M s

def

Care is taken as usual to avoid variable capture ie in line of the denition

for subst z x z y and z M

L M is dened as fol lows Denition The function

varx x

def

appx L yL substx L yL

def

xL xL

def

Our denition is for untyp ed terms we can easily extend it to typ ed terms and

consider it as a map from cutfree sequent calculus derivations to normal natural

deductions in Herb elins notation

WesaythatL determines the term L and similarly for the derivation

represented by L and the deduction represented by LWe reserve the name

as in for the corresp onding function intro duced but not named in

p REMARK from L to N dened by

varx vrx

def

appx L yL apx L y L

def

xL xL

def

Note that is just the comp osite of with the bijection from M to N Details

are in

Denition Anequation L L is trivial i L L similarly

for trivial and similarly for permutations and transformations

The corresp ondence from M to L

Denition The function M L is denedbyrecursion on the size of

terms of M as fol lows

x varx

def

xM Ms appx M z z Ms z new

def

xM xM

def

P

n

where siz exM M siz eM and siz exM siz eM

n i

i

Lemma M M for any M

The denition is based on the construction in which in fact describ ed a

See x of for a detailed account rightinverse to rather than to

Our denition is for untyp ed terms we can easily extend it to typ ed terms and

consider it as a map from normal natural deductions in Herb elins notation to

cutfree sequent calculus derivations

Example

Consider the usual natural deduction essentially the S combinator of the se

quent A B C A B A C in intuitionistic logic where the two

o ccurrences of A form an assumption class

A B C A A B A

B C B

C

This deduction is represented in the context hz A B C y A B x Ai

by the term apapvrz anvrxanapvry anvrx of N and bythe

term z x y x of M

Many dierent cutfree sequent calculus derivations determine this deduc

tion for example those represented in the same context by the terms

S appz x w appw appy x v v uu

def

S appz x w appw appy x v v uappy x v u

def

S appz x w appy x v appw v uu

 def

S appz appy x vx w appy x v appw v uu

 def

S appy x v appz x w appw v uu

 def

Commonly these derivations are regarded as the same b ecause they are p er

mutation variants of each other The terms are related in the following ways

using the p ermutation reduction rules describ ed in detail b elow

S S S S S

 


i ii i ii

There are in fact innitely many cutfree derivations with the same image S

by use of the p ermutation rule in reverse

i

The purp ose of this pap er is to makesuch observations b oth precise and

general Kleene discussed such p ermutations in the context of LK and LJ

without discussing the relationship with natural deductions gives a more

detailed presentation of the theory of p ermutations

Normality

In this section wegiveanintrinsic denition of the notion of normalityfor

derivations which will turn out to b e equivalent b oth to irreducibility wrt our

p ermutation reduction rules and to b eing canonical as elements of the bres

of the mapping

Denition Let L beatermofL L is normal i in any subterm of the form

appx L yL L is either vary or of the form appy L zL with y L

  

and y L



Example The term S appz x w appw appy x vvuu of xisnor

def

mal the other terms in that section are not

A normal term of the form appx L x appx L x appx L x v ar x

    

is interpreted in N as x N N N where N interprets L similarly for longer

 i i

terms

M is normal Lemma Normality Lemma For each term M of M

Pro of By induction on the size of M

Case M is x then M isjustvarx which is normal

M isappx M zz Ms new z by Case M is xM Ms then

induction M and z Ms are normal In fact z Ms is either varz

or of the form appz L wL withz since it was new not free in L or

  

L Any application subterm of M must b e M itself or a subterm



either of M orof z Ms in the rst case wehave shown it has the

desired form in the second case we use the normalityof M in the

third case we use the normalityof z Ms

Case M is xM then M isxM by induction M is normal and

obviously the abstraction of a normal term is normal

We will show the converse that all normal terms L are of the form M First

weidentify a set of p ermutation reduction rules for reducing terms L to normal

form

Permutation reductions

Permutation reducibility is a relation b etween terms of L formalised bymeans

of the new judgment form L L readasL and L are terms of L and the

rst reduces to the second by a single p ermutation reduction This relation is

inductively generated by

L L

xL xL

L L L L

appx L yL appx L yL appx L y L appx L y L

and the following p ermutation reduction rules

i appx L yL L if y L

ii appx L y appz L wL



appz appx L yL w appx L yL if y z



ii appx L y appy L wL



appx L y appy appx L yL w appx L yL



iii appx L y z L z appx L yL

with the constraintiniithaty is new and the constraints that in ii and

ii y is free in L or in L since otherwise appz L wL in the LHS of ii




matches L in the LHS of i or resp ectively the RHS of ii reduces by i

back to the LHS

Note iandiimaybecombined when y z and y L but y L to



yield the elegant p ermutation

v appx L y appz L wL appz L w appx L yL




y ii to appz appx L yL w appx L yL which The LHS reduces b



reduces by i to the RHS Note that scop e rules for the LHS imply that w x

and w L so if w L v can b e used again and again



Note We could also use the rule

iv appx L yL appx L y appx L zjzyjL

where z is new and jzyjL indicates L in which zero or more o ccurrences of y

are replaced by z Using iv ii i and ii we obtain ii

Although iv seems more primitive our main theorem is most naturally proved

using ii and establishes by induction that instances of iv are obtainable

using i ii ii and iii

From nowonwe use the symbol for the p ermutation reducibility relation

and for its transp ose and denote as usual the reexive transitive

closures of the relations and denotes the reexive symmetric transitive

closure of Wesay that L and L are interpermutable when L L We

say that L reduces toL or that L is reducible to L iL L

Rule i simplies the derivation byremoving an unnecessary step ii p er

mutes instances of L past each other as in ii roughly achieves the

eect of ii when one principal formula originates in the other iii p ermutes

L past Rasin Rules i and ii are not p ermutations in Kleenes

sense b ecause the principal formula of the top rule o ccurs as an activeformula

of the lower rule Kleene however allowed structural rules of whichwehave

none Rules i and iv from which ii can b e derived corresp ond to his

mo dication of derivations with structural rules

Prop osition Each of these permutation reduction rules is and trivial

Pro of Routine consider for example ii with y new

appx L yappy L wL



apx L y apy L w L



apapx L apx L y L w apx L y L



apx L y apy apx L y L w apx L y L



appx L y appy appx L yL w appx L yL



We shall see in x examples of p ermutation rules from thatinvolve disjunc

tion and are not trivial

Irreducibil i ty

Here weshow that normal terms are irreducible later weshow the converse

Denition L is irreducible i no reduction is applicable to L

y Lemma Each normal term L is irreducible Lemma Irreducibilit

Pro of Since subterms of normal terms are normal we need only check for

each rule normal instances L of the LHS We consider the cases in turn

Rule i L is of the form appx L yL fory L By normality L is

either vary orappy L zL contrary to y L

 

Rule ii L is of the form appz L y appz L wL for y z By normal



ity y z a contradiction

Rule ii L is of the form appz L y appy L wL with y free in L or



L By normality y is not free in L or L a contradiction




Rule iii L is of the form appz L y z L By normality z L must b e

vary or an application which are imp ossible

Normalisability

The argument here is based on Herb elins calculus to make the induction easier

One might also use the description xxN N N of normal terms but

n

this description is not so convenientinamechanical verication and it is not

easy to handle connectives such as disjunction

Lemma Permutability Lemma Let M and M be terms of MThen

M yM substx M yM appx

Pro of By induction on the size of M When y is not free in M the LHS

reduces bypermutation i to M towhich the RHS is identical by simplica

tion so wemay assume that y M

Case siz eM so M is z for some variable z whichby our as

sumption must b e y So the LHS is appx M yy

appx M yvary by denition of

xM by denition of

substx M yy by denition of subst

which is the RHS So in this case the LHS and the RHS are identical

Case siz eM we supp ose the lemma is true for all M of lesser size

Then M is either of the form z M Msoroftheformz M andin

the former case two sub cases arise according to whether z y or z y

Case ii M z M Ms when z y by assumption y is free in M Ms

So the LHS is appx M y z M Ms

appx M y appz M z z Ms

where z is new by denition of

M yM z appx M yz Ms appz appx

by p ermutation reduction rule ii

appz substx M yMz appx M y z Ms

by induction since siz eM sizez M Ms

appz substx M yMz substx M yz Ms

by induction since siz ez Ms sizez M Ms

substx M yMz z substsx M yMs appz

by denition of using z y

z substx M yMsubstsx M yMs

by denition of subst since z is new

substx M yz M Ms

by denition of subst since z y

which is the RHS

Case ii M y M Ms Two sub cases arise y free in M Ms and

otherwise The rst sub case is routine similar to ii but using rule

ii In the second sub case where y is not free in M Msby direct

computation

M yy M Ms substx M yy M Ms appx

Case iii M z M routine using rule iii

Theorem For every term L of L L L

Pro of By induction on the structure of L First supp ose L isavariable x

then trivially the LHS and RHS are identical using the denitions of and

Second the case when L xL is a routine use of the induction hyp othesis

Third if L appx L yL then L is by induction twice reducible to

appx L y L and by the p ermutability lemma this reduces to

substx L yL ie to appx L yL whichisjust L

Corollary For every term L of L L L and for every pair L L

L L i L L and ii L L i L L of terms of Li

Pro of i L L implies that L L L L the

converse follows by Prop osition

Theorem Let L be a term of L The fol lowing areequivalent

L is normal

L is irreducible

L L

L is of the form M for some M

Pro of follows by the irreducibility lemma is from theo

rem is trivial follows by the normality lemma

Thus theorem is a weak normalisability result every term L can b e reduced

to a normal form and the normal forms are the irreducible terms

Conuence and Strong Normalisation

Theorem The rewriting system i ii ii iii is conuent on L

Pro of Supp ose L L and L L ThenL L L since

the reductions are trivial So all of L L and L reduce to the same normal

form L

Without further restrictions the system of rules is nonterminating eg rule v

can b e used rep eatedly and v dep ends on unrestricted ii and i Note that

ii can b e used rep eatedly on its own b ecause eg assuming y z w L



and y L



appx L y appz L wL



yL appz appx L yL w appx L



appx appz appx L yL wL y appz appx L yjyyjL wL



where the second reduction is allowed b ecause x y implicitly b ecause of

the scoping rules To restrict this while at the same time allowing enough

reductions for the pro of of the p ermutability lemma to work is tricky

The instances of the p ermutation reduction rules used in the pro of have their

L arguments of the form M whichwesaw in Theorem to b e exactly the nor

mal terms Thus the pro of of the lemma incorp orates an innermost reduction

strategy this suggests one should conjecture that the system is strongly nor

malising if one makes restrictions such as normality of the arguments of terms

b eing reduced Let x be a variable wesay that a term L is xnormal i L

is either varxorisappx L yL withx L and x L and L being

y normal Clearly terms of the form z Msare z normal for z Ms

Conjecture The rewriting system i ii ii iii is SN if

a rules ii ii arerestrictedtocases wheretheargument L of the LHS



appx L y appz L wL is w normal and



b rules ii ii arerestrictedtocases wherethearguments L L and L



of the LHS appx L y appz L wL are normal



Note that with these restrictions the pro of of the p ermutability lemma still

works

Schwichtenb erg outlines a pro of of this conjecture strengthened by

omission of condition b as follows He develops a new notation binary se

quent terms in which M fy Lg corresp onds to our appy L vM hinting at

v

the translation to natural deduction terms M y L whichwewould write as

v

apy Lv M More generally there are multiary sequent terms suchas

M fy L L g corresp onding to our appy L w appw L vM where w L

v

and w M and similarly for vectors L of terms in place of L L Our rule ii

restricted by condition a and with for ease of exp osition a very restricted

form appw N w w of the argument L is translated to the reduction



w fw N g fz L g fx L gw fw N g fx L g fz L fx L gg

w w y

w y

y

w

in which N L and L mayinfactbevectors and thus w fw N g represents

w

the general form of the L argument allowed by the strengthened form of the



conjecture The other rules are represented similarly eg ii by For

example our reduction by ii of S to S from x is simulated by the reduction

 

u fw vg fz xg fy xgu fw vg fy xg fz x fy xgg

u u v

w v

v w

Termination of the rule set f g and of some similar rule sets

is shown in using a decreasing measure on terms The termination of our

rule set fi ii ii iiig with the restrictions mentioned ab ove therefore

follows thus establishing the strengthened version of our conjecture It would

b e of interest to have a full and direct pro of of this without using the multiary

notation on which the measure function dep ends of

Extension to other logical constants

This section considers the extension of the theory to cover the other intuitionistic

logical constants We refer to the full pap er for details The main p ointof

interest is that some of the Kleenestyle p ermutations are not trivial

Kleenes analysis was for a system with primitive structural rules Wecan

consider the following table in whichtheintersection of the row R and the

column C refers to the p ermutable pair RC in which R lies ab ove C and may

b e p ermuted to b elowit

L R L R L R

L X X

R XX XX X XX

L

R XX XX X XX

L N N N N

R XX XX X XX

In this table indicates that there is a single p ermutation reduction rule

indicates that there is a pair of reduction rules indicates that there is a

p ermutable pair but it is not used in the pro of of the p ermutability theorem

b ecause it is the reverse of a p ermutable pair that is used X indicates that

the p ermutation is forbidden XX that there is no p ermutable pair b ecause

b oth R and C are right rules and N indicates that the p ermutation is not

trivial essentially b ecause the notion of normality used in NJ do es not allow

intro duction rules to b e p ermuted up into minor premisses of elimination rules

Eachpermutation that is marked N in the table eg

LR xw heny z L z L w heny z xL z xL x y

using the notation of is not trivial if we apply to the two sides of

LR then we get normal terms representing I andE steps resp ectively

Related work

Theorem of xof for the negative fragmentofintuitionistic logic is similar

to ii of our corollary but for the systems with cut Zuckers argument

showing that twoderivations with the same image under are interp ermutable

is a case analysis on the last steps of the two derivations for example the

case of b oth last steps b eing L is dealt with by use of derivations with cut

Thus his notion of interp ermutable uses p ermutations involving the cut rule

Moreover there is no reference in to Kleenes theory of p ermutations See

for further discussion but still for the systems with cut of Zuckers results

Mints available to us after our own pro of of an early version of theorem

using rather than proves the same theorem but without clarifying

whether or not the p ermutations are directed and which p ermutations are re

quired by means of an induction on the structure of derivations in the general

case not just prop ositional logic our use of the term notation for derivations

allows in contrast the nature of the p ermutations to b e made precise and

amenable to mechanical treatment His work applies to Gentzens system

LJ with explicit weakening and contraction rules rather than as in our case

to Kleenes G where these rules are built into the logical rules Our iv

corresp onds to his use of transformations to movecontraction similarlyour

i corresp onds to his transformations to moveweakening down to wards the

ro ot He also describ es the normal forms using constraints on the structure of

derivations similar to ours

Tro elstra has proved a similar weak normalisation theorem for a Gentzen

calculus based on Gi with the normal derivations b eing in corresp on

dence with natural deductions in long normal form under the complete discharge

convention This calculus lacks the term lab els that wehave used b oth to fa

cilitate the naming of derivations and their p ermutations and b ecause of the

connections with logic programming viewed as a search for normal terms inhab

iting formulae viewed as typ es alsomentions some diculties in Mints

treatmentofcontraction

Bellin and van de Wiele prove a similar result for a multiplicativelinear

logic without prop ositional constants relating sequent calculus derivations to

pro of nets Andreolis work on fo cusing pro ofs in linear logic seems to b e

related in its stringent normality conditions on pro ofs but there is no p er

mutability theorem yet Pym and Wallen prove a theorem showing

howany derivation mayb e illtyp ed of the calculus can b e p ermuted to

obtain a welltyp ed derivation

Schwichtenb erg develops a new notation multiary sequent terms rep

resenting derivations of LJ a notion of multiary normal form permutative

conversions and a measure function with resp ect to which the conversion rules

are decreasing Our x discusses the use of this theory to prove a result ab out

strong termination for our rules

Conclusion

Wehave made precise for intuitionistic prop ositional logic the idea that two

pro ofs are really the same i they are interp ermutable moreover wehavepre

sented a rewriting system conuent and weakly normalising for reduction of

terms representing cutfree sequent calculus derivations to normal form That

this can b e made SN by appropriate restrictions for the implicational fragment

follows from Schwichtenb ergs results in For all the prop ositional connec

tives wehave identied precisely which of the Kleenestyle p ermutations are

required and p ointed out some that are inappropriate Our metho ds illus

trate the utility of Herb elins representation of lamb daterms whichbringsthe

head variable to the outside We are condent that the metho ds generalise to

rstorder logic see and its successors for details in due course

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and J Seldin eds Curry Festschrift Academic Press

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D Miller G Nadathur F Pfenning and A Scedrov Uniform pro ofs as a

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G Mints Normal forms of sequent derivations in P Odifreddi ed

Kreiseliana A K Peters W ellesley Massachusetts also

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G Pottinger Normalization as a homomorphic image of cutelimination

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D Prawitz Natural deduction Almquist Wiksell Sto ckholm

D Pym and L Wallen Pro of searchinthecalculus in G Huet and

G Plotkin eds Logical frameworks Cambridge University Press

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H Schwichtenb erg Termination of p ermutativeconversions in intuitionistic

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A S Tro elstra Marginalia on sequent calculi Studia Logica toappear

also University of Amsterdam Inst for Logic Language and Computation

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er The corresp ondence b etween cutelimination and normalization J Zuck

Annals of Mathematical Logic