Strong normalization of lambda-bar-mu-mu-tilde-calculus with explicit substitutions

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Strong normalization of

lambda-bar-mu-mu-tilde-calculus with explicit

substitutions

Emmanuel Polonovski

To cite this version:

Emmanuel Polonovski. Strong normalization of lambda-bar-mu-mu-tilde-calculus with explicit

sub-stitutions. FOSSACS, 2004, Barcelona, Spain. pp.423-437. �hal-00004321�

with Expli it Substitutions

EmmanuelPolonovski

PPS,CNRS-UniversiteParis7 Emmanuel.Polonovskipps.jussieu.fr

Abstra t. The~ - al ulus, dened by Curienand Herbelin [7℄, isa variantofthe- al ulusthatexhibitssymmetriessu hasterm/ ontext and all-by-name/ all-by-value. Sin e it is a symmetri , and hen e a non-deterministi al ulus,usualproofte hniquesofnormalizationneeds someadjustmentstobemadetoworkinthissetting.Hereweprovethe strong normalization (SN) of simply typed- al ulus~ with expli it substitutions.Forthatpurpose,werstproveSNofsimplytyped-~ al ulus(byavariant oftheredu ibilityte hniquefromBarbaneraand Berardi[2℄),thenweformalizeaproofte hniqueofSNviaPSN (preser-vationofstrongnormalization),andweprovePSNbythe perpetuality te hnique,asformalizedbyBonelli[5℄.

1 Introdu tion

1.1 ~- al ulus and Expli itSubstitutions

The ~- al ulus, dened by Curien and Herbelin [7℄, is asymmetri variant ofParigot's- al ulus[11℄ thatprovidesatermnotationfor lassi alsequent al ulus. Itexhibits symmetriessu h asterms/ ontextsand all-by-name/ all-by-value. Its two main redu tion rules form a symmetri riti al pair, whi h makes the al ulus non-deterministi (non- on uent) and raises diÆ ulties in normalizationproofs:anaivedenitionofredu ibility andidateswouldfallin asymmetri loopofmutualindu tion.

On the other hand, al uli with expli it substitutions were introdu ed [1℄ as a bridge between - al ulus [6℄ and on rete implementations of fun tion-nal programminglanguages.Those al uliintend to rene the evaluation pro- ess byproposingredu tion rulesto dealwith thesubstitution me hanism{a meta-operationin the traditionnal - al ulus.In thestudy ofthose al uli,an importanttaskwastoestablishgoodpropertiessu has:

 Simulation of  redu tion,whi h says that aterm that an be redu ed to anotherin thetraditionnal - al ulus analso beredu ed tothe sameone inthe al uluswithexpli itsubstitutions.

-stronglynormalizing(i.e. annotbeinnitelyredu ed),itisalsostrongly normalizingwithrespe ttothe al uluswithexpli itsubstitutions.

 Strong normalization (SN), whi h says that, with respe t to a typing sys-tem,everytyped termis stronglynormalizingin the al uluswith expli it substitutions.

Itwasremarked,at on e, that expli itsubstitutions raisesmorediÆ ulties in normalization proofs, due to the fa t that redu tions an now take pla e in an argument substituted in a term to a variable whi h is not free in that term. Su h redu tions produ e no tra e in the original al ulus, be ause the substitution is bounded to disappear. Therefore we annot easily inferSN for expli itsubstitutionsfromstrongnormalizationoftheoriginal al ulus.

1.2 The ~- al ulus with Expli itSubstitutions:x~

Here we work on  ~x, an expli it substitutions version \a la" x [4℄ of the ~ - al ulus. Its syntax was introdu ed in [9℄ and, in the same paper, there wasanattempttoprovestrongnormalizationofthedeterministi all-by-name fragmentdire tlybytheredu ibilityte hnique.Unfortunately,thete hniquedid notworksoni ely,andtheproofofakeylemma(Weakeninglemma)turnedout tobebugged...Wekeepthiste hniqueforthepure al ulus(i.e.withoutexpli it substitutions),and,inordertoliftittothesymmetri al ulus,weadjustitlike Barbanera andBerardididfor theirsymmetri - al ulus [2℄.Wewill see that redu ibilitysets onstru tedbyxedpointensurethat theirdenition willnot fall inthesymmetri inniteloopoftermsdened by ontextsandvi e versa.

ToproveSN,weformalizeate hniqueinitiallysuggestedbyHerbelin,whi h onsists inexpanding substitutionsinto pure~-redexesand toinherit SN of thewhole al ulusbySNofthepure al ulusandbyPSN.

Finally, to provePSN, we usethe perpetualityte hnique, asformalizedby Bonelli[5℄. Themain point ofthis te hniqueis to exhibita strategywi h pre-servesinniteredu tions.Thistogetherwithsomematerialto tra ethe substi-tutionsba kwards,allowsusto establishPSNby ontradi tion.

Inthesequel,wewillnoteSN R

forthesetofstronglynormalizingtermsin the al ulusR .WewilluseFV(t)todenotethesetoffreevariablesoft,dened in theusualway.

1.3 Organization

We rst present the (simply typed) ~- al ulus and we prove SN bythe re-du ibilityte hnique(se tion2).Inse tion3,weusetheperpetualityte hnique to establishPSN.Se tion4formalizestheproofte hniqueofSN via PSN,and givesthematerial to useitfor  ~x. Finally,we givetheproofof SN of~x

Werstre allthedenitionofthe ~- al ulus,thenwedeneredu ibilitysets andnallyweestablishstrongnormalizationofthepure al ulus.

2.1 Denition

Therearethreesynta ti ategories:terms, ontextsand ommands,respe tively noted v, e and . We take twovariable sets: Var is the set of term variables, notedx,y,z et .;Var

?

isthesetof ontextvariables,noted,,et .Wewill notet anobje t,i.e.oneofv,eor .Thesyntaxofthe~- al ulusis:

::=hvjei

v::=x jx:v je
vj : e::=j:ejv
ejx: e

Redu tionrulesaregivenbelow.Therules()and(e )form a riti alpair:

() hx:vjv 0
ei!hv 0 jex:hvjeii ( e ) he 0
vj:ei!h:hvjeije 0 i () h: jei! [e=℄ (e ) hvje x: i! [v=x℄ (sv) :hvji!v if62FV(v) (se) x:hxjeie !e ifx62FV(e)

Typesareusual simpletypesplus theminus typeA B whi his the sym-metri ounterpartof thearrow typeA!B, itsmeaningis Aand notB.We workherein lassi alsequent al ulus,withanotationtoexhibitaformulaina sequent: `Aj
isthesamesequentas `A;
buttheformulaAisexhibited asa tiveformula.Forfurtherdetailsaboutthisframeworkandtheisomorphism withobje tsofthe~- al ulus,see [7℄.

Threesequentformsareusedtotypethesynta ti ategories:the ommands are typed by ( `
), the termsby ` Aj
and the ontexts by jA `
. Herearethetypingrules:

:( ;x:A`
) jex: :A`
`v:Aj
je:A`
hvjei:( `
) :( `:A;
) `: :Aj
j:A`
;:A ;x:A`
jx:A je:B`:A;
j:e:A B`
;x:A`v:Bj
`x:v:A!Bj
`v:Aj
je:B`
`v:Bj
je:A`

Wesimultaneouslydene, byindu tion ofthetypestru ture: { theoperators: Lambda (X 1 ;X 2 )= Def fx:v j 8v 0 2X 1 ;e2X 2 hv[v 0 =x℄jei2[[`℄℄ g Cons (X 1 ;X 2 ) = Def fv
e j v2X 1 ande2X 2 g ^ Lambda (X 1 ;X 2 )= Def f:e j 8e 0 2X 1 ;v2X 2 hvje[e 0 =℄i2[[`℄℄g ℄ Cons (X 1 ;X 2 ) = Def fe
v j e2X 1 andv2X 2 g Mu(X) = Def f: j 8e2X [e=℄2[[`℄℄ g f Mu(X) = Def fex: j 8v2X [v=x℄2[[`℄℄g Remark1. Muand f

Muarede reasingoperators:thegreaterX is,thelesser one annd: 's (resp.x: 's)e that normalizeagainstalleinX.

Then  ifAisatomi Neg [[`A℄℄ (Y)=Var[Mu(Y) Neg [[A`℄℄ (X)=Var ? [ f Mu(X)  ifA=A 1 !A 2 Neg [[`A℄℄ (Y)=Var[Mu(Y)[Lambda ([[`A 1 ℄℄ ;[[A 2 `℄℄) Neg [[A`℄℄ (X)=Var ? [ f Mu(X)[Cons([[`A 1 ℄℄;[[A 2 `℄℄)  ifA=A 1 A 2 Neg [[`A℄℄ (Y)=Var[Mu(Y)[ ℄ Cons([[A 1 `℄℄;[[`A 2 ℄℄) Neg [[A`℄℄ (X)=Var ? [ f Mu(X)[ ^ Lambda ([[A 1 `℄℄;[[`A 2 ℄℄) Sin eMuand f

Muarede reasingoperators,Negisalsoade reasingoperator. SoNeg

[[`A℄℄ ÆNeg

[[A`℄℄

is anin reasing operator,and byTarski'stheorem it hasaxedpointX

0 ; { theredu ibilitysets:

[[`℄℄=SN ~ and [[`A℄℄=X 0 and [[A`℄℄=Neg (X 0 ):

(i) Var[[`A℄℄ (ii) Var

?

[[A`℄℄

(iii)v2[[`A℄℄ () eitherv=x orv=e
v 0 with A=A 1 A 2 ; e2[[A 1 `℄℄ andv 0 2[[`A 2 ℄℄ orv=: and 8e2[[A`℄℄ [e=℄2[[`℄℄ orv=x:v 0 withA=A 1 !A 2 and 8v 00 2[[`A 1 ℄℄ ;e2[[A 2 `℄℄ hv 0 [v 00 =x℄jei2[[`℄℄ (iv) e2[[A`℄℄ () eithere= ore=v
e 0 with A=A 1 !A 2 ; v2[[`A 1 ℄℄ ande 0 2[[A 2 `℄℄ ore=x: e and 8v2[[`A℄℄ [v=x℄2[[`℄℄ ore=:e 0 with A=A 1 A 2 and 8e 00 2[[A 1 `℄℄;v2[[`A 2 ℄℄ hvje 0 [e 00 =℄i2[[`℄℄

Proof. From the denition of the redu ibility sets, we have[[`℄℄ =SN  ~

and thepoints(i)and(ii).Weprovethepoints(iii)and(iv).Duetothesymmetry, itsuÆ estoprove(iii).

v2[[`A℄℄ () v2Neg [[`A℄℄

ÆNeg [[A`℄℄

([[`A℄℄ ):

We then onsider the dierent shapes of A and we inline the orresponding denition ofNeg [[`A℄℄ ÆNeg [[A`℄℄ ([[`A℄℄). 2.3 Strong Normalization

Herearethetwotraditionnallemmasofstrongnormalizationoftheredu ibility sets(RS)and losurebyredu tion.

Lemma1 (SN of RS). Let A be a type. Then [[`A℄℄ SN ~ (1), [[A`℄℄ SN ~ (2)and[[`℄℄SN ~ (3).

Proof. Byindu tion onthestru tureofA.

1. We onsider thedierentformsofv2[[`A℄℄: { v=x:thenv2SN ~ . { v = e
v 0 : then A =A 1 A 2

and we on lude by using the indu tion hypothesis twi e.

{ v=: :bythepoint(ii)ofproposition1,2[[A`℄℄,then,bythepoint (iii)ofproposition1, [=℄2[[`℄℄,that givesus 2[[`℄℄(=SN

~ ).We thenhave: 2SN .

{ v=x:v,thenA=A 1 !A 2 :togetv2SN ~ ,weneedv 2SN ~ . Byredu ibilityofx:v 0 ,wehave8v 00 2[[`A 1 ℄℄;e2[[A 2 `℄℄ hv 0 [v 00 =x℄jei2 [[`℄℄ (=SN  ~

).Bythepoints(i)and(ii) ofproposition1,we antake xforv

00

andfore,andthatgivesushv 0 [x=x℄ji2SN  ~ .Wededu e v 0 2SN  ~ and on lude.

2. Theproofforeissimilarto theproofforv bysymmetry. 3. Bydenition[[`℄℄=SN

 ~ .

Lemma2 (Closure byredu tion). 1. v2[[`A℄℄ ; v!v 0 =) v 0 2[[`A℄℄ . 2. e2[[A`℄℄; e!e 0 =) e 0 2[[A`℄℄. 3. 2[[`℄℄; ! 0 =) 0 2[[`℄℄ .

Proof. Byindu tion onA, onsideringthedierentshapesofv, e,and . 1.1. v=x:thennomoreredu tion ano ur.

1.2. v = e 1

v 1

: we must onsider two possible redu tions e 1
v 1 ! e 2
v 1 or e 1
v 1 !e 1
v 2

.Ineither ase,we on ludebyindu tionhypothesis. 1.3. v=: : we onsiderthefollowingtwo ases.

 The redu tionis : !: 0

. Bydenition of : 2 [[`A℄℄ wehave 8e2[[A`℄℄ [e=℄2SN ~ .Then weget 0 [e=℄2SN  ~ (alwaysfor anye2[[A`℄℄)andwe on ludewiththepoint(iii)ofproposition1.  The redu tionis :hvji ! v with 62 FV(v). Weknowby

hypoth-esis that :hvji 2 [[`A℄℄ , then, by the point (iii) of proposition 1, 8e 2 [[A`℄℄ hvji[e=℄ 2 SN

~

, i.e. hvjei 2 SN ~

. If v is a vari-able, then we on lude immediately. if v = : , h: jei 2 SN

~ implies that [e=℄ 2 SN

 ~

, whi h gives us : 2 [[`A℄℄ by the point (iii) of proposition 1. If v = x:v

0 , hx:v 0 jei 2 SN ~ gives us, for e = v 1
e 1 , hv 1 jex:hv 0 je 1 ii 2 SN  ~ then hv 0 je 1 i[v 1 =x℄ 2 SN ~ and hv 0 [v 1 =x℄je 1 [v 1 =x℄i 2 SN  ~

and nally, sin e x is not free in e 1 , hv 0 [v 1 =x℄je 1 i2 SN  ~

, whi h is enough,by thepoints (iv)and (iii) of proposition 1,to on lude. 1.4. v=x:v 0 :A=A 1 !A 2

and theredu tionisx:v 0

!x:v 00

. Bythepoint (iii)ofproposition1,weknowthat8v

000 2[[`A 1 ℄℄;e2[[A 2 `℄℄ hv 0 [v 000 =x℄jei2 [[`℄℄ =SN ~ ,so8v 000 2[[`A 1 ℄℄;e2[[A 2 `℄℄ hv 00 [v 000 =x℄jei2 [[`℄℄ =SN ~ , andwearedone.

2.x. Sameas1.x.bysymmetry(wherexrangesfrom1to 4). 3. 2[[`℄℄:then 2SN ~ and ! 0 impliesthat 0 2SN  ~ =[[`℄℄. Herearenowsomelemmas to\indu tivelybuild"themembershipofaRS. Lemma3.

v2[[`A℄℄; e2[[A`℄℄ =) hvjei2[[`℄℄:

Proof. Toshowthat hvjei2[[`℄℄is, bydenition,to showthat hvjei2SN ~

. Wetakeallpossiblepairsforv andeandwereasonbyindu tiononthestrong normalisationofvande(whi hwegetbylemma1)andonthelengthofv and e. We onsider allthe possible redu tionsof hvjei.If theredu tiono urs in v

of: 2[[`A℄℄,

 ife=ex: ,we on ludesymmetri allytothelastpoint,  ifv=x:v 0 ande=v 00
e 0 (withA=A 1 !A 2 ),theredu tionishx:vjv 00
e 0 i!hv 00 jex:hv 0 je 0

ii.We onsiderthepossibleredu tionsofhv 00 jex:hv 0 je 0 ii. Byredu ibilityofvande,wehavev

00 2SN ~ andhv 0 [v 00 =x℄je 0 i2SN ~ . Consequently,sin etheredu tions annoto urinnitelyinthoseterms,we willgettoredu eoneofthefollowing(wherev

00 !  v 1 ,hv 0 je 0 i!  hv 2 je 2 i): { hv 1 jex:hxje 2 ii ! hv 1 je 2

i : by indu tion hypothesis, we have hv 00 je 0 i 2 SN  ~ andhv 1 je 2

iisoneofitsredu ts. { hv 1 jex:hv 2 je 2 ii ! hv 2 [v 1 =x℄je 2 [v 1

=x℄i : this term is also a redu t of hv 0 [v 00 =x℄je 0 [v 00 =x℄i whi h is in SN ~

by redu ibility of v, due to the fa tthat sin ex isnotfreein e

0 ,hen ein e 2 ,e 2 [v 1 =x℄=e 2 . { h: 1 jex:hv 2 je 2 ii! 1 [ex:hv 2 je 2 i=℄ with v 1 =: 1 . Byredu ibility of e and by the lemma 2 we have :

1

2 [[`A 1

℄℄, that gives us, by denition,that 1 [ex:hv 2 je 2 i=℄ belongs to [[`℄℄ ifx:hve 2 je 2 i belongs to [[A 1

`℄℄. Andthis last onditionis satised,bydenition, ifand onlyif 8v 3 2[[`A 1 ℄℄ wehavehv 2 [v 3 =x℄je 2 [v 3

=x℄i2[[`℄℄, whi his a onsequen e of the redu ibility of v (with e

2 [v

3

=x℄ = e 2

, by the sameargument as above).  Ife=:e 0 andv=e 00
v 0

,we on ludesymmetri allytothelastpoint.  Inallother ases,noredu tion ano ur.

Lemma4. If v[v 0 =x℄ 2 [[`B℄℄ for all v 0 2 [[`A℄℄ then x:v 2 [[`A!B℄℄ . If e[e 0 =℄2[[B`℄℄for all e 0 22[[A`℄℄then:e2[[`A B℄℄ .

Proof. Bysymmetry,weneedonlyto proveoneof theimpli ations,letus take the rst one. To prove that x:v 2 [[`A!B℄℄ , we need, by the point (iii) of proposition1,to provethat for allv

0 2[[`A℄℄ ;e2[[B`℄℄, hv[v 0 =x℄jei2[[`℄℄.By hypothesis,wehavev[v 0

=x℄2[[`B℄℄ . We on ludewiththelemma3.

Hereistheadequa ylemma.

Lemma5 (Adequa y). Let A be a type and t an obje t su h that FV(t)  X 1 [X 2 (X 1 VarandX 2 Var ?

)and thevariables x i

2X 1

are oftype B i and the variables 

j 2X

2

are of typeC j

. For all set of obje tsv i ;e j su h that 8i v i 2[[`A i ℄℄ and8j e j 2[[B j

`℄℄ wehave, a ordingly tothe shapeof t,

1. ifX 1 :B`v:AjX 2 :C then v[v 1 =x 1 ;:::;v n =x n ;e 1 = 1 ;:::;e m = m ℄2[[`A℄℄ 2. ifX 1 :Bje:A`X 2 :C then e[v 1 =x 1 ;:::;v n =x n ;e 1 = 1 ;:::;e m = m ℄2[[A`℄℄ 3. if :(X 1 :B `X 2 :C)then [v 1 =x 1 ;:::;v n =x n ;e 1 = 1 ;:::;e m = m ℄2[[`℄℄ Remark 2. We note X 1 : B theenumerationfx i : B i

ji 2 [1;n℄g(the samefor X

2 :C).

Proof. Wenote[==℄thesubstitution[v 1 =x 1 ;:::;v n =x n ;e 1 = 1 ;:::;e m = m ℄.We

rea-i i i { v =e
v

0

: by indu tion hypothesis on e and v 0

, and by the point (iii) of proposition1,we on ludeimmediately.

{ v=x:v 0 :wethenhaveA=A 0 !A 00

.Sin ewe anrenameboundvariables, we ansupposethatx62fx

1 ;:::;x n g,whi hgivesus(x:v 0 )[==℄=x:(v 0 [==℄). Byindu tion hypothesis, for all v

00 2[[`A 0 ℄℄ wehavev 0 [v 00 =x;==℄ 2[[`A 00 ℄℄ andbythelemma4,wearedone.

{ v = : : sin e we an rename bound variables, we an suppose that  62 f

1 ;:::;

m

g. Now, by the point (iii) of proposition 1, to prove that (: )[==℄=:( [==℄)2[[`A℄℄weneedonlytoprovethat,foralle2[[A`℄℄, [e=;==℄2[[`℄℄whi h isdonebyindu tion hypothesis.

{ e:the asesforearesimilartothoseforv bysymmetry.

{ = hvjei. By indu tion hypothesis on v and e, and by the lemma 3, we on ludeimmediately.

We an nowestablishthemaintheorem ofthisse tion.

Theorem1. Every typed ~obje tisstronglynormalizing.

Proof. Let t be an obje t of the  ~- al ul typed by and
,i.e. su h that the on lusion of itstyping judgement is either ` t : Aj
, or jt : A `
, or t : ( `
). Suppose that its freevariables are f

1 ;:::; m ;x 1 ;:::;x n g,ea h one typed x i : A i and  i : B i

. By the points (i) and (ii) of proposition 1, we get that for all i, x

i 2 [[`A i ℄℄ and  i 2 [[B i

`℄℄. Then, by the lemma 5, t[x 1 =x 1 ;:::;x n =x n ; 1 = 1 ;:::; m = m

℄=tisinaredu ibilityset.Bythelemma1, wegett2SN

~ .

3 PSN of ~- al ulus with Expli it Substitutions

Werstdenethe ~- al uluswithexpli itsubstitutions.Thenweshowsome usefulresultsonthesubstitution al ulus.Andnally,weestablishtheproperty ofpreservationofstrongnormalization.

3.1 Denition

Tothethreesynta ti ategoriespresentedin thelast se tion,weaddafourth, regardingexpli itsubstitutions, noted. Inthesequel,willstandforeithera termora ontextvariable.Thesyntaxofthe ~x- al ulusis:

::= hvjeij 

v ::= x jx:v je
vj : jv e ::= j:ej v
ejx: e je  ::=[x v℄j[ e℄

Thesour eDom() of isx if =[x v℄and if =[ e℄.Thebody S()of isvintherst aseandeinthese ond.Wewillsaythatasubstitution belongstoSN ifitssubstituenditselfbelongstoSN .

( 0

`
0

).Herearethetypingrulesforexpli itsubstitutions:

`v:Aj
[x v℄:( ;x:A`
))( `
) je:A`
[ e℄:( `
;:A))( `
) je:A`
 :( `
))( 0 `
0 ) 0 je:A`
0 `v:Aj
 :( `
))( 0 `
0 ) 0 `v :Aj
0 :( `
)  :( `
))( 0 `
0 )  :( 0 `
0 ) Theredu tionrulesarethefollowing:

() hx:vjv 0
ei!hv 0 jex:hvjeii ( e ) he 0
vj:ei!h:hvjeije 0 i (mu) h: jei! [ e℄ (gmu) hvjex: i! [x v℄

(sv) :hvji!v if62FV(v) (se) x:hxjeie !e ifx62FV(e)

( ) hvjei !hvjei (x1) x !S() ifx2Dom() (x2) x !x ifx62Dom() (1)  !S() if2Dom() (2)  ! if62Dom() (
) (v
e) !(v)
(e) (e
) (e
v) !(e)
(v) () (x:v) !x:(v) ( e  ) (:e) !:(e) () (: ) !:( ) (e) (ex: ) !x:( e )

Wereasonmodulo- onversionontheboundvariableintherules(),(e), ()and(

e  ).

3.2 Substitution Cal ulus

Wewillnote:

 x the set of rules on erning the propagation of substitutions, namely , x1,x2,1,2,
,e
,,

e

 , ande,

 :xthesetofrulesnotinx,namelythose on erningredu tionsoftheoriginal al ulus:,

e

normalforms arepure obje ts(i.e. withoutsubstitutions). Proof. Wedenethefollowingmeasureh:

h() =1 h(hvjei)=h(v)+h(e)+1 h(v
e) =h(v)+h(e)+1 h(e
v) =h(v)+h(e)+1 h(x:v) =h(v)+1 h(:e) =h(e)+1 h(: ) =h( )+1 h(ex: ) =h( )+1 h(t[ t 0 ℄)=h(t)(h(t 0 )+1)

Weeasily he kthatea hx-redu tionstri tlyde reasesh.Weproveby ontra-di tionthatthenormalformsarepureobje ts:ifthereisasubstitution,welook to theobje tto whi hitisapplied andwendaredu tiontoperform.

Wewillnote x(t)thex-normalformofanobje tt. Lemma7 (Con uen e of x). xis on uent.

Proof. All riti alpairs have disjoint redexes, whi h gives us lo al on uen e. ByNewmanlemma andlemma6weget on uen e.

Lemma8 (Substitution). x(t[ t 0

℄)=x(t)f x(t 0

)g. Proof. Weprove,byindu tionontheheightoftand ofthet

i ,that x(t[ 1 t 1 ℄:::[ n t n ℄)=x(t)f 1 x(t 1 )g:::f n x(t n )g:

Lemma9 (Simulation of the ~- al ulus). For all t and upure obje ts, if t! ~ uthent!  ~ x u.

Proof. Byindu tion onthestru ture of t.Theonlyinteresting asesare those in whi htheredu tiono ursattheroot.

{ h: jei!  f eg:wehave h: jei! mu [ e℄! x x( [ e℄) lemma8 = x( )f x(e)g:

Sin eh: jeiis apureobje t,x( )= ,x(e)=eandwearedone. { hvje x: i!



f vg:this aseissimilar tothepreviousbysymmetry. { Theotherrulesaresimulatedinonestepbytheirhomonymesin ~x.

We say that a redu tion is void if it o urs in the body of a substitution t[ t

0

℄su hthat62x(t).Wenoteit v !. Lemma10 (Proje tion). 1. Ift!  ~ x uthenx(t)!  ~ x(u). 2. Ift! :x

uisnot avoid redu tion,thenx(t)! + ~

x(u).

Proof. We onsiderthree ases: { theredu tionist! x u.Thenx(t)=x(u). { theredu tionist v ! :x u.Thenx(t)=x(u). { theredu tionist! :x

We use the perpetuality te hnique, formalised by Bonelli [5℄. In fa t, we use only the rst part of the te hnique, whi h is enough to prove preservation of strongnormalisation.Wegivesomelemmastoextra t avoidsubstitutionwith aninnitederivationinside,andto tra ethissubstitutionba kwards.

Lemma11. Let t 0 ! ~ x t 1 !  ~x t 2 ! ~ x ::: be an innite redu tion. If x(t 0 )2SN ~

, thenthere exists an integer k su h that for all i >k,we have t i v ! ~ t i+1 .

Proof. Sin e x is strongly normalizing, the redu tionmust be t 0 !  x t 1 ! :x t 2 !  x t 3 ! :x t 4 ::: Bylemma 10,wehavex(t 0 )!   ~ x(t 1 )!  ~ x(t 2 ) !  ~ x(t 3 )!   ~ x(t 4

)::: Furthermore,for all even i, if t i+1 ! :x t i+2 isnot a void redu tion,then x(t i )! + ~ x(t i+2 ).From x(t 0 )2SN  ~

wededu ethat there existsksu hthatforallevenigreaterthankwehavet

i+1 v ! :x t i+2 .Wemust now provethat from a ertain point, both :x and x redu tions are void. For that,wedenethefollowingmeasure:

h() =1 h(hvjei)=h(v)+h(e)+1 h(: ) =h( )+1 h(x: )e =h( )+1 h(t[ t 0 ℄)=  h(t)(h(t 0 )+1)if2FV(x(t)) h(t)2 else

Thelast lauseguaranteesthatavoidredu tionleavesthemeasureun hanged. Weeasily satises that allother redu tions stri tlyde raese this measure,and we on lude.

Thenextnotionisusefultoisolateavoidsubstitution.

Denition1 (Skeleton). The skeleton of an obje t, noted SK(t), is indu -tively denedasfollows:

SK() = SK(hvjei)=hSK(v)jSK(e)i SK(: ) =:SK( ) SK(ex: ) =x:SK( )e SK(t[ u℄)=SK(t)[ ℄ Weremark thatif t v !u, thenSK(t)=SK(u).

Thefollowinglemma says that ifthere isan innite derivation, then there existsasubstitutioninwhi hthereis aninnitederivation.

Lemma12. Let an innite derivation be t 0 ! ~ x t 1 !  ~x t 2 ! ~ x ::: If x(t 0 )2SN ~

,thenthereexistsanintegerk,anobje tt,avariable,a ontext C andan obje tsequen eu

i su hthat t 0 !   x~ t k = C[t[ u k ℄℄ v !   x~ C[t[ u k +1 ℄℄ v !   x~ C[t[ u k +2 ℄℄::: with u k ! u k +1 ! u k +2 ! u k +3 :::

Proof. Bylemma11,thereexistsksu hthatforalli>k,t i ! ~ x t i+1 .Then, wehaveSK(t k )=SK(t i

)forallik. Thederivationtreeoft k

beinginnite, bythepigeon holeprin iple,aninnitederivation musttakepla e inthesame substitutionofSK(t

k

),andwearedone.

Lemma13 (Substitutiontra ing-1step). Lettandubetwoobje tssu h that t! ~ x uandu=C[u 1 [ u 2 ℄℄.Then 1. eithert=C 0 [u 0 1 [ u 2 ℄℄, 2. ort=C 0 [u 0 1 [ u 0 2 ℄℄with u 2 !u 0 2 , 3. oru 1 is a ommandand if=then t=C[h:u 1 ju 2

i℄elset=C[hu 2

je x:u 1

i℄.

Proof. Wereasonbyindu tionont andwe onsiderthefollowingtwo ases:

 Theredu tiontakespla eattheroot.Firstnotethatifu 1

[ u 2

℄appears inasub-termofu,whi hisalso asub-term oft,then fora ontextC

0 and u 0 1 =u 1

therstitemholds.Thisappliesalsowhentheruleusedtoredu e attheroot isoneofx or.Elseiftherule ismuorgmu, thenthethird itemholds, elseifitisanotherrule,thentherstitemholds,inboth ases, weusetheempty ontext.

 Theredu tionisinternal.

{ t=.Theresultholdstrivially. { t = hvjei with either v !

~ x v 0 ore ! ~ x e 0 . We onsider the rst ase,sin ethese ondoneissimilar.Wehaveu=hv

0 jeiand: ? if thesub-term u 1 [ u 2 ℄ o urs in v 0

, then weuse indu tion hy-pothesis.

? else thesub-term u 1

[ u 2

℄o ursin e;thentherstitemholds. { t=v
eort=e
v witheither v !  x~ v 0 ore! ~ x e 0 .We on lude similarlytothepreviouspoint.

{ t=: orx: e orx:v or:e.Weuseindu tionhypothesis. { t=t

1 [ t

2

℄. Therearetwo ases: ? t 1 !  ~ x t 0 1 and u = t 0 1 [ t 2 ℄. Then if u 1 [ u 2 ℄ o urs in t 0 1 we use indu tion hypothesis. If it o urs in t

2

the rst item holds trivially.Finally,ifu=u

1 [ u

2

℄then wetaketheempty ontext forC 0 ,u 0 1 =t 1

andtherstitemholds. ? t 2 !  ~x t 0 2 and u=t 1 [ t 0 2 ℄. Thenifu 1 [ u 2 ℄o ursin t 1 the rstitemholdstrivially.Ifito ursint

0 2

weuseindu tionhypothesis. Finally, if u=u

1 [ u

2

℄ then wetake theempty ontext for C 0 , u 0 1 =t 1 andu 0 2 =t 2

andthese onditemholds.

Thisresultisnaturallyextendedto many-stepsredu tions.

Lemma14 (Substitutiontra ing). Let t 1

;:::;t n

beobje ts su hthat,for all i,t i !  ~ x t i+1 andt n =C[u 1 [ u 2 ℄℄. Then

1. either=andthere isisu hthat t i =C 0 [h:u 0 1 ju 0 2 i℄ withu 2 !  ~ x u 0 2 , 2. or=x andthereisi su hthatt

i =C 0 [hu 0 2 jex:u 0 1 i℄with u 2 !  u 0 2 ,

3. ort 1 =C[u 1 [ u 2 ℄℄ withu 2 !  ~x u 2 .

Proof. Byindu tion onthenumberofredu tionsteps,usinglemma13. Weformalisethenotionofderivationordering.

Denition2. Let  and be two innite derivations starting form an obje t t

1

.Then is alled smallerthan if theyredu ethe same redexes for the rst n 1steps, andthe nthredex redu edby isastri tsubterm ofthe nth redex redu edby .

Hereisthemaintheoremofthisse tion. Theorem2 (PSN). t2SN

~

)t2SN  ~x

.

Proof. By ontradi tion.Supposethat there existsapure termtwhi h anbe innitely redu ed in the ~x- al ulus. We take a minimal derivation of this term. Bylemma 12, at a ertain point,we an exhibit a innitederivation in a void substitution. By lemma 14, we an go ba kwards until we rea h the redu tionwhi h reatesthissubstitutionwhilekeepingtheinniteredu tionin it. This reationpoint ( hosenby theminimalderivation) isaproperprex of theredu tionpointoftheinnitederivation insidethe futurebodyofthe void substitution.This ontradi tstheminimalityofthederivation.

4 PSN Implies SN

4.1 Proof Te hnique

Thete hniquewepresenthereisverygeneraland anbeappliedtomany al uli withexpli itsubstitutions. Theideaof thiste hniqueis thefollowing:lett be atypedtermwithexpli itsubstitutions,with itstypingjudgement,webuilda typedtermt

0

ofthepure al ulusbyexpandingthesubstitutionsoftinredexes. We allthisexpansionAteb.Werequirethefollowingtwoproperties,whi hare enoughtoestablishtheorem3.

Property 1 (Preservation of typability). Ift is typable in the al uluswith ex-pli itsubstitution, thenAteb(t)istypablein thepure al ulus.

Property 2 (Initialization).Ateb(t)redu estotin0ormorestepsinthe al ulus withexpli itsubstitutions.

We annowestablishthetheorem.

Theorem3. Foralltypingsystemsu hthatalltypabletermsarestrongly nor-malizing,if thereexistsafun tionAteb fromexpli it substitutiontermstopure termssatisfying properties1and2thenPSNimpliesSN.

Proof. For all typed term t of the al ulus with expli it substitution, Ateb(t) isapuretyped term(byproperty1).Byhypothesis ofstrongnormalizationof the pure typed al ulus, we haveAteb(t) 2SN (in the present ase SN

 ~ ). By hypothesis of PSN we obtain that Ateb(t) is in SN (in the present ase SN

~ x

).Byproperty2,we getAteb(t) ! 

t, whi h givesus dire tly t2SN (inthepresent aseSN ).

Here is the denition of Ateb. It is obvious that for all t, Ateb(t) ontains no substitutions. We then he k that this fun tion satises the twoproperties we mentionabove.

Denition3.

Ateb(x) =x Ateb() =

Ateb(x:v) =x:Ateb(v) Ateb(:e)=:Ateb(e) Ateb(: ) =:Ateb( ) Ateb(ex: )=x:Ateb( )e Ateb(e
v) =Ateb(e)
Ateb(v) Ateb(v
e) =Ateb(v)
Ateb(e) Ateb(hvjei) =hAteb(v)jAteb(e)i

Ateb( [x v℄) =hAteb(v)jex:Ateb( )i Ateb( [ e℄) =h:Ateb( )jAteb(e)i Ateb(v[x v

0

℄)=:hx:Ateb(v)jAteb(v 0

)
i With freshvariable Ateb(v[ e℄) =:h:hAteb(v)jijAteb(e)i With freshvariable Ateb(e[x v℄) =y:hAteb(v)jee  x:hyjAteb(e)ii With y freshvariable Ateb(e[ e

0

℄)=x:hAteb(ee 0

)
xj:Ateb(e)i Withx freshvariable

Proof. (ofproperty1) Easybyindu tionontheproofofthetyping judgement oft.

Proof. (ofproperty2)Wepro eedbyindu tionont.Onlythe asesfor substi-tutionsarenoteasy.Bythesymmetryofthesystem,we onsideronlyonehalf ofit.

{ WehaveAteb( [x v℄)=hAteb(v)jex:Ateb( )iand

hAteb(v)je x:Ateb( )i!  Ateb( )[x Ateb(v)℄: { WehaveAteb(v[x v 0 ℄)=:hx:Ateb(v)jAteb(v 0 )
i and :hx:Ateb(v)jAteb(v 0 )
i !  :hAteb(v 0 )jex:hAteb(v)jii ! e  :(hAteb(v)ji[x Ateb(v 0 )℄) !  :hAteb(v)[x Ateb(v 0 )℄j[x Ateb(v 0 )℄i ! 2 :hAteb(v)[x Ateb(v 0 )℄ji! sv Ateb(v)[x Ateb(v 0 )℄:

{ WehaveAteb(v[ e℄)=:h:hAteb(v)jijAteb(e)iand

:h:hAteb(v)jijAteb(e)i ! 

:(hAteb(v)ji[ Ateb(e)℄) !

:hAteb(v)[ Ateb(e)℄j[ Ateb(e)℄i !

2

:hAteb(v)[ Ateb(e)℄ji! sv

Ateb(v)[ Ateb(e)℄:

We olle ttogetherourresultstoprovethemaintheoremofthiswork. Theorem 4. The typed ~x- al ulusisstronglynormalizing.

Proof. ByTheorem1(SNforpure al ulus),Theorem2(PSN) andTheorem3 (PSN impliesSN).

5 A hievements and Perspe tives

Using various proof te hniques, we haveestablished that the ~x- al ulus is stronglynormalizing.Forthatpurpose,wehaveformalizedaproofte hniqueof SN via PSN. Let us mentionthat wehave su essfullyapplied this te hnique, with some adjustments, to proveSN of the - al ulus(introdu ed in [3℄) for thersttime,asfarasweknow.Wealsoused ittoestablishthatPSNimplies SNforthe- al ulus[1℄,forwhi hPSNisknowntofail[10℄,showingthat,for this al ulus,theonlyproblem ofSNisin PSN.

Itremainsanopenproblemtobuildadire tproof,bytheredu ibility te h-nique, of SN forasymmetri non-deterministi al uluswithexpli it substitu-tions.Anotherdire tion ofwork ouldbetorepla esubstitutions \ala"xby substitutions\ala"

ws

[8℄,whi hyields,throughtheadditionofexpli it weak-enings,amorepowerfulsubstitutionsystem.Itmayevenhelpustondadire t proofofSN.Atlast,weplantoworkonase ondorderversionof~x.

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1. Abadi,M., Cardelli,L.,Curien,P.-L.,Levy,J.-J.:Expli it Substitutions.Journal ofFun tionalProgramming(1991).

2. Barbanera, F., Berardi, S.: A symmetri lambda- al ulus for lassi al program extra tion.Pro eedingsofTACS'94(1994),Springer-VerlagLNCS789,495{515. 3. Benaissa, Z.-E.-A., Briaud, D., Les anne,P., Rouyer-Degli,J.:  , a al ulus of expli itsubstitutionswhi hpreservesstrongnormalisation.JournalofFun tional Programming(1996).

4. Bloo,R.,Geuvers,H.:Expli itSubstitution:ontheEdgeofStrongNormalisation. Theoreti alComputerS ien e(1999),211,375{395.

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6. Chur h,A.:TheCal uliofLambdaConversion.Prin etonUniv.Press(1941). 7. Curien, P.-L.,Herbelin, H.:Thedualityof omputation.Pro eedings ofICFP'00

(2000),ACMPress, 233{243.

8. Guillaume, B.: Un al ul desubstitution ave etiquettes.PhDthesis, Universite deSavoie(1999).

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10. Mellies,P.-A.:Typed- al uliwithexpli itsubstitutionsmaynotterminate. Pro- eedingsofTLCA'95(1995),SpringerLNCS,902,328{334.

11. Parigot,M.:- al ulus:Analgorithmi interpretationof lassi alnatural dedu -tion.Pro eedingsofLICS'93(1993),ComputerSo ietyPress,39{46.