Sound and Complete Typing for lambda-mu

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by  Steffen van Bakel

Sound and Complete Typing for λµ
Steffen vanBakel
DepartmentofComputing,ImperialCollegeLondon,180Queen’sGate,LondonSW72BZ,UK
[email protected]
InthispaperwedefineintersectionanduniontypeassignmentforParigot’scalculusλµ. Weshow
that this notion is complete (i.e. closed under subject-expansion), and show also that it is sound
(i.e.closedundersubject-reduction). Thisimpliesthatthisnotionofintersection-uniontypeassign-
mentissuitabletodefineasemantics.
Introduction
TheIntersectionTypeDisciplinehasproventobeanexpressivetoolforstudyingterminationandseman-
tics for the λ-calculus [12, 10]. Intersection type assignment is defined as an extension of the standard,
implicative type assignment known as Curry’s system [15] (see also [24]), which expresses function
composition and application; the extension made consists of relaxing the requirement that a parameter
forafunctionshouldhaveasingletype,addingthetypeconstructor ∩ nextto→. Thissimpleextension
allowsforagreatleapincomplexity: notonlycana(filter)modelbebuiltfortheλ-calculususinginter-
sectiontypes,alsostrongnormalisation(termination)canbecharacterisedviaassignabletypes;however,
typeassignmentbecomesundecidable. Theliteratureonintersection typesisvast;itwasfirstdefinedby
Coppo and Dezani-Ciancaglini in [13] and its development took place over a number of years, culmi-
natinginthepaperbyBarendregt, Coppo,andDezani-Ciancaglini [11],andhasbeenexploredbymany
peoplesince.
It is natural to ask if these results can be achieved for other calculi (reduction systems) as well,
andinpreviouspaperstheauthorinvestigated(incollaboration) TermRewritingSystems[6],andObject
OrientedCalculi[5];Maffeislookedatintersectiontypesinthecontextoftheπ-calculus[26]. Inorderto
cometoacharacterisation ofstrongnormalisation forCurienandHerbelin’s(untyped) sequent calculus
λµµ˜ [14], Dougherty, Ghilezan and Lescanne presented System M∩∪ [17], that defines a notion of
intersection anduniontypingforthatcalculus; inalaterpaper[18],theypresented animprovedversion
oftheiroriginal system.
In[3],theauthorrevisited SystemM∩∪,andshowedthatthatsystemwasneithersound (i.e.closed
under reduction), nor complete (i.e. closed under reverse reduction); the same holds for the system pre-
sented in [18]. To address completeness, [3] adds ⊤ as the maximal and ⊥ as the minimal type, and
extends the set of derivation rules; however, soundness is shown to be impossible to achieve without
restricting typeability (effectively making less terms typeable). In[4], the author attempted tosolve the
sameissue,butthistimeinthecontextofthesequentcalculus X,asdefinedbyLengrand[25],andlater
studied by Lescanne and the author [7, 8]; X is a sequent calculus in that it enjoys the Curry-Howard
isomorphism withrespect totheimplicative fragment ofGentzen’s LK [20]. Theadvantage ofusing the
sequent approach isthatitisnowpossible toexplore theduality ofintersection andunionfully, through
which we can study and explain various anomalies of union type assignment [32, 9] and quantification
[22,27]. AlsoforX,thecompleteness resultfollowsrelatively easily,butsoundness canonlybeshown
E.Pimentel,B.VenneriandJ.Wells(Eds.):Workshopon
IntersectionTypesandRelatedSystems2010(ITRS2010).
EPTCS45,2011,pp.31–44,doi:10.4204/EPTCS.45.3
32 SoundandCompleteTypingforλµ
forrestrictedsystems(effectivelycall-by-name andcall-by-value reduction, butitmightbepossiblethat
othersound restrictions existaswell). Themainconclusion ofthosepapersisthat, insymmetriccalculi
(like λµµ˜ and X) it is inevitable that intersection and union are truly dual, and that the very nature of
thosecalculimakesasoundandcompletesystemunachievable.
In this paper we will continue on this path and bring intersection types to the context of classical
logic, by presenting a notion of intersection and union type assignment for the (untyped) calculus λµ,
thatwasfirstdefinedbyParigotin[31],andwaslaterextensively studied byvariousauthors.
Intersectionanduniontypeshavealsobeenstudiedinthecontextoftheλ-calculusin[9];alsoforthe
systemdefinedinthatpapersoundnessislost,whichcanonlyberecoveredbylimitingtoparallelreduc-
tion, i.e.allresiduals ofaredexneedtobecontracted inparallel. Theproblem oflossofsoundness also
appearsinothercontexts,suchasthatofML withside-effects [22,33,27],andthatofusingintersection
anduniontypesinanoperationalsetting[16,19]. Ashere,alsotherethecauseoftheproblemisthatthe
type-assignment rules are not fully logical, making the context calls (which form part of the reduction
in X) unsafe; this has, in part, already been observed in [23] in the context of Curien and Herbelin’s
calculus λµµ˜ [14]. This also explains why, for ML with side-effects, quantification is no longer sound
[22,27]: alsothe(∀I)and(∀E)rulesofML arenotlogical.
In the view of those failures, the result presented here comes as a surprise. We will define a notion
of type assignment for λµ that uses intersection and union types, and show that it is both sound and
complete. The system presented is a natural extension of the strict intersection type assignment system
as defined in[2]; this implies that intersection models the distribution of arguments in aparameter call.
Butitisalsoanaturalextensionofthesystemforλµ,andinordertoachievecompletenessforstructural
reduction, as in the papers mentioned above, union types are added. However, the union types are no
longerdualtointersection types;uniontypesplayonlyamarginalrole,aswasalsotheintentionof[18].
Contrary to that paper, however, we do not see union as negated intersection, but see a union type as a
stricttype;inparticular, wedonotallowthenormal(∪I)and(∪E)rulesasusedin[9],whichweknow
create thesamesoundness problem. Moreover, although onecanlinkintersection typeswiththelogical
connector and, the union types we use here have no relation with or; one could argue that therefore
perhapsunionisnottherightnametouseforthistypeconstructor, butwewillstickwithitnonetheless.
The limited view of union types is mirrored by λµ’s limited (with respect to λµµ˜ and X) notion
of context1. In λµ, we distinguish control structures as those terms that start with a context switch
µα.[β]M, followed by a number of arguments; since union types allow us to express that the various
continuations (all called α) need not have the same type, weuse adifferent formulation for rule (→E),
which has an implicit use of union elimination (see Definition 3.2). The type system defined here will
beshowntobethenaturalone,inthatintersection andunionplaytheirexpectedrolesforcompleteness.
Becausetheuseofintersection andunionislimitedinthatacontextvariablecannothaveanintersection
type,andalthoughweallowuniontypesfortermvariables,wedonothavethenormalunionelimination
rule;thankstothesetworestrictions, wecanshowsoundness aswell.
1 The calculus λµ
Parigot’s λµ-calculus [29] is a proof-term syntax for classical logic, expressed in Natural Deduction,
definedasanextension oftheCurrytypeassignment systemfortheλ-calculus. Wequicklyrevisesome
basicnotions:
1Inparticular,λµµ˜’sµ˜x.cisnotrepresented.
SteffenvanBakel 33
Definition1.1 (LAMBDA TERMS AND β-CONTRACTION [10]) 1. λ-termsaredefinedby:
M,N ::= x|λx.M| MN
2. Thereduction relation → isdefinedasthecontextual closureoftherule:
β
(λx.M)N → M[N/x]
β
Curry(orsimple)typeassignmentfortheλ-calculus isdefinedas:
Definition1.2 1. Let ϕrangeoveracountable(infinite)setoftype-variables. ThesetofCurry-types
isdefinedbythegrammar:
A,B ::= ϕ|(A→B)
2. Curry-type assignment isdefinedbythefollowingnatural deduction system.
Γ,x:A⊢M:B Γ⊢M:A→B Γ⊢N:A
(Ax): (→I): (→E):
Γ,x:A⊢x:A Γ⊢λx.M:A→B Γ⊢ MN:B
With λµ Parigot created amulti-conclusion typing system which corresponds to classical logic; the
derivable statements have the shape Γ ⊢ M:A|∆, where A is the main conclusion of the statement,
λµ
expressed as the active conclusion, and ∆ contains the alternative conclusions, consisting of pairs of
Greek characters and types; the left-hand context Γ, as usual, contains pairs of Roman characters and
types, andrepresents thetypesofthefreetermvariables of M. AswithImplicative Intuitionistic Logic,
thereductionrulesforthetermsthatrepresenttheproofscorrespondtoproofcontractions; thedifference
is that the reduction rules for the λ-calculus are the logical reductions, i.e. deal with the elimination of
atype construct that hasbeen introduced directly above. Inaddition tothese, Parigot expresses also the
structuralrules,whereeliminationtakesplaceforatypeconstructorthatappearsinoneofthealternative
conclusions (the Greek variable is the name given to a subterm): he therefore needs to express that the
focus of the derivation (proof) changes, and this is achieved by extending the syntax with two new
constructs [α]M and µα.M that act as witness to deactivation and activation, which together move the
focusofthederivation.
Wewillnowpresentthevariantofλµweconsiderinthispaper,asconsideredbyParigotin[30];for
convenience, wesplittermsintotwocategories: wedefineterms,andcontrolstructure:
Definition1.3 (SYNTAX OFλµ) Theλµ-termsweconsider are:
M,N ::= x|λx.M| MN |µα.[β]M.
Wealsodefinecontrolstructure asasubsetofterms: C ::= µα.[β]M |CM.
Toshortenproofsandnotation, wewilltreat µα.Masatermaswell,wheneverconvenient.
Asusual,λx.MbindsxinM,andµα.[β]MbindsαinM,andthenotionsoffreeandboundvariables
are defined accordingly; the notion of α-conversion extends naturally to bound names, and we assume
Barendregt’s convention onfreeandboundvariables.
In λµ, reduction of terms is expressed via implicit substitution; as usual, M[N/x] stands for the
substitution ofalloccurrences of x in M by N, and M[N·γ/α] stands fortheterm obtained from M in
whichevery(pseudo) sub-term oftheform [α]M′ issubstituted by [γ](M′N) (γ isafreshvariable) (in
Parigot’snotation: (µα.[β]M)N →µγ.[β]M[[γ]PN/[α]P]).
Wedefineformallyhowtopreform theµ-substitution; thisisconvenient inlaterproofs.
34 SoundandCompleteTypingforλµ
Definition1.4 Wedefine M[N·γ/α]byinduction overthestructure oftermsby:
∆
x[N·γ/α] = x
∆
(λx.M)[N·γ/α] = λx.(M[N·γ/α])
∆
(M1M2)[N·γ/α] = M1[N·γ/α] M2[N·γ/α]
∆
(µδ.[α]M)[N·γ/α] = µδ.[γ](M[N·γ/α]N)
∆
(µδ.[β]M)[N·γ/α] = µδ.[β](M[N·γ/α]) β6=α
Wehavethefollowingrulesofcomputation inλµ:
Definition1.5 (λµREDUCTION) Parigotdefinesanumberofreduction rules: twocomputational rules
logical (β): (λx.M)N → M[N/x]
structural (µ): (µα.[β]M)N → µγ.([β]M[N·γ/α])
aswellasthesimplification rules:
renaming: µα[β](µγ.[δ]M) → µα.[δ]M[β/γ]
erasing: µα.[α]M → M ifαdoesnotoccurin M.
ηµ: µα.[β]M → λxµγ.[β]M[x·γ/α]
whichareaddedmainlytosimplifythepresentation ofhisresults2.
Reduction onλµ-termsisdefinedasthecompatible closureoftheserules.
Itispossible todefinemorereduction rules, butParigotrefrained fromthatsince heaimedatdefininga
confluentreduction system.
The intuition behind the structural rule is given by de Groote [21]: “in a λµ-term µα.M of type
A→B,onlythesubtermsnamedbyαarereallyoftype A→B(...);hence,whensuchaµ-abstractionis
applied toanargument, thisargument mustbepassedovertothesub-termsnamedbyα.” Inthispaper,
we will only deal with the logical, structural and renaming rule; this is also the restriction made by de
Grootein[21].
Type assignment for λµ is defined by the following natural deduction system; there is a main, or
active,conclusion, labelled byatermofthiscalculus, andthealternativeconclusions arelabelledbythe
setofGreekvariables α,β,etc.
Definition1.6 (TYPING RULES FORλµ) Our types are those of Definition 1.2, extended with the type
constant ⊥thatisessentially addedtoexpressnegation, i.e.:
A,B ::= ϕ|⊥|(A→B) (A6=⊥)
Thetypeassignmentrulesare:
(Ax): Γ ⊢ x:A|∆ (x:A∈Γ) (µ): ΓΓ⊢⊢µMα.[:βB]M|α::AA,|ββ:B:B,∆,∆ ΓΓ⊢⊢µMα.[:αA]M|α::AA,|∆∆
Γ,x:A ⊢ M:B|∆ Γ ⊢ M:A→B|∆ Γ ⊢ N:A|∆
(→I): (→E):
Γ ⊢ λx.M: A→B|∆ Γ ⊢ MN:B|∆
2Infact,Parigotformulatestherenamingruleas[β](µγ.M)→M[β/γ];since[β](µγ.M)isnotaterm,wewritetherule
differently.
SteffenvanBakel 35
We can think of [α]M as storing the type of M amongst the alternative conclusions by giving it the
nameα-thesetofGreekvariablesiscalledthesetofcontext variables(ornames).
Asanexampleillustratingthefactthatthissystemismorepowerfulthanthesystemfortheλ-calculus,
hereisaproofofPeirce’sLaw(duetoOngandSteward[28]):
(Ax)
x:(A→B)→A,y:A ⊢ y:A|α:A,β:B
(⊥)
x:(A→B)→A,y:A ⊢ [α]y:⊥|α:A
(µ)
x:(A→B)→A,y:A ⊢ µβ.[α]y:B|α:A
(Ax) (→I)
x:(A→B)→A ⊢ x:(A→B)→A|α:A x:(A→B)→A ⊢ λy.µβ.[α]y:A→B|α:A
(→E)
x:(A→B)→A ⊢ x(λy.µβ.[α]y):A|α:A
(⊥)
x:(A→B)→A ⊢ [α](x(λy.µβ.[α]y)):⊥|α:A
(µ)
x:(A→B)→A ⊢ µα.[α](x(λy.µβ.[α]y)):A|
(→I)
⊢ λx.µα.[α](x(λy.µβ.[α]y)):((A→B)→A)→A|
Noticethat⊥playsnopartinthisproof. Indeed,wecandefinethesetoftypeswithout⊥;theunderlying
logicofsuchasystemthencorresponds tominimalclassical logic[1].
Since we allow ⊥ as a proper type, we can even express negation (of course, it is also implicitly
present in the right-hand type environment), and can give a derivation for ¬¬A→A, so can express
doublenegation elimination; soinλµwecanrepresent fullClassicalLogic.
(Ax)
x:A ⊢ x:A|α:A,δ:⊥
(µ)
x:A ⊢ µδ.[α]x:⊥|α:A
(Ax) (→I)
y:(A→⊥)→⊥ ⊢ y:(A→⊥)→⊥| β:⊥ ⊢ λx.µδ.[α]x:A→⊥|α:A
(→E)
y:(A→⊥)→⊥ ⊢ y(λx.µδ.[α]x):⊥|α:A,β:⊥
(µ)
y:(A→⊥)→⊥ ⊢ µα.[β]y(λx.µδ.[α]x):A| β:⊥
(→I)
⊢ λy.µα.[β]y(λx.µδ.[α]x):((A→⊥)→⊥)→A| β:⊥
Noticethatthistermisnotclosed, since βisfree,albeitoftype⊥.
DeGroote[21]considers avariantof λµ whichseparates thenamingand µ-binding features3. This
gives a considerable different system, that allows for ¬¬A→A to be inhabited via (the closed term)
λy.µα.y(λx.[α]x). DeGroote’svariantofλµ [21]usesthesyntax
M,N ::= x|λx.M| MN |µα.M|[β]M
andsplitsrule(µ)into
Γ ⊢ M:⊥|α:A,∆ Γ ⊢ M:A| β:A,∆
(µ): (⊥):
Γ ⊢ µα.M:A|∆ Γ ⊢ [β]M:⊥| β:A,∆
3NoticethatthenParigot’srenamingruleiscorrect. Wecouldhavepresentedourresultsforthismorepermissivesystem,
butwouldhavehadtosacrificesoundnessandcompletenessfortherenamingrule. Noticethatwewouldstillhavesoundness
andcompletenessforthetwocomputationalrules,whicharearguablythemostimportant.
36 SoundandCompleteTypingforλµ
Inthissystemwecanderive
(Ax)
x:A ⊢ x:A|α:A
(⊥)
x:A ⊢ [α]x:⊥|α:A
(Ax) (→I)
y:(A→⊥)→⊥ ⊢ y:(A→⊥)→⊥| ⊢ λx.[α]x:A→⊥|α:A
(→E)
y:(A→⊥)→⊥ ⊢ y(λx.[α]x):⊥|α:A
(µ)
y:(A→⊥)→⊥ ⊢ µα.y(λx.[α]x):A|
(→I)
⊢ λy.µα.y(λx.[α]x):((A→⊥)→⊥)→A|
Forthemoment,wewilldealwithParigot’soriginalsystemonly;weaimtoextendourresultstode
Groote’svariantinfuturework.
2 The Strict Intersection Type Assignment System for the λ-calculus
Theremainder ofthis paper will be dedicated anotion of intersection/union typing on λµ. This willbe
defined as a natural extension of the Strict Intersection System [2] for the λ-calculus. Before we come
tothat,wewillbrieflysummarisethelatter.
Definition2.1 (STRICT TYPES) 1. Let ϕrangeoveraninfinite,enumerablesetoftypevariables. The
setT ofstricttypes,rangedoverby A,B,etcisdefinedthroughthegrammar:
s
A,B ::= ϕ|⊤→B|(A1∩···∩An)→B (n≥1)
The set T of intersection types is defined as the union of {⊤} and the closure of T under inter-
s
section; wewill use A,B,etc for intersection types as well, and mention which set they belong to
whennecessary.
2. A statement is an expression of the form M:A, with M∈Λ and A∈T. M is the subject and A
thepredicate of M:A.
3. A type-environment Γ is apartial mapping from term variables to intersection types, and wewrite
x:A∈ΓifΓ(x)= A.
Soifwewriteatypeas A→B,then B∈T ,and A∈T .
s
In the notation of types, asusual, right-most outer-most parentheses in arrow types willbe omitted,
andweassume ∩ tobindstronger than→. Fromhereon, wewillwritenfortheset{1,...,n}.
Wewillconsiderapre-orderontypeswhichtakesintoaccounttheidem-potence, commutativityand
associativity oftheintersection typeconstructor, anddefines ⊤tobethemaximalelement.
Definition2.2 1. Therelation‘≤’isdefinedastheleastpre-order onT suchthat:
A1∩···∩An ≤ Ai, foralli∈n,n≥1
B≤ Ai, foralli∈n ⇒ B≤ A1∩···∩An, n≥0
2. OnT ,therelation ‘∼’isdefinedby:
A≤B≤A ⇒ A∼B A∼B&C∼D ⇐⇒ A→C∼B→D
SteffenvanBakel 37
3. The relations ‘≤’ and ‘∼’, are extended to contexts by: Γ≤Γ′ if and only if for every x:A′∈Γ′
thereisan x:A∈Γsuchthat A≤ A′,and: Γ∼Γ′ ⇐⇒ Γ≤Γ′≤Γ.
T will be considered modulo ∼; then ≤ becomes a partial order. It is easy to show that both
(A∩B)∩C∼ A∩(B∩C)and A∩B∼B∩A,sothetypeconstructor ∩ isassociativeandcommutative,
and we will write ∩nAi for A1∩···∩An, and consider ⊤ to be the empty intersection: ⊤ = ∩0Ai.
Moreover, wewillassume,unlessstatedexplicitlyotherwise, thatin∩ A each A isstrict.
n i i
Definition2.3 The strict type assignment is defined by the following natural deduction system (where
alltypesmentionedarestrict,withtheexception of Ainrule(→I)and(→E)):
(∩E): Γ,x:∩nAi⊢x:Aj (n≥1,j∈n) (∩I): Γ⊢ΓM⊢:AMi:∩(∀Ai∈n) (n≥0)
n i
Γ,x:A⊢ M:B Γ⊢ M:A→B Γ⊢N:A
(→I): (→E):
Γ⊢λx.M:A→B Γ⊢MN:B
WewillwriteΓ⊢ M:Aforstatementsthatarederivedusingtheserules.
NoticethatΓ⊢ M:⊤forallΓ,Mbyrule(∩I).
Properties ofthissystem havebeenstudiedin[2].
3 Intersection and union type assignment for λµ
Wewillnowdefineanotionoftypeassignment forλµ thatusesintersection anduniontypes.
We see the context variables α as names for possible continuations that in the philosophy of inter-
section types need not all be typed with the same type; we therefore allow multiple types for a context
variable intheenvironment ∆,groupedusinganewtypeconstructor, whichwecallunion.
Bindingacontextvariablethengeneratesacontextswitchµα.[β]M,whichnaturallyhasauniontype
∪ A ;reductionoftheterm(µα.[β]M)N willbringtheoperand N toeachofthepseudosubtermsin M
n i
oftheshape[α]Q(‘named’withα),whereQhastype A ;sinceNgetsplacedbehindQ,thisimpliesthat
i
A =C→B andthatthereforethetypeforαshouldbe∪ (C→B);thisthenalsoimpliesthatNshould
i i i n i i
have all the types C (∀i∈n); rule (→E) as below expresses exactly that. The only ‘functionality’ we
i
needforuniontypestherefore istheabilitytochoose acollection oftypesforαamongstthosestoredin
∆;thisisrepresented byrule(∪E).
Definition3.1 (THE SYSTEM ⊢∩∪) 1. The set of strict types we consider for the intersection-union
λµ
typeassignmentis:
A,B ::= ϕ|B1∪···∪Bm |(A1∩···∩An)→B (n,m≥0)
As above, we call A1∩···∩An (with n ≥ 0) an intersection type, and call B1∪···∪Bm (with
m≥0)auniontype;weuse⊤fortheemptyintersection type,and⊥fortheemptyuniontype.
2. Therelation ≤ofDefinition2.2isextended tointersection-union typesby:
A1∩···∩An ≤ Ai, foralli∈n,n≥1
B≤Ai, foralli∈n ⇒ B≤ A1∩···∩An, n≥0
Bj ≤ B1∪···∪Bm, forall j∈m,m≥1
Bj≤A, forall j∈m ⇒ B1∪···∪Bm≤ A, m≥0
38 SoundandCompleteTypingforλµ
OnT ,therelation ‘∼’isdefinedbythesamewayasinDefinition2.2.
3. A left environment Γ is a partial mapping from term variables to intersections of strict types, and
wewrite x:A∈Γ if Γ(x)= A. Similarly, aright environment ∆ contains only strict types, which
canbeuniontypes.
4. Therelations‘≤’and‘∼’,areextended toleftandrightenvironments by: Γ≤Γ′ ifandonlyiffor
every x:A′∈Γ′ there isan x:A∈Γ such that A≤ A′,and Γ∼Γ′ ⇐⇒ Γ≤Γ′≤Γ,and ∆≤∆′ if
foreveryα:A∈∆thereexists α:A′∈∆′ suchthat A≤ A′,and∆∼∆′ ⇐⇒ ∆≤∆′≤∆.
Notice that we consider union types to be strict as well; this implies that we allow an intersection of
uniontypes,aunionofuniontypes,butnotaunionofintersection types.
Definition3.2 (THE SYSTEM ⊢∩∪) Intersection-union typeassignment forλµisdefinedvia:
λµ
Γ ⊢ M:A |∆ (∀i∈n)
(∩E): Γ,x:∩nAi ⊢ x:Ai |∆ (∩I): Γ ⊢ Mi:∩ A |∆ (n≥0,n6=1)
n i
Γ,x:A ⊢ M:B|∆ Γ ⊢ M:∪ (A →B)|∆ Γ ⊢ N:A |∆(∀i∈n)
(→I): (→E): n i i i (n≥1)
Γ ⊢ λx.M:A→B|∆ Γ ⊢ MN:∪ B |∆
n i
Γ ⊢ M:∪ B | β:∪ A,α:B,∆ Γ ⊢ M:∪ B | β:∪ A,∆
(∪E): Γ ⊢ µα.[βm]Mj :B| βn:∪i A,∆ (∪mBj≤∪nAi) Γ ⊢ µβ.[βm]Mj :∪ An |i∆ (∪mBj≤∪nAi)
n i n i
WewriteΓ ⊢∩∪ M:A|∆ifthisstatementisderivableusingtheserules.
λµ
Wewillnormallynotdistinguish betweenthetwovariants of(∪E).
Noticethatthetraditional (→E)ofDefinition2.3isobtained bytaking n=1. Moreover, all A can
i
beintersectiontypes,soeachcanbe⊤;thisiswhythatruleisnotformulatedusingΓ ⊢∩∪ N:∩ A |∆.
λµ n i
If x:∪ B ∈Γ, then we can only derive Γ ⊢∩∪ x:∪ B |∆, i.e. we have no way of eliminating a union
m j λµ m j
assigned to a term variable. Moreover, we have no traditional rules (∪I) and (∪E) on terms, which
wouldbeformulated (asin[9]),via
Γ ⊢ M:A|∆ Γ ⊢ N:A∪B|∆ Γ,x:A ⊢ M:C|∆ Γ,x:B ⊢ M:C|∆
(∪I): (∪E):
Γ ⊢ M:A∪B|∆ Γ ⊢ M[N/x]:C|∆
Thesecreatethesubject-reduction problem dealtwithinthatpaperbylimitingtoparallelreduction.
Notice that both the strict system for the λ-calculus and the system for λµ are true subsystems; the
firstbynotallowinguniontypes, oralternative conclusions, thesecondbylimitingtoCurrytypes.
Lemma3.3 (GENERATIONLEMMA) • IfΓ ⊢∩∪ x:A|∆,thenthereexists x:B∈Γsuchthat B≤ A.
λµ
• If Γ ⊢∩∪ λx.M:A|∆, then there exists B,C (∀i∈n) such that A = ∩ (B→C), and, for all
λµ i i n i i
i∈n,Γ,x:B ⊢∩∪ M:C |∆.
i λµ i
• If Γ ⊢∩∪ MN:A|∆, then A = ∪ A, and for every i∈n there exists B ∈T such that Γ ⊢∩∪
λµ n i i λµ
M:∪ (B→A )|∆andΓ ⊢∩∪ N:B |∆.
n i i λµ i
• If Γ ⊢∩∪ µα.[β]M:A|∆, then there are A (∀i∈n) such that A = ∩ A , and, for every i∈n,
λµ i n i
therearemi,m′i withm′i≤mi and Bij (∀j∈mi)suchthatΓ ⊢λ∩µ∪ M:∪m′iBki |β:∪miBij,α:Ai,∆.
Proof. Byeasyinduction.
SteffenvanBakel 39
The system ⊢∩∪ does not have choice, i.e. we cannot show that, if Γ ⊢∩∪ M:A∪B|∆, then either
λµ λµ
Γ ⊢∩∪ M:A|∆ofΓ ⊢∩∪ M:B|∆aswouldholdinanintuitionistic system. Take:
λµ λµ
(∩E)
x:A ⊢ x:A| β:B,δ:A∪(A→B)
(∪E)
x:A ⊢ µβ.[δ]x:B|δ:A∪(A→B)
(→R)
⊢ λx.µβ.[δ]x:A→B|δ:A∪(A→B)
(∪E)
⊢ µδ.[δ](λx.µβ.[δ]x): A∪(A→B)|
Notice that we cannot derive ⊢∩∪ µδ.[δ](λx.µβ.[δ]x) : A|, nor ⊢∩∪ µδ.[δ](λx.µβ.[δ]x) : A→B|,
λµ λµ
since the twooccurrences of [δ] need tobe typed differently, but withrelated types. Thisiscomparable
toboth Aand A→Btobeneededasassumption for xtotypeλx.xx.
Wecanshowthatageneral (∩E)(forallterms)isadmissible.
Lemma3.4 IfΓ ⊢∩∪ M:∩ A |∆,thenΓ ⊢∩∪ M:A |∆,foralli∈n.
λµ n i λµ i
Proof. Easy.
Thefollowingresultisstandard.
Lemma3.5 (THINNING &WEAKENING) 1. Let Γ ⊢∩∪ M:A|∆; take Γ′ = {x:B∈Γ | x∈fv(M)}
λµ
and∆′ ={α:B∈∆|α∈fv(M)},thenΓ′ ⊢∩∪ M:A|∆′.
λµ
2. LetΓ ⊢∩∪ M:A|∆,andΓ′≤Γand∆≤∆′,thenΓ′ ⊢∩∪ M:A|∆′.
λµ λµ
Proof. Byeasyinduction.
Asaconsequence, thefollowingrulesareadmissible:
Γ ⊢ M:A|∆
(T):
{x:B∈Γ|x∈fv(M)} ⊢ M:A|{α:B∈∆|α∈fv(M)}
Γ ⊢ M:A|∆
(W): (Γ′≤Γ,∆≤∆′)
Γ′ ⊢ M:A|∆′
4 Subject reduction and expansion
Wewillnowshowourmainresults,byshowingthatournotionoftypeassignmentissoundandcomplete.
Westartbyshowingtwovariantsofthesubstitution lemma.
Lemma4.1 (TERMSUBSTITUTION LEMMA) Let A be strict; Γ ⊢∩∪ M[N/x]:A|∆ if and only if there
λµ
exists C∈T suchthatΓ,x:C ⊢∩∪ M:A|∆andΓ ⊢∩∪ N:C|∆.
λµ λµ
Proof. Byinduction on M.
M≡x: ⇒: IfΓ ⊢∩∪ x[N/x]:A,thenΓ,x:A ⊢∩∪ x:AandΓ ⊢∩∪ N:A.
λµ λµ λµ
⇐: If Γ ⊢∩∪ x:A|∆, then there exists A (∀i∈n) such that A = A from some k∈n, and
λµ i k
Γ=Γ′,x:∩ A ,soΓ′,x:∩ A ⊢∩∪ x:A |∆. FromΓ ⊢∩∪ N:∩ A |∆andLemma3.4,we
n i n i λµ k λµ n i
haveΓ ⊢∩∪ N:A|∆,soΓ ⊢∩∪ x[N/x]:A|∆.
λµ λµ
40 SoundandCompleteTypingforλµ
M≡y6=x: ⇒: ByLemma3.5,since y[N/x]≡y,and x6∈fv(y).
⇐: Γ ⊢∩∪ y[N/x]:A|∆⇒Γ ⊢∩∪ y:A|∆. TakeC=⊤;byLemma3.5,Γ,x:⊤ ⊢∩∪ y:A|∆.
λµ λµ λµ
M′ = M1M2 : Let A=∪rAj,withr≥1. Noticethat(M1M2)[N/x]= M1[N/x]M2[N/x].
⇒: Then,byLemma3.3,thereareDj∈T (∀j∈r)suchthatΓ ⊢λ∩µ∪ M1[N/x]:∪r(Dj→Aj)|∆
and Γ ⊢λ∩µ∪ M2[N/x]:Dj |∆, for all j∈r. Thenby induction, there are C1,C21,...,C2r such
that:
• Γ,x:C1 ⊢λ∩µ∪ M1:∪r(Dj→Aj)|∆andΓ ⊢λ∩µ∪ N:C1|∆,aswellas
• Γ,x:C2j ⊢λ∩µ∪ M2:Dj |∆andΓ ⊢λ∩µ∪ N:C2j |∆,forall j∈r.
TakeC=C1∩C21∩···∩C2r;thenbyweakeningand(→E),wegetΓ,x:C ⊢λ∩µ∪ M1M2:A|∆;
noticethat Γ ⊢∩∪ N:C|∆by(∩I).
λµ
⇐: If Γ,x:C ⊢λ∩µ∪ M1M2:∪rAj |∆, then by Lemma 3.3 there exists Dj ∈T (j∈r) such that
Γ,x:C ⊢λ∩µ∪ M1:∪r(Dj→Aj)|∆ and Γ,x:C ⊢λ∩µ∪ M2:Dj |∆, for j∈r. Then, by induction,
Γ ⊢λ∩µ∪ M1[N/x]:∪r(Dj→Aj)|∆and Γ ⊢λ∩µ∪ M2[N/x]:Dj |∆forall j∈r;theresultfol-
lowsby(→E).
M≡λy.M′; M≡µα.[β]M′: Byinduction.
Because of Lemma 3.4, we can extend the above results also to the case that A is an intersection
type;noticethatthisisimplicitly usedinthethirdcase,where D canbeanintersection type.
j
Dually,wehave:
Lemma4.2 (STRUCTURAL SUBSTITUTION LEMMA) Γ ⊢λ∩µ∪ M′[N·γ/α]:C|γ:∪nBi,∆ifandonlyifthere
are A (∀i∈n) such that for every A there exists a B such that, for all i∈n, Γ ⊢∩∪ N:A |∆, and
i i i λµ i
Γ ⊢∩∪ M′:C|α:∪ (A →B),∆.
λµ n i i
Proof. Weonlyshowtheinteresting cases.
M′ =x: Then x[N·γ/α]=x;asabovetheresultfollows,ineitherdirection, bythinning andweaken-
ing.
M′ =λx.M: Byinduction.
M′ = M1M2 : Then M1M2[N·γ/α]= M1[N·γ/α] M2[N·γ/α];assumeCisstrict.
⇒: Let C = ∪ C , with r ≥ 1. Then, by Lemma 3.3, there exists D ∈T (∀j∈r) such that
r j j
Γ ⊢λ∩µ∪ M1[N·γ/α]:∪r(Dj→Cj)|γ:∪mBj,∆ and Γ ⊢λ∩µ∪ M2[N·γ/α]:Dj |γ:∪mBj,∆, for
j∈r. Thenbyinduction, thereare A (∀i∈k)and A′ (∀i∈l)withk+l=msuchthat
i i
• Γ ⊢λ∩µ∪ M1:∪r(Dj→Cj)|α:∪k(Ai→Bi),∆ and, for all i∈k, Γ ⊢λ∩µ∪ N:Di |∆, as well
as
• Γ ⊢λ∩µ∪ M2:Dj |α:∪l(A′i→Bi),∆forall j∈r,and,foralli∈l,Γ ⊢λ∩µ∪ N:A′i |∆.
Thenbyweakeningand(→E),wegetΓ ⊢λ∩µ∪ M1M2:∪rCj |α:∪k(Ai→Bi)∪∪l(A′i→Bi),∆;
noticethat Γ ⊢∩∪ N:F|∆forall F∈{A (∀i∈k), A′ (∀i∈l)}.
λµ i i
⇐: If Γ ⊢λ∩µ∪ M1M2:∪rCj |α:∪n(Ai→Bi),∆, then A = ∪nAi, and there are Dj ∈T (∀j∈r)
such that Γ ⊢λ∩µ∪ M1:∪r(Dj→Cj)|α:∪n(Ai→Bi),∆ and Γ ⊢λ∩µ∪ M2:Dj |α:∪n(Ai→Bi),∆,
for j ∈r. Then, by induction, Γ ⊢λ∩µ∪ M1[N·γ/α]:∪r(Dj→Cj)|γ:∪mBj,∆ and Γ ⊢λ∩µ∪
M2[N·γ/α]:Dj |γ:∪mBj,∆forall j∈r;theresultfollowsby(→E).