Higher-Order Termination: from Kruskal to Computability

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by  Frédéric Blanqui

Higher-Order Termination:
From Kruskal to Computability
Fre´de´ricBlanqui1,Jean-PierreJouannaud2⋆,andAlbertRubio3
1 INRIA&LORIA,BP101,54602Villiers-le´s-NancyCEDEX,France
7 2 LIX,E´colePolytechnique,91400Palaiseau,France
0 3 TechnicalUniversityofCatalonia,PauGargallo5,08028Barcelona,Spain
0
2

n
1 Introduction
a
J
5 Terminationisamajorquestioninbothlogicandcomputerscience.Inlogic,termina-
tionisattheheartofprooftheorywhereitisusuallycalledstrongnormalization(ofcut

] elimination).Incomputerscience,terminationhasalwaysbeenanimportantissuefor
O
showingprogramscorrect.Intheearlydaysoflogic,strongnormalizationwasusually
L
shownbyassigningordinalstoexpressionsinsuchawaythateliminatingacutwould
.
s yieldan expressionwith a smallerordinal.Inthe earlydaysofverification,computer
c
scientists used similar ideas, interpreting the arguments of a program call by a natu-
[
ralnumber,suchastheirsize.Showingthesizeoftheargumentstodecreaseforeach

2 recursivecallgivesaterminationproofoftheprogram,whichishoweverratherweak
v
sinceitcanonlyyieldquitesmallordinals.Inthesixties,Taitinventedanewmethod
9
forshowingcuteliminationofnaturaldeduction,basedona predicateovertheset of
3
0 terms, such that the membership of an expression to the predicate implied the strong
9 normalizationpropertyforthatexpression.Thepredicatebeingdefinedbyinductionon
0 types,orevenasafixpoint,thismethodcouldyieldmuchlargerordinals.Latergener-
6
alizedbyGirardunderthenameofreducibilityorcomputabilitycandidates,itshowed
0
very effective in proving the strong normalization property of typed lambda-calculi
/
s withpolymorphictypes,dependenttypes,inductivetypes,andfinallyacumulativehi-
c
: erarchyof universes. On the programmingside, research on termination shifted from
v
programming to executable specification languages based on rewriting, and concen-
i
X tratedonautomatablemethodsbasedontheconstructiononwell-foundedorderingsof
r thesetofterms.ThemilestonehereisDershowitz’srecursivepathordering(RPO),in
a thelateseventies,whosewell-foundednessproofisbasedonapowerfulcombinatorial
argument, Kruskal’s tree theorem, which also yields rather large ordinals. While the
computabilitypredicatesmustbedefinedforeachparticularcase,andtheirproperties
provedbyhand,therecursivepathorderingcanbeeffectivelyautomated.
These two methods are completely different. Computability arguments show ter-
mination, that is, infinite decreasing sequences of expressions e0 ≻ e1 ≻ ...en ≻
en+1...donotexist.Kruskal’sbasedargumentsshowwell-orderedness:foranyinfi-
nitesequenceofexpressions{e } ,thereisapairj <ksuchthate (cid:22)e .Itiseasyto
i i j k
seethatwell-orderednessimpliestermination,buttheconverseisnottrue.
⋆ProjectLogiCal,PoˆleCommundeRechercheenInformatiqueduPlateaudeSaclay,CNRS,
E´colePolytechnique,INRIA,Universite´Paris-Sud.
In the late eighties, a new question arose: termination of a simply-typed lambda-
calculus language in which beta-reduction would be supplemented with terminating
first-orderrewriterules.Breazu-TannenandGallierontheonehand[12],andOkada[23]
ontheotherhand,showedthatterminationwassatisfied bythecombinationbyusing
computability arguments. Indeed, when rewriting operates at basic types and is gen-
eratedbyfirst-orderrewriterules, beta-reductionandrewritingdonotinterfere.Their
result, proved for a polymorphic λ-calculus, was later generalized to the calculus of
constructions[1].Thesituationbecomesradicallydifferentwithhigher-orderrewriting
generatedby rulesoperatingonarrow-types,or involvinglambda-bindingsor higher-
order variables. Such an example is providedby Go¨del’s system T, in which higher-
orderprimitiverecursionfornaturalnumbersgeneratedbyPeano’sconstructors0and
sisdescribedbythefollowingtwohigher-orderrules:
rec(0,U,V)→U
rec(s(X),U,V)→@(V,X,rec(X,U,V))
whererecisafunctionsymboloftypeN→T →(N→T →T)→T,U isahigher-
ordervariableoftypeT andV ahigher-ordervariableoftypeN→T →T,foralltype
T.JouannaudandOkadainventedtheso-calledgeneral-schema[17],apowerfulgener-
alizationofGo¨del’shigher-orderprimitiverecursionofhighertypes.Followingthepath
initiatedbyBreazu-TannenandGallierontheonehand,andOkadaontheotherhand,
terminationofcalculibasedonthegeneralschemawasprovedbyusingcomputability
argumentsas well [2,17,18].The generalschema was then reformulatedby Blanqui,
JouannaudandOkada[3,4]inordertoincorporatecomputabilityargumentsdirectlyin
its definition,openingthe way to new generalizations.Go¨del’ssystem T can be gen-
eralized in two ways, by introducingtype constructorsand dependenttypes, yielding
theCalculusofConstructions,andbyintroducingstrictlypositiveinductivetypes.Both
togetheryield theCalculusofInductiveConstructions[24],thetheoryunderlyingthe
Coqsystem[14],inwhichrewriteruleslikestrongeliminationoperateontypes,raising
newdifficulties.Blanquigaveageneralizationofthegeneralschemawhichincludesthe
CalculusofInductiveConstructionsasaparticularcaseunderthenameofCalculusof
AlgebraicConstructions[6,7].
Thegeneralschema,however,istoosimpletoanalyzecomplexcalculidefinedby
higher-order rewrite rules such as encodings of logics. For that purpose, Jouannaud
and Rubio generalized the recursive path ordering to the higher-order case, yielding
the higher-order recursive path ordering (HORPO) [19]. The RPO well-foundedness
prooffollowsfromKruskal’stree theorem,butno such theoremexistsin presenceof
a binding construct, and it is not at all clear that such a theorem may exist. What is
remarkable is that computability argumentsfit with RPO’s recursive structure. When
applied to RPO, these arguments result in a new, simple, well-foundedness proof of
RPO.Onecouldevenarguethatthisisthefirstwell-foundednessproofofRPO,since
Dershowitzshowedmore:well-orderedness.
Combining the general schema and the HORPO is indeed easy because their ter-
minationpropertiesarebothbasedoncomputabilityarguments.Theresultingrelation,
HORPO withclosure,combinesanorderingrelationwitha membershippredicate.In
thispaper,wereformulateandimprovearecentideaofBlanqui[9]bydefininganew
versionoftheHORPOwithclosurewhichintegratessmoothlytheideaofthegeneral
schemaintoHORPOintheformofaneworderingdefinition.
Sofar,wehaveconsideredthekindofhigher-orderrewritingdefinedbyusingfirst-
order pattern matching as in the calculus of constructions. These orderings need to
containβ-andη-reductions.Showingterminationofhigher-orderrewriterulesbasedon
higher-orderpatternmatching,thatis,rewritingmoduloβandηnowusedasequalities,
turnsouttorequiresimplemodificationsofHORPO[20].Wewillthereforeconcentrate
hereonhigher-orderorderingscontainingβ-andη-reductions.
We introducehigher-orderalgebrasin Section 2. In Section 3, we recallthe com-
putability argument for this variation of the simply typed lambda calculus. Using a
computability argument again, we show in Section 4 that RPO is well-founded. We
introducethegeneralschemainsection5,andtheHORPOinSection6beforetocom-
binebothinSection7.Weendupwithrelatedworkandopenproblemsinthelasttwo
sections.
2 Higher-Order Algebras
The notion of a higher-order algebra given here is the monomorphic version of the
notion of polymorphic higher-order algebra defined in [21]. Polymorphism has been
ruledoutforsimplicity.
2.1 Types,SignaturesandTerms
GivenasetSofsortsymbolsofafixedarity,denotedbys:∗n ⇒∗,thesetT oftypes
S
isgeneratedfromthesesetsbythearrowconstructor:
T :=s(Tn)|(T →T )
S S S S
fors:∗n ⇒∗ ∈S
Typesheadedby→arearrowtypeswhiletheothersarebasictypes.Typedeclarations
are expressions of the form σ1 × ··· × σn → σ, where n is the arity of the type
declaration,andσ1,...,σn,σaretypes.Atypedeclarationisfirst-orderifitusesonly
sorts,otherwisehigher-order.
Weassumegivenasetoffunctionsymbolswhicharemeanttobealgebraicopera-
tors.Eachfunctionsymbolf isequippedwithatypedeclarationf:σ1×···×σn →σ.
WeuseF forthesetoffunctionsymbolsofarityn.F isafirst-ordersignatureifall
n
itstypedeclarationsarefirst-order,andahigher-ordersignatureotherwise.
ThesetofrawtermsisgeneratedfromthesignatureF andadenumerablesetX of
variablesaccordingtothegrammar:
T :=X |(λX.T)|@(T,T)|F(T,...,T).
Termsgeneratedbythefirsttwogrammarrulesarecalledalgebraic.Termsoftheform
λx.u are called abstractionswhile terms of the form @(u,v) are called applications.
Theterm@(u,v)iscalleda(partial)left-flatteningof@(...@(@(u,v1),v2),...,vn),
withubeingpossiblyanapplicationitself.Termsotherthanabstractionsaresaidtobe
neutral.WedenotebyVar(t)(BVar(t))thesetoffree(bound)variablesoft.Wemay
assumeforconvenience(andwithoutfurthernotice)thatboundvariablesinatermare
alldifferent,andaredifferentfromthefreeones.
Termsareidentifiedwithfinitelabeledtreesbyconsideringλx.,foreachvariablex,
asaunaryfunctionsymbol.Positionsarestringsofpositiveintegers,theemptystring
Λ denotingthe rootposition.Thesubterm of t at positionp is denotedby t| , andby
p
t[u] the result of replacingt| at positionp in t by u. We write s(cid:3)u if u is a strict
p p
subtermofs.We uset[] foratermwithahole,calledacontext.Thenotationswill
p
beambiguouslyusedtodenotealist,amultiset,orasetoftermss1,...,sn.
2.2 TypingRules
Typingrulesrestrictthesetoftermsbyconstrainingthemtofollowaprecisediscipline.
Environmentsaresetsofpairswrittenx : σ,wherexisavariableandσ isatype.Let
Dom(Γ) = {x | x : σ ∈ Γ forsometypeσ}. We assume there is a unique pair of
the formx : σ for everyvariablex ∈ Dom(Γ). Ourtypingjudgmentsare written as
Γ ⊢ M : σ ifthetermM canbeprovedtohavethetypeσ intheenvironmentΓ.A
termM hastypeσ in theenvironmentΓ if Γ ⊢ M : σ isprovablein theinference
system of Figure 1. A term M is typable in the environmentΓ if there exists a type
σ such that M has type σ in the environmentΓ. A term M is typable if it is typable
in some environmentΓ. Note that function symbols are uncurried,hence must come
alongwithalltheirarguments.
Functions:
Variables: f :σ1×...×σn→σ
x:σ∈Γ
Γ ⊢ t1 :σ1 ... Γ ⊢ tn :σn
Γ ⊢ x:σ
Γ ⊢ f(t1,...,tn):σ
Abstraction: Application:
Γ ∪{x:σ} ⊢ t:τ Γ ∪{x:σ} ⊢ s:σ→τ Γ ⊢ t:σ
Γ ⊢ (λx:σ.t):σ→τ Γ ⊢ @(s,t):τ
Fig.1.Typingjudgmentsinhigher-orderalgebras
2.3 Higher-OrderRewriteRules
Substitutionsarewrittenasin{x1 : σ1 7→ (Γ1,t1),...,xn : σn 7→ (Γn,tn)}where,
foreveryi ∈ [1..n],t isassumeddifferentfromx andΓ ⊢ t :σ .Wealsoassume
i i i i i
that S Γ is an environment. We often write x 7→ t instead of x : σ 7→ (Γ,t), in
i i
particular when t is ground.We use the letter γ for substitutionsand postfix notation
fortheirapplication.Substitutionsbehaveasendomorphismsdefinedonfreevariables.
A(possiblyhigher-order)termrewritingsystemisasetofrewriterulesR={Γ ⊢ l →
i i
r : σ } ,wherel andr arehigher-ordertermssuchthatl andr havethesametype
i i i i i i i
σ intheenvironmentΓ .GivenatermrewritingsystemR,atermsrewritestoaterm
i i
p
t at position p with the rule l → r and the substitution γ, written s−→t, or simply
l→r
s→ t,ifs| =lγandt=s[rγ] .
R p p
p
Atermssuchthats−→tiscalledR-reducible.Thesubterms| isaredexins,and
p
R
tisthereductofs.IrreducibletermsaresaidtobeinR-normalform.Asubstitutionγ
∗
isinR-normalformifxγisinR-normalformforallx.Wedenoteby−→thereflexive,
R
transitiveclosureoftherewriterelation−→.
R
Givenarewriterelation−→,atermsisstronglynormalizingifthereisnoinfinite
sequence of rewrites issuing from s. The rewrite relation itself is strongly normaliz-
ing, or terminating, if all terms are strongly normalizing, in which case it is called a
reduction.
Threeparticular higher-orderequation schemasoriginate fromthe λ-calculus, α-,
β-andη-equality:
λx.v = λy.v{x7→y}ify 6∈BVar(v) ∪ (Var(v)\{x})
α
@(λx.v,u)−→ v{x7→u}
β
λx.@(u,x) −→ u ifx6∈Var(u)
η
Asusual,wedonotdistinguishα-convertibleterms.β-andη-equalitiesareusedas
reductions,whichisindicatedbythelong-arrowsymbolinsteadoftheequalitysymbol.
The above rule-schemasdefine a rewrite system which is known to be terminating,a
resultprovedinSection3.
2.4 Higher-OrderReductionOrderings
Wewillmakeintensiveuseofwell-foundedorderings,usingthevocabularyofrewrite
systemsfororderings,forprovingstrongnormalizationproperties.Forourpurpose,an
ordering,usuallydenotedby≥,isareflexive,symmetric,transitiverelationcompatible
withα-conversion,thatis,s= t≥u= vimpliess≥v,whosestrictpart>isitself
α α
compatible.We will essentially use strictorderings,and hence,the wordorderingfor
them too. We will also make use of order-preservingoperationson relations, namely
multisetandlexicographicextensions,see[15].
Rewrite orderings are monotonic and stable orderings,reduction orderings are in
addition well-founded, while higher-order reduction orderings must also contain β-
andη-reductions.Monotonicityof>isdefinedasu > v impliess[u] > s[v] forall
p p
contextss[] .Stabilityof>isdefinedasu > v impliessγ > tγ forallsubstitutions
p
γ. Higher-orderreduction orderingsare used to prove termination of rewrite systems
includingβ-andη-reductionsbysimplycomparingthelefthandandrighthandsides
oftheremainingrules.
3 Computability
Simplymindedargumentsdonotworkforshowingthestrongnormalizationproperty
ofthesimplytypedlambda-calculus,forβ-reductionincreasesthesizeofterms,which
precludesaninductionontheirsize,andpreservestheirtypes,whichseemstopreclude
aninductionontypes.
Tait’sideais togeneralizethe strongnormalizationpropertyinorderto enablean
inductionontypes.Toeachtypeσ,weassociateasubset[[σ]]ofthesetofterms,called
the computabilitypredicateof type σ, orset of computableterms of typeσ. Whether
[[σ]]containsonlytypabletermsoftypeσ isnotreallyimportant,althoughitcanhelp
intuition.Whatisessentialarethepropertiesthatthefamilyofpredicatesshouldsatisfy:
(i) computabletermsarestronglynormalizing;
(ii) reductsofcomputabletermsarecomputable;
(iii) aneutraltermuiscomputableiffallitsreductsarecomputable;
(iv) u:σ →τ iscomputableiffsois@(u,v)forallcomputablev.
A(non-trivial)consequenceofallthesepropertiescanbeaddedtosmooththeproof
ofthecomingMainLemma:
(v) λx.uiscomputableiffsoisu{x7→v}forallcomputablev.
Apartfrom(v),theabovepropertiesrefertoβ-reductionviathenotionsofreduct
and strong normalization only. Indeed, various computability predicates found in the
literatureusethesamedefinitionparametrizedbytheconsideredreductionrelation.
Thereare severalwaysto define a computabilitypredicate bytaking as its defini-
tion some of the propertiesthat it should satisfy. For example, a simple definition by
inductionontypesisthis:
s:σ ∈[[σ]]forσbasiciffsisstronglynormalizing;
s:θ →τ ∈[[σ →τ]]iff@(s,u):τ ∈[[τ]]foreveryu:θ ∈[[θ]].
Analternativeforthecaseofbasictypeis:
s:σ ∈[[σ]]iff∀t:τ .s−→tthent∈[[τ]].
This formulation defines the predicate as a fixpoint of a monotonic functional. Once
the predicate is defined, it becomes necessary to show the computability properties.
Thisusesaninductionontypesinthefirstcaseoraninductiononthedefinitionofthe
predicateinthefixpointcase.
Tait’sstrongnormalizationproofisbasedonthefollowingkeylemma:
Lemma1 (Main Lemma). Let s be an arbitrary term and γ be an arbitrary com-
putablesubstitution.Thensγiscomputable.
Proof. Byinductiononthestructureofterms.
1. sisavariable:sγiscomputablebyassumptiononγ.
2. s = @(u,v). Since uγ and vγ are computable by induction hypothesis, sγ =
@(uγ,vγ)iscomputablebycomputabilityproperty(iv).
3. s = λx.u.Bycomputabilityproperty(v),sγ = λx.uγ iscomputableiffuγ{x 7→
v} iscomputableforan arbitrarycomputablev. Letnowγ′ = γ ∪{x 7→ v}.By
definition of substitutions for abstractions, uγ{x 7→ v} = uγ′, which is usually
ensuredbyα-conversion.Byassumptionsonγandv,γ′iscomputable,anduγ′is
thereforecomputablebythemaininductionhypothesis. ⊓⊔
Sinceanarbitrarytermscanbeseenasitsowninstancebytheidentitysubstitution,
which is computable by computabilityproperty (iii), all terms are computableby the
MainLemma,hencestronglynormalizingbycomputabilityproperty(i).
4 The Recursive PathOrdering and Computability
Inthissection,werestrictourselvestofirst-orderalgebraicterms.Assumingthattheset
offunctionsymbolsisequippedwithanorderingrelation≥ ,calledprecedence,anda
F
statusfunctionstat,writingstat forstat(f),werecallthedefinitionoftherecursive
f
pathordering:
Definition1. s≻ tiff
rpo
1. s=f(s)withf ∈F,andu(cid:23)tforsomeu∈s
rpo
2. s=f(s)withf ∈F,andt=g(t)withf > g,andA
F
3. s=f(s)andt=g(t)withf = g ∈Mul,ands(≻) t
F mul
rpo
4. s=f(s)andt=g(t)withf = g ∈Lex,ands(≻) tandA
F lex
rpo
whereA=∀v ∈t.s ≻ v and s (cid:23) tiffs ≻ tors=t
rpo rpo rpo
Wenowshowthewell-foundednessof≻ byusingTait’smethod.Computability
rpo
isdefinedhereasstrongnormalization,implyingcomputabilityproperty(i).Weprove
thecomputabilityproperty:
(vii)Letf ∈F andsbecomputableterms.Thenf(s)iscomputable.
n
Proof. Therestrictionof≻ totermssmallerthanorequaltothetermsinsw.r.t.≻
rpo rpo
is a well-foundedorderingwhich we use for building an outer induction on the pairs
(f,s)orderedby(> ,(≻ ) ) .Thisorderingiswell-founded,sinceitisbuilt
F rpo statf lex
fromwell-foundedorderingsbyusingmappingsthatpreservewell-foundedorderings.
Wenowshowthatf(s)iscomputablebyprovingthattiscomputableforalltsuch
thatf(s) ≻ t. Thispropertyis itself provedbyan (inner)inductionon |t|, and by
rpo
caseanalysisupontheproofthatf(s)≻ t.
rpo
1. subtermcase:∃u ∈ ssuchthatu ≻ t.Byassumption,uiscomputable,hence
rpo
soisitsreductt.
2. precedencecase:t=g(t),f > g,and∀v ∈t,s≻ v.Byinnerinduction,vis
F rpo
computable,hencesoist.Byouterinduction,g(t)=tiscomputable.
3. multiset case: t = f(t) with f ∈ Mul, and s(≻ ) t. By definition of the
rpo mul
multisetextension,∀v ∈t, ∃u∈ssuchthatu(cid:23) v.Sincesisavectorofcom-
rpo
putabletermsbyassumption,soist.Weconcludebyouterinductionthatf(t)=t
iscomputable.
4. lexicographiccase:t=f(t)withf ∈Lex,s(≻ ) t,and∀v ∈t, s≻ v.By
rpo lex rpo
innerinduction,tisstronglynormalizing,andbyouterinduction,soisf(t)=t.
⊓⊔
Thewell-foundednessof≻ followsfromcomputabilityproperty(vii).
rpo
5 The General Schema and Computability
Asintheprevioussection,weassumethatthesetoffunctionsymbolsisequippedwith
aprecedencerelation≥ andastatusfunctionstat.
F
Definition2. ThecomputabilityclosureCC(t = f(t)),withf∈F,isthesetCC(t,∅),
s.t. CC(t,V), withV ∩Var(t) = ∅,isthesmallestsetoftypabletermscontainingall
variablesinV andtermsint,closedunder:
1. subterm of basic type: let s ∈ CC(t,V), and u be a subterm of s of basic type σ
suchthatVar(u)⊆Var(t);thenu∈CC(t,V);
2. precedence:letf > g,ands∈CC(t,V);theng(s)∈CC(t,V);
F
3. recursive call: let f(s) be a term such that terms in s belong to CC(t,V) and
t(−→ ∪(cid:3)) s;theng(s)∈CC(t,V)foreveryg = f;
β statf F
4. application:lets : σ1 → ... → σn → σ ∈ CC(t,V)andui : σi ∈ CC(t,V)for
everyi∈[1..n];then@(s,u1,...,un)∈CC(t,V);
5. abstraction:lets∈CC(t,V∪{x})forsomex∈/ Var(t)∪V;thenλx.s∈CC(t,V);
6. reduction:letu∈CC(t,V),andu−→ v;thenv ∈CC(t,V).
β∪(cid:3)
WesaythatarewritesystemRsatisfiesthegeneralschemaiff
r ∈CC(f(l))forallf(l)→r ∈R
We now considercomputabilitywith respectto the rewrite relation−→ ∪−→ ,
R β
and add the computability property(vii) whose proof can be easily adapted from the
previousone.WecanthenaddanewcaseinTait’sMainLemma,fortermsheadedby
analgebraicfunctionsymbol.Asaconsequence,therelation−→ ∪−→ isstrongly
β R
normalizing.
Example1 (System T). We show the strong normalization of Go¨del’s system T by
showingthatitsrulessatisfy thegeneralschema.Thisisclear forthefirstrulebythe
baseCaseofthedefinition.Forthesecondrule,wehave:V ∈CC(rec(s(X),U,V))by
baseCase;s(X)∈CC(rec(s(X),U,V))bybaseCaseagain,andX ∈CC(rec(s(X),U,V))
byCase2,assumingrec > s;U ∈ CC(rec(s(X),U,V))bybaseCase,henceallar-
F
gumentsoftherecursivecallareinCC(rec(s(X),U,V)).Sinces(X)(cid:3)X holds,we
haverec(X,U,V)∈CC(rec(s(X),U,V)).Therefore,weconcludewith@(V,X,rec(X,U,V))∈CC(rec(s(X),U,V))
byCase4.
6 The Higher-Order Recursive PathOrdering
6.1 TheIngredients
– A quasi-ordering on types ≥ called the type ordering satisfying the following
TS
properties:
1. Well-foundedness:> iswell-founded;
TS
2. Arrowpreservation:τ →σ = αiffα=τ′ →σ′, τ′ = τ andσ = σ′;
TS TS TS
3. Arrow decreasingness: τ → σ > α implies σ ≥ αorα = τ′ →
TS TS
σ′,τ′ = τ andσ > σ′;
TS TS
4. Arrowmonotonicity:τ ≥ σ impliesα → τ ≥ α → σ andτ → α ≥
TS TS TS
σ →α;
Aconvenienttypeorderingisobtainedbyrestrictingthesubtermpropertyforthe
arrowintheRPOdefinition.
– Aquasi-ordering≥ onF,calledtheprecedence,suchthat> iswell-founded.
F F
– Astatusstat ∈{Mul,Lex}foreverysymbolf ∈F.
f
Thehigher-orderrecursivepathordering(HORPO)operatesontypingjudgments.
Toeasethereading,wewillhoweverforgettheenvironmentandtypeunlessnecessary.
Let
A=∀v ∈t s ≻ voru ≻ vforsomeu∈s
horpo horpo
Definition3. GiventwojudgmentsΓ ⊢ s:σandΣ ⊢ t:τ,
Σ Σ
s ≻ tiffσ ≥ τ and
TS
horpo
1. s=f(s)withf ∈F,andu (cid:23) tforsomeu∈s
horpo
2. s=f(s)withf ∈F,andt=g(t)withf > g,andA
F
3. s=f(s)andt=g(t)withf = g ∈Mul,ands( ≻ ) t
F mul
horpo
4. s=f(s)andt=g(t)withf = g ∈Lex,ands( ≻ ) tandA
F lex
horpo
5. s=@(s1,s2),ands1 (cid:23) tors2 (cid:23) t
horpo horpo
6. s=λx:α.uwithx6∈Var(t),andu (cid:23) t
horpo
7. s=f(s)withf ∈F,t=@(t)isapartialleft-flatteningoft,andA
8. s=f(s)withf ∈F,t=λx:α.vwithx6∈Var(v)ands ≻ v
horpo
9. s=@(s1,s2),t=@(t),and{s1,s2}( ≻ )mul t
horpo
10. s=λx:α.u, t=λx:β.v,α= β,andu ≻ v
TS
horpo
11. s=@(λx:α.u,v)andu{x7→v} (cid:23) t
horpo
12. s=λx:α.@(u,x), x6∈Var(u)andu (cid:23) t
horpo
Example2 (System T). The new proof of strong normalization of System T is even
simpler.Forthefirstrule,weapplyCase1.Forthesecond,weapplyCase7,andshow
recursively that rec(s(X),U,V)≻ V by Case 1, rec(s(X),U,V)≻ X by
horpo horpo
Case1appliedtwice,andrec(s(X),U,V)≻ rec(X,U,V)byCase3,assuminga
horpo
multisetstatusforrec,whichfollowsfromthecomparisons(X)≻ X byCase1.
horpo
The strong normalization proof of HORPO is in the same style as the previous
strongnormalizationproofs,althoughtechnicallymorecomplex[21].Thisproofshows
that HORPO and the general schema can be combined by replacing the membership
u∈susedincase1bythemoregeneralmembershipu∈CC(f(s)).Itfollowsthatthe
HORPOmechanismisinherentlymoreexpressivethantheclosuremechanism.
BecauseofCases11and12,HORPO isnottransitive.Indeed,thereareexamples
forwhichtheproofofs≻+ trequiresguessingamiddletermusuchthats≻ u
horpo horpo
andu≻ t.Guessingamiddletermwhennecessaryisautomatedintheimplemen-
horpo
tations of HORPO and HORPO with closure available fromthe web page of the first
twoauthors.
7 Unifying HORPOandthe Computability Closure
A major advantage of HORPO over the general schema is its recursive structure. In
contrast,themembershiptothecomputabilityclosureisundecidableduetoitsCase3,
butdoesnotinvolveanytypecomparison.Tocombinetheadvantagesofboth,wenow
incorporatetheclosureconstructionintotheHORPOasanordering.Besides,wealso
incorporatethepropertythatargumentsofatypeconstructorarecomputablewhenthe
positivity condition is satisfied as it is the case for inductive types in the Calculus of
InductiveConstructions[7,24].
s:σ ≻ t:τ iff
X
horpo s=f(s) ≻ t iff
Var(t)⊆Var(s) and comp
∅ 1.t∈X
1.s=f(s)ands ≻ t
comp 2.∃si ∈Acc(s): si(cid:23)Xcompt
2.s=f(s)andσ ≥TS τ and 3.t=g(t), f >TS gand
(a)t=g(t), f > gandA
F X
(b) t=g(t), f = g, ∀v ∈t: s ≻ v
F comp
s(ho≻rpo)statftandA 4.t=g(t), f =TS g,
(c) t=@(t1,t2)andA ∀v ∈t :s ≻X vand
3.s=@(s1,s2),σ ≥TS τ and comp
(a)t=@(t1,t2)and Acc(s)(ho≻rpo)statfλX.t
{s1,s2}( ≻ )mul{t1,t2} 5.t=@(u,v),
horpo
(b) s1 (cid:23) tors2 (cid:23) t s ≻X uands ≻X v
horpo horpo comp comp
(c) s1 =λx.uand 6.t=λx:α.uand
u{x7→s2} (cid:23) t X·{x:α}
horpo s ≻ u
4.s=λx:α.u,σ ≥ τ and comp
TS
(a)t=λx:β.v,α= β
TS wheres ∈Acc(f(s))
andu ≻ v i
horpo (siisaccessibleins)
(b) x6∈Var(t)andu (cid:23) t iff
horpo
1.sisthelefthandsideof
(c) u=@(v,x),x6∈Var(v)
anancestorgoals≻ u
andv (cid:23) t horpo
horpo 2.sisthelefthandsideofthe
currentgoals≻ u, and,
comp
whereA=∀v ∈t: eitherf :σ →σand
s ≻ vor∃u∈s: u ≻ v σoccursonlypositivelyinσ .
i
horpo horpo
Example3. We considernowthetypeofBrouwer’sordinalsdefinedfromthetypeN
bytheequationOrd=0⊎s(Ord)⊎lim(N→Ord).NotethatOrdoccurspositively