Rewriting modulo in Deduction modulo

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Rewriting modulo in Deduction modulo

Frédéric Blanqui

To cite this version:

Frédéric Blanqui. Rewriting modulo in Deduction modulo. Rewriting Techniques and Applications,

14th International Conference, RTA 2003, Jun 2003, Valencia, Spain. �inria-00105625�

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inria-00105625, version 1 - 11 Oct 2006

FrédériBlanqui

Laboratoired'Informatiquedel'ÉolePolytehnique

91128 PalaiseauCedex,Frane

Abstrat. Westudytheterminationofrewritingmoduloasetofequa-

tionsintheCalulusofAlgebraiConstrutions,anextensionoftheCal-

ulusofConstrutionswithfuntionsandprediatesdenedbyhigher-

order rewrite rules. In a previous work, we dened general syntati

onditionsbasedonthenotionofomputabilitylosureforensuringthe

terminationoftheombinationofrewritingandβ^-redution.

Here,weshowthatthisresultispreservedwhenonsideringrewriting

moduloasetof equationsifthe equivalenelasses generated by these

equationsarenite,theequationsare linearandsatisfy generalsynta-

ti onditions also based onthe notion of omputability losure. This

inludesequationslikeassoiativityandommutativityandprovidesan

originaltreatmentofterminationmoduloequations.

1 Introdution

TheCalulusofAlgebraiConstrutions(CAC)[2,3℄isanextensionoftheCal-

ulusofConstrutions(CC)[9℄withfuntionsandprediatesdenedby(higher-

order)rewriterules.CCembodiesin thesameformalismGirard'spolymorphi

λ^-alulusând^De^Bruijn's^dependent^types,^whihâllowsône^to^formalize^propo-

sitionsandproofsof(imprediative)higher-orderlogi.Inaddition,CACallows

funtionsandprediatestobedenedbyanysetof(higher-order)rewriterules.

And, in ontrast with (rst-order) Natural Dedution Modulo [13℄, proofs are

partoftheterms.

Verygeneral onditionsare studied in [2,4℄ for preserving the deidability

oftype-hekingandthelogialonsistenyof suh asystem.Butthese ondi-

tionsdonottakeintoaountrewritingmoduloequationslikeassoiativityand

ommutativity (AC), whih would be very useful in proof assistantslike Coq

[22℄ sine itinreasesautomation and dereasesthe size of proofs. Wealready

used the rewritingengine ofCiME [8℄, whih allowsrewriting modulo AC, for

a prototype implementation of CAC, and now work on a new version of Coq

inludingrewritingmoduloAC.Inthispaper,weextendtheonditionsgivenin

[2℄todealwithrewritingmoduloequations.

2 The Calulus of Algebrai Construtions

Weassume thereaderfamiliar with typed λ^-aluli^[1℄^and ^rewriting^[11℄.^The

CalulusofAlgebraiConstrutions(CAC)[2℄simplyextendsCCbyonsidering

asetF ôf^symbolsând â^setRôf^rewrite ^rules.^The^termsôf^CACâre:

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t, u∈ T ::=s|x|f |[x:t]u|tu|(x:t)u

where s∈ S ={⋆,2}îsâ^sort,x∈ X â^variable, f ∈ F^,[x:t]uânabstration,

tuânappliation, and(x:t)uâ^dependent ^produt,^writtent⇒uîfx^does^not

freelyourin u^.

Thesort ⋆ ^denotes ^the ûniverse ôf ^typesând propositions, and the sort 2

denotestheuniverseofprediatetypes(alsoalledkinds).Forinstane,thetype

nat ôf^natural^numbersîsôf^type⋆^, ⋆îtself îsôf^type2and nat⇒⋆^,^the^type

ofprediatesovernat^,^is^of^type2.

Weuse bold fae letters fordenotingsequenes of terms.Forinstane, t ^is

thesequenet1. . . tn ^where n=|t|îs^the^lengthôf t^,ând(x:T)U îs^the^term (x1:T1). . .(xn:Tn)U ^(weîmpliitlyâssume^that|x|=|T|=n^).

WedenotebyFV(t)^the^setôf^free^variablesôft^,^bydom(θ)^the^domainôfâ

substitutionθ^,^byPos(t)^the^setôf^Dewey's^positionsôft^,^byt|p^the^subtermôf

t ât^position p^,ând^byt[u]p^the^replaementôft|p ^byu^.

Everysymbolf îsêquipped^withâ^sortsf^,ânârityαf ândâ^typeτf ^whih

maybe any losed term of theform (x : T)U ^with |x| =αf^. ^The ^terms^only

builtfromvariablesandappliationsoftheform ft^with|t|=αf ^are^algebrai.

AtypingenvironmentΓ îs ânôrdered^list ôf^typedelarationsx:T^. Îff îs

asymboloftypeτf = (x:T)U^,^we^denote^byΓf ^theenvironmentx:T^.

ArulefortypingsymbolsisaddedtothetypingrulesofCC:

(symb)

⊢τf :sf

⊢f :τf

Arewriteruleisapairl→r^suh^that⁽¹⁾lîsâlgebrai,⁽²⁾lîs^notâ^variable,

and(3)FV(r)⊆FV(l)^.Ônlyl ^has^to ^beâlgebrai:r ^mayôntainappliations, abstrations andproduts.This isapartiularaseofCombinatoryRedution

System(CRS)[18℄whihdoesnotneedhigher-order pattern-mathing.

IfG ⊆ F^,RG îs ^the^setôf^rules^whose ^left-hand ^sideîs^headed^byâ^symbol

in G^.Â ^symbolf ^withR{f} =∅ îsônstant,ôtherwiseîtîs(partially)dened.

Aruleis left-linear (resp.right-linear)if novariableours morethanone

in the left-hand side (resp. right-hand side). A rule is linear if it is both left-

linear and right-linear.A rule is non-dupliating ifno variable ours morein

theright-handsidethanin theleft-hand side.

Atermt R^-rewrites^toâ^termt^′^,^written t→R t^′^, îf^there êxistsâ^position pⁱⁿt^,â^rulel→r∈ Rândâsubstitutionσ ^suh^that t|p=lσ ândt^′=t[rσ]p^.

A termt β^-rêwrites^to â^termt^′^,^written t→β t^′^, îf^thereêxists â^position pⁱⁿ t ^suh ^that t|p = ([x:U]v u)ândt^′ =t[v{x7→u}]p^.^Givenâ^relation→ândâ

termt^,^let→(t) ={t^′∈ T |t→t^′}^.

Finally, in CAC, βR-equivalent typesare identied. Morepreisely, in the typeonversionruleof CC,↓β ^is^replaed^by↓βR^:

(onv)

Γ ⊢t:T T ↓βRT^′ Γ ⊢T^′:s Γ ⊢t:T^′

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whereu↓βRvⁱ^thereêxistsâ^termw^suh^thatu→^∗_βRwândv→^∗_βRw^,→^∗_βR

being the reexiveand transitive losure of →β ∪ →R^. ^This ^rule ^means ^that

anytermt ôf^typeT ⁱⁿ ^theenvironmentΓ îsâlso ôf^type T^′ îfT ândT^′ ^have

aommonredut(andT^′ îsôf^type^some^sorts^).^Fôrînstane,îft îsâ^proofôf P(2 + 2)^thentîsâlsoâ^proofôf P(4)îfRôntains^the^following^rules:

x+ 0 → x x+ (s y) → s(x+y)

Thisdereasesthesize ofproofsandinreasesautomationaswell.

Asubstitutionθ^preserves ^typing^fromΓ ^to∆^, ^writtenθ:Γ ;∆^,îf,^forâll x∈dom(Γ)^, ∆ ⊢xθ :xΓ θ^, ^where xΓ îs^the ^typeâssoiated ^to xⁱⁿ Γ^. ^Type-

preservingsubstitutionsenjoythefollowingimportantproperty:ifΓ ⊢t:T ^and θ:Γ ;∆ ^then∆⊢tθ:T θ^.

For ensuring the subjet redution property (preservation of typing under

redution), everyrulefl→r îsêquipped^withânenvironmentΓ ândâ^substi-

tution ρ^suh ^that,¹ îff : (x:T)U ând γ={x7→l} ^thenΓ ⊢flρ:U γρ ând Γ ⊢r:U γρ^.^Thesubstitution ρâllows^to êliminatenon-linearitiesonlydue to typingand thus makesrewriting moreeientand onuene easierto prove.

Forinstane, the onatenation on polymorphi lists (type list : ⋆ ⇒ ⋆ ^with

onstrutors nil : (A : ⋆)listA ând cons: (A : ⋆)A ⇒ listA⇒ listA⁾ôf ^type (A:⋆)listA⇒listA⇒listAân^be^dened^by:

app A(nil A^′)l^′ → l^′

app A(cons A^′ x l)l^′ → cons A x (app A x l l^′) app A(app A^′ l l^′)l^′′ → app A l(app A l^′ l^′′)

with Γ = A : ⋆, x : A, l : listA, l^′ : listA ând ρ = {A^′ 7→ A}^. ^Forînstane, app A (nil A^′) îs ^not ^typableⁱⁿ Γ ^(sine A^′ ∈/ dom(Γ)⁾ ^but ^beomes ^typable

if weapply ρ^. ^This ^does ^not ^matter ^sine, îf ân înstane app Aσ (nil A^′σ) îs

typablethenAσ ^is^onvertible^toA^′σ^.

3 Rewriting Modulo

Now, weassumegiven asetE ôf êquationsl =r ^whih ^will ^be^seenâsâ^set ôf

symmetrirules,thatis, asetsuhthatl→r∈ E ⁱr→l∈ E^.^The^onditions

onrulesimplythat,ifl=r∈ E^,^then⁽¹⁾^bothl ândrâreâlgebrai,⁽²⁾^bothl

andrâre^headed^byâ^funtion^symbol,⁽³⁾lândr^have^the^same^(free)^variables.

Examplesof equationsare:

x+y ⁼y+x (ommutativityof+)

x+ (y+z)⁼(x+y) +z (assoiativityof+)

x×(y+z)⁼(x×y) + (x×z) (distributivityof×⁾ x+ 0 ⁼x (neutralityof0⁾

1

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add A x(add A^′ y S)⁼add A y (add A^′ x S) union A S S^′ ⁼union A S^′ S

union A S (union A^′ S^′ S^′′)⁼union A(union A^′ S S^′)S^′′

where set : ⋆ ⇒ ⋆^, empty : (A : ⋆)setA^, add : (A : ⋆)A ⇒ setA⇒ setA ând union : (A : ⋆)setA ⇒ setA ⇒ setA ^formalize^nite ^sets ôf êlements ôf ^type A^. Êxept ^for distributivity whih is not linear, and the equation x+ 0 = x

whose equivalene lasses are innite, all the other equations will satisfy our

strongnormalizationonditions.Notehoweverthatdistributivityandneutrality

analwaysbeused as rules when orientedfrom left to right. Hene,the word

problemforabeliangroupsorabelianringsforinstaneanbedeidedbyusing

normalizedrewriting[19℄.

Ontheotherhand,thefollowingexpressionsarenotequationssineleftand

right-handsideshavedistintsetsofvariables:

x×0 ⁼0 ⁽0^is^absorbing^for×⁾ x+ (−x)⁼0 ^(inverse)

Let ∼ ^be ^the ^reexive ând ^transitive ^losure ôf →E ⁽∼ îs ân êquivalene

relation sine E îs ^symmetri). ^We âre ^now înterested ⁱⁿ ^the termination of

=→β∪ ∼→R^(instead^of→β∪ →R ^before).^In^the^following,^we^may^denote

→E ^byE^,→R ^byR^and→β ^byβ^.

Inordertopreserveallthebasipropertiesofthealulus,wedonothange

the shapeof the relationused in the typeonversionrule (onv): twotypesT

and T^′ âre ônvertibleîfT ↓T^′ ^with→=→β ∪ →R ∪ →E^.^But^this ^raises^the

questionofhowtohekthisondition,knowingthat→^may^be^notterminating.

Westudythisproblemin Setion6.

4 Conditions of strong normalization

Inthestrongnormalizationonditions,wedistinguishbetweenrst-ordersym-

bols(set F1⁾ând higher-order symbols (set Fω^). ^Tô ^preisely^dene ^what îsâ

rst-order symbol, we need a little denition before. We say that a onstant

prediatesymbolisprimitiveifitisnotpolymorphiandifitsonstrutorshave

no funtional arguments. This inludes in partiular any rst-order data type

(naturalnumbers,listsofnaturalnumbers,et.).Now,asymbolf ^is^rst-order

ifitisaprediatesymbolofmaximalarity, 2

orifitisafuntion symbolwhose

outputtypeis aprimitiveprediatesymbol.Anyother symbolis higher-order.

LetRι=RFι ^and Eι=EFι ^forι∈ {1, ω}^.

Sine the pioneer works on the ombination of λ^-alulus ^and ^rst-order

rewriting [7,20℄, it is well known that the addition at the objet level of a

strongly normalizingrst-order rewritesystem preservesstrong normalization.

Thisomesfrom thefat thatrst-orderrewritingannotreateβ^-redexes.^On

2

Aprediate symbolf ôf^type(x:T)U îsôf^maximalârity îfU =⋆^,^that îs,îf^the

elementsoftypeft^are^not^funtions.

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the other hand, higher-order rewriting an reate β^-redexes. ^This ^is ^why ^we

have other onditions on higher-order symbols than merely strong normaliza-

tion.Furthermore,in orderfor thetwosystemstobeombinedwithoutlosing

strongnormalization[23℄,wealsorequirerst-orderrulestobenon-dupliating

[21℄.Notehoweverthatarst-ordersymbolanalwaysbeonsideredashigher-

order(butthestrongnormalizationonditionsonhigher-ordersymbolsmaynot

bepowerfulenoughforprovingthetermination ofitsdening rules).

Thestrongnormalizationonditionsonhigher-orderrewriterulesarebased

on the notionof omputability losure [5℄. Weare going to use this notionfor

theequationstoo.

Typedλ^-aluli^are^generally^proved^stronglynormalizingbyusingTaitand Girard's tehnique of omputability prediates/reduibility andidates [14℄. In-

deed, a diret proof of strong normalization by indution on the struture of

termsdoesnotwork.TheideaofTait,laterextendedbyGirardtothepolymor-

phi λ^-alulus, îs ^to ^strengthen ^the îndution ^hypothesis âs^follows.^Toêvery

typeT^,ôneâssoiatesâ^set [[T]]⊆ SN ^(setôf^stronglynormalizingterms),and provesthat everytermoftypeT îsômputable,^that îs,^belongs^to [[T]]^.

Now,ifweextendsuhaaluluswithrewriting,forpreservingstrongnor-

malization,arewriterulehastopreserveomputability.Theomputabilitylo-

sureofatermt îsâ^set ôf^terms^that âreômputable^whenevert îtselfîsôm-

putable.So,iftheright-handsiderôfâ^rulefl→r^belongs^to^theomputability losureofl^,âônditionâlled^the^General^Shema,^thenrîsômputable^when-

everthetermsin l^are^omputable.

Formally,theomputabilitylosure fora rule(fl→ r, Γ, ρ) ^with τf = (x: T)U ândγ={x7→l}îs^the^setôf^termst^suh^that^the^judgment⊢ct:U γρân

bededued fromtherules ofFigure 1,wherethe variablesofdom(Γ)^are ^on-

sideredassymbols(τx=xΓ^),>F îsâwell-foundedquasi-ordering(preedene) onsymbols,withx <Ff ^forâllx∈dom(Γ)^,>f îs^the^multisetôrlexiographi extension

3

ofthesubtermordering

4,andT ↓f T^′ ⁱT ândT^′^haveâômmon

redutby→f=→β∪ →_R^<

f

whereR^<_f ={gu→v∈ R |g <Ff}^.

Inaddition,everyvariable x∈dom(Γ)îs ^required^to ^beâessible ⁱⁿ ^some li^, ^that îs, xσ îs ômputable ^whenever liσ îs ômputable. ^The ârgumentsôf â

onstrutor-headed term are always aessible.Fora funtion-headed term ft

with f : (x:T)Cv ândC ônstant,ônly^the ti^'s^suh^that C ôurs ^positively

in Ti âreâessible⁽X ôurs^positivelyⁱⁿY ⇒X ând^negativelyⁱⁿX ⇒Y^).

Therelation⊢cîs^similar^to^the^typing^relation⊢ôf^CACêxept^that^symbol

appliations are restritedto symbolssmallerthan f^, ^or^to ^arguments^smaller

than l ⁱⁿ ^theâse ôf ân âppliation ôf â^symbolêquivalent^to f^.^So, ^verifying

that arule satises theGeneral Shema amountsto hekwhether r ^has^type U γρ ^with^the^previousrestritionson symbolappliations. Itthereforehasthe sameomplexity.

3

Orasimpleombinationthereof,dependingonthestatusoff^.

4

We use a more powerful ordering for dealing with reursive denitions on types

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Fig.1.Computabilitylosure for(fl→r, Γ, ρ)

(ax)

⊢^c⋆:2

(symb

<

)

⊢cτg:sg

⊢cg:τg

(g <Ff)

(symb

=

)

⊢^cτg:sg δ:Γg ;_c∆

∆⊢^cgyδ:V δ

(τg= (y:U)V, g=^Ff ^andyδ <f l)

(var)

∆⊢cT :s

∆, x:T ⊢cx:T (x /∈dom(∆))

(weak)

∆⊢^cT :s ∆⊢^cu:U

∆, x:T⊢cu:U (x /∈dom(∆))

(abs)

∆, x:U⊢^cv:V ∆⊢^c(x:U)V :s

∆⊢c[x:U]v: (x:U)V

(app)

∆⊢^ct: (x:U)V ∆⊢^cu:U

∆⊢^ctu:V{x7→u}

(prod)

∆, x:U⊢cV :s

∆⊢^c(x:U)V :s

(onv)

∆⊢ct:T ∆⊢cT :s ∆⊢cT^′:s

∆⊢ct:T^′ (T ↓f T^′)

Now, how the omputability losure an help us in dealingwith rewriting

moduloequations?Whenonetriestoprovethateverytermisomputable,inthe

aseofatermft^,îtîs^suient^to^prove^thatêvery^redutôfftîsômputable.

Intheaseofahead-redutflσ→rσ^,^this^follows^from^the^fat^thatr^belongs

totheomputabilitylosureofl^sine,^by^indutionhypothesis,thetermsin lσ

areomputable.

Now, with rewriting modulo, a R^-step ^an^be^preeded ^by E^-steps: ft →^∗_E gu→Rt^′^.^To^apply^the^previous^method^withgu^,^we^must^prove^that^the^terms

inu âreômputable.^Thisân^beâhieved^byâssuming^that^theêquationsâlso

satisfytheGeneralShemainthefollowingsense:anequation(fl→gm, Γ, ρ)

with τg = (x: T)U ând γ={x7→m} ^satises^the ^General^Shemaîf, ^forâll i^,⊢c mi:Tiγρ^,^that îs,^the^termsⁱⁿm ^belong^to^theomputabilitylosureofl^.

Bysymmetry,thetermsinl^belong^to^theomputabilitylosureofm^.

Oneaneasilyhekthatthisonditionissatisedbyommutativity(what-

everthetypeof+îs)ândassoiativity(ifbothy ândz âreâessibleⁱⁿy+z^):

x+y = y+x x+ (y+z) = (x+y) +z

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Forommutativity,thisisimmediateanddoesnotdependonthetypeof+^:

bothy ^andx^belong^to^theomputabilitylosureofx^andy^.

For assoiativity,wemust provethat both x+y ^andz ^belong ^to ^the^om-

putabilitylosureCCôfxândy+z^.Îf^weâssume^that^bothyândzâreâessible

in y+z^(whih îs^theâse^forînstaneîf+ :nat⇒nat⇒nat^),^thenz^belongs

to CCând, ^byûsingâ^multiset^status^forômparing ^theârgumentsôf+^,x+y

belongstoCC^too^sine{x, y}_mul{x, y+z}^.

Wenowgiveallthestrongnormalizationonditions.

Theorem1 (Strongnormalization ofβ∪ ∼R^).^Let∼1 ^be^the ^reexive^and

transitivelosureof E1^. ^The^relation=→β ∪ ∼→R ^is^stronglynormalizing if the followingonditions adaptedfrom [2℄aresatised:

• →=→β∪ →R∪ →E ^is^onuent,⁵

• ^the ^rules^ofR1 ^arenon-dupliating,

6 R1∩ Fω=E1∩ Fω=∅⁷^and∼1→R₁ ^is

stronglynormalizing onrst-orderalgebrai terms,

• ^the ^rules ôfRω ^satisfy^the ^General ^Shemaândâre ^safe,⁸

• ^rules ^on^prediate ^symbols ^have ^no^ritial^pair, ^satisfy ^the^General ^Shema⁹

andare small, 10

andif the following new onditionsaresatisedtoo:

• ^thereîs^no êquationôn ^prediate ^symbols,

• E ^is^linear,

• ^the ^equivalene ^lasses^modulo ∼^are^nite,

• ^every ^rule(fl→gm, Γ, ρ)∈ E ^satises^the ^General ^Shemaⁱⁿ ^the ^following

sense: ifτg = (x:T)U ^andγ={x7→m} ^then, ^for^all i^,⊢cmi:Tiγρ^.

Notallowingequationsonprediatesymbolsisanimportantlimitation.How-

ever, one annot have equations on onnetors if one wants to preserve the

Curry-Howardisomorphism.Forinstane, withommutativityon∧^, ^one^looses

subjetredution.Take∧:⋆⇒⋆⇒⋆^,pair: (A:⋆)(B :⋆)A⇒B⇒A∧B ând π1: (A :⋆)(B : ⋆)A∧B ⇒A ^dened ^by π1 A B (pair A^′ B^′ a b)→a^. ^Then, π1B A (pair A B a b)îsôf ^typeB ^but aîs ^not.

5 Strong normalization proof

Thestrongnormalizationprooffollowstheonegivenin[6℄verylosely.

11

Weonly

givethedenitionsandlemmasthatmustbemodied.Aspreviouslyexplained,

5

Iftherearetype-levelrewriterules.

6

Iftherearehigher-orderrules.

7

First-orderrules/equationsonlyontainrst-ordersymbols.

8

Nopattern-mathingonprediates.

9

Thereareotherpossibilities.See[2℄formoredetails.

10

Arulefl→rîs^smallîfêvery^prediate^variableⁱⁿrîsêqual^toôneôf^theli^'s.

11