A PROOF OF THE STANDARD REDUCTION THEOREM IN THE LAMBDA CALCULUS

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1 A PROOF OF THE STANDARD REDUCTION THEOREM IN THE LAMBDA CALCULUS WILLIAM RICHTER ABSTRACT. Fellesen s Standard Reducton Theorem for the λ v Calculus yelds an algorthm that models the Scheme nterpreter. The β-nf λ Calculus analogue s Barendregt s Normalzaton Theorem. We gve a smple proof of ths by portng and smplfyng Fellesen s proof.. INTRODUCTION Barendregt [Ba, Thm ] proves a Normalzaton Theorem for the λ Calculus: f a Λ expresson has a β-nf. Plotkn [Fe, Thm. 3..4] proved the analogue for hs λ-value Calculus (hch models Scheme): hs standard reducton algorthm alays produces a value, f an expresson has one. Barendregt deduces the Normalzaton Theorem from a Standardzaton Theorem [Ba, Thm..4.7] [Ha, Thm. 3.37] (Fellesen and Plotkn follo sut [Fe, Thm. 3..8]). We gve a drect proof of the Normalzaton Theorem, by portng and smplfyng Fellesen s proof. We addtonally smplfy Fellesen s proof by replacng hs sze estmate Lemma [Fe, Lem. 3..3] by Lemma 2.4 belo. We found Fellesen s Lemma dffcult because t as not clear hat the doman of hs sze functon as, or hat the sze functon as measurng. We thnk e have a sgnfcantly shorter than Barendregt s proof. Hoever, Barendregt s proof s of ndependent nterest, as hs proof of the Standardzaton Theorem gros out of a longer but more perspcacous proof of the Church-Rosser theorem [Ba, Thm...0] than the usual Tat and Martn-Löf proof. Barendregt reduces all the resduals of a gven redex, from the nsde out, n order to prove a damond-lke property. We begn th the defnton of our λ Calculus β-nf seekng leftmost reducton algorthm. Follong Fellesen [Fe, Def. 3..3], e ll call each step of the reducton process the standard reducton arro. Thanks to Paul Burchard for the dagram package, hch uses XY-pc arros.

2 2 WILLIAM RICHTER Defnton.. The standard reducton arro relaton satsfyng s defned to be the smallest () M N f MβN () λx.m λx.n f M N () MN M N f M M and M s not a λ abstracton (v) MN MN f N N and M s a β-nf but not a λ abstracton The follong lemma, hch e leave as an exercse, shos that the standard reducton arro s an algorthm. Lemma.2. For any Λ term M hch s not a β-nf, there exsts a unque N such that M N. We ll no rte M β N rather than MβN, snce the relaton contans β. Recall that β, no rtten β, s the relaton, or arro (λx.m)u β M[x := U ] The arro s defned to be the syntactc closure of β. The arro s defned to be the reflexve transtve closure of. The reflexve transtve closure of the standard reducton arro ll be denoted by. The λ Calculus Normalzaton Theorem s no: Theorem A. If M N, th N a β-nf, then M N. Thus the standard reducton arro gves an algorthm for evaluaton of expressons n the λ Calculus, returnng β-nfs. 2. REPLACEMENT FOR FELLEISEN S SIZE ESTIMATE LEMMA The arro s also the reflexve transtve closure of a relaton hch follong Hankn e ll call grand reducton: Defnton 2.. The grand reducton arro s defned to be the smallest relaton gven by () (2) (3) (4) x x λx.m λx.n f M N MN M N f M N and M N (λx.m)u N[x := V ] f M N and U V Remark 2.2. Grand reducton s the key defnton of Tat and Martn-Löf s proof of the Church-Rosser theorem. Barendregt [Ba, Def ], Hankn [Ha, Def. 3.4] and Fellesen [Fe, Def ] defne axom () to be M M. Our apparently more restrctve defnton s equvalent to thers, because e can deduce M M from axoms ( 3), by nducton on M.

3 LAMBDA CALCULUS STANDARD REDUCTION 3 Let β mean the reflexve transtve closure of β, orβ. Snce β s a subset of the Standard Reducton arro, the arro β s a subset of. We gve a mld extenson of a lemma [Ba, Lem ] [Fe, Lem ] hch Hankn [Ha, p. 37] states as Property (). Lemma 2.3. () Assume that M N and U V. Then M[x := U ] N [x := V ] and ths grand reducton has the same type as the grand reducton M N, unless the type of M N s (). (2) If M β N, then for any Λ expresson U M[x := U ] β N [x := U ] Proof. The proof of part follos mmedately from the argument of Barendregt [Ba, Lem ] and Fellesen [Fe, Lem ]. A smple verson of ther argument proves part 2, hch e gve for completeness. We re gven the β-reducton M =(λy.p )V β P [y := V ]=N. Then M[x := U ]=(λy.p [x := U ]) V [x := U ] β P [x := U ][y := V [x := U ]] = P [y := V ][x := U ]=N[x:= U ] by the Substton Lemma [Ha, Lem. 2.], snce by the varables conventon, y/ FV(U). No e gve the λ Calculus port for a replacement for Fellesen s sze estmate Lemma [Fe, Lem. 3..3]. Our replacement orks because the purpose of Fellesen s lemma s to deal th grand reductons of type (4). We thnk that our Lemma 2.4 s a much cleaner ay to do so. Lemma 2.4. For any grand reducton M N, there exsts L th M β L here the grand reducton L N s of type (), (2) or (3). Proof. We use nducton on M. IfM Ns a grand reducton of type (), (2) or (3), e re done. So assume that M N s of type (4), say M =(λx.p )U Q[x := V ]=N, th P Q and U V. By nducton, there exsts K th P β K, and a grand reducton K Q of type (), (2) or (3). By Lemma 2.3, e have M =(λx.p )U β P [x := U ] β K[x := U ] Q[x := V ]=N N

4 4 WILLIAM RICHTER and the last grand reducton has the same type as K Q, unless K Q has type (). So f K Q s of type (2) or (3), then e re done: e can take L = K[x := U ]. No let s assume that K Q s of type (), that s, K = Q = y, for some varable y. We have to cases no. If y = x, then K[x := U ] Q[x := V ] s U V. The nductve hypothess apples to U V, so there exsts L th U β L V = N, and L N a grand reducton of type (), (2) or (3), and e re done: M =(λx.p )U β P [x := U ] β U β L N No f nstead y x, then K[x := U ] Q[x := V ] s y y. And so e re done, e have M =(λx.p )U β P [x := U ] β y y = N, here the grand reducton s of type (). Usng our replacement for Fellesen s sze estmate Lemma 3..3, e port one of hs to lemmas [Fe, Lem ] to the λ Calculus. Lemma 2.5. (a) If M x, then M β x. (b) Gven M λx.n, there exsts a grand reducton L N such that M β λx.l λx.n (c) Gven M V, th V a β-nf, then M V. Proof. Gven M λx.n, by Lemma 2.4 there exsts L such that M β L λx.n, th the second arro a grand reducton of type (2), snce () and (3) are mpossble. So by defnton, there exsts K such that L = λx.k λx.n, th K N. Ths proves (b). The proof of (a) s smlar, but even easer. For (c), assume M V, th V a β-nf. We argue by nducton on V. By Lemma 2.4 there exsts L such that M β L V, th the 2nd arro a grand reducton of type (), (2) or (3). If type (), e re done, snce L = V. If type (2), then by defnton, L = λx.k λx.u = V, th K U. U s then a β-nf, and by nducton e have that K U, and e re done. If type (3), then L =(HK) (AB) =V, th H A and K B, and A, Bβ-nf, th A not a λ abstracton. By nducton e have that

5 LAMBDA CALCULUS STANDARD REDUCTION 5 H A and K B, and snce A s not a λ abstracton, no term n the sequence H A can be a λ abstracton, ncludng H, so e have that (HK) (AK) (AB). For the λ v Calculus, Lemma 2.4 suffces to replace Fellesen s sze estmate Lemma Hoever, n n the λ Calculus t s convenent to take Lemma 2.4 one step farther, to smplfy the grand reductons of type (3). For ths e need some termnology suggested by Hankn s sketch of Barendregt s proof of the Standardzaton Theorem [Ha, Thm. 3.37]. Any applcaton M can be rtten unquely as M = M M 2 M k, for some k 2, th M ether a varable or a λ abstracton. If M s a varable, then M s called a eak head normal form [Ha, p. 49]. If M s a λ abstracton, then M M 2 s called the head redex of M. We defne a eak head reducton, rtten,by (λx.m) UM 2 M k M[x:= U ] M 2 M k The transtve reflexve closure of ll be rtten as usual. We defne the eak nternal reducton arro by M M 2 M k N N 2 N k f M t N t for t =,...k and f M N a grand reducton of type () or (2). We collect some smple propertes of eak head reducton and leave the proof as an exercse. Lemma 2.6. () Gven M N and H, e have (MH) (NH). (2) Gven M N and H K, e have (MH) (NK). (3) s a subset of. No e have our extenson of Lemma 2.4: Lemma 2.7. For any grand reducton M N, there exsts L such that M L N Proof. We use nducton on N. If N s a varable, then the result follos from Lemma 2.5(a), snce β s a subset of. If N s a λ abstracton, then then the result follos from Lemma 2.5(b). If N =(QK) s an applcaton, then Lemma 2.4 mples that there exsts L =(PH) th M β L N, the second grand reducton of type (3). No by nducton

6 6 WILLIAM RICHTER (on N), e have the left-hand dagram: P R=R R k Q = Q Q k L=PH N = QK RH here R Q and R ether a varable or a λ abstracton. Then by Lemma 2.6, e have the rght-hand dagram. 3. PROOF OF THEOREM A Usng Lemma 2.7, e can port Fellesen s man Lemma [Fe, Lem. 3..] to the λ Calculus. Lemma 3.. () Gven M N L, there exsts N such that M N L. (2) Gven M N L, there exsts N such that M N L. Proof. Part 2 follos easly from part by buldng ladders. We no prove part by nducton on N. By Lemma 2.7, there exsts M N = N N 2 N k F =F F 2 F k L th grand reductons F t N t, and th F ether a λ abstracton or a varable. We consder the to cases separately. If F s a λ abstracton, then F = λx.p λx.q = N, for some grand reducton P Q. We consder the to cases k =and k>separately. If k =, then L = λx.r, th a standard reducton Q R. Noby nducton (on N) ehaveq th P Q R, and e re done. If k>, then call U = F 2, V = N 2, F = F 3 F k, and N = N 3 N k. Then by Lemma 2.3, e have a grand reducton P [x := U ] Q[x := V ], and therefore by reducng the head redexes M (λx.p ) U F (λx.q) V N = N N = P [x := U ] F Q[x:= V ] N = L Ths fnshes the case here F a λ abstracton.

7 LAMBDA CALCULUS STANDARD REDUCTION 7 If nstead F = x = N, then suppose that N a D, and that N t s a β-nf for <t<a,so N L=x N a DN a+ N k. We apply nducton to F a N a D. So there exsts Z such that F a Z D. Furthermore by Lemma 2.5(c), F t N t for <t<a,so F =xf 2 F k xn 2 F k xn 2 N a F a F k xn 2 N a ZF a+ F k xn 2 N a DN a+ N k, here the last grand reducton s obtaned from the grand reductons Z D and F t N t for a<t k. No fnally e have Proof of Theorem A. Gven M N, for a β-nf N, there s a sequence of grand reductons M = M 0 M M 2 M l N We use nducton on l. Forl =0, Lemma 2.5(c) proves our result. For l>0, by nducton M M U. By Lemma 3.(b), there exsts N such that M N U. By Lemma 2.5(c), N U, and e re done. REFERENCES [Ba] Barendregt: The Lambda Calculus: ts Syntax and Semantcs. (Studes n Logc and the Foundatons of Math., Vol. 03). North Holland 98 [Fe] Fellesen, M.: Programmng Languages and Lambda Calcul. eprnt, 27 pages, matthas/4eb/mono.ps [Ha] Hankn: Lambda Calcul: A Gude for Computer Scentsts. (Grad. Texts n CS, Vol. 3). Oxford Unv. Press 994 WILLIAM RICHTER, MATHEMATICS DEPARTMENT, NORTHWESTERN UNIVERSITY, EVANSTON IL E-mal address: rchter@math.nu.edu