Standardization of a call-by-value calculus

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1 Standardization of a call-by-value calculus Giulio Guerrieri 1, Luca Paolini 2, and Simona Ronchi Della Rocca 2 1 Laboratoire PPS, UMR 7126, Université Paris Diderot, Sorbonne Paris Cité F Paris, France giulio.guerrieri@pps.univ-paris-diderot.fr 2 Dipartimento di Informatica, Università degli Studi di Torino C.so Svizzera 185, Torino, Italia {paolini,ronchi}@di.unito.it Abstract We study an extension of Plotkin s call-by-value lambda-calculus by means of two commutation rules (sigma-reductions). Recently, it has been proved that this calculus can provide elegant characterizations of many semantic properties. We prove a standardization theorem for this calculus by generalizing Takahashi s approach of parallel reductions. The standardization property allows us to prove that our calculus enjoys some meaningful conservation properties with respect to Plotkin s one. In particular, we show that the notion of solvability for this calculus coincides with that for Plotkin s call-by-value lambda-calculus ACM Subject Classification D.3.1 Formal Definitions and Theory, F.3.2 Semantics of Programming Language, F.4.1 Mathematical Logic. Keywords and phrases standardization, sequentialization, lambda-calculus, sigma-reduction, parallel reduction, call-by-value, head reduction, ernal reduction, solvability, potential valuability, observational equivalence. Digital Object Identifier /LIPIcs.xxx.yyy.p 1 Introduction The λ v -calculus has been roduced by Plotkin in [16], in order to give a formal account of the call-by-value evaluation, which is the most commonly used parameter passing policy for programming languages. λ v shares the syntax with the classical, call-by-name, λ-calculus, but its reduction rule, β v, is a restriction of the β one, firing only in case the argument is a value (i.e., a variable or an abstraction). While β v is enough for evaluation, it is turned out to be too weak in order to study operational properties of terms. For example, in λ, the β-reduction is sufficient to characterize solvability and (in addition with η) separability, but, in order to characterize similar properties for λ v, it has been necessary to roduce different notions of reduction unsuitable for a correct call-by-value evaluation (see [14, 15]), which is disappoing and requires complex reasoning. In this paper we study λ σ v, the extension of λ v proposed in [3]. It keeps the λ v syntax and it adds to the β v -reduction two commutation rules, called σ 1 and σ 3, which unblock β v -redexes which are hidden by the hyper-sequential structure of terms. It is well-known (see [15, 1]) that in λ v there are normal forms that are unsolvable, e.g. (λyx.xx)(zz)(λx.xx). The more evident benefit of λ σ v is that the commutation rules make all normal forms solvable (indeed (λyx.xx)(zz)(λx.xx) This work was partially supported by LINTEL TO_Call1_2012_0085, i.e. a Research Project funded by the Compagnia di San Paolo. Giulio Guerrieri, Luca Paolini and Simona Ronchi Della Rocca; licensed under Creative Commons License CC-BY Conference title on which this volume is based on. Editors: Billy Editor and Bill Editors; pp Leibniz International Proceedings in Informatics Schloss Dagstuhl Leibniz-Zentrum für Informatik, Dagstuhl Publishing, Germany

2 2 Standardization of a call-by-value calculus is not a λ σ v normal form). More generally, the so obtained language, allows to characterize operational properties, like solvability and potential valuability, in an ernal and elegant way (see [3]). In this paper we prove a standardization property in λ σ v, and some of its consequences, namely its soundness with respect to the semantics of λ v. Let us recall the notion of standardization, which has been first studied in the ordinary λ-calculus (see, for example [5, 8, 2]). A reduction sequence is standard if its redexes are ordered in a given way, and the corresponding standardization theorem establishes that every reduction sequence can constructively be transformed o a standard one. Standardization is a key tool to grasp the way in which reductions works that sheds some light on redexes relationships and their dependencies. It is useful for characterize semantic properties through reduction strategies (the proof of operational semantics adequacy is a typical use). In the λ v setting standardization theorems have been proved by Plotkin [16], Paolini and Ronchi Della Rocca [15, 13] and Crary [4]. The definition of standard sequence of reductions considered by Plotkin and Crary coincides, and it imposes a partial order between redexes, while Paolini and Ronchi Della Rocca define a total order between them. All these proofs are developed by using the notion of parallel reduction roduced by Tait and Martin-Löf (see Takahashi [19] for details and eresting technical improvements). We emphasize that this method does not involve the notion of residual of a redex, on which many classical proofs for the λ-calculus are based (see for example [2, 8]). As in [16, 19, 15, 4], we use a suitable notion of parallel reduction for developing our standardization theorem for λ σ v. In particular we consider two groups of redexes, head β v -redexes and head σ-redexes (putting together σ 1 and σ 3 ), and we induce a total order between head redexes of the two groups, without imposing any order between head σ-redexes themselves. More precisely, when σ-redexes are missing, this notion of standardization coincides with that presented in [15]. Moreover, we show that it is not possible to strength our standardization by (locally) ordering σ 1 -reduction to σ 3 -reduction (or viceversa). As usual, our standardization proof is based on a sequentialization result: inner reductions can always be postponed to the head ones, for a non-standard definition of head reduction. Sequentialization has eresting consequences: it allows us to prove that fundamental operational properties in λ σ v, like observational equivalence, potential valuability and solvability, are conservative with respect to the corresponding notions of λ v. This fully justify the project in [3] where λ σ v has been roduced as a tool for studying the operational behaviour of λ v. Other variants of λ v have been roduced in the literature for modeling the call-by-value computation. We would like here to cite at least the contributions of Moggi [10, 11], Felleisen and Sabry [17, 18], Maraist et al. [9], Herbelin and Zimmerman [7], Accattoli and Paolini [1]. All these proposals are based on the roduction of new constructs to the syntax of λ v, so the comparison between them is not easy with respect to syntactical properties (some detailed comparison is given in [1]). We po out that the calculi roduced in [10, 11, 17, 18, 9, 7] present some variants of our σ 1 and/or σ 3 rules, often in a setting with explicit substitutions. Outline. In Section 2 we roduce the language λ σ v and its operational behaviour; in Section 3 the sequentialization property is proved; Section 4 contains the main result, i.e., standardization; in Section 5 some conservativity results with respect to Plotkin s λ v -calculus are proved. Section 6 concludes the paper, with some h for a future work. 2 The call-by-value lambda calculus with sigma-rules In this section we present λ σ v, a call-by-value λ-calculus roduced in [3] that adds two σ-reduction rules to pure (i.e. without constants) call-by-value λ-calculus defined by Plotkin in [16].

3 G. Guerrieri, L. Paolini and S. Ronchi Della Rocca 3 The syntax of terms of λ σ v [3] is the same as the one of ordinary λ-calculus and Plotkin s call-by-value λ-calculus λ v [16] (without constants). Given a countable set V of variables (denoted by x, y, z,... ), the sets Λ of terms and Λ v of values are defined by mutual induction: (Λ v ) V, U ::= x λx.m values (Λ) M, N, L ::= V MN terms Clearly, Λ v Λ. All terms are considered up to α-conversion. The set of free variables of a term M is denoted by fv(m). Given V 1,..., V n Λ v and pairwise distinct variables x 1,..., x n, M{V 1 /x 1,..., V n /x n } denotes the term obtained by the capture-avoiding simultaneous substitution of V i for each free occurrence of x i in the term M (for all 1 i n). Note that, for all V, V 1,..., V n Λ v and pairwise distinct variables x 1,..., x n, V {V 1 /x 1,..., V n /x n } Λ v. Contexts (with exactly one hole ), denoted by C, are defined as usual via the grammar: C ::= λx.c CM MC. We use C M for the term obtained by the capture-allowing substitution of the term M for in the context C. Notation. For now on, we set I = λx.x and = λx.xx. The reduction rules of λ σ v consist of Plotkin s β v -reduction rule, roduced in [16], and two simple commutation rules called σ 1 and σ 3, studied in [3]. Definition 1 (Reduction rules). We define the following binary relations on Λ (for any M, N, L Λ and any V Λ v ): (λx.m)v βv M{V/x} (λx.m)nl σ1 (λx.ml)n with x / fv(l) V ((λx.l)n) σ3 (λx.v L)N with x / fv(v ). For any r {β v, σ 1, σ 3 }, if M r M then M is a r-redex and M is its r-contractum. In this sense, a term of the shape (λx.m)n (for any M, N Λ) is a β-redex. We set σ = σ1 σ3 and v = βv σ. The side conditions on σ1 and σ3 in Definition 1 can be always fulfilled by α-renaming. Obviously, any β v -redex is a β-redex but the converse does not hold: (λx.z)(yi) is a β-redex but not a β v -redex. Redexes of different type may overlap: for example, the term I is a σ 1 -redex and it contains the β v -redex I; the term (I )(xi) is a σ 1 -redex and it contains the σ 3 -redex (I ) which contains in turn the β v -redex I. According to the Girard s call-by-value boring translation ( ) v of terms o Intuitionistic Multiplicative Exponential Linear Logic proof-nets, defined by (A B) v =!A v!b v (see [6]), the images under ( ) v of a σ 1 -redex (resp. σ 3 -redex) and its contractum are equal modulo some bureaucratic steps of cut-elimination. Notation. Let R be a binary relation on Λ. We denote by R (resp. R + ; R = ) the reflexivetransitive (resp. transitive; reflexive) closure of R. Definition 2 (Reductions). Let r {β v, σ 1, σ 3, σ, v}. The r-reduction r is the contextual closure of r, i.e. M r M iff there is a context C and N, N Λ such that M = C N, M = C N and N r N. The r-equivalence = r is the reflexive-transitive and symmetric closure of r.

4 4 Standardization of a call-by-value calculus Let M be a term: M is r-normal if there is no term N such that M r N; M is r-normalizable if there is a r-normal term N such that M r N; M is strongly r-normalizing if there is no sequence (N i ) i N such that M = N 0 and N i r N i+1 for any i N. Finally, r is strongly normalizing if every N Λ is strongly r-normalizing. Patently, σ v and βv v. Remark 3. For any r {β v, σ 1, σ 3, σ, v} (resp. r {σ 1, σ 3, σ}), values are closed under r- reduction (resp. r-expansion): for any V Λ v, if V r M (resp. M r V ) then M Λ v ; more precisely, V = λx.n and M = λx.n for some N, N Λ with N r N (resp. N r N). Proposition 4 (see [3]). The σ-reduction is confluent and strongly normalizing. The v-reduction is confluent. The λ σ v -calculus, λ σ v for short, is the set Λ of terms endowed with the v-reduction v. The set Λ endowed with the β v -reduction βv is the λ v -calculus (λ v for short), i.e. the Plotkin s call-by-value λ-calculus [16] (without constants), which is thus a sub-calculus of λ σ v. Example 5. M = (λy. )(xi) σ1 (λy. )(xi) βv (λy. )(xi) βv... and N = ((λy. )(xi)) σ3 (λy. )(xi) βv (λy. )(xi) βv... are the only possible v-reduction paths from M and N respectively: M and N are not v-normalizable, and M = v N. Meanwhile, M and N are β v -normal and different, hence M βv N (by confluence of βv, see [16]). Informally, σ-rules unblock β v -redexes which are hidden by the hyper-sequential structure of terms. This approach is alternative to the one in [1] where hidden β v -redexes are reduced thanks to a rule acting at a distance (through explicit substitutions). It can be shown that the call-by-value λ-calculus with explicit substitution roduced in [1] can be embedded in λ σ v. 3 Sequentialization In this section we aim to prove a sequentialization theorem (Theorem 22) for the λ σ v -calculus by adapting the Takahashi s method [19, 4] based on parallel reductions. Notation. From now on, we always assume that V, V Λ v. Note that the generic form of a term is V M 1... M m for some m N (in particular, values are obtained when m = 0). The sequentialization result is based on a partitioning of v-reduction between head and ernal reduction. Definition 6 (Head β v -reduction). We define inductively the head β v -reduction βv the following rules (m N in both rules): by β v (λx.m)v M 1... M m βv M{V/x}M 1... M m N βv N right V NM 1... M m βv V N M 1... M m The head β v -reduction βv reduces exactly the same redexes (see [14]) as the left reduction defined in [16, p. 136], called evaluation in [17, 18, 4]. The head β v -reduction is deterministic and does not reduce values. Definition 7 (Head σ-reduction). We define inductively the head σ-reduction σ by the following rules (m N in all the rules, x / fv(l) in the rule σ 1, x / fv(v ) in the rule σ 3 ): σ 1 (λx.m)nlm 1... M m σ (λx.ml)nm 1... M m N σ N right V NM 1... M m σ V N M 1... M m V ((λx.l)n)m 1... M m σ (λx.v L)NM 1... M m σ 3

5 G. Guerrieri, L. Paolini and S. Ronchi Della Rocca 5 The head (β v σ-)reduction is βvσ = βv σ. The ernal reduction is v = v βvσ. Notice that βv βv βv and σ σ σ and v βvσ v. Values are normal forms for the head reduction, but the converse does not hold: xi / Λ v is head-normal. Informally, if N βv N (resp. N σ N ) then N is gotten from N by reducing the leftmost-outermost β v -redex (resp. σ 1 - or σ 3 -redex), not in the scope of a λ. Differently from βv, the head σ-reduction σ is not deterministic, because the leftmostoutermost σ 1 - and σ 3 -redexes may overlap: for example, if M = (λy.y )( (xi))i then M σ (λy.y I)( (xi))=n 1 by applying the rule σ 1 and M σ (λz.(λy.y )(zz))(xi)i =N 2 by applying the rule σ 3. Note that N 1 contains only a head σ 3 -redex and N 1 σ (λz.(λy.y I)(zz))(xI)= N which is normal for βvσ; meanwhile N 2 contains only a head σ 1 -redex and N 2 σ (λz.(λy.y )(zz)i)(xi) which is normal for βvσ: the head reduction βvσ is not confluent and a term may have several head-normal forms (this example does not contradict the confluence of σ-reduction because N 2 σ N but by performing an ernal reduction step). Anyway, next Corollary 26.2 implies that if a term M has a head normal form N Λ v then N is the unique head normal form of M. Definition 8 (Parallel reduction). We define inductively the parallel reduction by the following rules (m N in all the rules, x / fv(l) in the rule σ 1, x / fv(v ) in the rule σ 3 ): V V M i M i (0 i m) β v (λx.m 0)VM 1... M m M 0{V /x}m 1... M m N N L L M i M i (0 i m) (λx.m 0)NLM 1... M m (λx.m 0L )N M 1... M m σ 1 V V N N L L M i M i (1 i m) σ3 V ((λx.l)n)m 1... M m (λx.v L )N M 1... M m M i M i (0 i m) λ (λx.m 0)M 1... M m (λx.m 0)M 1... M m M i M i (1 i m) x M 1... M m x M 1... M m In Definition 8 the rule var has no premises when m = 0: this is the base case of the inductive definition of. The rules σ 1 and σ 3 have exactly three premises when m = 0. Intuitively, M M means that M is obtained from M by reducing a number of β v -, σ 1 - and σ 3 -redexes (existing in M) simultaneously. Definition 9 (Strong and ernal parallel reduction). We define inductively the ernal parallel reduction by the following rules (m N in the rule right): M M λx.m λx.m λ x x var var V V N N M i M i (1 i m) right V NM 1... M m V N M 1... M m The strong parallel reduction is defined by: M N iff M N and there exist M, M Λ such that M β v M σ M N. Notice that the rule right for has exactly two premises when m = 0. Remark 10. The relations, and are reflexive. The reflexivity of follows immediately from the reflexivity of and. The proofs of reflexivity of and are both by structural induction on a term: in the case of, remind that every term is of the form (λx.n)m 1... M m or x M 1... M m for some m N and then apply the rule λ or var respectively, together to the inductive hypothesis; in the case of, remind that every term is of the form λx.m or x or V NM 1... M m for some m N and then apply the rule λ (together to the reflexivity of ) or var or right (together to the reflexivity of and the inductive hypothesis) respectively.

6 6 Standardization of a call-by-value calculus One has (first, prove that by induction on the derivation of M M, the other inclusions follow from the definition of ) and, since is reflexive (Remark 10), βv and σ. Observe that R for any R { βv, βv,,, }, even if for different reasons: for example, by reflexivity of (Remark 10), whereas βv by reducing the (leftmost) β v -redex. Remark The head β v -reduction βv does not reduce a value (in particular, does not reduce under λ s), i.e., for any M Λ and any V Λ v, one has V βv M. 2. The head σ-reduction σ does neither reduce a value nor reduce to a value, i.e., for any M Λ and any V Λ v, one has V σ M and M σ V. 3. Variables and abstractions are preserved under ( -expansion), i.e., if M x (resp. M λx.n ) then M = x (resp. M = λx.n for some N Λ such that N N ). 4. If M M then λx.m R λx.m for any R {,, }. Indeed, for R {, } apply the rule λ to conclude, then λx.m λx.m according to the definition of. 5. For any V, V Λ v, V V iff V V. The left-to-right direction holds because ; conversely, assume V V : if V is a variable then necessarily V = V and hence V V by applying the rule var for ; otherwise V = λx.n for some N Λ, and then necessarily V = λx.n with N N, so V V by applying the rule λ for. See Appendix, p. 16 See Appendix, p. 16 Remark If M M and N N then MN M N. For the proof, it is sufficient to consider the last rule of the derivation of M M. 2. If R { βv, σ } and M R M, then MN R M N for any N Λ. For the proof, it is sufficient to consider the last rule of the derivation of M R M, for any R { βv, σ }. 3. If M M and N N where M / Λ v, then MN M N : indeed, the last rule in the derivation of M M can be neither λ nor var because M / Λ v. The hypothesis M / Λ v is crucial: for example, x x and I but I and thus x(i ) x. 4. v v. As a consequence, = v and (by Proposition 4) is confluent. 5. v v, so = v. Thus, by Remark 11.3, variables and abstractions are preserved under v -expansion, i.e., if M v x (resp. M v λx.n ) then M = x (resp. M = λx.n with N v N ). 6. For any R { βv, σ }, if M R M then M{V/x} R M {V/x} for any V Λ v. The proof is by straightforward induction on the derivation of M R M for any R { βv, σ }. As expected, a basic property of parallel reduction is the following: Lemma 13 (Substitution lemma for ). If M M and V V then M{V/x} M {V /x}. See Appendix, p. 16 Proof. By straightforward induction on the derivation of M M. The following lemma will play a crucial role in the proof of Lemmas and shows that the head σ-reduction σ can be postponed to the head β v -reduction βv. Lemma 14 (Commutation of head reductions). 1. If M σ L βv N then there exists L Λ such that M βv L = σ N. 2. If M σ L β v N then there exists L Λ such that M β v L σ N. 3. If M β vσ M then there exists N Λ such that M β v N σ M. Proof. 1. By induction on the derivation of M σ L. Let us consider its last rule r. If r = σ 1 then M = (λx.m 0 )N 0 L 0 M 1... M m and L = (λx.m 0 L 0 )N 0 M 1... M m where m N and x / fv(l 0 ). Since L βv N, there are only two cases:

7 G. Guerrieri, L. Paolini and S. Ronchi Della Rocca 7 either N 0 βv N 0 and N = (λx.m 0 L 0 )N 0M 1... M m (according to the rule right for βv ), then M βv (λx.m 0 )N 0L 0 M 1... M m σ N; or N 0 Λ v and N =M 0 {N 0 /x}l 0 M 1... M m (by the rule β v as x / fv(l 0 )), so M βv N. If r = σ 3 then M = V ((λx.l 0 )N 0 )M 1... M m and L = (λx.v L 0 )N 0 M 1... M m with m N and x / fv(v ). Since L βv N, there are only two cases: either N 0 βv N 0 and N = (λx.v L 0 )N 0M 1... M m (according to the rule right for βv ), then M βv V ((λx.l 0 )N 0)M 1... M m σ N; or N 0 Λ v and N =VL 0 {N 0 /x}m 1... M m (by the rule β v, as x / fv(v )), so M βv N. Finally, if r = right then M = V N 0 M 1... M m and L = V N 0M 1... M m with m N and N 0 σ N 0. By Remark 11.2, N 0 / Λ v and thus, since L βv N, the only possibility is that N 0 βv N 0 and N = V N 0 M 1... M m (according to the rule right for βv ). By induction hypothesis, there exists N 0 Λ such that N 0 βv N 0 = σ N 0. Therefore, M βv V N 0 M 1... M m = σ N. 2. Immediate consequence of Lemma 14.1, using standard techniques of rewriting theory. See Appendix, p Immediate consequence of Lemma 14.2, using standard techniques of rewriting theory. See Appendix, p. 17 We are now able to travel over again the Takahashi method [19, 4] in our setting with β v - and σ-reduction. The next four lemmas govern the strong parallel reduction. Lemma 15. If M M and N N and M / Λ v, then MN M N. Proof. One has MN M N by Remark 12.1 and since M M. By hypothesis, there exist m, n N and M 0,... M m, N 0,..., N n such that M = M 0, M m = N 0, N n M, M i βv M i+1 for any 0 i < m and N j σ N j+1 for any 0 j < n; by Remark 12.2, M i N βv M i+1 N for any 0 i < m and N j N σ N j+1 N for any 0 j < n. As M / Λ v, one has N n N M N by Remark Therefore, MN M N. Lemma 16. If M M and N N then MN M N. Proof. If M / Λ v then MN M N by Lemma 15 and since N N. Assume M Λ v : MN M N by Remark 12.1, as M M and N N. By hypothesis, there are m, m, n, n N and M 0,..., M m, M 0,..., M m, N 0,..., N n, N 0,..., N n such that: M = M 0, M m = M 0, M m for any 0 i < m, N = N 0, N n = N 0, N n any 0 j < n. M, M i βv M i+1 for any 0 i < m, and M i σ M i +1 N, N j βv N j+1 for any 0 j < n and N j σ N j +1 for By Remark 11.3, M m Λ v since M Λ v, therefore m = 0 by Remark 11.2, and thus M m = M 0 M (in particular, M m M ) and M m Λ v. Thanks to the rules right for βv and σ, one has M m N j βv M m N j+1 for any 0 j < n, and M m N j σ M m N j +1 for any 0 j < n. By Remark 12.2, M i N 0 βv M i+1 N 0 for any 0 i < m. By applying the rule right for, one has M m N n M N. Therefore, MN = M 0 N 0 β v M m N 0 β v M m N n = M m N 0 σ M m N n M N and hence MN M N. Lemma 17. If M M and V V, then M{V/x} M {V /x}. Proof. By Lemma 13, one has M{V/x} M {V /x} since M M and V V. We proceed by induction on M Λ. Let us consider the last rule r of the derivation of M M. If r = var then there are two cases: either M = x and then M{V/x}=V V =M {V /x}; or M = y x and then M{V/x} = y = M {V /x}, so M{V/x} M {V /x} by Remark 10.

8 8 Standardization of a call-by-value calculus If r = λ then M = λy.n and M = λy.n with N N ; we can suppose without loss of generality that y / fv(v ) {x}. One has N{V/x} N {V /x} according to Lemma 13. By applying the rule λ for, one has M{V/x} = λy.n{v/x} λy.n {V /x} = M {V /x} and thus M{V/x} M {V /x}. Finally, if r = right then M = UNM 1... M m and M = U N M 1... M m for some m N with U, U Λ v, U U, N N and M i M i for any 1 i m. By induction hypothesis, U{V/x} U {V /x} (indeed U U according to Remark 11.5) and N{V/x} N {V /x}. By Lemma 16, U{V/x}N{V/x} U {V /x}n {V/x} and hence, by Lemma 15 since U {V /x}n {V/x} / Λ v, one has M{V/x} = U{V/x}N{V/x}M 1 {V/x}... M m {V/x} U {V /x}n {V /x}m 1{V /x}... M m{v /x} = M {V /x}. In the proof of the two next lemmas, as well as in the proof of Corollary 21 and Theorem 22, our Lemma 14 plays a crucial role: indeed, since the head σ-reduction well eract with the head β v -reduction, the Takahashi method [19, 4] is still working when adding the rules σ 1 and σ 3 to Plotkin s β v -reduction. Lemma 18 (Substitution lemma for ). If M M and V V then M{V/x} M {V /x}. Proof. By Lemma 13, one has M{V/x} M {V /x} since M M and V V. By hypothesis, there exist m, n N and M 0,... M m, N 0,..., N n such that M = M 0, M m = N 0, N n M, M i βv M i+1 for any 0 i < m and N j σ N j+1 for any 0 j < n; by Remark 12.6, M i {V/x} βv M i+1 {V/x} for any 0 i < m, and N j {V/x} σ N j+1 {V/x} for any 0 j < n. By Lemma 17, one has N n {V/x} M {V /x}, thus there exist L, N Λ such that M{V/x} β v N 0 {V/x} σ N n {V/x} β v N σ L M {V /x}. By Lemma 14.2, there exists N Λ such that M{V/x} β v N 0 {V/x} β v N σ N σ L M {V /x}, therefore M{V/x} M {V /x}. Now we are ready to prove the main lemma, which means that parallel reduction coincides with strong parallel reduction (the inclusion is trivial). Lemma 19 (Main Lemma). If M M then M M. Proof. By induction on the derivation of M M. Let us consider the last rule r. If r = var then M = x M 1... M m and M = x M 1... M m where m N and M i M i for any 1 i m. By reflexivity of (Remark 10), x x. By induction hypothesis, M i M i for any 1 i m. Therefore, M M by applying Lemma 16 m times. If r = λ then M = (λx.m 0 )M 1... M m and M = (λx.m 0)M 1... M m where m N and M i M i for any 0 i m. By induction hypothesis, M i M i for any 1 i m. According to Remark 11.4, λx.m 0 λx.m 0. So M M by applying Lemma 16 m times. If r = β v then M = (λx.m 0 )V M 1... M m and M = M 0{V /x}m 1... M m where m N, V V and M i M i for any 0 i m. By induction hypothesis, V V and M i M i for any 0 i m. By applying the rule β v for βv, one has M βv M 0 {V/x}M 1... M m ; moreover M 0 {V/x}M 1... M m M by Lemma 18 and by applying Lemma 16 m times, thus there are L, N Λ such that M βv M 0 {V/x}M 1... M m β v L σ N M. So M M. If r = σ 1 then M = (λx.m 0 )N 0 L 0 M 1... M m and M = (λx.m 0L 0)N 0M 1... M m where m N, L 0 L 0, N 0 N 0 and M i M i for any 0 i m. By induction hypothesis, N N and M i M i for any 1 i m. By applying the rule σ 1 for σ, one has M σ (λx.m 0 L 0 )N 0 M 1... M m. By Remark 12.1, M 0 L 0 M 0L 0 and thus λx.m 0 L 0 λx.m 0L 0 according to Remark So (λx.m 0 L 0 )N 0 M 1... M m M by applying Lemma 16 m + 1 times, hence there are L, N Λ such that M σ (λx.m 0 L 0 )N 0 M 1... M m β v L σ N M. By Lemma 14.2, there is L Λ such that M β v L σ L σ N M and thus M M.

9 G. Guerrieri, L. Paolini and S. Ronchi Della Rocca 9 Finally, if r = σ 3 then M = V ((λx.l 0 )N 0 )M 1... M m and M = (λx.v L 0)N 0M 1... M m with m N, V V, L 0 L 0, N 0 N 0 and M i M i for any 1 i m. By induction hypothesis, N 0 N 0 and M i M i for any 1 i m. By the rule σ 3 for σ, one has M σ (λx.v L 0 )N 0 M 1... M m. By Remark 12.1, V L 0 V L 0 and thus λx.v L 0 λx.v L 0 according to Remark So (λx.v L 0 )N 0 M 1... M m M by applying Lemma 16 m + 1 times, hence there are L, N Λ such that M σ (λx.(λy.m 0 )L 0 )N 0 M 1... M m β v L σ N M. By Lemma 14.2, there is L Λ such that M β v L σ L σ N M, so M M. Next Lemma 20 and Corollary 21 show that ernal parallel reduction can be shifted after head reductions. Lemma 20 (Postponement). If M L and L βv N (resp. L σ N) then there exist L Λ such that M βv L (resp. M σ L ) and L N. Proof. By induction on the derivation of M L. Let us consider its last rule r. If r = var, then M = x = L which contradicts L βv N and L σ N by Remarks If r = λ then L = λx.l for some L Λ, which contradicts L βv N and L σ N by Remarks Finally, if r = right then M = V M 0 M 1... M m and L = V L 0 L 1... L m where m N, V V (so V V by Remark 11.5), M 0 L 0 (so M 0 L 0 ) and M i L i for any 1 i m. If L βv N then there are two cases, depending on the last rule r of the derivation of L βv N. If r = β v then V = λx.n 0, L 0 Λ v and N = N 0{L 0 /x}l 1... L m, thus M 0 Λ v and V = λx.n 0 with N 0 N 0 by Remark By Lemma 13, one has N 0 {M 0 /x} N 0{L 0 /x}. Let L = N 0 {M 0 /x}m 1... M m : so M = (λx.n 0 )M 0 M 1... M m βv L (apply the rule β v for βv ) and L N by applying Remark 12.1 m times. If r = right then N = V N 0 L 1... L m with L 0 βv N 0. By induction hypothesis, there exists L 0 Λ such that M 0 βv L 0 N 0. Let L = V L 0M 1... M m : so M βv L (apply the rule right for βv ) and L N by applying Remark 12.1 m + 1 times. If L σ N then there are three cases, depending on the last rule r of the derivation of L σ N. If r = σ 1 then m > 0, V = λx.n 0 and N = (λx.n 0L 1 )L 0 L 2... L m, thus V = λx.n 0 with N 0 N 0 by Remark Thanks to Remarks 12.1 and 11.4, one has λx.n 0 M 1 λx.n 0L 1. Let L = (λx.n 0 M 1 )M 0 M 2... M m : so M = (λx.n 0 )M 0 M 1... M m σ L (apply the rule σ 1 for σ ) and L N by applying Remark 12.1 m times. If r = σ 3 then L 0 = (λx.l 01 )L 02 and N = (λx.v L 01 )L 02 L 1... L m. Since M 0 (λx.l 01 )L 02, necessarily M 0 = (λx.m 01 )M 02 with M 01 L 01 and M 02 L 02 (so M 02 L 02 ). Thanks to Remarks 12.1 and 11.4, one has λx.v M 01 λx.v L 01. Let L = (λx.v M 01 )M 02 M 1... M m : therefore M = V ((λx.m 01 )M 02 )M 1... M m σ L (apply the rule σ 3 for σ ) and L N by applying Remark 12.1 m + 1 times. If r = right then N = V N 0 L 1... L m with L 0 σ N 0. By induction hypothesis, there exists L 0 Λ such that M 0 σ L 0 N 0. Let L = V L 0M 1... M m : so M σ L (apply the rule right for σ ) and L N by applying Remark 12.1 m + 1 times. Corollary 21. If M L and either L βv such that M β v L σ L N. N or L σ N, then there exist L, L Λ Proof. Immediate by Lemma 20 and Lemma 19, applying Lemma 14.2 if L σ N.

10 10 Standardization of a call-by-value calculus Now we obtain our first main result (Theorem 22): any v-reduction sequence can be sequentialized o an head β v -reduction sequence followed by an head σ-reduction sequence, followed by an ernal reduction sequence. In ordinary λ-calculus, the well-known result corresponding to our Theorem 22 says that a β-reduction sequence can be factorized in a head reduction sequence followed by an ernal reduction sequence (see for example [19, Corollary 2.6]). Theorem 22 (Sequentialization). If M v M then there exist L, N Λ such that M β v L σ N v M. Proof. By Remark 12.4, M M and thus there are m N and M 0,..., M m Λ such that M = M 0, M m = M and M i M i+1 for any 0 i < m. We prove by induction on m N that there are L, N Λ such that M β v L σ N M, so N v M by Remark If m = 0 then M = M 0 = M and hence we conclude by taking L = M = N. Finally, suppose m > 0. By induction hypothesis applied to M 1 M, there exist L, N Λ such that M 1 β v L σ N M. By applying Lemma 19 to M, there exist L 0, N 0 Λ such that M β v L 0 σ N 0 M 1. By applying Corollary 21 repeatedly, there exists N Λ such that N 0 β N vσ N and hence M β N vσ M. According to Lemma 14.3, there exists L Λ such that M β v L σ N M. It is worth noticing that in Definition 7 there is no distinction between head σ 1 - and head σ 3 -reduction steps, and, according to it, the sequentialization of Theorem 22 imposes no order between head σ-reductions. We note σ1 and σ3 respectively the reductions relations σ1 σ and σ3 σ. So, a natural question arises: is it possible to sequentialize them? The answer is negative, as proved by the next two counterexamples. Let M = x((λy.z )(zi)) and N = (λy.xz )(zi): M σ3 (λy.xz )(zi) σ1 N, but there exists no L such that M σ 1 L σ 3 N. In fact M contains only a head σ 3 -redex and (λy.xz )(zi) contains only a head σ 1 -redex Let M = x((λy.z )(zi) ) and N = (λy.x(z ))(zi): M σ1 x((λy.z )(zi)) σ3 N but there is no L such that M σ 3 L σ 1 N. In fact M contains only a head σ 1 -redex and x((λy.z )(zi)) contains only a head σ 3 -redex. So, the impossibility of sequentializing a head σ-reduction sequence is due to the fact that a head σ 1 -reduction step can create a head σ 3 -redex, and viceversa. This is not a problem, since head σ-reduction is strongly normalizing (by Proposition 4 and since σ σ ). Our approach does not force a strict order of head σ-reductions. 4 Standardization Now we are able to prove the main result of this paper, i.e., a standardization theorem for λ σ v (Theorem 25). In particular we provide a notion of standard reduction sequence that avoid any auxiliary notion of residual redexes, by closely following the definition given in [16]. Notation. For any k, m N with k m, we note M 0,, M k, M m head a sequence of terms such that M i βv M i+1 when 0 i < k, and M i σ M i+1 when k i < m. It is easy to check that M head for any M Λ. The notion of standard sequence of terms is defined by using the previous notion of head-sequence. Our notion of standard reduction sequence is mutually defined together with the notion of inner-sequence of terms (Definition 23). This definition allows us to avoid non-deterministic cases remarked in [7] (we provide more details at the end of this section). We denote by M 0,, M m std (resp. M 0,, M m in ) a standard (resp. inner) sequence of terms.

11 G. Guerrieri, L. Paolini and S. Ronchi Della Rocca 11 Definition 23 (Standard and inner sequences). Standard and inner sequences of terms are defined by mutual induction as follows: 1. if M 0,, M m head and M m,, M m+n in then M 0,, M m+n std, where m, n N; 2. M in, for any M Λ; 3. if M 0,, M m std then λz.m 0,, λz.m m in, where m N; 4. if V 0,, V h std and N 0,, N n in then V 0 N 0,, V 0 N n,, V h N n in, where h, n N; 5. if N 0,, N n in, M 0,, M m std and N 0 Λ v, then N 0 M 0,, N n M 0,, N n M m in, where m, n N. For instance, let M = (λy.ix)(z( I))(II): M v (λy.ix)(z( I))I v (λy.x)(z( I))I and M v (λy.ix(ii))(z( I)) v (λy.ix(ii))(z(ii)) are not standard sequences; M v (λy.ix)(z( I))I and M v (λy.ix)(z(ii))(ii) v (λy.ix)(zi)(ii) v (λy.ix(ii))(zi) v (λy.x(ii))(zi) v (λy.xi)(zi) are standard sequences. Remark 24. For any n N, if N 0,, N n in (resp. N 0,, N n head ) then N 0,, N n std. Indeed, N 0 head (resp. N n in by Definition 23.2), so N 0,, N n std by Definition In particular, N std for any N Λ: apply Definition 23.2 and Remark 24 for n = 0. Theorem 25 (Standardization). 1. If M v M then there is a sequence M,, M std. 2. If M v M then there is a sequence M,, M in. Proof. Both statements are proved simultaneously by induction on M Λ. 1. If M = z then, by Theorem 22, M β v L σ N v z for some L, N Λ. By Remarks 12.5 and 11.2, L = N = z; therefore M β v z and hence there is a sequence M,, z head. Thus, M,, z std by Remark 24. If M = λz.n then, by Theorem 22, M β v L σ L v λz.n for some L, L Λ. By Remarks 12.5 and 11.2, L = L = λz.n with N v N. So M β v λz.n and hence there is a sequence M,, λz.n head. By induction on (1), there is a sequence N,, N std, thus λz.n,, λz.n in by Definition Therefore M,, λz.n,, λz.n std by Definition If M = N L then, by Theorem 22, M β v M σ M 0 v N L for some M, M 0 Λ. By Remark 3, M 0 = NL for some N, L Λ, since v v and M / Λ v. Thus there is a sequence M,, M,, NL head. By Remark 12.5, NL N L ; clearly, each step of is an instance of the rule right of Definition 9. There are two sub-cases. If N Λ v then N N and L L, so N v N and L v L by Remarks By induction respectively on (1) and (2), there are sequences N,, N std and L,, L in, thus NL,, NL,, N L in by Definition Therefore M,, M,, NL,, NL,, N L std by Definition If N / Λ v (i.e., N =VM 1... M m with m>0) then N N and L L, so N v N and L v L by Remarks By induction respectively on (2) and (1), there are sequences N,, N in and L,, L std. Hence NL,, N L,, N L in by Definition Thus M,, M,, NL,, N L,, N L std by Definition If M = z then M = z by Remark 12.5, hence z in by Definition If M = λz.l then M = λz.l and L v L by Remark Hence there is a sequence L,, L std by induction on (1). By Definition 23.3, λz.l,, λz.l in. If M = N L then M = NL for some N, L Λ by Remark 3, since v v and M / Λ v. By Remark 12.5, NL N L ; clearly, each step of is an instance of the rule right of Definition 9. There are two sub-cases.

12 12 Standardization of a call-by-value calculus If N Λ v then N N and L L, so N v N and L v L by Remarks Thus there are sequences N,, N std and L,, L in by induction respectively on (1) and (2). Therefore, by Definition 23.4, NL,, NL,, N L in. If N / Λ v (i.e. N = VM 1... M m with m > 0) then N N and L L, thus N v N and L v L by Remarks By induction respectively on (2) and (1), there are sequences N,, N in and L,, L std. So NL,, N L,, N L in by Definition Due to non-sequentialization of head σ 1 - and head σ 3 -reductions, several standard sequences may have the same starting term and ending term: for instance, if M = I( I)I and N = (λz.(λx.xi)(zz))i then M v (λx.xi)( I) v N and M v (λz.i(zz))ii v (λz.i(zz)i)i v N are both standard sequences from M to N. Finally, we can compare our notion of standardization with that given in [16]. To make the comparison possible we avoid σ-reductions and we recall that βv is exactly the Plotkin s left-reduction [16, p. 146]. As remarked in [7, 1.5 p. 149], both sequences (λz.ii)(ii) v (λz.i)(ii) v (λz.i)i and (λz.ii)(ii) v (λz.ii)i v (λz.i)i are standard according to [16]. On the other hand, only the second sequence is standard in our sense. It is easy to check that collapsing the two notions of inner and standard sequence given in Definition 23, we get a notion of standard sequence that accept both the above sequences. 5 Some conservativity results The sequentialization result (Theorem 22) has some eresting semantic consequences. It allows us to prove that the λ σ v -calculus is sound with respect to the call-by-value observational equivalence roduced by Plotkin in [16]. Moreover we can prove that some notions, like that of potential valuability and solvability, roduced in [15] for λ v, coincide with the same notions in λ σ v. This justify the idea that λ σ v is an useful tool for studying the properties of λ v, which was inspiring its definition in [3]. Our starting po is the following corollary. Corollary If M v V Λ v then there exists V Λ v such that M β v V v V. 2. For any V Λ v, M β v V if and only if M β V. vσ Proof. The first po is proved by observing that, by Theorem 22, there are N, L Λ such that M β v L σ N v V. By Remarks 12.5, N Λ v and thus L = N according to Remark The second po is a consequence of Lemma 14.3 and Remark Let us recall the notion of observational equivalence defined by Plotkin [16]. Definition 27 (Halting, observational equivalence). Let M Λ. We say that ( the evaluation of) M halts if there exists V Λ v such that M β v V. The ( call-by-value) observational equivalence is an equivalence relation = on Λ defined by: M = N if, for every context C, one has that C M halts iff C N halts. 1 Clearly, similar notions can be defined for λ σ v using βvσ instead of βv. Corollary 26 proves that head σ-reduction plays no role neither in deciding the halting problem for evaluation, nor in reaching a particular value. So we can conclude that the observational equivalence in λ σ v is conservative with respect to that of λ v. 1 Original Plotkin s definition of call-by-value observational equivalence (see [16]) also requires that C M and C N are closed terms, according to the tradition identifying programs with closed terms.

13 G. Guerrieri, L. Paolini and S. Ronchi Della Rocca 13 Theorem 28 (Adequacy of v-reduction). If M v M then M halts iff M halts. Proof. If M β v V Λ v then M v M v V since βv v. By Corollary 26.1, there exists V Λ v such that M β v V. Thus M halts. Conversely, if M β v V Λ v then M v V since βv v. By confluence of v (Proposition 4, since M v M ) and Remark 3 (since V Λ v ), there is V Λ v such that V v V and M v V. By Corollary 26.1, there is V Λ v such that M β v V. So M halts. Corollary 29 (Soundness). If M = v N then M = N. Proof. Let C be a context. By confluence of v (Proposition 4), M = v N implies that there exists L Λ such that M v L and N v L, hence C M v C L and C N v C L. By Theorem 28, C M halts iff C L halts iff C N halts. Therefore, M = N. Plotkin [16] has already proved that M = βv N implies M = N, but our Corollary 29 is not obvious since our λ σ v -calculus equates more than Plotkin s λ v -calculus (see Example 5). The converse of Corollary 29 does not hold since λx.x(λy.xy) = but λx.x(λy.xy) and are different v-normal forms and hence λx.x(λy.xy) v. A further remarkable consequence of Corollary 26.1 is that the notions of potential valuability and solvability for λ σ v -calculus (studied in [3]) can be shown coincide with the ones for Plotkin s λ v -calculus (studied in [14, 15]), respectively. Let us recall their definition. Definition 30 (Potential valuability, solvability). Let M be a term: M is v-potentially valuable (resp. β v -potentially valuable) if there are m N, pairwise distinct variables x 1,..., x m and V, V 1,..., V m Λ v such that M{V 1 /x 1,..., V m /x m } v V (resp. M{V 1 /x 1,..., V m /x m } β v V ); M is v-solvable (resp.β v -solvable) if there are n,m N, variables x 1,..., x m and N 1,...,N n Λ such that (λx 1... x m.m)n 1 N n v I (resp. (λx 1... x m.m)n 1 N n β v I). Theorem 31. Let M be a term: 1. M is v-potentially valuable if and only if M is β v -potentially valuable; 2. M is v-solvable if and only if M is β v -solvable. Proof. In both pos, the implication from right to left is trivial since βv v. Let us prove the other direction. 1. Since M is v-potentially valuable, there are variables x 1,..., x m and V, V 1,..., V m Λ v (with m 0) such that M{V 1 /x 1,..., V m /x m } v V ; then, there exist V Λ v such that M{V 1 /x 1,..., V m /x m } β v V by Corollary 26.1 and because βv βv, therefore M is β v -potentially valuable. 2. Since M is v-solvable, there exist variables x 1,..., x m and terms N 1,..., N n (for some n, m 0) such that (λx 1... x m.m)n 1 N n v I; then, there exists V Λ v such that (λx 1... x m.m)n 1 N n β v V v I by Corollary 26.1 and because βv βv. According to Remarks 12.5, V = I and therefore M is β v -solvable. So, thanks to Theorem 31, the semantic (via a relational model) and operational (via two sub-reductions of v ) characterization of v-potential valuability and v-solvability given in [3, Theorems 24-25] is also a semantic and operational characterization of β v -potential valuability and β v -solvability. The difference is that in λ σ v these notions can be studied inside the calculus, while it has been proved in [14, 15] that the β v -reduction is too weak in order to characterize them. So λ σ v is a useful tool for studying semantic properties of λ v.

14 14 Standardization of a call-by-value calculus 6 Conclusions In this paper we have proved a standardization theorem for the λ σ v -calculus, roduced in [3]. The used technique is a notion of parallel reduction. Let us recall that parallel reduction in λ-calculus has been defined by Tait and Martin-Löf in order to prove confluence of the β-reduction, without referring to the difficult notion of residuals. Takahashi in [19] has simplified this technique and showed that it can be successfully applied to standardization. We would like to remark that our parallel reduction cannot be used to prove confluence of v. Indeed, take M = (λx.l) ( (λy.n)((λz.n )N ) ) L, M 1 = (λx.ll ) ( (λy.n)((λz.n )N ) ) and M 2 = ( (λy.(λx.l)n)((λz.n )N ) ) L : then M M 1 and M M 2 but there is no term M such that M 1 M and M 2 M. To sum up, does not enjoy the Diamond Property. The standardization result allows us to formally verify the correctness of λ σ v with respect to the semantics of λ v, so we can use λ σ v as a tool for studying properties of λ v. This is a remarkable result: in fact some properties, like potential valuability and solvability, cannot be characterized in λ v by means of β v -reduction (as proved in [14, 15]), but they have a natural operational characterization in λ σ v (via two sub-reductions of v ). We plan to continue to explore the call-by-value computation, using λ σ v. As a first step, we would like to revisit and improve the Separability Theorem given in [12] for λ v. Still the issue is more complex than in the call-by-name, indeed in ordinary λ-calculus different βη-normal forms can be separated (by the Böhm Theorem), while in λ v there are different normal forms that cannot be separated, but which are only semi-separable (e.g. I and λz.(λu.z)(zz)). We hope to completely characterize separable and semi-separable normal forms in λ σ v. This should be a first step aimed to define a semantically meaningful notion of approximants. Then, we should be able to provide a new insight on the denotational analysis of the call-by-value, maybe overcoming limitations as that of the absence of fully abstract filter models [15, Theorem ]. Last but not least, an unexplored but challenging research direction is the use of commutation rules to improve the call-by-value evaluation. We do not have concrete evidence supporting such possibility, but since λ σ v is strongly related to the calculi presented in [7, 1], which are endowed with explicit substitutions, we are confident that a sharp use of commutations can have a relevant impact in the evaluation. References 1 Beniamino Accattoli and Luca Paolini. Call-by-Value Solvability, Revisited. In Tom Schrijvers and Peter Thiemann, editors, Functional and Logic Programming, volume 7294 of Lecture Notes in Computer Science, pages Springer Berlin Heidelberg, Henk Barendregt. The Lambda Calculus: Its Syntax and Semantics, volume 103 of Studies in logic and the foundation of mathematics. North Holland, Amsterdam, Alberto Carraro and Giulio Guerrieri. A Semantical and Operational Account of Call-by- Value Solvability. In Anca Muscholl, editor, Foundations of Software Science and Computation Structures, volume 8412 of Lecture Notes in Computer Science, pages Springer Berlin Heidelberg, April An extended version (with all the proofs) is available at 4 Karl Crary. A Simple Proof of Call-by-Value Standardization. Technical Report CMU-CS , Carneige Mellon University, Haskell B. Curry and Robert Feys. Combinatory Logic, volume 1. North Holland, Amsterdam, Jean-Yves Girard. Linear logic. Theoretical Computer Science, 50(1):1 102, Hugo Herbelin and Stéphane Zimmermann. An Operational Account of Call-by-Value Minimal and Classical lambda-calculus in "Natural Deduction" Form. In Pierre-Louis Curien,

15 G. Guerrieri, L. Paolini and S. Ronchi Della Rocca 15 editor, Typed Lambda Calculi and Applications, 9th International Conference, TLCA 2009, Lecture Notes in Computer Science, pages , Roger Hindley. Standard and normal reductions. Transactions of the American Mathematical Society, pages , John Maraist, Martin Odersky, David N. Turner, and Philip Wadler. Call-by-name, call-byvalue, call-by-need and the linear lambda calculus. Theoretical Computer Science, 228(1 2): , Eugenio Moggi. Computational lambda-calculus and monads. Technical report, Edinburgh University, Tech. Report ECS-LFCS Eugenio Moggi. Computational Lambda-Calculus and Monads. In Proceedings of the Fourth Annual Symposium on Logic in Computer Science (LICS 89), pages IEEE Computer Society, Luca Paolini. Call-by-Value Separability and Computability. In Antonio Restivo, Simona Ronchi Della Rocca, and Luca Roversi, editors, Proceedings of the 7th Italian Conference in Theoretical Computer Science - ICTCS 01 (Torino, Italy, October 4-6, 2001), volume 2202 of Lecture Notes in Computer Science, pages Springer-Verlag, Luca Paolini and Simona Ronchi Della Rocca. Parametric parameter passing lambdacalculus. Information and Computation, 189(1):87 106, February Luca Paolini and Simona Ronchi Della Rocca. Call-by-value Solvability. Theoretical Informatics and Applications, 33(6): , November RAIRO Series, EDP-Sciences. 15 Luca Paolini and Simona Ronchi Della Rocca. The Parametric λ-calculus: a Metamodel for Computation. Texts in Theoretical Computer Science: An EATCS Series. Springer-Verlag, Berlin, Gordon D. Plotkin. Call-by-name, call-by-value and the lambda-calculus. Theoretical Computer Science, 1(2): , Amr Sabry and Matthias Felleisen. Reasoning About Programs in Continuation-passing Style. SIGPLAN Lisp Poers, V(1): , January Amr Sabry and Matthias Felleisen. Reasoning about programs in continuation-passing style. Lisp and symbolic computation, 6(3-4): , Masako Takahashi. Parallel reductions in lambda-calculus. Information and Computation, 118(1): , April 1995.

16 16 Standardization of a call-by-value calculus A Technical appendix A.1 Proofs omitted in the article Stated at p. 6 Remark Certainly, v v. 5. Certainly, v v. Proof. 4. The proof that M v M implies M M is by induction on M Λ. Recall that M = V N 1... N n for some n N, V Λ v and N i Λ for any 1 i n. By reflexivity of (Remark 10), N i N i for any 1 i n. There are five cases: either n > 0, V = λx.n, N 1 Λ v and M = N{N 1 /x}n 2... N n, then N N by reflexivity of (Remark 10), so M M by applying the rule β v. or n > 1, V = λx.n and M = (λx.nn 2 )N 1 N 3... N n, then N N by reflexivity of (Remark 10), so M M by applying the rule σ 1. or n > 0, N 1 = (λx.l)n and M = (λx.v L)NN 2... N n, then V V, L L and N N (Remark 10), so M M by applying the rule σ 3. or M = V N 1... N n with V v V, then V = λx.n and V = λx.n with N v N by Remark 3, so N N by induction hypothesis; thus M M by the rule λ; or n > 0 and M = V N 1... N i... N n with N i v N i for some 1 i n, then N i N i by induction hypothesis, V V by reflexivity of (Remark 10); hence M M by applying the rule var or λ, according as V is a variable or an abstraction, respectively. The proof that M M implies M v M is by straightforward induction on the derivation of M M. 5. The proof that M v M implies M M is by induction on M Λ. Remember that M = V N 1... N n for some n N, V Λ v and N i Λ for any 1 i n. As M v M and M βvσ M, one has: either M = V N 1... N n with V v V, then V = λx.n and V = λx.n with N v N by Remark 3, so N N according to Remark 12.4, thus M M by the rule λ for ; or n > 0 and M = V N 1N 2... N n with N 1 v N 1, then N 1 N 1 by induction hypothesis, and V V and N i N i for any 2 i n by reflexivity of (Remark 10); hence M M by applying the rule right for ; or n > 1 and M = V N 1... N i... N n with N i v N i for some 2 i n, then N i N i by Remark 12.4, V V and N j N j for any 2 j n with j i and N 1 N 1 by reflexivity of and (Remark 10); hence M M by applying the rule right for. The proof that M M implies M derivation of M M. v M is by straightforward induction on the Stated at p. 6 Lemma 13 (Substitution lemma for ). If M M and V V then M{V/x} M {V /x}. Proof. By induction on the derivation of M M. Let us consider the last rule r. If r = var then M = y M 1... M m and M = y M 1... M m with m N and M i M i for any 1 i m. By induction hypothesis, M i {V/x} M i {V /x} for any 1 i m. If y x