Kirsten Lackner Solberg. Dept. of Math. and Computer Science. Odense University, Denmark

Transcription

1 Inference Systems for Binding Time Analysis Kirsten Lackner Solberg Dept. of Math. and Computer Science Odense University, Denmark June 21, 1993

2 Contents 1 Introduction 4 2 Review of Binding Time Analysis Well-formedness of types : : : : : : : : : : : : : : : : : : : : : : : : : : : : Well-formedness of expressions : : : : : : : : : : : : : : : : : : : : : : : : : The algorithms for binding time analysis : : : : : : : : : : : : : : : : : : : An algorithm for binding time analysis of types : : : : : : : : : : : An algorithm for binding time analysis of terms : : : : : : : : : : : 10 3 A Constraint based Binding Time Analysis Types and their well-formedness : : : : : : : : : : : : : : : : : : : : : : : : Expressions and their well-formedness : : : : : : : : : : : : : : : : : : : : : The assumption list : : : : : : : : : : : : : : : : : : : : : : : : : : : The well-formedness of expressions : : : : : : : : : : : : : : : : : : Properties of the well-formedness relation : : : : : : : : : : : : : : : 20 4 Incorporating [up] and [down] [up] and [down] on function types : : : : : : : : : : : : : : : : : : : : : : : [up] and [down] on non-function types : : : : : : : : : : : : : : : : : : : : The [up-down]-rule : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : Making the [up-down]-rule implicit : : : : : : : : : : : : : : : : : : : : : : 26 5 Generating the Constraint Set Properties of the algorithms : : : : : : : : : : : : : : : : : : : : : : : : : : 36 6 Solving the Constraint Set Properties of the algorithms : : : : : : : : : : : : : : : : : : : : : : : : : : 43 7 Recursion, Constants and Lists Recursion and constants : : : : : : : : : : : : : : : : : : : : : : : : : : : : A constraint based binding time analysis : : : : : : : : : : : : : : : Incorporating [up] and [down] : : : : : : : : : : : : : : : : : : : : : 47 1

3 7.1.3 Generating the constraint set : : : : : : : : : : : : : : : : : : : : : Lists : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : A constraint based binding time analysis : : : : : : : : : : : : : : : Incorporating [up] and [down] : : : : : : : : : : : : : : : : : : : : : Generating the constraint set : : : : : : : : : : : : : : : : : : : : : 53 8 Conclusion 56 A Proofs 58 A.1 Proofs from Section 3 : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 58 A.1.1 Lemma 3 : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 58 A.1.2 Lemma 4 : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 58 A.1.3 Lemma 6 : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 59 A.1.4 Lemma 7 : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 61 A.1.5 Proportion 8 : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 61 A.1.6 Proportion 9 : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 63 A.2 Proofs from Section 4 : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 65 A.2.1 Lemma 11 : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 65 A.2.2 Lemma 12 : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 66 A.2.3 Lemma 13 : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 68 A.2.4 Lemma 14 : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 68 A.2.5 Lemma 15 : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 69 A.2.6 Lemma 16 : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 71 A.2.7 Lemma 17 : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 72 A.2.8 Lemma 18 : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 73 A.3 Proofs from Section 5 : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 74 A.3.1 Lemma 23 : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 74 A.3.2 Lemma 24 : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 77 A.4 Proofs from Section 6 : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 82 A.4.1 Fact used in the proofs of Lemma 29 and 30 : : : : : : : : : : : : : 82 A.4.2 Lemma 29 : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 82 A.4.3 Lemma 30 : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 83 2

4 B A Miranda implementation of the algorithms 84 B.1 The T2 types : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 84 B.2 The E2 terms : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 85 B.3 Collecting the constraints : : : : : : : : : : : : : : : : : : : : : : : : : : : 85 B.3.1 Substitutions : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 87 B.3.2 Unication : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 89 B.3.3 The algorithm L : : : : : : : : : : : : : : : : : : : : : : : : : : : : 91 B.4 Solving the constraints : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 93 B.5 An example : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 96 3

5 1 Introduction We consider the problem of introducing a distinction between binding times (e.g. compiletime and run-time) into functional languages. It is well-known that such a distinction is important for the ecient implementation of imperative languages [1] and more recent results show that the performance of functional languages may be improved by using binding time information (e.g. [11, 6]). There are several approaches to the specication of binding time analysis. Some approaches are based on variants of abstract interpretation (e.g. [2, 3, 5]), others are based on projection analysis (e.g. [7]) and yet others (e.g. [9, 10]) use non-standard type systems and develop corresponding type inference algorithms. In this paper we shall take a logical approach and aim at constructing an algorithm for generating a set of constraints to be solved. In this way we will be able to make full use of substitutions as in ordinary type inference [8] this is contrary to other algorithms (e.g. [9]) where extra recursive calls have to be performed. The starting point for our work is the inference system for binding times of the simply typed -calculus as specied in [9]. This is reviewed in Section 2. However, we shall reformulate it in a style motivated by the inference system for linear types given in [13] using the so-called \use" types. This is described in Section 3. We construct an algorithm for binding time analysis from this inference system. This algorithm is O (n 4 ) where n is the size of the given term where the algorithm of [9] is exponential in the size of the term. We proceed in a couple of stages. First we get rid of the two rules [up] and [down] to get a simpler inference system. This is done in Section 4. In Section 5 we present an algorithm for nding the constraints that has to be fullled in order to turn a 1-level term into a term in the 2-level -calculus and in Section 6 we solve the constraints. In Section 7 we extend the types with lists and the terms with recursion and constants. Section 8 concludes. 4

6 [A] `0 A i : r [] `0 t 1 : r `0 t 2 : r `0 t 1 t 2 : r [!] `0 t 1 : r `0 t 2 : r `0 t 1! t 2 : r [A] `0 A i : c [] `0 t 1 : c `0 t 2 : c `0 t 1 t 2 : c [!] `0 t 1 : c `0 t 2 : c `0 t 1! t 2 : c [up] `0 t 1! t 2 : r `0 t 1! t 2 : c Figure 1: Well-formedness of the 2-level types 2 Review of Binding Time Analysis In this section we review the binding time analysis of Nielson and Nielson ([9]). In a 2-level -calculus the binding times are explicitly marked on each construction. For us a type of the 2-level -calculus is either a base type, a function type, or a product type t 2 T2 t ::= A i j A i j t t j t t j t! t j t! t where the A i 's are the base types. Here the underlined constructions are those of run-time kind and the non-underlined are those of compile-time kind. The terms are e 2 E2 e ::= he, ei j he, ei j fst e j fst e j snd e j snd e j x.e j x.e j e ( e ) j e ( e ) j x Notice that there is only one sort of variable, x. The overall binding time of a variable is determined by the -binding of it. 2.1 Well-formedness of types We rst introduce rules for annotating types. First we say that a type t is well-formed of binding time b where b is either r or c denoting run-time and compile-time, respectively, if `0 t : b. This well-formedness relation is given in Figure 1. A run-time function type can be thought of as a piece of code. The compiler, which generates code, can manipulate this piece of code. Therefore a run-time function type can be both of run-time kind and compile-time kind. This fact is expressed by the rule [up], which allows us to turn a runtime function type of kind run-time into a run-time function type of kind compile-time. Only the [up] rule allows us to transform a run-time type into a compile-time type and furthermore this is only possible for function types. 5

7 Example 1 An example of using Figure 1 is to show that the type ((A! A)! (A! A))! ((A! A)! (A! A)) is well-formed of compile-time kind for some base type A. First we have that `0 from [A]. From [!] we get `0 A : r `0 A : r `0 A! A : r A : r (1) Applying [!] to two copies of (1) we get `0 A! A : r `0 A! A : r `0 ((A! A)! (A! A)) : r Now we apply [up] to get the binding time c `0 ((A! A)! (A! A)) : r `0 ((A! A)! (A! A)) : c (2) From (1) we get by applying [up] `0 A! A : r `0 A! A : c (3) Now apply [!] to two copies of (3) to get `0 A! A : c `0 A! A : c `0 (A! A)! (A! A): c (4) The result is now obtained by using [!] to combine (2) and (4) `0 ((A! A)! (A! A)) : c `0 ((A! A)! (A! A)) : c `0 ((A! A)! (A! A))! ((A! A)! (A! A)) : c Well-formedness of expressions Next we say that the term e has type t and binding time b under the assumptions tenv if tenv `0 e : t : b. The type environment tenv is a function from variables to 2-level types and binding times. That is tenv x = (t, b) where t is the type of the variable x and b is the binding time of x. Given tenv then the ( function tenv[(t, b)/x] is dened by (t, b), if x = y (tenv[(t, b)/x]) y = tenv y, otherwise The well-formedness relation for 2-level terms is given in Figure 2. Basically we have two copies of the traditional inference system for typing the -calculus, one for the run-time 6

8 level and one for the compile-time level. Furthermore we have the two rules [up] and [down] allowing the two binding times to mix. The idea behind [up] is that in order to turn a compile-time term into a run-time term (i.e. to allow it to be evaluated at runtime) it has to express some computation, i.e. it must have a run-time function type. In order to turn a run-time term into a compile-time term (i.e. to talk about its evaluation at compile-time) its type must not only be a run-time function type, but the term is also not allowed to reference \free" run-time objects. Example 2 As an example of using Figure 2 we show that the term x.y.x ( y ) with type ((A! A)! (A! A))! ((A! A)! (A! A)) is well-formed of compile-time kind for some base type A. Let tenv be given by tenv x = (((A! A)! (A! A)), c) tenv y = (A! A, c) tenv z = undened if z 6= x and z 6= y From [x] we get tenv `0 x : ((A! A)! (A! A)) : c and tenv `0 y : A! A : c. Applying [down] on both of them gives tenv `0 x : ((A! A)! (A! A)) : c tenv `0 x : ((A! A)! (A! A)) : r and Now we can apply [()] to get tenv `0 y : A! A : c tenv `0 y : A! A : r tenv `0 x : ((A! A)! (A! A)) : r tenv `0 y : A! A : r tenv `0 x ( y ) : A! A : r Since tenv only contains variables of compile-time kind and the type of x ( y ) is a run-time function type of run-time kind it is possible to apply [up] to obtain tenv `0 x ( y ) : A! A : r tenv `0 x ( y ) : A! A : c After applying [] two times we obtain the desired result tenv 00 tenv 0 tenv `0 x ( y ) : A! A : c `0 y.x ( y ) : (A! A)! (A! A) : c `0 x.y.x ( y ) : ((A! A)! (A! A))! ((A! A)! (A! A)) : c 7

9 [x] tenv `0 x i : t : b, if tenv x i = (t, b) ^ `0 t : b [hi] tenv `0 e 1 : t 1 : r tenv `0 e 2 : t 2 : r tenv `0 h e 1, e 2 i : t 1 t 2 : r [hi] tenv `0 e 1 : t 1 : c tenv `0 e 2 : t 2 : c tenv `0 h e 1, e 2 i : t 1 t 2 : c [fst] tenv `0 e : t 1 t 2 : r tenv `0 fst e : t 1 : r [fst] tenv `0 e : t 1 t 2 : c tenv `0 fst e : t 1 : c [snd] tenv `0 e : t 1 t 2 : r tenv `0 snd e : t 2 : r [snd] tenv `0 e : t 1 t 2 : c tenv `0 snd e : t 2 : c [] tenv[(t 2, r)/x] `0 e : t 1 : r tenv `0 x.e : t 2! t 1 : r, if `0 t 2 : r [] tenv[(t 2, c)/x] `0 e : t 1 : c tenv `0 x.e : t 2! t 1 : c, if `0 t 2 : c [()] tenv `0 e 1 : t 2! t 1 : r tenv `0 e 2 : t 2 : r tenv `0 e 1 ( e 2 ) : t 1 : r [()] tenv `0 e 1 : t 2! t 1 : c tenv `0 e 2 : t 2 : c tenv `0 e 1 ( e 2 ) : t 1 : c [down] tenv `0 e : t : c tenv `0 e : t : r, if `0 t : r [up] tenv `0 e : t : r tenv `0 e : t : c, if `0 t : c ^ 8i: tenv x i = (t i, c) Figure 2: Well-formedness of the 2-level -calculus 8

10 where tenv 0 and tenv 00 are given by and tenv 0 x = (((A! A)! (A! A)), c) tenv 0 z = undened if z 6= x tenv 00 z = undened for all variables This inference system is part of the one used in [9] to construct a binding time analysis The algorithms for binding time analysis In [9] the algorithm for binding time analysis is in two parts, one for types and one for terms An algorithm for binding time analysis of types The algorithm T BTA for binding time analysis of types presented in [9] calculates an annotated type t and its overall binding time b (r or c) given a type t 0 and the overall binding time b 0 of the type. The calculated type is the type with as few underlined constructions as possible and it is well-formed of kind b (i.e. `0 t : b can be inferred from Figure 1). This annotation expresses that as many computations as possible are performed at compile-time. For base types: if the type is underlined and the overall binding time is r or if the type is not underlined and the overall binding time is c, then the type is allready well-formed. For an underlined type and overall binding time c or a non-underlined type and overall binding time r the well-formed type is A i with overall binding time r. This can be summarized by: T BTA [[ A i : c ]] = A i : c T BTA [[ A i : r ]] = A i : r T BTA [[ A i : r ]] = A i : r T BTA [[ A i : c ]] = A i : r Writing A i b we mean A i if b is c and A i if b is r. The set fr, cg forms a partially ordered set with the partial order given by r c. Now T BTA for base types can be written as T BTA [[ A b 1 i : b 2 ]] = let b = b 1 u b 2 in A b i : b where u is the meet operation on (fr, cg, ). For product types it is a bit more complicated: if the two subtypes do not have the same overall binding time after the recursive call, then a new recursive call will be performed. Writing b we mean if b is c and if b is r. 9

11 T BTA [[ t 1 b1 t 2 : b 2 ]] = let t 0 1 : b 0 1 = T BTA [[ t 1 : b 1 u b 2 ]] t 0 2 : b 0 2 = T BTA [[ t 2 : b 1 u b 2 ]] b = b 0 1 u b 0 2 in if b 0 = then 1 b0 2 t t 0 b0 2 b0 1 else T BTA [[ t 0 1 b t 0 2 : b ]] For function types the denition is analogous to product types, except that the [up]-rule of Figure 1 has to be taken into account. Writing! b we mean! if b is c and! if b is r. T BTA [[ t 1! b1 t 2 : b 2 ]] = let t 0 1 : b 0 1 = T BTA [[ t 1 : b 1 u b 2 ]] t 0 2 : b 0 2 = T BTA [[ t 2 : b 1 u b 2 ]] b = b 0 u 1 b0 2 in if b 0 = then 1 b0 2 t 0 1! b0 1 t 0 2 : b 2 else T BTA [[ t 0!b 1 t 0 : b 2 2 ]] The dierence between product types and function types is in the result; for product types the overall binding time of the two subtypes and the product is the same, whereas for function types the subtypes has the same overall binding time, but this need not be the overall binding time of the function type. This corresponds to the application of the rule [up]. It is shown in [9] that for all types t in T2 and binding times b `0 T BTA [[ t : b ]] can be inferred in Figure 1 and furthermore (T BTA [[ t : b ]]) t : b holds and (T BTA [[ t : b ]]) is the biggest among those t 0 : b 0 satisfying t 0 : b 0 t : b. Here the partial order is extended to pairs t : b by dening t 0 : b 0 t : b to mean t 0 t ^ b 0 b. On types, is dened as t 0 t if and only if t 0 and t are the same type if we ignore the underlinings and every underlining in t is also in t An algorithm for binding time analysis of terms The algorithm E BTA for binding time analysis of terms presented in [9] calculates an annotated term e, its type t, and its overall binding time b given a term e 0, a type t 0 and an overall binding time b 0. The annotated term is the term with as few underlined constructions as possible and it is well-formed of type t and binding time b (i.e. `0 e : t : b can be inferred from Figure 2). 10

12 When doing the analysis the variables will be annotated with their types and binding times to make the formulation easier. Writing x b we mean x if b is c and x if b is r. For pairs the analysis is straight forward: perform recusive calls on the subterms and if they do not have the same overall binding time, then try with a \smaller" binding time b 0. Writing h b, i we mean h, i if b is c and h, i if b is r. E BTA [[ h b 1 e 1, e 2 i : t 1 b 2 t 2 : b 0 ]] = let in e 0 1 : t 0 1 : b 0 1 = E BTA [[ e 1 : t 1 : b 1 u b 2 u b 0 ]] e 0 2 : t 0 2 : b 0 2 = E BTA [[ e 2 : t 2 : b 1 u b 2 u b 0 ]] b 0 = b 0 u 1 b0 2 if b 0 1 = b 0 2 then h b0 1 e 0 1, e 0 2i : t 0 1 b0 2 t 0 2 : b 0 0 else E BTA [[h e b0 0 1, e 0 2i : t 0 1 t b0 0 2 : b 0 ]] For the projections we rst have to get the missing type-information. We know the type of one component and have to infer the other. E BTA [[ fst b1 e : t 0 : b 0 ]] = let t 0 : = T 0 b0 0 BTA [[ t 0 : b 0 ]] t 1 be given by e has type t 0 t 1 when e and t 0 have no underlinings e 0 : t 00 0 b0 t 00 1 : b 0 = E BTA [[ e : t 0 0 b 1 t 1 : b 1 ]] in UD (fst b0 e 0 : t 00 0 : b 0, b 0 0) E BTA [[ snd b1 e : t 0 : b 0 ]] = let t 0 0 : b 0 0 = T BTA [[ t 0 : b 0 ]] t 1 be given by e has type t 1 t 0 when e and t 0 have no underlinings e 0 : t 00 1 b0 t 00 0 : b 0 = E BTA [[ e : t 1 b 1 t 0 0 : b 1 ]] in UD (snd b0 e 0 : t 00 0, b b0 0 ) 0 The function UD is used to check the applicability of the rules [up] and [down] in Figure 2. UD (e : t : b, b 0 ) = if (b = c) ^ (b 0 = r) ^ (`0 t : b 0 ) then e : t : b 0 else if (b = r) ^ (b 0 = c) ^ (`0 t : b 0 ) ^ all variables x in e have the form x c then e : t : b 0 else e : t : b down up For variables the type annotation on the variable and the type have to be the same. Again UD is used because [up] or [down] may be applicable. 11

13 E BTA [[ x b1 [t 1 ] : t 0 : b 0 ]] = let t 0 0 : b 0 0 = T BTA t 0 : b 0 t 0 1 : b 0 1 = T BTA (t 1 : b 1 ) u (t 0 0 : c) in UD (x b0 1 [t 0 1] : t 0 1 : b 0 1, b 0 0) For -abstractions all the occurrences of x in e have to have the same type and binding time annotation. Writing b. we mean. if b is c and. if b is r. E BTA [[ b 1 x[t 1 ].e t 2! b 2 t 3 : b 0 ]] = let t 0 2! b0 2 t 0 3 : b 0 0 = T BTA [[ t 2! b 2 t 3 : b 0 ]] e 0 : t 0 : b 0 = E BTA [[ e : t 0 : 3 b0 2 ]] Y = f(t 1, b 1 ), (t 0, 2 )g [ b0 f(t, b) x b [t] occurs in e 0 g t : b = u Y in if all members of Y is the same then UD ( b1 x[t 1 ].e 0 t 1! b1 t 0 : b 1, b 0 0) else E BTA [[ b x[t].[x b [t]/x]e 0 t! b t 0 : b 0 0 ]] For application we have to infer the missing type information for the operand. We start with a type containing no underlinings, then we use a recursive call to get more information about the type. By doing this we cannot throw away the information we have gained, therefore we need the auxiliary function E BTA which do not throw it away. Writing ( b ) we mean ( ) if b is c and ( ) if b is r. E BTA [[ e 1 ( b 1 e 2 ) : b 0 ]] = let t 1 be given by e 2 has type t 1 when e 2 has no underlinings in E BTA [[ e 1 ( e 2 ) : t 1! b 1 t 0 : b 0 ]] E BTA [[ e 1 ( e 2 ) : t 1! b 1 t 2 : b 0 ]] = let t 0 : = T 2 b0 0 BTA [[ t 2 : b 0 ]] e 0 1 : t 0 1! b0 1 t 00 2 = E BTA [[ e 1 : t 1! b 1 t 0 2 : b 1 ]] e 0 2 : t 00 1 : b 0 2 = E BTA [[ e 2 : t 1 : b 1 ]] t 0 : b 0 = (t 0 1, b 0 1) u (t 00 1, b 0 2) in if (t 0 1, b 0 1) = (t 00 1, b 0 2) then UD (e e 0 ) : :, ) (b0 2 t00 2 b0 1 b0 0 else E BTA [[ e 0 1 ( e 0 2 ) : t 0! t 00 2 : b 0 0 ]] Let Fe be a function from all the variables in e to T2fr, cg satisfying that for all x if Fe x = (t, b), then `0 t : b can be infered. It is shown in [9] that if e has type t, then Fe `0 E BTA [[ e : t : b ]] 12

14 can be inferred in Figure 2 and furthermore (E BTA [[ e : t : b ]]) e : t : b holds and (E BTA [[ e : t : b ]]) is the greatest among those e 0 : t 0 : b 0 satisfying e 0 : t 0 : b 0 e : t : b provided that Fe 0 `0 e 0 : t 0 : b 0 can be infered. Here the partial order is extended to triples by dening e 0 : t 0 : b 0 e : t : b to mean e 0 e ^ t 0 t ^ b 0 b. On terms is dened as e 0 e if and only if e 0 and e are the same term if we ignore the underlinings and every underlining in e is also in e 0 This algorithm is exponential in the size of the term. 13

15 W(! b A i, p) = [b = p] W(! b (U V), p) = [W(U, b), W(V, b), b = p] W(! b (U ( V), p) = [W(U, b), W(V, b), b p] Figure 3: Well-formedness for types. 3 A Constraint based Binding Time Analysis Before dening the new well-formedness relation, we shall write the types and terms in a dierent way, so that they correspond more closely to the types and terms of linear logic. In this way we can write the rule for e.g. [hi] and [hi] as one rule. The new system we get in this section corresponds in a one-to-one manner to the analysis of Section Types and their well-formedness The new types are U 2 T U ::=! b A i j! b (U U) j! b (U ( U) where b is r (for run-time) or c (for compile-time) or it is a so-called binding time variable. The set fr, cg forms a partially ordered set with the partial order given by r c. The types of T and T2 correspond to one another: T T2! r A i A i! c A i A i! r (U U) t t! c (U U) t t! r (U ( U) t! t! c (U ( U) t! t Just as the T2 types have to be well-formed the new types have to be well-formed. We do this by means of constraints on the values a binding time variable can take. The constraints are a list of inequalities between binding times of the form p = q, p < q, or p q; later we shall also allow constraints of other forms. The constraints can be solved if there exists a mapping from all the binding time variables to fr, cg such that all the inequalities are satised. From this follows that the constraint set is unsolvable if its transitive closure contains inequalities of the form c r, c = r, r < r, c < c, or c < r. The function W(U, p), dened in Figure 3, is used to determine constraints so that the type U with overall binding time p is well-formed. A base type! b A i is well-formed of kind p if p = b. A product type! b (U V) is well-formed of kind p if U and V are well-formed of kind b and b = p. The same should be true for function types but a run-time function 14

16 type can be of both run-time kind and compile-time kind. The relation between the function W in Figure 3 and the well-formedness relation for types `0 in Figure 1 is given by Lemma 3 and 4. The proofs of the lemmas are in Appendix A on page 58. Lemma 3 If `0 t : b (Figure 1) and U is the type corresponding to t, then W(U, b) (Figure 3) is solvable. 2 Lemma 4 If W(U, b) (Figure 3) is solvable by M and t is the type corresponding to U M, then `0 t : M b (Figure 1) can be derived; U M is the type U where all binding time variables b i are replaced by M b i. 2 Example 5 As an example of using W we will nd the constraints for! b 1 (! b 2 (! b 3 (! b 4 A (! b 5 A) (! b 8 (! b 6 A (! b 7 A)) (! b 9 (! b 10 (! b 11 A (! b 12 A) (! b 13 (! b 14 A (! b 15 A))) to be well-formed of binding time p. We calculate W(! b 1 (! b 2 (! b 3 (! b 4 A (! b 5 A) (! b 8 (! b 6 A (! b 7 A)) (! b 9 (! b 10 (! b 11 A (! b 12 A) (! b 13 (! b 14 A (! b 15 A))), p) = [W(! b 1 (! b 2 (! b 3 (! b 4 A (! b 5 A) (! b 8 (! b 6 A (! b 7 A)),b 1 ), W(! b 9 (! b 10 (! b 11 A (! b 12 A) (! b 13 (! b 14 A (! b 15 A)), b 1 ), b 1 p] = [W(! b 3 (! b 4 A (! b 5 A), b 2 ), W(! b 8 (! b 6 A (! b 7 A), b 2 ), b 2 b 1, W(! b 10 (! b 11 A (! b 12 A), b 9 ), W(! b 13 (! b 14 A (! b 15 A), b 9 ), b 9 b 1, b 1 p] = [W(! b 4 A, b 3 ), W(! b 5 A, b 3 ), b 3 b 2, W(! b 6 A, b 8 ), W(! b 7 A, b 8 ), b 8 b 2, b 2 b 1, W(! b 11 A, b 10 ), W(! b 12 A, b 10 ), b 10 b 9, W(! b 14 A, b 13 ), W(! b 15 A, b 13 ), b 13 b 9, b 9 b 1, b 1 p] = [b 4 = b 3, b 5 = b 3, b 3 b 2, b 6 = b 8, b 7 = b 8, b 8 b 2, b 2 b 1, b 11 = b 10, b 12 = b 10, b 10 b 9 ], b 14 = b 13, b 15 = b 13, b 13 b 9, b 9 b 1, b 1 p] = [b 3 = b 4 = b 5, b 3 b 2, b 6 = b 7 = b 8, b 8 b 2, b 2 b 1, b 10 = b 11 = b 12, b 10 b 9, b 13 = b 14 = b 15, b 13 b 9, b 9 b 1, b 1 p] This means that the type has the form! b 1 (! b 2 (! b 3 (! b 3 A (! b 3 A) (! b 6 (! b 6 A (! b 6 A)) (! b 9 (! b 10 (! b 10 A (! b 10 A) (! b 13 (! b 13 A (! b 13 A))) with binding time p and constraints [b 3 b 2, b 6 b 2, b 2 b 1, b 10 b 9, b 13 b 9, b 9 b 1, b 1 p] 15

17 b 1 b 2 b 3 b 6 b 9 b 10 b 13 p r r r r r r r r r r r r r r r c c r r r r r r c c r r r c r r c c r r r c r c c c r r r c c r c c r r r c c c c c c r r r r r c c c r r c r r c c c r r c r c c c c r r c c r c c c r r c c c c c c r c r r r c c c r c c r r c b 1 b 2 b 3 b 6 b 9 b 10 b 13 p c c r c c r c c c c r c c c r c c c r c c c c c c c c r r r r c c c c r c r r c c c c r c r c c c c c r c c r c c c c r c c c c c c c c r r r c c c c c c r r c c c c c c r c c c c c c c c r c c c c c c c c c Figure 4: Solutions to the constraints in Example 5 All the solutions to the constraints are listed in Figure 4. Notice that the constraints have more than one solution. This means that if we want to nd all the well-formed annotations of a given T2 type, then we rst assign binding time variables to the corresponding T type. Now we can nd the constraints using W, and solving them give all the solutions which can be translated back to T2. That we have found all the possible annotations of the T2 type follows from Lemma 3 and Expressions and their well-formedness The terms are now u 2 E u ::= ( b u, u) j fst b u j snd b u j b x.u j ( b u u) j x where b is r or c or a binding time variable. Note that there is no annotations on variables and that the binding time of a variable is determined by its -binding. The terms correspond to the E2 terms in a one-to-one manner: E ( r u, u) ( c u, u) fst r u fst c u snd r u snd c u r x.u c x.u ( r u u) ( c u u) x E2 he, ei he, ei fst e fst e snd e snd e x.e x.e e ( e ) e ( e ) x 16

18 [weakening] [contraction] A:I `1 u : U : p [C] A:I, x : t : b `1 u : U : p [C, W(t, b)] A:I, x : V : p, y : V : q `1 u : U : b [C] A:I, z : V : p `1 u[z/x, z/y] : U : b [C, p = q] and x not in A [exchange] A:I, x : V 1 : p, y : V 2 : q, B:J `1 u : U : b [C] A:I, y : V 2 : q, x : V 1 : p, B:J `1 u : U : b [C] and z not in A Figure 5: The structural rules The assumption list An assumption list has the form x 1 : t 1 : b 1, : : :, x n : t n : b n where the x i 's are variables, the t i 's are the T type of the i'th variable, and the b i 's are the binding time of the i'th variable. We shall assume throughout that all the x i 's are distinct. This means that when combining two assumption lists they have to be disjoint as opposed to the type environment in Figure 2 which is only extended for -abstraction. When making a proof in the inference system of Figure 2 we start with a \big" type environment whereas here we start with only one variable in the assumption list. When the assumption list is written in the form A:I, then I is the list of all the binding times of all the variables, and A is the list of all variables and their types. To combine two assumption lists A:I and B:J we use the notation A:I, B:J. When two lists of all the binding times I and J are to be combined we write (I, J). A list of containing only one binding time b is written as (b). A list of all the binding times containing b as the rst element and I as the rest is written as (b, I). One may note that [weakening] allows us to extend an assumption list, [contraction] allows us to identify variables in an assumption list, and [exchange] allows us to reorder the assumption list. The three structural rules [weakening], [contraction], and [exchange] in Figure 5 for handling the assumption list have no analogy in Figure 2 because there the type environment is a function The well-formedness of expressions Now the well-formedness relation has the form A:I `1 u : U : b [C] and says that the term u has type U and binding time b under the assumptions A:I, and provided that the constraint set C can be solved. 17

19 [(I] [id] x : U : b `1 x : U : b [W(U, b)] A:I, x : U : p `1 v : V : q [C] A:I `1 p x.v :! p (U ( V) : p [C, p = q] [(E] A:I `1 v :! b (U ( V) : p [C] B:J `1 u : U : q [D] A:I, B:J `1 ( b v u) : V : b [C, D, p = q = b] [I] A:I `1 u : U : p [C] B:J `1 v : V : q [D] A:I, B:J `1 ( p u, v) :! p (U V) : p [C, D, p = q] [E 1 ] [E 2 ] [down] [up] A:I `1 u :! p (U V) : q [C] A:I `1 fst p u : U : p [C, p = q] A:I `1 v :! p (U V) : q [C] A:I `1 snd p v : V : p [C, p = q] A:I `1 u : U : p [C] A:I `1 u : U : q [C, D(U, p, q)] A:I `1 u : U : p [C] A:I `1 u : U : q [C, U(U, p, I, q)] D(! b A i, p, q) = [r = c] D(! b (U V), p, q) = [r = c] D(! b (U ( V), p, q) = [q < p, b q] U(! b A i, p, I, q) = [r = c] U(! b (U V), p, I, q) = [r = c] U(! b (U ( V), p, I, q) = [p < q, I q, b p] Figure 6: The well-formedness relation for the 2-level -calculus 18

20 The [id]-rule of Figure 6 says that with the assumption that the variable x has type U and binding time b, then x has type U and kind b if U is well-formed of kind b. This rule is much the same as [x]. The [(I]-rule says that if with the assumption list A:I, x : U : p the term v has type V and binding time q, and constraints C, then with the assumption list A:I the term p x.v has type! p (U ( V) with binding time p and constraints C and [p = q]. By comparing this rule with [!] and [!] in Figure 2 the rules say that the variable and the body of the abstraction have to have the same binding time and that the -abstraction itself has exactly this binding time. The rule [(E] says that if with the assumption list A:I a term v has type! b (U ( V) and binding time p and constraints C, and with the assumption list B:J a term u has type U and binding time q and constraints D, then with the assumption list A:I and B:J the term ( b v u) has type V and binding time b and constraints C and D and [p = q = b]. By comparing this rule with [()] and [()] in Figure 2 the rules say that if with the same type environment the two terms have the same binding time, then the new term has this binding time. The dierence between the two inference systems is that in Figure 6 the assumption lists are dierent and in Figure 2 the type environments are equal. It is here the rules [weakening], [contraction], and [exchange] are to be applied. The rules [I], [E 1 ], and [E 2 ] can be explained and compared with Figure 2 in much the same way. The [down]-rule is used to transform a term of run-time function type of kind compile-time into a term of run-time function type of kind run-time. The function D is used to generate the constraints to ensure the correct use of the [down]-rule. To explain the denition of the function consider the [down]-rule of Figure 2. We have to change the binding time from c to r, this is achieved by the constraint [q < p]; the only solution to this is q = r and p = c. It is required that the rule is only applied on run-time function types and not to compile-time function types; that is we must ensure that b = r is the only possible solution for b. This is achieved by the constraint [b q] (= [b r]). The side condition of [down] in Figure 2 says that the type has to be well-formed of the new binding time q (= r); this can be ensured by the constraints generated by W(! b (U ( V), q). But we know that! b (U ( V) is well-formed of binding time p (= c) and that the type is a run-time function type. Then we also know that! b (U ( V) is well-formed of binding time r. So we can omit to generate the constraints W(! b (U ( V), q). It should only be possible to apply [down] in the case where the term has a function type and therefore D will generate unsolvable [r = c] constraints in all other cases. The [up]-rule is used to transform a term of run-time function type of kind run-time into a term of run-time function type of kind compile-time. The function U is used to generate constraints to ensure the correct use of the [up]-rule. Again we have to ensure that the binding time is changed from r to c, this is done by the constraint [p < q], which has one solution, p = r and q = c. Furthermore in the assumption list all the binding times have to be compile-time and this is ensured by the constraints [I q] because q = c. The operation is point-wise on I and can be written like this [(b, I) q] = [q b, I q] [I q] = [ ] if I is empty 19

21 But we will write it as [I q] for simplicity. We have to ensure that the rule is only applied on run-time function types; that is we must ensure that b is r: here the constraint [b p] (= [b r]) will do. We have to ensure that the type is well-formed of the new binding time q (= c); this can be ensured by the constraints generated by W(! b (U ( V), q). Again we use the fact that! b (U ( V) is well-formed of binding time p (= r) and that it is a run-time function type and therefore the type is also well-formed of binding time c. Finally, in the cases where the term does not have a function type we let U generate an unsolvable constraint [r = c] Properties of the well-formedness relation All the types constructed in `1 are well-formed. This property is given by Lemma 6. The proof of the lemma is in Appendix A on page 59. Lemma 6 If then A:I `1 u : U : b [C] (Figure 6) and the constraints C are solvable by M W(U, b) is solvable by M and W(V, p) is solvable by M for all x : V : p 2 A:I. 2 A variable can be renamed. The proof of the lemma is in Appendix A on page 61. Lemma 7 If A:I, x : V : b `1 u : U : p [C] and y 62A:I, then A:I, y : V : b `1 u[y/x] : U : p [C, W(V, b)] and C solvable by M, [C, W(V, b), b = b] solvable by M. 2 The relation between the well-formedness relation in Figure 2 and the one dened in Figure 6 is given by Proportion 8 and 9. The proofs of the proportions are in Appendix A on page 61 and 63. Proportion 8 If A:I `1 u : U : b [C] (Figure 6) and the constraints C are solvable by M and e is the term corresponding to u M and t is the type corresponding to U M, then there exists tenv so that tenv `0 e : t : M b (Figure 2) and if x i : U i : b i 2 A:I, then tenv x i = (t i, M b i ) and t i is the type corresponding to U M i ; u M and U M are term u and the type U, respectively, where all binding time variables b i are replaced by M b i. 2 20

22 b 1 = p 1 = p 2 b 2 = b 3 = b 4 b 5 = b 6 = b 7 r r r c c r c c c Figure 7: Solutions to constraints in Example 10 Proportion 9 If tenv `0 e : t : b (Figure 2), then there exists constraints C and A:I such that A:I `1 u : U : b [C] (Figure 6) and C is solvable and t is the type corresponding to U, and e is the term corresponding to u. Whenever tenv x i = (t i, b i ) and `0 t i : b i, then x i : U i : b i 2 A:I provided t i is the type corresponding to U i. 2 Example 10 As an example of using the system in Figure 6 we use the same example as for Figure 2. We want to show that the term c x. c y.( r x y) with type! c (! r (! r (! r A (! r A) (! r (! r A (! r A)) (! c (! r (! r A (! r A) (! r (! r A (! r A))) and binding time c is well-formed for some base type! r A. In doing this we rst place binding time variables everywhere we can to make the proof more general. The question is now how can we annotate the term b 1 x. b 2 y.( b 3 x y) given that it has binding time p? We start by using [id] twice and we have and x : Ux : p 1 `1 x : Ux : p 1 [W(Ux, p 1 )] y : Uy : p 2 `1 y : Uy : p 2 [W(Uy, p 2 )] where Ux is! b 1 (! b 2 (U y (! b 5 (! b 6 A (! b 7 A)) and Uy =! b 2 (! b 3 A (! b 4 A). Applying [(E] we get x : Ux : p 1, y : Uy : p 2 `1 ( b 1 x y) :! b 5 (! b 6 A (! b 7 A) : b 1 [W(Ux, p 1 ), W(Uy, p 2 ), p 1 = p 2 = b 1 ] Applying [(I] we get x : Ux : p 1 `1 p 2 y.( b 1 x y) :! p 2 (Uy (! b 5 (! b 6 A (! b 7 A)) : p 2 [W(Ux, p 1 ), W(Uy, p 2 ), p 1 = p 2 = b 1, p 2 = b 1 ] Applying [(I] once more we get to ; `1 p 1 x. p 2 y.( b 1 x y) :! p 1 (Ux (! p 2 (Uy (! b 5 (! b 6 A (! b 7 A))) : p 1 [W(Ux, p 1 ), W(Uy, p 2 ), p 1 = p 2 = b 1, p 2 = b 1, p 1 = p 2 ] The constraints from W(Ux, p 1 ) are [b 3 = b 2, b 4 = b 2, b 2 b 1, b 6 = b 5, b 7 = b 5, b 5 b 1, b 1 p 1 ] and the constraints from W(Uy, p 2 ) are [b 3 = b 2, b 4 = b 2, b 2 p 2 ]. 21

23 All the constraints are [b 3 = b 2, b 4 = b 2, b 2 b 1, b 6 = b 5, b 7 = b 5, b 5 b 1, b 1 p 1, b 3 = b 2, b 4 = b 2, b 2 p 2, p 1 = p 2 = b 1, p 2 = b 1, p 1 = p 2 ]. The solutions to the constraints are in Figure 7 and the term is b 1 x. b 1 y. ( b 1 x y) with type! b 1 (! b 1 (! b 2 (! b 2 A (! b 2 A) (! b 5 (! b 5 A (! b 5 A)) (! b 1 (! b 2 (! b 2 A (! b 2 A) (! b 5 (! b 5 A (! b 5 A))) and binding time b 1. None of the solutions are the one we are looking for. In the proof we did not apply the rule [up] and [down] as we did in the proof in example 2. Now we try to copy what we did in example 2. Again we start by using [id] twice as above, but before we apply [(E] we apply [down] on the results from [id]. Then we get and x : Ux : p 1 `1 x : Ux : p 1 [W(Ux, p 1 )] x : Ux : p 1 `1 x : Ux : p 2 [W(Ux, p 1 ), D(Ux, p 1, p 2 )] y : Uy : p 3 `1 y : Uy : p 3 [W(Uy, p 3 )] y : Uy : p 3 `1 y : Uy : p 4 [W(Uy, p 3 ), D(Uy, p 3, p 4 )] Again Ux is! b 1 (! b 2 (U y [(E] to get (! b 5 (! b 6 A (! b 7 A)) and Uy =! b 2 (! b 3 A (! b 4 A). Now we apply x : Ux : p 1, y : Uy : p 3 `1 ( b 1 x y) :! b 5 (! b 6 A (! b 7 A) : b 1 [W(Ux, p 1 ), D(Ux, p 1, p 2 ), W(Uy, p 3 ), D(Uy, p 3, p 4 ), p 2 = p 4 = b 1 ] Now we apply [up] and get x : Ux : p 1, y : Uy : p 3 `1 ( b 1 x y) :! b 5 (! b 6 A (! b 7 A) : p 5 [W(Ux, p 1 ), D(Ux, p 1, p 2 ), W(Uy, p 3 ), D(Uy, p 3, p 4 ), p 2 = p 4 = b 1, U(! b 5 (! b 6 A (! b 7 A), b 1, I, p 5 )] where I = (p 1, p 3 ). Finally we apply [(I] twice to get x : Ux : p 1 `1 p 3 y.( b 1 x y) :! p 3 (Uy (! b 5 (! b 6 A (! b 7 A)) : p 3 [W(Ux, p 1 ), D(Ux, p 1, p 2 ), W(Uy, p 3 ), D(Uy, p 3, p 4 ), p 2 = p 4 = b 1, U(! b 5 (! b 6 A (! b 7 A), b 1, I, p 5 ), p 3 = p 5 ] ; `1 p 1 x. p 3 y.( b 1 x y) :! p 1 (Ux (! p 3 (Uy (! b 5 (! b 6 A (! b 7 A))) : p 1 [W(Ux, p 1 ), D(Ux, p 1, p 2 ), W(Uy, p 3 ), D(Uy, p 3, p 4 ), p 2 = p 4 = b 1, U(! b 5 (! b 6 A (! b 7 A), b 1, I, p 5 ), p 3 = p 5, p 1 = p 3 ] The constraints are now [b 3 = b 2, b 4 = b 2, b 2 b 1, b 6 = b 5, b 7 = b 5, b 5 b 1, b 1 p 1, p 2 < p 1, b 1 p 2, b 3 = b 2, b 4 = b 2, b 2 p 3, p 4 < p 3, b 2 p 4, p 2 = p 4 = b 1, b 1 < p 5, p 5 p 1, p 5 p 3, b 5 b 1, p 3 = p 5, p 1 = p 3 ]. There is one solution which is the one we are looking for (b 1 = p 2 = p 4 = r, b 2 = b 3 = b 4 = r, b 5 = b 6 = b 7 = r, p 1 = p 3 = p 5 = c). 2 22

24 D 0 (! b A i, p, q) = [r = c] D 0 (! b (U V), p, q) = [r = c] D 0 (! b (U ( V), p, q) = [q p, b q] U 0 (! b A i, p, I, q) = [r = c] U 0 (! b (U V), p, I, q) = [r = c] U 0 (! b (U ( V), p, I, q) = [p q, p t I q, b p] Figure 8: [up] and [down] on function types 4 Incorporating [up] and [down] Now we want to build the two rules [up] and [down] into all the other logical rules and the axiom [id]. Then all what is left will be the structural and logical rules. This makes it easier to make a proof in the inference system, since we do not have to think explicitly about using [up] and [down] as we had to do in Example 10. This implies that when making a proof, we just apply the logical rules to get all the solutions with one proof instead of two or even more proofs as in Example 10. To do this we proceed in four stages: Stage 1: Modify the denition of U and D such that [up] and [down] may leave the binding time unchanged (in the case of function types). Stage 2: Modify the denition of U and D such that [up] and [down] may succeed on base types and product types as well as function types. Stage 3: Combine the [up] and [down] rule into one rule called [up-down]. Stage 4: Integrate the [up-down]-rule with all the other rules. Again the new system we get in this section corresponds in a one-to-one manner to the system in Section 3 and therefore also to the analysis in Section [up] and [down] on function types In Stage 1 we shall modify the denitions of U and D such that [up] and [down] may leave the binding time unchanged (in the case of function types). This is done by dening two new functions D 0 and U 0 with this property and use them together with the rules in Figure 6 instead of D and U. This new well-formed relation is called `2. The two functions D 0 and U 0 are dened in Figure 8. The idea is to allow q p instead of q < p in D 0 and p q instead of p < q in U 0. This works ne for D 0 but not for U 0. To see this consider the case where the type is a function type,! b (U ( V) and both p and q are c, that means no change in binding time. Then the constraints generated by U would be 23

25 U 0 (! b (U ( V), p, I, q) = U 0 (! b (U ( V), c, I, c) = [c c, I c, b c] Thus the assumption list is restricted to contain only compile-time variables. To avoid this we allow the constraints to also contain inequalities of the form p t I q which means that the least upper bound of p and every binding time of I has to be greater than or equal to q. It is an abbreviation for [p t (b, I) q] = [p t b q, p t I q] [p t I q] = [ ] if I is empty Instead of using [I q] in the denition of U 0 we use [p t I q]. This solves the problem since c t b is c for all b. If the [up]-rule is going to change the binding time, then p is r (and q is c) and r t b is b for all b and [r t I q] is equivalent to [I q]. In the case of no change in binding time for a compile-time function type the constraints that ensure that the type is a run-time function type, that is [b p] in U 0 and [b q] in D 0, are both solvable because both p and q have to be c since otherwise the type is not well-formed. If both p and q are r, then the type is not well-formed and the constraints are unsolvable because [b p] = [c r] and [b q] = [c r]. The relation between the inference systems of Figure 6 and 8 is given by the next two lemmas. The proofs of the lemmas are in Appendix A on page 65 and 66. Lemma 11 If A:I `1 u : U : b [C] in Figure 6 and the constraints C are solvable by M, then C 0 exists so that A:I `2 u : U : b [C 0 ] in Figure 8 and the constraints C 0 are solvable by M. 2 Lemma 12 If A:I `2 u : U : b [C] in Figure 8 and the constraints C are solvable by M, then C 0 exists so that A:I `1 u : U : b [C 0 ] in Figure 6 and the constraints C 0 are solvable by M [up] and [down] on non-function types In Stage 2 we modifying the [up] and [down] rules to succeed on base type and product type as well as function types. The two new functions D 00 and U 00 are dened in Figure 9 and are used in Figure 6 instead of D and U. The new well-formed relation is now called `3. The constraints generated for a base type or a product type U have to ensure that the binding time is not changed ([p = q]). The relation between the inference systems of Figure 8 and 9 is given by the next two lemmas. The proofs of the lemmas are in Appendix A on page 68. Lemma 13 If A:I `2 u : U : b [C] in Figure 8 and the constraints C are solvable by M, then C 0 exists so that A:I `3 u : U : b [C 0 ] in Figure 9 and the constraints C 0 are solvable by M. 2 24

26 D 00 (! b A i, p, q) = [p = q] D 00 (! b (U V), p, q) = [p = q] D 00 (! b (U ( V), p, q) = [q p, b q] U 00 (! b A i, p, I, q) = [p = q] U 00 (! b (U V), p, I, q) = [p = q] U 00 (! b (U ( V), p, I, q) = [p q, p t I q, b p] Figure 9: [up] and [down] on non-function types [up-down] A:I `4 u : U : p [C] A:I `4 u : U : q [C, UD(U, p, I, q)] UD(! b A i, p, I, q) = [p = q] UD(! b (U V), p, I, q) = [p = q] UD(! b (U ( V), p, I, q) = [b q, b p, p t I q] Figure 10: The [up-down]-rule Lemma 14 If A:I `3 u : U : b [C] in Figure 9 and the constraints C are solvable by M, then C 0 exists so that A:I `2 u : U : b [C 0 ] in Figure 8 and the constraints C 0 are solvable by M The [up-down]-rule In Stage 3 we combine [up] and [down] into one rule [up-down], this can be achieved by having one rule combining [up] and [down] and then generate constraints with a new function UD. The new well-formed relation `4 is dened as in Figure 6 but instead of using the two rules [up] and [down] we use the rule [up-down] dened in Figure 10. For base types and product types there are no change from D 00 and U 00 : we still generate the constraint [p = q]. We want the constraints to be as follows on function types: b p q UD(! b (U ( V), p, I, q) r r r id r r c up r c r down r c c id c r r unsolvable c r c unsolvable c c r unsolvable c c c id 25

27 Here id means that the generated constraints have to be solvable and do not change the binding time. up and down mean that the constraints have to be solvable and they change the binding time according to the application of respectively the rule [up] or [down]. The three rows marked with unsolvable correspond to the fact that a compile-time function type cannot be of run-time kind. Both D 00 and U 00 behaves just like this but for one case each: D 00 cannot cope with the case up and U 00 not with down. The problem comes from the bond between p and q. Clearly we cannot include both q p and p q as then p = q would follow and rule [up-down] would always act as the identity, contrary to what we are aiming for. In the following table the constraints from D 00 and U 00 are summarized. In the last column there is a X if all the constraints are solvable and nothing if they are unsolvable. If the solvablity of the constraints depends on I then this is showed by the constraints involving I. This only happens in the case of up. b p q [b q] [b p] [p t I q] solvable r r r [r r] [r r] [r t I r] X r r c [r c] [r r] [r t I c] [r t I c] r c r [r r] [r c] [c t I r] X r c c [r c] [r c] [c t I c] X c r r [c r] [c r] [r t I r] c r c [c c] [c r] [r t I c] c c r [c r] [c c] [c t I r] c c c [c c] [c c] [c t I c] X Notice that the constraints of UD(! b (U ( V), p, I, q) are those of D(! b (U ( V), p, q) and U(! b (U ( V), p, I, q) except that we have not included q p from D and p q from U. The only case where it is necessary to look at I is when p is r and q is c. This knowledge can be used when solving the constraints, so we collect the constraints in the form p t I q rather than writing it out. The relation between the inference systems of Figure 9 and 10 is given by the next two lemmas. The proofs of the lemmas are in Appendix A on page 69 and 71. Lemma 15 If A:I `3 u : U : b [C] in Figure 9 and the constraints C are solvable by M, then C 0 exists so that A:I `4 u : U : b [C 0 ] in Figure 10 and the constraints C 0 are solvable by M. 2 Lemma 16 If A:I `4 u : U : b [C] in Figure 10 and the constraints C are solvable by M, then C 0 exists so that A:I `3 u : U : b [C 0 ] in Figure 9 and the constraints C 0 are solvable by M Making the [up-down]-rule implicit The new rules can now be formed like [old rule] [up-down] 26

28 [(I] [id] x : U : b `5 x : U : q [W(U, b), UD(U, b, (b), q)] A:I, x : U : p `5 v : V : q [C] A:I `5 p x.v :! p (U ( V) : b [C, UD(! p (U ( V), p, I, b), p = q] [(E] A:I `5 v :! b (U ( V) : p [C] B:J `5 u : U : q [D] A:I, B:J `5 ( b v u) : V : s [C, D, UD(V, b, (I, J), s), p = q = b] [I] A:I `5 u : U : p [C] B:J `5 v : V : q [D] A:I, B:J `5 ( p u, v) :! p (U V) : p [C, D, p = q] [E 1 ] [E 2 ] [weakening] [contraction] A:I `5 u :! p (U V) : q [C] A:I `5 fst p u : U : b [C, UD(U, p, I, b), p = q] A:I `5 v :! p (U V) : q [C] A:I `5 snd p v : V : b [C, UD(V, p, I, b), p = q] A:I `5 u : U : p [C] A:I, x : V : b `5 u : U : p [C, W(V, b)] A:I, x : V : p, y : V : q `5 u : U : b [C] A:I, z : V : p `5 u[z/x, z/y] : U : b [C, p = q] and x not in A [exchange] A:I, x : V 1 : p, y : V 2 : q, B:J `5 u : U : b [C] A:I, y : V 2 : q, x : V 1 : p, B:J `5 u : U : b [C] W(! b A i, p) = [b = p] W(! b (U V), p) = [W(U, b), W(V, b), b = p] W(! b (U ( V), p) = [W(U, b), W(V, b), b p] UD(! b A i, p, I, q) = [p = q] UD(! b (U V), p, I, q) = [p = q] UD(! b (U ( V), p, I, q) = [b q, b p, p t I q] and z not in A Figure 11: The well-formedness relation for the 2-level -calculus without [up] and [down] 27

29 for every logical rule of Figure 6. This is illustrated by the rule [(I]. The new rule [(I] is obtained by [up-down] A:I, x : U : p `5 v : V : q [C] [(I] A:I `5 p x.v :! p (U ( V) : p [C, p = q] A:I `5 p x.v :! p (U ( V) : b [C, UD(! p (U ( V), p, I, b), p = q] Figure 11 denes the well-formed relation with the [up-down]-rule integrated in all the logical rules, and therefore no explicit [up-down] rule. There is one exception in the rule [I] because the type in the conclusion is of product type and it makes no sense to apply the [up-down]-rule. The relation between the inference systems of Figure 10 and 11 is given by the next two lemmas. The proofs of the lemmas are in Appendix A on page 72 and 73. Lemma 17 If A:I `4 u : U : b [C] in Figure 10 and the constraints C are solvable by M, then C 0 exists so that A:I `5 u : U : b [C 0 ] in Figure 11 and the constraints C 0 are solvable by M. 2 Lemma 18 a If A:I `5 u : U : b [C] in Figure 11 and the constraints C are solvable by M, then C 0 exists so that A:I `4 u : U : b [C 0 ] in Figure 10 and the constraints C 0 are solvable by M. 2 From the proportions 8 and 9 and the lemmas 12, 11, 14, 14, 16, 17, and 18 follows: Theorem 19 and If A:I `5 u : U : b [C] (Figure 11) and the constraints C are solvable by M and e is the term corresponding to u M and t is the type corresponding to U M, then there exists tenv so that tenv `0 e : t : M b (Figure 2) and if x i : U i : b i 2 A:I, then tenv x i = (t i, M b i ) and t i is the type corresponding to U M i ; u M and U M are term u and the type U, respectively, where all binding time variables b i are replaced by M b i. If tenv `0 e : t : b (Figure 2), then there exists constraints C and A:I such that A:I `5 u : U : b [C] (Figure 11) and C is solvable and t is the type corresponding to U, and e is the term corresponding to u. Whenever tenv x i = (t i, b i ) and `0 t i : b i, then x i : U i : b i 2 A:I provided t i is the type corresponding to U i. Example 20 As an example of using the inference system in Figure 11 we use the same term as for Example. 10. We want to show that the term c x. c y.( r x y) with type! c (! r (! r (! r A (! r A) (! r (! r A (! r A)) (! c (! r (! r A (! r A) (! r (! r A (! r A))) and binding time c is well-formed for some base type! r A. In doing this we rst place binding time variables everywhere we can to make the proof more general. The question is now how can we annotate the term b 1 x. b 2 y.( b 3 x y) given that it has binding time p? We start with using [id] twice and we have x : Ux : p 1 `5 x : Ux : p 2 [W(Ux, p 1 ), UD(Ux, p 1, (p 1 ), p 2 )] 28 2