Proofs from Section 4.2

We first prove the Weakening Lemma. The result for systems, stated in the text, relies on similar results for threads and values.

PROPOSITION (4.5).

(a) IfΓ^Â P and ∆^; Γ then ∆^Â P.

(b) IfΓ^Âwp and ∆^; Γ then ∆^Âwp.

(c) IfΓ^ÂwV:ζ and ∆^; Γ then ∆^ÂwV:ζ.

Proof. All three results are proved, in a straightforward manner, by judgment in-duction (i.e. by inin-duction on the length of the type inference). We give one example for each result.

(a) (^A -^{aÕ 5&×} ) SupposeΓ^ÂÝ06C p^D becauseΓ^·0 :^{'( )} andΓ^Âæ p. Using the auxiliary results we obtain∆^{ 0} :^{'( )} and∆^Âæ p. Using^A -^{aÖÕ 5&×} , we have∆^—06C p^D . (b) (^A -^a

Ô

) SupposeΓ^Âwu?^GX:ζ^I p because:

Γ^Âwu:^{) 3 495 7} ζ⁸ and Γ⁺ wX:ζ^Âwp

Since we identify terms up to alpha-equivalence, the variables in X can also be chosen to be new to ∆, in which case ∆⁺ wX:ζ is well-defined, and it is easy to see that ^G∆⁺ wX:ζ^{Iõ; G} Γ⁺ wX:ζ^I . So we may apply induction to the above two statements to obtain:

∆^Âwu:^{) 3 495 7} ζ⁸ and ∆⁺ wX:ζ^Âwp

The rule^A -^aÔ may now be employed to infer∆^Âwu?^GX:ζ^I p as required.

(c) (^A -^>² ) SupposeΓ^Âwu:ζbecauseΓ^Gw+ uI£; ζ. Since∆^; Γthen by transitivity we have∆^Gw+uI•; ζ. Using^A -^>² , one can the infer∆^Âwu:ζ, as required. ^¾ As corollaries we immediately have the following:

COROLLARY A.1. (a) IfΓ^Â P thenΓ⁺ wV:ζ^Â P.

(b) IfΓ⁺ wV:ξ^Â P andζ^; ξthenΓ⁺ wV:ζ^Â P. ^¾ Proposition 4.5 states that well-typing is preserved when the typing environ-ment is augenviron-mented. It is also preserved when the typing environenviron-ment is decreased by omitting all occurrences of identifiers that do not occur free in the system being typed. LetΓ^> u denote the result of eliminating u from Γ, i.e. ^G Γ^> u^{I G} u^I is unde-fined and ^GΓ^> uI G w+ uI is undefined for every w. For any syntactic element t, let

“fid^GtI ” return the free identifiers in t.

LEMMA A.2 (RESTRICTION).

(a) IfΓ^Â P and u^zL fid^G PI thenΓ^> u^Â P.

(b) IfΓ^Âwp and u ^zL fid^GpI É *

w/ thenΓ^> u^Â_wp.

(c) IfΓ^ÂwU:ξand u ^zL fid^GUI É *

w/ thenΓ^> u^Â_wU:ξ.

Proof. In each case the result follows by a straightforward judgment induction. We

leave the details to the interested reader. ^¾

As a corollary we have that typing is preserved by scope extrusion:

COROLLARY A.3. Suppose e does not appear free in Q. ThenΓ^ÂìG νe^{I G}Q^E P^I if and only ifΓ^Â Q^EGνe^I P

Proof. We examine the case when e is a channel; the case in which e is a location is similar. Suppose Γ^{Â G} νa:Λ^{I G}Q^E PI . Then using ^A -^52B_`)h× and ^A -^{Q Aa ×} , we have thatΓ⁺ Λa ^Â Q andΓ⁺Λa^Â P. ApplyingLemma A.2to the first of these we obtain

GΓ⁺ Λa^{I >} a^Â Q, i.e. Γ ^Â Q since a is new toΓ. Applying ^A -^52B_`)× to the second statement we obtainΓ^{Â G}νa:Λ^I and therefore^A -^{Q Aa×} givesΓ^Â Q^EG νa:Λ^I P.

The converse uses the same arguments, in the reverse direction. ^¾ As a step toward proving subject reduction, note that closed terms are preserved by reduction.

LEMMA A.4. If P is closed and P^{d f} P^J then P^J is closed.

Proof. By induction on the judgment P ^{d f} P^J. ^¾

The proof of subject reduction for the typing system depends, as is often the case, on a substitution lemma. However in this case before the appropriate version can be proved we need the following technical Lemma.

LEMMA A.5.

(a) IfΓ^Â k:K and Γ⁺ z:K+ zX:ζ^Âwp then Γ⁺ kX:ζ^Â_w^³Úk^Ûz´^Ú

p^{* EkL} z^E/ . (b) IfΓ^Â k:K and Γ⁺ z:K+ zX:ζ^ÂwU:ξ then Γ⁺ kX:ζ^Â_w^³Úk^Ûz^´Ú U

* E

k^L z^E/ :ξ.

Proof. For both results the proof is similar. Informally the proof proceeds, in the case of threads, by taking a derivation of the judgmentΓ⁺ z:K+ zX:ζ^Âwp, substitut-ing k for z throughout and thereby obtainsubstitut-ing a derivation of Γ⁺ kX:ζ^Â_w^³Úk^Ûz^´Ú p^*EkL z/^E . Formally it is a straightforward induction on type judgments. We omit the details.

¾

Proof of the Substitution Lemma

We present the proof for the extended type system of Section 6. For this proof only, we write ^Â as shorthand for ^{Â JËJ}. The proofs for the other type systems are somewhat simpler.

LEMMA (4.7). For any closed value V:

(a) IfΓ^ÂvV:ζ and Γ⁺ vX:ζ^Âwp then Γ^Â_w^³ÚV^ÛX^´Ú p^{* EVL} X^E/ . (b) IfΓ^ÂvV:ζ and Γ⁺ vX:ζ^ÂwU:ξ then Γ^Â_w^³ÚV^ÛX^´Ú U^*EVL X^E/ :ξ.

Note that there is no corresponding substitution result for systems, because values must be typed at a specific location.

Throughout the proof we use primes to indicate terms in which the substitution has been performed; i.e. for t an element of any syntactic category, t^J denotes t^{* EVL} X^E/ .

We first prove the result (b) for values. The proof proceeds by induction on the structure of X. There are four cases: X may be w, X may be some identifier other than w, or X may have the the formX or z^{S NS}x^O.

First, suppose that X^@ w. Because X^@ w it must be that w^{J @} V ^@ k for some k. We proceed by induction on U to show thatΓ^ÂkU^J:ξ.

½ Suppose that U ^@ w and therefore U^{J @} V ^@ k. The second premise may be writtenΓ⁺ vw:ζ^Âww:ξ. Here we know that ζ^; ξand therefore the result follows by applying weakening to the first premise (Γ^Â k:ζ).

½ Suppose that U ^@ u ^W@ w. The second premise may be writtenΓ⁺ vw:ζ^Âwu:ξ.

There are two possibilities. Ifξis a location type, we must have thatΓ^GuIY; ξ and thusΓ^Âku:ξ. Otherwiseζmust be of the form^{'( )2*} u:ξ^{J +-,,,./} whereξ^{J ;} ξ.

SinceΓ^Âvk:ζwe can therefore conclude thatΓ^Âku:ξ.

½ In the other cases, U ^{@ S}U:^Sξand U ^@{06Nb^O:L^NSB^O, the result follows using the innermost induction.

Suppose, instead, that X^@ x^@W w. In this case it must be that w^{J @} w. Again we proceed by induction on U to show thatΓ^ÂwU^J:ξ.

½ Suppose that U ^@ x and therefore U^{J @} V. Eitherξis a location type and so by the first premiseΓ^Â V:ξ, orξis another type and so v must be equal to w and again the first premise give the required resultΓ^ÂwV:ξ.

½ Suppose that U ^@ u ^W@ x. The result is immediate by applying the Restriction Lemma (Lemma A.2) to the second premise.

½ Again, the other cases follow by straightforward induction.

Suppose X ^@ X:^{S S}ζ. Therefore V must have the formV and by assumption we^S have that:

Γ ^ÂvV:^{S S}ζ and Γ⁺ vX:^{S S}ζ ^ÂwU:ξ We can rewrite this as:

Γ ^ÂvV1:ζ1^+,$,,.+ V_n:ζn and Γ⁺ vX1:ζ1^+,,$,.+ vX_n:ζn ^ÂwU:ξ

Using induction we have:

Γ⁺ vX1:ζ1^+-,,,=+ v^GX_n? 1:ζn^? 1^{I Â}w^³ÚVnÛXn^´Ú U^{* EVnL} Xn^E/ :ξ

Repeating this process n times yieldsΓ^Âw^á U^J:ξ, as desired.

Finally, suppose X ^@ z^NSx^O:K^NA^{S O}. Therefore V must have the form k^NSa^O and by assumption we have that:

Γ ^Âvk^NSa^O:K^NSa^O and Γ⁺ vk^NSa^O:K^NSa^O|Â_wU:ξ We can rewrite this as:

Γ ^Â k:K and Γ ^Âk^Sa:A^S and Γ⁺ z:K+ z^Sx:A^{S Â}_wU:ξ UsingLemma A.5we have:

Γ⁺ k^Sx:A^{S Â}_w³Úk^Ûz^´Ú U

*E

k^L z/^E :ξ Applying induction yieldsΓ^Â_wá U^J:ξ, as desired.

Having established the result for values, we now prove the result (a) for threads:

Γ^ÂvV:ζ and Γ⁺ vX:ζ^Âwp imply Γ^Âw^á p^J

Again we proceed by induction on the structure of X. The inductive cases are as before, so we only present the base case where X is an identifier x. This case is established by a secondary induction on the judgmentΓ⁺ vX:ζ^Âwp. Most of the cases in the secondary induction are straightforward, the exceptions being the cases for input and channel restriction. We show these two cases.

First consider the case for^A -^aJËJÔ . Our proof obligation is to show:

Γ^ÂvV:ζ and Γ⁺ vx:ζ^Âwu?^GY:ξ^I q imply Γ^Âw^á u^J?^G Y:ξ^I q^J (*) There are two cases to consider, x^@ w and x ^W@ w. First suppose that x^@ w. Hereζ must be a location type, sayKand therefore V must be a location name, say k. The premises in (*) may therefore be written:

Γ^Â k:K and Γ⁺ w:K^Â_wu?^GY:ξ^I q From^A -^aJËJÔ we have:

Γ⁺ w:K ^Â_wu:^{)3 4656* a 7}ξ^{8 /} and Γ⁺ w:K+ wY:ξ ^Âwq

UsingLemma A.5twice, we obtain:

Γ ^Â_ku^J:^{)3 4656* a 7}ξ^{8 /} and Γ⁺ kY:ξ ^Â_kq^J

Finally,^A -^ašJËJÔ can be applied to arrive at the desired conclusion,Γ^Âku^J?^G Y:ξ^I q^J. Continuing the case for^A -^aJƒJÔ , suppose x^W@ w. This case is a standard application of induction. The details are as follows. Using the second premise of (*) and^A -^aJËJÔ , we can conclude that: Now we may use the inner induction to conclude:

Γ ^Âwu^J:

)3 4656*

a 7

ξ^{8 /} and Γ⁺ wY:ξ ^Âwq^J Therefore using^A -^aÔJËJ we have, as desired,Γ^Âwu^J?^G Y:ξ^I q^J.

Now consider the case for channel restriction ^A -^{52B_`) JƒJÔ} . In this case the proof obligation is:

Γ^ÂvV:ζ and Γ⁺ vx:ζ^Âw ^G νa:AI q imply Γ^Âw^{á G}νa:AI q^J (**) Using the second premise of (**) and^A

-52B_|)

JƒJ

Ô , we can conclude that:

Γ⁺ vx:ζ⁺ wa:A^Â_wq (***)

At this point we must consider two cases, either x ^W@ w or x^@ w. First suppose that x^@W w. Then (***) can be rewritten asΓ⁺ wa:A⁺ vx:ζ^Âwq and we can apply induction to getΓ⁺ wa:A^Â_wá q^J and then^A -^{52B_`) J¬JÔ} to getΓ^Â_wá Gνa:AI q^J, as required.

On the other hand if x^@ w, then we must useLemma A.5. Since x^@ w, it must be that V is a location name and thus V ^@ k and ζ^{@ K} for some k, K. We can therefore rewrite the first premise of (**) and the statement (***) as:

Γ^Â k:K and Γ⁺ w:K+ wa:A^Â_wq

These can be applied toLemma A.5to yieldΓ⁺ ka:A^Â_kq^J and thus, using^A -^{52B_|) JƒJÔ} , Γ^Âk ^G νa:A^I q^J, as required.

Proof of the Subject Reduction Theorem THEOREM (4.6).

(a) If Pⁿ P^J thenΓ^Â P if and only ifΓ^Â P^J. (b) If P ^{d f} P^J thenΓ^Â P impliesΓ^Â P^J.

The first statement is proved by induction on the proof of Pⁿ P^J. The main axiom, scope extrusion ^Q -^{By Aa} , is covered by theCorollary A.3. The other axioms and rules are straightforward calculations left to the interested reader.

The second statement is proved by induction on the proof of P ^d2f P^J. The rule ^a-^{Q Aa} follows from the first part the remaining rules are, again, straightforward calculations. We give two examples.

½ a-^{b (c2B} states ^06Cu :: p^D›d2f k^C p^D . By supposition Γ ^‰09Cu :: p^D . Then using hypothesis, which entailsΓ^Âæ a!

7

V⁸ p. Using the hypothesis and the rules for typing it must also be that:

Γ^Âæ V:ζ Γ^Âæ a:^{)3 46517} ζ⁸ Γ^Âæ a:^{)3 46517} ζ⁸ Γ⁺

o

X:ζ^Âæ q

Note here that V is a closed value. We can apply the Substitution Lemma to obtainΓ^Âæ q^*EVL X^E/ , as required.