Proof of Theorem 7: (3) Completeness Completeness is by induction on the derivation of reduction in the global calculus As before, we consider adapted reduction rules which are taken

modulo structural equality and reduction contexts, which are equivalent to the reduction rules in Fig- ure 18 (page 39). We list these adapted rules in the following, using the same names as the original rules in Figure 18. As before, we assume all variables are distinct, including across participants, so that we can write e.g.σ`e⇓Vinstead ofσ`e@A⇓V. We again writeCr[ ]for a reduction context

in the grammar of interactions.

(INIT)I≡Cr[A→B:b(νννs)˜ .I0] I 0_≡C r[(νννs)˜ I0] (σ,I) → (σ,I0₎ (COMM) I≡Cr[A→B:shop,e,xi.I0] I 0_≡C r[I0] σ`e⇓V (σ,I) → (σ[x@B7→v],I0<sub>)</sub> (ASSIGN)I≡Cr[x@A:=e.I0] I 0<sub>≡C</sub> r[I0] σ`e⇓V (σ,I) → (σ[x@B7→v],I0₎ (IFTRUE)I≡Cr[ifethenI0telseI0f] I

0_≡C r[I0t] σ`e⇓tt (σ,I) → (σ,I0₎ (SUM)I≡Cr[ifethenI0lelseI0r] I 0_≡C r[I0l] (σ,I) → (σ,I0₎

(REC)I≡Cr[recX.I0] (σ, Cr[I[(recX.I0)/X]]) → (σ 0_{, I}0₎ (σ,I) → (σ0_,I0₎

We omit(IFFALSE)and the symmetric case. Note the rules(PAR)and(RES)are no longer necessary since they are absorbed in the above rules. Up to the application of the rules of≡, all rules above except(REC)are the base cases. In the following reasoning, we use the obvious annotated version of these rules (which preserve thread labels across reduction, except when a new top-level parallel composition arises as a result of reduction, we take off its label).

In the following, by induction on the height of derivations, we show if

(σ,A)→(σ0,A0)

then

EPP(I, σ)→(P0_,σ0₎

where

(128) EPP(

A

0, σ0) = (P₀0,σ0)such thatP0l _≡recP₀0_.

Above, as in the proof of soundness, we neglect participants information in the endpoint processes, and aggregate the local states intoσ, assuming all local variables are distinct. For simplicity we also abbreviate (128) to:

(129) (P,σ0) l ≡rec EPP(

A

0, σ0)

We set

A

≡ (νννt)˜Π0≤i≤n

A

i

where each

A

iis a prime interaction (i.e. an interaction which does not contain a non-trivial top-level

parallel composition). Henceforth we safely neglect(νννt)˜. As before, we let

T

to be the set of threads andΨto be the partition of the family of thread projections w.r.t these threads. We writeS,S0, . . .for the elements ofΨ.

For(INIT), we can set:

A0

def= A→:τ0Bτ1_ch(ννν_{s)˜ .}

A

0

0 and consider the reduction:

(130) (

A

,σ) → ((νννs)˜

A

0

0|Π1≤i≤n

A

i,σ)

The endpoint projection of(

A

, σ)contains a pair of an input and an output corresponding to the redex of this reduction:

!ch(s)˜.Q def= t₀≤i≤n!ch(s)˜.Qi ≡!ch(s)˜.t0≤i≤nQi ({!ch(s)˜.Qi}0≤i≤n∈Ψ)

and an output:

Then we can write down

A

as !ch(s˜.Q|ch(νννs)˜.R|S. Thus we have a reduction: (131) (!ch(s)˜.Q|ch(νννs)R˜ |S,σ) → (!ch(s)˜.Q|(νννs)(Q˜ |R)|S,σ)

By the exactly identical reasoning as in the corresponding case in the proof of soundness, the residual in (130) and that in e (131) are related in the way:

EPP((νννs)˜

A

0

0|Π1≤i≤n

A

i, σ) l (!ch(s)˜.Q|(νννs)(Q˜ |R)|S,σ)

hence as required.

For(COMM), assume without loss of generality we have

A

0 def= Aτ0→Bτ1:shop,e,xi.A00 and consider the reduction:

(σ,

A

) → (σ0,

A

₀0|Π1≤i≤n

A

i)

whereσ0_{=σ[x@A7→V]with:}

σ@A`e⇓V.

The thread projection of

A0

toτ0has the formsop_j(e).R(when the branching is a singleton we omit the symbolΣ, similarly henceforth) such that{sopj(e).R} ∈Ψ, while the one ontoτ1has the form

sop(y).Q. Without loss of generality (cf. Proposition 11) we regardτ0,1is used only in

A0

. Thus we can set:

EPP(

A,

σ) ≡ (sop(y).Q|sopj(e).R|S,σ)

hence we have:

(132) EPP(

A

, σ) → (Q|R|S,σ0)

(in (132), the update of the store is safely done due to our stipulation that all local variables are distinct.) By the same reasoning as in the corresponding case in the proof of soundness, we know

EPP(

A

₀0|Π1≤i≤n

A

i, sigma0) l (Q_|R_|S_,σ0₎ as required.

For(ASSIGN), we can set

A

0 def= xτ@A:=e.A00. We consider the reduction:

(133) (σ,

A

) → (σ0,

A

₀0|Π1≤i≤n

A

i)σ0

with appropriateσ0_{. The thread projection ontoτhas the shapex:=e.TP(}

_A

0

0, τ), hence we have the reduction:

(134) (x@A:=e.TP(

A

₀0, τ)|R,σ) → (TP(

A

₀0, τ)|R,σ0)

As in the corresponding case in the proof of soundness, (134) shows that all thread projections of

A

0

except atτremain invariant from that of

A, whose aggregate is

R; and the projection ontoτprecisely matches that of the residual of (133), hence as required.

For(IFTRUE), we can set

A0

def= ifeτ@Athen

A

0

0telse

A

00f

with which we have the reduction:

(135) (

A,

σ) → (

A

₀0_t|Π1≤i≤n

A

i,σ)

Observing

TP(

A

₀, τ) def= ifethenTP(

A

₀0_t, τ)elseTP(

A

₀0_f, τ)

we have the reduction for the endpoint projection: (136) (i f thenelseeTP(

A

0

0t, τ)TP(

A

00f, τ)|R,σ) → (TP(

A

00t, τ)|R,σ)

whereeevaluates to true inσ. By the reasoning for the corresponding case in the soundness proof,

Rin (136) may contain replicated inputs which are the result of merging complete threads from

A

0

0f.

Thus we obtain:

EPP(

A

₀0_t|Π1≤i≤n

A

i, σ) l (TP(

A

₀0_t, τ)|R,σ)

as required.

(IFFALSE)and(SUM)are similarly reasoned. For(REC), let:

A

0 def= recX.A00. Further assume we have:

(137) (

A

,σ) → (

A

₀00|Π₁≤i≤n

A

i,σ0)

The reduction (137) comes from, by the recursion rule above: (138) (

A

₀0[(recX.

A

₀0)/X],σ) → (

A

₀00,σ0)

Now the endpoint projections of

A0

has the form:

(139) EPP(

A0

, σ) def= ((ΠPi0)|R,σ)

whereRis a collection of replicated processes and eachP0

iis not replicated and has the shape:

P0

i def= recX.Pi.

We then consider the endpoint projection of the unfolding of

A0

:

(140) EPP(

A

₀0[(recX.A₀0)/X], σ) def= ((ΠPi[Pi0/X])|R,σ)

Note the right-hand side of (140) is then-times unfoldings of (139). Thus by induction hypothesis and applying the recursion rule in the endpoint processesn-times we obtain:

(141) EPP(

A,

σ) → EPP(

A

₀00|Π1≤i≤n

A

i, σ0)

as required. This exhausts all cases, establishing completeness. This concludes the proof of Theorem 7.

16.7. An Example of Endpoint Projection. In the following we present an example of the

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