Lemmas and Corollaries

Lemma 1(Right Inversion). IfΣ^I;∆^I C,env(T_c, T_s, k, t) ::

Σ^O;∆^O, c:S then proc

r P

c

∈ CwhereΣ;∆ _T

s,T_c k,t P ::

c: [S]^+(kTs+t) .

Demonstração. Inversion onprocandcompose

Lemma 2(Left Inversion). IfΣ^I;∆^I, c:S C,env(T_c, T_s, k, t) ::

Σ^O;∆^O then hproc

r_i P_i

x_ii

∀i∈[1,n]∈ CwhereΣ_i;∆_i, c:S_{i T}

s,T_c k,t P_i ::

x_i :S_i

andQn

i=1S_i = [S]^+(kTs+t).

Demonstração. Inversions onprocandcompose

Lemma 3(Resource Inversion). IfΣ^I;∆^I C,env(T_c, T_s, k, t) ::

Σ^O, r:R;∆^O

then either I. idle(r)∈ C, or

II. proc r

P

x

∈ C

Demonstração. Inversions onproc,idle, andcompose

Lemma 4(Combreduction). If−;∆ Closed{ C }:: (r:R;x:A)with proc

q

Comb(f , d, z←(y₁,· · ·, y_n)) z

∈ C, for anyq,f,d,y₁,· · ·, y_nandz,x, then either 1. C−−−−−^Closed→ D, for someD, or

2. Cis communicating throughc∈∆orx.

Demonstração. This lemma is proved using a recursive proof that only terminates because theh-calculus does not permit cyclical references.

Σ_C;∆∆_C C:: (Σ_C, r :R;∆_C, x:A) (by inversion on (Closed) and substitution) for someΣ_C and∆_C

(by inversion onComb)

−;y₁:^•d1T₁^s−(t+d1)1,· · ·, y_n:^•dnTn^s−(t+dn)1 ^k,t_s,c Comb(f , p, z←(y₁,· · ·, y_n)) ::

z:^{•max(d1,···,dn)•p}T_out^{s−(t+max(d1,···,dn)+p)}1

wheres > t+max(d₁,· · ·, dn) +pandp >0

∀i ∈[1, n].y_i ∈∆∆_C (By (proc))

∀i ∈[1, n].eithery_i ∈∆ory_i∈∆_C (By Lemma 2)

We analyse two subcases: If there is ak∈[1, n]such thaty_k ∈∆, and if for alli ∈[1, n].y_i ∈

∆_C.

Case I(∃k ∈[1, n].y_k ∈ ∆). By definition of communication, C is communicating through y_k ∈∆so case 2 applies.

Case II(∀i ∈[1, n].y_i ∈∆_C).

C=C⁰⁰,h proc

q^y_i P_i^y

y_ii

∀i∈[1,n] (by applications of Lemma 1)

whereΣ_i;∆_{i s,c}^k,t P_i^y::

y_i:^•diT_i^s−(t+di)1 for someC⁰⁰,q^y_i,Σ_i and∆_i

(by inversions on (Comb) and (Signal-1)) P_i^y=Comb(g_i, p_i, y_i ←(w₁,· · ·, w_m)) orP_i^y=Sig(d_i, y_i ←e_i)

for somem, p_i, g_i, w₁,· · ·, w_mande_i

We analyse two subcases: If there is ak∈[1, n]such thatP_k^y=Comb(g_k, p_k, y_k ←(w₁,· · ·, w_m)), and if for alli ∈[1, n].P_i^y=Sig(d_i, y_i ←e_i).

Case II.I(∃k∈[1, n].P_k^y=Comb(g_k, pk, yk ←(w₁,· · ·, wm))).

C=C⁰⁰,proc q_k

Comb(g_k, p_k, y_i ←(w₁,· · ·, w_m)) y_k

(by substitution and generalization) for someC⁰⁰

By applying Lemma 4 recursively we know that eitherC−^Closed−−−−→ D, for someD(case 1), orC is communicating throughc∈∆orx(case 2).

Case II.II(∀i∈[1, n].P_i^y=Sig(d_i, y_i←e_i)).

(by substitution and generalization) C=C⁰⁰⁰,proc

q

Comb(f , p, z←(y₁,· · ·, y_n)) z

,h proc

q_i^y

Sig(d_i, y_i ←e_i) y_ii

∀i∈[1,n]

for someC⁰⁰⁰

C−−−−−^Closed→ C⁰⁰⁰,proc q

Sig(max(d₁,· · ·, d_n) +p, z←f (e₁,· · ·, e_n)) z

(by Comb) Case 1 applies.

Lemma 5(Regreduction). If−;∆ Closed{ C }:: (r:R;x:A)withproc

q

Reg(z←y) z

∈ C, for anyC⁰,q,y andz,x, then either 1. C−−−−−^Closed→ D, for someD, or

2. Cis communicating throughc∈∆orx.

Demonstração.

Σ_C;∆∆_C C:: (Σ_C, r :R;∆_C, x:A) (by inversion on (Closed) and substitution) for someΣ_C and∆_C

−;y:^•dT^s−(t+d)1 ^k,t_s,c Reg(z←y) ::

z:^•s−tT^s1

(by inversion onReg) wheres > t+d

y∈∆∆_C (By (proc))

Eithery∈∆ory∈∆_C (By Lemma 2)

Case I(y∈∆). By definition of communication,Cis communicating throughy∈∆so case 2 applies.

Case II(y∈∆_C).

C=C⁰⁰,proc q

Reg(z←y) z

,proc q^y

P^y

y

(by applications of Lemma 1) for someC⁰⁰ andq^y

(by inversions on (Comb) and (Signal-1)) P^y=Comb(g, p, y←(w₁,· · ·, w_m)) orP^y=Sig(d, y←e)

for somem, p, g, w₁,· · ·, w_mande

Case II.I(P^y=Comb(g, p, y←(w₁,· · ·, wm))).

C=C⁰⁰,proc q_k

Comb(g, p, y←(w₁,· · ·, w_m)) y_k

(by substitution and generalization) for someC⁰⁰

By Lemma 4, either C−−−−−^Closed→ D, for some D(case 1), orC is communicating through c∈∆ orx (case 2).

Case II.II(P^y=Sig(d, y←e)).

(by substitution and generalization) C=C⁰⁰⁰,proc

q

Reg(z←y) z

,proc q^y

Sig(d, y←e) y

,env(c, s, t, k) whereC⁰⁰=C⁰⁰⁰,env(c, s, t, k)

(by Reg) C−−−−−^Closed→ C⁰⁰,idle(r),proc

−

ticks−d;clock; a←Sig(0, a⁰←e)← {−;−}; y←a y freshaanda⁰

Case 1 applies.

Lemma 6(Internal resource request). IfΣ_C;∆∆_C C:: (Σ_C;∆_C, x:A)andCis requesting resourcem∈Σ_C thenC−−−−−^Closed→ D, for someD.

Demonstração.

Σ_C;∆∆_C C:: (Σ_C;∆_C, x:A) (from main assumption) C=C⁰,proc

p

b←m← {Σ^p;∆^p};P d

(from Definition 38) for someC⁰,b,Σ^p,∆^p,P andd

C=C⁰⁰,idle(mBM[Σ^m] [∆^m] [y]) orC=C⁰,proc m

M[Σ^m] [∆^m] [y]

y

(by Lemma 3) for someC⁰⁰,M,Σ^m,∆^mandy

We proceed analysing both cases:

Case I(C=C⁰⁰,idle(mBM[Σ^m] [∆^m] [y])).

(from Definition of context) C=E,idle(mBM[Σ^m] [∆^m] [y]),proc

p

b←m← {Σ^p;∆^p};P d

(by(inst-1)) C−−−−−^Closed→ E,proc

m

M[Σ^p

Σ^m] [∆^p

∆^m] [a y]

a

,proc p

P[a

b]

d (fresha)

Case II(C=C⁰,proc m

M[Σ^m] [∆^m] [y]

y

).

(from Definition of context) C=E,proc

m

M[Σ^m] [∆^m] [y]

y

,proc p

b←m← {Σ^p;∆^p};P d

(by(ext-2)) C−−−−−^Closed→ E,proc

m

M[(Σ^p×Σ^m)

Σ^m] [(∆^p×∆^m)

∆^m] [a y]

y

,proc p

P[a

b]

d (fresha)

Lemma 7(Internal communication). IfΣ_C;∆∆_C C:: (Σ_C, r:R;∆_C, x:A)andCis com-municating throughc∈∆_C then either

1. C−−−−−^Closed→ D, for someD, or

2. Cis communicating throughc∈∆orx.

Demonstração.

Σ_C;∆∆_C C:: (Σ_C, r :R;∆_C, x:A) (from main assumption)

(c:S)∈∆_C (generalization)

for someS

C=C⁰,env(T_c, T_s, k, t) (by Def. of Well-Formed Configuration) for someC⁰, Ts, Tc, k andt

C⁰=C⁰⁰,proc r^R

P^R

c

(by Lemma 1) whereΣ^R;∆^R _T

s,T_c

k,t P^R::

c:S for someC⁰⁰, r^R, P^R,∆^R andΣ^R C⁰=C⁰⁰⁰,h

proc r_i^L

P_i^L

x_ii

∀i∈[1,n] (by Lemma 2)

whereΣ^L_i;∆^L_i, c:S_{i T}

s,T_c

k,t P_i^L::

x_i:A_i and

Yn

i=1

S_i=S for anyC⁰⁰⁰, n, r_i^L, P_i^L, x_i, A_i,Σ^L_i and∆^L_i

C⁰=C^∗,h proc

r_i^L P_i^L

x_ii

∀i∈[1,n],proc r^R

P^R

c

(by Def. of Well-Formed Configuration) since∀i ∈[1, n].x_i,c

for someC^∗ C=C^∗,h

proc r_i^L

P_i^L

x_ii

∀i∈[1,n],proc r^R

P^R

c

,env(T_c, T_s, k, t) (by substitution) Σ^I∗;∆^I∗ C^∗,env(T_c, T_s, k, t) ::

Σ^O∗;∆^O∗

(by generalization) for someΣ^I∗,∆^I∗,Σ^O∗and∆^O∗

Σ^L;∆^L, c:S h proc

r_i^L P_i^L

xi

i

∀i∈[1,n],env(T_c, Ts, k, t) (by (proc) and (compose)) ::r₁^L:R^L₁,· · ·, r_n^L:R^L_n;x₁ :A₁,· · ·, x_n:A_n

where

n

Y

i=1

Σ^L_i =Σ^Land

n

Y

i=1

∆^L_i =∆^L for someR^L₁· · ·R^L_n

Σ^R;∆^R proc r^R

P^R

c

,env(T_c, T_s, k, t) ::

r^R:R^R;c:S

(by (proc) and (compose)) for someR^R

Σ^I∗×Σ^L×Σ^R;∆^I∗×∆^L×∆^R, c:S (by (proc) and (Compose) C^∗,h

proc r_i^L

P_i^L

x_ii

∀i∈[1,n],proc r^R

P^R

c

,env(T_c, T_s, k, t) ::

Σ^O∗, r^R:R^R, r₁^L:R^L₁,· · ·, r_n^L:R^L_n;∆^O∗, c:S, x₁:A₁,· · ·, x_n:A_n

Σ_C =Σ^I∗×Σ^L×Σ^R (by Substitution)

Σ_C =Σ^O∗, r^R:R^R, r₁^L:R^L₁,· · ·, r_n^L:R^L_n

∆_C, x:A=∆^O∗, c:S, x₁:A₁,· · ·, x_n:A_n

∆∆_C =

∆^I∗×∆^L×∆^R , c:S

Now we analyse each case of c : S covering all possible cases in which C is commu-nicating through c ∈ ∆_C. We will, for each case of c :S, use inversion steps to infer the possible values of P^R, P_i^L and more generally C^∗ (excluding cases where C is not communicating through c), to prove that, for every case, either C is communicating externally or an operational semantics rule applies.

Case I(S=→− α^τS⁰).

P^R=c←putv;P^R0 (by inversion of→R)

Yn

i=1

S_i =→−

α^τS⁰ (by substitution)

S_i =→−

α^τS_i⁰ orS_i=^•τS_i⁰ (by definition of Merge (×)) with at least one→−

α^τS_i⁰ instance

P_i^L=v_i ←getc;P_i^L0 orP_i^L=P_i^tick (by inversion of (→L) and generalization) with at least onev_i←getc;P_i^L0 instance

for anyv_i andP_i^ticksuch thatΣ^L_i;∆^L_i, c:^•τS_{i T}⁰

s,T_c

k,t P_i^tick::

x_i :A_i C=E,h

proc r_i^L

v_i ←getc;P_i^L0 x_ii

∀i∈I^get,proc r^R

c←putv;P^R0 c

(by substitution) forE=C^∗,h

proc r_i^L

P_i^tick x_ii

∀i∈I^tick,env(T_c, T_s, k, t)

and anyI^•andI^→such that{I^tick, I^get}is a partition of [1,n]

Σ^R;∆^R0, v :α^τ1 _T

s,Tc

k,t c←putv;P^R0::

c:→− α^τS_i

(by inversion on (→R)) for any∆^R0

(v:α^τ1)∈∆^R (by definition of context)

(v:α^τ1)∈

∆^I∗×∆^L×∆^R

(by Lemma X) (v:α^τ1)∈

∆^I∗×∆^L×∆^R

, c:S (by definition of Merge (×))

(v:α^τ1)∈∆∆_C (by substitution)

Either(v:α^τ1)∈∆or(v:α^τ1)∈∆_C (by Lemma 2)

Case I.I((v:α^τ1)∈∆).

C=A,proc r^R

c←putv;P^R0 c

(by generalization) for someA

(v:α^τ1)∈∆ (Case I.I)

Cis communicating throughv∈∆ (by definition of communication) Case 2 applies.

Case I.II((v:α^τ1)∈∆_C).

C=A,proc rsig

Sig(ρ, v←e) v

(by Lemma 1 and inversion on (Signal−2)) for anye, and someAandρ≤0

(by substitution and definition of Well-Formed Configuration) C=E⁰,h

proc r_i^L

vi ←getc;P_i^L0 xi

i

∀i∈I^get,proc r^R

c←putv;P^R0 c ,proc

r_sig

Sig(ρ, v←e) v whereE=E⁰,proc

r_sig

Sig(ρ, v←e) v C−−−−−^Closed→ E⁰,h

proc r_i^L

P_i^L0[v v_i]

x_ii

∀i∈I^get,proc r^R

P^R0

c

(by (→1)) ,proc

r_sig

Sig(ρ, v←e) v Case 1 applies.

Case II(S=←− α^τS⁰).

P^R=v←getc;P^R0 (by inversion of (←R))

Yn

i=1

S_i =←−

α^τS⁰ (by substitution)

S_i =→−

α^τS_i⁰ orS_i=←−

α^τS_i⁰orS_i =^•τS_i⁰ (by definition of Merge (×)) with exactly one←−

α^τS_i⁰ instance

(by inversion of (→L) (←L) and generalization) P_i^L=v_i ←getc;P_i^L0 orP_i^L=c←putv_i;P_i^L0 orP_i^L=P_i^tick

with exactly onec←putvi;P_i^L0 instance for anyv_i andP_i^ticksuch thatΣ^L_i;∆^L_i, c:^•τS_{i T}⁰

s,T_c

k,t P_i^tick::

x_i :A_i C=E,h

proc r_i^L

vi ←getc;P_i^L0 xi

i

∀i∈I^get,proc r_k^L

c←putvk;P_k^L0 xk

(by substitution) ,proc

r^R

v←getc;P^R0 c forE=C^∗,h

proc r_i^L

P_i^tick x_ii

∀i∈I^tick,env(T_c, T_s, k, t)

for anyk∈[1, n]andI^•andI^→such that{I^tick, I^get,{k}}is a partition of [1,n]

Σ^L_k;∆^L_k⁰, v_k :α^τ1, c:←− α^τS_{k T}⁰

s,T_c

k,t c←putv_k;P_k^L0::

x_k :A_k

(by inversion on (←L)) for any∆^L_k⁰

(v_k :α^τ1)∈∆^L_k (by definition of context)

(v_k :α^τ1)∈∆^L (becauseQn

i=1∆^L_i =∆^Land Lemma X) (v_k :α^τ1)∈

∆^I∗×∆^L×∆^R

(by Lemma X) (v_k :α^τ1)∈

∆^I∗×∆^L×∆^R

, c:S (by definition of Merge (×))

(v_k :α^τ1)∈∆∆_C (by substitution)

Either(v_k :α^τ1)∈∆or(v_k :α^τ1)∈∆_C (by Lemma 2) Case II.I((v_k :α^τ1)∈∆).

C=A,proc r_k^L

c←putvk;P^L0_k xk

(by generalization)

for someA

(v_k :α^τ1)∈∆ (Case II.I)

Cis communicating throughv_k ∈∆ (by definition of communication) Case 2 applies.

Case II.II((v_k:α^τ1)∈∆_C).

C=A,proc r_sig

Sig(ρ, v_k ←e) v_k

(by Lemma 1 and inversion on (Signal−2)) for anyA,eandρsuch thatρ≤0

(by substitution and definition of Well-Formed Configuration) C=E⁰,h

proc r_i^L

v_i ←getc;P_i^L0 x_ii

∀i∈I^get,proc r_k^L

c←putv_k;P_k^L0 x_k ,proc

r^R

v←getc;P^R0 c

,proc r_sig

Sig(ρ, v_k←e) v_k whereE=E⁰,proc

r_sig

Sig(ρ, v_k ←e) v_k C−−−−−^Closed→ C^∗0,h

proc r_i^L

P_i^L0[v_k v_i]

x_ii

∀i∈I^get,proc r_k^L

P_k^L0

x_k

(by (→2)) ,proc

r^R

P^R0[v_k v]

c

,proc r_sig

Sig(ρ, v_k ←e) v_k Case 1 applies.

Case III(S=α^τS⁰).

P^R=Sig(τ, c←e) (by inversion of (Signal-2))

Yn

i=1

S_i =α^τS⁰ (by substitution)

S_i =α^τS_i⁰ (by definition of Merge (×))

where Yn

i=1

S_i⁰=S⁰

P_i^L=Comb(f_i, p_i, x_i ←(y_i1,· · ·, y_im)), or (by inversion of (Comb), (Reg), (→R) and (←L))

=Reg(x_i ←c), or

=z_i ←putc;P_i^0L

where there is onek∈[1, m]such thaty_ik=c for anyf_i,p_i,m,y_ij,y_i,z_i andP_i^0L

C=E,h proc

r_i^L

Comb(f_i, p_i, x_i ←(y_i1,· · ·, y_im)) x_ii

∀i∈I^comb (by substitution) ,h

proc r_i^L

Reg(x_i ←c) x_ii

∀i∈I^reg,h proc

r_i^L

z_i←putc;P_i^0L x_ii

∀i∈I^put

,proc r^R

Sig(τ, c←e) c for someE=C^∗,env(T_c, T_s, k, t)

for anyρ,I^comb,I^regandI^put where{I^comb, I^reg, I^put}is a partition of [1,n]

If I^comb is not empty Lemma 4 is applied and if I^reg is not empty Lemma 5 is applied. In both of these cases, C −−−−−^Closed→ D, for some D, and Case 1 applies. The remaining case is the one where bothI^combandI^reg are empty andI^put= [1, n]:

C=E,h proc

r_i^L

z_i ←putc;P_i^0L ρi

∀i∈[1,n],proc r^R

Sig(τ, c←e) c For simplification, we choose to focus on only one specifick∈[1, m]

C=E⁰,proc r_k^L

z_k ←putc;P_k^0L x_k

,proc r^R

Sig(τ, c←e) c

(by substitution) for someE⁰

Now we analyse separately the case wherex_k =z_k and the case wherex_k,z_k:

Case III.I(x_k =z_k).

Σ_k;∆_k, c:α^τS_{k T}⁰

s,T_s k,t

x_k←putc;P_k^0L ::

x_k :→− α^τA_k

(by inversion on (→R)) x_k:→−

α^τA_k

∈∆_C Sincex_k ,x C=E⁰⁰,h

proc q_i

P_i^∗

w_ii

∀i∈[1,l] (by Lemma 2)

wherew_i :A_i and Yl

i=1

B_i=→− α^τA_k

B_i=→−

α^τB⁰_i orB_i =^•τB⁰_i (by Definition of merge (×)) for anyB⁰_i

P_i^∗=v_i←getx_k;P_i^∗0orP_i^∗=tickτ;P_i^∗0 (by inversion of (→L) and (tick)) for anyv_i andP_i^∗0

(by substitution and Definition of Configuration) C=A,h

proc q_i

v_i←getx_k;P_i^∗0 w_ii

∀i∈I^get

,proc r_k^L

x_k ←putc;P_k^0L x_k

,proc r^R

Sig(τ, c←e) c for someAandI^get⊆[1, l]

(by (→1)) C−−−−−^Closed→ A,h

proc q_i

P_i^∗0[c

v_i] w_ii

∀i∈I^get,proc r_k^L

P_k^0L

x_k

,proc r^R

Sig(τ, c←e) c Case 1 applies.

Case III.II(x_k ,z_k).

Σ_k;∆_k, c:α^τS_k⁰, z_k:←− α^τA_{k T}

s,T_s k,t

z_k ←putc;P_k^0L ::

x_k :C

(by inversion on (←L)) z_k :←−

α^τA_k

∈∆^I∗ z_k :←−

α^τA_k

∈∆∆_C (by substitution)

Eitherz_k ∈∆orz_k ∈∆_C (by Lemma X)

If z_k ∈∆then Case 2 applies. Now we continue analysing the case where z_k ∈ ∆_C. In this case, there could be other processes consumingz_k with other types. By Definition of merge, for the configuration to be well-typed, the types are either z_k :←−

α^τA⁰_k orz_k :^•αA⁰_k, and the result of their merge is zk :←−

α^τA^R_k. Now we proceed applying Lemma 2 on zk :←−

α^τA⁰_k and Lemma 1 onz_k :←−

α^τA^R_k, leaving us with the configuration:

C=E⁰⁰,proc q

v←getz_k;Q⁰ z_k

,h proc

q_i

v_i ←getz_k;Q_i w_ii

∀i∈I^get

for anyv,Q⁰,q_i,v_i,w_i andI^get

(by substitution and Definition of Configuration) C=A,proc

q

v←getz_k;Q⁰ z_k

,h proc

q_i

v_i ←getz_k;Q_i w_ii

∀i∈I^get

,proc r_k^L

z_k ←putc;P_k^0L x_k

,proc r^R

Sig(τ, c←e) c for someAandI^get⊆[1, l]

C−−−−−^Closed→ A,proc q

Q⁰[c/v]

z_k

,h proc

q_i Q_i[c

v_i] w_ii

∀i∈I^get (by (→2)) ,proc

r_k^L P_k^0L

x_k

,proc r^R

Sig(τ, c←e) c Case 1 applies.

Case IV(S=→−

⊕_c{`:S`}_`∈L).

P^R=c.k;P^R0 (by inversion of (→−

⊕R)) for anyP^R0andk∈L

Yn

i=1

Si =→−

⊕_c{`:S`}_`∈L (by substitution)

S_i =→−

⊕_c{`:S<sub>i`</sub>}_`∈LorS_i =any other type (by definition of Merge (×)) with at least one→−

⊕_c{`:Si`}_`∈Linstance P_i^L=casecof n

`⇒P<sub>i`</sub>^L0o

`∈L orP_i^L=any other process (by inversion of (→−

⊕ L)) with at least onecasecof n

`⇒P<sub>i`</sub>^L0o

`∈Linstance for anyP<sub>i`</sub>^L0

C=E,proc r^R

c.k;P^R0 c

,h proc

r_i^L

casecof n

`⇒P<sub>i`</sub>^L0o

`∈L

x_ii

∀i∈I^⊕ (by substitution) whereE=C^∗,env(T_c, T_s, k, t)andI^⊕⊆[1, n]

C−−−−−^Closed→ E,proc r^R

P^R0

c

,h proc

r_i^L P_ik^L0

x_ii

∀i∈I^⊕ (by (⊕1))

Case 1 applies.

Case V(S =←−

⊕_c{`:S`}_`∈L).

P^R=casecof n

`⇒P<sub>`</sub>^R0o

`∈L (by inversion of (←−

⊕R)) for anyP_`^R0

Yn

i=1

S_i =←−

⊕_c{`:S<sub>`</sub>}_`∈L (by substitution)

S_i =←−

⊕_c{`:S<sub>i`</sub>}_`∈Lor→−

⊕_c{`:S<sub>i`</sub>}_`∈LorS_i =any other type (by definition of Merge (×)) with exactly one←−

⊕_c{`:S<sub>i`</sub>}_`∈Linstance

(by inversion of (←−

⊕ L), (→−

⊕ L)) P_i^L=c.korP_i^L=casecof n

`⇒P<sub>i`</sub>^L0o

`∈L orP_i^L=any other process with exactly onec.kinstance

for anykandP_i`^L0 C=E,proc

r_j^L

c.k;P^R0_j x_j

,proc r^R

casecofn

`⇒P<sub>`</sub>^R0o

`∈L

c

(by substitution) ,h

proc r_i^L

casecof n

`⇒P<sub>i`</sub>^L0o

`∈L

x_ii

∀i∈I^⊕

whereE=C^∗,env(T_c, T_s, k, t),j<I^⊕and{I^⊕,{j}} ⊆[1, n]

C−−−−−^Closed→ E,proc r_j^L

P^R0_j

x_j

,proc r^R

P_k^R0

c

,h proc

r_i^L P_ik^L0

x_ii

∀i∈I^⊕ (by (⊕2)) Case 1 applies.

Case VI(S=S₁⊗S₂).

P^R= (c→(c₁, c₂)). P₁^R

P₂^R

(by inversion of (⊗R)) for anyc₁,c₂,P₁^RandP₂^R

Yn

i=1

S_i =S₁⊗S₂ (by substitution)

S_i =S_i1⊗S_i2 (by definition of Merge (×))

for anyS_i1 andS_i2

P_i^L= (ci1, ci2)←c;P_i^L0 (by inversion of (⊗L)) for anyc_i1,c_i2,P_i^L0

C=E,proc r^R

(c→(c₁, c₂)). P₁^R

P₂^R

c

(by substitution) ,h

proc r_i^L

(c_i1, c_i2)←c;P_i^L0 x_ii

∀i∈[1,n]

whereE=C^∗,env(T_c, T_s, k, t) C−−−−−^Closed→ E,proc

r₁^R

P₁^R[a₁

c₁] [a₂ c₂]

a₁

,proc r₂^R

P₂^R

a₂

(by (⊗2)) ,h

proc r_i^L

P_i^L0[a₁

c_i1] [a₂ c_i2]

x_ii

∀i∈[1,n]

(freshr₁^R,r₂^R,a₁ anda₂) Case 1 applies.

Case VII(S=^•τ1T^τ21).

P^R=Comb(f , p, c←w₁,· · ·, w_n), or (by inversion of (Comb), (Reg) and (Signal-1))

=Reg(c←w), or

=Sig(τ₁, c←e).

wherep >0andmax(delay(w₁),· · ·,delay(w_n)) +p=τ₁ for anyf,n,w_i,wande

(It is important to note another possibility istickτ1;aux←Sig(0, aux←e) ;c ←aux but from process equivalence we have that tick τ₁;aux ← Sig(0, aux←e) ;c ← aux ≡ Sig(τ₁, aux←e)so we only consider the latter version)

We analyse each case separately:

Case VII.I(P^R=Comb(f , p, c←w₁,· · ·, w_n)).

C−−−−−^Closed→ D, for someD (by Lemma 4)

Case 1 applies.

Case VII.II(P^R=Reg(c←w)).

C−−−−−^Closed→ D, for someD (by Lemma 5)

Case 1 applies.

Case VII.III(P^R=Sig(τ₁, c←e)).

Yn

i=1

S_i =^•τ1T^τ21 (by substitution)

S_i =^•τ1T^τ21 (by definition of Merge (×)) P_i^L=Comb(f_i, pi, zi←(u₁,· · ·, um)) orP_i^L=Reg(c←u) (by inversion of (Comb) and (Reg))

where there is onej ∈[1, m]such thaty_j=c for somem,f_i,p_i,z_i,u_i andu.

In the case of Combwe apply Lemma 4 and in the case of Regwe apply Lemma 5. In both cases we get thatC−−−−−^Closed→ Dfor someD. Case 1 applies.

Case VIII(S=µz.S).

P^R=L:P^R0 (by inversion of (µR))

for anyLandP^R0 C=E,proc

r^R

L:P^R0 c

(by substitution) for someE

C−−−−−^Closed→ E,proc r^R

P^R0h

L:P^R0. Li

c

(by (Loop)) Case 1 applies.

Case IX(S= 1).

P^R=endc (by inversion of (1R))

for anyLandP^R0 C=E,proc

r^R end

(by substitution)

for someE C−−−−−^Closed→ E,idle

r^R

(by (1)) Case 1 applies.

Lemma 8. IfΣ^I;∆^I C::

Σ^O;∆^O

, then either I. C −→ C⁰, for someC⁰, or

II. Cis communicating throughc∈∆^I, or III. Cis communicating throughc∈∆^O, or IV. Cis requesting resource fromr∈Σ^I, or

V. Cdoes not haveprocobjects (computation is over).

Demonstração. By induction on the configuration type rules that formC. Case X(One semantic object).

Case X.I(idle).

inst(DEF(P), R)

idle

−;− idle(rBP),env(s, c, k, t) :: (r:R;−) Cdoes not haveprocobjects (computation is over).

Case X.II(proc).

[Σ]^+T; [∆]^+T s,c k,t P ::

x: [A]^+T

inst((Σ;∆;A), R) (T =k×s+t) proc Σ;∆ proc

r P

x

,env(s, c, k, t) :: (r:R;x:A) For every processP, one of the cases apply. For instance:

• IfP =x←y, thenC −→ C⁰, for someC⁰,

• ifP =x←gety, thenCis communicating throughc∈∆^Oorc∈∆^I,

• ifP =tickτ, thenC −→ C⁰, for someC⁰,

• ifP =x←r← {Σ;∆}, thenCis requesting resource fromr∈Σ^I. Case XI(compose).

Σ^I₁;∆^I₁ C₁,env(s, c, k, t) ::

Σ^O₁;∆^O₁

Σ^I₂;∆^I₂ C₂,env(s, c, k, t) ::

Σ^O₂;∆^O₂

Compose Σ^I₁×Σ^I₂;∆^I₁×∆^I₂ C₁C₂,env(s, c, k, t) ::

Σ^O₁Σ^O₂;∆^O₁∆^O₂ By induction hypothesis, we have thatIH1andIH2simultaneously:

(IH1) Either

I. C₁,env(c, s, k, t)→ C− ⁰

1,env(c, s, k⁰, t⁰), or II. C₁is communicating throughc∈∆^I₁, or III. C₁is communicating throughc∈∆^O₁, or

IV. C₁is requesting resource fromr∈Σ^I₁, or

V. C₁does not haveprocobjects (computation is over).

(IH2) Either

I. C₂,env(c, s, k, t)→ C− ⁰

2,env(c, s, k⁰, t⁰), or II. C₂is communicating throughc∈∆^I₂, or

III. C₂is communicating throughc∈∆^O₂, or IV. C₂is requesting resource fromr∈Σ^I₂, or

V. C₂does not haveprocobjects (computation is over).

We proceed analysing all possible cases provided by (IH1) and(IH2). We start analysing cases of(IH1)for any kind ofC₂, using(IH2)only in the last case:

Case XI.I(C₁,env(c, s, k, t)→ C− ⁰

1,env(c, s, k⁰, t⁰), for someC⁰

1).

C₁C₂→ C− ⁰

1C₂ (by Lemma 7)

Case XI.II(C₁is communicating throughc∈∆^I₁).

C₁C₂ is communicating throughc∈∆^I₁×∆^I₂ (by Lemma 8) Case XI.III(C₁ is communicating throughc∈∆^O₁).

C₁C₂ is communicating throughc∈∆^O₁∆^O₂ (by Lemma 9) Case XI.IV(C₁ is requesting resource fromr∈Σ^I₁).

C₁C₂ is requesting resource fromr∈Σ^I₁×Σ^I₂ (by Lemma 10) Case XI.V(C₁does not haveproc objects). ByIH2:

Case XI.V.I(C₂→ C− ⁰

2, for someC⁰

2).

C₁C₂→ C− ₁C⁰

2 (by Lemma 7)

Case XI.V.II(C₂ is communicating throughc∈∆^I₂).

C₁C₂ is communicating throughc∈∆^I₁×∆^I₂ (by Lemma 8) Case XI.V.III(C₂ is communicating throughc∈∆^O₂).

C₁C₂ is communicating throughc∈∆^O₁∆^O₂ (by Lemma 9) Case XI.V.IV(C₂is requesting resource fromr∈Σ^I₂).

C₁C₂ is requesting resource fromr∈Σ^I₁×Σ^I₂ (by Lemma 10) Case XI.V.V(C₂ does not have proc objects (computation is over)).

C₁C₂does not have proc objects (computation is over) (by Definition 3)

No documento Universidade de Brasília (páginas 120-134)