Design and Verification of Adaptive Cache Coherence Protocols ...

Lemma 38 D is strongly terminating and con uent, that is, rewriting a Migratory term with
respect to the draining rules always terminates and reaches the same normal form, regardless
of the order in which the rules are applied.
Proof The termination is obvious because throughout the draining process, only one new
message (Purge or Flush) can be generated for each cache cell, and each message is consumed
when it is received. The con uence follows from the fact that the draining rules do not interfere
with each other. 2
Lemma39isobvious since the draining rules cannot modify the processor-to-memory bu ers,
memory-to-processor bu ers and processors. Lemma 40 ensures that all the caches and message
queues are empty in a drained term. The proof follows from Lemma 36, which guarantees that
all the messages can be consumed when they are received.
Lemma 39 Given a Migratory term s,
(1) Pmb id(s) = Pmb id(dr(s))
(2) Mpb id(s) = Mpb id(dr(s))
(3) Proc id(s) = Proc id(dr(s))
Lemma 40 Given a Migratory term s,
(1) Cache id(dr(s)) =
(2) Min(dr(s)) =
(3) Mout(dr(s)) =
(4) Cin id(dr(s)) =
(5) Cout id(dr(s)) =
Lemma 41 ensures that, given a Migratory term, if an address is cached in some cache, then
the memory cell of the address in the corresponding drained term contains the same value. The
proof simply follows from Lemma 36 and the con uence property of the draining rules.
Lemma 41 Given a Migratory term s,
(1) Cell(a,v,Clean) 2 Cache id(s) ) Cell(a,v,C[ ]) 2 Mem(dr(s))
(2) Cell(a,v,Dirty) 2 Cache id(s) ) Cell(a,v,C[ ]) 2 Mem(dr(s))
De nition 42 (Mapping from Migratory to CRF) Given a Migratory term s, the corre-
sponding CRF term f(s) is the drained term dr(s) with all message queues and cache identi ers
removed.
6.4.3 Simulation of Migratory in CRF
Theorem 43 (CRF Simulates Migratory) Given Migratory terms s1 and s2,
s1 ! s2 in Migratory ) f(s1) ! f(s2) in CRF
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