Design and Verification of Adaptive Cache Coherence Protocols ...

is true if and only if P (hs2, s3, :::i) is true. Intuitively, \ P " means that \P will be true in
the next term of the sequence. Note that this operator cannot appear in liveness speci cations
it can only be used in liveness proofs. In the following theorems, let P , Q , F , G and H be
predicates regarding a sequence of terms.
Theorem-A: PQ if
(P ^G ) Q
2(P ):Q )
2((:P ^ P ) ) G )
P (s0) ) G (s0) where s0 is the initial term of the system.
Proof Let be a sequence of terms hs0, s1, s2, :::i where s0 is the initial term. Without
losing generality, we assume 9i 2f0 1:::g P (si). There are two cases:
8j 2 f0 1:::ig P (sj). This implies G (s0) and 8j 2 f0 1:::ig :Q (sj). Therefore,
9k 2fi +1i+2:::g Q (sk).
9m 2f0 1:::i; 1g (:P (sm) ^8j 2fm +1:::ig P (sj)). This implies G (sm+1) and
8j 2fm +1m+2:::ig:Q (sj). Therefore, 9k 2fi +1i+2:::g Q (sk). 2
Theorem-B: P(P ^Q ) if
PQ
2((P ^ :P ) ) Q ).
Proof Let be a sequence of terms hs1, s2, :::i. Without losing generality, we assume P (s1).
This implies 9i 2f1 2:::g Q (si). There are two cases:
8j 2f1 2:::ig P (sj). Therefore, P (si) ^ Q (si).
9k 2f1 2:::i; 1g (P (sk) ^:P (sk+1)). This implies P (sk) ^ Q (sk). 2
Theorem-C: (P ^G ^F )((Q ^G ^F )_H ) if
PQ
2((G ^ :G ) ) H )
2((G ^F ) ) F ).
Proof Let be a sequence of terms hs1, s2, :::i. Without losing generality, we assume P (s1)
^ G (s1) ^ F (s1). This implies 9i 2f1 2:::g Q (si). There are two cases:
8j 2f1 2:::ig G (sj). This implies 8j 2f1 2:::ig F (sj). Therefore, Q (si) ^ G (si)
^ F (si).
9k 2f1 2:::i; 1g (G (sk) ^:G (sk+1)). This implies H (sk+1). 2
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