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Concurrent Systems/notes/13 - Weak Bisimilarity.md

@@ -24,4 +24,18 @@ $\approx$ is a
 1. equivalence
 2. congruence
 3. weak bisimulation
-4. $\sim a$
+4. $\sim \subset \approx$
+
+#### Examples of weakly bisimilar processes
+![](../../Pasted%20image%2020250428083727.png)
+
+**Theorem:** given any process P and any sum M, N, then:
+1. $P \approx \tau.{P}$
+2. $M+N+\tau.N \approx M + \tau.N$
+3. $M+\alpha.P+\alpha.(N+\tau.P) \approx M + \alpha.(N + \tau.P)$
+
+*Proof:*
+take the symmetric closure of the following relations, that can be easily shown to be weak simulations:
+1. $S = \{ P,\tau.P \}\cup Id$
+2. $S=\{ M+N+\tau.N,M+\tau.N \}\cup Id$
+3. $S=\\{ (M+\alpha.P+) \}$