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Concurrent Systems/notes/13 - CCS cose varie.md

@@ -29,3 +29,16 @@ Which challenges can (a.P)\a have?
 ## Idempotency of Sum
 **Proposition:** $α.P+α.P+M ∼ α.P+M$
 *Proof:*
+$$S = \{ (α.P+α.P+M , α.P+M) \}$$
+Is it a bisimulation?
+NO: the problem is that, for example:
+- α.P+α.P+M –α–> P
+- α.P+M –α–> P
+- BUT (P,P) in general does NOT belong to S!
+
+So we can try with $$S = \{ (α.P+α.P+M , α.P+M) \} ∪ \{(P,P)\}$$
+But it is not yet a bisimulation. 
+P –β–> P’ (challenge and reply), but we don't have (P', P') in S.
+
+So let's try with: $$S = \{ (α.P+α.P+M , α.P+M) \} ∪ Id$$
+Let's go!