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Concurrent Systems/notes/14 Checking bisimilarity, an inference system.md

@@ -7,6 +7,8 @@ Inference system = axioms + inference rules
 - completeness: whatever is bisimilar, it can be inferred
 
 #### Axioms & Rules for Strong Bisimilarity
+just what we've seen so far, literally.
+
 ![350](images/Pasted%20image%2020250429082812.png)
  quite obvious.
 
@@ -15,11 +17,19 @@ basically we can let the left or the right process evolve, leaving the other unc
 
 ![350](images/Pasted%20image%2020250429083129.png)
 - if a process does not perform any action, a restriction won't do anything
+- restriction is distributive
 - if the process does not do the restricted action, the restriction won't do anything
 	- what if we restrict on the actual action?
 		- well, then the process becomes 0
 
 ![350](images/Pasted%20image%2020250429083455.png)
+- if $P = P \land P = Q => Q = P$ (it's transitive :o)
+- the same happens with contexts
+
+
+#### Soundness theorem: $\vdash P=Q \implies P \sim Q$
+*Proof:*
+If $\vdash LHS=RHS$e need to consider the relation $\{ LHS,RHS \}$
 
 ![](images/Pasted%20image%2020250429083535.png)
 $P$ is in standard form if and only if $P \triangleq \sum_{i}\alpha_{i}P_{i}$ and $\forall_{i}P_{i}$ is in standard form.