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9
Concurrent Systems/notes/14 Checking bisimilarity, an inference system.md
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@ -29,12 +29,15 @@ basically we can let the left or the right process evolve, leaving the other unc
#### Soundness theorem: $\vdash P=Q \implies P \sim Q$
*Proof:*
If $\vdash LHS=RHS$e need to consider the relation $\{ LHS,RHS \}$
If $\vdash LHS=RHS$, we need to consider the relation $\{ LHS,RHS \} \cup Id$ and prove it's a bisimulation (spoiler: it is).
![](images/Pasted%20image%2020250429083535.png)
Since bisimilarity is an equivalence and a congruence, the inference rules holds.
>[!def] Standard form
$P$ 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.
**Lemma:** $\forall P \exists P'$* in standard form such that $\vdash P = P'$
**Lemma:** $\forall P \space \exists \space P'$ in standard form such that $\vdash P = P'$
*Proof:* by induction on the structure of P.
**Base case:** $P \triangleq 0$. It suffices to consider $P' \triangleq 0$ and conclude reflexivity.