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Concurrent Systems/notes/12b - CCS cose varie.md
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## Congruence
One of the main aims of an equivalence notion between processes is to make equational reasonings of the kind: “if P and Q are equivalent, then they can be interchangeably used in any execution context”.
This feature on an equivalence makes it a *congruence*
**This feature on an equivalence makes it a *congruence***
Not all equivalences are necessarily congruences (even though most of them are).
To properly define a congruence, we first need to define an execution context, and then what it means to run a process in a context. Intuitively:
![200](../../Pasted%20image%2020250415090109.png)
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An equivalence relation $R$ is a congruence if and only if
$$\forall (P, Q) \in R, \forall C.(C[P], C[Q]) \in R$$
Is bisimilarity a congruence? Yes.
**Is bisimilarity a congruence? Yes.**
**Theorem:**
$$if \space P ∼ Q \space then \space \forall C.C[P] ∼ C[Q]$$