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Concurrent Systems/notes/15 - A formal language for LTSs.md

@@ -34,4 +34,14 @@ hence, P |= ☐aFF holds true only if P cannot perform any action a
 ![](../../Pasted%20image%2020250505085454.png)
 
 Let us now define the set of formulae satisfied by a process as $$L(P) = \{ \phi \in Form:P \models \phi \}$$
-To simplify the proof, let us modify the set of formulae 
+To simplify the proof, let us modify the set of formulae by allowing conjunctions over a numerable set of formulae
+
+##### Theorem $$P \sim Q \space iif \space L(P)=L(Q)$$
+(=>) By induction on the syntax tree of the formula, let's show that $$φ∈L(P) \space iff \space φ∈L(Q)$$
+*Base case:* The only possible case is with φ = TT.
+- by the satisfiability relation, $\phi$ belongs to the set of formulae of every process
+- so also to L(P) and L(Q)
+
+*Inductive step:* Let's assume the thesis for every tree of height at most h. Let h+1 be the height of $\phi$. Let's distinguish on the outmost operator in $\phi$
+![](../../Pasted%20image%2020250505090603.png)
+