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Concurrent Systems/notes/11 - LTSs and Bisimulation.md
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@ -91,4 +91,12 @@ where S1 and S2 are bisimulations, then we have to show that S is a bisimulation
The proof is done by showing that ∼ is a simulation. The proof is done by showing that ∼ is a simulation.
By definition of similarity, we have to show that $$∀(p,q)∈∼ ∀p –a–> p′ ∃q –a–> q′ s.t.(p′,q′)∈∼$$ By definition of similarity, we have to show that $$∀(p,q)∈∼ ∀p –a–> p′ ∃q –a–> q′ s.t.(p′,q′)∈∼$$
Let us fix a pair (p,q) ∈∼
Bisimilarity of p and q implies the existence of a bisimulation S such that (p,q) ∈ S.
Hence, for every transition p –a–> p′, there exists a transition q –a–> q′ such that (p′, q′) ∈ S.
So, (p', q') ∈ ∼
**Theorem:** For every bisimulation S, it holds that S ⊆ ∼.
*Proof:*
Let (p,q) ∈ S. Then, there exists a bisimulation that contains the pair (p, q); thus, (p, q) ∈ ∼.