feat: definition branching bisimulation

Cslib/Foundations/Semantics/LTS/Bisimulation.lean

@@ -42,11 +42,18 @@ we prove to be sound and complete.
 - `SWBisimilarity lts` is the binary relation on the states of `lts` that relates any two states
 related by some sw-bisimulation on `lts`.
 
+- `lts.IsBranchingBisimulation r`: the relation `r` on the states of the LTS `lts` is a branching
+bisimulation.
+- `BranchingBisimilarity lts` is the binary relation on the states of `lts` that relates any two
+states related by some branching bisimulation on `lts`.
+
+
 ## Notations
 
 - `s1 ~[lts] s2`: the states `s1` and `s2` are bisimilar in the LTS `lts`.
 - `s1 ≈[lts] s2`: the states `s1` and `s2` are weakly bisimilar in the LTS `lts`.
 - `s1 ≈sw[lts] s2`: the states `s1` and `s2` are sw bisimilar in the LTS `lts`.
+- `s1 ≈br[lts] s2`: the states `s1` and `s2` are branching bisimilar in the LTS `lts`.
 
 ## Main statements
 
@@ -1090,3 +1097,41 @@ theorem SWBisimilarity.eqv [HasTau Label] {lts : LTS State Label} :
   }
 
 end WeakBisimulation
+
+section BranchingBisimulation
+
+/-! ## Branching bisimulation and branching bisimilarity -/
+
+/-- A branching bisimulation is similar to a `WeakBisimulation` in that it allows for a saturation
+of internal transitions, but it is a stronger equivalence in that it is able to distinguish
+when internal work constitutes a choice. There are several (equivalent) definitions used in
+literature. The original one can be found in [R.J. van Glabbeek and W.P. Weijland,
+"Branching Time and Abstraction in Bisimulation Semantics", 1990]. But we follow the slightly
+more popular definition of [de Nicola and Vaandrager, 1995, https://doi.org/10.1145/201019.201032].
+-/
+def LTS.IsBranchingBisimulation [HasTau Label] (lts : LTS State Label)
+  (r : State → State → Prop) :=
+  ∀ ⦃s1 s2⦄, r s1 s2 → ∀ μ,
+    /- In case of a transition on the left -/
+    (∀ s1', lts.Tr s1 μ s1' →
+       /- A single tau transition can be mimicked by doing nothing. -/
+       (μ = HasTau.τ ∧ r s1' s2) ∨
+       /- Any transition can be mimicked by first performing a number of
+       τ transitions while relating to the original state, and then mimicking
+       the intended transition. -/
+       ∃ s2' s2'', lts.STr s2 HasTau.τ s2' ∧ lts.Tr s2' μ s2'' ∧ r s1 s2' ∧ r s1' s2'')
+    ∧
+    /- The symmetric case of a transition on the right -/
+    (∀ s2', lts.Tr s2 μ s2' →
+       (μ = HasTau.τ ∧ r s1 s2') ∨
+       ∃ s1' s1'', lts.STr s1 HasTau.τ s1' ∧ lts.Tr s1' μ s1'' ∧ r s1' s2 ∧ r s1'' s2')
+
+/-- Two states are branching bisimilar if they are related by some branching bisimulation. -/
+def BranchingBisimilarity [HasTau Label] (lts : LTS State Label) : State → State → Prop :=
+  fun s1 s2 =>
+    ∃ r : State → State → Prop, r s1 s2 ∧ lts.IsBranchingBisimulation r
+
+/-- Notation for branching bisimilarity. -/
+notation s:max " ≈br[" lts "] " s':max => BranchingBisimilarity lts s s'
+
+end BranchingBisimulation