diff --git a/.obsidian/workspace.json b/.obsidian/workspace.json
index 5c30757..836497b 100644
--- a/.obsidian/workspace.json
+++ b/.obsidian/workspace.json
@@ -226,10 +226,11 @@
       "companion:Toggle completion": false
     }
   },
-  "active": "46d9dfa1750fc4db",
+  "active": "9d3f25b0eb9b478f",
   "lastOpenFiles": [
-    "Concurrent Systems/notes/13 -.md",
     "Concurrent Systems/slides/class 12.pdf",
+    "Concurrent Systems/notes/13 -.md",
+    "Pasted image 20250415082906.png",
     "Concurrent Systems/notes/12 - Calculus of communicating system.md",
     "Concurrent Systems/notes/11 - LTSs and Bisimulation.md",
     "Concurrent Systems/notes/images/Pasted image 20250414164549.png",
@@ -242,7 +243,6 @@
     "Concurrent Systems/notes/images/Pasted image 20250414152234.png",
     "Concurrent Systems/notes/images/Pasted image 20250414152221.png",
     "Concurrent Systems/notes/images/Pasted image 20250414104010.png",
-    "Concurrent Systems/notes/images/Pasted image 20250414103800.png",
     "HCIW/notes/3 - Beacons.md",
     "HCIW/slides/Zooming interfaces.pdf",
     "HCIW/slides/Gestural interaction.pdf",
diff --git a/Concurrent Systems/notes/13 -.md b/Concurrent Systems/notes/13 -.md
index a60a1d1..3f11c4c 100644
--- a/Concurrent Systems/notes/13 -.md	
+++ b/Concurrent Systems/notes/13 -.md	
@@ -7,3 +7,9 @@ The specification of a binary semaphore is the following:
 $$S_{}^{(2)} \triangleq p \cdot S_{1}^{(2)}$$
 $$S_{1}^{(2)} \triangleq p \cdot S_{1}^{(2)}+v\cdot S^{(2)}$$
 $$S_{2}^{(2)} \triangleq v \cdot S_{1}^{(2)}$$
+
+If we consider S(2) as the specification of the expected behavior of a binary semaphore and S(1) | S(1) as its concrete implementation, we can show that $$S^{(1)}|S^{(1)} \space \textasciitilde \space S^{2}$$
+This means that the implementation and the specification do coincide. To show this equivalence, it suffices to show that following relation is a bisimulation:
+![](../../Pasted%20image%2020250415082906.png)
+
+## Congruence
diff --git a/Pasted image 20250415082906.png b/Pasted image 20250415082906.png
new file mode 100644
index 0000000..eb41a52
Binary files /dev/null and b/Pasted image 20250415082906.png differ