diff --git a/.obsidian/community-plugins.json b/.obsidian/community-plugins.json
index 0871a1f..dc1b1be 100644
--- a/.obsidian/community-plugins.json
+++ b/.obsidian/community-plugins.json
@@ -1,9 +1,9 @@
 [
   "obsidian-ocr",
-  "pdf-plus",
   "obsidian-git",
   "mathlive-in-editor-mode",
   "smart-second-brain",
   "local-gpt",
-  "companion"
+  "companion",
+  "pdf-plus"
 ]
\ No newline at end of file
diff --git a/.obsidian/workspace.json b/.obsidian/workspace.json
index 7197b2b..d2db11a 100644
--- a/.obsidian/workspace.json
+++ b/.obsidian/workspace.json
@@ -34,10 +34,10 @@
               "type": "pdf",
               "state": {
                 "file": "Concurrent Systems/slides/class 6.pdf",
-                "page": 5,
-                "left": -19,
-                "top": 437,
-                "zoom": 0.7998812351543944
+                "page": 4,
+                "left": -23,
+                "top": 143,
+                "zoom": 0.6627078384798101
               },
               "icon": "lucide-file-text",
               "title": "class 6"
@@ -100,8 +100,7 @@
       }
     ],
     "direction": "horizontal",
-    "width": 307.5,
-    "collapsed": true
+    "width": 307.5
   },
   "right": {
     "id": "bc4b945ded1926e3",
@@ -215,10 +214,10 @@
       "companion:Toggle completion": false
     }
   },
-  "active": "88f2e3a5b973712d",
+  "active": "51157f32453cba69",
   "lastOpenFiles": [
-    "Concurrent Systems/notes/6 - Atomicity.md",
     "Concurrent Systems/slides/class 6.pdf",
+    "Concurrent Systems/notes/6 - Atomicity.md",
     "Concurrent Systems/notes/images/Pasted image 20250318090733.png",
     "Concurrent Systems/notes/images/Pasted image 20250318090909.png",
     "Concurrent Systems/slides/class 5.pdf",
diff --git a/Concurrent Systems/notes/6 - Atomicity.md b/Concurrent Systems/notes/6 - Atomicity.md
index d54f470..a2a7530 100644
--- a/Concurrent Systems/notes/6 - Atomicity.md	
+++ b/Concurrent Systems/notes/6 - Atomicity.md	
@@ -39,13 +39,16 @@ Let $\to$ denote $\to_{H} \cup \bigcup_{X \in H} \to _{X}$
 We now show that $\to$ is acyclic.
 1. It cannot have cycles with 1 edge (i.e. self loops): indeed, if $op \to op$, this would mean that $res(op) < inv(op)$, which of course does not make any sense.
 
-2. it cannot have cycles with 2 edges:
+2. It cannot have cycles with 2 edges:
 	- let's assume that $op \to op' \to op$
 	- both arrows cannot be $\to_H$ nor $\to_X$ (for some X), otw. it won't be a total order (and would be cyclic)
 	- it cannot be that one is $\to_X$ and the other $\to_Y$ (for some $X \neq Y$), otherwise op/op' would be on 2 different objects.
-	- **So it must b**e $op \to_X op' \to_H op$ (or vice versa)
+	- **So it must b**e $op \to_X op' \to_H op$ *(or vice versa)*
 		- then, $op' \to op$ means that $res(op') <_H inv(op)$
-		- Since $\hat{S}_X$ is a linearization of $\hat{H}|_X$ and op/op' are on X, this implies $res(op') <_X inv(op)$, which means that $op' \to_X op$, and so $\to_X$ would be cyclic.
+		- Since $\hat{S}_X$ is a linearization of $\hat{H}|_X$ and op/op' are on X (literally because we have op ->x op'), this implies $res(op') <_X inv(op)$, which means that $op' \to_X op$, and so it won't be a total order... So this is not possible either.
+
+3. It cannot have cycles with more than 2 edges:
+	1. 
 
 
 > [!PDF|red] class 6, p.6>  we would have a cycle of length