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Concurrent Systems/notes/6 - Atomicity.md

@@ -60,7 +60,13 @@ We now show that $\to$ is acyclic.
 			-  $op3 \to_H op4$ means that $res(op3) <_H inv(op4)$
 
 			- We would then have that $res(op1) <_H inv(op2) <_H res(op3) <_H inv(op4)$
-			- So by transitivity $res(op1) <_H inv(op4)$
+			- So by transitivity $res(op1) <_H inv(op4)$, i.e. $op1 \to_H op4$
+				- IN CONTRADICTION WITH HAVING CHOSEN A SHORTEST CYCLE
+				- as if op4 = op1, then this could not happen as $\to$ is a total order.
+
+This said, we can say that **every DAG admits a topological order** (a total order of its nodes that respects the edges), we will call $\to'$ the topological order for $\to$
+
+Let us define a linearization of $\hat{H}$ as follows: $$\hat{S}=inv(op1)res(op1)$$
 
 
 > [!PDF|red] class 6, p.6>  we would have a cycle of length