proof format updates using gfm

content/CSE347/CSE347_L8.md

@@ -139,7 +139,8 @@ We could first upper bound the size of the optimal cut is at most $|E|$.
 
 We will then prove that solution we found is at least half of the optimal cut $\frac{|E|}{2}$ for any graph $G$.
 
-Proof:
+<details>
+<summary>Proof</summary>
 
 When we terminate, no vertex could be moved
 
@@ -153,7 +154,7 @@ Summing over all vertices, the total number of crossing edges is at least $\frac
 
 So the total number of non-crossing edges is at most $\frac{|E|}{2}$.
 
-QED
+</details>
 
 #### Set cover
 
@@ -226,9 +227,10 @@ Need to prove its:
 
 We claim that the size of the set cover found is at most $H_n\log n$ times the size of the optimal set cover.
 
-###### First bound:
+Proof of first bound:
 
-Proof:
+<details>
+<summary>Proof</summary>
 
 If the optimal picks $k$ sets, then the size of the set cover found is at most $(1+\log n)k$ sets.
 
@@ -264,15 +266,16 @@ So $n(1-\frac{1}{k})^{|C|-1}=1$, $|C|\leq 1+k\ln n$.
 
 So the size of the set cover found is at most $(1+\ln n)k$.
 
-QED
+</details>
 
 So the greedy set cover is not too bad...
 
-###### Second bound:
+Proof of second bound:
 
 Greedy set cover is a $H_d$-approximation algorithm of set cover.
 
-Proof:
+<details>
+<summary>Proof</summary>
 
 Assign a cost to the elements of $X$ according to the decisions of the greedy set cover.
 
@@ -350,4 +353,4 @@ $$
 
 So the approximation ratio for greedy set cover is $H_d$.
 
-QED
+</details>