proof format updates using gfm

content/CSE347/CSE347_L7.md

@@ -142,7 +142,8 @@ How many digits are in each integer?
 
 Claim 1: If Subset Sum has a solution, then $\Psi$ is satisfiable.
 
-Proof:
+<details>
+<summary>Proof</summary>
 
 Say $S'$ is a solution to Subset Sum. Then there exists a subset $S' \subseteq S$ such that $\sum_{a_i\in S'} a_i = t$. Here is an assignment of truth values to variables in $\Psi$ that satisfies $\Psi$:
 
@@ -154,11 +155,12 @@ This is a valid assignment since:
 - We pick either $v_i$ or $\overline{v_i}$
 - For each clause, at least one literal is true
 
-QED
+</details>
 
 Claim 2: If $\Psi$ is satisfiable, then Subset Sum has a solution.
 
-Proof:
+<details>
+<summary>Proof</summary>
 
 If $A$ is a satisfiable assignment for $\Psi$, then we can construct a subset $S'$ of $S$ such that $\sum_{a_i\in S'} a_i = t$.
 
@@ -174,7 +176,7 @@ Say $t=\sum$ elements we picked from $S$.
   - If $q_j=2$, then $z_j\in S'$
   - If $q_j=3$, then $y_j\in S'$
 
-QED
+</details>
 
 ### Example 2: 3 Color
 
@@ -210,15 +212,16 @@ Key for dangler design:
 
 Connect to all $v_i$ with true to the same color. and connect to all $v_i$ with false to another color.
 
-'''
-TODO: Add dangler design image here.
-'''
+> [!TIP]
+>
+> TODO: Add dangler design image here.
 
 #### Proof of reduction for 3-Color
 
 Direction 1: If $\Psi$ is satisfiable, then $G$ is 3-colorable.
 
-Proof:
+<details>
+<summary>Proof</summary>
 
 Say $\Psi$ is satisfiable. Then $v_i$ and $\overline{v_i}$ are in different colors.
 
@@ -228,13 +231,16 @@ For each dangler color is connected to blue, all literals cannot be blue.
 
 ...
 
-QED
+</details>
 
 Direction 2: If $G$ is 3-colorable, then $\Psi$ is satisfiable.
 
-Proof:
+<details>
+<summary>Proof</summary>
 
-QED
+
+
+</details>
 
 ### Example 3:Hamiltonian cycle problem (HAMCYCLE)
 
@@ -242,9 +248,7 @@ Input: $G(V,E)$
 
 Output: Does $G$ have a Hamiltonian cycle? (A cycle that visits each vertex exactly once.)
 
-Proof is too hard.
-
-but it is an existing NP-complete problem.
+Proof is too hard. But it is an existing NP-complete problem.
 
 ## On lecture