Prove the following using only the rules of natural deduction for propositional logic given in the lecture notes. Your proof should be written in the formal proof style. You can use Lurch to check your proofs. If you finish these and want more practice there are more problems in the lecture notes you can try. Feel free to ask an instructor for help if you get stuck.

**1. The Most Beautiful Tautology?**$P \Rightarrow (P \Leftrightarrow P \text{ or } (\neg P \text{ and }P))$Bonus: Do you see why is it beautiful?**2. Alternate form of implication.**$(P \Rightarrow Q) \Leftrightarrow (\neg P \text{ or } Q)$**3. Associativity of or.**$P \text{ or } (Q \text{ or } R) \Leftrightarrow (P \text{ or } Q) \text{ or } R$**4. Pierce’s Law**$((P \Rightarrow Q) \Rightarrow P) \Rightarrow P$