Our top priority is always the safety and well-being of our students. Given the COVID-19 pandemic, we can no longer hold our in-person residential program for 2020 as originally planned. Instead, in order to do our part as we all navigate this crisis together, the Prove it! team will be offering a free online program this summer.

All spots are currently filled and we are no longer accepting additional applications. You can find information about additional summer math programs at the AMS website.

• 1. Carefully read section 6.10 of the Lecture Notes linked to belown on deriving rules of inference from theorems and definitions. See if you can derive the template form of the rule of inference for induction on page 30 from a rule of inference that has no premises and concludes axiom N4, using the rules in section 6.10.
• 2. Prove the following using only the rules of natural deduction for predicate logic given in the lecture notes. Your proof should be written in the semiformal proof style using any of the shortcuts we went over. All quantified variables represent natural numbers. Note that Lurch does not have the Peano Axioms built in (though you could define them if you wanted to.) Feel free to ask an instructor for help if you get stuck.
• a. Nonzero implies successor. $\forall n,n\neq 0\Rightarrow \exists m, n=\sigma(m)$
• b. No number is it’s own successor. $\forall n,n\neq \sigma(n)$
• c. Associativity of Addition. $\forall n,\forall m, \forall p, (n+m)+p=n+(m+p)$