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### Exercise 1:

• We define binary trees in the following way:

• Define a function eq_bintree such that the lemma eq_bintreeP holds, and prove it.

• We define the depth of a tree as a function of type bintree -> nat as follows:

• We define balanced trees as the following predicate:

• prove that balanced trees of equal depth are in fact equal:

### Exercise 2:

Let $$a \ge 0$$ and $$n \ge 1$$ be natural numbers.

• Show that $a ^ n − 1 = (a - 1) \sum_{i = 0}^{n-1} a^i.$
• Hint Search _ sum in MC.
• Hint do as many have as needed.

Let $$a , n \ge 2$$ be natural numbers.

• Show that if $$a ^ n − 1$$ is prime, then $$a = 2$$ and $$n$$ is prime. Complete the following proof script

• We write $$M_n = 2 ^ n − 1$$ the $$n^{\textrm{th}}$$ Mersenne number. Show that $$M_{100}$$ is not prime.

WARNING: do not substitute n = 100 in an expression where you have 2 ^ n, otherwise the computation will take forever and possibly crash your jscoq (in which case you will need to Reset worker or reload the page), hence you must use the previous lemma.