Please give answer in Latex format if possible!

1. (10 points] Let Simple Tree be the inductive set defined by the following construc- tors: a. Leaf : N Simple Tree. b. Branch1 : Simple Tree + Simple Tree. c. Branch2 : Simple Tree x Simple Tree Simple Tree. The function leaves : Simple Tree + N is defined by recursion and pattern matching as: a. leaves(Leaf(n)) = 1. b. leaves(Branch1(t)) = leaves(t). c. leaves(Branch2(ti, t2)) = leaves(ti) + leaves(t2). The function branches : Simple Tree + N is defined by recursion and pattern matching as: a. branches(Leaf(n)) = 0. b. branches(Branch1(t)) = 1 + branches(t). c. branches(Branch2(t1, t2)) = 1 + branches(ti) + branches(t2). Prove that, for all t e Simple Tree, leaves(t)