(2 pages). Please help me by showing in Latex format if possible. Much appreciated!

2. (10 points) Let BinNum be the inductive set defined by the following constructors: Zero : Bin Num. One : Bin Num. JoinZero : Bin Num Bin Num. Join One : BinNum Bin Num. The members of Bin Num represent binary numerals like 1011 and 010. Zero represents 0: One represents 1; and if u represents U, then JoinZero(u) represents UO and Join One(u) represents U1. For example, Join One (JoinZero (JoinOneOne)) represents the binary number 1101. The function len : Bin Num N maps a member of BinNum to its length. len is defined by the following equations using recursion and patter matching: len (Zero) = 1 len(One) = 1. len (JoinZero(u)) = len(u) +1. len (Join One()) = len(u) + 1 The function val: BinNum N maps a member of Bin Num to the value of the binary numeral it rep- resents. For example, val(Join One (JoinZero(Join One (One)))) = (1101)2 = 13. val is defined by the following equations using recursion and pattern matching val(Zero) = 0. val(One) = 1. val(JoinZero(u) = 2 *valu). val(Join One(u) = (2 val()) + 1. The function add : Bin Num x Bin Num Bin Num is intended to implement addition on members of BinNum. It is defined by the following equations using recursion and pattern matching add(u, Zero) = u. add (Zero.u) = u add(One, One) = JoinZero(One). add(JoinZero(u), One) = JoinOne(u). add(One, JoinZero(u)) = Joinoneu). add (JoinOneu), One) = JoinZero (add(u, One). add(One, Join One()) = JoinZero (add(u, One). add (JoinZero(u), JoinZero(u)) = JoinZero(add(u, v). add(JoinOne(u), JoinZero(u)) = Join One(add(u, v). add(JoinZero(u), Join One(u)) = Join One(add(u, v). add(JoinOne(u), JoinOne()) = JoinZero(add(add(u, v), One)). Notice that the algorithm behind the definition is essentially the same algorithm that children learn to add numbers represented as decimal numerals. The last equation is a bit complicated because it involves a carry of 1. Lemma 1. For all u, v E BinNum, len(add(u, v))