Let L be the language over the alphabet {0,1} whose elements are exactly the words produced by the following rules: (1) 0L (2) If uL then u0 L (3) If uL then 1u0 L

(a) Prove that every word w L has the form 1^k0^k+a where k>=0 and a>=1. (b) Prove that if w is a word of the form 1^k0^k+a where k>=0 and a>=1, then w L.