CSC 236 , Summer 2005 , Assignment 2 sample solution

Proof (complete induction on n): If n = 0, then P (0) asserts that there is f(0) = 1 distinct output starting with an empty string. This is certainly true, since the unique empty string is output, so the base case holds. Induction step: Assume that P (f0; : : : ; n 1g) is true for some arbitrary natural number n. I need to prove that this implies P (n) is true. If n = 0 there is nothing to prove, since this was veri ed in the base case. Otherwise the IH assume P (i) and P (n 1 i) for every 0 i n 1. WLOG, assume that the rst character of the original sequence of length n is the character x, and partition the output sequences according to where x occurs in the output | at position i of the output, where 0 i n 1. This partition counts all possible outputs, and has no duplicates, since a particular output is speci ed by the position of the character x. Since this is a LIFO stack, the i characters that are output before x, in positions f0; : : : ; i 1g, must have been pushed onto the stack after x was pushed, and popped from the stack before x was popped. Thus these characters are the next i characters pushed following x in the original sequence, that is characters f1; : : : ; ig of the original sequence. Since they are pushed and