Eliminating left recursion from example

LeftRecursionElimination

To begin with, the wikipedia left recursion algorithm is much eaiser to read: http://en.wikipedia.org/wiki/Left_recursion#Removing_indirect_left_recursion

So, we have the non-terminals A_1, and A_2:
	A_1 -> A_2a | b
	A_2 -> A_1c | ab

let's step through the algorithm:

i = 1
	j = 1
		j > i-1, break
	do nothing - no direct left recursions
	
i = 2
	j = 1
		so, we know that A_j = 1, and A_i = 2
		at the moment:
			A_1 -> A_2a | b
			A_2 -> A_1c | ab
		-------------------------
		The algorithm wants to replace all instances where we have:
			A_2 -> A_1$ (where $ is some string of terminals and non terminals)
		with the direct version, ie, replacing A_2 -> A_1$ with the direct A_1.

		This gives us:
			A_2 -> A_2ac | bc | ab
		To be extra clear (direct from A_1 bits in brackets):
			A_2 -> (A_2a)c | (b)c | ab
	j = 2
		j > i-1 break
	We then need to eliminate the direct left recursions in A_2:
		A_2 -> A_2ac | bc | ab
			|	
		       \_/
		A_2 -> bcA_3 | abA_3
		A_3 -> acA_3 | E

So, our completed grammar is:

A_1 -> A_2a | b
A_2 -> bcA_3 | abA_3
A_3 -> acA_3 | E