I am trying to prove (without converting it into Automata, through just simplification) that the following two regular expressions are equal:
$ (\epsilon + 0^*1^+0)^* = \epsilon + (0+1)^*10 $
here is what I have got so far:
$$ (\epsilon + 0^*1^+0)^* = \\ (0^*1^+0)^* = \\ (0^*1^*10)^* = \\ \epsilon + (0^*1^*10)^+ = \\ \epsilon + (0^*1^*10)(0^*1^*10)^* = \\ \epsilon + 0^*1^*10(0^*1^*10)^*$$
I seem to be stuck, I am not sure if I am doing it the correct way, or if there is a better way to prove this.
I know that $(a+b)^* = (a^*b^*)^* $ is true and I think this will certainly come in handy for the problem above but I am not sure how to prove $(a+b)^* = (a^*b^*)^* $ either.