Why can I transform the regular expression 1*0 + 1*0(0+1)*(0+1) to 1*0(0+1)*?

Question

I can't quite understand, why I can transform the regex 1*0 + 1*0(0+1)*(0+1) to 1*0(0+1)*. Anyone able to help me?

Answer 1

You can use the law of distributivity:

  (1*0)+(1*0(0+1)*(0+1))
= (1*0ε)+(1*0(0+1)*(0+1))
= (1*0)(ε+(0+1)*(0+1))

and then apply the definition of the the Kleene star a* = ε+a*a:

= (1*0)((0+1)*)
= 1*0(0+1)*