(0+ 11*0) + (0 +11*0)(1 +01*0)* (1 +01*0) =1*0(1 +01*0)* [closed]

Question

I needed help with the following question which belongs to Arden's theorem as I am having a hard time solving it. Please find the question below

Prove: $$(0+ 11^*0) + (0 +11^*0)(1 +01^*0)^* (1 +01^*0) = 1^*0(1 +01^*0)^*$$

Answer 1

Call $(0+11^*0)$ as A and $(1+01^*0)$ as B. Then what you have on left-hand side is just: $$ R = A + AB^*B$$ Using Arden's Theorem we know that in this case: $$R = AB^*$$

Then what we have simplified is: $$(11^*0+0)(1+01^*0)^*$$ Another simplification using Arden's Theorem would give us: $$(11^*+ \epsilon)0(1+01^*0)^*$$ Which is finally equivalent to: $$1^*0(1+01^*0)^*$$