Solution
= Solution
No transition joins distinct $A_i$, so
$$
Q(A,A^c)=\sum_iQ(A_i,A_i^c).
$$
Consequently
$$
\Phi(A)=\sum_i\frac{\pi(A_i)}{\pi(A)}\Phi(A_i)
\geq\inf_i\Phi(A_i).
$$
= Solution
No transition joins distinct $A_i$, so
$$
Q(A,A^c)=\sum_iQ(A_i,A_i^c).
$$
Consequently
$$
\Phi(A)=\sum_i\frac{\pi(A_i)}{\pi(A)}\Phi(A_i)
\geq\inf_i\Phi(A_i).
$$