What is the relationship between the disjuncts of formulae $a,b$ ( in DNF) when $ \neg a$ is not a tautology and $(a \rightarrow b)$ is a tautology?
Say (up to logical equivalence) $ a= a_1 \vee a_2 ... \vee a_n$ and $b= b_1 \vee ... \vee b_m$ for conjunctive clauses $a_1,... ,a_n,b_1,...,b_m$. Then a bit of calculation gives $\forall i \leq n \exists j\leq m \ (a_i \rightarrow b_j)$, which isn't particularly enlightening to me. So I am wondering if there is something better one could say. Any suggestions would be appreciated!