We propose a generic method for deciding the language inclusion problem between context-free languages and deterministic contextfree languages. Our method extends a given decision procedure for a subclass to another decision procedure for a more general subclass called a refinement of the former. To decide $\mathcal{L}_0 \subseteq \mathcal{L}_1$, we take two additional arguments: a language $\mathcal{L}_2$ of which $\mathcal{L}_1$ is a refinement, and a proof of $\mathcal{L}_0 \subseteq \mathcal{L}_2$. Our technique then refines the proof of $\mathcal{L}_0 \subseteq \mathcal{L}_2$ to a proof or a refutation of $\mathcal{L}_0 \subseteq \mathcal{L}_1$. Although the refinement procedure may not terminate in general, we give a sufficient condition for the termination. We employ a type-based approach to formalize the idea, inspired from Kobayashi’s intersection type system for model-checking recursion schemes. To demonstrate the usefulness, we apply this method to obtain simpler proofs of the previous results of Minamide and Tozawa on the inclusion between context-free languages and regular hedge languages, and of Greibach and Friedman on the inclusion between context-free languages and superdeterministic languages.