# Observability = Typability + Inhabitation

We define an observability property for a calculus with pattern matching which is inspired by the notion of solvability for the lambda-calculus. We prove that observability can be characterized by means of typability and inhabitation in an intersection type system P based on non-idempotent types. We show first that the system P characterizes the set of terms having canonical form, i.e. that a term is typable if and only if it reduces to a canonical form. But the set of observable terms is properly contained in the set of canonical. Thus, typability alone is not sufficient to characterize observability, in contrast to the solvability case for lambda-calculus. We then prove that typability, together with inhabitation, provides a full characterization of observability, in the sense that a term is observable if and only if it is typable and the types of all its arguments are inhabited. We complete the picture by providing an algorithm for the inhabitation problem of P.

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