Section: New Results

Characterizing Independence in Type Theory

Participants : Kaustuv Chaudhuri, Yuting Wang.

In formal proof languages based on type theory, it is often the case that a theorem is proved for a certain kind of typing context, but needs to be used in a different context. For example, theorems about natural numbers may be proved in an empty typing context, since the type of natural numbers contains no higher-order features (i.e., natural numbers are closed), but we may need to use these properties of natural numbers when reasoning about $\lambda$-terms in De Bruijn notation, where the typing context is non-empty. In such a situation, it is useful to automatically transport the existing theorems to the new kinds of contexts, since we know that the theorem in question cannot depend on the properties of $\lambda$-terms. While this example is rather trivial, it becomes non-trivial when theorems are proved about higher-order data structures, which are commonly encountered when reasoning about syntax with binding constructs.

One way to achieve such reuse automatically is a technique called subordination, which is based on analyzing the constructors for a certain type and defining syntactic criteria under which certain normal terms of one type can have subterms of another type. Unfortunately, the classical definition of subordination lacks a proof-theoretic justification, and has surprising properties in third-order (and higher) signatures.

In [36] (TLCA'15), we propose a proof-theoretic characterization of a kind of dual to subordination, called independence, that characterizes when normal terms of one type cannot contain subterms of another type. This is achieved by means of proving an inductive strengthening lemma about the signatures in the two-level logic approach. We also show how to automatically prove such lemmas in certain commonly encountered situations in the theorem prover Abella.