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Indexed families of types are a way of associating run-time data with compile-time abstractions that can be used to reason about them. We propose an extensible theory of indexed types, in which programmers can define the index data appropriate to their programs and use them to track properties of run-time code. The essential ingredients in our proposal are (1) a logical framework, which is used to define index data, constraints, and proofs, and (2) computation with indices, both at the static and dynamic levels of the programming language. Computation with indices supports a variety of mechanisms necessary for programming with extensible indexed types, including the definition and implementation of indexed types, meta-theoretic reasoning about indices, proof-producing run-time checks, computed index expressions, and run-time actions of index constraints. Programmer-defined index propositions and their proofs can be represented naturally in a logical framework such as LF, where variable binding is used to model the consequence relation of the logic. Consequently, the central technical challenge in our design is that computing with indices requires computing with the higher-order terms of a logical framework. The technical contributions of this paper are a novel method for computing with the higher-order data of a simple logical framework and the integration of such computation into the static and dynamic levels of a programming language. Further, we present examples showing how this computation supports indexed programming.