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We dene the notion of an inductively dened type in the Calculus of Constructions and show how inductively dened types can be represented by closed types. We show that all primitive recursive functionals over these inductively dened types are also representable. This generalizes work by B ohm & Berarducci on synthesis of functions on term algebras in the second-order polymorphic -calculus (F 2 ). We give several applications of this generalization, including a representation of F 2 -programs in F 3 , along with a denition of functions reify, reflect, and eval for F 2 in F 3 . We also show how to dene induction over inductively dened types and sketch some results that show that the extension of the Calculus of Construction by induction principles does not alter the set of functions in its computational fragment, F ! . This is because a proof by induction can be realized by primitive recursion, which is already denable in F ! .