Guarded recursion is a form of recursion where recursive calls are guarded by delay modalities. Previous work has shown how guarded recursion is useful for reasoning operationally about programming languages with advanced features including general references, recursive types, countable non-determinism and concurrency.Guarded recursion also offers a way of adding recursion to type theory while maintaining logical consistency. In previous work we initiated a programme of denotational semantics in type theory using guarded recursion, by constructing a computationally adequate model of the language PCF (simply typed lambda calculus with fixed points). This model was intensional in that it could distinguish between computations computing the same result using a different number of fixed point unfoldings.In this work we show how also programming languages with recursive types can be given denotational semantics in type theory with guarded recursion. More precisely, we give a computationally adequate denotational semantics to the language FPC (simply typed lambda calculus extended with recursive types), modelling recursive types using guarded recursive types. The model is intensional in the same way as was the case in previous work, but we show how to recover extensionality using a logical relation.All constructions and reasoning in this paper, including proofs of theorems such as soundness and adequacy, are by (informal) reasoning in type theory, often using guarded recursion.
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