An infinitary rewriting interpretation of coinductive types
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We introduce an infinitary rewriting semantics for strictly positive nested higher-order (co)inductive types. This may be seen as a refinement and generalisation of the notion of productivity in term rewriting to a setting with higher-order functions and with data specifed by nested higher-order inductive and coinductive definitions. We prove an approximation theorem which essentially states that if a term reduces to an arbitrarily large finite approximation of an infinite object in the interpretation of a coinductive type, then it infinitarily reduces to an infinite object in the interpretation of this type. We introduce a sufficient syntactic correctness criterion, in the form of a type system, for finite terms decorated with type information. Using the approximation theorem we show that each well-typed term has a well-defined interpretation in our infinitary rewriting semantics. This gives an operational interpretation of typable terms which takes into account the "limits" of infinite reduction sequences.
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