https://hal.inria.fr/hal-00778975Dagand, Pierre-EvaristePierre-EvaristeDagandMathematically Structured Programming Group - Mathematically Structured Programming Group - University of Strathclyde [Glasgow]Mcbride, ConorConorMcbrideMathematically Structured Programming Group - Mathematically Structured Programming Group - University of Strathclyde [Glasgow]Elaborating inductive definitionsHAL CCSD2013Inductive definitionType theory[INFO.INFO-PL] Computer Science [cs]/Programming Languages [cs.PL]Inria Saclay, IstDamien Pous and Christine Tasson2013-01-21 15:48:362020-11-25 16:42:032013-01-21 16:43:38enConference papersapplication/pdf1We present an elaboration of inductive de nitions down to a universe of datatypes. The universe of datatypes is an internal presentation of strictly positive types within type theory. By elaborating an inductive de nition { a syntactic artefact { to its code { its semantics { we obtain an internalised account of inductives inside the type theory itself: we claim that reasoning about inductive de nitions could be carried in the type theory, not in the meta-theory as it is usually the case. Besides, we give a formal speci cation of that elaboration process. It is therefore amenable to formal reasoning too. We prove the soundness of our translation and hint at its completeness with respect to Coq's Inductive de nitions. The practical bene ts of this approach are numerous. For the type theorist, this is a small step toward bootstrapping, i.e. implementing the inductive fragment in the type theory itself. For the programmer, this means better support for generic programming: we shall present a lightweight deriving mechanism, entirely de nable by the programmer and therefore not requiring any extension to the type theory.