Strong Normalization and Equi-(Co)Inductive Types

Abstract

heprtment of gomputer ieneD niversity of wunih yettingenstrFTUD hEVHSQV wünhenD qermny abel@tcs.ifi.lmu.deAbstract. e type system for the lmdElulus enrihed with reurE sive nd oreursive funtions over equiEindutive nd Eoindutive types is presented in whih ll wellEtyped progrms re strongly normlizingF he hoie of equiEindutive typesD insted of the more ommon isoE indutive typesD in)uenes oth redution rules nd the strong normlE iztion proofF fy emedding isoE into equiEtypesD the ltter ones re reognized s more fundmentlF e model sed on orthogonlity is onE struted where semntil type orresponds to set of oservtionsD nd soundness of the type system is provenF

Mixed Inductive/Coinductive Types and Strong Normalization

“…This does not lead to stuck terms, since in such a case the recursive function constituting the frame can be unfolded instead. In previous work [4], we considered a corecursive value in a recursive frame as stuck, leading to an unsatisfactory treatment of mixed induction/coinduction. The present work overcomes this flaw.…”

“…In contrast, equi -inductive types come with the type equation µF = F (µF ), so wrapping and unwrapping is silent on the term level. Recently [4], I have put forth a type system for strongly normalizing terms with equi -(co)inductive types, but it behaves badly for so-called mixed inductive/coinductive types.…”

“…Hughes, Pareto, and Sabry [11] present sized types in the equi-style, yet they consider only finitely branching data types and explicitly exclude a type of stream processors. In my previous attempt at equi-(co)inductive types [4] I constructed a semantics based on biorthogonals, which are due to Girard and have been successfully applied at interpreting languages based on classical logic (see, e. g., Parigot [14]). However, I had to consider a recursive function applied to a corecursive value blocked, preventing the use of mixed inductive/coinductive types.…”