A unifying theory of dependent types: the schematic approach

Abstract

Zhaohui Luo Department of Computer Science, University of Edinburgh JCMB, KB, Mayfield Rd., Edinburgh EH9 3JZ, U.K. 1 Introdu tion We present a theory of dependent types whi h uni es oherently Martin-L of's type theory with universes [ML75, ML84, NPS90℄ and Coquand-Huet's Cal ulus of Constru tions [CH88, Coq89℄. The theory an be seen as an extension of the Extended Cal ulus of Constru tions [Luo89, Luo90a℄ by a large lass of indu tive data types. It is a further development of the idea to enhan e in type theory a on eptual distin tion between the notions of logi al formula (proposition) and data type, and is another step aiming at the development of a unifying language for modular development of programs, spe i ations, and proofs ( .f., [Luo91b℄). The presentation here is parti ularly inspired by Martin-Lof's presentation of type theory by logi al framework [NPS90℄ and Coquand and Mohring's work on indu tive types [CPM90℄. The type theory is formulated in a logi al framework extended by kind s hemata. It onsists of an impredi ative universe of logi al propositions, a lass of indu tive data types overed by a general form of s hemata, and predi ative type universes. Our presentation is di erent from that of [CPM90℄ in the following aspe ts. Using a logi al framework allows us to de ne a purely intensional type theory with a general s hema for indu tive types. In parti ular, besides distinguishing the notions of logi al proposition and data type, we pay spe ial attention to the intensionality of omputational equality and how to obtain a learly nonir ular re e tion prin iple in introdu ing predi ative universes. Furthermore, our approa h is more moderate in that, for example, our s hemata only over indu tive types but not indu tive relations. There are two reasons in favor of su h a moderate approa h. First, it seems that the ne essity to over general indu tive relations by the s hemata is still to be justi ed; in our setting, logi al indu tive relations an be obtained by impredi ative de nitions and many indu tive families of data types an be de ned using predi ative universes. Se ond, by taking su h a moderate approa h, we hope that the resulting type theory allows a mole ular or ompositional understanding, as we shall brie y explain in the on lusion. Several basi ideas are elaborated below to explain our motivations and points of view.