Mutual and Higher Inductive Types in Homotopy Type Theory
نویسنده
چکیده
Inductive types can be cleanly represented internally as W-types [14] [20], that is, as initial algebras of containers [1]. In this paper, we give a similar presentation that extends the notion of W-type to more general forms of induction, including mutually defined data types and higher inductive types.
منابع مشابه
Computational Higher Type Theory IV: Inductive Types
This is the fourth in a series of papers extending Martin-Löf’s meaning explanation of dependent type theory to higher-dimensional types. In this installment, we show how to define cubical type systems supporting a general schema of cubical inductive types, inductive types whose constructors may take dimension parameters and may have specified boundaries. Using this schema, we are able to speci...
متن کاملType theory in a type theory with quotient inductive types
Type theory (with dependent types) was introduced by Per Martin-Löf with the intention of providing a foundation for constructive mathematics. A part of constructive mathematics is type theory itself, hence we should be able to say what type theory is using the formal language of type theory. In addition, metatheoretic properties of type theory such as normalisation should be provable in type t...
متن کاملHigher Inductive Types in Cubical Computational Type Theory
In homotopy type theory (HoTT), higher inductive types provide a means of defining and reasoning about higher-dimensional objects such as circles and tori. The formulation of a schema for such types remains a matter of current research. We investigate the question in the context of cubical type theory, where the homotopical structure implicit in HoTT is made explicit in the judgmental apparatus...
متن کاملHomotopical Patch Theory ( Expanded
Homotopy type theory is an extension of Martin-Löf type theory, based on a correspondence with homotopy theory and higher category theory. In homotopy type theory, the propositional equality type becomes proof-relevant, and corresponds to paths in a space. This allows for a new class of datatypes, called higher inductive types, which are specified by constructors not only for points but also fo...
متن کاملHomotopical Patch Theory (expanded Version)
Homotopy type theory is an extension of Martin-Löf type theory, based on a correspondence with homotopy theory and higher category theory. In homotopy type theory, the propositional equality type becomes proof-relevant, and corresponds to paths in a space. This allows for a new class of datatypes, called higher inductive types, which are specified by constructors not only for points but also fo...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 2014