Type theory

Inductive type

In type theory, a system has inductive types if it has facilities for creating a new type from constants and functions that create terms of that type. The feature serves a role similar to data structures in a programming language and allows a type theory to add concepts like numbers, relations, and trees. As the name suggests, inductive types can be self-referential, but usually only in a way that permits structural recursion. The standard example is encoding the natural numbers using Peano's encoding. Inductive nat : Type := | 0 : nat | S : nat -> nat. Here, a natural number is created either from the constant "0" or by applying the function "S" to another natural number. "S" is the successor function which represents adding 1 to a number. Thus, "0" is zero, "S 0" is one, "S (S 0)" is two, "S (S (S 0))" is three, and so on. Since their introduction, inductive types have been extended to encode more and more structures, while still being predicative and supporting structural recursion. (Wikipedia).

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Higher Inductive Types - Peter Lumsdaine

Peter Lumsdaine Dalhousie University; Member, School of Mathematics October 1, 2012 For more videos, visit http://video.ias.edu

From playlist Mathematics

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http://mathispower4u.wordpress.com/

From playlist Introduction to Proof

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From playlist Sequences

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From playlist GEOMETRY CH 2 PROOFS & REASONING

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From playlist cs273a

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From playlist Mathematics

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From playlist Functions

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Michael Shulman University of California, San Diego; Member, School of Mathematics November 14, 2012 For more videos, visit http://video.ias.edu

From playlist Mathematics

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From playlist Mathematics

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From playlist 2023 Machine Assisted Proofs Workshop

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Related pages

Initial algebra | Univalence axiom | Intuitionistic type theory | Natural deduction | Coinduction | Homotopy type theory | Isomorphism | Number | Polynomial functor (type theory) | Structural induction | Induction-induction | Peano axioms | Relation (mathematics) | Arity | Tree (graph theory) | Dependent type | Coq | Curry–Howard correspondence | Successor function | Universe (mathematics) | Natural number | Dual (category theory) | Type theory | Parity (mathematics) | Well-founded relation | Polynomial functor