Lesson plan
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. It can be defined in Rocq (formerly named Coq) as follows: Inductive nat : Type := | O : nat | S : nat -> nat. Here, a natural number is created either from the constant "O" (representing zero) or by applying the function "S" to another natural number. "S" is the successor function which represents adding one to a number. Thus, "S O" is one, "S (S O)" is two, "S (S (S O))" 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.
Article summary
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. It can be defined in Rocq (formerly named Coq) as follows: Inductive nat : Type := | O : nat | S : nat -> nat. Here, a natural number is created either from the constant "O" (representing zero) or by applying the function "S" to another natural number. "S" is the successor function which represents adding one to a number. Thus, "S O" is one, "S (S O)" is two, "S (S (S O))" 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.
Key claims
No claims recorded.
Discussion questions
- Summarise the article in your own words. Which single claim would you verify first, and why?
- Which claim in this article is most likely to be revised as the sources update? How would you check its current status?
- If two sources disagreed about one of these facts, what would you need in order to trust one over the other?
Bibliography
No sources recorded.
Key claims and bibliography are generated from the same tracked claims and sources as the article page.