# Dependent type

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== Illative Combinatory Logic == | == Illative Combinatory Logic == | ||

− | To see how Illative [[ | + | To see how Illative [[Combinatory logic]] deals with dependent types, see combinator '''G''' described in [http://citeseer.ist.psu.edu/246934.html Systems of Illative Combinatory Logic complete for first-order propositional and predicate calculus] by Henk Barendregt, Martin Bunder, Wil Dekkers. |

It seems to me that the dependent type construct | It seems to me that the dependent type construct | ||

<math>\forall x : S \Rightarrow T</math> | <math>\forall x : S \Rightarrow T</math> |

## Revision as of 00:05, 2 March 2006

## Contents |

# 1 The concept of dependent types

## 1.1 General

Dependent Types in Programming abstract in APPSEM'2000

## 1.2 Type Theory

Simon Thompson: Type Theory and Functional Programming. Section 6.3 deals with dependent types, but because of the strong emphasis on Curry-Howard isomorphism and the connections between logic and programming, the book seemed cathartic for me even from its beginning.

## 1.3 Illative Combinatory Logic

To see how Illative Combinatory logic deals with dependent types, see combinator **G** described in Systems of Illative Combinatory Logic complete for first-order propositional and predicate calculus by Henk Barendregt, Martin Bunder, Wil Dekkers.
It seems to me that the dependent type construct
of Epigram corresponds to
in Illative Combinatory Logic. I think e.g. the followings should correspond to each other:

# 2 Dependently typed languages

## 2.1 Epigram

Epigram is a full dependently typed programming language see especially

- Epigram Tutorial by Conor McBride
- and Why dependent types matter by Thorsten Altenkirch, Conor McBride and James McKinna).

Dependent types (of this language) also provide a not-forgetful concept of **views** (already mentioned in the Haskell Future;
the connection between these concepts is described in p. 32 of Epigram Tutorial (section *4.6 Patterns Forget; Matching Is Remembering*).

## 2.2 Other techniques

APPSEM Workshop on Subtyping & Dependent Types in Programming

# 3 Dependent types in Haskell programming

- John Hughes: Dependent Types in Haskell (some ideas).
- SimulatingDependentTypes of HaWiki