# Difference between revisions of "Dependent type"

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# The concept of dependent types

## General

Dependent Types in Programming abstract in APPSEM'2000

## 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.

## 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 $\forall x : S \Rightarrow T$ of Epigram corresponds to $\mathbf G\;S\;(\lambda x . T)$ in Illative Combinatory Logic. I think e.g. the followings should correspond to each other:

• $\mathrm{realNullvector} :\;\;\;\forall n: \mathrm{Nat} \Rightarrow \mathrm{RealVector}\;n$
• $\mathbf G\;\,\mathrm{Nat}\;\,\mathrm{RealVector}\;\,\mathrm{realNullvector}$

# Dependently typed languages

## Epigram

Epigram is a full dependently typed programming language see especially

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).

# Dependent types in Haskell programming

## Proposals

John Hughes: Dependent Types in Haskell (some ideas).