# Dependent type

(Difference between revisions)
 Revision as of 14:30, 1 March 2006 (edit)m (table of contents)← Previous diff Revision as of 15:27, 1 March 2006 (edit) (undo)m (Link to Combinatory Logic wiki page)Next diff → Line 5: Line 5: [http://www-sop.inria.fr/oasis/Caminha00/abstract.html Dependent Types in Programming] abstract in APPSEM'2000 [http://www-sop.inria.fr/oasis/Caminha00/abstract.html Dependent Types in Programming] abstract in APPSEM'2000 - 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. + To see how Illative [[CombinatoryLogic]] 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 $\forall x : S \Rightarrow T$ $\forall x : S \Rightarrow T$

## 1 The concept of dependent types

Dependent Types in Programming abstract in APPSEM'2000

To see how Illative CombinatoryLogic 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:

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

## 2 Dependently typed languages

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).