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Dependent type

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1 The concept of dependent types

1.1 General


Dependent Types in Programming abstract in APPSEM'2000

1.2 Type Theory


1.3 Illative Combinatory Logic

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

2.1 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).

2.2 Other techniques

[APPSEM Workshop on Subtyping & Dependent Types in Programming

3 Dependent types in Haskell programming