# Dependent Type Research Materials

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In computer science and logic, a dependent type is a type that depends on a value. Dependent types play a central role in intuitionistic type theory and in the design of functional programming languages like ATS, Agda and Epigram.

An example is the type of n-tuples of real numbers. This is a dependent type because the type depends on the value n.

Deciding equality of dependent types in a program may require computations. If arbitrary values are allowed in dependent types, then deciding type equality may involve deciding whether two arbitrary programs produce the same result; hence type checking may become undecidable.

The Curry–Howard correspondence implies that types can be constructed that express arbitrarily complex mathematical properties. If the user can supply a constructive proof that a type is inhabited (i.e., that a value of that type exists) then a compiler can check the proof and convert it into executable computer code that computes the value by carrying out the construction. The proof checking feature makes dependently typed languages closely related to proof assistants. The code-generation aspect provides a powerful approach to formal program verification and proof-carrying code, since the code is derived directly from a mechanically verified mathematical proof.

## Systems of the lambda cube

Henk Barendregt developed the lambda cube as a means of classifying type systems along three axes. The eight corners of the resulting cube-shaped diagram each correspond to a type system, with simply typed lambda calculus in the least expressive corner, and calculus of constructions in the most expressive. The three axes of the cube correspond to three different augmentations of the simply typed lambda calculus: the addition of dependent types, the addition of polymorphism, and the addition of higher kinded type constructors (functions from types to types, for example). The lambda cube is generalized further by pure type systems.

### First order dependent type theory

The system $\lambda \Pi$ of pure first order dependent types, corresponding to the logical framework LF, is obtained by generalising the function space type of the simply typed lambda calculus to the dependent product type.

Writing $\mbox{Vec}({\mathbb R},n)$ for $n$-tuples of real numbers, as above, $\Pi n:{\mathbb N}.\mbox{Vec}({\mathbb R},n)$ stands for the type of functions which given a natural number n returns a tuple of real numbers of size n. The usual function space arises as a special case when the range type does not actually depend on the input, e.g. $\Pi n:{\mathbb N}.{\mathbb R}$ is the type of functions from natural numbers to the real numbers, written as ${\mathbb N}\to{\mathbb R}$ in the simply typed lambda calculus.

### Second order dependent type theory

The system $\lambda \Pi 2$ of second order dependent types is obtained from $\lambda \Pi$ by allowing quantification over type constructors. In this theory the dependent product operator subsumes both the $\to$ operator of simply typed lambda calculus and the $\forall$ binder of System F.

### Higher order dependently typed polymorphic lambda calculus

The higher order system $\lambda \Pi \omega$ extends $\lambda \Pi 2$ to all four forms of abstraction from the lambda cube: functions from terms to terms, types to types, terms to types and types to terms. The system corresponds to the Calculus of constructions whose derivative, the calculus of inductive constructions is the underlying system of the Coq proof assistant.

### Object-oriented programming

Some recent research[1] has been directed at combining dependent type theory with object-oriented programming.

## Comparison

Language Actively developed Paradigm[fn 1] Tactics Proof terms Termination checking Types can depend on[fn 2] Universes Proof irrelevance Program extraction Extraction erases irrelevant terms
Agda Yes[2] Purely functional Few/limited[fn 3] Yes Yes (optional) Any term Yes (optional)[fn 4] Proof-irrelevant arguments (experimental)[4] Haskell, Javascript Yes[4]
ATS Yes[5] Functional / imperative No[6] Yes Yes ? ? ? Yes ?
Cayenne No Purely functional No Yes No Any term No No ? ?
Coq Yes[7] Purely functional Yes Yes Yes Any term Yes[fn 5] No Haskell, Scheme and OCaml Yes
Dependent ML No[fn 6] ? ? Yes ? Natural numbers ? ? ? ?
Epigram 2 Yes[8] Purely functional No Coming soon[dated info] By construction Any term Coming soon[dated info] Coming soon[dated info] Coming soon[dated info] Coming soon[dated info]
Guru No[9] Purely functional[10] hypjoin[11] Yes[10] Yes Any term No Yes Carraway Yes
Idris Yes[12] Purely functional[13] Yes[14] Yes Yes (optional) Any term No No Yes Yes, aggressively[14]
Matita Yes[15] Purely functional Yes Yes Yes Any term Yes ? OCaml ?
NuPRL No Purely functional Yes Yes Yes Any term Yes ? Yes ?
F* Yes Functional /imperative ? ? ? ? ? ? ? ?
PVS Yes ? Yes ? ? ? ? ? ? ?
Typed Racket Yes No ? ? ? ? ? ? ? ?
Sage ? Hybrid typechecking ? ? ? ? ? ? ? ?
Twelf Yes Logic programming ? Yes Yes (optional) Any (LF) term No No ? ?
Xanadu No[16] Imperative ? ? ? ? ? ? ? ?

## Footnotes

1. ^ This refers to the core language, not to any tactic or code generation sublanguage.
2. ^ Subject to semantic constraints, such as universe constraints
3. ^ Ring solver[3]
4. ^ Optional universes, optional universe polymorphism, and optional explicitly specified universes
5. ^ Universes, automatically inferred universe constraints (not the same as Agda's universe polymorphism) and optional explicit printing of universe constraints
6. ^ Has been superseded by ATS

## References

1. ^ Anton Setzer (2007). "Object-oriented programming in dependent type theory". In Henrik Nilsson. Trends in Functional Programming, vol. 7. Intellect. pp. 91–108.
2. ^
3. ^
4. ^ a b
5. ^
6. ^
7. ^
8. ^
9. ^
10. ^ a b Aaron Stump (6 April 2009). "Verified Programming in Guru". Retrieved 28 September 2010.
11. ^ Adam Petcher (1 April 2008). "Deciding Joinability Modulo Ground Equations in Operational Type Theory". Retrieved 14 October 2010.
12. ^
13. ^
14. ^ a b
15. ^
16. ^

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