Dependent type

 ( Foundations Of Mathematics ) 

In computer science and logic, a dependent type is a type whose definition depends on a value. It is an overlapping feature of type theory and . In intuitionistic type theory, dependent types are used to encode logic's quantifiers like "for all" and "there exists". In functional programming languages like Agda, ATS, Rocq (previously known as Coq), F*, Epigram, Idris, and Lean, dependent types help reduce bugs by enabling the programmer to assign types that further restrain the set of possible implementations.

Two common examples of dependent types are dependent functions and dependent pairs. The return type of a dependent function may depend on the value (not just type) of one of its arguments. For instance, a function that takes a positive integer $n$ may return an array of length $n$, where the array length is part of the type of the array. (Note that this is different from polymorphism and generic programming, both of which include the type as an argument.) A dependent pair may have a second value, the type of which depends on the first value. Sticking with the array example, a dependent pair may be used to pair an array with its length in a type-safe way.

Dependent types add complexity to a type system. Deciding the 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 the decidability of type checking may depend on the given type theory's semantics of equality, that is, whether the type theory is intensional or extensional.

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History In 1934, Haskell Curry noticed that the types used in typed lambda calculus, and in its combinatory logic counterpart, followed the same pattern as axioms in propositional logic. Going further, for every proof in the logic, there was a matching function (term) in the programming language. One of Curry's examples was the correspondence between simply typed lambda calculus and intuitionistic logic.

Predicate logic is an extension of propositional logic, adding quantifiers. Howard and de Bruijn extended lambda calculus to match this more powerful logic by creating types for dependent functions, which correspond to "for all", and dependent pairs, which correspond to "there exists".

Because of this, and other work by Howard, propositions-as-types is known as the Curry–Howard correspondence.

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Formal definition Loosely speaking, dependent types are similar to the type of an Indexed family. More formally, given a type $A:\backslash mathcal\{U\}$ in a universe of types $\backslash mathcal\{U\}$, one may have a family of types $B:A\backslash to\backslash mathcal\{U\}$, which assigns to each term $a:A$ a type $B(a):\backslash mathcal\{U\}$. We say that the type $B(a)$ varies with $a$.

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Π type A function whose type of return value varies with its argument (i.e. there is no fixed codomain) is a dependent function and the type of this function is called dependent product type, pi-type ( type) or dependent function type. From a family of types $B:\; A\; \backslash to\; \backslash mathcal\{U\}$ we may construct the type of dependent functions $\backslash prod\_\{x\; :\; A\}\; B(x)$, whose terms are functions that take a term $a\; :\; A$ and return a term in $B(a)$. For this example, the dependent function type is typically written as $\backslash prod\_\{x:A\}\; B(x)$ or $\backslash prod\; \{(x:A)\}\; B(x)$.

If $B:A\backslash to\backslash mathcal\{U\}$ is a constant function, the corresponding dependent product type is equivalent to an ordinary function type. That is, $\backslash prod\_\{x:A\}B$ is judgmentally equal to $A\backslash to\; B$ when $B$ does not depend on $x$.

The name 'Π-type' comes from the idea that these may be viewed as a Cartesian product of types. Π-types can also be understood as model theory of universal quantifiers.

For example, if we write $\backslash operatorname\{Vec\}(\backslash mathbb\{R\},n)$ for n-tuples of real numbers, then $\backslash prod\_\{n:\backslash mathbb\{N\}\}\; \backslash operatorname\{Vec\}(\backslash mathbb\{R\},n)$ would be the type of a function which, given a natural number , returns a tuple of real numbers of size . The usual function space arises as a special case when the range type does not actually depend on the input. E.g. $\backslash prod\_\{n:\backslash mathbb\{N\}\}\; \{\backslash mathbb\{R\}\}$ is the type of functions from natural numbers to the real numbers, which is written as $\backslash mathbb\{N\}\backslash to\backslash mathbb\{R\}$ in typed lambda calculus.

For a more concrete example, taking $A$ to be the type of unsigned integers from 0 to 255 (the ones that fit into 8 bits or 1 byte) and $B(a)\; =\; X\_a$ for $a\; :\; A$, then $\backslash prod\_\{x:A\}\; B(x)$ devolves into the product of $X\_0\backslash times\; X\_1\backslash times\; X\_2\backslash times\backslash ldots\backslash times\; X\_\{253\}\backslash times\; X\_\{254\}\backslash times\; X\_\{255\}$.

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Σ type

The dual of the dependent product type is the dependent pair type, dependent sum type, sigma-type, or (confusingly) dependent product type. Sigma-types can also be understood as existential quantifiers. Continuing the above example, if, in the universe of types $\backslash mathcal\{U\}$, there is a type $A:\backslash mathcal\{U\}$ and a family of types $B:A\backslash to\backslash mathcal\{U\}$, then there is a dependent pair type $\backslash sum\_\{x:A\}\; B(x)$. (The alternative notations are similar to that of types.)

The dependent pair type captures the idea of an ordered pair where the type of the second term is dependent on the value of the first. If $(a,b):\backslash sum\_\{x:A\}\; B(x),$ then $a:A$ and $b:B(a)$. If $B$ is a constant function, then the dependent pair type becomes (is judgementally equal to) the product type, that is, an ordinary Cartesian product $A\backslash times\; B$.

For a more concrete example, taking $A$ to again be type of unsigned integers from 0 to 255, and $B(a)$ to again be equal to $X\_a$ for 256 more arbitrary $X\_a$, then $\backslash sum\_\{x:A\}\; B(x)$ devolves into the sum $X\_0+X\_1+X\_2+\backslash ldots+X\_\{253\}+X\_\{254\}+X\_\{255\}$.

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Example as existential quantification

Let $A:\backslash mathcal\{U\}$ be some type, and let $B:A\backslash to\backslash mathcal\{U\}$. By the Curry–Howard correspondence, $B$ can be interpreted as a logical predicate on terms of $A$. For a given $a\; :\; A$, whether the type $B(a)$ is inhabited indicates whether $a$ satisfies this predicate. The correspondence can be extended to existential quantification and dependent pairs: the proposition $\backslash exists\{a\}\{\backslash in\}A\backslash ,B(a)$ is true if and only if the type $\backslash sum\_\{a:A\}B(a)$ is inhabited.

For example, $m:\backslash mathbb\{N\}$ is less than or equal to $n:\backslash mathbb\{N\}$ if and only if there exists another natural number $k:\backslash mathbb\{N\}$ such that $m\; +\; k\; =\; n$. In logic, this statement is codified by existential quantification:

$$m\backslash le\; n\; \backslash iff\; \backslash exists\{k\}\{\backslash in\}\backslash mathbb\{N\}\backslash ,m+k=n.$$

This proposition corresponds to the dependent pair type:

$$\backslash sum\_\{k:\backslash mathbb\{N\}\}\; m+k=n.$$

That is, a proof of the statement that $m$ is less than or equal to $n$ is a pair that contains both a non-negative number $k$, which is the difference between $m$ and $n$, and a proof of the equality $m\; +\; k\; =\; n$.

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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.

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First order dependent type theory

The system $\backslash lambda\; \backslash Pi$ 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.

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Second order dependent type theory

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

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Higher order dependently typed polymorphic lambda calculus

The higher order system $\backslash lambda\; \backslash Pi\; \backslash omega$ extends $\backslash lambda\; \backslash Pi\; 2$ 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 Rocq.

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Simultaneous programming language and logic

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 . 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.

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Comparison of languages with dependent types

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See also

• Typed lambda calculus
• Intuitionistic type theory
• Design by contract

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Further reading

• (1990). 9780198538141, Oxford University Press. . ISBN 9780198538141
• Brandl, Helmut (2022). Calculus of Constructions
• (2025). 9781595930279 ISBN 9781595930279
• (2025). 9781595939197 ISBN 9781595939197
• (2025). 9783642046513, Springer. ISBN 9783642046513
• (2025). 9781723139413, Lean Publishing. ISBN 9781723139413
• (2025). 9789811242380, World Scientific. ISBN 9789811242380

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External links

• Dependently Typed Programming 2008
• Dependently Typed Programming 2010
• Dependently Typed Programming 2011
• "Dependent type" at the Haskell Wiki

Categories: Foundations Of Mathematics, Type Theory, Type Systems

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