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 (Π type) or dependent function type.^[4] From a family of types ${\displaystyle B:A\to {\mathcal {U}}}$ we may construct the type of dependent functions ${\textstyle \prod _{x:A}B(x)}$, whose terms are functions that take a term ${\displaystyle a:A}$ and return a term in ${\displaystyle B(a)}$. For this example, the dependent function type is typically written as ${\textstyle \prod _{x:A}B(x),}$ ${\textstyle \prod _{x:A}B(x),}$ or ${\textstyle \prod {(x:A)}B(x)}$.

If ${\displaystyle B:A\to {\mathcal {U}}}$ is a constant function, the corresponding dependent product type is equivalent to an ordinary function type. That is, ${\textstyle \prod _{x:A}B}$ is judgmentally equal to ${\displaystyle A\to B}$ 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 models of universal quantifiers.

For example, if we write ${\displaystyle \operatorname {Vec} (\mathbb {R} ,n)}$ for n-tuples of real numbers, then ${\textstyle \prod _{n:\mathbb {N} }\operatorname {Vec} (\mathbb {R} ,n)}$ would be the type of a function 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. ${\textstyle \prod _{n:\mathbb {N} }{\mathbb {R} }}$ is the type of functions from natural numbers to the real numbers, which is written as ${\displaystyle \mathbb {N} \to \mathbb {R} }$ 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 ${\textstyle \prod _{x:A}B(x)}$ devolves into the product of X₀ × X₁ × X₂ × ... × X₂₅₃ × X₂₅₄ × X₂₅₅.

Σ type

The dual of the dependent product type is the dependent pair type, dependent sum type, sigma-type, or (confusingly) dependent product type.^[4] Sigma-types can also be understood as existential quantifiers. Continuing the above example, if, in the universe of types ${\displaystyle {\mathcal {U}}}$, there is a type ${\displaystyle A:{\mathcal {U}}}$ and a family of types ${\displaystyle B:A\to {\mathcal {U}}}$, then there is a dependent pair type ${\textstyle \sum _{x:A}B(x)}$. (The alternative notations are similar to that of Π 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 ${\textstyle (a,b):\sum _{x:A}B(x),}$ then ${\displaystyle a:A}$ and ${\displaystyle b:B(a)}$. 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 ${\displaystyle A\times B}$.^[4]

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 ${\textstyle \sum _{x:A}B(x)}$ devolves into the sum X₀ + X₁ + X₂ + ... + X₂₅₃ + X₂₅₄ + X₂₅₅.

Example as existential quantification

Let ${\displaystyle A:{\mathcal {U}}}$ be some type, and let ${\displaystyle B:A\to {\mathcal {U}}}$. By the Curry–Howard correspondence, B can be interpreted as a logical predicate on terms of A. For a given ${\displaystyle a:A}$, 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 ${\displaystyle \exists {a}{\in }A\,B(a)}$ is true if and only if the type ${\textstyle \sum _{a:A}B(a)}$ is inhabited.

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

$${\displaystyle m\leq n\iff \exists {k}{\in }\mathbb {N} \,m+k=n.}$$

This proposition corresponds to the dependent pair type:

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

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.

First order dependent type theory

The system ${\displaystyle \lambda \Pi }$ 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.

Second order dependent type theory

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

Higher order dependently typed polymorphic lambda calculus

The higher order system ${\displaystyle \lambda \Pi \omega }$ extends ${\displaystyle \lambda \Pi 2}$ 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.