nLab dependent product type

Dependent product types


Type theory

natural deduction metalanguage, practical foundations

  1. type formation rule
  2. term introduction rule
  3. term elimination rule
  4. computation rule

type theory (dependent, intensional, observational type theory, homotopy type theory)

syntax object language

computational trinitarianism =
propositions as types +programs as proofs +relation type theory/category theory

logicset theory (internal logic of)category theorytype theory
predicatefamily of setsdisplay morphismdependent type
proofelementgeneralized elementterm/program
cut rulecomposition of classifying morphisms / pullback of display mapssubstitution
introduction rule for implicationcounit for hom-tensor adjunctionlambda
elimination rule for implicationunit for hom-tensor adjunctionapplication
cut elimination for implicationone of the zigzag identities for hom-tensor adjunctionbeta reduction
identity elimination for implicationthe other zigzag identity for hom-tensor adjunctioneta conversion
truesingletonterminal object/(-2)-truncated objecth-level 0-type/unit type
falseempty setinitial objectempty type
proposition, truth valuesubsingletonsubterminal object/(-1)-truncated objecth-proposition, mere proposition
logical conjunctioncartesian productproductproduct type
disjunctiondisjoint union (support of)coproduct ((-1)-truncation of)sum type (bracket type of)
implicationfunction set (into subsingleton)internal hom (into subterminal object)function type (into h-proposition)
negationfunction set into empty setinternal hom into initial objectfunction type into empty type
universal quantificationindexed cartesian product (of family of subsingletons)dependent product (of family of subterminal objects)dependent product type (of family of h-propositions)
existential quantificationindexed disjoint union (support of)dependent sum ((-1)-truncation of)dependent sum type (bracket type of)
logical equivalencebijection setobject of isomorphismsequivalence type
support setsupport object/(-1)-truncationpropositional truncation/bracket type
n-image of morphism into terminal object/n-truncationn-truncation modality
equalitydiagonal function/diagonal subset/diagonal relationpath space objectidentity type/path type
completely presented setsetdiscrete object/0-truncated objecth-level 2-type/set/h-set
setset with equivalence relationinternal 0-groupoidBishop set/setoid with its pseudo-equivalence relation an actual equivalence relation
equivalence class/quotient setquotientquotient type
inductioncolimitinductive type, W-type, M-type
higher inductionhigher colimithigher inductive type
-0-truncated higher colimitquotient inductive type
coinductionlimitcoinductive type
presettype without identity types
set of truth valuessubobject classifiertype of propositions
domain of discourseuniverseobject classifiertype universe
modalityclosure operator, (idempotent) monadmodal type theory, monad (in computer science)
linear logic(symmetric, closed) monoidal categorylinear type theory/quantum computation
proof netstring diagramquantum circuit
(absence of) contraction rule(absence of) diagonalno-cloning theorem
synthetic mathematicsdomain specific embedded programming language

homotopy levels


Dependent product types


In dependent type theory, a dependent product type x:AB(x)\prod_{x\colon A} B(x), for a dependent type x:AB(x):Typex\colon A\vdash B(x)\colon Type is the type of “dependently typed functions” assigning to each x:Ax\colon A an element of B(x)B(x).

In a model of the type theory in categorical semantics, this is a dependent product. In set theory, it is an element of an indexed product.

It includes function types as the special case when BB is not dependent on AA, product types as a special case when AA is the type of Booleans, and dependent sequence types as a special case when AA is the natural numbers type.


The inference rules for dependent function types (aka “dependent product types” or “Π\Pi-types”):


Like any type constructor in type theory, a dependent product type is specified by rules saying when we can introduce it as a type, how to construct terms of that type, how to use or “eliminate” terms of that type, and how to compute when we combine the constructors with the eliminators.

The type formation rule for dependent product type is:

A:Typex:AB(x):Type x:AB(x):Type\frac{A\colon Type \qquad x\colon A \vdash B(x) \colon Type}{\prod_{x\colon A} B(x)\colon Type}

As a negative type

Dependent product types are almost always defined as negative types. In this presentation, primacy is given to the eliminators. The natural eliminator of a dependent product type says that we can apply it to any input:

f: x:AB(x)a:Af(a):B(a)\frac{f\colon \prod_{x\colon A} B(x) \qquad a\colon A}{f(a) \colon B(a)}

The constructor is then determined as usual for a negative type: to construct a term of x:AB(x)\prod_{x\colon A} B(x), we have to specify how it behaves when applied to any input. In other words, we should have a term of type B(x)B(x) containing a free variable x:Ax\colon A. This yields the usual “λ\lambda-abstraction” constructor:

x:Ab:B(x)λx.b: x:AB(x)\frac{x\colon A\vdash b\colon B(x)}{\lambda x.b\colon \prod_{x\colon A} B(x)}

The beta-reduction rule is the obvious one, saying that when we evaluate a λ\lambda-abstraction, we do it by substituting for the bound variable.

(λx.b)(a) βb[a/x](\lambda x.b)(a) \;\to_\beta\; b[a/x]

If we want an eta-conversion rule, the natural one says that every dependently typed function is a λ\lambda-abstraction:

λx.f(x) ηf \lambda x.f(x) \;\to_\eta\; f

As a positive type

It is also possible to present dependent product types as a positive type. However, this requires a stronger metatheory, such as a logical framework. We use the same constructor (λ\lambda-abstraction), but now the eliminator says that to define an operation using a function, it suffices to say what to do in the case that that function is a lambda abstraction.

(x:Ab:B(x))c:Cf: x:AB(x)funsplit(c,f):C\frac{(x\colon A \vdash b\colon B(x)) \vdash c\colon C \qquad f\colon \prod_{x\colon A} B(x)}{funsplit(c,f)\colon C}

This rule cannot be formulated in the usual presentation of type theory, since it involves a “higher-order judgment”: the context of the term c:Cc\colon C involves a “term of type B(x)B(x) containing a free variable x:Ax\colon A”. However, it is possible to make sense of it. In dependent type theory, we need additionally to allow CC to depend on x:AB(x)\prod_{x\colon A} B(x).

The natural β\beta-reduction rule for this eliminator is

funsplit(c,λx.g) βc[g/b] funsplit(c, \lambda x.g) \;\to_\beta c[g/b]

and its η\eta-conversion rule looks something like

funsplit(c[λx.b/g],f) ηc[f/g]. funsplit(c[\lambda x.b / g], f) \;\to_\eta\; c[f/g].

Here g: x:AB(x)c:Cg\colon \prod_{x\colon A} B(x) \vdash c\colon C is a term containing a free variable of type x:AB(x)\prod_{x\colon A} B(x). By substituting λx.b\lambda x.b for gg, we obtain a term of type CC which depends on “a term b:B(x)b\colon B(x) containing a free variable x:Ax\colon A”. We then apply the positive eliminator at f: x:AB(x)f\colon \prod_{x\colon A} B(x), and the η\eta-rule says that this can be computed by just substituting ff for gg in cc.

Positive versus negative

As usual, the positive and negative formulations are equivalent in a suitable sense. They have the same constructor, while we can formulate the eliminators in terms of each other:

f(a) funsplit(b[a/x],f) funsplit(c,f) c[f(x)/b] \begin{aligned} f(a) &\coloneqq funsplit(b[a/x], f)\\ funsplit(c, f) &\coloneqq c[f(x)/b] \end{aligned}

The conversion rules also correspond.

In dependent type theory, this definition of funsplitfunsplit only gives us a properly typed dependent eliminator if the negative dependent product type satisfies η\eta-conversion. As usual, if it satisfies propositional eta-conversion then we can transport along that instead—and conversely, the dependent eliminator allows us to prove propositional η\eta-conversion. This is the content of Propositions 3.5, 3.6, and 3.7 in (Garner).

As types of dependent anafunctions

In the same way that one could define equivalence types as types of one-to-one correspondences and function types as types of anafunctions, one could define dependent function types as types of dependent anafunctions. This requires both identity types and heterogeneous identity types being defined first, which we shall write as a= Aba =_A b and x= B pyx =_{B}^{p} y respectively for a:Aa:A, b:Ab:A, p:a= Abp:a =_A b, x:B(a)x:B(a), and y:B(b)y:B(b). We use Agda notation (x:A)B(x)(x:A) \to B(x) for dependent function types rather than the dependent product type notation x:AB(x)\prod_{x:A} B(x) or Π(x:A).B(x)\Pi(x:A).B(x) in this section.

Rules for dependent function types

ΓAtypeΓ,x:AB(x)typeΓ(x:A)B(x)typeΓAtypeΓ,x:AB(x)typeΓ,f:(x:A)B(x),x:A,y:B(x) A,B(f,x,y)type\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma, x:A \vdash B(x) \; \mathrm{type}}{\Gamma \vdash (x:A) \to B(x) \; \mathrm{type}} \qquad \frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma, x:A \vdash B(x) \; \mathrm{type}}{\Gamma, f:(x:A) \to B(x), x:A, y:B(x) \vdash \mathcal{F}_{A, B}(f, x, y) \; \mathrm{type}}
ΓAtypeΓ,x:AB(x)typeΓ,x:Af(x):B(x)Γλx.f(x):(x:A)B(x)\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma, x:A \vdash B(x) \; \mathrm{type} \quad \Gamma, x:A \vdash f(x):B(x)}{\Gamma \vdash \lambda x.f(x):(x:A) \to B(x)}
ΓAtypeΓ,x:AB(x)typeΓ,x:Af(x):B(x)Γ,x:Aα(x): A,B(λx.f(x),x,f(x))\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma, x:A \vdash B(x) \; \mathrm{type} \quad \Gamma, x:A \vdash f(x):B(x)}{\Gamma, x:A \vdash \alpha(x):\mathcal{F}_{A, B}(\lambda x.f(x), x, f(x))}
ΓAtypeΓ,x:AB(x)typeΓ,f:(x:A)B(x),x:Aev(f,x):B(x)\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma, x:A \vdash B(x) \; \mathrm{type}}{\Gamma, f:(x:A) \to B(x), x:A \vdash \mathrm{ev}(f, x):B(x)}
ΓAtypeΓ,x:AB(x)typeΓ,f:(x:A)B(x),x:Aβ(f,x): A,B(f,x,ev(f,x))\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma, x:A \vdash B(x) \; \mathrm{type}}{\Gamma, f:(x:A) \to B(x), x:A \vdash \beta(f, x):\mathcal{F}_{A, B}(f, x, \mathrm{ev}(f, x))}
ΓAtypeΓ,x:AB(x)typeΓ,f:(x:A)B(x),x:A,y:B(x),u: A,B(f,x,y)κ(f,x,y,u):ev(f,x)= B(x)y\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma, x:A \vdash B(x) \; \mathrm{type}}{\Gamma, f:(x:A) \to B(x), x:A, y:B(x), u:\mathcal{F}_{A, B}(f, x, y) \vdash \kappa(f, x, y, u):\mathrm{ev}(f, x) =_{B(x)} y}
ΓAtypeΓ,x:AB(x)typeΓ,f:(x:A)B(x),x:A,y:B(x),u: A,B(f,x,y)η(f,x,y,u):β(f,x)= A,B(f,x) κ(f,x,y,u)u\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma, x:A \vdash B(x) \; \mathrm{type}}{\Gamma, f:(x:A) \to B(x), x:A, y:B(x), u:\mathcal{F}_{A, B}(f, x, y) \vdash \eta(f, x, y, u):\beta(f, x) =_{\mathcal{F}_{A, B}(f, x)}^{\kappa(f, x, y, u)} u}

By the rules for dependent pair types and dependent function types, one could derive that

ΓAtypeΓ,x:AB(x)typeΓ,f:(x:A)B(x)η(f):(x:A)isContr((y:B)× A,B(f,x,y))\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma, x:A \vdash B(x) \; \mathrm{type}}{\Gamma, f:(x:A) \to B(x) \vdash \eta(f):(x:A) \to \mathrm{isContr}\left((y:B) \times \mathcal{F}_{A, B}(f, x, y)\right)}

which is precisely the statement that A,B(f)\mathcal{F}_{A, B}(f) is a dependent anafunction for all dependent functions f:(x:A)B(x)f:(x:A) \to B(x).


The standard rules for type-formation, term introduction/elimination and computation of dependent product type are listed for instance in part I of

See also:

  • Richard Garner, On the strength of dependent products in the type theory of Martin-Löf, arXiv.

On the categorical semantics of dependent product types as relative right adjoints to context extension in comprehension categories:

Last revised on May 19, 2023 at 22:08:25. See the history of this page for a list of all contributions to it.