nLab dependent product type

Idea

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

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 $B$ is not dependent on $A$, product types as a special case when $A$ is the type of Booleans, and dependent sequence types as a special case when $A$ is the natural numbers type.

Overview

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

Definition

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:

$$\frac{A\colon Type \qquad x\colon A \vdash B(x) \colon Type}{\prod_{x\colon A} B(x)\colon Type}$$

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:

$$\frac{f\colon \prod_{x\colon A} B(x) \qquad a\colon A}{f(a) \colon B(a)}$$

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

$$\frac{x\colon A\vdash b\colon B(x)}{\lambda x.b\colon \prod_{x\colon A} B(x)}$$

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

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

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

$$\lambda x.f(x) \;\to_\eta\; f$$

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 ($\lambda$-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.

$$\frac{(x\colon A \vdash b\colon B(x)) \vdash c\colon C \qquad f\colon \prod_{x\colon A} B(x)}{funsplit(c,f)\colon C}$$

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

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

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

and its $\eta$-conversion rule looks something like

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

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

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:

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

The conversion rules also correspond.

In dependent type theory, this definition of $funsplit$ only gives us a properly typed dependent eliminator if the negative dependent product type satisfies $\eta$-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 $\eta$-conversion. This is the content of Propositions 3.5, 3.6, and 3.7 in (Garner).

Dependent product types a la Russell and a la Tarski

In dependent type theory, there are two different ways to interpret the term $f:\prod_{x:A} B(x)$:

1. $f$ is literally a family of terms in the family of types $B(x)$ indexed by $A$

2. $f$ is a term representation for a family of terms in the family of types $B(x)$ indexed by $A$

This situation is similar to how there are two notions of type universe where small types of a universe are interpreted a la Russell, literally as types, or a la Tarski, as a term representation of types. Thus, in analogy to type universes, we can refer to dependent product types a la Russell and function types a la Tarski.

Dependent product types a la Russell and a la Tarski are expressed respectively via the elimination rule of function types:

• given type $A$ and the type family $x:A \vdash B(x)$ and an element $f:\prod_{x:A} B(x)$, one could form the family of terms $x:A \vdash f(x):B$

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma, x:A \vdash B(x) \; \mathrm{type} \quad \Gamma \vdash f:\prod_{x:A} B(x)}{\Gamma, x:A \vdash f(x):B(x)}$$

• given type $A$ and the type family $x:A \vdash B(x)$ one could form the family of terms $f:\prod_{x:A} B(x), x:A \vdash \mathrm{eval}(f, x):B(x)$

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma, x:A \vdash B(x) \; \mathrm{type}}{\Gamma, f:\prod_{x:A} B(x), x:A \vdash \mathrm{eval}(f, x):B(x)}$$

Dependent product types a la Tarski corresponds to the notion of dependent product in category theory where the dependent product $\Pi(A, B)$ literally comes with a morphism $\mathrm{eval}:\Pi(A, B) \times A \to \Sigma(A, B)$ in the slice category $C/A$, but dependent product types a la Russell are the one most commonly used in dependent type theory.

The conversion rules for dependent product types a la Russell are as follows:

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma, x:A \vdash B(x) \; \mathrm{type} \quad \Gamma, x:A \vdash b(x):B(x)}{\Gamma, x:A \vdash (\lambda x:A.b(x))(x) \equiv b(x):B(x)}$$

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma, x:A \vdash B(x) \; \mathrm{type} \quad \Gamma \vdash f:\prod_{x:A} B(x)}{\Gamma \vdash f \equiv \lambda x:A.f(x):\prod_{x:A} B(x)}$$

and the conversion rules for dependent product types a la Tarski are as follows:

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma, x:A \vdash B(x) \; \mathrm{type} \quad \Gamma, x:A \vdash b(x):B(x)}{\Gamma, x:A \vdash \mathrm{eval}(\lambda x:A.b(x), x) \equiv b(x):B(x)}$$

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma, x:A \vdash B(x) \; \mathrm{type}}{\Gamma, f:\prod_{x:A} B(x) \vdash f \equiv \lambda x:A.\mathrm{eval}(f, x):\prod_{x:A} B(x)}$$

For the rest of the article we shall assume the use of dependent product types a la Russell.

Weak and strict dependent product types

In dependent type theory, weak dependent product types are dependent product types for which the computation rules ($\beta$-conversion) and uniqueness rules ($\eta$-conversion) are propositional rather than judgmental:

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma, x:A \vdash B(x) \; \mathrm{type} \quad \Gamma, x:A \vdash b(x):B(x)}{\Gamma, a:A \vdash \beta_{A \to B}^{x:A.b(x)}(a):(\lambda(x:A).b(x))(a) =_{B(x)} b(a)}$$

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma, x:A \vdash B(x) \; \mathrm{type}}{\Gamma, f:\prod_{x:A} B(x) \vdash \eta_{A \to B}(f):f =_{\prod_{x:A} B(x)} \lambda(x:A).f(x)}$$

Weak dependent product types appear in weak type theories.

This contrasts with strict dependent product types which use judgmental computation and uniqueness rules:

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma, x:A \vdash B(x) \; \mathrm{type} \quad \Gamma, x:A \vdash b(x):B(x)}{\Gamma, a:A \vdash (\lambda(x:A).b(x))(a) \equiv b(a):B(x)}$$

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma, x:A \vdash B(x) \; \mathrm{type}}{\Gamma, f:\prod_{x:A} B(x) \vdash f \equiv \lambda(x:A).f(x):\prod_{x:A} B(x)}$$

Strict dependent product types appear in most dependent type theories such as Martin-Löf type theory, observational type theory, and cubical type theory.

For strict dependent product types, the judgmental computation and uniqueness rules automatically imply the propositional computation and uniqueness rules, as by the rules for judgmental equality and identity types, judgmental equality of two terms always implies propositional equality of the two terms.

In terms of function types

Given a dependent type theory with function types, dependent sum types, and identity types, the dependent product type of a type family $B(x)$ indexed by $x:A$ can be defined as the type of functions $f:A \to \sum_{x:A} B(x)$ from $A$ to the dependent sum type $\sum_{x:A} B(x)$ such that the composite of $f$ with the first projection function $\pi_1:\left(\sum_{x:A} B(x)\right) \to A$ is the identity function on $A$

$$\prod_{x:A} B(x) \coloneqq \sum_{f:A \to \sum_{x:A} B(x)} \lambda x:A.\pi_1(f(x)) =_{A \to A} \mathrm{id}_A$$

The underlying family of elements is then given by the composite of $f:A \to \sum_{x:A} B(x)$ with the second projection function of the dependent sum type:

$$x:A \vdash \pi_2(f(x)):B(x)$$

Similarly, given a family of elements $x:A \vdash b(x):B(x)$, one could construct the function

$$\lambda x:A.(x, b(x)):A \to \sum_{x:A} B(x)$$

such that given $x:A$, $\pi_1(\lambda x:A.(x, b(x))(x)) \equiv x$. By lambda abstraction, one has

$$\lambda x:A.\pi_1(\lambda x:A.(x, b(x))(x)) \equiv \mathrm{id}_A$$

and so the dependent function is given by

$$(\lambda x:A.(x, b(x)), \mathrm{refl}_{A \to A}(\mathrm{id}_A)):\sum_{f:A \to \sum_{x:A} B(x)} \lambda x:A.\pi_1(f(x)) =_{A \to A} \mathrm{id}_A$$

One also has $\pi_2(\lambda x:A.(x, b(x))(x)) \equiv b(x)$ which is the associated computation rule for dependent function types. Meanwhile, from the judgmental $\eta$-conversion rules for negative dependent sum types and function types, one could prove the judgmental $\eta$-conversion rule for dependent function types. Given

$$f:\sum_{f:A \to \sum_{x:A} B(x)} \lambda x:A.\pi_1(f(x)) =_{A \to A} \mathrm{id}_A$$

one has

$$f \equiv (\lambda x:A.(x, \pi_2(f(x))), \pi_2(f))$$

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 $a =_A b$ and $x =_{B}^{p} y$ respectively for $a:A$, $b:A$, $p:a =_A b$, $x:B(a)$, and $y:B(b)$. We use Agda notation $(x:A) \to B(x)$ for dependent function types rather than the dependent product type notation $\prod_{x:A} B(x)$ or $\Pi(x:A).B(x)$ in this section.

Rules for dependent function types

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma, x:A \vdash B(x) \; \mathrm{type}}{\Gamma \vdash (x:A) \to B(x) \; \mathrm{type}} \qquad \frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma, x:A \vdash B(x) \; \mathrm{type}}{\Gamma, f:(x:A) \to B(x), x:A, y:B(x) \vdash \mathcal{F}_{A, B}(f, x, y) \; \mathrm{type}}$$

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma, x:A \vdash B(x) \; \mathrm{type} \quad \Gamma, x:A \vdash f(x):B(x)}{\Gamma \vdash \lambda x.f(x):(x:A) \to B(x)}$$

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma, x:A \vdash B(x) \; \mathrm{type} \quad \Gamma, x:A \vdash f(x):B(x)}{\Gamma, x:A \vdash \alpha(x):\mathcal{F}_{A, B}(\lambda x.f(x), x, f(x))}$$

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma, x:A \vdash B(x) \; \mathrm{type}}{\Gamma, f:(x:A) \to B(x), x:A \vdash \mathrm{ev}(f, x):B(x)}$$

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma, x:A \vdash B(x) \; \mathrm{type}}{\Gamma, f:(x:A) \to B(x), x:A \vdash \beta(f, x):\mathcal{F}_{A, B}(f, x, \mathrm{ev}(f, x))}$$

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma, x:A \vdash B(x) \; \mathrm{type}}{\Gamma, f:(x:A) \to B(x), x:A, y:B(x), u:\mathcal{F}_{A, B}(f, x, y) \vdash \kappa(f, x, y, u):\mathrm{ev}(f, x) =_{B(x)} y}$$

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma, x:A \vdash B(x) \; \mathrm{type}}{\Gamma, f:(x:A) \to B(x), x:A, y:B(x), u:\mathcal{F}_{A, B}(f, x, y) \vdash \eta(f, x, y, u):\beta(f, x) =_{\mathcal{F}_{A, B}(f, x)}^{\kappa(f, x, y, u)} u}$$

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

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

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

Properties

Universal property of dependent product types

The universal property of dependent product types states that for all types $A$ and type families $x:A \vdash B(x)$, there is a family of functions

$$\pi_A:\prod_{x:A} \left(\prod_{x:A} B(x)\right) \to B(x)$$

such that for any other type $C$ and family of functions $f:\prod_{x:A} C \to B(x)$, there is a unique function $c:C \to \prod_{x:A} B(x)$ such that

$$\prod_{x:A} \prod_{y:C} \pi_A(x)(c(y)) =_{B(x)} f(x)(y)$$

If there is a type universe $U$, then one could wrap this into a single axiom.

$$\mathrm{up}_{A, B}:\prod_{C:U} \prod_{f:\prod_{x:A} C \to B(x)} \exists!c:C \to \prod_{x:A} B(x).\prod_{x:A} \prod_{y:C} \pi_A(x)(c(y)) =_{B(x)} f(x)(y)$$

Typal computation and uniqueness rules

The typal computation rule for function types is provable from the other four typal type formers of function types. Given type $A$, type family $x:A \vdash B(x)$ and dependent function $f:\prod_{x:A} B(x)$, we have, by the elimination rule and the introduction rule, a dependent function $\lambda x:A.f(x):\prod_{x:A} B(x)$, which by the uniqueness rules of dependent product types are equal to each other

$$\eta_{\prod_{x:A} B(x)}(f):f =_{\prod_{x:A} B(x)} \lambda x:A.f(x)$$

By the inductively defined function $\mathrm{idtohomotopy}$ which takes identifications between dependent functions to homotopies between dependent functions, we have that

$$\mathrm{idtohomotopy}(f, \lambda x:A.f(x))(\eta_{\prod_{x:A} B(x)}(f)):\prod_{x:A} f(x) =_{B(x)} (\lambda x:A.f(x))(x)$$

which is the typal computation rule for dependent function types.

Typal congruence rules

These are called typal congruence rules because they are the analogue of the judgmental congruence rules which use identity types and weak equivalence types instead of judgmental equality.

Strict dependent product types

Since dependent product types are negative types, we first present the typal congruence rule for the elimination rule of dependent product types

Theorem

Given a type $A$ and a type family $x:A \vdash B(x)$, dependent functions $f:\prod_{x:A} B(x)$ and $g:\prod_{x:A} B(x)$ and an identification $p:f =_{\prod_{x:A} B(x)} g$ there are families of identifications $x:A \vdash \mathrm{compelim}(f, g, p)(x):f(x) =_{B(x)} g(x)$.

Proof

We simply define the dependent function $\mathrm{compelim}$ to be happly, which is inductively defined on identity types.

The next is the typal congruence rule for the introduction rule of dependent function types. However, unlike the case for the other two rules, one needs dependent function extensionality.

Theorem

Assuming dependent function extensionality, given type $A$ and family of types $x:A \vdash B(x)$, families of elements $x:A \vdash b(x):B(x)$ and $x:A \vdash b'(x):B(x)$, and families of identifications $x:A \vdash p(x):b(x) =_{B(x)} b'(x)$, there is a identification

$$\mathrm{congintro}_{x:A.p(x)}:\lambda (x:A).b(x) =_{\prod_{x:A} B(x)} \lambda (x:A).b'(x)$$

Proof

By the computation rule of strict dependent function types, there are families of judgmental equalities

$$x:A \vdash ((\lambda x:A.b(x))(x) \equiv b(x):B(x)$$

$$x:A \vdash ((\lambda x:A.b'(x))(x) \equiv b'(x):B(x)$$

Thus, by the structural rules of judgmental equality, there are families of identifications

$$x:A \vdash p(x):(\lambda x:A.b(x))(x) =_{B(x)} (\lambda x:A.b'(x))(x)$$

and by $\lambda$-abstraction, one gets the dependent function

$$\lambda (x:A).p(x):\prod_{x:A} (\lambda x:A.b(x))(x) =_{B(x)} (\lambda x:A.b'(x))(x)$$

By dependent function extensionality, there is an equivalence of types

$$\mathrm{ext}_{\prod_{x:A} B(x)}^{-1}:(\lambda x:A.b(x)) =_{\prod_{x:A} B(x)} (\lambda x:A.b'(x)) \simeq \prod_{x:A} (\lambda x:A.b(x))(x) =_{B(x)} (\lambda x:A.b'(x))(x)$$

which yields an identification

$$\mathrm{ext}_{\prod_{x:A} B(x)}^{-1}^{-1}(\lambda (x:A).p(x)):(\lambda x:A.b(x)) =_{\prod_{x:A} B(x)} (\lambda x:A.b'(x))$$

We define

$$\mathrm{congintro}_{x:A.p(x)} \coloneqq \mathrm{ext}_{\prod_{x:A} B(x)}^{-1}^{-1}(\lambda (x:A).p(x)):(\lambda x:A.b(x)) =_{\prod_{x:A} B(x)} (\lambda x:A.b'(x))$$

Finally, we present the typal congruence rule for the formation rule of function types, which relies upon the previous two results. The theorem and proof differs significantly whether one uses definitional isomorphisms or some notion of equivalences of types.

Using definitional isomorphisms

Theorem

Given types $A$ and $A'$ and type families $x:A \vdash B(x)$, $x:A' \vdash B'(x)$ and definitional isomorphisms $e_A:A \cong A'$ and dependent function $e_B:\prod_{x:A} B(x) \cong B'(e_A(x))$ consisting of a family of definitional isomorphisms, there is a definitional isomorphism

$$\mathrm{congform}(e_A, e_B):\left(\prod_{x:A} B(x)\right) \cong \left(\prod_{x:A'} B'(x)\right)$$

Proof

Since for definitional isomorphism $e_A:A \cong A'$, we have judgmental equalities $e_A(e_A^{-1}(x)) \equiv x:A'$ and $e_A^{-1}(e_A(x)) \equiv x:A$, so we do not need to transport across identifications. Instead, we define the function

$$\mathrm{congform}(e_A, e_B):\left(\prod_{x:A} B(x)\right) \to \left(\prod_{x:A'} B'(x)\right)$$

by

$$\mathrm{congform}(e_A, e_B) \coloneqq \lambda (f:\prod_{x:A} B(x)).\lambda x:A'.e_B(e_A^{-1}(x))(f(e_A^{-1}(x)))$$

and the inverse function by

$$\mathrm{congform}(e_A, e_B)^{-1} \coloneqq \lambda (f:\prod_{x:A'} B'(x)).\lambda x:A.e_B(x)^{-1}(f(e_A(x)))$$

Now it suffices to construct judgmental equalities

$$f:\prod_{x:A} B(x) \vdash \mathrm{congform}(e_A, e_B)^{-1}(\mathrm{congform}(e_A, e_B)(f)) \equiv f:\prod_{x:A} B(x)$$

$$g:\prod_{x:A'} B'(x) \vdash \mathrm{congform}(e_A, e_B)(\mathrm{congform}(e_A, e_B)^{-1}(g)) \equiv g:\prod_{x:A'} B'(x)$$

from where it implies that $\mathrm{congform}(e_A, e_B)$ is thus a definitional isomorphism.

By definition, we have

$$\mathrm{congform}(e_A, e_B)^{-1}(\mathrm{congform}(e_A, e_B)(f)) \equiv \lambda x:A.e_B(x)^{-1}((\lambda x:A'.e_B(e_A^{-1}(x))(f(e_A^{-1}(x))) )(e_A(x)))$$

and by the computation rules of strict dependent product types, we have

$$(\lambda x:A'.e_B(e_A^{-1}(x))(f(e_A^{-1}(x))) )(e_A(x)) \equiv e_B(e_A^{-1}(e_A(x)))(f(e_A^{-1}(e_A(x))))$$

and because $e_A$ is a definitional isomorphism, we have

$$e_B(e_A^{-1}(e_A(x)))(f(e_A^{-1}(e_A(x)))) \equiv e_B(x)(f(x))$$

By the congruence rules for substitution of judgmental equality, we have

$$\lambda x:A.e_B(x)^{-1}((\lambda x:A'.e_B(e_A^{-1}(x))(f(e_A^{-1}(x))) )(e_A(x))) \equiv \lambda x:A.e_B(x)^{-1}(e_B(x)(f(x)))$$

Since for all $x:A$ each $e_B(x)$ is also a strict equality, we have

$$e_B(x)^{-1}(e_B(x)(f(x))) \equiv f(x)$$

by the congruence rules for substitution of judgmental equality, we have

$$\lambda x:A.e_B(x)^{-1}(e_B(x)(f(x))) \equiv \lambda x:A.f(x)$$

and by the uniqueness rule of dependent function types we have

$$\lambda x:A.f(x) \equiv f$$

thus by the transitive rule for judgmental equality we have

$$\mathrm{congform}(e_A, e_B)^{-1}(\mathrm{congform}(e_A, e_B)(f)) \equiv f$$

for all $f:\prod_{x:A} B(x)$

Similarly, by definition, we have

$$\mathrm{congform}(e_A, e_B)(\mathrm{congform}(e_A, e_B)^{-1}(f)) \equiv \lambda x:A'.e_B(e_A^{-1}(x))(( \lambda x:A.e_B(x)^{-1}(f(e_A(x))) )(e_A^{-1}(x)))$$

and by the computation rules for strict dependent product types, we have

$$( \lambda x:A.e_B(x)^{-1}(f(e_A(x))) )(e_A^{-1}(x)) \equiv e_B(e_A^{-1}(x))^{-1}(f(e_A(e_A^{-1}(x))))$$

and because $e_A$ is a definitional isomorphism, we have

$$e_B(e_A^{-1}(x))^{-1}(f(e_A(e_A^{-1}(x)))) \equiv e_B(e_A^{-1}(x))^{-1}(f(x))$$

By the congruence rules for substitution of judgmental equality, we have

$$\lambda x:A'.e_B(e_A^{-1}(x))(( \lambda x:A.e_B(x)^{-1}(f(e_A(x))) )(e_A^{-1}(x))) \equiv \lambda x:A'.e_B(e_A^{-1}(x))(e_B(e_A^{-1}(x))^{-1}(f(x)))$$

Since for all $x:A$ each $e_B(x)$ is also a strict equality, we have

$$e_B(e_A^{-1}(x))(e_B(e_A^{-1}(x))^{-1}(f(x))) \equiv f(x)$$

by the congruence rules for substitution of judgmental equality, we have

$$\lambda x:A'.e_B(e_A^{-1}(x))(e_B(e_A^{-1}(x))^{-1}(f(x))) \equiv \lambda x:A'.f(x)$$

and by the uniqueness rule of dependent function types we have

$$\lambda x:A.f(x) \equiv f$$

thus by the transitive rule for judgmental equality we have

$$\mathrm{congform}(e_A, e_B)(\mathrm{congform}(e_A, e_B)^{-1}(f)) \equiv f$$

for all $f:\prod_{x:A'} B'(x)$.

Since we have functions

$$\mathrm{congform}(e_A, e_B):\left(\prod_{x:A} B(x)\right) \to \left(\prod_{x:A'} B'(x)\right)$$

$$\mathrm{congform}(e_A, e_B)^{-1}:\left(\prod_{x:A'} B'(x)\right) \to \left(\prod_{x:A} B(x)\right)$$

and families of judgmental equalities

$$f:\prod_{x:A} B(x) \vdash \mathrm{congform}(e_A, e_B)^{-1}(\mathrm{congform}(e_A, e_B)(f)) \equiv f:\prod_{x:A} B(x)$$

$$g:\prod_{x:A'} B'(x) \vdash \mathrm{congform}(e_A, e_B)(\mathrm{congform}(e_A, e_B)^{-1}(g)) \equiv g:\prod_{x:A'} B'(x)$$

we could form the definitional isomorphism

$$\mathrm{toEquiv}(\mathrm{congform}(e_A, e_B), \mathrm{congform}(e_A, e_B)^{-1}):\left(\prod_{x:A'} B'(x)\right) \cong \left(\prod_{x:A} B(x)\right)$$

By a common abuse of notation we denote the definitional isomorphism by the same name as the underlying function $\mathrm{congform}(e_A, e_B)$; thus we have

$$\mathrm{congform}(e_A, e_B):\left(\prod_{x:A} B(x)\right) \cong \left(\prod_{x:A'} B'(x)\right)$$

Using weak equivalences of types

Theorem

Given types $A$ and $A'$ and type families $x:A \vdash B(x)$, $x:A' \vdash B'(x)$ and equivalence $e_A:A \simeq A'$ and dependent function $e_B:\prod_{x:A} B(x) \simeq B'(e_A(x))$, there is an equivalence

$$\mathrm{congform}(e_A, e_B):\left(\prod_{x:A} B(x)\right) \simeq \left(\prod_{x:A'} B'(x)\right)$$

Proof

We define the function

$$\mathrm{congform}(e_A, e_B):\left(\prod_{x:A} B(x)\right) \to \left(\prod_{x:A'} B'(x)\right)$$

by

$$\mathrm{congform}(e_A, e_B) \coloneqq \lambda (f:\prod_{x:A} B(x)).\lambda x:A'.\mathrm{transport}_{x:A'.B'(x)}(e_A(e_A^{-1}(x)), x, \mathrm{sec}_{e_A}(x), e_B(e_A^{-1}(x))(f(e_A^{-1}(x))))$$

and the inverse function by

$$\mathrm{congform}(e_A, e_B)^{-1} \coloneqq \lambda (f:\prod_{x:A'} B'(x)).\lambda x:A.e_B(x)^{-1}(f(e_A(x)))$$

where the equivalence $e_A:A \simeq A'$ has families of identifications

$$x':A \vdash \mathrm{sec}_{e_A}(x):e_A(e_A^{-1}(x)) =_{A'} x$$

$$x:A \vdash \mathrm{ret}_{e_A}(x):e_A^{-1}(e_A(x)) =_A x$$

witnessing that $e_A^{-1}$ is a section and retraction of $e_A$ respectively.

Now it suffices to construct families of identifications

$$f:\prod_{x:A} B(x) \vdash \mathrm{ret}_{\mathrm{congform}(e_A, e_B)}(f):\mathrm{congform}(e_A, e_B)^{-1}(\mathrm{congform}(e_A, e_B)(f)) =_{\prod_{x:A} B(x)} f$$

$$g:\prod_{x:A'} B'(x) \vdash \mathrm{sec}_{\mathrm{congform}(e_A, e_B)}(g):\mathrm{congform}(e_A, e_B)(\mathrm{congform}(e_A, e_B)^{-1}(g)) =_{\prod_{x:A'} B'(x)} g$$

from where it implies that $\mathrm{congform}(e_A, e_B)$ has a coherent inverse and contractible fibers and is thus an equivalence of types.

---

By definition,

$$\mathrm{congform}(e_A, e_B)^{-1}(\mathrm{congform}(e_A, e_B)(f)) \equiv \lambda x:A.e_B(x)^{-1}((\lambda x:A'.\mathrm{transport}_{x:A'.B'(x)}(e_A(e_A^{-1}(x)), x, \mathrm{sec}_{e_A}(x), e_{B'}(e_A^{-1}(x))(f(e_A^{-1}(x)))))(e_A(x)))$$

By the computation rules of strict dependent function types, there is a family of judgmental equalities

$$x:A \vdash (\lambda x:A'.\mathrm{transport}_{x:A'.B'(x)}(e_A(e_A^{-1}(x)), x, \mathrm{sec}_{e_A}(x), e_B(e_A^{-1}(x))(f(e_A^{-1}(x)))))(e_A(x)) \equiv \mathrm{transport}_{x:A'.B'(x)}(e_A(e_A^{-1}(e_A(x))), e_A(x), \mathrm{sec}_{e_A}(e_A(x)), e_B(e_A^{-1}(x))(f(e_A^{-1}(e_A(x)))))$$

and thus by the structural rules of judgmental equalities and the judgmental congruence rules for dependent function types, a judgmental equality

$$\mathrm{congform}(e_A, e_B)^{-1}(\mathrm{congform}(e_A, e_B)(f)) \equiv \lambda x:A.e_B(x)^{-1}(\mathrm{transport}_{x:A'.B'(x)}(e_A(e_A^{-1}(e_A(x))), e_A(x), \mathrm{sec}_{e_A}(e_A(x)), e_B(e_A^{-1}(e_A(x)))(f(e_A^{-1}(e_A(x))))))$$

In addition, the equivalence $e_A:A \simeq A'$ has the coherence condition

$$\mathrm{coh}_{e_A}(x):\mathrm{sec}_{e_A}(e_A(x)) =_{e_A(e_A^{-1}(e_A(x))) =_{A'} e_A(x)} \mathrm{ap}_{e_A}(e_A^{-1}(e_A(x)), x, \mathrm{ret}_{e_A}(x))$$

So we have the family of elements

$$x:A \vdash \mathrm{transport}_{x:A'.B'(x)}(e_A(e_A^{-1}(e_A(x))), e_A(x), \mathrm{sec}_{e_A}(e_A(x)), e_B(e_A^{-1}(e_A(x)))(f(e_A^{-1}(e_A(x))))):B'(e_A(x))$$

and by applying the function

$$\lambda p:e_A(e_A^{-1}(e_A(x))) =_{A'} e_A(x).\mathrm{transport}_{x:A'.B'(x)}(e_A(e_A^{-1}(e_A(x))), e_A(x), p, e_B(e_A^{-1}(e_A(x)))(f(e_A^{-1}(e_A(x)))))$$

across $\mathrm{coh}_{e_A}(x)$ one gets the family of identifications

$$x:A \vdash \mathrm{ap}_{\lambda p:e_A(e_A^{-1}(e_A(x))) =_{A'} e_A(x).\mathrm{transport}_{x:A'.B'(x)}(e_A(e_A^{-1}(e_A(x))), e_A(x), p, e_B(e_A^{-1}(e_A(x)))(f(e_A^{-1}(e_A(x)))))}(\mathrm{sec}_{e_A}(e_A(x)), \mathrm{ap}_{e_A}(e_A^{-1}(e_A(x)), x, \mathrm{ret}_{e_A}(x)), \mathrm{coh}_{e_A}(x)$$

in type

$$\mathrm{transport}_{x:A'.B'(x)}(e_A(e_A^{-1}(e_A(x))), e_A(x), \mathrm{sec}_{e_A}(e_A(x)), e_B(e_A^{-1}(e_A(x)))(f(e_A^{-1}(e_A(x))))) =_{B'(e_A(x)} \mathrm{transport}_{x:A'.B'(x)}(e_A(e_A^{-1}(e_A(x))), e_A(x), \mathrm{ap}_{e_A}(e_A^{-1}(e_A(x)), x, \mathrm{ret}_{e_A}(x)), e_B(e_A^{-1}(e_A(x)))(f(e_A^{-1}(e_A(x)))))$$

From the family of elements

$$x:A \vdash \mathrm{transport}_{x:A'.B'(x)}(e_A(e_A^{-1}(e_A(x))), e_A(x), \mathrm{ap}_{e_A}(e_A^{-1}(e_A(x)), x, \mathrm{ret}_{e_A}(x)), e_B(e_A^{-1}(e_A(x)))(f(e_A^{-1}(e_A(x))))):B(e_A(x))$$

one could define the family of dependent functions

$$x:A \vdash \lambda y:A.\lambda p:y =_A x.\mathrm{transport}_{x:A'.B'(x)}(e_A(y), e_A(x), \mathrm{ap}_{e_A}(y, x, p), e_B(y)(f(y))):\left(\sum_{y:A} y =_A x\right) \to B(e_A(x))$$

This means by applying the above dependent function to $e_A^{-1}(e_A(x))$, $\mathrm{ret}_{e_A}(x)$, $x$, $\mathrm{refl}_A(x)$, $\mathrm{ret}_{e_A}(x)$, and $\mathrm{apd}_{y:A.y =_A x}(e_A^{-1}(e_A(x)), x, \mathrm{ret}_{e_A}(x), \mathrm{ret}_{e_A}(x), \mathrm{refl}_A(x))$, one gets the family of identifications

$$x:A \vdash \mathrm{apbinary}_{\lambda y:A.\lambda p:y =_A x.\mathrm{transport}_{x:A'.B'(x)}(e_A(y), e_A(x), \mathrm{ap}_{e_A}(y, x, p), e_B(y)(f(y)))}(e_A^{-1}(e_A(x)), \mathrm{ret}_{e_A}(x), x, \mathrm{refl}_A(x), \mathrm{ret}_{e_A}(x), \mathrm{apd}_{y:A.y =_A x}(e_A^{-1}(e_A(x)), x, \mathrm{ret}_{e_A}(x), \mathrm{ret}_{e_A}(x), \mathrm{refl}_A(x))$$

in type

$$\mathrm{transport}_{x:A'.B'(x)}(e_A(e_A^{-1}(e_A(x))), e_A(x), \mathrm{ap}_{e_A}(e_A^{-1}(e_A(x)), x, \mathrm{ret}_{e_A}(x)), e_B(e_A^{-1}(e_A(x)))(f(e_A^{-1}(e_A(x))))) =_{B(e_A(x))} \mathrm{transport}_{x:A'.B'(x)}(e_A(x), e_A(x), \mathrm{ap}_{e_A}(x, x, \mathrm{refl}_A(x)), e_B(x)(f(x)))$$

By the judgmental computation rules of identity types, we have $\mathrm{ap}_{e_A}(x, x, \mathrm{refl}_A(x)) \equiv \mathrm{refl}_{A'}(e_A(x))$, which means that we have

$$\mathrm{transport}_{x:A'.B'(x)}(e_A(x), e_A(x), \mathrm{ap}_{e_A}(x, x, \mathrm{refl}_A(x)), e_B(x)(f(x))) \equiv \mathrm{transport}_{x:A'.B'(x)}(e_A(x), e_A(x), \mathrm{refl}_{A'}(e_A(x)), e_B(x)(f(x)))$$

and we have

$$\mathrm{transport}_{x:A'.B'(x)}(e_A(x), e_A(x), \mathrm{refl}_{A'}(e_A(x)), e_B(x)(f(x))) \equiv e_B(x)(f(x))$$

Thus by concatenation we have identification

$$\begin{array}{c} \mathrm{ap}_{\lambda p:e_A(e_A^{-1}(e_A(x))) =_{A'} e_A(x).\mathrm{transport}_{x:A'.B'(x)}(e_A(e_A^{-1}(e_A(x))), e_A(x), p, e_B(e_A^{-1}(e_A(x)))(f(e_A^{-1}(e_A(x)))))}(\mathrm{sec}_{e_A}(e_A(x)), \mathrm{ap}_{e_A}(e_A^{-1}(e_A(x)), x, \mathrm{ret}_{e_A}(x)), \mathrm{coh}_{e_A}(x) \\ \bullet \mathrm{apbinary}_{\lambda y:A.\lambda p:y =_A x.\mathrm{transport}_{x:A'.B'(x)}(e_A(y), e_A(x), \mathrm{ap}_{e_A}(y, x, p), e_B(y)(f(y)))}(e_A^{-1}(e_A(x)), \mathrm{ret}_{e_A}(x), x, \mathrm{refl}_A(x), \mathrm{ret}_{e_A}(x), \mathrm{apd}_{y:A.y =_A x}(e_A^{-1}(e_A(x)), x, \mathrm{ret}_{e_A}(x), \mathrm{ret}_{e_A}(x), \mathrm{refl}_A(x)) \end{array}$$

in type

$$\mathrm{transport}_{x:A'.B'(x)}(e_A(e_A^{-1}(e_A(x))), e_A(x), \mathrm{sec}_{e_A}(e_A(x)), e_B(e_A^{-1}(e_A(x)))(f(e_A^{-1}(e_A(x))))) =_{B'(e_A(x))} e_B(x)(f(x))$$

and by application of $e_B(x)^{-1}$ to the above identification, we have

$$\begin{array}{c} \mathrm{ap}_{e_B(x)^{-1}}(\mathrm{transport}_{x:A'.B'(x)}(e_A(e_A^{-1}(e_A(x))), e_A(x), \mathrm{sec}_{e_A}(e_A(x)), e_B(e_A^{-1}(e_A(x)))(f(e_A^{-1}(e_A(x))))), e_B(x)(f(x)), \\ \mathrm{ap}_{\lambda p:e_A(e_A^{-1}(e_A(x))) =_{A'} e_A(x).\mathrm{transport}_{x:A'.B'(x)}(e_A(e_A^{-1}(e_A(x))), e_A(x), p, e_B(e_A^{-1}(e_A(x)))(f(e_A^{-1}(e_A(x)))))}(\mathrm{sec}_{e_A}(e_A(x)), \mathrm{ap}_{e_A}(e_A^{-1}(e_A(x)), x, \mathrm{ret}_{e_A}(x)), \mathrm{coh}_{e_A}(x) \\ \bullet \mathrm{apbinary}_{\lambda y:A.\lambda p:y =_A x.\mathrm{transport}_{x:A'.B'(x)}(e_A(y), e_A(x), \mathrm{ap}_{e_A}(y, x, p), e_B(y)(f(y)))}(e_A^{-1}(e_A(x)), \mathrm{ret}_{e_A}(x), x, \mathrm{refl}_A(x), \mathrm{ret}_{e_A}(x), \mathrm{apd}_{y:A.y =_A x}(e_A^{-1}(e_A(x)), x, \mathrm{ret}_{e_A}(x), \mathrm{ret}_{e_A}(x), \mathrm{refl}_A(x)) \end{array}$$

in type

$$e_B(x)^{-1}(\mathrm{transport}_{x:A'.B'(x)}(e_A(e_A^{-1}(e_A(x))), e_A(x), \mathrm{sec}_{e_A}(e_A(x)), e_B(e_A^{-1}(e_A(x)))(f(e_A^{-1}(e_A(x)))))) =_{B(x)} e_B(x)^{-1}(e_B(x)(f(x)))$$

Similarly, The family of equivalences

$$x:A \vdash e_B(x):B(x) \simeq B'(e_A(x))$$

has a family of identifications

$$x:A, y:B(x) \vdash \mathrm{ret}_{e_B}(x, y):e_B(x)^{-1}(e_B(x)(y)) =_{B(x)} y$$

witnessing that $e_B(x)^{-1}$ is a retraction of $e_B(x)$ for each $x:A$.

By substituting $f(x)$ in for $y$ we get the family of identifications

$$x:A \vdash \mathrm{ret}_{e_B}(x, f(x)):e_B(x)^{-1}(e_B(x)(f(x))) =_{B(x)} f(x)$$

and thus the family of identifications

$$\begin{array}{c} \mathrm{ap}_{e_B(x)^{-1}}(\mathrm{transport}_{x:A'.B'(x)}(e_A(e_A^{-1}(e_A(x))), e_A(x), \mathrm{sec}_{e_A}(e_A(x)), e_B(e_A^{-1}(e_A(x)))(f(e_A^{-1}(e_A(x))))), e_B(x)(f(x)), \\ \mathrm{ap}_{\lambda p:e_A(e_A^{-1}(e_A(x))) =_{A'} e_A(x).\mathrm{transport}_{x:A'.B'(x)}(e_A(e_A^{-1}(e_A(x))), e_A(x), p, e_B(e_A^{-1}(e_A(x)))(f(e_A^{-1}(e_A(x)))))}(\mathrm{sec}_{e_A}(e_A(x)), \mathrm{ap}_{e_A}(e_A^{-1}(e_A(x)), x, \mathrm{ret}_{e_A}(x)), \mathrm{coh}_{e_A}(x) \\ \bullet \mathrm{apbinary}_{\lambda y:A.\lambda p:y =_A x.\mathrm{transport}_{x:A'.B'(x)}(e_A(y), e_A(x), \mathrm{ap}_{e_A}(y, x, p), e_B(y)(f(y)))}(e_A^{-1}(e_A(x)), \mathrm{ret}_{e_A}(x), x, \mathrm{refl}_A(x), \mathrm{ret}_{e_A}(x), \mathrm{apd}_{y:A.y =_A x}(e_A^{-1}(e_A(x)), x, \mathrm{ret}_{e_A}(x), \mathrm{ret}_{e_A}(x), \mathrm{refl}_A(x)) \\ \bullet \mathrm{ret}_{e_B}(x, f(x)) \end{array}$$

in type

$$e_B(x)^{-1}(\mathrm{transport}_{x:A'.B'(x)}(e_A(e_A^{-1}(e_A(x))), e_A(x), \mathrm{sec}_{e_A}(e_A(x)), e_B(e_A^{-1}(e_A(x)))(f(e_A^{-1}(e_A(x)))))) =_{B(x)} f(x)$$

and the dependent function

$$\begin{array}{c} \lambda x:A.\mathrm{ap}_{e_B(x)^{-1}}(\mathrm{transport}_{x:A'.B'(x)}(e_A(e_A^{-1}(e_A(x))), e_A(x), \mathrm{sec}_{e_A}(e_A(x)), e_B(e_A^{-1}(e_A(x)))(f(e_A^{-1}(e_A(x))))), e_B(x)(f(x)), \\ \mathrm{ap}_{\lambda p:e_A(e_A^{-1}(e_A(x))) =_{A'} e_A(x).\mathrm{transport}_{x:A'.B'(x)}(e_A(e_A^{-1}(e_A(x))), e_A(x), p, e_B(e_A^{-1}(e_A(x)))(f(e_A^{-1}(e_A(x)))))}(\mathrm{sec}_{e_A}(e_A(x)), \mathrm{ap}_{e_A}(e_A^{-1}(e_A(x)), x, \mathrm{ret}_{e_A}(x)), \mathrm{coh}_{e_A}(x) \\ \bullet \mathrm{apbinary}_{\lambda y:A.\lambda p:y =_A x.\mathrm{transport}_{x:A'.B'(x)}(e_A(y), e_A(x), \mathrm{ap}_{e_A}(y, x, p), e_B(y)(f(y)))}(e_A^{-1}(e_A(x)), \mathrm{ret}_{e_A}(x), x, \mathrm{refl}_A(x), \mathrm{ret}_{e_A}(x), \mathrm{apd}_{y:A.y =_A x}(e_A^{-1}(e_A(x)), x, \mathrm{ret}_{e_A}(x), \mathrm{ret}_{e_A}(x), \mathrm{refl}_A(x)) \\ \bullet \mathrm{ret}_{e_B}(x, f(x)) \end{array}$$

in type

$$\prod_{x:A} e_B(x)^{-1}(\mathrm{transport}_{x:A'.B'(x)}(e_A(e_A^{-1}(e_A(x))), e_A(x), \mathrm{sec}_{e_A}(e_A(x)), e_B(e_A^{-1}(e_A(x)))(f(e_A^{-1}(e_A(x)))))) =_{B(x)} f(x)$$

By dependent function extensionality, we get the identification

$$\begin{array}{c} \mathrm{ext}_{\prod_{x:A} B(x)}^{-1}(\lambda x:A.\mathrm{ap}_{e_B(x)^{-1}}(\mathrm{transport}_{x:A'.B'(x)}(e_A(e_A^{-1}(e_A(x))), e_A(x), \mathrm{sec}_{e_A}(e_A(x)), e_B(e_A^{-1}(e_A(x)))(f(e_A^{-1}(e_A(x))))), e_B(x)(f(x)), \\ \mathrm{ap}_{\lambda p:e_A(e_A^{-1}(e_A(x))) =_{A'} e_A(x).\mathrm{transport}_{x:A'.B'(x)}(e_A(e_A^{-1}(e_A(x))), e_A(x), p, e_B(e_A^{-1}(e_A(x)))(f(e_A^{-1}(e_A(x)))))}(\mathrm{sec}_{e_A}(e_A(x)), \mathrm{ap}_{e_A}(e_A^{-1}(e_A(x)), x, \mathrm{ret}_{e_A}(x)), \mathrm{coh}_{e_A}(x) \\ \bullet \mathrm{apbinary}_{\lambda y:A.\lambda p:y =_A x.\mathrm{transport}_{x:A'.B'(x)}(e_A(y), e_A(x), \mathrm{ap}_{e_A}(y, x, p), e_B(y)(f(y)))}(e_A^{-1}(e_A(x)), \mathrm{ret}_{e_A}(x), x, \mathrm{refl}_A(x), \mathrm{ret}_{e_A}(x), \mathrm{apd}_{y:A.y =_A x}(e_A^{-1}(e_A(x)), x, \mathrm{ret}_{e_A}(x), \mathrm{ret}_{e_A}(x), \mathrm{refl}_A(x)) \\ \bullet \mathrm{ret}_{e_B}(x, f(x))) \end{array}$$

in type

$$\lambda x:A.e_B(x)^{-1}(\mathrm{transport}_{x:A'.B'(x)}(e_A(e_A^{-1}(e_A(x))), e_A(x), \mathrm{sec}_{e_A}(e_A(x)), e_B(e_A^{-1}(e_A(x)))(f(e_A^{-1}(e_A(x)))))) =_{\prod_{x:A} B(x)} f$$

and since we defined

$$\mathrm{congform}(e_A, e_B)^{-1}(\mathrm{congform}(e_A, e_B)(f)) \equiv \lambda x:A.e_B(x)^{-1}(\mathrm{transport}_{x:A'.B'(x)}(e_A(e_A^{-1}(e_A(x))), e_A(x), \mathrm{sec}_{e_A}(e_A(x)), e_B(e_A^{-1}(e_A(x)))(f(e_A^{-1}(e_A(x))))))$$

we have

$$\begin{array}{c} \mathrm{ext}_{\prod_{x:A} B(x)}^{-1}(\lambda x:A.\mathrm{ap}_{e_B(x)^{-1}}(\mathrm{transport}_{x:A'.B'(x)}(e_A(e_A^{-1}(e_A(x))), e_A(x), \mathrm{sec}_{e_A}(e_A(x)), e_B(e_A^{-1}(e_A(x)))(f(e_A^{-1}(e_A(x))))), e_B(x)(f(x)), \\ \mathrm{ap}_{\lambda p:e_A(e_A^{-1}(e_A(x))) =_{A'} e_A(x).\mathrm{transport}_{x:A'.B'(x)}(e_A(e_A^{-1}(e_A(x))), e_A(x), p, e_B(e_A^{-1}(e_A(x)))(f(e_A^{-1}(e_A(x)))))}(\mathrm{sec}_{e_A}(e_A(x)), \mathrm{ap}_{e_A}(e_A^{-1}(e_A(x)), x, \mathrm{ret}_{e_A}(x)), \mathrm{coh}_{e_A}(x) \\ \bullet \mathrm{apbinary}_{\lambda y:A.\lambda p:y =_A x.\mathrm{transport}_{x:A'.B'(x)}(e_A(y), e_A(x), \mathrm{ap}_{e_A}(y, x, p), e_B(y)(f(y)))}(e_A^{-1}(e_A(x)), \mathrm{ret}_{e_A}(x), x, \mathrm{refl}_A(x), \mathrm{ret}_{e_A}(x), \mathrm{apd}_{y:A.y =_A x}(e_A^{-1}(e_A(x)), x, \mathrm{ret}_{e_A}(x), \mathrm{ret}_{e_A}(x), \mathrm{refl}_A(x)) \\ \bullet \mathrm{ret}_{e_B}(x, f(x))):\mathrm{congform}(e_A, e_B)^{-1}(\mathrm{congform}(e_A, e_B)(f)) =_{\prod_{x:A} B(x)} f \end{array}$$

We can define the witness that $\mathrm{congform}(e_A, e_B)^{-1}$ is a retraction of $\mathrm{congform}(e_A, e_B)$ as

$$\begin{array}{c} \mathrm{ret}_{\mathrm{congform}(e_A, e_B)}(f) \coloneqq \\ \mathrm{ext}_{\prod_{x:A} B(x)}^{-1}(\lambda x:A.\mathrm{ap}_{e_B(x)^{-1}}(\mathrm{transport}_{x:A'.B'(x)}(e_A(e_A^{-1}(e_A(x))), e_A(x), \mathrm{sec}_{e_A}(e_A(x)), e_B(e_A^{-1}(e_A(x)))(f(e_A^{-1}(e_A(x))))), e_B(x)(f(x)), \\ \mathrm{ap}_{\lambda p:e_A(e_A^{-1}(e_A(x))) =_{A'} e_A(x).\mathrm{transport}_{x:A'.B'(x)}(e_A(e_A^{-1}(e_A(x))), e_A(x), p, e_B(e_A^{-1}(e_A(x)))(f(e_A^{-1}(e_A(x)))))}(\mathrm{sec}_{e_A}(e_A(x)), \mathrm{ap}_{e_A}(e_A^{-1}(e_A(x)), x, \mathrm{ret}_{e_A}(x)), \mathrm{coh}_{e_A}(x) \\ \bullet \mathrm{apbinary}_{\lambda y:A.\lambda p:y =_A x.\mathrm{transport}_{x:A'.B'(x)}(e_A(y), e_A(x), \mathrm{ap}_{e_A}(y, x, p), e_B(y)(f(y)))}(e_A^{-1}(e_A(x)), \mathrm{ret}_{e_A}(x), x, \mathrm{refl}_A(x), \mathrm{ret}_{e_A}(x), \mathrm{apd}_{y:A.y =_A x}(e_A^{-1}(e_A(x)), x, \mathrm{ret}_{e_A}(x), \mathrm{ret}_{e_A}(x), \mathrm{refl}_A(x)) \\ \bullet \mathrm{ret}_{e_B}(x, f(x))):\mathrm{congform}(e_A, e_B)^{-1}(\mathrm{congform}(e_A, e_B)(f)) =_{\prod_{x:A} B(x)} f \end{array}$$

---

Similarly, by definition,

$$\mathrm{congform}(e_A, e_B)(\mathrm{congform}(e_A, e_B)^{-1}(f)) \equiv \lambda x:A'.\mathrm{transport}_{x:A'.B'(x)}(e_A(e_A^{-1}(x)), x, \mathrm{sec}_{e_A}(x), e_B(e_A^{-1}(x))(( \lambda x:A.e_B(x)^{-1}(f(e_A(x))) )(e_A^{-1}(x))))$$

By the computation rules of strict dependent function types, there is a family of judgmental equalities

$$x:A \vdash (\lambda x:A.e_B(x)^{-1}(f(e_A(x))))(e_A^{-1}(x)) \equiv e_B(e_A^{-1}(x))^{-1}(f(e_A(e_A^{-1}(x))))$$

and thus by the structural rules of judgmental equalities and the judgmental congruence rules for dependent function types, a judgmental equality

$$\lambda x:A'.\mathrm{transport}_{x:A'.B'(x)}(e_A(e_A^{-1}(x)), x, \mathrm{sec}_{e_A}(x), e_B(e_A^{-1}(x))((\lambda x:A.e_B(x)^{-1}(f(e_A(x))) )(e_A^{-1}(x)))) \equiv \lambda x:A'.\mathrm{transport}_{x:A'.B'(x)}(e_A(e_A^{-1}(x)), x, \mathrm{sec}_{e_A}(x), e_B(e_A^{-1}(x))(e_B(e_A^{-1}(x))^{-1}(f(e_A(e_A^{-1}(x))))))$$

From the family of elements

$$x:A' \vdash \mathrm{transport}_{x:A'.B'(x)}(e_A(e_A^{-1}(x)), x, \mathrm{sec}_{e_A}(x), e_B(e_A^{-1}(x))(e_B(e_A^{-1}(x))^{-1}(f(e_A(e_A^{-1}(x))))))$$

one could define the family of dependent functions

$$x:A' \vdash \lambda y:A.\lambda p:y =_A x.\mathrm{transport}_{x:A'.B'(x)}(y), x, p, e_B(e_A^{-1}(x))(e_B(e_A^{-1}(x))^{-1}(f(y))))$$

This means by applying the above dependent function to $e_A(e_A^{-1}(x))$, $\mathrm{sec}_{e_A}(x)$, $x$, $\mathrm{refl}_A(x)$, $\mathrm{sec}_{e_A}(x)$, and $\mathrm{apd}_{y:A.y =_A x}(e_A(e_A^{-1}(x)), x, \mathrm{sec}_{e_A}(x), \mathrm{sec}_{e_A}(x), \mathrm{refl}_A(x))$, one gets the family of identifications

$$x:A' \vdash \mathrm{apbinary}_{\lambda y:A.\lambda p:y =_A x.\mathrm{transport}_{x:A'.B'(x)}(y), x, p, e_B(e_A^{-1}(x))(e_B(e_A^{-1}(x))^{-1}(f(y))))}(e_A(e_A^{-1}(x)), \mathrm{sec}_{e_A}(x), x, \mathrm{refl}_A(x), \mathrm{sec}_{e_A}(x), \mathrm{apd}_{y:A.y =_A x}(e_A(e_A^{-1}(x)), x, \mathrm{sec}_{e_A}(x), \mathrm{sec}_{e_A}(x), \mathrm{refl}_A(x)))$$

in type

$$\mathrm{transport}_{x:A'.B'(x)}(e_A(e_A^{-1}(x)), x, \mathrm{sec}_{e_A}(x), e_B(e_A^{-1}(x))(e_B(e_A^{-1}(x))^{-1}(f(e_A(e_A^{-1}(x)))))) =_{B'(x)} \mathrm{transport}_{x:A'.B'(x)}(x, x, \mathrm{refl}_A(x), e_B(e_A^{-1}(x))(e_B(e_A^{-1}(x))^{-1}(f(x))))$$

By the judgmental computation rules of identity types, we have

$$\mathrm{transport}_{x:A'.B'(x)}(x, x, \mathrm{refl}_A(x), e_B(e_A^{-1}(x))(e_B(e_A^{-1}(x))^{-1}(f(x)))) \equiv e_B(e_A^{-1}(x))(e_B(e_A^{-1}(x))^{-1}(f(x)))$$

Similarly, The family of equivalences

$$x:A \vdash e_B(x):B(x) \simeq B'(e_A(x))$$

has a family of identifications

$$x:A, y:B'(e_A(x)) \vdash \mathrm{sec}_{e_B}(x, y):e_B(x)(e_B(x)^{-1}(y)) =_{B(x)} y$$

witnessing that $e_B(x)^{-1}$ is a section of $e_B(x)$ for each $x:A$.

By substituting $f(e_A(e_A^{-1}(x)))$ in for $y$ and $e_A^{-1}(x)$ in for $x$ in the above expression we get the family of identifications

$$x:A' \vdash \mathrm{sec}_{e_B}(e_A^{-1}(x), f(e_A(e_A^{-1}(x)))):e_B(e_A^{-1}(x))(e_B(e_A^{-1}(x))^{-1}(f(e_A(e_A^{-1}(x))))) =_{B'(e_A(e_A^{-1}(x)))} f(e_A(e_A^{-1}(x)))$$

and by transport across $\mathrm{sec}_{e_A}(x)$ we get

$$x:A' \vdash \mathrm{transport}_{z:A'.e_B(x)(e_B(x)^{-1}(f(z))) =_{B'(z)} f(z)}(e_A(e_A^{-1}(x)), x, \mathrm{sec}_{e_A}(x), \mathrm{sec}_{e_B}(e_A^{-1}(x), f(e_A(e_A^{-1}(x))))):e_B(e_A^{-1}(x))(e_B(e_A^{-1}(x))^{-1}(f(x))) =_{B'(x)} f(x)$$

By concatenation of identifications we get

$$\begin{array}{c} \mathrm{apbinary}_{\lambda y:A.\lambda p:y =_A x.\mathrm{transport}_{x:A'.B'(x)}(y), x, p, e_B(e_A^{-1}(x))(e_B(e_A^{-1}(x))^{-1}(f(y))))}(e_A(e_A^{-1}(x)), \mathrm{sec}_{e_A}(x), x, \mathrm{refl}_A(x), \mathrm{sec}_{e_A}(x), \mathrm{apd}_{y:A.y =_A x}(e_A(e_A^{-1}(x)), x, \mathrm{sec}_{e_A}(x), \mathrm{sec}_{e_A}(x), \mathrm{refl}_A(x))) \\ \bullet \mathrm{transport}_{z:A'.e_B(x)(e_B(x)^{-1}(f(z))) =_{B'(z)} f(z)}(e_A(e_A^{-1}(x)), x, \mathrm{sec}_{e_A}(x), \mathrm{sec}_{e_B}(e_A^{-1}(x), f(e_A(e_A^{-1}(x))))) \end{array}$$

in type

$$\mathrm{transport}_{x:A'.B'(x)}(e_A(e_A^{-1}(x)), x, \mathrm{sec}_{e_A}(x), e_B(e_A^{-1}(x))(e_B(e_A^{-1}(x))^{-1}(f(e_A(e_A^{-1}(x)))))) =_{B'(x)} f(x)$$

and by $\lambda$-abstraction we get

$$\begin{array}{c} \lambda x:A'.\mathrm{apbinary}_{\lambda y:A.\lambda p:y =_A x.\mathrm{transport}_{x:A'.B'(x)}(y), x, p, e_B(e_A^{-1}(x))(e_B(e_A^{-1}(x))^{-1}(f(y))))}(e_A(e_A^{-1}(x)), \mathrm{sec}_{e_A}(x), x, \mathrm{refl}_A(x), \mathrm{sec}_{e_A}(x), \mathrm{apd}_{y:A.y =_A x}(e_A(e_A^{-1}(x)), x, \mathrm{sec}_{e_A}(x), \mathrm{sec}_{e_A}(x), \mathrm{refl}_A(x))) \\ \bullet \mathrm{transport}_{z:A'.e_B(x)(e_B(x)^{-1}(f(z))) =_{B'(z)} f(z)}(e_A(e_A^{-1}(x)), x, \mathrm{sec}_{e_A}(x), \mathrm{sec}_{e_B}(e_A^{-1}(x), f(e_A(e_A^{-1}(x))))) \end{array}$$

in type

$$\prod_{x:A'} \mathrm{transport}_{x:A'.B'(x)}(e_A(e_A^{-1}(x)), x, \mathrm{sec}_{e_A}(x), e_B(e_A^{-1}(x))(e_B(e_A^{-1}(x))^{-1}(f(e_A(e_A^{-1}(x)))))) =_{B'(x)} f(x)$$

By dependent function extensionality, we get the identification

$$\begin{array}{c} \mathrm{ext}_{\prod_{x:A} B(x)}^{-1}(\lambda x:A'. \\ \mathrm{apbinary}_{\lambda y:A.\lambda p:y =_A x.\mathrm{transport}_{x:A'.B'(x)}(y), x, p, e_B(e_A^{-1}(x))(e_B(e_A^{-1}(x))^{-1}(f(y))))}(e_A(e_A^{-1}(x)), \mathrm{sec}_{e_A}(x), x, \mathrm{refl}_A(x), \mathrm{sec}_{e_A}(x), \mathrm{apd}_{y:A.y =_A x}(e_A(e_A^{-1}(x)), x, \mathrm{sec}_{e_A}(x), \mathrm{sec}_{e_A}(x), \mathrm{refl}_A(x))) \\ \bullet \mathrm{transport}_{z:A'.e_B(x)(e_B(x)^{-1}(f(z))) =_{B'(z)} f(z)}(e_A(e_A^{-1}(x)), x, \mathrm{sec}_{e_A}(x), \mathrm{sec}_{e_B}(e_A^{-1}(x), f(e_A(e_A^{-1}(x)))))) \end{array}$$

in type

$$\lambda x:A'.\mathrm{transport}_{x:A'.B'(x)}(e_A(e_A^{-1}(x)), x, \mathrm{sec}_{e_A}(x), e_B(e_A^{-1}(x))(e_B(e_A^{-1}(x))^{-1}(f(e_A(e_A^{-1}(x)))))) =_{\prod_{x:A'} B'(x)} f$$

and since we defined

$$\mathrm{congform}(e_A, e_B)(\mathrm{congform}(e_A, e_B)^{-1}(f)) \equiv \lambda x:A'.\mathrm{transport}_{x:A'.B'(x)}(e_A(e_A^{-1}(x)), x, \mathrm{sec}_{e_A}(x), e_B(e_A^{-1}(x))(( \lambda x:A.e_B(x)^{-1}(f(e_A(x))) )(e_A^{-1}(x))))$$

we have

$$\begin{array}{c} \mathrm{ext}_{\prod_{x:A} B(x)}^{-1}(\lambda x:A'. \\ \mathrm{apbinary}_{\lambda y:A.\lambda p:y =_A x.\mathrm{transport}_{x:A'.B'(x)}(y), x, p, e_B(e_A^{-1}(x))(e_B(e_A^{-1}(x))^{-1}(f(y))))}(e_A(e_A^{-1}(x)), \mathrm{sec}_{e_A}(x), x, \mathrm{refl}_A(x), \mathrm{sec}_{e_A}(x), \mathrm{apd}_{y:A.y =_A x}(e_A(e_A^{-1}(x)), x, \mathrm{sec}_{e_A}(x), \mathrm{sec}_{e_A}(x), \mathrm{refl}_A(x))) \\ \bullet \mathrm{transport}_{z:A'.e_B(x)(e_B(x)^{-1}(f(z))) =_{B'(z)} f(z)}(e_A(e_A^{-1}(x)), x, \mathrm{sec}_{e_A}(x), \mathrm{sec}_{e_B}(e_A^{-1}(x), f(e_A(e_A^{-1}(x)))))):\mathrm{congform}(e_A, e_B)(\mathrm{congform}(e_A, e_B)^{-1}(f)) =_{\prod_{x:A'} B'(x)} f \end{array}$$

We can define the witness that $\mathrm{congform}(e_A, e_B)^{-1}$ is a section of $\mathrm{congform}(e_A, e_B)$ as

$$\begin{array}{c} \mathrm{sec}_{\mathrm{congform}(e_A, e_B)}(f) \coloneqq \mathrm{ext}_{\prod_{x:A} B(x)}^{-1}(\lambda x:A'. \\ \mathrm{apbinary}_{\lambda y:A.\lambda p:y =_A x.\mathrm{transport}_{x:A'.B'(x)}(y), x, p, e_B(e_A^{-1}(x))(e_B(e_A^{-1}(x))^{-1}(f(y))))}(e_A(e_A^{-1}(x)), \mathrm{sec}_{e_A}(x), x, \mathrm{refl}_A(x), \mathrm{sec}_{e_A}(x), \mathrm{apd}_{y:A.y =_A x}(e_A(e_A^{-1}(x)), x, \mathrm{sec}_{e_A}(x), \mathrm{sec}_{e_A}(x), \mathrm{refl}_A(x))) \\ \bullet \mathrm{transport}_{z:A'.e_B(x)(e_B(x)^{-1}(f(z))) =_{B'(z)} f(z)}(e_A(e_A^{-1}(x)), x, \mathrm{sec}_{e_A}(x), \mathrm{sec}_{e_B}(e_A^{-1}(x), f(e_A(e_A^{-1}(x)))))):\mathrm{congform}(e_A, e_B)(\mathrm{congform}(e_A, e_B)^{-1}(f)) =_{\prod_{x:A'} B'(x)} f \end{array}$$

---

Since we have functions

$$\mathrm{congform}(e_A, e_B):\left(\prod_{x:A} B(x)\right) \to \left(\prod_{x:A'} B'(x)\right)$$

$$\mathrm{congform}(e_A, e_B)^{-1}:\left(\prod_{x:A'} B'(x)\right) \to \left(\prod_{x:A} B(x)\right)$$

and families of identifications

$$f:\prod_{x:A} B(x) \vdash \mathrm{ret}_{\mathrm{congform}(e_A, e_B)}(f):\mathrm{congform}(e_A, e_B)^{-1}(\mathrm{congform}(e_A, e_B)(f)) =_{\prod_{x:A} B(x)} f$$

$$g:\prod_{x:A'} B'(x) \vdash \mathrm{sec}_{\mathrm{congform}(e_A, e_B)}(g):\mathrm{congform}(e_A, e_B)(\mathrm{congform}(e_A, e_B)^{-1}(g)) =_{\prod_{x:A'} B'(x)} g$$

we could form the equivalence

$$\left(\mathrm{congform}(e_A, e_B), \mathrm{qInvToIsEquiv}\left(\mathrm{congform}(e_A, e_B)^{-1}, \lambda f:\prod_{x:A} B(x).\mathrm{ret}_{\mathrm{congform}(e_A, e_B)}(f), \lambda g:\prod_{x:A'} B'(x).\mathrm{sec}_{\mathrm{congform}(e_A, e_B)}(g)\right)\right):\left(\prod_{x:A} B(x)\right) \simeq \left(\prod_{x:A'} B'(x)\right)$$

By a common abuse of notation we denote the equivalence by the same name as the underlying function $\mathrm{congform}(e_A, e_B)$; thus we have

$$\mathrm{congform}(e_A, e_B):\left(\prod_{x:A} B(x)\right) \simeq \left(\prod_{x:A'} B'(x)\right)$$

Weak dependent product types

Theorem

Assuming dependent function extensionality, given types $A$ and $A'$ and type families $x:A \vdash B(x)$ and $x:A \vdash B'(x)$ and equivalences $e_A:A \simeq A'$ and dependent function of equivalences $e_B:\prod_{x:A} B(x) \simeq B'(e(x))$, there is an equivalence

$$\mathrm{congform}(e_A, e_B):\left(\prod_{x:A} B(x)\right) \simeq \left(\prod_{x:A'} B'(x)\right)$$

Since dependent function types are negative types, we first present the typal congruence rule for the elimination rule of dependent function types

Theorem

Given a type $A$ and a type family $x:A \vdash B(x)$, dependent functions $f:\prod_{x:A} B(x)$ and $g:\prod_{x:A} B(x)$ and an identification $p:f =_{\prod_{x:A} B(x)} g$ there are families of identifications $x:A \vdash \mathrm{compelim}(f, g, p)(x):f(x) =_{B(x)} g(x)$.

Proof

We simply define the dependent function $\mathrm{compelim}$ to be happly, which is inductively defined on identity types.

The next is the typal congruence rule for the uniqueness rule of dependent function types.

Theorem

For weak dependent product types with dependent function

$$\eta:\prod_{f:\prod_{x:A} B(x)} f =_{\prod_{x:A} B(x)} \lambda x:A.f(x)$$

given

• a type $A$

• a type family $x:A \vdash B(x)$

• dependent functions $f:\prod_{x:A} B(x)$ and $f':\prod_{x:A} B(x)$

• an identification $p:f =_{\prod_{x:A} B(x)} f'$,

there is a family of identifications

$$\mathrm{etaCong}_{\prod}(f, f', p):\mathrm{transport}(f, f', p)(\eta_{\prod_{x:A} B(x)}(f)) =_{f =_{\prod_{x:A} B(x)} \lambda x:A.f(x)} \eta_{\prod_{x:A} B(x)}(f')$$

Proof

We simply define $\mathrm{etaCong}_{\prod}(f, f', p)$ to be the dependent function application to identifications $\mathrm{apd}(\eta, f, f' p)$.

The next is the typal congruence rule for the introduction rule of function types. However, unlike the case for the other two rules, one needs dependent function extensionality.

Theorem

Assuming dependent function extensionality, given a type $A$ and a type family $x:A \vdash B(x)$, families of elements $x:A \vdash b(x):B(x)$ and $x:A \vdash b'(x):B(x)$, and families of identifications $x:A \vdash p(x):b(x) =_{B(x)} b'(x)$, there is an identification

$$\mathrm{congintro}_{x:A.p(x)}:\lambda (x:A).b(x) =_{\prod_{x:A} B(x)} \lambda (x:A).b'(x)$$

Proof

By the computation rule of weak dependent function types, there are families of identifications

$$x:A \vdash \beta_{\prod_{x:A} B(x)}^{x:A.b(x)}(b(x)):((\lambda x:A.b(x))(x) =_{B(x)} b(x)$$

$$x:A \vdash \beta_{\prod_{x:A} B(x)}^{x:A.b'(x)}(b'(x)):((\lambda x:A.b'(x))(x) =_{B(x)} b'(x)$$

Thus, there are families of identificaitons

$$x:A \vdash \beta_{\prod_{x:A} B(x)}^{x:A.b(x)}(b(x)) \bullet p(x) \bullet \beta_{\prod_{x:A} B(x)}^{x:A.b'(x)}(b'(x))^{-1}:(\lambda x:A.b(x))(x) =_{B(x)} (\lambda x:A.b'(x))(x)$$

and by $\lambda$-abstraction, one gets the dependent function

$$\lambda (x:A).\beta_{\prod_{x:A} B(x)}^{x:A.b(x)}(b(x)) \bullet p(x) \bullet \beta_{\prod_{x:A} B(x)}^{x:A.b'(x)}(b'(x))^{-1}:(\lambda x:A.b(x))(x) =_{B(x)} (\lambda x:A.b'(x))(x)$$

By dependent function extensionality, there is an equivalence of types

$$\mathrm{ext}_{\prod_{x:A} B(x)}^{-1}:(\lambda x:A.b(x)) =_{\prod_{x:A} B(x)} (\lambda x:A.b'(x)) \simeq \prod_{x:A} \mathrm{Id}_{B(x)}((\lambda x:A.b(x))(x), (\lambda x:A.b'(x))(x))$$

which yields an identification

$$\mathrm{ext}_{\prod_{x:A} B(x)}^{-1}^{-1}(\lambda (x:A).\beta_{\prod_{x:A} B(x)}^{x:A.b(x)}(b(x)) \bullet p(x) \bullet \beta_{\prod_{x:A} B(x)}^{x:A.b'(x)}(b'(x))^{-1}):(\lambda x:A.b(x)) =_{\prod_{x:A} B(x)} (\lambda x:A.b'(x))$$

We define

$$\mathrm{congintro}_{x:A.p(x)} \coloneqq \mathrm{ext}_{\prod_{x:A} B(x)}^{-1}^{-1}(\lambda (x:A).\beta_{\prod_{x:A} B(x)}^{x:A.b(x)}(b(x)) \bullet p(x) \bullet \beta_{\prod_{x:A} B(x)}^{x:A.b'(x)}(b'(x))^{-1}):(\lambda x:A.b(x)) =_{\prod_{x:A} B(x)} (\lambda x:A.b'(x))$$

Application in logic

In logic, dependent functions types express universal quantifications. More precisely, for $x:A \vdash \phi(x)$ a predicate on a type $A$, under propositions as types the universal quantification $\forall x:A.\phi(x)$ is the dependent product type $\prod_{x:A} \phi(x)$ (or rather the bracket type of that if one wishes to force this to be of type $Prop$ again ).

Graph of a dependent function

Given a type $A$ and a type family $x:A \vdash B(x)$, there is a function

$$\mathrm{graph}:\left(\prod_{x:A} B(x)\right) \to \left(A \to \sum_{x:A} B(x)\right)$$

which takes a dependent function $f:\prod_{x:A} B(x)$ and returns the graph of a dependent function

$$\mathrm{graph}(f):A \to \sum_{x:A} B(x)$$

defined by $\mathrm{graph}(f)(x) \equiv (x, f(x))$ for all $x:A$. As a dependent anafunction the graph of the dependent function is represented by the identity type family

$$x:A, y:B(x) \vdash f(x) =_{B(x)} y$$

Relation to sections

A family of type $x:A \vdash B(x)$ is equivalently a type $B$ with a function $f:B \to A$. Then each $B(x)$ is defined as the fiber of $f$ at element $x:A$. Then the dependent product of a function is defined as the dependent product type

$$\Pi_{A, B}(f) \coloneqq \prod_{x:A} \sum_{y:B} f(y) =_A x$$

or equivalently, due to the type theoretic axiom of choice, as the dependent sum type

$$\Pi_{A, B}(f) \coloneqq \sum_{g:A \to B} \prod_{x:A} f(g(x)) =_A x$$

which says that $g$ is a section of $f$. One could eliminate the use of the dependent product type entirely by using the definition of dependent product type from function types:

$$\Pi_{A, B}(f) \coloneqq \sum_{g:A \to B} \sum_{h:A \to \sum_{x:A} f(g(x)) =_A x} \mathrm{pr}_{\sum}^{A} \circ h =_{A \to A} \mathrm{id}_A$$

Categorical interpretation

In categorical semantics, the dependent product types are relative right adjoints to context extension in comprehension categories.

There is also another interpretation in category theory of the dependent product type over $x:A \vdash B(x) \; \mathrm{type}$ as the terminal $A$-indexed wide span under $B(x)$, the object $\prod_{x:A} B(x)$ with a family of morphisms

$$\pi_A(x):\left(\prod_{x:A} B(x)\right) \to B(x)$$

such that for any other object $C$ with a family of morphisms $f(x):C \to B(x)$, there exists a unique morphism $u_C:C \to \prod_{x:A} B(x)$ such that

$$\pi_A(x) \circ u_C = f(x)$$

Related concepts

• dependent product
• dependent sum type
• dependent pullback type
• function type
• dependent sequence type
• dependent extension type

References

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

• Nicola Gambino, Lectures on dependent type theory (pdf)

Another textbook account could be found in section 2.1 of:

• Egbert Rijke, Introduction to Homotopy Type Theory, Cambridge Studies in Advanced Mathematics, Cambridge University Press (arXiv:2212.11082)

as well as sections 1.4 and 2.9 of:

• Univalent Foundations Project, Homotopy Type Theory – Univalent Foundations of Mathematics (2013)

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:

• Michael Lindgren, Dependent products as relative adjoints, Stockholm (2021) [pdf]