nLab extensional type theory

Idea

Extensional type theory denotes the flavor of type theory in which identity types satisfy the reflection rule, saying that if two terms are typally equal then they are also judgmentally equal.

In particular, this implies that all identity types are propositions / of h-level 1, and thus equivalently that all types are required to be h-sets. Therefore, extensional type theory is a set-level type theory, and hence a form of set-level foundations. However, there are other set-level type theories, such as those obtained by adding UIP as an axiom.

Note: For a while, the nLab incorrectly used “extensional type theory” to refer to what we now call set-level type theory. If you encounter uses of this sort, please correct them.

Extensional type theory is poorly behaved metatheoretically, and very difficult to implement in a proof assistant. However, it is sometimes more convenient to work with informally, and there are conservativity theorems relating it to other set-level type theories that are better-behaved.

Type theory which is not extensional is called intensional type theory.

Remark

The word “extensional” in type theory (even when applied to identity types) sometimes refers instead to the axiom of function extensionality. In general this property is orthogonal to the one considered here: function extensionality can hold or fail in both extensional and intensional type theory.

In particular, homotopy type theory is intensional in that identity types are crucially not demanded to be propositions, but function extensionality is often assumed (in terms of these intensional identity types, of course) — in particular, it follows from the univalence axiom. Indeed, univalence itself is arguably an extensionality principle for the universe (Hofmann and Streicher originally introduced it under the name “universe extensionality”), but it is inconsistent with “extensional type theory” in the sense considered here.

Remark

The origin of the names “extensional” and “intensional” is somewhat confusing. In fact they refer to the behavior of the definitional equality. The idea is that the identity type is always an “extensional” notion of equality (although it can be more or less extensional, depending on whether further extensionality principles like function extensionality and univalence hold). Thus, if the definitional equality coincides with the identity type, as it does under the reflection rule, the former is also extensional, and so we call the type theory “extensional” — while if the two equalities do not coincide, then the definitional equality has room to be more intensional than the identity type, and so we call the type theory “intensional”.

Definition

The Martin-Lof definition of identity types as an inductive type family makes them intensional. To make the type theory extensional, we add a rule that any inhabitant of an identity type $p:Id_A(x,y)$ induces a definitional equality between $x$ and $y$. In other words, we have an “equality reflection rule” of the form

$$\frac{p:Id_A(x,y)}{x\equiv y}$$

At first, this may appear to be only a “skeletality” assumption, since it does not assert explicitly that $p$ is reflexivity rather than a nontrivial loop. However, we can derive this with the induction rule for identity types. Consider the dependent type

$$(x:A),(y:A),(p:Id_A(x,y)) \;\vdash\; Id_{Id_A(x,y)}(p,refl(x)).$$

This is well-typed because the reflection rule applied to $p$ yields a judgmental equality $x\equiv y$, so that we have $refl(x):Id_A(x,y)$. Moreover, substituting $x$ for $y$ and $refl(x)$ for $p$ yields the type $Id_{Id_A(x,x)}(refl(x),refl(x))$, which is inhabited by $refl(refl(x))$.

Thus, by induction on identity, we have a term in the above type, witnessing a typal equality between $p$ and $refl(x)$. Finally, applying the equality reflection rule again, we get a definitional equality $p\equiv refl(x)$.

Remark

On the other hand, if in addition to the equality reflection rule we postulate that all equality proofs are definitionally equal to reflexivity, then we can derive the induction rule for identity types. Extensional type theory is often presented in this form.

A different, also equivalent, way of presenting extensional type theory is with a definitional eta-conversion rule for the identity types; see here.

Using the interval type

If the dependent type theory has an interval type $\mathbb{I}$ with judgmental computation rules for the point constructors, it suffices for the two point constructors $0:\mathbb{I}$ and $1:\mathbb{I}$ to be definitionally equal:

$$\frac{\Gamma \; \mathrm{ctx}}{\Gamma \vdash 0 \equiv 1:\mathbb{I}}$$

Proof

By the recursion principle of the interval type, for elements $x:A$, $y:A$, and $p:\mathrm{Id}_A(x, y)$, we have a function $\mathrm{rec}_\mathbb{I}^A(x, y, p):\mathbb{I} \to A$ such that $\mathrm{rec}_\mathbb{I}^A(x, y, p, 0) \equiv x$ and $\mathrm{rec}_\mathbb{I}^A(x, y, p, 1) \equiv y$. If we have $0 \equiv 1:\mathbb{I}$, then by the congruence rules of definitional equality we have

$$x \equiv \mathrm{rec}_\mathbb{I}^A(x, y, p, 0) \equiv \mathrm{rec}_\mathbb{I}^A(x, y, p, 1) \equiv y$$

which is precisely equality reflection.

Alternatively, one can add inference rules for definitional $\mathbb{I}$-localization:

$$\frac{\Gamma \vdash A \; \mathrm{type}}{\Gamma, p:\mathbb{I} \to A \vdash \mathrm{const}_{A, \mathbb{I}}^{-1}(p):A}$$

$$\frac{\Gamma \vdash A \; \mathrm{type}}{\Gamma, x:A \vdash \mathrm{const}_{A, \mathbb{I}}^{-1}(\lambda i:\mathbb{I}.x) \equiv x:A}$$

$$\frac{\Gamma \vdash A \; \mathrm{type}}{\Gamma, p:\mathbb{I} \to A \vdash \lambda i:\mathbb{I}.\mathrm{const}_{A, \mathbb{I}}^{-1}(p) \equiv p:\mathbb{I} \to A}$$

The usual notion of $\mathbb{I}$-localization, provable in any dependent type theory with an interval type, implies definitional $\mathbb{I}$-localization in extensional type theory by equality reflection. On the other hand, one can prove equality reflection from definitional $\mathbb{I}$-localization.

Theorem

Definitional $\mathbb{I}$-localization implies equality reflection.

Proof

Given $x:A$, $y:A$, and $p:x =_A y$, by the recursion principle of the interval type, we have a function $\mathrm{rec}_{\mathbb{I}, A}(x, y, p):\mathbb{I} \to A$. By definitional $\mathbb{I}$-localization and the recursion principle of the interval type, we have an element

$$\mathrm{const}_{A, \mathbb{I}}^{-1}(\mathrm{rec}_{\mathbb{I}, A}(x, y, p)):A$$

such that

$$\mathrm{const}_{A, \mathbb{I}}^{-1}(\mathrm{rec}_{\mathbb{I}, A}(x, y, p)) \equiv \mathrm{const}_{A, \mathbb{I}}(\mathrm{const}_{A, \mathbb{I}}^{-1}(\mathrm{rec}_{\mathbb{I}, A}(x, y, p)), 0) \equiv \mathrm{rec}_{\mathbb{I}, A}(x, y, p, 0) \equiv x$$

$$\mathrm{const}_{A, \mathbb{I}}^{-1}(\mathrm{rec}_{\mathbb{I}, A}(x, y, p)) \equiv \mathrm{const}_{A, \mathbb{I}}(\mathrm{const}_{A, \mathbb{I}}^{-1}(\mathrm{rec}_{\mathbb{I}, A}(x, y, p)), 1) \equiv \mathrm{rec}_{\mathbb{I}, A}(x, y, p, 1) \equiv y$$

hence, we can derive by the inference rules for judgmental equality that $x \equiv y$, which is precisely equality reflection.

Theorem

In the presence of the large recursion principle of the interval type, definitional isomorphism types imply equality reflection.

Proof

In the presence of definitional isomorphism types and the inductively defined identity types, transport can be defined as definitional isomorphism via the J-rule, since the identity function or identity equivalence is a definitional isomorphism and thus one can apply the J-rule to reflexivity to get the identity as a definitional isomorphism:

$$x:A, y:A, p:x =_A y \vdash \mathrm{ind}_{\mathrm{Id}}^{A, B(x) \cong B(y)}(p):B(x) \cong B(y)$$

$$x:A \vdash \mathrm{ind}_{\mathrm{Id}}^{A, B(x) \cong B(x)}(\mathrm{refl}_A(x)) \equiv \mathrm{toDefIso}(\chi:A.\chi, \chi:A.\chi):B(x) \cong B(x)$$

By large recursion of the interval, given two types $A$ and $B$ and an equivalence of types $e:A \simeq B$, one can construct an interval-indexed type family $(\mathrm{rec}_{\mathbb{I}}^{A, B, e}(x))_{x:\mathbb{I}}$ such that $\mathrm{rec}_{\mathbb{I}}^{A, B, e}(0) \equiv A$, $\mathrm{rec}_{\mathbb{I}}^{A, B, e}(1) \equiv B$, and $q:\mathrm{tr}_{\mathrm{rec}_{\mathbb{I}}^{A, B, e}}(p) =_{A \simeq B} e$. With definitional isomorphism types, the equivalences can be strictified to definitional isomorphisms by

$$\mathrm{ind}_{\mathrm{Id}}^{\mathbb{I}, \mathrm{rec}_{\mathbb{I}}^{A, B, e}(0) \cong \mathrm{rec}_{\mathbb{I}}^{A, B, e}(1)}(p):A \cong B$$

This implies that for all types $A$, the equivalence

$$\mathrm{const}_{\mathbb{I}, A} \coloneqq \lambda x:A.\lambda i:\mathbb{I}.x:A \simeq (\mathbb{I} \to A)$$

in $\mathbb{I}$-localization can be turned into the definitional isomorphism

$$\mathrm{ind}_{\mathrm{Id}}^{\mathbb{I}, \mathrm{rec}_{\mathbb{I}}^{A, \mathbb{I} \to A, \mathrm{const}_{\mathbb{I}, A}}(0) \cong \mathrm{rec}_{\mathbb{I}}^{A, \mathbb{I} \to A, \mathrm{const}_{\mathbb{I}, A}}(1)}(p):A \cong (\mathbb{I} \to A)$$

which is definitional $\mathbb{I}$-localization, and definitional $\mathbb{I}$-localization implies equality reflection as proven in the definitional $\mathbb{I}$-localization and extensional type theory articles.

Using dependent sum types of identity types

In extensional type theory, there are ways of defining certain dependent sum types of identity type as a negative type, which all result in equality reflection.

The idea is that using the inference rules for dependent sum types, the standard J-rule states that the dependent sum type $\sum_{x:A} \sum_{y:A} x =_A y$ is a positive copy of $A$ with respect to the diagonal function

$$\Delta_{A}(x) \coloneqq (x, (x, \mathrm{refl}_A(x))):\sum_{x:A} \sum_{y:A} x =_A y$$

However, there is also a negative version of copy types, whose elimination, computation, and uniqueness rules state that the diagonal function is a definitional isomorphism:

• Elimination rules for negative identity types:

$$\frac{\Gamma \vdash A \; \mathrm{type}}{\Gamma, z:\sum_{x:A} \sum_{y:A} x =_A y \vdash \Delta_A^{-1}(z):A}$$

• Computation rules for negative identity types:

$$\frac{\Gamma \vdash A \; \mathrm{type}}{\Gamma, x:A \vdash \Delta_A^{-1}(\Delta_{A}(x)) \equiv x:A}$$

• Uniqueness rules for negative identity types:

$$\frac{\Gamma \vdash A \; \mathrm{type}}{\Gamma, z:\sum_{x:A} \sum_{y:A} x =_A y \vdash \Delta_{A}(\Delta_A^{-1}(z)) \equiv z:\sum_{x:A} \sum_{y:A} x =_A y}$$

The uniqueness rules yield equality reflection. By the properties of judgmental equality and dependent sum types, from

$$(\Delta_A^{-1}(x, y, p), \Delta_A^{-1}(x, y, p), \mathrm{refl}_A(\Delta_A^{-1}(x, y, p))) \equiv (x, y, p)$$

one can derive $x \equiv \Delta_A^{-1}(x, y, p)$ and $y \equiv \Delta_A^{-1}(x, y, p)$ and thus $x \equiv y$.

Similarly, the Paulin-Mohring J-rule states that given $x:A$, the dependent sum type $\sum_{y:A} x =_A y$ is a positive unit type of $A$ with specified element

$$(x, \mathrm{refl}_A(x)):\sum_{y:A} x =_A y$$

However, there are also negative unit types, which states that $\sum_{y:A} x =_A y$ is a definitional singleton with judgmental center of contraction $(x, \mathrm{refl}_A(x))$

• Uniqueness rules for negative identity types:

$$\frac{\Gamma \vdash A \; \mathrm{type}}{\Gamma, x:A, z:\sum_{y:A} x =_A y \vdash (x, \mathrm{refl}_A(x)) \equiv z:\sum_{y:A} x =_A y}$$

Similarly to the case for the standard Martin-Lof identity types, the uniqueness rule yields equality reflection, because by the properties of judgmental equality and dependent sum types, one can derive $x \equiv y$ from $(x, \mathrm{refl}_A(x)) \equiv (y, p)$.

Properties

Decidability

Extensional Martin-Löf type theory does not have decidable type checking. See intensional type theory for more on this.

Theorem

In addition, extensional type theory does not have decidable judgmental equality.

Proof

Already, the equality reflection rule for the natural numbers type implies undecidability of judgmental equality in the dependent type theory.

$$\frac{\Gamma \vdash p:\mathrm{Id}_\mathbb{N}(x,y)}{\Gamma \vdash x \equiv y}$$

In order to show that, in an arbitrary context $\Gamma$, $0 \equiv 1$ in $\mathbb{N}$, we need to determine whether the empty type is a pointed type in $\Gamma$, which is undecidable in general.

Related concepts

• axiom K, axiom UIP

• intensional type theory, homotopy type theory

• observational type theory, higher observational type theory

Examples

• NuPRL

• Andromeda?

References

• Martin Hofmann, Extensional concepts in intensional type theory, Ph.D. thesis, University of Edinburgh, (1995) (ECS-LFCS-95-327, pdf)

• Ieke Moerdijk, E. Palmgren, Wellfounded trees in categories. Ann. Pure Appl. Logic, 104:189-218 (2000)

• Ieke Moerdijk, E. Palmgren, Type theories, toposes and constructive set theory: predicative aspects of AST, Ann. Pure Appl. Logic, 114:155-201, (2002)

• Jonathan Sterling, Carlo Angiuli, Daniel Gratzer, Cubical syntax for reflection-free extensional equality. In Herman Geuvers, editor, 4th International Conference on Formal Structures for Computation and Deduction (FSCD 2019), volume 131 of Leibniz International Proceedings in Informatics (LIPIcs), pages 31:1-31:25. (arXiv:1904.08562, doi:10.4230/LIPIcs.FSCD.2019.31)

• Jonathan Sterling, Carlo Angiuli, Daniel Gratzer, A Cubical Language for Bishop Sets, Logical Methods in Computer Science, 18 (1), 2022. (arXiv:2003.01491).

Extensional type theory is discussed in chapter 2 of:

• Carlo Angiuli, Daniel Gratzer, Principles of Dependent Type Theory (pdf)