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 propositionally equal then they are also definitionally 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 definitional 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 propositional 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.

Properties

Decidability

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

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.FCSD.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).