Paper in proceedings, 2018

The univalence axiom expresses the principle of extensionality for dependent type theory. How-

ever, if we simply add the univalence axiom to type theory, then we lose the property of canonicity

— that every closed term computes to a canonical form. A computation becomes ‘stuck’ when it

reaches the point that it needs to evaluate a proof term that is an application of the univalence

axiom. So we wish to find a way to compute with the univalence axiom. While this problem has

been solved with the formulation of cubical type theory, where the computations are expressed us-

ing a nominal extension of lambda-calculus, it may be interesting to explore alternative solutions,

which do not require such an extension.

As a first step, we present here a system of propositional higher-order minimal logic (PHOML).

There are three kinds of typing judgement in PHOML. There are terms which inhabit types, which

are the simple types over Ω. There are proofs which inhabit propositions, which are the terms

of type Ω. The canonical propositions are those constructed from ⊥ by implication ⊃. Thirdly,

there are paths which inhabit equations M = A N , where M and N are terms of type A. There are

two ways to prove an equality: reflexivity, and propositional extensionality — logically equivalent

propositions are equal. This system allows for some definitional equalities that are not present

in cubical type theory, namely that transport along the trivial path is identity.

We present a call-by-name reduction relation for this system, and prove that the system

satisfies canonicity: every closed typable term head-reduces to a canonical form. This work has

been formalised in Agda.

ever, if we simply add the univalence axiom to type theory, then we lose the property of canonicity

— that every closed term computes to a canonical form. A computation becomes ‘stuck’ when it

reaches the point that it needs to evaluate a proof term that is an application of the univalence

axiom. So we wish to find a way to compute with the univalence axiom. While this problem has

been solved with the formulation of cubical type theory, where the computations are expressed us-

ing a nominal extension of lambda-calculus, it may be interesting to explore alternative solutions,

which do not require such an extension.

As a first step, we present here a system of propositional higher-order minimal logic (PHOML).

There are three kinds of typing judgement in PHOML. There are terms which inhabit types, which

are the simple types over Ω. There are proofs which inhabit propositions, which are the terms

of type Ω. The canonical propositions are those constructed from ⊥ by implication ⊃. Thirdly,

there are paths which inhabit equations M = A N , where M and N are terms of type A. There are

two ways to prove an equality: reflexivity, and propositional extensionality — logically equivalent

propositions are equal. This system allows for some definitional equalities that are not present

in cubical type theory, namely that transport along the trivial path is identity.

We present a call-by-name reduction relation for this system, and prove that the system

satisfies canonicity: every closed typable term head-reduces to a canonical form. This work has

been formalised in Agda.

canonicity

type theory

univalence

Chalmers, Computer Science and Engineering (Chalmers), Computing Science (Chalmers)

University of Bergen

Chalmers, Computer Science and Engineering (Chalmers), Computing Science (Chalmers)

22nd International Conference on Types for Proofs and Programs, TYPES 2016

Novi Sad, Serbia,

Novi Sad, Serbia,

Algebra and Logic

Computer Science

Information and Communication Technology

Basic sciences

10.4230/LIPIcs.TYPES.2016.3

9783959770651