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Abstract
Cubical type theory is an extension of Martin-Löf type theory recently proposed by Cohen, Coquand, Mörtberg, and the author which allows for direct manipulation of n-dimensional cubes and where Voevodsky’s Univalence Axiom is provable. In this paper we prove canonicity for cubical type theory: any natural number in a context build from only name variables is judgmentally equal to a numeral. To achieve this we formulate a typed and deterministic operational semantics and employ a computability argument adapted to a presheaf-like setting.
Highlights
Cubical type theory as presented in [7] is a dependent type theory which allows one to directly argue about n-dimensional cubes, and in which function extensionality and Voevodsky’s Univalence Axiom [15] are provable
This establishes that the judgmental equalities of the theory are sufficient to compute closed naturals to numerals; we have even given a deterministic reduction relation to do so
It should be noted that we could have worked with the corresponding untyped reduction relation A B and take I A B to mean I A = B, I A, I B, and A B etc
Summary
Cubical type theory as presented in [7] is a dependent type theory which allows one to directly argue about n-dimensional cubes, and in which function extensionality and Voevodsky’s Univalence Axiom [15] are provable. We devise an operational semantics given by a typed and deterministic weak-head reduction included in the judgmental equality of cubical type theory. We will follow Tait’s computability method [12,13] and devise computability predicates on typed expressions in name contexts and corresponding computability relations (to interpret judgmental equality) These computability predicates are indexed by the list of free name variables of the involved expressions and should be such that substitution induces a cubical set structure on them. This poses a major difficulty given that the reduction relation is in general not closed under name substitutions. The present paper is part of the author’s PhD thesis [11]
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