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arXiv · 1904.08562

Cubical Syntax for Reflection-Free Extensional Equality

Jonathan Sterling·Carlo Angiuli·Daniel Gratzer
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Abstract

We contribute XTT, a cubical reconstruction of Observational Type Theory which extends Martin-L\"of's intensional type theory with a dependent equality type that enjoys function extensionality and a judgmental version of the unicity of identity types principle (UIP): any two elements of the same equality type are judgmentally equal. Moreover, we conjecture that the typing relation can be decided in a practical way. In this paper, we establish an algebraic canonicity theorem using a novel cubical extension (independently proposed by Awodey) of the logical families or categorical gluing argument inspired by Coquand and Shulman: every closed element of boolean type is derivably equal to either 'true' or 'false'.

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Jonathan Sterling, Carlo Angiuli, Daniel Gratzer. 2019-04-18. Cubical Syntax for Reflection-Free Extensional Equality. https://doi.org/10.4230/lipics.fscd.2019.31

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