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A Cubical Language for Bishop Sets (2003.01491v5)
Published 3 Mar 2020 in cs.LO and math.LO
Abstract: We present XTT, a version of Cartesian cubical type theory specialized for Bishop sets `a la Coquand, in which every type enjoys a definitional version of the uniqueness of identity proofs. Using cubical notions, XTT reconstructs many of the ideas underlying Observational Type Theory, a version of intensional type theory that supports function extensionality. We prove the canonicity property of XTT (that every closed boolean is definitionally equal to a constant) using Artin gluing.