Cubical Type Theory
ثبت نشده
چکیده
The equality on the inverval I is the equality in the free bounded distributive lattice on generators i, 1− i. The equality in the face lattice F is the one for the free distributive lattice on formal generators (i = 0), (i = 1) with the relation (i = 0) ∧ (i = 1) = 0. We have [(r ∨ s) = 1] = (r = 1) ∨ (s = 1) and [(r∧s) = 1] = (r = 1)∧ (s = 1). An irreducible element of this lattice is a face, a conjunction of elements (i = 0) and (j = 1) and any element is a disjunction of irreducible elements (unique up to the absorption law).
منابع مشابه
Cubical sets and the topological topos
Coquand’s cubical set model for homotopy type theory provides the basis for a computational interpretation of the univalence axiom and some higher inductive types, as implemented in the cubical proof assistant. This paper contributes to the understanding of this model. We make three contributions: 1. Johnstone’s topological topos was created to present the geometric realization of simplicial se...
متن کاملOn Higher Inductive Types in Cubical Type Theory
Cubical type theory provides a constructive justification to certain aspects of homotopy type theory such as Voevodsky’s univalence axiom. This makes many extensionality principles, like function and propositional extensionality, directly provable in the theory. This paper describes a constructive semantics, expressed in a presheaf topos with suitable structure inspired by cubical sets, of some...
متن کاملCubical Assemblies and Independence of the Propositional Resizing Axiom
We construct a model of cubical type theory with a univalent and impredicative universe in a category of cubical assemblies. We show that the cubical assembly model does not satisfy the propositional resizing axiom.
متن کاملHigher Inductive Types in Cubical Computational Type Theory
In homotopy type theory (HoTT), higher inductive types provide a means of defining and reasoning about higher-dimensional objects such as circles and tori. The formulation of a schema for such types remains a matter of current research. We investigate the question in the context of cubical type theory, where the homotopical structure implicit in HoTT is made explicit in the judgmental apparatus...
متن کاملA Model Of Type Theory In Cubical Sets With Connections
In this thesis we construct a new model of intensional type theory in the category of cubical sets with connections. To facilitate this we introduce the notion of a nice path object category, a simplification of the path object category axioms of [vdBG12] that nonetheless yields the full path object category structure. By defining cubical n-paths and contraction operators upon them we exhibit t...
متن کاملCanonicity for Cubical Type Theory
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 a...
متن کامل