Implementing mathematics with the Nuprl proof development system
Implementing mathematics with the Nuprl proof development system
Programming in Martin-Lo¨f's type theory: an introduction
Programming in Martin-Lo¨f's type theory: an introduction
Handbook of logic in computer science
Normalization and Partial Evaluation
Applied Semantics, International Summer School, APPSEM 2000, Caminha, Portugal, September 9-15, 2000, Advanced Lectures
Extensional Equality in Intensional Type Theory
LICS '99 Proceedings of the 14th Annual IEEE Symposium on Logic in Computer Science
System F with type equality coercions
TLDI '07 Proceedings of the 2007 ACM SIGPLAN international workshop on Types in languages design and implementation
PLPV '07 Proceedings of the 2007 workshop on Programming languages meets program verification
The identity type weak factorisation system
Theoretical Computer Science
Weak ω-Categories from Intensional Type Theory
TLCA '09 Proceedings of the 9th International Conference on Typed Lambda Calculi and Applications
Two-dimensional models of type theory
Mathematical Structures in Computer Science
Univalent foundations of mathematics
WoLLIC'11 Proceedings of the 18th international conference on Logic, language, information and computation
TLDI '12 Proceedings of the 8th ACM SIGPLAN workshop on Types in language design and implementation
A Computational Interpretation of Parametricity
LICS '12 Proceedings of the 2012 27th Annual IEEE/ACM Symposium on Logic in Computer Science
Equality proofs and deferred type errors: a compiler pearl
Proceedings of the 17th ACM SIGPLAN international conference on Functional programming
Proceedings of the 18th ACM SIGPLAN international conference on Functional programming
System FC with explicit kind equality
Proceedings of the 18th ACM SIGPLAN international conference on Functional programming
Calculating the Fundamental Group of the Circle in Homotopy Type Theory
LICS '13 Proceedings of the 2013 28th Annual ACM/IEEE Symposium on Logic in Computer Science
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Higher-dimensional dependent type theory enriches conventional one-dimensional dependent type theory with additional structure expressing equivalence of elements of a type. This structure may be employed in a variety of ways to capture rather coarse identifications of elements, such as a universe of sets considered modulo isomorphism. Equivalence must be respected by all families of types and terms, as witnessed computationally by a type-generic program. Higher-dimensional type theory has applications to code reuse for dependently typed programming, and to the formalization of mathematics. In this paper, we develop a novel judgemental formulation of a two-dimensional type theory, which enjoys a canonicity property: a closed term of boolean type is definitionally equal to true or false. Canonicity is a necessary condition for a computational interpretation of type theory as a programming language, and does not hold for existing axiomatic presentations of higher-dimensional type theory. The method of proof is a generalization of the NuPRL semantics, interpreting types as syntactic groupoids rather than equivalence relations.