Modular correspondence between dependent type theories and categories including pretopoi and topoi
Mathematical Structures in Computer Science
Proceedings of the 3rd workshop on Programming languages meets program verification
A Modular Type-Checking Algorithm for Type Theory with Singleton Types and Proof Irrelevance
TLCA '09 Proceedings of the 9th International Conference on Typed Lambda Calculi and Applications
Refinement Types as Proof Irrelevance
TLCA '09 Proceedings of the 9th International Conference on Typed Lambda Calculi and Applications
Irrelevance in type theory with a heterogeneous equality judgement
FOSSACS'11/ETAPS'11 Proceedings of the 14th international conference on Foundations of software science and computational structures: part of the joint European conferences on theory and practice of software
Dependent session types via intuitionistic linear type theory
Proceedings of the 13th international ACM SIGPLAN symposium on Principles and practices of declarative programming
Towards concurrent type theory
TLDI '12 Proceedings of the 8th ACM SIGPLAN workshop on Types in language design and implementation
Proof-Carrying code in a session-typed process calculus
CPP'11 Proceedings of the First international conference on Certified Programs and Proofs
Inductive Types in Homotopy Type Theory
LICS '12 Proceedings of the 2012 27th Annual IEEE/ACM Symposium on Logic in Computer Science
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Image factorizations in regular categories are stable under pullbacks, so they model a natural modal operator in dependent type theory. This unary type constructor [A] has turned up previously in a syntactic form as a way of erasing computational content, and formalizing a notion of proof irrelevance. Indeed, semantically, the notion of a support is sometimes used as surrogate proposition asserting inhabitation of an indexed family. We give rules for bracket types in dependent type theory and provide complete semantics using regular categories. We show that dependent type theory with the unit type, strong extensional equality types, strong dependent sums, and bracket types is the internal type theory of regular categories, in the same way that the usual dependent type theory with dependent sums and products is the internal type theory of locallyCartesian closed categories. We also show how to interpret first-order logic in type theory with brackets, and we make use of the translation to compare type theory with logic. Specifically, we show that the propositions-as-types interpretation is complete with respect to a certain fragment of intuitionistic first-order logic, in the sense that a formula from the fragment is derivable in intuitionistic first-order logic if, and only if, its interpretation in dependent type theory is inhabited. As a consequence, a modified double-negation translation into type theory (without bracket types) is complete, in the same sense, for all of classical first-order logic.