Propositions as [Types]

Abstract

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 locally cartesian 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.

Dependent session types via intuitionistic linear type theory

“…Of course, the system as we have presented so far has really no way of determining if it is really the case that a proof object is not used for its computational content. Luckily, proof theory can help us, with the concept of proof irrelevance [3,29].…”

Proof-Carrying Code in a Session-Typed Process Calculus

“…But we do not want to presuppose or "bake in" any particular analysis or strategy, but formulate the type theory so that we can seamlessly move between different specifications. This is what a modality for proof irrelevance [15,14,2] in the type theory allows us to do.…”

“…But we do not want to erase this requirement entirely, of course, just avoid sending a proof term. We can do this by using the type-theoretic concept of proof irrelevance [15,14,2]. Generally, a type [A] (pronounced "bracket A") is the type inhabited by proofs of A, all of which are identified.…”

Irrelevance in Type Theory with a Heterogeneous Equality Judgement

“…Awodey and Bauer [8] give a categorical treatment of proof irrelevance which is very similar to Pfenning and Reed's. However, they work in the setting of Extensional Type Theory with undecidable type checking, I could not directly use their results for this work.…”