Substitution up to isomorphism
Fundamenta Informaticae - Special issue: lambda calculus and type theory
TYPES '95 Selected papers from the International Workshop on Types for Proofs and Programs
Proceedings of the Symposium on Lambda-Calculus and Computer Science Theory
On the Interpretation of Type Theory in Locally Cartesian Closed Categories
CSL '94 Selected Papers from the 8th International Workshop on Computer Science Logic
Normalization by Evaluation for Martin-Löf Type Theory with One Universe
Electronic Notes in Theoretical Computer Science (ENTCS)
Normalization by Evaluation for Martin-Lof Type Theory with Typed Equality Judgements
LICS '07 Proceedings of the 22nd Annual IEEE Symposium on Logic in Computer Science
A formalisation of a dependently typed language as an inductive-recursive family
TYPES'06 Proceedings of the 2006 international conference on Types for proofs and programs
Electronic Notes in Theoretical Computer Science (ENTCS)
Electronic Notes in Theoretical Computer Science (ENTCS)
Categorial semantics of a solution to distributed dining philosophers problem
FAW'10 Proceedings of the 4th international conference on Frontiers in algorithmics
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This note is about work in progress on the topic of ''internal type theory'' where we investigate the internal formalization of the categorical metatheory of constructive type theory in (an extension of) itself. The basic notion is that of a category with families, a categorical notion of model of dependent type theory. We discuss how to formalize the notion of category with families inside type theory and how to build initial categories with families. Initial categories with families will be term models which play the role of canonical syntax for dependent type theory. We also discuss the formalization of the result that categories with finite limits give rise to categories with families. This yields a type-theoretic perspective on Curien's work on ''substitution up to isomorphism''. Our formalization is being carried out in the proof assistant Agda 2 developed at Chalmers.