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Questions tagged [dependent-type]

An overlapping feature of type theory and type systems.

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40 votes
4 answers
8k views

Why does Coq have Prop?

Coq has a type Prop of proof irrelevant propositions which are discarded during extraction. What are the reason for having this if we use Coq only for proofs. Prop is impredicative, so Prop : Prop, ...
Konstantin Solomatov's user avatar
48 votes
5 answers
7k views

What is the most intuitive dependent type theory I could learn?

I am interested in getting a really solid grasp on dependent typing. I've read most of TaPL and read (if not fully absorbed) 'Dependent Types' in ATTaPL. I've also read and skimmed a bunch of articles ...
John Salvatier's user avatar
21 votes
1 answer
2k views

Why it's impossible to declare an induction principle for Church numerals

Imagine, we defined natural numbers in dependently typed lambda calculus as Church numerals. They might be defined in the following way: ...
Konstantin Solomatov's user avatar
16 votes
2 answers
465 views

How to show that a type in a system with dependent types is not inhabited (i.e. formula not provable)?

For systems without dependent types, like Hindley-Milner type system, the types correspond to formulas of intuitionistic logic. There we know that its models are Heyting algebras, and in particular, ...
Petr's user avatar
  • 2,601
11 votes
2 answers
887 views

Formalizing the theory of finite sets in type theory

Most proof assistants have a formalization of the concept of "finite set". These formalizations, however, differ wildly (although one hopes that they are all essentially equivalent!). What I don't ...
Jacques Carette's user avatar
10 votes
1 answer
365 views

Example of where violation of strict positivity condition in inductive types leads to inconsistency

Most dependent typed systems have a strict positivity conditions for inductive types. Does anybody know an example where violation of the condition leads to inconsistency in the system?
Konstantin Solomatov's user avatar
8 votes
1 answer
829 views

Fixed points in dependent type theories

Most dependent type theories aim for some notion of correctness in two respects: The type system must be decidable. The type system must be consistent. e.g. $\forall \tau. \tau$ should not be ...
Jake's user avatar
  • 1,203
7 votes
2 answers
650 views

"Correctness" of type theory

How to "proof" that type theory is correct? Or at least explain that it's meaningful in some sense. In what extent is this a mathematical question and in what is a philosophical one? When type ...
Konstantin Solomatov's user avatar
4 votes
0 answers
169 views

Hereditary Substitution with Inductives and Eliminators?

I'm wondering, is there any existing work on hereditary substitution with inductive type families and dependent eliminators? In particular, normalizing the application of an eliminator to an ...
Joey Eremondi's user avatar
4 votes
1 answer
774 views

Dependent Sums and Products

I'm trying to understand the connections between a few different concepts fundamental to dependent type theory. Dependent functions ($\Pi$-types) Including non-dependent functions ($A \rightarrow B$)...
Joel Burget's user avatar
3 votes
1 answer
210 views

Type of induction principle for fixpoint types

To the Calculus of Constructions we could add a general fixpoint type constructor (accepting inconsistencies or assuming F is a ...
Labbekak's user avatar
  • 701
1 vote
1 answer
146 views

Formulation of Tarski-style universes in LF

Lately I've been asking questions on type theory on MSE, and I've been getting great answers, but I decided to give a try to this site and see if it will be helpful as well. I'm looking at this note ...
user175254's user avatar