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Questions tagged [dependent-types]

An overlapping feature of type theory and type systems.

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Why does Coq include let-expressions in its core language

Coq includes let-expressions in its core language. We can translate let-expressions to applications like this: let x : t = v in b ~> (\(x:t). b) v I understand ...
49 views

Calculus of constructions, type-in-type and recursion

Does adding type-in-type to the calculus of constructions lead to (general) recursion? Such that one can write the Y combinator.
135 views

What does canonicity property mean in Type Theory?

The "Computational Component" section of the Type Theory - Wikipedia (as well as a few papers about cubical type theory and 2d type theory) talk about canonicity property. Would you please explain ...
79 views

Difference between computation in proposition proof and definitional computation?

As stated in equality at nLab, "computational equality" is about computational steps which take for example, $s(s(0))+ s(0)$ to $s(s(s(0)))$ and it acts exactly and can be considered same as ...
61 views

Why values can not be replaced with their extensionally equal values in an intensional system?

Thomas Streicher states in Investigations into Intensional Type Theory(§Introduction p.5) that: Although in Intensional constructive set theory (Intensional Type Theory) one can do most of the ...
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Definition of extensional and propositional equality in Martin-Lof extensional type theory

Martin Hofmann states in Extensional Concepts in Intensional Type Theory (§1.1 p.[4-5]) that: A similar situation occurs in extensional Martin-Lof type theory where propositional and definitional ...
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Definitional equality of two propositions about propositional equality

Martin Hofmann states in Extensional Concepts in Intensional Type Theory (§1.1 p.3) that: It is important that definition equality is pervasive so if M and N are definitionally equal then P(M) is ...
49 views

What untyped term inhabits induction on natural numbers in CoC?

Induction on Church-encoded natural numbers (which I will call indNat) can not be defined within the Calculus of Constructions. If we assumed ...
64 views

Propositional truncation of excluded middle

It is clear to me that it should be impossible to prove : exclMidl = isProp A → ((A) ⊎ (¬ A)) Because it would give deciding oracle for every Proposition. My ...
44 views

Does dependent type checkers need to store lambda parameter type in their core language?

"Core language" refers to the exported well-typed terms that can be evaluated (or reduced). In the core language of MiniAgda, a dependently-typed language, the parameter type of a lambda is not ...
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Is it possible to interpret some Martin-Löf types as abelian monoids in such a way that any abelian monoid can be represented as a type?

For instance, I can interpret the unit type as the trivial monoid with one element. Non-dependent pairs $A \times B$ can be interpreted as the direct sum $A ⊕ B$ when $A$ and $B$ can both be ...
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308 views

Is there any difference between extensible records and dependent maps

In a typed setting, records can be thought of as a map from field to type. If there is a well-typed record merge operation (which allows overlapping fields), is there any real difference between the ...
283 views

About the Identity function in Agda

I've defined the identity function in Agda as follows: idd : (∀ {ℓ} {A: Set ℓ}) → A → A idd a = a I want to ask you if the ...
682 views

Examples of Dependent Types

I'm gathering examples from everything that I've read about Dependent Type Theory mainly from Dependent types at Work paper. This is my list so far of some dependent types (with abbreviations of ...