Questions tagged [dependent-type]
An overlapping feature of type theory and type systems.
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Definitional equality of recursive function definition by “infinite unfolding”
The context is checking definitional equality in dependent type theory implementations.
Consider in Coq
...
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57 views
Proof that CIC or Dybjer-style eliminators are strongly-normalizing?
Related to this question
I'm wondering, what is the standard technique for showing that dependent types with eliminators are strongly normalizing? I'm thinking something like the Calculus of ...
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0answers
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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 ...
6
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3answers
267 views
When a type is a value?
In functional programming and in the theoretical setting of the $\lambda$-calculus it is standard to consider a lambda abstraction $\lambda x.M$ as a value. In my understanding, the intuitive reason ...
5
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1answer
97 views
Strong Normalization of Extended Calculus of Constructions (CC with cumulative universes)
There are some proofs around to prove the strong normalization of the calculus of constructions (i.e. that all type systems in the lambda cube are strongly normalizing). I have analyzed the proof ...
6
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1answer
123 views
General Induction Principle
Let us suppose that we want to provide for each inductive type an axiom describing the associated elimination/induction principle. For example, given a definition for the naturals:
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5
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2answers
302 views
Preservation under Substitution with Telescopes
In the simply typed lambda calculus, one can show the following result, known as "preservation under substitution":
If $\Gamma \vdash v : \tau_1$ and $(x : \tau_1) \vdash t : \tau_2$,
then $\Gamma \...
9
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1answer
407 views
Proof techniques for showing that dependent type checking is decidable
I'm in a situation where I need to show that typechecking is decidable for a dependently-typed calculus I'm working on. So far, I've been able to prove that the system is strongly normalizing, and ...
7
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1answer
155 views
If the untyped language is terminating, can we still derive a contradiction from `Type : Type`?
Question
If a pure type system has a terminating proof language, can we have Type : Type at the logic level without causing paradoxes (i.e., without causing ...
7
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1answer
227 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 ...
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2answers
184 views
What are the limitations of dependent typing?
It seems dependent types can provide lots of desirable guarantees about program behavior. What kinds of program properties can they NOT guarantee (besides what's computationally infeasible)? Is there ...
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0answers
116 views
What is the proof for the inconsistency of impredicaitivity + excluded middle + large elimination in type theory
Why is the combination of
impredicativity + excluded middle + large elimination
inconsistent in dependent type theory?
My understanding of large elimination is I am doing large elimination if I am ...
8
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2answers
189 views
How can you build a coinductive memoization table for recursive functions over binary trees?
The StreamMemo library for Coq illustrates how to memoize a function f : nat -> A over the natural numbers. In particular when ...
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0answers
173 views
Using Dependent Type Theory for Categories that are not LCCC
I have recently been working with polynomial functors and monads based mostly on Gambino-Kock. There they define polynomial functors in a Locally Cartesian Closed Category (LCCC) and extensively use ...
15
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2answers
515 views
In the Hott book, are the most of the type formers redundant? And if so, why?
In chapter 1 and Appendix A of the Hott book, several primitive type families are presented (universe types, dependent function types, dependent pair types, Coproduct types, Empty Type, Unit type, ...
4
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0answers
116 views
Completeness of realizability semantics for higher-order type theory
In this answer I mention a paper by Geuvers in which he describes a class of models for a type theory $\lambda P_2$ which is a sub-system of the CoC and roughly corresponds to 2nd order predicate ...
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2answers
296 views
Explicit set of types and terms in MLTT
Whenever I read a presentation of MLTT, especially in the context of the correspondence of MLTT with LCCCs (eg. Seely's paper), they say "the type constructors/formation rules are..." and then list a ...
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3answers
229 views
Proving running time upper bounds for algorithms in dependent type theory
Proof assistants are a valuable tool for verifying
the correctness of proofs of mathematical theorems.
When dealing with proofs of correctness of algorithms,
one is not only interested on showing ...
6
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0answers
125 views
Relationship between Pataraia's theorem and inductive-recursive definitions?
Pataraia's fixed point theorem gives a constructive proof of the fact that if you have a monotone function $f$ on a DCPO, then it has a least fixed point. I've frequently used this fixed point theorem ...
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2answers
283 views
Motivation for Dependent Type
By the Curry-Howard Isomorphism we view propositions (or theorems if true/inhabited) as types.
But take the type $\mathbb{N} \implies \mathbb{N} \implies \mathbb{N} $.
We have as witnesses to this ...
6
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1answer
121 views
How to mechanically derive the recursor of a type from its constructors?
In Martin-Löf Dependent Type Theory a type is commonly prescribed by how to construct its canonical terms and how to show that its canonical terms are definitionally equal. This means that the ...
6
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1answer
212 views
Type checking, Hypothetical judgments, meaning explanations and computational type theory
We say that a system is a computational type theory if it is a type theory defined by not a bunch of inference rules, but some sort of Martin-Löfian meaning explanations (e.g. in the sense of NuPRL). ...
4
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1answer
239 views
Is it possible to verify a typechecker for a total dependently-typed language in that language's logic?
I understand the diagonalization argument against implementing an eval function in a total language, and that typechecking in a dependently typed language requires ...
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1answer
135 views
What's the effect of imposing the following restriction on inductive type families?
Let a simple expression be either:
A free variable
A data constructor of an inductive type family, applied to 0 or more simple expressions
What would be the effect of imposing the following ...
4
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1answer
125 views
What is a canonical term of $\text{Id}_A(x,y)$ if $x$ is not jugdmentally identical to $y$?
In the context of constructive type theory, a term inhabiting some type is said to be in canonical form if it is explicitly built up using the constructors of that type.
Particularly, the only ...
8
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1answer
1k views
Dependent Types and Compile Time Types
I am trying to grok dependent types, and there's something that I find unclear.
In C++, templates can have non-type (template) parameters. The values of these parameters have to be specified at ...
6
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1answer
313 views
Examples of Universe inconsistency in normal use of dependent types
In dependent types, Type : Type results in inconsistency (Girard's or Hurken's paradox).
Are there examples of universe inconsistency (where assuming ...
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1answer
354 views
PiSigma: why does 'unfold' bind a variable?
I'm trying to understand the paper ΠΣ: Dependent Types without the Sugar by implementing an interpreter and type checker for the language. In doing so, I've seen that the ...
7
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1answer
166 views
Is there a theory of overloading types?
There is a sound theory of overloading operators and functions realized by type classes in Haskell, and to rougher extent by traits in Rust, etc.
In mathematics however, there are many situations ...
12
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1answer
315 views
Dependent types over Church-encoded type in PTS/CoC
I'm experimenting with pure type systems in Barendregt's lambda cube, specifically with the most powerfull one, the Calculus of Constructions. This system has sorts ...
13
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3answers
569 views
What are the negative consequences of extending CIC with axioms?
Is it true that adding axioms to the CIC might have negative influences in the computational content of definitions and theorems? I understand that, in the theory's normal behavior, any closed term ...
13
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2answers
312 views
Church-Rosser property for dependently typed lambda calculus?
It is well-known that the Church-Rosser property holds for $\beta \eta$-reduction in simply-typed lambda calculus. This implies that the calculus is consistent, in the sense that not all equations ...
4
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1answer
113 views
Rendering of type-level computation
Programming languages with dependent types and/or higher-kinded types feature what might be called compile-time computation at the type-level. This is usually defined as follows (I'm omitting some ...
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1answer
822 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:
...
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1answer
245 views
Reference for the fact that (0=1) implies false requires a universe in MLTT
It's a fairly well-known fact that deriving a contradiction from a disequality (for example, $(0=1) \to \bot$) in Martin-Loef type theory requires a universe.
The proof is also fairly ...
9
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1answer
122 views
Relating univalence for a theory of cateogries to the skeleton concept
Say I work in homotopy type theory and my sole objects of study are conventional categories.
Equivalences here are given by functors $F:{\bf D}\longrightarrow{\bf C}$ and $G:{\bf C}\longrightarrow{\...
2
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1answer
325 views
Dependent Sums and Products
I'm trying to understand the connections between a few different concepts fundamental to dependent type theory.
Dependent functions ($\Pi$-types)
Including non-dependent functions ($A \rightarrow B$)...
2
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2answers
192 views
How to translate the axiom schema of induction by Curry-Howard?
I'm trying to understand the Curry-Howard correspondence. I am comfortable with it for propositional logic, but get confused when $\forall, \exists$ quantifiers come in the picture.
The axiom schema ...
18
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2answers
592 views
Why an infinite type hierarchy?
Coq, Agda, and Idris have an infinite type hierarchy (Type 1 : Type 2 : Type 3 : ...). But why not do it instead like λC, the system in the lambda cube that's closest to the calculus of constructions, ...
4
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1answer
317 views
Rules about Prop and Set in UTT
In Luo's UTT (type theory which is used in Agda, Idris, and other dependently typed programming languages), there're are two rules for $\Pi$ types. One for $\mathsf{Prop}$ and one for $\mathsf{Set}$. ...
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1answer
226 views
What does consistency mean for “computational theories” corresponding to inductive types?
I am currently reading the book by Luo on computation and reasoning. In the book he contrasts inductive types considered as computational theories with axiomatic theories widespread in "standard" ...
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2answers
390 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 ...
9
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1answer
225 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?
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3answers
3k 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, ...
3
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2answers
360 views
Good description of Calculus of Inductive Construction
I want to learn more about Calculus of Inductive Constructions. What can you recommend to read on this topic? All the materials which I found are either in French or too basic (the Coq'Art book).
The ...
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2answers
160 views
How to use Prop from UTT in Agda
In Ulf Norell's thesis he mentions that Agda is based on Luo's UTT. However, I can't find a way to use Prop there. Is there any way to do so?
4
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2answers
141 views
Well-formedness condition for inductive types
I work on implementing a simple dependently typed language. I want to implement inductive types there. However, I want them to be well formed. From what I've seen in Coq not all types are acceptable. ...
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2answers
632 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 ...
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2answers
1k views
Is compiler for dependent type much harder than an intepreter?
I have been learning something about implementing dependent types, like this tutorial, but most of them is implementing interpreters. My question is, it seems that implementing a compiler for ...
4
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2answers
188 views
dependent types and higher-order logic applied in the realm of DSLs
(this is a beginner question and English is not my first language)
I am searching for references on using theorem proves based on dependent type theory (or Martin Löf type-theory) and higher-order ...
