Questions tagged [dependent-type]
An overlapping feature of type theory and type systems.
8
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2answers
161 views
Intuition Behind Strict Positivity?
I'm wondering if someone can give me the intuition behind why strict positivity of inductive data types guarantees strong normalization.
To be clear, I see how having negative occurrences leads to ...
5
votes
1answer
111 views
What will go wrong if a recursive record type has a negative eta rule?
In the context of Agda like dependent type theory:
This short paper https://jesper.sikanda.be/files/vectors-are-records-too.pdf says some inductive type can be seen as records, for example ...
0
votes
1answer
115 views
First-order multi arity functions in dependent type?
(cross posted from Reddit https://www.reddit.com/r/dependent_types/comments/b1ts8b/firstorder_multi_arity_functions_in_dependent_type/?
Take Agda for example, functions of multi arity is "encoded" as ...
5
votes
0answers
113 views
An axiom for John Major's Equality
In the the standard library of Coq, there is the axiom:
Axiom JMeq_eq : forall (A:Type) (x y:A), JMeq x y -> x = y.
Why isn't it provable? Can it be reduced ...
1
vote
1answer
103 views
Dependent C-style types with subtyping rule
I'm looking for previous work regarding an extension of a C-style type system in which types may have constraints and have a defined subtyping rule. In particular, I'm interested in defining algebra-...
0
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1answer
99 views
Definitional equality of recursive function definition by “infinite unfolding”
The context is checking definitional equality in dependent type theory implementations.
Consider in Coq
...
3
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0answers
70 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 ...
4
votes
0answers
128 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 ...
6
votes
3answers
280 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
votes
1answer
110 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
votes
1answer
133 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:
...
5
votes
2answers
307 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 \...
11
votes
1answer
416 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 ...
8
votes
1answer
161 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
votes
1answer
248 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 ...
1
vote
2answers
208 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 ...
2
votes
0answers
131 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
votes
2answers
197 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 ...
15
votes
0answers
176 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
votes
2answers
526 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
votes
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 ...
6
votes
2answers
298 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 ...
6
votes
3answers
242 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
votes
0answers
130 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 ...
3
votes
2answers
305 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
votes
1answer
126 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
votes
1answer
215 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
votes
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 ...
1
vote
1answer
141 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
votes
1answer
126 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
votes
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
votes
1answer
335 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 ...
9
votes
1answer
355 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
votes
1answer
170 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
votes
1answer
330 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
votes
3answers
578 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
votes
2answers
318 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
votes
1answer
117 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 ...
17
votes
1answer
853 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:
...
10
votes
1answer
246 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
votes
1answer
124 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 $G:{\bf C}\longrightarrow{\...
2
votes
1answer
344 views
Dependent Sums and Products
I'm trying to understand the connections between a few different concepts fundamental to dependent type theory.
Dependent functions ($\Pi$-types)
Including non-dependent functions ($A \rightarrow B$)...
2
votes
2answers
196 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
votes
2answers
612 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
votes
1answer
324 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 $\mathsf{Set}$. ...
5
votes
1answer
228 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" ...
6
votes
2answers
401 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
votes
1answer
233 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?
34
votes
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
votes
2answers
371 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 ...
