Questions tagged [dependent-type]

An overlapping feature of type theory and type systems.

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Is just one W-type enough for formalizing mathematics?

We work in intensional Martin-Löf type theory with $0$, $1$, $2$, $\Pi$, $\Sigma$, $W$ and a cumulative hierarchy of universes. Suppose our goal is to formalize constructive mathematics. Now if we ...
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Set-theoretic encoding of functions in type theory

Functions usually get encoded in set theory as follows. A function $A\to B$ is a subset $f\subset A\times B$ such that $\pi_1:f\to A$ is a bijection. In type theory to give a function $A\to B$ is to ...
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Dependent eliminator for empty type in intensional Martin-Löf type theory

In calculus of inductive constructions you can just say that the empty type is the type with no constructors and it automatically builds the dependent eliminator. But let's say I'm setting up ...
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Explicit type system with infinite non-cumulative universe hierarchy

Is there an open-source proof assistant or at least an explicit set of rules written down somewhere for a type system with an infinite non-cumulative universe hierarchy and unique typing? I want to ...
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Complexity of finding typing derivation trees

In type theories where type checking is decidable do we have estimates for how much time/space it takes to find a typing derivation tree of a valid typing judgment? Do any published references do this ...
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$\mathbb{N}$ in intensional MLTT with judgmentally commutative $+$ and $\times$

Is there a way to implement natural numbers in intensional Martin-Löf type theory so that addition and multiplication is judgmentally commutative?
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Model of MLTT with $\eta$ rule where function extensionality fails

Consider intensional Martin-Löf type theory with judgmental $\eta$ rule for dependent product types. Is there a model of it where function extensionality fails?
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Model of homotopy type theory where propositional & judgmental equality coincide for closed terms

In intensional Martin-Löf type theory we can prove the metatheorem that two closed terms are propositionally equal iff they are judgmentally equal. Is there a non-empty model of homotopy type theory ...
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71 views

Normal term of double negation of W-type

Consider the intensional Martin-Löf type theory without axiom of choice or the law of excluded middle. Let $A:U_0$ be a type and $B:A\to U_0$ be a function such that $\Sigma_{a:A}(B(a)\to 0)$is ...
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1answer
264 views

Defining inductive types in intensional type theory purely in terms of type-theoretic data

To define a (non-indexed) W-type all we need is a type $A:U$ and a function $B:A\to U$ and we get a type $W_{a:A}B(a)$. To check that this definition is valid we only need to check that the ...
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Defining binary natural numbers without quotient types

Let's say we work in a dependent type theory with W-types and we want to have a type for binary natural numbers. We don't want to add quotient types or higher inductive types to the system. How to ...
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1answer
207 views

Impredicativity + large eliminations (with no excluded middle) in Coq

It is known that impredicativity + large eliminations + excluded middle is inconsistent. Prop is impredicative and consistent with excluded middle, but does not ...
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138 views

What logic do refinement types correspond to?

I'm interested in applicability of refinement types to theorem-proving hence the questions about their logical expressiveness. Let's say, we have a type system which corresponds to some logic ...
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106 views

“Interesting” categories whose internal logic is a dependent-linear type theory

Dependent-linear type theories may be a functional programmer's dream, but is it categorically interesting, i.e. is it the internal language of an "interesting" category? By "...
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Intuition behind nested positivity and counterexamples

I'm looking at the nested positivity conditions for inductive types stated in the Coq manual. First off, are there any other references (not necessarily for Coq, but in dependent type theories ...
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Is there a known notion of “stochastic dependent pair”?

I came upon this when thinking about the semantics of probabilistic programs. Say you have a generative model N ~ Poisson() for n = 1:N X[i] ~ Normal() Then the ...
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Category theory lambda cube?

If simply typed lambda calculus corresponds to cartesian closed categories, what types of categories do other calculi in the lambda cube correspond to? https://en.m.wikipedia.org/wiki/Lambda_cube
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Would it be possible to derive `transp` natively from Path, Interval and typecase?

Assume for a moment that we extended Agda with an Interval and a Path type, but not transp (which is a primitive currently). I'm ...
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1answer
167 views

Defining finite sets inductively in a proof assistant?

To represent finite sets within coq, we either use something like ListSet, which are just definitions on top of list, or we ...
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Postulating self types in a proof assistant

Self types introduce two typing new rules (simplified): $ \frac{\Gamma \; \vdash \; t : [t/x]T}{\Gamma \; \vdash \; t : \iota x. T}$ I-self and $ \frac{\Gamma \; \vdash \; t : \iota x. T}{\Gamma \; \...
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1answer
196 views

Non-termination, strict positivity and free monads

Using the standard encoding of a free monad in Haskell and its fmap instance: ...
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1answer
126 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 ...
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142 views

Set:Set or Negative Inductives in a Total Language?

In total dependently typed languages, general recursion is forbidden, since this can allow for non-terimination. However, dependently typed language can still describe Turing-complete computations (...
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166 views

Implementations of Dependent Type Theory

I am trying to find a minimal implementation of dependent type theory that supports Pi Types (obviously) Modules containing records Inductive data types Universe Hierarchy A notion of equality ...
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140 views

Lack of atomic propositions in the Calculus of Constructions from ATTAPL textbook

I am working through the Dependent Types chapter from Advanced Topics in Types and Programming Languages (ATTAPL) by Benjamin Pierce et al. I am confused with the calculus presented Fig 2-7 (Calculus ...
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Proposition terms vs types in Coq

Consider the following div function written in Coq. It takes in a proof that the divider is non-zero. Definition div (n d:nat) (pf: ~(d = 0)) := n/d. Focus on <...
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PHOAS with extrinsic typing?

Parameterized Higher Order Abstract Syntax (PHOAS) is a representation of syntax trees that allows the host language's binding to be used to represent binding in the language being modelled, while ...
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1answer
254 views

Small kernel (i.e. proof-verifier) for Agda?

Proof-assistants usually include a lot of machinery that assists in the creation of proofs. The creation process may be unsound without risking the soundness of the proof-assistant if the alleged ...
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1answer
285 views

Fixed set of type constructors to simulate all intensional inductive families?

I'm wondering, are there small dependent calculi that can simulate a language with inductive families (that is, has a type isomorphic to each inductive family, at least as powerful of induction ...
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203 views

Structural equality of Pi Types with heterogeneous equality?

I'm trying to implement a proof of the following type: ...
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Formalization of dependent record types/kinds in MLTT or variant thereof?

Has anyone formalized Pollack's Dependently Typed Records in Type Theory ? Agda would be preferred, but anyone close to MLTT would work. Weaker versions would be fine too, i.e. what Luo dubs '...
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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 ...
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1answer
206 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 ...
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1answer
150 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 ...
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482 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 ...
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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-...
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138 views

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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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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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 ...
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345 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 ...
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1answer
204 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 ...
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1answer
202 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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365 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 \...
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600 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 ...
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1answer
191 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 ...
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440 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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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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What is the proof for the inconsistency of impredicativity + 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 ...
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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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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 ...