Questions tagged [type-theory]
Type structure is a syntactic discipline for enforcing levels of abstraction.
346
questions
2
votes
2answers
162 views
How does axiom K contradict univalence?
I have seen it claimed several times that axiom K is inconsistent with univalence (e.g. here and here), but I have never seen a proof sketch. Specifically, I'm curious about how this manifests in the ...
10
votes
3answers
377 views
Does the order of declarations in an inductive type matter?
I was wondering if the order of declarations of an inductive type can matter.
For example in Coq you can define Nat either by:
...
3
votes
2answers
426 views
Constructing terms of function types out of the empty type
If a function $f$ is understood as its graph, i.e. a set of pairs $\langle x,y\rangle$ where $x$ is input and $y$ is output, then the empty set $\emptyset$ is a valid function, and for any set $A$, we ...
1
vote
1answer
81 views
Are coproduct types redundant in presence of natural numbers and $\Sigma$-types?
In the homotopy type theory book section A.2.5 defines $\Sigma$-types, A.2.6 coproduct types and A.2.9 the natural numbers type.
If we already have $\Sigma$-types and the natural numbers type can we ...
43
votes
4answers
4k views
How would I go about learning the underlying theory of the Coq proof assistant?
I'm going over the course notes at CIS 500: Software Foundations and the exercises are a lot of fun. I'm only at the third exercise set but I would like to know more about what's happening when I use ...
7
votes
0answers
157 views
$\lambda$-definability and structure preserved by homomorphisms
I imagine there are some standard results that bear on this, but I'm having trouble finding a proof or refutation of it.
Some prelimary definitions.
A Henkin structure $A = (A^\cdot, ⟦\cdot⟧_A)$ for ...
7
votes
2answers
276 views
What's the categorical semantics of definitional equality?
The categorical semantics of a dependent type theory is normally described as a CwA/CwF/CompCat/etc. and in these models, we can talk about propositional equality by interpreting an 'identity type'. ...
6
votes
3answers
1k views
How does type theory change how one thinks about programming?
I have been dabbling in HoTT and I am convinced that dependent type theory is much more suitable than set theory for proof assistants.
Now, this made me wonder - how fundamental is Type Theory ...
2
votes
2answers
168 views
What is the point of the eliminator for the unit type?
In the HoTT book p. 436 A.2.8 the eliminator $\mathrm{ind}_{\mathbf{1}}$ for the unit type is described.
What is the point of it? What if you did not introduce it and instead just replaced all the ...
5
votes
1answer
104 views
$\eta$-reduction not locally confluent on well-typed terms
This paper says: "In the presence of a unit type, $\eta$-reduction is not even locally confluent on well-typed terms [20]."
[20] is a reference to a 300-page book with no further details and ...
2
votes
0answers
67 views
Reference request: characterisation of simultaneous substitution
For simply typed λ-calculus, a simultaneous substitution from $\Gamma$ to $\Delta$ is concretely a type-preserving map from variables in $\Delta$ to terms in $\Gamma$. See, for example, Programming ...
8
votes
0answers
152 views
Constructive Strong Normalization of the Extended Calculus of Constructions
The extended calculus of constructions (ECC) is basically the calculus of constructions with cumulative universes. I use the definition which Zhaohui Luo used in his PhD theses which contained a proof ...
3
votes
1answer
133 views
2
votes
1answer
97 views
External failure of law of excluded middle in Martin-Löf type theory
Is there an explicit type $T$ in Martin-Löf type theory such that $(T\to \mathbf{0})\to\mathbf{0}$ has an explicit closed term and $T$ can be shown externally to not have closed terms?
11
votes
1answer
433 views
Is there a good notion of non-termination and halting proofs in type theory?
Constructive type theory with its basic interpretation under the curry howard correspondence consists only of total, computable functions. In the literature, some has been said on using "computational ...
0
votes
0answers
81 views
CubicalTT: successor of add proof?
Lecture 2 of the cubical type theory lectures provide a proof of
(suc a) + b = suc (a + b):
...
2
votes
1answer
156 views
Alternatives to Normalization by Evaluation
Reading about lambda calculus I got the impression that normalization is evaluation.
So I don't understand what is meant by Normalization by Evaluation (used e.g. in several publications of A. Abel).
...
6
votes
1answer
180 views
What are the pros and cons for type cases in dependent type theories?
Pattern matching on $\cal U$ is allowed in XTT and Idris2 (for unerased types), and that implies the injectivity of type constructors (that's just my intuition, though -- I also wonder how do I prove ...
1
vote
0answers
69 views
MLTT/MiniTT: why do normal forms of sum types carry environments?
I am learning how to implement MiniTT: a simple type theoretic language, which is a dependently typed language with sum types, mutual recursive/inductive definitions and a universe of small types.
A ...
9
votes
4answers
919 views
Explaining monad transformers in categorical terms
Most resource regarding categorical notions in programming describe monads, but I've never seen a categorical description of monad transformers.
How could monad transformers be described in the terms ...
2
votes
2answers
424 views
Does univ : univ always lead to a contradiction in a dependently typed language?
I am currently following Checking Dependent Types with Normalization by Evaluation: A Tutorial by David Christiansen, where we consider the type of U (the universe ...
3
votes
2answers
357 views
What technique is used to implement type checking for CoC?
I am studying David Christiansen's tutorial on implementing a dependently typed language, where it says:
Typed normalization by evaluation is far from the only way to implement conversion checking ...
10
votes
1answer
541 views
Why is regularity a problem in cubical type theory?
In my current understanding, regularity in cubical type theory is the following definitional equality (I'm using $A~\textbf{type}$ to emphasize the fact that $i \notin FV(A)$):
$$
\cfrac{A~\textbf{...
3
votes
0answers
78 views
Equality reflection without $\eta$
Are type theories with equality reflection (i.e. having a term of an identity type between two objects allows us to freely swap them) but not the $\eta$-rule for $\Pi$-types interesting?
Does function ...
5
votes
1answer
120 views
Definitional function extensionality for functions out of $\mathbf{2}$
Denote by $a$ and $b$ the canonical terms of $\mathbf{2}$.
For any map $f:\mathbf{2}\to \mathcal{U}$ we have the eliminator$$\mathrm{ind}_{\mathbf{2}}(f) : \big(f(a) \times f(b)\big) \to \prod_{x:\...
3
votes
1answer
216 views
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 ...
0
votes
2answers
80 views
Pi-type over a list in dependent type theory
In a formalization I need to create an inductive type which has a term for each element in a list, something like this (I'll use Agda in the following, but everything here is standard dependent type ...
3
votes
1answer
147 views
Are uniqueness rules converse to introduction rules?
I've seen many people connecting introduction rules and elimination rules, saying that they're dual notion. Indeed, like in the categorical model, these rules are symmetric morphisms. However, if we ...
0
votes
1answer
88 views
Reference for context-free grammar for Martin-Löf type theory
Are the terms and the types of Martin-Löf type theory described by context-free grammars? Have such grammars been written down somewhere?
1
vote
2answers
115 views
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 ...
3
votes
0answers
85 views
Boltzmann sampling for containers/dependent polynomials?
I’d like to randomly sample from dependently-typed data structures.
Has anyone looked at extending Boltzmann sampling to containers or dependent polynomials?
-1
votes
1answer
94 views
How much type information do Hindley-Milner proof assistants need to remain sound?
A known benefit of the HM type system is that you can usually infer a term's most general type with no user-provided type annotations. For example, if my theory contains the standard axiom: $$\forall ...
2
votes
0answers
141 views
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 ...
2
votes
1answer
106 views
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 ...
0
votes
0answers
48 views
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 ...
2
votes
0answers
75 views
What is the relation of parametricity and function extensionality?
In Agda function extensionality can be defined like this:
funExt = {A : Set} {B : Set} {f1 f2 : A → B} → (∀ x1 x2 → x1 ≡ x2 -> f1 x1 ≡ f2 x2) → f1 ≡ f2
One may ...
2
votes
2answers
135 views
$\mathbb{N}$ 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?
13
votes
3answers
932 views
"Guarded" negative occurrences in definition of inductive types, always bad?
I know how some negative occurrences can definitively be bad:
...
4
votes
1answer
133 views
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?
4
votes
0answers
149 views
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 ...
2
votes
1answer
81 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 ...
5
votes
1answer
296 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 ...
3
votes
1answer
153 views
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 ...
2
votes
1answer
94 views
Typing inference as a map on abstract syntax trees
Is there a reference that explains typing inference for Martin-Löf type theory as a computable map from abstract syntax trees of terms to abstract syntax trees of types? I don't want to identify non-...
3
votes
1answer
86 views
Relating functors to relational functors with the parametricity translation
$\newcommand{\Type}{\text{Type}}\newcommand{\id}{\text{id}}\newcommand{\map}{\text{map}}$
In attempting to answer the question: Rigorous proof that parametric polymorphism implies naturality using ...
2
votes
1answer
71 views
Surjection from a type to a universe
We work in homotopy type theory. Can there be a type $A:U_m$ and a map $f:A\to U_n$ for some $n\geq m$ such that the type $\prod_{T:U_n} \|\mathrm{fib}_f(T)\|$ is inhabited?
5
votes
1answer
176 views
Choose term of coproduct type
We work in homotopy type theory. Denote the propositional truncation of a type $A$ by $\|A\|$ and the function type between types $A$ and $B$ by $A \to B$.
Can you construct a term of the following ...
7
votes
1answer
395 views
Strongly normalizing type theory beyond induction-recursion
Are there known type theories in the literature, which have strong normalization proofs and their proof-theoretical strength goes beyond strength of type theories with induction-recursion?
4
votes
0answers
79 views
Relative consistency of various Martin-Löf style type theories
I am wondering about relative consistency of various Martin-Löf type theories, when compared to one another, I will use MLTT for the intensional Martin-Löf type theory with $\Pi$, $\Sigma$, $\mathbb{N}...
4
votes
0answers
412 views
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 ...
