Unanswered Questions
15 questions with no upvoted or accepted answers
22
votes
0
answers
568
views
Categorical semantics of Agda
I would like to know the state of the art regarding the categorical semantics of the type theory implemented by Agda — or at least some approximation of that type theory that is amenable to ...
- 3,190
12
votes
0
answers
169
views
Rules for mutual inductive/coinductive types
Some proof assistants, like Agda and maybe Coq, allow families of mutually defined types, or nested definitions of types, in which some are inductive and others are coinductive. I have no idea what ...
- 3,190
10
votes
0
answers
187
views
Is there a proof assistant (or an embedding) for the (co)end calculus?
A Higher-Order Calculus for Categories describes a system where you can conveniently perform manipulations with categories, functors, Yoneda embeddings etc. An example of the rules is: $$\frac{\Gamma ,...
- 2,846
9
votes
0
answers
308
views
Alternatives to universe levels
All of the type theory based proof assistants that I have seen have an infinite hierarchy of type universes to avoid the type of types being a term of itself. Are there alternative systems which could ...
- 91
6
votes
0
answers
114
views
Is every logical theory embeddable in a dependently typed extensional type theory?
In category theory by the Yoneda embedding every category $\mathcal{C}$ is a subcategory of a category of presheafs $[\mathcal{C}^{\text{op}}, \text{Set}]$. Every category of presheafs is a topos and ...
- 2,868
5
votes
0
answers
244
views
Mere propositions and Consistency with Impredicativity, Excluded Middle and Large Elimination
Setup
Current Understanding
I've recently been trying to learn about the interaction of Impredicative Polymorphism, Large Elimination and Excluded Middle (notably, inconsistency). Notably, this is ...
- 73
5
votes
0
answers
95
views
What is the relation of $\lambda^\to$ and $\lambda^{\to\times}$ to cartesian closed categories?
I am learning about the categorical semantics of type theory. I've written some preliminary results in Agda. In particular, I partially proved that the contexts and substitutions of simply-typed ...
- 4,025
4
votes
0
answers
120
views
Proving "proof methods" as theorems in type-theory based proof systems
For example, suppose we have proved associativity of some binary operator $+ : T \to T$ as
add_assoc : forall (x y : T), x + y + z = x + (y + z).
We can thus prove ...
- 41
4
votes
0
answers
106
views
Independence of function extensionality
Who first realized that function extensionality cannot be proved within vanilla MLTT, or some variations of it? Now to my knowledge the simplest way to show this is by syntactic models. But surely ...
- 4,025
3
votes
0
answers
95
views
Limitations of simple type theory proof assistants for undergraduate-level mathematics?
I'm interested in understanding the practical power and limitations of simple type theory, particularly as compared with dependent type theory, in supporting formalized proofs of theorems liable to ...
- 345
2
votes
0
answers
68
views
Tool for typing mathematical physics, e.g. differential geometry
Many expressions in mathematical physics use a rather sloppy notation, e.g. the Lie bracket on a vector field is defined using ambiguous notation, where $X : M \rightarrow TM$ is first defined as a ...
- 4,025
1
vote
0
answers
105
views
Proving Quine's notion that identity belongs to logic within type-constrained proof assistants
I'm having difficulty generalizing the following proof to permit predicates of any finite arity.
Consider the following axioms of identity consistent with W. V. O. Quine's argument that relative ...
- 111
0
votes
0
answers
119
views
MLTT with first-order reasoning and equality-reasoning information preservation
Terms in Extensional MLTT don't contain equality-reasoning information (implicit transports), effectively meaning data is lost, which is bad. But at the same time, higher-order reasoning (reasoning ...
- 345
0
votes
0
answers
106
views
Formalization of partial functions for combinatorial counting
I need assistance in defining axioms for partial functions in total function theory that is available in Coq.
Specifically, I'm looking for a constructive definition of a partial function that ...
- 9,722
0
votes
0
answers
96
views
Existential variables in dependent type theory
For the kernel of a proof assistant free variables/universal quantification may be sufficient. In higher level languages such as Coq's tactic language indeterminate variables (not sure of the wording ...
- 2,868