Community Digest

Top new questions this week:

Why are impredicative constructions used less in type theory than in material set theory?

Many infinitary objects in (say) ZFC are constructed with impredicative principles. The natural numbers are formed by intersecting every inductive set (whose existence is given by the axiom of ...

mathematics impredicativity  
user avatar asked by Trebor Score of 6
user avatar answered by Meven Lennon-Bertrand Score of 5

Unclarity about Preorder class in Lean4

I realize the port of Mathlib to Lean4 is not finished yet, but I've run into a definition that I do not quite understand. I'm quite new at using theoremprovers as well as stackexchange, so please be ...

beginner lean4 definition  
user avatar asked by Pieter Cuijpers Score of 5
user avatar answered by Andrej Bauer Score of 5

Is 'subsingleton elimination' the same concept as 'function comprehension'?

I saw: subsingleton elimination from lean-forward, which, I so far understood as "eliminate a type in Prop to a type in whatever universe that we know has at ...

irrelevance prop  
user avatar asked by ice1000 Score of 3
user avatar answered by Meven Lennon-Bertrand Score of 6

How to unfold definitions in Lean / find the right theorems to apply?

After a few days playing around with Lean4, I notice I keep running into the problem of how to find the right theorems to apply. The situation below is one I run into particularly often, so perhaps I ...

beginner lean4 strategy  
user avatar asked by Pieter Cuijpers Score of 2
user avatar answered by Pieter Cuijpers Score of 2

Equality of records with members whose types are dependent (Lean 4)

(Previously, I asked about converting a term a: A to a term of type B provided that A = B. ...

type-theory lean4  
user avatar asked by Jozef Mikušinec Score of 2
user avatar answered by Jason Rute Score of 3

Question about default definitions in fields

In Unclarity about Preorder class in Lean4 I asked why the third and fourth field (lt and lt_iff_le_not_le) in the definition of MyPreorder below would both be necessary, as one follows from the other ...

beginner lean4 definition  
user avatar asked by Pieter Cuijpers Score of 1
user avatar answered by Pieter Cuijpers Score of 1

Greatest hits from previous weeks:

What difficulties are there in basing a proof assistant on the $\lambda\mu$-calculus and has any proof assistant tried?

The $\lambda\mu$-calculus is a variant of the $\lambda$ calculus introduced by Parigot to capture classical logic. The Wikipedia article describing it in more detail is here. In modern proof ...

design implementation  
user avatar asked by Andrew Marmaduke Score of 19
user avatar answered by Trebor Score of 16

Do you need a Hilbert style Epsilon operator for definitions in set theory?

I've started to play with mechanizing some set theory stuff. I'm not sure if I want a constructive flavor or not yet. Anyhow you can do stuff like axiomize the empty set $$ \top \vdash \exists P. \...

foundations set-theory axiom-of-choice  
user avatar asked by Molossus Spondee Score of 14
user avatar answered by Andrej Bauer Score of 17

What is the current state of proof assistants?

I am a high school senior going into college and I am applying to a scholarship in which I must write an essay about a potential future technology that would dramatically impact humans. I immediately ...

beginner  
user avatar asked by gmod Score of 7
user avatar answered by Jason Rute Score of 12

What set-theoretic definitions can't easily be formalized in a type theory?

Most proof assistants (with some exceptions like Isabelle/ZF or the B method) rely on type theory. See also the MathOverflow question What makes dependent type theory more suitable than set theory for ...

mathematics type-theory set-theory  
user avatar asked by ErikMD Score of 28
user avatar answered by Andrej Bauer Score of 32

Tutorial implementations of NbE

For a project I'm planning on doing with some students this summer, I'm looking for an implementation of normalization-by-evaluation whose code the students can read, understand, and extend. So I'd ...

implementation recommendations  
user avatar asked by Mike Shulman Score of 13
user avatar answered by Jonathan Sterling Score of 12

In Lean, contradiction tactic failed but actually goal accomplished

I've been playing with Lean, trying to prove the next lemma: lemma l1_cl (A B C : Prop) : ((A → B) → C) → ((A ∧ ¬ B) ∨ C) := ...

lean proof-review lean3  
user avatar asked by Paul Snopov Score of 7
user avatar answered by Mario Carneiro Score of 11

How to access local definitions in Agda

I'm working in cubical agda. I am wondering how to access local definitions once outside of the local environment. For example, suppose I have the following code: ...

agda cubical-agda  
user avatar asked by IsAdisplayName Score of 9
user avatar answered by András Kovács Score of 6
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