Stack Exchange Network

Stack Exchange network consists of 174 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

The tag has no usage guidance.

5
votes
0answers
142 views

Independence of characters with respect to polynomials

I came across the following property : Let $\mathfrak{g}$ be a Lie algebra over a ring $k$ without zero divisors, $\mathcal{U}=\mathcal{U}(\mathfrak{g})$ be its enveloping algebra. As such, $\...
1
vote
0answers
50 views

Proof — swapping sum with integral

Problem In Ceperley's 95 article on path integral Monte Carlo approach I have encountered $\hat{\rho}:L^{2}(R^{3N})\to L^{2}(R^{3N})$ $\hat{\rho} = e^{-\beta \hat{H}}$, where $\hat{H}$ is a ...
4
votes
0answers
151 views

formalization of coordinate-free linear algebra in a proof assistant

I am aware of projects that formalize linear algebra in existing proof assistants (i.e. Coq), but it seems like most of them are based on matrices. I was wondering if it's done in a coordinate-free ...
1
vote
0answers
185 views

Are all solutions to an ordinary differential equation continuous solutions to the associated implied differential equation and vice versa?

Now I have to heavily emphasize the fact that I have never studied differential algebra or the concept of other types of differentiation (which is what I believe is the concept behind a differential ...
5
votes
0answers
267 views

Are there any recent advances in formalizing the undecidability of $\mathit{CH}$?

The website Formalizing 100 Theorems by Freek Wiedijk contains a list of some theorems that were chosen at some point as good candidates for formalization (because of their complexity, their ...
1
vote
0answers
96 views

Expansion of prolate spheroidal harmonics

For two coordinate frames $O'$ and $O''$ both offset along the $z$-axis by $\pm R$ respectively, with corresponding offset spherical coordinates $r'$, $\theta'$, $r''$ and $\theta''$, and with prolate ...
5
votes
4answers
321 views

Binomial ID $\sum_{k=m}^p(-1)^{k+m}\binom{k}{m}\binom{n+p+1}{n+k+1}=\binom{n+p-m}{n}$

Could I get some help with proving this identity? $$\sum_{k=m}^p(-1)^{k+m}\binom{k}{m}\binom{n+p+1}{n+k+1}=\binom{n+p-m}{n}.$$ It has been checked in Matlab for various small $n,m$ and $p$. I have a ...
5
votes
2answers
645 views

What exactly is a judgement?

Before formulating my question, let me briefly sum up what I know about the topic (feel free to correct me if something I claimed is false!). This is for you good to see what my state of knowledge is, ...
1
vote
0answers
203 views

How to prove this Gaussian Mixture theorem? (Fitting/Overfitting)

Note from OP: I gave up and reposted this Question with a Bounty on Cross Validated HERE. In certain applications, we approximate an unknown pdf by placing uniformly weighted Gaussian terms at each ...
32
votes
1answer
3k views

How much mathematics has been formally verified?

That's a vague question so allow me to tighten it up a bit. I recently noticed that there is a formal machine verified proof of the Central Limit Theorem (CLT) implemented with Isabelle. This ...
9
votes
2answers
812 views

Why is there a need for ordinal analysis?

Consider the Peano axioms. There exists a model for them (namely, the natural numbers with a ordering relation $<$, binary function $+$, and constant term $0$). Therefore, by the model existence ...
4
votes
2answers
421 views

Uniform Convergence of Moment Generating Function

In the article, "The Empirical Moment Generating Function" by Csörgö, the author defines the empirical moment generating function for a sample of $n$ variables $X_1,X_2, \dots, X_n$ as: $$ \begin{...
44
votes
5answers
3k views

How much of the ATLAS of finite groups is independently checked and/or computer verified?

In a recent talk Serre made some comments about proofs that rely on the classification of finite simple groups (CFSG) and on the ATLAS of Finite Groups. Namely, he said that a proof that relied on the ...
26
votes
0answers
861 views

Next steps on formal proof of classification of finite simple groups

While people are steaming ahead on finessing the proof of the classification of finite simple groups (CFSG), we have a formal proof in Coq of one of the first major components: the Feit-Thompson odd-...
2
votes
1answer
280 views

existence of multiplicity of roots [closed]

Im confuse..I read in an article that in dealing with polynomials, a quadratic equation can have either 2 real roots, 1 equal real root or 2 complex roots...but in dealing with random polynomials only ...
70
votes
3answers
4k views

How do I verify the Coq proof of Feit-Thompson?

I probably don't have the appropriate background to even ask this question. I know next to nothing about formal or computer-aided proof, and very little even about group theory. And this question is ...
14
votes
1answer
2k views

Where can I find Gonthier's Coq code proving the four color theorem?

In a 2008 article in the Notices, Georges Gonthier announced a computer-checked proof of the four color theorem using Coq: Gonthier, Georges. Formal proof—the four-color theorem. Notices Amer. ...
0
votes
1answer
743 views

Is there any danger far from home? (Edited & Revised Version) [closed]

The notion of formal proof is defined by finite sequences ($<\omega$ - sequences) of sentences. In some sense if a sentence $\sigma$ is (finitely) provable from the theory $T$ it is very "near" to ...
2
votes
1answer
1k views

Hilbert style axiomatic proof or sequent Calculus?

I am puzzling with the question which of the two proof systems (Hilbert style axiomatic proofs or substructural Sequent Calculi) is the most discriminatory? With discriminatory I mean is which proof ...
1
vote
1answer
995 views

Where is a proof of “2 is more than 1 plus 1” said by Saunders Mac Lane? [closed]

I came across this statement in the autobiography by Saunders Mac Lane. It was the interaction between solenoids and group extension that got our collaboration started, and this first work of ...
1
vote
1answer
679 views

Since an inconsistent system can prove its own consistency…

Say a proof for the consistency of a formal system (proved within the formal system) is known. There are two possible cases: 1. the formal system is consistent and it can be and has been proven to be, ...
2
votes
1answer
404 views

Sequent calculus: is there a complete linear reasoning (i.e., no trees)?

In Gentzen's sequent calculus, a formal proof is described by a tree, with each node representing the sequent obtained from the child(ren) by applying a given inference rule. If no inference rule has ...
16
votes
3answers
914 views

Finite versions of Godel' s incompleteness

Assume you have some notion of proof complexity: for instance, at the basic level, the length of a proof, or the number of symbols used, take your pick (there are more involved measures, but for sake ...
8
votes
1answer
415 views

Proving that a combinatorial sequence has no compact formula

Suppose we have a sequence $a_n$ given by some combinatorial formula, e.g. involving a sum of n terms (like ${n \choose k}^{10}3^{-k}$ etc.). Sometimes it is plausible that there is no compact ...
13
votes
2answers
3k views

Consequences of technically proving anything in Coq (on at least Linux) exploiting a bug? [closed]

Technically, it is possible to prove anything in Coq proof assistant [1] (on at least Linux) due to a programming feature (or bug). This seems tractable when validating large proofs. Human analysis ...
42
votes
6answers
8k views

How true are theorems proved by Coq?

Less tongue in cheek, is it known what the relative consistency is for theorems proved with an automatic theorem prover? Of course this depends somewhat on what assumptions one makes with respect to ...