After posting an answer on this MathOverflow question https://mathoverflow.net/questions/450930/is-there-an-identity-between-the-associative-identity-and-the-constant-identity , I wonder if it might be a suitable graduate research project to see if current generation #ProofAssistant / #MachineLearning / #AI tools can be used to determine the logical relationship between various universal equational laws that could be satisfied by a single binary operation + on a set (i.e., by a magma). For instance, in the answer to this related question https://mathoverflow.net/questions/450890/is-there-an-identity-between-the-commutative-identity-and-the-constant-identity?noredirect=1&lq=1 it was shown (by a slightly intricate argument) that the law (𝑥+𝑥)+𝑦=𝑦+𝑥 implies the commutative law 𝑥+𝑦=𝑦+𝑥, but not conversely, while I showed that the law 𝑥+(𝑦+𝑧)=(𝑥+𝑦)+𝑤 is strictly intermediate between the triple constant law 𝑥+(𝑦+𝑧)=(𝑤+𝑢)+𝑣 and the associative law 𝑥+(𝑦+𝑧)=(𝑥+𝑦)+𝑧. It seems that this is a restrictive enough fragment of #mathematics (or even of #UniversalAlgebra) that automated tools should function rather well, without being so trivial as to be completely solvable by brute force.

Ask #ChatGPT to prove something and you'll understand how #ProofAssistant feels when you prove something :)

Liquid Tension Experiment
https://en.wikipedia.org/wiki/Liquid_Tension_Experiment

"Liquid Tension Experiment" (self-titled first studio album)
https://songwhip.com/liquid-tension-experiment/liquid-tension-experiment

I discovered this group when I read about
#PeterScholze “Liquid Tensor Experiment”: https://www.quantamagazine.org/lean-computer-program-confirms-peter-scholze-proof-20210728/
https://www.nature.com/articles/d41586-021-01627-2

#nowlistening #music #InstrumentalRock #ProgressiveRock
#LiquidTensionExperiment
#mathematics #ProofAssistant

@wen Thank you! All these comments are very helpful for checking whether our designs are cool and general enough, even if we might not support some of them in the end. (For example, we might not implement something outside the current LSP + common extensions.)

PS: from the POV of reusable components, showing a list of visible definitions at any position is equivalent to auto-completion. 😛 A fast scope checker is indeed on the agenda. @totbwf @jonmsterling #ProofAssistant

Mathematicians welcome computer-assisted proof in ‘grand unification’ theory

Proof-assistant software handles an abstract concept at the cutting edge of research, revealing a bigger role for software in mathematics .

…

A grand unification

Around 2018, Scholze and Clausen began to realize that the conventional approach to the concept of topology led to incompatibilities between these three mathematical universes — geometry, functional analysis and p-adic numbers — but that alternative foundations could bridge those gaps. Many results in each of those fields seem to have analogues in the others, even though they apparently deal with completely different concepts. But once topology is defined in the ‘correct’ way, the analogies between the theories are revealed to be instances of the same ‘condensed mathematics’, the two researchers proposed. “It is some kind of grand unification” of the three fields, Clausen says.
…

#mathematics #computing #ProofAssistant #software

https://www.nature.com/articles/d41586-021-01627-2?error=cookies_not_supported&code=bc3aeb22-729c-47ee-bea5-27423a45cde4

Mathematicians welcome computer-assisted proof in ‘grand unification’ theory
https://www.nature.com/articles/d41586-021-01627-2

#Lean #ProofTheory #ProofAssistant #Math

#Programming #Book
> The #SoftwareFoundations series is a broad introduction to the #mathematical underpinnings of #reliable #software.
> The principal novelty of the series is that every detail is one hundred percent formalized and machine-checked: the entire text of each volume, including the exercises, is literally a "proof script" for the #Coq #ProofAssistant.
https://softwarefoundations.cis.upenn.edu/

#ComputerScience #Justice
> #Catala: A #Programming #Language for the #Law
> We have implemented a compiler for Catala, and have proven the correctness of its core compilation steps using the F* #ProofAssistant. We evaluate #Catala on several #legal texts that are #algorithms in disguise, notably section 121 of the #US federal income #tax and the byzantine #French family benefits; in doing so, we uncover a #bug in the official implementation.
https://arxiv.org/abs/2103.03198