Thank you, Steven, for formalizing oeis-a1! I think, there will be a general counting function assigning to any extensible structure class š¯’˛ and a given cardinalityĀ ק the cardinality of the quotient of the subclass of š¯’˛ having carrier sets with cardinality ק . The restriction of this function to finite cardinalities will give the OEIS sequences.
But first I have a few questions about using Metamath and set.mm: (1) Is set.mm (without proofs) available as pdf? The book describes how I can create LaTeX input for individual statements or proofs, but not how the entire structure, including the important introductory paragraphs, can be integrated. Cross references to definitions would be nice. (2) Either is there another tool to get the definition of a symbol in a quick way? (also vim macros are welcome!) (3) I would like to test your definition of a1. But I do not understand, how to do this without changing set.mm. There is no second read possible. These are not the only questions I have when dealing with Metamath. But I hope that answering them will make it a little easier for me to get started. Am 23.11.2024 um 19:51 schrieb Steven Nguyen:
As a current description, by default people work on their own often-overlapping topics, with the help of the community answering any questions. I'm unsure so I'll leave the question of contributors open. The project proposed here seems highly interesting. I think the main part of the work is simply defining all the functions, since many concepts haven't been defined yet. One potential difficulty is that Metamath currently does not have much tooling. So computer-generated proofs are ironically tedious and manual to prove by default (not sure if https://us.metamath.org/mpeuni/631prm.html was done manually; https://us.metamath.org/mpeuni/ax-bgbltosilva.htmlĀ is also illustrative). Computer-generated proofs are rare in math, luckily. But if the plan is to prove a very large amount of sequences then I imagine it would be more efficient to make the tooling for it first. Or (dun dun dun?) ask the Lean community. Teamwork makes the dream work, in this case the sequences are math-community-wide so I'd imagine the larger amount of people doing Lean proofs would make it easier. I don't see any other strategy than starting with some sequence, and start seeing what needs to be done (a lot). As a start, here's A1, informally the (number of groups of order N), the definition is (number of groups of order N, mod equivalence) a1 = ( n e. NN |-> ( # ` ( { g e. Grp | ( od ` g ) = n } /. ~=g ) ) ) On Thursday, November 21, 2024 at 10:54:30ā€ÆAM UTC-6 Peter Dolland wrote: Hello Metamath Community, I'm still new here. So let me explain my intention, what I want here. My original motivation comes from combinatorics. I am interested in the relationship between the specification of an algebraic structure (such as directed or undirected graph, magma resp. groupoid, semigroup, group, topological space, partition, etc.) and the sequence that indicates how many non-isomorphic instances of the structure there are over a set with n elements, where n=0,1,2,3,.... . These sequences are published on oeis.org <http://oeis.org>: A273, A88, A1329, A27851, A1, A1930, A41,... . For some of them, a generating algorithm (in some programming language) is also given. However, in none of these sequences is a formal specification of the underlying algebraic structure given, although this is often much simpler than the generating algorithm. My impression so far is that the desired specifications can be created with MetaMath without any effort, as long as they are not already contained in set.mm <http://set.mm>. A real challenge would probably be to verify the specified algorithms against the specifications. It seems to me that a basis of theorems would have to be created first. In particular, Polya's counting theorem would need to be formalised. But perhaps one of you knows what already exists on this topic and can be used. Irrespective of this, it also seems worthwhile to me to create a bridge between MetaMath and OEIS in such a way that MetaMath specifications are added to corresponding sequences. The aim here could be to improve the findability of sequences. For example, the additional criterion that the elements of the carrier set are to be regarded as different (labelled) could be achieved by adding an independent complete order relation. For example, compare the sequences A88 with A53763, A273 with A2416, A41 with A110 or A1930 with A798. So you can see the first sequence is for each pair the unlabelled version. And the second ones yield the labelled version. I see a need for action that goes far beyond my areas of different expertises. The question now arises as to whether there are people in your community who are willing and able to support me in my search of a bridge as mentioned above. I have to admit that I have not yet delved very deeply into MetaMath and set.mm <http://set.mm>. If I find someone here, I really will have chance in order to acquire the necessary knowledge and skills. I am hopefully very convinced that MetaMath makes a decisive contribution to the further development of mathematics and computer science by formalising important statements. With best wishes, Peter Dolland -- You received this message because you are subscribed to the Google Groups "Metamath" group. To unsubscribe from this group and stop receiving emails from it, send an email to [email protected]. To view this discussion visit https://groups.google.com/d/msgid/metamath/a333331f-693f-4d44-86c0-74f8c7a09611n%40googlegroups.com <https://groups.google.com/d/msgid/metamath/a333331f-693f-4d44-86c0-74f8c7a09611n%40googlegroups.com?utm_medium=email&utm_source=footer>.
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