leanprover-community / leanprover-community/mathlib4

Define `Asymptotics.IsSubpolynomial`

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Description

I think we should have a definition for "along the filter l, f is a big O of 1 + ‖g‖^k for some natural k", along with some basic lemmas such as:

  • stability under addition, multiplication, power, composition
  • equivalence with some other definitions (e.g using (1 + ‖g‖)^k)
  • if you have a finite set of functions which are subpolynomial in g, you can pick the constants in the big O and the power uniformly.

This is motivated by proofs about functions with temperate growth, which I claim could be simpler with such an API. But this would also be useful more generally for people studying asymptotic behaviours (e.g to define groups with polynomial growth). In fact, maybe something like this exists already in downstream projects?

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First steps

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Research direction

Start by searching the existing Asymptotics API and downstream projects for related definitions. Establish the proposed IsSubpolynomial definition and its intended filter and norm parameters, then add the requested stability, equivalence, and uniform-constants lemmas; done means the API supports the motivating temperate-growth proofs.

Written by the indexing model from the issue text.

Assessment

Domain
tooling
Issue type
Feature
Difficulty
5/5
Estimated time
Over a week
Activity status
Quiet
Clarity
Mostly clear
Newbie friendliness
38/100

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