Fix a few minor Lean misformalisations #238
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The error was a simple typo: `<` instead of `=`.
The problem was that `(a * b : Fin 3 → ℝ)` first coerces `a` and `b` to vectors and then uses the multiplication on vectors (which is pointwise multiplication). Thus the group structure declared in `hGgrp` was completely ignored. This could have been fixed by writing `(↑(a * b) : Fin 3 → ℝ)` but I think it is much clearer to use an embedding and not depend upon arcane details Lean's coercion elaboration.
eric-wieser
reviewed
Oct 22, 2024
eric-wieser
approved these changes
Oct 22, 2024
This is a slightly subjective fix. The issue is that `f > 0` does not mean that `f` is positive everywhere; it means that `f` is non-negative everywhere and positive at at least one point. I believe this is what is intended in the informal statement. The issue is slightly confounded by the fact that if for all `x`, `deriv f x > f x` and `f x` is non-negative then the derivative is strictly positive, and thus `f` is strictly monotone, and thus being non-negative implies being strictly positive everywhere. So there is no mathematical change to the meaning here but I think this is a more faithful representation of the question. It should also be noted that the fact that there is no mathematical change depends upon not-quite-totally-trivial facts. I have also taken the opportunity to rewrite question with slightly improved style.
The statement should be that the sum diverges, rather than that it has no finite limit (even though these are easily equivalent given the latent positivity hypotheses).
GeorgeTsoukalas
approved these changes
Oct 23, 2024
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