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Fix a few minor Lean misformalisations #229

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merged 5 commits into from
Oct 29, 2024
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ocfnash
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@ocfnash ocfnash commented Oct 16, 2024

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The informal solution provided at:
  https://kskedlaya.org/putnam-archive/2022s.pdf
contains some typos. The condition:
$$
  b < \ln(1 - r_{-})^2 - |a|r_{-}
$$
should be:
$$
  b < \ln(1 + r_{-}^2) - ar_{-}
$$
Similarly for the $r_{+}$ case.

The argument given in the PDF correctly derives that for $0 < a < 1$ one
should have:
$$
  b < g(r_{-})
$$
where $g(x) = \ln(1 + x ^ 2) - ax$ (and similarly for the $r_{+}$ case.
However there are some typos when transcribing this to the headline solution
(given higher up).
@GeorgeTsoukalas
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Nice catch for noticing a mistake. I think the case for the $r_+$ case may still be inaccurate, in particular I think the term may need to be $\dots - |a|r_+$. Here is a graph of an example: https://www.desmos.com/calculator/e90qfxhkr9.

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ocfnash commented Oct 22, 2024

Sorry I meant to get back to you about this: will do so tomorrow.

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ocfnash commented Oct 24, 2024

I think the case for the r + case may still be inaccurate, in particular I think the term may need to be ⋯ − | a | r + . Here is a graph of an example: https://www.desmos.com/calculator/e90qfxhkr9.

In fact I believe all is well, though I nearly went mad trying to understand why your plot made things look wrong. I somehow FINALLY realised that in this desmos site, log is base 10 so we need to use ln. (There is also one other sign error in that calculator you provided but this was easy to spot.) Here is a new calculator which I believe demonstrates that all is well: https://www.desmos.com/calculator/mykkqq1f2t

@ocfnash ocfnash changed the title Fix 2022 A1 Fix a few minor Lean misformalisations Oct 24, 2024
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ocfnash commented Oct 24, 2024

In the meantime, I have added three more problems where we have identified misformalisations to this PR.

As always, thanks for review and maintenance.

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In fact I believe all is well, though I nearly went mad trying to understand why your plot made things look wrong. I somehow FINALLY realised that in this desmos site, log is base 10 so we need to use ln. (There is also one other sign error in that calculator you provided but this was easy to spot.) Here is a new calculator which I believe demonstrates that all is well: https://www.desmos.com/calculator/mykkqq1f2t

Whoops! Sorry for causing confusion, I was trying to avoid it with the graph but had the opposite effect. I am now confident in the change.

This was disprovable because the integral was using the two-dimensional
measure from the plane in the integral on the one-dimensional set that
is the circle (and thus was always zero). We could fix this by supplying
the one-dimensional Hausdorff measure but this is too heavy-weight
given that we can just parameterise and use the interval integral.

A second error was the inclusion of the typeclasses `[AddCommGroup V]
[Module ℝ V]`. The subset already has this algebraic structure and by
supplying these there is ambiguity of which structure applies, as well
as the fact that these are arbitrary typeclasses with no relation to the
ambient structure on `MvPolynomial (Fin 2) ℝ`.
@ocfnash ocfnash force-pushed the main branch 2 times, most recently from f5290ac to 76e3114 Compare October 29, 2024 14:33
This was disprovable because of division-by-zero silliness. The fix is
to restate the problem in the language of rational functions.
This was disprovable because `y` could fail to be differentiable at `1`.
E.g., one could define `y` such that:
`y x = ∫ t in (1 : ℝ)..x, putnam_1963_a3_solution f n x t` for `x ≥ 1`
and `y x = 0` for `x < 1`. This will satisfy the RHS of the iff in the
goal but not the left.

I see two obvious ways to resolve the issue:
 1. Demand that `y` is C^n on `[1, ∞)`
 2. Require that `y x = ∫ t in (1 : ℝ)..x, putnam_1963_a3_solution f n x t`
    holds on a neighbourhood of `[1, ∞)`

The informal statement is very vague so I have somewhat arbitrarily
opted for the first of these resolutions.

This was previously "fixed" in 7a92152 but unfortunately an error remained.
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Everything looks good to me, thanks for the contribution!

@GeorgeTsoukalas GeorgeTsoukalas merged commit 5947cf6 into trishullab:main Oct 29, 2024
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3 participants