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I've been trying to understand why FullSimplify is giving a condition that is implied by the assumptions (otherwise the set is empty). However, I couldn't understand why:

Clear["Global`*"]
l[v_] := v;
q[\[Beta]_, v_] := (\[Beta] - n*c) /(1 + (l[v] + n^2)*c);
Assuming[a > 0 && \[Beta] > 0 && c > 0 && v > 0 && n > 0 && 
  c < (\[Beta]*n - 1)/(l[v] + n^2*(a + 1)), 
 FullSimplify@Reduce[D[q[\[Beta], v], v] < 0]]

It yields:

c n < \[Beta]

The inequality is reversed if I reverse the inequality in the derivative, when I'd expect (reply to comment below explains why):

False

Any ideas about what could possibly be the problem?

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  • $\begingroup$ Why do you expect it to return False in the later case? As far as I can tell, it is producing the right inequality. The denominator you have (once the partial derivative is evaluated) is a square of a real number (ie. it's positive), which then leads the inequality for the numerator that you're seeing. $\endgroup$
    – ConservedCharge
    May 10, 2021 at 7:05
  • $\begingroup$ $\beta - nc>\beta- \frac{n(\beta n - 1)}{l(v) + n^2(a + 1)} $, which is greater than 0 if $\beta[l(v) + n^2(a + 1)]>n(\beta n - 1)\Leftrightarrow \beta l(v) + \beta n^2 a+n>0$. Right? $\endgroup$
    – Ararat
    May 10, 2021 at 8:19
  • $\begingroup$ That's correct, but it appears as if the assumptions aren't . Notice (after removing only the FullSimplify@ part) that we're still seeing results that are not restricted by the specified assumptions. For example, we still get a $n < 0$ case in the results. I would also be curious to see what the proper way to impose the constraints in this situation is. As it is, it seems to be doing something like Simplify[Sign[\!\( \*SubscriptBox[\(\[PartialD]\), \(v\)]\(q[\[Beta], v]\)\)], a > 0 && \[Beta] > 0 && c > 0 && v > 0 && n > 0] $\endgroup$
    – ConservedCharge
    May 10, 2021 at 18:55
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    $\begingroup$ Assumptions is not an option for Reduce and FullSimplify is a way to incorporate them. Omitting FullSimplify will cause the assumptions to be ignored (mathematica.stackexchange.com/questions/179820/…). It works, but in my case it didn't. I just figured out the reason. The maximum number of non-linear variables is set to 4 by default. Thus I had to increase it (mathematica.stackexchange.com/questions/245199/…) $\endgroup$
    – Ararat
    May 10, 2021 at 20:29
  • $\begingroup$ Nice; perhaps you can answer your own question in that case. $\endgroup$
    – ConservedCharge
    May 10, 2021 at 21:23

1 Answer 1

Reset to default
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Not really a solution of my own but, for sake of completeness I'll reproduce here with due credit.

As explained here, the problem is caused by exceeding the number of non-linear variables. By expanding it, we'd get the expected result:

Clear["Global`*"]
SetSystemOptions["SimplificationOptions" -> {"AssumptionsMaxNonlinearVariables" ->  10}];
l[v_] := v;
q[\[Beta]_, v_] := (\[Beta] - n*c) /(1 + (l[v] + n^2)*c);
Assuming[a > 0 && \[Beta] > 0 && c > 0 && v > 0 && n > 0 && 
  c < (\[Beta]*n - 1)/(l[v] + n^2*(a + 1)), 
 FullSimplify@Reduce[D[q[\[Beta], v], v] > 0]]

It yields:

False
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