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Is it possible to add some pattern to $Assumptions


I apologize if it is a too elementary question but I could not find the appropriate documentation so far.



My goal is simple. I would like to add some assumptions that are defined in terms of patterns rather than symbols. For example, I would like to something like this


$Assumptions = { a[ ___ ] > 0 };

In my ideal world, this should set every expression with the Head a should be considered positive. Is it possible in Mathematica?


EDIT: Thanks. Following the first comment and the first answer, I did the following experiment. I still got puzzled about the result. Maybe it is just because of the intricate interaction between Integrate and $Assumptions.


$Assumptions = {A[___] > 0, B > 0};

Integrate[ Exp[ - A[x] t] , {t, 0, \[Infinity]}]
Integrate[ Exp[ - B t] , {t, 0, \[Infinity]}]
(* output *)

ConditionalExpression[1/A[x], Re[A[x]] > 0]
1/B

In this example, Integrate does not make use of the fact A[___]>0.



Answer



This is correct; you can also use the following forms, which are generally considered better suited to functional-programming style (in that you don't change the variables globally, just locally):


Block[
  {$Assumptions = (a[___] > 0)},
  code]


or, better yet,


Assuming[
  a[___] > 0,
  code]

Edit:


(In respond to the OP's code sample) I'm a bit surprised that the code treats A[x] and B differently, but the problem seems to come from an assumption that A[x] could be a complex number. This can be fixed with an additional assumption (also pointed out by march):


Assuming[
  {A[___] > 0, B > 0, A[___] \[Element] Reals},
  {Integrate[Exp[-A[x] t], {t, 0, \[Infinity]}],

   Integrate[Exp[-B t], {t, 0, \[Infinity]}]}]


{1/A[x], 1/B}



I would be interested to hear if anyone else knows why A[x] and B are treated differently by the integration system. I initially thought it might be the assumption that x is a variable, but this example is slightly more illustrative:


Assuming[
  {c[___] > 0, c[x] > 0, f > 0},
  {Integrate[Exp[-c[1]*t], {t, 0, \[Infinity]}],
   Integrate[Exp[-c[x]*t], {t, 0, \[Infinity]}],

   Integrate[Exp[-f*t], {t, 0, \[Infinity]}]}]


{ConditionalExpression[1/c[1], Re[c[1]] > 0], 1/c[x], 1/f}



It seems to me that assumptions containing patterns are treated differently than explicit assumptions, yet it is clear from my previous example (with A[___] \[Element] Reals) that assumptions with patterns are not just being ignored.


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