Assumptions with patterns

Question

I want that Mathematica assumes every symbol like Subscript[s,i] to be evaluate as a real and positive number. I have tried

$Assumptions = {Element[{h, k}, Integers], h > -1, k > -1, 
  Element[{Subscript[m, _], Subscript[s, _]}, Reals], Subscript[s, _] > 0}

but

Subscript[g, i_][x_] := 
  1/√(2 π Subscript[s, i]) Exp[-((x - Subscript[m, i])^2/(2 Subscript[s, i]))];

Simplify@Integrate[Subscript[g, i][x] Subscript[g, j][x], {x, -∞, ∞}]

still gives conditional expression for the integrand. Any suggestion? Thanks!

Answer 1

Patterned assumptions seem to need to match the ConditionalExpression's condition exactly to work out for some cases.

The ∈ Reals-assumptions do work as you gave them, while the inequality Subscript[s,_]>0 does not, but observe the different behavior of Subscript[s,_]>=0:

Evaluating without any assumptions first:

f = 1/√(2 π Subscript[s, 
   i]) Exp[-((x - Subscript[m, i])^2/(2 Subscript[s, i]))]
expr=Integrate[f, {x, -∞, ∞}]
(* ConditionalExpression[1, Re[Subscript[s, i]] >= 0] *) 

(Note, that the assumptions about h, k and m have no effect and are therefore superfluous.)

Now, bringing in the patterned assumptions in two versions:

greaterOnly = And @@ {Subscript[s, _] > 0, Element[Subscript[s, _], Reals]}
greaterEqual = And @@ {Subscript[s, _] >= 0, Element[Subscript[s, _], Reals]}

Refine[expr,greaterOnly]
(* ConditionalExpression[1, Subscript[s, i] >= 0] *) 
Refine[expr,greaterEqual]
(* 1 *)

So, the Element-part of assumptions is used in any case, while the inequality requires a "perfect" match.

Hope this helped!

Note: The behavior seems to depend on the Mathematica version. The code above works in 10.1, but behaves differently in 9.x and 8.x (see comments).

Answer 2

The document never promises that pattern-matching is supported inside Assumptions. (Though in some cases it does seem to be!) So the only stable way I can think of is as following:

$Assumptions = {};(*This is to guarentee the Join below will work*)
Subscript[g, i_][x_] := 
    ($Assumptions = Union[$Assumptions~Join~
     {{Subscript[m, i], Subscript[s, i]} ∈ Reals, Subscript[s, i] > 0}]; 
   Exp[-((x - Subscript[m, i])^2/(2 Subscript[s, i]))]/Sqrt[2 π Subscript[s, i]]);

Simplify[Integrate[Subscript[g, i][x] Subscript[g, j][x], {x, -∞, ∞}]]

Tested in v9.0.1 and v12.3.1, Windows 10.