Mdtryhht

I have a list which looks like this

list = 0, Subscript[x,7], -Subscript[x,3]-Subscript[x,9], -Subscript[x,9];

and all the $x_i$'s are positive semidefinite (i.e. nonnegative) real numbers. I would like to be able to sort this into

sortedlist = -Subscript[x,3]-Subscript[x,9], -Subscript[x,9], 0, Subscript[x,7]

How do I achieve this? I tried

Assuming[Subscript[x,3] > 0 && Subscript[x,7] > 0 && Subscript[x,9] > 0, Sort[list]]

But this obviously does not work. In general, I'd like to be able to impose more constraints on the $x_i's$ when they're being sorted.

Here is a possibility:

sortWithAssumptions[list_, assum_] := Module[order,
 order[a_, b_] := Simplify[a < b, assum];
 Sort[list, order]
]

For your example:

sortWithAssumptions[
 0,Subscript[x,7],-Subscript[x,3]-Subscript[x,9],-Subscript[x,9],
 Subscript[x,3]>0&&Subscript[x,7]>0&&Subscript[x,9]>0
] //TeXForm

$left-x_3-x_9,-x_9,0,x_7right$

Another example:

sortWithAssumptions[
 0,Subscript[x,7],-Subscript[x,3]-Subscript[x,9],-Subscript[x,9], Subscript[x,9],
 Subscript[x,3]>0&&Subscript[x,7]>0&&Subscript[x,9]>0&&Subscript[x,7]<Subscript[x,9]
] //TeXForm

$left-x_3-x_9,-x_9,0,x_7,x_9right$

How about:

list[[Ordering[list /. _Subscript -> 1]]]

-Subscript[x, 3] - Subscript[x, 9], -Subscript[x, 9], 0, Subscript[x, 7]

So basically we sort it the way it would be sorted with all subscripts == 1.

Sort[list, TrueQ@Simplify[#1 < #2, _Subscript > 0] &]

(* Out: -Subscript[x, 3] - Subscript[x, 9], -Subscript[x, 9], 0, Subscript[x, 7] *)

In this case, we can use RankedMin and FullSimplify to get the answer you seek

Assuming[
 Subscript[x, 3] > 0 && Subscript[x, 7] > 0 && Subscript[x, 9] > 0, 
 FullSimplify[Table[RankedMin[list, i], i, 1, Length[list]]]]
(* -Subscript[x, 3] - Subscript[x, 9], -Subscript[x, 9], 0, Subscript[x, 7] *)

This has the advantage of not returning a (potentially) wrong answer if the sort order is uncertain.