       Re: Changing the Natural Sort Order

• To: mathgroup at smc.vnet.net
• Subject: [mg49184] Re: Changing the Natural Sort Order
• From: ab_def at prontomail.com (Maxim)
• Date: Tue, 6 Jul 2004 03:34:28 -0400 (EDT)
• References: <ccb5hr\$ekg\$1@smc.vnet.net>
• Sender: owner-wri-mathgroup at wolfram.com

```"David Park" <djmp at earthlink.net> wrote in message news:<ccb5hr\$ekg\$1 at smc.vnet.net>...
> Dear MathGroup,
>
> Is it possible to change the natural sort order of symbols that is used in Sort?
>
> I would like something like the following statement (that does not work).
>
> Assuming[d < b < c, Sort[{a, b, c, d, f}]]
>
> giving the desired output
>
> {a,d,b,c,f}
>
> I won't be sorting simple lists of symbols, but lists of similar, but unspecified, expressions that contain the symbols. For example...
>
> {h[x,g[a]], h[x,g[b]], h[x,g[c]], h[x,g[d]], h[x,g[f]]}
>
> which should give
>
> {h[x,g[a]], h[x,g[d]], h[x,g[b]], h[x,g[c]], h[x,g[f]]}
>
> Is there any way to do this?
>
> David Park
> djmp at earthlink.net

The first thing to try when dealing with assumptions is to use Refine:

In:=
Assuming[b < a && b > c,
Sort[{a, b, c}, Refine[# < #2]& ]]

Out=
{c, b, a}

In general, this won't work if the test is undefined for some pairs as

In:=
Assuming[a > b,
Sort[{a, x, b}, Refine[# < #2]& ]]

Out=
{a, x, b}

If you have h[x,g[a]] instead of a, then you can simply use
#[[2,1]]<#2[[2,1]] instead of #<#2, or you can pull out the occurences
of a,b,c with Cases.

However, it is hard to guarantee anything with Refine; even a
simplification of the form Refine[Sign[expr],expr>0] may fail:

In:=
Refine[Sign[x1*y1+x2*y2+x3*y3], x1*y1+x2*y2+x3*y3 > 0]

Out=
Sign[x1*y1+x2*y2+x3*y3]

So you're probably better off just defining a function which will
return the desired ordering of the elements, without relying on
assumptions.

You can use Reduce for more complicated cases:

In:=
Assuming[1 < E^x < E,
Sort[{x, 0, 1},
Reduce[ForAll[x, \$Assumptions, # < #2], Reals]& ]]

Out=
{0, x, 1}

Maxim Rytin
m.r at inbox.ru

```

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