Re: Flat: Problems & Workarounds Sweden.

• To: mathgroup at smc.vnet.net
• Subject: [mg8492] Re: Flat: Problems & Workarounds Sweden.
• From: Robert Villegas <villegas>
• Date: Tue, 2 Sep 1997 16:15:36 -0400
• Organization: Wolfram Research
• Sender: owner-wri-mathgroup at wolfram.com

```> This is the result from my work trying to construct a associative function.
> Assume I want a associative function h[] which treats numerical arguments
> specially. Starting from the template in Roman Maeders "Programming in
> Mathematica", 3rd ed. I define h[] as
>
> In[3]:= SetAttributes[h,{Flat,OneIdentity}]
> h[x_,y_?NumericQ] := f[y,x]
> h[x_?NumericQ,y_] := f[x,y]
> h[x_] := x
> h[] = 1

When I want an associative function h that acts as the identity on
singletons, I ditch the use of attributes like Flat and OneIdentity
and write a definition to make h self-flattening.  If I want h to
recognize certain argument arrangements and re-group them, I control
this myself with additional definitions, since the pattern-matcher under
the influence of Flat might try groupings and orderings of them different
from what I want.

Here is how I recommend formulating your h:

ClearAll[h];

h[elems___] /; MemberQ[Unevaluated[{elems}], _h] :=
Flatten[Unevaluated @ h[elems], Infinity, h];

h[elems__, n_?NumberQ] := f[n, h[elems]];
h[n_?NumberQ, elems__] := f[n, h[elems]];

h[singleton_] := singleton;
h[] = 1;

This h does what you asked for the examples you cited, and its behavior
seems reasonable in related cases:

In[37]:=
{h[], h[1], h[a], h[1, a], h[a, 1], h[a, b], h[2, a, 3], h[2, 3, a],
h[a, 2, 3]}

Out[37]=
{1, 1, a, f[1, a], f[1, a], h[a, b], f[3, f[2, a]], f[2, f[3, a]],

f[3, f[2, a]]}

Your function isn't precisely specified, so this may not be exactly what
you want, but it shouldn't be difficult to modify this to do something a
bit different.

The key definition in h was the following

h[elems___] /; MemberQ[Unevaluated[{elems}], _h] :=
Flatten[Unevaluated @ h[elems], Infinity, h]

which canonicalizes the arguments of h so that other definitions have the
luxury of knowing that none of h's arguments have head h.  From there, make
a list of the precise arrangements you want to recognize and re-group, e.g.

(1)  A bunch of arguments followed by a final NumericQ argument

(2)  An initial NumericQ argument followed by a bunch of arguments

then add the definitions for these.

It is occasionally necessary to use HoldPattern on the left-hand sides of
rules, or to notice the automatic ordering of rules, but not usually.

Robby Villegas

```

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