evaluation - Recursion depth exceeded in pattern matching with attribute Flat

I am defining a linear associative multiplication operation Mult[]:

ClearAll[Mult]
SetAttributes[Mult, {Flat, OneIdentity}];
Mult[A___, a_, B___] := a Mult[A, B] /; NumberQ[a]

Then I try to evaluate a simple expression, which (I believe) does not match the rule written above:

Mult[X, Y]

I got the error Recursion depth of 1024 exceeded. It is definitely related to the attribute Flat, but I do not see how. What am I doing wrong?

Answer

I find Flat hard to work with. Despite seeing it in use numerous times I'm never quite sure what it will do until I try it, so maybe I'm not the person to answer your question. Nevertheless I shall try.

Consider this related, reduced example:

Attributes[foo] = {Flat, OneIdentity};
foo[a_] := "inert" /; Print @ HoldForm[a]

foo[X, Y];

foo[X,Y]


X

Y

Critically the entire expression foo[X, Y] is substituted for the pattern a_ in one of the match attempts. This leads to infinite recursion.

One way to get around this is to prevent evaluation of that expression, which is what I used HoldForm for above. Applied to your original function:

ClearAll[Mult]
Attributes[Mult] = {Flat, OneIdentity};
Mult[A___, a_, B___] := a Mult[A, B] /; NumberQ[Unevaluated @ a]


Mult[X, Y]

Mult[3, x, 7, y, z]

Mult[X, Y]

21 Mult[x, y, z]