Krdytkyu

I am just beginning to learn about attributes of function in mathematica.

I saw the example "Flat". But there is something I don't get :

SetAttributes[fonction, Flat]



fonction[fonction[x]]



(*fonction[x]*)



fonction[x_] := x^2;



fonction[fonction[x]]



(*$RecursionLimit::reclim2: Recursion depth of 1024 exceeded during evaluation of fonction[x].



Hold[fonction[fonction[x]]]*)

Why do I have an error ? Shouldn't it returns me fonction[x]=x^2 because of the flat attribute ?

The main point to understand here is that the kernel does reduce your fonction[fonction[x]] to fonction[x]. You get into troubles after this reduction has been made, when just fonction[x] is being evaluated, and because of a different reason.

To illustrate this, we can look at the Trace of the evaluation:

ClearAll[f];

SetAttributes[f, {Flat}];

f[x_] := x^2;

Trace[f[f[1]]][[1, 1 ;; 2]]

(*{f[1], f[1]^2}*)

As you can see, there is only one f here but still there is a recursion problem. We can just call f[1] and get the same issue.

To understand what has happened, let's just define:

ClearAll[f];

SetAttributes[f, {Flat}];

f[x_] := Hold[x];

f[1]

(*Returns Hold[f[1]]*)

The reason for this extra f in the output is that for a Flat symbol expressions f[x] and f[f[x]] are identical. So, when a pattern-matcher encounters f[1] it treats the expression as f[f[1]] and consequently substitutes f[1], not 1, instead of x in the rhs of the definition. The pattern matcher prefers f[f[1]] over f[1] when matching x_ to allow for matching a sequence of arguments as a whole:

f[1, 2]

(*Returns Hold[f[1, 2]]*)

Here the pattern matcher treated f[1, 2] as f[f[1, 2]] and replaced x by f[1, 2] accordingly.

As chuy has already mentioned in the comments, you can add OneIdentity attribute to a symbol. Then the pattern-mathcer will prefer f[1] over f[f[1]] when matching f[x_] if there is only one argument inside the expression:

ClearAll[f];

SetAttributes[f, {Flat, OneIdentity}];

f[x_] := Hold[x];

f[1]

f[1, 2]

(*Returns Hold[1] and Hold[f[1, 2]]*)

Note, however, that OneIdentity attribute will not save your form recursion when there are more than one argument: f[1, 2] will be matched as f[f[1, 2]], f[1, 2] will be squared, f[1, 2]^2, and the f[1, 2] inside the square will again be matched as f[f[1, 2]]. So, basically, use Flat attribute only for symbols which really stand for some associative operators or you are likely to get into trouble.

ClearAll[f];

SetAttributes[f, {Flat, OneIdentity}];

f[x_] := x^2;

f[1]

f[1, 2]

(*1

  $RecursionLimit::reclim2 bla-bla-bla

  Hold[f[1, 2]^2]

*)

The following example may help:

SetAttributes[f, Flat];

Hold[f[f[x]]] /. HoldPattern[f[x_]] :> x^2

The result is:

Hold[f[f[x]]^2]

To see what's happening, we may run

MatchQ[f[a, b], f[_]]

The result is True. Thus we see that f[a,b] is identified as f[f[a,b]]. This is what the attribute Flat does to a function.