[Axiom-math] Re: iterators and cartesian product.

[Francois Maltey]

> I'm not being facetious, I'm trying to understand your perspectives :

For me it's an exercice I give to my students in maple :
create the list of all matrices which can be magic. 
They are a subset of this list :

[matrix [[a,b,15-a-b],[c,d,15-c-d],[15-a-c,15-b-d,15-a-d]] 
  for << all the (a,b,c,d) in 2..8 x 1..9 x 1..9 x 2..8 >> ]

My first query was about this real exercice : create this list of matrices.
(and then extract the magic ones)

mupad syntax is 
[matrix [[a,b,15-a-b],[c,d,15-c-d],[15-a-c,15-b-d,15-a-d]] 
  $a=2..8 $b=1..9 $c=1..9 $d=2..8]

maple syntax is 
['''matrix [[a,b,15-a-b],[c,d,15-c-d],[15-a-c,15-b-d,15-a-d]] 
  $a=2..8' $b=1..9' $c=1..9' $d=2..8]

So I'm looking for a pretty syntax in order to enumerate a cartesian Product
of lists or of segments.

There are 3 possible methods :

// A //

Today we can use

L := [1,3,5,7,9]
concat [concat [[100*a+10*b+c for a in 1..2] for b in 0..9] for c in L]

I don't find fair many concat.

Of corse 
   Union_{c in C} (Union_{b in B} {F(a,b,c) for c in C}) 
=  {F(a,b,c) for (a,b,c) in AxBxC} 

I write the second terms in mathematics,
and so I prefer to have this second syntax in axiom, 
and the concat command looks like an union.

// B //

Waldek writes :
> Concerning more general problem: I think that we also need other approaches.
> More precisely, it would be good to add better iteration constructs, which
> make iterating over products easier.  For example, something like:

> [[a, b] for a in 1..10 repeat for b in 1..5] 

> Well, concat version creates intermediate lists and concatenates them.
> Psychologically double iteration is an atomic operation, 

> so it is easier (at least for some folks) to think about. 

// C //

Martin seems to prefer :

L := [1,3,5,7,9]
[100*a+10*b+c for (a,b,c) in Product (1..2,0..9,L)]

I don't disapprove this point of view. 
The magic [function|domain] Product generates the list 
of all the possibilities.

but I'm really waiting for automatic coerces in order to have 
a light command in the interpreter because this construct is (for me) 
very common. So I don't want to add a lot of $... and @... everytime.

In this case the "keyword" Product has various arguments : 
List or Segment. All terms don't have necessary the same type.

An other example :
LP is a list of Lagrange polynoms, Lx is the list of associated values.
I could test : 
    test (set [eval (P, x) for (P,x) in Product (LP, Lx)] = set [0,1])
    test (set [eval (P, x) for (P,x) in Product (LP, 1..8)] = set [0,1])
even if it is better to test with a matrix.
    test (matrix [[eval (P, x) for P in LP] for x in Lx] = identity(n))

F.