It is often very useful to be able to refer to a sequence currently under construction, for example to define the sequence recursively. For this purpose the Self operator is available.
This operator enables the user to refer to an already defined previous entry s[n] of the enumerated sequence s inside the sequence constructor, or the sequence s itself.
> s := [ i gt 2 select Self(i-2)+Self(i-1) else 1 : i in [1..100] ]; > &+s; 927372692193078999175
Instead of using a loop to apply the same binary associative operator to all elements of a complete enumerated sequence, it is possible to use the reduction operator &.
Given a complete enumerated sequence S = [ a1, a2, ..., an ] of elements belonging to an algebraic structure U, and an (associative) operator : U x U -> U, form the element a1 a2 a3 ... an.Currently, the following operators may be used to reduce sequences: +, *, and, or, join, meet, cat. An error will occur if the operator is not defined on U.
If S contains a single element a, then the value returned is a. If S is the null sequence (empty and no universe specified), then reduction over S leads to an error; if S is empty with universe U in which the operation is defined, then the result (or error) depends on the operation and upon U. The following table defines the return value:
EMPTY NULL &+ U!0 error &* U!1 error &and true true &or false false &join empty null &meet error error &cat empty null
The iteration operator in can be used in loops or in the set and sequence constructors to enumerate the defined elements of a sequence.
When multiple range sequences are used, it is important to know in which order the range are iterated over; the rule is that the repeated iteration takes place as nested loops where the first range forms the innermost loop, etc. See the examples below.
Enumerate the defined elements of the sequence S. x takes on the successive defined values in the sequence; if the dual iteration form is used then i is the corresponding numeric index of x.
> [ <number, letter> : number in [1..5], letter in ["a", "b", "c"]]; [ <1, a>, <2, a>, <3, a>, <4, a>, <5, a>, <1, b>, <2, b>, <3, b>, <4, b>, <5, b>, <1, c>, <2, c>, <3, c>, <4, c>, <5, c> ] > r := []; > for letter in ["a", "b", "c"] do > for number in [1..5] do > Append(~r, <number, letter>); > end for; > end for; > r; [ <1, a>, <2, a>, <3, a>, <4, a>, <5, a>, <1, b>, <2, b>, <3, b>, <4, b>, <5, b>, <1, c>, <2, c>, <3, c>, <4, c>, <5, c> ]This explains why the first construction below leads to an error, whereas the second leads to the desired sequence.
> // The following produces an error:
> [ <x, y> : x in [0..5], y in [0..x] | x^2+y^2 lt 16 ];
^
User error: Identifier 'x' has not been declared
> [ <x, y> : x in [0..y], y in [0..5] | x^2+y^2 lt 16 ];
[ <0, 0>, <0, 1>, <1, 1>, <0, 2>, <1, 2>, <2, 2>, <0, 3>, <1, 3>, <2, 3> ]
Note the following! In the last line below there are two different things
with the name x. One is the (inner) loop variable,
the other just an identifier with value 1000
that is used in the bound for the other (outer) loop variable y:
the limited scope of the inner loop variable x makes it invisible to y,
whence the error in the first case.
> // The following produces an error:
> #[ <x, y> : x in [0..5], y in [0..x] | x^2+y^2 lt 100 ];
^
User error: Identifier 'x' has not been declared
> x := 1000;
> #[ <x, y> : x in [0..5], y in [0..x] | x^2+y^2 lt 100 ];
59
> letters := Eltseq("abc");
> [ <i, x> : i -> x in letters ];
[ <1, "a">, <2, "b">, <3, "c"> ]
> for i -> x in letters do
> printf "letters[%o] is %o n", i, x;
> end for;
letters[1] is a
letters[2] is b
letters[3] is c
Dual iteration allows us to easily get the indices of defined elements;
note the use of an underscore in the dual iteration below to ignore the
values, since we do not use them.
> S := [ 7, 3 ]; > S[5] := 2; > S; [ 7, 3, undef, undef, 2 ] > [ i : i -> _ in S ]; [ 1, 2, 5 ]