Emulating Sequence with my own function

Question

I just considered if/how one could implement Sequence in Mathematica if it were not predefined. It turned out that the following simple definition has in all my tests exactly the right behaviour:

myseq /: f_[x___, myseq[y___], z___] := f[x, y, z]

Now my question: Does this already correctly reproduce the full behaviour of Sequence, or is there something Sequence does but myseq doesn't which I missed in my tests?

Here's what I tested:

foo[myseq[a, b]]
(*
==> foo[a, b]
*)
Hold[mysec[a,b]]
(*
==> Hold[a, b]
*)
HoldComplete[myseq[a,b]]
(*
HoldComplete[myseq[a, b]]
*)
Hold[f[myseq[a,b]]]
(*
==> Hold[f[myseq[a, b]]]
*)
f[myseq[myseq[a,b],c,d],e,myseq[f,g,myseq[]]]
(*
==> f[a, b, c, d, e, f, g]
*)

Answer 1

Ok, my two cents. The answer seems to be - you can't. There are 3 "magic" symbols which are wired into the core evaluator much deeper than the rest: Evaluate, Unevaluated, and Sequence. You can't fully emulate any of those without essentially writing your own version of Mathematica evaluator on top of the built-in one.

For the record, I first read about it in the book of David Wagner, "Power programming with Mathematica - the Kernel", p.207. Which means - if this is correct, I take the credit, but if it is wrong, he is the one to blame :). But, seriously, there was nothing in my experience to contradict this. You may emulate some aspects of Sequence, but I would be very surprised if you could make a complete emulation (without writing your own evaluator on top of the system one).

Let me also add that, while it is hidden, you do use Sequence in your approach, since the y___ pattern is internally destructured as Sequence. Check this out:

ClearAll[myseq];
myseq /: f_[x___, myseq[y___], z___] := f[x, Head[Unevaluated[y]], z]

and now

f[1, myseq[], 5]

f[1, Sequence, 5]

Answer 2

I think an issue that you will never be able to completely solve is one of priorities. Sequence is flattened before upvalues. So, for starters

ClearAll[f];
f /: g[f, _] := 9

So

g[f, myseq[2, 3]]

9

but

g[f, Sequence[2, 3]]

g[f, 2, 3]

Answer 3

There are functions related to Sequence: BlankSequence, BlankNullSequence and SlotSequence. These can be used to perform operations similar to Sequence but they are not identical.

As pointed out in the comments Sequence obeys the attribute SequenceHold:

SetAttributes[test, SequenceHold]

test[ Sequence[1, 2, 3] ]

test[Sequence[1, 2, 3]]

The others do not:

{1, 2, 3} /. _[x__] :> test[x]

test[##] &[1, 2, 3]

test[1, 2, 3]

test[1, 2, 3]

In fact these operations succeed even if test has HoldAllComplete.

You could emulate this behavior at least in part by modifying your definition to check for SequenceHold:

myseq /: f_[x___, myseq[y___], z___] /; 
  FreeQ[Attributes @ Unevaluated @ f, SequenceHold] := f[x, y, z]

Unlike the functions above Unevaluated uses Sequence:

test[1, Evaluate@Unevaluated[5, 6], 3]

test[1, Sequence[5, 6], 3]