I'm looking for a notation that indicates the elements in a set should be expanded in some form in an expression.
For example:
Let set $X=\{x_1, x_2, ..., x_k\}$ be the set of all Boolean inputs to a Boolean function $F(X)$.
I'd like to define a Boolean differential calculus expression like the following:
$$\bigvee_{I\in\mathcal{P}(\{1, 2, ..., k\})}\left(\frac{\partial F(X)}{\partial((x_{I})...)}\land\left(\bigwedge_{i \in I}dx_i\right)\right)$$
... where the "$(x_I)...$" in $\partial((x_I)...)$ indicates that the elements in the set I (which in this example are being using as indices) should be expanded in the expression, as follows:
If $F(X)$ were to be specified as $F(x_1, x_2)$, then the expression above should equate to:
$$ \left(\frac{\partial F(X)}{\partial(x_1)}\land(dx_1)\right) \lor\left(\frac{\partial F(X)}{\partial(x_2)}\land(dx_2)\right) \lor\left(\frac{\partial F(X)}{\partial(x_1, x_2)}\land(dx_1 \land dx_2)\right) $$
I'm guessing using "$(x_I)...$" isn't the right way to do it (?).
In this case, it's almost like a "big comma" operator is needed, similar to $\sum$, $\prod$, etc., though I'm also hoping for something more compact and generic.
Edit: Changed my straw man expansion syntax slightly from "$x_{I...}$" to "$(x_I)...$"