<s>[INST] A question regarding case notation Consider the parameter $P$ and a property $S$ associated with this parameter represented as $P_S$. For example, consider that the height of a tree (“H”) is my parameter and the property is the tree’s name. Therefore, $H_{T1}$ refers to the height of the tree named $T1$.
Now, consider that $H_{T1}$ depends on 2 inputs, named $A$ and $B$ respectively. So, for the above example we have:
$$
H_{T1}^{A,B}
$$
My first concern is whether the above is a valid notation or should I have written something else like $H_{T1}(A,B)$?
I am then using the cases environment in order to define the different value that $H_{T1}$ takes for a number of differetn $A, B$ inputs:
$$
\begin{eqnarray}
H_{T1}^{A,B} =
    \begin{cases}
        H_{T1}^{5,8}          & \text{if $A=5$, $B=8$}\\
        H_{T1}^{9,1}          & \text{if $A=5$, $B=1$}\\
        H_{T1}^{3,3}          & \text{if $A=3$, $B=3$}\\
    \end{cases}
\end{eqnarray}
$$
For a particular set of $A,B$ inputs, e.g. for the set $(A=3, B=3)$, $H_{T1}$ depends on $2$ extra inputs, named $C$ and $D$ respectively like that:
$$
\begin{eqnarray}
H_{T1}^{3,3,C,D} =
    \begin{cases}
        H_{T1}^{3,3,1,1}          & \text{if $A=3$, $B=3$, $C=1$, $D=1$}\\
        H_{T1}^{3,3,4,1}          & \text{if $A=3$, $B=3$, $C=4$, $D=1$}\\
        H_{T1}^{3,3,6,2}          & \text{if $A=3$, $B=3$, $C=6$, $D=2$}\\
    \end{cases}
\end{eqnarray}
$$
Since only for $H_{T1}^{3,3}$ I have the 2 extra inputs, what would be the best way of writing that? I thought of writing something like that:
$$
\begin{eqnarray}
H_{T1}^{A,B} =
    \begin{cases}
        H_{T1}^{5,8}          & \text{if $A=5$, $B=8$}\\
        H_{T1}^{9,1}          & \text{if $A=5$, $B=1$}\\
        H_{T1}^{3,3,C,D}      & \text{if $A=3$, $B=3$}\\
    \end{cases}
\end{eqnarray}
$$
where
$$
\begin{eqnarray}
H_{T1}^{3,3,C,D} =
    \begin{cases}
        H_{T1}^{3,3,1,1}          & \text{if $A=3$, $B=3$, $C=1$, $D=1$}\\
        H_{T1}^{3,3,4,1}          & \text{if $A=3$, $B=3$, $C=4$, $D=1$}\\
        H_{T1}^{3,3,6,2}          & \text{if $A=3$, $B=3$, $C=6$, $D=2$}\\
    \end{cases}
\end{eqnarray}
$$
but I am not sure whether it’s correct or not.
Thanks in advance.
 [/INST] You could use $H_{T1,A,B}$, or $H_{T1;A,B}$. Your idea for your function is almost right, but you've included some superfluous information. Try 
$H_{T1,A,B,C,D}=\cases{a, \text{if } A=1,B=1,∀C,D,\\b, \text{if }A=2,B=2, ∀ C,D,\\c, \text{if } A=3,B=3,C=1,D=2,\\d, \text{if } A=3,B=3,C=1,D=3}$
or use your simplifying trick and say
$H_{T1,A,B,C,D}=\cases{a, \text{if } A=1,B=1,\\b, \text{if }A=2,B=2, \\ H_{T1,3,3,C,D}, \text{if } A=3,B=3, \text{where}}$
$H_{T1,3,3,C,D}=\cases{c, \text{if } C=1,D=2,\\d, \text{if } C=1,D=3.}$
However, you can make clearer what is going on by presenting $H_{T1,A,B},A,B\not=3$ as a special case of $H_{T1,A,B,C,D}. Thus,
$H_{T1,A,B,C,D}=\cases{H_{T1,A,B}, \text{if } (A,B)\not=(3,3),\\c, \text{if } A=B=3,C=1,D=2,\\d, \text{if } A=B=3,C=1,D=3.}$
</s>