What type of widely accepted mathematical notations are available to access tuple components?
For exmaple, let suppose that I want to define a predicate $P(t,a)$ to be satisfied when the 3rd element of the 8-dimensional tuple $t$ is equal to $a$.
Are all of the following notations widely accepted?
Is there any additional widely accepted notation that I have not listed?
- $P(t,a) \equiv t[3]=a$
- $P(t,a) \equiv t(3)=a$
- $P(t,a) \equiv \exists b,c,e,f,g,h,i \in N:(b,c,a,e,f,g,h,i)=t$
- $P(t,a) \equiv t = (\ldots,\ldots,a,\ldots,\ldots,\ldots,\ldots,\ldots)$
- $P((b,c,d,e,f,g,h,i),a) \equiv t=d$
I also thought about using the following notation. First I define the operator $.third$ which look similar to object oriented programming syntax, then I use the operator in the definition of the predicate:
$\begin{align*} &\forall t=(b,c,d,e,f,g,h,i) \in N^8 \\ & \quad t.third \equiv d \\ & \\ &P(t,a) \equiv t.third = a \end{align*}$
Is the notation above correct and used anywhere?
If not, is there a way to refer to tuple parameters by "name" rather than index?