What's a sequential part of an n-tuple called?

Say I have an ordered tuple $T = (x_1,x_2,x_3,x_4)$. I want to create new tuples that are only valid if they satisfy the following properties:

1. The first element of the tuple must be $x_1$.
2. The successor of each element $x_i$ in the tuple must be the successor of element $x_i$ in $T$.

For example, $T' = (x_1)$ and $T'' = (x_1,x_2,x_3)$ are valid tuples that satisfy the above properties. However, $T''' = (x_1,x_3,x_4)$ is an example that does not satisfy the properties because the successor of $x_1$ should be $x_2$, not $x_3$.

Is there a name or formal definition for these types of tuples?

The word you are looking for is "prefix".