What are the widely recognised notations to define a function that replaces the n-th element in tuple?
Let suppose that I want to formally define a function $r_3: N^8 \times N \rightarrow N^8$ informally defined as $r_3(t,a)$ is equal to the tuple $t$ with the 3rd element replaced by $a$.
Are all of the following notations widely accepted?
Is there any additional widely accepted notation that I have not listed?
- $f(t,a) \equiv (t[1],t[2],a,t[4],t[5],t[6],t[7],t[8])$
- $f(t,a) \equiv (\pi_1(t),\pi_2(t),a,\pi_4(t),\pi_5(t),\pi_6(t),\pi_7(t),\pi_8(t))$
- $f(t,a) \equiv (t_1, t_2, a, t_4, t_5, t_6, t_7, t_8)\quad \text{if } t=(t_1, t_2, t_3, t_4, t_5, t_6, t_7, t_8) $