I am formalizing a system that support pair of tuple and nat as input for function. However, I have some problem with the pair type in Coq.
From what I have learnt about Coq, pair is defined to be left-associative. i.e. (1, 2, 3)and ((1, 2), 3) have the same type of nat * nat * nat. Sometimes, it can be treated as a 3-tuple of nat. One the other hand, (1, (2, 3)) has type nat * (nat * nat), which is a pair of nat and pair (or 2-tuple).
Idealy, I want to have a system that allows pair of nat and tuple and pair of tuple and nat. i.e. Both (1, (2, 3)) and ((1, 2), 3) are differ from (1, 2, 3). Is there any suggested datatype in Coq for such model?
(1, 2, 3) is just a nice notation for ((1, 2), 3), and nat * nat * nat is just a nice notation for (nat * nat) * nat. So this is already what you want.
If you want to see the parentheses, you should tick View -> Display parentheses in CoqIDE.
Note that if you wanted triple not encoded as pairs, you could define them as:
Record triple (A B C : Type): Type :=
{
t_fst : A;
t_snd : B;
t_trd : C
}