I just realised that I have a very trivial/basic question concerning the syntax of Lambda Calculus.
Question 1: Is possible to have brackets between a lambda symbol, i.e., "$\lambda$" and a dot "$.$"?
That is, is possible to have such an expression: $$\lambda x (yz) p. M?$$
I would say yes, since we have to read the $(yz)$ in the example as one term. Thus, by considering two examples, in the first we have $$(\lambda x (yz) p. xyxzp)abc \rightarrow_\beta ayazc,$$ while in the second $$(\lambda x (yz) p. x(yz)p)abc \rightarrow_\beta abc,$$ since we are treating $yz$ as a single term, i.e., with $N \equiv yz$, we have $$(\lambda x (yz) p. x(yz)p)abc = (\lambda x N p. xNp)abc.$$
However, this leads me to a second question, namely:
Question 2: Is $$(\lambda x (yz) p. x(yz)p)abc$$ the same as $$(\lambda x (yz) p. xyzp)abc?$$
This time I would answer no, since an ideal machine would consider the $y$ and the $z$ as two separate terms in $(\lambda x (yz) p. xyzp)abc$. Hence, we would have $$(\lambda x (yz) p. xyzp)abc \rightarrow_\beta ayzc.$$
Thus, is my reasoning correct?
As always, any feedback will be greatly appreciated.
Thank you for your time.