definitions of the terms "1:1" and "onto" can be found at http://www.mathreference.com/set,onto.html
or http://www.cs.utk.edu/~booth/311-02/notes/notesonfns.html.
In short, a relation R between two sets A={ai} and B={bj} is 1:1 if
for any value ai, there is exactly one value R(ai) = bi.
The relation is "onto" if it has an inverse function R' and the
inverse function is 1:1 (for any bj, R'(bj) = aj), and we can further
say that R is 1:1 and onto if ai = R'(R(ai)).
From that perspective, if the NAT66 device is configured with the
same "inside" and "outside" prefixes, the algorithm is 1:1 and onto -
a datagram sent from Alice through an ISP Bob and converted to Bob's
prefix to any system Carol, and to which Carol replies, will go
through one of the NAT66 devices between Bob and Alice (due to the
prefix), and will ultimately get to Alice using the same address Alice
originally chose.
If the NAT66 devices have different "inside" and "outside" prefixes -
Bob and Bobby are different ISPs and Alice can connect via either -
that observation applies to the set of NAT66 devices between Alice and
Bob, and separately applies to the set of NAT66 devices between Alice
and Bobby. Due to the nature of routing, we cannot presume that the
reply to a datagram bearing Bob's prefix would go to Bobby; the
routing system will deliver it to Bob. So the fact that the relation
doesn't work going out through Bob and returning through Bobby is
irrelevant.
So I will argue that with respect to any communication between Alice
and Carol, the relation defined in NAT66 is 1:1 and onto.
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