Good morning,

> * Sub-payment - one or more attempts, each of which semantically pay
for "the same thing" for "the same value", set up in parallel.
> a sub-payment may have multiple attempts running in parallel, but only
one attempt will be claimable.
> * Payment - one or more sub-payments, each of which semantically pay
for "different parts of an entire thing", set up in parallel.
> a payment may have multiple sub-payments, and all sub-payments will be
claimed atomically.

This can be also thought of as:

Payment = ONE-OF(attempt_11, ..., attempt_m1) AND ... AND
ONE-OF(attempt_n1, ..., attempt_m'n)

Its dual also deserves some thought:

Payment = ONE-OF(attempt_11 AND ... AND attempt_m1), ..., (attempt_n1
AND ... AND attempt_m'n))

or in words, "A payment is an atomic value transfer through many paths,
each of which carry a part of the entire value -- many alternative
groups of paths are available to be used, but only one of the groups
eventually goes through."

Is there a reason to design in preference of one of the two?

Speaking of generalization, it would be nice to have arbitrary AND-OR
combinations, but this needs further exploration:

> If we want to create more complex access structures then we use
verifiable secret sharing where the discrete log of B is split up into
shares and distributed according the the desired structure.

One possible milestone of this generalisation would be to enable atomic
payments where the paying wallet says "there are all these known paths,
each with such and such capacity; I want some to go through such that
the desired value is transferred in aggregate, no more, no less
(possibly within a margin of error)".

Kindly ignore me if I'm regurgitating already discussed stuff.

Best,
Orfeas

-- 
The University of Edinburgh is a charitable body, registered in
Scotland, with registration number SC005336.

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