# The shuffle pasting, IV

This is the fourth in a series of talks investigating the shuffle
pasting. The goal is to show that shuffles form a well-formed
loop-free pasting scheme, and to characterize well-formed subpasting
schemes.
For an n-stage (p,q)-shuffle collection
,
the following are equivalent:

- fills
- is fit
- R ()
is well formed.

After recalling the two order relations, I proved (most of) the
difficult implication, that fitness implies filling. The crucial
thing to show is that if an n-kind is needed for one bounding
(n-1)-kind then it is needed for all. In this, a lemma characterizing
the <|-order of elements in a filling and fit collection in terms
of whether the relevant swap has or has not been carried out in
*one* of them is instrumental.
See sections 8-9 of the paper ``The shuffle pasting'' (in preparation).

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