8. Nodes access data by providing index bindings for all indices they destroy.
9. Nodes set data by providing index bindings for all indices they create.
+
+Case Study
+----------
+
+An inner product between an entirely dense 3D data-set and one stored in PPD format.
+
+1. Let's say that an inner product doesn't entirely destroy its indices.
+
+2. Instead, the indices it produces are a composite of its input indices.
+
+3. Now both inputs generate indices that are not destroyed (sort of).
+
+4. We can now fuse provided that we can generate loops for the fused
+iteration.
+
+4. However, we only fuse on common prefixes, but the Function Set
+has the PPD-iteration loop as its outermost loop.
+
+5. We chose the common-prefix law in order to avoid redundant
+computation of values, but the PPD loop doesn't do this. The subsets the
+PPD loop iterates over are disjoint.
+
+6. Both block matrices and the PPD-representation have these external
+sorts of indices. Both iterate over disjoint subsets of the data. It
+would be nice to have more examples to fully characterise them.
+
+7. So when are we allowed to move the PPD iteration loop? Why is this
+such an issue? If in the integrals_kinetic example, we were able to move
+the PPD iteration loop outside our assignment loop (fixed # of PPDs per
+sphere), it would destroy everything. Clearly, it must be constrained
+somehow.
+
+8. The work by Kotlyar et al. focusses on how to construct the set of
+tuples required for iterations, but now how they are to be reduced. The
+PPD-loop cannot be moved because the reductions are non-trivial for a
+number of the quantum chemistry operators (e.g. FFT).
+
+9. All loop re-arrangements must respect two restrictions:
+ - Loops cannot move outside the loops they depend on (the obvious one).
+ - Loops cannot move above a node that destroys them. Why?
+ It represents some form of implicit reduction over that index?
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