More idempotents

The main new result is that the type of retracts of `X` is a retract of the type of partially-coherent idempotents on `X` (here called "quasi-idempotents" by analogy with "quasi-inverse"), with the retraction being the splitting construction.  The witness `I : f o f == f` can be recovered from the splitting (`f` with this datum is here called a "pre-idempotent"), but not the additional coherence `J` in a quasi-idempotent.  A counterexample to the latter is in progress.  I'm planning a second blog post to go along with this, but probably not until I have that counterexample formalized too.
Hmm.  I find it easier to think of the whole thing as one theorem.  For an intermediate lemma with a path between types rather than an equivalence, the types of the other data `rp`, `sp`, and `Hp` would "mix" paths with functions, and I'd have to think a while to figure out what they should be.  I find it easier to write down what the end result should be and then let Coq tell me how it computes out under the transport lemmas.