Section splits_and_perms.
Context {T: Type}.
Fixpoint splits (l: list T): list (T * list T) :=
match l with
| nil => nil
| h :: t => (h, t) :: map (fun xy => (fst xy, h :: snd xy)) (splits t)
end.
Proof with auto.
induction l...
simpl.
rewrite map_length.
intuition.
Qed.

Lemma splits_are_perms l p: In p (splits l) -> Permutation (fst p :: snd p) l.