Set Implicit Arguments.
Unset Strict Implicit.
Unset Printing Implicit Defensive.
(* LL: TODO: Add proof obligation that the result paths be prefix free? *)
Class CoArbitrary (A : Type) : Type :=
{
coarbitrary : A -> positive;
coarbReverse : positive -> option A;
coarbCorrect : forall a, coarbReverse (coarbitrary a) = Some a
}.
Instance coArbPos : CoArbitrary positive.

Lemma nat_lemma :   forall a : nat,
Some (Init.Nat.pred (Pos.to_nat (Pos.of_nat (S a)))) = Some a.