Section PBTREE2BTREE.
Variable A : Type.
Variable eqA_dec : forall a b : A, {a = b} + {a <> b}.
Variable empty : A.
(* Turn a partial tree into a binary tree *)
Fixpoint to_btree (a : pbtree A) : btree A :=
match a with
| pbleaf b => leaf b
| pbleft l1 => to_btree l1
| pbright l1 => to_btree l1
| pbnode l1 l2 => node (to_btree l1) (to_btree l2)
end.

(* Computing the binary tree preserves the leaves*)
Theorem to_btree_inb :
forall a b, inpb (pbleaf a) b -> inb (leaf a) (to_btree b).