Proof using.
intros c; elim c; unfold pbbuild in |- *; simpl in |- *; auto.
intros (a1, l1) l; case l; auto.
simpl in |- *; intros H H0; apply distinct_pbleaves_pbadd_prop1; auto.
intros (a2, l2) l3 H0 H1; apply distinct_pbleaves_pbadd_prop2; auto.
change (~ inpb (pbleaf a1) (pbbuild ((a2, l2) :: l3))) in |- *.
red in |- *; intros H2; case inpb_pbbuild_inv with (2 := H2).
red in |- *; intros H; discriminate.
intros x H; case unique_key_in with (b2 := x) (1 := unique_prefix2 _ _ H1);
auto.
apply H0.
apply unique_prefix_inv with (1 := H1); auto.
Qed.

(* The composition of building and computing on trees with distinct leaves is the identity *)
Theorem pbbuild_compute_peq :
forall t, distinct_pbleaves t -> pbbuild (compute_pbcode t) = t.