Proof using.
intros l1; elim l1; simpl in |- *; auto.
intros a1 a2 l [H| H]; auto; case H.
intros a; case a.
intros l H a1 a2 l0; case l0; simpl in |- *; auto.
intros a0; elim l; simpl in |- *; auto.
intuition.
intros a3; case a3; simpl in |- *; auto.
intros p H0; case in_app_or with (1 := H0); auto.
elim l; simpl in |- *; auto.
intuition.
intros a3; case a3; simpl in |- *; auto.
intros p p0 H0; case in_app_or with (1 := H0); auto with datatypes.
intros H1; case H with (1 := H1); intuition.
intros l H a1 a2 l0; case l0; simpl in |- *; auto.
intros a0; elim l; simpl in |- *; auto.
intuition.
intros a3; case a3; simpl in |- *; auto.
intros p H0; case in_app_or with (1 := H0); auto.
elim l; simpl in |- *; auto.
intuition.
intros a3; case a3; simpl in |- *; auto.
intros p p0 H0; case in_app_or with (1 := H0); auto with datatypes.
intros H1; case H with (1 := H1); intuition.
Qed.

(* Specifications: *)
(* Adding in a leaf just creates a singleton list *)
Theorem all_pbleaves_pbleaf :
forall l a1 a2, all_pbleaves (pbadd a1 (pbleaf a2) l) = a1 :: nil.