(* 
  A predicate that associates an initial height, a list of
  height, a cover and a tree
*)
Inductive height_pred : nat -> list nat -> list (btree A) -> btree A -> Prop :=
  | height_pred_nil :
      forall (n : nat) (t : btree A), height_pred n (n :: nil) (t :: nil) t
  | height_pred_node :
      forall (n : nat) (ln1 ln2 : list nat) (t1 t2 : btree A)
        (l1 l2 : list (btree A)),
      height_pred (S n) ln1 l1 t1 ->
      height_pred (S n) ln2 l2 t2 ->
      height_pred n (ln1 ++ ln2) (l1 ++ l2) (node t1 t2).(* 
  A predicate that associates an initial height, a list of
  height, a cover and a tree
*)
Inductive height_pred : nat -> list nat -> list (btree A) -> btree A -> Prop :=
  | height_pred_nil :
      forall (n : nat) (t : btree A), height_pred n (n :: nil) (t :: nil) t
  | height_pred_node :
      forall (n : nat) (ln1 ln2 : list nat) (t1 t2 : btree A)
        (l1 l2 : list (btree A)),
      height_pred (S n) ln1 l1 t1 ->
      height_pred (S n) ln2 l2 t2 ->
      height_pred n (ln1 ++ ln2) (l1 ++ l2) (node t1 t2).(* 
  A predicate that associates an initial height, a list of
  height, a cover and a tree
*)
Inductive height_pred : nat -> list nat -> list (btree A) -> btree A -> Prop :=
  | height_pred_nil :
      forall (n : nat) (t : btree A), height_pred n (n :: nil) (t :: nil) t
  | height_pred_node :
      forall (n : nat) (ln1 ln2 : list nat) (t1 t2 : btree A)
        (l1 l2 : list (btree A)),
      height_pred (S n) ln1 l1 t1 ->
      height_pred (S n) ln2 l2 t2 ->
      height_pred n (ln1 ++ ln2) (l1 ++ l2) (node t1 t2).

Lemma height_pred_disj_larger_aux : forall (n : nat) (ln : list nat) (t : btree A) (l : list (btree A)), height_pred n ln l t -> forall ln1 ln2 a, ln = ln1 ++ a :: ln2 -> (forall n1 : nat, In n1 ln1 -> n1 < a) -> (forall n1 : nat, In n1 ln2 -> n1 <= a) -> (exists ln3 : _, ln2 = a :: ln3) \/ ln = n :: nil /\ l = t :: nil.