(* 
  A list of trees is a cover of a tree if all the trees in the
  list delimit a frontier in the tree
*)
Inductive cover : list (btree A) -> btree A -> Prop :=
  | cover_one : forall t, cover (t :: nil) t
  | cover_node :
      forall l1 l2 t1 t2 t3,
      permutation l1 (t1 :: t2 :: l2) ->
      cover (node t1 t2 :: l2) t3 -> cover l1 t3.(* 
  A list of trees is a cover of a tree if all the trees in the
  list delimit a frontier in the tree
*)
Inductive cover : list (btree A) -> btree A -> Prop :=
  | cover_one : forall t, cover (t :: nil) t
  | cover_node :
      forall l1 l2 t1 t2 t3,
      permutation l1 (t1 :: t2 :: l2) ->
      cover (node t1 t2 :: l2) t3 -> cover l1 t3.(* 
  A list of trees is a cover of a tree if all the trees in the
  list delimit a frontier in the tree
*)
Inductive cover : list (btree A) -> btree A -> Prop :=
  | cover_one : forall t, cover (t :: nil) t
  | cover_node :
      forall l1 l2 t1 t2 t3,
      permutation l1 (t1 :: t2 :: l2) ->
      cover (node t1 t2 :: l2) t3 -> cover l1 t3.

Lemma cover_inv_app_aux : forall t t1 t2 l, cover l t -> t = node t1 t2 -> l = node t1 t2 :: nil \/ (exists l1 : _, (exists l2 : _, (cover l1 t1 /\ cover l2 t2) /\ permutation l (l1 ++ l2))).