Section Cover.
Variable A : Type.
Variable eqA_dec : forall a b : A, {a = b} + {a <> b}.
Variable empty : A.
(* 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.
Hint Constructors cover : core.
Proof using.
intros t l1 l2 H; generalize l2; elim H; clear H t l1 l2; auto.
intros t l2 H; rewrite (permutation_one_inv _ _ _ H); auto.
intros l1 l2 t1 t2 t3 H H0 H1 l0 H2.
apply cover_node with (2 := H0).
apply permutation_trans with (2 := H).
apply permutation_sym; auto.
Qed.
intros t1 t2 l1 H; elim H; clear t1 l1 H; simpl in |- *; auto.
intros t; apply cover_node with (l2 := nil (A:=btree A)) (t1 := t2) (t2 := t);
auto.
intros l1 l2 t1 t0 t3 H H0 H1.
apply cover_node with (l2 := t2 :: l2) (t1 := t1) (t2 := t0); auto.
apply permutation_trans with (t2 :: t1 :: t0 :: l2); auto.
apply permutation_trans with (t1 :: t2 :: t0 :: l2); auto.
apply cover_permutation with (1 := H1); auto.
Qed.
Proof using.
intros l t H; case H; simpl in |- *; auto.
intros t0; discriminate.
intros l1 l2 t1 t2 t3 H0 H1; red in |- *; intros H2;
absurd (length l1 = length (t1 :: t2 :: l2)); auto.
rewrite H2; simpl in |- *; intros; discriminate.
apply permutation_length with (1 := H0); auto.
Qed.
Proof using.
intros l; elim l; simpl in |- *; auto.
intros H; case H; auto.
intros a l0; case l0; auto.
intros H H0; exists a; auto.
intros b l1 H H0; case H.
red in |- *; intros; discriminate.
intros x H1; exists (node a x).
apply cover_cons_l; auto.
Qed.
Proof using.
intros l t1 t2 t3 H; generalize t2 t3; elim H; clear H l t1 t2 t3;
auto with datatypes.
simpl in |- *; intros t t2 t3 [H1| H1]; auto.
rewrite H1; auto.
case H1.
intros l1 l2 t1 t2 t3 H H0 H1 t0 t4 H2 H3.
cut (In t0 (t1 :: t2 :: l2)); auto.
simpl in |- *; intros [H4| [H4| H4]]; auto with datatypes.
apply (H1 (node t1 t2)); simpl in |- *; auto.
rewrite H4; auto.
apply (H1 (node t1 t2)); simpl in |- *; auto.
rewrite H4; auto.
apply (H1 t0); simpl in |- *; auto.
apply permutation_in with (1 := H); simpl in |- *; auto.
Qed.
Proof using.
intros l t1 t2 H H0; apply cover_in_inb_inb with (1 := H) (2 := H0); auto.
Qed.
Let cover_inv_leaf_aux :
forall t l, cover l t -> forall a : A, t = leaf a -> l = leaf a :: nil.
Proof using.
intros t l H; elim H; simpl in |- *; auto.
intros t0 a H0; apply f_equal2 with (f := cons (A:=btree A)); auto.
intros.
generalize (H2 a H3); intros; discriminate.
Qed.
Proof using.
intros a l H; (apply cover_inv_leaf_aux with (t := leaf a); auto).
Qed.
Proof using.
intros t1 t2 H; inversion H; auto.
absurd (length (t1 :: nil) = length (t0 :: t3 :: l2)).
simpl in |- *; intros; discriminate.
apply permutation_length with (1 := H0); auto.
Qed.
Proof using.
intros t t1 t2 l H; elim H.
intros t0 Ht0; rewrite Ht0; auto with datatypes.
intros l1 l2 t0 t3 t4 H0 H1 H2 H3; right.
case H2; auto.
intros H4.
exists (t1 :: nil); exists (t2 :: nil); simpl in |- *; repeat (split; auto).
injection H4; intros H5 H6 H7; rewrite <- H5; rewrite <- H6; rewrite <- H7;
auto.
clear H2 H3; intros (l3, (l4, ((Hl1, Hl2), Hl3))).
case (in_app_or l3 l4 (node t0 t3)).
apply permutation_in with (1 := Hl3); auto with datatypes.
intros H2; case in_ex_app with (1 := H2).
intros l5 (l6, Hl5).
exists (t0 :: t3 :: l5 ++ l6); exists l4; repeat (split; auto).
apply cover_node with (l2 := l5 ++ l6) (t1 := t0) (t2 := t3); auto.
apply cover_permutation with (1 := Hl1).
rewrite Hl5.
apply permutation_trans with (node t0 t3 :: l6 ++ l5); auto.
apply (permutation_app_swap _ l5 (node t0 t3 :: l6)).
apply permutation_trans with (1 := H0).
simpl in |- *; repeat apply permutation_skip.
apply permutation_inv with (a := node t0 t3).
apply permutation_trans with (1 := Hl3).
rewrite Hl5; auto.
change
(permutation ((l5 ++ node t0 t3 :: l6) ++ l4)
((node t0 t3 :: l5 ++ l6) ++ l4)) in |- *.
apply permutation_app_comp; auto.
apply permutation_trans with ((node t0 t3 :: l6) ++ l5); auto.
simpl in |- *; auto.
intros H2; case in_ex_app with (1 := H2).
intros l5 (l6, Hl5).
exists l3; exists (t0 :: t3 :: l5 ++ l6); repeat (split; auto).
apply cover_node with (l2 := l5 ++ l6) (t1 := t0) (t2 := t3); auto.
apply cover_permutation with (1 := Hl2).
rewrite Hl5.
apply permutation_trans with (node t0 t3 :: l6 ++ l5); auto.
apply (permutation_app_swap _ l5 (node t0 t3 :: l6)).
apply permutation_trans with (1 := H0).
apply permutation_trans with ((t0 :: t3 :: l5 ++ l6) ++ l3); auto.
simpl in |- *; repeat apply permutation_skip.
apply permutation_inv with (a := node t0 t3).
apply permutation_trans with (1 := Hl3).
rewrite Hl5.
apply permutation_trans with ((l5 ++ node t0 t3 :: l6) ++ l3); auto.
change
(permutation ((l5 ++ node t0 t3 :: l6) ++ l3)
((node t0 t3 :: l5 ++ l6) ++ l3)) in |- *; auto.
apply permutation_app_comp; auto.
apply permutation_trans with ((node t0 t3 :: l6) ++ l5); auto.
simpl in |- *; auto.
Qed.
Proof using.
intros t1 t2 l H; apply cover_inv_app_aux with (t := node t1 t2); auto.
Qed.
Proof using.
intros t1 t2 l1 l2 H1; generalize t2 l2; elim H1; clear t1 t2 l1 l2 H1;
simpl in |- *; auto.
intros t t2 l2 H; apply cover_cons_l; auto.
intros l1 l2 t1 t2 t3 H H0 H1 t0 l0 H2.
apply cover_node with (l2 := l2 ++ l0) (t1 := t1) (t2 := t2); auto.
apply permutation_trans with ((t1 :: t2 :: l2) ++ l0); auto.
Qed.

(* A cover gives a direct account of the number of the tree *)
Theorem cover_number_of_nodes :
forall t l,
cover l t ->
number_of_nodes t =
fold_left (fun x y => x + number_of_nodes y) l 0 + pred (length l).