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.
Proof using.
intros t l H; elim H; clear H t l; simpl in |- *; auto.
intros l1 l2 t1 t2 t3 H H0 H1.
apply trans_equal with (1 := H1).
rewrite fold_left_permutation with (2 := H); simpl in |- *; auto.
rewrite permutation_length with (1 := H); simpl in |- *; auto.
rewrite fold_left_init with (h := S); simpl in |- *; auto.
intros a b1 b2; repeat rewrite plus_assoc_reverse.
apply f_equal2 with (f := plus); auto; apply plus_comm.
Qed.
(* A local function to compute all covers of a certain length *)
Fixpoint all_cover_aux (l : list (btree A)) (n : nat) {struct n} : list (btree A) :=
match n with
| O => nil
| S n1 =>
flat_map
(fun l1 =>
match l1 with
| nil => nil
| a :: nil => a :: nil
| a :: b :: l2 => all_cover_aux (node a b :: l2) n1
end) (all_permutations l)
end.
(* A function to compute all covers *)
Definition all_cover l := all_cover_aux l (length l).
Proof using.
intros n; elim n; simpl in |- *; auto.
intros l t H H0; elim H0.
intros n0 H l t H0 H1.
case in_flat_map_ex with (1 := H1); clear H1.
intros x; case x; clear x.
simpl in |- *; intros (H1, H2); case H2.
intros b x; case x; clear x.
simpl in |- *; intros (H1, [H2| H2]).
rewrite <- H2.
rewrite permutation_one_inv with (a := b) (l := l); auto.
apply all_permutations_permutation; auto.
case H2.
intros b1 l1 (H1, H2).
apply cover_node with (l2 := l1) (t1 := b) (t2 := b1); auto.
apply permutation_sym; apply all_permutations_permutation; auto.
apply H; auto.
apply eq_add_S; apply trans_equal with (1 := H0).
apply trans_equal with (length (b :: b1 :: l1)); auto.
apply permutation_length.
apply permutation_sym; apply all_permutations_permutation; auto.
Qed.
Proof using.
intros l t H; apply all_cover_aux_cover with (n := length l); auto.
Qed.
intros n; elim n; simpl in |- *; auto.
intros l; case l; simpl in |- *; auto.
intros t H H0; inversion H0.
generalize (permutation_nil_inv _ _ (permutation_sym _ _ _ H1)); intros;
discriminate.
intros; discriminate.
intros n0 H l t H0 H1; inversion H1.
simpl in |- *; auto.
apply in_flat_map with (t1 :: t2 :: l2); auto.
apply H; auto.
apply eq_add_S; apply trans_equal with (1 := H0).
apply trans_equal with (length (t1 :: t2 :: l2)); auto.
apply permutation_length; auto.
apply permutation_all_permutations; auto.
apply permutation_sym; auto.
Qed.
Proof using.
intros l t H; unfold all_cover in |- *; apply cover_all_cover_aux; auto.
Qed.
(* Covers are decidable *)
Definition cover_dec :
(forall a b : A, {a = b} + {a <> b}) ->
forall l t, {cover l t} + {~ cover l t}.
intros H l t; case (In_dec (btree_dec _ H) t (all_cover l)).
intros H1; left; apply all_cover_cover; auto.
intros H1; right; Contradict H1; apply cover_all_cover; auto.
Defined.

(* All the leaves of the tree gives a cover to the tree. *)
Theorem cover_all_leaves :
forall t : btree A, cover (map (fun x : A => leaf x) (all_leaves t)) t.