Section PBTree.
Variable A : Type.
Variable empty : A.
Variable eqA_dec : forall a b : A, {a = b} + {a <> b}.
(* Partial Binary Tree (no more than 2 sons *)
Inductive pbtree : Type :=
| pbleaf : A -> pbtree
| pbleft : pbtree -> pbtree
| pbright : pbtree -> pbtree
| pbnode : pbtree -> pbtree -> pbtree.
Proof using.
intros p; case p; simpl in |- *;
try (intros; right; red in |- *; intros; discriminate).
intros a; left; exists a; simpl in |- *; auto.
Qed.
(* Predicate for belonging *)
Inductive inpb : pbtree -> pbtree -> Prop :=
| inpb_leaf : forall t : pbtree, inpb t t
| inpb_left : forall t t1 t2 : pbtree, inpb t t1 -> inpb t (pbleft t1)
| inpb_right : forall t t1 t2 : pbtree, inpb t t1 -> inpb t (pbright t1)
| inpb_node_l : forall t t1 t2 : pbtree, inpb t t1 -> inpb t (pbnode t1 t2)
| inpb_node_r : forall t t1 t2 : pbtree, inpb t t2 -> inpb t (pbnode t1 t2).
Hint Constructors inpb : core.
(* Equality on partial trees is decidable *)
Definition pbtree_dec : forall a b : pbtree, {a = b} + {a <> b}.
intros a; elim a; simpl in |- *; auto.
intros a0 b; case b; try (intros; right; red in |- *; intros; discriminate).
intros a1; case (eqA_dec a0 a1); intros H1.
left; rewrite H1; auto.
right; red in |- *; Contradict H1; inversion H1; auto.
intros p H b; case b; try (intros; right; red in |- *; intros; discriminate).
intros p1; case (H p1).
intros e; rewrite e; auto.
intros H1; right; Contradict H1; inversion H1; auto.
intros p H b; case b; try (intros; right; red in |- *; intros; discriminate).
intros p1; case (H p1).
intros e; rewrite e; auto.
intros H1; right; Contradict H1; inversion H1; auto.
intros p H p0 H0 b; case b;
try (intros; right; red in |- *; intros; discriminate).
intros p1 p2; case (H p1); intros H1.
case (H0 p2); intros H2.
left; rewrite H1; rewrite H2; auto.
right; Contradict H2; injection H2; auto.
right; Contradict H1; injection H1; auto.
Defined.
(* Belonging is decidable *)
Definition inpb_dec : forall a b, {inpb a b} + {~ inpb a b}.
intros a b; elim b.
intros a0; case a;
try (intros; right; red in |- *; intros HH; inversion HH; auto; fail).
intros a1; case (eqA_dec a1 a0); intros HH.
left; rewrite HH; auto.
right; Contradict HH; inversion HH; auto.
intros p Hp; case Hp; auto; intros HH.
case (pbtree_dec a (pbleft p)); intros HH1.
left; rewrite HH1; auto.
right; red in |- *; intros HH2; generalize HH HH1; inversion HH2; auto.
intros p Hp; case Hp; auto; intros HH.
case (pbtree_dec a (pbright p)); intros HH1.
left; rewrite HH1; auto.
right; red in |- *; intros HH2; generalize HH HH1; inversion HH2; auto.
intros p H p0 H0.
case H; auto; intros H1.
case H0; auto; intros H2.
case (pbtree_dec a (pbnode p p0)); intros HH1.
left; rewrite HH1; auto.
right; red in |- *; intros HH2; generalize H1 H2 HH1; inversion HH2; auto.
Defined.
Proof using.
intros t1 t2 t3 H H1; generalize t1 H; elim H1; clear H H1 t1 t2 t3; auto.
Qed.
Proof using.
intros t; elim t; simpl in |- *; auto.
intros a; exists a; auto.
intros b (a, H); exists a; auto.
intros b (a, H); exists a; auto.
intros b (a, H) b0 H0; exists a; auto.
Qed.
(* Predicate for Partial Trees with Different Leaves *)
Definition distinct_pbleaves (t : pbtree) : Prop :=
forall t0 t1 t2 : pbtree,
inpb (pbnode t1 t2) t -> inpb t0 t1 -> inpb t0 t2 -> False.
Proof using.
intros a; red in |- *.
intros a0 t1 t2 H; inversion H.
Qed.
Hint Resolve distinct_pbleaves_Leaf : core.
Proof using.
intros t1 t2 H; red in |- *.
intros a t0 t3 H0 H1 H2.
apply (H a t0 t3); auto.
Qed.
Proof using.
intros t1 t2 H; red in |- *.
intros a t0 t3 H0 H1 H2.
apply (H a t0 t3); auto.
Qed.
Proof using.
intros t1 H; red in |- *.
intros a t0 t3 H0 H1 H2.
apply (H a t0 t3); auto.
Qed.
Proof using.
intros t1 H; red in |- *.
intros a t0 t3 H0 H1 H2.
apply (H a t0 t3); auto.
Qed.
Proof using.
intros a; red in |- *.
intros a0 t1 t2 H; inversion H.
Qed.
Hint Resolve distinct_pbleaves_pbleaf : core.
Proof using.
intros t H; red in |- *.
intros a t1 t2 H0 H1 H2; apply (H a t1 t2); auto.
inversion H0; auto.
Qed.
Proof using.
intros t H; red in |- *.
intros a t1 t2 H0 H1 H2; apply (H a t1 t2); auto.
inversion H0; auto.
Qed.
Hint Resolve distinct_pbleaves_pbleft distinct_pbleaves_pbright : core.
(* Transform a tree in a code *)
Fixpoint compute_pbcode (a : pbtree) : code A :=
match a with
| pbleaf b => (b, nil) :: nil
| pbleft l1 =>
map
(fun v : A * list bool =>
match v with
| (a1, b1) => (a1, false :: b1)
end) (compute_pbcode l1)
| pbright l1 =>
map
(fun v : A * list bool =>
match v with
| (a1, b1) => (a1, true :: b1)
end) (compute_pbcode l1)
| pbnode l1 l2 =>
map
(fun v : A * list bool =>
match v with
| (a1, b1) => (a1, false :: b1)
end) (compute_pbcode l1) ++
map
(fun v : A * list bool =>
match v with
| (a1, b1) => (a1, true :: b1)
end) (compute_pbcode l2)
end.
Proof using.
intros p; elim p; simpl in |- *; auto;
try (intros p0; case (compute_pbcode p0); simpl in |- *; auto);
intros; red in |- *; intros HH1; discriminate.
Qed.
Hint Resolve compute_pbcode_not_null : core.
Proof using.
intros t; elim t; simpl in |- *; auto.
intros a a0 l [H1| H1]; try (case H1; fail).
injection H1; intros H2 H3; rewrite H3; auto.
intros p H a l H0; apply inpb_left; auto.
case (in_map_inv _ _ _ _ _ H0).
intros (a1, l1) (Ha1, Hl1); apply (H a l1); auto.
injection Hl1; intros HH1 HH2; rewrite HH2; auto.
intros p H a l H0; apply inpb_right; auto.
case (in_map_inv _ _ _ _ _ H0).
intros (a1, l1) (Ha1, Hl1); apply (H a l1); auto.
injection Hl1; intros HH1 HH2; rewrite HH2; auto.
intros h H h0 H0 a l H1.
case in_app_or with (1 := H1); auto; intros H2.
case (in_map_inv _ _ _ _ _ H2).
intros (a1, l1) (Ha1, Hl1); auto.
apply inpb_node_l; apply (H a l1).
injection Hl1; intros Hl2 Hl3; rewrite Hl3; auto.
case (in_map_inv _ _ _ _ _ H2).
intros (a1, l1) (Ha1, Hl1); auto.
apply inpb_node_r; apply (H0 a l1).
injection Hl1; intros Hl2 Hl3; rewrite Hl3; auto.
Qed.
Proof using.
intros a p; elim p; simpl in |- *; auto.
intros a0 H; inversion H.
exists (nil (A:=bool)); auto.
intros p0 H H0; case H; auto.
inversion H0; auto.
intros x1; elim (compute_pbcode p0); simpl in |- *; auto.
intros HH; case HH.
intros a0 l H1 [H2| H2]; auto.
exists (false :: x1); left; rewrite H2; auto.
case H1; auto.
intros x H3; exists x; auto.
intros p0 H H0; case H; auto.
inversion H0; auto.
intros x1; elim (compute_pbcode p0); simpl in |- *; auto.
intros HH; case HH.
intros a0 l H1 [H2| H2]; auto.
exists (true :: x1); left; rewrite H2; auto.
case H1; auto.
intros x H3; exists x; auto.
intros p0 H p1 H0 H1; inversion H1.
case H; auto.
intros x Hx; exists (false :: x).
apply in_or_app; left; auto.
change
(In ((fun v => match v with
| (a1, b1) => (a1, false :: b1)
end) (a, x))
(map (fun v => match v with
| (a1, b1) => (a1, false :: b1)
end) (compute_pbcode p0))) in |- *;
apply in_map; auto.
case H0; auto.
intros x Hx; exists (true :: x).
apply in_or_app; right; auto.
change
(In ((fun v => match v with
| (a1, b1) => (a1, true :: b1)
end) (a, x))
(map (fun v => match v with
| (a1, b1) => (a1, true :: b1)
end) (compute_pbcode p1))) in |- *;
apply in_map; auto.
Qed.

(* The computing code has the first property to be prefix *)
Theorem pb_unique_prefix1 :
forall (t : pbtree) (a1 a2 : A) (lb1 lb2 : list bool),
In (a1, lb1) (compute_pbcode t) ->
In (a2, lb2) (compute_pbcode t) -> is_prefix lb1 lb2 -> a1 = a2 :>A.