Section SubstPred.
Variable A : Type.
(* Take two covers an substitute the elements of one by the element of the other *)
Inductive subst_pred :
list (btree A) -> list (btree A) -> btree A -> btree A -> Prop :=
| subst_pred_id :
forall (t1 t2 : btree A) (l1 l2 : list (btree A)),
subst_pred (t1 :: nil) (t2 :: nil) t1 t2
| subst_pred_node :
forall (t1 t2 t3 t4 : btree A) (l1 l2 l3 l4 l5 l6 : list (btree A)),
subst_pred l1 l2 t1 t2 ->
subst_pred l3 l4 t3 t4 ->
subst_pred (l1 ++ l3) (l2 ++ l4) (node t1 t3) (node t2 t4).
Hint Resolve subst_pred_id subst_pred_node : core.
Proof using.
intros t1 t2 l1 l2 H; elim H; auto.
Qed.
Proof using.
intros t1 t2 l1 l2 H; elim H; auto.
Qed.
Proof using.
intros t1 t2 l1 l2 H; elim H; auto.
intros t0 t3 t4 t5 l0 l3 l4 l5 l6 l7 H0 H1 H2 H3; repeat rewrite length_app; auto.
Qed.
Proof using.
intros t1 l1 l2 H; generalize l2; elim H; clear t1 l1 l2 H.
intros t l l2; case l2.
simpl in |- *; intros; discriminate.
intros b l0; case l0; simpl in |- *; auto.
intros H; exists b; auto.
intros; discriminate.
intros t1 t2 l1 l2 l3 H H0 H1 H2 l0 H3.
case (H0 (first_n l0 (length l1))); auto.
rewrite first_n_length; auto;
(rewrite <- H3; rewrite length_app; auto with arith).
intros t4 HH1.
case (H2 (skip_n l0 (length l1))); auto.
rewrite skip_n_length; auto;
(rewrite <- H3; rewrite length_app; rewrite minus_plus; auto with arith).
intros t5 HH2.
exists (node t4 t5); auto.
rewrite <- (first_n_skip_n_app _ (length l1) l0); auto.
Qed.

(* A height predicate can be completed in a substitution *)
Theorem height_pred_subst_pred :
forall (n : nat) (ln : list nat) (t1 : btree A) (l1 l2 : list (btree A)),
height_pred n ln l1 t1 ->
length l1 = length l2 ->
exists t2 : btree A, height_pred n ln l2 t2 /\ subst_pred l1 l2 t1 t2.