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(* 02110-1301 USA                                                     *)
(********************************************************************** Proof of Huffman algorithm: SubstPred.v  Definition of the substitution in a tree with respect to a cover  Definition: subst_pred  Laurent.Thery@inria.fr (2003) **********************************************************************)
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.

(* Specifications: *)
(* The first cover of the substitution is an ordered cover *)
Theorem subst_pred_ordered_cover_l :
forall (t1 t2 : btree A) (l1 l2 : list (btree A)),
subst_pred l1 l2 t1 t2 -> ordered_cover l1 t1.