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(* 02110-1301 USA                                                     *)
(********************************************************************** Proof of Huffman algorithm: CoverMin.v  Definition of a minimal tree for a cover  Defintion: cover_min Laurent.Thery@inria.fr (2003) **********************************************************************)
From Huffman Require Export Cover.
From Huffman Require Export WeightTree.
Section CoverMin.
Variable A : Type.
Variable f : A -> nat.
(* To be a tree of minimum weight for a cover *)
Definition cover_min (l : list (btree A)) (t1 : btree A) : Prop :=
cover l t1 /\
(forall t2 : btree A, cover l t2 -> weight_tree f t1 <= weight_tree f t2).

(* Minimum tree for a singleton cover *)
Theorem cover_min_one : forall t : btree A, cover_min (t :: nil) t.
Proof using.