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(* 02110-1301 USA                                                     *)
(********************************************************************** Proof of Huffman algorithm: PBTree2BTree.v  Definition of translation from partial trees to binary trees  Definition: to_btree  Laurent.Thery@inria.fr (2003) **********************************************************************)
From Huffman Require Export Aux.
From Huffman Require Export Code.
From Huffman Require Export Build.
From Huffman Require Export ISort.
Require Export Compare_dec.
From Huffman Require Export Permutation.
From Huffman Require Export UniqueKey.
From Huffman Require Export PBTree.
From Huffman Require Export BTree.
Section PBTREE2BTREE.
Variable A : Type.
Variable eqA_dec : forall a b : A, {a = b} + {a <> b}.
Variable empty : A.
(* Turn a partial tree into a binary tree *)
Fixpoint to_btree (a : pbtree A) : btree A :=
match a with
| pbleaf b => leaf b
| pbleft l1 => to_btree l1
| pbright l1 => to_btree l1
| pbnode l1 l2 => node (to_btree l1) (to_btree l2)
end.

(* Computing the binary tree preserves the leaves*)
Theorem to_btree_inb :
forall a b, inpb (pbleaf a) b -> inb (leaf a) (to_btree b).