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(* 02110-1301 USA                                                     *)
(********************************************************************** Proof of Huffman algorithm: PBTree.v  Definition of partial binary trees (nodes have upto 2 sons)  Definitions: pbtree, inpb, pbfree, compute_pbcode, pbadd pbbuild, all_pbleaves, distinct_pbleaves, compute_pbcode  Laurent.Thery@inria.fr (2003) **********************************************************************)
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.

(* Predictate for belonging *)
Theorem pbleaf_or_not :
forall p, (exists a : _, p = pbleaf a) \/ (forall a : A, p <> pbleaf a).