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(* 02110-1301 USA                                                     *)
(********************************************************************** Proof of Huffman algorithm: BTree.v  Definition and some properties of binary trees  Definitions: btree, inb, all_leaves, distinct_leaves, compute_code, number_of_nodes  Laurent.Thery@inria.fr (2003) **********************************************************************)
From Huffman Require Export Code.
From Huffman Require Export ISort.
Require Export Compare_dec.
From Huffman Require Export Weight.
From Huffman Require Export UniqueKey.
Section Tree.
Variable A : Type.
Variable eqA_dec : forall a b : A, {a = b} + {a <> b}.
Variable empty : A.
(* Definition of binary trees *)
Inductive btree : Type :=
| leaf : A -> btree
| node : btree -> btree -> btree.
(* Predicate that defines belonging *)
Inductive inb : btree -> btree -> Prop :=
| inleaf : forall t : btree, inb t t
| innodeL : forall t t1 t2 : btree, inb t t1 -> inb t (node t1 t2)
| innodeR : forall t t1 t2 : btree, inb t t2 -> inb t (node t1 t2).
Hint Constructors inb : core.

(* inb is transitive *)
Theorem inb_trans : forall t1 t2 t3, inb t1 t2 -> inb t2 t3 -> inb t1 t3.
Proof using.