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(* 02110-1301 USA                                                     *)
(* Contribution to the Coq Library   V6.3 (July 1999)                    *)
(*     A development written by Pierre Castéran for Coq V6.1  Coq : A product of inria : "http://pauillac.inria.fr/coq/coq-eng.html"  Pierre Castéran : A product of LaBRI :   LaBRI, Universite Bordeaux-I      |    12 place Puy Paulin 351 Cours de la Liberation        |    33000 Bordeaux F-33405 TALENCE Cedex             |    France France                            |    (+ 33) 5 56 81 15 80 tel : (+ 33) 5 56 84 69 31 fax : (+ 33) 5 56 84 66 69 email: casteran@labri.u-bordeaux.fr www: http://dept-info.labri.u-bordeaux.fr/~casteran  "Les rêves sont aussi beaux que la réalité, mais ils ne sont pas mieux". ( J.L.Borges )    *)
(* binary nat trees *)
Global Set Asymmetric Patterns.
Require Export Peano_dec.
(*       II The binary tree data structure ****************************************** ******************************************   We define here the "nat_tree" inductive data type. In the remaining of this file, the term "binary tree" will denote a complete binary tree, the internal nodes of which are labeled by natural integers. *)
Inductive nat_tree : Set :=
| NIL : nat_tree
| bin : nat -> nat_tree -> nat_tree -> nat_tree.
(* Example:   3 /  \ /    \ 7      8          Coq representation: / \    / \ NIL \  NIL NIL      (bin three (bin seven NIL (bin nine NIL NIL)) 9                         (bin eight NIL NIL)) / \             where three is the term (S (S (S O))) and so on ... /   \ NIL  NIL *)
(* II.1   Some definitions and lemmas ************************************)
(* (binp t) means "t is not NIL"  *)
Inductive binp : nat_tree -> Prop :=
binp_intro : forall (n : nat) (t1 t2 : nat_tree), binp (bin n t1 t2).
Hint Resolve binp_intro: searchtrees.
(******************************)
Hint Resolve NIL_not_bin: searchtrees.
(************************************************************)
Hint Resolve diff_nil_bin: searchtrees.
(* II.2 membership ******************************************* (occ t p) means: "t has an occurrence of p" *)
Inductive occ : nat_tree -> nat -> Prop :=
| occ_root : forall (n : nat) (t1 t2 : nat_tree), occ (bin n t1 t2) n
| occ_l :
forall (n p : nat) (t1 t2 : nat_tree), occ t1 p -> occ (bin n t1 t2) p
| occ_r :
forall (n p : nat) (t1 t2 : nat_tree), occ t2 p -> occ (bin n t1 t2) p.
Hint Resolve occ_root occ_l occ_r: searchtrees.
Definition member (n : nat) (t : nat_tree) := occ t n.
Derive Inversion_clear OCC_INV with
(forall (n p : nat) (t1 t2 : nat_tree), occ (bin n t1 t2) p).
(* Coq < Check OCC_INV. OCC_INV : (P:nat->nat->nat_tree->nat_tree->Prop) (n,p:nat)(t1,t2:nat_tree) (P p p t1 t2) ->((occ t1 p)->(P n p t1 t2)) ->((occ t2 p)->(P n p t1 t2)) ->(occ (bin n t1 t2) p)->(P n p t1 t2) *)

Lemma occ_inv :
forall (n p : nat) (t1 t2 : nat_tree),
occ (bin n t1 t2) p -> n = p \/ occ t1 p \/ occ t2 p.