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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 )    *)
(* V:        INSERTION ********************************** **********************************   V.1   Definition of an INSERT predicate (a la  Prolog) ***************************************************   We begin with the definition of a predicate:  (INSERT n t t') if t' is a binary search tree containing exactly n and the elements of t *)
Inductive INSERT (n : nat) (t t' : nat_tree) : Prop :=
insert_intro :
(forall p : nat, occ t p -> occ t' p) ->
occ t' n ->
(forall p : nat, occ t' p -> occ t p \/ n = p) ->
search t' -> INSERT n t t'.
Hint Resolve insert_intro: searchtrees.
(* ZZZZ ! insert_rec is defined ! yep ! *)
(* the specification of an insertion algorithm *)
Definition INSERT_SPEC (n : nat) (t : nat_tree) :=
{t' : nat_tree | INSERT n t t'}.
(* technical lemmas preparing our algorithm *)

Lemma insert_nil : forall n : nat, INSERT n NIL (bin n NIL NIL).