(**************************************************************************)
(*       ___                                                              *)
(*      ||M||                                                             *)
(*      ||A||       A project by Andrea Asperti                           *)
(*      ||T||                                                             *)
(*      ||I||       Developers:                                           *)
(*      ||T||         The HELM team.                                      *)
(*      ||A||         http://helm.cs.unibo.it                             *)
(*      \   /                                                             *)
(*       \ /        This file is distributed under the terms of the       *)
(*        v         GNU General Public License Version 2                  *)
(*                                                                        *)
(**************************************************************************)

(* This file was automatically generated: do not edit *********************)

include "Coq.ma".

(*#***********************************************************************)

(*  v      *   The Coq Proof Assistant  /  The Coq Development Team     *)

(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *)

(*   \VV/  **************************************************************)

(*    //   *      This file is distributed under the terms of the       *)

(*         *       GNU Lesser General Public License Version 2.1        *)

(*#***********************************************************************)

(*i $Id: Heap.v,v 1.3.2.1 2004/07/16 19:31:19 herbelin Exp $ i*)

(*#* A development of Treesort on Heap trees *)

(* G. Huet 1-9-95 uses Multiset *)

include "Lists/List.ma".

include "Sets/Multiset.ma".

include "Sorting/Permutation.ma".

include "Relations/Relations.ma".

include "Sorting/Sorting.ma".

(* UNEXPORTED
Section defs
*)

(* UNEXPORTED
cic:/Coq/Sorting/Heap/defs/A.var
*)

(* UNEXPORTED
cic:/Coq/Sorting/Heap/defs/leA.var
*)

(* UNEXPORTED
cic:/Coq/Sorting/Heap/defs/eqA.var
*)

(* UNAVAILABLE OBJECT: cic:/Coq/Sorting/Heap/defs/gtA.con *****************)

inline procedural "cic:/Coq/Sorting/Heap/defs/gtA.con" "defs__" as definition.

(* UNEXPORTED
cic:/Coq/Sorting/Heap/defs/leA_dec.var
*)

(* UNEXPORTED
cic:/Coq/Sorting/Heap/defs/eqA_dec.var
*)

(* UNEXPORTED
cic:/Coq/Sorting/Heap/defs/leA_refl.var
*)

(* UNEXPORTED
cic:/Coq/Sorting/Heap/defs/leA_trans.var
*)

(* UNEXPORTED
cic:/Coq/Sorting/Heap/defs/leA_antisym.var
*)

(* UNEXPORTED
Hint Resolve leA_refl.
*)

(* UNEXPORTED
Hint Immediate eqA_dec leA_dec leA_antisym.
*)

(* UNAVAILABLE OBJECT: cic:/Coq/Sorting/Heap/defs/emptyBag.con ************)

inline procedural "cic:/Coq/Sorting/Heap/defs/emptyBag.con" "defs__" as definition.

(* UNAVAILABLE OBJECT: cic:/Coq/Sorting/Heap/defs/singletonBag.con ********)

inline procedural "cic:/Coq/Sorting/Heap/defs/singletonBag.con" "defs__" as definition.

inline procedural "cic:/Coq/Sorting/Heap/Tree.ind".

(*#* [a] is lower than a Tree [T] if [T] is a Leaf
    or [T] is a Node holding [b>a] *)

inline procedural "cic:/Coq/Sorting/Heap/leA_Tree.con" as definition.

inline procedural "cic:/Coq/Sorting/Heap/leA_Tree_Leaf.con" as lemma.

inline procedural "cic:/Coq/Sorting/Heap/leA_Tree_Node.con" as lemma.

(* UNEXPORTED
Hint Resolve leA_Tree_Leaf leA_Tree_Node.
*)

(*#* The heap property *)

inline procedural "cic:/Coq/Sorting/Heap/is_heap.ind".

(* UNEXPORTED
Hint Constructors is_heap.
*)

inline procedural "cic:/Coq/Sorting/Heap/invert_heap.con" as lemma.

(* This lemma ought to be generated automatically by the Inversion tools *)

inline procedural "cic:/Coq/Sorting/Heap/is_heap_rec.con" as lemma.

inline procedural "cic:/Coq/Sorting/Heap/low_trans.con" as lemma.

(*#* contents of a tree as a multiset *)

(*#* Nota Bene : In what follows the definition of SingletonBag
    in not used. Actually, we could just take as postulate:
    [Parameter SingletonBag : A->multiset].  *)

inline procedural "cic:/Coq/Sorting/Heap/contents.con" as definition.

(*#* equivalence of two trees is equality of corresponding multisets *)

inline procedural "cic:/Coq/Sorting/Heap/equiv_Tree.con" as definition.

(*#* specification of heap insertion *)

inline procedural "cic:/Coq/Sorting/Heap/insert_spec.ind".

inline procedural "cic:/Coq/Sorting/Heap/insert.con" as lemma.

(*#* building a heap from a list *)

inline procedural "cic:/Coq/Sorting/Heap/build_heap.ind".

inline procedural "cic:/Coq/Sorting/Heap/list_to_heap.con" as lemma.

(*#* building the sorted list *)

inline procedural "cic:/Coq/Sorting/Heap/flat_spec.ind".

inline procedural "cic:/Coq/Sorting/Heap/heap_to_list.con" as lemma.

(*#* specification of treesort *)

inline procedural "cic:/Coq/Sorting/Heap/treesort.con" as theorem.

(* UNEXPORTED
End defs
*)