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(*         *      The Rocq Prover / The Rocq Development Team           *)
(*  v      *         Copyright INRIA, CNRS and contributors             *)
(* <O___,, * (see version control and CREDITS file for authors & dates) *)
(*   \VV/  **************************************************************)
(*    //   *    This file is distributed under the terms of the         *)
(*         *     GNU Lesser General Public License Version 2.1          *)
(*         *     (see LICENSE file for the text of the license)         *)
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(** This file is deprecated, for a tree on list, use [Mergesort.v]. *)
(** A development of Treesort on Heap trees. It has an average complexity of O(n.log n) but of O(n²) in the worst case (e.g. if the list is already sorted) *)
(* G. Huet 1-9-95 uses Multiset *)
#[local]
Set Warnings "-deprecated".
Section defs.
(** * Trees and heap trees *)
(** ** Definition of trees over an ordered set *)
Variable A : Type.
Variable leA : relation A.
Variable eqA : relation A.
Let gtA (x y:A) := ~ leA x y.
Hypothesis leA_dec : forall x y:A, {leA x y} + {leA y x}.
Hypothesis eqA_dec : forall x y:A, {eqA x y} + {~ eqA x y}.
Hypothesis leA_refl : forall x y:A, eqA x y -> leA x y.
Hypothesis leA_trans : forall x y z:A, leA x y -> leA y z -> leA x z.
Hypothesis leA_antisym : forall x y:A, leA x y -> leA y x -> eqA x y.
#[local]
Hint Resolve leA_refl : core.
#[local]
Hint Immediate eqA_dec leA_dec leA_antisym : core.
Let emptyBag := EmptyBag A.
Let singletonBag := SingletonBag _ eqA_dec.
Inductive Tree :=
| Tree_Leaf : Tree
| Tree_Node : A -> Tree -> Tree -> Tree.
(** [a] is lower than a Tree [T] if [T] is a Leaf or [T] is a Node holding [b>a] *)
#[deprecated(since="8.3", note="Use mergesort.v")]
Definition leA_Tree (a:A) (t:Tree) :=
match t with
| Tree_Leaf => True
| Tree_Node b T1 T2 => leA a b
end.
#[deprecated(since="8.3", note="Use mergesort.v")]

Lemma leA_Tree_Node :
forall (a b:A) (G D:Tree), leA a b -> leA_Tree a (Tree_Node b G D).