(** ** From trees to multisets *)

  (** 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].  *)

  #[deprecated(since="8.3", note="Use mergesort.v")]
  Fixpoint contents (t:Tree) : multiset A :=
    match t with
      | Tree_Leaf => emptyBag
      | Tree_Node a t1 t2 =>
        munion (contents t1) (munion (contents t2) (singletonBag a))
    end.(** equivalence of two trees is equality of corresponding multisets *)
  #[deprecated(since="8.3", note="Use mergesort.v")]
  Definition equiv_Tree (t1 t2:Tree) := meq (contents t1) (contents t2).(** * From lists to sorted lists *)

  (** ** Specification of heap insertion *)

  Inductive insert_spec (a:A) (T:Tree) : Type :=
    insert_exist :
    forall T1:Tree,
      is_heap T1 ->
      meq (contents T1) (munion (contents T) (singletonBag a)) ->
      (forall b:A, leA b a -> leA_Tree b T -> leA_Tree b T1) ->
      insert_spec a T.(** * From lists to sorted lists *)

  (** ** Specification of heap insertion *)

  Inductive insert_spec (a:A) (T:Tree) : Type :=
    insert_exist :
    forall T1:Tree,
      is_heap T1 ->
      meq (contents T1) (munion (contents T) (singletonBag a)) ->
      (forall b:A, leA b a -> leA_Tree b T -> leA_Tree b T1) ->
      insert_spec a T.

#[deprecated(since="8.3", note="Use mergesort.v")] Lemma insert : forall T:Tree, is_heap T -> forall a:A, insert_spec a T.