(** ** Merging two sorted lists *)
Inductive merge_lem (l1 l2:list A) : Type :=
merge_exist :
forall l:list A,
Sorted leA l ->
meq (list_contents _ eqA_dec l)
(munion (list_contents _ eqA_dec l1) (list_contents _ eqA_dec l2)) ->
(forall a, HdRel leA a l1 -> HdRel leA a l2 -> HdRel leA a l) ->
merge_lem l1 l2.
Instance: Equivalence (@meq A).
(** ** 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.
#[deprecated(since="8.3", note="Use mergesort.v")]

Lemma insert : forall T:Tree, is_heap T -> forall a:A, insert_spec a T.
Inductive build_heap (l:list A) : Type :=
heap_exist :
forall T:Tree,
is_heap T ->
meq (list_contents _ eqA_dec l) (contents T) -> build_heap l.