Module P := Raw.Proofs.
Definition t := MapS.t D.t.
Definition cmp e e' :=
match D.compare e e' with EQ _ => true | _ => false end.
(** One step of comparison of elements *)
Definition compare_more x1 d1 (cont:R.enumeration D.t -> comparison) e2 :=
match e2 with
| R.End _ => Gt
| R.More x2 d2 r2 e2 =>
match X.compare x1 x2 with
| EQ _ => match D.compare d1 d2 with
| EQ _ => cont (R.cons r2 e2)
| LT _ => Lt
| GT _ => Gt
end
| LT _ => Lt
| GT _ => Gt
end
end.
(** Comparison of left tree, middle element, then right tree *)
Fixpoint compare_cont s1 (cont:R.enumeration D.t -> comparison) e2 :=
match s1 with
| R.Leaf _ => cont e2
| R.Node l1 x1 d1 r1 _ =>
compare_cont l1 (compare_more x1 d1 (compare_cont r1 cont)) e2
end.
(** Initial continuation *)
Definition compare_end (e2:R.enumeration D.t) :=
match e2 with R.End _ => Eq | _ => Lt end.
(** The complete comparison *)
Definition compare_pure s1 s2 :=
compare_cont s1 compare_end (R.cons s2 (Raw.End _)).
(** Correctness of this comparison *)
Definition Cmp c :=
match c with
| Eq => LO.eq_list
| Lt => LO.lt_list
| Gt => (fun l1 l2 => LO.lt_list l2 l1)
end.
Hint Resolve cons_Cmp : core.
(** The dependent-style [compare] *)
Definition eq (m1 m2 : t) := LO.eq_list (elements m1) (elements m2).
Definition lt (m1 m2 : t) := LO.lt_list (elements m1) (elements m2).
Definition compare (s s':t) : Compare lt eq s s'.

Lemma lt_trans : forall m1 m2 m3 : t, lt m1 m2 -> lt m2 m3 -> lt m1 m3.