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.

Lemma cons_Cmp : forall c x1 x2 d1 d2 l1 l2,
X.eq x1 x2 -> D.eq d1 d2 ->
Cmp c l1 l2 -> Cmp c ((x1,d1)::l1) ((x2,d2)::l2).