(*========================================================== ============================================================ TOPOLOGICAL FMAPS, HMAPS -  WITH TAGS AND EMBEDDINGS  PART 3:  EQUIVALENCES, CHARACTERISTICS, GENUS THEOREM, EULER_POINCARE FORMULA, PLANARITY CRITERIA  COMPLETE: OK!! ============================================================ ==========================================================*)
Require Export Jordan2.
(*========================================================== CONNECTIVITY IN HMAPS : ==========================================================*)
(* Connectivity: Definition in Prop "A� la" Warshall *)
Fixpoint eqc(m:fmap)(x y:dart){struct m}:Prop:=
match m with
V => False
|  I m0 x0 _ _ => x=x0 /\ y=x0 \/ eqc m0 x y
|  L m0 _ x0 y0 =>
eqc m0 x y
\/ eqc m0 x x0 /\ eqc m0 y0 y
\/ eqc m0 x y0 /\ eqc m0 x0 y
end.
(* Decidability of eqc: *)
(* Reflexivity of eqc: *)
(* Symmetry of eqc: *)
(* Transitivity of eqc: *)
(* OK: *)
(* OK: *)
(* OK: *)
(* OK: *)
(* OK: *)
(* OK: *)
(* OK: *)
Definition expf(m:fmap)(x y:dart):Prop:=
inv_hmap m /\ MF.expo m x y.

Lemma expf_dec : forall(m:fmap)(x y:dart),
{expf m x y} + {~expf m x y}.