(*========================================================== ============================================================ TOPOLOGICAL FMAPS, QHMAPS, HMAPS -  PROOFS OF THE GENUS THEOREM AND THE EULER RELATION  PART 3:  EQUIVALENCES, CHARACTERISTICS, GENUS, EULER_POINCARE, CONSTRUCTIVE CONDITION OF PLANARITY  (J.-F. Dufourd - June 2006 - FOR PUBLICATION completed in October 2007 / January 2008)  (* COMPLETE, APART FROM LEMMA expf_clos_symm, WHICH IS ADMITTED TO PROVE THE NECESSARY CONDITION OF PLANARITY *)
============================================================
==========================================================*)
(*========================================================== DART EQUIVALENCES IN QHMAPS ==========================================================*)
Require Export Euler2.
(* Face equivalence: *)
Definition eqf(m:fmap)(x y:dart):=
expf m x y \/ expf m y x.
(* Reflexivity of eqf: *)
(* Symmetry of eqf: *)
(* Transitivity of eqf: *)
(* Decidability of eqf: *)
(* 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: *)
(* LEMMAS, USED IN THE FOLLOWING: *)
(* Idem, for A_1: *)
(* OK: *)
(* INVERSES, Idem: *)
(* OK: *)
(* Face path implies equivalence: with the LEMMAS above: *)
(* OK: *)
(* IDEM : *)

Lemma expf_A_r_eqc : forall(m:fmap)(k:dim),
inv_qhmap m ->
forall (x y:dart), expf m x (A m k y) -> eqc m x y.