(*========================================================== ============================================================  TOPOLOGICAL FMAPS, QHMAPS, HMAPS -  PROOFS OF THE GENUS THEOREM AND THE EULER RELATION  PART 1: BASES, FMAPS, QHMAPS, HMAPS, CLOSURES OF A, A_1  (J.-F. Dufourd, June 2006 - FOR PUBLICATION completed in October 2007)  ============================================================ ==========================================================*)
(* coqtop -opt *)
Global Set Asymmetric Patterns.
(*======================================================== DIMENSIONS ========================================================*)
(* Dimensions: *)
Inductive dim:Set := di0 : dim | di1 : dim.
(* Action of B_1 on exd: *)
induction m.
simpl in |- *.
tauto.
simpl in |- *.
intros.
generalize (IHm k x y).
tauto.
simpl in |- *.
intros k x y.
elim (eq_dim_dec k d).
elim (eq_dart_dec x d1).
generalize (IHm k x y).
tauto.
simpl in |- *.
generalize (IHm k x y).
tauto.
simpl in |- *.
intro.
apply IHm.
Qed.

Lemma exd_B_1_l : forall (m:fmap)(k:dim)(x y:dart),
exd (B_1 m k x) y ->  exd m y.