(*=========================================================== expe AND bottom ============================================================*)
(* expo and between in an edge: *)
Definition expe(m:fmap)(z t:dart):=
MA0.expo m z t.
Definition betweene(m:fmap)(z v t:dart):=
MA0.between m z v t.
(* TO HAVE expe_bottom: *)

(* Specifications: *)
Lemma cA0_MA0:forall(m:fmap)(z:dart),
cA m zero z = MA0.f m z.
Lemma bottom_cA: forall(m:fmap)(k:dim)(z:dart),
inv_hmap m -> exd m z ->
bottom m k (cA m k z) = bottom m k z.
Lemma expe_bottom_ind: forall(m:fmap)(z:dart)(i:nat),
inv_hmap m -> exd m z ->
let t:= Iter (cA m zero) i z in
bottom m zero z = bottom m zero t.
Lemma expe_bottom: forall(m:fmap)(z t:dart),
inv_hmap m -> MA0.expo m z t ->
bottom m zero z = bottom m zero t.