(*===========================================================
                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.

Lemma bottom_B0_bis: forall(m:fmap)(z:dart)(i j:nat), inv_hmap m -> exd m z -> ~pred m zero z -> let zi := Iter (MA0.f m) i z in let zj := Iter (MA0.f m) j z in let m0 := B m zero zi in (i < j < MA0.Iter_upb m z)%nat -> bottom m0 zero zj = A m zero zi.