(*===========================================================
                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_quad: forall(m:fmap)(z v:dart)(j:nat), inv_hmap m -> exd m z -> ~pred m zero z -> let m0 := B m zero v in let t := Iter (MA0.f m) j z in (j < MA0.Iter_upb m z)%nat -> ~MA0.expo m z v -> bottom m0 zero t = bottom m zero t.