Definition cnfZ (Annot: Type) (TX : Tauto.kind -> Type)  (AF : Type) (k: Tauto.kind) (f : TFormula (Formula Z) Annot TX AF k) :=
  rxcnf Zunsat Zdeduce normalise negate true f.Definition ZweakTautoChecker (w: list ZWitness) (f : BFormula (Formula Z) Tauto.isProp) : bool :=
  @tauto_checker (Formula Z)  (NFormula Z) unit Zunsat Zdeduce normalise negate  ZWitness (fun cl => ZWeakChecker (List.map fst cl)) f w.Definition ceiling (a b:Z) : Z :=
  let (q,r) := Z.div_eucl a b in
    match r with
      | Z0 => q
      | _  => q + 1
    end.

Lemma narrow_interval_lower_bound a b x : a > 0 -> a * x >= b -> x >= ceiling b a.