Definition makeCuttingPlane (p : PolC Z) : PolC  Z * Z :=
  let (g,c) := Zgcd_pol p in
    if Z.gtb g Z0
      then (Zdiv_pol (PsubC Z.sub p c) g , Z.opp (ceiling (Z.opp c) g))
      else (p,Z0).Definition genCuttingPlane (f : NFormula Z) : option (PolC Z * Z * Op1) :=
  let (e,op) := f in
    match op with
      | Equal => let (g,c) := Zgcd_pol e in
        if andb (Z.gtb g Z0) (andb (negb (Zeq_bool c Z0)) (negb (Zeq_bool (Z.gcd g c) g)))
          then None (* inconsistent *)
          else (* Could be optimised Zgcd_pol is recomputed *)
            let (p,c) := makeCuttingPlane e  in
              Some (p,c,Equal)
      | NonEqual => Some (e,Z0,op)
      | Strict   =>  let (p,c) := makeCuttingPlane (PsubC Z.sub e 1) in
        Some (p,c,NonStrict)
      | NonStrict => let (p,c) := makeCuttingPlane e  in
        Some (p,c,NonStrict)
    end.Definition nformula_of_cutting_plane (t : PolC Z * Z * Op1) : NFormula Z :=
  let (e_z, o) := t in
    let (e,z) := e_z in
      (padd e (Pc z) , o).Definition is_pol_Z0 (p : PolC Z) : bool :=
  match p with
    | Pc Z0 => true
    |   _   => false
  end.

(* Specifications: *)
Lemma is_pol_Z0_eval_pol : forall p, is_pol_Z0 p = true -> forall env, eval_pol env p = 0.