Fixpoint max_var_psatz (w : Psatz Z) : positive :=
  match w with
  | PsatzIn _ n => xH
  | PsatzSquare p => max_var xH (Psquare 0 1 Z.add Z.mul Zeq_bool p)
  | PsatzMulC p w => Pos.max (max_var xH p) (max_var_psatz w)
  | PsatzMulE w1 w2 => Pos.max (max_var_psatz w1) (max_var_psatz w2)
  | PsatzAdd w1 w2  => Pos.max (max_var_psatz w1) (max_var_psatz w2)
  | _   => xH
  end.Fixpoint max_var_prf (w : ZArithProof) : positive :=
  match w with
  | DoneProof => xH
  | RatProof w pf | CutProof w pf => Pos.max (max_var_psatz w) (max_var_prf pf)
  | SplitProof p pf1 pf2 => Pos.max (max_var xH p) (Pos.max (max_var_prf pf1) (max_var_prf pf1))
  | EnumProof w1 w2 l => List.fold_left
                           (fun acc prf => Pos.max acc (max_var_prf prf)) l
                           (Pos.max (max_var_psatz w1) (max_var_psatz w2))
  | ExProof _ pf => max_var_prf pf
  end.Fixpoint ZChecker  (l:list (NFormula Z)) (pf : ZArithProof)  {struct pf} : bool :=
  match pf with
    | DoneProof => false
    | RatProof w pf =>
      match eval_Psatz l w  with
        | None => false
        | Some f =>
          if Zunsat f then true
            else ZChecker (f::l) pf
      end
    | CutProof w pf =>
      match eval_Psatz l w with
        | None => false
        | Some f =>
          match genCuttingPlane f with
            | None => true
            | Some cp => ZChecker (nformula_of_cutting_plane cp::l) pf
          end
      end
    | SplitProof p pf1 pf2 =>
      match genCuttingPlane (p,NonStrict) , genCuttingPlane (popp p, NonStrict) with
      | None , _ | _ , None => false
      | Some cp1 , Some cp2 =>
        ZChecker (nformula_of_cutting_plane cp1::l) pf1
        &&
        ZChecker (nformula_of_cutting_plane cp2::l) pf2
      end
    | ExProof x prf =>
      let fr := max_var_nformulae l in
      if Pos.leb x fr then
      let z    := Pos.succ fr in
      let t    := Pos.succ z in
      let nfx  := xnnormalise (mk_eq_pos x z t) in
      let posz := xnnormalise (bound_var z) in
      let post := xnnormalise (bound_var t) in
      ZChecker (nfx::posz::post::l) prf
      else false
      | EnumProof w1 w2 pf =>
       match eval_Psatz l w1 , eval_Psatz l w2 with
         |  Some f1 , Some f2 =>
           match genCuttingPlane f1 , genCuttingPlane f2 with
             |Some (e1,z1,op1) , Some (e2,z2,op2) =>
               if (valid_cut_sign op1 && valid_cut_sign op2 && is_pol_Z0 (padd e1 e2))
                 then
                   (fix label (pfs:list ZArithProof) :=
                   fun lb ub =>
                     match pfs with
                       | nil => if Z.gtb lb ub then true else false
                       | pf::rsr => andb (ZChecker ((psub e1 (Pc lb), Equal) :: l) pf) (label rsr (Z.add lb 1%Z) ub)
                     end)   pf (Z.opp z1)  z2
                  else false
              |   _    ,   _   => true
           end
          |   _   ,  _ => false
    end
end.Fixpoint bdepth (pf : ZArithProof) : nat :=
  match pf with
    | DoneProof  => O
    | RatProof _ p =>  S (bdepth p)
    | CutProof _  p =>   S  (bdepth p)
    | SplitProof _ p1 p2 => S (Nat.max (bdepth p1) (bdepth p2))
    | EnumProof _ _ l => S (List.fold_right (fun pf x => Nat.max (bdepth pf) x)   O l)
    | ExProof _ p   => S (bdepth p)
  end.

(* Specifications: *)
Lemma in_bdepth : forall l a b y, In y l -> ltof ZArithProof bdepth y (EnumProof a b l).
Lemma ltof_bdepth_split_l : forall p pf1 pf2, ltof ZArithProof bdepth pf1 (SplitProof p pf1 pf2).
Lemma ltof_bdepth_split_r : forall p pf1 pf2, ltof ZArithProof bdepth pf2 (SplitProof p pf1 pf2).
Lemma eval_Psatz_sound : forall env w l f', make_conj (eval_nformula env) l -> eval_Psatz l w = Some f' -> eval_nformula env f'.
Lemma makeCuttingPlane_ns_sound : forall env e e' c, eval_nformula env (e, NonStrict) -> makeCuttingPlane e = (e',c) -> eval_nformula env (nformula_of_cutting_plane (e', c, NonStrict)).
Lemma cutting_plane_sound : forall env f p, eval_nformula env f -> genCuttingPlane f = Some p -> eval_nformula env (nformula_of_cutting_plane p).
Lemma genCuttingPlaneNone : forall env f, genCuttingPlane f = None -> eval_nformula env f -> False.
Lemma eval_nformula_mk_eq_pos : forall env x z t, env x = env z - env t -> eval_nformula env (xnnormalise (mk_eq_pos x z t)).
Lemma eval_nformula_bound_var : forall env x, env x >= 0 -> eval_nformula env (xnnormalise (bound_var x)).