Library Float.Paux


Require Export Digit.
Require Export Option.
Require Export Inverse_Image.
Require Export Wf_nat.
Require Import BinPos.

Fixpoint exp (n m : nat) {struct m} : nat :=
  match m with
  | O ⇒ 1
  | S m'n × exp n m'
  end.

Theorem expPlus : n p q : nat, exp n (p + q) = exp n p × exp n q.
intros n p; elim p; simpl in |- *; auto with arith.
intros n0 H' q; rewrite mult_assoc_reverse; rewrite <- H'; auto.
Qed.

Fixpoint positive_exp (p n : positive) {struct n} : positive :=
  match n with
  | xHp
  | xO n1
      match positive_exp p n1 with
      | r
          (fun (x : positive) (_ : positivepositive)
             (y : positive) ⇒ (x × y)%positive) r (
            fun xx) r
      end
  | xI n1
      match positive_exp p n1 with
      | r
          (fun (x : positive) (_ : positivepositive)
             (y : positive) ⇒ (x × y)%positive) p (
            fun xx)
            ((fun (x : positive)
                (_ : positivepositive)
                (y : positive) ⇒ (x × y)%positive) r (
               fun xx) r)
      end
  end.

Theorem positive_exp_correct :
  p n : positive,
 nat_of_P (positive_exp p n) =
 exp (nat_of_P p) (nat_of_P n).
intros p n; elim n; simpl in |- *; auto.
intros p0 H.
repeat
 rewrite
  (fun (x y : positive) (_ : positivepositive) ⇒
   nat_of_P_mult_morphism x y); simpl in |- *;
 auto.
rewrite ZL6; rewrite expPlus; rewrite H; auto.
intros p0 H.
repeat
 rewrite
  (fun (x y : positive) (_ : positivepositive) ⇒
   nat_of_P_mult_morphism x y); simpl in |- *;
 auto.
rewrite H; rewrite <- expPlus; rewrite <- ZL6; auto.
rewrite mult_comm; simpl in |- *; auto.
Qed.

Fixpoint Pdiv (p q : positive) {struct p} :
 Option positive × Option positive :=
  match p with
  | xH
      match q with
      | xH(Some _ 1%positive, None _)
      | xO r(None _, Some _ p)
      | xI r(None _, Some _ p)
      end
  | xI p'
      match Pdiv p' q with
      | (None, None)
          match (1 - Zpos q)%Z with
          | Z0(Some _ 1%positive, None _)
          | Zpos r'(Some _ 1%positive, Some _ r')
          | Zneg r'(None _, Some _ 1%positive)
          end
      | (None, Some r1)
          match (Zpos (xI r1) - Zpos q)%Z with
          | Z0(Some _ 1%positive, None _)
          | Zpos r'(Some _ 1%positive, Some _ r')
          | Zneg r'(None _, Some _ (xI r1))
          end
      | (Some q1, None)
          match (1 - Zpos q)%Z with
          | Z0(Some _ (xI q1), None _)
          | Zpos r'(Some _ (xI q1), Some _ r')
          | Zneg r'(Some _ (xO q1), Some _ 1%positive)
          end
      | (Some q1, Some r1)
          match (Zpos (xI r1) - Zpos q)%Z with
          | Z0(Some _ (xI q1), None _)
          | Zpos r'(Some _ (xI q1), Some _ r')
          | Zneg r'(Some _ (xO q1), Some _ (xI r1))
          end
      end
  | xO p'
      match Pdiv p' q with
      | (None, None)(None _, None _)
      | (None, Some r1)
          match (Zpos (xO r1) - Zpos q)%Z with
          | Z0(Some _ 1%positive, None _)
          | Zpos r'(Some _ 1%positive, Some _ r')
          | Zneg r'(None _, Some _ (xO r1))
          end
      | (Some q1, None)(Some _ (xO q1), None _)
      | (Some q1, Some r1)
          match (Zpos (xO r1) - Zpos q)%Z with
          | Z0(Some _ (xI q1), None _)
          | Zpos r'(Some _ (xI q1), Some _ r')
          | Zneg r'(Some _ (xO q1), Some _ (xO r1))
          end
      end
  end.

Definition oZ h := match h with
                   | None ⇒ 0
                   | Some pnat_of_P p
                   end.

Theorem Pdiv_correct :
  p q,
 nat_of_P p =
 oZ (fst (Pdiv p q)) × nat_of_P q + oZ (snd (Pdiv p q))
 oZ (snd (Pdiv p q)) < nat_of_P q.
intros p q; elim p; simpl in |- *; auto.
3: case q; simpl in |- *; try intros q1; split; auto;
    unfold nat_of_P in |- *; simpl in |- *;
    auto with arith.
intros p'; simpl in |- *; case (Pdiv p' q); simpl in |- *;
 intros q1 r1 (H1, H2); split.
unfold nat_of_P in |- *; simpl in |- ×.
rewrite ZL6; rewrite H1.
case q1; case r1; simpl in |- ×.
intros r2 q2. rewrite Z.pos_sub_spec; unfold Pos.compare.
CaseEq (Pcompare (xI r2) q Datatypes.Eq);
 simpl in |- *; auto.
intros H3; rewrite <- (Pcompare_Eq_eq _ _ H3); simpl in |- *;
 unfold nat_of_P in |- *; simpl in |- ×.
apply f_equal with (f := S); repeat rewrite (fun x ymult_comm x (S y));
 repeat rewrite ZL6; unfold nat_of_P in |- *;
 simpl in |- *; ring.
intros H3; unfold nat_of_P in |- *; simpl in |- *;
 repeat rewrite ZL6; unfold nat_of_P in |- *;
 repeat rewrite (fun x yplus_comm x (S y)); simpl in |- *;
 apply f_equal with (f := S); ring.
intros H3; case (Pminus_mask_Gt _ _ H3); intros h (H4, (H5, H6));
 unfold Pminus in |- *; rewrite H4.
apply
 trans_equal
  with
    (nat_of_P q + nat_of_P h +
     Pmult_nat q2 2 × Pmult_nat q 1);
 [ rewrite <- nat_of_P_plus_morphism; rewrite H5; simpl in |- *;
    repeat rewrite ZL6; unfold nat_of_P in |- *;
    apply f_equal with (f := S)
 | unfold nat_of_P in |- × ]; ring.
intros r2. rewrite Z.pos_sub_spec; unfold Pos.compare.
CaseEq (Pcompare 1 q Datatypes.Eq); simpl in |- *; auto.
intros H3; rewrite <- (Pcompare_Eq_eq _ _ H3); simpl in |- *;
 repeat rewrite ZL6; unfold nat_of_P in |- *; simpl in |- *;
 apply f_equal with (f := S);ring.
intros H3; unfold nat_of_P in |- *; simpl in |- *;
 repeat rewrite (fun x yplus_comm x (S y));
 simpl in |- *; apply f_equal with (f := S); repeat rewrite ZL6;
   unfold nat_of_P in |- *; simpl in |- *; ring.
intros H3; case (Pminus_mask_Gt _ _ H3); intros h (H4, (H5, H6));
 unfold Pminus in |- *; rewrite H4;
 apply
  trans_equal
   with
     (nat_of_P q + nat_of_P h +
      Pmult_nat r2 2 × Pmult_nat q 1);
 [ rewrite <- nat_of_P_plus_morphism; rewrite H5; simpl in |- *;
    repeat rewrite ZL6; unfold nat_of_P in |- *;
    apply f_equal with (f := S)
 | unfold nat_of_P in |- × ]; ring.
intros r2. rewrite Z.pos_sub_spec; unfold Pos.compare.
 CaseEq (Pcompare (xI r2) q Datatypes.Eq); simpl in |- *;
 auto.
intros H3; rewrite <- (Pcompare_Eq_eq _ _ H3); simpl in |- *;
 unfold nat_of_P in |- *; simpl in |- *; repeat rewrite ZL6;
 unfold nat_of_P in |- *; apply f_equal with (f := S);
 ring.
intros H3; unfold nat_of_P in |- *; simpl in |- *;
 repeat rewrite ZL6; unfold nat_of_P in |- *;
 apply f_equal with (f := S); ring.
intros H3; case (Pminus_mask_Gt _ _ H3); intros h (H4, (H5, H6));
 unfold Pminus in |- *; rewrite H4;
 apply trans_equal with (nat_of_P q + nat_of_P h);
 [ rewrite <- (nat_of_P_plus_morphism q); rewrite H5;
    unfold nat_of_P in |- *; simpl in |- *;
    repeat rewrite ZL6; unfold nat_of_P in |- *;
    apply f_equal with (f := S)
 | unfold nat_of_P in |- × ]; ring.
case q; simpl in |- *; auto.
generalize H2; case q1; case r1; simpl in |- *; auto.
intros r2 q2. rewrite Z.pos_sub_spec; unfold Pos.compare.
CaseEq (Pcompare (xI r2) q Datatypes.Eq);
 simpl in |- *; auto.
intros; apply lt_O_nat_of_P; auto.
intros H H0; apply nat_of_P_lt_Lt_compare_morphism; auto.
intros H3 H7; case (Pminus_mask_Gt _ _ H3); intros h (H4, (H5, H6));
 unfold Pminus in |- *; rewrite H4;
 apply plus_lt_reg_l with (p := nat_of_P q);
 rewrite <- (nat_of_P_plus_morphism q); rewrite H5;
 unfold nat_of_P in |- *; simpl in |- *; repeat rewrite ZL6;
 unfold nat_of_P in |- *;
 apply le_lt_trans with (Pmult_nat r2 1 + Pmult_nat q 1);
 auto with arith.
intros r2 HH; case q; simpl in |- *; auto.
intros p2; case p2; unfold nat_of_P in |- *; simpl in |- *;
 auto with arith.
intros p2; case p2; unfold nat_of_P in |- *; simpl in |- *;
 auto with arith.
intros r2 HH. rewrite Z.pos_sub_spec; unfold Pos.compare.
CaseEq (Pcompare (xI r2) q Datatypes.Eq);
 simpl in |- ×.
intros; apply lt_O_nat_of_P; auto.
intros H3; apply nat_of_P_lt_Lt_compare_morphism; auto.
intros H3; case (Pminus_mask_Gt _ _ H3); intros h (H4, (H5, H6));
 unfold Pminus in |- *; rewrite H4;
 apply plus_lt_reg_l with (p := nat_of_P q);
 rewrite <- (nat_of_P_plus_morphism q); rewrite H5;
 unfold nat_of_P in |- *; simpl in |- *; repeat rewrite ZL6;
 unfold nat_of_P in |- *;
 apply le_lt_trans with (Pmult_nat r2 1 + Pmult_nat q 1);
 auto with arith.
intros HH; case q; simpl in |- *; auto.
intros p2; case p2; unfold nat_of_P in |- *; simpl in |- *;
 auto with arith.
intros p2; case p2; unfold nat_of_P in |- *; simpl in |- *;
 auto with arith.
intros p'; simpl in |- *; case (Pdiv p' q); simpl in |- *;
 intros q1 r1 (H1, H2); split.
unfold nat_of_P in |- *; simpl in |- *; rewrite ZL6; rewrite H1.
case q1; case r1; simpl in |- *; auto.
intros r2 q2. rewrite Z.pos_sub_spec; unfold Pos.compare.
CaseEq (Pcompare (xO r2) q Datatypes.Eq);
 simpl in |- *; auto.
intros H3; rewrite <- (Pcompare_Eq_eq _ _ H3); simpl in |- *;
 unfold nat_of_P in |- *; simpl in |- *; repeat rewrite ZL6;
 unfold nat_of_P in |- *; ring.
intros H3; unfold nat_of_P in |- *; simpl in |- *;
 repeat rewrite ZL6; unfold nat_of_P in |- *;
 ring.
intros H3; case (Pminus_mask_Gt _ _ H3); intros h (H4, (H5, H6));
 unfold Pminus in |- *; rewrite H4;
 apply
  trans_equal
   with
     (nat_of_P q + nat_of_P h +
      Pmult_nat q2 2 × Pmult_nat q 1);
 [ rewrite <- (nat_of_P_plus_morphism q); rewrite H5;
    unfold nat_of_P in |- *; simpl in |- *;
    repeat rewrite ZL6; unfold nat_of_P in |- ×
 | unfold nat_of_P in |- × ]; ring.
intros H3; unfold nat_of_P in |- *; simpl in |- *;
 repeat rewrite ZL6; unfold nat_of_P in |- *;
 ring.
intros r2. rewrite Z.pos_sub_spec; unfold Pos.compare.
CaseEq (Pcompare (xO r2) q Datatypes.Eq); simpl in |- *;
 auto.
intros H3; rewrite <- (Pcompare_Eq_eq _ _ H3); simpl in |- *;
 unfold nat_of_P in |- *; simpl in |- *; repeat rewrite ZL6;
 unfold nat_of_P in |- *; ring.
intros H3; unfold nat_of_P in |- *; simpl in |- *;
 repeat rewrite ZL6; unfold nat_of_P in |- *;
 ring.
intros H3; case (Pminus_mask_Gt _ _ H3); intros h (H4, (H5, H6));
 unfold Pminus in |- *; rewrite H4;
 apply trans_equal with (nat_of_P q + nat_of_P h);
 [ rewrite <- (nat_of_P_plus_morphism q); rewrite H5;
    unfold nat_of_P in |- *; simpl in |- *;
    repeat rewrite ZL6; unfold nat_of_P in |- ×
 | unfold nat_of_P in |- × ]; ring.
generalize H2; case q1; case r1; simpl in |- ×.
intros r2 q2. rewrite Z.pos_sub_spec; unfold Pos.compare.
CaseEq (Pcompare (xO r2) q Datatypes.Eq);
 simpl in |- *; auto.
intros; apply lt_O_nat_of_P; auto.
intros H H0; apply nat_of_P_lt_Lt_compare_morphism; auto.
intros H3 H7; case (Pminus_mask_Gt _ _ H3); intros h (H4, (H5, H6));
 unfold Pminus in |- *; rewrite H4;
 apply plus_lt_reg_l with (p := nat_of_P q);
 rewrite <- (nat_of_P_plus_morphism q); rewrite H5;
 unfold nat_of_P in |- *; simpl in |- *;
 unfold nat_of_P in |- *; simpl in |- *; repeat rewrite ZL6;
 unfold nat_of_P in |- *;
 apply lt_trans with (Pmult_nat r2 1 + Pmult_nat q 1);
 auto with arith.
intros; apply lt_O_nat_of_P; auto.
intros r2 HH. rewrite Z.pos_sub_spec; unfold Pos.compare.
CaseEq (Pcompare (xO r2) q Datatypes.Eq);
 simpl in |- ×.
intros; apply lt_O_nat_of_P; auto.
intros H3; apply nat_of_P_lt_Lt_compare_morphism; auto.
intros H3; case (Pminus_mask_Gt _ _ H3); intros h (H4, (H5, H6));
 unfold Pminus in |- *; rewrite H4;
 apply plus_lt_reg_l with (p := nat_of_P q);
 rewrite <- (nat_of_P_plus_morphism q); rewrite H5;
 unfold nat_of_P in |- *; simpl in |- *; repeat rewrite ZL6;
 unfold nat_of_P in |- *;
 apply lt_trans with (Pmult_nat r2 1 + Pmult_nat q 1);
 auto with arith.
auto.
Qed.

Section bugFix.
Variable
  PdivAux :
    positive
    positiveOption positive × Option positive.

Fixpoint PdivlessAux (bound p base length : positive) {struct length}
 : Option positive × Option positive × nat :=
  match Pcompare bound p Datatypes.Eq with
  | Datatypes.Gt(Some _ p, None _, 0)
  | _
      match PdivAux p base with
      | (None, None)(None _, None _, 1)
      | (None, Some r1)(None _, Some _ r1, 1)
      | (Some q1, None)
          match length with
          | xH(Some _ q1, None _, 0)
          | xO length'
              match PdivlessAux bound q1 base length' with
              | (s2, None, n)(s2, None _, S n)
              | (s2, Some r2, n)
                  (s2,
                  Some _
                    ((fun (x : positive)
                        (_ : positivepositive)
                        (y : positive) ⇒ (x × y)%positive) r2
                       (fun xx) base), S n)
              end
          | xI length'
              match PdivlessAux bound q1 base length' with
              | (s2, None, n)(s2, None _, S n)
              | (s2, Some r2, n)
                  (s2,
                  Some _
                    ((fun (x : positive)
                        (_ : positivepositive)
                        (y : positive) ⇒ (x × y)%positive) r2
                       (fun xx) base), S n)
              end
          end
      | (Some q1, Some r1)
          match length with
          | xH(Some _ q1, None _, 0)
          | xO length'
              match PdivlessAux bound q1 base length' with
              | (s2, None, n)(s2, Some _ r1, S n)
              | (s2, Some r2, n)
                  (s2,
                  Some _
                    ((fun (x : positive)
                        (_ : positivepositive)
                        (y : positive) ⇒ x × y) r2 (
                       fun xx) base + r1)%positive,
                  S n)
              end
          | xI length'
              match PdivlessAux bound q1 base length' with
              | (s2, None, n)(s2, Some _ r1, S n)
              | (s2, Some r2, n)
                  (s2,
                  Some _
                    ((fun (x : positive)
                        (_ : positivepositive)
                        (y : positive) ⇒ x × y) r2 (
                       fun xx) base + r1)%positive,
                  S n)
              end
          end
      end
  end.
End bugFix.

Definition Pdivless := PdivlessAux Pdiv.

Theorem Pdivless1 :
  bound p q base,
 Pcompare bound p Datatypes.Eq = Datatypes.Gt
 Pdivless bound p base q = (Some _ p, None _, 0).
intros bound p q base H; case q; simpl in |- *; auto; intros; rewrite H; auto.
Qed.

Theorem Pdivless2 :
  bound p length base,
 Pcompare bound p Datatypes.Eq Datatypes.Gt
 Pdivless bound p base length =
 match Pdiv p base with
 | (None, None)(None _, None _, 1)
 | (None, Some r1)(None _, Some _ r1, 1)
 | (Some q1, None)
     match length with
     | xH(Some _ q1, None _, 0)
     | xO length'
         match Pdivless bound q1 base length' with
         | (s2, None, n)(s2, None _, S n)
         | (s2, Some r2, n)
             (s2,
             Some _
               ((fun (x : positive)
                   (_ : positivepositive)
                   (y : positive) ⇒ (x × y)%positive) r2 (
                  fun xx) base), S n)
         end
     | xI length'
         match Pdivless bound q1 base length' with
         | (s2, None, n)(s2, None _, S n)
         | (s2, Some r2, n)
             (s2,
             Some _
               ((fun (x : positive)
                   (_ : positivepositive)
                   (y : positive) ⇒ (x × y)%positive) r2 (
                  fun xx) base), S n)
         end
     end
 | (Some q1, Some r1)
     match length with
     | xH(Some _ q1, None _, 0)
     | xO length'
         match Pdivless bound q1 base length' with
         | (s2, None, n)(s2, Some _ r1, S n)
         | (s2, Some r2, n)
             (s2,
             Some _
               ((fun (x : positive)
                   (_ : positivepositive)
                   (y : positive) ⇒ x × y) r2 (
                  fun xx) base + r1)%positive,
             S n)
         end
     | xI length'
         match Pdivless bound q1 base length' with
         | (s2, None, n)(s2, Some _ r1, S n)
         | (s2, Some r2, n)
             (s2,
             Some _
               ((fun (x : positive)
                   (_ : positivepositive)
                   (y : positive) ⇒ x × y) r2 (
                  fun xx) base + r1)%positive,
             S n)
         end
     end
 end.
intros bound p length base; case length; simpl in |- *;
 case (Pcompare bound p Datatypes.Eq); auto;
 (intros H; case H; auto; fail) || (intros p' H; case H; auto).
Qed.

Theorem compare_SUP_dec :
  p q : positive,
 Pcompare p q Datatypes.Eq = Datatypes.Gt
 Pcompare p q Datatypes.Eq Datatypes.Gt.
intros p q; case (Pcompare p q Datatypes.Eq); auto; right; red in |- *;
 intros; discriminate.
Qed.
Hint Resolve lt_O_nat_of_P: arith.

Theorem odd_even_lem : p q, 2 × p + 1 2 × q.
intros p; elim p; auto.
intros q; case q; simpl in |- ×.
red in |- *; intros; discriminate.
intros q'; rewrite (fun x yplus_comm x (S y)); simpl in |- *; red in |- *;
 intros; discriminate.
intros p' H q; case q.
simpl in |- *; red in |- *; intros; discriminate.
intros q'; red in |- *; intros H0; case (H q').
replace (2 × q') with (2 × S q' - 2).
rewrite <- H0; simpl in |- *; auto.
repeat rewrite (fun x yplus_comm x (S y)); simpl in |- *; auto.
simpl in |- *; repeat rewrite (fun x yplus_comm x (S y)); simpl in |- *;
 auto.
case q'; simpl in |- *; auto.
Qed.

Theorem Pdivless_correct :
  bound p q base,
 1 < nat_of_P base
 nat_of_P p nat_of_P q
 nat_of_P p =
 oZ (fst (fst (Pdivless bound p base q))) ×
 exp (nat_of_P base) (snd (Pdivless bound p base q)) +
 oZ (snd (fst (Pdivless bound p base q)))
 (oZ (fst (fst (Pdivless bound p base q))) < nat_of_P bound
  oZ (snd (fst (Pdivless bound p base q))) <
  exp (nat_of_P base) (snd (Pdivless bound p base q)))
 ( bound',
  nat_of_P bound = nat_of_P base × bound'
  nat_of_P bound nat_of_P p
  nat_of_P bound
  nat_of_P base × oZ (fst (fst (Pdivless bound p base q)))).
intros bound p q base Hb; generalize q; pattern p in |- *;
 apply
  well_founded_ind
   with (R := fun a bnat_of_P a < nat_of_P b);
 auto; clear p q.
apply wf_inverse_image with (R := lt); auto.
exact lt_wf; auto.
intros p Rec q Hq.
case (compare_SUP_dec bound p); intros H1.
rewrite Pdivless1; auto; simpl in |- ×.
repeat (split; auto with arith).
ring; auto.
apply nat_of_P_lt_Lt_compare_morphism; apply ZC1; auto.
intros bound' H'1 H2; Contradict H2; apply lt_not_le;
 apply nat_of_P_lt_Lt_compare_morphism; apply ZC1;
 auto.
rewrite Pdivless2; auto; simpl in |- ×.
generalize (Pdiv_correct p base); case (Pdiv p base); simpl in |- ×.
intros o1; case o1; simpl in |- ×.
intros o1' o2; case o2; simpl in |- ×.
intros o2' (Ho1, Ho2).
generalize Hq; case q; simpl in |- *; auto.
intros p0 Hq0;
 (cut (nat_of_P o1' nat_of_P p0); [ intros Hrec | idtac ]).
cut (nat_of_P o1' < nat_of_P p); [ intros Hrec1 | idtac ].
generalize (Rec _ Hrec1 _ Hrec).
CaseEq (Pdivless bound o1' base p0); simpl in |- ×.
intros p1; case p1; simpl in |- ×.
intros o3; case o3; simpl in |- *; auto.
intros o3' o4; case o4; simpl in |- *; auto.
intros o4' n Eq1; rewrite nat_of_P_plus_morphism;
 rewrite nat_of_P_mult_morphism.
intros (H'1, ((H'2, H'3), H'4)); repeat (split; auto).
rewrite Ho1; rewrite H'1; ring.
apply lt_le_trans with (S (nat_of_P o4') × nat_of_P base).
simpl in |- *; rewrite (fun x yplus_comm x (nat_of_P y));
 auto with arith.
rewrite (fun x ymult_comm x (nat_of_P y)); auto with arith.
intros bound' H'5 H'6;
 case (le_or_lt (nat_of_P bound) (nat_of_P o1'));
 intros H'7; auto.
apply (H'4 bound'); auto.
rewrite Pdivless1 in Eq1; auto.
discriminate.
apply nat_of_P_gt_Gt_compare_complement_morphism; auto.
intros n Eq1 (H'1, ((H'2, H'3), H'4)); repeat (split; auto).
rewrite Ho1; rewrite H'1; ring.
apply lt_le_trans with (nat_of_P base × 1); auto with arith.
rewrite mult_comm; simpl in |- *; auto with arith.
intros bound' H'5 H'6;
 case (le_or_lt (nat_of_P bound) (nat_of_P o1'));
 intros H'7; auto.
apply (H'4 bound'); auto.
rewrite Pdivless1 in Eq1; auto.
inversion Eq1.
rewrite <- H0.
case (le_or_lt bound' (nat_of_P o1')); intros H'8; auto.
rewrite H'5; auto with arith.
Contradict H'6; auto.
apply lt_not_le; rewrite Ho1; auto.
apply
 lt_le_trans
  with
    (nat_of_P o1' × nat_of_P base + 1 × nat_of_P base);
 auto with arith.
simpl in |- *; auto with arith.
rewrite <- mult_plus_distr_r.
replace (nat_of_P o1' + 1) with (S (nat_of_P o1'));
 auto with arith.
rewrite H'5; auto with arith.
rewrite <- (fun x ymult_comm (nat_of_P x) y); auto with arith.
rewrite plus_comm; auto with arith.
apply nat_of_P_gt_Gt_compare_complement_morphism; auto with arith.
intros o4; case o4; simpl in |- ×.
intros o4' n Eq1; rewrite nat_of_P_plus_morphism;
 rewrite nat_of_P_mult_morphism.
intros (H'1, ((H'2, H'3), H'4)); repeat (split; auto).
rewrite Ho1; rewrite H'1; ring.
apply lt_le_trans with (S (nat_of_P o4') × nat_of_P base).
simpl in |- *; rewrite (fun x yplus_comm x (nat_of_P y));
 auto with arith.
rewrite (fun x ymult_comm x (nat_of_P y)); auto with arith.
intros bound' H H0; apply (H'4 bound'); auto.
case (le_or_lt (nat_of_P bound) (nat_of_P o1')); intros H'8;
 auto.
rewrite Pdivless1 in Eq1; auto.
discriminate.
apply nat_of_P_gt_Gt_compare_complement_morphism; auto.
intros n Eq1 (H'1, ((H'2, H'3), H'4)); repeat (split; auto).
rewrite Ho1; rewrite H'1; ring.
apply lt_le_trans with (nat_of_P base × 1); auto with arith.
rewrite mult_comm; simpl in |- *; auto with arith.
intros bound' H H0; apply (H'4 bound'); auto.
case (le_or_lt (nat_of_P bound) (nat_of_P o1')); intros H'8;
 auto.
rewrite Pdivless1 in Eq1; auto.
discriminate.
apply nat_of_P_gt_Gt_compare_complement_morphism; auto.
rewrite Ho1; auto.
apply lt_le_trans with (nat_of_P o1' × 1 + nat_of_P o2');
 auto with arith.
rewrite mult_comm; simpl in |- *; auto with arith.
apply le_lt_trans with (nat_of_P o1' + 0); auto with arith.
apply plus_le_lt_compat; auto with arith.
apply mult_S_le_reg_l with (n := pred (nat_of_P base)).
replace (S (pred (nat_of_P base))) with (nat_of_P base).
apply
 (fun p n m : natplus_le_reg_l n m p) with (p := nat_of_P o2').
rewrite (plus_comm (nat_of_P o2')); simpl in |- *; auto with arith.
rewrite (mult_comm (nat_of_P base)); simpl in |- *; auto with arith.
rewrite <- Ho1; auto with arith.
apply le_trans with (1 := Hq0); auto with arith.
replace (nat_of_P (xI p0)) with (1 + 2 × nat_of_P p0);
 auto with arith.
apply plus_le_compat; auto with arith.
unfold nat_of_P in |- *; simpl in |- *; (rewrite ZL6; auto).
generalize (lt_O_nat_of_P base); case (nat_of_P base);
 simpl in |- *; auto; intros tmp; inversion tmp.
intros p0 Hq0;
 (cut (nat_of_P o1' nat_of_P p0); [ intros Hrec | idtac ]).
cut (nat_of_P o1' < nat_of_P p); [ intros Hrec1 | idtac ].
generalize (Rec _ Hrec1 _ Hrec).
CaseEq (Pdivless bound o1' base p0); simpl in |- ×.
intros p1; case p1; simpl in |- ×.
intros o3; case o3; simpl in |- *; auto.
intros o3' o4; case o4; simpl in |- *; auto.
intros o4' n Eq1; rewrite nat_of_P_plus_morphism;
 rewrite nat_of_P_mult_morphism.
intros (H'1, ((H'2, H'3), H'4)); repeat (split; auto).
rewrite Ho1; rewrite H'1; ring.
apply lt_le_trans with (S (nat_of_P o4') × nat_of_P base).
simpl in |- *; rewrite (fun x yplus_comm x (nat_of_P y));
 auto with arith.
rewrite (fun x ymult_comm x (nat_of_P y)); auto with arith.
intros bound' H'5 H'6;
 case (le_or_lt (nat_of_P bound) (nat_of_P o1'));
 intros H'7; auto.
apply (H'4 bound'); auto.
rewrite Pdivless1 in Eq1; auto.
discriminate.
apply nat_of_P_gt_Gt_compare_complement_morphism; auto.
intros n Eq1 (H'1, ((H'2, H'3), H'4)); repeat (split; auto).
rewrite Ho1; rewrite H'1; ring.
apply lt_le_trans with (nat_of_P base × 1); auto with arith.
rewrite mult_comm; simpl in |- *; auto with arith.
intros bound' H'5 H'6;
 case (le_or_lt (nat_of_P bound) (nat_of_P o1'));
 intros H'7; auto.
apply (H'4 bound'); auto.
rewrite Pdivless1 in Eq1; auto.
inversion Eq1.
rewrite <- H0.
case (le_or_lt bound' (nat_of_P o1')); intros H'8; auto.
rewrite H'5; auto with arith.
Contradict H'6; auto.
apply lt_not_le; rewrite Ho1; auto.
apply
 lt_le_trans
  with
    (nat_of_P o1' × nat_of_P base + 1 × nat_of_P base);
 auto with arith.
simpl in |- *; auto with arith.
rewrite <- mult_plus_distr_r.
replace (nat_of_P o1' + 1) with (S (nat_of_P o1'));
 auto with arith.
rewrite H'5; auto with arith.
rewrite <- (fun x ymult_comm (nat_of_P x) y); auto with arith.
rewrite plus_comm; auto with arith.
apply nat_of_P_gt_Gt_compare_complement_morphism; auto with arith.
intros o4; case o4; simpl in |- ×.
intros o4' n Eq1; rewrite nat_of_P_plus_morphism;
 rewrite nat_of_P_mult_morphism.
intros (H'1, ((H'2, H'3), H'4)); repeat (split; auto).
rewrite Ho1; rewrite H'1; ring.
apply lt_le_trans with (S (nat_of_P o4') × nat_of_P base).
simpl in |- *; rewrite (fun x yplus_comm x (nat_of_P y));
 auto with arith.
rewrite (fun x ymult_comm x (nat_of_P y)); auto with arith.
intros bound' H H0; apply (H'4 bound'); auto.
case (le_or_lt (nat_of_P bound) (nat_of_P o1')); intros H'8;
 auto.
rewrite Pdivless1 in Eq1; auto.
discriminate.
apply nat_of_P_gt_Gt_compare_complement_morphism; auto.
intros n Eq1 (H'1, ((H'2, H'3), H'4)); repeat (split; auto).
rewrite Ho1; rewrite H'1; ring.
apply lt_le_trans with (nat_of_P base × 1); auto with arith.
rewrite mult_comm; simpl in |- *; auto with arith.
rewrite Ho1; auto.
intros bound' H H0; apply (H'4 bound'); auto.
case (le_or_lt (nat_of_P bound) (nat_of_P o1')); intros H'8;
 auto.
rewrite Pdivless1 in Eq1; auto.
discriminate.
apply nat_of_P_gt_Gt_compare_complement_morphism; auto.
apply lt_le_trans with (nat_of_P o1' × 1 + nat_of_P o2');
 auto with arith.
rewrite mult_comm; simpl in |- *; auto with arith.
apply le_lt_trans with (nat_of_P o1' + 0); auto with arith.
apply plus_le_lt_compat; auto with arith.
rewrite Ho1; auto with arith.
apply mult_S_le_reg_l with (n := pred (nat_of_P base)).
replace (S (pred (nat_of_P base))) with (nat_of_P base).
apply
 (fun p n m : natplus_le_reg_l n m p) with (p := nat_of_P o2').
rewrite (plus_comm (nat_of_P o2')); simpl in |- *; auto with arith.
rewrite (mult_comm (nat_of_P base)); simpl in |- *; auto with arith.
rewrite <- Ho1; auto with arith.
apply le_trans with (1 := Hq0); auto with arith.
replace (nat_of_P (xO p0)) with (0 + 2 × nat_of_P p0);
 auto with arith.
apply plus_le_compat; auto with arith.
unfold nat_of_P in |- *; simpl in |- *; (rewrite ZL6; auto).
generalize (lt_O_nat_of_P base); case (nat_of_P base);
 simpl in |- *; auto; intros tmp; inversion tmp.
replace (nat_of_P 1) with 1; auto with arith.
rewrite Ho1; generalize (lt_O_nat_of_P o2');
 (case (nat_of_P o2'); simpl in |- *).
intros tmp; Contradict tmp; auto with arith.
generalize (lt_O_nat_of_P o1');
 (case (nat_of_P o1'); simpl in |- *).
intros tmp; Contradict tmp; auto with arith.
generalize (lt_O_nat_of_P base);
 (case (nat_of_P base); simpl in |- *).
intros tmp; Contradict tmp; auto with arith.
intros n H n0 H0 n1 H01; rewrite (fun x yplus_comm x (S y));
 simpl in |- ×.
intros tmp; Contradict tmp; auto with arith.
generalize Hq; case q; simpl in |- *; auto.
intros p0 Hq0 (Ho1, Ho2);
 (cut (nat_of_P o1' nat_of_P p0); [ intros Hrec | idtac ]).
cut (nat_of_P o1' < nat_of_P p); [ intros Hrec1 | idtac ].
generalize (Rec _ Hrec1 _ Hrec).
CaseEq (Pdivless bound o1' base p0); simpl in |- ×.
intros p1; case p1; simpl in |- ×.
intros o3; case o3; simpl in |- *; auto.
intros o3' o4; case o4; simpl in |- *; auto.
intros o4' n Eq1; rewrite nat_of_P_mult_morphism.
intros (H'1, ((H'2, H'3), H'4)); repeat (split; auto).
rewrite Ho1; rewrite H'1; ring.
rewrite (fun x ymult_comm x (nat_of_P y)); auto with arith.
intros bound' H'5 H'6;
 case (le_or_lt (nat_of_P bound) (nat_of_P o1'));
 intros H'7; auto.
apply (H'4 bound'); auto.
rewrite Pdivless1 in Eq1; auto.
discriminate.
apply nat_of_P_gt_Gt_compare_complement_morphism; auto.
intros n Eq1 (H'1, ((H'2, H'3), H'4)); repeat (split; auto).
rewrite Ho1; rewrite H'1; ring.
apply lt_le_trans with (nat_of_P base × 1); auto with arith.
rewrite mult_comm; simpl in |- *; auto with arith.
intros bound' H'5 H'6;
 case (le_or_lt (nat_of_P bound) (nat_of_P o1'));
 intros H'7; auto.
apply (H'4 bound'); auto.
rewrite Pdivless1 in Eq1; auto.
inversion Eq1.
rewrite <- H0.
case (le_or_lt bound' (nat_of_P o1')); intros H'8; auto.
rewrite H'5; auto with arith.
Contradict H'6; auto.
apply lt_not_le; rewrite Ho1; auto.
apply
 lt_le_trans
  with
    (nat_of_P o1' × nat_of_P base + 1 × nat_of_P base);
 auto with arith.
simpl in |- *; auto with arith.
rewrite <- mult_plus_distr_r.
replace (nat_of_P o1' + 1) with (S (nat_of_P o1'));
 auto with arith.
rewrite H'5; auto with arith.
rewrite <- (fun x ymult_comm (nat_of_P x) y); auto with arith.
rewrite plus_comm; auto with arith.
apply nat_of_P_gt_Gt_compare_complement_morphism; auto with arith.
intros o4; case o4; simpl in |- ×.
intros o4' n Eq1; rewrite nat_of_P_mult_morphism.
intros (H'1, ((H'2, H'3), H'4)); repeat (split; auto).
rewrite Ho1; rewrite H'1; ring.
rewrite (fun x ymult_comm x (nat_of_P y)); auto with arith.
intros bound' H H0; apply (H'4 bound'); auto.
case (le_or_lt (nat_of_P bound) (nat_of_P o1')); intros H'8;
 auto.
rewrite Pdivless1 in Eq1; auto.
discriminate.
apply nat_of_P_gt_Gt_compare_complement_morphism; auto.
intros n Eq1 (H'1, (H'2, H'3)).
generalize (lt_O_nat_of_P o1'); rewrite H'1; intros tmp; inversion tmp.
rewrite Ho1; auto.
apply le_lt_trans with (nat_of_P o1' × 1 + 0); auto with arith.
rewrite mult_comm; simpl in |- *; auto with arith.
repeat rewrite (fun xplus_comm x 0); simpl in |- *; auto with arith.
apply mult_S_le_reg_l with (n := pred (nat_of_P base)).
replace (S (pred (nat_of_P base))) with (nat_of_P base).
rewrite (mult_comm (nat_of_P base)); simpl in |- *; auto with arith.
rewrite (fun xplus_comm x 0) in Ho1; simpl in Ho1; rewrite <- Ho1.
generalize Hq0; clear Hq0;
 replace (nat_of_P (xI p0)) with (2 × nat_of_P p0 + 1);
 try intros Hq0.
case (le_lt_or_eq _ _ Hq0); auto.
rewrite (fun x yplus_comm x (S y)); intros Hl1.
apply le_trans with (2 × nat_of_P p0); auto with arith.
generalize Hb Ho1; case (nat_of_P base); auto.
intros tmp; Contradict tmp; auto with arith.
intros base'; case base'.
intros tmp; Contradict tmp; auto with arith.
intros base''; case base''.
replace (nat_of_P (xI p0)) with (1 + 2 × nat_of_P p0);
 auto with arith.
intros Hb0 Ho0 H; Contradict H; rewrite Ho0.
rewrite (fun x ymult_comm x (S y)).
apply Compare.not_eq_sym.
apply odd_even_lem.
unfold nat_of_P in |- *; simpl in |- *; (rewrite ZL6; auto).
intros n Hb0 Ho0 H; rewrite H.
apply le_trans with (S (S n) × nat_of_P p0 + nat_of_P p0);
 auto with arith.
apply plus_le_compat; auto with arith.
rewrite plus_comm; simpl in |- *; auto with arith.
rewrite plus_comm; unfold nat_of_P in |- *; simpl in |- *;
 (rewrite ZL6; unfold nat_of_P in |- *; auto).
generalize (lt_O_nat_of_P base); case (nat_of_P base);
 simpl in |- *; auto; intros tmp; inversion tmp.
intros p0 Hq0 (Ho1, Ho2);
 (cut (nat_of_P o1' nat_of_P p0); [ intros Hrec | idtac ]).
cut (nat_of_P o1' < nat_of_P p); [ intros Hrec1 | idtac ].
generalize (Rec _ Hrec1 _ Hrec).
CaseEq (Pdivless bound o1' base p0); simpl in |- ×.
intros p1; case p1; simpl in |- ×.
intros o3; case o3; simpl in |- *; auto.
intros o3' o4; case o4; simpl in |- *; auto.
intros o4' n Eq1; rewrite nat_of_P_mult_morphism.
intros (H'1, ((H'2, H'3), H'4)); repeat (split; auto).
rewrite Ho1; rewrite H'1; ring.
rewrite (fun x ymult_comm x (nat_of_P y)); auto with arith.
intros bound' H'5 H'6;
 case (le_or_lt (nat_of_P bound) (nat_of_P o1'));
 intros H'7; auto.
apply (H'4 bound'); auto.
rewrite Pdivless1 in Eq1; auto.
discriminate.
apply nat_of_P_gt_Gt_compare_complement_morphism; auto.
intros n Eq1 (H'1, ((H'2, H'3), H'4)); repeat (split; auto).
rewrite Ho1; rewrite H'1; ring.
replace 0 with (0 × exp (nat_of_P base) n); auto with arith.
intros bound' H'5 H'6;
 case (le_or_lt (nat_of_P bound) (nat_of_P o1'));
 intros H'7; auto.
apply (H'4 bound'); auto.
rewrite Pdivless1 in Eq1; auto.
inversion Eq1.
rewrite <- H0.
case (le_or_lt bound' (nat_of_P o1')); intros H'8; auto.
rewrite H'5; auto with arith.
Contradict H'6; auto.
apply lt_not_le; rewrite Ho1; auto.
apply
 lt_le_trans
  with
    (nat_of_P o1' × nat_of_P base + 1 × nat_of_P base);
 auto with arith.
simpl in |- *; auto with arith.
rewrite <- mult_plus_distr_r.
replace (nat_of_P o1' + 1) with (S (nat_of_P o1'));
 auto with arith.
rewrite H'5; auto with arith.
rewrite <- (fun x ymult_comm (nat_of_P x) y); auto with arith.
rewrite plus_comm; auto with arith.
apply nat_of_P_gt_Gt_compare_complement_morphism; auto with arith.
intros o4; case o4; simpl in |- ×.
intros o4' n Eq1; rewrite nat_of_P_mult_morphism.
intros (H'1, ((H'2, H'3), H'4)); repeat (split; auto).
rewrite Ho1; rewrite H'1; ring.
rewrite (fun x ymult_comm x (nat_of_P y)); auto with arith.
intros bound' H H0; apply (H'4 bound'); auto.
case (le_or_lt (nat_of_P bound) (nat_of_P o1')); intros H'8;
 auto.
rewrite Pdivless1 in Eq1; auto.
discriminate.
apply nat_of_P_gt_Gt_compare_complement_morphism; auto.
intros n Eq1 (H'1, ((H'2, H'3), H'4)); repeat (split; auto).
rewrite Ho1; rewrite H'1; ring.
replace 0 with (0 × exp (nat_of_P base) n); auto with arith.
intros bound' H H0; Contradict H0; rewrite Ho1; rewrite H'1; simpl in |- *;
 auto with arith.
rewrite Ho1; auto with arith.
apply le_lt_trans with (nat_of_P o1' × 1 + 0); auto with arith.
rewrite mult_comm; simpl in |- *; auto with arith.
apply mult_S_le_reg_l with (n := pred (nat_of_P base)).
replace (S (pred (nat_of_P base))) with (nat_of_P base).
apply le_trans with (nat_of_P p); auto.
rewrite Ho1; rewrite (fun xplus_comm x 0); simpl in |- *; auto.
rewrite (mult_comm (nat_of_P base)); simpl in |- *; auto with arith.
apply le_trans with (1 := Hq0); auto with arith.
replace (nat_of_P (xO p0)) with (2 × nat_of_P p0);
 auto with arith.
unfold nat_of_P in |- *; simpl in |- *; (rewrite ZL6; auto).
generalize (lt_O_nat_of_P base); case (nat_of_P base);
 simpl in |- *; auto; intros tmp; inversion tmp.
replace (nat_of_P 1) with 1; auto with arith.
intros Hq0 (H, H0); Contradict Hq0.
apply lt_not_le.
rewrite H.
generalize (lt_O_nat_of_P o1'); case (nat_of_P o1');
 simpl in |- *; auto.
intros tmp; Contradict tmp; auto with arith.
intros n H2; generalize Hb.
case (nat_of_P base); simpl in |- *; auto.
intros tmp; Contradict tmp; auto with arith.
intros base'; case base'; simpl in |- *; auto with arith.
intros tmp; Contradict tmp; auto with arith.
intros o2; case o2; simpl in |- *; auto with arith.
intros o2' (Ho1, Ho2); repeat (split; auto with arith).
rewrite mult_comm; simpl in |- *; auto with arith.
intros bound' H H0; Contradict Ho2.
apply le_not_lt.
rewrite <- Ho1.
generalize H; case bound'.
rewrite mult_comm; simpl in |- *; auto.
intros Eq2; generalize (lt_O_nat_of_P bound); rewrite Eq2; intros tmp;
 Contradict tmp; auto with arith.
intros n Eq2; apply le_trans with (nat_of_P bound); auto with arith.
rewrite Eq2; auto with arith.
rewrite mult_comm; simpl in |- *; auto with arith.
intros (Ho1, Ho2); generalize (lt_O_nat_of_P p); rewrite Ho1; intros tmp;
 Contradict tmp; auto with arith.
Qed.

Definition PdivBound bound p base := Pdivless bound p base p.

Theorem PdivBound_correct :
  bound p base,
 1 < nat_of_P base
 nat_of_P p =
 oZ (fst (fst (PdivBound bound p base))) ×
 exp (nat_of_P base) (snd (PdivBound bound p base)) +
 oZ (snd (fst (PdivBound bound p base)))
 (oZ (fst (fst (PdivBound bound p base))) < nat_of_P bound
  oZ (snd (fst (PdivBound bound p base))) <
  exp (nat_of_P base) (snd (PdivBound bound p base)))
 ( bound',
  nat_of_P bound = nat_of_P base × bound'
  nat_of_P bound nat_of_P p
  nat_of_P bound
  nat_of_P base × oZ (fst (fst (PdivBound bound p base)))).
intros; unfold PdivBound in |- *; apply Pdivless_correct; auto.
Qed.

Theorem PdivBound_correct1 :
  bound p base,
 1 < nat_of_P base
 nat_of_P p =
 oZ (fst (fst (PdivBound bound p base))) ×
 exp (nat_of_P base) (snd (PdivBound bound p base)) +
 oZ (snd (fst (PdivBound bound p base))).
intros bound p base H; generalize (PdivBound_correct bound p base); intuition.
Qed.

Theorem PdivBound_correct2 :
  bound p base,
 1 < nat_of_P base
 oZ (fst (fst (PdivBound bound p base))) < nat_of_P bound.
intros bound p base H; generalize (PdivBound_correct bound p base); intuition.
Qed.

Theorem PdivBound_correct3 :
  bound p base,
 nat_of_P p < nat_of_P bound
 PdivBound bound p base = (Some _ p, None _, 0).
intros bound p base H; (unfold PdivBound in |- *; apply Pdivless1; auto).
apply nat_of_P_gt_Gt_compare_complement_morphism; auto with arith.
Qed.

Theorem PdivBound_correct4 :
  bound p base bound',
 1 < nat_of_P base
 nat_of_P bound = nat_of_P base × bound'
 nat_of_P bound nat_of_P p
 nat_of_P bound
 nat_of_P base × oZ (fst (fst (PdivBound bound p base))).
intros bound p base bound' H H1 H2; case (PdivBound_correct bound p base);
 auto; intros H'1 (H'2, H'3); apply (H'3 bound'); auto with arith.
Qed.
Transparent Pdiv.