(****************************************************************************  IEEE754  :  NDiv  Laurent Thery  ***************************************************************************** Definition of the quotient and divisibility for natural and relative numbers*)
Require Export Omega.
Require Export Paux.
Definition oZ1 (x : Option positive) :=
match x with
| None => 0%Z
| Some z => Zpos z
end.
(* We use the  function Pdiv function to build our Zquotient*)
Definition Zquotient (n m : Z) :=
match n, m with
| Z0, _ => 0%Z
| _, Z0 => 0%Z
| Zpos x, Zpos y => match Pdiv x y with
| (x, _) => oZ1 x
end
| Zneg x, Zneg y => match Pdiv x y with
| (x, _) => oZ1 x
end
| Zpos x, Zneg y => match Pdiv x y with
| (x, _) => (- oZ1 x)%Z
end
| Zneg x, Zpos y => match Pdiv x y with
| (x, _) => (- oZ1 x)%Z
end
end.
intros z; case z; simpl in |- *;
try (intros; apply sym_equal; apply inject_nat_convert; auto);
auto.
Qed.
intros z; case z; simpl in |- *; auto with arith.
Qed.
Hint Resolve Zero_le_oZ: arith.
(* It has the required property *)
intros m n; unfold Zquotient in |- *; case n; simpl in |- *.
intros H; case H; auto.
intros n' Hn'; case m; simpl in |- *; auto.
exists 0%Z; repeat split; simpl in |- *; auto with zarith.
intros m'; generalize (Pdiv_correct m' n'); case (Pdiv m' n'); simpl in |- *;
auto.
intros q r (H1, H2); exists (oZ1 r); repeat (split; auto with zarith).
rewrite <- (inject_nat_convert (Zpos m') m'); auto.
rewrite H1.
rewrite inj_plus; rewrite inj_mult.
rewrite <- (inject_nat_convert (Zpos n') n'); auto.
repeat rewrite inj_oZ1; auto.
rewrite inj_oZ1; rewrite Z.abs_eq; auto with zarith.
rewrite <- (inject_nat_convert (Zpos m') m'); auto with zarith.
rewrite <- (inject_nat_convert (Zpos n') n'); auto with zarith.
rewrite inj_oZ1; rewrite Z.abs_eq; auto with zarith.
rewrite <- (inject_nat_convert (Zpos n') n'); auto with zarith.
intros m'; generalize (Pdiv_correct m' n'); case (Pdiv m' n'); simpl in |- *;
auto.
intros q r (H1, H2); exists (- oZ1 r)%Z; repeat (split; auto with zarith).
replace (Zneg m') with (- Zpos m')%Z; [ idtac | simpl in |- *; auto ].
rewrite <- (inject_nat_convert (Zpos m') m'); auto.
rewrite H1.
rewrite inj_plus; rewrite inj_mult.
rewrite <- (inject_nat_convert (Zpos n') n'); auto.
repeat rewrite inj_oZ1; auto with zarith.
ring.
rewrite <- Zopp_mult_distr_l; rewrite Zabs_Zopp.
rewrite inj_oZ1; rewrite Z.abs_eq; auto with zarith.
rewrite <- (inject_nat_convert (Zpos m') m'); auto with zarith.
rewrite <- (inject_nat_convert (Zpos n') n'); auto with zarith.
rewrite Zabs_Zopp.
rewrite inj_oZ1; rewrite Z.abs_eq; auto with zarith.
rewrite <- (inject_nat_convert (Zpos n') n'); auto with zarith.
intros n' Hn'; case m; simpl in |- *; auto.
exists 0%Z; repeat split; simpl in |- *; auto with zarith.
intros m'; generalize (Pdiv_correct m' n'); case (Pdiv m' n'); simpl in |- *;
auto.
intros q r (H1, H2); exists (oZ1 r); repeat (split; auto with zarith).
replace (Zneg n') with (- Zpos n')%Z; [ idtac | simpl in |- *; auto ].
rewrite <- (inject_nat_convert (Zpos m') m'); auto.
rewrite H1.
rewrite inj_plus; rewrite inj_mult.
rewrite <- (inject_nat_convert (Zpos n') n'); auto.
repeat rewrite inj_oZ1; auto with zarith.
ring.
replace (Zneg n') with (- Zpos n')%Z; [ idtac | simpl in |- *; auto ].
rewrite Zmult_opp_opp.
rewrite inj_oZ1; rewrite Z.abs_eq; auto with zarith.
rewrite <- (inject_nat_convert (Zpos n') n'); auto with zarith.
rewrite <- (inject_nat_convert (Zpos m') m'); auto with zarith.
rewrite inj_oZ1; rewrite Z.abs_eq; auto with zarith.
rewrite <- (inject_nat_convert (Zpos n') n'); auto with zarith.
intros m'; generalize (Pdiv_correct m' n'); case (Pdiv m' n'); simpl in |- *;
auto.
intros q r (H1, H2); exists (- oZ1 r)%Z; repeat (split; auto with zarith).
replace (Zneg m') with (- Zpos m')%Z; [ idtac | simpl in |- *; auto ].
rewrite <- (inject_nat_convert (Zpos m') m'); auto.
replace (Zneg n') with (- Zpos n')%Z; [ idtac | simpl in |- *; auto ].
rewrite H1.
rewrite inj_plus; rewrite inj_mult.
rewrite <- (inject_nat_convert (Zpos n') n'); auto.
repeat rewrite inj_oZ1; auto with zarith.
ring.
replace (Zneg n') with (- Zpos n')%Z; [ idtac | simpl in |- *; auto ].
rewrite <- Zopp_mult_distr_r; rewrite Zabs_Zopp.
rewrite inj_oZ1; rewrite Z.abs_eq; auto with zarith.
rewrite <- (inject_nat_convert (Zpos m') m'); auto with zarith.
rewrite <- (inject_nat_convert (Zpos n') n'); auto with zarith.
rewrite Zabs_Zopp.
rewrite inj_oZ1; rewrite Z.abs_eq; auto with zarith.
rewrite <- (inject_nat_convert (Zpos n') n'); auto with zarith.
Qed.
intros z1 z2 H H0; case (Z.eq_dec z2 0); intros Z1.
rewrite Z1; red in |- *; case z1; simpl in |- *; auto; intros; red in |- *;
intros; discriminate.
case (ZquotientProp z1 z2); auto; intros r (H1, (H2, H3)).
case (Zle_or_lt 0 (Zquotient z1 z2)); auto; intros Z2.
Contradict H3; apply Zle_not_lt.
replace r with (z1 - Zquotient z1 z2 * z2)%Z;
[ idtac | pattern z1 at 1 in |- *; rewrite H1; ring ].
repeat rewrite Z.abs_eq; auto.
pattern z2 at 1 in |- *; replace z2 with (0 + 1 * z2)%Z; [ idtac | ring ].
unfold Zminus in |- *; apply Z.le_trans with (z1 + 1 * z2)%Z; auto with zarith.
apply Zplus_le_compat_l.
rewrite Zopp_mult_distr_l.
apply Zle_Zmult_comp_r; auto with zarith.
unfold Zminus in |- *; rewrite Zopp_mult_distr_l; auto with zarith.
Qed.
(* The property of a  number to divide another one (Ndivides n m) shoud be read as m divides n *)
Definition Zdivides (n m : Z) := exists q : Z, n = (m * q)%Z.
intros n m H' H'0.
case H'0; intros z1 Hz1.
case (ZquotientProp n m); auto; intros z2 (Hz2, (Hz3, Hz4)).
cut (z2 = 0%Z);
[ intros H1; pattern n at 1 in |- *; rewrite Hz2; rewrite H1; ring | idtac ].
cut (z2 = ((z1 - Zquotient n m) * m)%Z); [ intros H2 | idtac ].
case (Z.eq_dec (z1 - Zquotient n m) 0); intros H3.
rewrite H2; rewrite H3; ring.
Contradict Hz4.
replace (Z.abs m) with (1 * Z.abs m)%Z; [ idtac | ring ].
apply Zle_not_lt; rewrite H2.
rewrite Zabs_Zmult; apply Zle_Zmult_comp_r; auto with zarith.
generalize H3; case (z1 - Zquotient n m)%Z;
try (intros H1; case H1; auto; fail); simpl in |- *;
intros p; case p; simpl in |- *; auto; intros; red in |- *;
simpl in |- *; auto; red in |- *; intros; discriminate.
rewrite Zmult_minus_distr_r; rewrite (Zmult_comm z1); rewrite <- Hz1;
(pattern n at 1 in |- *; rewrite Hz2); ring.
Qed.
intros n m H'; red in |- *.
exists (Zquotient n m); auto.
pattern n at 1 in |- *; rewrite H'; auto with zarith.
Qed.
intros n m p H'; red in H'.
elim H'; intros q E.
red in |- *.
exists q.
rewrite E.
auto with zarith.
Qed.
intros a b c H H0.
case (Zle_or_lt c 0); intros Zl1.
apply Zle_antisym; apply Zmult_le_reg_r with (p := (- c)%Z); try apply Z.lt_gt;
auto with zarith; repeat rewrite <- Zopp_mult_distr_r;
rewrite H0; auto with zarith.
apply Zle_antisym; apply Zmult_le_reg_r with (p := c); try apply Z.lt_gt;
auto with zarith; rewrite H0; auto with zarith.
Qed.
intros n m p H' H'0.
case H'0; intros q E.
exists q.
apply Zeq_mult_simpl with (c := p); auto.
rewrite (Zmult_comm n); rewrite E; ring.
Qed.
Definition ZdividesP : forall n m : Z, {Zdivides n m} + {~ Zdivides n m}.
intros n m; case m.
case n.
left; red in |- *; exists 0%Z; auto with zarith.
intros p; right; red in |- *; intros H; case H; simpl in |- *; intros f H1;
discriminate.
intros p; right; red in |- *; intros H; case H; simpl in |- *; intros f H1;
discriminate.
intros p; generalize (Z_eq_bool_correct (Zquotient n (Zpos p) * Zpos p) n);
case (Z_eq_bool (Zquotient n (Zpos p) * Zpos p) n);
intros H1.
left; apply ZdividesZquotientInv; auto.
right; Contradict H1; apply sym_equal; apply ZdividesZquotient; auto.
red in |- *; intros; discriminate.
intros p; generalize (Z_eq_bool_correct (Zquotient n (Zneg p) * Zneg p) n);
case (Z_eq_bool (Zquotient n (Zneg p) * Zneg p) n);
intros H1.
left; apply ZdividesZquotientInv; auto.
right; Contradict H1; apply sym_equal; apply ZdividesZquotient; auto.
red in |- *; intros; discriminate.
Defined.
(* Eval Compute in (ZdividesP (POS (xO (xO xH))) (POS (xO xH))). *)
intros m.
case (ZquotientProp m 1); auto.
red in |- *; intros; discriminate.
intros z (H1, (H2, H3)).
pattern m at 2 in |- *; rewrite H1; replace z with 0%Z; try ring.
generalize H3; case z; simpl in |- *; auto; intros p; case p;
unfold Z.lt in |- *; simpl in |- *; intros; discriminate.
Qed.
intros m; exists m; auto with zarith.
Qed.
intros z1 z2; case z1; case z2; simpl in |- *; auto;
try (intros; discriminate); intros p1 p2 H1; injection H1;
(intros H2; rewrite H2); auto.
Qed.
intros z1 z2; case z1; case z2; try (simpl in |- *; auto with zarith; fail).
intros p1 p2;
apply
Zabs_intro with (P := fun x => (x <= Z.abs (Zpos p2) + Z.abs (Zneg p1))%Z);
try rewrite Zopp_plus_distr; auto with zarith.
intros p1 p2;
apply
Zabs_intro with (P := fun x => (x <= Z.abs (Zpos p2) + Z.abs (Zneg p1))%Z);
try rewrite Zopp_plus_distr; auto with zarith.
Qed.
Hint Resolve Zabs_tri: zarith.
intros m n q r H' H'0 H'1 H'2.
case (ZquotientProp m n); auto; intros z (H0, (H1, H2)).
case (Zle_or_lt (Z.abs q) (Z.abs (Zquotient m n))); intros Zl1; auto with arith.
case (Zle_lt_or_eq _ _ Zl1); clear Zl1; intros Zl1; auto with arith.
Contradict H1; apply Zlt_not_le.
pattern m at 1 in |- *; rewrite H'0.
apply Z.le_lt_trans with (Z.abs (q * n) + Z.abs r)%Z; auto with zarith.
apply Z.lt_le_trans with (Z.abs (q * n) + Z.abs n)%Z; auto with zarith.
repeat rewrite Zabs_Zmult.
replace (Z.abs q * Z.abs n + Z.abs n)%Z with (Z.succ (Z.abs q) * Z.abs n)%Z;
[ auto with zarith | unfold Z.succ in |- *; ring ].
case (Zabs_eq_case _ _ Zl1); auto.
intros H;
(cut (Zquotient m n = 0%Z);
[ intros H3; rewrite H; repeat rewrite H3; simpl in |- *; auto | idtac ]).
cut (Z.abs (Zquotient m n) < 1)%Z.
case (Zquotient m n); simpl in |- *; auto; intros p; case p;
unfold Z.lt in |- *; simpl in |- *; intros; discriminate.
apply Zlt_mult_simpl_l with (c := Z.abs n); auto with zarith.
case (Zle_lt_or_eq 0 (Z.abs n)); auto with zarith.
intros H3; case H'; auto.
generalize H3; case n; simpl in |- *; auto; intros; discriminate.
rewrite <- Zabs_Zmult; rewrite (Zmult_comm n).
replace (Z.abs n * 1)%Z with (Z.abs n); [ idtac | ring ].
apply Z.le_lt_trans with (1 := H1).
apply Zlt_mult_simpl_l with (c := (1 + 1)%Z); auto with zarith.
replace ((1 + 1) * Z.abs m)%Z with (Z.abs (m + m)).
replace ((1 + 1) * Z.abs n)%Z with (Z.abs n + Z.abs n)%Z; [ idtac | ring ].
pattern m at 1 in |- *; rewrite H'0; rewrite H0; rewrite H.
replace (- Zquotient m n * n + r + (Zquotient m n * n + z))%Z with (r + z)%Z;
[ idtac | ring ].
apply Z.le_lt_trans with (Z.abs r + Z.abs z)%Z; auto with zarith.
rewrite <- (Z.abs_eq (1 + 1)); auto with zarith.
rewrite <- Zabs_Zmult; apply f_equal with (f := Z.abs); auto with zarith.
Contradict H'1; apply Zlt_not_le.
pattern m at 1 in |- *; rewrite H0.
apply Z.le_lt_trans with (Z.abs (Zquotient m n * n) + Z.abs z)%Z;
auto with zarith.
apply Z.lt_le_trans with (Z.abs (Zquotient m n * n) + Z.abs n)%Z;
auto with zarith.
repeat rewrite Zabs_Zmult.
replace (Z.abs (Zquotient m n) * Z.abs n + Z.abs n)%Z with
(Z.succ (Z.abs (Zquotient m n)) * Z.abs n)%Z;
[ auto with zarith | unfold Z.succ in |- *; ring ].
Qed.
intros m n; case (Z.eq_dec n 0); intros Z1.
rewrite Z1; unfold Zquotient in |- *; case n; case m; simpl in |- *; auto.
case (ZquotientProp m n); auto; intros r1 (H'2, (H'3, H'4)); auto with zarith.
apply sym_equal;
apply ZquotientUnique with (q := (- Zquotient m n)%Z) (r := (- r1)%Z);
auto.
pattern m at 1 in |- *; rewrite H'2; ring.
rewrite <- Zopp_mult_distr_l; repeat rewrite Zabs_Zopp; auto.
rewrite Zabs_Zopp; auto.
Qed.
intros n m q H; case (Zle_lt_or_eq _ _ H); intros Z0.
case (Z.eq_dec q 0); intros Z1.
rewrite Z1; unfold Zquotient in |- *; case n; case m; simpl in |- *;
auto with zarith.
case (Zle_or_lt (Z.abs (Zquotient n q)) (Z.abs (Zquotient m q))); auto;
intros H'1.
case (ZquotientProp m q); auto; intros r1 (H'2, (H'3, H'4)); auto with zarith.
case (ZquotientProp n q); auto; intros r2 (H'5, (H'6, H'7)); auto with zarith.
Contradict H'6.
apply Zlt_not_le.
apply Z.lt_le_trans with (1 := Z0).
rewrite H'2.
apply Z.le_trans with (Z.abs (Zquotient m q * q) + Z.abs r1)%Z; auto with zarith.
apply Z.le_trans with (Z.abs (Zquotient m q * q) + Z.abs q)%Z; auto with zarith.
repeat rewrite Zabs_Zmult.
replace (Z.abs (Zquotient m q) * Z.abs q + Z.abs q)%Z with
(Z.succ (Z.abs (Zquotient m q)) * Z.abs q)%Z;
[ idtac | unfold Z.succ in |- *; ring ].
cut (0 < Z.abs q)%Z; auto with zarith.
case (Zle_lt_or_eq 0 (Z.abs q)); auto with zarith.
intros H'6; case Z1; auto.
generalize H'6; case q; simpl in |- *; auto; intros; discriminate.
case (Zabs_eq_case _ _ Z0); intros Z1; rewrite Z1; auto with zarith.
rewrite ZquotientZopp; rewrite Zabs_Zopp; auto with zarith.
Qed.
intros r v H H'; red in |- *; intros H'0; case H'0; intros q E.
absurd (Z.abs v < Zpower_nat r (digit r v))%Z; auto with zarith.
apply Zle_not_lt.
case (Z.eq_dec q 0); intros Z1.
case H'; rewrite E; rewrite Z1; ring.
pattern v at 2 in |- *; rewrite E.
rewrite Zabs_Zmult.
pattern (Zpower_nat r (digit r v)) at 1 in |- *;
replace (Zpower_nat r (digit r v)) with (Zpower_nat r (digit r v) * 1)%Z;
[ idtac | ring ].
rewrite (fun x y => Z.abs_eq (Zpower_nat x y)); auto with zarith.
apply Zle_Zmult_comp_l; auto with zarith.
generalize Z1; case q; simpl in |- *; try (intros H1; case H1; auto; fail);
intros p;
(case p; unfold Z.le in |- *; simpl in |- *; intros; red in |- *; intros;
discriminate).
Qed.
intros n m H' H'0; case H'0; intros q E; rewrite E.
rewrite Zabs_Zmult.
pattern (Z.abs m) at 1 in |- *; replace (Z.abs m) with (Z.abs m * 1)%Z;
[ idtac | ring ].
apply Zle_Zmult_comp_l; auto with zarith.
generalize E H'; case q; simpl in |- *; auto;
try (intros H1 H2; case H2; rewrite H1; ring; fail);
intros p; case p; unfold Z.le in |- *; simpl in |- *;
intros; red in |- *; discriminate.
Qed.
intros m n p Z1; case (Z.eq_dec n 0); intros Z2.
rewrite Z2; unfold Zquotient in |- *; case (m * p)%Z; case m; simpl in |- *;
auto.
case (ZquotientProp m n); auto; intros r (H1, (H2, H3)).
apply sym_equal; apply ZquotientUnique with (r := (r * p)%Z);
auto with zarith.
pattern m at 1 in |- *; rewrite H1; ring.
rewrite Zmult_assoc.
repeat rewrite (fun x => Zabs_Zmult x p); auto with zarith.
repeat rewrite Zabs_Zmult; auto with zarith.
apply Zmult_gt_0_lt_compat_r; auto with zarith.
apply Z.lt_gt; generalize Z1; case p; simpl in |- *;
try (intros H4; case H4; auto; fail); unfold Z.lt in |- *;
simpl in |- *; auto; intros; red in |- *; intros;
discriminate.
Qed.
intros n m p H' H'0.
case H'; intros z1 Hz1.
case H'0; intros z2 Hz2.
exists (z1 + z2)%Z; rewrite Hz1; rewrite Hz2; ring.
Qed.
intros n m p H' H'0.
case H'; intros z1 Hz1.
case H'0; intros z2 Hz2.
exists (z1 - z2)%Z; rewrite Hz1; rewrite Hz2; ring.
Qed.
intros n m p q H' H'0.
case H'; intros z1 Hz1.
case H'0; intros z2 Hz2.
exists (z1 * z2)%Z; rewrite Hz1; rewrite Hz2; ring.
Qed.
intros n m p H' H'0.
case H'; intros z1 Hz1.
case H'0; intros z2 Hz2.
exists (z1 * z2)%Z; rewrite Hz1; rewrite Hz2; ring.
Qed.

Theorem ZdividesLessPow :
forall (n : Z) (m p : nat),
m <= p -> Zdivides (Zpower_nat n p) (Zpower_nat n m).